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https://theses.hal.science/tel-01743851/file/2017IMTA0031_BartAnicet.pdf

MONFROY M Éric Charlotte Mme Truchet Professeur M Salvador Abreu Mme Béatrice Professeur B Érard Professeur M Nicolas Beldiceanu Professeur M Philippe Codognet Maître M Benoît Delahaye Keywords: Constraint Modelling -Constraint Solving -Program Verification -Abstract Interpretation -Model Checking viii modélisation par contraintes -résolution par contraintes -vérification de programmes -interprétation abstraite -vérification de modèles Constraint Modelling, Constraint Solving, Program Verification, Abstract Interpretation, Model Checking Constraint Modelling and Solving of some Verification Problems Short abstract: Constraint programming offers efficient languages and tools for solving combinatorial and computationally hard problems such as the ones proposed in program verification. In this thesis, we tackle two families of program verification problems using constraint programming. In both contexts, we first propose a formal evaluation of our contributions before realizing some experiments. The first contribution is about a synchronous reactive language, represented by a block-diagram algebra. Such programs operate on infinite streams and model real-time processes. We propose a constraint model together with a new global constraint. Our new filtering algorithm is inspired from Abstract Interpretation. It computes over-approximations of the infinite stream values computed by the block-diagrams. We evaluated our verification process on the FAUST language (a language for processing real-time audio streams) and we tested it on examples from the FAUST standard library. The second contribution considers probabilistic processes represented by Parametric Interval Markov Chains, a specification formalism that extends Markov Chains. We propose constraint models for checking qualitative and quantitative reachability properties. Our models for the qualitative case improve the state of the art models, while for the quantitative case our models are the first ones. We implemented and evaluated our verification constraint models as mixed integer linear programs and satisfiability modulo theory programs. Experiments have been realized on a PRISM based benchmark. Introduction Computer scientists started to write programs in order to produce softwares realizing dedicated tasks faster and more efficiently than a human could perform. However, in adhoc developments the more complex is the problem to solve the longer it takes to write its corresponding solving program. Moreover, few modifications in the description of the problem to solve may impact many changes in the program. The artificial intelligence research domain tries to develop more generic approaches such that a single artificially intelligent program may solve a wide variety of heterogenous problems. Constraint programming is a research axis in the artificial intelligence community where constraints are sets of rules to be satisfied and the intelligent program must find a solution according to these rules. Thus, the objective of the constraint programming community is to produce languages and tools for solving constraint based problems. Such problems are expressed in a declarative manner where programs consist in a set of rules (called constraints) to be satisfied. Thus, a constraint programming user enumerates his/her rules and uses a black-box tool (called solver) for solving his/her problem. These are two major research activities in constraint programming: modelling and solving. The modelling activity works on the expressiveness of the constraint language and manipulates constraint programs in order to improve the resolution process. The solving activity consists in developing algorithms, tools, and solvers for improving the efficiency of the resolution process. For the last decades computers and information systems have been highly democratized for private and company usages. In both contexts, more and more complex systems are developed in order to realize a wide variety of applications (smart applications, embedded systems for air planes, medical robot assistants, etc.). As for many other production fields, writing systems must respect quality rules such as conformity, efficiency, and robustness. In this thesis, we are concerned by the verification problem consisting in verifying if an application, a program, a system matches its specifications (i.e., its expected behavior). This concern gained an important interest after social or business impacts are identified, or after past failures. One of the most remarkable examples is the crash of the Ariane 5 missile, 36 seconds after its launch on June 4, 1996. The accident was due to a conversion of a 64-bit floating point number into a 16-bit integer value. Another example is the bug in Intel's Pentium II floating-point division unit in the 90's, which forced to replace faulty processors, severely damaged Intel's reputation, and implied a loss of about 475 million US dollars. These events happened in the 90', and software are now more and more used to automatically control critical systems such as nuclear power plants, chemical plants, trafficc o n t r o ls y s t e m s ,a n ds t o r ms u r g eb a r r i e r s . F u r t h e r m o r e ,e v e np r o g r a m sw i t hl e s s critical impact require attention, since the competition between products gives benefits to the systems with less bugs, a better reactivity, etc. Thus, the verification objective is to attest the validity of a system according to an expected behavior. While such problems may be solved using add-hoc techniques or proper tools, it appears that by nature or by reformulation of the problem, constraint programming offers effective solutions. For instance, the system/program can be formulated as a set of rules and the expected behavior as a set of constraints. Thus, verifying the validity of the system/program behavior consists in determining if satisfying the rules implies to satisfy the expected behavior. On the other hand, some verification considerations may produce combinatorial problems. In this context constraint programming clearly appears as a suitable solution. In this thesis we tackle program verification problems as applications to be treated using constraint programming. Scientific Context As said before this thesis concerns constraint programming modelling and solving for some program verification problems. In this section we briefly present all the scientific context of the thesis by identifying separately the various scientific thematics tackled in this manuscript. We start by presenting constraint modelling and solving. Then, we continue with the two program verification approaches used in this thesis, and we conclude by presenting the two programming paradigms to be verified in this thesis. Constraint Modelling. A Constraint Satisfaction Program (CSP for short) is a set of constraints over variables each one associated with a domain. Thus, constraint modelling consists in formulating a given problem into a CSP. There exist various research communities each one dedicated to model families of CSPs. Recall first that the general problem of satisfying a CSP (i.e., finding a valuation of the variables satisfying all the constraints in the CSP) is a hard problem. There exists CSP families being tractable in exponential, polynomial, or ever linear time. In this thesis we consider constraint modelling ranging from mathematical programming such as continuous and mixed integer linear programing (respectively LP and MILP for short), finite and continuous domains programs without linearity restrictions on the constraints (respectively named FD and CD for short), and Satisfiability Modulo Theory (SMT for short) mixing Boolean and theories such as arithmetics. See Section 2.2 for more details. Constraint Solving. Various tools, named solvers, have been developed for solving CSP instances. Each one is mainly specialized to solve specific CSP families (e.g.,u nbounded integer linear arithmetics, constraints over variables with finite domain, continuous constraints). The combinatorics implied by the relations between the variables in the constraints makes a CSP hard to solve. This requires to explore search spaces composed of all the valuations possibly candidate for solving the problem. However, the size of such search space is exponential in terms of the problem sizes (number of variables) in general. Huge research efforts has been put into solvers in order to propose tools for (intelligently) explore huge search spaces and solve CSP instances. See Section 2.3 for more details. Program Verification A program describes the behavior of a possibly infinite process by defining possible transitions from states to states. Due either to the runtime environment or to the non determinism of the state successions, one program may have a finite or even an infinite number of possible runs. Also, according to the nature of a program its runs may encounter either a finite or an infinite number of states in theory. Thus, program verification consists in determining if the program traces (i.e., the state sequences realized by the runs) respect a given property. These properties may be time dependent (e.g., for each run the state A must be encountered after the state B, the state A must not be encountered before a given time t) or time independent (e.g., for each run all the variables are bounded by given constants). There are two main approaches for verifying properties on program: dynamic analysis vs. static analysis. Dynamic analysis requires to run the program to attest the validity of the property. On the contrary, static analysis performs verification at compilation time without running the program/system (roughly speaking dynamic analysis can be considered as an online process compared to static analysis which is an offline process). See Section 3.1 for more details. In this thesis we only consider complete static analyzes of programs with infinite runs (i.e.,w ed on o t consider dynamic and bounded analyzes). Abstract Interpretation. Abstract Interpretation is a program verification technique for static program analysis. In this context, we consider programs with unbounded running times and infinite state systems. Recall that in such cases the general program verification problem is undecidable since this class of problems contains the halting problem. Thus, Abstract Interpretation provides a verification process, which terminates, using over-approximations of the semantics of the program to verify. Indeed, well chosen abstractions produce semi-decidable problems. Thus, verification tools based on abstract interpretation either prove the validity of the property or may not conclude. Hence, such method cannot find counter-examples and falsify properties. See Section 3.2 for more details. Model Checking. Model Checking is a program verification technique for static program analysis. As for Abstract Interpretation, programs/models to be verified may have unbounded running times and infinite state space. Thus, model checking is a verification method that explores all possible system states. In this way, it can be shown that a given system model truly satisfies a certain property. Hence, such method proves the validity or the non validity of the property. More specifically, it can return a counter-example in non validity case. See Section 3.3 for more details. Synchronous Reactive Language. Motivated by the nature of embedded controllers requiring to be reactive to the environment at real-time, synchronous languages have been designed for programming reactive control systems. These languages naturally deal with 1.2. Problems and Objectives the complexity of parallel systems. Indeed, parallel computations are realized in a lockstep such that all computations are synchronized reactions. Hence, this synchronization ensures by construction a guarantee of determinism and deadlock freedom. Finally, these languages abstract away all architectural and hardware issues of embedded, distributed systems such that the programmer can only concentrate on the functionalities. Instances of such languages are Faust, Lustre, and Esterel and have been successfully used in the context of critical systems requiring strong verification (e.g., space applications, railway, and avionics) using certified compiler (e.g., Scade [Sca] tool from Esterel Technologies providing a DO-178B level A certified compiler). Chapter 4 concerns the verification of synchronous reactive languages. Probabilistic Programming Language. Various systems are subject to phenomena of a stochastic nature, such as randomness, message loss, probabilistic behavior. Probabilistic programming languages are used to describe such systems using probabilities to define the sequence of states in the program. One of the most popular probabilistic models for representing stochastic behaviors are the discrete-time Markov Chains (MCs for short). Instance of probabilistic programming languages for writing MCs are Problog and Prism. Chapter 5 concerns the verification of models extending the Markov chain model describing parametrized probabilistic systems. Problems and Objectives As presented in the previous sections, program verification is a computationally hard problem with major issues. Recall first that a program describes the behavior of a possibly infinite process by defining possible transitions from states to states. However, the verification is performed on an abstraction of the program named model 1 . In this thesis, we consider finite models with infinite state spaces and infinite runs. Even if a program admits a priori an infinite state space its executions may encounter a( p o t e n t i a l l yi n fi n i t e )s u b s e to ft h ed e c l a r e ds t a t es p a c e . T h u s ,o n ew o u l dl i k et od e t e rmine this smaller state space in order to verify the non reachability of undesired states. This problem is reducible to the search of program over-approximations, i.e.,b o u n ding all the program variables. This is an objective of Abstract Interpretation where the program describing precisely the system evolution from a state to another, named the concrete program, is abstracted. This abstracted construction is related to the concrete program in such a manner that if an over-approximation holds for the abstraction then, this approximation also holds for the concrete program. Furthermore, constraint programs allow to describe over-approximations such as convex polyhedrons using linear constraints, ellipsoids using quadratic constraints, etc. Thus, since constraint programming is a generic declarative programing paradigm it may be seen as a verification process for over-approximating variable in declarative programs. In the first contribution, we consider a block-diagram language where executions are infinite streams and the objective is to bound the stream values using constraint programming. However, bounding the state space is not enough for some verification requirements. In our second problem, the objective is to determine if a specific state is reachable at execution time. Indeed, abstractions can only determine if a specific state is unreachable. For this verification problem, we consider programs representable as finite graph structures where the nodes form the state space and the edges give state to state transitions. Thus, verifying the reachability of a state in such a structure is performed by activating or deactivating transitions in order to reach the target state. However, these activations can be restricted by guards, or other structural dependent rules. Clearly, this corresponds to acom binatoricproblemtosolv e. F orthisreason,sinceoneoftheobjectiv esofconstrain t programming is to solve highly combinatorial problems, the verification community is interested in the CP tools. Some links between constraint programming and program verification are presented in Section 3.4. To conclude, constraint programming proposes languages to model and solve problems by focusing on the problem formulation instead of the resolution process. Program verification leads to problems which by definition or by nature are close to constraint programs. Thus, the verification community uses constraint programming tools for developing analyzers instead of producing ad-hoc algorithms. In this thesis, we position ourself as constraint programmers and we consider verification problems as applications. Thus, our objective is to present how the constraint programming advances in modelling and solving helps to answer some verification problems. Contributions The contributions are split into two distinct chapters, and they are related to different verification research axes, but both using constraint programming. The first contribution applies constraint programming to verify some properties of a real-time language, while the second one is about verification of extensions of Markov chains. Here are the abstracts of these two contributions. Verifying a Synchronous Reactive Language with constraints (Chapter 4). Formal verification of real time programs in which variables can change values at every time step, is difficult due to the analyses of loops with time lags. In our first contribution, we propose a constraint programming model together with a global constraint and a filtering algorithm for computing over-approximation of real-time streams. The global constraint handles the loop analysis by providing an interval over-approximation of the loop invariant. We apply our method to the FAUST language which is a language for processing real-time audio streams. We conclude with experiments that show that our approach provides accurate results in short computing times. This contribution has been published in a national conference [1], an international conference [2], and a journal [3]. Verifying a Parametric Probabilistic Language with constraints (Chapter 5). Parametric Interval Markov Chains (pIMCs) are a specification formalism that extends Markov Chains (MCs) and Interval Markov Chains (IMCs) by taking into account imprecision in the transition probability values: transitions in pIMCs are labeled with parametric intervals of probabilities. In this work, we study the difference between pIMCs and other Markov Chain abstractions models and investigate the three usual semantics for IMCs: once-and-for-all, interval-Markov-decision-process, and at-every-step. In particular, we prove that all these semantics agree on the maximal/minimal reachability probabilities of agivenIMC.W etheninvestigatesolutionstoseveralparametersynthesisproblemsinthe context of pIMCs -consistency, qualitative reachability, and quantitative reachabilitythat rely on constraint encodings. Finally, we conclude with experiments by implementing our constraint encodings with promising results. This contribution has been published in a national conference [4], an international workshop without proceedings [5], and an international conference [6] (to appear). Outline The thesis in organized in four main chapters. Chapter 2 presents the constraint programming paradigm. Chapter 3 introduces program verification/model checking problems. We conclude this chapter by briefly introducing the two verification methods named Abstract Interpretation and Model Checking in order to motivate the two following chapters which respectively use these verification methods. Chapter 4 contains our first contribution. This chapter proposes a constraint programming model together with a global constraint and a filtering algorithm inspired from abstract interpretation for computing over-approximation of real-time streams. This chapter is also illustrated and validated by some experiments. Chapter 5 contains our second contribution. This chapter proposes constraint programming models for verifying qualitative and quantitative properties of parametric interval Markov chains with a model checking objective. This chapter also concludes with experiments. Note that both contribution chapters are self-contained including introduction, motivation, background, state of the art, contributions, and bibliography. Finally, Chapter 6 concludes this thesis document. Chapter 2 Constraint Programming Introduction Computer scientists started to write programs in order to produce softwares realizing dedicated tasks faster and more efficiently than a human could perform. However, in ad-hoc developments the more complex is the problem to solve the longer it is to write its corresponding solving program. Moreover, few changes in the description of the problem to solve may impact many changes in the program. Thus, the artificial intelligence research domain tries to develop more generic approaches such that a single artificially intelligent program may solve a wide variety of heterogeneous problems. Among all possible artificial intelligences, we focus in this thesis on those dealing with constraint based problems. In such problems, one can enumerate a set of objects with possibly many different states for each object and a set of accepted configurations over these objects w.r.t. the states (cf. Definition 2.1.1). Definition 2.1.1 (Constraint Based Problem). Let A be a set of objects, and S be a set of object states. A Constraint Based Problem P over objects A with states S represents a set of configurations (i.e., a set of associations between objects from A and states in S). Formally P ⌘ L s.t. L ✓ S A . In this chapter, we first present constraint modelling (i.e., variables, domains, constraints, and constraint programs definitions) and various research axes dedicated to constraint modelling (SAT, CP, LP, etc.). Then, we present these research axes dealing with constraint programs by describing their common resolution processes and their specific strategies developed in each one. Restrictions. In this thesis we consider modelling with real, integer, and Boolean variables with finite or infinite domains without restrictions on the constraints (e.g., enumerations, linear and non-linear inequations, Boolean compositions, global constraints) using the existential quantification of variables and being time-independent 1 . Finally, we consider complete methods for solving such models. Constraint Modelling Constraint modelling is the action of formulating a given constraint based problem into a constraint based program. Definition 2.1.1 recalled that a constraint based problem is described by a set of objects, a set of object states, and gathered into a set of configurations. Constraint programming uses a dedicated vocabulary. In the following we take care to well separate the constraint based problems from constraint based programs. Indeed constraint based problems are commonly expressed in a natural language while constraint programs are expressed in a mathematical (or mathematical-like) language or a programming language. A constraint program uses variables associated with domains linked by constraints. Roughly speaking, the variables with their domains will model the objects with their states in the constraint based program while the constraints will model the configurations in the constraint based program. We now present a wide landscape of variables, domains, and constraints encountered in constraint modelling while encoding a constraint based problem into a constrained program. 5 Z0Z0Z 4 0Z0Z0 3 Z0Z0Z 2 0Z0Z0 1 Z0Z0Z abcde Q Q Q Q Q (a) Example 1 (n-Queens Problem). The n-Queens problem will be our backbone example for illustrating constraint modelling and solving in this section. Let n be a natural number. Thus we consider an n ⇥ n chessboard and n queens. The n-Queens Problem objective is to place the n queens on the chessboard such that no two queens threaten each other (i.e., no two queens share the same row, column, or diagonal). In this example, objects are the n queens and states are the n ⇥ n cells of the chessboard. Thus, a configuration is an arrangement of the n queens on the chessboard. Variables and Domains A constraint based problem is described by a set of variables, each variable being associated with a non-empty set called its domain. From now on in this section, X will refer to as e to fn variables x 1 ,...,x n , D x will be the domain associated to the variable x 2 X, and D will contain all the domains associated to the variables in X. We identify four variable types according to their domain. We say that a variable x with domain D x is: • aB oo l e a nv a r i a b l ei ff its domain is a binary set (i.e., D x = {true, false}) • an integer variable iff its domain only contains integers (i.e., D x ✓ Z) • ar a t i o n a lv a r i a b l ei ff its domain only contains rational numbers (i.e., D x ✓ Q) • ar e a lv a r i a b l ei ff its domain only contains real numbers (i.e., D x ✓ R) A domain can be given in extension by enumerating all the elements composing it or in intension using an expression representing all its elements. One common compact representation is the interval representation together with the union of intervals. Formally, let E ✓ R be a non-empty totally ordered set and a, b 2 R2 be two interval endpoints. We write I E ([a, b]) for the set containing all the (closed, semi-opened, opened) intervals subsets of the interval [a, b] ✓ E. When modelling, we usually separate real variables with interval domains from others. The first ones are called continuous variables while the remaining are called discrete variables. Furthermore, we separate finite variables (i.e., variables whose domains have a finite number of elements) from infinite variables. For instance afi n i t ev a r i a b l ec a nb ei n t r o d u c e db yd o m a i ne n u m e r a t i o n( e.g.,d o m a i n{1, 2,...,50}) and infinite variables can be defined by interval domain (e.g., domain [-1, 1] subset of R). Note that there exists other domains such that the symbolic domains where each domain may contain an infinite number of possibly ordered symbols, or the set domains where each domain element is a set of values. In this thesis we perform constraint modelling with Boolean, integer, rational, and real-number domains. Finally, a valuation of the variables in X 0 ✓ X is a mapping v associating to each variable in X 0 av a l u ei ni t sd o m a i n( i.e., v : X 0 ! D s.t. v(x) 2 D x for all x 2 X 0 ). Example 2. Here are some domain instances: • {0, 1,...,100} =[0, 100] ✓ N finite domain over integers • {0, 1, 2, 3, 5, 7, 11} finite discontinuous domain enumeration • [0, 100] ✓ R infinite continuous domain • {0} [ [1, 100] ✓ R infinite semi-continuous domain Constraints A constraint is defined over a set of variables and represents a set of accepted valuations. Formally a constraint c over the variables X with domains D is semantically equivalent to a set of valuations from X to D: c ⌘ V such that V ✓ D X . Constraints can be represented in extension by enumerating accepted valuations or in intension by a predicate over the variables in the constraint. With Boolean variables, the atomic constraints are the logical predicates such as the negation (¬), the conjunction (^), the disjunction (_), the implication ()), the equivalence (,). For other domains, we consider atomic constraints as equations or inequations where their left-hand side and right-hand side are arithmetic expressions (i.e., any mathematical expressions such that polynomials, trigonometric functions, logarithms, exponentials, etc.). In the context of finite variables, the Constraint Programming community proposes a catalogue of constraints with a high level semantics called global constraints (cf. Section 2.2.5). According to the domains considered in this thesis (i.e., B, Z, Q,a n dR)o n ei m po r t a n tc o n s t r a i n tc h a r a c t e r i z a t i o n is the linearity. We say that a constraint is linear iff its arithmetic expressions are linear arithmetic expressions (i.e., not containing products of variables). Less importantly one may also consider the convexity properties of the arithmetic expressions. Furthermore, recall that there exist two quantifiers: the existential and the universal quantifiers. Thus, in quantified constraints, variables are associated to quantifiers and the CSP is satisfiable iff the quantifiers hold for the given domains (e.g., 9x 2 [-1, 1], 8y 2 [0, 1] : x + y  1i s satisfiable). In this thesis we only consider constraints with the existential quantifier (i.e., the universal quantifier is not allowed). Finally, a constraint problem is the composition of atomic constraints with logical operators. Example 3. Figure 2.2 describes three constraints c 1 , c 2 ,andc 3 over two variables x and y. Geometrically speaking, constraint c 1 defines a disc, c 2 is an upper half-space, and c 3 is a rectangle. Thus, c 1 can be expressed by a quadratic inequality, c 2 by a non-linear inequality, and c 3 by a conjunction of four linear inequalities. The pink zone contains all the solutions of the CSP C 1 with constraints c 1 , c 2 , c 3 over variables x, y with respective domains [0, 8] ✓ R,a n d[ 0 , 6] ✓ R. One may also consider the CSPs C 2 and C 3 which respectively contain the constraints c 1 ^(c 2 _ c 3 )a n dc 1 , (¬c 2 ^c3 ) producing different solution areas (i.e., solution spaces/feasible regions). Satisfaction and Optimization Problems A constraint satisfaction problem consists in determining if a constraint satisfaction program (cf. Definition 2.2.1)i se i t h e rsatisfiable or unsatisfiable. Formally, a valuation v satisfies a CSP P =(X, D, C)i ff there exists a valuation v over variables X satisfying all the constraints in C (i.e., the set of constraints in C is interpreted as a conjunction of constraints). If such a valuation v exists we say that P is satisfiable (and v is named a solution of P), otherwise we say that P is unsatisfiable. In the following, we call CSP family a set of CSPs sharing properties (e.g.,o n l y using integer variables, only considering linear constraints). Thus, according to a CSP family its theoretical complexity for the satisfaction problem may be polynomial or not, and either be undecidable. Table 2.1 from [7] synthesizes theoretical complexities for solving the satisfaction problem according to variables and constraints types. For instance the satisfaction of: a conjunction of linear constraints over real variables can be solved in polynomial time [START_REF] Gács | Khachiyan's algorithm for linear programming[END_REF]; a conjunction of constraints over integer finite variables is an NP-complete problem [START_REF] Thomas | The Complexity of Satisfiability Problems[END_REF]; non-linear constraints over unbounded integer variables is an undecidable problem [7]. Constraint Modelling Given a problem to answer, the objective of constraint modelling is to encode the problem to be solved into a constraint program such that a solution of the constraint program can be translated into a solution of the original problem. Definition 2.2.2 recalls the concept of CSP modelling. Constraint modelling is presented in Definition 2.2.3. In order to construct a constrain program P 0 modelling a constraint based problem P one must find a correspondence relation linking the valuations satisfying P 0 with the configurations belonging to P. Thus, if one is able to satisfy the CSP the correspondence relation ensures the existence of a configuration belonging to the constraint based problem. Furthermore, if the correspondence relation is decidable (ideally in polynomial time), one can construct at least one valid configuration from a solution given by the CSP. Definition 2.2.2 (Model). Let A be a set of objects, S be a set of object states, and P be a constraint based problem. We say that the CSP (X, D, C) models P iff there exists a correspondence relation R ✓ D X ⇥ S A s.t. 1. for each (v, v 0 ) 2 R, the valuation v satisfies C and the configuration v 0 belongs to P Modelling a constraint based problem as a constraint satisfaction program can be characterized in four steps. Definition 2.2.3 requires the existence of a correspondence relation. Thus, the programmer mainly builds the CSP while taking into account the correspondence between the original problem and the developed CSP. Firstly, the programmer identifies the decisions variables: i.e., the variables with a clear semantics in the problem to be solved. Secondly, she/he determines the auxiliary variables (i.e.,n o n decision variables used for intermediates constraints/computations). Thirdly, she/he sets the domain of each variable, also called the limits of each variable in the context of interval based domains. Fourthly, she/he adds the constraints that must be satisfied by the variables. These four steps are not necessarily straightforward and the programmer usually refines each step until a fix point is reached: the constructed CSP models the problem to solve. The following example proposes a modelling for the n-Queens problem. Note that this is a first modelling and that we are going to improve it in the following sections. Example 4 (Example 1 continued). We propose a first CSP modelling M 0 for solving the n-Queens problem where the decision variables model the columns and the lines chosen for the queens. Formally, let L be the set of all the n-Queens problems with n 2 N. M 0 is the mapping associating to each n-Queens problem in L the CSP (X, D, C)s u c ht h a tX contains one variable c i and one variable `i per queen index i 2 {1,...,n}. These variables respectively indicate the column and the line position of the ith queen on the chessboard. Furthermore, the domain for all these variables is {1,...,n} and the constraints are the followings ones for each pair (i, j)oftwodifferent queen indexes: 1. queens i and j are not on the same line: `i 6 = `j; 2. queens i and j are not on the same column: c i 6 = c j ; 3. queens i and j are not on the same diagonal: |(`i -`j)/(c ic j )| 6 = 1. Note the abstraction difference between the modelling M 0 and the CSP produced by M 0 which models the n-Queens problem in L with n 2 N fixed. The CSPs produced by M 0 have a quadratic size in terms of n (cf. the 3 ⇥ n 2 constraints) and use non linear constraints over integer variables. As in other programing paradigms (functional programming, object oriented programming, etc.) one problem can be written as many (syntactically) different constraint programs with equivalent semantics (i.e., they are all equivalently satisfiable or unsatisfiable or they all find the same optimal solution). We discuss this problematic in Section 2.3.2. Modeller. According to the type of variables (e.g., Boolean, integer, continuous variables) and the type of the constraints (e.g., linear, convex, non-linear, global constraints) one may look for the most appropriate research axes for modelling its problem. With an objective to share a common modelling language the mathematical programming community proposed A Mathematical Programming Language (AMPL for short) [START_REF] Fourer | Algorithms and Model Formulations in Mathematical Programming[END_REF] as an algebraic modelling language for describing CSPs. AMPL is supported by dozens of state-of-the-art tools for constraint program solving (e.g., CPLEX [11], Couenne [START_REF] Belotti | Mixed-integer nonlinear optimization[END_REF], Gecode [13], JaCoP [START_REF] Kuchcinski | JaCoP -Java Constraint Programming Solver[END_REF]). However, each research axe (each one specialized on dedicated families of CSPs) developed its proper modelling languages and tools. We present a landscape of CSPs families with their respective modelling languages and tools. 1. SAT (for Boolean Satisfiability Problem) contains CSPs with contraints over Boolean variables. The Conjunctive Normal Form (CNF for short) which consists of conjunctions of disjunctions of literals (e.g.,( x 1 _ ¬x 2 ) ^(x 2 _ x 3 )w h e r ex 1 , x 2 ,a n d x 3 are three Boolean variables) is the main practical modelling language used in this community. The DIMACS [15] format is the standard text format for CNF representation. See [START_REF] David | CNF Encodings[END_REF] for more details about CNF encodings. 2. LP, IP, MILP (respectively for Linear Programming, Integer Programming, and Mixed-Integer Linear Programming) contain constraint programs with respectively: linear constraints over continuous variables; linear constraints over integer variables; and linear constraints over continuous and integer variables. These three families are identified as mathematical programming languages. Formally the constraint programs of these families are presented in the following form: Ax  b where x is ac o l u m nv e c t o ro fv a r i a b l e sw i t hh e i g h tn,a n dA 2 R m,n and b 2 R m,1 are two matrices of coefficients. This encodes m inequalities over n variables. [START_REF] Aris | Mathematical modelling techniques[END_REF] recalls various modelling technics and use for these CSPs families. 3. FD (for Finite Domain Programming) contains constraint programs with constraints over variables with finite domains. There is no restriction on the constraints types. They can be linear, convex, non convex, non-linear such as trigonometric functions, exponential. There are also richer constraints expressed in the form of predicates, known as global (or meta) constraints which have been identified for their expressiveness (e.g., all-different, element, global-cardinality) and help the solution process (cf. Section 2.3.1). See the Global Constraint Catalogue for more details [START_REF] Beldiceanu | Global Constraint Catalog: Past, Present, and Future[END_REF]. Finally, there are two main formats for representing CSPs (XCSP3 [START_REF] Boussemart | XCSP3: An Integrated Format for Benchmarking Combinatorial Constrained Problems[END_REF]a n d FlatZinc [START_REF] Becket | Specification of FlatZinc[END_REF]) each one associated with a constraint modeller (resp. MCSP3 [START_REF] Lecoutre | MCSP3 : la modélisation pour tous[END_REF]and MiniZinc [START_REF] Nethercote | MiniZinc: Towards a Standard CP Modelling Language[END_REF]). While this CSP family allows any logical combination of constraints (negation, disjunction, implications, etc.) FD solvers are called propagation based solvers and are specialized for solving conjunction of constraints [START_REF] Rossi | Handbook of Constraint Programming[END_REF]. 4. SMT (for Satisfiability Modulo Theory) allows any logical combinations of constraints over continuous and integer variables. The satisfiability stands for the logical combination of constraints while the theory stands for the semantics of the combined constraints. The logical combination of constraints can use any logical constraints (i.e., conjunction, disjunction, negation, implication, equivalence). Theories range from linear-constraints to non-linear constraints with integer, real-number, Boolean, or any combination of these types (even bit vectors and floating-point numbers). The SMT-LIB format [START_REF] Barrett | The SMT-LIB Standard: Version 2.0[END_REF] is the standard format for representing CSP from this family. This norm also describes all the standard theories and their dependencies. SMT is more general than SAT, IP, LP, MILP. See [START_REF] De | Satisfiability Modulo Theories: Introduction and Applications[END_REF] for an introduction to and applications of SMT. Example 5 (n-Queens Problem Continued). We proposed in Example 4 the modelling M 0 for solving the n-Queens problem. This modelling can be transformed into a linear integer modelling M 1 producing CSPs from the IP family by replacing the non-linear constraints |(`i -`j)/(c ic j )| 6 = 1 by the constraints `i -`j 6 = c ic j and `i -`j 6 = c jc i . Furthermore, this modelling can be transformed into a FD modelling M 2 by replacing the 2 ⇥ n 2 constraints ensuring the "no threat" requirement by lines and columns with the only two following constraints: all-different(`1,...,`n)a n dall-different(c 1 ,...,c n ). Thus, M 2 models are smaller than those from M 0 and M 1 Consider the 5-Queens problem. M 0 , M 1 , and M 2 respectively produces 30, 40, and 22 constraints and have 10 variables. In addition to having less constraints, the models produced by M 2 use the all-different global constraint which ensures faster resolution than the use of a clique of binary inequality constraints. Example 5 illustrates how our n-Queens problem can be supported by the IP and the FD families. Thus, a constraint based problem can be modelled as many constraint programs such that each one can be targeted to possibly different constraint satisfaction program families. In the next section we explain how the different CSPs families are solved. Constraint Solving In this section we give an overview of CSP solving. While we presented in the previous section various CSP families for modelling constraint based problem, we present in this section how these families are solved in practice. Remark We present in this section some general methods for solving the CSPs families presented previously. However, before using the general solution one may also check if its problem does not belong to a subfamilies with practical/theoretical advantages. For instance, SAT community uses the Post's lattice for differentiating clones of Boolean functions for whose the satisfaction problem is in P or in NP [START_REF] Harry | Satisfiability Problems for Propositional Calculi[END_REF]. In FD these complexity differentiation are dichotomy theorems, one famous is the Schaefer's dichotomy theorem [START_REF] Thomas | The Complexity of Satisfiability Problems[END_REF]. Finally, in the non-linear programming context, the quadratic convex programming is in lower complexity class than non-linear programming [START_REF] Kozlov | Polynomial Solvability of Convex Quadratic Programming[END_REF]. Satisfaction In the general case, the combinatorics implied by the relations between the variables in the constraints makes a CSP hard to solve. In practice, complete solvers need to explore the search space (i.e., the set containing all the valuations). This is performed by branch and reduce algorithm where the search space is explored by developing a dynamic tree construction. Each node in the tree corresponds to a state in the exploration process (i.e., a succession of choices/decisions leading to a partial valuation of the variables and/or a reduction of the domains size and/or the adjunction of learned knowledges, etc.). Thus, ap a t hf r o mt h er o o tt oal e a fr e c u r s i v e l yc u tt h es e a r c hs p a c ei n t os m a l l e rs e a r c hs p a c e s until the satisfiability or the unsatisfiability is proven [START_REF] Land | An automatic method for solving discrete programming problems[END_REF][START_REF] Clausen | Branch and Bound Algorithms -Principles And Examples[END_REF]. The search starts from the root node which consists of the original CSP to solve (i.e.,a l lt h ev a l u a t i o n sa r e candidates possibly satisfying the CSP). Then, for each node in the tree exploration process the algorithm starts by reducing the current search space. This mainly consists in applying inferences rules such as resolution rules, computing consistency in order to reduce the search space while preserving all the valuations satisfying the CSP. Thanks to these reductions the next step checks if the reduced CSP is trivially satisfiable or unsatisfiable (e.g., the CSP has been syntactically reduced to a tautology, a contradiction, an empty set of constraints, etc.). If the CSP is trivially satisfiable, then a valid valuation can be found (mainly by reading the domains which has been reduced thanks to the successive cuts). containing the decisions history. If the CSP is trivially unsatisfiable, then this exploration path is closed and the exploration process carries on in an other opened exploration path. Otherwise, the current state space is split into possibly 2, 3, . . . , n smaller search spaces and the exploration process will be evaluated for each smaller CSP instances. Algorithm 1 recalls this generic search strategy. The two main generic functions are reduce and splitSearchSpace which respectively reduces the search of the CSP while preserving all the valuations satisfying it (i.e., this function may only remove unsatisfying valuations), and split the current CSP in many k CSPs (with a possibly different k 2 N at every loop iteration) such that the union of their search spaces cover the whole search space of the split CSP (it is not required to perform a partitioning and sub-problems may share valuations). Also, we considered here a queue as a CSP buffer but a more sophisticated object may be used to select dynamically the next buffered CSP to the treated. The algorithm stops when it finds a valuation satisfying one sub-problem. We now discuss how this generic is implemented for treating CSPs from various famillies. queue.add(P) 6: while not(queue.empty()) do # Reduces the CSP while preserving all solutions 9: P 0 reduce(P 0 ) 10: 11: # Case the CSP is trivially satisfiable after reduction: returns a sat valuation • In the SAT community the DPLL algorithm [START_REF] Davis | A Machine Program for Theorem-proving[END_REF] corresponds to the instantiation of splitSearchSpace by the selection of a non-instantiated variable x (i.e.,a variable x with domain {true, false})a n dt oth ec o n s tru c tio no ft w oC S P ss u c hth a t the first one contains the clause x and the second one contains the clause ¬x. Then, the reduce function performs unit propagation and pure literal elimination. The isTriviallySat function checks if constraints form a consistent set of literals and isTriviallyUnsat function checks the emptiness of the set of constraints. • In the FD community, the constraint propagation with backtracking method consists in instantiating splitSearchSpace and reduce in the following way. In general, splitSearchSpace starts by selecting a non-instantiated variable x. Then, it constructs one CSP per value k in the domain of x such that each constructed CSP is derived from the current CSP by setting the domain of x to the singleton domain {k}. We call search strategy a heuristic returning for a given CSP the next variables and domain values to use in order to realize the split search. On the other hand, the reduce function performs propagations by calling filtering algorithms and computing consistencies (e.g., node consistency, arc consistency, path consistency). Informally, a filtering algorithm removes values that do not appear in any solution. Global constraints usually come with dedicated filtering algorithms empowering the propagation process. Finally, the isTriviallySat function checks if the instantiated variables satisfy all the constraints and the isTriviallyUnsat function checks if a constraint is violated or if a variable domain becomes empty. See [START_REF] Rossi | Handbook of Constraint Programming[END_REF] for more details. • The branch-and-reduce framework used for solving non-linear programs with continuous variables, for instance HC4 [START_REF] Benhamou | Revising Hull and Box Consistency[END_REF], corresponds to instantiate in Algorithm 1 the function splitSearchSpace by the branching function (e.g., select a variable x with domains [a, b] ✓ R and a real number c 2 [a, b] in order to construct two CSPs which respectively contain the constraints x  c and x ≥ c), the reduce uses reducing consistencies in order to filter domain variables while preserving solutions. Finally, the isTriviallySat function guesses a valuation satisfying all the constraints for tight domains and the isTriviallyUnsat function checks if a constraint is violated or if a variable domain became empty. • The SMT community gathers solving techniques from various CSP families. Indeed, a SMT instance is considered as the generalization of a Boolean SAT instance in which various sets of variables are replaced by predicates (e.g., linear or non-linear expressions for continuous variables, integer constraints). Thus, the splitSearchSpace enumerates solutions of the SAT instance abstracting the CSP to solve (i.e., each constraint which is not a Boolean function is replaced by a unique Boolean variable). Then, each solution of the SAT instance is translated into a set of constraints which leads to a conjunction of constraints, each one being a sub-problem to be solved. According to the theory of this sub-problem (linear programming, integer programming, etc.) proper methods from the corresponding CSP family are used. This approach, which is referred to as the eager approach loses the high-level semantics encoded in the predicates. Actual SMT solvers now use a lazy approach solving partial SAT sub-problem and then answering their corresponding predicate parts while constructing a global solution [START_REF] Monniaux | A Survey of Satisfiability Modulo Theory[END_REF]. In SMT, we call strategy an implementation/construction of the splitSearchSpace and reduce functions. Furthermore each community (i.e., SAT, FD, etc.) provides many tools which implement the generic Algorithm 1. The performance of these tools is mainly due to their implementation of the reduce and splitSearchSpace functions inherited from years of research next to the practical resolution of real world problems. This includes the study and the availability of wide variety of concrete heuristics [START_REF] David | Branch-and-bound algorithms: A survey of recent advances in searching, branching, and pruning[END_REF] and search strategies [START_REF] Peter Van Beek | Backtracking Search Algorithms[END_REF] eventually branched with an offline or online learning (e.g., no good, learned clauses). In this thesis we focus on constraint modelling of constraint based problems and on domain reducing functions called propagators implemented in the reduce function. We presented independently how various CSPs families tackle the solving problem. However, some research has been realized in order to make them collaborate. For instance, the finite domains CSP family met continuous domains CSP family while preserving global constraints by linking CHOCO and IBEX solvers [START_REF] Fages | Combining finite and continuous solvers[END_REF]. Also, the integration of both IP and FD has been discussed helping to design a system such as SIMPL [START_REF] Tallys | An Integrated Solver for Optimization Problems[END_REF]. On the other hand, we already mentioned the fact that the SMT community uses solving techniques for clearly identified theories. In the same time, they started to include global constraints from FD and [START_REF] Bankovic | An Alldifferent Constraint Solver in SMT[END_REF] shows how the all-different constraint can be supported by SMT solvers which offers promising results. Finally, in [START_REF] Arbab | Coordination of Heterogeneous Distributed Cooperative Constraint Solving[END_REF] the authors develop cooperative constraint solver systems using a control-oriented coordination language. This work has been used for solving non-linear problems [START_REF] Monfroy | Implementing Non-linear Constraints with Cooperative Solvers[END_REF] and interval problems [START_REF] Granvilliers | Symbolic-interval Cooperation in Constraint Programming[END_REF] as well. We presented here a generic complete algorithm answering the constraint satisfaction problem. Such complete method always returns a valuation satisfying the given CSP if it exists and returns none if such valuation does not exist. Thus, an incomplete algorithm may not be able to indicate if the CSP is unsatisfiable but may find a valuation satisfying the constraint program. We consider complete solvers in this thesis. Improving Models As said in the modelling section there is more than one CSP which encodes a given constraint based problem. Furthermore, the time required for solving these equivalent CSPs may differ from one to another with possibly an exponential gap. We present here various methods exploring how CSPs can be improved for reducing solving time: reformulation, symmetry-breaking, redundant constraints, relaxation, and over-constraint. In all cases these improvements can be performed by hand. However, solvers may implement them for automatic uses. There exists model transformations for transposing a CSP from a family to another one. Definition 2.3.1 recalls the concepts of reformulation, i.e., how to produce a new CSP from an existing CSP modelling the same contraint based problem. A low level constraint language such that the Conjunctive Normal Form (i.e., a Boolean functions expressed as conjunctions of disjunctions of literals) language used in the SAT community is now tractable with SAT state-of-the-art solvers for millions of variables and constraints [START_REF] Marijn | Solving and Verifying the Boolean Pythagorean Triples Problem via Cube-and-Conquer[END_REF]. In some cases reformulating a satisfaction problem into a lower constraint language may offer better resolution times. For instance in [START_REF] Lardeux | Expressively Modeling the Social Golfer Problem in SAT[END_REF], the authors reformulate their modelling from CP to SAT which highly increases the sizes of the CSP. But the first modelling is not solved by CP solvers while the second one is solved by SAT solvers. Example 6 (n-Queens Refomulation). Consider the modelling M 2 presented in Example 5. Recall that the "no threat" rule for diagonals is managed by the constraints `i-`j 6 = c i -c j and `i -`j 6 = c jc i considered for each pair of two different queen indexes i and j. These constraints are equivalent to `ic i 6 = `jc j and `i + c i 6 = `j + c j . Since they must hold for each pair of two different queen indexes, all these constraints can be replaced by the two all-different global constraints. Since the all-different constraint support variables as inputs we create the auxiliary variables x i and y i for all i 2 {1,...,n} such that x i = `ic i and y i = `i + c i . Thus, the constraints expressing the "no threat" rule for diagonals can be replaced by the two constraints all-different(x 1 ,...,x n )a n d all-different(y 1 ,...,y n ). We call M 3 this modelling derived from M 2 . M 3 is called a reformulation of M 2 . M 3 models contain n +3 c o n s t r a i n t s w h e r e a s M 2 models contain a quadratic number of constraints in term of the number of queens n. Note that a reformulation may also change the variables and their domains and is not restricted to constraint modifications. Definition 2.3.2 (Symmetry). Le P be a constraint based problems over a set of objects A with states S. We say that P contains symmetries iff there exists a permutation σ of the set of configurations s.t. P is stable by σ.( i.e., σ(c) 2 L, for all c 2 L ✓ S A s.t. L ⌘ P.) A symmetry in a constraint based problem is a permutation of the configurations in the problem (cf. Definition 2.3.2). Thus, symmetry breaking consists in taking advantages of symmetry detection in constraint based problem to only model a subset of all the configurations in the problem, i.e., to only model the configurations which can not be obtained by symmetries. Symmetry breaking reduces the size of the search space and therefore, the time wasted in visiting valuations which are symmetric to the already visited valuations. The solution time of a combinatorial problem can be reduced by adding new constraints, referred as symmetry breaking constraints. We invite the reader to consider [START_REF] Cohen | Symmetry Definitions for Constraint Satisfaction Problems[END_REF] for more details. Example 7 (n-Queens Symmetry Breaking). Consider the modelling M 3 presented in Example 6. Let n be a fixed number of queens and C be the CSP produced by M 3 for the n queens problem. Note that in C the n queens are unordered: i.e., all the queens are totally identical. Thus, for any valuation solution of C one may interchange the values (`i,c i )r e p r e s e n t i n gt h ep o s i t i o no ft h eith queen with the values (`j,c j )r e p r e s e n t i n g the position of the jth queen to obtain another valuation solution of C. We construct a new modelling, named M 4 , ordering the queens and realizing some symmetry breaking. M 4 is such that for each n 2 N its corresponding CSP model (X, D, C) for solving the n-Queens problem contains the c i variables with domain {1,...,n} and no variable `i. Similarly to the previous modellings the c i variables represent the column position of the queens. However, in this modelling each c i with i 2 {1,...,n} is fixed with a line, the ith line. Thus, c i contains the column position of the queen on the ith line and there are no more `i variables in M 4 . The constraints x i = `ic i and y i = `i + c i from M 3 are respectively replaced by x i = ic i and y i = i + c i in M 4 for all i 2 {1,...,n}. To sum up, the constraints in C are the following ones: all-different(c 1 ,...,c n ), all-different(x 1 ,...,x n ), all-different(y 1 ,...,y n ), x i = ic i and y i = i + c i for all i 2 {1,...,n}. Note that this modelling still contains symmetries (e.g., chessboard rotations, chessboard plane symmetries). For instance setting the domain of the variable c 1 to {1,...,dn/2e} removes some chessboard plane symmetries. Definition 2.3.3 (Redundancy). Let C be a CSP and c be a constraint over a set of variables X. We say that c is a redundant constraint for C iff adding the constraint c to the CSP C does not change the solution space of C. In a general context minimizing the number of constraints (and/or the number of variables) in a CSP does not necessary implies lower solving time in practice. We call redundant constraint a constraint which does not change the set of valuations satisfying a given CSP when adding this constraint to the CSP (cf. Definition 2.3.3). In practice adding well chosen redundant constraints may speed up the solving process, or obtain as c a l eu p( s e e [START_REF] Asarin | Using Redundant Constraints for Refinement[END_REF] for instance). However, in the linear programming case detecting redundant constraints in order to remove them may accelerate the resolution process (see [START_REF] Paulraj | A comparative study of redundant constraints identification methods in linear programming problems[END_REF] for more details). Definition 2.3.4 (Relax & Over-constrain). Let P be constraint based problem over a set of objects A and a set of states S. Let C be a CSP. We say that C is: • a relaxed modelling of problem P iff there exists a constraint based problem P 0 s.t. CSP C models problem P 0 and all the configurations in P belongs to P 0 ; • an over-constrained modelling of problem P iff there exists a constraint problem P 0 s.t. CSP C models problem P 0 and all the configurations in P 0 belongs to P. As last modelling strategies we present the relaxation and the over-constrain cases (cf. Definition 2.3.4). A relaxation is an over-approximation of a difficult problem by a nearby problem that is easier to solve. For instance a relaxation may transform integer variables into real variables (indeed this produces a greater solution space), or may only consider a convex hull of the problem by using only linear inequalities (e.g., [START_REF] Telgen | On relaxation methods for systems of linear inequalities[END_REF]). An over-constrain model is an under-approximation. This consists in modelling a constraint based problem contained by the original problem to solve. Thus, by reducing the size of the search space, one may hope a gain in term of resolution time (see [START_REF] Lardeux | Set constraint model and automated encoding into SAT: application to the social golfer problem[END_REF] for instance). Real numbers vs. Floating-point numbers As presented previously, one common variable domain for modelling is the real-numbers domain, written R. When using this domain one expects that the solver takes into account the classical arithmetic properties verified by R (e.g., associativity, commutativity, infinite limits, etc.). In practice, computer scientists use floating-point numbers for simulating real numbers. Floating-point numbers represent a finite numbers of real numbers with finite (binary) representation. The IEEE 754 norm [START_REF]IEEE Standard for Binary Floating-Point Arithmetic[END_REF] is now considered as the norm for representing floating-point numbers in programs. This norm encodes 2 32 finite real numbers where the smallest non-zero positive number that can be represented is 1⇥10 -101 and the largest is 9.999999 ⇥ 10 96 , the full range of numbers is -9.999999 ⇥ 10 96 through 9.999999 ⇥ 10 96 ; it contains two signed zeros +0 and -0, two infinities +1 and -1,a n d two kinds of N aN s. In the following we write F for the set of real-numbers representable in the IEEE 754 norm. 3 The first notable fact is that floating-point arithmetic (i.e., arithmetic over F)i sn o te q u i v a l e n tt or e a ln u m b e ra r i t h m e t i c( i.e., arithmetic over R). Precision limitation with floating point numbers implies rounding: a real-number x 2 R which is not in F is rounded to one of the neartest floating-point number. The IEEE 754 norm describes five rounding rules (two rules round to a nearest value while the others are called directed roundings and round to a nearest value in a direction such as 0, +1, -1). For instance: 0.1 10 (number 0.1r e p r e s e n t e di nb a s e1 0 )d o e sn o th a v eafi n i t e representation in base 2 and thus, it does not belong to F; there exists floating-point numbers x 2 F s.t. x +1=x in the floating-point arithmetic. Implementing real number arithmetic in CSP solvers is challenging. Recall that we presented CSP valuation solutions as a mapping from the variables to their respective domains. Since some valuations to R may not be representable with floating-point numbers, solvers like RealPaver [START_REF] Granvilliers | Algorithm 852: RealPaver: an interval solver using constraint satisfaction techniques[END_REF] find reliable characterizations with boxes (Cartesian product of intervals) of sets implicitly defined by constraints such that intervals with floating point bounds contain real-number solutions. Thus, the real number valuation solutions are bounded by the interval valuation solutions. Since softwares take more and more control over complex systems with possibly critical impact on the society (e.g., car driving software, automatic action placements, ...) the verification community develops methods for ensuring the validity of such programs. After introducing in a first section the main objectives of program verification, we go deeper into the two verification fields concerned by our contributions. Our first contribution relates to Abstract Interpretation and the second one considers Model Checking for Markov chains. Finally, we present a brief overview concerning how constraint programming meets verification problems. Warning. We choose the word "program" for system change descriptions while the verification community also uses the word "model" with the same signification. Recall that we already introduced the word "model" in the constraint programming background (cf. Section 2.2). Thus we reserve it for the constraint context. We focus in this thesis on program verification problems, i.e., we do not consider hardware verification problems. In this context, the word "program" refers to a computer science program written in a dedicated programming language [START_REF] Knuth | The Art of Computer Programming[END_REF]. There is a wide variety of programming languages which can be grouped by programming paradigms: functional programming (e.g., Javascript, Python), object oriented programming (e.g., C++, Java), reactive programming (e.g., FAUST, LUSTRE), probabilistic programming (e.g., ProbLog, RML), etc. Even if these languages may have different programming approaches they all share the same verification expectations. Indeed, whatever the language, a program is designed to be executed (in our concern we consider that programs are executed on a machine with memory). We briefly recall some vocabulary proper to program verification. We call run an execution of a program. During a run the machine memory varies over the time. We call state as n a p s h o to ft h em e m o r ya tag i v e nt i m e . Finally, a trace is the succession of states corresponding to a run of the program. Thus, program verification consists in analyzing traces in order to determine if a given property is satisfied. We name concrete semantics the set of all the traces of a given program. The variations for a single program between its traces may come from user inputs given at running time, non determinism of the program, probabilistic transitions in the program, etc. Introduction Example 8. Figure 3.1 describes a simple program using one variable x as memory. Here, the concrete semantics of this program contains exactly four traces. Each trace is represented as a curve on the graphics such that each time step t corresponds to the value of x at this time. Verification consists in verifying properties on the concrete semantics of programs. Recall first that these semantics are an "infinite" mathematical object (i.e.,a ni n fi n i t e set of potentially infinite sequences of states) which is not computable: it is not possible to write a program able to represent and to compute all the possible traces of any program. Otherwise, one may also solve the halting problem [START_REF] Turing | On Computable Numbers, with an Application to the Entscheidungsproblem[END_REF]. Thus, in the general case, questions about the concrete semantics of a program are undecidable. In practice the program traces may be finite. However note that in this thesis both contributions only consider infinite traces. Properties on such traces may be expressed using different formalisms. First, they can be time independent. In such case the property must hold during each execution time (e.g.,t h ev a r i a b l ex must never be equal to zero, the variable x must always take its values between -1a n d1 ) : s u c hp r o p e r t i e sa r ec a l l e di n v a r i a n t s . S e c o n d l y ,t i m e dependent properties express how the memory must vary over time for each trace (e.g., the variable x must not be equal to zero before a given line of the program, the variable x must be bounded by -1 and 1 at the end of the execution, using temporal modalities such as the linear temporal logic [START_REF] Pnueli | The Temporal Logic of Programs[END_REF]). Finally, a verification process has the objective to determine according to a given program and a property if this program satisfies this property. Recall that the program verification problem is undecidable in the general case (the halting problem can be turned into a verification problem). Thus, such verification process may validate, invalidate, or be non conclusive concerning the satisfaction of the property by the program. We now present two main approaches, named static analysis and dynamic analysis, for tackling verification problems. Roughly speaking, the first one may be seen as offline verification and the second one as online verification. We call static analysis av erification process working without explicitly running the programs. On the other hand, a dynamic analysis verification process requires to run the program in order to validate or invalidate a property. Furthermore, both approaches also consider bounded verification vs. unbounded verification. Bounded verification only checks the validity of the properties for sub traces (i.e., for a bounded time range of execution of the program) while unbounded verification checks the validity of the properties for all traces regardless of their length. Recall that the total set of traces of a given program is called the concrete semantics of this program. Given a concrete semantics an abstraction is a mathematical model (possibly a program) representing at least all the traces in the concrete semantics. Thus, the verification process may be performed with an abstraction of the program instead of the original program itself. We say that a verification process is sound w.r.t. to a program abstraction iff it agrees with the verification of the concrete domain. Otherwise this process is called unsound. Example 10. Figure 3.3a contains a program described in pseudo code using two variables x and y.V a r i a b l e y is initialized to 1 and variable x is initialized to an integer randomly selected between 1 and 5. Then, the loop body is evaluated while the condition y<3 ^x<5i st r u e ,a n dfi n a l l y ,t h ev a r i a b l ex is incremented by 1. Clearly, this simple program admits 5 traces presented in Figure 3.3b. These different traces come from the random function on line 3 allowing 5 possible outputs. Note that traces may share states/state sequences (e.g., state (4, 2) is in traces trace 1 and trace 3 ), traces may have different lengths (e.g., |trace 1 | = |trace 2 | = |trace 3 | =6w h i l e|trace 4 | = 4 ). Also the number of traces may be exponential or even infinite. Recall that computing this number is an undecidable problem in the general case. As said previously, a verification process is performed on an abstraction of the concrete semantic. In order to bound all the variables at every program step, AI uses a connexion between the concrete semantics and the so called abstract semantics. Regarding a program concrete semantic and a variable in this program, we call concrete domain the values taken by the variables at each program step. On the other hand, for the same program and variable an abstract domain contains a super set of the the variable concrete domain at each program step (i.e., an abstract domain is an over-approximation of a concrete domain). While the concrete domain is unique there exist many possible abstract domains. Thus, AI proposes a variety of (mathematical) abstractions such that each one has advantage and disadvantage in term of precision, representativity, computability, etc. The problem of determining tight abstract domains becomes harder when the program contains loops. Indeed, this problem can be reduced to the search of inductive invariant (ideally the smallest). AI uses widening and narrowing operators in order to approach such solution. This is performed by considering an abstraction of the loop transfer function. Then, using this transfer function for testing over-approximation candidates and following successive widening and narrowing iterations terminates and converges to an inductive invariant. We invite the reader to consider [START_REF] Cousot | Static determination of dynamic prop erties of programs[END_REF][START_REF] Cousot | Abstract interpretation: a unified lattice model for static analysis of programs by construction or approximation of fixpoints[END_REF] for a formal and exhaustive presentation of AI since no more background is required for our contributions. describes the link between concrete and abstract semantics and presents the interest of tight abstract domains to possible false positive. This example presents the interval and the polyhedron abstractions. There exists other abstract domains such as the sign and the octogone abstract domains. Furthermore, a precision gain with an abstract domain increases the theoretical complexity for maintaining it while progressing in the verification process. However the use of abstractions may offer a guarantee of termination of the verification process. Indeed, well chosen abstractions produce semi-decidable problems. Thus verification tools based on abstract interpretation either proves the satisfaction of the bounding property or cannot conclude. Example 11. Assume that a program using two variables x and y admits at a program step the concrete semantics presented in Figure 3.4a. Thus, each dot in the figure corresponds to a state encountered by one or more program traces at the given program step. One possible abstraction for this concrete domain is the use of intervals to create a box (i.e., a cartesian product of intervals). Figure 3.4b presents the smallest interval domain containing the concrete semantics. On the other hand, Figure 3.4c presents the smallest polyhedron domain containing the concrete domain. According to this concrete domain and these abstract domains consider the unsafe region as the red region presented in Figure 3.4d. Since none of the states in the concrete domain overlap the unsafe region, the program satisfies the property. However, note that the interval domain intersects with the unsafe region in Figure 3.4e. Thus, this abstract domain is not able to prove the validity of the property on the concrete semantics. Conversely, the polyhedron abstract domain presented in Figure 3.4f successfully proves the validity of the property on the concrete semantics. Model Checking Model Also, in our setting Markov chains do not have final states. Thus, we consider that the accepted runs are all the infinite sequences of states with non zero transition probabilities. Indeed, each run is associated with a probability corresponding to the product of all the probabilities encountered on the transitions. Figure 3.5b present the prefixes with size 4 of five infinite runs with their corresponding probability. The probability for the runs not starting from the initial state or including a missing transition is set to zero. As presented in section 3.1 we consider system/program verification based on trace properties. Model checking (also named property checking) consists in exhaustively and automatically checking whether the model of a system meets a given specification/property. In the context of finite-state machines, such properties are expressed in order to discriminate infinite runs (e.g., all the runs meeting state B before state C, all the runs reaching D before 5 transitions). Since the number of states is finite, and runs are infinite, the number of different runs is infinite but countable. The Linear Temporal Logic [START_REF] Pnueli | The Temporal Logic of Programs[END_REF] (LTL for short) uses temporal operators (e.g., next, until)a l l o w i n gt od e fi n es u c h sets of traces. Finally, qualitative verification and quantitative verification take into account a measure over the traces and consist in checking that the set of runs accepted by Example 13 (Example 12 continued). Consider the property asserting that the runs must encounter state D. This property does not hold for the Markov chain presented in Figure 3.5a. Indeed, there exist runs infinitely looping in state B or C which never encounter state D (more precisely there exists an infinite number of such runs). On the other hand, consider the property asserting that the probability of encountering state D equals 1: this property holds. Indeed, the probability of looping infinitely in state B or C equals to zero. Thus, all the runs with a non zero probability reach D and the probability of reaching D equals to 1. Recall, that runs are infinite and the set of state is finite. Thus, model checking explores all possible system states in a brute-force manner. This way, it can be shown that a given system model formally satisfies or falsifies a certain property. Hence, such method proves the validity of the property or returns a counter-example otherwise. Constraints meet Verification Considerable improvements in the efficiency and expressive power of constraint program solvers allowed to tackle problems more and more difficult to answer. In this section, after motivating the use of constraint programming for answering the two mains program verification problems considered in this thesis we present various verification processes using constraint programming. Even if a program admits a priori an infinite state space its executions may encounter a (potentially infinite) subset of the declared state space. Thus, one would like to determine this smaller state space in order to verify the non reachability of undesired states. This problem is reducible to the search of program over-approximations, i.e.,b o u n ding all the program variables. This is an objective of Abstract Interpretation where the program describing precisely the system evolution from a state to another, named the concrete program, is abstracted. This abstracted construction is related to the concrete program in such a manner that if an over-approximation holds for the abstraction then, this approximation also holds for the concrete program. Furthermore, constraint programs allow to describe over-approximations such as convex polyhedrons using linear constraints, ellipsoids using quadratic constraints, etc. Thus, since constraint programming is a generic declarative programing paradigm it may be seen as a verification process for over-approximating variable in declarative programs. In the first contribution, we consider a block-diagram language where executions are infinite streams and the objective is to bound the stream values using constraint programming. However, bounding the state space is not enough for some verification requirements. In our second problem, the objective is to determine if a specific state is reachable at execution time. Indeed, abstractions can only determine if a specific state is unreachable. For this verification problem, we consider programs representable as finite graph structures where the nodes form the state space and the edges give state to state transitions. Thus, verifying the reachability of a state in such a structure is performed by activating or deactivating transitions in order to reach the target state. However, these activations can be restricted by guards, or other structural dependent rules. Clearly, this corresponds to a combinatoric problem to solve. For this reason, since one of the objectives of constraint programming is to solve highly combinatorial problems, the verification community is interested in the CP tools. Dynamic Analysis. Software testing consists in checking the validity of a property on ag i v e np r o g r a mb yr u n n i n gs i m u l a t i o n s . T h ec l a s s i c a lboo kThe Art of Software Testing [START_REF] Glenford | The Art of Software Testing[END_REF] defines software testing as "the process of executing a program with the intent of finding errors" (i.e., finding runs which do not satisfy the specification). Thus, Constraint-Based Testing is the process of generating program test cases by using the constraint programming technology [START_REF] Demillo | Constraint-Based Automatic Test Data Generation[END_REF]. The test cases are not written by hand but constraint programming solvers are used to produce them. A recent survey for this research field can be found in [START_REF] Gotlieb | Constraint-Based Testing: An Emerging Trend in Software Testing[END_REF]. Static Analysis. Static program analysis is the automatic determination of runtime properties of programs. This consists in finding run-time errors at compilation time without code instrumentation or user interaction. K. R. Apt formalized the link between chaotic iterations such as used in abstract interpretation for moving between fixed points or inductive invariants and the resolution process used in contraint programming [START_REF] Krzysztof | From Chaotic Iteration to Constraint Propagation[END_REF][START_REF] Krzysztof | The Essence of Constraint Propagation[END_REF]. More recently, in [START_REF] Pelleau | A Constraint Solver Based on Abstract Domains[END_REF], the authors integrate abstract domains into a constraint programming solver by developing a tool named Absolute. Bounded Model Checking. Binary Decision Diagrams (BDDs) have been used for formal verification of finite state systems with success since their introduction in the beginning of the 90's. However, in [START_REF] Biere | [END_REF] the authors proposed CNF modellings instead of BDD modellings for realizing Bounded Model Checking (BMC for short). This CNF based verification process takes advantages of the efficiency of the SAT solvers which are now considered as the state-of-the-art techniques for bounded model checking. This formulation in the BMC context led to a scale up in the size of the verified programs and also replaced the dedicated methods developed with BDDs. CP solvers for testing applications More generally, constraint programming solvers have been used to test applications thanks to the expressiveness and the efficiency of constraint programming languages. For instance, [START_REF] Gerault | Constraint Programming Models for Chosen Key Differential Cryptanalysis[END_REF] uses constraint propagation to check a cryptanalysis problem: by providing a better solution, they proved that a solution claimed to be optimal in two cryptanalysis papers was not optimal. Constraints in formal verification More and more the verification community uses SMTst o o l si n s t e a do fd e d i c a t e da l g o r i t h m sf o rp e r f o r m i n gv e r i fi c a t i o np r o c e s s . I n d e e d , the actual best state-of-the-art SMT solvers are able to handle linear, non-linear, and even quantified constrained programs which may appear in program verification problems. For instance, see [START_REF] Bjørner | Program Verification as Satisfiability Modulo Theories[END_REF] for symbolic software model checking or the Extended Static Checker (ESC) tool using the Simplify SMT solver [START_REF] Detlefs | Simplify: A Theorem Prover for Program Checking[END_REF]. This concludes our overview of constraint programming and program verification. After having put in perspective both research fields we now propose two chapters, each one self contained, about solving some program verification problems using constraint modelling and solving. Chapter 4 Verifying a Real-Time Language with Constraints Contents This chapter treats the verification of synchronous languages modelled as blockdiagrams. The verification problem consists in bounding all the variables in the program. This problem is called the stream over-approximation problem. We present a global constraint in the spirit of Constraint Programming designed to deal with this problem. We propose filtering algorithms inspired from abstract interpretation and prove their validity. These algorithms are inspired from both continuous constraint programming and abstract interpretation. Finally, we propose an implementation of our modellings and discuss the results. This chapter is self-contained including introduction, motivation, background, state of the art, and contributions. Introduction Constraint programing (CP) [START_REF] Montanari | Networks of Constraints: Fundamental Properties and Applications to Picture Processing[END_REF]o ffers a set of efficient methods for modelling and solving combinatorial problems. One of its key ingredients is the propagation mechanism, which reduces the search space by over-approximating the solution set. For continuous constraints [START_REF] Benhamou | Applying interval arithmetic to real, integer and Boolean constraints[END_REF][START_REF] Chabert | Contractor programming[END_REF], propagation is defined in a generic way on a given constraint language, usually containing equalities, inequalities, and many operators (arithmetic operations, mathematical functions, etc). In this chapter, we present a method using this generic propagation scheme, combined with a new solving algorithm, for the resolution of av e r i fi c a t i o np r o b l e m . Our problem consists in checking the range of the outputs of programs written as block-diagrams, a common model for many real-time languages. More precisely, we are interested in DSP (Digital Signal Process) programs, based on a block-diagram algebra, which contains both typical real-time operations (split, merge, delay, ...) and mathematical functions [START_REF] Orlarey | An Algebra for Block Diagram Languages[END_REF]. All the variables are infinite streams over the reals. A stream represents the values taken by a variable at each time step. All the variables/streams are synchronized and they all receive a value at each tick of the clock. All the loops are thus, in theory, infinite by construction: the programs do never stop by themselves. Of course, they may stop computing in practice when all the signals are constant, or alternatively they can be killed by the user. The problem we tackle is the following: considering a block-diagram, which comes from a real-time program on streams, can we compute or approximate at a good precision the range of the stream output by this program? This problem is, in a more general form, at the core of another research area, Abstract Interpretation, as introduced in [START_REF] Cousot | Abstract interpretation: a unified lattice model for static analysis of programs by construction or approximation of fixpoints[END_REF][START_REF] Cousot | Static determination of dynamic prop erties of programs[END_REF]. Abstract Interpretation offers a great variety of tools to over-approximate traces of programs to prove the absence of some runtime errors, such as overflows. It relies on abstractions of the program traces, i.e. the possible values the variables may take during an execution. The set of all the possible program traces cannot be computed in the general case. In Abstract Interpretation, they are represented by an abstract element, easier to compute, which must both include all the program traces and be reasonably easy to compute. One of the first examples of such abstraction is the interval abstract domain [START_REF] Cousot | Static determination of dynamic prop erties of programs[END_REF], which is used in this work. An abstraction comes with several operators to mimic the program execution. Abstract Interpretation has been successfully applied to a wide range of applications the most famous one being the analysis of the flight-control commands of the Airbus A380 aircraft. In this work, we use tools from constraint programming to compute precise abstractions of all the stream variables of a real-time program. Our method is generic and can be applied to any language based on a block-diagram algebra for bounding the values taken by the input, output, or inner streams of the program. We present three applications of our method: compiler assistance, refactoring, and verification. We chose the verification application for the experiment section that we applied on the FAUST (Functional Audio Stream)1 language. This language has been designed for sound design and analysis [START_REF] Orlarey | Syntactical and semantical aspects of Faust[END_REF] of audio streams. FAUST is a functional language with a proper semantic based on block-diagrams, which makes it a language quite similar to LUSTRE [START_REF] Halbwachs | The synchronous dataflow programming language Lustre[END_REF]o ri t s commercial version SCADE. In practice, the compiler automatically generates a blockdiagram for each program. The outputs of these block-diagrams are real-valued streams which represent audio signals. These signals can for instance be sent to loudspeakers or other applications. By convention, digital audio signals must stay in the range [-1, 1]. In case the signal takes values out of this range, it can either damage the loudspeakers, or, more currently, be arbitrarily cut by the sound driver. In this case, the shape of the sound is modified and this produces a very audible sound effect called saturation. For this purpose, we compute bounds for the values taken by all the signals in the program and then we verify that the output streams stay in [-1 ; 1]. Verifying FAUST programs is essential since this language is intended for non computer scientists. An overflowing program not only produces a corrupted sound, but in practice, it often has conception mistakes. Moreover, FAUST programs are now used in concerts or commercial applications [START_REF][END_REF]. FAUST already embarks a static analyzer based on Abstract Interpretation, using the interval abstract domain. The analyzer computes the outputs of each operator from its inputs, with the bottom-up top-down (or HC4) algorithm. When the programs do not have loops or delays, this works very well. However, as soon as the programs have loops, the interval analysis cannot provide precise over-approximation (returning [-1, +1]) and the analysis fails. In this chapter, we first propose a model of the verification of a block-diagram as a constraint problem. Propagation based on the constraints allows us to compute an over-approximation of the range of the computed streams. But as soon as the program has loops, the approximations are too large. We present a specific solving method which identifies the loops in the constraint graph, and propagates these constraints in a specific way to find over-approximation with a better precision. We implemented this method on FAUST block-diagrams, using IBEX [START_REF] Araya | Upper Bounding in Inner Regions for Global Optimization under Inequality Constraints[END_REF][START_REF] Chabert | Contractor programming[END_REF] a constraint programming solver over continuous domains. We tested it on several programs from the FAUST library. Most of the times, the over-approximation returned by our method is optimal, in the sense that it is the best interval approximation. We have tested our method on the programs given as examples in the standard FAUST library, with good results: we were able to detect errors in two of these programs, and in general to fastly compute precise intervals over-approximating the program outputs. This chapter is organized as follows: Section 4.2 introduces the notion of blockdiagrams. Section 4.3 presents the conversion of block-diagrams into a first constraint model. Section 4.4 defines the global constraint used for a more efficient model. Different applications of our optimized model are presented in Section 4.5 and we consider one of them in Section 4.6 with our application language and present the results of the experimentation followed by related works. Finally, Section 4.7 discusses the contribution and future works. Related Works The research on Constraint Programming and Verification has always been rich, and gained a great interest in the past decade. Constraint Programming has been applied to verification for test generation (see [START_REF] Gotlieb | Constraint-Based Testing: An Emerging Trend in Software Testing[END_REF] for an overview), constraintbased model-checking [START_REF] Podelski | Static Analysis: 7th International Symposium, SAS 2000[END_REF], control-flow graph analysis [START_REF] Lee | Compiler Construction: 16th International Conference[END_REF] or even worst-execution time estimations [START_REF] Bygde | An Efficient Algorithm for Parametric WCET Calculation[END_REF]. More recently, detailed approaches have been presented by [START_REF] Collavizza | Generating Test Cases Inside Suspicious Intervals for Floating-point Number Programs[END_REF]o r [START_REF] Ponsini | Verifying floating-point programs with constraint programming and abstract interpretation techniques[END_REF] to carefully analyze floating-points conditions with continuous constraint methods. Other approaches mix CP and Abstract Interpretation. It has been known for a long time that both domains shared a lot of ideas, since for instance [START_REF] Krzysztof | The Essence of Constraint Propagation[END_REF] which expresses the constraint consistency as chaotic iterations. A key remark is the following: Abstract Interpretation is about over-approximating the traces of a program, and Constraint Programming uses propagation to over-approximate a solution set. It is worth mentioning that one of the over-approximation algorithms used in Abstract Interpretation, the bottom-up top-down algorithm for the interval abstraction [START_REF] Cousot | Abstract interpretation: a unified lattice model for static analysis of programs by construction or approximation of fixpoints[END_REF][START_REF] Cousot | Abstract Interpretation Frameworks[END_REF], is the same as the HC4 constraint propagator [START_REF] Benhamou | Applying interval arithmetic to real, integer and Boolean constraints[END_REF] (in the following, we will refer to this algorithm as HC4), which shows how close CP and Abstract Interpretation can sometimes be. More recent works explored these links in both ways, either to refine CP techniques [START_REF] Denmat | An abstract interpretation based combinator for modeling while loops in constraint programming[END_REF][START_REF] Pelleau | A Constraint Solver Based on Abstract Domains[END_REF] or to improve the Abstract Interpretation analysis [START_REF] Ponsini | Refining Abstract Interpretation-based Approximations with Constraint Solvers[END_REF][START_REF] Di | Worst-Case Scheduling of Software Tasks -A Constraint Optimization Model to Support Performance Testing[END_REF]. Finally GATeL [START_REF] Blanc | Handling State-Machines Specifications with GATeL[END_REF] uses Constraint Logic Programming for verifying real-time programs by test cases generation. In some sense, our work can be seen as solving a constraint problem on streams. There have been other works on stream constraints in the literature (e.g., [START_REF] Jasperc | Towards Practical Infinite Stream Constraint Programming: Applications and Implementation[END_REF][START_REF] Lallouet | Constraint programming on infinite data streams[END_REF]). However, this approach radically differs from ours because their stream constraints are meant to build an automaton whose paths are solutions of the constraints. In particular, we would not be able to analyze infinite streams in a non-regular language with these b 1 op := x 2 b 2 op := - b 3 op := ⇥ Figure 4.1: A block-diagram in BD(R) stream constraints. On the contrary, our constraints are expressed on infinite streams, and generated in order to compute hulls of the streams. Background This section introduces the block-diagram algebra for representing real-time programs. Syntax A block is a function that applies an operator on some ordered inputs, and generates one or more ordered outputs. For any block, we say that input i (respectively output j)e x i s t si ff i (resp. j)i sa n integer between 1 and the number of inputs (resp. outputs) of the block. Throughout this chapter, given a nonempty set E, Block(E)d e n o t e st h es e to fa l lt h eb l oc k so v e rE. (b 1 [1], [1]b 2 )fromblockb 1 to block b 2 ;(b 1 [1], [1]b 3 )fromb 1 to b 3 and (b 2 [1], [2]b 3 ) from b 2 to b 3 . This block-diagram has two inputs (i.e.,[ 1 ] b 1 and [2]b 2 )a n do n eo u t p u t (i.e., b 3 [1]). Semantics After the syntax, it is natural to define the block-diagram semantics: block-diagram interpretation, and block-diagram model. An interpretation is any valuation of all the inputs and all the outputs. We introduce the notion of model to highlight the interpretations considering the operators (i.e., such that the outputs correspond to the image of the inputs by the operators) and the connectors. Stream Up to here, we built block-diagrams over arbitrary sets. Now, we consider the set of streams. A stream is an infinite discrete sequence of values possibly different at each time step. We abbreviate streams using bracket notation. For instance the stream s starting with the values 2, 4.5, and -3( i.e., s(0 ) = 2, s(1) = 4.5, s(2) = -3) is abbreviated in [2, 4.5, -3,...]. In the following, it is important to remind that all the streams are infinite. Considering block-diagrams over streams reveals two categories of blocks: functional blocks,a n dtemporal blocks. Functional blocks can be computed independently at each time step, whereas temporal blocks have time dependencies. Functional blocks are introduced in Definition 4.2.7. Temporal blocks are blocks which are not functional blocks (i.e., ab l o c ki se i t h e rf u n c t i o n a lo rt e m p o r a l ) . W ee x h i b i to n eb l o c ka m o n ga l lt h et e m p o r a l blocks: the fby block (cf. Definition 4.2.8). This block has two inputs and one output. The output at time zero is the value given by its first input at time zero. For the following times, the fby operator outputs its second input delayed by one time step. BD(S(R) ). d has no input and no output. d contains 5 functional blocks: 0, 0.1, 0.9, +, and ⇥. Blocks 0, 0.1a n d0 .9u s ec o n s t a n to p e r a t o r s( i.e., 8t 2 N :0 .9(t)=0 .9). Blocks + (resp. ⇥)w i t hr e a l -n u m b e ri n p u ts t r e a m sa, b and real-number output stream c is such that c(t)=a(t)+b(t)( r e s p . c(t)=a(t) ⇥ b(t)). d contains one temporal block: the fby block (note that temporal blocks are hatched). Block-diagrams over streams are used for programming real-time applications. In this context we name execution trace or simply trace amodelofabloc k-diagram. Acycleina 0.1 + 0.9 ⇥ fby 0 [0.9; 0.9; 0.9; 0.9] [0; 0; 0; 0] [0. block-diagram is equivalent to a loop in a classic programming language. In practice, in order to be runnable (i.e., to compute a trace in real-time) a block-diagram over streams needs to satisfy two properties [START_REF] Oppenheim | Signals & Systems[END_REF]: no value must be depending on future values (called the causality property); infinite computation in cycle must be avoided (any cycle must contain temporal blocks to avoid infinite computation at each time step). In our contribution we only allow to use the fby block as temporal block. Under this condition, this implies that for any runnable block-diagram over streams each cycle contains at least one fby block. From now on, we only consider block-diagram verifying this statement. We will see in Section 4.6 that this restriction is not poor and that the fby block allows to represent many other temporal blocks. Stream Over-Approximation Problem Block-diagrams over streams can express the semantics of real-time programs. In such cases, the programmer could be interested in the verification of some properties of his/her program. These properties can concern outputs or internal streams (i.e.,o u t p u t so r local variables). We propose here a logical constraint model for the following problem: determine bounds of the streams of a block-diagram. Temporal and Interval Abstractions We illustrate the problematic on our running example. Figure 4.4 presents the first 21 values for the output streams of blocks ⇥, +, and fby model of our running example from Figure 4.3. For the first 21 time steps, one can see that the values are strictly increasing (i.e., streams are strictly increasing) between 0 and 1. Furthermore the same observation is still correct for the first 100, 1, 000, 1, 000, 000, and more, time steps (in our example, model streams are "infinitely" strictly increasing). Clearly, the greedy algorithm running the block-diagram time by time and gathering the accessible states (a state is a tuple composed of the values of all the streams at one time step) until convergence (i.e.,n o new state is reached) may not halt. Furthermore, a block-diagram can admit an infinite uncountable set of models/traces. Thus, there is no hope to run all these traces for gathering all the reachable states. In this context, Abstract Interpretation [START_REF] Cousot | Abstract interpretation: a unified lattice model for static analysis of programs by construction or approximation of fixpoints[END_REF][START_REF] Cousot | Static determination of dynamic prop erties of programs[END_REF]o ffers a great variety of tools for over-approximating traces of programs. It relies on abstractions of the program traces. The set of all the possible program traces is undecidable in the general case. In Abstract Interpretation, they are represented by an abstract element, easier to compute, which must both include all the program traces and be reasonably easy to compute. One of the first examples of such abstraction is the interval abstract domain [START_REF] Cousot | Static determination of dynamic prop erties of programs[END_REF]. While finding one over-approximation of the traces is easy (i.e., returning [-1, +1]) the objective is to find over-approximations with good quality (i.e., as small as possible intervals). Following paragraphs formally introduce the over-approximation problem with the over-approximation quality comparator. set may be infinite, discontinuous and even uncountable (i.e., a representation in extension is thus not possible). Thereby, given a stream we consider an interval superset of the temporal abstraction for representing this stream (i.e., this corresponds to the use of the interval abstract domain in Abstract Interpretation [START_REF] Cousot | Static determination of dynamic prop erties of programs[END_REF]). The best over-approximation (in the intervals) of a stream, is the smallest interval containing its temporal abstraction. Definition 4.3.1 (Temporal Abstraction). The temporal abstraction of a stream s, written ṡ, is the set of all its values. Any superset of ṡ is called an over-approximation of s and ṡ is the smallest over-approximation of s. ṡ = [ t2N s(t) For each interval I, we write dIe its upper bound and bIc its lower bound. In the following, we assume that D is a totally ordered set and we write I(D)t h es e to fa l lt h e intervals over D. Furthermore, D is called the extended set of D and it is equal to the union of D and its limits (e.g., R = R [ {-1, +1} and I(R)ist h es e tc o n ta in in gallth e intervals with finite and infinite bounds). Finally, given A ✓ D we write [A] the smallest interval in I(D)c o n t a i n i n gA. Model in Constraint Programming Constraint programming (CP for short) is a declarative programming paradigm, in which a program consists of a list of variables (each one declared with a domain) together with a list of constraints over these variables. Firstly, we do constraint programming modelling with variables domains over streams. Secondly, we focus on interval constraint programming [START_REF] Benhamou | Continuous and Interval Constraints[END_REF], i.e., constraint programming with variable domains over set of intervals. is the set of all the domains associated to the variables in X; C is a set of constraints over variables from X. A constraint is defined over a set of variables x 1 ,...,x k from X with k 2 N and is a subset of D x 1 ⇥ ...⇥ D x k . A valuation v of P is a map from X 0 ✓ X to D s.t. v(x) 2 D x for all x 2 X 0 . A valuation v satisfies a constraint c ✓ D x 1 ⇥ ...⇥ D x k iff (v(x 1 ),...,v(x k )) 2 c . Finally, a valuation satisfies P iff it satisfies all the constraints in P. We recall that the over-approximation problem presented in the previous section asks for over-approximations of all the traces of the block-diagram. Since one solution of the naive model corresponds to one trace of the block-diagram, one must find all the solutions of the naive model to solve the over-approximation problem. In the previous section we motivated the use of over-approximations in the intervals for representing set of traces. We now present a second model with variables over intervals for solving the over-approximation problem. This model, called medium model, is derived from the naive model. It consists in: 1) the same variables where domains are over intervals instead of streams (i.e., over I(D) instead of S(D)); 2) the same signatures of constraints where the operators over streams are replaced by their corresponding constraint for interval propagation. Interval propagation combines various technics from interval arithmetic, interval constraint propagation, domain filtering with partial consistency algorithms. Note that these extensions are not trivial and continue to motivate researchers (see [START_REF] Benhamou | Continuous and Interval Constraints[END_REF][START_REF] Lhomme | Consistency techniques for numeric csps[END_REF][START_REF] Collavizza | A Note on Partial Consistencies over Continuous Domains[END_REF]). In the following, we will particularly use interval arithmetic and interval (constraint) propagation. Example 20 (Interval Arithmetic). Instances of interval computation : [2, 6] + [-1, 3] = [1, 9] [2, 6] -[-1, 3] = [-1, 7] [2, 6] -[2, 6] = [-4, 4] [2, 6] ⇥ [-1, 3] = [-6, 18] [-1, 3] ⇥ [-1, 3] = [-3, 9] [-1, 3] 2 =[ 0 , 9] Note that some properties in real-number arithmetic are not true in interval arithmetic. Examples above illustrate that in general A -A 6 =[0, 0] and A 2 6 = A ⇥ A. [f ] from (I(D)) n to (I(D)) m such that [f ](X 1 ,...,X n )=Y 1 ,...,Y m where Y i = [{ ẏi | 9x j 2 X j ,y 1 ,...,y m = f (x 1 ,...,x n )}]. Interval arithmetic received big interest since Moore [START_REF] Edgar | Interval Analysis[END_REF] and the developments of interval analysis. We focus on interval arithmetic with interval extension of real-valued functions. Interval arithmetic concerns how classical functions from real-number arithmetic operates on intervals (see Example 20). We propose Definition 4.3.3 for transposing function over streams to interval functions which extends the definition from [START_REF] Edgar | Interval Analysis[END_REF] for extending real-number functions to interval functions. Table 4.1 presents standard arithmetic functions over real-numbers next to their corresponding stream functions and interval extension functions. When it is not ambiguous (i.e., in the context of intervals) we omit the brackets over the function names in order to keep the expressions simpler. Real Function Stream Function Interval Extension Function a, b 7 ! a + b a, b 7 ! c, s.t. c(t)=a(t)+b(t), 8n 2 N [a 1 ,a 2 ], [b 1 ,b 2 ] 7 ! [a 1 + b 1 ,a 2 + b 2 ] a, b 7 ! a -b a, b 7 ! c, s.t. c(t)=a(t) -b(t), 8n 2 N [a 1 ,a 2 ], [b 1 ,b 2 ] 7 ! [a 1 -b 2 ,a 2 -b 1 ] a, b 7 ! a ⇥ b a, b 7 ! c, s.t. c(t)=a(t) ⇥ b(t), 8n 2 N [a 1 ,a 2 ], [b 1 ,b 2 ] 7 ! [c 1 ,c 2 ] s.t. c 1 = min(a 1 ⇥b 1 ,a 1 ⇥b 2 ,a 2 ⇥b 1 ,a 2 ⇥b 2 ) c 2 = max(a 1 ⇥b 1 ,a 1 ⇥b 2 ,a 2 ⇥b 1 ,a 2 ⇥b 2 ) a 7 ! a 2 a 7 ! c, s.t. c(t)=a(t) 2 , 8n 2 N [a, b] 7 ! [c, max(a 2 ,b 2 )] s.t. c =0,ifa  0  b c = min(a 2 ,b 2 ), otherwise Table 4.1: Real-number functions, stream functions, and interval extension functions for the addition, the subtraction, the multiplication and the square functions The interval extension function of an operator is not unique but the functions with smallest images will by preferred (i.e., the function always returning D is a universal interval extension function). Constraint propagation is one of the key ingredient for CSP resolution [START_REF] Benhamou | Continuous and Interval Constraints[END_REF]. This consists in explicitly removing values in some variables domains which cannot satisfy the CSP, while preserving all the solutions. A function performing such operation over one constraint is called a propagator (cf. Definition 4. • for all x 2 X \ X 0 : In such cases interval propagation will contract the domains blocks after blocks (i.e., constraint after constraint). However, the convergence may appear after a huge number of interval propagations. In order to reach the gap to better over-approximation in less time we introduce a new constraint: the real-time-loop constraint. D 00 x = D 0 x • for all x 2 X 0 : D 00 x ✓ D 0 x • for all valuations v of x 1 ,...,x n in D 0 x 1 ,...,D 0 xn : if v satisfies c, then v(x i ) 2 D 00 x i for all x i 2 X 0 . Definition The real-time-loop constraint will model cycles2 in block-diagrams. A cycle in a blockdiagram corresponds to a directed cycle in the directed graph representing it. A cycle is a sub block-diagram in a block-diagram. The real-time-loop constraint takes three arguments: the cycle itself as a block-diagram/list of constraints, the cycle inputs as a vector of variables and the cycle outputs as a vector of variables. Let d be a blockdiagram cycle, inputs be its inputs, and outputs be its outputs, we instantiate the real-time-loop constraint as: real-time-loop(d,inputs,outputs) . An interpretation satisfies a real-time-loop constraint if and only if it satisfies the list of constraints (i.e., all the constraints). According to this new constraint, we propose two propagators in the following sections. The first one propagates from input domains to output domains and the second one do the opposite way. Optimized Model This section describes how we exploit the structure of block-diagrams to improve the precision of the over-approximations using our real-time-loop constraint in an optimized model. Even if there is a thin syntactical difference between the medium model and this optimized model, there is a big gap in terms of deduction power. From a constraint programming point of view, these graphs are the constraints dependency graphs (where nodes are the CSP constraints), except that the arcs are directed by the dependencies implied by the blocks. Figure 4.8 draws the dependency graph of the block-diagram in Figure 4.5. Again, temporal block nodes are hatched. The optimized model is derived from the first one presented in section 4.3.2. Note that each strongly connected component (i.e., set of nodes such that it exists a path between any two nodes from this set) in the dependency graphs is related to a loop in the block-diagram. Thereby, regarding the dependency graph of the block-diagram, we compute its strongly connected components and we replace for each one all its corresponding constraints in the medium model by one real-time-loop constraint taking the strongly connected component as argument. Figure 4.9 models the block-diagram in Figure 4.5 using the real-time-loop constraint. Note that in this model the real-time-loop constraint has three variables as inputs and none as outputs. Inputs to outputs propagator We now present how to propagate the real-time-loop constraints from inputs to outputs: according to over-approximations of the inputs of the loop, we want to determine overapproximations for the outputs of the loop. Remember that the block-diagram is evaluated over infinite discrete time. Given a cycle/loop, we extract a transfer function for this loop and then, we consider the interval extension function of this transfer function in order to find over-approximations. F (I(x 1 )(t),...,I(x k )(t)) = I(x 1 )(t +1),...,I(x k )(t +1) Given a set of block inputs or outputs, a loop transfer function computes values at the next time according to values at a given time. Real-time languages must ensure the causality property [START_REF] Oppenheim | Signals & Systems[END_REF]( i.e., it must not exist a stream computing its values according to future values). Due to this property, it is clear that each cycle block-diagram admits at least one loop transfer function and even admits at least one loop transfer function with an argument of minimal size. This problem can be reduced to a "covering graph problem". Let d be a cycle block-diagram, G =(V, A)bethedependencygraphofd,and S ✓ V be a set of vertices. The set S 0 such that S 0 ◆ S, for all s 2 S 0 all its predecessors are in S 0 ,a n dS 0 is minimal, is named the cover of G by S. Furthermore, we say that S is a causal set of G if the cover of G by S equals to V . Thus, finding a loop transfer function with an argument of minimal size can be reduced to finding a minimal causal set and then performing a breadth-first search from this set for constructing the transfer function. We propose a greedy algorithm, Algorithm 2, for computing a minimal causal set of a dependency graph. It enumerates the subsets of V by starting from the subsets with minimal size and stops when it has found a causal set. In our benchmark presented in Table 4. 4, we can see that this greedy algorithm does not run out-of-time (i.e.,i n practice in our benchmark it does not enumerate all the subsets of V but only a small amount). Finally, we use the Definition 4.3.3 to get a loop transfer function extended to the intervals (in the following, we simply call it a loop transfer function too). Once we get this function, we want to over-approximate associated streams. Proposition 1 allows to do so by finding stable intervals such as defined in Definition 4.4.3. An example is given below. Definition 4.4.3 (Interval Stability). Let D be a non-empty set, F be an interval function with arity n 2 N, and S 1 ,...,S n be n intervals from I(D). We say that S 1 ,...,S n is stable by # Look for the first subset of V which is a causal set 7: F iff F (S 1 ,...,S n ) = S 0 1 ,...,S 0 n s.t. S 0 i ✓ S i for all 1  i  n. for each A ✓ V enumerated by increasing size do . Let f and g be two functions from R to R such that f (y)=fby(0,y ⇥ 0.9+0.1) and g(y)=0 .9 ⇥ fby(0,y)+0.1f o ra l ly 2 R. Remind from Figure 4.5 that the symbol a stands for the fby block output and the ⇥ block first input. We have that f with argument {a} and g with argument {c} are two loop transfer functions for the cycle in our block-diagram running example. and that the symbol c stands for the + block output and the fby block second input. Thus, for any model I of the block-diagram and for all time step t in N, the value associated to a (resp. c)b yI at time t + 1 corresponds to the image by f (resp. g)o ft h ev a l u ea s s o c i a t e dt oa (resp. c) by I at time t (i.e.,i th o l d st h a tI(a)(t +1)=f (I(a)(t)) and I(c)(t +1)=g(I(c)(t))). Let F and G be two functions from I(R)t oI(R)s u c ht h a tF (Y )=fby([0],Y ⇥ [0.9] + [0.1]) and G(Y )=[ 0 .9] ⇥ fby([0],Y)+[0.1] ( here in the context of intervals the function "⇥", "+", and "fby" are not the real valued functions but their respective interval extension functions). Function F extends f to the intervals and function G extends g to the intervals. We have that F with argument {a} and G with argument {c} are two loop transfer functions (extended to the intervals) for the cycle in our block-diagram running example. Considering the loop transfer function F . Intervals [0; 1], [-1; 1] and [-4; 3] are stable by F . (the images are respectively [0; 1], [-0.8; 1] and [-3.5; 2.8]). Thus, by Proposition 1 all these intervals are valid over-approximations for stream c (i.e., the argument of F ). On the contrary intervals ; and [0, 0] are not stable (their images are respectively [0; 0] and [0; 0.1]). We conclude that ; and [0, 0] are not valid over-approximations for stream c. One of our main contributions is Algorithm 3. We propose a method inspired by abstract interpretation techniques viewed as a constraint program to determine stable sets of intervals. Proposition 2 states the correctness of the algorithm. Note that this algorithm may not systematically return the minimal over-approximation, but in practice it gives acceptable ones (see experiments in Section 4.6). This algorithm starts by associating each argument element of the function to a search space bounded by the intervals min[i] and max[i] which are respectively initialized with the empty set and the extended set of the considered domain. Then, at each iteration current[i] is selected such that it contains min[i] and it is contained in max[i]( i.e., min[i] ✓ current[i] ✓ max[i]i sa n invariant of the loop). Also, the variable state takes its values between "Increasing" and "Decreasing" and is initialized to "Increasing". It switches from increasing to decreasing when the interval current[i] is stable by the transfer function and switches from decreasing to increasing when the contrary occurs. Finally, functions selectIntervalBetween and continueLooping are heuristics (resp. selection heuristic and looping heuristic). If S in I(D) k is stable by F then, S is an over-approximation of the elements in X. Proposition 2. [Algorithm 3 Correctness] Let (u n ) n2N , (v n ) n2N , and (w n ) n2N be the sequences of values taken respectively by the variables "min", "current", and "max" at each evaluation of the loop condition (line 8) during an execution of Algorithm 3 over a function F . The following statements hold: for each i from 1 to k do return max 38: end function Algorithm 3: Over-approximation random search function Proof for Statements 2 and 3 are obtained by induction on the number of evaluations of the loop condition. We first check the validity of both statements at the first evaluation of the loop condition (line 8). We have that u 0 1. (u n ) is increasing and (w n ) is decreasing 2. for all n 2 N: u n ✓ v n ✓ w n 3. for all n 2 N: w n is stable by F . Proof. Let (u n ) n2N ,(v n ) n2N , [i]=v 0 [i]=;,a n dw 0 [i]=D for all i 2 {1,...,k}. Clearly, u 0 [i] ✓ v 0 [i] ✓ w 0 [i] for all i 2 {1,...,k}. This implies that u 0 ✓ v 0 ✓ w 0 (Statement 2). Furthermore w 0 equals to D k makes F (w 0 ) [ w 0 = D k = [i]=u n [i] ✓ v n+1 [i] ✓ v n [i]=w n+1 [i] (Statement 2). Moreover, the value v n set to w n+1 (cf. max[i] current[i], for all i 2 {1,...,k}) verify F (v n ) [ v n = v n in the considered case. Thus w n+1 is stable by F (statement 3). Proofs for cases 2, 3, and 4a r es i m i l a r . Proof for Statement 1. Let i be in {1,...,k}. Note that min[i]( i.e., u n [i]) is only updated at line 20 and that max[i](i.e., w n [i]) is only updated at line 22. Both are updated with the value of current[i]( i.e., v n [i]). Thus, we get that for all n 2 N: u n+1 [i]=v n [i] or u n+1 [i]=u n [i]; and w n+1 [i]=v n [i]o rw n+1 [i]=w n [i]. We get by statement 2 that u n [i] ✓ u n+1 [i]a n dw n+1 [i] ✓ w n [i] and this is correct for all i in {1,...,n}. We conclude that u n ✓ u n+1 and that w n+1 ✓ w n , i.e., u n is increasing and w n is decreasing. Example 23 (Example 22 continued). Table 4.2 details a trace of Algorithm 3 for the transfer function F in Example 22. Each column corresponds to one iteration of the "while" loop. Each line gives the values of the variables at the end of each iteration, except for current which contains the value when starting the iteration. Outputs to inputs propagator For this section, outputs are given and we want to over-approximate with an interval (as small as possible) the set containing all the inputs that could generate those outputs. This is done by propagating all the constraints in the real-time-loop constraint until a fixpoint is reached. Indeed, since the outputs are fixed, propagating the constraints either reduce the input domains or do not change any domain. Since an input domain of a block can be an output domain of another block, we continue propagating the domains until no domain is modified. This procedure corresponds to the standard HC4 algorithm [START_REF] Benhamou | Applying interval arithmetic to real, integer and Boolean constraints[END_REF] from interval constraint programming. Applications We present three generic applications using our model for real-time programs that can be represented as Block-Diagrams. Verification Program verification consists in checking properties of a given program written in a specific language. Block-Diagram programs are designed to run on a definite (possibly infinite) duration. Users may be interested in ensuring that no problem will occur during execution (especially if the software failure can impact damages). From a programmer point of view, one of the classic properties that can be checked is to ensure that some strategic or critical variables will stay into a specific interval. This problem is usually known as overflow checking. In our CP approach, fixing the input and then solving our model makes it possible to compute over-approximation for each stream/variable. Refactoring Usually, a single semantics meaning can be implemented by many different syntactical writings. It is well known that the same result (even for a given algorithm) can be obtained by different implementations. Refactoring consists in restructuring an existing implementation without changing its external behavior. On Block-Diagrams, refactoring consists in removing or adding blocks or connections without changing the output values. For instance, an if-then-else condition which is always evaluated to "true" can be replaced by its "then" statement. This is a particular case of refactoring: removing dead code. In our CP approach, fixing inputs and outputs before solving enables removing blocks leading to empty over-approximations. Compilation Assistant Block-Diagram is a high-level programming language designed to create Real-Time programs in an elegant and human readable way. As seen in the previous sections, such languages can manipulate delays. Note that these delays can be the result of a complex computation. This implies that the maximum delay may be unknown at compilation time. Thus it must be given at run time and at each-time step (i.e., the delay can change during execution). Hence, if the compiler is able to estimate the maximal delay, no value will be missing at execution time. With our CP model we can bound maximum delay for temporal blocks: such information can be given to the compiler in order to allocate appropriate arrays for saving delays. Application to FAUST and Experiments For the application section, we chose the Real-Time language FAUST and we focused on a verification problem. FAUST allows us to manipulate audio streams. To illustrate this section, we selected the volume-controller program (a real-world program) from the official set of examples as the running example. We first introduce the FAUST language, then the constraint programming model for verification problem, and finally we conclude with experiments over a set of real-world FAUST programs. Model FAUST Programs FAUST (Functional Audio Stream) has been designed for real-time signal processing and synthesis. Figure 4.10 presents the compilation scheme for creating FAUST applications. First, it needs a program, called the source program, written in the dedicated FAUST language (this language is not significant for our contribution and is similar to other languages designed for digital signal processing). See [START_REF] Orlarey | An Algebra for Block Diagram Languages[END_REF] for more details. Then, this source program must be compiled by the FAUST compiler. This produces a C++ program that can finally be compiled with a usual C++ compiler by targeting the desired device. This hatched process, allows a single FAUST program to run on phones, web browsers, concert devices, etc. The goal of the FAUST compiler is to produce a C++ optimized code (i.e., a code with good performances and well managed memory in order to run efficiently in real-time, even on small devices). The actual FAUST compiler already contains various technics from the compilation research field for tackling this objective. As shown on Figure 4.10, it operates in four steps: Block Semantics Constraint Model b = mem(a) ( b(0) = 0 b(t)= a(t -1), if t>0 b =[a S 0] c = delay(a, b) ( c(t)= 0 , i f t<b(t) c(t)= a(t -b(t)), otherwise c =[a S 0] c = prefix(a, b) ( c(0) = a(0) c(t)= b(t -1), if t>0 c =[a S b] Table 4.3: FAUST temporal blocks • it loads the source program in an internal representation, easy to manipulate (i.e., block-diagram) • it rewrites this block-diagram to a normal form by syntactic analyzis (e.g., simplifying redundant forms such as xx by 0) • it performs a static analysis in order to compute approximations of the semantics of the program (e.g., estimate the maximal size for a delay) • and finally it produces the C++ program thanks to all the gathered information In our experiments, we use the model proposed in the previous sections to improve the static analysis inside the FAUST compiler. To do so, we will consider the blockdiagram just before the C++ code generation. Note that the normalization and the static analyzis made by the actual FAUST compiler helps working on expressions with few occurrences of the same variable: this is important for the constraint programming model since propagation over continuous variables performs poorly on variables occurring in many constraints [START_REF] Edgar | Interval Analysis[END_REF]( e.g., the stream "s 0 = ss"e q u a l st oz e r of o ra l lt i m ew h i l ei t s constraint model over intervals "S 0 = S -S"i sn o te q u i v a l e n tt o[ 0 ;0 ] ) . The FAUST language for writing source code has a formally well defined semantics in the Block-Diagram language [START_REF] Orlarey | An Algebra for Block Diagram Languages[END_REF] and is expressive thanks to: three temporal blocks (prefix, mem,a n ddelay); common arithmetic functions (e.g., addition, subtraction, ...); many C++ imported functions (e.g., sin, cos, exp, ...); relational and conditional operators. 3 All these block operators admit an interval extension (as defined in Definition 4.3.3)w i t han a t u r a lt r a n s l a t i o nt oi n t e r v a lc o n s t r a i n t s . I np a r t i c u l a r ,T a b l e4.3 presents the semantics and the models of the temporal blocks. Example 24. Figure 4.11 is our running example in FAUST (the FAUST Volume Controller Program) while Figure 4.5 is its equivalent representation in block-diagram (note that this block-diagram is not in normal form since the constant expression 1 -0.999 has not been reduced to 0.001). When running this program with FAUST, the graphical interface presents a slider (vslider in the FAUST source code stands for "vertical slider") allowing to control the output volume (left sliding reduces the volume and right sliding increases the volume). CP problems are formatted in three parts. The first one contains the variable declaration: it introduces the variables with their corresponding type (e.g., integer, real-number). The second one precises a domain as an interval for each declared variable. The third one contains the constraints. Figure 4.13 depicts these parts for our running example using the Medium model presented in Section 4.3 and the optimized model presented in Section 4.4. We can read that: only Variables 10, 14, and 18 are over integers; Variables 8, 10, 12, 14, 16, 18 correspond to constants from the block-diagram; Variable 17 models the vslider with range/domain [-70; 4]; and Variable 2 stands for the input audio stream 4 . Note that the normalization performed by FAUST and used for our CP modelling replaced the constant expression "1 -0.999" by the constant 0.001 (cf. Variable 12 in Figure 4.13b); replaced the expression "vslider / 20" by the expression "vslider ⇥ 0.05" (cf. in Figure 4.13b the constraint [15] =mul [START_REF] David | CNF Encodings[END_REF][START_REF] Aris | Mathematical modelling techniques[END_REF]); and introduced identity operators (cf. identity constraints over the Variables 4 and 5 in Figure 4.13b). Even if the identity operators increase the size of the CP model, they will not affect the quality of the over-approximations (i.e., identity propagation can be done without loss of precision). We discuss about this point and possible improvements in Section 4.7. The block-diagram contains one loop, and thus, it is not surprising to find out the corresponding real-time-loop constraint in the CP model (see Figure 4.13d). Verifying FAUST Programs We described how to model FAUST programs in CP. We now discuss about the CP solver. The solver has been implemented using IBEX. It is able to deal with two types of variables: real-numbers (i.e., in practice approximated by floating-point numbers intervals) and integers (i.e., C++ int). Table 4.4 presents the results for our benchmark programs. It is composed of some pathological DSP programs, and of real world programs from the FAUST standard library. They have been selected for their interest since they are basics for many bigger FAUST compositions. From left to right, columns of the table represent: the name of the FAUST program; the number of constraints followed by the number of variables in the medium model; the number of constraints followed by the number of variables in the optimized model; the number of real-time-loop constraints with the maximum number of constraints and the maximum number of arguments for the transfer function; the average time for compiling a FAUST program into the medium model; the average time for compiling the medium model into the optimized model; the average time for solving the optimized model; the over-approximation returned by the solver for the output stream associated with an indicator of reachability of the smallest p "s t a n d sf o rverified over-approximation by human while "?" stands for unverified over-approximation mainly due to the program complexity in term of number of blocks/streams. In order to get readable outputs, intervals are given in decimal format with a fixed precision of 10 -2 . Among those programs, counter is an incremental infinite loop starting at 0; noise generates a random noise (i.e., sequence of random numbers) ; oscillator generates an oscillating sound wave, freeverb generates a reverb on the input stream, firstorder-filter is well named and corresponds to a first-order filter. Note that benchmarks from counter to freeverb presented in Table 4.4 are fundamental block-diagrams for building more complex programs by composition. As instances of aggregation, we propose af a m i l yo f6be n c h m a r k sf o ra d d i t i v es y n t h e s i s [START_REF] Smith | Spectral Audio Signal Processing[END_REF] concataning from 5 to 1, 000 of these fundamental block-diagrams (cf. add-synth-X-oscs benchmarks in Table 4.4). The whole benchmark description, with the detailed information (DSP, block-diagram, and models) for each program, can be found at http://anicet.bart.free.fr/benchmarks/ FAUST. Each call to the real-time-loop constraint propagator runs five times Algorithm 3 and returns the intersection of the computed over-approximations. Each call to Algorithm 3 is limited to 500 loop iterations/transfer function evaluations. The selection heuristic in Algorithm 3 does intelligent search by selecting a new bound for the moving/changing bound (e.g., if the application of the transfer function does not change the lower bound of an interval, it will only select a new upper bound for the next evaluation). The selected precision for interval is 10 -5 . The solver has been launched 10 times for each benchmark and the averages of computation times and solutions on the 10 runs are presented in Table 4. 4. Experimentation has been done on a 2.4 GHz Intel Core i5 processor with a memory limit set to 16 Go. Results and Discussion Results in Table 4.4 can be partitioned into three sets. • counter, paper-example, sinus, noise, allpass-filter, volume, combfilter, echo, stereo-echo, oscillator add-synth-X-oscs are benchmark programs for which the returned solution is the smallest overapproximation of the output stream, i.e., the smallest interval containing all the possible values at any runtime execution. It is well known in abstract interpretation that first order filters, cannot generally be over-approximated efficiently using intervals. However, the first-order-filter benchmark is a special case (nevertheless a standard in FAUST) for which the floating-point interval abstraction is contracting. • pink-noise, capture, karplus-strong, band-filter are benchmark programs for which the returned solution is the smallest over-approximation of the output stream using interval analyses. Indeed, the analysis/propagation is made block by block/constraint by constraint and some patterns cannot give small overapproximation without knowing local semantics such as e.g., xfloor(x)c o r r esponds to the decimal part of x. • for the other programs (see lines in Table 4.4 containing the "?" symbol), the returned solution may not be the smallest over-approximation of the output stream but we were not able to prove it by hand. In order to be included into the FAUST compiler, the verification must be executed in averyshorttime(moreorlessasecond). F orourexperiments,thesolverperformswellon that matter: even in rather complex programs (such as freeverb or harpe)itisableto answer quickly. For most of the programs, the longest task is to compile the medium model into the optimized model. This is due to the use of an external library to represent graph data structures and compute strongly connected components. However it seems to have good scalability: Table 4.4 shows that even when the size of the benchmark program is multiplied by more than 20, the execution time is only multiplied by 2. Finally, according to the over-approximations computed with our method, a FAUST user expert confirmed the existence of saturation in 3 programs: volume, pitch-shifter, and mixer. The saturation came from the fundamental volume FAUST program, which contained an incorrectly set constant (i.e., a vslider ranging from [-70; 4] instead of [-70 ; 2]). Due to the execution time of our method and the quality of the returned solutions, the FAUST developers shown a big interest for integrating our contribution in a future version of the official FAUST compiler. Nevertheless, note that our add-synth-X-oscs benchmarks ranging from 5 to 1,000 oscillators illustrates an exponentiel tendance in compiling and solving time, compared to the block-diagram size. Related works The research on Constraint Programming and Verification has always been rich, and gained a great interest in the past decade. Constraint Programming has been applied to verification for test generation (see [START_REF] Gotlieb | Constraint-Based Testing: An Emerging Trend in Software Testing[END_REF] for an overview), constraint-based model-checking ( [START_REF] Podelski | Static Analysis: 7th International Symposium, SAS 2000[END_REF]), control-flow graph analysis [START_REF] Lee | Compiler Construction: 16th International Conference[END_REF], or even worst-execution time estimations ( [START_REF] Bygde | An Efficient Algorithm for Parametric WCET Calculation[END_REF]). More recently, detailed approaches have been presented by [START_REF] Collavizza | Generating Test Cases Inside Suspicious Intervals for Floating-point Number Programs[END_REF]or [START_REF] Ponsini | Verifying floating-point programs with constraint programming and abstract interpretation techniques[END_REF] to carefully analyze floating-points conditions with continuous constraint methods. Our contribution mixes CP and Abstract Interpretation. It has been known for a long time that both domains shared a lot of ideas, since for instance [START_REF] Krzysztof | The Essence of Constraint Propagation[END_REF] which expresses the constraint consistency as chaotic iterations. A key remark is the following: Abstract Interpretation is about over-approximating the traces of a program, and Constraint Programming uses propagation to over-approximate a solution set. It is worth mentioning that one of the over-approximation algorithms used in Abstract Interpretation, the bottom-up top-down algorithm for the interval abstraction [START_REF] Cousot | Abstract interpretation: a unified lattice model for static analysis of programs by construction or approximation of fixpoints[END_REF][START_REF] Cousot | Abstract Interpretation Frameworks[END_REF], is the same as the HC4 constraint propagator [START_REF] Benhamou | Applying interval arithmetic to real, integer and Boolean constraints[END_REF]. Recent works explored this links in both ways, either to refine CP techniques [START_REF] Pelleau | A Constraint Solver Based on Abstract Domains[END_REF], or to improve the Abstract Interpretation analysis [START_REF] Di | Worst-Case Scheduling of Software Tasks -A Constraint Optimization Model to Support Performance Testing[END_REF]. As a close work in the Constraint Programming community, GATeL [START_REF] Blanc | Handling State-Machines Specifications with GATeL[END_REF] is a software based on logical constraint programming verifying real-time programs. This tool first translates a Lustre program (representable as a block-diagram) and the specification of its environment in an equivalent Prolog representation, i.e., in a Constraint Logic Program (CLP). Then, it adds the user defined test objective in the CLP and solves it, computing at e s ti n p u ts a t i s f y i n gt h eo b j e c t i v ef o rt h eg i v e nL u s t r ep r o g r a m . T h i sw o r ka l r e a d y gathers the CP and the verification of real-time programs communities. However, while Gatel performs test cases generation for real-time programs we are interesting in finding precise over-approximations. As a close work in the Abstract Interpretation community, ReaVer5 is a state-of-theart software for safety verification of data-flow languages, like Lustre, Lucid Synchrone or Zelus (all are close to FAUST), providing time-unbounded analysis based on abstract interpretation techniques. It features partitioning techniques and several analysis methods [START_REF] Schrammel | Logico-Numerical Verification Methods for Discrete and Hybrid Systems[END_REF] (e.g., Kleene iteration based methods with increasing and descending iterations, abstract acceleration, max-strategy iteration, and relational abstractions; logico-numerical product and power domains with convex polyhedra, octagons, intervals, and template polyhedra). Considering our problem of over-approximating stream in block-diagrams, while a solver like ReaVer embarks many technics from Abstract Interpretation to answer this problem, in our approach we focus on how a slightly modified Constraint Programming solver can be turned into a verification tool with good performances (i.e., in computation time and in over-approximations qualities). Experiments in the previous section show that our realtime-loop constraint together with the proposed propagators achieve these objectives. However, it is clear that this approach is not competitive when the interval abstract domain cannot tightly fit the concrete domain (i.e., in these cases, polyhedra, octagons, or an other domains may provide better over-approximations). In some cases the interval [-1, +1] is returned as over-approximation of the output streams, which is indeed the smallest over-approximation of the output streams in the interval abstract domain while [-1, 1] is a valid over-approximation of the concrete output stream. Conclusion and Perspectives Conclusion We proposed a constraint model using a global constraint for overapproximation of real-time streams represented with block-diagrams. The experiments show that our approach can reach very good, nearly always optimal, over-approximations in a short running time. Our method has been taken in consideration for a future implementation into the FAUST compiler. In addition, we showed that constraint programming can handle block-diagram analyses in an elegant and natural way. The concept of digital signal processing is not proper to FAUST nor to audio processing. Indeed, it also appears in a lot of applications receiving and processing digital signals: modems, multimedia devices, GPS, video processing; which empower this model. Thus, this gives good perspectives for this work. Perspectives The results of our experiments are fast and of good quality. However, we would like to point out some possible improvements. A common way to improve performances is to consider pre-processing. This consists in taking advantage of some knowledge about the semantics of the problem in order to find faster a solution. According to the application (e.g., verification, refactoring) it could be interesting to propagate variables with respect to a global order. For instance for verification, it will be faster to propagate from inputs to outputs instead of a totally arbitrary order. Algorithm 3 for stable interval search applies many times the transfer function of the loop. Thus, reducing the number of blocks per transfer function would have two impacts: decreasing the time needed by the solver, and decreasing the number of variable multiple occurrences. Factoring sets of blocks with specific semantics would lead to better models from which faster and better over-approximation would be computed. For example, removing identity constraints, factoring sub block-diagram with specific meaning such as filter would lead to better models. This chapter treats model checking of qualitative and quantitative properties over abstractions of Markov chains. In particular, we show in the qualitative context how constraint modellings produce better models in terms of size and resolution time. We also present a formal theorem allowing to produce a first practical approach for verifying some quantitative properties on the considered Markov chain abstractions. Finally, we propose an implementation of our modellings and discuss the results. This chapter is self-contained including introduction, motivation, background, state of the art, and contributions. Introduction Discrete time Markov chains (MCs for short) are a standard probabilistic modelling1 formalism that has been extensively used in the litterature to reason about software [START_REF] James | A Markov chain model for statistical software testing[END_REF] and real-life systems [START_REF] Husmeier | Probabilistic Modeling in Bioinformatics and Medical Informatics[END_REF]. However, when modelling real-life systems, the exact value of transition probabilities may not be known precisely. Several formalisms abstracting MCs have therefore been developed. Parametric Markov chains [START_REF] Alur | Parametric Real-time Reasoning[END_REF] (pMCs for short) extend MCs by allowing parameters to appear in transition probabilities. In this formalism, parameters are variables and transition probabilities may be expressed as polynomials over these variables. A given pMC therefore represents a potentially infinite set of MCs, obtained by replacing each parameter by a given value. pMCs are particularly useful to represent systems where dependencies between transition probabilities are required. Indeed, a given parameter may appear in several distinct transition probabilities, therefore requiring that the same value is given to all its occurences. Interval Markov chains [START_REF] Jonsson | Specification and Refinement of Probabilistic Processes[END_REF] (IMCs for short) extend MCs by allowing precise transition probabilities to be replaced by intervals, but cannot represent dependencies between distinct transitions. IMCs have mainly been studied with three distinct semantics interpretations. Under the once-andfor-all semantics, a given IMC represents a potentially infinite number of MCs where transition probabilities are chosen inside the specified intervals while keeping the same underlying graph structure. The interval-Markov-decision-process semantics (IMDP for short), such as presented in [START_REF] Chatterjee | Model-Checking omega-Regular Properties of Interval Markov Chains[END_REF][START_REF] Sen | Model-Checking Markov Chains in the Presence of Uncertainties[END_REF], does not require MCs to preserve the underlying graph structure of the original IMC but instead allows an "unfolding" of the original graph structure: new probability values inside the intervals can be chosen each time a state is visited. Finally, the at-every-step semantics, which was the original semantics given to IMCs in [START_REF] Jonsson | Specification and Refinement of Probabilistic Processes[END_REF], does not preserve the underlying graph structure too while allowing to "aggregate" and "split" states of the original IMC in the manner of probabilistic simulation. Model-checking algorithms and tools have been developed in the context of pMCs [START_REF] Dehnert | PROPhESY: A PRObabilistic ParamEter SYnthesis To ol[END_REF][START_REF] Moritz Hahn | PARAM: A Model Checker for Parametric Markov Models[END_REF][START_REF] Kwiatkowska | PRISM 4.0: Verification of Probabilistic Real-Time Systems[END_REF] and IMCs with the once-and-for-all and the IMDP semantics [START_REF]Model Checking of Open Interval Markov Chains[END_REF][START_REF] Benedikt | LTL model checking of interval Markov chains[END_REF]. State of the art tools [START_REF] Dehnert | PROPhESY: A PRObabilistic ParamEter SYnthesis To ol[END_REF] for pMC verification compute a rational function on the parameters that characterizes the probability of satisfying a given property, and then use external tools such as SMT solving [START_REF] Dehnert | PROPhESY: A PRObabilistic ParamEter SYnthesis To ol[END_REF] for computing the satisfying parameter values. For these methods to be viable in practice, the allowed number of parameters is quite limited. On the other hand, the model-checking procedure for IMCs presented in [START_REF] Benedikt | LTL model checking of interval Markov chains[END_REF] is adapted from machine learning and builds successive refinements of the original IMCs that optimize the probability of satisfying the given property. This algorithm converges, but not necessarilly to a global optimum. It is worth noticing that existing model checking procedures for pMCs and IMCs strongly rely on their underlying graph structure (i.e., respect the onceand-for-all semantics). However, in [START_REF] Chatterjee | Model-Checking omega-Regular Properties of Interval Markov Chains[END_REF] the authors perform model checking of !-PCTL formulas on IMCs w.r.t. the IMDP semantics and they show that model checking of LTL formulas can b e solved for the IMDP semantics by reduction to the mo del checking problem of !-PCTL on IMCs with the IMDP semantics. For all that, to the best of our knowledge, no solutions for model-checking IMCs with the at-every-step semantics have been proposed yet. In this thesis chapter, we focus on Parametric interval Markov chains [START_REF] Delahaye | Parameter Synthesis for Parametric Interval Markov Chains[END_REF] (pIMCs for short), that generalize both IMCs and pMCs by allowing parameters to appear in the endpoints of the intervals specifying transition probabilities, and we provide four main contributions. First, we formally compare abstraction formalisms for MCs in terms of succinctness: we show in particular that pIMCs are strictly more succinct than both pMCs and IMCs when equipped with the right semantics. In other words, everything that can be expressed using pMCs or IMCs can also be expressed using pIMCs while the reverse does not hold. Second, we prove that the once-and-for-all, the IMDP, and the at-every-step semantics are equivalent w.r.t. reachability properties, both in the IMC and in the pIMC settings. Notably, this result gives theoretical backing to the generalization of existing works on the verification of IMCs to the at-every-step semantics. Third, we study the parametric verification of fundamental properties at the pIMC level: consistency, qualitative reachability, and quantitative reachability. Given the expressivity of the pIMC formalism, the risk of producing a pIMC specification that is incoherent and therefore does not model any concrete MC is high. We therefore propose constraint encodings for deciding whether a given pIMC is consistent and, if so, synthesizing parameter values ensuring consistency. We then extend these encodings to qualitative reachability, i.e., ensuring that given state labels are reachable in all (resp. none)o f the MCs modelled by a given pIMC. Finally, we focus on the quantitative reachability problem, i.e., synthesizing parameter values such that the probability of reaching given state labels satisfies fixed bounds in at least one (resp. all) MCs modelled by a given pIMC. While consistency and qualitative reachability for pIMCs have already been studied in [START_REF] Delahaye | Parameter Synthesis for Parametric Interval Markov Chains[END_REF], the constraint encodings we propose are significantly smaller (linear instead of exponential). To the best of our knowledge, our results provide the first solution to the quantitative reachability problem for pIMCs. Our last contribution is the implementation of all our verification algorithms in a prototype tool that generates the required constraint encodings and can be plugged to any SMT solver for their resolution. Background In this section we introduce notions and notations that will be used throughout this chapter. Given a finite set of variables X = {x 1 ,...,x k }, we write D x for the domain of the variable x 2 X and D X for the set of domains associated to the variables in X.A valuation v over X is a set v = {(x, d)|x 2 X, d 2 D x } of elementary valuations (x, d) where for each x 2 X there exists a unique pair of the form (x, d)i nv. When clear from the context, we write v(x)=d for the value given to variable x according to valuation v. A rational function f over X is a division of two (multivariate) polynomials g 1 and g 2 over X with rational coefficients, i.e., f = g 1 /g 2 .W ew r i t eQ for the set of rational numbers and Q X for the set of rational functions over X. The evaluation v(g)o fap o l y n o m i a lg under the valuation v replaces each variable x 2 X by its value v(x). An atomic constraint over X is a Boolean expression of the form f (X) ./ g(X), with ./ 2 {, ≥,<,>,=} and f and g two functions over variables in X. An atomic constraint is linear if the functions f and g are linear. A constraint over X is a Boolean combination of atomic constraints over X. Given a finite set of states S, we write Dist(S)f o rt h es e to fp r o b a b i l i t yd i s t r i b u t i o n s over S, i.e., the set of functions µ : S ! [0, 1] such that P s2S µ(s)=1. W ewriteI for the set containing all the interval subsets of [0, 1]. In the following, we consider a universal set of symbols A that we use for labelling the states of our structures. We call these symbols atomic propositions. We will use Latin alphabet in state context and Greek alphabet in atomic proposition context. Constraints. Constraints are first order logic predicates used for modelling and solving combinatorial problems [START_REF] Rossi | Handbook of Constraint Programming (Foundations of Artificial Intelligence)[END_REF]. A problem is described with a list of variables, each in a given domain of possible values, together with a list of constraints over these variables. Such problems are then sent to solvers which decide whether the problem is satisfiable, i.e., if there exists a valuation of the variables satisfying all the constraints, and in this case compute a solution. Recall that checking satisfiability of constraint problems is difficult in general (cf. Chapter 2). Formally, a Constraint Satisfaction Problem (CSP) is a tuple Ω =(X, D, C)w h e r eX is a finite set of variables, D = D X is the set of all the domains associated to the variables from X,a n dC is a set of constraints over X. We say that a valuation over X satisfies Ω if and only if it satisfies all the constraints in C. We write v(C)f orthes atisfac tionre sult of the valuation of the constraints C according to v (i.e., true or false). In the following we call CSP encoding as c h e m ef o rf o r m u l a t i n gag i v e np r o b l e mi n t oaC S P .T h es i z eo f a CSP corresponds to the number of variables and atomic constraints appearing in the problem. Note that, in constraint programming, having less variables or less constraints during the encoding does not necessarily imply faster solving time of the problems. Discrete Time Markov Chains. A Discrete Time Markov Chain (DTMC or MC for short) is a tuple M =( S, s 0 , p, V ), where S is a finite set of states containing the initial state s 0 , V : S ! 2 A is a labelling function, and p : S ! Dist(S)i sap r o b a b i l i s t i c transition function. We write MC for the set containing all the discrete time Markov chains. A Markov Chain can be represented as a directed graph where the nodes correspond to the states of the MC and the edges are labelled with the probabilities given by the transition function of the MC. In this representation, a missing transition between two states represents a transition probability of zero. As usual, given an MC M, we call a path of M asequenceofstatesobtainedfromexecutingM, i.e., a sequence ! = s 1 ,s 2 ,... such that the probability of taking the transition from s i to s i+1 is strictly positive, p(s i )(s i +1)> 0, for all i.Ap a t h! is finite iff it belongs to S ⇤ , i.e., it represents a finite sequence of transitions from M. Example 25. Figure 5.1 illustrates the Markov chain M 1 =( S, s 0 , p, V ) 2 MC where the set of states S is given by {s 0 ,s 1 ,s 2 ,s 3 ,s 4 }, the atomic proposition are restricted to {↵, β}, the initial state is s 0 , and the labelling function V corresponds to {(s 0 , ;), (s 1 , {↵}), (s 2 , {β}), (s 3 , {↵, β}), (s 4 , ↵)}. The sequences of states (s 0 ,s 1 ,s 2 ), (s 0 ,s 2 ), and (s 0 ,s 2 ,s 2 ,s 2 ), are three (finite) paths from the initial state s 0 to the state s 2 . Reachability. A Markov chain M defines a unique probability measure P M over the paths from M. According to this measure, the probability of a finite path ! = s 0 ,s 1 ,...,s n in M is the product of the probabilities of the transitions executed along this path, i.e., P M (!)=p(s 0 )(s 1 ) • p(s 1 )(s 2 ) • ... • p(s n-1 )(s n ). This measure naturally extends to infinite paths (see [START_REF] Baier | Principles of Model Checking (Representation and Mind Series[END_REF]) and to sequences of states over S that are not paths of M by giving them a zero probability. Given an MC M, the overall probability of reaching a given state s from the initial state s 0 is called the reachability probability and written P M s 0 (3s)o rP M (3s)w h e nc l e a r from the context. This probability is computed as the sum of the probabilities of all finite paths starting in the initial state and reaching this state for the first time. Formally, let reach s 0 (s)={! 2 S ⇤ | ! = s 0 ,...s n with s n = s and s i 6 = s 80  i<n } be the set of such paths. We then define P M (3s)= P ω2reachs 0 (s) P M (!)i fs 6 = s 0 and 1 otherwise. This notation naturally extends to the reachability probability of a state s from a state t that is not s 0 , written P M t (3s)a n dt ot h ep r o b a b i l i t yo fr e a c h i n gal a b e l↵ ✓ A written P M s 0 (3↵). In the following, we say that a state s (resp. a label ↵ ✓ A)i sr e a c h a b l ei nM iff the reachability probability of this state (resp. label) from the initial state is strictly positive. Example 26 (Example 25 continued). In Figure 5.1 the probability of the path (s 0 ,s 2 , s 1 ,s 1 ,s 3 )is0.3•0.5•0.5•0.5=0.0375 and the probability of reaching the state s 1 from s 0 is P M 1 s 0 (3s 1 )=p(s 0 )(s 1 )+Σ +1 i=0 p(s 0 )(s 2 )•p(s 2 )(s 2 ) i •p(s 2 )(s 1 )=p(s 0 )(s 1 )+p(s 0 )(s 2 )•p(s 2 )(s 1 )• (1/(1 -p(s 2 )(s 2 ))) = 1. Furthermore, the probability of reaching β corresponds to the probability of reaching the state s 2 , which is 0.3h e r e . Markov Chain Abstractions Modelling an application as a Markov Chain requires knowing the exact probability for each possible transition of the system. However, this can be difficult to compute or to measure in the case of a real-life application (e.g., because of precision errors or limited knowledge). In this section, we start with a generic definition of Markov chain abstraction models. Then we recall three abstraction models from the literature, respectively pMC, IMC, and pIMC, and finally we present a comparison of these existing models in terms of succinctness. A Markov chain Abstraction Model is a specification theory for MCs. It consists in as e to fa b s t r a c to b j e c t s ,c a l l e dspecifications, each of which representing a (potentially infinite) set of MCs -implementations -togetherwithasatisfactionrelationdefiningthe link between implementations and specifications. As an example, consider the powerset of MC (i.e., the set containing all the possible sets of Markov chains). Clearly, (2 MC , 2) is a Markov chain abstraction model, which we call the canonical abstraction model. This abstraction model has the advantage of representing all the possible sets of Markov chains but it also has the disadvantage that some Markov chain abstractions are only representable by an infinite extension representation. Indeed, recall that there exists subsets of [0, 1] ✓ R which cannot be represented in a finite space (e.g., the Cantor set [START_REF] Cantor | Uber unendliche, lineare Punktmannigfaltigkeiten V (On infinite, linear point-manifolds)[END_REF]). We now present existing MC abstraction models from the literature. Existing MC Abstraction Models Parametric Markov Chain is an MC abstraction model from [START_REF] Alur | Parametric Real-time Reasoning[END_REF] where a transition can be annotated by a rational function over parameters. We write pMC for the set containing all the parametric Markov chains. Definition 5.3.2 (Parametric Markov Chain). A Parametric Markov Chain ( pMC for short) is a tuple I =(S, s 0 ,P,V,Y) where S, s 0 , and V are defined as for MCs, Y is a set of variables (parameters), and P : S ⇥ S ! Q Y associates with each potential transition a parameterized probability. Let M =(S, s 0 , p, V )beanMCandI =(S 0 ,s 0 0 ,P,V 0 ,Y)beapMC. Thesatisfaction relation |= p between MC and pMC is defined by M| = p I iff S = S 0 , s 0 = s 0 0 , V = V 0 ,a n d there exists a valuation v of Y such that p(s)(s 0 )e q u a l sv(P (s, s 0 )) for all s, s 0 in S. Example 27. Figure 5.2 shows a pMC I 0 =( S, s 0 ,P,V,Y)w h e r eS, s 0 ,a n dV are identical to those of the MC M from Figure 5.1, the set Y contains only one variable p, and the parametric transitions in P are given by the edge labelling (e.g., P (s 0 ,s 1 )=0.7, P (s 1 ,s 3 )=p,a n dP (s 2 ,s 2 )=1-p). Note that the pMC I 0 is a specification containing the MC M from Figure 5.1. Interval Markov Chains extend MCs by allowing to label transitions with intervals of possible probabilities instead of precise probabilities. We write IMC for the set containing all the interval Markov chains. Definition 5.3.3 (Interval Markov Chain [START_REF] Jonsson | Specification and Refinement of Probabilistic Processes[END_REF]). An Interval Markov Chain ( IMC for short) is a tuple I =( S, s 0 ,P,V), where S, s 0 , and V are defined as for MCs, and P : S ⇥ S ! I associates with each potential transition an interval of probabilities. Example 28. Figure 5.3 illustrates IMC I =(S, s 0 ,P,V)whereS, s 0 ,a n dV are similar to the MC given in Figure 5.1. By observing the edge labelling we see that P (s 0 ,s 1 )= [0, 1], P (s 1 ,s 1 )=[ 0 .5, 1], and P (s 3 ,s 3 )= [ 1 , 1]. On the other hand, the intervals of probability for missing transitions are reduced to [0, 0], e.g., P (s 0 ,s 0 )=[0, 0], P (s 0 ,s 3 )= [0, 0], P (s 1 ,s 4 )=[0, 0]. In the litterature, IMCs have been mainly used with three distinct semantics: atevery-step, interval-Markov-decision-process and once-and-for-all. All these semantics are associated with distinct satisfaction relations which we now introduce. The once-and-for-all IMC semantics [START_REF] Dehnert | PROPhESY: A PRObabilistic ParamEter SYnthesis To ol[END_REF][START_REF] Wongpiromsarn | TuLiP: A Software Toolbox for Receding Horizon Temp oral Logic Planning[END_REF][START_REF] Puggelli | Polynomial-Time Verification of PCTL Properties of MDPs with Convex Uncertainties[END_REF] is alike to the semantics for pMC, as introduced above. The associated satisfaction relation |= o I is defined as follows: An MC M =(T,t 0 , p, V M )s a t i s fi e sa nI M CI =(S, s 0 ,P,V I )i ff (T,t 0 ,V M )=(S, s 0 ,V I )a n df o r all reachable state s and all state s 0 2 S, p(s)(s 0 ) 2 P (s, s 0 ). In this sense, we say that MC implementations using the once-and-for-all semantics need to have the same structure as the IMC specification. Next, the interval-Markov-decision-process IMC semantics (IMDP for short) [START_REF] Chatterjee | Model-Checking omega-Regular Properties of Interval Markov Chains[END_REF][START_REF] Sen | Model-Checking Markov Chains in the Presence of Uncertainties[END_REF] operates as an "unfolding" of the original IMC by picking each time a state is visited a possibly new probability distribution which respects the intervals of probabilities. Thus, this semantics allows to produce MCs satisfying IMCs with a different structure. Formally, the associated satisfaction relation |= d I is defined as follows: An MC M =( T,t 0 , p, V M ) satisfies an IMC I =(S, s 0 ,P,V I )iff there exists a mapping ⇡ from T to S s.t. ⇡(t 0 )=s 0 , V I (⇡(t)) = V M (t)f o ra l ls t a t et 2 T , p(t)(t 0 ) 2 P (⇡(t), ⇡(t 0 )) for all pair of states t, t 0 in T , and for all state t 2 T and all state s 2 S there exists at most one state t 0 2 Succ(t) such that ⇡(t 0 )=s. Thus, we have that |= d I is more general than |= o I (i.e., whenever M| = o I I we also have M| = d I I). Note that in [START_REF] Chatterjee | Model-Checking omega-Regular Properties of Interval Markov Chains[END_REF][START_REF] Sen | Model-Checking Markov Chains in the Presence of Uncertainties[END_REF] the authors allows the Markov chains satisfying the IMCs w.r.t. |= d I to have an infinite state space. In this work we consider Markov chains with a finite state space. Finally, the at-every-step IMC semantics, first introduced in [START_REF] Jonsson | Specification and Refinement of Probabilistic Processes[END_REF], operates as a simulation relation based on the transition probabilities and state labels, and therefore allows MC implementations to have a different structure than the IMC specification. Compared to the previous semantics, in addition to the unfoldings this one allows to "aggregate" and "split" states from the original IMC. Formally, the associated satisfaction relation |= a I is defined as follows: An MC M =( T,t 0 , p, V M )s a t i s fi e sa nI M CI =(S, s 0 ,P,V I ) iff there exists a relation R ✓ T ⇥ S such that (t 0 ,s) 2 R and whenever (t, s) 2 R,w e have 1. the labels of s and t correspond: V M (t)=V I (s), 2. there exists a correspondence function δ : T ! (S ! [0, 1]) s.t. a) 8t 0 2 T if p(t)(t 0 ) > 0t h e nδ(t 0 )i sad i s t r i b u t i o no n S b) 8s 0 2 S :(Σ t 0 2T p(t)(t 0 ) • δ(t 0 )(s 0 )) 2 P (s, s 0 ), and c) 8(t 0 ,s 0 ) 2 T ⇥ S,i fδ(t 0 )(s 0 ) > 0, then (t 0 ,s 0 ) 2 R. Example 29 illustrates the three IMC semantics and Proposition 3 compares them. We say that an IMC semantics |= 1 is more general than another IMC semantics |= 2 iff for all IMC I and for all MC M if M| = 2 I then M| = 1 I. Also, |= 1 is strictly more general than |= 2 iff |= 1 is more general than |= 2 and |= 2 is not more general than |= 1 . Note that two probability distributions have been chosen for the state s 1 from I. This produces two states t 1 and t 0 1 in M 2 and changes the structure of the graph. Thus, M 2 6 |= o I I and M 2 |= a I I. Finally, in the MC M 3 with state space T the state s 3 from I has been "split" into two states t 3 and t 3 0 and the state t 1 "aggregates" the states s 1 and s 4 from I. The relation R ✓ T ⇥ S containing the pairs (t 0 ,s 0 ), (t 1 ,s 1 ), (t 1 ,s 4 ), (t 2 ,s 2 ), (t 3 ,s 3 ), and (t 3 0 ,s 3 )i sas a t i s f a c t i o nr e l a t i o nbe t w e e nM 2 and I such as defined by |= a I . Thus, M 3 |= a I I. On the other hand, M 3 6 |= d I I since there exist probabilities on transitions that cannot belong to their respective interval of probabilities on the IMC (e.g., p(t 2 ,t 1 )=0.8 6 2 [0, 0.6] = P (s 2 ,s 1 )). Proposition 3. The at-every-step satisfaction relation is (strictly) more general than the interval-markov-decision-process satisfaction relation which is (strictly) more general than the once-and-for-all satisfaction relation. I : ⇡(t 0 )=t 0 = s 0 , V 0 (s)=V (s)=V (⇡(s)) for all state s 2 S,a n dp(s)(s 0 ) 2 P (⇡(s), ⇡(s 0 )) since P (⇡(s), ⇡(s 0 )) = P (s, s 0 )a n dp(s)(s 0 ) 2 P (s, s 0 )f o ra l ls, s 0 2 S. Thus, M| = d I I. (2) If M| = d I I then there exists a mapping ⇡ from T to S s.t. ⇡(t 0 )=s 0 , V 0 (t)= V (⇡(t)) for all state t 2 T ,a n dp(t)(t 0 ) 2 P (⇡(t), ⇡(t 0 )) for all pair of states t, t 0 in T . The relation 3 . Consider the correspondence function δ from T to (S ! [0, 1]) such that δ(t 1 )(s 1 )=4 /5, δ(t 1 )(s 2 )=1 /5, δ(t 2 )(s 2 )=1,δ(t 0 )(s 0 )=1,andδ(t)(s)=0otherwise. Ontheotherhand,sincethe outgoing probabilities from state t 0 in M 0 3 do not belong to their respective interval on probabilities in I,w eh a v et h a tM 0 3 6 |= d I I. s 0 ; s 1 α s 2 α [0, 0.4] [0.6, 1] [0, 1] [0, 1] [0, 1] [0, 1] IMC I t 0 ; t 1 α t 2 α 0.4 0.6 0.5 0.5 0.5 0.5 MC M 0 1 t 0 ; t 1 α t 2 α t 2 0 α 0.4 0.6 1 1 MC M 0 2 t 0 ; R = {(t, ⇡(t)) | t 2 T } is such Parametric Interval Markov Chains, as introduced in [START_REF] Delahaye | Parameter Synthesis for Parametric Interval Markov Chains[END_REF], abstract IMCs by allowing (combinations of) parameters to be used as interval endpoints in IMCs. Under a given parameter valuation the pIMC yields an IMC as introduced above. pIMCs therefore allow the representation, in a compact way and with a finite structure, of a potentially infinite number of IMCs. Note that one parameter can appear in several transitions at once, requiring the associated transition probabilities to depend on one another. Let Y be a finite set of parameters and v be a valuation over Y . By combining notations used for IMCs and pMCs the set I(Q Y )c o n t a i n sa l lp a r a m e t r i z e di n t e r v a l so v e r[ 0 , 1], and for all s 0 s 1 α s 2 β s 3 α, β s 4 α [0, 1] [0, 1] [q, 1] [0.3,q] [0,p] [0.2,p] [0, 0.5] 1 [0, 0.5] [0.5, 1] I =[ f 1 ,f 2 ] 2 I(Q Y ), v(I)d e n o t e st h ei n t e r v a l[ v(f 1 ),v(f 2 )] if 0  v(f 1 )  v(f 2 )  1 and the empty set otherwise2 . We write pIMC for the set containing all the parametric interval Markov chains. Definition 5.3.4 (Parametric Interval Markov Chain [START_REF] Delahaye | Parameter Synthesis for Parametric Interval Markov Chains[END_REF]). A Parametric Interval Markov Chain ( pIMC for short) is a tuple P =(S, s 0 ,P,V,Y), where S, s 0 , V and Y are defined as for pMCs, and P : S ⇥ S ! I(Q Y ) associates with each potential transition a (parametric) interval. In [START_REF] Delahaye | Parameter Synthesis for Parametric Interval Markov Chains[END_REF] the authors introduced pIMCs where parametric interval endpoints are limited to linear combination of parameters. In our contribution we extend the pIMC model by allowing rational functions over parameters as endpoints of parametric intervals. Given ap I M CP =( S, s 0 ,P,V,Y)a n dav a l u a t i o nv, we write v(P)f o rt h eI M C( S, s 0 ,P v ,V) obtained by replacing the transition function P from P with the function P v : S ⇥ S ! I defined by P v (s, s 0 )=v(P (s, s 0 )) for all s, s 0 2 S. The IMC v(P)i sc a l l e da ninstance of pIMC P. Finally, depending on the semantics chosen for IMCs, three satisfaction relations can be defined between MCs and pIMCs. Finally, the parametric intervals from the transition function P are given by the edge labelling (e.g., P (s 1 ,s 3 )=[ 0 .3,q], P (s 2 ,s 4 )=[ 0 , 0.5], and P (s 3 ,s 3 )= [ 1 , 1]). Note that the IMC I from Figure 5 In the following, we consider that the size of a pMC, IMC, or pIMC corresponds to its number of states plus its number of transitions not reduced to 0, [0, 0] or ;. We will also often need to consider the predecessors (Pred), and the successors (Succ)o fs o m eg i v e n states. Given a pIMC with a set of states S, a state s in S, and a subset S 0 of S, we write: • Pred(s)={s 0 2 S | P (s 0 ,s) / 2 {;, [0, 0]}} • Succ(s)={s 0 2 S | P (s, s 0 ) / 2 {;, [0, 0]}} • Pred(S 0 )= S s 0 2S 0 Pred(s 0 ) • Succ(S 0 )= S s 0 2S 0 Succ(s 0 ) Abstraction Model Comparisons IMC, pMC, and pIMC are three Markov chain Abstraction Models. In order to compare their expressiveness and compactness, we introduce the comparison operators v and ⌘. Let (L 1 , |= 1 )and(L 2 , |= 2 ) be two Markov chain abstraction models containing respectively L 1 and L 2 . We say that L 1 is entailed by L 2 , written L 1 v L 2 ,i ff all the MCs satisfying L 1 satisfy L 2 modulo bisimilarity. (i.e., 8M| = 1 L 1 , 9M 0 |= 2 L 2 s.t. M is bisimilar to M 0 ). Definition 5.3.5 recalls the bisimilarity property from [START_REF] Baier | Principles of Model Checking (Representation and Mind Series[END_REF]. We say that L 1 is (semantically) equivalent to L 2 , written L 1 ⌘ L 2 ,i ff L 1 v L 2 and L 2 v L 1 . Definition 5.3.6 introduces succinctness based on the sizes of the abstractions. Definition 5.3.5 (Bisimulation [START_REF] Baier | Principles of Model Checking (Representation and Mind Series[END_REF]). Let M =( S, S 0 , p, V ) be an MC possibly containing more than one initial state (i.e., S 0 ✓ S). A probabilistic bisimulation on M is an equivalence relation R on S such that for all states (s 1 ,s 2 ) 2 R: V (s 1 )=V (s 2 ) and σ t2T p(s 1 ,t)=σ t2T p(s 2 ,t) for all T 2 S/R. We say that two MCs M 1 and M 2 are bisimilar iff there exists a probabilistic bisimulation over their union containing the pair (s 0 ,s 0 0 ) where s 0 and s 0 0 ) are respectively the initial state of M 1 and M 2 . Definition 5.3.6 (Succinctness). Let (L 1 , |= 1 ) and (L 2 , |= 2 ) be two Markov chain abstraction models. L 1 is at least as succinct as L 2 , written L 1  L 2 ,i ff there exists a polynomial p such that for every L 2 2 L 2 , there exists L 1 2 L 1 s.t. L 1 ⌘ L 2 and |L 1 |  p(|L 2 |). 3 Thus, L 1 is strictly more succinct than L 2 , written L 1 < L 2 ,i ff L 1  L 2 and L 2 6  L 1 . We start with a comparison of the succinctness of the pMC and IMC abstractions. Since pMCs allow the expression of dependencies between the probabilities assigned to distinct transitions while IMCs allow all transitions to be independant, it is clear that On the other hand, IMCs imply that transition probabilities need to satisfy linear inequalities in order to fit given intervals. However, these types of constraints are not allowed in pMCs. It is therefore easy to exhibit IMCs that, regardless of the semantics considered, do not have any equivalent pMC specification. As a consequence, pMC 6  (IMC, |= o I ), pMC 6  (IMC, |= d I ), and pMC 6  (IMC, |= a I ). We now compare pMCs and IMCs to pIMCs. Recall that the pIMC model is a Markov chain abstraction model allowing to declare parametric interval transitions, while the pMC model allows only parametric transitions (without intervals), and the IMC model allows interval transitions without parameters. Clearly, any pMC and any IMC can be translated into a pIMC with the right semantics (once-and-for-all for pMCs and the chosen IMC semantics for IMCs). This means that (pIMC, |= o pI )i sm o r es u c c i n c tt h a npMC and pIMC is more succinct than IMC for the three semantics. Furthermore, since pMC and IMC are not comparable due to the above results, we have that the pIMC abstraction model is strictly more succinct than the pMC abstraction model and than the IMC abstraction model with the right semantics. Clearly, any pMC and any IMC can be translated into a pIMC with the right semantics (once-and-for-all for pMCs and the chosen IMC semantics for IMCs). This means that (pIMC, |= o pI ) is more succinct than pMC and that pIMC is more succinct than IMC for the three semantics. Furthermore since pMC and IMC are not comparable (cf Lemma 1), we have that the pIMC abstraction model is strictly more succinct than the pMC abstraction model and than the IMC abstraction model with the right semantics. s 1 α s 0 β [0, 1] [0, 1] 1 IMC I s 1 α s 0 β p 1-p 1 pMC P 1 s 1 α s 2 α s 0 β p 1 q 1 1-p 1 -q 1 p 2 1-p 2 1 pMC P 2 s 1 α s 2 α ... ... sn α s 0 β p 1 q 1 1-p 1 -q 1 p 2 q 2 1-p 2 -q 2 pn 1-pn 1 pMC P n 1 α 0 β 1-p p p 1-p pMC P 1 α 0 β [0, 1] [0, 1] [0, 1] [0, 1] IMC I 1 α 0 β 1/4 3/4 3/4 1/4 MC M 1 1 α 0 β 1/4 3/ Note that (pMC, |= p )  (IMC, |= o I )c o u l db ea c h i e v e db yc o n s i d e r i n gad o m a i nf o re a c h parameter of a pMC, which is not allowed here. However, this would not have any impact on our other results. Qualitative Properties As seen above, pIMCs are a succinct abstraction formalism for MCs. The aim of this section is to investigate qualitative properties for pIMCs, i.e., properties that can be evaluated at the specification (pIMC) level, but that entail properties on its MC implementations. pIMC specifications are very expressive as they allow the abstraction of transition probabilities using both intervals and parameters. Unfortunately, as it is the case for IMCs, this allows the expression of incorrect specifications. In the IMC setting, this is the case either when some intervals are ill-formed or when there is no probability distribution matching the interval constraints of the outgoing transitions of some reachable state. In this case, no MC implementation exists that satisfies the IMC specification. Deciding whether an implementation that satisfies a given specification exists is called the consistency problem. In the pIMC setting, the consistency problem is made more complex because of the parameters which can also induce inconsistencies in some cases. One could also be interested in verifying whether there exists an implementation that reaches some target states/labels, and if so, propose a parameter valuation ensuring this property. This problem is called the consistent reachability problem. Both the consistency and the consistent reachability problems have already been investigated in the IMC and pIMC setting [START_REF] Delahaye | Consistency for Parametric Interval Markov Chains[END_REF][START_REF] Delahaye | Parameter Synthesis for Parametric Interval Markov Chains[END_REF]. In this section, we briefly recall these problems and propose new solutions based on CSP encodings. Our encodings are linear in the size of the original pIMCs whereas the algorithms from [START_REF] Delahaye | Consistency for Parametric Interval Markov Chains[END_REF][START_REF] Delahaye | Parameter Synthesis for Parametric Interval Markov Chains[END_REF] are exponential. Existential Consistency A pIMC P is existential consistent iff there exists an MC M satisfying P (i.e., there exists an MC M satisfying an IMC I instance of P). As seen in Section 5. In [START_REF] Delahaye | Consistency for Parametric Interval Markov Chains[END_REF], the author firstly proved that |= a pI and |= o pI semantics are equivalent w.r.t. existential consistency, and proposed a CSP encoding for verifying this property which is exponential in the size of the pIMC. Now, by Proposition 3 we also get that the three s with domain [0, 1] per transition (s, s 0 )i n{{s} ⇥ Succ(s) | s 2 S}; and one Boolean variable ⇢ s per state s in S. These Boolean variables will indicate for each state whether it appears in the MC solution of the CSP (i.e., in the MC satisfying the pIMC P). For each state s 2 S, Constraints are as follows: ρ 0 ρ 1 ρ 2 ρ 3 ρ 4 πp 2 [0, 1] πq 2 [0, 1] θ 1 0 θ 2 0 θ 1 1 θ 3 1 θ 1 2 θ 2 ( 1) ⇢ s ,i f s = s 0 (2) ¬⇢ s , Σ s 0 2Pred(s)\{s} ✓ s s 0 =0, ifs 6 = s 0 (3) ¬⇢ s , Σ s 0 2Succ(s) ✓ s 0 s =0 (4) ⇢ s , Σ s 0 2Succ(s) ✓ s 0 s =1 (5) ⇢ s ) ✓ s 0 s 2 P (s, s 0 ), for all s 0 2 Succ(s) Recall that given a pIMC P the objective of the CSP C 9c (P)istoconstructanMCM satisfying P. Constraint (1) states that the initial state s 0 appears in M. Constraint (2) ensures that for each non-initial state s,v a r i a b l e⇢ s is set to false iff s is not reachable from its predecessors. Constraint (4) ensures that if a state s appears in M, then its outgoing transitions form a probability distribution. On the contrary, Constraint (3) propagates non-appearing states (i.e., if a state s does not appear in M then all its outgoing transitions are set to zero). Finally, Constraint (5) states that, for all appearing states, the outgoing transition probabilities must be selected inside the specified intervals. Example 31. Consider the pIMC P given in Figure 5.7. Figure 5.10 describes the variables in C 9c (P): one variable per transition (e.g., ✓ 1 0 , ✓ 2 0 , ✓ 1 1 ), one Boolean variable per state (e.g., ⇢ 0 , ⇢ 1 ), and one variable per parameter (⇡ p and ⇡ q ). The following constraints correspond to the Constraints (2), ( 3), (4),a n d(5) generated by our encoding C 9c for the state 2 of P: ¬⇢ 2 , ✓ 2 0 =0 ¬⇢ 2 , ✓ 1 2 + ✓ 2 2 + ✓ 4 2 =0 ⇢ 2 , ✓ 1 2 + ✓ 2 2 + ✓ 4 2 =1 ⇢ 2 ) 0  ✓ 1 2  ⇡ p ⇢ 2 ) 0.2  ✓ 2 2  ⇡ p ⇢ 2 ) 0  ✓ 4 2  0.5 Finally, Figure 5.11 describes a solution for the CSP C 9c (P). Note that given a solution of a pIMC encoded by C 9c , one can construct an MC satisfying the given pIMC w.r.t. |= o I by keeping all the states s such that ⇢ s is equal to true and considering the transition function given by the probabilities in the ✓ s 0 s variables. We now show that our encoding works as expected. Proposition 5. A pIMC P is existential consistent iff C 9c (P) is satisfiable. Proof. Let P =(S, s 0 ,P,V,Y)beap I M C . [)] The CSP C 9c (P)=(X, D, C)i ss a t i s fi a b l ei m p l i e st h a tt h e r ee x i s t sav a l u a t i o nv of the variables in X satisfying all the constraints in C. Consider the MC M =(S, s 0 , p, V ) such that p(s, s 0 )=v(✓ s 0 s ), for all ✓ s 0 s 2 Θ and p(s, s 0 )=0otherwise. Firstly, we show by induction that for any state s in S:" i fs is reachable in M then v(⇢ s ) equals to true". This is correct for the initial state s 0 thanks to the Constraint (1). Let s be a state in S and assume that the property is correct for all its predecessors. By the Constraints (2), v(⇢ s )e q u a l strue if there exists at least one predecessor s 00 6 = s reaching s with a non-zero probability (i.e., v(✓ s s 00 ) 6 =0 ) . T h i si so n l yp o s s i b l eb yt h e Constraint (4) if v(⇢ s 00 )e q u a l strue. Thus v(⇢ s )e q u a l strue if there exists one reachable state s 00 s.t. v(✓ s s 00 ) 6 =0. Secondly, we show that M satisfies the pIMC P w.r.t. |= o I . We use Theorem 4 from [START_REF] Delahaye | Parameter Synthesis for Parametric Interval Markov Chains[END_REF] stating that |= a pI and |= o pI are equivalent w.r.t. qualitative reachability. We proved above that for all reachable states s in M,w eh a v ev(⇢ s )e q u a l st otrue. By the Constraints (5) it implies that for all reachable states s in M: p(s)(s 0 ) 2 P (s, s 0 )f o ra l l s and s 0 .4 [(] The pIMC P is consistent implies by the Theorem 4 from [START_REF] Delahaye | Parameter Synthesis for Parametric Interval Markov Chains[END_REF] stating that |= a pI and |= o pI are equivalent w.r.t. qualitative reachability, that there exists an implementation of the form M =(S, s 0 , p, V )where,forallreachablestatess in M,itholdsthatp(s)(s 0 ) 2 P (s, s 0 )foralls 0 in S. Consider M 0 =(S, s 0 ,p 0 ,V)s.t. foreachnonreachablestates in S: p 0 (s)(s 0 )=0,foralls 0 2 S. The valuation v is s.t. v(⇢ s )equalstrue iff s is reachable in M, v(✓ s 0 s )=p 0 (s)(s 0 ), and for each parameter y 2 Y av alidv aluecanbeselectedaccordingto p and P when considering reachable states. Finally, by construction, v satisfies the CSP C 9c (P). Our existential consistency encoding is linear in the size of the pIMC instead of exponential for the encoding from [START_REF] Delahaye | Parameter Synthesis for Parametric Interval Markov Chains[END_REF] which enumerates the powerset of the states in the pIMC resulting in deep nesting of conjunctions and disjunctions. Qualitative Reachability Let P =( S, s 0 ,P,V,Y)b eap I M Ca n d↵ ✓ A be a state label. We say that ↵ is existential reachable in P iff there exists an implementation M of P where ↵ is reachable (i.e., P M (3↵) > 0). In a dual way, we say that ↵ is universal reachable in P iff ↵ is reachable in any implementation M of P. As for existential consistency, we use a result from [START_REF] Delahaye | Consistency for Parametric Interval Markov Chains[END_REF] that states that the |= a I and the |= o I pIMC semantics are equivalent w.r.t. existential (and universal) reachability. As for the consistency problem, we get by Proposition 3 that the three IMC semantics are equivalent w.r.t. existential (and universal) reachability. Note first that in our C 9r encoding each ⇢ s variable indicates if the state s appears in the constructed Markov chain. However, the ⇢ s variable does not indicate if the state s is reachable from the initial state, but only if it is reachable from at least one other state (i.e., possibly different from s 0 ). Indeed, if the graph representation of the constructed Markov chain has strongly connected components (SCCs for short), then all the ⇢ s variables in one SCC may be set to true while this SCC may be unreachable from the initial state. This is not an issue in the case of the consistency problem. Indeed, if a Markov chain containing an unreachable SCC is proved consistent, then it is also consistent without this unreachable SCC. However, in the case of the reachability problem, these SCCs are an issue. The following encoding therefore takes into account these isolated SCCs such that ⇢ s variables are set to true if and only if they are all reachable from the initial state. This encoding will solve the qualitative reachability problems (i.e., checking qualitative reachability from the initial state). We propose a new CSP encoding, written C 9r , that extends C 9c , for verifying these properties. Formally, CSP C 9r (P)= (X [ X 0 ,D[ D 0 ,C[ C 0 )issuchthat(X, D, C)=C 9c (P), X 0 contains one integer variable ! s with domain [0, |S|] per state s in S, D 0 contains the domains of these variables, and C 0 is composed of the following constraints for each state s 2 S: (6) ! s =1, ifs = s 0 (7) ! s 6 =1, ifs 6 = s 0 (8) ⇢ s , (! s 6 =0) (9) ! s > 1 ) W s 0 2Pred(s)\{s} (! s = ! s 0 +1)^(✓ s 0 s > 0), if s 6 = s 0 (10) ! s =0, V s 0 2Pred(s)\{s} (! s 0 =0)_ (✓ s 0 s =0), ifs 6 = s 0 Recall first that CSP C 9c (P )c o n s t r u c t saM a r k o vc h a i nM satisfying P w.r.t. |= o I . Informally, for each state s in M the Constraints (6), ( 7), ( 9) and (10) in C 9r ensure that ! s = k iff there exists in M ap a t hf r o mt h ei n i t i a ls t a t et os of length k -1w i t h non zero probability; and state s is not reachable in M from the initial state s 0 iff ! s equals to 0. Finally, Constraint (8) enforces the Boolean reachability indicator variable ⇢ s to bet set to true iff there exists a path with non zero probability in M from the initial state s 0 to s (i.e., ! s 6 =0). Let S α be the set of states from P labeled with ↵. C 9r (P)thereforeproducesaMarkov chain satisfying P where reachable states s are such that ⇢ s = true. As a consequence, ↵ is existential reachable in P iff C 9r (P)a d m i t sas o l u t i o ns u c ht h a t W s2Sα ⇢ s ;a n d↵ is universal reachable in P iff C 9r (P)a d m i t sn os o l u t i o ns u c ht h a t V s2Sα ¬⇢ s . This is formalised in the following proposition. Proposition 6. Let P =( S, s 0 ,P,V,Y) be a pIMC, ↵ ✓ A be a state label, S α = {s | V (s)=↵}, and (X, D, C) be the CSP C 9r (P). • CSP (X, D, C [ W s2Sα ⇢ s ) is satisfiable iff ↵ is existential reachable in P • CSP (X, D, C [ V s2Sα ¬⇢ s ) is unsatisfiable iff ↵ is universal reachable in P Proof. Let P =(S, s 0 ,P,V,Y)b eap I M C ,↵ ✓ A be a state label, S α = {s | V (s)=↵}, and (X, D, C)bet h eC S PC 9r (P). Recall first, that by Proposition 5 the constraints (1) to (5) in C 9r (P)a r es a t i s fi e di ff there exists an MC M satisfying P w.r.t. |= a I . • [)] If CSP (X, D, C [ W s2Sα ⇢ s ) i ss a t i s fi a b l et h e nt h e r ee x i s t sav a l u a t i o nv solution of this CSP and a corresponding MC M satisfying P w.r.t. |= a I such as presented in the proof of Proposition 5. Furthermore, the constraints ( 6) to [START_REF] Fourer | Algorithms and Model Formulations in Mathematical Programming[END_REF] ensure by induction that for all state s 2 S: v(! s )=k with k ≥ 1i f there exists a path from the initial state s 0 to the state s of size k -1w i t h non zero probability in M,a n dv(! s )=0o t h e r w i s e . B yc o n s t r a i n t(8) we have that v(⇢ s )=true iff state s is reachable in M from the initial state s 0 . Finally, constraint W s2Sα ⇢ s ensures that at least one state labeled with ↵ must be reachable in M. Thus, ↵ is existential reachable in P. [(] If ↵ is existential reachable in P, then by [START_REF] Delahaye | Parameter Synthesis for Parametric Interval Markov Chains[END_REF] there exists an MC M satisfying P w.r.t. |= o I s.t. ↵ is reachable in M. By construction of our encoding, one can easily construct from M av a l u a t i o nv satisfying all the constraints in C [ W s2Sα ⇢ s s.t. v(! s ) contains the size (plus one) of an existing path in M from the initial state to the state s with a non zero probability, and v(! s )=0 if s is not reachable in M. • Note that ↵ is universal reachable in P iff there is no MC M satisfying P w.r.t. |= a I s.t. none of the states labelled with ↵ is reachable in M. "CSP (X, D, C[ V s2Sα ¬⇢ s ) is unsatisfiable" encodes this statement. As for the existential consistency problem, we have an exponential gain in terms of size of the encoding compared to [START_REF] Delahaye | Parameter Synthesis for Parametric Interval Markov Chains[END_REF]: the number of constraints and variables in C 9r is linear in terms of the size of the encoded pIMC. Remark. In C 9r Constraints (2) inherited from C 9c are entailed by Constraints ( 8) and [START_REF] Fourer | Algorithms and Model Formulations in Mathematical Programming[END_REF] added to C 9r . Thus, in a practical approach one may ignore Constraints (2) from C 9c if they do not improve the solver performances. Quantitative Properties We now move to the verification of quantitative reachability properties in pIMCs. Quantitative reachability has already been investigated in the context of pMCs and IMCs with the once-and-for-all semantics. In this section, we propose our main theoretical contribution: a theorem showing that the three IMC semantics are equivalent with respect to quantitative reachability, which allows the extension of all results from [START_REF] Wongpiromsarn | TuLiP: A Software Toolbox for Receding Horizon Temp oral Logic Planning[END_REF][START_REF] Benedikt | LTL model checking of interval Markov chains[END_REF]t o the at-every-step semantics. Based on this result, we also extend the CSP encodings introduced in Section 5.4 in order to solve quantitative reachability properties on pIMCs regardless of their semantics. t quantitative reachability Given an IMC I =( S, s 0 ,P,V)a n das t a t el a b e l↵ ✓ A, a quantitative reachability property on I is a property of the type P I (3↵)⇠p, where 0 <p<1a n d⇠2{,<,> , ≥}. Such a property is verified iff there exists an MC M satisfying I (with the chosen semantics) such that P M (3↵)⇠p. As explained above, existing techniques and tools for verifying quantitative reachability properties on IMCs only focus on the once-and-for-all and the IMDP semantics. However, to the best of our knowledge, there are no works addressing the same problem with the at-every-step semantics or showing that addressing the problem in the onceand-for-all and IMDP setting is sufficiently general. The following theorem fills this theoretical gap by proving that the three IMC semantics are equivalent w.r.t quantitative reachability. In other words, for all MC M such that M| = a I I or M| = d I I and for all state label ↵, there exist MCs M  and M ≥ such that M  |= o I I, M ≥ |= o I I and P M  (3↵)  P M (3↵)  P M ≥ (3↵). This is formalised in the following theorem. Theorem 1. Let I =( S, s 0 ,P,V) be an IMC, ↵ ✓ A be a state label, ⇠2{,<,> , ≥} and 0 <p<1. I satisfies P I (3↵)⇠p with the once-and-for-all semantics iff I satisfies P I (3↵)⇠p with the IMDP semantics iff I satisfies P I (3↵)⇠p with the at-everystep semantics. The proof presented in the following is constructive: we use the structure of the relation R from the definition of |= a I in order to build the MCs M  and M ≥ . In the following, when it is not specified the IMC satisfaction relation considered is the at-every-step semantics (i.e.,t h e|= a I satisfaction relation). As said previously, we use the structure of the relation R from the definition of |= a I in order to build the MCs M  and M ≥ presented in Theorem 1. Thus, we introduce some notations relative to R. Let I =(S, s 0 ,P,V I )b ea nI M Ca n dM =(T,t 0 , p, V M )b ea nM Cs u c ht h a tM| = a I I. Let R ✓ T ⇥ S be a satisfaction relation between M and I. For all t 2 T we write R(t)f o r the set {s 2 S | t R s}, and for all s 2 S we write R -1 (t)f o rt h es e t{t 2 T | s R t}. Furthermore we say that M satisfies I with degree n (written M n |= a I I)i fM satisfies I with a satisfaction relation R such that each state t 2 T is associated by R to at most n states from S (i.e., |R(t)|  n); M satisfies I with the same structure than I if M satisfies I with a satisfaction relation R such that each state t 2 T is associated to at most one state from S and each state s 2 S is associated to at most one state from T (i.e., |R(t)|  1f o ra l lt 2 T and |R -1 (s)|  1f o ra l ls 2 S). Proposition 7. Let I be an IMC. If an MC M satisfies I with degree n 2 N then there exists an MC M 0 satisfying I with degree 1 such that M and M 0 are bisimilar. The main idea for proving Proposition 7 is that if an MC M with states space T satisfies an IMC I with a states space S according to a satisfaction relation R then, each state t related by R to many states s 1 ,...,s n (with n>1) can be split in n states t 1 ,...,t n . The derived MC will satisfy I with a satisfaction relation R 0 where each t i is only associated by R 0 to the state s i (i  n). This M 0 will be bisimilar to M and it will satisfy I with degree 1. Note that by construction the size of the resulting MC is in O(|M| ⇥ |I|). Furthermore, we will use the until temporal modality (abbreviated U ) as presented in [START_REF] Baier | Principles of Model Checking (Representation and Mind Series[END_REF]. Let M be an MC and ↵, β be two state labelings. The probability of the property ↵ U β is given by the sum of the probabilities of all the finite paths starting in the initial state containing only states labeled with ↵ excepted for the last state which is labeled with β. Formally, let until s 0 (s)={! 2 S ⇤ | ! = s 0 ,...s n with V (s n )=β and V (s i )= ↵ 80  i<n} be the set of such paths. Thus, P M (↵ U β)= P ω2reachs 0 (s) P M (!). As for the reachability property, this notation naturally extends to states instead of labels, as well as conjunctions and disjunctions of states/labels. Proof for Proposition 7. Let I =( S, s 0 ,P,V I )b ea nI M Ca n dM =( T,t 0 , p, V )b ea n MC. If M satisfies I (with degree n)t h e nt h e r ee x i s t sas a t i s f a c t i o nr e l a t i o nR verifying the |= a I satisfaction relation. For each association (t, s) 2 R, we write δ s t the correspondence function chosen for this pair of states. M satisfies I with degree n means that each state in M is associated by R to at most n states in I. To construct an MC M 0 satisfying I with degree 1 we create one state in M 0 per association (t, s)inR. Formally, let M 0 be equal to (U, u 0 ,p 0 ,V 0 )suchthatU = {u s t | (t, s) 2 R}, u 0 = u s 0 t 0 , V 0 = {(u s t ,v) | v = V (t)}, and p 0 (u s t )(u s 0 t 0 )=p(t)(t 0 ) ⇥ δ s t (t 0 )(s 0 ). The following computation shows that the outgoing probabilities given by p 0 form a probability distribution for each state in M 0 and thus that M 0 is an MC. X t 0 Rs 0 p 0 (u s t )(u s 0 t 0 )= X t 0 Rs 0 p(t)(t 0 ) ⇥ δ s t (t 0 )(s 0 ) = X t 0 2T p(t)(t 0 ) ⇥ X s 0 2S δ s t (t 0 )(s 0 )= X t 0 2T p(t)(t 0 ) ⇥ 1=1 Finally, by construction of M 0 based on M which satisfies I, we get that R 0 = {(u s t ,s) | t 2 T,s 2 S} is a satisfaction relation between M 0 and I. Furthermore |{s | (u, s) 2 R 0 }| equals at most one. Thus, we get that M 0 satisfies I with degree 1. Consider the relation B 0 = {(u s t ,t) ✓ U ⇥ T | (t, s) 2 R}.W en o t eB the closure of B 0 by transitivity, reflexivity, and symmetry (i.e., B is the minimal equivalence relation based on B 0 ). We prove that B is a bisimulation relation between M and M 0 . By construction each equivalence class from B contains exactly one state t from T and all the states u s t such that (t, s) 2 R. Let (u s t ,t)b ei nB, t 0 be a state in T ,a n dB be the equivalence class from B containing t 0 (i.e., B is the set {t 0 } [ {u s 0 t 0 2 U | s 0 2 S and (t 0 ,s 0 ) 2 R}). Firstly note that by construction the labels agree on u s t and t: V 0 (u s t )=V (t). Secondly the following computation shows that p 0 (u s t )(B \ U )e q u a l st op(t)(B \ T ) and thus that u s t and t are bisimilar: p 0 (u s t )(B \ U )= X u s 0 t 0 2B\U p 0 (u s t )(u s 0 t 0 )= X u s 0 t 0 2B\U p(t)(t 0 ) ⇥ δ s t (t 0 )(s 0 ) = X {s 0 2S|s 0 Rt 0 } p(t)(t 0 ) ⇥ δ s t (t 0 )(s 0 )=p(t)(t 0 ) ⇥ X {s 0 2S|s 0 Rt 0 } δ s t (t 0 )(s 0 ) = p(t)(t 0 ) ⇥ 1=p(t)({t 0 })=p(t)(B \ T ) Corollary 1. Let I be an IMC, M be an MC satisfying I, and γ be a PCTL ⇤ formulae. There exists an MC M 0 satisfying I with degree 1 such that the probability P M 0 (γ) equals the probability P M (γ). Corollary 1 is derived from Proposition 7 joined with the probability preservation of the PCTL* formulae on bisimilar Markov chains (see [START_REF] Baier | Principles of Model Checking (Representation and Mind Series[END_REF], Theorem 10.67, p.813). Corollary 1 allows to reduce to one the number of states in the pIMC I satisfied by each state in the MC M while preserving probabilities. Thus, one can construct from an MC M satisfying an IMC I another MC M 0 satisfying the same IMC I where the states in M 0 are related to at most one state in I. However, some states in I may still be related to many states in M 0 . The objective of Lemma 2 is to reduce these relations to an "at most one" in both directions (I to M 0 and M 0 to I). Lemma 2. Let I =( S, s 0 ,P,V) be an IMC, M =( T,t 0 , p, V ) be an MC satisfying I with degree 1, and α ✓ A be a proposition. If M does not have the same structure than I then there exists an MC M 1 (resp. M 2 ) satisfying I with a set of states S 1 (resp. S 2 ) s.t. S 1 ⇢ S and P M 1 (3↵)  P M (3↵) (resp. S 2 ⇢ S and P M 2 (3↵) ≥ P M (3↵)). s 0 s 1 s 2 s 3 α 0.5 0.5 [0, 1] [0, 1] [0, 1] [0, 1] 1 IMC I t 0 t 1 t 0 1 t 2 t 3 α t 0 2 0.4 0.5 0.1 0.5 0.5 1 1 1 1 MC M 1 t 0 t 1 t 0 1 t 3 α t 0 2 Lemma 2 reduces the number of states in M while preserving the maximal or minimal reachability probability. This lemma has a constructive proof. The main idea of the proof is that we select one state s from the IMC I which is satisfied by many states t 1 ,...,t n in M. Thus, the MC M 0 keeping the state t k maximizing the probability of reaching ↵ in M and removing all the other states t i (i.e., remove the states t i such that i 6 = k and move the transitions to a state t i such that i 6 = k to arrive to the state t k )w i l lh a v el e s s states than M and will verify P M 1 (3↵) ≥ P M (3↵). Figure 5.12 contains an IMC I and three MCs M 1 , M 2 ,a n dM 3 . This illustrates how Lemma 2 operates for reducing the state space. We describe how to obtain M 2 from M 1 . Consider the state s 2 from I. This state is related to the states t 2 and t 0 2 in M 1 . Since P M 1 t 2 (3↵)=0andP M 1 t 0 2 (3↵)=1 we remove t 2 and we keep t 0 2 which has an higher probability to reach ↵. Then, all the transitions going to t 2 are changed in order to go to t 0 2 . This creates M 2 . Next, the same mechanism can be iterated to produce M 3 : consider s 1 from I and remove t 0 1 and keep t 1 from M 2 to produce M 3 . This allows to reduce the number of states in the constructed Markov chain while preserving the maximal/minimal reachability probability. Before proving Lemma 2, we introduce Lemma 3 which will be used for proving Lemma 2. Lemma 3. Let M =( S, s 0 , p, V ) be an MC, ↵ ✓ A be a proposition, and s be a state from S. Then P M s (3↵)= P M s (¬s U ↵) 1 -P M s (¬↵ U s) Proof. Let S 0 be the subset of S containing all the states labeled with ↵ in M. We write Ω n with n 2 N ⇤ the set containing all the paths ! starting from s s.t. state s appears exactly n times in ! and no state in ! is labeled with ↵.F o r m a l l yΩ n contains all the ! = s 1 ,...,s k 2 S k s.t. k 2 N, s 1 is equal to s, |{i 2 [1,k] | s i = s}| = n,a n d↵ 6 ✓ V (s i ) for all i 2 [1,k]. Given two sets of paths Ω and Ω 0 , we write Ω⇥Ω 0 their Cartesian product which is the set of paths {!! 0 | ! 2 Ω and ! 0 2 Ω 0 }. We get by (a) that (P M s (Ω n ⇥ S 0 )) n≥1 is a geometric series. For (⇤)recallthatgiv enanMCM and two non-empty paths ! and ! 0 on M s.t. s and s 0 are respectively the first state in ! and ! 0 we have by definition that P M s (!! 0 )=P M s (!s 0 ) • P M s 0 (! 0 ). In (b) we partition the paths reaching ↵ according to the Ω n sets and we use the geometric series of the probabilities to retrieve the required result. (a) P M s (Ω n ⇥ S 0 )=P M s (Ω 1 ⇥ Ω n-1 ⇥ S 0 ) (⇤) = P M s (Ω 1 ⇥ {s}) • P M s (Ω n-1 ⇥ S 0 ) = P M s (¬↵ U s) • P M s (Ω n-1 ⇥ S 0 ) (b) P M s (3↵)=Σ +1 n=1 P M s (Ω n ⇥ S 0 ) = P M s (Ω 1 ⇥ S 0 ) 1 -P M s (¬↵ U s) = P M s (¬s U ↵) 1 -P M s (¬↵ U s) Proof for Lemma 2. Let I =( S, s 0 ,P,V I )b ea nI M Ca n dM =( T,t 0 , p, V )b ea nM C satisfying I with degree 1. We write R the satisfaction relation between M and I with degree 1. The following proves in 3 steps the P M 1 (3↵)  P M (3↵)c a s e . 1. We would like to construct an MC M 0 satisfying I with less states than M 0 such that P M 0 (3↵)  P M (3↵). Since the degree of R equals to 1 each state t in T is associated to at most one state s in S. Furthermore, since M does not have the same structure than I then there exists at most one state from S which is associated by R to many states from T . Let s be a state from S such that |R -1 (s)| ≥ 2, T = {t 1 ,...,t n } be the set R -1 (s)w h e r et h et i are ordered by decreasing probability of reaching ↵ (i.e., P M t i (3↵) ≥ P M t i+1 (3↵)f o ra l l1 i<n). In the following we refer t as t n .W e produce M 0 from M by replacing all the transitions going to a state t 1 ,...,t n-1 by a transition going to t n , and by removing the corresponding states. Formally M 0 =(T 0 ,t 0 ,p 0 ,V 0 )s . t . T 0 =(T \ T ) [ { t}, V 0 is the restriction of V on T 0 , and for all t, t 0 2 T 0 : p 0 (t)(t 0 )=p(t)(t 0 )i ft 0 6 = t and p 0 (t)(t 0 )= P t 0 2 T p(t)(t 0 )o t h e r w i s e . X t 0 2T 0 p 0 (t)(t 0 )= X t 0 2T 0 \{ t} p 0 (t)(t 0 )+p 0 (t)( t) (1) = X t 0 2T 0 \{ t} p(t)(t 0 )+ X t 0 2 T p(t)(t 0 ) = X t 0 2T 0 \{ t}[ T p(t)(t 0 ) (2) = X t 0 2T p(t)(t 0 )=1 The previous computation holds for each state t in M 0 . It shows that the outgoing probabilities given by p 0 form a probability distribution for each state in M 0 and thus that M 0 is an MC. Note that step (1) comes from the definition of p 0 with respect to p and that step (2) comes from the definition of T 0 according to T and t. 2. We now prove that M 0 satisfies I. M satisfies I implies that there a exists a satisfaction relation R between M and I. Let R 0 ✓ T ⇥ S be s.t. t R 0 s if t R s and t R 0 s if there exists a state t 0 2 T s.t. t 0 R s.W e p r o v e t h a t R 0 is a satisfaction relation between M 0 and I. For each pair (t, s) 2 R we note δ (s,t) the correspondence function given by the satisfaction relation R. Let (t, s) be in R 0 and δ 0 : T 0 ! (S ! [0, 1]) be s.t. δ 0 (t 0 )(s 0 )=δ (t,s) (t 0 )(s 0 )i ft 0 6 = t and δ 0 (t 0 )(s 0 )=max t 0 2 T (δ (t,s) (t 0 )(s 0 )) otherwise. δ 0 is a correspondence function for the pair (t, s)i nR 0 such as required by the |= a I satisfaction relation: a) Let t 0 be in T . If t 0 6 = t then δ 0 (t 0 )i se q u i v a l e n tt oδ (t,s) (t 0 )(s 0 ) which is by definition a distribution on S. Otherwise t 0 = t and the following computation proves that δ 0 ( t)i sad i s t r i b u t i o no nS. For the step (1) remind that R is a satisfaction relation with degree 1 and that t R s. This implies that δ (t,s) ( t)(s 0 ) equals to zero for all s 0 6 =s. For the step (2), R is a satisfaction relation with degree 1 implies that δ (t,s) (t 0 )(s 0 )e q u a l st o0o r1f o ra l lt 0 2 T and s 0 2 S. Finally the recursive definition of the satisfaction relation R implies that there exists at least one state t 00 2 T s.t. δ (t,s) (t 00 )(s)d o e sn o te q u a lt oz e r o( i.e., equals to one). X s 0 2S δ 0 ( t)(s 0 )= X s 0 2S\{s} δ 0 ( t)(s 0 )+δ 0 ( t)(s) = X s 0 2S\{s} δ (t,s) ( t)(s 0 )+max t 00 2 T (δ (t,s) (t 00 )(s)) (1) = max t 00 2 T (δ (t,s) (t 00 )(s 0 )) =1 b) Let s 0 be in S. Step (1) uses the definition of p 0 according to p. Step (2) uses the definition of δ 0 according to δ (t,s) . Step (3) comes from the fact that for all t, t 0 2 T ⇥ T , we have by the definition of the satisfaction relation R with degree 1 and by construction of T that if p(t, t 0 ) 6 =0t h e nδ (t)(s) (t 0 , s)=1a n d δ (t,s) (t 0 )(s 0 )=0foralls 0 6 =s. Finally, step (4) comes from the definition of the correspondence function δ (t,s) for the pair (t, s)i nR. X t 0 2T 0 p 0 (t)(t 0 ) ⇥ δ 0 (t 0 )(s 0 ) = X t 0 2T 0 \{ t} p 0 (t)(t 0 ) ⇥ δ 0 (t 0 )(s 0 )+p 0 (t, t) ⇥ δ 0 ( t)(s 0 ) (1) = X t 0 2T 0 \{ t} p(t)(t 0 ) ⇥ δ 0 (t 0 )(s 0 )+ X t 0 2 T p(t)(t 0 ) ⇥ δ 0 ( t)(s 0 ) (2) = X t 0 2T 0 \{ t} p(t)(t 0 ) ⇥ δ (t,s) (t 0 )(s 0 )+ X t 0 2 T p(t)(t 0 ) ⇥ max t 00 2 T (δ (t,s) (t 00 )(s 0 )) (3) = X t 0 2T 0 \{ t} p(t)(t 0 ) ⇥ δ (t,s) (t 0 )(s 0 )+ X t 0 2 T p(t)(t 0 ) ⇥ δ (t,s) (t 0 )(s 0 ) = X t 0 2T 0 \{ t}[ T p(t)(t 0 ) ⇥ δ (t,s) (t 0 )(s 0 )= X t 0 2T p(t)(t 0 ) ⇥ δ (t,s) (t 0 )(s 0 ) (4) 2 P (s, s 0 ) c) Let t 0 be in T 0 and s 0 be in S. We have by construction of R 0 from R that if δ 0 (t 0 )(s 0 ) > 0t h e n( t 0 ,s 0 ) 2 R. 3. Ne now prove that the probability of reaching ↵ from t is lower in M 0 than in M. We consider the MC M 00 from M where the states containing the label ↵ are replaced by absorbing states. Formally M 00 =( T,t 0 ,p 00 ,V)s u c ht h a tf o ra l l t, t 0 2 T : p 00 (t, t 0 )=p(t, t 0 )i f↵ 6 ✓ V (t)e l s ep 00 (t, t 0 )=1i ft = t 0 and p 00 (t, t 0 )=0 otherwise. By definition of the reachability property we get that P M 00 t (3↵)e q u a l s to P M t (3↵)f o ra l ls t a t et in T 0 . Following computation concludes the proof. Step (1) comes from Lemma 3. Step (2) comes by construction of M 0 from M. Step (3) comes by construction of M 00 from M where states labeled with ↵ are absorbing states. Step (4) comes from the fact that P M 00 tn (3↵)i se q u a lt oP M 00 tn (¬(t 1 _ ... _ t n )U↵)+Σ 1in P M 00 tn (¬(t 1 _ ..._ t n )Ut i ) ⇥ P M 00 t i (3↵). Step (5) uses the fact that P M t i (3↵) ≥ P M tn (3↵)f o ra l l1 i  n and by construction this is also correct in M 00 . Last steps are straightforward. P M 0 t (3↵) (1) = P M 0 t (¬ t U ↵) 1 -P M 0 t (¬↵ U t) (2) = P M tn (¬(t 1 _ ..._ t n )U↵) 1 -P M tn (¬↵ U( t 1 _ ..._ t n )) (3) = P M 00 tn (¬(t 1 _ ..._ t n )U↵) 1 -P M 00 tn (3(t 1 _ ..._ t n )) (4) = P M 00 tn (3↵) - P 1in P M 00 tn (¬(t 1 _ ..._ t n )Ut i ) ⇥ P M 00 t i (3↵) 1 -P M 00 tn (3(t 1 _ ..._ t n )) (5)  P M 00 tn (3↵) - P 1in P M 00 tn (¬(t 1 _ ..._ t n )Ut i ) ⇥ P M 00 tn (3↵) 1 -P M 00 tn (3(t 1 _ ..._ t n )) (6) = P M 00 tn (3↵) ⇥ (1 - P 1in P M 00 tn (¬(t 1 _ ..._ t n )Ut i )) 1 -P M 00 tn (3(t 1 _ ..._ t n )) (7) = P M 00 tn (3↵) ⇥ (1 -P M 00 tn (3(t 1 _ ..._ t n ))) 1 -P M 00 tn (3(t 1 _ ..._ t n )) = P M 00 tn (3↵) = P M t (3↵) The same method can be used for proving that P M 2 (3↵) ≥ P M (3↵)b yd e fi n i n g T = {t 1 ,...,t n } to be the set R -1 (s)s.t. thestatest i are ordered by increasing probability of reaching ↵. Thereby the symbol  at step (5) for the computation of P M 0 t (3↵)i s replaced by the symbol ≥. Next, Lemma 4 is a consequence of Corollary 1 and Lemma 2 and states that the maximal and the minimal probability of reaching a given proposition is realized by Markov chains with the same structure than the IMC. Proof. Let I be an IMC and M be an MC satisfying I w.r.t. |= a I . Consider the sequence of MCs (M n ) n2N s.t. M 0 is the MC satisfying I with degree 1 obtained by Corollary 1 and for all n 2 N, M n+1 is the MC satisfying I with strictly less states than M n and verifying P M n+1 (3↵)  P Mn (3↵)g i v e nb yL e m m a2 if M n does not have the same structure than I and equal to M n otherwise. By construction (M n ) n2N is finite and its last element is aM a r k o vc h a i nM 0 with the same structure than I s.t. P M 0 (3↵)  P M (3↵). Thus, M 0 satisfies I w.r.t. |= o I s.t. P M 0 (3↵)  P M (3↵). The same method can be used for proving the other side of the inequality (i.e., there exists an MC M 0 s.t. M 0 |= o I I and P M (3↵)  P M 0 (3↵)). Finally, the following proves our Theorem 1 using Lemma 4 and Proposition 3. Proof for Theorem 1. Let I =(S, s 0 ,P,V)beanIMC,↵ ✓ A be a state label, ⇠2{,< ,>,≥} and 0 <p<1. Recall that according to an IMC satisfaction relation the property P I (3↵)⇠p holds iff there exists an MC M satisfying I (with the chosen semantics) such that P M (3↵)⇠p. Constraint Encodings Note that the result from Theorem 1 naturally extends to pIMCs. We therefore exploit this result to construct a CSP encoding for verifying quantitative reachability properties in pIMCs. As in Section 5.4, we extend the CSP C 9c , that produces a correct MC implementation for the given pIMC, by imposing that this MC implementation satisfies the given quantitative reachability property. In order to compute the probability of reaching state label ↵ at the MC level, we use standard techniques from [START_REF] Baier | Principles of Model Checking (Representation and Mind Series[END_REF] that require the partitioning of the state space into three sets S > , S ? ,a n dS ? that correspond to states reaching ↵ with probability 1, states from which ↵ cannot be reached, and the remaining states, respectively. Once this partition is chosen, the reachability probabilities of all states in S ? are computed as the unique solution of a linear equation system (see [START_REF] Baier | Principles of Model Checking (Representation and Mind Series[END_REF], Theorem 10.19, p.766). We now explain how we identify states from S ? ,S > and S ? and how we encode the linear equation system, which leads to the resolution of quantitative reachability. Let P =( S, s 0 ,P,V,Y)b eap I M Ca n d↵ ✓ A be a state label. We start by setting S > = {s | V (s)=↵}. We then extend C 9r (P)i no r d e rt oi d e n t i f yt h es e tS ? . Let ↵ s =1, if↵ = V (s) (12) ↵ s 6 =1, if↵ 6 = V (s) (13) λ s , (⇢ s ^(↵ s 6 =0)) (14) ↵ s > 1 ) W s 0 2Succ(s)\{s} (↵ s = ↵ s 0 +1)^(✓ s 0 s > 0), if ↵ 6 = V (s) (15) ↵ s =0, V s 0 2Succ(s)\{s} (↵ s 0 =0)_ (✓ s 0 s =0), if↵ 6 = V (s) Note that variables ↵ s play a symmetric role to variables ! s from C 9r : instead of indicating the existence of a path from s 0 to s, they characterize the existence of a path from s to a state labeled with ↵. In addition, due to Constraint (13), variables λ s are set to true iff there exists a path with non zero probability from the initial state s 0 to a state labeled with ↵ passing by s. Thus, ↵ cannot be reached from states such that λ s = false. Therefore, S ? = {s | λ s = false}, which is formalised in Proposition 8. First, note that s 3 is the only state labelled by {↵, β} in P. By considering the MC M built from the valuation of the transition variables in Figure 5.13 we have that: ↵ 0 =3 ,w h i c hi m p l i e st h a tt h e r ee x i s t sap a t hi nM with size 2 reaching ↵ from s 1 ; ↵ 1 = 2, which implies that there exists a path in M with size 1 reaching ↵ from s 1 ;a n d↵ 2 =0 ,w h i c hi m p l i e st h a tt h e r ei sn op a t hi nM reaching ↵ from s 1 , etc. Finally, by Constraint (13) we have that: λ 0 , λ 1 ,a n dλ 3 are true which implies that the states s 0 , s 1 ,a n ds 3 are reachable in M and they can reach ↵; λ 2 and λ 4 are false which implies that the states s 2 and s 4 cannot reach ↵ in M. Finally, we encode the equation system from [START_REF] Baier | Principles of Model Checking (Representation and Mind Series[END_REF] in a last CSP encoding that extends C 0 9r .L e t C 9r (P, ↵)=( X [ X 0 ,D [ D 0 ,C [ C 0 )b es u c ht h a t( X, D, C)=C 0 9r (P, ↵), X 0 contains one variable ⇡ s per state s in S with domain [0, 1], D 0 contains the domains of these variables, and C 0 is composed of the following constraints for each state s 2 S: (16) ¬λ s ) ⇡ s =0 (17) λ s ) ⇡ s =1, if↵ = V (s) (18) λ s ) ⇡ s = Σ s 0 2Succ(s) ⇡ s 0 ✓ s s 0 ,i f ↵ 6 = V (s) As a consequence, variables ⇡ s encode the probability with which state s eventually reaches ↵ when s is reachable from the initial state and 0 otherwise. Proposition 9. Let P =( S, s 0 ,P,V,Y) be a pIMC and ↵ ✓ A be a proposition. There exists an MC M| = a pI P iff there exists a valuation v solution of the CSP C 9r (P, ↵) s.t. v(⇡ s ) is equal to P M s (3↵) if s is reachable from the initial state s 0 in M and is equal to 0 otherwise. Proof. Let P =( S, s 0 ,P,V,Y)b eap I M Ca n d↵ ✓ A be a state label. C 9r extends the CSP C 0 9r that produces a MC M satisfying P (cf. Proposition 8)b yc o m p u t i n gt h e probability of reaching ↵ in M. In order to compute this probability, we use standard techniques from [START_REF] Baier | Principles of Model Checking (Representation and Mind Series[END_REF] that require the partitioning of the state space into three sets S > , S ? ,a n dS ? that correspond to states reaching ↵ with probability 1, states from which ↵ cannot be reached, and the remaining states, respectively. Once this partition is chosen, the reachability probabilities of all states in S ? are computed as the unique solution of an equation system (see [START_REF] Baier | Principles of Model Checking (Representation and Mind Series[END_REF], Theorem 10.19, p.766). Recall that for each state s 2 S variable ↵ s is equal to true iff s is reachable in M and s can reach ↵ with a non zero probability. Thus we consider S ? = {s | ↵ s = false}, S > = {s | V (s)=↵},a n d S ? = S \ (S ? [ S > ). Finally constraints in C 9r encodes the equation system from [START_REF] Baier | Principles of Model Checking (Representation and Mind Series[END_REF] according to chosen S ? , S > ,a n dS ? . Thus, ⇡ s 0 equals the probability in M to reach ↵. Example 33 (Example 32 continued). Consider the valuation given in Figure 5.13 as a partial solution to the CSP C 9r (P, {↵, β}). Let M be the MC built from this partial valuation. Since s 2 and s 4 cannot reach {↵, β} in M we have that S ? contains s 2 and s 4 . Furthermore, s 3 is the only state labelled by {↵, β} in M. Thus, S > contains s 3 and the remaining states s 0 and s 1 are in S ? . Finally, Constraints ( 16), [START_REF] Aris | Mathematical modelling techniques[END_REF],a n d( 18) encode the following system to compute for each state the quantitative reachability of {↵, β} in M: 8 > > > > > > < > > > > > > : ⇡ 0 =0 .7⇡ 1 +0.3⇡ 2 ⇡ 1 =0 .5⇡ 1 +0.5⇡ 3 ⇡ 2 =0 ⇡ 3 =1 ⇡ 4 =0 , 8 > > > > > > < > > > > > > : ⇡ 0 =0 .7⇡ 1 +0 ⇡ 1 =0 .5⇡ 1 +0.5 ⇡ 2 =0 ⇡ 3 =1 ⇡ 4 =0 , 8 > > > > > > < > > > > > > : ⇡ 0 =0 .7 ⇡ 1 =1 ⇡ 2 =0 ⇡ 3 =1 ⇡ 4 =0 Let p 2 [0, 1] ✓ R Prototype Implementation and Experiments Our results have been implemented in a prototype tool5 which generates the above CSP encodings, and CSP encodings from [START_REF] Delahaye | Parameter Synthesis for Parametric Interval Markov Chains[END_REF] as well. In this section, we first present our benchmark, then we evaluate our tool for the qualitative properties, and we conclude with the quantitative properties. Benchmark MCs have been used for many decades to model real-life applications. PRISM [START_REF] Kwiatkowska | PRISM 4.0: Verification of Probabilistic Real-time Systems[END_REF]i sa reference for the verification of probabilistic systems. In particular, it is able to verify properties for MCs. As said in Section 5.2, pIMCs correspond to abstractions of MCs. PRISM references several benchmarks based on MCs 6 . Note first that we only consider pIMCs with linear parametric expressions. In this context all the CSPs encodings for verifying the qualitative properties only use linear constraints while the CSPs encodings for verifying the quantitative properties produce quadratic constraints (i.e., non-linear constraints). This produces an order of magnitude between the time complexity for solving the qualitative properties vs the quantitative properties w.r.t. our encodings. Thus, we consider two different benchmarks presented in Table 5.1 and 5.2. In both cases, pIMCs are automatically generated from the PRISM model in a text format inspired from [START_REF] Wongpiromsarn | TuLiP: A Software Toolbox for Receding Horizon Temp oral Logic Planning[END_REF]. For the first benchmark used for verifying qualitative properties, we constructed the pIMCs from existing MCs by randomly replacing some exact probabilities on transitions by (parametric) intervals of probabilities. Our pIMC generator takes 4 arguments: the MC transition function; the number of parameters for the generated pIMC; the ratio of the number of intervals over the number of transitions in the generated pIMC; the ratio of the number of parameters over the number of interval endpoints for the generated pIMC. The benchmarks used are presented in Table 5.1. We selected 5 applications from PRISM [START_REF] Kwiatkowska | PRISM 4.0: Verification of Probabilistic Real-time Systems[END_REF]: herman -the self-stabilisation protocol of Herman from [START_REF] Kwiatkowska | Probabilistic Verification of Herman's Self-Stabilisation Algorithm[END_REF]; egl -t h e contract signing protocol of Even, Goldreich & Lempel from [START_REF] Norman | Analysis of Probabilistic Contract Signing[END_REF]; brp -t h eb o u n d e d retransmission protocol from [START_REF]Reachability analysis of probabilistic systems by successive refinements[END_REF]; crowds -t h ec r o w d sp r o t oc o lf r o m [START_REF] Shmatikov | Probabilistic Model Checking of an Anonymity System[END_REF]; and nand -the nand multiplexing from [START_REF] Norman | Evaluating the Reliability of NAND Multiplexing with PRISM[END_REF]. Each one is instantiated by setting global constants (e.g., N for the application herman, L and N for the application egl)l e a d i n gt om o r e or less complex MCs. We used our pIMC generator to generate an heterogeneous set of benchmarks: 459 pIMCs with 8 to 15, 102 states and 28 to 21, 567 transitions not reduced to [0, 0]. The pIMCs contain from 2 to 250 parameters over 0 to 7772 intervals. For the second benchmark used for verifying quantitative properties we extended the nand model from [START_REF] Norman | Evaluating the Reliability of NAND Multiplexing with PRISM[END_REF]. The original MC nand model has already been extended as a pMC in [START_REF] Dehnert | PROPhESY: A PRObabilistic ParamEter SYnthesis To ol[END_REF], where the authors consider a single parameter p for the probability that each of the N nand gates fails during the multiplexing. We extend this model to pIMC by considering one parameter for the probability that the initial inputs are stimulated and we have one parameter per nand gate to represent the probability that it fails. We consider 4 pIMCs with 104 to 7, 392 states and 147 to 11, 207 transitions not reduced to [0, 0]. The pIMCs contain from 4 to 12 parameters appearing over 82 to 5, 698 transitions. Constraint Modelling Given a pIMC in a text format our tool produces the desired CSP according to the selected encoding (i.e., one from [START_REF] Delahaye | Parameter Synthesis for Parametric Interval Markov Chains[END_REF], C 9c , C 9r ,orC 9r ). Recall that our benchmark only consider linear parametric expressions on transitions. The choice of the contraint programming language for implementing a CSP encoding depends on its nature (e.g.,t h et y p eo ft h e variables: integer vs. continuous, the kind of the contraints: linear vs. non-linear). Table 5.3 summarizes the natures the encodings where SotA stands the encoding from [START_REF] Delahaye | Parameter Synthesis for Parametric Interval Markov Chains[END_REF] answering the existential consistency problem. Thus, SotA, C 9c ,a n dC 9r can be implemented as Mixed Integer Linear Programs (MILP) [START_REF] Pablo | Mixed Integer Linear Programming Formulation Techniques[END_REF] and as Satisfiability Modulo Theory (SMT)p r o g r a m s [START_REF] Clark | Satisfiability Modulo Theories[END_REF] with QF LRA logic (Quantifier Free linear Real-number Arithmetic). This logic deals with Boolean combinations of inequations between linear polynomials over real variables. Note that, QF NRA does not deal with integer variables. Indeed logics mixing integers and reals are harder than those over reals only. However, all the integer variables in our encodings can be replaced by real-number variables. 7 Each integer variable x can be declared as a real variable whose real domain bounds are its original integer domain bounds; we also add the constraint x<1 ) x =0 . S i n c ew e only perform incrementation of x this preserves the same set of solutions (i.e., ensures integer integrity constraints). Finally, due to the non-linear constraints in C 9r ,t h e s e encodings are implemented as SMT programs [START_REF] Clark | Satisfiability Modulo Theories[END_REF] with the QF NRA logic (Quantifier Free Non linear Real-number Arithmetic). We use the same technique than for C 9c and C 9r for replacing integer variables by real-number variables. We chose the programming language Python for implementing our CSP modeller. We do not evaluate any arithmetic expression while generating CSPs, and numbers in the interval endpoints of the pIMCs are read as strings and no trivial simplification is performed while modelling. We do so to avoid any rounding of the interval endpoints when using floating point numbers. Experiments have been realized on a 2.4 GHz Intel Core i5 processor. Time out has been set to 10 minutes. Memory out has been set to 2Gb. Table 5.4 presents the size of the instances (i.e., the number of variables and the number of constraints) for solving the existential consistency problem on our benchmark using (1) SMT SotA encoding, (2) SMT C 9c encoding, and (3) MILP C 9c encoding. First, note that all the pIMCs are successfully compiled when using our C 9c encoding while the SotA encoding produces out of memory errors for 4 sets of benchmarks: more than 20% of the instances (see OM cells in Table 5.4). We recall that the SotA encoding is defined inductively and that it iterates over the power set of the states. In practice, this implies deep recursions joined with enumeration over the power set of the states. The exponential gain exposed in Section 5.4 is visible in terms of number of variables and constraints in Table 5.4, and in terms of encoding time in Figure 5.15. Each dot in Figure 5.15 corresponds to one instance of our benchmark. While the encoding time ranges between 0 and 1s when using the C 9c encoding, it varies between 0 and 500s when using the SotA encoding (if it does not run out of memory). MILP formulation of logical constraints (e.g., conjunction, disjunction, implication, equivalence) implies the introduction of binary variables called indicator variables [START_REF] Belotti | On handling indicator constraints in mixed integer programming[END_REF]. Each indicator variable is associated to one or more constraints. The valuation of the indicator variable activates or deactivates its associated constraints. We tried to formulate the SotA encoding into MILP. Unfortunately, the nested conjunctions and disjunctions imply the introduction of a huge number of indicator variables, leading to giant instances giving bad encoding and solving time. However, since the Boolean variables in C 9c exactly correspond to indicator variables, the MILP formulation of the C 9c encoding does not Solving We chose Z3 [START_REF] De | Z3: An Efficient SMT Solver[END_REF] in its last version (v. 4.4.2) as SMT solver. We chose CPLEX [11]i n its last version (v. 12.6.3.0) as MILP solver. Both solvers have not been tuned and we use their default strategies. Experiments have been realized on a 2.4 GHz Intel Core i5 processor. Time out has been set to 10 minutes. Table 5.4 presents the resolution time for the existential consistency problem on our first benchmark using (1) SMT SotA encoding, (2) SMT C 9c encoding, and (3) MILP C 9c encoding. While the SotA CSPs are larger than the C 9c CSPs, the solving time for the SotA CSPs appears to be competitive compared to the solving time for the C 9c CSPs. The scatter plot in Figure 5.16 (logarithmic scale) compares solving times for the SMT SotA encoding and SMT C 9c encoding. However when considering the resolution time of the problem (i.e., the encoding time plus the solving time) the C 9c encoding clearly answers faster than the SotA encoding. Finally, the comparison between the solving time using SMT C 9c encoding and MILP C 9c encoding is illustrated in Figure 5.17. It shows that the loss of safety by passing from real numbers with Z3 SMT resolution to floating point numbers with CPLEX MILP resolution leads to a non negligible gain in terms of resolution time (near to an exponential gain in our benchmark). Indeed the SMT C 9c encoding requires 50 seconds to complete the solving process while the MILP C 9c encoding needs less than 5 seconds for the same instances. Table 5.5 summarizes the results w.r.t. our second benchmark: the pIMC sizes (in terms of states, transitions, and parameters), the CSP sizes (in terms of number of variables and constraints), and the resolution time using the Z3 solver. Note first that we perform pre-processing when verifying reachability properties: i.e., we eliminate all the states that cannot reach the goal states. This explains why C 9r has less variables and constraints than C 9c . Finally, note the order of magnitude between the resolution time required for solving the qualitative properties vs the quantitative properties w.r.t. our encodings. Indeed, we did not succeed in solving pIMCs with more than 300 states and 400 transitions for quantitative properties while we verified pIMCs with more than 10,000 states and 20,000 transitions in the qualitative context. Conclusion and Perspectives In this chapter, we have compared several Markov Chain abstractions in terms of succinctness and we have shown that Parametric Interval Markov Chain is a strictly more succinct abstraction formalism than other existing formalisms such as Parametric Markov Chains and Interval Markov Chains. In addition, we have proposed constraint encodings for checking several properties over pIMC. In the context of qualitative properties such as existential consistency or consistent reachability, the size of our encodings is significantly smaller than other existing solutions. In the quantitative setting, we have compared the three semantics for IMCs and pIMCs and showed that the semantics are equivalent with respect to quantitative reachability properties. As a side effect, this result ensures that all existing tools and algorithms solving reachability problems in IMCs under the once-and-for-all semantics can safely be extended to the at-every-step semantics with no Based on this result, we have then proposed CSP encodings addressing quantitative reachability in the context of pIMCs regardless of the chosen semantics. Finally, we have developed a prototype tool that automatically generates our CSP encodings and that can be plugged to any constraint solver accepting the SMT-LIB format as input. Our tool for pIMC verification could be extended in order to manage other, more complex, properties (e.g., supporting the LTL-language in the spirit of what Tulip [START_REF] Wongpiromsarn | TuLiP: A Software Toolbox for Receding Horizon Temp oral Logic Planning[END_REF] does). Also one could investigate a practical way of computing and representing the set of all solutions to the parameter synthesis problem. Chapter 6 Conclusion and Perspectives In this thesis we tackled two families of program verification problems. In both cases we first investigated the nature of the verification problem in order to propose an accurate constraint resolution. Since we had no a priori restrictions on the constraint language, we proposed constraint models using non-linear constraints with unbounded continuous variables, mixed integer/continuous domain variables over linear constraints, and also quadratic constraints over mixed variables. We first present in this chapter the conclusions and perspectives of both contributions. We close this thesis with a general conclusion on the benefits of considering constraint programming for program verification. Block-Diagram Verification Block-diagrams are used to model real-time systems such as digital signal processes. Such systems appear in many applications receiving and processing digital signals: modems, multimedia devices, GPS, audio and video processing. We proposed a constraint model using our global constraint called real-time-loop for computing over-approximations of real-time streams, based on their block-diagrams representations. We introduced a global constraint and presented a dedicated filtering algorithm inspired by Abstract Interpretation. The experiments show that our approach can reach very good over-approximations in a short running time. Thus, our proposal has been taken in consideration for a future implementation into the FAUST compiler. More generally, our method shows that constraint programming can handle block-diagram analyses in an elegant and natural way. However we point out some perspectives. Firstly, the propagation loop may be improved according to the tackled verification problem (for instance, when verifying output streams one should favor input to output propagations instead of an arbitrary scheme). Secondly, some constraint patterns offer poor over-approximations when considering interval extensions (e.g., the interval extension of the pattern |xint(x)| which should compute the decimal part of x returns the interval abstraction [0; 20] with [-10; 10] as input domain whereas the concrete output domain of this pattern is [0; 1]). Thus, managing such patterns would improve the quality of the computed over-approximations. Thirdly, Chapter 6. Conclusion and Perspectives one should test our approach on a language which is not dedicated to audio processing in order to test the practical robustness of our approach with respect to the nature of the programs. Finally, over-approximations intervals ensure that there are no stream values outside the intervals but cannot conclude if they contain at least one stream value. Thus, one may investigate inner-approximations in order to certify the presence of stream values. This could be used to partition the space in three: the intervals that contain only stream values, the intervals that may contain stream values, and the intervals that do not contain any stream value. Markov Chain Abstraction Verification Markov chains model softwares and reallife systems for which probabilities play a fundamental role. We considered the Parametric Interval Markov Chain (pIMC for short) specification formalism for abstracting Markov chains. We first presented a formal theorem proving the equivalence of the three main pIMC semantics with respect to the reachability property. Then, we exploited this result for proposing constraint modellings answering consistency, qualitative, and quantitative reachability properties. For consistency and qualitative reachability, the state-of-the-art constraint models had an exponential size in terms of the verified pIMCs. We proposed constraints models with a linear size in terms of the pIMC size for solving the same problems using the same type of constraints and variables. For quantitative reachability, there was no existing verification process. We thus proposed the first verification process as constraint models in order to answer this problem. Furthermore, we took benefits of the constraint programming paradigm to propose modular constraints models: i.e., the quantitative models extend the qualitative models which extend the consistency models. We implemented our constraint models and we evaluated our prototype over a pIMC benchmark generated from PRISM programs. Constraint models have been generated as mixed linear integer programs and satisfiability modulo theory programs and we obtained promising results. In practice, these results lead to pIMCs closer to the effective resolution of real-word problems. We now present some perspectives. Firstly, parameters in pIMCs may correspond to possibly controllable inputs in the probabilistic systems or may model a cost to minimize or maximize. Thus, by adding an optimization function to our constraint encodings one may investigate such problems. Secondly, parameters may correspond to decisions to be taken for implementing the pIMC as an IMC in the real-world. The visualization of the parameters state space according to the satisfiability with respect to the property to verify helps to select accurate parameter valuations. However, while some research has been realized with 2 or 3 parameters, one should also investigate cases with more parameters. Finally, the pIMC specification formalism allows to abstract sets of Markov chains. It appears that our constraint encoding may offer another specification formalism. Indeed, one should take benefit of our constraint modellings for expressing guards, relations between parameters, constraints over some probabilities on transitions, etc. Thus, all the expresivness of the constraint tools could be used for modelling and solving the verification of "constrained Markov chains". To conclude, program verification is such a rich domain that it can potentially use many constraint tools. The theoretical complexity of program verification such as constraint satisfaction may belong to high complexity levels. Nevertheless, constraint programming solvers may offer a practical resolution to some hard problems However, several constraint programming communities chose different directions and they develop solvers dedicated to separate constraint languages. In this thesis, we considered verification problems through the prism of constraint programming. We proposed constraint programming approaches using various constraint languages for solving the considered verification problems. Our contributions help both fields of constraint programming and program verification to move closer together. Introduction La programmation par contraintes est un champ de recherche rattaché à l'intelligence artificielle. Un des objectifs de l'intelligence artificielle est de proposer des méthodes et des outils permettant de réaliser des tâches considérées comme complexes tant à un niveau logique, qu'à un niveau algorithmique. Ainsi le Graal de l'intelligence artificielle est de trouver une solution, un outil capable de résoudre une variété la plus grande possible de problèmes hétérogènes. C'est avec cet objectif que la programmation par contraintes se propose de résoudre tout un ensemble de problèmes qui peuvent être formulés à partir de contraintes. Une contrainte est une relation posée sur un ensemble de variables restreignant les affectations possibles entre les variables et leurs valeurs. En effet, une variable est un objet mathématique associé à un ensemble de valeurs pouvant lui être affectées. Ainsi, nous appelons valuation le choix de valeurs pour les variables, et satisfaire une contrainte revient à trouver une valuation qui satisfasse toutes les relations variables/valeurs établies par les contraintes. La modélisation en contraintes regroupe les différentes techniques utilisées pour passer d'un problème présenté en langage naturel vers un problème formellement décrit mathématiquement ou sous la forme d'un programme en contraintes (LP format, DIMACS format, XCSP format) appelé modèle. Une fois l'étape de modélisation terminée, le modèle est envoyé dans le système intelligent, appelé solver, pour être résolu. La première étape est appelée modélisation (en : modelling)e tl as e conde résolution (en : solving). Dans cette thèse, nous nous intéressons à la modélisation et à la résolution d'applications ciblées : la vérification de programmes. Lors de ces dernières décennies, l'informatique s'est démocratisée tant dans les usages privés que professionnels. Ainsi, ordinateurs et systèmes d'informations réalisent des applications les plus variées : application intelligentes, systèmes embarqués d'avions, robots médicaux, etc. Comme c'est le cas dans le cadre des chaînes de production, l'écriture de ces systèmes/applications doit respecter un certain nombre de règles de qualité telles que la conformité, l'efficacité, la robustesse. La vérification de programmes a pour objectif de s'assurer qu'une application, un programme, un système réponde aux spécifications données, c'est-à-dire que son comportement soit correct, qu'il ne contienne pas de "bug". En effet, l'histoire a démontré la nécessité de la mise en oeuvre de telles vérifications. Qu'il s'agisse de la fusée Ariane 5 qui a explosé 36 secondes après son décalage ou du défaut de l'unité de calcul du Pentium II d'Intel qui a causé une perte de 475 millions de dollars et qui a nui gravement à l'image de la marque, ces deux évènements auraient pu être évités s'ils avaient été certifiés, vérifiés formellement d'un point de vue logiciel/programme informatique. Pour autant, la vérification de programme est une tâche difficile car c'est un problème indécidable : en général, il n'est pas possible de construire un système capable de déterminer en temps fini si un programme est correct ou non. Pour autant, indécidable ne veut pas dire infaisable en pratique. Dans cette thèse, nous modélisons et résolvons via la programmation par contraintes des problèmes de vérification de programmes. Nous présentons dans les deux sections suivantes un résumé des deux chapitres contributions de la thèse. Chacun porte sur un problème de vérification de programmes et propose une résolution en contraintes. Vérification en contraintes d'un langage temps réel La programmation par contraintes s'attaque en général à des problèmes statiques, sans notion de temps. Cependant, les méthodes de réduction de domaines pourraient par exemple être utiles dans des problèmes portant sur des flux. C'est le cas de la vérification de programmes temps réel où les variables peuvent changer de valeur à chaque pas de temps. Pour cette contribution, nous nous intéressons à la vérification de domaines de variables (flux) dans le cadre d'un langage de diagrammes de blocs. La première contribution de cette thèse (Chapitre 4) propose une méthode de réduction de domaines de ces flux, pour encadrer finement les valeurs prises au cours du temps. En particulier, nous proposons une nouvelle contrainte globale real-time-loop, nous présentons une application au langage FAUST (un langage fonctionnel temps réel pour le traitement audio) et nous testons notre approche sur différents programmes FAUST. Contexte et problématique Comme précisé en introduction de cette section, nous souhaitons vérifier un langage temps réel. Plus précisément, ce langage se positionne dans la famille des diagrammes de blocs que nous présentons ci-dessous. Nous terminons cette section par présenter la problématique de vérification traitée dans cette contribution. Un bloc est une fonction appliquant un opérateur à un ensemble d'entrées ordonnées et produisant une ou plusieurs sorties ordonnées. A partir de là, un connecteur relie une sortie d'un bloc à une entrée d'un bloc. Nous appelons un diagramme de blocs un ensemble de blocs reliés par des connecteurs. Formellement, notons E un ensemble non vide. Un 0.1 + 0.9 ⇥ fby 0 [0.9; 0.9; 0.9; 0.9] [0; 0; 0; 0] [0. pour tout t 2 N. d contient également un bloc temporel : le bloc followed by abrégé par fby Par convention, nous hachurons les blocs temporels. La présence du bloc temporel permet de casser la dépendance cyclique. Ainsi, ce diagramme de blocs admet une seule exécution. Les valeurs pour les quatre premiers temps sont inscrites entre crochets à côté des connecteurs (remarquez le délai dû au bloc fby). Après avoir décrit les diagrammes de blocs et leurs exécutions, nous présentons la problématique de vérification sous-jacente. La présence de circuits dans la représentation graphique des diagrammes de blocs ainsi que les branchements de flux d'entrées faiblement bornés peuvent générer des exécutions plus ou moins variées. Un des souhaits récurrents de la vérification est de borner les valeurs prises par les variables d'un programme. Traduit dans le contexte des diagrammes de blocs, cela revient à chercher quelles valeurs peuvent passer par les entrées et les sorties des blocs qui composent un diagramme de blocs. Ainsi, le problème de vérification que nous considérons est celui de trouver pour chaque entrée et sortie de bloc un intervalle, appelé sur-approximation, qui contienne l'ensemble des valeurs prises sur cette entrée ou sortie pour toutes les exécutions possibles du diagramme de blocs. Nous appelons ce problème, le problème de sur-approximation de flux. Modélisation et résolution en contraintes Nous proposons une modélisation et une résolution en contraintes du problème de surapproximation de flux dans les diagrammes de blocs. La première remarque est que nous restreignons l'usage des blocs temporels au bloc fby uniquement. Ainsi, sous cette condition, tout cycle dans un diagramme de blocs admet au moins un bloc fby. 1Nous rappelons que modéliser en contraintes revient à transformer un problème à résoudre en un problème de satisfaction/d'optimisation de contraintes. Pour le problème qui nous intéresse, nous souhaitons trouver pour chaque entrée et sortie de bloc un intervalle qui sur-approxime l'ensemble des valeurs des flux pouvant passer par cette entrée/sortie. L'approche que nous avons choisie est de trouver une sur-approximation la plus petite possible sans imposer le critère de minimalité. Nous verrons par la suite que les résultats que nous obtenons sont très satisfaisants, proches du minimum. Pour rappel, un CSP est défini comme un ensemble de variables X chacune associée à un domaine D x (x 2 X)e tu ne n s e m b l ed ec o n t r a i n t e sC. Les contraintes définissent les valuations des variables acceptées par le CSP (ex : la contrainte x =2+y porte sur deux variables x et y et impose l'égalité entre la valeur de x et la valeur de y plus 2). Dans cette thèse, nous procédons en trois étapes de raffinement de nos modèles en contraintes pour atteindre notre modélisation finale. Le premier modèle, appelé modèle naïf, transforme le diagramme de blocs en un réseau de contraintes où les variables du CSP correspondent aux entrées et aux sorties des blocs du diagramme de blocs, et les contraintes sont exactement les opérateurs des blocs. Ainsi, toute solution de ce CSP correspond à une exécution du diagramme de blocs (c'est-à-dire les domaines des variables sont l'ensemble des flux à valeurs dans D). Les flux solutions pouvant être infinis et le nombre de solutions pouvant également être infini, il y a peu d'espoir de parvenir à synthétiser par ce modèle en contraintes l'ensemble des flux solutions pour chaque variable via une sur-approximation dans les intervalles. De fait, en s'inspirant de l'interprétation abstraite, nous considérons une abstraction du problème (qui peut être vue comme une relaxation pour la communauté contrainte) pour construire un deuxième modèle en contraintes. Ce second modèle, appelé modèle intermédiaire, considère comme domaine des variables l'ensemble des intervalles fermés à bornes dans D. 2 Les opérateurs des blocs sur les flux sont remplacés par une de leurs extensions aux intervalles. Ce modèle est tel que toute solution répond au problème de la sur-approximation de flux. Cependant, ce modèle intermédiaire retourne des solutions de faible qualité et il reste facilement bloqué aux infinis lors de la propagation de contraintes. Dès lors, nous proposons un dernier modèle, appelé modèle final, prenant en compte ce défaut de filtrage. Pour ce faire, nous introduisons une nouvelle contrainte globale : la contrainte real-time-loop . Nos recherches ont montré que la difficulté se posait au niveau des circuits. Chaque contrainte real-time-loop contient l'ensemble de la sémantique d'un circuit. C'est grâce à la connaissance de la sémantique haut niveau de cette contrainte que nous avons proposé un algorithme de filtrage dédié offrant des sur-approximations de meilleure qualité. L'algorithme de filtrage proposé est inspiré de l'interprétation abstraite et plus particulièrement de la technique de recherche d'invariant inductif par la méthode des montées et descentes dans le treillis des sur-approximations (widening and narrowing technique). L'algorithme proposé n'assure pas de retourner une sur-approximation minimale. Cependant il possède des heuristiques permettant d'augmenter les possibilités de recherche et donc d'augmenter les chances de s'approcher de la solution minimale. L'exemple 2 illustre les différences entre le modèle naïf, intermédiaire et final. Pour évaluer notre modélisation en contraintes, nous avons choisi le langage FAUST permettant de faire la synthèse et le traitement temps réel de flux audio. Ce langage possède une sémantique bien définie de telle sorte que le compilateur FAUST peut générer pour chaque programme le diagramme de blocs qui en est la sémantique. Ainsi, nous avons considéré un ensemble de programmes de la bibliothèque FAUST et nous les avons modélisés en contraintes en utilisant notre encodage final. Nous avons ensuite utilisé le solveur de contraintes continues IBEX pour y implanter notre contrainte globale real-time-loop . L'objectif était de vérifier que les flux de sorties des programmes ne provoquaient pas de saturation (c'est-à-dire qu'ils étaient sur-approximés par l'intervalle [-1; 1]). Nos résultats montrent que nous parvenons dans la majeure partie des cas à trouver la plus petite sur-approximation dans les intervalles et dans des temps très courts (de l'ordre de la seconde). Ainsi notre approche a été prise en considération par les développeurs de FAUST pour une éventuelle intégration dans une version future du logiciel. Pour autant, il est important de noter que la plus petite sur-approximation dans les intervalles par le calcul intervalle est parfois très éloignée du plus petit intervalle contenant les valeurs prises par les flux. En effet, les calculs à partir de sur-approximations produisent des marges de sur-approximations qui se répètent et s'amplifient. Il existe également des opérateurs de blocs produisant de grandes imprécisions lorsqu'ils sont étendus aux intervalles (ex : X[-]X 6 = {0}). Pour conclure, nous avons proposé plusieurs modélisations en contraintes pour le problème de la sur-approximation de flux dans les diagrammes de blocs. Nous avons proposé la contrainte globale real-time-loop pour répondre au problème donné. Puis nous avons évalué avec succès notre approche en considérant le problème de recherche de saturation dans des programmes FAUST. Ces travaux ont montré l'intérêt de l'utilisation de techniques de programmation par contraintes dans des cadres exotiques (la vérification de programmes utilisant des variables de flux). Ces travaux ont fait l'objet de trois communications/publications [1,2,3]. Vérification en contraintes de systèmes probabilistes Les chaînes de Markov (MCs) sont largement utilisées pour modéliser une très grande variété de systèmes basés sur des transitions probabilistes (ex : protocoles aléatoires, systèmes biologiques, environnements financiers). D'un autre côté, les chaînes de Markov à i n t e r v a l l e s p a r a m é t r é s ( p I M C s ) s o n t u n f o r m a l i s m e d e s p é c i fi c a t i o n p e r m e t t a n t d e représenter de façon compacte des ensembles infinis de chaînes de Markov. En effet, les PIMCs prennent en compte l'imprécision ou le manque de connaissances quant à la probabilité exacte de chaque évènement/transition du système en considérant des intervalles paramétrés de probabilités. Dans la seconde contribution de cette thèse (Chapitre 5), nous proposons d'abord une comparaison formelle de trois sémantiques existantes pour les PIMCs. Ensuite, nous proposons des encodages en contraintes pour vérifier des propriétés d'accessibilité qualitative et quantitative sur les pIMCs. En particulier, l'étude formelle des diffé r e n t e s s é m a n t i q u e s d e s p I M C s a p e r m i s d e p r o p o s e r d e s e n c o d a g e s e n contraintes succincts et performants. Enfin, nous concluons avec des expériences montrant l'amélioration de nos encodages en contraintes par rapport à ceux de l'état de l'art résolvant les mêmes problèmes sur les pIMCs. Contexte et problématique Un processus aléatoire est un système dans lequel le passage d'un état à un autre état est probabiliste : chaque état successeur a une certaine probabilité d'être choisi. Une chaîne de Markov à temps discret est un processus aléatoire dont le passage d'un état à un autre Modéliser une application comme une chaîne de Markov suppose de connaître exactement les probabilités pour chaque transition du système. Cependant, ces quantités peuvent être difficiles à calculer ou à mesurer pour des applications réelles (ex : erreurs de mesure, connaissance partielle). Les chaînes de Markov à intervalles (IMCs) étendent les chaînes de Markov en autorisant les probabilités de transition à varier dans des intervalles donnés. Ainsi, à chaque transition d'état à état est associé un intervalle au lieu d'une probabilité exacte. Enfin, les chaînes de Markov à intervalles paramétrés (pIMCs) autorisent l'utilisation d'intervalles dont les bornes sont variables. Ces bornes variables sont alors représentées par des paramètres (ou des combinaisons de paramètres), ce qui permet notamment l'expression de dépendances entre plusieurs transitions du système. Ainsi, les pIMCs représentent, d'une manière compacte et avec une structure finie, un ensemble potentiellement infini d'IMCs. Par transitivité, les pIMCs permettent de représenter potentiellement une infinité d'ensembles de chaînes de Markov. La propriété que nous allons vérifier est celle de l'accessibilité (en : reachability)d a n s les MCs. Formellement, la probabilité d'atteindre un état dans une MCs est donnée par la somme de la probabilité de tous les chemins atteignant l'état désiré (c'est-à-dire tous les chemins finis partant de l'état initial, terminant par l'état désiré et ne rencontrant pas cet état avant). De plus, la probabilité d'un chemin correspond aux produits des probabilités rencontrées sur les transitions état à état. Nous notons P M (3s)laprobabilit éd'atteindre un état s dans une MC M. Exemple 3. La figure 5 représente une MC M avec 5 états s 0 , s 1 , s 2 , s 3 et s 4 où s 0 est l'état initial et où nous pouvons lire par exemple que la probabilité de passer de l'état s 0 à l ' é t a t s 1 vaut 0.7e tq u ec e l l ed ep a s s e rd el ' é t a ts 0 à s 2 vaut 0.3. Ainsi les séquences d'états (s 0 ,s 1 ,s 3 ), (s 0 ,s 3 )e t( s 0 ,s 2 ,s 1 ,s 3 )s o n tt r o i sc h e m i n s( fi n i s )p a r t a n td el ' é t a t initial s 0 et terminant dans l'état s 3 ayant pour probabilités respectives 0.7 • 0.5=0 .35, 0.7•0=0et0.3•0.5•0.5=0.075 . Enfin, la probabilité d'atteindre l'état s 1 vaut p(s 0 )(s 1 )+ Σ +1 i=0 p(s 0 )(s 2 )•p(s 2 )(s 2 ) i •p(s 2 )(s 1 )=p(s 0 )(s 1 )+p(s 0 )(s Les pIMCs et les IMCs sont appelées des modèles d'abstractions de chaînes de Markov. En effet, comme dit précédemment, tout pIMC ou IMC représente/abstrait un ensemble de chaînes de Markov. Ainsi, nous disons qu'une chaîne de Markov satisfait une abstraction de chaînes de Markov ssi la chaîne de Markov appartient à l'ensemble des MCs représentées par l'abstraction. De plus, les IMCs sont formellement définies avec trois sémantiques d'abstractions : 1) once-and-for-all,2)IMDP et 3) at-every-step. La première sémantique définit que l'ensemble des MCs qui satisfont une IMC sont celles qui ont la même structure que l'IMC et dont la probabilité p de passer d'un état s accessible à un état s 0 appartient à l ' i n t e r v a l l e d e p r o b a b i l i t é s s u r l a t r a n s i t i o n d e s vers s 0 dans l'IMC. Nous disons que pour chaque intervalle de l'IMC une et une seule probabilité est sélectionnée. La seconde sémantique définit que l'ensemble des MCs qui satisfont une IMC sont celles qui autorisent de choisir plusieurs probabilités pour un même intervalle d'une IMC. Nous disons que l'IMC originale est "dépliée". Ainsi, un état d'une IMC peut se retrouver "copié" plusieurs fois dans la MC qui satisfait l'IMC. Enfin, la troisième sémantique autorise sous certaines conditions que certains états de l'IMC peuvent être fusionnés ou scindés en plusieurs états tout en autorisant le dépliage de l'IMC. Cette sémantique correspond à la sémantique originelle donnée aux IMCs. Nous montrons dans cette thèse que la sémantique at-everystep est plus générale que la IMDP, qui est plus générale que la once-and-for-all. Toutes ces sémantiques s'étendent aux pIMCs. Ainsi, la partie contribution aborde trois problèmes majeurs de vérification sur les pIMCs : la consistance (existentielle),l ' accessibilité qualitative (existentielle) et l'accessibilité quantitative (existentielle). Le problème de la consistance d'une pIMC détermine si une pIMC admet au moins une MC qui la satisfait. Le problème de l'accessibilité qualitative détermine si pour un ensemble d'états à atteindre il existe une MC qui satisfait la pIMC où un des états but peut être atteint (c'est-à-dire qu'il existe un chemin avec une probabilité non nulle qui part de l'état initial de cette MC et atteint l'état but). Le problème de l'accessibilité quantitative détermine si, pour un ensemble d'états à atteindre et un seuil d'accessibilité, il existe une MC qui satisfait la pIMC où la probabilité d'atteindre les états buts est supérieure ou inférieure au seuil. Modélisation et résolution en contraintes Dans un premier temps, nous avons prouvé que les valeurs de probabilités maximales et minimales d'accessibilité d'états sont atteintes par les MCs de même structure que les IMCs/pIMCs. C'est grâce à ce théorème fort que nous avons pu proposer des modèles en contraintes succincts pour vérifier les problèmes présentés dans la précédente section. En effet, il n'est plus nécessaire de considérer toutes les MCs avec "dépliages" pour vérifier la consistance et les propriétés d'accessibilité qualitatives et quantitatives, mais uniquement les MCs de même structure que la pIMC. Nous présentons maintenant les modèles en contraintes proposés. Nos modélisations en contraintes sont modulaires. C'est-à-dire qu'un premier lot de contraintes résout le problème de la consistance, puis l'ajout d'un second lot de contraintes vient répondre au problème de l'accessibilité qualitative et l'ajout d'un dernier lot permet de répondre au problème de l'accessibilité quantitative. L'objectif de nos modèles en contraintes est de construire une chaîne de Markov qui satisfasse l'IMC vérifiant la propriété désirée (consistance, accessibilité qualitative ou accessibilité quantitative). Ainsi, nos modèles en contraintes encodent de telles MCs. Formellement, étant donnée une pIMC à vérifier et T un ensemble d'états à atteindre, nos modèles en contraintes définissent les variables ⇢ s , ! s , ↵ s , λ s , ⇡ s pour chaque état s de la pIMC, une variable φ p par paramètre p de la pIMC et une variable θ s 0 s par intervalle paramétré dans la pIMC. Rappelons que ces variables ont pour objectif de construire une MC. Chaque variable θ s 0 s détermine la probabilité de la transition allant de l'état s vers l'état s 0 dans la MC. Pour tout état s,l av a r i a b l eρ s est une variable booléenne indiquant si l'état s est accessible depuis l'état initial ; la variable ω s est une variable entière qui vaut k s'il existe un chemin de taille k -1d e p u i sl ' é t a ti n i t i a lv e r ss, et qui vaut 0 sinon ; la variable α s est une variable entière qui vaut k s'il existe un chemin de taille k -1d e p u i ss vers un état but s 0 dans T , et qui vaut 0 sinon ; la variable λ s est une variable booléenne qui vaut true ssi il existe un chemin depuis l'état initial vers un état but de T passant par s ; et la variable π s vaut la probabilité d'atteindre l'état s depuis l'état initial si s est accessible et qui vaut 0 sinon. Voici les contraintes à considérer pour chaque état s de la pIMC : (1) ρ s , si s = s 0 (2) ¬ρ s , Σ s 0 2Pred(s)\{s} θ s s 0 =0, sis 6 = s 0 (3) ¬ρ s , Σ s 0 2Succ(s) θ s 0 s =0 (4) ρ s , Σ s 0 2Succ(s) θ s 0 s =1 (5) ρ s ) θ s 0 s 2 P (s, s 0 ), pour tout s 0 2 Succ(s) Nous montrons dans cette thèse que ces lots de contraintes permettent de répondre aux problèmes de la consistance existentielle (contraintes (1) à ( 5)), l'accessibilité qualitative existentielle (contraintes (1) à ( 10))e tl ' a c c e s s i b i l i t éq u a n t i t a t i v ee x i s t e n t i e l l e (contraintes (1) à ( 18)). De plus, nos modèles sont linéaires en taille par rapport à la taille de la pIMC là où les modèles de l'état de l'art sont exponentiels en taille. Nous terminons la contribution par une évaluation pratique de nos modèles en contraintes. Les contraintes sont linéaires (sauf la contrainte (18) qui est quadratique) et utilisent des expressions logiques comme l'équivalence et l'implication. Quant aux variables, nous sommes dans le cas mixte avec la présence de variables booléennes, entières et réelles. Ainsi, la communauté Satisfiability Modulo Theory se propose de résoudre ce genre de problèmes. Dans le cas linéaire il y a également la communauté Mixed Integer Linear Programming qui accepte nos CSPs. Nous sommes allés chercher un jeu de tests dans la communauté des MCs. Nous avons étendu ces MCs à des pIMCs et avons vérifié dessus les propriété de consistance, d'accessibilités qualitative et quantitative. Notre outil est disponible en ligne. 3 Nos résultats montrent que nos modèles en contraintes sont plus performants que ceux de l'état de l'art. En effet, nos modèles en contraintes gagnent un ordre de complexité en terme de taille ce qui permet de s'attaquer à des pIMCs beaucoup plus grandes (c'est-à-dire avec des dizaines de milliers d'états). Enfin, nous proposons un premier outil pour réaliser la vérification d'accessibilité quantitative sur des pIMCs. Pour cette propriété, nous parvenons à traiter des pIMCs ayant une centaine d'états. ( Pour conclure, dans cette partie de la thèse, nous avons réalisé une analyse formelle de propriétés sur les abstractions de chaînes de Markov. Dans cette analyse, nous avons montré que les différentes sémantiques données aux IMCs sont équivalentes par rapport à l'accessibilité quantitative de probabilité maximale et minimale (ce qui s'étend aux pIMCs). Grâce à ce résultat, nous avons présenté des modèles en contraintes qui forment la première solution pratique au problème de l'accessibilité quantitative dans les pIMCs. Dans le même temps, nous avons amélioré les encodages en contraintes existants pour résoudre la consistance existentielle et l'accessibilité qualitative. Enfin, nous avons proposé un outil implémentant nos divers encodages en contraintes. Ces travaux ont fait l'objet de trois communications/publications [4,5,6]. Contents 1 . 1 5 1. 2 7 1. 3 8 1. 4 11527384 Scientific Context ............................. Problems and Objectives ......................... Contributions ............................... Outline ................................... 9 Empty 5⇥5 chessboard with 5 Queens. Queens positioning respecting "no threat" rules. Queens positioning violating twice the diagonal "no threat" rule. Figure 2 . 1 : 5 - 215 Figure 2.1: 5-Queens problem illustrated with: (a) its 5⇥5 empty chessboard and its 5 queens; (b) a queen configuration satisfying the 5-Queens problem; and (c) a queen configuration violating the 5-Queens problem. Figure 2 . 2 : 22 Figure 2.2: Three constraints c 1 , c 2 and c 3 over two variables x and y. 1 : 1 function satisfaction(P =(X, D, C):CSP) return Map<X,D> Definition 2 . 3 . 1 ( 231 Reformulation). Le C be a CSP. A reformulation ⇢ transforms a CSP C into a CSP C 0 s.t. all the solutions of C can be mapped to a solution in C 0 and all the solutions of C 0 are translatable as a solution C. Thus, C 0 models the same problem than C. Figure 2 . 3 : 23 Figure 2.3: Box Paving for a Constraint Satisfaction Problem. Gray boxes only contain solutions. Pink boxes contains at least one solution. The union of the gray and the pink boxes covers all the solutions. ................................ 3.2 Abstract Interpretation .......................... 3.3 Model Checking .............................. 3.4 Constraints meet Verification ....................... Figure 3 . 1 : 31 Figure 3.1: Instance of four possible traces of a variable x while executing the same program. Figure 3 . 2 : 32 Figure 3.2: Instance of four possible traces of a variable x while executing the same program. Figure 3 . 3 : 33 Figure 3.3: Running program example with its five traces. Polyhedron AbstractProving the no forbidden zone overlapping. Figure 3 . 4 : 34 Figure 3.4: A concrete domain (a) over variables x and y abstracted by an interval abstract domain (b) and a polyhedron abstract domain (c) respectively without (a,b,c) and with (d,e,f) a forbidden zone s.t. interval abstraction produces a false alarm (e) while polyhedron abstraction proves safety (f). pref ix 1 : ABBD 0.168 pref ix 2 : ACBD 0.09 pref ix 3 : ACCB 0.075 pref ix 4 : ACEE 0 pref ix 5 : EEEE 0 (b) 5 trace prefixes with their respective probability to occur. Figure 3 . 5 : 35 Figure 3.5: A Markov chain next to 5 trace prefixes with size 4, associated with their respective probability to occur. Definition 4 . 2 . 1 ( 421 Block). Let E be a nonempty set. A block over E is a triple b = (op, n, m) such that: n 2 N is the number of inputs of the block, m 2 N is the number of outputs, and op : E n ! E m is the operator of the block. The n inputs and the m outputs are ordered: [i]b refers to the ith input (1  i  n) and b[j] to the jth output (1  j  m). Definition 4 . 2 . 2 ( 422 Connector). Let B be a set of blocks. A connector over B is a pair (b[i], [j]b 0 ) such that: b and b 0 are blocks from B; output i exists for block b and input j exists for block b 0 . Definition 4.2.3 (Block-Diagram). Let E be a nonempty set. A block-diagram over E is a pair d =( B, C) such that: B is a set of blocks over E and C is a set of connectors over B. An input (respectively output) of a block in B that does not appear in a connector of C is an input (respectively an output) of the block-diagram d. Similarly to the blocks, if a block-diagram d has n inputs and m outputs, we can order them and: [i]d refers to the ith input (1  i  n), and d[j]tothejth output (1  j  m). Finally, we denote BD(E)t h es e to fa l lt h eb l oc k -d i a g r a m so v e rE. Example 14. Figure 4 . 4 1 depicts a block-diagram over real numbers in Block(R)c o ntaining three blocks: block b 1 has the square function as operator; block b 2 has the subtraction, and block b 3 has the multiplication. Connectors are represented by arrows: connector Definition 4 . 2 . 4 ( 424 Interpretation). Let E be a nonempty set, b =( op, n, m) a block in Block(E), and d =(B, C) a block-diagram in BD(E).A ninterpretation I of block b is a mapping from each input i to an element in E (noted I([i]b)), and a mapping from each output j to an element in E (noted I(b[j])). An interpretation I of the block-diagram d is an interpretation of each block in B. Definition 4 . 2 . 5 ( 425 Model). Let E be a nonempty set, b =(op, n, m) a block in Block(E), and d =(B, C) a block-diagram in BD(E). • An interpretation I of block b is a model of b iff op(I([1]b),...,I([n]b)) = (I(b[1]),...I(b[m])) • An interpretation I of block-diagram d is a model of d iff 8b 2 B : I is a model of b and 8(b[i], [j]b 0 ) 2 C : I(b[i]) = I([j]b 0 ) Example 15 (Example 14 continued). A block-diagram interpretation is presented in Figure 4.2. The interpretation is given by labeling all the inputs and all the outputs. For instance, I([1]b 2 )equals4andI(b 3 [1]) equals 12. Moreover, this interpretation is a model of the block-diagram expressing that the input 2, 1 produces the output 12. Note that a block-diagram can have one or many models. The model presented in Example 15 is one among an infinity of possible ones. Figure 4 . 2 : 42 Figure 4.2: A block-diagram in BD(R) labeled with an interpretation Definition 4 . 2 . 7 ( 427 Functional Block). Let D be a nonempty set, and b =( op, n, m) in Block(S(D)). b is a functional block iff 9f : D n ! D m such that 8s 1 ,...,s n ,s 0 1 ,...,s 0 m 2 S(D): op(s 1 ,...,s n )=(s 0 1 ,...,s 0 m ) implies the following: 8t 2 N,f(s 1 (t),...,s n (t)) = (s 0 1 (t),...,s 0 m (t)) Definition 4.2.8 (Followed-by Block). Let D be a nonempty set. The followed-by block over D (written fby) is the block (op, 2, 1) in Block(S(D)) such that op is the function from S(D) ⇥ S(D) to S(D) where op(a, b)=c, c(0) = a(0), and c(t)=b(t -1), for all t>0.Example 16. Figure 4 . 4 3 shows a block-diagram d over real-number streams: d 2 Figure 4 . 3 : 43 Figure 4.3: A block-diagram over streams from BD(S(R)). In brackets, the first values of the model for t =0, 1, 2, 3. Example 17 ( 17 Example 18 continued). Block-diagram d over real-number streams in Figure 4.3 contains one cycle which contains one fby block. This block-diagram admits only one model/trace. Indeed, using the three constant blocks 0, 0.1a n d0 .9fi x e sa l lt h e values in the cycle. Values for the 4 first time steps of the model are attached to each connector (note the delay due to the fby block). Values for the 21 first time steps are presented in Figure 4.4. Note the delay between the output of the block + and the output of the fby block: the height of the circle corresponds to the height of the square at the previous time step. Problem Definition. Let D be a nonempty set and d be a block-diagram in BD(S(D)). Associate to each block input/output s in d as u b s e tS of D s.t. for each model/trace I of d and for each time step t in N the value s(t)isinS. S is called an over-approximation of s in d. Over-Approximation Quality. Let D be a nonempty set, d be a block-diagram in BD(S(D)), s be a block input/output in d,a n dS, S 0 subsets of D be two overapproximations of s in d. If S ✓ S 0 then the over-approximation S is preferred to the over-approximation S 0 . Example 18 (Example 17 continued). Interval [0, 1] contains all the values taken by the streams model of the outputs for the blocks +, ⇥ and fby for the first 21 time steps in the block-diagram in Figure 4.3. Interval [0.1, 0.9] is a better over-approximation than [0, 1] for the output of the block + for the 21 first time steps.We introduce the temporal abstraction of streams in Definition 4.3.1. The temporal abstraction of a stream returns the set of all values taken by this stream. This set (and any superset) is called an over-approximation of the stream. As said previously, the size of this Figure 4 . 4 : 44 Figure 4.4: Values of the streams model of the block-diagram in Figure 4.3 for the outputs of blocks ⇥,+ ,a n dfby for the 21 first time steps. Figure 4 . 5 : 45 Figure 4.5: A block-diagram over streams from BD(S(R)) with connectors labelled by variables Figure 4 . 6 : 46 Figure 4.6: Naive constraint model for the block-diagram in Fig 4.5 Figure 4 . 7 : 47 Figure 4.7: Medium constraint model for the block-diagram in Fig 4.5 Definition 4 . 4 3.2 introduces Constraint Satisfaction Problems. We propose to model as aC S Pt h es t r e a mo v e r -a p p r o x i m a t i o np r o b l e m . B l o c k -d i a g r a m sc o m p u t eo u t p u t sf r o m inputs. To determine over-approximations of the streams in a block-diagram (B, C)i nBD(S(D)), we associate to each input and to each output from the blocks in B a variable with domain S(D). Then, for each block in B we consider its operator as a constraint linking the block outputs to the block inputs. Furthermore, for each connector in C, we add a constraint to ensure the equality of its streams. We name naive model this model using variables over streams. Example 19 presents the naive model on our running example.Example 19 (Example 18 continued). Figure 4.5 contains the same block-diagram as in Example 18 with constraint variables associated to the inputs and the outputs. Note that in our example, variables have been unified per connectors (e.g.,[ 1 ] +=⇤[1] = b). About the constraint programming model, the block with the operator + computes c as af u n c t i o no fb and d, yielding the constraint: c = b + d. Figure 4 . 4 6 shows the constraint model over streams for our example. Definition 4.3.3 (Interval Extension Function). Let D be a nonempty set and f be a function from S(D) n to S(D) m with n, m 2 N. An interval extension function of f , is a function 3.4). For instance consider the constraint a = b+c over intervals. The function f (A, B, C)=(A\(B+C),B\(A-C),C\(A-B)) is apropagatorforthisconstraint. Ifthedomainsforthevariablesa, b,andc are respectively [-1, 4], [-1, 3], and [0, +1] then the propagator f reduces the domains for the variable a, b,a n dc to respectively [-1, 4], [-1, 3] and [0, 5]. Definition 4.3.4 (Constraint Propagator). Let (X, D, C) be a CSP with X = {x 1 ,...,x n }, and let c be a constraint in C defined over the set of variables X 0 ✓ X. A propagator f for the constraint c is a function from P(D) to P(D) such that f (D 0 x 1 ,...,D 0 xn )=D 00 x 1 ,...,D 00 xn with Figure 4 . 4 Figure 4.7 shows the medium constraint model for the block-diagram in Figure 4.5 constructed from its naive constraint model over streams in Figure 4.6.S o l v i n g t h i s constraint model computes an interval over-approximation of each stream, provided that the interval extensions of the functions are correct. Therefore, this translation of blockdiagrams into a constraint problem allows to compute hulls (over-approximations) of the streams. Figure 4 . 8 : 48 Figure 4.8: Dependency graph of the block-diagram in Figure 4.5 where strongly connected components are surrounded with dashed lines Figure 4 . 9 : 49 Figure 4.9: Optimized model of the block-diagram in Figure 4.5 Definition 4 . 4 . 1 ( 441 Dependency Graph). Let E be a nonempty set, and d =( B, C) be a block-diagram in BD(E). The dependency graph of d is the directed graph G =( V, A) in which each node of V corresponds to a different block from B such that |V | = |B| and each arc of A corresponds to a different connector from C such that |A| = |C|. Definition 4 . 4 . 2 ( 442 Loop Transfer Function). Let d be a cycle block-diagram in BD(S(D)) and X = {x 1 ,...,x k } be a set of blocks inputs or outputs from d called argument. F : D k 7 ! D k is a loop transfer function of d for argument X,i ff for all I model of d and for all t in N: 1 : 1 function minimalCausalSet(G : Graph) return Set<Vertex> 2: Proposition 1 . 1 Let D be a nonempty set, d =( B, C) be a block-diagram in BD(S(D)) and F be a loop transfer function (extended to intervals) of d with argument X of size k. and(w n ) n2N be the sequences of values taken respectively by the variables "min", "current", and "max"ateac hev aluationoftheloopcondition(line 8) during an execution of Algorithm 3 with a function F from I(D) k to I(D) k (k 2 N). Note that values for u n , v n ,a n dw n are in I(D) k . Let n 2 N and i 2 {1,...,k} we write u n [i], v n [i], and w n [i]t h eith interval in u n , v n , and w n respectively. Then we have u n [i], v n [i], and w n [i] belonging to I(D). 1 : 1 function overApproximation(f : I(D) k ! I(D) k ) return List<I(D)> , max, image : List<I(D)> 4: Figure 4 . 10 : 410 Figure 4.10: FAUST Compilation Scheme ----------------------------------------------// Title : Volume control in dB // Remark : extracted from Faust examples //----------------------------------------------import (" music . lib" ); smooth ( c ) = * (1-c) : +˜ * (c) ; // vslider :default value : 0 // range between : -70 and +4 // range with a step of : 0.1 gain = vslider (" [1]" ,0 ,-70, +4, 0.1) :d b 2 l i n e a r:s m o o t h ( 0 . 9 9 9 ) ; process = * (gain); Figure 4 Figure 4 . 12 :0=R 4412 Figure 4.11: FAUST Volume Controller Source Program Figure 5 . 1 : 51 Figure 5.1: MC M 1 Definition 5 . 3 . 1 ( 531 Markov chain Abstraction Model). A Markov chain abstraction model (an abstraction model for short) is a pair (L, |=) where L is a nonempty set and |= is a relation between MC and L. Let P be in L and M be in MC we say that M implements P iff (M, P) belongs to |= (i.e., M| = P). When the context is clear, we do not mention the satisfaction relation |= and only use L to refer to the abstraction model (L, |=). Figure 5.2: pMC I 0 Figure 5 . 5 Figure 5.3: IMC I Figure 5 . 4 :Figure 5 . 5 : 5455 Figure 5.4: MC M 2 satisfying the IMC I from Figure 5.3 w.r.t. |= d I ( 1 ) 1 Proof. Let I =( S, s 0 ,P,V)b ea nI M Ca n dM =( T,t 0 , p, V 0 )b ea nM C .W es h o wt h a t (1) M| = o I I ) M| = d I I;( 2 )M| = d I I ) M| = a I I; (3) in general M| = d I I 6 ) M| = o I I; (4) in general M| = a I I 6 ) M| = d I I. This will prove that |= a I is strictly more general than |= d I and that |= d I is strictly more general than |= o I . At the same time, note that the following examples also illustrates that even if a Markov chain satisfies an IMC with the same graph representation w.r.t. the |= a I relation it may not verify the |= o I relation. If M| = o I I then by definition of |= o I we have that T = S, t 0 = s 0 , V (s)=V 0 (s)f o r all s 2 S,a n dp(s)(s 0 ) 2 P (s, s 0 )f o ra l ls, s 0 2 S. The mapping ⇡ from T = S to S being the identity function is such as required by definition of |= d 3 Figure 5 . 6 : 356 Figure 5.6: IMC I, MCs M 0 1 , M 0 2 ,a n dM 0 3 s.t. M 0 1 |= a I I, M 0 1 |= d I I and M 0 1 |= o I I; M 0 2 |= d I I and M 0 2 6 |= o I I; M 0 3 |= a I I and M 0 3 6 |= d I I; the graph representation of I, M 0 1 , and M 0 3 are isomorphic; as required by definition of |= a I (consider for each state in T the correspondence function δ : T ! (S ! [0, 1]) s.t. δ(t)(s)=1if⇡(t)=s and δ(t)(s)=0otherwise). Th usM| = a I I. (3) Consider IMC I and MC M 0 2 from Figure 5.6. By definition of |= Figure 5 . 5 Figure 5.7: pIMC P Example 30 . 30 They are written |= a pI , |= d pI ,a n d|= o pI and defined as follows: M| = a pI P (resp. |= d pI , |= o pI )i ff there exists an IMC I instance of P such that M| = a I I (resp. |= d I , |= o I ). Consider the pIMC P =( S, s 0 ,P,V,Y) given in Figure 5.7. The set of states S and the labelling function are the same as in the MC and the IMC presented in Figures 5.1 and 5.3 respectively. The set of parameters Y has two elements p and q. Figure 5 . 8 : 58 Figure 5.8: IMC I with three pMCs P 1 , P 2 ,a n dP n entailed by I w.r.t. |= a I . Figure 5 . 10 :Figure 5 . 5105 Figure 5.10: Variables in the CSP produced by C 9c for the pIMC P from Fig. 5.7 5. 5 . 1 51 Equivalence of |= o I , |= d I and |= a I w.r. 3 Figure 5 . 12 : 3512 Figure 5.12: An IMC I and three MCs M 1 , M 2 ,a n dM 3 satisfying I w.r.t. |= a I s.t. P M 1 (3↵)  P M 2 (3↵)  P M 3 (3↵)a n dM 3 has the same structure as I. Lemma 4 . 4 Let I =( S, s 0 ,P,V) be an IMC, M be an MC satisfying I w.r.t. |= a I , and ↵ ✓ A be a proposition. There exist MCs M 1 and M 2 satisfying I w.r.t. |= o I such that P M 1 (3↵)  P M (3↵)  P M 2 (3↵). C 0 9r (P, ↵)=( X [ X 0 ,D [ D 0 ,C [ C 0 )b es u c ht h a t( X, D, C)=C 9r(P), X 0 contains one Boolean variable λ s and one integer variable ↵ s with domain [0, |S|] per state s in S, D 0 contains the domains of these variables, and C 0 is composed of the following constraints for each state s 2 S: state ρ s ω s α s λ s s 0 true 13true s 1 true 22true s 2 true 20false s 3 true 31true s 4 true 30false Figure 5 . 7 ( 11 ) 30false5711 Figure 5.13: A solution to the CSP C 0 9r (P, {↵, β})f o rt h ep I M CP from Fig. 5.7 Proposition 8 . 8 Let P =( S, s 0 ,P,V,Y) be a pIMC and ↵ ✓ A be a state label. There exists an MC M| = a pI P iff there exists a valuation v solution of the CSP C 0 9r (P, ↵) s.t. for each state s 2 S: v(λ s ) is equal to true iff P M s (3↵) 6 =0. Example 32. Figure 5.13 presents a solution to the CSP C 0 9r (P, {↵, β})f o rt h ep I M C P from Figure 5.7. Figure 5.14: nand K=1; N=3 benchmark formulated in the PRISM adapted from[START_REF] Kwiatkowska | PRISM 4.0: Verification of Probabilistic Real-time Systems[END_REF]. Figure 5 . 15 : 515 Figure 5.15: Comparing encoding time for the existential consistency problem Figure 5 . 16 : 516 Figure 5.16: Comparing solving time for the existential consistency problem Figure 5.17: Comparing solving time between SMT and MILP formulations Figure 1 :Exemple 1 . 11 Figure 1: A block-diagram over streams from BD(S(R)). In brackets, the first values of the model for t =0, 1, 2, 3. Figure 2 : 2 Figure 2: CSP C 1 produit par notre modèle en contraintes naïf pour le diagramme de blocs de la figure 1 Figure 3 : 3 Figure 3: CSP C 2 produit par notre modèle en contraintes intermédiaire pour le diagramme de la figure 1 Figure 4 : 4 Figure 4: CSP C 3 produit par notre modèle en contraintes final pour le diagramme de blocs de la figure 1 Exemple 2 . 2 La figure 2 contient le CSP C 1 résultant de notre modélisation en contrainte naïve pour le diagramme de blocs d présenté dans la figure 1. Notez en premier, que chaque connecteur a été associé à une variable. Par exemple, les sorties des blocs constants 0.1, 0 et 0.9 correspondent aux variables d, e et f dans le CSP. De plus, chaque bloc produit une contrainte dans le CSP (nous utilisons une notation préfixée pour l'écriture des opérateurs en contraintes). Par exemple le bloc fby du diagramme de blocs d produit la contrainte a =fby(e, c)dansC 1 . Ainsi, nous avons que toute solution de C 1 correspond à une exécution de d. Ensuite, la figure 3 contient le CSP C 2 produit par notre modélisation en contrainte intermédiaire pour le diagramme de blocs d. Étant donné un opérateur sur des flux, la notation [f ] correspond à l'extension aux intervalles en tant que contrainte de la fonction f . Par exemple, a =[ 0 , 1], b =[ 2 , 3] et c =[ 2 , 4] satisfait la contrainte a[+]b = c.D e fait, le passage de C 1 à C 2 ar e m p l a c él e sd o m a i n e sd efl u xàd e sd o m a i n e sài n t e r v a l l e s , et les opérateurs dans les contraintes sont remplacés par leurs extensions aux intervalles. Ainsi, pour toute variable x de C 1 , pour toute solution de C 2 l'intervalle choisi pour x contient toutes les valeurs des flux solutions de C 1 pour la variable x (c'est-à-dire pour toute valuation v 2 solution de C 2 et pour toute valuation v 1 solution de C 1 nous avons v 1 (x)(t) 2 v 2 (x)p o u rt o u tt 2 N). Enfin, la figure 4 contient le CSP C 3 utilisant notre contrainte globale real-time-loop . Cette contrainte prend trois arguments qui sont, dans l'ordre : les contraintes formant le circuit, les variables d'entrées du circuit, les variables de sorties du circuit. Ainsi dans C 1 la contrainte real-time-loop contient les blocs fby, ⇥ et +, les entrées d, e et f et n'a pas de sorties. Figure 5: Exemple de MC Figure 6 : 6 Figure 6: Exemple d'IMC s 0 Figure 7 : 7 Figure 7: Exemple d'IMC 2 )•p(s 2 )(s 1 )• (1/(1-p(s 2 )(s 2 ))) = 1. A côté, la figure 6 représente une IMC I. Puisque M al am ê m es t r u c t u r eq u eI et que les probabilités des transitions de M appartiennent aux intervalles correspondants dans I nous disons que M satisfait I. Pour terminer, la figure 7 représente une pIMC utilisant deux paramètres p et q. Notons que choisir les valeurs 0.6p o u rp et 0.5p o u rq produit l'IMC I. Nous disons que I implémente P. De fait, puisque M satisfait I et que I implémente P nous disons que M satisfait P. components are surrounded with dashed lines ............................................ 4.9 Optimized model of the block-diagram in Figure 4.5 ............... 4.10 FAUST Compilation Scheme ............................ 4.11 FAUST Volume Controller Source Program ................... 4.12 FAUST volume controller block-diagram before normalization. Edges are labeled with their corresponding variables in the CSP in Fig. 4.13b . . . . . . . 63 4.13 CSP for the volume benchmark ......................... 5.1 MC M 1 ....................................... 5.2 pMC I 0 ....................................... 5.3 IMC I ........................................ 5.4 MC M 2 satisfying the IMC I from Figure 5.3 w.r.t. |= d I ............ 5.5 MC M 3 satisfying the IMC I from Figure 5.3 w.r.t. |= a I ............ 5.6 IMC I, MCs M 0 1 , M 0 2 ,a n dM 0 3 s.t. M 0 1 |= a I I, M 0 1 |= d I I and M 0 1 |= o I I; M 0 2 |= d I I and M 0 2 6 |= o I I; M 0 3 |= a I I and M 0 3 6 |= d I I; the graph representation of I, M 0 1 ,a n dM 0 3 are isomorphic; ........................ 5.7 pIMC P ....................................... 5.8 IMC I with three pMCs P 1 , P 2 ,a n dP n entailed by I w.r.t. |= a I . ....... 5.9 pMC P, IMC I, MC M 1 , and MC M 2 s.t. M 1 |= p P and M 1 |= a I I but M 2 6 |= p P and M 2 |= a I I while P is entailed by I w.r.t. |= a I . .............................. 5.10 Variables in the CSP produced by C 9c for the pIMC P from Fig. 5.7 ..... Table 2 . 2 1: Complexity for the Constraint Satisfaction Problem Classes containing Linear and Non-Linear Constraints Problems over Real, Integer, Mixed, and Finite variables. CSP for short) is a triplet P =(X, D, C) where X is a set of variables, D contains the domains associated to the variables in X, and C is a finite set of contraints over variables from X. Real var. Integer var. Mixed var. Finite var. Linear P NP-complete NP-complete NP-complete Non-linear decidable undecidable undecidable NP-complete Definition 2.2.1 (Constraint Satisfaction Program). A Constraint Satisfaction Program ( Table 4 . 4 2: A trace table of Algorithm 3 for the transfer function F . still correct for the n +1th iteration. There are 4 cases depending on the variable states and current (i.e., v n ): 1. state =" I n c r e a s i n g "a n dF (v n ) ✓ v n 2. state =" I n c r e a s i n g "a n dF (v n ) 6 ✓ v n 3. state =" D e c r e a s i n g "a n dF (v n ) ✓ v n 4. state =" D e c r e a s i n g "a n dF (v n ) 6 ✓ v n Consider the first case. Condition in line 12 is true. This sets the variable state to "Decreasing" and the variable switch to "true". Next, in the for statement only the condition in line 21 is true. Thus, for all i 2 {1,...,k}: max[i] is updated to current[i] (i.e., w n+1 [i]=v(n)[i]); current[i] is updated to an interval between min[i] and its current value (i.e., u n [i] ✓ v n+1 [i] ✓ v n [i] ) and such interval exists by the inductive hypothesis v n ✓ u n ✓ w n ;a n dmin[i] is unchanged (i.e., u n+1 [i]=u n [i]). Finally we obtain by aggregation that u n+1 Table 4 . 4 44 #cstrs real-time-loop Time (in ms) Verification medium optim. max. max. comp. comp. solver Program name counter paper-example sinus first-order-filter noise allpass-filter volume comb-filter echo stereo-echo pink-noise capture karplus-strong oscillator band-filter #var 8 11 9 15 16 16 19 20 29 37 40 45 49 49 55 model 63 model # 141 cstrs args 73 151 74 141 10 6 151 10 6 151 11 6 161 11 7 151 15 5 1 11 1 19 15 151 26 18 251 28 15 2 10 1 34 21 361 35 18 381 35 23 361 42 34 191 medium 16 17 15 35 16 18 25 18 27 28 27 27 30 30 38 optim. solve 460 7 458 7 462 7 473 9 454 8 470 9 473 7 462 7 482 8 495 12 493 7 488 14 484 9 497 11 546 11 output [0; MAX] [0; 1] [-1; 1] [-1; 1] [-1; 1] [-3; 3] [-1.58; 1.58] [-oo; +oo] [-oo; +oo] [-oo; +oo] [-oo; +oo] [-oo; +oo] [-oo; +oo] [-1; 1] [-oo; +oo] p p p p p p p p p p p p p p p lowboost 59 46 38 191 33 508 10 [-oo; +oo] ? pitch-shifter 60 50 46 151 32 510 8 [-59902;59902] ? smooth-delay 100 85 25 3 43 4 40 789 17 [-oo; +oo] ? mixer 356 310 234 19 5 1 65 824 49 [-20.01;20.01] ? freeverb 371 335 103 24 13 1 69 994 41 [-oo; +oo] ? harpe add-synth-5-oscs add-synth-10-oscs add-synth-50-oscs add-synth-100-oscs 1,530 1,320 814 102 407 348 197 24 106 85 54 761 8 1 181 150 94 12 6 1 780 670 414 52 6 1 6 1 add-synth-250-oscs 3,780 3,270 2,014 252 6 1 add-synth-500-oscs 7,530 6,520 4,014 502 6 1 add-synth-750-oscs 11,280 9,770 6,014 752 6 1 add-synth-1000-oscs 15,030 13,020 8,014 1,002 6 1 76 84 110 244 1,108 86 935 52 605 15 689 17 609 1.6s 314 2.5s 4.5s 1.6s 12.5s 17.3s 10.1s 39.8s 1'18s 48.8s 1'25s 2'43s 2'34s [-oo; +oo] ? [-1; 1] p [-1; 1] p [-1; 1] p [-1; 1] p [-1; 1] p [-1; 1] p [-1; 1] p [-1; 1] p : Experimental results on a benchmark of FAUST programs )=s 2 , and ⇡(t 2 0 )=s 2 . Let p be the transition function of M 0 2 and P be the interval probability transition function of I. Clearly, we have that p(t)(t 0 ) 2 P (⇡(t), ⇡(t 0 )). Indeed, the relation R containing (t 0 ,s 0 ), (t 1 ,s 1 ), (t 1 ,s 2 ), (t 2 ,s 1 )a n d (t 2 ,s 2 )i sas a t i s f a c t i o nr e l a t i o nb e t w e e nI and M 0 d I we have that M 0 2 |= d On the other hand, it is clear that M 0 2 6 |= o I I since M 0 2 and I do not share the same state space. (4) Consider IMC I and MC M 0 3 from Figure 5.6. By definition of |= a I we have that M 0 3 |= a I I. I I. Indeed, consider the mapping ⇡ s.t. ⇡(t 0 )=s 0 , ⇡(t 1 )=s 1 , ⇡(t 2 .3 is an instance of P (by assigning the value 0.6totheparameter p and 0.5t oq). Furthermore, as said in Example 29, the Markov Chains M 1 and M 2 (from Figures 5.1 and 5.5 respectively) satisfy I w.r.t. |= a I , therefore M 1 and M 2 satisfy P w.r.t. |= a pI . Our comparison results are presented in Proposition 4. Firstly, Lemma 1 states that IMC and pMC are not comparable w.r.t. satisfaction relations Proof. We give a sketch of proof for each statement. Let (L 1 , |= 1 )a n d( L 2 , |= 2 )b et w o Markov chain abstraction models. Recall that according to the succinctness definition (cf. Definition 5.3.6) L 1 6  L 2 if there exists L 2 2 L 2 s.t. L 1 6 ⌘ L 2 for all L 1 2 L 1 . |= o I , |= d I ,a n d|= a I . Lemma 1. pMC and IMC abstraction models are not comparable in terms of succinctness: (1) pMC 6  (IMC, |= a I ), (2) pMC 6  (IMC, |= d I ), (3) pMC 6  (IMC, |= o I ), (4) (IMC, |= a I ) 6  pMC, (5) (IMC, |= d I ) 6  pMC, and (6) (IMC, |= o I ) 6  pMC. (1) Consider IMC I and pMCs P 1 , P 2 ,a n dP n (with n 2 N) from Figure 5.8. IMC I verifies the case (1). Indeed, the pMCs P 1 , P 2 ,an dP n (with n 2 N)areallen tailed by I w.r.t. |= a I but none of them is equivalent to I. Indeed one needs an infinite Figure 5.9: pMC P, IMC I, MC M 1 , and MC M 2 s.t. M 1 |= p P and M 1 |= a I I but M 2 6 |= p P and M 2 |= a I I while P is entailed by I w.r.t. Same example than from case (1) using Figure 5.8 can be used since all the pMCs P 1 , P 2 ,a n dP n (with n 2 N)a r ee n t a i l e db yt h eI M CI w.r.t. |= d Recall that the pIMC model is a Markov chain abstraction model allowing to declare parametric interval transitions, while the pMC model allows only parametric transitions (without intervals), and the IMC model allows interval transitions without parameters. 4 1/3 2/3 MC M 2 |= a I . I . However countable number of states for creating a pMC equivalent to I w.r.t. |= a state spaces must be finite. (2) I (3) Consider IMC I 0 similar to I from Figure 5 .8 excepted that the transition from s 1 to s 0 is replaced by the interval [0.5, 1]. Since the pMC definition does not allow to bound values for parameters there is no equivalent I 0 w.r.t. |= a I . (4) Note that parameters in pMCs enforce transitions in the concrete MCs to receive the same value. Since parameters may range over continuous intervals there is no hope of modelling such set of Markov chains using a single IMC. Figure 5.9 illustrates this statement. (5) Same remark than item (4) (6) Same remark than item (4) Proposition 4. The Markov chain abstraction models can be ordered as follows w.r.t. succinctness: (pIMC, |= o pI ) < (pMC, |= p ), (pIMC, |= o pI ) < (IMC, |= o I ), (pIMC, |= d pI ) < (IMC, |= d I ), and (pIMC, |= a pI ) < (IMC, |= a I ). Proof. 1. We first prove the equivalence w.r.t. |= o I and |= a I . Recall also that |= a I is more general than |= o I : for all MC M if M| = o I I then M| = o I I (Proposition 3). P I (3↵)⇠p with the at-every-step semantics implies that there exists an MC M s.t. M| = a I I and M⇠p. Thus by Lemma 4 we get that there exists an MC M 0 s.t. M 0 |= o I I and M 0 ⇠p. P I (3↵)⇠p with the IMDP semantics implies that there exists an MC M s.t. M| = d I I and M⇠p. Since |= a I is more general than |= d I we have that M| = a I I. Thus by Lemma 4 we get that there exists an MC M 0 s.t. M 0 |= o I I and M 0 ⇠p. [)] Direct from the fact that |= a I is more general than |= o I (Proposition 3) [(] 2. We now prove the equivalence w.r.t. |= o I and |= d I [)] Direct from the fact that |= a I is more general than |= o I . (Proposition 3) [(] be a probability bound. Adding the constraint ⇡ s 0  p (resp. ⇡ s 0 ≥ p)t ot h ep r e v i o u sC 9r encoding allows to determine if there exists a MC M| = a pI P #intervals #paramInBounds Set of benchmarks #pIMCs #nodes #edges min avg max min avg max #parameters herman N=3 herman N=5 herman N=7 egl L=2; N=2 egl L=2; N=4 egl L=4; N=2 egl L=4; N=4 brp M=3; N=16 brp M=3; N=32 brp M=4; N=16 brp M=4; N=32 crowds CS=10; TR=3 crowds CS=5; TR=3 nand K=1; N=10 nand K=1; N=5 nand K=2; N=10 nand K=2; N=5 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 8 32 128 238 6,910 7,165 696 1,897 3,619 55 444 1,405 28 0 7 18 0 3 11 244 19 50 87 0 12 38 2,188 37 131 236 3 31 74 253 16 67 134 0 15 57 494 509 47 136 276 3 32 115 15,102 15,357 1448 4,068 7,772 156 951 3,048 886 1,155 16 64 135 1 15 45 1,766 2,307 40 128 275 3 32 129 1,095 1,443 22 80 171 0 20 62 2,183 2,883 49 164 323 3 39 139 6,563 15,143 1,466 3,036 4,598 57 235 535 1,198 2,038 190 410 652 8 31 76 7,392 11,207 497 980 1,416 109 466 1,126 {50, 100, 250} {2, 5, 10} {2, 5, 10} {5, 15, 30} {2, 5, 10} {2, 5, 10} {2, 5, 10} {2, 5, 10} {2, 5, 10} {2, 5, 10} {2, 5, 10} {2, 5, 10} {5, 15, 30} {5, 15, 30} 930 1371 60 121 183 9 58 159 {50, 100, 250} 14,322 21,567 992 1,863 2,652 197 866 2,061 {50, 100, 250} 1,728 2,505 114 217 329 23 101 263 {50, 100, 250} Table 5 . 5 1: Benchmarks comp osed of 459 pIMCs over 5 families used for verifying qualitative properties such that P M (3↵)  p (resp ≥ p). Formally, let ⇠2{,<,≥,>} be a comparison operator, we write 6 ⇠ for its negation (e.g., 6  is >). This leads to the following theorem.Theorem 2. Let P =(S, s 0 ,P,V,Y) be a pIMC, ↵ ✓ A be a label, p 2 [0, 1], ⇠2{,< , ≥,>} be a comparison operator, and (X, D, C) be C 9r (P, ↵):Proof. Let P =( S, s 0 ,P,V,Y)b eap I M C ,↵ ✓ A be a state label, p 2 [0, 1], and ⇠2{,<,≥,>} be a comparison operator. Recall that C 9r (P, ↵)i saC S Ps . t . e a c h solution corresponds to an MC M satisfying P where ⇡ s 0 is equal to P M (3↵). Thus adding the constraint ⇡ s 0 ⇠ p allows to find an MC M satisfying P such that P M (3↵) ⇠ p. This concludes the first item presented in the theorem. For the second item, we use Theorem 1 with Proposition 9 which ensure that if the CSP C 9r (P, ↵)t ow h i c hi sa d d e d the constraint ⇡ s 0 6 ⇠ p is not satisfiable then there is no MC satisfying P w.r.t. |= a pI such that P M (3↵) 6 ⇠ p;t h u sP M (3↵) ⇠ p for all MC satisfying P w.r.t. |= a pI . • CSP (X, D, C [ (⇡ s 0 ⇠ p)) is satisfiable iff 9M| = a pI P s.t. P M (3↵) ⇠ p • CSP (X, D, C [ (⇡ s 0 6 ⇠ p)) is unsatisfiable iff 8M| = a pI P: P M (3↵) ⇠ p Table 5 . 5 2: Benchmarks comp osed of 4 pIMCs used for verifying quantitative prop erties Size of the Boolean Integer Real-number Boolean Linear Quadratic Encoding produced CSPs var. var. var. constr. constr. constr. SotA exponential no no yes yes yes no C ∃c linear yes no yes yes yes no C ∃r linear yes yes yes yes yes no C ∃r linear yes yes yes yes yes yes Table 5 . 5 3: Characteristics of the four CSP enco dings SotA, C 9c , C 9r ,a n dC 9r . Table 5 . 5 4: Comparison of sizes, modelling, and solving times for three approaches: (1) SotA encoding implemented in SMT, (2) C 9c encoding implemented in SMT, and (3) C 9c encoding implemented in MILP (times are given in seconds). N=10 7,392 11,207 12 18,611 111,366 9.46s 9,978 89,705 13.44s 17,454 147,015 T.O. pIMC C ∃c C ∃r C ∃r Benchmark #states #trans. #par. #var. #cstr. time #var. #cstr. time #var. #cstr. time nand K=1; N=2 104 147 4 255 1,526 0.17s 170 1,497 0.19s 296 2,457 69.57s nand K=1; N=3 252 364 5 621 3,727 0.24s 406 3,557 0.30s 703 5,828 31.69s nand K=1; N=5 930 1,371 7 2,308 13,859 0.57s 1,378 12,305 0.51s 2,404 20,165 T.O. nand K=1; Table 5 . 5 5: Comparison of solving times b etween qualitative and quantitative enco dings. 6) ω s =1, sis = s 0 123 (7) ! s 6 =1, sis 6 = s 0 (8) ⇢ s , (! s 6 =0) (9) ! s > 1 ) W s 0 2Pred(s)\{s} (! s = ! s 0 +1)^(✓ s 0 s > 0), si s 6 = s 0 (13) λ s , (⇢ s ^(↵ s 6 =0)) (14) ↵ s > 1 ) W s 0 2Succ(s)\{s} (↵ s = ↵ s 0 +1)^(✓ s 0 s > 0), si s 6 2 T (15) ↵ s =0, V s 0 2Succ(s)\{s} (↵ s 0 =0)_ (✓ s 0 (10) ! s =0, V s 0 2Pred(s)\{s} (! s 0 =0)_ (✓ s 0 s =0), sis 6 = s 0 (11) ↵ s =1, sis 2 T (12) ↵ s 6 =1, sis 6 2 T s =0), sis 6 2 T (16) ¬λ s ) ⇡ s =0 (17) λ s ) ⇡ s =1, sis 2 T (18) λ s ) ⇡ s = Σ s 0 2Succ(s) ⇡ s 0 ✓ s s 0 , si s 6 2 T We already introduced the word model in the constraint programming context. The reader must be careful that this word has an important role in both contexts. Note that Chapter 4 models a problem with time-dependency (verification of a reactive synchronous programming language). However the constraint modelling is time-independent. R is the extended set of R and equals the union of R and its limits (i.e., R = R [ {-1, +1}). We chose the IEEE 754 norm but any set of floating-point numbers can be considered. Be careful that in this thesis we use AI for Abstract Interpretation and not Artificial Intelligence. In the case of the presence of state labelling, the trace is the succession of labels associated to the states encountered at running time (cf. Chapter 5). FAUST is open source and available at http://faust.grame.fr For reading facilities, we simply write cycle instead of directed cycle while working with directed structures such as block-diagrams. See http://faust.grame.fr/index.php/documentation/references for listing and description. In music, a numeric audio stream is a sequence of values between -1 and 1 https://www.cs.ox.ac.uk/people/peter.schrammel/reaver/ In this chapter we use modelling with the verification meaning and we call encoding a CSP modelling. Indeed, when 0  v(f 1 )  v(f 2 )  1 is not respected, the interval is inconsistent and therefore empty. |L 1 | and |L 2 | are the sizes of L 1 and L 2 ,r e s p e c t i v e l y . As illustrated in Example 31, M is not a well formed MC since some unreachable states do not respect the probability distribution property. However, one can correct it by simply setting one of its outgoing transition to 1 for each unreachable state. All resources, benchmarks, and source code are available online as a Python library at https: //github.com/anicet-bart/pimc_pylib see the category discrete-time Markov chains on the PRISM website Note that this is not always free to obtain integer integrity constraints over real-numbers. Nous expliquons dans la thèse que cette restriction n'est pas trop forte et que les blocs temporels usuels (delay, memory, n-delay) peuvent être réécrits avec le bloc fby. D appelé ensemble étendu de D correspond à l'union de D et de ses limites (e.g., R = R[{-1, +1}) https://github.com/anicet-bart/pimc_pylib Dans cette thèse, nous avons abordé deux familles de problèmes traitant de la vérification de programmes. Pour chaque cas, nous avons d'abord étudié formellement la nature des problèmes de vérification concernés avant de proposer une résolution par les contraintes. Puisque nous ne nous imposions aucune restriction concernant le langage de contraintes, nous avons proposé des modélisations par contraintes utilisant des contraintes non-linéaires sur des variables non bornées, des variables mixtes entières/linéaires sur des contraintes linéaires, mais aussi des contraintes quadratiques sur des variables mixtes. Ainsi, la vérification de programmes est un champ de recherche riche pouvant faire appel à divers outils de la programmation par contraintes. La complexité théorique du problème de la vérification de programmes, comme celle du problème de la satisfaction de contraintes, peut s'avérer élevée. Cependant, les solveurs de la programmation par contraintes peuvent résoudre en partie ces problèmes difficiles. Pour autant, les communautés de la programmation par contraintes avancent sur des axes de recherches séparés en développant des solveurs dédiés à des langages de contraintes spécifiques. Finalement, dans cette thèse nous avons abordé la vérification de programmes sous l'angle de la programmation par contraintes. Cela nous a permis d'apporter de nouvelles idées dans les processus de vérification de programmes et de rapprocher ces deux domaines de recherche. Thèse de Doctorat Anicet BART Modélisation et résolution par contraintes de problèmes de vérification Constraint Modelling and Solving of some Verification Problems Résumé La programmation par contraintes offre des langages et des outils permettant de résoudre des problèmes à forte combinatoire et à la complexité élevée tels que ceux qui existent en vérification de programmes. Dans cette thèse nous résolvons deux familles de problèmes de la vérification de programmes. Dans chaque cas de figure nous commençons par une étude formelle du problème avant de proposer des modèles en contraintes puis de réaliser des expérimentations. La première contribution concerne un langage réactif synchrone représentable par une algèbre de diagramme de blocs. Les programmes utilisent des flux infinis et modélisent des systèmes temps réel. Nous proposons un modèle en contraintes muni d'une nouvelle contrainte globale ainsi que ses algorithmes de filtrage inspirés de l'interprétation abstraite. Cette contrainte permet de calculer des sur-approximations des valeurs des flux des diagrammes de blocs. Nous évaluons notre processus de vérification sur le langage FAUST, qui est un langage dédié à la génération de flux audio. La seconde contribution concerne les systèmes probabilistes représentés par des chaînes de Markov à intervalles paramétrés, un formalisme de spécification qui étend les chaînes de Markov. Nous proposons des modèles en contraintes pour vérifier des propriétés qualitatives et quantitatives. Nos modèles dans le cas qualitatif améliorent l'état de l'art tandis que ceux dans le cas quantitatif sont les premiers proposés à ce jour. Nous avons implémenté nos modèles en contraintes en problèmes de programmation linéaire en nombres entiers et en problèmes de satisfaction modulo des théories. Les expériences sont réalisées à partir d'un jeu d'essais de la bibliothèque PRISM. Abstract Constraint programming offers efficient languages and tools for solving combinatorial and computationally hard problems such as the ones proposed in program verification. In this thesis, we tackle two families of program verification problems using constraint programming. In both contexts, we first propose a formal evaluation of our contributions before realizing some experiments. The first contribution is about a synchronous reactive language, represented by a block-diagram algebra. Such programs operate on infinite streams and model real-time processes. We propose a constraint model together with a new global constraint. Our new filtering algorithm is inspired from Abstract Interpretation. It computes over-approximations of the infinite stream values computed by the block-diagrams. We evaluated our verification process on the FAUST language (a language for processing real-time audio streams) and we tested it on examples from the FAUST standard library. The second contribution considers probabilistic processes represented by Parametric Interval Markov Chains, a specification formalism that extends Markov Chains. We propose constraint models for checking qualitative and quantitative reachability properties. Our models for the qualitative case improve the state of the art models, while for the quantitative case our models are the first ones. We implemented and evaluated our verification constraint models as mixed integer linear programs and satisfiability modulo theory programs. Experiments have been realized on a PRISM based benchmark. Mots clés modélisation par contraintes, résolution par contraintes, vérification de programmes, interprétation abstraite, vérification de modèles.

01743857

en

2024/03/05 22:32:07

2017

https://theses.hal.science/tel-01743857/file/65201_BLANCO_MARTINEZ_2017_archivage.pdf

Quentin Heath Sonia Marin Ulysse Gérard Matteo Manighetti Maico Leberle And Taus Brock-Nannestad Danko Ilik Tomer Libal Giselle Reis Eric Goubault Frank Benjamin Werner Jessica Gameiro Valérie Berthou Laura Kovács Chantal Keller Beniamino Gabriel Matteo Kaustuv introduction rules Like any scientific undertaking, a thesis does not develop in a vacuum, and personal as it is, it is informed by the indirect as well as the direct influence of many people. In the roll call of important actors that must of need accompany such a document, regrettable omissions may be nigh inevitable. Indeed, as with test cases and program bugs, a mention in a preface may show the presence of an acknowledgment, not its absence-and in the flurry of writing and its companion events, in keeping with the software analogy, slips are not altogether unlikely to occur. In the world of computer science, at least, there is hope for getting things just right; this work marks the point at which I start walking in that direction, resolutely and in earnest. First and foremost in importance and intensity, I am indebted to Dale for the pleasure and the privilege of working with him. Much more than a directeur, he is a teacher and a mentor, knowledgeable and wise, and the nicest person you will ever meet; words do not do justice. May our paths continue to meet. From the moment I entered the scene in April 2014, arriving on the plateau de Saclay after the summer, Inria and École polytechnique have been wonderful places to work and study. My employment at Inria was funded by the ERC Advanced Grant ProofCert in Dale's team Parsifal, which takes most of the credit for being friendly and collegial to the utmost. Much has changed on the plateau in the intervening three years, the primordial constant Résumé La confiance formelle en une propriété abstraite provient de l'existence d'une preuve de sa correction, qu'il s'agisse d'un théorème mathématique ou d'une propriété du comportement d'un logiciel ou processeur. Il existe de nombreuses définitions différentes de ce qu'est une preuve, selon par exemple qu'elle est écrite soit par des humains soit par des machines, mais ces définitions traitent toutes du problème d'établir qu'un document représente en fait une preuve correcte. En particulier, la question de comment vérifier, communiquer et réutiliser des preuves automatiques, provenant de logiciels divers et très complexes, a été abordée par plusieurs propositions de solutions générales. Dans ce contexte, le cadre des Certificats de Preuve Fondamentaux (Foundational Proof Certificates, FPC) est une approche proposée récemment pour étudier ce problème, fondée sur des progrès de la théorie de la démonstration pour définir la sémantique des formats de preuve, comme par exemple des preuves par réfutation utilisant le principe de résolution. Les preuves écrites dans un format de preuve défini dans ce cadre peuvent alors être vérifiées indépendamment par un noyau vérificateur de confiance codé dans un langage de programmation logique d'ordre supèrieur comme λProlog. Cette thèse étend des résultats initiaux sur la certification de preuves du premier ordre en explorant plusieurs dimensions logiques essentielles, organisées en combinaisons correspondant à leur usage pratique en démonstration de théorèmes. Précédemment, les certificats de preuve se limitaient à servir de représentation des preuves complètes pour leur vérification indépendante. Ces certificats ont illustré le pouvoir expressif des FPC en codant la sémantique idealisée de systèmes logiques divers accompagnés de certificats correspondant aux petites preuves manuelles pour ces systèmes. On se demandait cependant si cette approche peut passer à l'échelle de logiciels et preuves de grande taille (jusqu'à plutôt téraoctets) et si ce cadre était assez flexible pour d'autres usages. La première partie de cette thèse récapitule les principes de la théorie de la démonstration et de la programmation logique, sur lesquelles repose notre conception des certificats de preuve, et fournit un aperçu du cadre des FPC proprement dit et sa mise en oeuvre de référence. La deuxième partie explore une première famille de formalismes basés sur la logique classique, sans points fixes, dont les preuves sont générées par des démonstrateurs automatiques de théorème. Désormais, le rôle des certificats de preuve s'étend pour englober des transformations de preuve qui peuvent enrichir ou compacter leur représentation. Ces transformations, qui s'appuient sur une notion de combinateur de certificats et de vérification parallèle, peuvent rendre des certificats plus simples opérationellement, ce qui motive la construction d'une suite de vérificateurs et formats de preuve de plus en plus fiables et performants, dont le noyau vérificateur peut être écrit dans un langage de programmation fonctionelle comme OCaml et même formalisé et verifié avec l'aide d'un assistant de preuve comme Coq. Ces développements permettent la certification automatique de bout en bout de résultats générés par deux familles majeures de démonstrateurs automatiques de théorème, utilisant techniques de résolution et satisfaisabilité. Ensuite, la troisième partie explore la logique intuitionniste, avec points fixes, égalité et quantification nominale, dont les preuves sont générées par des assistants de preuve. Dans cet environnement, les certificats de preuve assument des fonctions nouvelles, notamment l'écriture d'aperçus de preuve de haut niveau. Ces aperçus expriment des schémas de preuve tels qu'ils sont employés dans la pratique des mathématiciens, ou bien dans l'écriture dans un assistant de preuves avec l'aide de tactiques de haut niveau, en notant la création et utilisation de lemmes dans la preuve. Cette technique s'applique aussi aux méthodes comme le propertybased testing, avec lequel un assistant de preuve cherche automatiquement des contre-exemples qui révèlent parfois des erreurs dans la déclaration des théorèmes. Finalement, ces avances permettent la création de langages programmables de description de preuves pour l'assistant de preuve Abella. Introduction The unifying theme of this thesis is how to establish trust that a certain property, i.e., a theorem about some formalizable artifact, holds. If the abstract answer to this question is "with a proof," the issue becomes then to establish how proofs are represented and how to trust them. A document or more generally an act of communication purporting to represent a proof of a theorem cannot be blindly trusted: it must be checkable independently from its origin. In a world of mechanized proofs of formidable size and complexity, in addition to a convenience, automation becomes a necessity. Thus, it would be desirable to represent pieces of proof evidence from very different provenances in a unified format: proofs would then be represented by a document, or certificate, to which would be attached the description of the syntax of the proof and its semantics. With this information, independent checking becomes possible. As it is proofs we are dealing with, the branch of logic that studies proofs as mathematical objects, proof theory, is an obvious candidate for such an undertaking. From these theoretical foundations, the ProofCert project has developed broad-spectrum proof formats called Foundational Proof Certificates (FPC). The resulting framework establishes how to define certificate formats and build simple, trustworthy checkers that consume them. A checker results from the combination of a proof checking kernel and the definition of a proof certificate format. Until now, research on FPCs has validated it and seen it applied to a range of proof systems in proofs of concept, but had yet to graduate to certificates "in the large." In addition, with the desirable theoretical properties of the FPC framework established, it must be considered how an implementation of the framework can be trusted: who watches the watchmen? A further point of inquiry is whether proof certificates can represent more than inert evidence of a proof: whether in addition to checking them, proof certificates can grow and shrink, whether . they can be queried and interacted with. We shall consider all these questions in turn. These investigations will be placed in relation to both great groups of tools: automated and interactive theorem provers. The document is structured in three parts. Part I collects the general background common to the two main parts that follow it; it is complemented by more specialized background at the beginning of each of Part II and Part III. It is composed of two chapters. Chapter 2 commences with an overview of the concept of proof and its abstract treatment by the discipline of proof theory. It concentrates on sequent calculi as the proof systems of choice and their treatment of canonical forms of proofs to the point to which normalization can be gained by the development of focused proofs. Chapter 3 continues where the preceding chapter leaves off by further motivating the use of proof evidence as the foundation of trust and developing the sequent calculus into a logically sound, programmable framework where proof construction can be flexibly steered by the Foundational Proof Certificates that are at the center of this research. It complements this discussion with a number of examples which play various roles in subsequent chapters; these are the kind of small, textbook systems with which the FPC framework has been used to date with small, handcrafted examples. Part II studies developments taking place in a classical logic without fixed points, where the open-world assumption holds. Logics such as these underlie standard logic programming languages and automated theorem provers. Chapter 4 completes the background of Part I with the technical setting specific to this part of the document. It briefly discusses the uses of logic in computation to concentrate on the use of logic itself as a paradigm for computation-logic programming. After reviewing the common foundations of Prolog, it presents the salient features of λProlog, the language used throughout this part, and demonstrates the implementation of the kernel of a proof checker for the FPC framework in that language, which can be used to execute the examples in Chapter 3 as well as the original developments of future chapters. Chapter 5 explores the idea of combining sources of proof evidence by defining a combinator that attempts to construct a proof of a formula using two separate pieces of proof evidence in parallel. This opens rich possibilities of combination of proof search strategies and of elaboration of proof evidence. Using standard features of logic programming, this pairing combinator can be used to enrich a proof certificate with additional information (making it more efficient), compacting it by removing inessential information (making it potentially slower), or querying and extracting information about proofs. If a certificate is enriched to be essentially a trace of the logical computation, the reproduction of this trace is performed as a determinate computation, which does not resort to the distinguishing features of logic programming Chapter 6 exploits traces of computation of a proof used as proof evidence, which constitute not a general logic computation, but a particular case of functional computation. This suggest that we can implement specialized proof checkers for the determinate fragment of the FPC framework as functional programs, which rely on simpler languages and runtimes and avoid the hard problems that a logic programming language must integrate. Going further, the operation of these simplified proof checkers can be modeled in a proof assistant like Coq and proved correct. The result is a verified implementation of a simplified proof. Since the FPC definitions from Chapter 5 always allow the extraction of a full trace, any proof certificate defined in the framework can be run in the simplified, verified setting by that intermediate translation step. Chapter 7 begins to move the FPC framework from the realm of small examples and prototypes into the domain of theorem provers and proof formats as they are used in practice. Here we look at a family of automated theorem provers that solve instances of the classical boolean satisfiability problem. These tools are widely used and have already recognized the necessity for independent validation of results by defining their own proof output formats. We develop a proof theoretic view of the proofs underlying those formats representing proofs of unsatisfiability of a propositional formula, and show how to interpret their information as a proof certificate capable of guiding the reconstruction of a proof in our proof checkers. Chapter 8 proceeds in the same vein and seeks to apply the FPC framework to certify proofs produced by a concrete automated theorem prover. The general recipe involves analyzing the proof evidence produced by the concrete tool and modeling it in terms of a proof calculus, which in turn can be described by an FPC definition. We perform this study for Prover9, a venerable prover based on a resolution proof calculus like many leading automated theorem provers. Starting from a corpus of publicly available proofs, we apply the methods of Chapter 5 to obtain several certificates corresponding to the same proof with increasing amounts of information and study the effects on certificate size and checking time. are given as proof certificates: exhaustive and random generation of test data is followed by testing of the properties with the generated data. The expressive logics considered in this part make it possible to apply these techniques to the domain of the metatheory of programming languages. Chapter 13 integrates these developments in Abella. It considers the technical and algorithmic extensions that go into thoroughly integrating the FPC framework in a proof assistant. Proof certificates become a complement and even a substitute for the built-in tactics language, which is replaced with another language, that of certificates, which is not only programmable but offers the formal guarantees of correctness of the framework. Often a subject is treated in the context of one of the two main parts while being in whole or in part applicable to the other part. Each chapter is completed by a Notes section that complements the preceding sections with extended discussions and background, as well as connections to other chapters. Part I Logical foundations 2 Structural proof theory Concept of proof Logic-the scientific study of reasoning and deduction-has the concept of proof at its core: how is new knowledge created from what is already known? One presents the fruit of deduction as a proof, but what exactly is a proof, and how can we recognize one, that is, ensure that a claimed proof is correct? Our modern understanding of the concept harkens back to David Hilbert's vigorous efforts to infuse all of mathematics with complete and absolute (metamathematical) rigor-formalized by means of the kind of axiomatic proof systems which were pioneered by such bodies of work as Gottlob Frege's Begriffsschrift and Alfred North Whitehead and Bertrand Russell's Principia Mathematica. These same efforts sparked foundational controversies with mathematicians like Frege and L. E. J. Brouwer, and spurred Kurt Gödel to develop his incompleteness theoremswhich demonstrated that Hilbert's lofty ambitions, at least in their original form, were unattainable. The study of proofs as first-class mathematical objects developed in close connection with these advances. Vis-à-vis the purely mathematical conception of proof are the philosophical and sociological faces of argumentation as a mental activity and an act of communication. It is recognized that proof has a dual nature by which it can be seen alternatively as proof-as-message or as proof-as-certificate. Taken as a message, the purpose of a proof is the transmission of insight and understanding between mathematicians: to convey the lines of argumentation followed to arrive at a conclusion and to convince of the truth of that conclusion-the focus is on meaning, on semantics. Seen instead as a certificate, the purpose of proofs is the transmission and mechanical verification of knowledge: the use of the symbols and rules of a formal language to unambiguously derive new, correct phrases-the focus here is on syntax. In practice, both functions are closely related, and although formal study tends to emphasize the side of proof-as-certificate, proof-as-message considerations will be pervasive in much of the work that follows (especially throughout Part III, where proof assistants and user interaction are key subjects). The rest of the chapter is organized as follows: Section 2.2 traces the development of proof theory and describes its major formal systems. Section 2.3, in parallel, outlines the most important division in the taxonomy of standard logics, that of separating classical and intuitionistic logics, both of which will be called upon over the course of our study. Section 2.4 presents the sequent calculus, one of the major deductive systems, on which our formal studies will be based. Section 2.5 introduces the discipline of focusing, used to structure the proofs of the sequent calculus in more organized, abstract forms. Section 2.6 summarizes some important considerations on the relation between proof systems and the logics they model. Section 2.7 concludes the chapter. Evolution of proof theory Proof theory is the branch of mathematical logic that studies proofs as objects of mathematics. It is widely acknowledged that the modern study of proof has its roots in the axiomatizing undertakings of Hilbert's Program. Among the disciplines of proof theory, structural proof theory studies the structure and properties of proofs (in the sense of proof-as-certificate of the previous section). The concrete objects of its study are proof calculi: formalized systems of deduction where formulas and proofs are inductively defined objects, and the steps of deduction are carried out syntactically by the application of inference rules which transform formulas and in the process construct proof objects. There are three principal families of proof calculi, each of which will be presented and placed in its proper context in this section. First, Hilbert systems take their name from the refined calculus developed by Hilbert for the advancement of his Program. They reflect the longstanding tradition of organizing mathematical developments as a sequence of steps which starts from instances of a collection of logical axioms-for example, that every property P implies itself: P ⊃ P -and follows by applications of inference rules which derive new facts from previously known ones. Logical reasoning has historically relied on this discursive style, from Aristotle to Gottlob Frege-after whom these calculi are sometimes named Frege systems, as we shall reference in Chapter 11. Often, a single inference rule, that of modus ponens, is used. This . . rule states that if we know that a fact Q is implied by another fact P , and we also know that P holds, then we can infer that Q holds. For this reason (and closer to the uniform terminology that will be used shortly) modus ponens is also called implication elimination and depicted schematically as follows: P ⊃ Q P Q Second, natural deduction systems arose as a response to the linear and unstructured proofs of Hilbert-style systems, to better reflect the "natural" way in which proofs are built by mathematicians-in fact, although Hilbert systems have later been inspected under the lens of structural proof theory, it is in natural deduction that the discipline has its proper genesis. Natural deduction was developed by Gerhard [START_REF] Gentzen | Investigations into logical deduction[END_REF] in his landmark dissertation with the goal of accurately reflecting the mental process of reasoning and its dependencies. Centrally, it employs the concept of assumptions made over the course of a proof attempt and which may be closed at some later point. An important question in this more structured system is whether there exist normal forms of natural deduction proofs, so that many shallowly different derivations of the same property may be given a common representation. In his dissertation, Gentzen attempted to prove such a normalization property, succeeding in the intuitionistic case but failing in the classical case. Eventually, Dag [START_REF] Prawitz | Natural Deduction[END_REF] succeeded in doing so within the edifice of natural deduction, and we now know Gentzen himself persevered in his efforts until the knot was untangled [START_REF] Von | Gentzen's proof of normalization for natural deduction[END_REF]. The third and last great family of deductive systems is the sequent calculus, which Gentzen developed to work around the difficulties of the proof of normalization for natural deduction. Its technical motivation was to serve as a sort of meta-calculus in which the derivability relation of natural deduction could be expressed and reasoned about: the sequent calculus is more pedantic, but also more practical. In this formalism, an analog of normalization for sequent calculus proofs called cut elimination could be proved, and this result in turn rendered the original pursuit-i.e., a proof of normalization for intuitionistic natural deduction-unnecessary. Sequent calculi proved to be fertile theoretical ground, more mechanistic than their forebears and far more relevant in the looming age of computer science. They form the immediate substrate of the present work and will be discussed extensively in the pages that follow; after a brief orthogonal discussion, they will be properly introduced in Section 2.4. . Classical and intuitionistic logics Perhaps the most fundamental division in modern logic concerns the distinction between the standard logics: classical logic and intuitionistic logic. Classical logic is a "logic of truth," concerned with the assignment of truth values (say, true or false) to formal statements. The traditional study of logic starting with Aristotle can be framed in this tradition. By contrast, intuitionistic logic is a "logic of construction." It was developed from L. E. J. Brouwer's philosophy, notably by Arend Heyting. In proving properties about mathematical objects, an intuitionistic proof provides a way to construct objects exhibiting the properties being proved. For example, to have a proof of "A and B," we need separately a proof of A and a proof of B. Under the classical interpretation, the negation of a statement is an assertion of its falsity. Conversely, the intuitionistic negation of a statement points to the existence of a counterexample. As an illustration, consider proof by contradiction, a standard proof technique commonly used in mathematics. A proof by contradiction of a property p starts by assuming that p does not hold and proceeds to arrive at a contradiction. Many commonplace results-such as the classic proof of the irrationality of √ 2-make use of this method. However, such non-constructive arguments are invalid in intuitionistic logic. In practice, classical logic and intuitionistic logic can be related by disallowing in the latter the non-constructive parts of the former-that is, the principle of the excluded middle, which asserts that for every statement either itself or its negation is true. By virtue of this constraint, intuitionism rejects the aberrant "proofs" of classical logic which rely on non-constructive arguments. A priori, because intuitionistic logic can be defined as a restriction of classical logic, not only can intuitionistic logic not be stronger than classical logic, but it would moreover seem to be weaker-every intuitionistically provable theorem is classically provable, but there exist classical theorems which are not intuitionistically provable. Nonetheless, the relation between the expressive powers of both logics is subtler than their hierarchical relation might suggest. In the settings of propositional and first-order logic, there exist translation functions such that, if a formula is a theorem of classical logic, its translation is a theorem of intuitionistic logic: such functions are called double-negation translations [START_REF] Ferreira | On the relation between various negative translations[END_REF]-in fact, what such a function does is make explicit each use of the excluded middle by means of an encoding based on double negations. (However, a formula is not intuitionistically equivalent to its translation, and no general mappings exist in the opposite direction.) This leads to the observation that-in these settingsintuitionistic logic is more expressive, or finer than classical logic: if a formula is classically but not intuitionistically provable, we can find another formula which is intuitionistically provable and classically indistinguishable from the original. If classical logic can be seen as the logic of traditional mathematics, intuitionistic logic takes on the mantle of the logic of computer science. In the context of theorem proving, classical logic serves as the foundation of choice for reasoning in automated theorem provers, where a computer program attempts to find proofs of theorem candidates. In interactive theorem provers (or proof assistants), where the user drives the search for proofs, the foundational role is assumed by intuitionistic logic. Both classical and intuitionistic logics will be employed extensively as the bases for Part II and Part III, respectively. Sequent calculus This section introduces Gentzen's original sequent calculus, the proof theoretical foundation of our investigations. We concentrate on the calculus for classical logic and reference intuitionistic logic when appropriate. Classical logic will be the focus of Part II, while Part III will adopt intuitionistic logic as its vehicle; additional background is given in Chapter 9. Classical logic contains the familiar set of logical constants or connectives. In the propositional fragment, we have the nullary constants true ( t ) and false ( f ); the unary constant negation (or not, ¬); and the binary constants conjunction (or and, ∧), disjunction (or or, ∨), and implication ( ⊃). First-order logic extends propositional logic with the universal quantifier (∀) and the existential quantifier (∃). The standard classical equivalences establish the interrelations between these constants. We make no attempt to reduce the set of connectives to a small, or minimal, functionally complete set of operators. Added to these logical constants will be a type signature of non-logical constants that will function as atomic propositions (or simply atoms). A literal is either an atom or a negated atom. In first-order logic, as quantifiers are introduced, atoms can be parameterized by terms, also part of the signature. Quantifiers bind names within their scope and can be instantiated by the operation of substitution-all the usual subtleties and caveats about free and bound variables, capture-avoiding substitution, etc., apply here. Given a formula A, [t /x]A designates the substitution of a term t for a (free) variable x in A. A more computational view of all these aspects is deferred until Chapter 4. Sequent calculi are formal logic systems organized around the concept of sequents. In its basic form, a sequent combines two arbitrary lists of formulas separated by a turnstile, say: A 1 , . . . , A m B 1 , . . . , B n The list of formulas to the left of the turnstile is called the left-hand side (LHS) or the antecedent of the sequent and abbreviated Γ. The list of formulas to the right of the turnstile is called the right-hand side (RHS) or succedent of the sequent and abbreviated ∆. The classical semantics of a sequent states that, if all the formulas in the left-hand side are true, at least one formula in the right-hand side is true, as well. Equivalently, the antecedent list represents a conjunction of formulas, and the succedent list represents a disjunction of formulas. Every sequent calculus is presented as a collection of inference rules on sequents. Each inference rule has one conclusion below the line and any number of premises above the line-possibly complemented by provisos and side conditions. The inference rules are actually rule schemata: they are essentially "universally quantified" over all their variables (representing arbitrary formulas, list of formulas, etc.), so that any combination of values for those syntactic variables constitutes an instance of that inference rule. The rules of the standard sequent calculus for classical logic are presented in Figure 2.1. (These rules do not cover the nullary logical constants true and false, which can be trivially defined in terms of the other connectives.) The rules of the calculus are organized in three distinct groups. First, logical rules, also called introduction rules, analyze each logical connective on both sides of the turnstile in the conclusion of the rule and relate the conclusion to the necessary premises for the introduction of the connective. Second, identity rules treat the (symmetric) cases where the same formula appears on both sides of the turnstile: the axiom rule in the conclusion; and the cut rule in the premises. And third, structural rules manipulate the structure of the sequent without inspecting its formulas: they are exchange, which reorders the components of the sequent; weakening, which introduces new formulas in the conclusion; and contraction, which makes copies of formulas in the premise. This presentation showcases the deep, remarkable symmetry of classical logic. An introduction rule corresponds to the analysis of a formula on a certain side of the conclusion sequent with a certain top-level connective. The remaining Γ ∆ Γ, A ∆ W L Γ ∆ Γ A, ∆ W R Γ, A, A ∆ Γ, A ∆ C L Γ A, A, ∆ Γ A, ∆ C R Figure The LK proof system for classical logic with two-sided sequents [START_REF] Gentzen | Investigations into logical deduction[END_REF]. Here, A and B are arbitrary formulas; Γ and ∆ are lists of formulas. The proviso marked as † is the usual eigenvariable restriction that y must not be free in the components of the premise ( Γ, A, and ∆). . introduction rules Γ, A, B ∆ Γ, A ∧ B ∆ ∧ L Γ A, B, ∆ Γ A ∨ B, ∆ ∨ R Γ 1 , A ∆ 1 Γ 2 , B ∆ 2 Γ 1 , Γ 2 , A ∨ B ∆ 1 , ∆ 2 ∨ L Γ 1 A, ∆ 1 Γ 2 B, ∆ 2 Γ 1 , Γ 2 A ∧ B, ∆ 1 , ∆ 2 ∧ R Figure The multiplicative fragment of the LK proof system for classical logic with two-sided sequents. These rules replace ∧ i L , ∨ i R (for i ∈ {1, 2}), ∨ L , and ∧ R in Figure 2.1; the rest of the system remains unchanged. Presentation conventions are shared with Figure 2.1. contents of the conclusion (the Γ and ∆ lists) are called the context. In Figure 2.1, we have given a modern presentation of the original LK calculus as developed by Gentzen. Such a presentation is said to be additive because of the relation between the parts of the conclusion sequent and the parts of the premise sequents-namely, the context is the same across all premises (and coincides with the context in the conclusion). In contrast, a multiplicative reading requires that the parts of the conclusion exactly match the sum of the parts of the premises-contexts are disjoint across premises and merged in the conclusion. Two kinds of changes are made to enforce this regime. First, the split one-premise and-right (resp. or-left) rules are merged by requiring both conjuncts (resp. disjuncts) to be available in the premise. Second, the two-premise rules split the lists of formulas Γ and ∆ so that the provenance of each is recorded. The alternative rules are shown in Figure 2.2. In classical logic, these two views are interadmissible in the presence of weakening and contraction-i.e., both resulting calculi prove exactly the set of theorems of classical logic. The initial rules determine how a single formula is introduced or eliminated from both sides of a sequent simultaneously. The axiom rule is the sole proper initial rule of the classical calculus, in that it has no premises and can be used at all times. Gentzen's main result is the admissibility of the rule of cut under the form of the crucial cut-elimination theorem, which states that any theorem whose proof makes use of the cut rule possesses another proof which does not employ the cut rule. This result is the cornerstone of the theoretical study of sequent calculi and among the first important properties to establish for a new calculus, from which other related results generally follow easily. It corresponds to the notion of normal forms of proofs and strong proof normalization in systems like natural deduction, and underlies the original motivation for the conception of the sequent calculus. Finally, structural rules are the administrative elements of the calculus. First, the exchange rules allow arbitrary reorderings of the components of a sequent. As the introduction rules are defined to analyze formulas at specific positions-here, the rightmost left-hand side formula or the leftmost right-hand side formula, i.e., on the immediate neighbors of the turnstile symbol-it is necessary to rearrange the formulas on which work is to be carried out. Second, the contraction rules state that if a formula is present in a side of the sequent, its cardinality does not matter, as we can always make more copies. Together with the ultimate irrelevance of ordering by way of exchange, these rules substantiate the view of the LHS and RHS as sets of formulas. More commonly, they are modeled as multisets, which removes the exchange rules-this makes the instantiation of rule schemata somewhat less concrete. Third and last, the weakening rules allow the insertion of arbitrary formulas in the conclusion sequent. In addition to the standard two-sided presentation of sequent calculus-where formulas appear on both sides of the turnstile-classical logic also admits a onesided presentation, where all formulas are located on the right-hand side. In this version of the classical sequent calculus, two significant changes take place relative to the two-sided calculus: First, implication is defined as its classical reading, so that: A ⊃ B ≡ ¬A ∨ B, for all A and B. Second, negation is only allowed to have atomic scope, i.e., as part of a literal. In consequence, all formulas are assumed to be in negation normal form, or NNF. To push negations down to the atomic level, the De Morgan dual equivalences are used in combination with the double-negation translation. The resulting rewrite system implementing the NNF translation is strongly normalizing and confluent-hence, straightforward application of the translation rules terminates and arrives at the unique NNF. For arbitrary formulas A and B: ¬(A ∧ B) → ¬A ∨ ¬B ¬∀x.A → ∃x.¬A ¬(A ∨ B) → ¬A ∧ ¬B ¬∃x.A → ∀x.¬A ¬¬A → A The one-sided calculus has the advantage of being more concise and therefore simpler to implement. Therefore, it will become the basis for the kernel of the proof checker presented in Section 4.4. Figure 2.3 shows this sequent calculus. . introduction rules A, Γ A ∨ B, Γ ∨ 1 B, Γ A ∨ B, Γ ∨ 2 A, Γ B, Γ A ∧ B, Γ ∧ [y/x]A, Γ ∀x.A, Γ ∀ † [t /x]A, Γ ∃x.A, Γ ∃ identity rules A, ¬A axiom A, Γ 1 ¬A, Γ 2 Γ 1 , Γ 2 cut structural rules Γ 1 , B, A, Γ 2 Γ 1 , A, B, Γ 2 E Γ A, Γ W A, A, Γ A, Γ C 2.3 Figure The LK proof system for classical logic with one-sided sequents [START_REF] Schütte | Schlussweisen-kalküle der prädikatenlogik[END_REF][START_REF] Tait | Normal derivability in classical logic[END_REF]. The system consists of versions of the right rules of the two-sided LK where formulas are by definition in negation normal form. Rules for negation and implication are therefore no longer part of the system, and the negation of a formula represents the NNF of its negation. Presentation conventions are shared with Figure 2.1. Gentzen observed that a sequent calculus for intuitionistic logic, called LJ, results as a particular case of the (two-sided) classical calculus LK by imposing the simple restriction that the right-hand side of each sequent contain at most one conclusion: this is called the intuitionistic restriction. Due to this fundamental asymmetry, a one-sided calculus for intuitionistic logic-where left-hand side and right-hand side are confused-cannot be formulated. Provided that this proviso is threaded throughout all inference rules, Figure 2.1 can be used to describe LJ (a rewrite with a small amount of syntactic simplification is also feasible). Finally, note that based on these rules of Figure 2.1, we can connect the stated semantics of sequents to their single-formula equivalent, also in sequent form: (A 1 ∧ • • • ∧ A m ) ⊃ (B 1 ∨ • • • ∨ B n ) . . In the one-sided calculus of Figure 2.3, the context is simpler and always disjunctive, and this is equivalent to: ¬A 1 , . . . , ¬A m , B 1 , . . . , B n A derivation in sequent calculus is a proof tree whose edges are (correct) applications of the inference rules of the calculus, and whose nodes are sequents. A proof corresponds to a complete derivation, whose root sequent at the bottom is the formula that is proved, and whose leaves are all vacuous and derived by the axiom rule. Two operational readings of a proof are possible. The top-down interpretation starts from the axiom and composes them into more complex sequents. The bottom-up interpretation starts from the sequent to be proved and decomposes it in smaller proof obligations until it arrives at instances of the axioms. Under this latter view, the cut rule corresponds to the application of a lemma inside the proof of a theorem: one premise provides a proof of the lemma (as the goal of the sequent); the other premise uses the lemma to further the proof of the theorem (as a new hypothesis). Cut elimination, in turn, corresponds to the inlining of lemmas: instead of proving them once, building them from scratch at each point where they are needed. Example Consider the two right introduction rules for ∨ and ∃ from Figure 2. [START_REF] Marijn | Proofs for satisfiability problems[END_REF] where the two separate rules for disjunction (one for each disjunct) are compacted into a single rule: Γ A i , ∆ Γ A 1 ∨ A 2 , ∆ ∨ R If one attempts to prove sequents by reading these rules from conclusion to premises, the rules need either additional information from some external source (e.g., an oracle providing the disjunct i ∈ {1, 2} or the term t ) or some implementation support for non-determinism (e.g., unification and backtracking search). Indeed, it is difficult to meaningfully use Gentzen's sequent calculus to directly support proof automation. Consider attempting to prove the following sequent: Γ ∃x.∃y.(p x y) ∨ ((q x y) ∨ (r x y)) Here, assume Γ contains one hundred formulas. The search for a (cut-free) proof of this sequent confronts the need to choose from among 101 potentially applicable introduction rules. If we choose the right-side introduction rule, we will again be left with 101 introduction rules to apply to the premise. Thus, reducing this sequent to Γ (q t s) requires picking one path of choices in a space of 101 4 possible choice combinations. Focusing It is clear from the most cursory inspection that the unadorned sequent calculus exhibits prodigious levels of bureaucracy and nondeterminism: on the one hand, copious applications of the structural rules are required to constantly transform the sequents into forms suitable for the other two groups of rules; on the other, work can take place anywhere in the sequent at any time, resulting in an exponential number of equivalent interleavings. The resulting proofs are profoundly non-canonical and too unstructured to offer any kind of realistic support for automation. The practical question becomes, then, how to remedy this chaotic situation and thereby bring more order into the operation of the calculus. A series of advances in proof theory-stemming from the study of proofs to describe the semantics of logic programming, notably the uniform proofs of [START_REF] Miller | Uniform proofs as a foundation for logic programming[END_REF]-have shown that imposing certain reasonable restrictions on the proofs of the sequent calculus allow these proofs to be structured in alternating phases of two distinct types. These results crystallized, in the linear logic setting, in the form of the discipline of focusing and its associated focused proof systems, introduced by [START_REF] Andreoli | Logic programming with focusing proofs in linear logic[END_REF]; in the practical plane, these strategies have been applied to describe computational aspects of theorem proving by [START_REF] Chaudhuri | A logical characterization of forward and backward chaining in the inverse method[END_REF]. Further developments by [START_REF] Liang | Focusing and polarization in linear, intuitionistic, and classical logics[END_REF] extended this discipline by obtaining focused sequent calculi for classical and intuitionistic logic. This will be our starting point. Again, for the time being we continue to focus on classical logic. Starting from the one-sided LK, its corresponding focused version is called LKF, and shown in Figure 2.4. decide Γ ⇑ N Γ ⇓ N release Figure The LKF focused proof system for classical logic [START_REF] Liang | Focusing and polarization in linear, intuitionistic, and classical logics[END_REF]. Here, P is a positive formula; N is a negative formula; P a is a positive literal; C is a positive formula or a negative literal; A and B are arbitrary formulas; and ¬B is the negation of B, itself in negation normal form. The proviso marked as † is the usual eigenvariable restriction that y must not be free in the components of the premise ( Γ, B, and Θ). More generally, a focused proof system operates on polarized formulas, obtained from regular, unpolarized formulas by replacing each logical connective with a polarized version of it, positive or negative, and by assigning a polarity, again positive or negative, to each non-logical constant. In the one-sided focused sequent calculus for classical logic, the four connectives ∧, ∨, t and f exist in both positive and negative versions, signified respectively by a + or -superscript. The universal quantifier ∀ is always negative, whereas the existential quantifier ∃ is always positive. The polarity of a non-atomic formula is that of its top-level connective, and the polarity of an atomic formula is assigned arbitrarily. Because formulas of the one-sided calculus are always in negation normal form, the scope of negation is always atomic, and negation is subsumed by polarity and corresponds to a simple polarity flip: if an atom is defined as positively polarized, its negation is defined as negative, and vice versa-the polarity of each atom is arbitrary, but all instances of an atom must share the chosen polarity. From now on, when we discuss formulas we will usually mean polarized formulas; the distinction between unpolarized and polarized formulas will be made clear when it is not clear from the context. Second, at the level of inference rules, the same general classification in groups of rules remains, but is built upon a more basic distinction that divides sequents into two separate sequent types. Both these types divide their collection of formulas in two: a storage zone (abbreviated Γ, like the original, unfocused RHS) and a workbench (sometimes abbreviated Θ)-the two zones are separated by an arrow sign. Proofs will now be structured in groups of inference rules of the same kind constituting distinct phases: 1. Up-arrow sequents Γ ⇑ Θ are related to the asynchronous phase (variously called negative, invertible, and up-arrow phase). Here, Γ is a multiset of formulas and Θ is a list of formulas. Each inference rule in this phase is invertible-i.e., its premises are true iff its conclusion is true, so that they can be moved at the end of a proof without loss of completeness. Moreover, these rules involve exclusively up-arrow sequents in both conclusion and premises. Owing to these properties, invertible rules can be applied to saturation indistinctly in any order. In fact, Θ is modeled as a list to enforce an order of evaluation in which asynchronous rules are always applied to the head of the list. This phase corresponds to don't-care nondeterminism. 2. Down-arrow sequents Γ⇓B are related to the synchronous phase (variously called positive, non-invertible, and down-arrow phase). Here, Γ is a multiset of formulas and B is a single formula. Inference rules in this phase are not necessarily invertible, and some are indeed non-invertible. Throughout a synchronous phase, the system is focused on the single formula in the workbench and sequentially applies these non-invertible choices until no more such rules are applicable (at which point the phase ends). Because these choices are not reversible, we may need to backtrack and try other possibilities if they choices are not good. This phase corresponds to don'tknow nondeterminism. The three groups of inference rules remain with the following changes to their structure and organization: 1. Introduction rules are now subdivided into two groups depending on the phase in which they operate: asynchronous introduction rules operate on negative connectives, and synchronous introduction rules operate on positive connectives. For each phase, there is one introduction rule for each propositional connective of matching polarity-except for f + -and one rule for the corresponding quantifier. Because each polarized connective is introduced by at most one rule, inference rule names are simply those of the connectives they introduce. 2. Initial rules do not experiment substantial modifications. The interesting change occurs in the atomic init rule, which replaces the axiom rule. The scope of this rule is now limited to a positive literal as a focused formula whose negated complement is contained in the storage zone. Indeed, Gentzen proved that all instances of init can be eliminated except for those that operate on atomic formulas, which is the criterion the focused calculus adopts. Whereas the init rule is part of the positive phase, the cut rule operates in the negative phase. 3. Structural rules undergo the most significant changes with respect to the unfocused calculus. First, the standard data structures eliminate the need for an exchange rule. More importantly, weakening and contraction are now integrated in other rules: contraction is used in the new decide rule, and weakening in the init rule, above. The structural rules streamline the flux of formulas between the zones of the sequent and arbitrate the phase transitions. Because they are conceived to enable proof search, they are best interpreted by a bottom-up reading, unlike the top-down we have followed until now. First, in the asynchronous phase, when the head of the workbench list is positive or a literal-and therefore no asynchronous rules apply to it-the store rule moves it into the storage zone. Second, when all formulas of the asynchronous workbench have been processed, the decide rule selects a positive formula from the storage zone as focus of the synchronous phase that begins. And third, when a negative formulas is encountered in the positive phase-and therefore no synchronous rules apply to it-the release rule removes the focus from it and starts the next asynchronous phase. (Note that the cut rule presents an alternative to the decide rule that prolongs the asynchronous phase instead of switching to the asynchronous phase.) The resulting focused proofs are therefore structured as an alternation of negative and positive phases. The combination of a positive phase followed by a negative phase is called a bipole. This aggregation of rules greatly decreases the amount of nondeterminism in the calculus and organizes proofs into larger coherent units. Under the bottom-up reading-representing the aspect of proof search which focusing is designed to automate-once the decide rule focuses on a formula, this formula (and the sequence of synchronous choices made during the positive phase) guides the evolution of the proof up to the boundaries between the branching negative phases that follow the positive phase and the next set of positive phases. For this reason, bipoles are described as synthetic inference rules. It remains to prove the connection between the original classical calculus LK and its focused version LKF. An important result that will be referenced in subsequent chapters is the following: 2.5.1 Theorem Let B be a formula of classical logic. B is a theorem of classical logic iff the entry point sequent • ⇑ B is provable in LKF, where • is the empty storage and B is an arbitrary polarization of B. Proof. Proved in [START_REF] Liang | Focusing and polarization in linear, intuitionistic, and classical logics[END_REF]. In other words, LKF is sound and complete w.r.t. classical logic. Moreover, an LKF proof reveals an underlying LK proof by removing all polarities and upand down-arrow sequent annotations-and possibly adapting structural rules to the presentation of choice. In addition, LKF enjoys the cut-elimination property. Finally, it must be noted that although polarization does not affect provability, it influences the shapes and sizes of the proofs that can be found for a given theorem. Examples of the choice of polarities and their consequences on proof size will be given in Chapter 3. In particular, compare Sections 3.3 and 3.4 and the quantitative analyses in Section 8.5. As a result, it is easy to see that-after introducing the focusing decorations in the previously considered sequents-reducing proving the original sequent Γ ∃x∃y[(p x y) ∨ ((q x y) ∨ (r x y))] ⇓ to its reduced form Γ (q t s) ⇓ involves only those choices related to the formula marked for focus: no interleaving of other choices needs to be considered. Example Soundness and completeness Theorem 2.5.1 in the previous section states the properties of soundness and completeness of the system LKF for classical logic. Because these concepts recur in presentations of subsequent systems and frameworks, we collect some essential remarks here. A proof system is sound with respect to a logic if every statement it proves is a theorem of the logic. Conversely, a proof system is complete with respect to a logic if for every theorem of the logic there exist proofs of its statement in the proof system. In sequent calculus, part of the interest of cut elimination lies on its connection to the related property of consistency. Consider again the system LK in Figure 2.1. Except for the cut rule, all inference rules satisfy the subformula property: that is, every formula in the conclusion sequent is a subformula of one of the premise sequents. If we can eliminate cut, we are left with a system that globally satisfies the subformula property. From this, a proof of consistency is simple: if it were possible to obtain proof of a contradiction (i.e., prove false), this contradiction should be found as part of a premise for some inference rule in the calculus; because none of the rules allow for this propagation of false from conclusion to premise, the calculus is consistent. (Furthermore, because it globally satisfies the subformula property, proof search can be easily automated.) By similar arguments, LJ is found to be sound and complete w.r.t. intuitionistic logic [START_REF] Liang | Focusing and polarization in linear, intuitionistic, and classical logics[END_REF]. Starting in Chapter 3, we will study methods to restrict completeness in such a way that soundness is preserved by construction. The motivation will be to start from a system that is, say, "complete enough" and sculpt out of it another system, possibly less complete, but also "complete enough" for a certain domain of interest, in the hope that this leads to increased expressiveness and efficiency. Notes The present work develops in the rich soils of computational logic in general and structural proof theory in particular. As general overviews of the field, a number of monographs serve as good references, expounding the bases of structural proof theory and covering the needed logical preliminaries in various styles of presentation: [START_REF] Takeuti | Proof Theory[END_REF]; Buss (1998); [START_REF] Troelstra | Basic Proof Theory[END_REF]; [START_REF] Negri | Structural Proof Theory[END_REF][START_REF] Von | Elements of Logical Reasoning[END_REF]. While we shall cover both classical and intuitionistic logic, a wealth of other logics exist. Linear logic, the "logic of resources" developed by [START_REF] Girard | Linear logic[END_REF], is especially relevant. Concepts like the distinction between additive and multiplicative inference rules (and connectives)-already present in relevant logics [START_REF] Anderson | Entailment: The Logic of Relevance and Necessity[END_REF][START_REF] Read | Relevant Logic[END_REF]-are ubiquitous in linear logic. In fact, the notions of polarity and focused proof developed by [START_REF] Andreoli | Logic programming with focusing proofs in linear logic[END_REF] extend cleanly to classical and intuitionistic settings, as well as to fixed points (Baelde and Miller, 2007;[START_REF] Liang | Focusing and polarization in linear, intuitionistic, and classical logics[END_REF] and serve to give a proof theoretical reading to the complementary discipline of model checking [START_REF] Heath | A proof theory for model checking: An extended abstract[END_REF]. Linear logic is one of the primary representatives of the family of substructural logics, so called because they limit the application of structural rules-and at least along this axis constitute a weakening of classical logic. This logic reoccurs during the introduction of fixed points in Chapter 9. The original sequent calculi by Gentzen use the homonymous structures as their fundamental building block. However, various extensions and refinements to the paradigm have been proposed, of which focusing is one of the most important. One way to obtain more general systems is to extend the data structures: calculi based on hypersequents [START_REF] Arnon Avron | Hypersequents, logical consequence and intermediate logics for concurrency[END_REF][START_REF] Arnon Avron | The method of hypersequents in the proof theory of propositional non-classical logics[END_REF] do this by replacing simple sequents with a list (or a multiset) of sequents, usually interpreted disjunctively (like the formulas in a one-sided sequent); the motivation is to study general frameworks in which to easily model many families of interesting logics. Another extension to the basic data structures of the calculus employs nested sequents [START_REF] Brünnler | Nested Sequents[END_REF], whose components can be not only formulas, but also other sequents, thus turning the latter into a recursive data type. This last idea is related to deep inference [START_REF] Guglielmi | A system of interaction and structure[END_REF][START_REF] Guglielmi | Deep inference[END_REF], a design methodology for proof systems which, applied to the sequent calculus, allows inference rules to be applied anywhere in a sequent instead of being limited to the top-level connective of a certain formula. Structural generalizations also extend to the arena of focused systems, where multi-focusing (Chaudhuri et al., 2008a) allows the decide rule to operate on a set of formulas, in this case to further increase the canonicity of sequent calculus proofs that is the hallmark of focused systems. The classic treatise on the nature of proof-as-message is the book by [START_REF] Lakatos | Proofs and Refutations[END_REF]. The proof-as-certificate counterpart is covered by [START_REF] Mackenzie | Mechanizing Proof[END_REF], which traces the development of mechanized proofs with which we are directly concerned. More recent overviews include [START_REF] Asperti | Proof, message and certificate[END_REF] and, under the prism of Foundational Proof Certificates, [START_REF] Miller | Communicating and trusting proofs: The case for broad spectrum proof certificates[END_REF]. Although we are primarily concerned with the certifying nature of proofs, it is arguable that work in the latter chapters-Part III, and even sections of Part II-addresses the proof-asmessage half by a flexible linkage between formally defined certificates and the messages conveyed by them, namely concise, readable descriptions of proofs made syntactically-and semantically-precise while being usable by mathematicians. The rule of cut serves very different purposes in proofs created for (or by) machines or mathematicians. In addition to the important theoretical results derived from it-bearing witness to certain desirable characteristics of a logic, like consistency-the existence of cut-free proofs is instrumental to the automation of proof search. In turn, for the mathematician, the ability to organize proofs using lemmas is an absolute conceptual necessity. Beyond cognitive and stylistic considerations, cuts are indispensable-also for machines-to manage the complexity and size of proofs, as the combinatorial explosion that results from inlining every auxiliary lemma at every point of use quickly becomes intractable [START_REF] Marcello | The taming of the cut. Classical refutations with analytic cut[END_REF]. Foundational Proof Certificates Proof as trusted communication Chapter 2 opens by examining the mathematical concept of proof and briefly charting its formal study. As we concentrate on the more solidified and structured states of the malleable substance and, through rigorous efforts, further seek to reduce entropy and obtain a flawless crystallization of mathematical truth, the dual functions of proof-as-message and proof-as-certificate seemingly begin to coalesce. In fact, inasmuch as computer programs can not only build proofs, but also write, read, and check them, those external representations of machine proofs-transmitted between computer processes-must ideally function both as messages and certificates. When the recipient of some proof evidence is a computer program, that evidence should be enough to enable the program to reconstruct the proof object which that evidence purports to represent; otherwise, it has to be discarded. Of course, there are many precisions to make, and an incomplete proof, even an incorrect one, is not always without value. Nevertheless, a complete proof object is ultimately needed to establish trust, and the proof-as-certificate aspect guarantees and safeguards the utility of the proof-as-message. The concept of proof as an object-materialized as some kind of documentthat attests to the truth of a statement made about a certain, formally defined theory finds ample practical support in the ever-growing array of software tools that facilitate the production and verification of proofs: automated theorem provers (Alt-Ergo, Vampire, Z3), proof assistants (Coq, Isabelle, Abella), model checkers (NuSMV, PRISM, Bedwyr), programming languages with sophisticated type systems (Idris, F*), etc. These tools are generally not designed with communication in mind, and as their number and sophistication increases, so does fragmentation. Yet different tools have different strengths, and it is natural to wonder how to combine them-and their proofs-gracefully. The issue is more than a theoretical curiosity, as complex developments may involve proofs which are more easily obtained through a mix of (compatible) formalisms. The ability to understand and admit proofs irrespective of their provenance is a prerequisite for this scenario, akin to the interoperation of computer programs. One may draw inspiration from the progressive application of formal methods in the related area of programming languages. The study and development of programming languages have been aided by the use of at least two important theoretical frameworks: on the one hand, context-free grammars, or CFG [START_REF] Hopcroft | Introduction to Automata Theory, Languages, and Computation[END_REF], are used to define the structure of programs; on the other, structural operational semantics, or SOS [START_REF] Plotkin | A structural approach to operational semantics[END_REF][START_REF] Plotkin | A structural approach to operational semantics[END_REF], are used to define the evaluation and behavior of programming languages. Both of these frameworks make it possible to define the syntax and semantics of a programming language in a way that is independent of a particular parser and compiler. Specifications in these frameworks are both mathematically rigorous and easily given prototype implementations using the logic programming paradigm [START_REF] Borras | Centaur: the system[END_REF][START_REF] Hannan | Extended natural semantics[END_REF][START_REF] Stuart | Principles and implementation of deductive parsing[END_REF][START_REF] Miller | Formalizing operational semantic specifications in logic[END_REF]. These techniques scale to the definition of practical programming languages, as demonstrated by the formal specification of ML [START_REF] Milner | The Definition of Standard ML[END_REF]. Similarly, work on automated and interactive reasoning systems can benefit from the introduction of frameworks that are capable of defining the meaning of proof descriptions that are output by proving tools, and representing these descriptions in a shared language-that of logic. Such formal semantics of proof languages make it possible to establish a separation of concerns between the production of proofs and the checking of proofs. On the one hand, production is carried out by theorem provers: the various families of tools (such as those listed above) capable of reasoning about formal specifications. These are complex, evolving pieces of software; they are potentially difficult to prove correct, and thus vulnerable to programming errors which can endanger their logical soundness. On the other hand, proof checkers could be small and persistent, as well as easy to trust and prove correct. In such a setting, the provenance of a proof should not be critical for trusting it-subject to its successful checking by a trusted tool. The key, then, is to define a logical framework where the syntax and, critically, the semantics of proofs can be defined in a clean, declarative fashion that is both universal and permanent. That is, the framework should be able to represent a broad spectrum of proof structures, such as resolution refutations, tableaux, (un)satisfiability proofs, superposition, etc. All of these kinds of proof evidence-. . given a definition of their semantics-are modeled through a unified representation of proofs, which much satisfy certain properties. Definition The term proof certificate denotes a document that should elaborate into a full formal proof by means of a proof checker. Any general-purpose framework implementing these concepts should satisfy the following four requirements [START_REF] Miller | Communicating and trusting proofs: The case for broad spectrum proof certificates[END_REF], which we shall call Miller's desiderata, referring to them by number: 1. A simple checker can, in principle, check if a proof certificate denotes a proof. 2. The format for proof certificates must support a wide range of proof options. 3. A proof certificate is intended to denote a proof in the sense of structural proof theory. 4. A proof certificate can simply leave out details of the intended proof. Harnessing recent advances in structural proof theory, Foundational Proof Certificates (FPCs) have been proposed as a general framework for the expression of proof evidence [START_REF] Miller | A proposal for broad spectrum proof certificates[END_REF][START_REF] Chihani | Foundational proof certificates in first-order logic[END_REF]Chihani et al., , 2016b)). In this framework, focused sequent calculi (for which see Section 2.5) act as foundations of a unifying proof system, which-as its parallels in the programming language world-is easily implemented in proof checking kernels in a logic programming language, as we will see in Section 4.4. In this context, an FPC is a machine-readable document which expresses a proof in terms of a series of synthetic inference rules. Those inference rules, along with their logical interpretation, are given by a certificate definition, which operates as nexus between the two sides of a checker: (a) for the kernel, it is a small logic program loaded and run by the kernel, used to guide proof search; and (b) for the client, a small domain-specific language in which proofs (i.e., proof certificates) can be written. The payload that a client must provide to a proof checker consists of both a certificate and its definition, although FPC definitions are modular and it is in their vocation to be reusable. The rest of the chapter is organized as follows: Section 3.2 introduces the extensions to the focused sequent calculus that form the theoretical basis for the framework. Section 3.3 presents the first of four simple FPC definitions which are used as recurring examples, here a simple decision procedure for propositional logic. Section 3.4 continues with the second of these examples by defining precise guidance information obtained from an oracle in the certificate. Section 3.5 complements previous certificate formats with a more flexible constraint on the size of proofs expressed as their depth in number of bipoles. Section 3.6 completes the series of examples with a certificate format for proofs by binary resolution, closer to the proofs produced by automated theorem proving tools. Section 3.7 discusses the relationships between the various parts and realizations of the framework, which will find concrete expression in Chapter 4. Section 3.8 concludes the chapter. Augmented sequent calculus The standard sequent calculus of Section 2.4, while undeniably interesting from a theoretical perspective, is not well suited for automation because it lacks structure. One modern criterion for what constitutes a well-behaved proof system is whether the logic admits a focused version, as illustrated in Section 2.5. Focused proofs substantially reduce the amount of nondeterminism by structuring the proof as an alternation of asynchronous and synchronous phases. Within each asynchronous phase, only invertible rules are used and their ordering is irrelevant; within each synchronous phase, the sequence of inference rules is fully determined. Although focused proofs exhibit a much larger degree of canonicity, the sources of meaningful nondeterminism remain: picking a formula to focus on in the decide rule, devising a lemma to cut into the proof, choosing a disjunct or an existential witness to instantiate the rules of the synchronous phase, etc. Focused phases are, in a sense, synthetic inference rules, but lack flexibility. The solution adopted by the FPC framework is to augment the focused sequent calculus to allow fine control to the point of making it programmable. Consider Figure 3.1, which presents the augmented calculus LKF a . Three main categories of changes are introduced: 1. Starting from a standard focused system (here, the LKF of Figure 2.4), every inference rules has all its premises and its conclusion enriched with certificate terms, denoted Ξ. 2. Moreover, an additional premise is added to every rule-except that for negative true, t -. The new premise represents a predicate that relates the certificate terms in the conclusion and the premises, as well as every piece of additional information required by the rule (disjunctive choices, existential witnesses, etc.). When fresh eigenvariables are involved (i.e., treating the universal quantifier) the certificate terms on the premises are parameterized over the new variable. 3. Finally, the storage zone Γ is transformed into a multiset of indexed formulas, and inference rules storing and pulling formulas from this zone are modified to reflect the new data structure. The operational reading of the augmented calculus parallels bottom-up proof reconstruction. Under this discipline, a concrete certificate term is expected in the conclusion. The new (client-supplied) relational premises-which will be presently characterized as clerks and experts-would then take the conclusion certificate as "input" and use it to constrain all other elements as the "outputs" at the premises. If the output certificate is related to continuation certificates for the premises (and ancillary information: disjuncts, etc.), each possible combination of values offers an opportunity to continue, and possibly finish, the proof. In most practical FPC definitions, the relation behaves like a partial function from conclusion certificate to premise certificates. By contrast, some forms of ancillary information can not only span several different values, but also be completely unconstrained and left to the checker to reconstruct-logic variables can be used to reflect these degrees of freedom. Similarly, clerk and expert relational premises refer to formulas in storage exclusively by index; the model does not enforce a functional mapping, so that an index may select an arbitrary number of formulas from storage. Example Let us revisit the interesting cases of the introduction rules in Example 2.4.1-albeit in their focused versions. Consider the introduction rule for the positive disjunction. In bottom-up proof search-once the inference rule is augmented with an expert-, one of the charges of an FPC definition is to have the expert dictate the disjunctive choice. Based on a certificate term and its own defining clauses, the expert may stipulate that, at a given point in the derivation, the left or the right disjunct should be used to proceed with the proof. It may also decree that no choice is acceptable-and in consequence a derivation may not be obtained, even if one exists-or that either choice may be attempted (hence, an implementation should choose one of the disjuncts and continue the proof attempt, and return to try the second disjunct if the first fails). The treatment of the existential quantifier generalizes that of the positive disjunction in that the number of possible terms may be zero (if the type is uninhabited), finite or infinite (in inductively defined types like the natural numbers): more complex selection strategies are possible, but remain variations on those . asynchronous introduction rules Ξ 0 Γ ⇑ t -, Θ Ξ 1 Γ ⇑ A, Θ Ξ 2 Γ ⇑ B, Θ ∧ c (Ξ 0 , Ξ 1 , Ξ 2 ) Ξ 0 Γ ⇑ A ∧ -B, Θ Ξ 1 Γ ⇑ A, B, Θ ∨ c (Ξ 0 , Ξ 1 ) Ξ 0 Γ ⇑ A ∨ -B, Θ (Ξ 1 y) Γ ⇑ (B y), Θ ∀ c (Ξ 0 , Ξ 1 ) Ξ 0 Γ ⇑ ∀x.B, Θ † Ξ 1 Γ ⇑ Θ f c (Ξ 0 , Ξ 1 ) Ξ 0 Γ ⇑ f -, Θ synchronous introduction rules t e (Ξ 0 ) Ξ 0 Γ ⇓ t + Ξ 1 Γ ⇓ B 1 Ξ 2 Γ ⇓ B 2 ∧ e (Ξ 0 , Ξ 1 , Ξ 2 ) Ξ 0 Γ ⇓ B 1 ∧ + B 2 Ξ 1 Γ ⇓ B i ∨ e (Ξ 0 , Ξ 1 , i) Ξ 0 Γ ⇓ B 1 ∨ + B 2 i ∈ {1, 2} Ξ 1 Γ ⇓ (B t ) ∃ e (Ξ 0 , Ξ 1 , t ) Ξ 0 Γ ⇓ ∃B identity rules l, ¬P a ∈ Γ init e (Ξ 0 , l ) Ξ 0 Γ ⇓ P a init Ξ 1 Γ ⇑ B Ξ 2 Γ ⇑ ¬B cut e (Ξ 0 , Ξ 1 , Ξ 2 , B) Ξ 0 Γ ⇑ • cut structural rules Ξ 1 Γ, l, C ⇑ Θ store c (Ξ 0 , Ξ 1 , l ) Ξ 0 Γ ⇑ C, Θ store Ξ 1 Γ ⇓ P l, P ∈ Γ decide e (Ξ 0 , Ξ 1 , l ) Ξ 0 Γ ⇑ • decide Ξ 1 Γ ⇑ N release e (Ξ 0 , Ξ 1 ) Ξ 0 Γ ⇓ N release Figure The augmented LKF a focused proof system for classical logic [START_REF] Chihani | Foundational proof certificates in first-order logic[END_REF]. Presentation conventions are shared with Figure 2.4. available in the (binary) disjunctive case. In a focused proof system, the decide rule is the primary source of proof structure. The choice of formula under focus is afforded the same flexibility with the peculiarity that selections are determined by the indexing scheme imposed by the certificate terms and the store clerks. A decision depends on both the indexes allowed by the decide expert and the sets of formulas filed under those indexes. Again in Example 2.4.1, if all formulas share a single index, the number of choices remains unchanged; if each formula is stored under a unique index, a certificate term can record the sequence of choices that lead to a single, directed path to success. In brief, the augmentation of LKF by certificates, indexes, and clerk and expert relations modifies the original system merely by restricting when inference rules can be applied: it affects completeness and leaves soundness untouched. Theorem The system LKF a is sound w.r.t. classical logic (Chihani et al., 2016b, Section 5). Proof. The LKF system can be recovered from LKF a by removing all the augmentations (marked in Figure 3.1), and therefore every proof of LKF a is also a proof of LKF. The result follows from Theorem 2.5.1. Theorem 3.2.2 implies that the soundness of the system cannot be compromised by the client. Note that the augmentations of LKF a are completely generic and can be assigned arbitrary meanings by the client: this is the function of FPC definitions. By furnishing declarations and definitions (i.e., syntax and semantics) to define a set of augmentations, the sequent calculus gains support for programmable proof reconstruction. Effectively, such a description defines a family of certification strategies. There are five groups of elements that constitute an FPC definition, each described in the following paragraphs: 1. Polarization. A polarization strategy determines how to translate (standard) unpolarized formulas into polarized versions of them-the other direction is direct by polarity erasure. As noted in Section 2.5, the choice of polarities by itself does not affect provability, but its interactions with the other members of an FPC definition may. Despite this seeming leniency, the role of polarities should not be underestimated: they have a strong impact on the proofs that can be found for a given theorem, and can represent the difference between brute force search (exponential in complexity) and purposeful navigation, guided by a certificate along a predetermined set of nondeterministic choices. For illustration, compare the certificate formats and corresponding examples in Sections 3.3 and 3.4. 2. Certificate terms. Conceptually, certificate terms represent the state of a proof, or the region of a proof in which we find ourselves at a given point in the derivation. State information can be arbitrarily complex and, as the proof evolves, so do the certificates, and the information contained in them can be used (by clerks and experts) to steer the derivation towards success. 3. Indexes. Index terms control two important aspects of guided proof reconstruction: naming and storage. As synchronous formulas are moved to the storage zone Γ, they are annotated with a data structure contained in an index term, supplied in the store rule. Like certificate terms, they may contain arbitrary information, and can later be called upon to selectively retrieve a subset of candidate formulas for selection by the decide rule (to treat during the positive phase inaugurated by the rule) or by the init rule (to attempt to finish a branch of the derivation). Lookup can be as loose or as tight as the designer chooses, offering great control over backtracking points, classification and selection of formulas to make the proof progress, etc. A good indexing scheme is critical to performant proof checking. 4. Clerks. These predicates, denoted by the c subscript in Figure 3.1, define the control semantics of asynchronous rules. While not explicitly responsible for the decisions that are characteristic of synchronous rules, they can nonetheless perform bookkeeping and record information that will eventually enable their synchronous counterparts to make their decisions when they are executed. These can be thought of as ordinary program clauses, making full use of the power of logic programming. Nontermination is allowed by the framework, as it concerns itself only with the soundness of successful derivations-for which termination of all instances of clerks and experts is necessary. 5. Experts. The synchronous counterparts of clerks, signified by the e subscript, are-like their asynchronous duals-arbitrary program clauses, but in addition to possibly recording information while processing certificate terms they will be required to make the decisions demanded by their phase and supply information about those decitions to the checker. How much or how little they commit to one or several possible courses of action is not dictated by the framework and remains purely a design issue. For example, ∨ e must select one of the disjuncts, B 1 or B 2 , to continue the derivation. A possible, nondeterministic course of action is to try to use B 1 first, backtracking to try B 2 if a complete derivation based on B 1 cannot be found. Similarly, ∃ e may provide a set of closed terms as witnesses or let an arbitrary term be instantiated over the course of the proof. The phase of the inference rule for a given logical connective is immediate from the polarity of the connective. Structural rules are more nuanced: store occurs during the asynchronous phase and is assigned a clerk, although it is charged with the critical operation of filing formulas into storage, in the process assigning them indexes. In turn, decide arbitrates the transition from asynchronous to synchronous phase by using those very indexes, and its complement release mediates (albeit trivially) the other phase transition, from synchronous to asynchronous. Both these transitions can be seen as operating "between two worlds," and are here both declared as experts. The last group of identity rules is supervised by experts. init is a standard expert of the synchronous phase: it operates on a positive literal and involves selecting a stored complementary literal. On the other hand, cut acts as an alternative to decide at the boundary of an asynchronous phase, and instead of ending the phase, it prolongs it, but there is no doubt that its duties and position correspond to those of an expert. In the next few sections, we present a number of FPC definitions that illustrate typical uses of the framework and will recur in subsequent chapters. The definitions are given in executable λProlog code. They are direct, declarative transcriptions of the mathematical relations they encode, comprising certificate and index term constructors, and clauses for clerks and experts; polarization conventions are implicitly defined from those. Only standard logic programming features are used; argument order is preserved from Figure 3.1. We refer to Chapter 4, specifically Section 4.4 for a study of the implementation of LKF a as a proof checking kernel and its interface with client-defined clerks and experts, as given by the kernel's API. Running example: CNF decision procedure An extreme minimalist use of the FPC framework is studied as our first example. Consider the propositional fragment of classical logic, where all logical constants are negatively polarized. In this situation, there is no opportunity to offer guidance during the synchronous phase, and our only recourse is to exhaustive search. While completely blind and exponentially inefficient, it is clearly a sound, if empty, brute force "strategy," a fact that can be clearly represented with a simple FPC definition. Each of its five components can be explained as follows: 1. Polarization is purely negative across all allowed logical connectives (i.e., no quantifiers). By convention, formulas are in conjunctive normal form, and negations translate into atoms of complementary polarities, e.g., negated atoms are given negative polarity; non-negated atoms are given positive polarity. This is the only allowed use of positive polarity. 2. Certificate terms carry no information. A singleton, nullary constructor, cnf, leaves no option but to propagate it from conclusion to each premise, unchanged. 3. Indexes also carry no information. A singleton, nullary constructor, idx, is used to file all formulas in the store rule. Conversely, the decide rule can only use this unique index to specify which formula(s) to choose. Hence, the storage zone acts as a simple bucket of formulas; it contains no information to help us discriminate the right formula to focus on. Each instance of the decide rule attempts to find a proof for every formula currently stored in Γ. 4. Clerks are declared for the negative connectives to enable proof search to proceed through them and store positive formulas and negative literals, i.e., atoms, without distinction. 5. Experts are declared only for phase transitions and for the init rule, to allow an atom to be matched with its stored complement and close a branch of the derivation. It can be seen that, by design, the other inference rules cannot occur, and therefore their definition would have no effect in proof search. The FPC definition is presented in full in Figure 3.2. This is, in fact, a decision procedure for classical propositional logic. Example With patience, the theoremhood of small propositional formulas can now be checked by using cnf with the entry point sequent. Let F be a formula of LK, and F -be its negative polarization. Then F is a theorem of LK iff cnf • ⇑ F -is provable. Figure The CNF decision procedure FPC (Chihani et al., 2016b, Section 7.1). Running example: oracle strings Still in the decidable setting of classical propositional logic, the polar opposite of the exhaustive exploration of the previous section-our sole recourse given the lack of meaningful decisions-is the concrete expression of those decisions and their representation of one instance of a successful search as the certificate term. Thus, the tree of decisions effectively constitutes an oracle. The shift finds reflection in the choice of polarities, which in turn affect the proofs that can be found-and, fundamentally, open an avenue of feeding the oracle via a proof certificate. We next define the components of the FPC definition, reproduced in Figure 3.3. 1. Polarization is purely positive across the propositional connectives; negative polarity is allowed only with atomic scope. 2. Certificate terms carry as information a tree representing an oracle of decisions represented by the kind oracle, with constructors for branching conjunction (c, with a continuation for each branch) and disjunction (l and r, respectively instructing to pick the left or the right disjunct, with a continuation for the choice), as well as a branch terminator (emp). Oracle information is wrapped in three different constructors: the principal consumer of decisions, consume; as well as start and restart, used to simulate the homonymous rules of the purely positive fragment of LK in the focused setting of full LKF via its structural rules. Figure The oracle string FPC (Chihani et al., 2016b, Section 7.2). 3. Indexes carry little information: as in the previous section, lit is used to store negative literals for further discharge by the initial rule. A second index, root, marks the formula on which the (re)start rule proceeds to the regular checking (i.e., consumption) phase. 4. Clerks are defined exclusively for the store rule: at the outset, to store the positive theorem candidate as root; after a release that brings the state to await a restart, to store the negative literals that caused the release of focus as lit. 5. Experts are defined for each synchronous propositional introduction rule as consuming the oracle corresponding to that exact connective, i.e., disjunctive choices, conjunctive branchings, as well as closing branches by looking up literals under lit or treating true; all this occurs in the consume phase. In addition, the decide rule implements the (re)start by focusing on the formula stored under root, and the release of focus (into restart) when a negative literal is encountered. It is instructive to compare the purely negative proofs of Section 3.3 with the purely positive proofs of this section. By polarizing a classical formula with negative or positive bias, we obtain different proofs of the same (unpolarized) theorems, though these proofs have vastly differing structures and computational behaviors. Running example: decide depth Another useful FPC definition provides a simple restriction on the proofs it allows by placing a certain bound on their size and admitting only "small" proofs which satisfy that bound. While of limited use by itself, it is representative of a common pattern of certificates as resources. Let us examine its components group by group: 1. Polarization does not impose any restrictions on what connectives can be used: this FPC definition concerns itself solely with a measure of the size of an arbitrary proof. 2. Certificate terms are limited to a single constructor, dd, containing a single piece of information: the maximum allowed decide depth from the present point in the proof (represented by a natural number, here of type nat as the standard inductive type). 3. Indexes carry no information: as in the previous example, a singleton, nullary constructor (called indx in this case) is used for all formulas in the storage zone. 4. Clerks are defined for each asynchronous inference rule. Each clerk propagates the decide depth bound unchanged from conclusion to premises. When treating the universal quantifier, the abstraction over the continuation certificate is vacuous, i.e., the eigenvariable plays no role in the new certificate. Storage of formulas uses the unique index at our disposal, indx. 5. Experts are defined for each synchronous inference rule-except cut. Like their clerk duals, they propagate the decide depth bound from conclusion to premises and select the singleton index indx whenever an index term is solicited by the interface. There are two points of importance. First, the decide depth bound is decremented on the decide rule while the current allowance permits it (i.e., is greater than zero). Second, disjunctive Figure The decide depth FPC (Blanco et al., 2017a). choices and existential witnesses are completely unconstrained by the FPC definition, and fresh logic variables are returned to the kernel. Figure 3.4 presents the FPC definition just described. An inductive definition of natural numbers with zero and successor constructors (the latter written as succ in the figure) is assumed present. These FPCs have fewer constraints and in fact encompass the proofs covered by the definitions in Sections 3.3 and 3.4. Decide depth bounds are less useful for proof search in isolation, but offer a useful way to express the size of focused proofs in terms of their essential high-level components-bipoles. Running example: binary resolution It is instructive to see how the FPC framework scales up to the core of a standard proof technique. To this end we turn to resolution refutations. Suppose we want to prove a formula of the form ¬C 1 ∨ . . . ∨ ¬C n . This is equivalent to refuting the negation of the formula, i.e., C 1 ∧ . . . ∧ C n -as usual, assumed in conjunctive normal form-where each clause C i a disjunction of literals closed by universal quantifiers. The key idea of the technique lies in the binary resolution rule (here, subscripted as and bs are arbitrary literals). a 1 ∨ • • • ∨ a i-1 ∨ P a ∨ a i+1 ∨ • • • ∨ a m b 1 ∨ • • • ∨ b j-1 ∨ ¬P a ∨ b j+1 ∨ • • • ∨ b n a 1 ∨ • • • ∨ a i-1 ∨ a i+1 ∨ • • • ∨ a m ∨ b 1 ∨ • • • ∨ b j-1 ∨ b j+1 ∨ • • • ∨ b n A proof by binary resolution is structured as a sequence of applications of this rule. Each of these steps proceeds by attempting to apply the resolution rule to a pair of clauses to generate a new clause, until the empty clause f is reached, at which point the refutation succeeds. By assigning names to the clauses that compose the formula to be proved, as well as to the new clauses that result from applications of the resolution rule, it becomes possible to represent compactly a proof by resolution by a list of triples, each denoting the two premises and the conclusion of each application of the rule, respectively. A certificate representing a proof by resolution must encode this information. Assuming clause names are natural numbers assigned incrementally and starting from one, the following triple of lists is a natural encoding of the information contained in the proof. 1. A list of clauses corresponding to the C 1 , . . . , C n of the formula whose proof will be attempted. These will receive identifiers 1 through n. 2. A list of clauses used in the proof but not included in the input clauses, i.e., derived by applications of the resolution rule on previous clauses. These will receive identifiers starting from n + 1. 3. A list of triples of numeric indexes i, j, k , where i and j are the indexes of the premises and k the index of the conclusion of an instance of the binary resolution rule. In order to design the FPC definition in detail, we determine one possible shape of a general proof by resolution in LKF a , and work around to sculpt it during proof reconstruction. A detailed description can be found, e.g., in Chihani et al. (2016b). Conceptually, a proof is divided in three types of phases, or regions: 1. The first phase starts the proof and asynchronously stores the clauses of the goal formula, each under its respective index. After all clauses have been stored, the second phase commences. 2. The second phase translates each use of the resolution rule (represented by a triple of indexes in the certificate) into an instance of the cut rule, where the derived clause acts as cut formula. One of the branches of this cut continues on to the next instance of the resolution rule, forming a backbone of cuts that finishes with the empty clause. 3. The third phase (type) branches off from each instance of the cut rule in the second phase. It reconstructs a shallow proof that the triple of indexes specified by the certificate is, indeed, a correct application of the binary resolution rule. The FPC definition is shown in Figure 3.5. The three phase types in a proof by resolution correspond, in order, to the three groups of clerk and expert clausesseparated by line comments-in Figure 3.6. (This structure is revisited and expanded upon in Chapter 7.) Example Suppose we wish to prove the following formula: r (z) ∧ (∀x.¬r (x) ∨ t (x)) ∧ ¬t (z) Here, z is a term of a certain type i, and r and t are relations of type i → o. To prove the formula, we instead attempt to find a refutation of its negation: ¬r (z) ∨ (∃x.r (x) ∧ ¬t (x)) ∨ t (z) We may do so by hand or by resorting to any of a number of automated theorem provers. A resolution-based prover should be able to provide evidence for the validity of the goal formula. Instead of relying on an informally specified proof script, it would be easy to adapt a proving tool to emit the information contained in the proof by resolution as a formally defined FPC as per the definition in Figure 3.5. According to this encoding, a possible certificate will have the following shape: [resol 1 2 4, resol 4 3 5] [1 → r (z), 2 → ∀x.¬r (x) ∨ t (x), 3 → ¬t (z)] [4 → t (z), 5 → f ] In this presentation, indexes and their mappings to subformulas are given explicitly, whereas the basic encoding detailed above relies on implicit numbering. Both are acceptable variations of the same family of certificates. An independent . . , , checker can use this certificate and the definition on which it is based to verify that the goal formula is, in fact, a theorem. The binary resolution FPC just presented illustrates the intricacies of precise encodings and their close connection to proof reconstruction. The FPC definition is designed in such a way that some natural and mostly independent extensionsdiscussed throughout Chapter 8-are straightforward, but other, slightly different encodings of resolution refutations do not share this property. Experience shows that complex FPC definitions risk brittle behavior in the face of changes and additions, and tests need to be maintained and pre-and postconditions carefully documented. Naturally, simpler "proof scaffolds" are more robust-commonly at the cost of some loss in efficiency. (Following this line of thought, for example, the shallow proofs of the third phase could be given a much more compact, implicit representation.) Checkers, kernels, clients and certificates At a basic level, there is a juxtaposition between the two parts that form a proof checker. On the one hand there is a trusted, sound kernel that implements a focused sequent calculus as a logic program. On the other, there is a client that specializes the kernel by providing an untrusted FPC definition and certificates built on it. Thus, the combination of the kernel and an FPC definition results in a concrete instance of the proof checker. Nonetheless, note that the client side is divided in two parts: the FPC definition and the FPC proper, the latter itself the client of the former. While the implementation of the kernel does not concern the client side, its semantics is of direct concern to the author of an FPC definition. On the other side, the writer of FPCs based on a definition is only interested in the higher-level semantics given by that definition, and not by its implementation of the kernel's. Moreover, the writer of FPCs is seldom interested in polarities, as they do not affect provability and polarization is often fixed by the semantics of the FPC definition, so that as a user one can easily write, say, formulas of LK and FPCs, thus staying at a high level of abstraction. From the time of the original FPC proposal by [START_REF] Miller | A proposal for broad spectrum proof certificates[END_REF], a motivating analogy has been advanced in the form of what are called synthetic inference rules. The addition of focusing to the sequent calculus establishes a first approximation in the form of alternating synchronous and asynchronous phases. Pursuing the standard analogy, the metatheory of focused proof systems becomes the "rules Figure The binary resolution FPC (continued): implementation. of chemistry" which allow us to take "atoms" of inference and compose them into more complex, higher-level "molecules" of inference-these correspond to the inference rules of the calculus and the phases of focusing, respectively. In the FPC framework, all certificates are ultimately expressed in terms of those rules of inference, and proof checkers implement and apply those rules. An alternative and much closer interpretation of the FPC framework views, rather, in computational terms. We liken the framework, rather, to a fantastic new assembly language, one that programs a machine that is based on a certain logic and whose instruction set, or ISA, is the set of inference rules of its corresponding sequent calculus. The definition of the semantics of a certain FPC definition is essentially the definition of a domain-specific language, or DSL, in which proof evidence of the theoremhood of a formula can be expressed. A concrete certificate acts as a program written in the language of the FPC definition and interpreted on the checker that implements the assembly of the chosen logic, i.e., the architecture of the logic computer. The analogy is apt insofar as it reflects the deep ties between certificate definitions and the intricacies of proof systems: until now, programming these FPC definitions has been the delicate domain of the expert. While some attention has been paid to the discipline of programming FPC definitions (Blanco andChihani, 2016, 2017), the sole appeal remains to the underlying logic. Concrete definitions to certify a certain proof family or a tool cannot be easily extended to other, superficially similar tools, given that each employs its own ad hoc DSL. The task of writing what amounts to a compiler to a language based on logic remains a nontrivial task. Notes The increased complexity of modern automated theorem provers has brought with it a need for proof certification. Potential sources of errors in claimed proofs range from bugs in the code to inconsistencies in the object theory. To address these problematics, various tools for proof certification have been implemented that can improve our confidence that the output from theorem provers constitutes in fact a proof. These tools can be classified into two groups according to the object of verification: 1. A given theorem prover could itself be proved formally correct. See, for example, [START_REF] Ridge | A mechanically verified, sound and complete theorem prover for first order logic[END_REF]. . . 2. The output of a theorem prover can be verified independently from the tool that produced it. This possibility can be further subdivided in two types based on the proof reconstruction strategy: (a) Systems for replaying proofs using external theorem provers for the verification of specific proof steps. Among these, we count generalpurpose tools like Sledgehammer [START_REF] Lawrence | Source-level proof reconstruction for interactive theorem proving[END_REF][START_REF] Böhme | Sledgehammer: judgement day[END_REF], PRocH [START_REF] Kaliszyk | PRocH: Proof reconstruction for HOL light[END_REF], and GDV [START_REF] Sutcliffe | Semantic derivation verification: Techniques and implementation[END_REF], as well as more specialized efforts such as the verification of proofs generated by the E prover using Metis [START_REF] Lawrence | Source-level proof reconstruction for interactive theorem proving[END_REF]. (b) Tools that comprise an encoding or a translation of the semantics of certain theorem provers, which is then used to replay proofs from those known provers. This group admits a further subdivision based on specificity criteria: i. Specific tools, such as Ivy [START_REF] Mccune | IVY: A preprocessor and proof checker for first-order logic[END_REF] and the encodings of MESON [START_REF] Loveland | Mechanical theorem-proving by model elimination[END_REF] in HOL Light, and of Metis [START_REF] Hurd | First-order proof tactics in higher-order logic theorem provers[END_REF] in Isabelle. ii. General-purpose tools like Dedukti (Boespflug et al., 2012) and ProofCert [START_REF] Miller | A proposal for broad spectrum proof certificates[END_REF]. Our interest and our efforts concentrate in this last subcategory. These various classes of tools represent different approaches to proof certification. While we can have a high level of trust in the correctness of formally verified provers in category 1, their performance cannot be compared to that of the leading theorem provers like E and Vampire [START_REF] Riazanov | The design and implementation of VAMPIRE[END_REF]. The remaining groups do not pose restrictions on the provers themselves but the generality and automation of those in category 2 group come with the cost of using an external theorem prover and translations, which might result in reduced confidence. The families of tools in subfamily 2b require an understanding of the semantics of a theorem prover so that one may guarantee the soundness of proofs by their reconstruction in a low-level formal logic. Working with an actual proof has several advantages-as one can apply procedures like proof transformations. Group 2(b)ii has additional advantages over its sibling 2(b)i: a single certifier can be written that should be able to check proofs from a range of different systems and the existence of a common language for proofs allows for the creation of proof libraries and marketplaces [START_REF] Miller | A proposal for broad spectrum proof certificates[END_REF]. The tools in this last target group have had, so far, only limited success in the theorem proving community at large. One reason for this is that understanding and specifying the semantics of proofs requires sophistication in the interplay between deduction and computation-whether via rewriting in functional style or by proof search. Separating theorem provers from proof checkers using a simple, declarative specification of proof certificates is not new: see [START_REF] Harrison | History of interactive theorem proving[END_REF] for a historical account. We give here a brief partial sketch. A common starting point is the dependently typed λ-calculus LF [START_REF] Harper | A framework for defining logics[END_REF], originally proposed as a framework for specifying natural deduction proofs; the Elf system [START_REF] Pfenning | Elf: A language for logic definition and verified metaprogramming[END_REF] provided both type checking and inference for this framework; and the proof-carrying code project of [START_REF] Necula | Proof-carrying code[END_REF] used LF as a target proof language. The dependently typed λ-calculus has been extended with side conditions by the LFSC system; an implementation of it has been successfully used to check proofs coming from SMT solvers like CLSAT and CVC4 [START_REF] Stump | SMT proof checking using a logical framework[END_REF]. Yet another extension to the dependently typed λ-calculus is Deduction Modulo [START_REF] Cousineau | Embedding pure type systems in the lambda-Pi-calculus modulo[END_REF][START_REF] Boespflug | Conception d'un noyau de vérification de preuves pour le λΠcalcul modulo[END_REF]: in this calculus, rewriting is available. The Dedukti proof checker (Boespflug et al., 2012), based on this latter extension, endeavors to answer similar questions as those posed in this chapter through different methods, adopting a more computational view of checking based a functional instead of a relational paradigm, and congruences generated by sets of rewrite rules in lieu of FPC definitions. Dedukti has been successfully used to check proofs from such systems as Coq [START_REF] Boespflug | CoqinE : Translating the calculus of inductive constructions into the λΠ-calculus modulo[END_REF] and HOL [START_REF] Assaf | Translating HOL to Dedukti[END_REF] among other systems. In the domain of higherorder classical logic, the GAPT system [START_REF] Ebner | System description: GAPT 2.0[END_REF] is capable of proof checking in sequent calculus, resolution, and expansion trees-a generalization of Herbrand disjunctions-; it supports both checking and transformation among proofs expressed in those supported formats. The Foundational Proof Certificate framework described in this chapter was recently proposed as a means of defining the semantics of a wide range of proof languages for first-order classical and intuitionistic logic [START_REF] Chihani | Foundational proof certificates in first-order logic[END_REF][START_REF] Chihani | Certification of First-order proofs in classical and intuitionistic logics[END_REF]Chihani et al., 2016b). Instead of starting with a dependently typed λ-calculus, the FPC framework is based on Gentzen's lower-level notion of sequent calculus proof. Previously, FPC definitions have been formulated to model diverse sources of proof evidence, among these resolution refutations [START_REF] Robinson | A machine-oriented logic based on the resolution principle[END_REF], expansion trees [START_REF] Miller | Expansion tree proofs and their conversion to natural deduction proofs[END_REF], Frege proof systems, matings [START_REF] Peter | Refutations by matings[END_REF], simply typed and dependently typed λ-terms, equality reasoning (Chihani and Miller, 2016), tableau proofs for some modal logics [START_REF] Miller | Focused labeled proof systems for modal logic[END_REF][START_REF] Libal | Certification of Prefixed Tableau Proofs for Modal Logic[END_REF][START_REF] Marin | A focused framework for emulating modal proof systems[END_REF], and decision procedures based on conjunctive normal forms, truth table evaluation, and the G4ip calculus [START_REF] Dyckhoff | Contraction-free sequent calculi for intuitionistic logic[END_REF][START_REF] Troelstra | Basic Proof Theory[END_REF]. Some simple examples have been covered in this chapter. New applications are described throughout Parts II and III. As with other declarative and high-level frameworks, proof checkers for FPC specifications can be implemented using the logic programming model of computation [START_REF] Chihani | The proof certifier Checkers[END_REF](Chihani et al., , 2016b;;[START_REF] Miller | Proof checking and logic programming[END_REF]. In addition to those standard applications in theorem proving, FPCs have been applied to model checking (Heath andMiller, 2015, 2017) given a richer logic than used in the last paragraph. µM ALL, i.e., multiplicative-additive linear logic with greatest and least fixed points as logical connectives instead of exponentials, is suitable for this purpose (Baelde and Miller, 2007). In a similar vein, the addition of fixed points to intuitionistic logic, µLJ , serves to reason about constructive proofs and their expression as outlines, in a note closer to the connection between theorem provers and certification [START_REF] Baelde | Focused inductive theorem proving[END_REF]. In this extended logical framework, further developments will be studied in Part III. As we have noted, the formula indexing mechanism of the FPC framework does not impose functionality, i.e., different formulas can have the same index. Previously, indexes have been identified with diverse structures, including de Bruijn numerals and formula occurrences [START_REF] Chihani | Foundational proof certificates in first-order logic[END_REF]. It is possible to conceive very sophisticated indexing structures that assign sets of properties to the stored lemmas (e.g., "associativity lemmas" or "lemmas about natural number addition"). These rich indexing schemes could then be used to greatly increase the expressiveness of the decide rules. The simple examples in this chapter serve already to showcase the existence of versions of proofs with very different properties and behaviors. Both extremes of very implicit proofs (as in Section 3.3, with constant certificate size and exponential checking time) and very explicit proofs (as in Section 3.4) can be expressed in the framework. Surely the nature and effectiveness of proof checkers can be greatly affected by the level of detail of a proof format. Well-designed FPC definitions will exhibit the standard tradeoff between certificate space and checking time. Chapters 5 and 6 will pursue this line of inquiry. Even as we guide users away from the cryptic assembly of the sequent calculus, letting them instead write abstract certificate terms based on an FPC definition, the establishment of a mapping between these abstract terms and their correspondence with the assembly level is an inevitable step of significant complexity that needs to be repeated, with variations, for each FPC definition. Despite this, the applicability of the framework to a number of representative and highly varied settings has been studied with satisfactory results. Some inroads have been made in the application of the FPC framework to certify the output of resolution-based, automated theorem provers like the E prover [START_REF] Schulz | System description: E 1.8[END_REF][START_REF] Chihani | The proof certifier Checkers[END_REF], and more recently and comprehensively for Prover9 [START_REF] Mccune | Prover9 and mace4[END_REF]Blanco et al., 2017a). On this topic, see esp. Chapters 7 and 8, but also Chapters 11 and 13. Certification applies to automated and interactive theorem provers alikesoftware with common foundations yet very different operating principles. There exist efforts to integrate exemplars across the various categories of tools. Typically, one starts from a proof assistant and calls an automated theorem prover through an interface (called a proof hammer) to try to finish parts of the proof on behalf of the user; see, for example, [START_REF] Blanchette | Hammering towards QED[END_REF]. As integration grows, the lines separating these categories blur to the point that classification becomes unclear. For example, dependent types are commonly at the intersection between programming languages and proof assistants: for instance, Agda defines itself as both. Proof assistants may further shift towards automation by applying machine learning to the task of obtaining proof scripts starting from corpora of existing proofs [START_REF] Kaliszyk | Learning-assisted theorem proving with millions of lemmas[END_REF]. Part II Logics without fixed points 4 Logic programming in intuitionistic logic Logic and computation The mathematical study of computation encompasses a diverse range of abstract models, of which Turing machines and λ-calculi are among the best known; their notions of computable functions coincide with the concept of general recursion. Moreover, the Church-Turing thesis conjectures that every effectively calculable function is computable by a Turing machine or an assimilable model. Hence, if a concrete computer or a programming language running on a computer can simulate a Turing machine (up to finite amounts of memory), it can compute any function computable by a Turing machine: in other words, it is Turing-complete. Many models of computation have this property, and some of these models are based directly on logical principles. In fact, logic can be seen as playing two kinds of roles in computation [START_REF] Miller | Observations about using logic as a specification language[END_REF]. On the one hand, it can be used externally as a tool to reason about mathematical structures used to model programs and their behavior; that is, computation-as-model. On the other hand, logic can be used internally: logical elements (formulas, etc.) can be used as the building blocks of computation; that is, computation-as-deduction. In this latter class, two different visions giving rise to two distinct paradigms exist. First, proof normalization models the state of computation as a proof term, and the act of computing as the reduction of that proof term to a normal form; this is the foundation of functional programming. Second, proof search sees the state of computation as a collection of hypotheses and a goal to prove from those, and the act of computing as the derivation of a proof of the goal; this is the foundation of logic programming as embodied by languages such as Prolog-and our primary interest. The automation of proof search at the center of logic programming relies on the cut elimination property, which in suitable logics asserts the existence of cut-free, analytical proofs. The resulting programming style is not functional, but relational: pure logic programming, like pure functional programming, are free from side effects; logic programming generalizes functional programming by adding nondeterminism to the model of computation. The core concepts of logic programming share with functional programming core concepts like terms and types, while replacing functions with formulas, relations, and (explicit) proofs. This chapter is not meant to be a comprehensive introduction to logic programming: for this, consult among others [START_REF] Sterling | The Art of Prolog: Advanced Programming Techniques[END_REF]; O' Keefe (1990); [START_REF] Miller | Programming with Higher-Order Logic[END_REF]; our presentation is based on the latter book. For a historical perspective on the development of λProlog, see also [START_REF] Nadathur | An Overview of λProlog[END_REF]. The rest of the chapter is organized as follows. Section 4.2 introduces some essential concepts of logic programming. Section 4.3 provides a succinct tutorial introduction to the higher-order logic programming language λProlog, which is used extensively throughout the document. Section 4.4 presents a proof checking kernel for the FPC framework (in particular, the LKF a logic) as a representative application of logic programming ideas in a concrete language like λProlog. Section 4.5 concludes the chapter. Logic programming In what follows, we shall be interested in logic programming languages with support for rich types-this is in contrast with standard Prolog, which is untyped. Typed terms are interpreted in the usual sense of the Simple Theory of Types of [START_REF] Church | A formulation of the Simple Theory of Types[END_REF]. Under this view, a logic programming language must provide syntactic support to write signatures Σ that permit us to write terms and formulas over their types and type constructors, logic programs P as collections of formulas over a given signature, and goal formulas G. In addition to those three syntactic elements, the language must implement the semantics of a proof calculus, hence enabling the construction (by searching) of proofs of a goal G given a program P, both over a signature Σ. The necessity to construct proofs points towards an intuitionistic interpretation whose informal reading is as follows: 1. The proof of t succeeds regardless of signature and program. . . In implementation, a placeholder or logic variable for a concrete term (of which there may be infinitely many) will be generated for t and instantiated by solving problems of term unification. The proof of a conjunctive goal The resulting operational semantics must correspond to the declarative reading of logic in the underlying sequent. Missing from this picture is the treatment of atomic goals, which will depend on the specific logic being implemented. Let us now consider the logical framework of Horn clauses, which-in their first-order variation-are the substrate of the Prolog programming language. The following recursive definition defines formulas for goals G and for program clauses D; A denotes atomic formulas: G ::= t | A | G ∧ G | G ∨ G | ∃x.G D ::= A | G ⊃ D | D ∧ D | ∀x.D This is one of several equivalent definitions of Horn clauses. Here, quantifiers are polymorphic at the type of the bound variable x. The resulting logic is said to be first-order if quantification is only allowed at types of order 0 or 1; it is higherorder if quantification is allowed at types of arbitrary order (while excluding the type of predicates, commonly written o). Equivalently, program clauses can be organized as formulas of the following form: That is, a prefix of universally quantified variables that bind an implication with a conjunctive antecedent of atoms and an atomic consequent. The definition of Horn clauses disallows implications and universal quantifiers in goals; in consequence, the signature and the program remain unaltered during proof search (and therefore during program execution). To complete the semantics of the resulting programming language, it needs to furnish the semantics of proof search on atomic goals. This process of backchaining analyzes the program to determine if the goal is a known fact (in which case the proof is completed) or may be the consequent of some other antecedent conditions, in which case proofs for those will be sought. ∀x 1 . • • • ∀x m .A 1 ∧ • • • ∧ A n ⊃ A 0 . nat z ∀N .nat N ⊃ nat (s N ) ∀N .plus z N N ∀K .∀M .∀N .plus K M N ⊃ plus (s K ) M (s N ) Horn clauses are a powerful framework for writing logical specifications, such as the program that contains clauses defining the construction and addition of natural numbers in Figure 4.1. Nevertheless, it is possible to generalize them by carefully allowing both signatures and programs to grow during proof search. Hereditary Harrop formulas extend the definition of Horn clauses as follows: G ::= t | A | G ∧ G | G ∨ G | ∃x.G | D ⊃ G | ∀x.G D ::= A | G ⊃ D | D ∧ D | ∀x.D In the extended logic, universal quantification is allowed in goals, as is implication subject to the restriction that the antecedent be a program clause. The syntax of program clauses remains unaltered, but is now mutually recursive with the definition of goals. This richer framework is one of the cornerstones of λProlog. The second enhancement is the replacement of first-order terms with higher-order λ-terms and quantification. This support for the application of abstractions to bound variables enables a powerful form of abstract syntax-i.e., the representation of expressions not by strings, but by data structures-where constructs like names and variable bindings are reflected directly as binders in the representation meta-language. This approach is called higher-order abstract syntax, or HOAS [START_REF] Pfenning | Higher-order abstract syntax[END_REF]. In (higher-order) logic programming, λ-tree syntax [START_REF] Miller | Foundational aspects of syntax[END_REF][START_REF] Miller | Abstract syntax for variable binders: An overview[END_REF] incarnates the ideas of HOAS in a practical way; λProlog offers support for it by its use of dynamic higher-order pattern unification. These problems are especially relevant to Section 13.2; the difficulty of their application in a functional setting are discussed in Section 6.6. Logic programming languages commonly implement some impure features, such as a "cut" operator that restricts backtracking search-not to be confused with the logical rule of cut-or a negation operator which purports to succeed if a given goal fails. These extra-logical features, which have no reflection in the logic that serves as the foundation of logic programming, may facilitate some programming tasks, but do so at the cost of risking soundness if they are not applied in very restricted cases and with great care. We will have little use for these and will avoid them whenever possible, instead strongly favoring pure, declarative programs with a clear mirror image in the underlying logic. λProlog Atomic types in λProlog are defined by the keyword kind. Typically, a kind thus defined represents a type like natural numbers. However, these kinds can also define families of types parameterized by other types, such as lists of elements of a given type (predefined by λProlog) or pairs of elements of two given types. Arrow types in kinds are used for these directives: kind nat type. kind pair type -> type -> type. These kind expressions determine how type expressions may be constructed from the concrete kinds that they define. Concrete type expressions are derived by the usual mechanism of application: nat is already a complete type expression, whereas pair must be given arguments, for instance (pair nat nat) for pairs of naturals. In general, in a type expression τ 1 → • • • → τ n → τ 0 with n ≥ 0, τ 0 is called the target type and the τ i with i > 0 (if any) are called argument types. The order of a type expression O(τ 1 → τ 2 ) is defined recursively as max(O(τ 1 ) + 1, O(τ 2 )), where the base case of types expressions without type arguments has order 0. A first-order language restricts types to be of order 0 or 1. For kinds to be populated by terms, they need to be endowed with type constructors, defined by the keyword type and characterized by type expressions whose target type is the kind in question. For instance, natural numbers can be defined by zero and the successor of another natural number, and generic pairs by the two types of their elements. Polymorphism is supported by the introduction of type variables which can be unified with any concrete type. In concrete syntax, arrow notation is unambiguously overloaded in the two contexts of kind and type expressions: type z nat. type s nat -> nat. type pr A -> B -> pair A B. Based on this, we can write typed terms such as (s z) of type nat or (pr z (s z)) of type (pair nat nat). These examples illustrate application; the second operator of the λ-calculus, abstraction, is written \. For instance, (x\ (s x)) represents an abstraction which, applied to a natural number, return its successor. The name x is thus bound in the expression that follows the abstraction operator. The type of formulas in λProlog is designated by o. It is populated by the logical constants that represent the connectives of the logic, therefore taking other formulas (i.e., terms of type o) as their arguments. In addition to these logical constants, the programmer defines predicates (also called relations) by writing type constructors with o as a target type: for example, a relation that takes two natural numbers and relates them to their addition can be typed as: type plus nat -> nat -> nat -> o. That is, plus is a relation between triples of inductively defined naturals that shall be defined to hold iff the sum of the two first arguments is equal to the third. Relation symbols stand in contrast to type constructors with target types other than i, which define function symbols. The full specification for plus is shown in Figure 4.2. Generally speaking, in writing the clauses for those relations, we will ordinarily resort to the logical constants. In λProlog, these are the following: 7. pi of type (A -> o) -> o, for ∀. Quantification operates on a formula F abstracted over an arbitrary but defined type T, written pi (x:T)\F; the type can often be uniquely determined by type inference and the annotation dropped, thus writing simply pi x\F, where x\F is the argument of the connective. In proof search, a fresh eigenvariable of the appropriate type is applied to the abstracted formula. 8. sigma of type (A -> o) -> o, for ∃. In proof search, a fresh logic variable of the appropriate type is applied to the abstracted formula. Existential variables are often implicitly quantified and represented by names starting with an uppercase letter. However, the abstraction operator can bind any name in an expression: (X\ (s X)) and (x\ (s x)) are equivalent. The anonymous logic variable _ can be used to represent a bound variable which goes unused in the body of the abstraction. λProlog programs are structured into modules. A module is composed two parts. First, a signature file, which opens with the keyword sig <name>. and declares the interface for the module, namely kind and type operators and their complementary definitions. Second, a module file, started with the keyword module <name>., and which contains clauses for the declared relations and other private declarations. Both signatures and modules can depend on others of the same type by signature accumulation (accum_sig <name>.) and module accumulation (accumulate <name>.), respectively. These programming language concepts have a clear logical interpretation and interact harmoniously with the other features of λProlog. One important practical point is to ensure that each module used by a program be accumulated at exactly one point: multiple accumulations of the same module correspond to the creation of multiple copies of the contained clauses and a combinatorial explosion in the number of possible backtracking points. Like other logic programming languages, λProlog has (limited) support for I/O, as well as support for some extra-logical features like built-in arithmetic, a backtracking cut operator ! and a negation operator not used to exercise negationas-failure. All these will be used sparingly and only in situations when it is logically sound to do so. Nowadays, two major implementations of λProlog coexist: the Teyjus compiler [START_REF] Nadathur | System description: Teyjus -A compiler and abstract machine based implementation of λProlog[END_REF] and the ELPI interpreter [START_REF] Dunchev | ELPI: fast, embeddable, λProlog interpreter[END_REF]. Both will be the subject of lengthy discussion in Chapter 8. These are in addition to the declarative core of λProlog implemented at the specification level in the Abella theorem prover [START_REF] Baelde | Abella: A system for reasoning about relational specifications[END_REF]. FPC kernels Logic programming is ideally suited to the implementation of proof search, where built-in mechanisms such as backtracking search and unification coincide with the requirements of many proof systems-among those the augmented sequent calculi that define the FPC framework (Miller and Nadathur, 2012, Chapter 9). A proof checking kernel, i.e., a program that implements one such calculus, is an interesting logic program that exhibits the characteristic features of λProlog while laying the technical foundations for subsequent chapters. In this section we concentrate on the FPC proof system for classical logic, LKF a , introduced in Section 3.2 as an evolution of the sequent calculi presented throughout Chapter 2. The kernel and its public interface are presented in full in Figure 4.3. For the client, the interest resides in the standard interfaces essentially shared by all kernel implementations for a given logic. The interface is divided in three parts: 1. The definition of the object logic of polarized classical formulas. The signature defines a type of atoms atm and a type of terms i which may appear in atoms; both must be defined by the client. In addition, the logical constants are defined: constructors to inject atoms into positive or negative literals, as well as the various logical connectives. Quantifier constructors build formulas from abstractions over formulas, resorting to the native support of λProlog for binding representation of manipulation and thus avoiding its thorny implementation. In addition to the standard connectives in both polarities, a pair of delay connectives that force a polarity on an arbitrary formula are introduced for practical purposes. Delays can be defined in terms of the standard connectives and their presence is therefore inessential. A number of related predicates are charged with the construction and deconstruction of formulas from and to their components, as well as various polarity checks used by the inference rules of the sequent calculus. 2. The client signature for the FPC framework defines the kinds of certificates and indexes (as well as disjunctive choices) and declares predicates for clerks and experts corresponding to the annotations for each inference rule in Figure 3.1. In order to instantiate the kernel into a proof checker, a client must provide type constructors for certificates and indexes-together with any ancillary declarations on which those constructors depend-and must determine the semantics of those constructors by defining the behavior of Figure The LKF a kernel in λProlog (continued): implementation. clerks and experts through clauses of their predicates. In doing so, an FPC definition is integrated with the kernel. 3. The kernel signature proper defines both types of sequents, synchronous and asynchronous, expressing the proof search for a given conclusion sequent. Both corresponding async and sync relations define parameters for the certificate and the workbench; the indexed storage is maintained via hypothetical reasoning, adding facts to the logic program by filing formulas with their indexes as clauses of the storage predicate. Ordinarily, the end user is only interested in the elementary operation that takes a formula and a certificate, forms the initial entry sequent with both and performs proof search guided by the certificate term; this is what is represented by the interface relation to the kernel, lkf_entry. The kernel module implements guided proof search as a direct encoding of the proof system of Figure 3.1. Each inference rule is turned sideways and written clause with the conclusion at the head and the (conjunctive) premises as the body. As a general rule, calls to clerks and experts precede recursive calls to the proof search predicates on asynchronous and synchronous sequents, which are only performed if the corresponding clerk or expert declares the inference rule as applicable according to the certificate term. As noted above, the store rule uses the λProlog implication to extend the clauses that define the storage zone, initially empty; this relation is then queried by the decide rule. The treatment of quantifiers is also of interest: the universal quantifier relies on λProlog to generate a fresh term eigenvariable used to instantiate the formula abstraction; the continuation certificate produced by the clerk is also abstracted over a term: the new eigenvariable is applied to this abstracted certificate to obtain a "plain" certificate. Aside from these interesting techniques in very specific places, the kernel code is remarkably simple. In addition to the polarized formulas used by the kernel, represented by the kind form, it is common to define a standard classical (unpolarized) logic with a full set of connectives including, say, implication and non-atomic negation. Let us call this type of unpolarized formulas bool. An unpolarized logic is useful on the client side to write formulas as they are commonly understood; the module that defines the unpolarized logic must also furnish predicates to translate unpolarized formulas into polarized formulas-including their conversion to negation normal form. This facilities are commonly available without loss of generality, given that in common FPC definitions-such as those presented in Chapter 3-the polarization scheme is fixed and leaves no choices to the user. Typically, these polarization predicates project unpolarized atoms to a standard encoding of polarized atoms. Over time, several kernels and programming environments for those kernels have been developed with varying degrees of client support. One such family is used in works such as Chihani et al. (2016b); with minimal changes, this is the basis of the kernel presented in Figure 4.3, which is used throughout Part II. A related effort is represented by the Checkers system by [START_REF] Chihani | The proof certifier Checkers[END_REF], which offers more (scripting) support for modular definition of problems and their use of FPC definitions. In principle, the structural differences between both principal families of kernels are primordially cosmetic: both are straightforward implementations of sequent calculi-with focusing and augmentations-with some differences in their concrete syntax. (In fact, some of the experiments in coming chapters have been adapted to run in Checkers without issue by means of a shallow mapping translation.) Some versions of these systems use hosting to implement some kernels on top of others, deemed more canonical in some sense. Typically, this involves hosting classical logic in intuitionistic logic (namely, LKF a in LJF a ) by double-negation translations-as discussed in Section 2.3, but also by appealing to the LKU system of [START_REF] Liang | Focusing and polarization in linear, intuitionistic, and classical logics[END_REF]; for the connection with focusing, see Chihani et al. (2016a). A similar effort has been recently undertaken by Libal and Volpe (2016a). In reality, the coexistence of multiple kernels, including computational models that diverge significantly from the direct implementation of the sequent calculus, is possible and desirable; the only requirement is that the client interface-chiefly the clerks and experts through which FPC definitions sculpt the semantics of proofs-remains unchanged. Notes Proof checking has been implemented many times over the past decades, ranging from Automath [START_REF] De Bruijn | A survey of the project AUTOMATH[END_REF] to the Edinburgh LCF system [START_REF] Michael | Edinburgh LCF: A Mechanised Logic of Computation[END_REF] and, more recently, to Dedukti (Dedukti). Although logic programming engines have seldom been used for such purposes, they make for rather natural and direct implementations of proof checkers, as Section 4.4 has shown. Logic programming foundations make it possible to naturally perform certain tasks that might be harder to do in these other proof checking frameworks. One such central task is to do "proof reconstruction" as an integral part of proof checking: in terms of the FPC framework, the designer of a proof certificate format can leave out details of a proof from a certificate if that designer feels confident that the missing details can be reconstructed with acceptable costs. In that way, there is an easy trade-off between the size of certificates and the costs of checking those certificates. Certificate pairing Implicit and explicit versions of proof A central issue in designing a proof certificate format (i.e., an FPC definition) involves choosing the level of proof detail that is stored within a certificate. If a lot of details (e.g., complete substitution instances and complete computation traces) are recorded within certificates, simple programs can be used to check certificates. Of course, such certificates may also be large and impractical to communicate between prover and checker. On the other hand, if many details are left out, then proof checking would involve elements of proof reconstruction that can increase the time to perform proof checking-and reconstruction-as well as increase the sophistication of the proof checking mechanism. One approach to this trade-off is to invoke the Poincaré principle [START_REF] Barendregt | Autarkic computations in formal proofs[END_REF], which states that traces of computations (such as that for 2 + 2 = 4) should be left out of a proof and reconstructed by the checker. This principle requires a checker to be complex enough to contain a-possibly smallprogramming language interpreter capable of filling the gaps in a proof skeleton and thus elaborating it into a full proof. In frameworks like LFSC [START_REF] Oe | Fast and flexible proof checking for smt[END_REF] and the Dedukti checker [START_REF] Cousineau | Embedding pure type systems in the lambda-Pi-calculus modulo[END_REF]Boespflug et al., 2012;[START_REF] Assaf | Dedukti: a logical framework based on the λΠ-calculus modulo theory[END_REF], such computations are performed using deterministic functional programs. The FPC framework goes a step beyond such systems by allowing nondeterministic computation carried out by a higher-order logic programming language. As in other settings like finite state machines, nondeterministic specifications can be exponentially smaller than deterministic ones: such a possibility for shortening specifications is an interesting option to exploit in specifying proof certificates in particular. Of course, deterministic computations are instances of nondeterminis-. tic computations: similarly, FPCs can be restricted to deterministic computation when desired. Example The following example illustrates a difference between requiring all details to be present in a certificate and allowing a certificate to elide some details. A proof checker for first-order classical logic could be asked to establish that a given disjunctive collection of literals, say, L 1 ∨ • • • ∨ L n is provable. An explicit certificate of such a proof could be an unordered pair {i, j } ⊆ {1, . . . , n} such that L i and L j are complementary. A proof certificate term for this could be written as (complementary i j). If we allow nondeterminism, then the indexes i, j do not need to be provided: instead, we could simply confirm that there exist guesses for i and j such that literal L i is the complement of L j . Compactly, a certificate term may be written as, for example, some_complementary. Of course, there may be more than one such pair of guesses. The use of nondeterminism here is completely sensible since a systematic and naive procedure for attempting a proof of such a disjunction can reconstruct the missing details. The cost of this nondeterminism is, in this case, a quadratic number of guesses in the size of the number of literals. Since the sequent calculus can be used as the foundation for both logic programming and theorem proving, the nature and structure of nondeterministic choices in the search for sequent calculus proofs have received a lot of attention. For example, the original LK and LJ sequent calculus proof systems by [START_REF] Gentzen | Investigations into logical deduction[END_REF] contain so many choices that it is hard to imagine performing meaningful proof search directly in those proof systems. Instead, those original proof systems can be replaced by focused sequent calculus proof systems in order to help structure nondeterminism-this development was covered in Section 2.5. In particular, recall the common dichotomy between don't-care and don'tknow nondeterminism and how it gives rise to two different phases of focused proof construction. Don't-know nondeterminism is employed in the positive phase, where significant choices affecting the evolution of the proof-choices determined by, say, an oracle or a proof certificate-are chained together. Don'tcare nondeterminism is employed in the negative phase and it is responsible for performing determinate (i.e., functional) computation. As we shall see, this second phase provides support for the Poincaré principle. The rest of the chapter is organized as follows: Section 5.2 introduces the pairing meta-FPC. Section 5.3 illustrates how it can be used to elaborate proof certificates (introduce more details) and to distil proof certificates (remove some details). Section 5.4 presents the maximally elaborate FPC as the limit of elaboration and introduces the discussion of how certificate transformations can be used to provide trust in proof checking, which is the subject of Chapter 6. Section 5.5 outlines some experimental transformations between proof formats enabled by pairing. Section 5.6 concludes the chapter. Pairing of FPCs Because FPC definitions of proof evidence are declarative (in contrast to procedural), some powerful, formal manipulations of proof certificates are easily enabled. In this section, we demonstrate how the formal combination of two certificates-their pairing-can be used to transform proof certificates into other certificates, either more or less explicit than the first. Example Consider checking a proof certificate for a resolution refutation that does not contain the substitutions used to compute a resolvent (as in Example 3.6.1). Since the checking process computes a detailed focused sequent in the background, that process must compute all the substitution terms required by sequent calculus proofs (in the above example, the bound variable x shall be instantiated with the concrete term z ). If we could check in parallel a second certificate that allows for storing such substitution terms, then those instances could be inserted into the second, more explicit certificate. For example, suppose the existential expert is defined for two different certificate constructors: one we shall call instan (providing an explicit instantiation term for the existential quantifier) and a second one designated simply by f (which contains no relevant information). These correspond to the following two expert clauses: ∃ e ((instan t Ξ), Ξ, t ). ∃ e ((f Ξ), (f Ξ), t ). A pairing constructor of certificates, •, • , could be defined for all clerks and experts by simply invoking the same clerk and expert on the components of the pair. For example, the existential expert would be defined as: ∃ e ( Ξ 1 , Ξ 2 , Ξ 1 , Ξ 2 , t ) :-∃ e (Ξ 1 , Ξ 1 , t ) ∧ ∃ e (Ξ 2 , Ξ 2 , t ). In this way, if we pair an implicit proof certificate with the more explicit version of that certificate, we can use the underlying logic programming engine to . record into the more explicit certificate information that was discovered during the proof reconstruction of the implicit certificate. Fortunately, it is a simple matter to do just such parallel checking of two proof certificates. The full specification (using λProlog syntax) of such a process is given in Figure 5.1. In the figure, <c> is an infix constructor of type cert -> cert -> cert and <i> is an infix constructor of type index -> index -> index. This pairing operation allows for the parallel checking of two certificates: at each step, both certificates must agree to allow the proof to progress. Essentially, the pairing FPC can be seen as a meta-FPC or an FPC combinator that takes two FPC definitions and combines them into one: each clerk and expert calls the corresponding clerk and expert from each certificate in the pair. The pairing construct is composable: each half of a pairing can itself be another, nested pairing. For a pairing to be useful, the two certificates included in it must eventually be able to expand into the same underlying sequent calculus proof, but those certificates could retain different amounts of detail from each other-or different kinds of information. At each application of an inference rule, the two halves of a pairing must agree to allow proof construction to proceed (through success of their respective clerks or experts). In addition, they must agree on the positive choices that certificates relay to the kernel in order to find a proof. This concrete interface manifests directly in three pieces of information: (a) substitution terms for existential quantifiers; (b) choices for (positive) disjunctions; and (c) cut formulas. A fourth piece is necessary through indirection: while paired certificates need not agree on the notion of index-the pairing constructor <i> is used to form an index out of two indexes-the pair of indexes must be able to agree on a single formula on which to focus. Example In Example 5.2.1, two variants of a binary resolution certificate are used: one that does not include substitution information and one that does. The key difference is their treatment of the existential expert, where the more explicit certificate provides a witness term t -in the example, the single addition of z for the bound x in clause 2-and the more implicit certificate leaves a hole in the form of a logic variable to be instantiated at a later point by unification. The definition of pairing for the existential expert ensures that both certificates agree on the witness. Agreement in this case is trivial: the logic variable in the implicit certificate gets immediately instantiated with the substitution term when both are unified by the pairing expert. Thus, it is possible to transform a proof certificate type <i> index -> index -> index. type <c> cert -> cert -> cert. infix <i>, <c> 5. cutE (A <c> B) (C <c> D) (E <c> F) Cut :- cutE A C E Cut, cutE B D F Cut. allC (A <c> B) (x\ (C x) <c> (D x)) :- allC A C, allC B D. andNegC (A <c> B) (C <c> D) (E <c> F) :- andNegC A C E, andNegC B D F. andPosE (A <c> B) (C <c> D) (E <c> F) :- andPosE A C E, andPosE B D F. decideE (A <c> B) (C <c> D) (I <i> J) :- decideE A C I, decideE B D J. falseC (A <c> B) (C <c> D) :- falseC A C, falseC B D. initialE (C <c> B) (I <i> J) :- initialE C I, initialE B J. orNegC (A <c> B) (C <c> D) :- orNegC A C, orNegC B D. orPosE (A <c> B) (C <c> D) E :- orPosE A C E, orPosE B D E. releaseE (A <c> B) (C <c> D) :- releaseE A C, releaseE B D. someE (A <c> B) (C <c> D) W :- someE A C W, someE B D W. storeC (A <c> B) (C <c> D) (I <i> J) :- storeC A C I, storeC B D J. trueE (A <c> B) :- trueE A, trueE B. Figure The pairing meta-FPC. Signature and module names and accumulations of kernel signatures are omitted. Core declarations are assumed. encoding resolution that does not contain substitution terms to one that does contain substitution terms. The reverse is also possible. In a pairing construct, the more restrictive choice, or subset of choices, dominates each step of proof checking. Nonetheless, uses of pairing are not limited to zooming in and out on more implicits or explicit versions of the "same" proof certificate, as in the last example. Proof certificates can also be thought of as more general proof search tactics implementing more or less general strategies, and pairing these certificates as a modular method of tactic composition. Example Consider the FPC definition that limits proof search by decide depth only, defined in Section 3.5. This constraint can be easily combined with other, independent strategies, as these two applications show: 1. The CNF search procedure for propositional logic from Section 3.3 can generate very large proofs. To find out whether a relatively shallow and quick proof exists, cnf and dd-at the desired decide depth-can be paired and checked together. This pairing works because the clerks and experts of both FPCs are compatible: they allow the same connectives to proceed without restrictions, and the indexes used to store and decide on formulas are also compatible. 2. In combination with the semantics of the kernel, dd performs depth-first search, which can lead to much longer search times when shallow proofs avail. It is a simple matter to obtain iterative deepening by pairing dd with a pseudo-FPC that generates integers in increasing order at the root of the proof and communicates those integers to dd through a logic variable. With some support from the logic and the programming language, along with a growing library of FPC definitions, a sophisticated treatment of FPCs as a general-purpose formalism arises. This perspective is explored in Chapter 13. While the transformations between proof certificates that can take place using the pairing FPC are useful-as we argue throughout the chapter-the extent of such transformations is also limited. For example, pairing cannot be used to transform a proof certificate based on, say, conjunctive normal forms, into one based on resolution, since the former makes no use of cut and the latter contains cuts. The pairing of two such certificates will (almost) always fail to succeed. The fundamental limitation of pairing as a means of transforming proofs lays within the spectrum of "many details, fewer details" and not between two different styles of proof. This is the topic of the next section. Elaboration and distillation When it checks a certificate, a kernel is building a formal sequent calculus proof which is not explicitly stored in the certificate, but is, in a very real sense, performed by the kernel. It is the fact that such a sequent calculus proof is being built that helps to provide trust in the kernel. If a certificate lacks necessary details for building such a sequent calculus proof-for example, substitution instances as in Example 5.2.2-a kernel could attempt to reconstruct those details. (Indeed, the kernel does exactly that in the case of binary resolution where substitution instances are not part of the certificate.) The formal pairing of certificates described in the previous section connects two certificates that lead to the performance of the same sequent calculus proof. In the logic programming setting, it is completely possible to see such linking of certificates as a means to transform one certificate into another certificate. We use the term elaboration to refer to the process of transforming an implicit proof certificate into a more explicit proof certificate. The converse operation, which we call distillation, can also be performed: during such an operation, certain proof details can be discarded and a more implicit proof is produced. Since a given proof certificate can be elaborated into a number of different sequent calculus proofs, certificates can be used to provide high-level descriptions of classes of proofs: cnf and dd are examples of such classes-informed, of course, by the formulas to be proved. Other important classes are discussed in Chapter 11. Example Following Part (2) of Example 5.2.3, we can illustrate the concept of classes of proofs by the following pairing: cnf <c> (dd N) If such a combined reconstruction is possible for a given formula, pairing the proof checking of the CNF decision procedure with a more explicit form of FPC (here, decide depth) would mean that the missing proof details-namely, the decide depth of the proof, signified by the logic variable N-could be recorded. In a similar fashion, the notion of obvious logical inference by [START_REF] Davis | Obvious logical inferences[END_REF] can be described easily as an FPC: here, an inference is "obvious" if all quantifiers are instantiated at most once; thus, using a kernel to attempt to check such an FPC against a specific formula essentially implements the check of whether or not an "obvious inference" can complete the proof. Distillation, the complement of elaboration, also plays an important role in the practical manipulation of proof certificates. Consider, for example, a proof certificate that contains substitution instances for all quantifiers that appear within a proof (such as the resolution certificate with substitution information in Example 5.2.1). In some situations, such terms might be large and their occurrences within a certificate could make the size of this certificate explode-but the usefulness of this substitution information may vary. Namely, in the first-order logic setting, if a certificate stores instead linkage or mating information between literals in a proof, then the implied unification problems can be used to infer the missing substitutions-assuming that the kernel contains a trusted implementation of unification (for a discussion of related matters, see Chapter 6). The resulting certificates, where derivable substitutions are omitted, could be much smaller: checking these compressed certificates could, however, involve possibly large unification problems to be performed. The usual space-time tradeoffs apply; experimental coverage can be found in Section 8.5. Aside from applications to proof compression such as those, distilling can provide an elegant way to answer questions such as: What lemmas have been used in this proof? How deep (counting decide rules) is a proof? What substitution terms are used in a certain subproof? That is, it can be used as a framework to formulate proof queries to extract information from a proof. Certificates that retain only some coarse information, like the ones studied so far, can be used to provide some high-level insights into the structure of a given proof. Moreover, the next section provides a means of composing general queries which may involve information that is missing from a given proof certificate. Maximally explicit FPCs We can define a maximally explicit FPC that contains all the information that is explicitly needed to fill in all details in the augmented inference rules of a sequent calculus that implements the Foundational Proof Certificate framework. For the LKF a proof system, the corresponding maximally explicit FPC (sometimes referred to as maximally elaborate) is given in Figure 5.2. Such an FPC represents an exhaustive trace of a proof tree. The FPC definition is structured as follows: A top-level constructor, max, pairs a natural index with the symbolic representation of a proof tree. This wrapper is propagated to all continuation proof sub-trees and serves to assign . . kind max type. type ix nat -> index. type max nat -> max -> cert. type max0 max. type max1 max -> max. type max2 max -> max -> max. type maxa index -> max. type maxi index -> max -> max. type maxv (tm -> max) -> max. type maxt tm -> max -> max. type maxf form -> max -> max -> max. type maxc choice -> max -> max. allC (max N (maxv C )) (x\ max N (C x)). andNegC (max N (max2 A B)) (max N A) (max N B). andPosE (max N (max2 A B)) (max N A) (max N B). cutE (max N (maxf F A B)) (max N A) (max N B) F. decideE (max N (maxi I A)) (max N A) I. storeC (max N (maxi (ix N) A)) (max (s N) A) (ix N). falseC (max N (max1 A)) (max N A). orNegC (max N (max1 A)) (max N A). releaseE (max N (max1 A)) (max N A). orPosE (max N (maxc C A)) (max N A) C. someE (max N (maxt T A)) (max N A) T. trueE (max N max0). initialE (max N (maxa I)) I. increasing indexes to formulas as they are stored, so that the storage zone is (along each branch from the root of the tree) a functional mapping from naturals to formulas, making decide rules unambiguous. max is also the name of the type of symbolic proof trees, each of whose constructors represent different types of nodes, holding all information needed by the clerks and experts without recording the actual proof derivation. Each constructor represents a type of node in a proof tree: max0 is a leaf node; max1 is a simple unary node; max2 is a simple binary node; maxv is a unary node used to bind an eigenvariable to the rest of the tree; Figure maxt is a unary node annotated with a term; maxf is a binary node annotated with a cut formula; maxc is a unary node annotated with with a (disjunctive) choice; and maxi is a unary node annotated with an index. The encoding of the maximally elaborate FPC mirrors exactly the structure of the sequent calculus in Figure 3.1: thus, there is a one-to-one correspondence between proofs of LKF a and these maximally elaborate certificates, modulo indexing conventions. The operation of maximal elaboration can also be seen as injecting or recording a trace of an LKF proof in the FPC framework. Typical usage will have a certificate variable paired with another certificate Ξ driving the proof proper via the standard idiom: Ξ <c> (max 0 Max) Here, the first index is given as zero-which we take the liberty of writing in standard numeric notation-, but this choice is inconsequential: unique indexes will be asigned to each stored formula starting from this value. It is fundamental that the proof tree to be recorded (represented by the logic variable Max) be injected into the family of certificate constructors for the maximally elaborate FPC via the top-level injector max, so that the appropriate clerks and experts may be selected. In short, such a fully explicit proof certificate can be automatically obtained through elaboration of any other proof certificate and the use of the pairing of certificates. An important second use of maximally explicit FPCs is the checking of proof certificates, here really representing full proofs. In exchange for a comparatively large certificate size, checking becomes a determinate operation that can be performed very efficiently. Since all choices are stored in the certificate, proof checking becomes a purely functional computation. This has significant implications on the trust model of proof checkers as well as their implementation; these aspects are discussed at length in Chapter 6. At the beginning of the section, we proposed the definition of "a" maximally explicit FPC. Indeed, Figure 5.2 is not the only possible definition that constitutes an accurate trace of a proof tree. A more direct, slightly less compact definition (in terms of declarations) will replace the conflated node constructors of type max with one constructor of the appropriate type for each logical connective. For example, instead of max2-shared by both conjunctions-we will have: type maxAndNegC, maxAndPosE max -> max -> max. All other conflated encodings are similarly unfolded. The resulting definition is equivalent to the original, maximally elaborate FPC. The former, while more pedantic in its recording of logical connectives in the tree, does not contain more information than the compact definition because the inference rule at a node is uniquely determined by the its conclusion. Other determinate FPCs can be given, say, recording only synchronous choices while leaving the asynchronous phase implicit. For a discussion of determinacy in FPC definitions, see Chapter 6. A third use of maximally elaborate FPCs is querying through distillation, introduced in the previous section. A certificate where all proof information is explicitly available allows the extraction of arbitrary information about the proof object which said certificate represents. A general workflow may involve elaborating the source certificate into its maximal form and distilling the desired information from this intermediate representation, where said information is guaranteed to be available. Maximally explicit certificates are trivially defined for other logics with minor variations based on the types of nodes in the proof tree, each annotated with each piece of information output by clerks and experts. Experiments We have experimented with various uses of certificate pairing and we report briefly on some of those experiments here. In particular, we have used pairing in our λProlog checker in order to distill and elaborate a number of matrix-style, i.e., cut-free, proofs. A few representative instances based on our case studies are: • Propositional CNFs elaborate to matings [START_REF] Peter | Theorem proving via general matings[END_REF]. • Decision depth bounds elaborate to oracles (Chihani et al., 2016b, Section 7). • Propositional CNFs, matings, decision depth bounds and oracles elaborate to maximal certificates. All these pairings work in reverse as distillations, with the proviso that a maximally elaborate certificate distills to certificate formats which are compatible with the proof from which they were generated. Thus, the maximal elaboration of an oracle certificate (which operates on the positive phase) cannot be distilled as a mating tree (which operates on the negative phase) even if proofs in both formats exist. This illustrates the fundamental limitation of pairing. The ensemble of operations works as expected, but the formats mentioned in this section cannot easily be employed beyond small numbers of moderately sized examples, since these formats are seldom used in actual theorem provers. More extensive experiments must involve proof certificates where the cut rule is allowed, which furthermore will allow us to check the output of real software provers, both automated and interactive. For more details, see Chapters 7, 8, and 11. Notes The developments in this chapter have been presented in Blanco et al. (2017a), though earlier work, including [START_REF] Blanco | Proof outlines as proof certificates: a system description[END_REF], anticipates the interest of combining and extracting information from proofs-expressed as certificates. An application of the pairing FPC involves the reconstruction of a certain focused proof by two separate proof certificates. The definition in Figure 5.1 creates a pairing from two separate terms of the uniform certificate type cert. Contingent on the definition of the paired FPC definitions, this confusion can create ambiguity when attempting to elaborate a certificate of a certain kind, say A, to another one, say B, represented by a logic variable. To avoid mixed certificates combining parts from separate definitions, the clerks and experts in the B family must unambiguously constrain continuation certificates to their own family. A more type-based result could be achieved by defining the pairing constructor <c> as combining two disjoint certificate types: its type would be certA -> certB -> cert. The language should offer the option to alias families of certificates to either of the paired types-recent versions of Teyjus implement this experimental feature. Of course, both certA and certB could themselves be pairs of other FPCs. The matings referred to in Section 5.3 are related to expansion trees, representations of proofs studied by [START_REF] Miller | A compact representation of proofs[END_REF]; [START_REF] Chaudhuri | A multi-focused proof system isomorphic to expansion proofs[END_REF]. This paradigm has also been given expression in the FPC framework. As we noted already in Chapter 2, the advent of computer-aided verification enables proofs which by their sheer complexity are far beyond what manual calculation can reasonably achieve. In particular, automated theorem proverscommonly based on resolution calculi (for which see Section 3.6) and encoding theorems as instances of satisfiability problems (discussed at length in Chapter 7)are prone to generate enormous, low-level proof descriptions. The current largest reported proof was obtained by such methods by [START_REF] Marijn | Solving and verifying the boolean pythagorean triples problem via cube-and-conquer[END_REF] and consists of an unsatisfiability certificate almost 200 TB in size, itself not a full elaboration of a proof object. Substantial effort goes not only into producing, but into mechanically checking the correctness of such claims [START_REF] Cruz-Filipe | Efficient certified resolution proof checking[END_REF]. While the discussion in this chapter is limited to classical first-order logic, the FPC framework is applied to other frameworks, such as intuitionistic logic and logics extended with least and greatest fixed points. Certificate pairing and maximally elaborate certificates are equally applicable to all these-and will be applied in various settings in successive chapters. Trust and determinate FPCs The FPC framework was originally proposed and designed to give answer to two needs: communicating and trusting proofs [START_REF] Miller | Communicating and trusting proofs: The case for broad spectrum proof certificates[END_REF]. The separation between untrusted provers and trusted checkers plays a vital role in this endeavor-as a matter of fact, it is the subject of the very first requirement in Miller's desiderata (Definition 3.1.1). It has been convincingly argued that a (higher-order) logic programming language like λProlog is a good choice to implement the augmented focused sequent calculi that embody the FPC framework. The resulting checkers are both elegant and powerful, and encode the proof systems they implement so faithfully and tersely that it is a simple matter to sanction them as correct by construction, and thus trustworthy. The present chapter is dedicated to studying and sharpening this assertion and the very concept of trusted checkers. Firstly, we consider the implications of the general FPC architecture on its necessary trusted computing base (TCB). In a wide sense, the TCB comprises such varied elements as a hardware platform, an operating system and a compiler or interpreter. In the present day, it is not yet feasible to assemble a system where every component is formally verified; exemplars are few and fairly specific. Moreover, conformance to a specification does it itself guarantee that the specification constitutes a correct and complete description of the system: that is, verification does not entail validation. Yet even admitting all these components into the TCBa practical necessity-, we may question the choice of programming environment and its impositions on the TCB. As a matter of fact, higher-order logic programming comes with a very characteristic set of features: sophisticated operations like variable binding and substitution, higher-order unification, and backtracking search, are all available as language primitives. One may eye these complex operations with suspicion and wonder whether these must be admitted into the primeval TCB as the unavoidable price to pay, or whether the simpler computational model of pure functional programming offers a viable alternative whose implementation is easier to come to trust. At the same time, we may note that it is precisely those features of logic programming that make the encoding of kernels so straightforward and easy to trust-provided that the implementation of said features in a logic programming language are, indeed, correct. Furthermore, Miller's desiderata appear to mandate the use of logic-like features in any full implementation of the FPC framework. Section 5.4 anticipated a solution to this problem in the form of determinate proof certificates, which contain enough information about the proof they represent so as to render proof checking determinate. Such certificates could be passed as inputs to a simplified checker implemented in a functional programming language where none of the complicating signature features of logic programming are present. A priori, such a simplified checker should be even simpler to trust while implementing the determinate fragment of the FPC framework. Full generality could be restored by checking an arbitrary certificate with a general, logic programming-based checker, while using pairing to elaborate a determinate certificate, which could be then exported and re-checked against the same formula using a determinate checker. An architecture like this will be discussed in Chapter 8. Before that, the construction of such simplified checkers needs to be addressed. Secondly-and orthogonally to the aforementioned minimization of the TCB of the FPC framework-the informal correctness argument by which an implementation of a checker may be declared sound and "obviously correct" may be brought under scrutiny. Given that the integrity of the entire framework rests on the correctness of this critical piece of software, close examination is warranted. While it has been posited that proof checkers are amenable to formalization, the informal argument has been treated as satisfactory, and no verification efforts have been undertaken. A hidden source of complexity lies precisely in the qualities of logic programming languages that would need to be modeled and proven correct inside a proof assistant. Nevertheless, a determinate checker would not face these obligations and could offer an approachable starting point. Finally, we note by way of illustration that the notion of determinacy has considerable depth, beyond the proof traces that are its clearest expression. It is possible-at least in some logical settings-to leave out certain details from a proof certificate while still providing for determinate proof checking. For example, consider the variant of the maximally explicit FPC in which no substitution terms type maxv max -> max. type maxt max -> max. allC (max N (maxv C)) (max N C). someE (max N (maxt A)) (max N A) T. Certificates of this modified format will not contain any reference to eigenvariables or to substitution terms (existential witnesses). A proof checker for such certificates can, however, use so-called logic variables instead of explicit witness terms and then perform unification during the implementation of the initial rule. Since the unification of first-order terms (even in the presence of eigenvariables and their associated constraints) is determinate, such proof checking will not involve the need to perform backtracking search. The main downside for this variant of the maximally explicit certificate is that checking will involve the somewhat more complex operation of unification. Of course, such unification must deal with either Skolem functions or eigenvariables in order to address quantifier alternation-λProlog treats eigenvariables directly since it implements unification under a mixed quantifier prefix [START_REF] Miller | Unification under a mixed prefix[END_REF]. Figure The rest of the chapter is organized as follows: Section 6.2 presents the implementation of a functional checker for determinate certificates in the OCaml programming language. Section 6.3 addresses the verification of a determinate checker specialized for the maximally elaborate certificates of Section 5.4 in the Coq proof assistant. Section 6.4 considers the interface of these checkers with outside tools. Section 6.5 discusses the use of proof checkers and proof certificates as an extension of the repertoire of tactics available to proof assistants. Section 6.6 concludes the chapter. Functional checkers in OCaml As noted in Section 5.4, a sufficiently detailed certificate turns certificate checking into a behaviorally determinate process, which can be performed by a kernel programmed in a functional language without side effects. Such a checker will be simpler and potentially easier to analyze and trust. A maximally explicit certificate contains all the information needed to build a proof in its focused sequent calculus. In particular, all don't-know nondeterminism is given a definite answer in the certificate, so that unification and backtracking search are not utilized by the checker. To demonstrate this possibility, we implement a determinate proof checker as an OCaml program, called MaxChecker. It can be used with any determinate certificate definition, for which the following module implementations need to be given by the client: 1. A certificate type defining the certificate constructors. 2. An index type defining the index constructors, together with a comparison function (used by the context module to file formulas into storage). 3. An implementation of clerks and experts according to the specification of a bureau module. Each clerk and expert takes a certificate as input and returns an option type formed by continuation certificates and all necessary information: indexes, existential witness terms, etc. (As a special case, the true expert returns a boolean instead of an option type of unit.) The maximally explicit certificate of Figure 5.2 is one such definition. In addition to an FPC definition, a concrete instance of the kernel-and its concrete certificates-rely on a problem specification on which formulas are parameterized, as well as certificates, insofar as they can include formulas or some of their components. A problem signature is characterized by the following pair of client-side definitinions: 1. A term type, which could be empty in propositional formulas, where only atoms are used. The term type can be recursive, and term constructors are only expected to have term parameters. 2. An atom type, parametric on a term type. Atom constructors are only expected to have parameters of this term type. The public interface of the checker is simply an entry function that takes a certificate and a formula as arguments and returns a boolean indicating whether the certificate represents a determinate proof of the formula. This is translated into a call of the main function, which takes a certificate and a sequent and returns a boolean if and only if the certificate is able to prove the sequent; Figure 6.2 shows this function. The structure of the checker closely resembles λProlog-based kernels, with the difference that all pattern matching operations are structured in the same function instead of divided in program clauses. Each inference rule is delegated to its associated auxiliary function (clerk or expert), which is charged with calling the corresponding clerk or expert from the FPC definition used to instantiate the kernel and, if successful, performs the requisite recursive calls to the main function (with which all these helpers are mutually recursive). Figure 6.3 presents an representative selection of some of the more interesting functions. These illustrate in more detail a number of representative design constraints: 1. In the logic programming encoding, universal and existential quantifiers envelop a formula abstraction that becomes a formula after a term is applied to it: an eigenvariable for the universal quantifier and a term (fully unified or not) for the existential quantifier. In OCaml, the argument of both quantifiers is a function that takes a term argument and returns a formula; moreover, term arguments are always fully defined. 2. Terms at the kernel level can be either client-side terms, defined by the user, or eigenvariables. The latter are a separate kind of term, created fresh when the kernel encounters a universal quantifier, then passed to the formula function. In this kernel, they are never unified. Elements provided by the user (i.e., formulas and certificates) cannot contain eigenvarible terms. 3. The storage zone becomes a functional context where each index may occur once. An attempt to store a formula under an already claimed index results in an error. It is the responsibility of the certificate to ensure functionality of the indexing scheme. Finally, an example function from the bureau that implements the maximally explicit FPC definition is given in Figure 6.4; all clerk and expert functions have simple definitions in the same style. There is an important precision to make: at the end of the asynchronous phase, the decide and cut rules are in conflict: that is, the sequent does not uniquely check is defined with mutually recursive clerk and expert handlers. determine a unique inference rule that can be applied-subject to the certificate term and the clerks and experts. In general, the kernel must look at both bureau functions and ascertain that at most one allows the proof to proceed with the present certificate and, if one such function avails, use it, otherwise the proof cannot continue. If the kernel is specialized to work with a fixed FPC definition, it is possible that the two rules are never in conflict, and the specialized checker integrating such an FPC definition could be simplified accordingly. The maximally explicit FPC satisfies this property of absence of conflict. The resulting program is remarkably compact: the complete kernel is slightly over 150 lines of code, and the bureau for the maximally explicit FPC definition about 50 lines of code, both formatted for legibility, not compactness. The handful of supporting type modules referred to in the discussion (formulas, contexts, sequents, etc.) are equally succinct. The ensemble of modules and usage examples adds up to 350 lines of code. This is an auspicious starting point for formalization efforts, given that (in particular) the maximally explicit certificate contains all the information needed to build a focused sequent calculus proof, and the proof checker is a terminating program performing purely functional computation without any complex operations (like unification and search). The next section continues this line of research by performing the verification of a checker for propositional classical logic. Verified checkers in Coq The Coq proof assistant [START_REF] Bertot | Interactive Theorem Proving and Program Development. Coq'Art: The Calculus of Inductive Constructions[END_REF]) can be used to program a determinate proof checker as a purely functional, terminating program. This program can then be verified by proving the appropriate theorems on it, in particular the soundness theorem. Since we will only be utilizing one particular FPC format with this checker (i.e., the maximally explicit FPC), we shall consider the presentation of the specialized checker where the FPC definition is embedded in the kernel. This will remove the obligation to prove the soundness of the kernel for any conceivable FPC definition; this choice will remove some inessential reasoning clutter from the development. In this section, we present a formalization from first principles with an aim to uncovering the fundamental complexity of the problem, without any dependencies on libraries and complex external results. The port from OCaml is generally straightforward; two specific design choices merit discussion. To commence, consider the inductive type of polarized formulas for the propositional fragment of LKF where non-logical constants, i.e., atoms, are represented by the type of propositions in Coq, Prop. The negation of formulas and both unfocused and focused sequents are defined in the usual manner. The choice of Prop as the type of atoms is the most flexible and it is advantageous from an internal perspective: if the checker proves a formula, its depolarization trivially yields a Coq proposition which can then be considered proven-as a matter of fact, certified-and used normally inside Coq. The tradeoff is that the checker can no longer yield a simple boolean as an answer, because the initial rule involves equality between atoms (i.e., Coq propositions) which is not decidable in general. Therefore, the checker must operate modulo equality between Props and return a proposition consisting of these terminal equalities. In successful runs, those equalities will be identities, and the proposition trivially true. The second important choice is the representation of the indexed storage in both types of sequents. As we know from the OCaml implementation in Section 6.2 and, further still, from the definition of the maximally elaborate FPC in Section 5.4, unique indexes can be drawn from an incrementing counter from the root to each point in the proof tree-that is, unicity is enforced along each branch. Instead of using a partial map library, we exploit these observations by modeling the storage as a list. The index of an element is its position in the list, so that a fresh index is assigned by appending a formula at the end of the list. This representation, while inefficient, can be easily swapped-as the proofs will make clear, the few lemmas required by the main result are all easy, a fact that the simple list representation makes particularly clear. Given these considerations, we can encode the determinate checker as a simple fixed point definition, presented in Figures 6.5 and 6.6. Unlike in the OCaml checker, the maximally elaborate FPC is fixed and embedded in the checker: there are no helper functions; instead, the pattern matching of each clerk and expert and subsequent processing are all inlined. The resulting code is remarkably succinct and predictably decreasing on the size of the certificate. To begin to prove the soundness of the checker, we need to relate the polarized formulas of the FPC framework with logical connectives of propositions in Coq, both in isolation and as part of a sequent. As we follow the one-sided presentation of LKF, the latter case will correspond to the disjunction of the parts-in any case, the proofs will guarantee the correctness of this connection. tinued). Presentation conventions are shared with Figure 6.5. Definition Let • 0 be a depolarization function that maps polarized to unpolarized formulas. It assumes that the negative polarity is reserved for negated atoms. It is defined as follows: t + 0 = t -0 = t f + 0 = f -0 = f P a 0 = a N a 0 = ¬a A ∧ + B 0 = A ∧ -B 0 = A 0 ∧ B 0 A ∨ + B 0 = A ∨ -B 0 = A 0 ∨ B 0 Here, P a is the atom a positively polarized, N a is the atom a negatively polarized, and A and B are arbitrary polarized formulas. The notion of depolarization function is easily extended to focused sequents. In LKF, one-sided sequents are interpreted as classical disjunctions, and their depolarization is defined as: Γ ⇑ Θ 0 = A∈Γ A 0 ∨ B ∈Θ B 0 Γ ⇓ B 0 = A∈Γ A 0 ∨ B 0 The same scheme is valid for LKF a , where the index assigned to each formula in the storage area Γ is quietly discarded. An implementation of a checker must adapt the depolarization function on sequents to operate on what data structures are used to implement the various zones. In the present case, only lists are used. In order to prove the main result of the soundness of the checker, we make use of a small number of helper lemmas. The first property is a sort of noncontradiction applied to the depolarization function. Lemma Let B be a polarized formula and ¬B its negation. It cannot be the case that both their depolarizations, B 0 and ¬B 0 , hold. Proof. By a simple case analysis on the structure of B. The remaining auxiliary results are technical lemmas used to relate a particular implementation of the storage zone, its interpretation as a set of formulas at the . . sequent level, and the depolarization of this fraction of the sequent as a disjunction of (unpolarized) formulas. While these lemmas are formulated in terms of lists in the present treatment, they are easily swappable with homologues for the indexed data structures of choice, as the abstract properties we require are simple. Lemma Given an indexed storage zone Γ and an index i, if the index addresses a formula B, (i, B) ∈ Γ, then the depolarization of the storage Γ 0 is logically equivalent to the depolarization where the formula B is added as a disjunct: Γ 0 ∨ B 0 . Proof. In our encoding, the index lookup corresponds to indexed access. The proof proceeds a simple induction on the structure of the storage zone Γ and the logical properties of disjunction. Lemma Given an indexed storage zone Γ and an index i, if the index addresses a formula B, (i, B) ∈ Γ, and if the depolarization of the formula, B 0 , holds, then so does the depolarization of the entire zone, Γ 0 . Proof. In our encoding, the index lookup corresponds to indexed access. The proof proceeds a simple induction on the structure of the storage zone Γ. Lemma Given an indexed storage zone Γ and a formula B, if the depolarization of the storage zone augmented with the formula Γ ∪ {B } 0 holds, then so does the disjunction of the depolarization of the parts: Γ 0 ∨ B 0 . Proof. In our encoding, storage corresponds to the append operation at the end of the list modeling Γ. The proof proceeds by a simple induction on the structure of the storage zone Γ. Armed with these results, we are ready to prove the principal theorem, which establishes the connection (the implication) between the checkability of LKF a sequents and their corresponding depolarization as members of Coq's Prop type of propositions, Prop. Theorem Let Ξ be a maximally elaborate certificate and S be a sequent. If Ξ successfully certifies the sequent S via checker, then the depolarization of the sequent S 0 holds. Proof. The proof proceeds by structural induction on the certificate Ξ. Most cases follow directly from the induction hypotheses. The remaining cases, in addition, make use of the auxiliary results: 1. The case of cut makes use of Lemma 6.3.2. 2. The case of the initial rule, where the focus is on a positive atom, makes use of Lemma 6.3.3 along with the logical properties of disjunction and the excluded middle. 3. The case of the decide rule uses Lemma 6.3.4. 4. All variations of the store rule-applied to the various storable formulas-use Lemma 6.3.5. This concludes the proof. In trying to prove properties of a classical proof system (like LKF) in a constructive system (like Coq), the appeal to the axioms of classical logic is to be expected. Predictably, the axiom of the excluded middle makes a single appearance in the proof of the initial rule, where we look at an atom in both positive and negative polarities: back in Coq terms, a proposition and its negation. In practice, a checker exposes a limited interface where a certificate is used to check a single formula, from which the initial sequent (for which recall Theorem 2.5.1) is derived. This property is a specialization from the general result: Corollary Let Ξ be a maximally elaborate certificate and B be a formula. If Ξ certifies the initial sequent • ⇑ [B], then the unpolarized formula B 0 holds. Proof. Immediate from Theorem 6.3.6. In the special case of certificates as proof traces, if the proof system is complete and if every proof has a trace in certificate form, a complementary theorem could be stated that, if a formula is provable, there must exist a certificate that checks it. This property goes beyond the full scope of the FPC framework and is not considered here. One can imagine two possibilities to make practical use of a formally verified checker. First, a second standalone checker-like that in the previous sectioncould be extracted from Coq into, say, OCaml. Second, the checker could be used natively inside Coq as an alternative method for proof building. The next two sections discuss each of these possibilities in turn. Extraction of verified checkers The development of the proof checker in Coq constitutes an instance of verified programming, in which the code has been proven to satisfy its specification. To move beyond the boundaries of the proof assistant and become an independent executable program, code can be extracted to a functional programming language like OCaml (Letouzey, 2008). In terms of trust, we obtain correctness guarantees about the code by admitting additional systems into the trusted computing base. Specifically, those guarantees rely on the correctness of the proof assistant and on the procedure of code extraction, which themselves are not formally verified. Moreover, the process of extraction comes with certain restrictions that interact with the design choices made in the previous section. In the first place, the use of Coq's native Prop renders code extraction inapplicable. In order to obtain extractable code, we need to, say, move from the world of propositions to the world of booleans-as in the original OCaml checker-, thus replacing logical connectives with boolean operations throughout the development. Furthermore, propositions-as-atoms must be replaced with a general model of atoms that can be extracted, say, based on strings (as is the case in some kernels written in λProlog). Such a model easily allows an extracted checker to be adapted to general problem signatures without having to translate those signatures into Coq code (the definition of an atom type in the native OCaml checker for each instance of the kernel is a representative of this approach). Secondly, the possible interaction between any axioms of classical logic and the generation of purely functional, constructive code must be assessed. To begin with, code cannot be extracted from theorems that involve classical axioms, although in a Prop-based formalization this possibility is also precluded by the encoding of atoms. In a checker based on boolean return values where atoms are represented by a type with decidable equality, it is possible for the proofs to progress in a weaker setting. Even if some properties of classical logic are used to simplify reasoning in the final stages of the proof, extracting compilable code from the fixed point definitions instead of the theorems remains possible; this last option is employed by other verified systems such as the CompCert compiler. Besides previous considerations-given a suitable target for extraction-for the extracted program to be usable, it remains to determine how the user is to interact with it: how formulas and certificates will be input and output. Separate from the kernel there must be a parser that reads from an input stream that contains three families of items: (a) a collection of non-logical constants and their types (the signature discussed above); (b) a polarized version of a formula (the proposed theorem); and (c) a proof certificate expressed as a maximally explicit FPC. The kernel is then asked to check whether or not the given certificate yields a proof of the proposed theorem, calling to this end the verified checker function. If this check is successful, the kernel depolarizes the theorem and prints it out as a means to confirm what formula it has actually checked. As [START_REF] Pollack | How to believe a machine-checked proof[END_REF] has argued, the printer and parser of our system must be trusted to be faithfully representing the formulas that they input and output. This concern can be addressed in standard ways: by using standard parser generating tools in order to link trust in the checker with trust in a well-engineered and frequently used tool. Further refinements would come from an obvious direction, by crafting and employing verified parsers and printers. Strictly speaking, only the printer needs to be trusted, in that whatever formula was checked, and whether this differs from the input, can be ascertained by inspecting the trusted output of the checker, be this a simple "yes/no" answer or the declared theorem. FPCs by reflection in Coq The second use case for a verified checker takes place directly within the proof assistant where verification is carried out. In this environment, we can push this issue of trust another step. Since the MaxChecker is a simple terminating functional program, it is-as has been demonstrated-a simple matter to implement it within Coq. Moving on, one could formally prove that a successful check leads to a formal proof in, say, Gentzen's LK and LJ proof systems. By reflecting [START_REF] Boutin | Using reflection to build efficient and certified decision procedures[END_REF][START_REF] Harrison | Metatheory and reflection in theorem proving: A survey and critique[END_REF] these weaker proof systems into Coq-including the axiom of the excluded middle for classical logic proofs-, the chaining of a flexible (logic programming-based) certificate elaborator with the Coq-based checker can then be used to get the proof assistant to accept proofs from a range of other proof systems. A more direct avenue exploits propositions-as-atoms to produce a similar result. Given a polarized formula, we know by application of Theorem 6.3.6 that if the "output proposition" from the checker is provable, so is the depolarization of the input formula. In the case of a successful check, the output proposition is trivially provable, as it only involves conjunctions of identity equalities-where both sides of each equality are the same proposition. This usage scheme works as follows: 1. Given a (polarized) formula, secure a proof of theoremhood by an external prover, say, a resolution proof, for which an FPC definition exists. 2. Check the formula against the proof certificate and elaborate it to its maximally explicit form in a logic programming-based kernel. 3. Check the formula against the maximally explicit certificate in Coq. 4. Apply the soundness theorem to obtain the depolarization of the formula as a proved proposition to be used in a Coq development. To apply this style of proof, it is necessary to move between Coq propositions and their polarized versions, to use the checker to prove the latter and then recover the original propositions. This process of reification corresponds to that of polarization in focused sequent calculus. The subject will be taken up and treated more at length in Section 13.3. Notes The original development of MaxChecker in OCaml was first introduced in Blanco et al. (2017a) as a consequence of the development of determinate FPC definitions, in particular the maximally elaborate FPC of Section 5.4. The natural role of logic programming in the implementation of proof calculi in general and the FPC framework in particular is well established in the literature. Among others, it is discussed by [START_REF] Felty | Implementing tactics and tacticals in a higher-order logic programming language[END_REF]; [START_REF] Miller | Programming with Higher-Order Logic[END_REF]; [START_REF] Chihani | Certification of First-order proofs in classical and intuitionistic logics[END_REF]; Chihani et al. (2016b). Functional programming serves as a reasonable simplification in the implementation mechanisms required-at the level of programming languages-in the TCB of a proof checker. To go further, we must turn to the layers of software below the (possibly verified) checker: compiler, operating system, hardware. At present, an end-to-end verification of the entirety of the systems on which a program (here, the proof checker) relies is impracticable; efforts like DeepSpec (DeepSpec) aim at exactly such ubiquitous application of the techniques described in this chapter. Still, questions about security and about the correctness of verified specifications abound. Nonetheless, the production of a formal proof of the ascribed properties of the checker represents a significant increase in the level of trust we may accord to a proof checker-whose trustworthiness is, indeed, the ultimate measure of its value. In the discussion of extracting an executable functional program from a specification written in Coq, we must note that the concrete proof checker that has been the object of the chapter implements classical logic; in consequence, the associated proofs of correctness rely (in some, very limited, measure) on non-constructive reasoning, namely the axiom of the excluded middle. We may therefore be suspecious of the results of constructively extracting a program from such a development. However, subject to less stringent requirements, such an extraction-performed on the definition of the checker about which the theorems reason, and not on the theorems themselves-can be performed successfully. Indeed, classical reasoning does not by itself invalidate the extraction of a verified specification. For example, the CompCert verified compiler [START_REF] Leroy | Formal verification of a realistic compiler[END_REF] employs the excluded middle to derive some of its corollaries while its executables are extracted from the fixed point definitions. A similar situation is seen in certain specialized checkers for unsatisfiability proofs-the subject of Chapter 7-, a standard problem in classical logic (Cruz-Filipe et al., 2017a). In this chapter, we have presented a determinate checker written in OCaml side by side with a formalization of this MaxChecker in Coq which was restricted to the propositional fragment. In extending this treatment to the quantifiers, and with them to full first-order logic, the handling of bindings predictably becomes the principal point of interest. In Coq, bindings are not first-class constructs of the language and must therefore be explicitly modeled and their metatheory proved; several Coq libraries facilitate facilitate work with bindings and mitigate the increase in the complexity of proofs. Our use of Prop as the type of atoms is a further complication that needs to be addressed. A simplifying factor lifted from the OCaml checker consists of fixing a single type of terms-over which quantification may occur-mimicking the kernel in Figure 4.3 and those to come in Part III. An aspect of the OCaml code which resists easy formalization is the representation of bindings by function spaces in the encoding of higher-order abstract syntax. Adopting functions leads to so-called exotic terms and are far more general than the limited operation of substitution they are expected to represent [START_REF] Despeyroux | Higher-order abstract syntax in Coq[END_REF]. An appealing alternative is to specify the checker and prove it correct not in Coq, but in another proof assistant with rich metaprogramming support, where we can reason directly variable bindings, eigenvariables, etc., using λ-tree syntax, therefore obtaining simple proofs like those we have come to expect from our exposition of the propositional fragment. Abella, used extensively in Part III, is one such system; an introduction to it is included in Chapter 10. In any case, it should be noted that the propositional checker alone covers many common sources of proofs, including the satisfiability refutations that are the topic of the next chapter. A complementary development of reduced complexity is the adaptation of MaxChecker in both unverified and verified forms to function as a checker for intuitionistic logic; the changes required for this closely related kernel are few and predictably simple. A more ambitious undertaking may seek to obtain a verified checker for the general FPC framework, and in so doing would avoid the intermediate state of imposing determinacy-performed to increase trust in the underlying model of computation. Such a development is substantially more complex than the one undertaken in this chapter. Fundamentally, it relies on a certified implementation of logic programming and consequently of the interesting and complex problems of unification and backtracking search. Boolean satisfiability The boolean satisfiability problem, or SAT, is one of the quintessential problems of logic and computer science. Given a classical propositional (i.e., boolean) formula, the decision problem asks whether there exists a boolean interpretation that satisfies the formula-i.e., an assignment of truth values to the finite set of variables in the formula such that the formula evaluates to true. SAT is the first decision problem to be proved NP-complete [START_REF] Cook | The complexity of theorem-proving procedures[END_REF] and in the intervening half-century has been instrumental in the study and classification of the complexity of decision problems. While it might seem that theoretical intractability precludes practical application, recent advances in heuristic algorithms have heralded spectacular progress in the size and sophistication of the problems that specialized programs, called SAT solvers, can process, pushing automated theorem provers based on these techniques out of the ivory tower of academia and into industrial practice. We are mainly interested in proof evidence of the satisfiability properties of boolean expressions. A positive proof of satisfiability is simple: the existence of truth values that satisfy a propositional formula is expressed in first-order logic by binding the set of variables with existential quantifiers; a proof certificate needs only to give witnesses (true or false) for each of those variables. Such a positive certificate is easy to write and trivial to check. In contrast, a negative proof of unsatisfiability, or UNSAT, is more challenging. In proving the negation of the SAT property, the existential variables turn into universals, and the onus is to show that no combination of values given to those variables satisfy the formula-brute force being clearly impracticable. Similarly to resolution (treated in Section 3.6), unsatisfiability is of interest in theorem proving as a potential means of providing a refutation: to prove a theorem, obtain instead a refutation of its negation. Our interest will be in these latter unsatisfiability certificates and their formal checking. Example Let variables be chosen from the set x 1 , x 2 , . . .. The boolean expression x 1 ∧ ¬x 2 is satisfiable because there exist values for x 1 (true) and x 2 (false) that make the expression, and its first-order formulation ∃x 1 .∃x 2 .x 1 ∧ ¬x 2 , true. Conversely, the boolean expression x 1 ∧ ¬x 1 is unsatisfiable because the negation of its first-order formulation ∀x 1 .¬x 1 ∨ x 1 is a tautology. The question of correctness of SAT solvers is particularly apt: they are complex prover programs-engineered for efficiency and heavily optimized-which carry out critical verification tasks. In order to trust their results, it is imperative to ensure that no unsound reasoning may occur-but maintaining a proof of soundness of such a program as it evolves may be unfeasible. Fortunately, the separation of concerns between prover and checker is recognized and enforced through the definition of standard UNSAT certificate formats to provide evidence of a proof of unsatisfiability. Most modern software is based on Conflict-Driven Clause Learning, or CDCL [START_REF] João | GRASP: a search algorithm for propositional satisfiability[END_REF][START_REF] Moskewicz | Chaff: engineering an efficient SAT solver[END_REF])-a refinement of the classical DPLL search algorithm for complete satisfiability checking [START_REF] Davis | A computing procedure for quantification theory[END_REF][START_REF] Davis | A machine program for theorem-proving[END_REF])-, and support for those certificate formats is easily added to the base algorithm. Hence, instead of checking the tools, a specialized checker checks the results emitted by the tools. Those specialized proof checkers are simpler, more stable programs, more amenable to a formal proof of their correctness. In fact, a series of recent formalization efforts has produced a number of verified checkers to cement the trust in a theorem by way of an UNSAT certificate that purports to refute the negation of said theorem. Similarly to our checker in Chapter 6, these tools are built in a proof assistant like Coq or Isabelle and proven correct against their specification, and then extracted as executable code. To aid efficiency, certificate formats are commonly extended with additional information to make checking more deterministic; lacking support from the SAT solvers, an intermediate processing step is then needed to enrich the standard certificates before they are passed to the verified checker. Though the certificates change, the objective is always the same: verifying that the input formula is unsatisfiable. Instead of building a checker and laboriously proving it correct-(i.e., verifying that it is sound and "complete enough" for the domain of interest)-, the FPC framework provides a direct and foundational attack on the problem. Now, by first understanding how a proof of the unsatisfiability of a formula is built from the information in an UNSAT certificate, we will craft FPC definitions to carve such proofs directly into the logic. An obvious advantage will be that, although (as 4 8 1 2 -3 0 -1 -2 3 0 2 3 -4 0 -2 -3 4 0 1 3 4 0 -1 -3 -4 0 -1 2 4 0 1 -2 -4 It comprises 4 variables, say x 1 , x 2 , x 3 , x 4 , and 8 clauses. The represented formula is: p cnf (x 1 ∨ x 2 ∨ ¬x 3 ) ∧ (¬x 1 ∨ ¬x 2 ∨ x 3 ) ∧ • • • ∧ (x 1 ∨ ¬x 2 ∨ ¬x 4 ). with any other checker) we may want to prove a metatheorem about the relative completeness of the procedure, the framework guarantees its soundness, and in consequence, the concrete checker-run on top of a trusted kernel-cannot ever declare the theoremhood of an impostor formula. The rest of the chapter is organized as follows: Section 7.2 introduces the standard formats and the properties on which UNSAT certificates are based. Section 7.3 studies the connection between resolution proofs and a primitive form of UNSAT certificate based on resolution traces. Section 7.4 undertakes the certification in the FPC framework of the current family of UNSAT proof evidence. Section 7.5 continues the developments of the previous section while relaxing its reliance in the cut rule. Section 7.6 concludes the chapter. Redundancy properties and shallow certificates A SAT (or UNSAT) problem consists of a boolean expression, generally written in DIMACS CNF format. A text file represents a boolean expression in clausal normal form as a series of lines. The first line includes two numbers: the number of variables v present in the formula (represented implicitly by the numbers 1, 2, . . . , v, and the number of clauses c of the formula in CNF form. The first line is followed by c lines, each representing a clause. A clause is given as a subset of literals drawn from the variable set 1, 2, . . . , v (unsigned if they are positive, prefixed by a minus sign if they are negated) and terminated by 0. An example is given in Figure 7.1. The key insight of the DPLL backtracking algorithm is its introduction of simplifications based on the properties of boolean variables while preserving the completeness of backtracking search. In particular if a clause contains exactly one literal, its satisfaction forces a fixed truth value, which can be used to simplify other occurrences of the involved variable. Formally: Definition A unit clause is a clause with a single literal. Unit propagation (also called boolean constraint propagation, or BCP) is a simplification procedure for formulas in conjunctive normal form which, given a literal clause l , performs the following operations: 1. It removes every clause containing the literal l except the unit clause. 2. It removes every negated instance of the literal, ¬l , across all clauses. The removal of negated instances may cascade and lead to the formation of new unit clauses; the process is repeated to saturation. Unit propagation derives a conflict if it results in a pair of complementary unit clauses l and ¬l . This operation is also related to the inference rule of unit resolution, which simplifies the general rule of binary resolution (discussed in Section 3.6) by imposing that one of the resolvents be a unit clause. Current UNSAT certificate formats are designed around the use of redundant clauses, that is, clauses which added to a formula in CNF preserve properties like unsatisfiability. If we want to prove that a CNF formula is unsatisfiable, a simple procedure involves adding clauses that preserve unsatisfiability and hopefully assist in adding further redundant clauses until arriving at the empty clause-at which point the formula can be declared trivially unsatisfiable. For example, accumulating a tautological formula to a conjunction of clauses is always redundant, though also uninformative. Let us consider the following, more interesting property: Definition A clause C = l 1 ∨ • • • ∨ l n is a reverse unit propagation (RUP) clause w. Example Starting from the problem in Figure 7.1, take the first lemma in Figure 7.2, negate it and attempt to derive a conflict by unit propagation on the resulting formula: 1 2 -3 | 2 -3 | -3 | -3 -1 -2 3 | | | 2 3 -4 | 2 3 -4 | 3 -4 | -4 -2 -3 4 | -2 -3 4 | | 1 3 4 | 3 4 | 3 4 | 4 -1 -3 -4 | | | -1 2 4 | | | 1 -2 -4 | -2 -4 | | -1 | -1 | -1 | -1 -2 | -2 | -2 | -2 First, the unit clauses -1 and -2 are propagated. This generates a new unit clause, -3, whose propagation derives a conflict between 4 and -4. Therefore, the lemma is a RUP clause that can be added while preserving logical equivalence. The process is repeated with each subsequent lemma. Once all lemmas have been added, only the empty clause remains. To finish the proof, unit propagation on the set of original clauses and added lemmas must derive a conflict. . 1 2 -3 | | | -1 -2 3 | -2 3 | 3 | 3 2 3 -4 | 2 3 -4 | | -2 -3 4 | -2 -3 4 | -3 4 | 4 1 3 4 | | | -1 -3 -4 | -3 -4 | -3 -4 | -4 -1 2 4 | 2 4 | | 1 -2 -4 | | | 1 2 | | | 1 | 1 | 1 | 1 2 | 2 | 2 | 2 Thus, the original formula is unsatisfiable. The RUP format is reasonably compact and easy to implement (both its production and its checking), but rather costly to verify. The main contributor to the inefficiency of checking is the accumulation and persistence of lemmas, even as they become unnecessary by the addition of newer lemmas. In fact, a defining characteristic of the more general CDCL algorithm that lies at the foundations of modern SAT solvers is its ability to both add and remove clauses by conflict analysis techniques. Clause elimination techniques, like their additive counterparts, can delete clauses whose removal preserves properties like satisfiability. The solver must provide this information in the certificate, which motivates a simple extension to the RUP format. DRUP certificates add the option to mark extant clauses (or lemmas) as deleted by prefixing them with a d marker; they are otherwise identical to RUP certificates. Figure 7.3 shows an example certificate. Intuitively, when we attempt to derive a conflict by unit propagation, it is irrelevant that we ignore a subset of deleted lemmas, as long as we can find the conflict with less information. Clause deletion alone results in significant performance improvements, but progress does not stop there. More sophisticated redundancy properties are powerful enough to express in terms of them all the processing techniques employed by standard SAT solvers. Namely, the following property: 123 2. C contains a literal l such that all the clauses that result from resolving C with a clause C i of F on l (i.e., C i contains the literal ¬l ) have the AT property w.r.t. F . Definition A clause C has the resolution asymmetric tautology property (RAT) w.r.t. a CNF formula F = C 1 ∧ • • • ∧ C n if either: 1 2 0 d Addition of RAT clauses to a CNF formula preserves satisfiability (and therefore its complement, unsatisfiability). It is the strongest redundancy property that preserves SAT (and UNSAT). It is clear that RAT generalizes AT (i.e., RUP). We consider directly the certificate format that combines the addition of RAT lemmas with clause deletion: the result is called DRAT. Syntactically, it coincides with DRUP. In order to simplify operation, each lemma to be added either has the RUP property or it has the RAT property on its first literal. An example is given in Figure 7.4. Example Again starting from the problem in Figure 7.1, we take the first lemma in the certificate, this time from Figure 7.4. By applying unit propagation, we note that its negation does not derive a conflict. Thus, it does not have the RUP property, so we select the first (here, only) literal of the lemma and generate a new set of clauses by resolving it with all applicable current clauses: 1 2 -3 | 2 -3 -1 -2 3 | 2 3 -4 | -2 -3 4 | 1 3 4 | 3 4 -1 -3 -4 | -1 2 4 | 1 -2 -4 | -2 -4 Each of the three resulting clauses can then be checked for the RUP property with respect to the current clause set. The checks succeed, and so the original lemma has the RAT property and can be added to the set of clauses. The addition and deletion of lemmas continues until the end, when only the empty clause remains and unit propagation should derive a conflict without assistance of any more lemmas. DRAT is the current de facto standard for UNSAT certificates, partly owing to its adequate expressive power: the standard toolbox of processing techniques at the disposal of modern SAT solvers can be formulated in terms of sequences of RAT lemmas-although not all techniques can be given short translations; for an overview and recent developments see [START_REF] Järvisalo | Inprocessing rules[END_REF]; Heule et al. (2015). We say that all these formats are "shallow" because they do not incorporate a definition of their semantics in the sense that an FPC definition does. Our objective will be to interpret these formats as proper foundational certificates and furnish the missing proof theoretical pieces. Resolution FPCs and traces We commence with a further exploration of the connections between resolution refutations and UNSAT certificates, which are another form of refuting a formula, and consequently proving the theoremhood of its negation. Both families of proofs operate on the same principle: starting from a formula in clausal form, prove and accumulate a series of lemmas (themselves clauses) until arriving at an extended collection of clauses from which unsatisfiability is immediate. It is only in the properties and proofs of those lemmas that the two methods differ. First, let us recall the binary resolution FPC of Section 3.6. The heart of a general proof by binary resolution, as presented by Chihani et al. (2016b), is a backbone of cuts. Each cut corresponds to an application of the resolution rule, which adds a new, derived clause on one cut branch and proves that the clause indeed follows by binary resolution on the other cut branch; these auxiliary proofs are simple. It keeps adding derived clauses until it can derive the empty clause, thus finishing the proof. The following general proof pattern is inspired by this style of reasoning. Definition Let F = C 1 ∧ • • • ∧ C m a CNF formula, and L 1 , . . . , L n a sequence of lemmas, each of them a clause, that lead to a refutation of F by a certain rule of inference (e.g., binary resolution, RUP, etc.). If F is unsatisfiable, then the DNF formula ¬F = ¬C 1 ∨ • • • ∨ ¬C m , where negated clauses are conjuncts, is a tautology. If lemmas preserve satisfiability when added to F , so do their conjunctive negations when added to ¬F . Given a procedure, Λ, to build proofs of redundancy of a new (negated) lemma with respect to a set of assumptions (clauses and prior lemmas, all negated), an LKF proof by lemma backbone proceeds by: 1. Storing the negated clauses as initial assumptions. 2. Cutting each negated lemma into the proof, building a derivation Λ that it follows from the current set of assumptions (and adding it as an assumption in the opposite branch). 3. The final lemma is the empty clause (negated, true). The final proof, Λ 0 , shows the tautology of the complete set of negated clauses and lemmas by its redundancy property, while the opposite branch is closed by true. The proof schema is shown in Figure 7.5. Disjunctions are polarized negatively and conjunctions are polarized positively. To ensure all negated clauses are positive, including unit clauses, they can be paired with the conjunctive unit, true. The resolution certificate in Figure 3.5 is an example of this pattern-and, as we begin to corroborate in the next section, the same organization applies to redundancy properties like RUP. Before doing that, in complement to proofs which explicitly exploit redundancy properties, we further our study of resolution by adapting one of its main variants: Definition Let A B designate a clause that results from resolving clauses A and B. Hyperresolution is a generalization of the binary resolution rule that takes a sequence of clauses C 1 , C 2 , . . . , C n (n ≥ 2) and yields a clause that results 3 0 0 2 -1 -2 3 0 0 3 2 3 -4 0 0 4 -2 -3 4 0 0 5 1 3 4 0 0 6 -1 -3 -4 0 0 7 -1 2 4 0 0 8 from the application of a left fold on the list with the binary resolution relation: . Λ 1 . . . . ¬C 1 , . . . , ¬C m ⇑ L 1 Λ n . . . . ¬C 1 , . . . , ¬C m , ¬L 1 , . . . , ¬L n-1 ⇑ L n Λ 0 . . . . ¬C 1 , . . . , ¬C m , ¬L 1 , . . . , ¬L n ⇑ ¬C 1 , . . . , ¬C m , ¬L 1 , . . . , ¬L n ⇑ f - . . . , t + ⇓ t + . . . , t + ⇑ decide . . . ⇑ t + store ¬C 1 , . . . , ¬C m , ¬L 1 , . . . , ¬L n ⇑ cut ¬C 1 , . . . , ¬C m , ¬L 1 , . . . , ¬L n-1 ⇑ ¬L n store ¬C 1 , . . . , ¬C m , ¬L 1 , . . . L n-1 ⇑ cut . . . . ¬C 1 , . . . , ¬C m , ¬L 1 ⇑ ¬C 1 , . . . , ¬C m ⇑ ¬L 1 store ¬C 1 , . . . , ¬C m ⇑ cut . . . . ¬C 1 ⇑ ¬C 2 ∨ - • • • ∨ - ¬C m ⇑ ¬C 1 , ¬C 2 ∨ - • • • ∨ - ¬C m store ⇑ ¬C 1 ∨ - ¬C 2 ∨ - • • • ∨ - (• • • (C 1 C 2 ) • • • ) C n . Hyperresolution avoids the creation of intermediate clauses and leads to more compact proofs. SAT solvers are easily adapted to emit proof certificates based on conflict analysis and redundancy criteria like RUP, but they cannot be so easily modified to produce proofs by resolution. Resolution certificates contain large amounts of proof evidence which also makes them much larger, but faster to check. It is important to note that a sequence of resolution steps may result in zero, one, or several possible solutions. In the context of a correct certificate, such a sequence will yield at least one solution, but backtracking is needed to ensure the one needed for the proof is eventually inspected. Hyperresolution derivations are used in a legacy UNSAT certificate format, TraceCheck. A certificate file contains two kinds of lines: original clauses and derived lemmas-as in any proof by resolution. Each line starts with a unique number which names each clause. First, the clauses of the original form are givenalways following DIMACS conventions-terminated by two zeroes. They are followed by a sequence of lemmas, terminated with a zero, followed by a sequence of existing clause identifiers which derive the clause by hyperresolution, finished with the second zero. As always, the empty clause concludes the proof evidence. An example is shown in Figure 7.6. Clause 9 states that the lemma x 1 ∨ x 2 follows by resolving clauses 1 and 3 ( x 1 ∨ x 2 ∨ ¬x 3 x 2 ∨ x 3 ∨ ¬x 4 = x 1 ∨ x 2 ∨ ¬x 4 ); then resolving the result with clause 5 ( x 1 ∨ x 2 ∨ ¬x 4 x 1 ∨ x 3 ∨ x 4 = x 1 ∨ x 2 ∨ x 3 ); and, finally, resolving with 1 again ( x 1 ∨ x 2 ∨ x 3 x 1 ∨ x 2 ∨ ¬x 3 = x 1 ∨ x 2 ) . In all these cases, there is only one literal on which to resolve; literals repeated in both resolvents are not duplicated. The rest of the lemmas proceed similarly. To define an FPC for hyperresolution proofs (for example, expressed in a transcription of the TraceCheck format), we shall assume use of the lemma backbone pattern. It will then suffice to exhibit a proof procedure, Λ, that builds a proof of a lemma, ¬C 1 , . . . , ¬C k ⇑ C k+1 by making use of the hyperresolution information associated to the lemma in the certificate. Theorem Let ¬C 1 , . . . , ¬C k be a set of negated clauses. If a negated lemma L = ¬l 1 ∧ • • • ∧ ¬l n follows by a hyperresolution sequence [i 1 , . . . , i m ] on the negated clauses, Figure 7.7 builds an LKF a proof that the lemma ¬L follows from the assumed set of clauses. Proof. The proof evidence (i.e., the certificate) is the hyperresolution sequence. Assume for brevity that each negated clause ¬C i is stored under a matching numeric index i; the mapping is not shown in the figure. Initially, the lemma L is seen as a disjunction of literals, all of which are stored. All literals are stored under a unique index for the literals of the lemma; also for brevity, the figure does not show indexes explicitly. Once all literals are stored, the proof creates a backbone of m -1 cuts unrolling the sequence of applications of binary resolution steps in the hyperresolution sequence. The two first indexes in the sequence are used to select the clauses indexed by them; the cut formula is the result of resolving both clauses. For the jth cut, the formula ¬C k+j is equal to ¬C k ¬C j+1 (except ¬C k+1 = ¬C i 1 ¬C i 2 ). Therefore, ¬C k+j = (• • • (¬C i 1 ¬C i 2 • • • ) ¬C i j +1 . Then: 1. In the positive branch, we have to prove that the derived clause C k+j follows from binary resolution. This is exactly what the FPC definition in Section 3.6 does. The task is delegated to one such certificate, abbreviated Ξ in the figure. . . 2. In the negative branch, the new negated clause ¬C k+j is stored and the construction of the backbone continues. At the end of the backbone, when the last formula has been stored, we have precisely the result of the hyperresolution sequence, namely the expansion: ¬C k+m-1 = (• • • (¬C i 1 ¬C i 2 • • • ) ¬C i m . If the hyperresolution sequence yields the negated clause that was expected, this is ¬l 1 ∧ • • • ∧ ¬l n . Since the complementary literals are all stored from the beginning of the proof, focusing on the C k+m-1 yields n literal branches, each of which can be closed, and with it the proof. That is, the full proof of a TraceCheck-style UNSAT certificate proceeds by a nested lemma backbone whose main procedure for the proofs of lemmas, Γ, is that given by Theorem 7.3.4. This, in turn, builds a binary resolution proof from the information contained in a hyperresolution sequence, and delegates the proofs of lemmas to the procedure given by the binary resolution FPC. The only difference is that Theorem 7.3.4 does not derive the empty clause, but the context of literals of the lemma derived by hyperresolution. The FPC definition follows directly from this result and the structure in Figure 3.5 through the insertion of the nested backbone, which mimics the structure of the main backbone. The salient feature of the hyperresolution backbone is that the cut formulas are not contained in the lean certificate, but rather they are extracted from the context of the sequent, namely the storage zone-compare this with Example 3.6.1, where all derived clauses are explicitly provided. There are two technical solutions to this necessity: 1. Allow the cut expert to inspect the storage zone Γ to assist in the programmatic composition of a cut formula B. 2. Add information in the certificate (and the clerks and experts) to replicate the relevant sections of the storage zone in the certificate, so as to derive the cut formulas without access to the state of the kernel. In standard kernel designs the state is not only unwriteable, but also unreadable by clerks and experts. However, there is an argument to be made that experts can occasionally be more "expertly" if they can read parts of the state; this is especially true of cut. The second possibility of a certificate reflecting parts of the kernel in lockstep is also a recurring pattern that is observed in Section 11.6 and Section 12.4, where it will be discussed at length. Study of resolution will resume in Chapter 8. For the rest of this chapter, we concentrate on redundancy-based UNSAT certificates. . Λ 1 . . . . Ξ l 1 , . . . , l n , ¬C 1 , . . . , ¬C k ⇑ C k+1 Λ m-1 . . . . Ξ l 1 , . . . , l n , ¬C 1 , . . . , ¬C k+m-2 ⇑ C k+m-1 [] l 1 , . . . , l n , . . . ⇓ ¬l 1 • • • [] l 1 , . . . , l n , . . . ⇓ ¬l n [] l 1 , . . . , l n , ¬C 1 , . . . , ¬C k+m-1 ⇓ ¬C k+m-1 ∧ + [] l 1 , . . . , l n , ¬C 1 , . . . , ¬C k+m-1 ⇑ decide [] l 1 , . . . , l n , ¬C 1 , . . . , ¬C k+m-2 ⇑ ¬C k+m-1 store [k + m - 2, i m ] l 1 , . . . , l n , ¬C 1 , . . . , ¬C k+m-2 ⇑ cut . . . . [k + 1, i 3 , . . . , i m ] l 1 , . . . , l n , ¬C 1 , . . . , ¬C k+1 ⇑ [k + 1, i 3 , . . . , i m ] l 1 , . . . , l n , ¬C 1 , . . . , ¬C k ⇑ ¬C k+1 store [i 1 , . . . , i m ] l 1 , . . . , l n , ¬C 1 , . . . , ¬C k ⇑ cut . . . . [i 1 , . . . , i m ] l 1 , ¬C 1 , . . . , ¬C k ⇑ l 2 ∨ - • • • ∨ - l n [i 1 , . . . , i m ] ¬C 1 , . . . , ¬C k ⇑ l 1 , l 2 ∨ - • • • ∨ - l n store [i 1 , . . . , i m ] ¬C 1 , . . . , ¬C k ⇑ l 1 ∨ - • • • ∨ - Unsatisfiability FPCs with cut In this section we adapt the lemma backbone pattern to the families of UNSAT certificates that are actually used in practice. As in the previous section for hyperresolution proofs and the TraceCheck format, the sole metatheoretic obligation is to prove the redundancy of a lemma with respect to the set of clauses and previously added lemmas (both negated). We start by considering how to obtain LKF a proofs of RUP certificates. Theorem Let ¬C 1 , . . . , ¬C k be a set of negated clauses. Given a negated clausal lemma L = ¬l 1 ∧ • • • ∧ ¬l n , if the lemma has the RUP property w.r.t. the formula formed by the set of clauses, then there is a simple LKF a proof of the lemma. Proof. The base context in a RUP proof step (after the literals of the lemma have been stored) is the union of the set of previous clauses and lemmas (both negated), ¬C 1 , . . . , ¬C k , and the literals of the RUP lemma, l 1 , . . . , l n . Those literals are essentially unit clauses which will serve as sources of RUP reductions, i.e., unit propagations. These operations will be modeled as a backbone of cuts. The proof proceeds by levels. Initially, at level 0, all the stored formulas are active: clauses can be addressed by number; literals can be addressed by a common index. Both index types are qualified by level. Each level consists of an application of unit propagation on an active literal. To progress from level i to level i + 1, a cut is used. To obtain the cut formula, select an arbitrary i-level literal (i.e., currently active), say l i . Assume its negation ¬l i holds, and for each i-level negated clause, ¬C i j : 1. If the negated clause contains ¬l i , its true occurrence is removed at the next level: ¬C i+1 j = ¬C i j -{¬l i }. 2. If the negated clause contains l i , it is removed-there is no ¬C i+1 j . 3. All other clauses are copied unchanged at level i + 1: ¬C i+1 j = ¬C i j . The cut formula is the disjunction of all these clauses. Then: 1. In the positive branch, the revised negated clauses are stored independently reflecting their new level. All unit clauses at level i except l i are stored at . level i + 1 together with any new unit clauses contained in the cut formula. Having stored all (negated) clauses, we proceed with the next cut in the backbone on a new unit clause. 2. In the negative side, the negated cut formula becomes a (positive) CNF formula, ¬B = j C i+1 j . It will be stored and immediately focused upon. In the positive phase, as many branches as there are clauses in ¬B appear. For each branch, there is a release on a C i+1 j = m 1 ∨ • • • ∨ m k , and in the negative phase its literals are stored. Immediately, we focus on the i-level predecessor ¬C i j , which contains at least the complementary literals ¬m 1 ∧ • • • ∧ ¬m k . The positive phase results, again, in a number of branches on literals whose complement is in storage, and which are therefore easily closed. If the assumed literal for unit propagation, ¬l i , appears as well, l i is available in storage to close the branch, as well. The backbone ends when all unit clauses have been processed-and with it unit propagation. If the lemma did satisfy the RUP property, unit propagation must result in a pair of complementary literals, which will be available and can be used to finish the proof. As the case of TraceCheck, the full proof of the RUP UNSAT certificate is an exemplar of the nested backbone pattern. Also as with TraceCheck, lemma derivations rely on a smart choice of formulas based on the context by the cut expert. In addition, indexing is used heavily to keep track of the provenance of formulas in storage-which never deletes data. Besides the sequence of RUP lemmas, the certificate only needs to keep track of the current level in each subproof. Example Consider the problem in Figure 7.1. The claim is that the following formula is unsatisfiable. For compactness, variables x i are represented by their numeric identifiers, as in DIMACS format, and negated variables are barred: (1∨2∨ 3)∧( 1∨ 2∨3)∧(2∨3∨ 4)∧( 2∨ 3∨4)∧(1∨3∨4)∧( 1∨ 3∨ 4)∧( 1∨2∨4)∧(1∨ 2∨ 4) Equivalently, the following formula is a tautology: ( 1∧ 2∧3)∨(1∧2∧ 3)∨( 2∧ 3∧4)∨(2∧3∧ 4)∨( 1∧ 3∧ 4)∨(1∧3∧4)∨(1∧ 2∧ 4)∨( 1∧2∧4) . . We want to prove that (1 ∨ 2) is a RUP lemma. If that is the case, adding the negated disjunct ( 1 ∧ 2) to the presumed theorem preserves its tautology. The interesting part of the proof is the side of the cut that shows that the lemma follows from the assumptions. The negated lemma is precisely the original RUP clause, (1 ∨ 2). Once everything is stored, the goal looks like this at level 0: 1 ∧ 2 ∧ 3, 1 ∧ 2 ∧ 3, 2 ∧ 3 ∧ 4, 2 ∧ 3 ∧ 4, 1 ∧ 3 ∧ 4, 1 ∧ 3 ∧ 4, 1 ∧ 2 ∧ 4, 1 ∧ 2 ∧ 4, 1, 2 Again, for succinctness, indexes and sequent notation-including polaritiesare abbreviated. We observe that there are unit clauses to propagate, so we pick one, say 2, and assume its negation 2 holds. We use this as the basis for unit propagation and its effect on the currently considered set of stored (negated) clauses to define the cut formula: ( 1 ∧ 3) ∨ ( 3 ∧ 4) ∨ ( 1 ∧ 3 ∧ 4) ∨ (1 ∧ 3 ∧ 4) ∨ (1 ∧ 4) Let us first consider what happens on the negative branch, where the negation of the cut formula will be stored and immediately focused upon: (1 ∨ 3) ∧ (3 ∨ 4) ∧ (1 ∨ 3 ∨ 4) ∧ ( 1 ∨ 3 ∨ 4) ∧ ( 1 ∨ 4) Synchronous treatment results in as many branches are there are clauses. Let us consider the first of these, (1 ∨ 3). As a negative formula, it will be released and its literals 1, 3 stored in the context, signaling the end of the negative phase. Noting that the clause comes from the negation of a negated clause at the current level, namely 1 ∧ 2 ∧ 3, we decide on the origin clause. Observe that for every literal in the formula we just stored its complement-and if the clause contains 2, that is the unit clause we are using for unit propagation. Hence, all branches can be closed. This extends to all clauses in the negative branch of cut. In the positive branch of cut, we store the revised set of negated clauses and mark the used unit clause, so that the effective contents at level 1 are: 1 ∧ 3, 3 ∧ 4, 1 ∧ 3 ∧ 4, 1 ∧ 3 ∧ 4, 1 ∧ 4, 1 We now perform unit propagation on the still untreated unit clause 1, assuming 1 and proceeding in the same manner. The new cut lemma, once stored-which corresponds with the level 2 state, is the following. We obviate the negative side . of cut as immediate by the same procedure exhibited above. 3, 3 ∧ 4, ∧ 3 ∧ 4 Unit propagation generates a new unit clause, 3. If the same process is applied to obtain level 3, we end up with a pair of unit clauses, 4 and 4, from which the positive backbone of lemmas ends. The scheme just presented is easily extended to support DRUP certificates simply by adding deletion instructions to the certificate; these will update levels normally, and will additionally remove from considerations those clauses marked for deletion. Cut-free unsatisfiability FPCs In Section 7.3 and Section 7.4, we have constructed LKF a proofs based on a double backbone pattern: a primary spine of cuts adding a series of lemmas, and in the proof of each of these lemmas a secondary spine of cuts attending to hyperresolution or unit propagation criteria, respectively. Now we turn to a more general proof scheme that for proving lemmas based on redundancy criteria, like RUP certificates. The overall proof still follows the lemma backbone pattern, of course, but the derivations of the lemmas will not rely on cuts as it did in previous proofs. Theorem Let ¬C 1 , . . . , ¬C k be a set of negated clauses. Given a negated clausal lemma L = ¬l 1 ∧ • • • ∧ ¬l n , if the lemma is redundant w.r.t. the formula formed by the set of clauses, then there is a cut-free LKF a proof of the lemma. Proof. The base context in a redundancy proof step (after the literals of the lemma have been stored) is the union of the set of previous clauses and lemmas (both negated), ¬C 1 , . . . , ¬C k , and the literals of the current lemma, l 1 , . . . , l n . Those literals are essentially unit clauses. The proof proceeds as a tree of decides on negated clauses from the set ¬C 1 , . . . , ¬C k . A negated clause is inactive if it contains a positive literal whose negation is not among the l 1 , . . . , l n : a focused proof on a positive literal can only end by an init rule, for which the negated literal needs to be in the storage area. Clauses all of whose positive literals have negative counterparts in storage are active. The decide rule may focus on any active clause, say, ¬C i = m 1 ∧ + • • • ∧ + m k . Processing the conjunctions results in k branches, as many as literals in ¬C i . By definition of active clause, all branches on positive literals are closed by init on their negative complements. Branches focused on negative literals m j proceed by releasing the focus and storing the negative literal, thus potentially growing the set of available negative literals-and with them the set of active (negated) clauses. For each branch, the proof can continue by selecting a locally active, still unused (negated) clause until all branches are closed, in which case the proof succeeds. In this proof scheme, as opposed to previous ones, the expert in charge of orchestrating the proof is the decide expert. Like in previous proof schemes, it benefits greatly from having read access to the storage area-or otherwise replicating the necessary information as part of the certificate term, at the cost of additional complexity. Example Consider once again the problem defined by Figure 7.1. The DRAT certificate in Figure 7.4 asserts that 1 is a RAT lemma. Indeed we saw in Example 7.2.5 that the clause has the RAT but not the RUP property. On the negative side of the main cut, we will have the following goal, where presentation conventions will be shared with Example 7.4.2 throughout the example: 1 ∧ 2 ∧ 3, 1 ∧ 2 ∧ 3, 2 ∧ 3 ∧ 4, 2 ∧ 3 ∧ 4, 1 ∧ 3 ∧ 4, 1 ∧ 3 ∧ 4, 1 ∧ 2 ∧ 4, 1 ∧ 2 ∧ 4, 1 Now, instead of cutting on a formula, we decide on one of the clauses to operate on. The clause has to be such that all positive literals can be closed with currently available negative literals; otherwise, it is impossible to continue the proof. Here, from the stock of 1, we have two possible choices: ( 1 ∧ 3 ∧ 4) and (1 ∧ 2 ∧ 4). Say we focus on the latter. This yields three branches, one for each literal. 1 is closed immediately and two remain. One of them (after release and store) is: 1 ∧ 2 ∧ 3, 1 ∧ 2 ∧ 3, 2 ∧ 3 ∧ 4, 2 ∧ 3 ∧ 4, 1 ∧ 3 ∧ 4, 1 ∧ 3 ∧ 4, 1 ∧ 2 ∧ 4, 1 ∧ 2 ∧ 4, 1, 2 Now we have a larger stock to close positive literals. If we next decide on (1 ∧ 2 ∧ 3), two of the branches are immediately closed, and the single remaining branch adds 3 to the set of negative literals: 1, 2, 3. For the next step, we can decide on (2 ∧ 3 ∧ 4), which like its predecessor results in a single continuation branch, to which the negative literal 4 is added. Finally, by deciding on (1 ∧ 3 ∧ 4), all branches can be closed, and with it the subproof. . The second sequent that remained open from the first decide is: 1 ∧ 2 ∧ 3, 1 ∧ 2 ∧ 3, 2 ∧ 3 ∧ 4, 2 ∧ 3 ∧ 4, 1 ∧ 3 ∧ 4, 1 ∧ 3 ∧ 4, 1 ∧ 2 ∧ 4, 1 ∧ 2 ∧ 4, 1, 4 This allows a new negated clause to be decided on effectively, ( 2 ∧ 3 ∧ 4), but all the (unused) applicable clauses allow us to close at most one branch out of three. Let us say that we decide on this last choice, which at least can close one, and proceed as follows: 1. 2 is added to the list of negative literals: 1, 2, 4. By deciding on (1 ∧ 2 ∧ 3), the 1 and 2 branches are closed, and 3 is added to the list of negative literals: 1, 2, 3, 4. By deciding on (1 ∧ 3 ∧ 4), all branches are closed. 2. 3 is added to the list of negative literals: 1, 3, 4. By deciding on (1 ∧ 3 ∧ 4), all branches are closed. 3. 4 is closed immediately with 4. The proof of the lemma is thus finished. We have glossed over the indexing schemes which are used to make sure that we do not decide several times on the same negated clause along the same branch, as well as the maintenance of the available negated literals, but these are both easily encoded. In addition to that, the decide expert can make use of this information to implement sophisticated heuristics to decide in which order to try the available clauses, for example, based on the number of branches that will be immediately closed-so as to reduce the branching factor. Notes The SAT problem is one of the most representative problems in computer science, not only because of its theoretical significance, but also due to the breadth of practical applications of modern SAT solvers. Good general treatments of the problem can be found in the monographs Biere et al. (2009); [START_REF] Knuth | The Art of Computer Programming[END_REF]. [START_REF] Marijn | Proofs for satisfiability problems[END_REF] present the state of the art of what is broadly understood of a proof of (un)satisfiability. The idea of using unsatisfiability proofs as certificates for use by an independent checker goes back at least to [START_REF] Goldberg | Verification of proofs of unsatisfiability for CNF formulas[END_REF], shortly after the rise of the current generation of SAT solvers and, with them, the large-scale applications of the problem; their ascent coincided with the appearance of the modern series of SAT solving competitions (SatComp). In contrast to an early line of proof formats more directly based on resolution, the first of the current family of proof formats, RUP, was proposed by Gelder ( 2008) and quickly made its appearance in a dedicated track in the SAT competitions; the trimming of clauses that leads to the refinement of DRUP appeared some years later [START_REF] Marijn | Trimming while checking clausal proofs[END_REF] along with the DRUP-trim checker. The RAT property is of great importance because all proof techniques utilized by modern SAT solvers can be expressed in terms of sequences of RAT lemmas [START_REF] Järvisalo | Inprocessing rules[END_REF]-even though this does not imply that it can express everything efficiently. Its use coupled with clause deletion led to the definition of the DRAT format (Heule et al., 2013a;[START_REF] Wetzler | DRAT-trim: Efficient checking and trimming using expressive clausal proofs[END_REF] and its associated checker, DRAT-trim, which remain the reigning standards to this day. Like its predecessor, DRAT-trim is a highly optimized C program in which bugs are occasionally found. Indeed, such a well-defined and stable checker is a prime target for verification as undertaken by a number of recent formalization efforts using proof assistants-as opposed to our pure logic-based approach. As it stands, UNSAT certificate checking is an expensive operation: common figures report checking times in the same order of magnitude as proving times. These costs are compounded by the penalty of modeling and extracting checkers in higher-level languages and assistants, which has motivated the definition of specialized proof formats that refine a DRAT source certificate. The GRIT format [START_REF] Cruz-Filipe | Efficient certified resolution proof checking[END_REF] identifies the costliest operation in DRAT checkingnamely, finding the unit clauses used during unit propagation-and includes this information in the certificate; it is formalized in Coq, but does not cover the RAT property in its entirety. The LRAT format (Cruz-Filipe et al., 2017a) extends GRIT to verify all of RAT by choosing a pivot element and checking the RAT property across all clauses containing the negation of that element; certified checkers are implemented in Coq and ACL2 that are roughly as fast as the unverified DRAT-trim. Independently, the GRAT format [START_REF] Lammich | Efficient verified (UN)SAT certificate checking[END_REF]) also extends the insight of GRIT from DRUP to DRAT by more explicit information about a particular way to execute the checking operation; a verified checker is implemented in Isabelle. All these refinements are geared towards the production of reasonably performant, specialized checkers. Since solvers do not emit any of these extended certificate formats, an additional unverified stage transforms DRAT certificates to each new format; only the checker operating on these augmented certificates is verified. However, some form of completeness property on the mapping from a DRAT certificate to one of the new formats-stating, say, that the original certificate is checkable iff its map is-is only empirically validated. It should be noted that the standard boolean satisfiability problem is limited to classical propositional logic. The generalization of SAT to include quantifiers is called the quantified boolean formula problem, or QBF. An important generalization replaces variables with predicate and function symbols drawn from various background theories-themselves expressed in first-order logic with equality-: this is the satisfiability modulo theories problem, or SMT. However, the lively activity in certification standards and checkers for SAT has as of yet no close parallel in either extended setting, although a certificate format for QBF, called QRAT, has been proposed (Heule et al., 2014). For overviews of both problems, refer to the appropriate chapters in Biere et al. (2009). In the context of the FPC framework, the proof formats proposed in this chapter are very easily implemented-provided that the kernel allows clerks and experts to inspect its state without modifying it, otherwise relatively extensive bookkeeping is required. In standard presentations, including those in Chapters 4 and 10, the kernel is completely opaque to all clerks and experts, which are simply called by the kernel without any information about their context. Some older kernel implementations have provided limited context information to clerks and experts, in particular the one formula that is being operated upon in an inference rule. A somewhat more generous kernel (which remains functionally sealed) is a reasonable possibility and well suited to the generation of smart cut formulas and to sophisticated bookkeeping based on the contents of the context. While everything can be simulated at the level of clerks and experts, the result is less modular and more cumbersome. Section 11.6 illustrates this alternative approach, which recurs later in Section 12.4. Certification of theorem provers 8.1 Towards FPCs in the large In previous chapters, we have developed techniques to add and subtract detail to proof certificates while representing the same proof (Chapter 5). If sufficient detail is added, the process of checking becomes determinate and can be delegated to a functional checker that is even simpler, faster and more reliable than standard proof checkers based on logic programming (Chapter 6). Until this point, a number of realistic if somewhat academic FPC definitions have been presented (Chapter 3), but previous related publications [START_REF] Chihani | Certification of First-order proofs in classical and intuitionistic logics[END_REF]Chihani et al., 2016b;[START_REF] Libal | Certification of Prefixed Tableau Proofs for Modal Logic[END_REF]) have yet to address the transition from small, handcrafted examples and idealized calculi to one of the purported goals of ProofCert: the emission of proof certificates by provers and their independent validation by trusted proof checkers in realistic scenarios. A stepping stone was the adaptation as foundational proof certificates of well defined, standard proof formats (Chapter 7). Completing this opening move towards proof certificates "in the large" is the subject of the present chapter. In Section 6.1, we anticipated a general architecture to support a spectrum of trust levels, which we exercise over the course of the chapter. Truly, a determinate checker can reduce the trusted computing base needed to come to trust the theoremhood of a formula, but the proof traces that constitute tractable proof certificates for MaxChecker are as artificial as their interest is purely mechanistic. A practical solution must involve both non-determinate and determinate checkers, combined in such a way that the resulting system enjoys the expressiveness of the former and the trustworthiness of the latter. With the components at our disposal, it is now a relatively easy matter to describe the architecture of a composite proof checker that we can use to check any proof certificate defined by the FPC framework while only needing to trust MaxChecker. First, use the general, say, λProlog-based checker to perform the formal checking of a formula based on an arbitrary proof certificate accompanied by its FPC definition; this checking operation must pair the certificate with a maximally explicit certificate that will result from the elaboration of the first (as explained in Example 5.3.1 and Section 5.4). Second, independently run MaxChecker on this explicit certificate and the same formula. Of course, we only need to trust the second of these checkers-with the proviso that the trusted checker must contain a trusted printer to output successfully checked formulas. (An independent matter is how to obtain confidence in the correctness of the trusted checker, but-as noted in Chapter 6-a verified implementation of the full checker is not yet within reach, thus motivating the two-tier architecture.) Our goal is to certify the proof evidence produced by a bona fide, complex theorem proving tool, that is sufficiently powerful to provide us with realistic, reasonably sized and publicly available proof corpora. To that end, we have selected Prover9 (McCune, 2010), a legacy, automated theorem prover of modest capabilities. An important feature for our experiments is that the output from the software exposes a relatively simple and well-documented resolution calculus, perhaps the simplest deductive model used in practice. We will refine and elaborate the original resolution FPC presented in Section 3.6, and study some sources of nondeterminism and their effects on checking. The rest of the chapter is organized as follows: Section 8.2 introduces Prover9 and describes a significant subset that is used to model the proofs selected for certification. Section 8.3 revisits the binary resolution certificates and presents various extensions and elaborations addressing significant sources of nondeterministic behavior. Section 8.4 describes in detail the elaboration workflow and the practical composition of the general and determinate provers, along with the requirements of each. Section 8.5 analyzes the results of the certification of all publicly available (non-equational) Prover9 proofs in the TPTP library, the effect of the various formats and checkers, and the systems on which they run. Section 8.6 discusses the extension of the techniques employed throughout the chapter to certify the results of more tools and achieve some level of interoperation across tools. Section 8.7 concludes the chapter. . . The automated theorem prover Prover9 Prover9, developed by McCune ( 2010) is an automated theorem prover for firstorder and equational logic based on a simple resolution calculus, successor to the Otter theorem prover. While no longer actively developed-its last version was released in November 2009-, it remains (in spite of its simplicity) a moderately competent prover to this day and a benchmark for newer tools, as evidenced by its continued placement in the CASC system competition for automated theorem provers (CASC): it remains a good baseline for new developments. Yet in that simplicity we find one of its virtues. Prover9 reports proofs in a well-documented format, close to its calculus, and offers tools to manipulate (and even verify) those proofs. This rare luxury among automated theorem provers paves the way to certification without excessive efforts in reverse engineering. The output format of Prover9 represents proofs by a sequence of steps, each of which is derived from previous steps by one of 17 primary tactics (of which 14 are used in standard proofs), followed by a sequence drawn from five secondary tactics. Proofs in standard format may contain non-clausal assumptions and goals together with clauses. These standard proofs can be transformed by external programs, notably Prover9's own Prooftrans, which can simplify justifications and produce more verbose proofs in a subset of the grammar of tactics (i.e., instances of the hyperresolution rule can be transformed into sequences of applications of the binary resolution rule). As a consequence, Prover9 proofs can be brought to close proximity with our model binary resolution FPC described in Section 3.6. In fact, it will suffice to cover only the following five tactics from Prover9's vocabulary: 1. assumption: primary tactic which annotates the input formula. 2. clausify: primary tactic which annotates the result of translating a nonclausal assumption to CNF. 3. resolve: primary tactic which performs binary resolution on two clauses. 4. factor: primary tactic which performs factoring on two literals of a clause. 5. merge: secondary tactic which removes a literal that is identical to a previous literal in the clause that results from the previous tactic. Support for factoring requires simple additions to the binary resolution FPC listed in Figure 8.1. These additions are completely modular and permit certi- fication of the non-equational fragment of Prover9; the remaining equational fragment makes use of paramodulation and a small number of ancillary tactics. Such additions have been coded as proof certificates by [START_REF] Chihani | The proof certifier Checkers[END_REF] and applied to the certification of a (very small) fragment of proofs produced by the E prover. We will instead concentrate on what the seemingly simple binary resolution FPC can achieve with no or small changes when applied to proofs generated by an automated theorem prover like Prover9. Resolution certificate elaboration In addition to certifying a sizeable number of resolution proofs produced by a real theorem prover, we will explore the effects of adding and removing details from proofs through the pairing combinator introduced in Section 5.2. We note that the binary resolution FPC given in Section 3.6-which closely reflects the semantics of Prover9-leaves two optional aspects of a resolution proof implicit: 1. The order in which two clauses are resolved to yield a third is unspecified. 2. Substitution terms used to instantiate quantifiers are not given. The fact that the order of the resolvents is left unspecified has by itself a potentially enormous impact on proof checking if standard techniques are used, given the exponential number of backtracking points it can generate in a degenerate proof. A more explicit proof could provide one or both kinds of information, thereby making checking more deterministic. These proof formats can be easily encoded as FPC definitions in the style of Figure 3.5. In fact, they are simple variations that can be formulated as simple changes on the original FPC. Figure 8.2 shows the minimal change required to impose a fixed ordering of the resolvents, replacing two possible schemes for the decide rule in the left premise with one, slightly simplified clause. Figure 8.3 presents the limited additions made to support explicit substitution information; these changes extend the resolution rule to the new constructors and simplify the treatment of the decide rule and the existential quantifier. Both sets of changes, together with the original FPC, can be described as modular additions to a common template for resolution FPCs. The preceding notes have addressed the addition of more information to a proof certificate capable of modeling proof evidence produced by Prover9 (i.e., its elaboration). Conversely, Prover9 tactics specify not only the names of the affected clauses, but also (the indexes of) the involved literals. This information is lost in the encoding of the binary resolution FPC: strictly speaking, it is a distillation of the-in some respects-more complete proof produced by the tool. Example Consider the resolution example in Example 3.6.1. In Prover9, an equivalent proof is expressed by the following script, once preprocessed and then simplified by Prooftrans. Clause numbering is slightly beautified to reflect the fact that the clausify tactic is used to replace universal quantifiers with fresh eigenvariables, so that said quantifiers are not directly visible in processed proof scripts. Thus, we get: In practice, the signatures of the various flavors of the binary resolution FPC must use disjunct sets of constructors (e.g., resolve, factor,. . . for the "base" FPC in Figure 8.1; resolve', factor', . . . for the variation in Figure 8.2, etc.). This requirement allows the top-level constructor of a proof certificate-and any constructor at any point in a certificate-to determine the certificate family to which it belongs. With this information, a missing certificate (represented by a logic variable) can be reconstructed via pairing drawing exactly from the correct set of constructors. Otherwise, mixed certificates could be constructed and checked-at the cost of an explosion in the number of choices and an erosion of the separation of semantics we endeavor to dictate. Example [(r z), % 1 (forall x\ or (ng (r x)) (t x)), % 2 (ng (t z))] % 3 [(t z), % 4 % Introduce an order-ambiguous resolvent subproof. %type resolveX int -> cert -> cert. % Eigenvariables allowed into certificate here. type rquant (i -> cert) -> cert. % Use the following substitution term. type subst i -> cert -> cert. Here, clause numbering is implicit in both clause lists. The certificate is properly the third list of inference (resolution) rules; any ordering of the resolvent clauses is allowed. As a first refinement, the order of the resolvents is fixed, so that only one combination of the exponentially many reorderings is accepted: Figure [resolve' 4 (res' 1 (rex' 2 done')), resolve' 5 (res' 4 (rex' 3 done'))] The clause lists remain unaltered; only the certificate fraction is adapted. Finally, substitution information can be added to the relevant rules: [resolve'' 4 (res'' 1 (rex'' 2 (subst'' z done''))), resolve'' 5 (res'' 4 (rex'' 3 done''))] While there is no standard feature to perform renaming while allowing for modular definitions and name reuse in λProlog, an experimental branch of Teyjus enables this kind of operations. In what follows, the requisite renamings will be assumed regardless of the technical means used to achieve them. Certification workflow In this section, we describe the certification of resolution proofs produced by an automated theorem prover based on a simple resolution calculus-namely, the non-equational fragment of Prover9. The most permissive of the binary resolution FPCs that have been discussed (with unordered resolution and no substitution information) subsumes the inference rules used by the fragment of interest of Prover9. In order to organize the experiments, we compose a module resolutionelab which accumulates the various versions of the resolution FPC and defines instances of pairing between the unordered binary resolution certificate without substitution information (Figure 8.1) and other, more explicit certificate placeholders (Figures 8.2 and 8.3), which are completed by means of elaboration. These form a framework for our experiments in certification of Prover9 proofs and "translating between implicit and explicit versions of proof." The module is actually a schema for other modules-it does not define any non-logical constants or any proofs by resolution based on those constants: it is syntactically correct and can be readily compiled, but is inert. To define resolution problems and their proofs, the following elements can be plugged in: 1. In the signature file, constructors for atoms (of type bool) and terms (of type i), both of which may have term arguments. Each of these must be complemented in the module definition by a clause of pred_pname or fun_pname, respectively, used by the polarized to translate formulas into the NNF format required by the kernel. This constitutes the user signature. 2. In the module file, problem definitions based on the signature. These are the combination of a problem identifier, and a triple of lists: (a) A list of the clauses that constitute the input formula (therefore given implicitly in clausal normal form). (b) A list of clauses derived from applications of proof steps. (c) A list of proof steps providing justifications for the derived clauses representing. This list is properly a schematic representation of a proof by binary resolution with factoring. These declarations are given a prover-specific prefix to avoid clashes with other names. The module contains the payload given in Example 8.3.2-adapted to reflect the prefixed identifiers-alongside auxiliary declarations for printing and counting. Example pred_pname (r_p9 X1) "r_p9" [X1]. size_bool (r_p9 X1) Size :-size_term X1 SizeX1, Size is SizeX1 + 1. pred_pname (t_p9 X1) "t_p9" [X1]. size_bool (t_p9 X1) Size :-size_term X1 SizeX1, Size is SizeX1 + 1. fun_pname z_p9 "z_p9" []. size_term z_p9 Size :-Size is 1, !. fun_pname c1_p9 "z_p9" []. size_term c1_p9 Size :-Size is 1, !. The module defines three possible pairings, all starting from the Prover9like FPC in Figure 8.1 and elaborating to: (a) binary resolution with ordering (Figure 8.2); (b) binary resolution with ordering and substitutions (Figure 8.3); and (c) maximally explicit elaborations (Figure 5.2). For all three pairings, predicates are given in triads of: (a) elaboration; (b) elaboration followed by checking of the new certificate; and (c) elaboration followed by reporting. The decomposition in steps is geared towards producing accurate measurements, from which exact checking times can be derived. This approach has two practical advantages. First, experiments can be carried out from a single λProlog program, without exporting and importing intermediate results. Second, it circumvents certain limitations in Teyjus; as a result, a direct comparison with other implementations of the language, like ELPI, is possible. To this end, utility predicates are defined in the module in the following fashion: 1. check_unordered: check the Figure 8.1 certificate specified by a problem description without any additional operations. 2. elab_to_sans: pair the Figure 8.1 certificate with, and elaborate into, a Figure 8.2 certificate through checking. 3. check_sans: perform the elaboration in elab_to_sans, followed by an independent check of the resulting Figure 8.2 certificate. . . 4. print_sans: perform the elaboration in elab_to_sans, followed by printing of problem size statistics. 5. elab_to_subst, check_subst, print_subst: like the *_sans predicates above, but pairing and elaborating the example with a Figure 8.3 certificate. 6. elab_to_max, check_max, print_max: like both the *_sans and *_subst predicates above, but pairing and elaborating the example with a Figure 5.2 certificate. 7. elab_and_export: pair the Figure 8.1 certificate specified by the example with a Figure 5.2 placeholder, as elab_to_max, and output the certified formula and maximally explicit certificate as OCaml code, ready to be imported in MaxChecker. Each of these predicates takes a problem identifier as argument. The check_* and elab_to_* predicates work without any further additions. Reporting predicates rely on auxiliary operations on user-defined atom and term constructors as follows. 1. print_* require a size_bool clause for each atom constructor and a cut-terminated size_term clause for each term constructor. The size of a constructor is defined as 1 (the constructor itself), plus the sum of the sizes of its term arguments. 2. elab_and_export requires a print_name clause for each atom constructor and a cut-terminated print_term clause for each term constructor. Atom names relate the string representation of atoms given in pred_pname and a valid OCaml identifier. Terms relate the constructor to a valid OCaml identifier and to the printed representations of its arguments for use by MaxChecker. All these additions can be easily generated automatically. To import a formula and certificate into MaxChecker, the atom and term signatures must agree with the identifiers given by the translation. Example Finalizing the sequence from Example 8.4.1, the MaxChecker instantiation relies on the definition of isomorphic, native OCaml types for terms: type 'a t = | R_p9 of 'a | T_p9 of 'a As well as atoms: type t = | Eigenvariable of int | Z_p9 | C1_p9 MaxChecker then takes the translation of the original formula as well as the maximally elaborate certificate in OCaml syntax and checks the combination of both; this final piece of information is omitted for brevity. The process of certifying a Prover9 proof comprises three main steps: 1. Extract the problem signature from the Prover9 proof script. From this, constructors and auxiliary clauses are extracted. Each proof step is mapped into the sets of clauses and justifications that define a problem. Having done this, the resolution-elab module is instantiated. If MaxChecker is used, identifiers and constructors for its atom and term types are derived. 2. Certify the extracted formula with the extracted certificate on the λProlog kernel. The baseline goal check_unordered checks the certificate as given in the problem description; further exploration is possible. 3. If MaxChecker is used, solve goal elab_and_export, export the translated formula and certificate to OCaml, and run the functional checker on the maximally explicit elaboration. The sequence can be fully automated. A translation from Prover9 to either kernel should generate valid identifiers for each language, and particularly in λProlog avoid clashes between names, say, by using a dedicated namespace. Note that renumbering of clauses with respect to the original clause numbers of Prover9 may be necessary, given that the FPC numbers its clauses by order as they are given in the certificate, first the base clauses and after those the derived clauses; contrast this with Prover9 proofs, in which assumptions can be introduced at any point in the proof. Analysis of results The study, optimization and comparison of theorem provers relies on the existence of widespread benchmarks. In the automated theorem proving community, the TPTP library-short for Thousands of Problems for Theorem Provers-fulfills this role [START_REF] Sutcliffe | The TPTP problem library and associated infrastructure[END_REF]. Its success is cemented in its syntactic conventions, easy to understand by mathematicians and to parse and process by programs alike, and close to the syntax of Prolog. An important part of TPTP is its syntactic support for the expression not only of problems but of proofs of those problems. For a large number of theorem prover, the library contains a collection of solved theorems along with the generated proofs, TSTP-short for Thousands of Solutions for Theorem Provers- [START_REF] Sutcliffe | TSTP data-exchange formats for automated theorem proving tools[END_REF]. Those proofs are expressed as sequences of inference rules whose dependencies satisfy a DAG structure (in essence, a form of Frege proof, for which see Section 11.1). The ready availability of a widely recognized corpus of proofs forms the basis of the experimental study. We have collected the full set of Prover9 refutations in the TPTP library-a total of 2668 in version 6.4.0-and excluded 52 files with irregular formatting (the resulting set of examples is precisely that of version 6.3.0). Of these, 978 fall in the fragment supported by the resolution FPCs; 27 are empty proofs that refute false. The two largest problems are extreme outliers, also excluded since they would be of limited utility to establish or confirm trends. Each problem is expanded into a detailed proof in the simplified binary resolution calculus via the homonymous expand option of Prover9's built-in Prooftrans tool. A further preprocessing step is required. Prover9 accepts as input arbitrary first-order formulas (i.e., they need not be given in clausal normal form), and transforms those non-clausal assumptions as necessary by way of the clausify tactic; the resulting translated clauses are added and the original assumption given as their provenance. This has two consequences: First, non-clausal assumptions are made redundant by this "clausification" process and will be removed; note that resolution certificates only describes problems problems expressed in clausal normal form. Second, suppose that not all clauses of the input formula (or its CNF translation) are used in the proof. There is no guarantee that every single clauseincluding unused ones-will be part of the proof script produced by Prover9, and thus in the proof certificate derived from the script. In consequence, such a certificate may indeed represent a stronger theorem than the original formulation q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q 0e+00 1e+05 2e+05 used by the theorem prover, albeit the derived theorem statement can be easily shown to imply original theorem statement. Having obtained the data and performed this preprocessing, the general workflow described in the previous section is applied. As part of the experiments, we run the λProlog-based checker on two separate implementations of the language: the more mature compiler Teyjus [START_REF] Nadathur | System description: Teyjus -A compiler and abstract machine based implementation of λProlog[END_REF] and the more modern interpreter ELPI [START_REF] Dunchev | ELPI: fast, embeddable, λProlog interpreter[END_REF]. The dataset presents us with ample amounts of problems encoded as logic programs, in sizes and numbers rarely seen in the language. q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q qq q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q qq q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q qq q q q q q q qq q q q q qq q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q qq q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q qq q qq q q qq q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0 2 4 6 0 25000 50000 75000 100000 Payload size of problem up to 100000 ELPI checking time in seconds We have successfully checked all resolution refutations produced by Prover9 involving binary resolution and factoring; no errors have been found in this set of Prover9 proofs. Quantitative information emanates naturally from the data. A first point of interest concerns the size of certificates and how it is defined. The natural approach is to define the size of a resolution certificate to be the sum of the sizes of the initial and derived clauses along with their justifications; and the size of a maximally explicit certificate as the size of the actual certificate term plus the size of the original set of clauses. In this way, we compare different certificate formats by the size of their full payload: how much it costs to express a problem and provide proof evidence for it-the theorem is implicit in, say, a resolution certificate, but not in a maximally elaborate certificate. In both binary resolution and maximally explicit certificates, natural numbers are used extensively as indexes and have a great bearing on their overall sizes. Here we consider various possible representations. First, we may use machine integers and count a natural number as one constructor. Second, we can use the standard q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q qq q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q qq q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q qq q q q q q q qq q q q q qq q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q qq q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q qq qq q q q q q q q q q q q q q q q q q q q q q qq q q qq q q q q q q q q q q q q q q q q q q q qq q qq q q q q q qq q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0 1000 2000 3000 4000 0 25000 50000 75000 100000 Payload size of problem up to 100000 Teyjus checking time in seconds inductive definition and extend the count of the number of constructors to the inductively defined naturals; this results is a size increase, possibly important, which can then be measured. Other reasonable alternative inductive encodings (like the ternary definition that mimics a binary representation, where a number is either zero, twice the value of a number, or twice the value of a number plus one) will fall somewhere between the two extremes considered here. Figure 8.5 presents the effect of elaboration on certificate size. Adding ordering information (from unordered-without to ordered-without) does not affect certificate size, and therefore that first uninformative data series it not shown in the figure. Certificate sizes grow as they are made more explicit, though the blowup here is bounded by small constants. Elaborating from the original unordered-without to ordered-with adds a linear multiplicative constant to the payload; sizes grow by 16% on average. Finally, elaborating to the maximally explicit certificate causes an increase by an average factor of 2.8, ranging between 1.02 and 6.54. We do not expect hard trends since results depend on the coverage of a wide space of problems by available data, and in particular the corpus of larger examples at our disposal is relatively limited. If we adopt the simple inductive definition of natural numbers and adjust counts accordingly, the increase in size is considerable. On average, this representation causes certificates to grow by an average factor of 5.8 as they are elaborated to their maximally explicit form. However, there is much greater variability, ranging between factors of 1.2 to 361. Concentrating on the more natural approximation of natural indexes as constants, Figure 8.6 presents the evolution of checking times under the various certificate formats as the size of said certificates grows. As a matter of fact, and as may be expected, the more detailed a certificate is, the faster it is to check. Overall, progress is fairly rapid: for example, a sizeable certificate in the unordered-without format about 75000 symbols large can be checked in approximately 6 seconds; a similarly sized certificate in the maximally explicit format can be checked in about a second. However, due to size blowup it cannot be asserted that a maximally explicit certificate will always check faster than its resolution equivalent, and in fact for some of the larger problems we notice an inversion of this naive hypothesis. Both extremes (i.e., unordered binary resolution without substitutions and maximally explicit elaboration) appear to exhibit behavior that is fitted well by a quadratic regression curve, although the proof corpus becomes sparser as problem sizes grow, and more data would be necessary to establish definite trends-if any avail. In addition, it should be noted that the more explicit resolution certificates gain a large part of the efficiency of the much simpler maximal elaborations by a very moderate increase in size and complexity; ordering of the resolvents, in particular, is the determining factor in avoiding backtracking points. The use of Teyjus as λProlog runtime, as depicted in Figure 8.7 yields overall qualitatively similar results. However, performance is significantly slower and more asymmetrical. Outliers are more frequent and more extreme, and the overhead of elaboration is substantial with respect to the much faster and more consistent ELPI; the intermediate formats show particularly erratic behavior. Conversely, the checking times for OCaml-based MaxChecker on the large, maximally explicit certificates running are completely negligible compared to both elaboration and checking times in λProlog: in particular, MaxChecker checks every example in less than 0.01 seconds. Taken as a benchmark, the Prover9 corpus enables comparison of the two principal implementations of λProlog. Functionally, both have behaved equiva-lently with some minimal rough edges in the ELPI parser being uncovered by our experiments. The principal differences are summarized in the following points: 1. There are moderate limits to the size of the terms Teyjus can parse, both in the compiler tjcc and in the simulator tjsim. While these have not impeded the list-based formulation of resolution certificates, exporting and importing large proofs and formulas is problematic. This motivates the additive composition of elaboration and checking steps given in the resolution-elab module, in which once the unordered certificate is read all computation is performed in-memory. 2. The intermediate compilation step in Teyjus, absent from the ELPI interpreter, has scalability issues of its own. Compilation times are seen to grow substantially once a certain threshold in the size of the proof translation is reached. For the very largest examples in the corpus, this grows to make Teyjus unusable, to the point of compilation possibly failing to terminate (and certainly not doing so in any reasonable amount of time). 3. In some of the larger examples, the process of elaboration has been observed to surpass the capacity of Teyjus' internal data structures, causing a premature stack overflow and a termination of checking. This can be observed first when combining elaboration and checking in the check_* predicates. 4. Teyjus does not implement a predicate to measure execution times inside the language, whereas ELPI reports the execution time of goals by default. Therefore Teyjus must rely on external tools and we need find a way to separate the time taken to load the program from the proper user time required by elaboration and checking operations. There are observable performance differences between the two systems. Generally speaking, ELPI runs as fast or faster and scales better, although the two systems show different patterns of behavior, especially in the relative cost of running elaboration through paired certificates, compared with the checking time of each individual certificate. In its favor, the interface of Teyjus is more complete and more amenable to scripting. Although workarounds can be found for ELPI, batch reporting in Teyjus remains more informative. The next 700 certificate formats In this and the previous chapter, we have successfully extended the areas of application of the FPC framework from small handcrafted examples to the practical domain of automated theorem provers. We have done so by designing certificates for the proof evidence produced by actual theorem provers, and then converting and checking the proofs produced by those tools. In Chapter 7, the objects of formalization were part of a series of well defined proof formats shared by a large family of software in wide use-still a rarer fortune. In previous sections of this chapter, we considered a relatively simple and well documented proof format which could be reduced to a standard calculus with minimal changes. In light of these encouraging results, it remains to consider how to extend them-and their adoption-further. There are two principal aspects of this push towards widespread use: first, the definition of certificate formats supporting additional proving tools; second, the recognition of FPCs by these tools and the import of proofs and interoperability between compatible provers. The first aspect is intricately related to the mechanisms offered by the FPC framework to program proof search in the "logical computer" of the augmented sequent calculus. This, the critical step, remains the domain of the specialist logician. Indeed, the main difficulty lies in translating the proof evidence produced by a certain tool into formal terms. Each format constitutes a sort of domainspecific language for the writing of proofs, which must be compiled into the assembly language of the underlying proof system: an FPC definition fulfills the role of such a compiler. Currently, there is no systematic methodology to move from an arbitrary description of a proof semantics into a lower-level description built upon the sequent calculus; the capability to embed high-level descriptions of semantics as part of an FPC would greatly simplify this task. Some advances have been made in streamlining the use of augmented sequent calculi as a bona fide programming environment, therefore offering some guarantees of completeness (in a loose sense) and of continuous integration of tests-since programming in this exotic logical assembly, where instructions are inference rules, involves behaviors that exceed the complexity of, say, the more conventional imperative and functional languages (Blanco andChihani, 2016, 2017). Metatheoretical results-e.g., soundness and completeness of provability by an FPC definition with respect to a proof system and a mapping between proofs in that source system and concrete FPCs expressed in that FPC definition-must be established separately. The second aspect of interoperability through proof certificates is fundamentally one of integration. A prover may not only write its output as a proof certificate, but also read certificates in an understood definition, in essence acting as a checker (or delegating the task to a dedicated checker) and reconstructing a proof for a given formula, which can then upon success be accepted as proved. The expression of proof semantics is self-contained in FPC definitions and provides support for the use of proofs generated by external tools. (A more challenging sort of interoperability involves translating a proof from one certificate format to another a given tool can understand; as we observed in Chapter 5 this is, in general, not possible.) On the whole, this second aspect follows easily from the first, which is in that sense its prerequisite. The preceding two aspects are fundamentally technical in nature: they may involve a certain amount of work to specify and give proof theoretical readings of the various proof calculi, as well as translations as FPC definitions and integration of proof checkers with the provers themselves. These questions are best solved by establishing a dialog between the authors of theorem provers and the authors of FPC definitions (if they differ); the work presented here makes a convincing case that such a dialog is not only possible but deeply beneficial. The principal challenge is social: making the authors of theorem provers aware of the existence of these Foundational Proof Certificates and collaborating with them to add support for certification and proof checking. The desirability of such formats is acknowledged by the theorem proving community, as the recent first ARCADE workshopcelebrated as part of CADE-26 in August 2017-made manifest (ARCADE). While "every problem is a people problem," the FPC framework has a strong claim towards becoming a canonical solution, if it is not the only one. Notes The original certification experiments on Prover9 were published in Blanco et al. (2017a). Two additional topics are mentioned here in passing and are elaborated elsewhere. In related work in Blanco et al. (2016), we proposed an extension of the TPTP format that integrates the semantics of inference rules as logic programming specifications as an intermediate step towards filling the semantic void of freeform proof output formats. On the side of the FPC framework, a programming methodology and assistant tools that aim to make the use of proof certificates more approachable is proposed in [START_REF] Blanco | An interactive assistant for the definition of proof certificates[END_REF]. In order to capture all of Prover9's proofs in the TPTP repository we need to add support for paramodulation: the FPC for paramodulation given in [START_REF] Chihani | The proof certifier Checkers[END_REF] is a starting point, and can be adapted with few modifications to work with other, related implementations of resolution calculi like the one employed in this chapter. The development in that paper of the paramodulation FPC follows that of the proof certifier Checkers. This system implements a pair of kernels for the standard calculi LKF a and LJF a in the same tradition of the kernel used throughout the present Part II and defined in Section 3.7. To these kernels it adds a structured module system for the definition of problem signatures and their composition with FPC definitions to yield complete instances of the checker. The plan was to use Checkers to certify the proofs produced by the E prover, which is based on a superposition calculus-itself a variant of resolution. However, only a minimal sample of proofs could be checked owing to difficulties in modeling the semantics of the relatively rich inference rules of the theorem prover out of limited documentation. In our view, this experiment showcases the necessity for communication and collaboration between the authors of provers and checkers, and a greater need for documentation. Until now, the process for certifying the output of a theorem prover has been for a independent effort on the checking side to understand the semantics of each inference rule of the object calculus. This approach generally suffers from missing documentation, naming conventions, changes across software versions, etc. One way to overcome this gap is to supply the implementers of theorem provers with an easy to use format in which to describe the semantics of their inference rules. This format should be general enough to allow specifications to range from precise, determinate definitions to more implicit, less specific hints that would instruct a checker on how to reconstruct a full proof of the object calculus-even if left partly (albeit inessentially) unspecified. The insight of the proposal in Blanco et al. (2016) is to reduce the gap mentioned above by employing a format that is already known to implementers of theorem provers, namely the TPTP format, enriching a syntactic base with semantic information expressed in a logic programming style. The resulting model is closer to the proof checkers of the FPC framework, although there remains to establish the connection between the logic programs that embed the semantics of a proof system and the expanded proof in an underlying sequent calculus. The computational model of the related project Dedukti, based on the λcalculus, is closer to the more type-driven paradigm of common proof assistants, themselves routinely founded upon the precepts of functional programming (rather than logic programming, as in the FPC framework). Libraries have been developed to translate proof evidence originating in the common OpenTheory format shared by the HOL family of proof assistants (which includes the standard configuration of Isabelle), fragments of Coq and its close relative Matita (both based on the calculus of inductive constructors), and theorem provers like iProver and Zenon (both extended from first-order logic to Deduction Modulo). In this setting, work towards interoperability has already been initiated [START_REF] Assaf | Mixing HOL and Coq in Dedukti (extended abstract)[END_REF][START_REF] Cauderlier | FoCaLiZe and Dedukti to the rescue for proof interoperability[END_REF]. In practice, when such frameworks as FPCs are not directly understood by the source theorem provers, it is necessary to convert the proof evidence output by those provers into equivalent proof certificates. This translation is not part of the trusted computing base, as both theorem prover and proof checker operate on the same input formula, and the checker reconstructs the full proof from the certificate. Part III Logics with fixed points 9 Fixed points in logic Fixed points and equality as logical connectives Logical frameworks based on the theory of intuitionistic logic (outlined in Chapter 4)-as well as linear logic-have been adopted as the foundations of higher-order logic programming and used to specify many aspects of programming languages and theorem provers. An important limitation of the logical frameworks used until now-notably λProlog-is that they do not offer a natural treatment of inductive definitions and proofs under the paradigm of higher-order abstract syntax that those frameworks employ widely and fruitfully (for reference on these topics, see Section 4.2). As inductive reasoning is the bread-and-butter of functional and logic programming languages, and of their metatheory, extending the logic with a clean treatment of these concepts is of the essence for the formal study of those paradigms and reason about them effectively. [START_REF] Mcdowell | A logic for reasoning with higher-order abstract syntax[END_REF][START_REF] Miller | Abstract syntax for variable binders: An overview[END_REF], 2002) developed in response a logic that incorporated the concept of definitions, whose use allows certain declarations to be treated as closed, i.e., as fixed point expressions representing (mutually) recursive predicates-along with rules for unfolding (similar to backchaining) and induction on definitions. These connections were matured through research efforts by Miller andTiu (2003, 2005)-with [START_REF] Tiu | Induction and co-induction in sequent calculus[END_REF] and further work by [START_REF] Tiu | A Logical Framework for Reasoning about Logical Specifications[END_REF], 2006)-and Gacek et al. (2008b, 2011). Baelde and Miller (2007), followed by work by [START_REF] Baelde | A linear approach to the proof-theory of least and greatest fixed points[END_REF][START_REF] Baelde | On the proof theory of regular fixed points[END_REF][START_REF] Baelde | Least and greatest fixed points in linear logic[END_REF], extended these developments to general least and greatest fixed points expressions as logical connectives in various logics. The systems Bedwyr and Abella have their roots in this line of work, which forms also the proof theoretical basis of the developments in this Part III. From the perspective of the sequent calculus, the two principal criteria that invest a connective with logical legitimacy are: those rules are folded into the proof system. Let us first consider a generic fixed point operator, µ, say, for now, in a simple, one-sided calculus reminiscent of LK. The introduction rule for such a connective would be: B(µB) t, Γ µB t, Γ Here, B is a formula, abstracted over a recursive predicate and an arbitrary number n of variables, called the body of the fixed point. The fixed point operators take the body and exposes the abstracted variables. To this combination can be applied a list of n terms, t , acting as the arguments to the fixed point. The unfolding operation applies to the body its own definition wrapped in the fixed point (recursion) and the list of variables. Two example fixed point expressionsthe inductive definition of natural numbers and the addition relation on these-are shown in Figure 9.1. A predicate thus becomes a name for a fixed point expression. This illustration motivates the need for equality as a logical connective. Fixed point definitions can very naturally encode recursive specifications such as those written in a programming language like Prolog-compare the encoding of the same relations in λProlog, in Figure 4.2. In a relational specification expressed in pure logic, it becomes necessary to relate the abstracted variables that act as parameters of a predicate with the values passed them as arguments. In a basic sense, the pattern matching at the head of the clause must take place inside the logic. This is achieved by defining the introduction rules for equality. In the same one-sided setting as above, these are: t = t, Γ {Γσ : σ ∈ CSU(s = t )} s t, Γ † The rule for equality applied on two instances of the same term t has the same effect as the initial rule: it finishes the current branch of the proof. The rule for inequality inspects two terms, s and t , and determines the conditions under which they are equal; these conditions are expressed by a set of substitutions σ, each representing conditions (the complete set of unifiers, or CSU) which make the two terms equal. If there are no such solutions, the set of premises is empty and the rule succeeds immediately (this endows the system with a notion of negationas-failure); otherwise, each possible solution to the equation becomes a premise whose context is the same context of the conclusion, Γ, once the substitutions have been applied-this proviso is represented by †. The treatment of equality marks also the introduction of unification-which computes the substitutions that make two terms equal-not only in the implementation details of a logic programming language, but as part and parcel of the logic proper. In first-order logic, the complete set of unifiers coincides with the more familiar most general unifier, or MGU, to which we shall return in Section 13.2. Example Consider the inductive definition of the type of natural numbers in Figure 9.1 and compact notation for constants. The sequent 2 = 2 is immediately provable the application of the rule of equality, whereas 1 = 2 is not. Conversely, 1 2 is proved by the inequality rule: because no unifiers of 1 = 2 exist, the empty set of premises is trivially satisfied. Finally, S a S b, Γ (where a and b are variables) is provable if ( Γ)σ is provable subject to the substitution σ which allows a = b to unify. Coming back to the fixed point connectives, we may advance that with the addition of focusing the operation of the introduction rule for µ, also called the unfolding rule, would not split in two, but be identical in both asynchronous and synchronous splits of the original introduction rule. The differences that materialize the division between least and greatest fixed points (resp. µ and ν ) arise from the addition of (respectively) induction and coinduction: without these principles, both connectives are indistinguishable. If induction and coinduction are added, the symmetry between the least and greatest fixed points is restored. Returning to the two-sided sequent calculus, the least fixed point µB is characterized by the following two introduction rules: Γ, S t R BS x S x Γ, µB t R induct Γ B(µB) t Γ µB t unfold . The right introduction rule is just an unfolding of the fixed point, which expresses that B(µB) t ⊃ µB t . The left introduction rule is the induction principle, where S is the inductive invariant; its right premise shows that the invariant is a prefixed point (i.e., BS ⊆ S ), whereas the left premise shows that the invariant can be used in lieu of the fixed point to prove the base goal. Note that the right premise operates on a fresh set of universal variables used as arguments, x. In addition, from induct it is possible to derive a left unfolding rule as a particular case: Γ, B(µB) t R Γ, µB t R The greatest fixed point with its introduction rules is the dual of the least fixed point, and completely symmetric with it: Γ S t S x BS x Γ ν B t coinduct Γ, B(ν B) t R Γ, ν B t R unfold And with them, the right unfolding rule as a particular case of coinduction: Γ B(ν B) t Γ ν B t The rest of the chapter is organized as follows: Section 9.2 structures the developments outlined in this section as part of an intuitionistic sequent calculus. Section 9.3 extends the FPC framework with kernels based on those rich intuitionistic proof systems. Section 9.4 presents the concept of nominal abstraction and integrates it in the proof system. Section 9.5 concludes the chapter. Focused sequent calculus Section 2.4 presented the foundations of sequent calculus proof systems applied to classical logic-the dominant paradigm in automated theorem proving and in all of Part II. It was noted then that an intuitionistic sequent calculus results from a simple restriction on standard two-side sequents: namely, that at most one formula appear on the right-hand side. The pervasive symmetry of the classical setting allows a presentation based on one-sided sequents, on which the discipline of focusing was presented. This addition entailed a redesign of the structural rules of the traditional LK which preserved the soundness and completeness of the resulting LKF with respect to classical logic. This chapter lays the necessary proof theoretical foundations for the remaining chapters. The present Part III is closer to the world of inductive definitions, proof scripts, etc., which is characteristic of proof assistants, themselves typically built upon constructive logics. From the point of view of the sequent calculus, it is a simple matter to streamline the presentation of LK into that of LJ by enforcing the intuitionistic restriction at the level of inference rules. An important detail concerns the division between additive and multiplicative connectives. In the classical sequent calculus, the two sets of rules are interadmissible-in fact, the focused calculus LKF integrates both-but as can be seen from Figure 2.2, the right rule for multiplicative disjunction violates the intuitionistic restriction. Ergo, only additive disjunction is intuitionistically valid. As focusing is added to LJ to obtain LJF, only the positive disjunction (written ∨ instead of ∨ + ) is present in the system; both positive and negative conjunctions continue to coexist. Figure 9.2 shows the focused sequent calculus LJF. In contrast with Figure 2.4, it presents a two-sided development of a focused calculus; in addition, it integrates the constructive constraints in the rules instead of maintaining them as side conditions. Like Figure 2.4 (as opposed to, say, Figure 2.1), it features directly the structural rules adapted to the focusing discipline (as opposed to the more traditional weakening, contraction, etc.). Both factors (intuitionistic, two-sided) contribute to a more complex taxonomy of sequents: 1. Unfocused sequents Γ ⇑ Θ R divide their left-hand side into two zones: storage, Γ, and workbench, Θ. The right-hand side R is more interesting, because it contains exactly one formula and must model a limited singleplace, storage-or-workbench division: the formula must be assigned to one of these "areas." Thus: (a) Γ ⇑ Θ B ⇑ has the RHS formula B in the workbench; and (b) Γ ⇑ Θ ⇑ B has B in storage. 2. Focused sequents on the left Γ ⇓ B R, where the left workbench contains exactly the formula under focus. On the RHS, the goal must be in storage, because otherwise the asynchronous phase would not have finished giving way to a focus, in this case on the left. Focused sequents on the right Γ B ⇓, where the right workbench contains the formula under focus, and the LHS workbench must be empty. The two-sided focused sequent calculus further imposes a deterministic order of evaluation of formulas across all workbenches: first the proper workbench on the left-hand side, then the right-hand side formula if it is in "workbench mode." In spite of the greater superficial variety, the division between asynchronous and synchronous phases is identical to that in the one-sided calculus. However, twosided calculi are bigger (in number of inference rules) than one-sided calculi, all situations-except the self-symmetrical cut-have to be treated on both sides of the sequent. In exchange, the two-sided presentation emphasizes the symmetries in the calculus and allows for more uniform presentation across logics. Note that (two-sided) sequents in the asynchronous phase have two storage-orworkbench divisions represented by two up-arrows. In the asynchronous phase, a sequent has a formula under focus in the workbench of one of the sides (marked by the usual down-arrow) while the other side is storage-only, and the second arrow that separates this zone from the empty workbench is omitted. The LJF proof system with added fixed points and equality will become µLJF, the basis of our subsequent study. In LJF, the connectives ∧ + , ∨, t + , f + , and ∃ are positive; the connectives ∧ -, ⊃, t -, f -, and ∀ are negative. As for the new connectives, equality, =, and the least fixed point, µ, are defined as positive; the greatest fixed point, ν, is defined as negative. These polarities are natural choices for the semantics of the fixed points, though it may be possible to assign them differently (Baelde, 2008b, Chapter 4). The set of focused inference rules that expand LJF into µLJF are given in Figure 9.3. There are now essentially three operations (and their corresponding inference rules) that can be used to treat a least fixed point formula on the left-hand side of the sequent: 1. The most substantial inference rule on least fixed points is the induction rule. In its general form, its premises involve an induction invariant (as we will see in Chapter 11, there exist common simplifications that apply in most situations). Like the cut rule, induction is non-analytic in the sense that its inference rule does not have the subformula property. 2. The least fixed point can be unfolded on the left as a direct consequence of the induction rule. 3. The fixed point can be frozen in the sense that when a dedicated version of the store-left rule is applied to it, the resulting occurrence of the fixed point in storage will never be unfolded again, nor will it be the site of an induction. Such frozen fixed points can only be used later in proof construction within an instance of the initial rule. All three rules take place in the asynchronous phase. They are completed by the right-unfold rule on least fixed points in the synchronous phase. Dually, greatest fixed points feature synchronous unfolding on the left and three asynchronous rules on the right: coinduction (also non-analytic), right unfolding as a consequence of coinduction, and freezing. The freezing rules deserve special mention. As we process a fixed point asynchronously (say, µ on the left), at a certain point in the proof we decide to fix it and never modify it again (i.e., by inductive rules). As the fixed point is effectively negative, the storage rules would never operate on in. The freezing rule moves the fixed point to a region of storage reserved for fixed points, which one stored (i.e., frozen) are never decided upon. Following [START_REF] Blanco | Proof outlines as proof certificates: a system description[END_REF], we model this behavior through a dedicated frozen zone, written Φ. In the case of greatest fixed points, the homologous process takes place with one important distinction: the single-formula slot on the RHS functions as a multi-purpose zone. Once a greatest fixed point is moved to right storage (i.e., frozen), the whole RHS is blocked until proof's end. Frozen fixed points come into play in the revised initial rules. These rule now search not an atom of complementary polarity on the opposite side, but an identical (i.e., unifiable, see above) and unalterable (i.e., frozen) atom on the opposite frozen zone. In short, fixed points play the role of atoms and replace the "undefined atoms"-not definitional-with fixed points. The other connective that intervenes in the finalization of branches is equality. Interestingly, both new types of connectives-fixed points and equality-are used as initial-style rules, and both involve unification problems at the logic level. Critically, the addition of focusing must preserve the set of theorems of the original unfocused system: Theorem The system µLJF is sound and complete w.r.t. µLJ. Proof. Proved in [START_REF] Baelde | A linear approach to the proof-theory of least and greatest fixed points[END_REF]. Augmentations and kernels C, Γ ⇑ Θ R Γ ⇑ C, Θ R store l Γ ⇑ ⇑ D Γ ⇑ D ⇑ store r Figure The LJF focused proof system for intuitionistic logic [START_REF] Liang | Focusing and polarization in linear, intuitionistic, and classical logics[END_REF]. Here, P is a positive formula; N is a negative formula; P a is a positive literal; N a is a negative literal; A and B are arbitrary formulas; C is a negative formula or a positive literal; and D is a positive formula or a negative literal. R represents an arbitrary right-hand side; in structural and initial rules, a mixed style that combines this generic symbol and some sequent arrows is used. The proviso marked as † is the usual eigenvariable restriction. Φ; Γ ⇑ S t, Θ R Φ; Γ ⇑ B S ȳ S ȳ ⇑ Φ; Γ ⇑ µB t, Θ R inductL † Φ; Γ ⇑ S t ⇑ Φ; ⇑S ȳ BS ȳ ⇑ Φ; Γ ⇑ ν B t ⇑ inductR † Φ; Γ ⇑ B(ν B) t ⇑ Φ; Γ ⇑ ν B t ⇑ unfoldL Φ; Γ ⇑ B(µB) t, Θ R Φ; Γ ⇑ µB t, Θ R unfoldR µB t, Φ; Γ ⇑ Θ R Φ; Γ ⇑ µB t, Θ R freezeL Φ; Γ ⇑ ⇑ Φ ν B t Φ; Γ ⇑ ν B t ⇑ freezeR {Φ; Γ ⇑ Θ R}σ σ ∈ CSU(s = t ) Φ; Γ ⇑ s = t, Θ R synchronous introduction rules Φ; Γ B(µB) t ⇓ Φ; Γ µB t ⇓ unfoldR Φ; Γ ⇓ B(ν B) t R Φ; Γ ⇓ ν B t R unfoldL Φ; Γ ⇓ t = t R identity rules µB t ∈ Φ Φ; Γ µB t ⇓ initR Φ; Γ ⇓ ν B t ⇑ Φ ν B t initL Figure The µLJF focused proof system for intuitionistic logic with fixed points [START_REF] Baelde | Focused inductive theorem proving[END_REF]. This figure contains the new inference rules for least and greatest fixed points and equality which are added to the proof system in Figure 9.2. The style of encoding follows [START_REF] Blanco | Proof outlines as proof certificates: a system description[END_REF]. A new storage zone for frozen (least) fixed points, Φ, is added to all sequents and threaded throughout all existing inference rules. When a greatest fixed point is frozen, the entire right-hand side becomes frozen and can no longer be manipulated; this new restriction is represented by the storage-only annotation ⇑ Φ . Two new initial rules replace atoms with fixed points: an initial rule applies to a fixed point under focus if a frozen copy of the same fixed point is available on the opposite side of the sequent. Besides these changes, presentation conventions are shared with Figure 9.2. Ξ 1 : Γ ⇑ A, B, Θ R ∧ + c (Ξ 0 , Ξ 1 ) Ξ 0 : Γ ⇑ A ∧ + B, Θ R Ξ 1 : Γ ⇑ Θ R t + c (Ξ 0 , Ξ 1 ) Ξ 0 : Γ ⇑ t + , Θ R Ξ 1 : Γ ⇑ A ⇑ Ξ 1 : Γ ⇑ B ⇑ ∧ - c (Ξ 0 , Ξ 1 , Ξ 2 ) Ξ 0 : Γ ⇑ A ∧ -B ⇑ t - c (Ξ 0 ) Ξ 0 : Γ ⇑ t -⇑ Ξ 1 : Γ ⇑ A, Θ R Ξ 2 : Γ ⇑ B, Θ R ∨ c (Ξ 0 , Ξ 1 , Ξ 2 ) Ξ 0 : Γ ⇑ A ∨ B, Θ R f + c (Ξ 0 ) Ξ 0 : Γ ⇑ f , Θ R Ξ 1 : Γ ⇑ A B ⇑ ⊃ c (Ξ 0 , Ξ 1 ) Ξ 0 : Γ ⇑ A ⊃ B ⇑ (Ξ 1 y) : Γ ⇑ [y/x]B ⇑ ∀ c (Ξ 0 , Ξ 1 ) Ξ 0 : Γ ⇑ ∀x.B ⇑ (Ξ 1 y) : Γ ⇑ [y/x]B, Θ R ∃ c (Ξ 0 , Ξ 1 ) Ξ 0 : Γ ⇑ ∃x.B, Θ R synchronous introduction rules Ξ 1 : Γ A ⇓ Ξ 2 : Γ B ⇓ ∧ + e (Ξ 0 , Ξ 1 , Ξ 2 ) Ξ 0 : Γ A ∧ + B ⇓ t + e (Ξ 0 ) Ξ 0 : Γ t + ⇓ Ξ 1 : Γ ⇓ A i R ∧ - e (Ξ 0 , Ξ 1 , i) Ξ 0 : Γ ⇓ A 1 ∧ -A 2 R Ξ 1 : Γ A i ⇓ ∨ e (Ξ 0 , Ξ 1 , i) Ξ 0 : Γ A 1 ∨ A 2 ⇓ Ξ 1 : Γ A ⇓ Ξ 2 : Γ ⇓ B R ⊃ e (Ξ 0 , Ξ 1 , Ξ 2 ) Ξ 0 : Γ ⇓ A ⊃ B R Ξ 1 : Γ ⇓ [t /x]B R ∀ e (Ξ 0 , Ξ 1 , t ) Ξ 0 : Γ ⇓ ∀x.B R Ξ 1 : Γ [t /x]B ⇓ ∃ e (Ξ 0 , Ξ 1 , t ) Ξ 0 : Γ ∃x.B ⇓ Figure The augmented LJF a focused proof system for intuitionistic logic (Chihani et al., 2016b). Presentation conventions are shared with Figure 9.2. . . identity rules initL e (Ξ 0 ) Ξ 0 : Γ ⇓ N a N a (l, P a ) ∈ Γ initR e (Ξ 0 , l ) Ξ 0 : Γ P a ⇓ Ξ 1 : Γ ⇑ F ⇑ Ξ 2 : Γ ⇑ F ⇑ R cut e (Ξ 0 , Ξ 1 , Ξ 2 , F ) Ξ 0 : Γ ⇑ ⇑ R structural rules l, N ∈ Γ Ξ 1 : Γ ⇓ N R decideL e (Ξ 0 , Ξ 1 , l ) Ξ 0 : Γ ⇑ ⇑ R Ξ 1 : Γ P ⇓ decideR e (Ξ 0 , Ξ 1 ) Ξ 0 : Γ ⇑ ⇑ P Ξ 1 : Γ ⇑ P ⇑ R releaseL e (Ξ 0 , Ξ 1 ) Ξ 0 : Γ ⇓ P R Ξ 1 : Γ ⇑ N ⇑ releaseR e (Ξ 0 , Ξ 1 ) Ξ 0 : Γ N ⇓ Ξ 1 : l, C , Γ ⇑ Θ R storeL c (Ξ 0 , Ξ 1 , l ) Ξ 0 : Γ ⇑ C, Θ R Ξ 1 : Γ ⇑ ⇑ D storeR c (Ξ 0 , Ξ 1 ) Ξ 0 : Γ ⇑ D ⇑ Figure The augmented LJF a focused proof system for intuitionistic logic (continued). Presentation conventions are shared with Figure 9.2. The identity of each inference rule is immediate from its corresponding clerk or expert; names are therefore omitted. . asynchronous introduction rules Ξ 1 : Φ; Γ ⇑ S t, Θ R (Ξ 2 ȳ) : Φ; Γ ⇑ B S ȳ S ȳ ⇑ inductL c (Ξ 0 , Ξ 1 , Ξ 2 , S) Ξ 0 : Φ; Γ ⇑ µB t, Θ R † Ξ 1 : Φ; Γ ⇑ B(µB) t, Θ R unfoldL c (Ξ 0 , Ξ 1 ) Ξ 0 : Φ; Γ ⇑ µB t, Θ R Ξ 1 : l, µB t , Φ; Γ ⇑ Θ R freezeL c (Ξ 0 , Ξ 1 , l ) Ξ 0 : Φ; Γ ⇑ µB t, Θ R Ξ 1 : Φ; Γ ⇑ S t ⇑ (Ξ 2 ȳ) : Φ; ⇑S ȳ BS ȳ ⇑ inductR c (Ξ 0 , Ξ 1 , Ξ 2 , S) Ξ 0 : Φ; Γ ⇑ ν B t ⇑ † Ξ 1 : Φ; Γ ⇑ B(ν B) t ⇑ unfoldR c (Ξ 0 , Ξ 1 ) Ξ 0 : Φ; Γ ⇑ ν B t ⇑ Ξ 1 : Φ; Γ ⇑ ⇑ Φ ν B t freezeR c (Ξ 0 , Ξ 1 ) Ξ 0 : Φ; Γ ⇑ ν B t ⇑ { Ξ 1 : Φ; Γ ⇑ Θ R}σ σ ∈ CSU(s = t ) = c (Ξ 0 , Ξ 1 ) Ξ 0 : Φ; Γ ⇑ s = t, Θ R Figure The augmented µLJF a focused proof system for intuitionistic logic with fixed points [START_REF] Blanco | Proof outlines as proof certificates: a system description[END_REF]. Presentation conventions are shared with Figure 9.3. . . synchronous introduction rules Ξ 1 : Φ; Γ B(µB) t ⇓ unfoldR e (Ξ 0 , Ξ 1 ) Ξ 0 : Φ; Γ µB t ⇓ Ξ 1 : Φ; Γ ⇓ B(ν B) t R unfoldL e (Ξ 0 , Ξ 1 ) Ξ 0 : Φ; Γ ⇓ ν B t R = e (Ξ 0 ) Ξ 0 : Φ; Γ ⇓ t = t R identity rules l, µB t ∈ Φ initR e (Ξ 0 , l ) Ξ 0 : Φ; Γ µB t ⇓ initL e (Ξ 0 ) Ξ 0 : Φ; Γ ⇓ ν B t ⇑ Φ ν B t Figure The augmented µLJF a focused proof system for intuitionistic logic with fixed points (continued). Presentation conventions are shared with Figure 9.3. . The extension follows along the lines of Section 3.2. Notice that the new frozen storage, Φ, is indexed like Γ, but each is independent from the other. In particular, frozen fixed points are only selected from storage by the new initial-right rulewhich, like decide-left on stored, non-frozen formulas, uses a non-deterministic index in the FPC framework. The single formula that may be frozen on the right requires no index because there is at most one choice. Implementing the FPC framework for intuitionistic logics as kernels also proceeds as a natural extesion of the treatment of LKF a in Section 4.4, although new technical considerations will come into play. They will be discussed in Section 10.3. A point of interest in using these kernels is the profusion of sources of nonterminating behavior-notably, by the inductive rules (including unfolding on both sides) of least and greatest fixed points. Relatedly, many more inference rules are in conflict with respect to LKF a -where only decide and cut could be applied under the same conditions-; again, fixed points are the new source of conflict with the set of three rules which can be applied asynchronously on each kind of fixed point. An FPC definition will need to carefully orchestrate the operation on fixed points and their operations or risk copious backtracking and even nontermination. As in the case of LKF a by Theorem 3.2.2, usage of the augmented system is justified by a simple soundness guarantee formulated in terms of erasure of the augmentations of the FPC framework. This protects the system from anomalous behavior in client-supplied FPC definitions, as in the example of nonterminating unfolding above. Theorem The system µLJF a is sound w.r.t. intuitionistic logic with fixed points ( µLJ). Proof. The µLJF system can be recovered from µLJF a by removing all the augmentations (marked in Figures 9.4 and 9.6), and therefore every proof of µLJF a is also a proof of µLJF: this is the soundness guarantee. The result follows from Theorem 9.2.1. Nominal abstraction The final extension to the logic, complementary to the addition of fixed points and equality, is nominal abstraction. In specifying and reasoning about structures, it is common to rely on a recursive traversal on inductive types; in many interesting cases, these constructs involve a notion of binding-pervasive, for example, in the naming effects of quantifiers in logic, and in foundational aspects programming language syntax such as variables and their scope. When inspecting terms of such types, we recurse inside the binders of a (globally) closed terms and consequently need to consider (locally) open terms. A standard technique to model the dynamic behavior of binders (as opposed to their static structure, reflected in the terms proper), involves the addition of an evaluation context recording open binders as the structure is recursed. At the level of the meta-logic used to write the aforementioned specifications, quantifiers are the primitives that model binding. Indeed, universal quantification displays some desirable traits to perform this kind of reasoning. In the intensional interpretation adopted in the sequent calculus since Gentzen's original designs (e.g., Figure 2.3), the corresponding introduction rule quantifier-read bottom-upstates that to prove the universal quantifier it suffices to prove the formula where the bound variable has been substituted with a new eigenvariable: a fresh, unused variable at the appropriate type, unused elsewhere in the proof, which represents a generic instance of the type. For Gentzen, eigenvariables are immutable and unaffected by variable substitutions. However, the introduction of fixed points and equality-and the style of direct reasoning on logic specifications-turns eigenvariables into sites for substitution (in particular, equality on the left-hand side involves unification of eigenvariables). Example Suppose there is a property P that takes two arguments. Under the reading of eigenvariables as fresh names, a proof of ∀x.∀y.P x y involves two different names, x and y. Furthermore, a proof of ∀z .P z z involves just one name in the proof of P . However, consider the following implication: ∀x.∀y.P x y ⊃ ∀z .P z z. Although this is logically valid, it can be interpreted under the conception of eigenvariables as fresh names as stating that if there is a proof of P using two different names there is a proof of P with a single name-which strays from the intended meaning of the specification. Hence, the treatment of logic until this point conflates the concepts of universal quantification and generic judgment-by having the universal quantifier assume these two sets of incompatible competencies. The distinction between the two can be addressed directly within the logic by various means. Among these, we use the concept of nominal abstraction embodied by a new nabla quantifier (signified ∇) developed by Miller andTiu (2002, 2005) to represent generic judgments, i.e., statements relying on the declaration of fresh local variables. The nabla quantifier extends sequents with an explicit representation of local context. Consider the standard sequent notation from Section 2.4 (with the intuitionistic restriction integrated): Σ : A 1 , . . . , A n A 0 Here we have made explicit the eigenvariable signature Σ containing the set of eigenvariables introduced, say, by the asynchronous rules on quantifiers in Figure 9.2 (routinely signaled by the proviso † used to represent the eigenvariable restriction)-Section 11.4 discusses an explicit encoding of this signature and its limitations. Each formula in the sequent must now be decorated with a local context σ, similar to the global context Σ but with scope limited to its corresponding formula and containing the set of locally fresh variables (in effect, turning sequents into binding structures): Σ : σ 1 A 1 , . . . , σ n A n σ 0 A 0 The introduction rules for nabla are shown, for intuitionistic sequent calculus, in Figure 9.8. Note that there is one introduction rule on the left and one on the right, both identical in their treatment of the affected formula and its context-and leaving the rest of the sequent intact. When focusing is applied, nabla exhibits a self-duality which is reflected in the duplication of both introduction rules in both asynchronous and synchronous phases. That is, nabla is unaffected by focusing, a fundamentally neutral connective. The additions are compatible with the extension to µLJF made in Figure 9.3. The same Figure 9.8 represents the additions made in LJF a (and µLJF a ). In its dedicated purpose of reasoning about generic judgments, the nominal quantifier is treated eagerly by the kernel. No clerks and experts are used, and its treatment always succeeds. In an implementation of the FPC framework, this process is carried out by the kernel, unbeknownst to the client side. Implementation issues are studied in Section 10.3. Notes The development of the proof theory of fixed points springs from a line of research that studies the use of definitions and induction (McDowell andMiller, 2000, 2002;[START_REF] Momigliano | Induction and co-induction in sequent calculus[END_REF][START_REF] Miller | A proof theory for generic judgments[END_REF][START_REF] Tiu | Cut elimination for a logic with induction and co-induction[END_REF]. The use of fixed points was originally studied in the context of linear logic, where it constitutes an alternative to the exponentials for the modeling of Γ ⇑ σ, (y : τ) [y/x]B ⇑ Γ ⇑ σ ∇(x : τ).B ⇑ † Γ ⇑ σ, (y : τ) [y/x]B, Θ R Γ ⇑ σ ∇(x : τ).B, Θ R † synchronous introduction rules Γ ⇓ σ, (y : τ) [y/x]B R Γ ⇓ σ ∇(x : τ).B R † Γ σ, (y : τ) [y/x]B ⇓ Γ σ ∇(x : τ).B ⇓ † Figure The LJF focused proof system for intuitionistic logic augmented with nominal quantification. In this presentation, the global context Σ is explicitly maintained, and local contexts are extended to all formulas. These changes are threaded throughout all other inference rules, although local contexts are only manipulated by the rules in this figure. All zones previously involving formulas are likewise augmented so that their members are pairs of formulas and their contexts. The freshness restriction † applies to all inference rules with respect to the freshly introduced and locally scoped nominal constant, y. Presentation conventions are shared with Figure 9.2. unbounded behavior (Baelde and Miller, 2007;[START_REF] Baelde | A linear approach to the proof-theory of least and greatest fixed points[END_REF][START_REF] Baelde | On the proof theory of regular fixed points[END_REF][START_REF] Baelde | Least and greatest fixed points in linear logic[END_REF]. These results-extended to intuitionistic logic-provide the justification for designing proof systems in this way, and form the proof theoretical core of the development of reasoning systems like Bedwyr and Abella, to which the next chapter is devoted. An interesting question is whether results such as Theorem 9.3.1 extend to classical logic as well. This point is unknown: it is certainly possible to obtain a classical system with added fixed points, say, µLK, but the extension of focusing to a tentative µLKF remains an open problem. The addition of least and greatest fixed points brings a pair of new initial rules: such formulas can function as "defined" atoms and eliminate the need for a separate category of proper ("undefined") atoms, such as has been used assumed throughout Part II. Nevertheless, undefined atoms can be associated a more natural notion of genericity. Both defined and undefined atoms with their corresponding sets of inference rules can coexist in the same proof system instead of discarding one for the other. The common kinds of sequents encountered in proofs involving undefined and defined atoms differ: in the former case, sequents with large numbers of small formulas abound; in the latter, sequents have fewer, substantially larger formulas. At the root is the definitional nature of fixed points: for example, suppose we define multiplication in terms of iterated addition. Figure 9.1 presents a least fixed point expression for addition; the corresponding least fixed point for multiplication follows a similar structure, but it moreover inlines the fixed point for addition. This growth in the size of individual expressions continues as more complex definitions are composed from existing ones. Purely positive fixed points, where every connective is positive, occur commonly in logic specifications-such as that in Figure 9.1. In this context, focusing on a purely positive (least) fixed point on the right typically corresponds to the concept of performing a computation as part of a proof. When this is the case, the focus is never released and the proof succeeds or fails based on the ability of the specification to perform the required computation. For example, according to the aforementioned specification, Γ plus 2 2 4⇓ will succeed-and in logic programming, since the third argument is functionally determined by the other two, the result of the addition operation can be computed via proof search-; conversely, Γ plus 2 2 5⇓ will fail. This relationship has been studied in closer detail by [START_REF] Gérard | Separating functional computation from relations[END_REF]. The nominal abstraction presented in this chapter is the full development of the nabla quantifier, which culminates in the work of [START_REF] Miller | A proof theory for generic judgments[END_REF]; Gacek et al. (2008b[START_REF] Gacek | Nominal abstraction[END_REF]: this is the theory used in systems like Abella. A minimal presentation, which removes some of the standard properties of the connective, is developed in Baelde (2008a); we shall not consider it further here. Binding can be modeled by other, alternative means, but nominal abstraction is both powerful and convenient; Baelde et al. (2014, Section 6) provides a practical overview. We revisit the problem in Section 12.3. Section 12.5 also returns to the question of negation-as-failure first observed in the rules for equality as a logical connective. The effects of equality in proof search [START_REF] Viel | Proof search when equality is a logical connective[END_REF] are closely related to unification, which itself will be a subject of further discussion in Section 13.2. Automating logic Traditionally, model checking has been seen as separate from theorem proving. Where theorem proving revolves around the concept of provability, model checking [START_REF] Clarke | Model Checking[END_REF][START_REF] Baier | Principles of model checking[END_REF][START_REF] Grumberg | 25 years of model checking: history, achievements, perspectives[END_REF] considers satisfiability under a certain model. However, the extension of standard logics with fixed points unifies both views and allows model checking to be interpreted in terms of deduction, i.e., as a specific kind of theorem proving activity. It does so by observing that the exploration of fixed points captures both finite success and finite failure. In Part II, predicates were formally undefined (i.e., they did not have an associated introduction rule); instead, atoms were defined by a theory which specified how to derive conclusions from them. The introduction of fixed points enables the definition of recursive definitions directly within the logic. The definitions thus embedded are constant throughout proof search; this fact marks the move from the open-world to the closed-world assumption. The Bedwyr system generalizes standard logic programming through the implementation of a fragment of the logics described in Chapter 9 that is nonetheless amenable to automation [START_REF] Tiu | Mixing finite success and finite failure in an automated prover[END_REF]Baelde et al., 2007). The logic is organized in two levels, in such a way that all rules on the left are invertible and, in consequence, proof search alternates between the left and the right sides while giving preference to the left side, whose nondeterminism is by construction of the don't-care variety. The resulting language can be expressed compactly by the following grammar: L 0 ::= t | A | L 0 ∧ L 0 | L 0 ∨ L 0 | ∃x.L 0 | ∇x.L 0 L 1 ::= t | A | L 1 ∧ L 1 | L 1 ∨ L 1 | ∃x.L 1 | ∇x.L 1 | ∀x.L 1 | L 0 ⊃ L 1 . Here, each predicate A (taking the place of atomic formulas in Chapter 4) is classified as belonging to either L 0 or L 1 . Moreover, predicates-encoded as fixed points-are stratified so that a definition may refer only to predicates at lower levels (of stratification, not to be confused with the two levels of the logic). Goals can be drawn from both L 0 and L 1 . The restriction that forbids nested implications splits proof reconstruction in two levels and consequently two specialized provers, one for each class of formulas. The L 0 prover corresponds to a simplified version of λProlog extended to allow nabla in the body of clauses. In addition to adopting the observations about the organization of proof search in phases made above, the L 1 prover must treat the case of the implication G 0 ⊃ D 1 . It does so in two steps: 1. First, attempt to prove the L 0 goal-where L 1 eigenvariables are treated as sites for substitution, i.e., L 0 logic variables, and L 1 logic variables are disallowed. 2. For every solution to G 0 , apply its set of substitutions to D 1 and proceed to find a proof under those. As with similar rules, and empty set of solutions vacuously results in success. Thus, in this logic, failure to prove a goal corresponds to a proof of the negation of the goal. Formulated in terms of standard, depth-first proof search, the state exploration associated to model checking properties can derive inefficient search and redundant treatment of goals. These particularities can be accommodated by the addition of tabling of proved goals reflecting the provability relation between entries in the tables [START_REF] Miller | Incorporating tables into proofs[END_REF][START_REF] Miller | Extracting proofs from tabled proof search[END_REF]. Both finite success and finite failure can be tabled. Other, more expressive logics may not lend themselves well to full automation, but their automation is regardless of great interest. In front of programs such as automated theorem provers and model checkers is the group of tools called proof assistants (also, interactive theorem provers). Ultimately, the objective of both families of tools is the same-proving theorems by building formal proofs in a given logic-, but proof assistants rely on their users for instructions on how to build proofs (and may attempt to discharge simple goals by automated proving techniques). Their uses range from the formal verification of softwareas undertaken in Section 6.3-to the rigorous proof of mathematical results so complex that they resist manual analysis, among these famously the four color theorem [START_REF] Gonthier | A computer checked proof of the four colour theorem[END_REF] and the Kepler conjecture [START_REF] Thomas | A formal proof of the kepler conjecture[END_REF]. A more powerful logic than the one implemented by Bedwyr is G [START_REF] Gacek | Nominal abstraction[END_REF], which is the core of the Abella proof assistant [START_REF] Gacek | The Abella interactive theorem prover (system description)[END_REF][START_REF] Baelde | Abella: A system for reasoning about relational specifications[END_REF]. It shares with Bedwyr the two-level approach, which distinguishes a reasoning level (presented in the next section) and a specification level, which is implemented as a subset of λProlog. The rich logic with support for the λtree syntax approach makes it well suited to model and study the metatheory of programming languages (Gacek et al., 2008a[START_REF] Gacek | A two-level logic approach to reasoning about computations[END_REF][START_REF] Wang | Reasoning about higher-order relational specifications[END_REF]. In fact, the G logic is very close to µLJF and its aggregated extensions from Chapter 9. Kernels and typical programs we will write in the reasoning level are compatible with the common subset of Abella and Bedwyr, which we call Bedwyr 0 , and can be written in such a way that they are valid specifications in both systems simultaneously. The rest of the chapter is organized as follows: Section 10.2 provides a tutorial introduction to Abella and Bedwyr and outlines a common dialect that can be used to write compatible specifications for both systems. Section 10.3 expounds the kernels that implement the FPC framework for the logics used in this part (namely, µLJF a ). Section 10.4 provides some additional examples. Section 10.5 concludes the chapter. Abella For the most part, the reasoning level of Abella coincides syntactically with Bedwyr-and semantically in the common fragment shared by both. Atomic types are defined by the keyword Kind and type constructors by the keyword Type; note that kind expressions make use of a lowercase type; currently, more complex kind expressions are not supported. For example: Kind nat type. Type z nat. Type s nat -> nat. The type of formulas at the reasoning level is called prop (as opposed to o at the specification level). The following logical constants are given: 1. true of type prop, for t. 8. nabla of type (A -> prop) -> prop, for ∇. 9. = of type A -> A -> prop, for =. Note that Abella's implication is the implication at L 1 in the previous section and not the hypothetical implication of Section 4.3. A key difference with respect to λProlog is that predicates in Abella are defined as definitions with target type prop, all of whose clauses must be given at definition time, with clauses being separated with ; and terminated with ., and the head and the body of a clause being separated with :=. Inductive definitions are given by Define and coinductive definitions are given by CoDefine declarations. Other syntactic conventions resemble those of λProlog. For example, for the inductive definition of natural numbers: In λProlog, a predicate always fails if the theory defines no clauses for it. In Abella, this behavior must be made explicit through a clause in the definition: Define undefined : nat -> nat -> prop by undefined X Y := false. Inductive and coinductive definitions correspond to fixed points, respectively least and greatest (indeed, we will make use of an explicit correspondence in Section 13.4). Theorems and proofs are introduced by the Theorem environment, which gives name to a formula and follows by the description of a proof by a script written in the language of tactics of Abella. These are not too relevant to the present discussion; refer to the tutorial [START_REF] Baelde | Abella: A system for reasoning about relational specifications[END_REF] for details. Modular support in Abella is very limited. At the beginning of a development, a specification written in a pure subset of λProlog can be loaded by the keyword Specification and specification-level predicates referenced by enclosing them in curly brackets. Although the specification is written in λProlog, the closedworld assumption is enforced on this level once Abella finishes loading it. However, composition of λProlog modules is supported by the standard mechanism of accumulation. For example, if the code in Figure 4.2 is contained in a module named test, it could be used as follows: Specification "test". Theorem zero_is_natural : { is_nat z }. search. The equivalent specification at the reasoning level is given in Figure 11.1. The programming languages defined by Abella and Bedwyr are very closely related. In order to write code that is valid under both systems, a few important differencesmostly additional restrictions in Abella-must be noted: 1. Existential variables that do not appear in the head of a clause must be explicitly declared in Abella by exists. Such variables can be left implicit in Bedwyr, as they commonly are in λProlog, by resorting to the syntactic convention of uppercase identifiers. 2. Anonymous variables are not supported in Abella, and instead explicit named identifiers must be given. In Bedwyr, they can be written _. 3. Polymorphic types are not supported in Abella beyond those predefined by the logical constants (quantifers and equality). Hence, for example, separate list types and constructors must be defined for each type of list-as defined by the type of its elements. In Bedwyr they are written as in full λProlog. 4. Type constructors in Abella are always prefix. Bedwyr has conventions to define infix operators out of sequences of special characters, such as |= and ++, which also determine associativity. 5. The predefined signature in Abella is essentially empty: everything needs to be built from scratch. Bedwyr defines string and natural literals with equality only. 6. Some meta-commands available in Bedwyr only support operations like file inclusion and assertions about finite success and finite failure-which have no direct correspondence in the stronger logic of Abella. These features need to be mimicked by external preprocessors and other Abella constructs. Of this list, only the lack of polymorphic types is profound, although an experimental extension by Yuting Wang is currently in development. These restrictions also extend to the λProlog interpreter in predictable ways, which does not at the moment implement the language in full-extended to the standard library of predefined types and their predicates, which is absent from the interpreter: everything in a development must be explicitly defined. We should also note that Bedwyr provides an incomplete implementation of higher-order pattern unification, which can lead to unexpected failures in correct code. Throughout Part III we shall sometimes, for the sake of conciseness in presentation, resort to the Bedwyr flavor of syntax, which more closely resembles the logic programs presented in λProlog. FPC kernels As an original development, adding to the existing kernels in λProlog, we implemented a family of kernels based on µLJF a and its extensions, on which to base further experimentation. As in Section 4.4, it also serves to present an interesting use case for Abella as a logic programming language. Here we present the basic version of the kernel. Figure 10.1 shows the encoding of the logic in Abella. There is no separation between the signature and the definition of the module, nor built-in mechanisms for module composition. Unlike in λProlog, composing a logic program from its constituent "modules" must carefully track the order of the dependencies between them-a more primitive endeavor. Based on the logic thus defined, Figure 10.2 shows how the fixed point definitions in Figure 9.1 can be encoded in Abella. These fixed points are here named as Definitions of Abella, which allows us to refer to them symbolically, say, when defining multiplication in terms of addition without having to write all definition strata every time. Thus, we could write: Needless to say, this style is unwieldy and we would ideally like to avoid it altogether, relying instead on writting something like Abella definitions and having the fixed points generated automatically. Section 13.4 discusses how to use Abella to program µLJF. Figure 10.3 shows the centerpiece of the development: the µLJF a kernel in its basic, declarative version; it relies on standard declarations and logic programmings while presenting some features of interest. First, note that-unlike in λProlog-storage zones must be explicitly represented as part of the sequents, since hypothetical judgments do not allow us to grow the model at runtime under the open-world assumption. Second, there is a single instance of implication in the kernel (as there is in the LKF a kernel); its role is not to file formulas in storage, but to treat equality on the left. In kernels with and without fixed points alike, these are the only instances of implication we have observed: practical clerks and experts are operationally simple programs which make no use of such advanced features-in fact, clerks and experts in Abella can be equivalently programmed in λProlog at the specification level with minimal changes, none profound. Unlike in λProlog, because of the closed-world assumption, a signature for clerks and experts independent from their definitions cannot be given, though it is implied by the kernel. If it were, or if clerks and experts were written in λProlog, it would be a variation on Figure 10.6. A small number of secondary decisions are more dependent on the design of the particular implementation. For example, the development version of the kernel extends sequents with bookkeeping structures and spy harnesses to facilitate debugging. The ordering of the inference rules, in particular those that are in conflict, has a potentially large performance impact on proof search-in the kernel of Figure 10.3 fixed points operations have the following priorities: initial, unfold, induction, and freezing (subject to concrete FPC restrictions); when lemmas are added in Section 11.4, they will be placed after the standard decide rule. An interesting detail is that the implication on the left is treated by two distinct experts, each recursing on the premises in a different order; we used these to assess their impact in proof search, commonly finding advantageous to treat the antecedent first-under Bedwyr's incomplete handling of unification, this was often the only feasible option. To conclude the section, we explore two extensions to the declarative kernel. In first place, in our encoding of the µLJF logic in Figure 10.1, we were able to devise a clever encoding that allowed us to express fixed points with arbitrary % Xi Phi Gamma % Delta Goal Define async : cert -> list_ctx -> list_ctx -> list_bool -> goal -> prop, syncL : cert -> list_ctx -> list_ctx -> bool -> goal -> prop, syncR : cert -> list_ctx -> list_ctx -> bool -> prop by async Xi Phi Gamma (cons_bool (and P Q) Delta) G := exists Xi', andClerk Xi Xi' /\ async Xi' Phi Gamma (cons_bool P (cons_bool Q Delta)) G ; async Xi Phi Gamma (cons_bool (or P Q) Delta) G := exists Xi' Xi'', orClerk Xi Xi' Xi'' /\ async Xi' Phi Gamma (cons_bool P Delta) G /\ async Xi'' Phi Gamma (cons_bool Q Delta) G ; async Xi Phi Gamma nil_bool (unk (imp P Q)) := exists Xi', impClerk Xi Xi' /\ async Xi' Phi Gamma (cons_bool P nil_bool) (unk Q) ; async Xi Phi Gamma (cons_bool ff Delta) G := ffClerk Xi ; async Xi Phi Gamma (cons_bool tt Delta) G := exists Xi', ttClerk Xi Xi' /\ async Xi' Phi Gamma Delta G ; async Xi Phi Gamma nil_bool (unk (all P)) := exists Xi', allClerk Xi Xi' /\ forall x, async (Xi' x) Phi Gamma nil_bool (unk (P x)) ; async Xi Phi Gamma (cons_bool (some P) Delta) G := exists Xi', someClerk Xi Xi' /\ forall x, async (Xi' x) Phi Gamma (cons_bool (P x) Delta) G ; async Xi Phi Gamma (cons_bool (eq P Q) Delta) G := exists Xi', eqClerk Xi Xi' /\ ((P = Q) -> async Xi' Phi Gamma Delta G) ; Figure The µLJF a kernel in Abella. async Xi Phi Gamma (cons_bool (mu B T) Delta) G := exists Xi', unfoldLClerk Xi Xi' /\ async Xi' Phi Gamma (cons_bool (B (mu B) T) Delta) G ; async Xi Phi Gamma (cons_bool (mu B T) Delta) G := exists Xi' Xi'' S, indClerk Xi Xi' Xi'' S /\ async Xi' Phi Gamma (cons_bool (S T) Delta ) G /\ forall x, async (Xi'' x) Phi Gamma (cons_bool (B S x) nil_bool) (unk (S x)) ; async Xi Phi Gamma (cons_bool (mu B T) Delta) G := exists Xi' Idx, freezeLClerk Xi Xi' Idx /\ async Xi' (cons_ctx (kvp Idx (mu B T)) Phi) Gamma Delta G ; async Xi Phi Gamma nil_bool (unk (nu B T)) := exists Xi', unfoldRClerk Xi Xi' /\ async Xi' Phi Gamma nil_bool (unk (B (nu B) T)) ; async Xi Phi Gamma nil_bool (unk (nu B T)) := exists Xi' Xi'' S, coindClerk Xi Xi' Xi'' S /\ async Xi' Phi Gamma nil_bool (unk (S T)) /\ forall x, async (Xi'' x) Phi nil_ctx (cons_bool (S x) nil_bool) (unk (B S x)) ; async Xi Phi Gamma nil_bool (unk (nu B T)) := exists Xi', freezeRClerk Xi Xi' /\ async Xi' Phi Gamma nil_bool (frz (nu B T)) ; syncR Xi Phi Gamma (and P Q) := exists Xi' Xi'', andExpert Xi Xi' Xi'' /\ syncR Xi' Phi Gamma P /\ syncR Xi'' Phi Gamma Q ; syncR Xi Phi Gamma (or P Q . syncL Xi Phi Gamma (all P) G := exists Xi' T, allExpert Xi Xi' T /\ syncL Xi' Phi Gamma (P T) G ; syncR Xi Phi Gamma (some P) := exists Xi' T, someExpert Xi Xi' T /\ syncR Xi' Phi Gamma (P T) ; syncR Xi Phi Gamma (eq T T) ) := exists Xi' C, orExpert Xi Xi' C /\ ( (C = left /\ syncR Xi' Phi Gamma P) \/ (C = right /\ syncR Xi' Phi Gamma Q) ) ; syncL Xi Phi Gamma (imp P Q) G := exists Xi' Xi'', impExpert Xi Xi' Xi'' /\ syncL Xi' Phi Gamma Q G /\ syncR Xi'' Phi Gamma P ; syncL Xi Phi Gamma (imp P Q) G := exists Xi' Xi'', impExpert' Xi Xi' Xi'' /\ syncR Xi'' Phi Gamma P /\ syncL Xi' Phi Gamma Q G ; syncR Xi := eqExpert Xi ; syncL Xi Phi Gamma (nu B T) (frz (nu B T)) := initLExpert Xi ; syncL Xi Phi Gamma (nu B T) G := exists Xi', unfoldLExpert Xi Xi' /\ syncL Xi' Phi Gamma (B (nu B) T) G ; syncR Xi Phi Gamma (mu B T) := exists Idx, initRExpert Xi Idx /\ member_ctx (kvp Idx (mu B T)) Phi ; syncR Xi Phi Gamma (mu B T) := exists Xi', unfoldRExpert Xi Xi' /\ syncR Xi' Phi Gamma (B (mu B) T) ; async Xi Phi Gamma (cons_bool C Delta) G := exists Xi' Idx, negative C /\ storeLClerk Xi Xi' Idx /\ async Xi' Phi (cons_ctx (kvp Idx C) Gamma) Delta G ; async Xi Phi Gamma nil_bool (unk G) := exists Xi', positive G /\ storeRClerk Xi Xi' /\ async Xi' Phi Gamma nil_bool (sto G) ; async Xi Phi Gamma nil_bool G := exists Xi' Idx C ?1, (G = (sto ?1) \/ G = (frz ?1)) /\ decideLClerk Xi Xi' Idx /\ member_ctx (kvp Idx C) Gamma /\ syncL Xi' Phi Gamma C G ; async Xi Phi Gamma nil_bool (sto G) := exists Xi', decideRClerk Xi Xi' /\ syncR Xi' Phi Gamma G ; syncL Xi Phi Gamma C G := exists Xi', positive C /\ releaseLExpert Xi Xi' /\ async Xi' Phi Gamma (cons_bool C nil_bool) G ; syncR Xi Phi Gamma G := exists Xi', negative G /\ releaseRExpert Xi Xi' /\ async Xi' Phi Gamma nil_bool (unk G). Figure The µLJF a kernel in Abella (finished). . . Type andClerk cert -> cert -> prop. Type impClerk cert -> cert -> prop. Type ffClerk cert -> prop. Type ttClerk cert -> cert -> prop. Type allClerk cert -> (i -> cert) -> prop. Type someClerk cert -> (i -> cert) -> prop. Type eqClerk cert -> cert -> prop. Type unfoldLClerk cert -> cert -> prop. Type unfoldRClerk cert -> cert -> prop. Type freezeRClerk cert -> cert -> prop. Type orClerk cert -> cert -> cert -> prop. Type freezeLClerk cert -> cert -> idx -> prop. Type indClerk cert -> cert -> (i -> cert) -> (i -> bool) -> prop. Type coindClerk cert -> cert -> (i -> cert) -> (i -> bool) -> prop. Type andExpert cert -> cert -> cert -> prop. Type orExpert cert -> cert -> choice -> prop. Type impExpert cert -> cert -> cert -> prop. Type impExpert' cert -> cert -> cert -> prop. Type ttExpert cert -> prop. Type allExpert cert -> cert -> i -> prop. Type someExpert cert -> cert -> i -> prop. Type eqExpert cert -> prop. Type initLExpert cert -> prop. Type unfoldLExpert cert -> cert -> prop. Type initRExpert cert -> idx -> prop. Type unfoldRExpert cert -> cert -> prop. Type decideLClerk cert -> cert -> idx -> prop. Type decideRClerk cert -> cert -> prop. Type releaseLExpert cert -> cert -> prop. Type releaseRExpert cert -> cert -> prop. Type storeLClerk cert -> cert -> idx -> prop. Type storeRClerk cert -> cert -> prop. Figure The FPC signature of µLJF a in Abella. Clerk and expert predicates must adhere to this hypothetical specification. If these predicates are given in λProlog, it suffices to change the type constructors to type and the target types to o; in this case, the separate signature can be given. numbers of arguments, therefore obtaining a universal encoding of the connectives. When it comes to quantification, we are not so lucky. In the kernel for the µLJF a logic-as with LKF a before it-the object logic features quantification over the type of terms only. Even if all primitive kinds are reflected on the type i, arrow types cannot be modeled by this device alone-in fact, simple types are required to study the metatheory of interesting languages, as well as formalisms like the π-calculus. In the absence of polymorphism in Abella, a succinct encoding is not possible. A workable if unsatisfying solution involves the definition of separate sets of quantifiers at different types (using exclusively i and the arrow), encoding the inference rules for quantifiers in the kernel once for each type at which quantification is supported, and likewise cloning clerk and expert definitions. A summary of changes is given in Figure 10.7. In second and last place, the addition of nominal abstraction to the logic, discussed in Section 9.4, must also be reflected in the kernel. For this we use a technique through which nominal variables are explicitly represented as local context with scope at the level of formulas. The encoding we used is based on the explicit representation of sequents developed by [START_REF] Miller | Encoding generic judgments[END_REF]; [START_REF] Mcdowell | Reasoning with higher-order abstract syntax in a logical framework[END_REF]-we retain the terminology even though in our context it is no longer used to record eigenvariables. The resulting kernel is shown in Figure 10.8. Under this regime, formulas are abstracted over a type which represents nominal variables as projections over the term of types of a counter stack: fst rst, fst rst rst, etc. When pattern matching a formula at the head (i.e., conclusion) of an inference rule, its components are themselves formula abstractions to which the abstraction variable is propagated. This extends to every inference rule in the system, with some representative examples detailed in Figure 10.9. All four inference rules for nabla, presented in Figure 9.8, are treated identically and transparently-in the sense that their are invisible to the FPC framework. In each case, a new nominal variable is generated by projecting the stack of rst to the type of terms through i, and a new rst is added to the stack for the continuation. However, pattern matching in this encoding generates unification problems that fall outside the fragment of pattern unification supported by Abella and Bedwyrwhich, failing to find a solution, will be unable to perform any checking with this kernel. Despite this difficulty, the problems have simple solutions, which an extension of the unification framework (presented in Section 13.2) restores. %% Logic % Argument coercions for argument lists Type arg_ii (i -> i) -> i. % Logical constants Type all_ii, some_ii ((i -> i) -> bool) -> bool. % Polarity snippets negative (all_ii P) ; positive (some_ii P) ; %% Mocked FPC signature Type allExpert_ii cert -> cert -> (i -> i) -> prop. Type someExpert_ii cert -> cert -> (i -> i) -> prop. Type allClerk_ii cert -> ((i -> i) -> cert) -> prop. Type someClerk_ii cert -> ((i -> i) -> cert) -> prop. %% Kernel syncL Xi Phi Gamma (all_ii P) G := exists Xi' T, allExpert_ii Xi Xi' T /\ syncL Xi' Phi Gamma (P T) G ; syncR Xi Phi Gamma (some_ii P) := exists Xi' T, someExpert_ii Xi Xi' T /\ syncR Xi' Phi Gamma (P T) ; syncL Xi Phi Gamma (all_ii P) G := exists Xi' T, allExpert_ii Xi Xi' T /\ syncL Xi' Phi Gamma (P T) G ; syncR Xi Phi Gamma (some_ii P) := exists Xi' T, someExpert_ii Xi Xi' T /\ syncR Xi' Phi Gamma (P T) ; 10.7 Figure Extensions to the encoding of the µLJF a logic to support polymorphic quantification in Abella. For each supported type, new connectives with their polarities, argument list contents, clerks and experts, and type-specific instances of the general inference rules are cloned. Here we show the additions for quantification over the arrow type i → i. % Object-level encoding of nabla, without polarity Type nabl (i -> bool) -> bool. % Kernel with treatment of nabla . . Define async : cert -> list_ctx -> list_ctx -> list_bool -> goal -> prop, syncL : cert -> list_ctx -> list_ctx -> (evs -> bool) -> goal -> prop, syncR : cert -> list_ctx -> list_ctx -> (evs -> bool) -> % Some standard and interesting cases syncR Xi Phi Gamma (l\ and (P l) (Q l)) := exists Xi' Xi'', andExpert Xi Xi' Xi'' /\ syncR Xi' Phi Gamma P /\ syncR Xi'' Phi Gamma Q ; async Xi Phi Gamma nil_bool (unk (l\ all (P l))) := exists Xi', allClerk Xi Xi' /\ forall x, async (Xi' x) Phi Gamma nil_bool (unk (l\ P l x)) ; syncR Xi Phi Gamma (l\ some (P l)) := exists Xi' T, someExpert Xi Xi' T /\ syncR Xi' Phi Gamma (l\ P l T) ; async Xi Phi Gamma (cons_bool (l\ mu (B l) (T l)) Delta) G := exists Xi', unfoldLClerk Xi Xi' /\ async Xi' Phi Gamma (cons_bool (l\ (B l) (mu (B l)) (T l)) Delta) G ; async Xi Phi Gamma (cons_bool (l\ mu (B l) (T l)) Delta) G := exists Xi' Xi'' S, indClerk Xi Xi' Xi'' S /\ async Xi' Phi Gamma (cons_bool (l\ S (T l)) Delta ) G /\ forall x, async (Xi'' x) Phi Gamma (cons_bool (l\ (B l) S x) nil_bool) (unk (l\ S x)) ; 10.9 Figure Extensions to the µLJF a kernel written in Abella to support nabla (continued). Examples An interesting example adapts the pairing combinator introduced in Section 5.2 for the LKF a system, to the present µLJF a in Figure 10.10. The port is completely straightforward and presents no difficulties. It is important to note that a definition of the pairing clerks and experts in Abella is, by itself, useless-it requires other certificate definitions upon which to operate. In this closed world, then, it is not possible to write a self-contained definition of the FPC that is at the same time capable of interacting with the FPCs that use it. Consequently FPC definitions in Abella appear more (syntactically) complex than they (semantically) are. The solution to this problem is to use the specification level and write and compose definitions as λProlog modules by one of two means: 1. Modifying the kernel to accept FPC definitions at the specification level instead of at the reasoning level. 2. Generating FPC definitions at the reasoning level from definitions at the specification level by a preprocessor. Either way, the specification in λProlog is not only much shorter, but also more legible and modular. Because µLJF a is a two-sided calculus, the size of an FPC definition roughly doubles that of a similar one-sided calculus; this point is taken up again in Section 12.4. Notes Some of our developments on FPC kernels originate in work on Bedwyr and then ported to Abella, which is the one system that remains in active development. Bedwyr supports automation of proof search to a greater degree; aspects of this behavior could be built into Abella, to which end Chapter 13 offers an advanced preliminary study. The µLJF a kernel was originally developed for the work presented in [START_REF] Blanco | Proof outlines as proof certificates: a system description[END_REF] and subsequently refined. Support for nabla was added with a view towards the work presented in Blanco et al. (2017b). In our discussion of quantifier polymorphism and the applicable workarounds, the battery of changes is limited to the µLJF a system. When we exercise the proof system indirectly by writing Abella programs and reifying them into µLJF a , this last step-for which refer to Section 13.4-will also need to be extended. Some versions of LKF a and LJF a kernels written in λProlog (Chihani et al., 2016b) make use of this language's polymorphic features, though all the versions contemplated in this work are monomorphic. A point by which we have set little store is the definition of a certificate transformer at the entry point of the kernel which is charged with performing marshaling. This functionality is supplied to streamline the definition of compact, initial forms of certificates, which are expanded to their full form before the start of checking proper. This is little more than a convenience, but removes from the user the burden of initializing bookkeeping structures in which they as clients have no direct interest. Marshaling will be used to define compact outline formats in Chapter 11. By default, the marshaling predicate can simply be taken to be the identity relation. Proof scripts in Abella apply tactics to indirectly invoke the inference rules of the underlying logic G. Upon success, a proof script generates a witness that records a trace of information that morally resembles the elaborations of Chapter 5specifically Section 5.4-and could serve as the bedrock of thorough certification of Abella proofs. Abella lacks a language of tacticals to compose tactics from other tactics. All these aspects can be handled inside the FPC framework and are treated in further detail in Chapter 13. If model checking can be seen as deduction, its proof evidence may also be expressed as proof certificates, as Heath andMiller (2015, 2017) have shown. Previous applications of model checking include the work of [START_REF] Mundhenk | The complexity of model checking for intuitionistic logics and their modal companions[END_REF]. Tabling has been studied, among others, by [START_REF] Ramakrishna | Efficient model checking using tabled resolution[END_REF]; [START_REF] Yang | A logical encoding of the picalculus: model checking mobile processes using tabled resolution[END_REF]; [START_REF] Tiu | Model checking for π-calculus using proof search[END_REF]. At the beginning we compared model checking and theorem proving in terms of the relation between satisfiability and provability. It is worth observing that a similar connection has been observed in Chapter 7 between satisfiability and unsatisfiability-and, more generally, the notion of refutation. . % Signature type idx2 idx -> idx -> idx. type pair# cert -> cert -> cert. % Module. ffClerk (pair# L0 R0) :- ffClerk L0, ffClerk R0. ttClerk (pair# L0 R0) (pair# L1 R1) :- ttClerk L0 L1, ttClerk R0 R1. andClerk (pair# L0 R0) (pair# L1 R1) :- andClerk L0 L1, andClerk R0 R1. orClerk (pair# L0 R0) (pair# L1 R1) (pair# L2 R2) :- orClerk L0 L1 L2, orClerk R0 R1 R2. impClerk (pair# L0 R0) (pair# L1 R1) :- impClerk L0 L1, impClerk R0 R1. eqClerk (pair# L0 R0) (pair# L1 R1) :- eqClerk L0 L1, eqClerk R0 R1. ttExpert (pair# L0 R0) :- ttExpert L0, ttExpert R0. andExpert (pair# L0 R0) (pair# L1 R1) (pair# L2 R2) :- andExpert L0 L1 L2, andExpert R0 R1 R2. orExpert (pair# L0 R0) (pair# L1 R1) C :- orExpert L0 L1 C, orExpert R0 R1 C. impExpert (pair# L0 R0) (pair# L1 R1) (pair# L2 R2) :- impExpert L0 L1 L2, impExpert R0 R1 R2. impExpert' (pair# L0 R0) (pair# L1 R1) (pair# L2 R2) :- impExpert' L0 L1 L2, impExpert' R0 R1 R2. eqExpert (pair# L0 R0) :- eqExpert L0, eqExpert R0. allClerk (pair# L0 R0) (x\ pair# (L1 x) (R1 x)) :- allClerk L0 L1, allClerk R0 R1. someClerk (pair# L0 R0) (x\ pair# (L1 x) (R1 x)) :- someClerk L0 L1, someClerk R0 R1. allExpert (pair# L0 R0) (pair# L1 R1) T :- allExpert L0 L1 T, allExpert R0 R1 T. someExpert (pair# L0 R0) (pair# L1 R1) T :- someExpert L0 L1 T, someExpert R0 R1 T. Figure The pairing meta-FPC in Abella implemented at the specification level. . . indClerk (pair# L0 R0) (pair# L1 R1) (x\ pair# (L2 x) (R2 x)) S :- indClerk L0 L1 L2 S, indClerk R0 R1 R2 S. coindClerk (pair# L0 R0) (pair# L1 R1) (x\ pair# (L2 x) (R2 x)) S :- coindClerk L0 L1 L2 S, coindClerk R0 R1 R2 S. unfoldLClerk (pair# L0 R0) (pair# L1 R1) :- unfoldLClerk L0 L1, unfoldLClerk R0 R1. unfoldRExpert (pair# L0 R0) (pair# L1 R1) :- unfoldRExpert L0 L1, unfoldRExpert R0 R1. unfoldLExpert (pair# L0 R0) (pair# L1 R1) :- unfoldLExpert L0 L1, unfoldLExpert R0 R1. unfoldRClerk (pair# L0 R0) (pair# L1 R1) :- unfoldRClerk L0 L1, unfoldRClerk R0 R1. freezeLClerk (pair# L0 R0) (pair# L1 R1) (idx2 IL IR) :- freezeLClerk L0 L1 IL, freezeLClerk R0 R1 IR. initRExpert (pair# L0 R0) (idx2 IL IR) :- initRExpert L0 IL, initRExpert R0 IR. freezeRClerk (pair# L0 R0) (pair# L1 R1) :- freezeRClerk L0 Frege proofs Consider the familiar notion of Frege proofs-also known as Hilbert proofs-: lists of formulas such that every formula in that list is either an axiom or follows from previously listed formulas using an inference rule. This notion of inference rule, as used in this and other styles of proof, is, usually, greatly restricted by limitations of human psychology, and by what skeptics are willing to trust. Typically, checking the application of inference rules involves simple syntactic checks. Example Take the following rule for set inclusion, which states that, given a set A and a strict subset B, we may conclude B: A A ⊃ B B The applicability of this rule requires deciding on whether or not two premises have the structure A and A ⊃ B, and the conclusion has the structure B. The introduction of automation into theorem proving has allowed us to engineer inference steps that are significantly more substantial, and can comprise both computation and deduction. As we note extensively, recent proof theoretic results allow us to extend the literature of theorem proving from being a study of minuscule inference rules-such as modus ponens in Hilbert-style systems, or Gentzen-style introduction rules-to a study of large-scale, formally defined, synthetic inference rules. In this chapter, we describe a particular way to specify and check such synthetic inference rules as a way to inductively prove lemmas from previous lemmas, in a style close to that in which proofs are written inside proof assistants like Coq, Isabelle or Abella. Let us return to the world of Frege proofs. In what follows, we will not speak of axioms, as the concept is redundant and can be subsumed by inference rules: an axiom can be generally described as an inference rule that depends on zero previous lemmas. Moreover, when we speak of a formula that is a member of a list of formulas comprising a Frege proof, we shall usually refer to it simply as a lemma. Presently, we will consider the relationship between this linearized style and more structured, arborescent schemes-closer to the proof trees of the sequent calculus-and its effects on proof checking. The rest of the chapter is structured as follows: Section 11.2 presents a motivating example of the kind of proof development used throughout the chapter to guide out designs. Section 11.3 introduces the concept of proof outlines as highlevel descriptions of proofs and discusses their logical interpretation. Section 11.4 translates that interpretation into the logic, augmenting the µLJF a proof system and the checkers that implement it. Section 11.5 describes the first of two families of proof outlines, a lightweight yet flexible collection of proof descriptors. Section 11.6 elaborates on the previous section in a second family of outlines that offers finer control akin to that found in the proof scripts of proof assistants. Section 11.7 revisits the case study of Section 11.2 and reviews a number of interesting applications of proof outlines. Section 11.8 concludes the chapter. Case study In this section, we will present a fundamental motivating example from which many other use cases follow. Consider defining the addition of natural numbers using the standard inductive relational specification in Abella, shown in Figure 11.1. Here, the expected inductive definitions are given together with a predicate for typing judgments about naturals, whose utility will immediately become apparent. Once these definitions are introduced, routinely we will find that we need to establish several properties of the addition relation immediately before progressing to more interesting work, e.g., that addition is determinate and total. Anyone familiar with proving such theorems knows that their proofs are simple: basically, the obvious induction leads quickly to a final proof. Figure 11.2 shows how this is done in Abella. The necessity for typing judgments now becomes clear: unlike systems like Coq, Abella can only induct on hypotheses, even if typed variables avail in the context. The direct Abella equivalent involves applying induction on a predicate that follows the full inductive structure of the kind in question, i.e., a typing judgment that takes an arbitrary member of its typed, validates its structure by exhaustive recursion through its inductively defined constructors, and succeeds. (Clearly, it is possible to derive these predicates mechanically, though Abella does not provide this facility.) Of course, if we wish to prove more facts about addition, we may need to come up with and prove some lemma before simple inductions will work. For example, proving the commutativity of addition makes use of two additional lemmas, as shown in Figure 11.3. These three theorems, as well as those in Figure 11.2, all have the same high-level proof outline: apply induction with the obvious invariant, apply some previously proved lemmas and the inductive hypothesis, and deal with any remaining branches by case analysis. The fact that many theorems can be proved by resorting to this pattern of induction-lemmas-cases-and, indeed, whole developments are routinely organized around it-is well known and built into existing theorem provers. For example, the waterfall model of the Boyer-Moore theorem prover [START_REF] Boyer | A Computational Logic[END_REF] proves such theorems in a similar fashion, but operates on inductive definitions of functions. In a similar relational style as that of Abella, the Twelf system [START_REF] Pfenning | System description: Twelf -A metalogical framework for deductive systems[END_REF] can often prove certain properties automatically, such as the statements that some relations are total and functional, using a series of similar steps to those described here [START_REF] Schürmann | A coverage checking algorithm for LF[END_REF]. The tactics and tacticals of LCF have also been used to implement procedures that attempt to find proofs using this kind of process [START_REF] Wilson | Inductive proof automation for Coq[END_REF]. Finally, and closer to the present approach, the TAC procedure of [START_REF] Baelde | Focused inductive theorem proving[END_REF] attempts to apply precisely such a scheme, although in a rather fixed and inflexible fashion, which has not continued beyond its original development. Moreover, this corresponds with standard mathematical practice, in which "simple proofs" may be described schematically and with minimum clutter, concentrating on interesting cases and application of useful lemmas. In this respect, proof outlines are not only of utility to machine checkers, but also to mathematicians. The pedantry inherent to formal developments spawns a large number of shallow, mostly unremarkable proof obligations, most of which can be mechanically dispatched by a small number of common proof patterns. In this section, we show how the FPC framework can formally specify such an algorithm. Following the paradigm of focused proof systems for first-order logic, there is a clear, high-level outline to follow for doing proof search for cut-free proofs: first do all invertible inference rules and then select a formula on which to do a series of non-invertible choices. This latter phase ends when one encounters invertible inference rules again or the proof ends. In the setting we describe here, there are two significant complicating features with which to be concerned. 1. Treating the induction rule. The invertible phase is generally treated as a place where no important choices in the search for a proof appear. When dealing with a formula that is a fixed point, however, this is no longer true. As described in Section 9.2, we treat a fixed point expression either by freezing-for which see also [START_REF] Baelde | Least and greatest fixed points in linear logic[END_REF]-, unfolding, or using an invariant to perform an induction-here, this will be extended with the possibility of deriving the "obvious" inductive invariant. These options are directly connected to the rules introduced in Figure 9.3. In particular: (a) We can choose to "freeze" the fixed point, meaning that we choose not to induct on it. (b) We can set up an inductive step. This second choice is in turn divided into three sub-choices: i. We can choose to simply unfold a fixed point definition. In fact, the concept of unfolding follows as a direct consequence of applying induction. ii. We can take an explicit induction offered by the author in the certificate. In the context of this discussion, a human actor will seldom need to make use of this option-very often, the obvious invariant (for which see below) is all one needs. However, the author of a certificate can also be a theorem prover or a proof checker. In this latter case of machine-generated proofs, an explicit invariant may be routinely inserted into a certificate. These questions are treated in Section 11.4. iii. We can select the surrounding sequent context to be the actual inductive invariant. This corresponds to the notion of obvious or immediate invariant. 2. Lemmas must be invoked. The application of lemmas into a proof outline is critical to the kind of linear proof development we have in mind. Although the focusing framework does not restrict the shape of lemmas, we consider here the effect of focused proof construction with a lemma that is a Horn clause. For example, the three lemmas addressing the commutativity of addition in Figure 11.2 are Horn clauses. Example Consider applying a Horn lemma of the form ∀x.[A 1 ⊃ A 2 ⊃ A 3 ] in proving the sequent Γ E. Since the formulas A 1 , A 2 , and A 3 are polarized positively, we can design the proof outline (simply by only allowing fixed points to be frozen during this part of the proof) so that Γ ⇓ ∀x.[A 1 ⊃ A 2 ⊃ A 3 ] E is provable if and only if there is a substitution θ for the variables in the list of variables x such that θ A 1 and θ A 2 are in Γ and the sequent Γ, θ A 3 E is provable. The application of such a lemma is then seen as forward chaining: if the context Γ contains two atoms (i.e., frozen fixed points), then add a third. The main issue that a certificate-as-proof-outline therefore needs to provide is some indication of what lemmas should be used during the construction of a proof. The following natural specifications of collections of supporting lemmasstarting from the least explicit to the most explicit-are easily written within our framework: 1. A bound on the number of lemmas that can be used to finish the proof, chosen freely from the collection of previously proven and known lemmas. 2. A list of possible lemmas to use in finishing the proof. These can be assumed as hypotheses during the proof; a separate proof for each lemma is required as well. 3. A tree of lemmas, indicating which lemmas are applied following the conjunctive structure of the remaining proof. Each of these three categories refine the previous one with additional information on which lemmas to use and where, thereby reducing the amount of nondeterminism and enabling faster proofs-or disproofs of the proof outline. Additional refinements are possible and can bring outlines even closer to the proof scripts that are commonly written as avatars for proofs in interactive theorem provers. Before we study the encoding of these procedures as proof certificates, we need to consider the extensions imposed on the proof system-and its implementation-by the two distinguishing features of this proposal: lemmas and obvious inductions. Augmenting contexts as illustrated in Example 11.3.1 is critical for eventually enabling obvious inductions to succeed in completing a proof. In this way, the focused proof system can easily be used to apply lemmas. All this will be the subject of the next section. Logic support In general, it appears that (co)inductive invariants are often complex, large, and tedious structures to build and use. Thus, it is most likely that we need to develop a number of techniques by which invariants are not built directly but are rather implied by alternative reasoning principles. For example, Abella allows the user to do induction not by explicitly entering an invariant but rather by performing a certain kind of guarded, circular reasoning. Closer to our approach, Coq automatically derives induction principles from inductive definitions, but also allows users to define their own custom inductions. In the present context, we consider a single approach to specifying invariants. Let us consider the case of induction on a least fixed point, i.e., on the right-hand side of the sequent during the asynchronous phase. Recall the associated inference rules in Figures 9.3 and 9.6, namely the µLJF rule for induction (i.e., on least fixed points; we omit the frozen zones for succinctness): Γ ⇑ S t, Θ R Γ ⇑ B S ȳ S ȳ ⇑ Γ ⇑ µB t, Θ R inductL The principle we now introduce involves taking the conclusion of that rule Γ ⇑ µB t, Θ R and abstracting out the fixed point expression to yield the obvious invariant, which we write Ŝ. This invariant is extracted from the conclusion in such a way that one of the premises, Γ ⇑ S t, Θ R, has an easy proof-in fact, it is made trivial by definition of Ŝ. As a result, only the second premise related to the induction rule needs to be properly proved. The following augmented rule is used to generate and check whether or not the obvious induction invariant can be used. The resulting extensions are presented in Figure 11.4. Sequents are strenghtened with a "zone" Σ which represents the list of eigenvariables in the sequent, required to compute obvious invariants. The development of obvious coinduction is completely symmetric. In both cases, there is the choice of whether to omit or maintain the branch that becomes redundant upon application of the obvious invariant. Pruning said branch from the sequent calculus requires (a) enriching the calculus with the computation of said obvious invariants (thereby becoming part of any kernel implementing this logic); and (b) furnishing evidence of the provability of the obvious branch in the general case (given in Figure 11.5). Figure 11.4 shows both possible formulations of the resulting calculus, and their augmentations with the corresponding clerks are given in Figure 11.6. From the point of view of the writer of proof certificates, the fact that the left premises of the full rules follow directly from the obvious invariants allows us to confuse both presentations. That is, without pruning the trivial branch, the kernel can perform proof search to check this premise-which is guaranteed to succeed-while clerks need only produce a continuation certificate for the non-trivial branch, effectively as if the obvious premise was discharged as a proof obligation. After developing the necessary logic support, all that remains is to integrate them in the proof checker developed in Section 9.3. The extensions are neatly divided in two groups of changes paralleling the discussion in Section 11.3: 1. The obvious (co)induction rules are added as variants of, respectively, standard (co)induction. The kernel must be extended with the Σ zone for eigenvariables, and the new rules come with specialized code to compute obvious (co)invariants. There are two technical disadvantages to this last addition. First, we inject relatively complex, non-declarative code into the trusted kernel-although we are under no obligation to trust it if we verify both premises for each involved rule, i.e., if we rule out the single-premise rules in Figure 11.6. Second, and more seriously, there is no clean, declarative way to maintain a list of eigenvariables in the presence of eigenvariable unification: under this discipline, the obvious rules are only meaningful in the initial stage of a proof, before any such operations (triggered by the equality rules) pollute the Σ zone. In particular, the scheme does not support nested obvious inductions. {y } ∪ Σ; Γ ⇑ [y/x]B ⇑ Σ; Γ ⇑ ∀x.B ⇑ {y } ∪ Σ; Γ ⇑ [y/x]B, Θ R Σ; Γ ⇑ ∃x.B, Θ R Σ; Γ ⇑ S t, Θ R {ȳ }; Γ ⇑ B S ȳ S ȳ ⇑ Σ; Γ ⇑ µB t, Θ R inductL Σ; Γ ⇑ Ŝ t, Θ R {ȳ }; Γ ⇑ B Ŝ ȳ Ŝ ȳ ⇑ † Σ; Γ ⇑ µB t, Θ R obviousL {ȳ }; Γ ⇑ B Ŝ ȳ Ŝ ȳ ⇑ † Σ; Γ ⇑ µB t, Θ R obviousL † Ŝ := λ x.∀Σ. x = t ⊃ Θ ⊃ R Σ; Γ ⇑ S t ⇑ {ȳ }; ⇑ S ȳ BS ȳ ⇑ Σ; Γ ⇑ ν B t ⇑ inductR Σ; Γ ⇑ Ŝ t ⇑ {ȳ }; ⇑ Ŝ ȳ B Ŝ ȳ ⇑ ‡ Σ; Γ ⇑ ν B t ⇑ obviousR {ȳ }; ⇑ Ŝ ȳ B Ŝ ȳ ⇑ ‡ Σ; Γ ⇑ ν B t ⇑ obviousR ‡ Ŝ := λ x.∃Σ. x = t ∧ Γ . . . . t = t ⇓ = e . . . , Θ Θ 1 ⇓ • • • . . . , Θ Θ n ⇓ Γ, Ŝ t, Θ Θ ⇓ ∧ e . . . ⇓ R R Γ, Ŝ t, Θ ⇓ ( Θ) ⊃ R R ⊃ e Γ, Ŝ t, Θ ⇓ t = t ⊃ ( Θ) ⊃ R R ⊃ e Γ, Ŝ t, Θ ⇓ ∀Σ. t = t ⊃ ( Θ) ⊃ R R ∀ e Γ, Ŝ t, Θ ⇓ Ŝ t R β Γ, Ŝ t, Θ ⇑ R decide Γ ⇑ Ŝ t, Θ R store Π . . . . Γ ⇑ B Ŝ ȳ Ŝ ȳ ⇑ Γ ⇑ µB t, Θ R obviousL 11. 5 Figure Proof schema of the obvious induction showing how the obvious branch follows directly from the obvious invariant, i.e., the goal follows from the obvious invariant and the remaining hypotheses. Because all the components of the obvious invariant are present in the sequent in complementary positions, all the branches in the subproof suggest immediate applications of the initial rules. A proof obligation for the non-trivial branch remains to be satisfied by a proof Π. . asynchronous introduction rules Ξ 1 : Σ; Γ ⇑ Ŝ t, Θ R (Ξ 2 ȳ) : {ȳ }; Γ ⇑ B Ŝ ȳ Ŝ ȳ ⇑ obviousL c (Ξ 0 , Ξ 1 , Ξ 2 ) † Ξ 0 : Σ; Γ ⇑ µB t, Θ R (Ξ 1 ȳ) : {ȳ }; Γ ⇑ B Ŝ ȳ Ŝ ȳ ⇑ obviousL c (Ξ 0 , Ξ 1 ) † Ξ 0 : Σ; Γ ⇑ µB t, Θ R Ξ 1 : Σ; Γ ⇑ Ŝ t ⇑ (Ξ 2 ȳ) : {ȳ }; ⇑ Ŝ ȳ B Ŝ ȳ ⇑ obviousR c (Ξ 0 , Ξ 1 , Ξ 2 ) ‡ Ξ 0 : Σ; Γ ⇑ ν B t ⇑ (Ξ 1 ȳ) : {ȳ }; ⇑ Ŝ ȳ B Ŝ ȳ ⇑ obviousR c (Ξ 0 , Ξ 1 ) ‡ Ξ 0 : Σ; Γ ⇑ ν B t ⇑ 11.6 Figure Extensions to the augmented µLJF a focused proof system needed to support obvious inductions in proof outlines [START_REF] Blanco | Proof outlines as proof certificates: a system description[END_REF]. The new inference rules introduced in Figure 11.4 are augmented with clerks in the usual way. Note that the premises that involve the body B of the fixed point take a continuation certificate abstracted over the fresh variable(s) ȳ. The side conditions † and ‡ that define the obvious (co)invariants are unchanged from Figure 11.4. 2. The treatment of lemmas presents us with two orthogonal choices. First, do we treat them as assumptions within a proof? Second, do we consider them separate from the standard storage zones on which the decide rules operate? Here, the answer to these two questions will be "yes." A separate zone Λ, i.e., a map from lemma names to formulas, is threaded throughout the kernel, and the checker's interface is enriched with an additional list of lemmas to populate Λ. The consequence of the first question is that lemma application will be modeled as a decide rule, and not as a cut rule (which must come with an inlined proof of the lemma). The consequence of the second section is the definition of lemma decide rules which operate like local decide rules, limited to the zone Λ, leaving the old rules unchanged. It is straightforward to see that all these additions preserve the soundness of the enriched µLJF a system-by the same elementary reasoning used, say, to prove the soundness of LKF a with respect to LKF in Theorem 3.2.2. The extensions to the Abella kernel are summarized in Figure 11.7. The new zones are threaded through existing code, new inference rules are added and an interface function composes the initial sequence and populates the zone of lemmas; all other rules remain otherwise unaltered. The new zones are populated only when new eigenvariables are generated (Σ) and when the kernel is called with a set of lemmas (Λ). Obvious inductions rely, for the generation of invariants, on a significant amount of nondeclarative code to implement the computation of Ŝ in Figure 11.4, which is not shown here. The rules depicted here are the brief, one-premise rules, which do not separately check the branch made trivial by the obvious invariants, given that these are proven to be built correctly, otherwise soundness is endangered. Of course, in the context of a development, say, in an interactive theorem prover, we will wish to enforce that a proof be supplied for every lemma used as a hypothesis at any point in the development. This work of integration will be discussed in Chapter 13. Certificate families: simple outlines With the client-side requirements for proof outlines established and reflected in the logic, we turn our attention to the actual FPC definitions in two related families of outline certificates. The first of these families defines what we call simple outlines. This format aims to be simple to read and write without limiting the proofs that it can find compared to other, denser representations. Simple outlines do not restrict polarities and use a simple indexing scheme for referring to formulas by simple, unstructured names, i.e., individual indexes for lemmas can be defined as nullary constructors of kind idx as needed. Here we are especially interested in the naming and selection of lemmas, as managed by the client side. The kernel handles its internal indexing of formulas generated during proof reconstruction by two related mechanisms: Define async : list_lemma -> cert -> list_i -> list_ctx -> list_ctx -> list_bool -> goal -> prop, syncL : list_lemma -> cert -> list_i -> list_ctx -> list_ctx -> bool -> goal -> prop, syncR : list_lemma -> cert -> list_i -> list_ctx -> list_ctx -> bool -> prop by async Lambda Xi Sigma Phi Gamma (cons_bool (mu B T) Delta) G := exists Xi' S, indClerk' Xi Xi' /\ indInvariant' Sigma Delta G T S /\ forall x, async Lambda (Xi' x) (cons_i x nil_i) Phi Gamma (cons_bool (B S 1. Atoms, i.e., frozen fixed points, are stored on the left-upon freezing-under a special index idxatom. The initial rule on the left always selects from this bucket. (Recall that freezing on the right-hand side is a singleton, and so freezing it does not require an index.) 2. All other formulas are stored on the left under a second dedicated index idxlocal. The decide rule on the left can either select a "local formula" or a lemma by any of the mechanisms described below. (Again, storing and deciding on the right do not require any indexing.) If greater granularity is required, it is a simple matter to implement it modularly and add it to simple outlines via certificate pairing. For this, see Chapter 5 as well as its adaptation in Section 10.4. The latter which would need to be extended with the decide rule on lemmas from Section 11.4; this external piece of information requires agreement between the two halves of a certificate pair-unlike local decides, made independent by paired indexes. The bulk of the complexity of the FPC definition rests on certificate constructors and their limited manipulations by clerks and experts. A first group of certificate constructors is meant to represent whole proof subtrees, from the root of the branch down to every leaf of the subtree. In this sense, they are self-contained and fully manage their own (limited) bookkeeping needs. The following four constructors are provided: 1. (induction B AU SU AC SC): asynchronously look for the closest fixed point, apply obvious (co)induction to it and perform bounded search through apply (as explained next). 2. (inductionS B AU SU AC SC S): asynchronously look for the closest fixed point, apply (co)induction with the supplied invariant and perform bounded search through apply. 3. (apply B AU SU AC SC): perform bounded search and try to finish the proof by search. 4. search: attempt to finish the proof by simple exploration of the tree and application of initial rules to all branches. These certificate constructors form a hierarchy that represents the evolution of the state of the proof: from induction to a number of bipoles-which comprise the application of lemmas-up to the posited end of the proof. All parameters are natural numbers representing decreasing counters. Those counters are used to bound the possible sources of non-terminating behavior in proof search: (a) decide rules, and (b) fixed point operations. Among the latter, freezing is bounded by definition and induction is controlled singly by induction and inductionS, leaving unfolding as the sole related source of unbounded behavior. Bounded proof search depends on the following set of parameters: 1. Bipole bound: after the current bipole, proof search may proceed up to B additional bipoles deeper. The release rule marks the end of a bipole; the rule is only enabled when the natural number of bipoles can be decremented. 2. Asynchronous unfolding: each bipole after the current one is allowed to perform up to AU unfolding operations along each branch during the asynchronous phase. 3. Synchronous unfolding: each bipole after the current one is allowed to perform up to SU unfolding operations along each branch during the synchronous phase. 4. Asynchronous unfolding, current bipole: until the end of the current bipole, up to AC asynchronous unfolding operations may be performed along each branch during the asynchronous phase. 5. Synchronous unfolding, current bipole: until the end of the current bipole, up to SC synchronous unfolding operations may be performed along each branch during the synchronous phase. The "current bipole" counters AC and SC are technical bookkeeping devices, initialized respectively to AU and SU at the beginning of each bipole (i.e., on the release rules). The only reason to write a certificate where the initial current bipole counters are unequal to the per-bipole bounds would be as a very limited form of optimization. This unnecessary verbosity is averted by defining marshalled forms of the certificate constructors where only per-bipole counters are given. These shorthand forms are closer to the minimal outlines the user may want to write, but may involve substantial backtracking search. In the first place, these certificates representing full subtrees leave the choice of which formula to focus on at each decide rule-lemma or local formula-to the kernel, which in turn must explore all possibly combinations allowed by its local context and the list of lemmas given to it. A second source of inefficiency comes from the unfolding bounds: these must be as large or greater than the needs of the costliest phases, even if in most cases much smaller bounds suffice. To address the inefficiencies that arise from the extreme conciseness of the certificates above-and the laxity they impose on costly backtracking search throughout the entire proof tree-, a second family of certificate constructors is presented. The new constructors are written in a continuation-passing style, or CPS, which can be used to describe the interesting features of a section of a proof : asynchronous branching, order and choice of lemmas, and local unfolding bounds. A local description of a region of a proof tree is followed by continuation certificates. The following constructors are defined: 1. (induction? C): in the asynchronous phase, look for the first available fixed point and do the obvious (co)induction, then continue the proof using the continuation certificate C. 2. (inductionS? CL CR S): do induction on the first asynchronously available fixed point using S as invariant, and continue the two resulting branches of the proof using continuation certificates CL and CR. This set of constructors represents the structure of a proof more faithfully. Their definition is completely local and their bookkeeping is self-contained in their own counters, which have on effect on continuation certificates. To finish a branch of the proof in this format, a certificate constructor of the first group is used. If a certificate reproduces the bipole structure of a proof closely enough, it will generally suffice to use the shallow search to complete each branch of the proof scaffold, or a limited form of apply in a slightly more general setting, after the general high-level structure of the proof has been established. (case Figure 11.8 presents the continuation-passing style parts of the FPC definition in the more compact notation of λProlog-i.e., at the specification level of Abellawhere some of the more verbose idioms of Abella are elided for compactness. The code is presented in pure pattern matching style. This level of presentation affords a modular organization of the development; thus, the counterpart definitions for whole proof subtrees (similar to those shown here for the CPS fragment) can be contained in a separate file and are combined only when a specification is loaded-instead of accumulating as definitions at the reasoning level, more massive and less organized. Certificate families: administrative outlines The second family of high-level FPC definitions is what we call administrative outlines-in reference to the greater amount of bookkeeping information they contain, geared towards more precise descriptions of proofs of large families of commonplace theorems. Specifically, simple outlines offer means of describing the high-level structure of a proof and specifying what lemmas it may employ; they do not offer a comparable level of control over the structure of formulas and local hypotheses, and how to refer to them-for example, all local decisions are treated through a singleton index idxlocal, even though knowing how to make use of which hypotheses, and when, is at the heart of the proof scripts of assistants like Abella. Administrative outlines will provide mechanisms to express this information in the FPC framework. The certificate constructors of administrative outlines allow for a significant amount of guidance to steer proof search more accurately, especially with respect to disambiguating choice points instead of relying on backtracking searchpotentially very costly, especially as proofs grow. In describing the general structure of a proof, the kind of structures found in simple outlines reoccur, with some important differences. These advanced features will be chiefly involved with refined forms of indexing. More specifically, we define a general-purpose index type for the storage of formulas, consisting of two parts which correspond to the two kinds of formulas we need to store-lemmas and local formulas: 1. A numeric index used as a unique identifier for formulas stored through the decide rule, and unused in the case of externally supplied lemmas. 2. A boolean index used to label and describe a formula. It holds a name through which to refer to lemmas, and maintains bookkeeping information in the case of locally stored formulas. Certificate constructors in this FPC definition form a single family of tactics in continuation-passing style, with a single terminal constructor. While they look similar to the simple outlines of the previous section, their control flow differs significantly. A common control structure, Ctrl, constrains the operation of all of them. The defined constructors are the following: 1. (induction Ctrl NamesB Cert): apply obvious (co)induction on the first (asynchronously) available fixed point, and continue the proof using Cert. Use NamesB to give names to the components of the fixed point, as explained below. 2. (inductionS Ctrl S NamesB NamesS Cert): a variant of the previous constructor, apply (co)induction on the first available fixed point using S as invariant, then continue the proof using bounded search constrained by Ctrl on the base case and Cert on the inductive case. Use NamesB to give names to the components of the fixed point and NamesS to give names to the components of S. 3. (case Ctrl CertL CertR): apply asynchronous case analysis, locating the first disjunction on the left. Continue the proof using CertL and CertR for the left and right branches, respectively. 4. (apply Ctrl Idx Names Cert): perform bounded search, using Idx to decide on the next lemma and use Names to give names to the components of the selected lemma.Continue the proof using Cert. 5. (search Ctrl): perform bounded search without applying induction or deciding on externally provided lemmas. All certificates share a common control structure that maintains bounds and bookkeeping parameters to constrain proof search. There is some variability in the way these parameters are used by the FPC, as will be seen. The structure has the form (ctrl Limits Names), where Names is the current naming structure and Limits contains the following bounds and bookkeeping information. The five parameters given (less uniformly) for the various constructors of simple outlines are always present through the control structure. The design is reminiscent of TAC [START_REF] Baelde | Focused inductive theorem proving[END_REF] and some of the terminology is biased towards least fixed points, so that asynchronous and synchronous unfolding are associated with the left-hand and the right-hand side rules, respectively. The additional control fields are the following: 1. Release right: a boolean flag that enables the release rule when active. In common proofs of "administrative lemmas" involving least fixed points, once we reach a focus on the right, it is often meant to represent a purely positive computation which either terminates or fails. In these situations, a release is considered a dead end, and thus ruled out. 2. Next local index: an incrementing counter that contains the next available unique index to store formulas along the current branch of the proof, starting from zero. 3. Current local index: a bookkeeping parameter that maintains information about the progress of local indexes for the decide rule as the range of possible choices cycles through the range of indexes in local storage (given by the previous parameter). We now turn our attention to the second component of the control structure. Consider the proof scripts written in a proof assistant like Abella. It is remarkable that a script consists of a sequence of decisions, and these can conceivably be turned into certificate constructors. In addition to selecting formulas and lemmas, an important part of the instructions in a proof script is the set of hypotheses that are used to instantiate each formula that the proof operates upon. This information has been absent from our certificates so far, which means backtracking search must be applied to find a right combination of values, if one indeed exists. In some cases this will be easy, but in others much time will be wasted applying, say, sequences of lemmas that make no sense, or using the wrong parameters, leading in the end to complex proof attempts that must be discarded. Syntactically, a formula can be seen as a tree whose leaves are drawn from the nullary logical constants, the fixed point operators and the equality operator; and its nodes from the remaining, recursive logical constants. (Interestingly, unfolding operations grow a fixed point leaf into a subtree.) Under this view, a naming structure associated to a formula is another tree that replicates the branching structure of the formula down to the fixed points contained in the formula, and attaches names to them. Example Consider the fixed point definition of the addition relation on natural numbers given in Figure 10.2. Suppose we want to refer to the recursive call to plus in its body as "H1." Mimicking the branching structure of the formula by split constructors and labeling nodes by name constructors, we may get: (split _ (split _ (split _ (name "H1")))) Note that linear branching (i.e., quantifiers) is omitted from the syntax. Nodes representing subtrees that do not contain any fixed points are in effect don't-care identifiers (because they will never be used to constrain the inference rules that need to name fixed points-as-atoms), here represented by anonymous variables _ (but see the discussion after Example 11.6.2). The object of naming structures is labeling the "atoms" in a formula so that they can be matched with the contents of the context, especially frozen fixed points acting as hypotheses. Formulas as commonly defined in the FPC framework are terms of a simple inductive type; they are not annotated and are mostly or completely opaque to clerks and experts-see Section 4.4 for a general discussion, and Section 7.4 for a use case that benefits from reflective inspection. In consequence, name annotations have to be provided separately and in parallel to formulas and sequents. At the cost of some redundancy, it is possible to furnish this information in the certificate without any changes to the kernel being needed. To this end, we use the second member of the control structure. This piece of information must shadow the structure of the sequent with a level of detail that allows us to give names to the components of interest. Boolean indexes, defined earlier, will hold this information, of which a simple name (such as is used to tag lemmas) is a particular case. Since formulas in storage already hold this information, the certificate must maintain name maps for the workbench zones on each side of the sequent turnstile, respectively: (names Delta Goal). The data mirroring just described creates a new requirement for code mirroring in the FPC definition: clerks and experts must be aware of the changes made in the workbench by the kernel on each inference rule and reflect those operations exactly in the naming structures for them to remain accurate. This lockstep requirement is a fairly strong dependency which, while incapable of compromising soundness, might lead to mangled information that impedes progress of the proofs. The issue is mitigated by the fact that the kernel's operation is stable and well documented, although this lockstep pattern reoccurs in other scenarios, including applications to test data generation described in Section 12.6. Example 11.6.2 illustrates the mimicry of the kernel by the clerks and experts. Example Observe the implementation of the inference rules of the system µLJF a in Figure 9.6 and companion figures as the Abella kernel in Figure 10.3. Figure 11.11 presents the implementation of the lockstep pattern for two representative rules: the introduction rule for negative conjunction and the full induction rule on least fixed points. The clerks and experts are essentially implemented by auxiliary predicates which generate the continuations from the designated input certificates. They achieve this by manipulating the control structure through other, modular auxiliary predicates. For example, the conjunctive clerk looks for a split in the naming structure at the head of the list of names that replicate the LHS of the sequent and replaces it with its two sub-components-exactly as the kernel decomposes the connective. If a split naming structure is not found at that position, the default is copied on both positions. The inductive clerk is somewhat more complicated-reflecting the manipulations that turn the conclusion of the inference rule into its premises. In particular, it must extract and combine naming structures for the fixed point and the invariant. The guiding principle is the same: the naming structure mirrors at all times the sequent to which the certificate is associated. However, the burden of naming must not be an obligation, and restoring the lost flexibility is indeed simple. A name leaf will give name to all relevant sub-formulas covered by it. In this way we can define "buckets" of homonymous formulas which can be stored and decided upon indifferently. If we do not wish to make any use of this functionality, it suffices to write a naming structure for the initial sequent where a single name is used-like idxatom is in simple outlines: (names nil (name "Dummy")). (Note that definite identifiers, and not anonymous variables as in Example 11.6.1, must be used if there are fixed points to be stored and decided upon.) 11.6.3 Example Consider a theorem statement of the form A ⊃ B ⊃ C . Suppose we want to write a proof outline where we refer to the formula A as a hypothesis named "H1," to the formula B as hypothesis "H2," and to the goal C as "G." In the top-level certificate constructor of the administrative outline, we will provide the following naming structure: (names nil (split (name "H1") (split (name "H2") (name "G"))) ) Similarly, suppose there is in our collection of lemmas one of the form D ⊃ E, such that, if we can equate A with D, we may infer E from it. We can guide the kernel towards this selection selection by a certificate that supplies the name of this lemma and establishes the correspondence between the hypothesis D and its match in the context, A, based the symbolic names given them above. Thus, we would pick "H1" and generate a new name for the conclusion E, say, "H3." We would write: (split (name "H1") (name "H3")) The last distinct role of naming structures concerns their role in inductive schemes (i.e., involving certificate constructors induction and inductionS). Two dedicated structures are needed for this: 1. Both induction and inductionS require a naming structure that assigns labels to the unfolding of the fixed point being induced upon. Because induction is a carefully guided "tactic," it is reasonable to assume that we know what our desired induction is-as well as the names of its parts. 2. In the general induction scheme of inductionS, the certificate must supply (along with the invariant) a matching structure naming the parts of said invariant. The FPC definition will then construct the naming structure that results from applying the induction rule. Therefore, the first naming structure (i.e., that of the unfolding, above) must confer a special annotation to the leaf corresponding to the place where the invariant will be injected. On a final note, continuation certificates inherit the state of the naming structures of their predecessors at the continuation point. Ultimately, all this scaffolding replicates at the FPC level what proof assistants do to generate and manage names. Much of it could be automated, although the problem of choosing and using good names is difficult and best left to the client. For each lemma, a self-contained induction with adequate allowances for decide depth and unfolding will arrive at the same proof as the scripts in Figure 11.3. Here, the most compact, single-constructor forms are given with tight bounds and in marshaled form. Experiments Let us return to the case study in Section 11.2. Suppose for the moment that Abella has been extended with a tactic, certify, which takes a proof outline and uses it to attempt to prove the current theorem. It is patently clear from, say, Figure 11.3 that the proofs expressed therein can be written as outlines. Inspection of the proofs or experimentation quickly lead to the simple outlines of Figure 11.12-or to the essentially identical administrative outlines. (The interface between Abella and the checker is detailed in Chapter 13.) Example We shall now revisit the case study in Section 11.2 and detail how such proofs can be expressed in terms of proof outlines, leading the way to the expression of proof scripts as proof certificates. Let us analyze in detail the proof script of plustotal in Figure 11. This corresponds to an application of the obvious induction, which implicitly includes a case analysis-which derives as many goals as the predicate has clauses: two in the case of addition, the zero case and the successor case. (Here, we assume a model in which the obvious induction does not require a certificate for the trivial branch.) Note that the induction certificate greedily inducts on the first available fixed point; inducing on others requires an easy generalization in the FPC definition, or reordering the hypotheses-so that H# becomes H1-in the theorem statement to account for that constraint. In certificate terms, this translates into the following certificate pattern: (induction? (case? 0 ... ... ) ) In it, there are two holes for the continuation certificates in each case. We now turn to consider each subgoal in turn. In the zero case, we have a simple goal, exists S, plus z M S, solved by a simple use of the search tactic. It is easy to see this involves focusing on the right for the positive phase, unfolding once on the right and applying initial. In certificate terms: (apply? 0 1 (idx "local") search) In the successor case, the process is a bit more complex. The goal starts from the following sequent: Whereas in Abella we need to appeal to the induction hypothesis explicitly, in the certificate outline it becomes a natural consequence of the asynchronous phase, and it need not be handled explicitly here. In Abella, a new hypothesis results from apply IH to H3: H4 : forall M, nat M -> (exists S, plus N1 M S) The rest of the sequent remains unchanged. At this point we have a collection of "atoms" and negative formulas on the left, and a positive goal on the right: if nothing is done to the atoms, they will be frozen, and we will have reached the end of asynchrony. The proof script instructs to operate on H4 using other hypotheses for assumptions. Therefore, fixed are indeed frozen and we move to the synchronous phase, and a bipole must be signaled through a local decide: (apply? 0 0 (idxlocal) ...) This local decide will reproduce the choice of H2. In Abella, at the end of the synchronous phase, a new hypothesis has been generated: H5 : plus N1 M S Finally, the search tactic finishes the goal, and with it the proof. This corresponds, as in the zero case, to a focus on the right hand side, with unfolding and initial. The final certificate is thus: (induction? (case? 0 (apply? 0 1 (idxlocal) search) (apply? 0 0 (idxlocal) (apply? 0 1 (idxlocal) search)) ) ) By a similar process, we-or an automated analyzer integrated in Abella-can reformulate the main result of Section 11.2 as a detailed proof outline. Example The proofs in Figure 11.3 establish the commutativity of the addition relation between natural numbers as the theorem pluscom. A proof by simple induction relies on two auxiliary lemmas, one for each case of the induction on natural numbers (zero and successor), themselves proved separately by simple inductions. We shall look at the certificate for the main theorem, which involves the application of lemmas (the proofs of those two lemmas are simpler inductions in the manner of Example 11.7.1). The complete certificate is given by the term: (induction? (case? 0 (apply? 1 0 (idx "plus0com") search) (apply? 2 0 (idx "local") (apply? 0 0 (idx "plusscom") (apply? 0 0 (idx "local") search))) ) ) Compare this with the proof script in Figure 11.3. In the certificate, decides on lemmas refer to those by their given names earlier in the Abella session. Again, the structure is simple: after starting with the obvious induction on the first argument of the addition, a case? branches out the proof certificate into the two cases for zero and successor. Both follow the same pattern of applying a lemma by focusing on it, from which the proof of the branch follows; the application of the corresponding lemma is preceded by a small amount of preprocessing. The focused discipline of the outline FPCs imposes additional structure to the proofs, which in turn results in some amount of automated processing. In Abella, the proof of the successor branch of the induction opens with the following sequent: Note that the first two steps in the proof script-i.e., the case analysis on "H3" and the application of the inductive hypothesis-are directly handled by the semantics of the FPC and not reflected in the outline: in a sense, there is nothing interesting in these two obvious steps. This is related to the concept of progressing unfolding as described by [START_REF] Baelde | Focused inductive theorem proving[END_REF]. After these, it is easy to identify which proof steps correspond to unfoldings and record these in the allowances made in the certificate terms. The translation from proof scripts to outline certificates is mechanical but cumbersome for the user of a proof assistant. In practice, it is simpler to fall back to the more implicit but slower outlines: compare the original proof scripts in Figure 11.3 with the much more readable outlines given in Figure 11.12-the latter compact the outlines studied in Example 11.7.2. Clearly, more generous bounds will also arrive at the same proofs-provided that the necessary lemmas are available-, but it would be desirable to refine said bounds to be as tight as possible, and to add additional details to make proofs more efficient. . . all this without imposing the burden on the user. Certificate pairing (for which see Chapter 5) can be easily applied to enable these kinds of manipulations, in particular in combination with the simple outlines of Section 11.5 for purposes of certificate elaboration. The essential component (alongside unfolding bounds) that separates the more implicit and more explicit forms of outlines is the specification of a skeleton for the proof tree decorated with the required decisions, in particular the necessary lemmas as they occur. An alternative representation that can be written succinctly (like implicit outlines) and paired easily may encapsulate the tree of decisions in a separate data structure, parallel to the unfolding bounds. The modus operandi will be to enrich the self-contained certificate constructors of simple outlines with such an abstract tree of decisions. This is achieved by extending the FPC definition of simple outlines with the following doubles of the self-contained constructors: The third kind of self-contained certificate, search, involves no such decisions and therefore remains unaltered. For brevity, the inductionS variant is not considered here; it experiments identical changes. Clerk and expert analogs for the new copies are declared with identical behavior except for the new checks that decision trees must match the structure of the proof, as illustrated in Figure 11.13. The reason for declaring a new family of constructors instead of performing elaboration, say, from self-contained to continuation-style simple outlines is that non-disjoint constructors make certificate elaboration ambiguous-as discussed in Section 5.6-noting that continuation-style certificates can at any point finish a branch by a self-contained certificate. The type of decision trees is defined by the following constructors: 1. A branching constructor, btbranch, used to represent a branching point in the proof with the decision trees for each branch. 2. Two decision constructors, btlocal and btlemma for local and lemma decides, respectively, each taking an index describing the decision and a continuation certificate. 3. A terminal constructor, btinit, representing a subtree where no decisions take place. 11.13 Figure Decision trees are implemented in the FPC family for simple outlines by a small set of additions implemented as clerk and expert clauses, whose interesting cases are summarized here. In the closed world of Abella, these new clauses are added to the existing definitions and cannot be accumulated separately. If the usual pairing combinator used to check two certificates in tandem is represented by (pair# C1 C2), a simple but useful elaboration is achieved by combining a self-contained outline (in the style of Section 11.5) with an extended outline (as given in this section), identical in every detail except that the decision tree is left unspecified as a logic variable. The standard outline drives proof reconstruction, while the extended outline-containing the same amount of information-is necessarily in agreement, but also records the branching points and decisions in its vacant structure. Upon a successful check, the decision tree will contain information about the number of decisions made, what lemmas were used, and the general structure of the proof. This data can be exploited to speed up subsequent checking, or even to further elaborate the new information in the form of a nested certificate in continuation-passing style. Example Suppose we find a valid compact certificate for pluscom-in the context of a session, where previously proved theorems are available as theoremswhich takes the following form: (induction 3 2 0 2 0) (We know from our analysis and Figure 11.12 that these bounds are indeed sufficient in the presence of the auxiliary lemmas for the zero and successor cases.) While this is easy to use, we may want to extract some more information from the proof, be it to learn something from it (i.e., the list of used lemmas) or to elaborate it into a derived certificate that is easier to check. To retrieve the tree of decisions, it suffices to pair it with its double as described above: (pair# (induction 3 2 0 2 0) (induction# 3 2 0 2 0 D)) The joint checking of these two certificates in lockstep leads to the recording of the decision points in D, as desired. Proof outlines can be applied with identical ease and flexibility to many problem domains and to degrees of complexity beyond the simple case study used in this chapter: arithmetic, data structures such as lists, metaprogramming concepts like program contexts and calculi like CCS and the π-calculus, alongside mathematical concepts like simulation and bisimulation, etc. Notes The certificates for outline families presented in this chapter were first published in [START_REF] Blanco | Proof outlines as proof certificates: a system description[END_REF]. The final technical step is to enable certificates such as these to be developed and executed directly within Abella. An end-to-end solution is described in Section 13.3. Our first system made use of the Bedwyr model checking system, itself closely related to Abella. The changes to Abella in this delegated approach were minimum, as the proof assistant sat at the top of an architecture designed to produce and check proof outlines by a simple linear workflow in three steps: 1. A theorem prover (Abella in this case) is the source of the concrete syntax of definitions, theorems and proofs. Its direct involvement in the workflow can be quite limited. We extended the tactics language with ship tactic which marked proof obligations that were to be discharged by an external checker by means of a provided proof outline. 2. A translator specific to each theorem prover converts the concrete syntax of the theorem prover into that of the proof checker. It may simply use proof certificates declared by a ship tactic or derive certificates automatically starting from proof scripts and other similar evidence. Such a translator may be built into the theorem prover, thus encapsulating this and the previous step. The translator generates input usable directly by a general-purpose checker. 3. A proof checker implements a proof system according to the precepts of the FPC framework. We used Bedwyr to implement and execute a µLJF a kernel-instantiated with the FPC definitions of outlines and the translations generated from previous steps. A similar process of certification by outlines can be adopted by other theorem provers. It suffices to provide a connection between those provers and the proof checker by means of a custom translator. These components are free to implement sophisticated refinement mechanisms to produce more efficient certificates from the information contained in proof scripts-or provide limited guidance and rely on the checker to reconstruct the missing details. The proof certificates described in this chapter differ from their use as an independent representation of proof evidence-rather, they represent decision procedures which reconstruct a proof from a collection of high-level descriptions of proofs. The abstract inference steps defined by those certificate constructors have been generally decidable; commonly, discussions about Frege proofs involve polynomially checkable inference rules. This restriction is superfluous, and proof reconstruction may involve large numbers of choices and undecidable procedures. None of these generalizations are a concern of the FPC framework, which only forbids-by construction-menaces to the soundness of whatever proofs result from reconstruction. How long this reconstruction takes, or whether it terminates, has no bearing on the soundness property. The minimalistic proof outlines explored in the chapter are, indeed, fairly expensive to check (and are executed in a programming environment which is not itself optimized for performance), but their elaboration to more detailed forms of proof evidence mitigates these costs in subsequent runs. As we have seen, obvious induction rules constitute an important exception to the dictum that it is easy to take the step of trusting the direct encoding of a sequent calculus in a proof checking kernel-provided that we are willing to trust the logic engine used to that end. Although the code involve in the computation of the obvious invariants may be more involved, even external to the kernel, said code does not need to be trusted, since we can always arrange to check the generated invariants. Property-based testing, or PBT [START_REF] Fink | Property-based testing: a new approach to testing for assurance[END_REF], is a methodology developed to assist in the testing of computer programs by automatically generating and executing test data. In its basic form, PBT pairs a function to be tested with two additional categories of information: (a) a collection of properties constituting an executable specification of the function that relates inputs and outputs; and (b) a set of data generators for the types of the inputs of the function. Given all these elements, it is possible to automate the use of generators to create a certain number of inputs, pass them to the function, and verify whether the outputs satisfy the required properties. While generation of test data comes in manifold methods [START_REF] Utting | A taxonomy of modelbased testing approaches[END_REF], the two principal strategies are random and (subject to certain constraints) exhaustive testing. Used well, the technique enables quick exploration of the state space of a problem and rapid discovery of errors-either in the specification or in the implementation-in the form of concrete and actionable inputs that falsify the posited properties of the function. A third component is commonly added to the framework: (c) a set of data shrinkers which, given an input that contradicts the specification, attempt to find smaller, derived inputs that continue to exhibit the error. Hence, we obtain small counterexamples that are easier to understand and quicker to lead to the root causes of the error. The concept of PBT was originally applied to programming languages and was notably pioneered by Haskell's QuickCheck [START_REF] Claessen | Quickcheck: a lightweight tool for random testing of haskell programs[END_REF], whence it spread to many other languages. Soon it made the jump from the world of programming languages to most major proof assistants, where simple ports of QuickCheck gave way to more specialized counterexample generators, such as Nitpick [START_REF] Blanchette | Nitpick: a counterexample generator for higher-order logic based on a relational model finder[END_REF] for Isabelle. In this latter milieu, it is used to complement theorem proving activities with a preliminary phase of conjecture testing-whose goal is to zero in on incorrect theorem statements before substantial effort is put into them until a dead end in the proof is found. (Although, as the Curry-Howard correspondence states, the difference between proofs and programs is not profound.) In this chapter, we will adapt the FPC framework to develop a proof theoretical reconstruction of this style of testing for relational specifications-such as those commonly used to describe the semantics of programming languages-and explore its benefits in this environment. We do this by revisiting the concept of proof outlines introduced in Chapter 11, noting that the operation PBT strongly resembles the complement of a very succinct outline which, instead of attempting to prove a theorem, seeks to disprove it. Under the relational lens, (co)inductive definitions correspond to functions, theorem statements correspond to properties, and FPCs implement property-based testing proper. The certificates we write will be used to describe the shape and size constraints of possible counterexamples to a property. If such an outline can be completed in the µLJF calculus, we will have obtained a certified counterexample to the property in question. The rest of the chapter is organized as follows: Section 12.2 presents and illustrates the techniques on standard, first-order (algebraic) data structures. Section 12.3 lifts those techniques to more interesting structures containing bindings, represented using λ-tree syntax. Section 12.4 defines disproof outlines as FPCs. Section 12.5 applies those outlines to various problems and benchmarks about the metatheory of programming languages-extracted from related tools such as PLT Redex [START_REF] Felleisen | Semantics Engineering with PLT Redex[END_REF][START_REF] Klein | Randomized testing in plt redex[END_REF]-encoding those problems in λProlog, i.e., the specification level of Abella. Section 12.6 looks at the same problem within pure Abella, i.e., at the meta level of the prover. Section 12.7 concludes the chapter. Standard property-based testing To commence, we introduce and advocate a relational view of property-based testing starting from a general scenario. Suppose we wish to prove instances of a class of formulas which obeys the following general pattern: ∀ (x : τ) .P (x) ⊃ Q(x) Here, both P and Q are given relational specifications (for clarity, application is denoted in functional notation). In this setting, it can be important to sometimesas discussed in Section 11.2-move the type judgment for x into the logic of a formula by turning the type τ into a predicate τ(•) and adding it to the premise of the theorem statement thus: ∀x.(τ(x) ∧ P (x)) ⊃ Q(x) In general, proving such theorems can involve significant work since their proofs may be complex and involve the clever invention of prior lemmas and induction invariants. Indeed, it will often be the case in practice that a conjecture of this form will be stated, yet its formal statement will not, in fact, be a theorem of the logic because of specification errors in the relational definition of either P or Q, regardless of whether the intuition of the conjecture is correct. Therefore, it can be valuable to first attempt to find counterexamples to such formulas before any proof attempts, in the hope of finding many typical but shallow errors, and possibly some deeper ones. We would instead attempt to prove the negation of a conjecture, i.e., a formula of the form: ∃x.(τ(x) ∧ P (x)) ∧ ¬Q(x) That is, if a term t of type τ can be discovered such that P (t ) holds while Q(t ) does not, then one can return to the specifications of P and Q and revise them using the concrete evidence in t to determine how the specifications are wrong. Let us call these formulas counterexample lemmas. The process of writing and revising relational specifications could be aided if proofs of such lemmas and their associated counterexamples were discovered quickly. Example Suppose that we wish to write a relational specification for reversal of lists (say, of natural numbers). There are many ways to write such a specification, but in every case the statement of the idempotency of reverse should be a theorem. If the specification is named rev, the statement can be written as: ∀(L : list nat).∀(R : list nat). ∀(L : list nat). rev L R ⊃ rev R L ⊃ L = L More compactly, L could be dropped and the condition on rev written as rev L R ⊃ rev R L. Here, we assume the standard definition for a polymorphic type of lists instantiated with natural numbers as the type of elements. (While λProlog supports such polymorphic types, this feature is not yet part of Abella's mainline; this limitation extends to the embedded λProlog interpreter of the specification logic.) The literature describes two interpretations of relational specifications written in the style of Horn-like clauses in proof theoretical terms. 1. In specifications written directly in a relational style, say, in a language like Prolog or λProlog, we can interpret some of those specifications directly as, say, Horn clauses-provided that they are in that fragment of the logic. For example, each purely positive clause for the specification of addition of natural numbers in Figure 4.2 corresponds to one of the Horn clauses in Figure 4.1. In this encoding, proof search is structured as a series of alternations between a goal reduction phase and a backchaining phase [START_REF] Miller | Uniform proofs as a foundation for logic programming[END_REF]. Focused proof systems generalize this style of proof reconstruction, where goal reduction and backchaining correspond to the asynchronous and synchronous phases, respectively, and proof search is thus organized in bipoles. LKF is a representative of such a system. 2. A complementary approach to the proof theory of Horn clauses involves encoding those clauses as fixed points, hence effecting the closed-world assumption. (For example, the same Prolog-style specification of Figure 4.1 can be transcribed as the fixed point definitions of Figure 9.1.) In a proof system enriched with fixed points-such as µLJF-proofs of counterexample lemmas span a single bipole: first, a synchronous generation phase followed on all its premises by a single, asynchronous testing phase that completes the proof. Consequently, implementing the testing phase is an easy job; the difficulties lie in efficiently steering the generation phase through potentially large amounts of nondeterminism. Complexity is thus concentrated in the design and combination of generators. Example The counterexample lemma for the specification in Example 12.2.1 is the following, where list_nat is a generator of lists of natural numbers: ∃L.∃R. (list_nat L ∧ list_nat R) ∧ (rev L R ∧ (rev R L ⊃ ⊥)) Here, we associate conjunctions in two groups. On the left is the test data generation phase, charged with the selection of inputs. On the right is the purely computational checking phase, which validates the combination of inputs (i.e., the reverse of L must be R) and attempts to prove the negation of the conclusion. Note a naive generation phase like the one shown here assigns independent values to variables which are functionally related, resulting in backtracking and wasted work. If we take the view that rev "computes" R from L, we can simply drop the superfluous generator list_nat R. In the above example, the property theorem is correct, and in consequence, failure to prove it (i.e., success in proving a counterexample lemma) must arise from an error in the specification of rev. Properties (and generators) can also be defective: if we claim that a list is always equal to its reverse (rev L L), we can test it by looking for instances of its counterexample lemma: ∃L.∃R. list_nat L ∧ (rev L R ∧ (L = R ⊃ ⊥)) Here, any non-symmetrical list of two or more elements, such as [0; 1], will falsify the pseudo-property. PBT should help us find both types of errors quickly. Treating metatheoretical properties Describing computational tasks using proof theory often allows us to lift descriptions based on first-order (i.e., algebraic) terms to descriptions based on higher-order abstract syntax, specifically the λ-tree syntax representation [START_REF] Miller | Foundational aspects of syntax[END_REF][START_REF] Miller | Abstract syntax for variable binders: An overview[END_REF], which gives clean, declarative readings of variable binders. These possibilities extend to the two logical frameworks used throughout this thesis and represented by the two interpretations of relational specifications studied in this chapter. 1. Once logic programming is described in terms of proof search, it is natural to extend the treatment of first-order terms and Horn clauses (in Prolog) to the general manipulation of λ-terms (in λProlog). 2. Similarly, a sequent calculus presentation of model checking and inductive theorem proving in a first-order logic with fixed points [START_REF] Baelde | Least and greatest fixed points in linear logic[END_REF][START_REF] Heath | A proof theory for model checking: An extended abstract[END_REF] leads to generalizations based on λ-terms like Bedwyr and Abella, respectively. The full treatment of λ-tree syntax in a logic with fixed points is usually accommodated by the addition of nominal quantification with the nabla quantifier, as treated in Section 9.4. An important result about nabla is the following: implementation-is in Figure 12.2. The direct treatment of λ-terms within the PBT setting will allow us to apply the same generate-and-test proofs to find bugs in implementations of programming languages-such as failure of expected properties like type preservation-seamlessly lifted here to terms with binders. The type preservation property can be given form as a candidate theorem: Example ∀E .∀E .∀T . step E E ⊃ wt E T ⊃ wt E T If is_exp is a generator of expressions, i.e., λ-terms, and is_ty is a generator of simple types, the following counterexample lemma can be used to attempt to uncover faults in the specification: ∃E .∃E .∃T . (is_exp E ∧is_exp E ∧is_ty T )∧(step E E ∧wt E T ∧(wt E T ⊃ ⊥)) As in previous examples, we note that both E and T should be functionally dependent on E and, if this fact has been established beforehand, the independent generation of dependent data can be substantially simplified. Before we show how to implement this framework in each of the two logical settings under consideration (in λProlog and Abella, respectively), we need to encode the framework inside these two systems together with the FPC definitions which implement PBT around generate-and-test disproof outlines. This last element, common to both frameworks, is presented in the next section. Disproof outlines Chapter 11 presented FPC definitions to describe classes of problems via high-level outlines, in particular describing the general shape of proofs and the application of lemmas-as is customary in, say, proof assistants. Likewise, previous sections in this chapter have described and given proof theoretical justification to another broad class of problems of practical interest: that of counterexample search whose mechanization has been popularized by PBT frameworks. We will now proceed to the description of these disproof outlines, which will support both exhaustive and random generation of inputs. By convention, the FPCs we define will operate on counterexample lemmas of the form: ∃X 1 . • • • ∃X n .Generate(X 1 , . . . , X n ) ∧ Test(X 1 , . . . , X n ) Here, the "top-level" conjunction separates the two phases of the process. The generation phase will be a conjunction of generators of ground terms assigning values to all or some of the existential variables. The testing phase will perform conjunctive computations that assign values to the remaining variables, verify conditions, and ultimately attempt to prove the negation of the conclusion of the original candidate theorem. A simple and complete class of generators for inductive types are the typing judgments discussed in Section 11.2, although many other schemes are possible. Let us first present the design for exhaustive testing, following the model of SmallCheck [START_REF] Claessen | Smallcheck and lazy smallcheck: automatic exhaustive testing for small values[END_REF]. The FPC definition provides a single top-level certificate constructor, qstart, which takes two certificates: one for the generation phase and one for the checking phase. As the proof starts, the formula immediately obtains focus on the right-hand side. The initial certificate is tasked with the traversal of the formula, generating logic variables for existential quantifiers until it reaches the top-level conjunction, and there the two continuation certificates are distributed, each to its respective branch. Each phase then draws from its own set of certificate constructors: 1. The generation phase is controlled by a certificate constructor, qgen, which takes a descriptor that places concrete bounds on the size of the terms to generate. These bounds, together with the generator predicates, precisely define the search space for the run. All the action takes place under focus on the right, as positive conjunctions branch out. Generator predicates are expected to embody purely positive, terminating computations-carried out by the unfold expert, ever on the right-hand side. Two standard definitions are contemplated: (a) A height bound, or qheight, which constraints the depth of the derivation trees for the generated terms. This bound is decremented as the generators perform right unfoldings and propagated unaltered on branching points. Notably, this implies that the initial bound applies to every term in the generation phase. (b) A size bound, or qsize, which limits the size of terms in number of constructors. As in the previous case, allowances are decremented on right unfoldings. However, the size bound is defined so as to span the sum of constructors of all generated terms. Therefore, sizes must be communicated across both sides of a branching point (i.e., a positive conjunction); an intermediate variable is used to this end. 2. The testing phase uses a simple constructor, qsearch. It follows the generation phase and is designed to be much simpler. As in the generation phase, a number of conjunctive subgoals are computed in separate branches; these subgoals, too, are purely positive, terminating computations. Commonly, the final goal is the negation of the original conclusion and must therefore also allow enough asynchrony to release the focus, move the original conclusion to the left-hand side and search until an application of equality on a unification problem without solutions finishes the proof. Figure 12.3 shows the FPC definition for this exhaustive search procedure. Note how, in conformance with the single-bipole proof outline, indexing is unusually absent from the picture. Vis-à-vis FPC definitions for LKF in λProlog, where the open-world assumption reigns, all the "clauses" of a given clerk or expert must be grouped together in an inductive definition, which limits modularity. In addition, it demands that all clerks and experts be part of the definition-though unused clerks and experts are absent from the figure. This results in somewhat less readable definitions, but does not detract from the simplicity of the design, as seen here. Of course, other generation schemes beyond the two expounded in this section, many other strategies are possible by making available more guidance information through the certificates. Example A standard certificate with a height bound of 5 can be written: (qstart (qheight 5) qsearch) And a certificate with a size bound of 5 can be written: (qstart (qsize 5 X) qsearch) Note that here 5 is given as the upper bound, whereas the lower bound is left open as a logic variable: it is indifferent how many constructors we use, as long as the bound is not surpassed. Bound propagation is carried out by andExpert and eqExpert in Figure 12.3. (In both examples, natural bounds are written in algebraic notation for clarity.) The second variant is random testing as implemented by QuickCheck-style tools [START_REF] Claessen | Quickcheck: a lightweight tool for random testing of haskell programs[END_REF]. The structure of proofs is not affected by the change in strategy, and indeed the checking phase remains unaltered: changes are limited to the dynamic behavior of the generation phase, that is, how test data are generated to attempt to progress through the checking phase and arrive at a proof. These modifications are reflected in the certificate terms and the clerks and experts which consume those certificates; the generators themselves are unaltered, but play an indirect and fundamental role in the design work that follows. Critically, recall that relational specifications of generators of inductive data types can be encoded as fixed points. The bodies of those fixed point expressions are built out of a series of disjuncts, one for each constructor-for instance, recall the transcription of the typing judgment on natural numbers in Figure 9.1. For each branching point (i.e., a disjunction, one of a series of choices that leads to the encoding of a constructor clause), we will assign a fixed weight to each branch, and select one of the branches proportionally and at random. This regime induces the following changes in the FPC design: 1. First, we need to fix the number of times the generation phase will be executed. Generation needs to satisfy a further requirement: if a set of generated values fails to complete the proof-thus falsifying the original property-the complete set of values must be discarded en bloc. That is, backtracking must roll the proof back to the start of the generation phase and randomly assemble a completely new, independent set of inputs. In consequence, qstart must be augmented with such a number of tries parameter, which can be decremented each time we split the initial certificate at the top-level conjunction. Block backtracking will be the responsibility of the generation phase, next. 2. Implementing random branching for generators, we encounter another instance of a lockstep requirement pattern (as observed in Section 11.6 for administrative outlines) whereby information about formulas which cannot be attached to formulas without annotations must become part of the certificate and mimic some of the behaviors of the kernel. Consequently, the generation certificate, qcert, is structured around an abstract description of generators, qform, which mimics their branching structure (i.e., conjunctions, qand, and weighted disjunctions, qor) down to named recursive generator calls, qname, and don't-care branches without generators, qnone. The generation certificate thus contains: . . (a) A map that assigns names of generators with their qform structure. All generators used in such descriptors must have corresponding entries in the map. (b) Two fields that mirror the workbench zones of the sequent with their qform descriptors and progress in lockstep with the kernel. The lockstep requirement is visible in particular in the synchronous rules for conjunction, disjunction, and unfolding. Moreover, the disjunctive expert must procure some source of randomness from which to determine which branch to take; it must also fail upon backtracking, so that the generation phase runs from beginning to end each time, without intermediate states. The certificate definition for random PBT is given in Figure 12.5. The presentation relies not only on Bedwyr syntax for readability, like Figure 12.3, but also on some handy features: shallow, like the predefined string type; and deep, like I/O predicates. In fact, both λProlog and Bedwyr offer system interfaces which, though rudimentary, allow the passing of information to a system call or coprocess to, say, generate a random number between a certain range, or directly pick left or right based on the weight of each branch-and the semantics of I/O predicates correspond to the desired block backtracking behavior. In the purely relational model of Abella, no such interactive facilities avail. For example, suppose a size bound carries over to the maximum number of random choices involved in data generation. In such a situation, a list of random numbers could be passed in the certificate and consumed by: (a) the various iterations of the generation phase; and (b) each instance of the generation phase. Treatment of these external sources of randomness would parallel the propagation of bounds seen in the qsize certificates of Figure 12.3, although the solution is not as satisfactory as extending Abella with some form of I/O predicate, following the example of Bedwyr-which was extended precisely for purposes such as this. The following certificate will guide the proof by generating 10 lists of naturals. An empty list is chosen with 50% probability, otherwise a natural and a continuation list are generated. In generating natural numbers, zero is chosen 90% of the time, otherwise we return the successor of the recursively generated number. Example (qstart 10 (qcert ((qmap "is_natlist" (qand qnone (qor 50 50 qnone (qand qnone (qand (qname "is_nat") (qname "is_natlist")))))) :: (qmap "is_nat" (qand qnone (qor 90 10 qnone (qand qnone (qname "is_nat"))))) :: nil) nil (qname "is_natlist") ) qsearch ) Note that, for example, the formula structure of list_nat is reproduced as a qmap entry labeled with the name "is_natlist", and the positions of fixed point generators are likewise marked with the names of the entries in the map. In this fashion, the certificate knows what it is generating and how to take random choices. As with other appearances of this lockstep pattern, the burdensome part of the certificates is an easy target for automation, and extends gracefully to other, possibly more sophisticated random generation strategies. For the remainder of the chapter, illustrations will be based on the cleaner exhaustive FPC, which runs in current versions of Abella without changes. . . λ Hosted PBT in λProlog The first of the two extension methods discussed in Section 12.2 makes use of some higher-order features of λProlog to work on signatures with binders using λ-tree syntax. Following [START_REF] Mcdowell | Reasoning with higher-order abstract syntax in a logical framework[END_REF], we introduce a simple specification logic, which in this case is basically a basic Prolog-like meta-interpreter whose sole peculiarity is its interpretation of nabla as λProlog's universal quantifier. The definition of the interpreter drives the derivation of our object logic, which represents Horn-style clauses through the two-place predicate prog which relates heads and bodies of the object clauses. These clauses are built out of object-level logical constants (tt, or, and, nabla) and user-defined predicate constructors. Figure 12.7 presents the kernel; in this purely positive presentation, each logical constant, as well as the unfolding of prog clauses, is controlled by a client-defined expert predicate-nabla remains transparent to the kernel, as usual. Two predicates are in charge of the computation, interp and check, differing only on the use in the latter of the expert predicates to steer search. Both interpret object-level connectives as λProlog code, and look up and unfold client-side constructors in program database of prog. Porting the FPC definition of Figure 12.3 to this embedded kernel is straightforward and much more succinct, only partly due to the slightly simplified model of the checker; Figure 12.8 presents this version of the FPC. Note that there is no need to handle the disjunctive expert: because prog-based specifications allow alternatives in the form of variant clauses, disjunctions can be dispensed with at this level. However, it is important to note that-at the specification level-there is no alternative to disjunction in recursive definitions, and therefore the proof theory must give the connective full consideration. Example We now revisit, in full, the disproof of the property in Example 12.2.2. Again, we want to generate lists of natural numbers and compute their reverse, and find out something about the combination of specification and property. Again, we want to falsify the assertion that the reverse of a list is equal to itself. Figure 12.9 gives the encoding of the problem as prog clauses in the hosted kernel. (Note how the specification makes full use of λProlog's type system.) To prove the counterexample lemma, it suffices to pose a goal like the following: cexrev XS YS :check (qgen (qheight 3)) (is_natlist XS), interp (rev XS YS), not (XS = YS). :-In > 0, In' is In -1. 12.8 Figure The SimpleCheck-style FPC for (bounded) exhaustive property-based testing in embedded λProlog. Signature and module names and accumulations of kernel signatures are omitted. Here, we determine that, since the generation phase alone require guidance, we generate candidate lists in accordance with the limits of a generation certificate, here up to a certain depth by means of qheight, pairing it to the goal via check. The testing phase performs deterministic computation, each of which components can be delegated to the meta-interpreter interp. Finally, the negated conclusion employs negation-as-failure (NAF). Given that this NAF involves ground terms exclusively, the call is logically sound. We are now ready to lift the full implementation-programming environment, kernel, FPC definition-to λ-tree syntax. Figure 12.10 translates the static semantics of Figure 12.2 to prog clauses in the hosted λProlog kernel, making use of a signature with the obvious type declarations for constants. The encoding in λProlog is standard: we declare constructors for terms, constants, and types, while carving out values via an appropriate predicate, is_value. Similarly to values, another predicate, is_err, characterizes the threading in the operational semantics of the special error expression-used to model runtime errors such as taking the head of an empty list. Third in line is the small-step evaluation relation, step. Progress is defined in terms of these three This version of the λ-calculus enjoys some usual properties, of which we will focus on two. First, it satisfies subject reduction and progress-where progress means, from a direct reading of the progress predicate, "being either a value, an error, or able to take an evaluation step." In fact we can easily prove those results 12.11 Figure Top-level ( λ-term) expression generators written in hosted λProlog. Figure For bound variables to appear in the generated expressions a dedicated context needs to be maintained and its contents drawn as valid expressions. This latter step is performed in raw λProlog outside the prog interpreter so that it does not wrongly correlate prog steps taken inside a generator with size bounds of generated terms. in a theorem prover like Abella. Furthermore, by defining λ-term generators in the usual manner PBT can be applied to this setting, as well. The only point of interest concerns the role of nabla and the use of variables generated by it in the resulting expressions, as illustrated in Figure 12.11. Example Complementing a proof of subject reduction, we may wonder whether the calculus enjoys the subject expansion property, as well. The sagacious reader will promptly note this is highly unlikely, but will also observe that, rather than waste time in a fruitless proof attempt, we can define a counterexample lemma and let the machine disprove the property for us: . . λ There are many other interesting queries we can explore in this fashion. Are there terms that do not have a type? Are there terms for which evaluation does not converge to a value? This, and many others, are useful applications of the PBT approach. As a more comprehensive validation we addressed the nine mutations proposed by the PLT Redex benchmark, to be identified as violations the preservation or progress properties. Example The first mutation in the PLT Redex benchmark introduces a bug in the typing rule for application, matching the range of the function type to the type of the argument. The change occurs in the T-APP rule in Figure 12.2: Γ Σ M 1 : A → B Γ Σ M 2 : B Γ Σ M 1 M 2 : B T-APP-B1 In the specification, the mistake translates into replacing the program clause for typing applications with the following faulty code in Figure 12.10: prog (wt Ga (app X Y) T) (and (wt Ga X (funTy H T)) (wt Ga Y T)). And, as we can verify, the mutation causes both properties to fail: The table in Figure 12.12 summarizes the results. In this comparison, αCheck is set to use negation-as-failure for fairness, although this is not always the optimal choice [START_REF] Cheney | Advances in propertybased testing for αProlog[END_REF]. In any case, the λProlog implementation completes all the problems in the alloted time, consistently maintaining a time advantage over αCheck which only grows larger as the challenges become more complex. Bug #6 requires exploring the state space up to a depth of 11, the largest of the sents: a bug number, the property that is the subject of the test (preservation or progress), checking times in αCheck and λProlog ( αC and λP, respectively, with a timeout threshold of 300 seconds), one of the smallest counterexamples found, and a description of each bug along with a rating of increasing difficulty (Shallow, Medium, Unnatural). . . problem suite. Under this discipline, it is trivial to adapt the pairing FPC of Section 5.2 to the hosted kernel and seamlessly combine both families of bounds. It is also interesting to note that, for this particular set of problems, Teyjus and ELPI perform indistinguishably from one another-in contrast to their showing in Section 8.5, where ELPI consistently beat Teyjus. The developments in this section can be run directly in λProlog or inside Abella by loading the λProlog kernel, FPC definition and programs by means of the Specification command (see Section 10.2). Native PBT in Abella The second of the two extension methods discussed in Section 12.2 encodes relational specifications directly as fixed points. These fixed point encodings can be written directly in a logic like µLJF-as implemented in Section 10.3, directly embedded in Abella. However, the deep connection between Abella and logic (itself compatible to a large extent with µLJF; see also Section 10.2) does imply that relational specifications can be programmed directly as Abella definitions and transparently reflected into the logic (this aspect is studied in Section 13.4). In order to accommodate rich specifications involving nominal quantification, the rich kernel described in Section 10.3, in particular Figure 10.8, must be used. From the point of view of the user, specifications are written in pure Abella and transparently reflected into the kernel; working directly with Abella has the added advantage of a richer type system for user terms than the kernel implements-and the disadvantage of working with closed inductive definitions, less modular than the λProlog code of the previous section (so an external element of composition would be needed). For example, compare Figure 12.13 with its isomorph in Figure 12.10, above. Other specification code obeys the same shallow syntactic translation rules and therefore requires no particular attention. To make the code in the figure embeddable, it suffices that contexts and bindings be implemented in terms of user-defined types instead of the predefined types o and olist. Example In the correct implementation of the semantics of Stlc of Figure 12.13, there will be a (relatively involved) proof of the progress theorem: Theorem prog : forall E T, wt nil E T -> progress M. Suppose now the typing relation wt is replaced with the version that contains bug #1 of the PLT Redex benchmark, as shown in Figure 12.14. In this case, prog is not a theorem, i.e., it can be falsified. One way to show this is to write the . . corresponding counterexample lemma, cexprog, and prove it. This could be done in Abella by coming up with witnesses manually, using them to instantiate the existentials and completing the proof. More interestingly, we can use PBT with a counterexample outline to certify the counterexample lemma-simply by adding generators to the statement of the manually proved lemma. Even more, by adding the typing judgments to the original prog non-theorem, we can falsify it by proving the associated counterexample lemma with the same certificate, i.e., a disproof outline. (c cons) X) XS)) XS := is_value X /\ is_value XS ; step (app (lam M T) V) (M V) := is_value V ; step (app M1 M2) (app M1' M2) := step M1 M1' ; step (app V M2) (app V M2') := is_value V /\ step M2 M2'. As opposed to Example 12.2.1, which pairs a correct implementation with an incorrect specification, this example combines a correct specification (i.e., property) with a buggy implementation. Noting that given a ground term E its type T is determined by the typing relation wt, it is possible to balance the generation phase by producing only an independent set of inputs from which others (in this case the type) are derived by computation. The resulting system is functionally equivalent to the hosted λProlog kernel of the previous section. However, the architecture in its current form it is not competitive with λProlog in terms of performance. This is principally due to the runtime behavior of the pure embedded Abella kernel. Executing the kernel as part of an Abella proof involves large numbers of redundant clause lookups. What is worse, pattern matching the arguments of a call to the main relations (async, etc.) with the various clauses representing inference rules creates multiple, also redundant, unification problems; even though each of these effectively matches formal parameters and actual arguments at the top, smaller, trivial subproblems are generated and vacuously solved after this step. If we use the kernel with nabla, the unification problems are slightly more complex, but even a simpler implementation of the unification algorithm-such as is described in Section 13.2has a negligible effect (a small penalty factor) compared to the grind imposed by Abella's search-for a discussion on these issues, see also Section 13.1. Nonetheless, as we note, this organization of the system yields equivalent results, even as significant redesigns are needed to make it usable in practice. Notes The present proof theoretical treatment of property-based testing was first presented in Blanco et al. (2017b). Property-based testing validates code against an executable specification by the automatic generation test data, typically following random or exhaustive regimes, or a combination of both. It was originally conceived for its use in testing applied to programming languages [START_REF] Claessen | QuickCheck: a lightweight tool for random testing of Haskell programs[END_REF] and has since to most major proof assistants [START_REF] Christian Blanchette | Automatic proof and disproof in Isabelle/HOL[END_REF][START_REF] Bibliography | Foundational property-based testing[END_REF] to complement theorem proving with a preliminary phase of conjecture testing. For a comprehensive review, refer, e.g., to [START_REF] Cheney | αCheck: A mechanized metatheory model checker[END_REF]. In the more specific arena of model checking applied to metatheoretical pursuits, one of the most representative initiatives is PLT Redex [START_REF] Felleisen | Semantics Engineering with PLT Redex[END_REF], an executable domain-specific language for mechanizing semantic models built on top of the Scheme dialect DrRacket. It supports QuickCheck-style random testing and its usefulness has been demonstrated in several impressive case studies [START_REF] Klein | Run your research: on the effectiveness of lightweight mechanization[END_REF]. However, PLT Redex offers limited support for relational specifications and none whatsoever for binding signatures. αCheck's role is to address those deficiencies [START_REF] Cheney | αCheck: A mechanized metatheory model checker[END_REF]. On top of the logic programming language αProlog, the tool adds a checker for relational in the same vein as has been done in this chapter. One of several possible implementation techniques is based as well on NAF-as far as testing of the conclusion is concerned. The generation phase is instead "wired in" via iterative-deepening search based on the height of derivations; in this regard αCheck is less flexible than our FPC-based architecture (and can be interpreted as offering a fixed choice of clerks and experts). Finally, more distant cousins in the logic programming world are declarative debugging [START_REF] Naish | A declarative debugging scheme[END_REF] and the Logic-Based Model Checking project at Stony Brook (LMP). As we have noted, the implementation of random PBT is not directly supported by Abella due to a lack of I/O functionality while in proof mode. Clerks and experts can be programmed and run in Bedwyr, as shown above, or equivalently in λProlog-note, however, that the I/O capabilities of λProlog are not available in the integrated interpreter that Abella implements, but ELPI integration is an appealing possibility. Neither λProlog nor Bedwyr offer a stable system interface nor a random number generator, so that a source of randomness must be obtained by a relatively rudimentary interface with the runtime system, e.g., via file descriptors or coprocesses. A possible "pure" workaround involves parameterizing the random disproof outlines by a list of random choices which would be obtained by some external means and passed as an argument. This alternative can be executed in pure Abella and in its embedded λProlog, but must be carefully threaded through clerks and experts so that different parts of the sequence are passed to different branches of a proof. Interesting examples of the metatheory of programming languages require the addition of nabla to the logic, presented in Section 9.4, but our problem domain also involves simplifications to the logic from features that are not required. In particular, the initial rules are absent and only equality is used to close proof branches; all atoms are defined as fixed points. If our model is limited to PBT, only a subset of the connectives is used for this purpose, and structural rules are severely restricted (i.e., no stores or decides). Furthermore, polarization is fixed. All these facts can be combined to provide a simplified kernel. As observed in Example 12.6.1, the dependencies between the various generators and computational predicates paves the way to a certain amount of shifting obligations between the generation and the testing phase. This is particularly important for performance, because generating independent data and filtering only those sets that satisfy certain judgments (e.g., generating terms and types independently and accepting only pairs such that the term has the selected type) is potentially very inefficient. In the chapter, we have considered generators that are complete for given inductive types, but this is not a strict requirement. It is possible and often interesting to define generators for specific subtypes such as, say, "small" integers. Independent notions such as exhaustive or random generation, combined with custom generators, can be flexibly combined into composite certificate definitions simply by the usual mechanism of module accumulation in logic programming. More in general, we may want to use a certificate not simply to witness a counterexample, but to point to the specific point where a proof fails, therefore avoiding the inspection of all the paths that (unhelpfully for the counterexample) succeed. The proof process can be seen as a dialogue between the doomed derivation of the impostor theorem and the proof of its negation-the counterexample lemma. Potentially, this could assist in repairing the specification or the property, and therefore the proof. The notion of productive use of failure [START_REF] Ireland | Productive use of failure in inductive proof[END_REF] may serve as inspiration. For example, in Example 12.6.1, we want to skip directly to the case of a list with two or more distinct elements, avoiding empty and singleton lists which offer no useful information. This would suggest modifying the generator of lists to account for these minimum size requirements. Another natural area of interest is in applications of model checking, for instance applied to graphs and properties like reachability in degenerate cases. This could be useful to find bad behaviors in program analysis. A proof theoretical presentation of this area has been initiated by Heath andMiller (2015, 2017). 13 Certificate integration in a proof assistant Abella architecture The Abella proof assistant is an OCaml program implemented as a collection of modules. Some of those modules define a public interface and encapsulate critical data structures, whereas others have a less obtuse structure. Nevertheless, the overall architecture is clearly laid out and welcomes modular extensions of the sort required to bring the developments of this part to executable form. The issues and technical considerations involved in this process-on which the systems described throughout Part III depend-are outlined in this chapter. In a preliminary overview, we introduce the primary components where development is concentrated. At the heart of Abella-and central also for our purposes-is the Term module that represents higher-order λ-terms and models the suspension calculus on which their manipulation is based [START_REF] Nadathur | The suspension notation for lambda terms and its use in metalanguage implementations[END_REF]. The interface type of terms is reproduced in Figure 13.1. Interestingly, the variables of Abella terms are simply memory cells, and their comparison is given by pointer equality: their attributes do not factor into such operations as substitution, and seemingly identical but separate variables can be created-that is, a variable is created only once. A problematic point is that this representation decision is neither hidden by the semantics of the module interface nor even by its publicly declared definition. Over the course of our developments, we have encountered and corrected several bugs and misbehaviors. Also related to term manipulation, but less critical for our purposes, are the modules Metaterm of meta-level terms, i.e., Abella's propositions, and Typing, where the still-untyped terms read in interactive use are defined. Other basic modules that we will manipulate include Prover, which acts as the main interface syncL and syncR. Calling the checker, say, on an unfocused sequent collects all clauses operating on unfocused sequents and pattern matches the arguments via unification on all of them. Since the kernel has few functions with "many" clauses, this leads to much repeated and wasted work. Furthermore, because most pattern matching consists in copying inputs around or decomposing at the top level of formulas only, the fairly costly unification algorithms spend their substantial runtime mostly on trivial identities. The rest of the chapter is organized as follows: Section 13.2 presents an extended, hierarchical unification model for Abella which allows more flexible use of higher-order unification and the effective embedding of kernels with nabla for use with the FPC framework. Section 13.3 begins to develop the integration of the FPC framework in Abella proper, introducing the FPC-based tactics used in previous chapters as a complement, or replacement, of the standard tactics of Abella-the sole limitation being that our tactics are limited to accepting certificate terms representing complete proofs in current versions of Abella instead of being used to build a proof interactively. Section 13.4 completes the proof theoretical integration of the framework by connecting the logic of Abella with the logic implemented by the embedded kernel. Section 13.5 considers the uniform organization of FPC definitions in λProlog and their modular composition extended, as well, to Abella. Section 13.6 concludes the chapter. Extended unification Abella, like Bedwyr (with some limitations) and all modern implementations of λProlog, implements (higher-order) pattern unification, a decidable fragment of full higher-order unification [START_REF] Huet | A unification algorithm for typed λ-calculus[END_REF]. In general, the extension of first-order unification [START_REF] Robinson | A machine-oriented logic based on the resolution principle[END_REF][START_REF] Martelli | An efficient unification algorithm[END_REF] to higher-order terms preserves none of the desirable qualities of the former: in particular, higher-order unification is undecidable (more precisely, semidecidable, so there are no guarantees of termination: we cannot know if a solution exists); it is non-determinate (so, if there may be arbitrarily many incomparable solutions which cannot be expressed as a single, most general unifier, or MGU); and it is typed (in the sense that term-level typing information plays an important role in the search for solutions). In contrast, unification problems on higher-order terms satisfying the pattern restriction preserve the good traits of first-order unification (decidability, determinacy and type-freeness), being the weakest extension from first-order to the higher-order setting in which the usual rules of α βη-conversion hold (Miller, 1991a). Typically, pattern unification is applied dynamically. That is, in solving a higher-order unification problem, we assume that the problem is in the pattern fragment; if at some point we find an equation which fails to satisfy the pattern restriction, it is set aside until-by the effect of substitutions dictated by other equations in the problem-it becomes a pattern equation. Although pattern unification is considered "almost complete in practice" with problems outside the fragment occurring very rarely, there are at least two reasons to look beyond the status quo. First, a controlled, interactive application of full higher-order unification inside a proof assistant can avoid its pitfalls while allowing a user to guide the solution of complex problems. Second, there are significant use cases which have simple, well-behaved solutions in spite of falling outside the pattern fragment in non-fundamental ways-the enriched kernel needed to represent nabla at the object level in Section 12.6 is one such case. The second case will be addressed by implementing a recent technique proposed by [START_REF] Libal | Functions-as-constructors higher-order unification[END_REF] called functions-as-constructors (FC) unification, or FCU. In pattern unification, applications with existential variables at the head require that all arguments be distinct variables, universally quantified within the scope of the head. FCU extends patterns (as well as some prior proposed generalizations) by observing that the requirement that bound variables be used as arguments can be extended to term constructors, which very often are acting as functions and do not alter the desirable properties of unification patterns-the traditional head and cons operations on lists are examples of this. The conditions of operation of the extended algorithm must be given first. Definition In a formula, an occurrence of a bound variable is essentially universal if it is bound by a positive occurrence of the universal quantifier, a negative occurrence of the existential quantifier or a term-level abstraction. All other bound variables, bound by a positive existential quantifier or a negative universal quantifier, are said to be essentially existential. Essentially universally quantified variables can be instantiated only with eigenvariables, and essentially existentially quantified variables can be instantiated with general terms (in logic programming, this involves logic variables and unification). In the context of a unification problem, an occurrence of term is rigid if its head is a (term-level) bound variable or an eigenvariable; it is flexible if its head is a logic variable. A unification equation can be classified according to the rigidity of its terms as rigid-rigid, rigid-flexible, flexible-rigid, or flexible-flexible. Here we shallow follow [START_REF] Libal | Functions-as-constructors higher-order unification[END_REF] for general notation and naming conventions. Given a signature of non-logical constants C and a signature of essentially universally quantified variables Σ, a restricted term in an equation e is defined as follows, where BV (e) is the set of bound variables of e: 1. A variable in Σ or BV (e) is a restricted term. 2. A (non-vacuous) application ( f t 1 • • • t n ) is a restricted term if f is either in C, Σ, or BV (e), and t 1 , . . . , t n are all restricted terms. A system of equations satisfies the FCU property iff it satisfies the following three restrictions: 1. Argument restriction: for all (non-vacuous) applications (X t 1 • • • t n ) where X is an essentially existentially quantified variable, t 1 , . . . , t n are all restricted terms. 2. Local restriction: for all (non-vacuous) applications (X t 1 • • • t n ) where X is an essentially existentially quantified variable, no argument t i is a subterm of a different argument t j . 3. Global restriction: for all pairs of (non-vacuous) applications (X t 1 • • • t n ) and (Y s 1 . . . s m ) where both X and Y are essentially existentially quantified variables, no argument t i of the first application is a strict subterm of an argument s j of the second application. Example By inclusion, all unification problems in the pattern fragment are also in FC. Given C = {cons, nil, fst, rst} and Σ = {l, z } at appropriate types: 1. cons (X (fst l )) (rst l ) = rst (Y (fst l ) (fst (rst l ))) is in FC. 2. X (cons z nil) = rst l breaks the argument restriction. 3. X (fst l ) l = cons z l breaks the local restriction. These examples were originally given by [START_REF] Libal | Functions-as-constructors higher-order unification[END_REF]. The explicit encoding of sequents explored by McDowell and Miller (2002, Section 4) and applied to the construction of proof checking kernels with nominal quantification in Section 10.3 and Figure 10.8 naturally results in unification problems involving the fst and rst constructors: these problems are conceptually simple, but strictly fall outside the pattern fragment, but are covered by FC unification. The restrictions of the FCU property are sufficient conditions to maintain determinacy. A series of results in [START_REF] Libal | Functions-as-constructors higher-order unification[END_REF], together with the algorithm given there, show that unification problems satisfying the FCU property are decidable, determinate and (essentially) type-free-since, although the algorithm works in a typed setting, it is the presence of a constructor or a bound variable that guides application of the algorithm (as part of the restricted terms of Definition 13.2.1), not the types of those constructors and variables. The original presentation of the FCU algorithm follows closely that of pattern unification in Miller (1991a), with minor changes in some of the rules. Like pattern unification, it uses a pruning substitution (also slightly modified from its original form) that assists in optimizing the term-traversal computations involved in the performance of the occurs-check-an operation called variable elimination. The pruning operation is applied to exhaustion before the applicable steps of the main algorithm; the paper makes no attempt at organizing the sequence of applications of pruning to avoid inefficiencies. In order to open up unification-both by allowing controlled use of full higher-order unification and by treating additional classes of solvable problems without user intervention-we integrate a hierarchy of unification algorithms in the Abella proof assistant. The first addition is a purely functional implementation of higher-order unification as presented by [START_REF] Huet | A unification algorithm for typed λ-calculus[END_REF], assuming the η-rule-as implied by the axiom of functional extensionality. The algorithm operates by building a matching tree where nodes correspond to unification problems-with the original problem at the root-and edges correspond to substitutions; leaves can be failure nodes indicating a failed solution attempt or success nodes, each indicating a solution to the original problem as the sequence of substitutions from root to success node. Solutions are preunifiers since outstanding flexible-flexible equations are underconstrained and left untreated by the algorithm. To ensure tractability, some technical changes are needed. First, termination is imposed by computing matching trees down to a certain, finite depth. Information is preserved by introducing a third kind of pseudo-leaf, the suspend node, from which tree construction can resume incrementally. Second, the pointer-based, shared term representation (see Figure 13.1) is ill-suited to the construction of matching trees, where several potential substitutions for the same variable need to be considered simultaneously. Instead of relying on Abella's side-effecting implementation (which cannot support multiple competing substitutions for a given variable) a purely functional implementation of term operations like normalization and substitution-in which new copy-terms are returned and variable equality is based on attributes and not on pointer equality-is provided. The development is encapsulated in a new module that follows the general structure of the original Unify module-Abella's implementation of pattern unification and its interface with the rest of the system. Like that module, it parameterizes essentially existential and universal variables by two sub-modules: universal (resp. existential) quantification represents either: (a) essential universal (resp. existential) quantification on the right; or (b) essential existential (resp. universal) quantification on the left. Nominal abstraction is easily added and treated in both cases as constant-like. The module signature is shown in Figure 13.2. Example In addition to serving as the basis for specialized unification algorithms, the Huet module can be more directly utilized to expose unification to the user of Abella through specialized tactics-as opposed to the common view of it as a black box that either succeeds or fails. This exposure allows one to judiciously apply the power of full higher-order unification when a problem requires it, all the while maintaining automation of a well-behaved fragment like patterns or FC. As a particularly useful illustration, we have implemented a tactic, match, which in its simplest mode of operation seeks to solve the unification problem defined by the current goal. For example, given the signature C = {a, }, a goal of the form F a = a a represents unification a problem with four distinct unifiers: { F , λ x. x x }, { F , λ x. a x }, { F , λ x. x a }, and { F , λ x. a a }. If we apply match to such a goal, we obtain a disjunction of goals generated by applying each of the unifiers to the original goal, from which we need to be able to finish the proof at hand. While the result in this example is simple, more complex behaviors arise in richer unification problems. To manage the possible nontermination of higher-order unification, external depth bounds must be added to the tactic. At a given point, construction of the matching tree may halt and the results associated to any success nodes returned. The only risk is that proofs that rely on a solution that has not yet been found cannot be finished without further, possibly incremental exploration-until the right disjunct of the goal is found. Applying the match tactic to the premises is trickier, as shown by the fact that for a unification problem with one solution at a certain depth, failing to go deep enough to find that solution and simply reporting an empty (if partial) list of solutions may be confused with a success by negation-as-failure. During case analysis, Abella performs a single step of Huet's algorithm covering the simple scenario of application of such a tactic. To complement match, a simpl tactic that simplifies and updates sequents operates as a complement to the former tactic and can be used to perform a manually controlled application of higher-order unification. The second addition to the unification framework is a functional implementation of the FCU algorithm. A line of attack that circumvents its still relatively undeveloped algorithmic presentation harnesses the general higher-order unification algorithm (just described) by the following observation: Huet's algorithm applied to a problem which satisfies the FCU condition must terminates by the properties of the FCU fragment; moreover, if the problem has a solution, the matching tree has a single success node which represents the preunifier of the most general unifier that is the solution of the problem. The hosted version of the FCU algorithm follows this high-level outline: 1. Check the FCU property for the problem. If it fails, notify that the problem falls outside the supported fragment. 2. If the check is successful, iteratively grow the matching tree by the bounded version of Huet's algorithm until the success node is reached, and extract the associated preunifier. 3. Turn the preunifier into the unifier by generating pruning substitutions followed by the FCU substitutions for flexible-flexible pairs, repeating the substitution to exhaustion. 4. Apply the substitutions of the most general unifier thus obtained to the original problem. Implementation is straightforward with two important technical notes: First, whereas Huet's algorithm uses η-expanded terms throughout its treatment, subsequent treatment (including by the FCU algorithm) relies on their η-contracted form, so the interface code must adjust term representation accordingly. Second, any extensions of unification based on a functional representation of terms (such as in Huet's matching trees) must translate the resulting unifier (i.e., a list of substitutions) into the corresponding side-effecting substitutions implemented by Abella's term library, taking care not to alter the original terms and their pointers-on which the standard substitution operation of Abella relies. In the case of FC unification, which strictly extends the functionality of pattern unification, the most flexible solution is to encapsulate the single unification operation in a module, Fcunify, which exactly reproduces Abella's native Unify interface (shown in Figure 13.3) with minimal changes. An appealing possibility is to make FCU the default unification algorithm in Abella-begin stronger than pattern unification while retaining its high-level computational properties. This poses several technical challenges: First, the figure makes clear that the interface to the unification algorithm depends on a particular unification algorithm, namely pattern unification. In particular, unification failures and errors are not generic enough. The negative impact of this coupling is that a full replacement of pattern unification by the more general FC unification cannot properly succeed without (fairly minor) changes in the interface. Secondly, and independently from the interface's reliance on a particular algorithm, Abella depends (implicitly) on a particular implementation of unification. Aspects like the generation of names of new existential variables or the order in which possible solutions to a case analysis step are generated depend implicitly on undocumented and unplanned behavior. Therefore, changes that preserve the semantics of unification can break existing proof scripts and modify Abella's interaction with the user. However, only a complete redesign at the core of the program can remove this limitation. In spite of these considerations, both versions of the tactics-using either pattern and FC unification-can coexist gracefully. Additionally, tactics which use unification in the course of attempting to complete a full subgoal-notably, search-can be completely replaced by the more powerful FCU algorithm, as their succeed-or-fail behavior leaves no partial proof state that may differ from Abella's standard. This extended, FCU-based search is exactly what is needed to run the enriched kernel natively in Abella, as required in Section 12.6. There is a performance penalty arising from the added complexity and unoptimized implementation, but the replacement does not significantly affect the trends observed in that part. To our knowledge, this is the first implementation and practical use of the FCU algorithm. This existing interface is adopted as a wrapper to expose our new developments to Abella, noting that error reporting generalizes the contents of the standard interface, which are specific to pattern unification. Certifying tactics in Abella In Section 4.4, we saw how a proof system like LKF a admits a natural, direct encoding as a logic program in a logic programming language like λProlog. As fixed points and equality are added to the logic, resulting in proof systems like µLJF a , the same techniques are applicable by selecting a logic programming language with fixed points whose semantics corresponds closely to those of the logic, such as the common core of both Bedwyr and Abella. In this latter context, a kernel is an Abella program-more precisely, a definition-operating on (polarized) formulas built on the kernel's object logic, itself defined as regular Abella data types; this approach was introduced in Section 10.3. In Chapter 11 and Chapter 12, we glossed over the applied aspects of applying certification in this environment. In fact, Chapter 11 is formulated in pure Abella and assumes that we can call a full checker by means of special tactics. Those tricks will be revealed now. Up to this point, we have a kernel and FPC definitions written as Abella programs. Thus, the checker can exist as part of an Abella development. At this level, we can define fixed points and natively-through the checker-prove properties about formulas based on those fixed points. Nonetheless, and insofar as Abella's G logic (or a fragment thereof) is compatible with the intuitionistic calculus µLJF, an intriguing possibility is to use Abella to certify properties defined at the reasoning level of Abella itself. In order to have Abella's hosted µLJF a kernel, also at the reasoning level, discharge proof obligations inside Theorem environments, we need to make Abella aware of the kernel and expose its functionality through new tactics, e.g., the certify of Chapter 11. In addition, Abella's theorems need to be reified into formulas of the kernel's object logic in order to serve as valid inputs. First, let us account for the additions to the tactics language. The certify tactic is easily defined in terms of a call to search. We start considering the scenario that maps directly to the kernel's interface, in which a single formula wrapped in the initial sequent is checked against a certificate. That is, a certificate will be given to discharge the entire proof of a theorem, with the language of FPCs as a full replacement of Abella's language of proof scripts (for full proofs). This is achieved by the following pseudo-algorithm: 1. Reify the statement of the full theorem in Abella into a (polarized) formula of the kernel's object logic. (This process is covered after the present discussion on tactics.) 2. Given a certificate term (of type cert), execute the kernel's entry point check on this certificate and the translated formula. This execution is carried out by a call to the search tactic without depth search restrictions. Under this scheme, we seek to use the sequent calculus embodied by the kernel to find a full proof of (the projection of) the original theorem. Indeed, given a certificate term and suitable FPC definitions, it suffices to allow the full checker to run and either succeed, thus proclaiming the theoremhood of the reified theorem statement-and, provided that the translation is adequate, of the original. The computational character of proof checking means that all we need to do behind the scenes is search on the embedded kernel. However, to obtain efficient proof checking we need to make the tactic aware that its execution is controlled not by an explicit bound, but implicitly by the nature of the kernel. By default, the tactic employs a stateless version of iterative deepening which attempts to find proofs of growing depth 1, 2, . . ., while repeating all the work at lower bounds. We enrich the tactic with the ability to perform depth-first search-the standard choice for deep proofs whose backtracking points are completely determined by the kernel. (This behavior is not always what interactive users want, and can have a negative effect on the performance of Abella proof scripts where finding short proofs with less control is more important. However, DFS can be toggled in the call to the extended search inside the certify tactic wrapper.) Example With this apparatus in place, sessions like the one in Figure 11.12 (prefixed by its corresponding declarations and definitions) can be executed natively in Abella. Likewise, property-based testing as described in Section 12.6 becomes directly executable (for this, see Example 13.3.4). A first generalization of certify consists in employing it as a first-class tactic. Instead of being a full replacement of the legacy tactical language of Abella and using it to prove full theorems, a certificate term can be provided at any point in the proof in an attempt to find a subproof, thus discharging the current goal. The reifier must now inspect an arbitrary Abella sequent-essentially, a collection of a goal, a list of hypotheses and a list of eigenvariables-and fold back the new components in the translation. Namely, given eigenvariables x 1 , . . . , x m , hypotheses H 1 , . . . , H n and goal G, it must provide a translation of the formula: ∀x 1 . • • • ∀x m .H 1 ⊃ • • • ⊃ H n ⊃ G . The translation of this representation of the sequent can then be passed to the checker. Suppose certify takes a certificate Ξ. It is a simple matter to define a certificate constructor which reconstructs the original sequent (or rather, its reified form) in the kernel by asynchronously introducing both eigenvariables and hypotheses and then applying the supplied certificate to the exact sequent for which it was written. Such a preprocessing wrapper can be written in exact form as (preprocess m n Ξ). Its definition is completely straightforward. Now, instead of resorting to standard Abella tactics, we may choose to prove the goal with a certificate. For example, from the original example, we may use: certify (apply? 1 0 (idx "plus0com") search). Example Which will allow us to progress to the second subgoal. On the other hand, a weak tactic like certify search. will fail to prove the goal, much like a regular search would. In this way we can use certificates at any point in a proof. A second orthogonal generalization of the basic certify tactic is the addition of interactivity. Instead of presuming a certificate-a description of an entire proof-is known in advance, it is possible to define certificates with holes in them. If those holes are undecorated logic variables, they are not in themselves useful, but adequate support from the FPC definition can be obtained. Suppose a certificate constructor, ask, is provided. The role of such a certificate is simply to as the client for a certificate, read it and use it to try to finish the current goal. Embedded as a continuation inside a more complex certificate, this simple extension implements the strategy to "use a certificate to continue the proof until a certain point, and when this point is reached ask for more information to proceed." In effect, in this way we reproduce in the FPC framework the interactive proving loop characteristic of proof assistants like Abella, and enables a flexible and complete revamp of its tactical language. This proving style can be implemented in Bedwyr-where I/O predicates were added precisely for this purpose-, but not yet in Abella, which currently lacks the ability for tactics to interact with the user-though, paralleling Bedwyr, they are a feasible design extension which could be used in Abella. Example Still illustrating these concepts from refinements of Example 11.7.2 and its associated Figure 11.3, suppose now that we wish to perform a proof through the FPC framework that resembles the information flow of the Abella proof script-in regular practice, we commonly lack a formal proof at the outset and build one through the use of the proof assistant. The first step is clear: perform an induction and case analysis to obtain the zero and successor cases. What to do in each subgoal is not yet clear. Therefore, we may leave these unspecified by writing a full certificate which is, nonetheless, missing crucial information, though these point are explicitly annotated as follows: (induction? (case? 0 ask ask ) ) In fact, this is a more concrete representation of the incremental construction of a certificate in Example 11.7.1. Applied to pluscom, this certificate in fact succeeds and stops at the same sequent shown in Example 13.3.2. At this point, we can continue writing suitable certificates for each branch, such as as noted for the zero case in the example: (apply? 1 0 (idx "plus0com") ask) Here, note that as a continuation certificate we decide to simply apply the lemma and then return for further instructions: this successive refinement closely parallels the common language of standard tactics. However, note that once we step into the world of FPCs we are committed to it: it is not considered at this point that one could drop back into the world of uncertified tactics. The complement of the certify family of tactics is the falsify tactic that is necessary to directly implement the treatment of property-based testing developed in Chapter 12-certainly, counterexample lemmas can themselves be explicitly written as theorems and proved (as they were given in the examples in that chapter), but this tedium can be averted by integrating the process in the proof environment, i.e., as a dedicated tactic. Said tactic never succeeds in proving the current goal; rather, it is of interest for its "side effects," i.e., its informative output: if a counterexample is found, instantiations for the variables are shownwhich is to prompt an interruption of the user's proof efforts until the problem is repaired. The most modular alternative is to define falsify as a wrapper around certify in the following manner: 1. From the current goal (or sequent), attempt to derive a counterexample lemma according to the patterns described in Section 12.2. 2. If a counterexample lemma avails, wrap it in a fresh sequent and apply to it the certify tactic (reification and checking). The falsify tactic is parameterized by a certificate term-a priori, typically a disproof outline, but this is not a strict requirement-, which is propagated to certify. 3. Upon success, inspect Abella's proof witness and extract the values of the prefix of existential variables, and output them to the console. Note that success in the counterexample lemma does not extend to the original goal. Example The disproof outlines thus enabled in both positive and negative form (i.e., through the tactics certify and falsify) permit the full range of experiments presented in Example 12.6.1 and Figure 12.14. Note that, in our discussion, tactics like certify take a full certificate term as argument and use it to guide the search for a complete proof by cleverly driving a native search-like tactic. These certifying tactics can be embedded in standard proof scripts. The addition of interactivity at the certificate level will eventually make it possible to dispense with legacy tactics and build proofs in Abella fully within the FPC framework. Connection between Abella and the kernel Second, after describing the new tactics added to Abella, we turn to the reification process by which Abella terms are reflected into terms of µLJF a as implemented by the embedded kernel. The core of the logic G on which Abella is based is an intuitionistic and predicative subset of Church's Simple Theory of Types extended with generic judgments via the nabla quantifier [START_REF] Gacek | Nominal abstraction[END_REF]. To those core rules are also added rules for definitions, induction and coinduction with the standard fixed point interpretation. The resulting logic can be given a sequent calculus presentation which largely coincides with µLJ-described in Section 9.1, and from which the proof systems that are the central subject of Part III derive through focusing ( µLJF) and augmentation ( µLJF a ). Owing to the relatedness of both systems, moving between them is, in this instance, quite simple, as established by the next definition. Definition The reification function from formulas of G to formulas of µLJ is defined as a function • . Connectives of each logic are distinguished by subscripts. For formulas in the core (first-order) fragment of G, map the top-level connective to its corresponding version in µLJ, and recurse on its subformulas. For example: A ∧ G B = A ∧ µLJ B ∀ G A = Πx.∀ µLJ Ax Here, Π denotes the meta-level quantifier; other connectives receive analogous treatment. Definitions in G are given by a finite set of clauses Π x.p x B p x. Here, p is a predicate constant that takes a number of arguments given by the length of x. A predicate is defined by exactly one clause with body B-with standard restrictions to guarantee the existence of fixed points-, whose interpretation is given by the unfolding rules (analogous to those in Section 9.1). Instead of the generic , we write It is immediate that the operation of reification thus defined is an isomorphism across the fragments of both logics-being essentially identical-and therefore preserves provability between G and µLJ, as supported by Baelde and Miller (2007, Proposition 3). In fact, the kernel developed in Section 9.3 implements the focused and augmented version of µLJ directly in terms of Abella's implementation of G. The connection between the unpolarized logic µLJ considered now and its extension µLJF a is given in two steps: from µLJ to µLJF by Baelde et al. (2010, Theorem 1), and from µLJF to µLJF a by Theorem 9.3.1. In a similar vein, the translation operation as implemented in tactics like certify proceeds in two steps, each requiring its own considerations: First, an Abella (i.e., G) formula is reified into µLJ, following Definition 13.4.1. Ostensibly, no attempts are made to reify Abella sequents in mid-(co)induction: inductive reasoning in Abella is modeled by size restriction annotations which sustain the cyclic reasoning used by induction and coinduction tactics (Baelde et al., 2014, Section 5). Second, the reified µLJ formula moves into the world of polarized formulas common to µLJF and µLJF a . To this end, a polarization function is needed. In our experience, a purely positive polarization (for least fixed points) is the most useful in practice and can be made the default, but we need to be able to specify other polarization strategies. For this, the certify tactic-and therefore falsify, which must supply adequate parameters to it-must accept a second (optional) parameter to convert from an unpolarized µLJ formula to a formula suitable for use by the kernel. Example In the LKF setting, a simple scheme that has been used in previous work, e.g., by Chihani et al. (2016b), parameterizes the relation that converts unpolarized client-side formulas into polarized kernel-side formulas (in negation normal form) by two logical connectives: a conjunction and a disjunction. In so doing, it allows the user to select the polarities of both connectives. Here, bool is the type of unpolarized formulas and form is the type of polarized formulas. This simple approach works in most cases, but more sophisticated conversions are possible-of course, in an intuitionistic logic like µLJF, the polarity of disjunction is fixed. Such relations can be loaded as part of the specification, written in λProlog, and referenced by name in the syntax of certifying tactics. After covering both reification and additions to the tactics language (in the previous section), all that remains is to connect the structure and evolution of an Abella development session to our logical framework-hence establishing the . . basis for trust in the embedded certification framework. At its core, a session is a sequence of three types of commands, each with a corresponding image in the certified world of µLJF a : 1. Kind and Type declarations of types and constructors build up the signature. All constructors are modeled on top of the term monotype used by the kernel, i, assisted as needed by typing judgments-as introduced in Section 11.2, and by means of which ambiguities are resolved. 2. Define and CoDefine definitions give names to (respectively, least and greatest) fixed point expressions. References to existing, stratified definitions inline their corresponding fixed points at the point where they occur in subsequent expressions. For the sake of efficiency, the certifying version of Abella maintains a shadow table that stores the µLJ fixed point alongside the native Abella definition, indexed by name. This mechanism is also used to define formulas and refer to them by name inside certificates-especially useful for complicated and possibly reoccurring formulas like induction invariants and cut formulas. 3. Theorem statements accompanied by full, succeeding proof scripts. These are trusted to be elaborated into full proofs by Abella-and now, alternatively by a combination of Abella and the embedded kernel if they rely on certify tactics. Previously defined theorems are admissible as lemmas under the assumption that a full, formal proof for them is available, i.e., they are available for use in a "lemma context" and need not be re-proved each time they are used. (The skip tactic admits an obligation without proof, unsoundly assuming the existence of an arbitrary proof; it must be disallowed in any serious development and will not be considered here.) By composing this sequence of signature extensions, definitions, and proofs, we can obtain a pure, proof theoretical view of an Abella development as a single proof, say, expressed in µLJF, displayed in Figure 13.4. This is yet another instance of the cut backbone proof pattern first observed in the resolution proofs of Section 3.6. In this reading, the session can always be finished by inspecting a trivial goal, or the proof can be continued by formulating a new result and prolonging the session by a cut. On one side, the proof of the theorem is given; on the other, the theorem is added to the lemma context and can be used in subsequent "proof branches." If every theorem is proved by tactics in the certify family, . such a "session proof tree" is actually being built; otherwise, Abella's soundness needs to be trusted as well. A tactics language based on the FPC framework solves this problem organically; the feasibility of this approach has been explored in Section 11.7 and is hereby given technical substance and logical legitimacy. The final question concerns the threading of seemingly isolated calls to the checker by certify within the larger context of the session proof tree: namely, all previously proved lemmas must be made available as lemma decides-motivated back in Section 11.4. The most faithful encoding extends the kernel interface with a check_with_lemmas call which populates the lemma map, Λ, with the available theorems. To offer finer control over lemma decides, the certify tactics are extended with a third optional parameter: in its basic form, a list of theorem names-a subset of the current context of proved theorems-whose use as lemmas is allowed within the proof. Example Once again in the context of Example 11.7.2, pluscom is preceded by lemmas plus0com and plusscom. Therefore, calls to the certify tactic correspond to calls to the kernel interface check_with_lemmas where the context of lemmas is populated by the two previous results. In this way, the backbone of theorems that compose an Abella session are connected to the formal proof represented in Figure 13.4. It should be noted that the proof of plusscom has plus0com (as well as any previous results) available as lemma decides. In large sessions, a wealth of uninformative decision points has potentially deleterious effects on performance, which motivates the introduction of filters on the list of previous (restricted to usable) lemmas. Clerks and experts as specifications Thus far, there has been a clear separation between kernels written in pure λProlog (in Part II) and kernels written in Abella (in Part III). Noticeably, the specification of an FPC definition in λProlog is not only more compact and readable-partly, because the open-world assumptions enables us to declare only what we need and elide the rest-, but also more modular-for the same reason, we can seamlessly compose definitions, whereas in Abella all clerks and experts need to be defined, and do so at once. (Compare, for example, the (abridged) presentation of the SimpleCheck FPC in Abella in Figure 12.3 with the code written for λProlog in Figure 12.8; the simplified execution model in the latter case hardly accounts The session starts with the default signature and a goal R that is trivially provable, e.g., by finishing the session. In terms of proofs, a session is a sequence of theorem statements represented by cuts: on one premise, a proof Π i for the theorem T i must be given; on the other premise, the proved lemma is stored (by freezing) and ready to be used in subsequent proofs. This picture elides definitions which augment the effective signature and does not contemplate the introduction of the unsoundness by skipped proofs. for the diminished complexity.) However, for all practical purposes, the code of an FPC definition-i.e., its clerks and experts-are simple logic programs that inhabit the common fragment of λProlog and Abella, and consequently, a unified treatment can be foreseen. In Abella, the existence of a specification logic in the form of a slightly trimmed down version of λProlog (for details, see Section 10.2) can be exploited to take advantage of the terseness of λProlog specifications. This change involves minor changes-or, alternatively, additions-to the kernel, where calls to clerks and experts are delegated to the specification level by the curly bracket notation. Specification-level predicates of type o thus replace reasoning-level predicates of type Prop, and a specification including those predicates must be loaded before the kernel, as usual. In particular: 1. Clerks and experts are defined as λProlog predicates in the style of Part II. 2. Clerks and experts are called from the kernel in curly brackets, thus representing their provenance at the specification level. Otherwise, the implementation remains unchanged and as obviously correctand amenable to formalization-as its counterpart in pure Abella. (This extension is not compatible with Bedwyr, however.) By using specifications, λProlog-defined FPC definitions can be combined in a flexible manner, but once imported into Abella by the Specification command, the open world is closed and cannot be modified. Moreover, the FPC declarations become part of whatever client-defined specifications are require by the user. A further improvement from modularity comes from allowing multiple names specifications to be imported and handled separately-this is achieved by a simple technical modification. The default, nameless namespace in Abella can be preserved for backwards compatibility. With this increased modularity, the user can load several FPC specifications separately, whether they cooperate or they conflict. All that is missing is to allow the certify family of tactics to (optionally) specify a subset of FPC specification namespaces to select clauses only from the designated names. This provides increased control to seamlessly change the style of certification within a single development session and without risk of conflicts. The modification is made at the level of the basic search tactic, though it can be extended to others, as well. To summarize, we have explored various levels of modularity and indirection and various connections between λProlog, Bedwyr and Abella and their connections to the Abella world, along with requisite background and extensions: 1. A kernel and FPC definitions can be defined in λProlog, loaded at the specification level and executed purely at that level, as was done in Section 12.5. 2. The kernel can be defined as Abella code, and FPC definitions can be given indistinctly as Abella of λProlog clauses-as discussed in this section. To prove properties written directly as Abella code, reification to the kernel logic is employed. If no λProlog code is utilized, proof obligations can be "shipped" to Bedwyr for execution as an alternative. 3. Given the contact surface between G and µLJF a , a third possibility involves executing a kernel which manipulates Abella formulas almost directly, with the sole addition of polarization. The lack of embedding then implies that a simple search no longer succeeds, and the language of tactics must be upgraded to feature a slightly more complete, Bedwyr-like search. Notes The FCU algorithm is one of several computationally simple restrictions of higherorder unification (and extensions of higher-order pattern unification) than can compute MGUs for the unification problems generated during the execution of a kernel with support for nominal abstraction. Previously, Tiu (2002) proposed a limited extension to the pattern unification algorithm of Miller (1991a,b) modified to account exclusively for the constructors of the lists that compose the local contexts in their explicit representation. FCU recently received another, independent implementation by [START_REF] Hamana | A functional implementation of function-as-constructor higherorder unification[END_REF]. In the event of success of the falsify tactic, it is desirable to obtain a printout of the witness terms that effectively falsify the target property (by proving the counterexample lemma, which sees the universal quantifiers as existentials which are instantiated by the generators). This information is easily extracted by the trace of the proof contained in Abella's witness terms, and is then trivially added to falsify's wrapper around certify, which produces said witness from the counterexample lemma. . Afterword "A thesis is not finished: it is abandoned." On the conduct and culmination of doctoral studies a professor once offered me this curious maxim. By that jocular koan he exorcized the powerful symbolism the artifact often holds in the eyes of the student: that one should not attempt in vain to square the circle in a quest for perfect completeness. There is-there should always be-more. Indeed, a thesis is not a living document as much as it is a travel photograph, laboriously developed at a milestone, just as we set for the next one. Ultimately, it is not an end in and of itself; rather, it is the first leg in a journey of scientific research-or is it really the first? Either way, it should not be the last. To conclude, I reproduce a poem by Antonio Machado, Spanish exile in France. It is very well known, likely because of its simple poignancy. Caminante, son tus huellas el camino, y nada más; caminante, no hay camino: se hace camino al andar. Al andar se hace camino, y al volver la vista atrás se ve la senda que nunca se ha de volver a pisar. Caminante, no hay camino, sino estelas en la mar. The journey continues. 4. 1 Figure 1 Logic specification of natural numbers and addition on them as Horn clauses. The specification is based on a type nat with two constructors representing the standard inductive definition of natural numbers: z of type nat, and s of type nat → nat. 4. ; of type o -> o -> o, for ∨. 5. :-of type o -> o -> o, for ⊂ in the sense of the implication ⊃ found in program clauses D, where the succedent (i.e., the head of a clause) is written before the antecedent (i.e., the body of the clause), following the standard syntax of proof search as implemented in logic programming languages. 6. => of type o -> o -> o, for ⊃, the implication found in goals G, whose succedent is a program clause D which is used to extend the program for the purposes of finding a proof of the succedent under hypothetical reasoning. A maximally elaborate FPC. Signature and module names and accumulations of kernel signatures are omitted. Core declarations are assumed. Indexes are drawn from the usual inductive definition of natural numbers, nat here, where s denotes the successor. A variation on the maximally explicit FPC of Figure5.2. The types and clauses given here replace the uses of maxv and maxt in that figure.In this version, first-order unification provides determinate computation of witness terms, and eigenvariables are managed fully by the kernel. are stored in the certificate. Specifically, we redefine two pieces of information: (a) the certificate constructors previously assigned to eigenvariable bindings and substitution terms, used respectively to treat the universal and the existential quantifier; and (b) the clerk and expert predicates that linked those constructors with their corresponding connectives. The changes with respect to Figure5.2 are shown in Figure6.1. Figure let rec check certificate = function | Unfocused(storage, workbench) -> (match workbench with | [] -> if decide_cut_conflict certificate then failwith "Decide and cut are in conflict" else let check_decide = decide_expert certificate storage and check_cut = cut_expert certificate storage in check_decide || check_cut | hd :: tl -> (match hd with | NegativeFalse -> false_clerk certificate storage tl | NegativeAnd(left, right) -> and_clerk left right certificate storage tl | NegativeOr(left, right) -> or_clerk left right certificate storage tl |The MaxChecker kernel written in OCaml. For display purposes, the store and release rules make use of catch-all matches instead of exhaustive listings. 6. 3 Figure 3 The MaxChecker interface to FPCs in OCaml. Shown here are some especially interesting cases. All definitions are defined in mutual recursion with the kernel as defined in Figure6.2. let decide_expert = function | Index(index, next) -> Some (next, index) | _ -> None 6.4 Figure Definition of clerks and experts of the maximally elaborate FPC against the MaxChecker bureau interface, as called in Figure 6.3. All definitions follow the pattern exemplified here. 0 7. 1 Figure 01 Standard example of propositional formula in the DIMACS CNF format. 0 7. 3 FigureFigure 03 Certificate of unsatisfiability for the formula in Figure 7.1 in the DRUP certificate format. Certificate of unsatisfiability for the formula in Figure 7.1 in the DRAT certificate format. Figure The lemma backbone proof pattern for CNF refutations. Figure Certificate of unsatisfiability for the formula in Figure7.1 in the TraceCheck certificate format. Consider the resolution proof given in TraceCheck format in Figure 7.6. Clauses 1 through 8 reproduce the formula to refute (namely, that of Figure 7.1), and lemmas 9 through 12 constitute a proof by (hyper)resolution. Figure Proof schema of a hyperresolution proof step in LKF a . % 1 Figure 1 List of (the) clause on which to factor. type factr int -> cert. % Used to start the sync phase in factor checking. type fsmall cert. %% MODULE cutE (rlist (factor I K ::Certs)) (factr I) (rlisti K Certs) Cut :lemma K Cut. % Describe the meaning of the factoring subproof. allC (factr I) (x\ factr I). orC (factr I) (factr I)Additions made to the binary resolution FPC (Figure 3.5) to support factoring. A new top-level proof step constructor is added to the resolution step, along with constructors and clauses for the (small) factoring subproof. (c1) | t(c1). [clausify(0)]. % Left premise allC (res I Cert) (x\ res I Cert). orC (res I Cert) (res I Cert). falseC (res I Cert) (res I Cert). storeC (res I Cert) (res I Cert) lit. %decideE (res I (rex J Cert)) (rex J Cert) (idx I). %decideE (res I (rex J Cert)) (rex I Cert) (idx J). decideE (res I Cert) Cert (idx I) :-Cert = (rex _ _). someE (rex J Cert) (rex J Cert) T. someE done done T. 8.2 Figure Addition of ordering of resolvents to the (unordered) binary resolution FPC (Figure 3.5). The only necessary change occurs in decide rule of the left premise, where instead of two clauses, one for each possible ordering of the resolvents, a single clause fixes said ordering. Changes are noted by showing, for each affected rule, the old clauses commented and immediately followed by their replacements. Continuing Example 8.3.1, we look at the various representations under consideration. The full standard payload (unordered, without substitution information) strongly closely resembles the description in Example 3.6.1: The ordered binary resolution FPC with added subtitution information. Changes to the signature are limited to the removal of resolvent sufproofs with ambiguous ordering (cut clauses involving the resolveX constructor are likewise removed) and the introduction of constructors for quantification and substitution. Changes in the module (see Figure8.3) add constructor condutions to the rules, which further constraint the system and allow for more natural elaboration. Presentation conventions are shared with Figure8.2. Figure orC (start Ct Certs) (start Ct Certs). falseC (start Ct Certs) (start Ct Certs). storeC (start Ct Certs) (start Ct' Certs) (idx Ct) :-inc Ct Ct'. cutE (start _ Certs) C1 C2 Cut :-cutE (rlist Certs) C1 C2 Cut.%cutE (rlist (resolve K Cert::Certs)) Cert (rlisti K Certs) Cut :-% lemma K Cut, Cert = (res _ _). cutE (rlist (resolve K Cert::Certs)) Cert (rlisti K Certs) Cut :lemma K Cut, (Cert = (res _ _); Cert = (rquant _)). cutE (rlist (factor I K ::Certs)) (factr I) (rlisti K Certs) Cut :-The ordered binary resolution FPC with added subtitution information (continued). Presentation conventions are shared with Figure 8.2. Figure Space complexity of resolution certificate elaborations. Payload sizes are defined as the sum of components of a problem: formula and certificate. The size is computed as the sum of constructors, where natural numbers (serving as indexes) are represented natively and counted as single constructors. In this and following figures, data series are represented by the following color codes: unordered-without (Figure8.1), ordered-without (Figure8.2), ordered-with (Figure8.3), maximal (Figure5.2). 8. 6 Figure 6 Time complexity of resolution certificate elaborations on ELPI. Presentation conventions are shared with Figure 8.5. 8. 7 Figure 7 Time complexity of resolution certificate elaborations on Teyjus. Presentation conventions are shared with Figure 8.5. 1 Figure 1 (a) the definition of introduction rules for the connective; (b) the preservation of the cut elimination property once . nat ≡µ(λNat. λn.n = 0∨ ∃n .n = S n ∧ + Nat n ) plus ≡µ(λPlus. λk.λm.λn.(k = 0 ∧ + m = n)∨ ∃k .∃n .n = S n ∧ + p = S p ∧ + Plus k m n ) 9.Logic specification of natural numbers and addition on them as a least fixed points in µLJF. The specification is based on a type nat with two constructors representing the standard inductive definition of natural numbers: 0 of type nat, and S of type nat → nat. 2. false of type prop, for f . 3. /\ of type prop -> prop -> prop, for ∧. . 4. \/ of type prop -> prop -> prop, for ∨.5. -> of type prop -> prop -> prop, for ⊃.6. forall of type (A -> prop) -> prop, for ∀.7. exists of type (A -> prop) -> prop, for ∃. Define nat : nat -> prop by nat z ; nat (s N) := nat N. Define times : (i -> bool) -> prop by times (mu Pred\Args\ or (some N\ (eq Args (zero ++ N ++ zero ++ argv))) (some K\ some M\ some N\ and (eq Args ((succ K) ++ M ++ N ++ argv)) (some N'\ and (Pred (K ++ M ++ N' ++ argv)) (Plus (N' ++ M ++ N ++ argv))))) := plus Plus. Figure The µLJF a kernel in Abella (continued). prop by async Xi Phi Gamma (cons_bool (l\ nabl (P l)) Delta) G := async Xi Phi Gamma (cons_bool (l\ P (rst l) (fst l)) Delta) G ; async Xi Phi Gamma nil_bool (unk (l\ nabl (P l))) := async Xi Phi Gamma nil_bool (unk (l\ P (rst l) (fst l))) ; syncL Xi Phi Gamma (l\ nabl (P l)) G := syncL Xi Phi Gamma (l\ P (rst l) (fst l)) G ; syncR Xi Phi Gamma (l\ nabl (P l)) := syncR Xi Phi Gamma (l\ P (rst l) (fst l)) ; % Kernel interface Define prove : cert -> bool -> prop by prove Cert Form := exists Cert', unmarshal Cert Cert' /\ async Cert' nil_ctx nil_ctx nil_bool (unk (l\ Form)). 10. 8 Figure 8 Extensions to the µLJF a kernel written in Abella to support nabla. L1, freezeRClerk R0 R1. initLExpert (pair# L0 R0) :-initLExpert L0, initLExpert R0. storeLClerk (pair# L0 R0) (pair# L1 R1) (idx2 IL IR) :-storeLClerk L0 L1 IL, storeLClerk R0 R1 IR. decideLClerk (pair# L0 R0) (pair# L1 R1) (idx2 IL IR) :-decideLClerk L0 L1 IL, decideLClerk R0 R1 IR. storeRClerk (pair# L0 R0) (pair# L1 R1) :-storeRClerk L0 L1, storeRClerk R0 R1. decideRClerk (pair# L0 R0) (pair# L1 R1) :-decideRClerk L0 L1, decideRClerk R0 R1. releaseLExpert (pair# L0 R0) (pair# L1 R1) :-releaseLExpert L0 L1, releaseLExpert R0 R1. releaseRExpert (pair# L0 R0) (pair# L1 R1) :-releaseRExpert L0 L1, releaseRExpert R0 R1. 10.11 Figure The pairing meta-FPC in Abella implemented at the specification level (continued). 1 Figure 1 Define nat : nat -> prop by nat z ; nat (s N) := nat N. Define plus : nat -> nat -> nat -> prop by plus z N N ; plus (s N) M (s P) := plus N M P.11.Relational specification of natural numbers and addition on them in standard Abella at the reasoning level. Compare this with the specification level of λProlog in Figure4.2. 11. 4 Figure 4 Extensions to the µLJF focused proof system needed to support obvious inductions in proof outlines, introduced informally in[START_REF] Baelde | Focused inductive theorem proving[END_REF]. A sequent is now prefixed by Σ, its set of eigenvariables. Asynchronous quantifiers add new eigenvariables and (not shown here) asynchronous equality (possibly) unifies them. Fixed points are endowed with a new obvious (co)induction rule, each with two variants: with or without checking of the obvious premise. The provisos define the obvious invariant that trivially satisfies the obvious branch. . x) nil_bool) (unk (S x)) ; async Lambda Xi Sigma Phi Gamma nil_bool (unk (nu B T)) := exists Xi' S, coindClerk' Xi Xi' /\ coindInvariant' Sigma Gamma T S /\ forall x, async Lambda (Xi' x) (cons_i x nil_i) Phi nil_ctx (cons_bool (S x) nil_bool) (unk (B S x)) ; async Lambda Xi Sigma Phi Gamma nil_bool G := exists Xi' Idx C ?1, (G = (sto ?1) \/ G = (frz ?1)) /\ decideLClerk' Xi Xi' Idx /\ member_lemma (lemma Idx C) Lambda /\ syncL Lambda Xi' Sigma Phi Gamma C G ; Define prove_with_lemmas : cert -> bool -> list_lemma -> prop by prove_with_lemmas Cert Form Lemmas := exists Cert', unmarshal Cert Cert' /\ async Lemmas Cert' nil_i nil_ctx nil_ctx nil_bool (unk Form). 11. 7 Figure 7 Extensions to the µLJF a kernel written in Abella to support both non-local decides on lemmas and obvious induction and coinduction. 1 . 1 (induction! B AU SU) is expanded to the unmarshalled certificate (induction B AU SU AU SU), and similarly for inductionS.2. (apply! B AU SU) is expanded to (apply B AU SU AU SU). ? A CL CR): in the current asynchronous phase, look for the first (branching) left disjunction. Apply the asynchronous unfolding rule at most A times to get to one such connective. To continue proof search, use continuation certificates CL and CR for the left and right branches. 4. (apply? A S I C): finish the current bipole, performing at most A asynchronous unfoldings and S synchronous unfoldings until the end of the bipole. To decide on a formula at the boundary of the asynchronous and synchronous phases, use index I. When the bipole ends at the release rule, use certificate C to continue proof search. nat M -> forall N, nat N -> forall P, plus M N P -> plus M (s N) (s P). certify (induction! 1 0 1).Theorem pluscom :forall N, nat N -> forall M, nat M -> forall S, plus N M S -> plus M N S. certify (induction! 2 1 0). 11.12 Figure (Simple) proof outlines for the commutativity of addition in Abella. 2. It begin by the common induction pattern:induction on #. intros. case H#. N, nat N * -> (forall M, nat M -> (exists S, plus N M S)) H2 : nat M H3 : nat N1 * ============================ exists S, plus (s N1) M S Variables: M S N1 IH : forall N, nat N * -> (forall M, nat M -> (forall S, plus N M S -> plus M N S)) H2 : nat M H3 : plus (s N1) M S H4 : nat N1 * ============================ plus M (s N1) S 1 . 1 (induction# B AU SU AC SC D): decorated version of the self-contained certificate (induction B AU SU AC SC) with an extra parameter, D, representing the tree of decisions made throughout the proof. 2. (apply# B AU SU AC SC D): decorated version of the self-contained certificate (apply B AU SU AC SC), extended with a decision tree D as for the above case. Let step a specification of the small-step evaluation relation and wt a specification of the typing relation-reflecting the language semantics in Figure 12.2, while the signature (with binders) obeys the syntax laid out in Figure 12.1. 12. 5 Figure 5 The QuickCheck-style FPC for random property-based testing in Bedwyr. Presentation conventions are shared with Figure 12.3; unchanged blocks are omitted as well. list_nat ≡µ(λListNat.λl . l = nil∨ ∃n.∃l .l = cons n l ∧ + nat n ∧ + ListNat l ) cexsexp M M' A :check (qgen (qsize 8 _)) (step M M'), interp (wt null M' A), not (interp (wt null M A)). A = listTy M' = c nl M = app (c hd) (app (app (c cns) (c nl)) (c _)) cexprog M A :check (qgen (qsize 6 _)) (wt null M A), not (interp (progress M)). A = intTy M = app (c hd) (c (toInt zero)) cexpres M M' A :check (qgen (qsize 8 _)) (wt null M A), interp (step M M'), not (interp (wt null M' A)). A = funTy listTy intTy M' = lam (x\ c hd) listTy M = app (lam (x\ lam (y\ x) listTy) intTy) (c hd) 0 ) 0 nil)) the type of cons returns int (. cons) nil vars do not match in lookup (S) 12.12 Figure Results of PBT on the Stlc benchmark. In order, each column repre- Define progress : exp -> prop by progress V := is_value V ; progress E := is_err E ; progress M := exists N, step M N. Define memb : o -> olist -> prop by memb X (X :: Gamma) ; memb X (Y :: Gamma) := memb X Gamma. Define tcc : cnt -> ty -> prop by tcc (toInt N) intTy; tcc nl listTy ; tcc hd (funTy listTy intTy) ; tcc tl (funTy listTy listTy) ; tcc cons (funTy intTy (funTy listTy listTy)). Define wt : olist -> exp -> ty -> prop by wt Ga M A := memb (bind M A) Ga ; wt Ga error T ; wt Ga (c M) T := tcc M T ; wt Ga (app X Y) T := exists H, wt Ga X (funTy H T) /\ wt Ga Y H ; wt Ga (lam F Tx) (funTy Tx T) := nabla x, wt (bind x Tx :: Ga) (F x) T. 12.13 Figure Dynamic semantics of the Stlc language, implemented in Abella. Define wt : olist -> exp -> ty -> prop by wt Ga M A := memb (bind M A) Ga ; wt Ga error T ; wt Ga (c M) T := tcc M T ; % wt Ga (app X Y) T := % exists H, wt Ga X (funTy H T) /\ wt Ga Y H ; wt E (app M N) U := exists T, wt E M (funTy T U) /\ wt E N U ; wt Ga (lam F Tx) (funTy Tx T) := nabla x, wt (bind x Tx :: Ga) (F x) T. Theorem cexprog : exists E T, wt nil E T /\ (progress E -> false). skip. Theorem cexprog : exists E T, gen_exp E /\ gen_ty T /\ wt nil E T /\ (progress E -> false). certify (qstart (qgen (qheight 5)) qsearch). Theorem prog : forall E T, gen_exp E -> gen_ty T -> wt nil E T -> progress M. falsify (qstart (qgen (qheight 5)) qsearch).12.14 Figure Buggy implementation of the wt typing relation, replacing that given in Figure12.13 according to bug #1 of the PLT Redex benchmark. In this version of the semantics, counterexample lemmas are provable and their associated nontheorems are falsifiable by the application of PBT techniques. 4. X (fst l ) = rst (Y (cons (fst l ) (rst l ))) breaks the global restriction. ( * Pairs of terms, esp. disagreement pairs *) type pair = Term.term * Term.term val pair_to_string : pair -> string (* Information about failures *) type huet_restriction = | Type_restriction of Term.term * Term.term exception Not_huet of huet_restriction (* Bounded matching trees *) type mtree = | Success | Failure | Suspend | Node of pair list * (pair * mtree) list val mtree_to_string : mtree -> string (* Disagreement set nodes for simplification *) type node = | NSuccess | NFailure | NPairs of pair list (* Basic signature of unification modules *) module type Unification = sig val is_flexible : Term.term -> bool val is_rigid : Term.term -> bool val simpl : node -> node val umatch : pair -> pair list val from_pairs : int -> pair list -> mtree val from_mtree : int -> mtree -> mtree val unifiers : mtree -> (pair list * pair list) list end (* Higher-order unification on the left and on the right *) module Left : Unification module Right : Unification 13.2 Figure Huet unification module in Abella. The two sub-modules implement a common unification interface for use on both sides of a two-sided sequent. unify_error -> string exception UnifyError of unify_error val right_unify : ?used:(id * term) list -> term -> term -> unit val left_unify : ?used:(id * term) list -> term -> term -> unit val try_with_state : fail:'a -> (unit -> 'a) -> 'a val try_right_unify : ?used:(id * term) list -> term -> term -> bool val try_left_unify : ?used:(id * term) list -> term -> term -> bool val try_left_unify_cpairs : used:(id * term) list -> term -> term -> (term * term) list option val try_right_unify_cpairs : term -> term -> (term * term) list option val left_flexible_heads : used:(id * term) list -> sr:Subordination.sr -> ((id*ty) list * term * term list) -> ((id*ty) list * term * term list) -> term list 13.3 Figure Unify unification module in Abella. It implements left and right unification in several variations (standard, returning success or a list of conflict pairs). Recall the proof by certificate of the commutativity of addition of natural numbers developed in Example 11.7.2. A hybrid proof may perform the induction in the standard Abella vernacular, reusing the first part of the script in Figure 11.3: induction on 1. intros. case H1.This brings us to the first subgoal: = for least and greatest fixed point definition clauses, respectively. Thus, we have:Π x.p x µ = B p x = Πp.Π x.µ B p x Π x.p x ν = B p x = Πp.Π x.ν B p x > form -> form -> o) -> (form -> form -> form -> o) -> bool -> form -> o. Figure Representation of an Abella session as a focused proof. Returning now to Example 2.4.1, consider the synchronous introduction rules for ∨ + and ∃ as refined by Figure2.4. Focused proof systems address the kinds of combinatorial explosions witnessed in the previous example by organizing introduction rules into two distinct phases. The two rules of interest are synchronous, i.e., they operate on a single formula marked as under focus. G 1 ∧ G 2 proceeds by finding proofs of G 1 and G 2 based on the unaltered signature and program. 3. The proof of a disjunctive goal G 1 ∨ G 2 proceeds by finding a proof of G 1 or a proof of G 2 based on the unaltered signature and program. 4. The proof of an implicational goal D ⊃ G proceeds by finding a proof of the consequent G based on the unaltered signature and the program extended with the program clause dictated by the antecedent, P ∪ {D }. 5. The proof of a universal goal ∀x.G proceeds by finding a proof of [y/x]G- where x is replaced by a fresh constant y (an eigenvariable)-on the signature extended with the new constant Σ ∪ {y } and the unaltered program. 6. The proof of an existential goal ∃x.G proceeds by finding a proof of [t /x]G- where x is replaced by a term t -on the unaltered signature and program. r.t. a CNF formula F if unit propagation on an assignment that falsifies C (that is, the addition to F of the unit clauses ¬l 1 , . . . , ¬l n ) derives a conflict.A RUP clause is said to have the asymmetric tautology property, or AT. Addition of RUP clauses to a CNF formula preserves logical equivalence. RUP is the strongest redundancy property that preserves equivalence.The simplest among the UNSAT certificate formats we will study is that called reverse unit propagation (RUP) certificate format. A RUP certificate consists of Figure Certificate of unsatisfiability for the formula in Figure7.1 in the RUP certificate format. a series of clauses, called "lemmas," each of which has the RUP property with respect to the accumulation of the initial formula and previous RUP lemmas. The certificate ends with the empty clause. Figure7.2 presents an example of a RUP certificate, with clauses represented in the same DIMACS CNF format as the input formula. 1 2 0 1 0 2 0 0 7.2 Continuing from Example 8.3.2, the translation of the problem to λProlog results in a series of term and atom declarations in the signature: type r_p9, t_p9 i -> bool. type z_p9, c1_p9 i. Consider the final formulation of the counterexample lemma for the obviously false property of list reversal in Example 12.2.2. In it, the generation phase consists of a single list of naturals, for which we may use the standard typing judgments. For natural numbers, this corresponds to the nat fixed point of Figure9.1; list_nat is in turned defined in terms of nat and the usual constructors: % Generation Kind qform type. Type qor nat -> nat -> qform -> qform -> qform. Type qand qform -> qform -> qform. Type qname string -> qform. Type qnone qform. Kind qmap type. Type qmap string -> qform -> qmap. Type qcert list qmap -> list qform -> qform -> cert. % Staging Type qstart numidx -> cert -> cert -> cert. % Number of attempts Define iterate : numidx -> prop by iterate (s _) ; iterate (s N) := iterate N. % Clerks and experts Define releaseRExpert : cert -> cert -> prop by releaseRExpert qsearch qsearch. Define decideRClerk : cert -> cert -> prop by decideRClerk (qcert Map Delta Goal) (qcert Map Delta Goal) ; decideRClerk (qstart Tries Gen Chk) (qstart Tries Gen Chk). Define storeRClerk : cert -> cert -> prop by storeRClerk (qcert Map Delta Goal) (qcert Map Delta Goal) ; storeRClerk (qstart Tries Gen Chk) (qstart Tries Gen Chk). Define unfoldRExpert : cert -> cert -> prop by unfoldRExpert (qcert Map Delta qnone) (qcert Map Delta qnone) ; unfoldRExpert (qcert Map Delta (qname Name)) (qcert Map Delta Form) := member (qmap Name Form) Map ; unfoldRExpert qsearch qsearch. Define unfoldLClerk : cert -> cert -> prop by unfoldLClerk qsearch qsearch. Figure Reversal of lists of natural numbers in hosted λProlog. Presentation conventions are shared with Figure 12.7.predicates in the homonymous predicate that embodies the dynamic semantics. The static semantics is contained by the typing predicate wt (for "with type"); this assigns an arbitrary type to error and types constants via a table encoded by tcc. Note that the encoding we have chosen uses explicit contexts as opposed to the hypothetical judgments of[START_REF] Mcdowell | Reasoning with higher-order abstract syntax in a logical framework[END_REF] (see also Section 4.3). This choice avoids implications in the body of the typing predicate and, as a result, allows us to use λProlog universal quantification to implement nabla at the reasoning level. % Signature kind nat type. type zero nat. type succ nat -> nat. type is_nat nat -> prolog. kind lst type -> type. type null lst A. type cons A -> lst A -> lst A. type is_natlist lst nat -> prolog. type append lst A -> lst A -> lst A -> prolog. type rev lst A -> lst A -> prolog. % Module prog (is_nat zero) (tt). prog (is_nat (succ N)) (is_nat N). prog (is_natlist null) (tt). prog (is_natlist (cons Hd Tl)) (and (is_nat Hd) (is_natlist Tl)). prog (append null K K) (tt). prog (append (cons X L) K (cons X M)) (append L K M). prog (rev null null) tt. prog (rev (cons X XS) RS) (and (rev XS SX) (append SX (cons X null) RS)). 12.9 Dynamic semantics of the Stlc language, implemented in hosted λProlog, corresponding to the rules presented in Figure12.2 under an appropriate type signature. % Simple generation prog (is_exp (c Cnt)) (is_cnt Cnt). prog (is_exp (app Exp1 Exp2)) (and (is_exp Exp1) (is_exp Exp2)). prog (is_exp (lam ExpX Ty)) (and (nabla x\ is_exp (ExpX x)) (is_ty Ty)). prog (is_exp error) (tt). % Maintaining a context of lambda variables prog (is_exp' _ (c Cnt)) (is_cnt Cnt). prog (is_exp' Ctx (app Exp1 Exp2)) (and (is_exp' Ctx Exp1) (is_exp' Ctx Exp2)). prog (is_exp' Ctx (lam ExpX Ty)) (and (nabla x\ is_exp' (cons x Ctx) (ExpX x)) (is_ty Ty)). prog (is_exp' _ error) (tt). prog (is_exp' Ctx X) (tt) :- memb_ctx Ctx X. Γ, B ∆ Γ, A ∨ B ∆ ∨ L Γ A, ∆ Γ B, ∆ Γ A ∧ B, ∆ ∧ R Γ 1 A, ∆ 1 Γ 2 , B ∆ 2 Γ 1 , Γ 2 , A ⊃ B ∆ 1 , ∆ 2 ⊃ L Γ, A B, ∆ Γ A ⊃ B, ∆ ⊃ R Γ A, ∆ Γ, ¬A ∆ ¬ L Γ, A ∆ Γ ¬A, ∆ ¬ R Γ, [t /x]A ∆ Γ, ∀x.A ∆ ∀ L Γ [y/x]A, ∆ Γ ∀x.A, ∆ ∀ R † Γ, [y/x]A ∆ Γ, ∃x.A ∆ ∃ L † Γ [t /x]A, ∆ Γ ∃x.A, ∆ ∃ R identity rules A A axiom Γ 1 A, ∆ 1 Γ 2 , A ∆ 2 Γ 1 , Γ 2 ∆ 1 , ∆ 2 cut structural rules Γ 1 , B, A, Γ 2 ∆ Γ 1 , A, B, Γ 2 ∆ E L Γ ∆ 1 , B, A, ∆ 2 Γ ∆ 1 , A, B, ∆ 2 E R First, at the level of formulas, the notion of polarity is central to focused systems. Recall how, in the previous section, the inference rules for conjunction and disjunction admit two interchangeable presentations: additive (in Figure2.1) and multiplicative (in Figure2.2). Instead of choosing one conjunction and one disjunction, making both variations of each connective available in the proof system can lead to greater control over the inferences. This control will be achieved by defining two conjunctions and two disjunctions, and assigning to each copy the corresponding additive or multiplicative rule.. .asynchronous introduction rulesΓ ⇑ t -, Θ Γ ⇑ A, Θ Γ ⇑ B, Θ Γ ⇑ A ∧ -B, Θ Γ ⇑ A, B, Θ Γ ⇑ A ∨ -B, Θ Γ ⇑ [y/x]B, Θ Γ ⇑ ∀x.B, Θ † Γ ⇑ Θ Γ ⇑ f -, Θ synchronous introduction rules Γ ⇓ t + Γ ⇓ B 1 Γ ⇓ B 2 Γ ⇓ B 1 ∧ + B 2 Γ ⇓ B i Γ ⇓ B 1 ∨ + B 2 i ∈ {1, 2} Γ ⇓ [t /x]B Γ ⇓ ∃x.B identity rules ¬P a , Γ ⇓ P a init Γ ⇑ B Γ ⇑ ¬B Γ ⇑ • cut structural rules Γ, C ⇑ Θ Γ ⇑ C, Θ store P, Γ ⇓ P P, Γ ⇑ • -t(z). [assumption]. t(z). [resolve(1,a,2,a)]. \$F. [resolve(3,a,4,a)]. In the same way LJF is extended to µLJF-and, before them, LJ is extended to µLJ-by the addition of fixed points and equality, the inference rules in Figure9.3, extended with clerks and experts by the usual method, are added to the augmented LJF a to form the full augmented system µLJF a : intuitionistic logic with fixed points (and equality) augmented with clerks and experts. The standard intuitionistic logic is augmented in Figure9.4, and the new rules in the system for the fragment comprising fixed points and equality are shown in Figure9.6..asynchronous introduction rulesΓ ⇑ A, B, Θ R Γ ⇑ A ∧ + B, Θ R Γ ⇑ Θ R Γ ⇑ t + , Θ R Γ ⇑ A ⇑ Γ ⇑ B ⇑ Γ ⇑ A ∧ -B ⇑ Γ ⇑ t -⇑ Γ ⇑ A, Θ R Γ ⇑ B, Θ R Γ ⇑ A ∨ B, Θ R Γ ⇑ f , Θ R Γ ⇑ A B ⇑ Γ ⇑ A ⊃ B ⇑ Γ ⇑ [y/x]B ⇑ Γ ⇑ ∀x.B ⇑ † Γ ⇑ [y/x]B, Θ R Γ ⇑ ∃x.B, Θ R † synchronous introduction rules Γ A ⇓ Γ B ⇓ Γ A ∧ + B ⇓ Γ t + ⇓ Γ ⇓ A i R Γ ⇓ A 1 ∧ -A 2 R Γ A i ⇓ Γ A 1 ∨ A 2 ⇓ Γ A ⇓ Γ ⇓ B R Γ ⇓ A ⊃ B R Γ ⇓ [t /x]B R Γ ⇓ ∀x.B R Γ [t /x]B ⇓ Γ ∃x.B ⇓ identity rules Γ ⇓ N a N a init l Γ, P a P a ⇓ init r Γ ⇑ B ⇑ Γ ⇑ B ⇑ R Γ ⇑ ⇑ R cut structural rules Γ, N ⇓ N R Γ, N ⇑ ⇑ R decide l Γ P ⇓ Γ ⇑ ⇑ P decide r Γ ⇑ P ⇑ R Γ ⇓ P R release l Γ ⇑ N ⇑ Γ N ⇓ release r being a pleasant and productive life thanks to my fellow Parsifalians: an eclectic group of talented and wonderful people, among whom I am honored to count myself. Type all, some (i -> bool) -> bool. Type eq i -> i -> bool. Type mu, nu ((i -> bool) -> i -> bool) -> i -> bool. % Polarities Define negative : bool -> prop by negative (imp P Q) ; negative (all P) ; negative (nu B T). Define positive : bool -> prop by positive tt ; positive ff ; positive (and P Q) ; positive (or P Q) ; positive (some P) ; positive (eq P Q) ; positive (mu B T). Figure The logic µLJF encoded in Abella. The types of formulas and terms, bool and i, are given; term constructors must be given by the signature of the full proof checker. When list types are needed, dedicated constructors need to be declared as well, possibly together with a member predicate. The fixed point connectives are declared as taking a single argument, by convention representing a list of arguments encoded as a term of the logic by the reserved constructors arg@ and argv. . Define is_nat : (i -> bool) -> prop by is_nat (mu Pred\Args\ (some N\ and (eq Args (N ++ argv)) (or (eq N zero) (some N'\ and (eq N (succ N')) (Pred (N' ++ argv)))))). Define plus : (i -> bool) -> prop by plus (mu Pred\Args\ (some K\ some M\ some N\ and (eq Args (K ++ M ++ N ++ argv)) (or (and (eq K zero) (eq M N)) (some K'\ some N'\ and (and Define is_nat' : (i -> bool) -> prop by is_nat' (mu Pred\Args\ or (eq Args (zero ++ argv)) (some N\ and (eq Args ((succ N) ++ argv)) (Pred (N ++ argv)))). Define plus' : (i -> bool) -> prop by plus' (mu Pred\Args\ or (some N\ (eq Args (zero ++ N ++ N ++ argv))) (some K\ some M\ some N\ and (eq Args ((succ K) ++ M ++ (succ N) ++ argv)) (Pred (K ++ M ++ N ++ argv)))). Figure Logic specification of natural numbers and addition on them as a least fixed points in µLJF encoded in Abella. For clarity, arg@ is written as infix ++, i.e., (M ++ N ++ argv) instead of (arg@ M (arg@ N argv)). Two versions of each fixed point are given: first (unprimed), with a global pattern match of the argument list and specializations in each clause; second (primed), with pattern matching entirely contained within each clause. Figure Simple properties of addition in Abella. Each of totality and determinism follow directly from an induction on the first argument of the plus relation and routine case analysis and applications on hypotheses in the context. Here, we endeavor to show how to describe the simple rules that can be used to prove a given lemma based on previously proved lemmas. Specifically, we will define proof certificates that describe the structure of the intended proof outlines that we expect and then run a proof checker on those certificates to see whether or not the certificate can be elaborated into a full proof of the candidate theorem. Since the design of the certificate language is based on the proof theory of synthetic connectives and since the proof checker we use employs both unification and backtracking search, this approach to describing high-level inference rules is both highly flexible and natural. . . Figure Commutativity of addition in Abella. A proof by simple induction relies on two auxiliary lemmas, one for each case of the induction on natural numbers (zero and successor). Each of the two lemmas is proved by simple induction, as is the main theorem-with the proviso that now not only hypotheses in the context, but also the auxiliary lemmas may be applied. Variable types are implicitly based on the typographic conventions for the various syntactic categories defined in Figure 12.1. Theorem If fixed point definitions do not contain implications and negations (i.e., they are essentially positive), then moving between the universal quantifier ∀ and the nabla quantifier ∇ does not affect the provability of atomic formulas. Proof. Follows from the properties of nabla in Miller and Tiu (2005, Section 7.2). Because the present study is limited to Horn style recursive definitions, there will be no observable differences between both quantifiers. In consequence, in the first setting, we will be able to use, say, the λProlog implementation of the universal quantifier, pi, to implement nabla. Among other applications, nabla has been used to formalize the metatheory of systems like the λ-calculus and the π-calculus. We illustrate its applications to property-based testing on a variation of the first: the simply-typed λ-calculus extended with primitives for natural numbers and lists of natural numbers, following the PLT Redex benchmark. We call this language Stlc, and its syntax is given in Figure 12.1; its static semantics-which shall become the basis of our for constants, bound variables, abstractions and applications, as well as suspended terms and pointers used for side effects. The representation relies on helper functions to inspect and manipulate terms, but these invariants are not strictly enforced by the Term module. to the core of the prover, and Tactics, where the language of tactics is defined. Those two modules will be our gateway to building the FPC framework inside the very nucleus of Abella. The last module of special interest is Unify, where a heavily customized implementation of unification on Abella terms-with substitutions as side effects on the imperative-style terms-is provided [START_REF] Nadathur | Practical higher-order pattern unification with on-the-fly raising[END_REF]. This unification module, critical Abella's execution of proofs, will be less amenable to the kinds of direct manipulation we have in store. (However, this impediment will not be as pervasive as the leaky representation of terms.) We must note that Abella's computational engine is not particularly efficient: When used to execute an embedded checker, it generates large numbers of redundant unification problems. At its core, a checker performs proof reconstruction by means of an implementation of an augmented sequent calculus. From the entry point, an inductively defined proof object is constructed by mutually recursive calls from the bottom up. A successful recursive call represents the application of an inference rule of the proof system on the conclusion and the generation of the premises. The relations and their clauses correspond to the inference rules organized by the kind of sequent of their conclusion: in µLJF a , these are async, Titre : Applications des Certificats de Preuve Fondamentaux à la démonstration automatique de théorèmes Mots clés : démonstration automatique de théorèmes, logique computationnelle, Certificats de Preuve Fondamentaux, assistants de preuve, théorie de la démonstration, aperçus de preuve Résumé : La confiance formelle en une propriété abstraite provient de l'existence d'une preuve de sa correction, qu'il s'agisse d'un théorème mathématique ou d'une qualité du comportement d'un logiciel ou processeur. Il existe de nombreuses définitions différentes de ce qu'est une preuve, selon par exemple qu'elle est écrite soit par des humains soit par des machines, mais ces définitions sont toutes concernées par le problème d'établir qu'un document représente en fait une preuve correcte. Le cadre des Certificats de Preuve Fondamentaux (Foundational Proof Certificates, FPC) est une approche proposée récemment pour étudier ce problème, fondée sur des progrès de la théorie de la démonstration pour définir la sémantique des formats de preuve. Les preuves ainsi définies peuvent être vérifiées indépendamment par un noyau vérificateur de confiance codé dans un langage de programmation logique. Cette thèse étend des résultats initiaux sur la certification de preuves du premier ordre en explorant plusieurs dimensions logiques essentielles, organisées en combinaisons correspondant à leur usage en pratique: d'abord, la logique classique sans points fixes, dont les preuves sont générées par des démonstrateurs automatiques de théorème; ensuite, la logique intuitionniste avec points fixes et égalité, dont les preuves sont générées par des assistants de preuve. Les certificats de preuve ne se limitent pas comme précédemment à servir de représentation des preuves complètes pour les vérifier indépendamment. Leur rôle s'étend pour englober des transformations de preuve qui peuvent enrichir ou compacter leur représentation. Ces transformations peuvent rendre des certificats plus simples opérationellement, ce qui motive la construction d'une suite de vérificateurs de preuve de plus en plus fiables et performants. Une autre nouvelle fonction des certificats de preuve est l'écriture d'aperçus de preuve de haut niveau, qui expriment des schémas de preuve tels qu'ils sont employés dans la pratique des mathématiciens, ou dans des techniques automatiques comme le property-based testing. Ces développements s'appliquent à la certification intégrale de résultats générés par deux familles majeures de démonstrateurs automatiques de théorème, utilisant techniques de résolution et satisfaisabilité, ainsi qu'à la création de languages programmables de description de preuve pour un assistant de preuve. Title : Applications of Foundational Proof Certificates in theorem proving Keywords : automated theorem proving, computational logic, foundational proof certificates, proof assistants, proof theory, proof outlines Abstract : Formal trust in an abstract property, be it a mathematical result or a quality of the behavior of a computer program or a piece of hardware, is founded on the existence of a proof of its correctness. Many different kinds of proofs are written by mathematicians or generated by theorem provers, with the common problem of ascertaining whether those claimed proofs are themselves correct. The recently proposed Foundational Proof Certificate (FPC) framework harnesses advances in proof theory to define the semantics of proof formats, which can be verified by an independent and trusted proof checking kernel written in a logic programming language. This thesis extends initial results in certification of first-order proofs in several directions. It covers various essential logical axes grouped in meaningful combinations as they occur in practice: first, classical logic without fixed points and proofs generated by automated theorem provers; later, intuitionistic logic with fixed points and equality as logical connectives and proofs generated by proof assistants. The role of proof certificates is no longer limited to representing complete proofs to enable independent checking, but is extended to model proof transformations where details can be added to or subtracted from a certificate. These transformations yield operationally simpler certificates, around which increasingly trustworthy and performant proof checkers are constructed. Another new role of proof certificates is writing high-level proof outlines, which can be used to represent standard proof patterns as written by mathematicians, as well as automated techniques like property-based testing. We apply these developments to fully certify results produced by two families of standard automated theorem provers: resolution-and satisfiability-based. Another application is the design of programmable proof description langages for a proof assistant.

01743886

en

2024/03/05 22:32:07

2017

https://theses.hal.science/tel-01743886/file/HUANG_Chao-Kun_2017_ED269.pdf

Chao-Kun Huang Keywords: Cavitation, Écoulement diphasique, HEM, TTV Cavitation, Two-phase flow, Homogeneous model, Transport equation model Turbulence and cavitation : applications in the NSMB and OpenFOAM solvers Résumé L'objectif de ce travail de thèse concerne l'étude et la mise en oeuvre de deux modèles de cavitation dans le solveur NSMB (Navier-Stokes-Multi-Blocks): les modèles HEM (Homogeneous Equilibrium Model) et une équation pour le taux de vide: le modèle à transport de taux de vide (TTV). Le phénom ène de cavitation est modélisé par différentes équations d'état de mélange liquide-vapeur (EOS). De s simulations numériques sont réalisées sur des écoulements diphasiques compressibles unidimensi onnels et bidimensionnels avec des conditions d'interface et comparées à des solutions de référence . De plus, la méthode TTV basée sur le taux de vide incluant les termes source pour la vaporisation et la condensation dans le logiciel libre open source OpenFOAM est également présentée sur la géom étrie Venturi pour capturer le phénomène du jet réentrant. La modélisation de la turbulence joue un r ôle majeur dans la capture des comportements instationnaires et un limiteur est introduit pour réduir e la viscosité turbulente afin de mieux prédire la structure à deux phases. Une comparaison de diver s modèles de cavitation couplés avec des modèles de turbulence est étudiée. Les résultats computat ionnels sont comparés aux données expérimentales existantes. RÉSUMÉ En général, la cavitation se réfère à des poches de gaz apparaissant dans un écoulement fluide. En d'autres termes, il s'agit d'un phénomène diphasique avec changement de phase. La cavitation se produit lorsque la pression d'écoulement est inférieure à la pression de vapeur saturante. Les structures ainsi formées sont entraînées par l'écoulement et lorsqu'elles atteignent une zone de pression plus élevée, elles se condensent et implosent violement. La cavitation conduit à des pertes importantes de performance de l'installation, à des problèmes d'instabilités de fonctionnement des machines et à l'erosion des parois du composant. C'est ainsi une source de problèmes techniques primordiaux dans le domaine des turbomachines hydrauliques et de la construction navale. Il existe différents types de cavitation selon la configuration d'écoulement, les propriétés du fluide et les géométries. Généralement, il y a quatre types de cavitation de base et c'est-à-dire traveling cavitation, sheet cavitation, cloud cavitation et tip-vortex cavitation. Il est classique de distinguer si l'écoulement est cavité ou non par le nombre de cavitation qui est défini par l'écart adimensionnel entre une pression de référence et la pression de vapeur saturante, noté σ ∞ = (P ∞ -P vap )/(0.5ρ ∞ U 2 ∞ ). P ∞ représente la pression absolue en un point de référence de l'écoulement, P vap est la pression de la vapeur saturante à la température d'essai, ρ ∞ est la masse volumique du liquide et U ∞ est la vitesse de référence. La prédiction numérique de la cavitation reste un défi pour plusieurs raisons. La modélisation du changement de phase (thermodynamique) et les interactions avec la turbulence n'est pas encour totalement établie. Du point de vue de la modélisation, la grande majorité des codes dédiés à la simulation de la cavitation est basée sur une approche moyennée à la fois pour l'écoulement diphasique et la turbulence. Une hiérarchie de modèles existe, du modèle simple à trois modèles d'équations (un fluide ou modèle homogène) jusqu'au modèle à sept équations (deux fluides) qui restent plus adaptés pour des géométries simples ou des fluides nonvisqueux. Les modèles deux fluids à sept équations sont les plus complets. Dans ce modèle, on suppose que les deux phases coexistent à chaque point du champ d'écoulement et sont exprimées en termes de deux ensembles d'équations de conservation qui développent l'équilibre de masse, de moment et d'énergie pour chaque phase. L'équation de transport pour la fraction de vide est introduite pour décrire la topologie de l'écoulement. Les modèles réduits à six équations sont similaires aux modèles de sept équations à l'exception sans tenir compte de l'équation d'évolution de la fraction de vide. Cependant, ils restent difficile à utiliser en écoulements industruels (turbomachines). La méthode à un fluide, ou méthode homogène, considère les écoulements comme un mélange de deux fluides se comportant comme un fluide qui est semblable au courant monophasé. De cette façon, un seul ensemble d'équations de conservation est employé pour exprimer l'interaction fluide pour le mélange. Compte tenu de sa simplicité et de son faible coût de calcul, la méthode homogène est plus intéressante pour les simulations numériques des écoulements cavitants. La plupart des phénomènes de cavitation impliquent une turbulence et l'interaction turbulencecavitation est un phénomène sous-connu et documenté (dû notamment à la difficulté d'effectuer v des mesures expérimentales dans les écoulements cavitants). Les effets de la compressibilité sur la turbulence et les effets de la phase dispersée sont également inconnus. La précision numérique de la cavitation turbulente dépend de la modélisation de la cavitation et de la turbulence. Ainsi, le choix d'une modélisation de la turbulence est une question importante pour la simulation de la cavitation. La simulation numérique directe (Direct Numerical Simulation (DNS)) a la capacité la plus élevée de résoudre toutes les échelles de turbulence. Toutefois, il nécessite une résolution de grille très fine et, par conséquent, il est encore assez difficile à appliquer en raison de la consommation élevée de performances informatiques. Bien que la simulation des grands échelles (Large Eddy Simulation (LES)) ait déjà été mise en oeuvre pour les écoulements turbulents de cavitation, les codes habituels sont formulés dans un modèle de Navier-Stokes (RANS) de Reynolds à tensor turbulent par une équation de transport kε (hypothèse de Boussinesq) Entre l'effort de calcul et la précision. Cependant, les modèles standards de viscosité par tourbillons basés sur l'hypothèse de Boussinesq tendent à sur-prédire la viscosité par tourbillonnement qui réduit l'effet du jet re-entrant et de la décomposition de structure biphasée. Ces modèles de turbulence sont inadéquats pour prédire correctement la dynamique des bulles de cavitation. Plusieurs solutions ont été proposées et testées pour réduire la viscosité des turbulences et améliorer le comportement des modèles de turbulence. Reboud a proposé une modification arbitraire en introduisant un limiteur de viscosité de turbulence assigné en fonction de la densité au lieu d'utiliser directement la densité du mélange. Une méthode basée sur le filtre (Filter-based Method (FBM)) qui combine le concept de filtre et le modèle RANS a été étudiée en imposant une échelle de filtre indépendante, généralement la taille de la grille, sur le calcul de la viscosité de Foucault. Une fois que l'échelle de longueur de turbulence est supérieure à la taille du filtre, la viscosité de turbulence peut être réduite par une fonction de filtrage linéaire. L'interaction entre la turbulence et la cavitation en ce qui concerne l'instabilité et la structure du flux est complexe et mal comprise. De plus, il ya moins d'études sur l'influence des modèles de turbulence sur le débit de cavitation. Dans cette étude, la correction de Reboud est mise en oeuvre en trois modèles de turbulence différents et simulée avec différents modèles de cavitation. L'objectif final est de fournir un aperçu de l'interaction entre les modèles de turbulence et de cavitation. Cette étude présente la mise en oeuvre et la validation des modèles de cavitation développés au LEGI (Laboratoire des Écoulements Géophysiques et Industriels) dans les solveurs NSMB (solveur compressible structuré multiblocks parallèle avec maillage chimère) et OpenFOAM (Open source Field Operation And Manipulation). Les modèles de mélange homogène ou un fluide avec une équation d'état de barotrope effectués au LEGI ont réalisé dans le solveur NSMB. Les modèles proposés ont été validés à l'aide de divers cas de test non invasifs, y compris le problème de mouvement de l'interface, le tube de choc eau-air et le tube d'expansion et l'interaction chocbulle. La possibilité d'obtenir des solutions correctes de ces cas de test a été étudiée. Les résultats obtenus à partir des cas de test indiquent que la mise en oeuvre de ces deux modèles de cavitation ne pouvait malheureusement pas être la panacée et être généralisée pour tous les cas de test. Bien que les validations aient montré la capacité des modèles à simuler le développement de la cavitation, les deux modèles souffrent toujours du problème de l'instabilité numérique. La principale différence entre ces deux modèles est que le modèle à trois équations a l'hypothèse d'un équilibre thermodynamique complet entre les phases; par conséquent, cela pourrait expliquer les écarts existant dans les cas de test ci-dessus. Puisque la mise en oeuvre et la validation dans le solveur NSMB avaient déjà pris trop de temps, afin d'atteindre les objectifs de cette étude, qui sont la turbulence et la cavitation, un autre logiciel open source libre, OpenFOAM, a été adopté pour effectuer les cavitations dans un venturi. Les modèles à quatre équations qui sont composés de trois lois de conservation pour le vi mélange plus une équation de transport pour le taux de vide dans le solveur OpenFOAM appelée interPhaseChangeFoam est étudié. Une comparaison de divers modèles de cavitation couplés à des modèles de turbulence sur la géométrie Venturi 2D et 3D a été proposée. Le solveur interPhaseChangeFoam a été utilisé pour simuler la poche de cavitation par la formulation de modèles de cavitation à équation de transport à rapport de vide, y compris les modèles Kunz, Merkle et SchnerrSauer. Pour la fermeture de la turbulence, trois modèles sont considérés: le modèle Spalart-Allmaras à une équation, le modèle kε à deux équations et le modèle Menter kω SST. Le limiteur de turbulence Reboud est introduit pour réduire la viscosité turbulente afin de capturer la dynamique du jet ré-entrant. Les résultats numériques ont été comparés à des données expérimentales concernant la ration de vide moyennée dans le temps et la vitesse longitudinale, la pression pariétale, les fluctuations de pression de paroi RMS et la viscosité tourbillonnaire turbulente. Les résultats ont montré que l'utilisation d'un limiteur de turbulence par turbulence permet au modèle de simuler correctement les comportements instables de la feuille, cependant de grandes différences apparaissent entre les modèles et l'effet de la réduction n'est pas assez fort. En général, les trois modèles de cavitation étaient capables de reproduire le phénomène de jet ré-entrant, mais la longueur de la cavité était sur-prédite. Parmi les résultats issus de la simulation qui ont été comparés aux données expérimentales, c'est le modèle de cavitation de Kunz couplé au modèle de turbulence kω SST qui pourrait avoir une meilleure prédiction pour la géométrie Venturi. De plus, l'effet 3D n'a pas beaucoup amélioré la prédiction en fonction des résultats numériques obtenus. Ceci peut être dû au problème d'étalonnage du terme de transfert de masse du taux de condensation et du coefficient de vitesse de vaporisation ou au manque de cohérence thermodynamique. Aussi, l'impact sur la valeur de l'exposant n utilisé dans cette correction doit être étudié. En outre, interPhaseChangeFoam est un solveur incompressible qui est moins capable de résoudre le type de géométrie interne. vii INTRODUCTION Background of cavitation C avitation is a phenomenon that occurs frequently in conventional hydraulic components such as pumps, valves, turbines and propellers. Over-speeds imposed by the local geometry, shear phenomena, acceleration or vibration may cause local pressure drops in the fluid. When the flow pressure is less than the vapor pressure of the fluid, there is a partial vaporization and vapor structures arise. The so formed structures are entrained by the flow and when they reach a higher pressure zone they condense and implode violently. Cavitation leads to significant loss of system performance, problems of instability of operation of machines and erosion of the component walls. It is thus a primary source of technical problems in the field of hydraulic turbomachinery, naval propulsion and space as well as in high pressure fuel injection. However, it should be noticed that in certain cases cavitation has a desired effect, for example, supercavitation for underwater vehicles such as torpedoes. The gaseous cavities enveloping the external body make it possible to reduce the friction drag. In addition, cavitation is used for the purpose of cleaning by the control of erosion. The mechanisms of the process of cavitation and boiling are similar except that in boiling, the vaporization occurs with only small pressure change. In contrast to boiling, the vaporization in cavitation occurs under only a minor temperature change (Figure 1.1). In the development of a space launcher, cavitation is one of the most limiting factor generated by the hydraulic because it requires from the design phase the introduction of safety margins resulting primarily from an increase in pressure in the reservoirs. This increase in pressure requires an increase in the wall thickness which generates an increase in the structure. The magnitude of this increase in dry weight is 100 k g for 100 mbar of additional pressure, which corresponds about to 2% of the total weight of the largest telecommunications satellite built. Cavitation appears in the ergol turbo pumps of the launcher propellant and it generates falls of performances, instability of operation as well as mechanical loads on structures. The consequences can be tragic as the failure of the Japanese H-II launch vehicle in 1999. As for the shipbuilding industry, cavitation is one of the major constraints in the design of marine propellers. Noise, vibration, erosion as issues resulting of cavitation are very tricky. The appearance and disappearance of bubbles on the propeller blades create local pressure fluctuations that can be compared to shock waves because of their violence. Moreover propeller produces a rotating flow in its wake. Sections of rudders that are placed behind the propeller are then in incidence and can cavitate violently at high speed. Cavitation is also very energetic and very noisy in the audible range. Depending on the type of cavitation frequencies and very specific signatures appear. This type of nuisance is obviously crucial for military vessels, as brought up to 100 km offshore by poorly controlled cavitation. The determination of cavitation instabilities regime is essential. In the hydraulic energy field, cavitation is a limiting phenomenon in the design phase of hydraulic machinery (pumps, turbines) and its consequences in terms of erosion of the walls are a very important nuisance (operating range and duration component life). Damage to solid walls (Figure 1.2) is caused by very short pressure spikes (10ns to 1µs), high amplitude (∼ 1GPa), attributed to the impact of pressure waves emitted during the collapse of vapor structures. Knowledge of the dynamics of pockets is therefore very important. Also operating machinery instabilities related to the hydrodynamic coupling between the inter-blade channels are observed. Types of cavitation There exists different patterns of cavitation according to the flow configuration, the properties of the fluid and the geometries. Generally, there are four basic types of cavitation and are described briefly below: • Traveling cavitation These bubbles are formed in the zone of low pressure, travel with the flow and implode after when they enter the region of higher pressure. This kind of cavitation is observed particularly in the blades of turbine or propeller (Figure 1.3). • Tip-vortex cavitation At the tips of the rotating blade or wing, the pressure may be very low locally which will generate a filament-looking cavitation (Figure 1.6). Cavitation inception It is conventional to distinguish whether the flow is cavitating or not by means of cavitation number, σ ∞ , which is defined as σ ∞ = P ∞ -P vap 0.5ρ ∞ U 2 ∞ (1.1) This parameter relates the vapor pressure, P vap , to the free-stream pressure, P ∞ , and the free-stream dynamic pressure. Once the cavitation number, σ ∞ , is reduced in the flow, cavitation will first be observed to appear at some particular value which can be called the incipient cavitation, σ i . The pressure coefficient, C P , is given by the relation: C P = P -P ∞ 0.5ρ ∞ U 2 ∞ (1.2) Therefore, cavitation number can be compared to the pressure coefficient and the following estimate is considered for cavitation inception, σ i σ i = -C P min = P min -P ∞ 0.5ρ ∞ U 2 ∞ (1.3) where C P min is the minimum pressure coefficient. With these definitions above, it is useful to consider that if P min = P vap or σ ∞ = -C P min , the incipient cavitation occurs which means the limiting regime between the non-cavitating and cavitating flow. If further reduction in cavitation number which implies that σ ∞ < -C P min , the developed cavitation happens with an increase in the size and number of bubbles. CHAPTER 1. INTRODUCTION Objectives and organization of this thesis Cavitation for most engineering applications is turbulent, and the interplay between cavitation and turbulence makes the cavitation dynamics even more complicated, and thus the detail dynamics of the phase change is not well understood. Specific issues to numerical techniques in this type of flow also persist. The objectives of this thesis are to implement several cavitation models in the NSMB solver. The emphasis is placed on the study and implement of the Homogeneous Equilibrium Models (HEM) coupled with a barotropic state law and a void ratio Transport-based Equation Model (TEM). The TEM based method for the void ratio including the source terms for vaporization and condensation in the free, open source software OpenFOAM (Open source Field Operation And Manipulation) is also presented on the Venturi geometry to capture the re-entrant jet phenomenon. For the turbulence closure, a density correction approach proposed by Reboud is imposed to several turbulence models. Besides the introduction, which presents the background of cavitation and the objectives of the study, the thesis is organized as follows. In Chapter 2, a literature review for the modeling of two-phase flows is investigated which presents the theory in the modeling of cavitating flow, including the different models used for the present work. In Chapter 3, the flow solvers, the NSMB and OpenFOAM, used in this study are described, including the essential elements of the governing equations, the modeling concepts and the numerical schemes. In Chapter 4, different test cases carried out by the NSMB solver are presented together with validations against exact solutions of the Euler equations and the models implemented in the solver. In Chapter 5, the 2D and 3D Venturi geometry are performed by OpenFOAM with the built-in solver interPhaseChangeFoam coupled with different turbulence models. Validation and comparisons are done with experimental measurements including time-averaged void ratio and velocity profiles, RMS wall pressure fluctuations. Finally, conclusions and future investigations are discussed in Chapter 6. C H A P T E R 2 REVIEW OF CAVITATION MODELING N umerical prediction of cavitation remains a challenge for several reasons. First the modeling of phase transition (thermodynamics) and the interactions with the turbulence is not fully established. In addition, it is a complicated task to deal with the large variations of density between the liquid and vapor phases. Specific issues to numerical techniques in this type of flow also persist. On the issue of numerical architecture (compressible or incompressible low Mach preconditioning extended to variable densities), the question remains open. However, several studies have shown better capture re-entrant jet of cavitation bubbles by compressible codes [START_REF] Venkateswaran | Computation of multiphase mixture flows with compressibility effects[END_REF]Goncalvès et al., 2010a;[START_REF] Park | Pressure-based solver for incompressible and isothermal compressible flows with cavitation[END_REF][START_REF] Skoda | Comparison of compressible explicit density-based and implicit pressure-based cfd methods for the simulation of cavitating flows[END_REF]. Modeling of two-phase flows In this chapter only the modeling of gas-liquid flows are presented. There exists two main approaches for the gas-liquid flows : • Direct or interface-based methods • The averaged or diffusion methods of the interface Direct resolution methods The so-called direct resolution methods allow to solve all the spatial and temporal scales of the two-phase flows. These kinds of methods reconstruct the interfaces and describe the propagation of the flow, while solving the Navier-Stokes equations. There are different ways of representing the spatial and temporal evolution of an interface : CHAPTER 2. REVIEW OF CAVITATION MODELING • Front tracking method (Lagrangian) • Level Set method (Eulerian) • Volume Of Fluid method (Eulerian) • Diffuse interface method ( [START_REF] Jamet | Methodes a interfaces diffuses pour la modelisation des ecoulements dihasiques[END_REF]) Because of the existence of various velocities at the interface i.e. liquid phase velocity, vapor phase velocity and interface velocity, phase changes are difficult to be taken into account in these kinds of methods. Moreover, the reconstruction of the interface in three-dimensional flows can be difficult and very time consuming. The average resolution methods In most of these problems, it is not necessary and would be extremely difficult to know the instantaneous values of the local variables of the flow due to the limitation of the capabilities of computers and the difficulty in predicting the position of the interfaces. The prediction of "averaged" properties are mostly interested in, such as the pressure drop in a bubble flow, the volume flow rate in a conduit etc... For this purpose, "averaged" forms of the equilibrium equations will be used to predict mean values of the flow parameters which are meaningful and experimentally accessible. Moreover, since the equations of equilibrium appear in the form of partial differential equations, it is desirable that the mean properties and their first derivatives, spatial and temporal, should be continuous. The presence of interfaces leads to serious difficulties for the mathematical formulation of the problem, in the same way as the shock waves in single phase. The concept of beginning with these methods is the use of instantaneous conservation laws of fluid mechanics for each phase. The interfaces appear as surfaces of discontinuity for the different properties of the fluid, so the fundamental equilibrium equations are expressed in the form of "averaged interface conditions". There are many ways to "average". Averaging of conservation laws can be carried out: • in space • in time • statistically from a set of measures • or by a combination of the preceding ones (space/time, statistics/space...). MODELING OF TWO-PHASE FLOWS Spatial averaging has been mainly used in the field of nuclear engineering (average over a section of a pipe). It allowed the development of 1D code for the safety analysis of nuclear reactors by averaging the equations on the section of a pipe. Similar to the use of the RANS approach for turbulent single-phase flows, the temprol averaging is widely used for two-phase flows, especially if they are turbulent. Indeed, since transport phenomena are highly dependent on local fluctuations of variables, it is easier in this case to link the laws of state and behavior needed to close the problem with experimental measurements [START_REF] Ishii | Thermo-Fluid Dynamics of Two-Phase Flow[END_REF] . Local time-averaged equations In single-phase turbulent regime, an approach in the sense of Reynolds averaged which treats the instantaneous Navier-Stokes equations statistically is used. For a steady flow, the overall average of equations (average obtained over a large number of realizations) can be replaced by a temporal averaging (ergodic hypothesis). In the two-phase flow; the location of the interface is unknown in time and space, the instantaneous equations can not be solved. The equations are averaged by decomposing each variable into an average part and a fluctuating part. The temporal averaging operator of the instantaneous equations reveals the presence rate α, defined by: α = T k T (2.1) which represents the time T k of the presence of the phase k, with respect to a duration T. After spatial discretization of the computational domain, the presence rate is averaged over each cell and is then expressed as the volume fraction: α = V k V (2.2) where V k is the volume of the phase k in a volume mesh V . The different models Different classes of models are present in the literature according to the number of conservation laws treated and the assumptions made: equilibrium model/relaxed model, homogeneous model/two-velocity model, two-fluid model/one-fluid model: • Two-fluid models The full seven-equation two-phase models proposed by Baer et Nunziato [START_REF] Baer | A two-phase mixture theory for the deflagration-to-detonation transition (DDT) in reactive granular materials[END_REF] are the most complete. These models take into account explicitly the nonequilibrium effects between phases (unequilibrium of pressure, velocity and temperature) CHAPTER 2. REVIEW OF CAVITATION MODELING but remain difficult to be used in industrial flows (turbomachinery). A seven-equation model has been used for supercavitation and expansion tube problems by Saurel [START_REF] Métayer | Modelling evaporation fronts with reactive riemann solvers[END_REF][START_REF] Saurel | A multiphase model for compressible flows with interfaces, shocks, detonation waves and cavitation[END_REF]. The two-fluid method remains more suited for inviscid and simple geometries [START_REF] Métayer | Modelling evaporation fronts with reactive riemann solvers[END_REF]Saurel et al., 2008a;[START_REF] Petitpas | Diffuse interface model for high speed cavitating underwater systems[END_REF][START_REF] Zein | Modeling phase transition for compressible two-phase flows applied to metastable liquids[END_REF][START_REF] Saurel | A multiphase model for compressible flows with interfaces, shocks, detonation waves and cavitation[END_REF]Yeom andChang, 2006, 2013]. • One-fluid homogeneous mixture models The models are composed of three conservation laws written for the mixture and are based on a assumption of non-slip between the phases. With the assumption of thermodynamic equilibrium, the Homogeneous Equilibrium Models (HEM) are constituted. The non-equilibrium effects can be introduced empirically [START_REF] Yoon | BIBLIOGRAPHY Choking flow modeling with mechanical and thermal non-equilibrium[END_REF]. Different equations of state for the mixture have been developed in cavitation in a thermosensitive fluid : barotropic law [START_REF] Cooper | Analysis of single and two-phase flow in turbopump inducers[END_REF][START_REF] Rapposelli | A barotropic cavitation model with thermodynamic effects[END_REF], algorithm for calculating temperature based on the equality of the free enthalpies between the phases [START_REF] Edwards | Low-diffusion flux splitting methods for real fluid flows with phase transition[END_REF]]. • Reduced models with five equations These models are obtained from a simplification of the complete two-fluid model. The archetype five-equation model is the one of Kapila [START_REF] Kapila | Two-phase modeling of deflagration-to-detonation transition in granular materials: Reduced equations[END_REF] which is composed of two conservation equations for masses, one conservation equation for the mixture momentum, one conservation equation for the mixture energy and one non-conservative equation for the void ration to describe the flow topology. They involve two temperature which makes it possible to reproduce thermal non-equilibrium effects, as proposed in the model of Saurel [START_REF] Saurel | Modelling phase transition in metastable liquids: application to cavitating and flashing flows[END_REF] for cavitation simulation in diesel injectors. Some formulations have been proposed to the simulation of interface between two fluids [START_REF] Allaire | A five-equation model for the simulation of interfaces between compressible fluids[END_REF][START_REF] Kreeft | A new formulation of kapila's five-equation model for compressible two-fluid flow, and its numerical treatment[END_REF][START_REF] Murrone | A five equation reduced model for compressible two phase flow problems[END_REF][START_REF] Tian | [END_REF]. • Relaxed models with four equations A four-equation model was developed for a flashing flows and ebullition applications : the Homogeneous Relaxation Model (HRM). It consists of three conservation laws for the mixture and one additional transport equation for the void ratio. The latter contains a relaxation source term. The source term involves a relation time that is the time for the system to regain its thermodynamic equilibrium state. This relaxation time is very difficult to determine and is estimated from experimental data [START_REF] Barret | Schemes to compute unsteady flashing flows[END_REF][START_REF] Downar-Zapolski | The non-equilibrium relaxation model for one-dimensional flashing liquid flow[END_REF]. Another formulation of the relaxation term was proposed by Helluy [START_REF] Helluy | Relaxation models of phase transition flows[END_REF], based on a constrained convex optimization problem on the mixture entropy. Another four-equation model which is very popular to simulate cavitating flows in cold water has been adapted to cryogenic application [START_REF] Hosangadi | Numerical study of cavitation in cryogenic fluids[END_REF][START_REF] Utturkar | Recent progress in modelling of cryogenic cavitation for liquid rocket propulsion[END_REF][START_REF] Zhang | Computational fluid dynamic study on cavitation in liquid nitrogen[END_REF] by adding a transport equation for the void ratio : the MODELING OF TWO-PHASE FLOWS Transport-based Equation Model (TEM). This equation includs a cavitation source term for the modeling of condensation and vaporization. The main difficulty is related to the formulation of the source term and the tunable parameters involved for the vaporization and condensation process. The calculation of the void fraction by an additional transport equation including the source terms for vaporization and condensation processes is increasingly used for this model. In this case, the term of mass transfer between phases must be treated explicitly. Several empirical formulations have been proposed to simulate cavitating flows [START_REF] Ahuja | Simulations of cavitating flows using hybrid unstructured meshes[END_REF][START_REF] Wang | Large eddy simulation of a sheet/cloud cavitation on a {NACA0015} hydrofoil[END_REF][START_REF] Merkle | Computational modeling of the dynamics of sheet cavitation[END_REF][START_REF] Singhal | Mathematical basis and validation of the full cavitation model[END_REF][START_REF] Venkateswaran | Computation of multiphase mixture flows with compressibility effects[END_REF][START_REF] Vortmann | Thermodynamic modeling and simulation of cavitating nozzle flow[END_REF][START_REF] Wu | Time-dependent turbulent cavitating flow computations with interfacial transport and filterbased models[END_REF][START_REF] Morgut | Comparison of mass transfer models for the numerical prediction of sheet cavitation around a hydrofoil[END_REF][START_REF] Kunz | A preconditioned navier-stokes method for two-phase flows with application to cavitation prediction[END_REF][START_REF] Senocak | A pressure-based method for turbulent cavitating flow computations[END_REF][START_REF] Hosangadi | Numerical study of cavitation in cryogenic fluids[END_REF] but still suffer from a calibration problem and thermodynamics inconsistency [START_REF] Goncalvès | Constraints on equation of state for cavitating flows with thermodynamic effects[END_REF]. Different sets of parameters are presented in [START_REF] Utturkar | Recent progress in modelling of cryogenic cavitation for liquid rocket propulsion[END_REF][START_REF] Frikha | Influence of the cavitation model on the simulation of cloud cavitation on 2d foil section[END_REF][START_REF] Agnieszka | Review of numerical models of cavitating flows with the use of the homogeneous approach[END_REF]. The different classes of models are summarized in The two-fluid model This model is about the Navier-Stokes equations for those phases involved. Here the case of two phases is considered, where k is the phase index, k=1, 2. This gives the following six conservation equations : ∂α k ρ k ∂t + ∇. α k ρ k u k = Γ k (2.3) ∂α k ρ k u k ∂t + ∇. α k ρ k u k ⊗ u k = -∇(α k p k ) + ∇.(α k τ k ) + α k ρ k F k + M k (2.4) ∂α k ρ k E k ∂t + ∇. α k ρ k E k u k = -∇. α k q k -∇. pu k + ∇. τ k .u k + α k ρ k F k .u k + Q k (2.5) CHAPTER 2. REVIEW OF CAVITATION MODELING E = e + 1 2 u 2 is the specific total energy. Γ k , M k , Q k are the source terms relating to transfers of mass, momentum and energy between phases. They represent the interfacial effects and must be modeled. M k = M Γ k + P kI ∇α k + F d k (2.6) The term M Γ k represents the momentum transfer due to the mass transfer. F d k corresponds to the interfical friction force exerted on the phase k. P kI is the pressure of phase k at the interface. Q k = H Γ k -p kI ∂α k ∂t + F d k .u kI + Q kI (2.7) H Γ k = L vap Γ k represents the energy transfer due to the mass transfer, where L vap is the latent heat of phase change. Q kI corresponds to the interfacial heat transfer. u kI is the vector of velocity of phase k at the interface. In addition: 2 k=1 M k = M m = 0 and 2 k=1 Q k = Q m = 0 (2.8) It should notice that these two terms are not necessary equal to zero although they are generally be taken like that. Indeed due to the variation of the curvature of the interface, the momentum and the energy provided by one phase are not equal to those received by the other. The one-fluid model This model, also known as homogeneous mixture approach of two-phase flow consists in writing the averaged Navier-Stokes equations for a "mixing" fluid. It is assumed that the two phases move at the same velocity (i.e. neglecting the drag term between phases). The exchanges and the unequilibrium between phases are then no longer directly modeled, but it is possible to represent them in the closure of the system. Actually, the equation of state of the mixture may introduce a difference at the saturation point (for example, the barotropic law). A physical property of the mixture is defined by a weighting of the void ratio to its value between each phase. For the weighting of the extensive properties, the density will be used. ρ m = αρ V + (1 -α) ρ L and ρ m e m = αρ V e V + (1 -α) ρ L e L (2.9) The conservation equations are as follows : ∂ρ m ∂t + ∇. ρ m u m = 0 (2.10) ∂ρ m u m ∂t + ∇. ρ m u m ⊗ u m = -∇(p m ) + ∇.(τ m ) + ρ m F m (2.11) ∂ρ m E m ∂t + ∇. ρ m E m u m = -∇. q m -∇. pu m + ∇. τ m .u m + ρ m F m .u m (2.12) It can be observed that the energy required for phase change, the latent heat, does not appear explicitly in the energy conservation equation. In fact, this term is treated implicitely for the mixture. Four-equation models These models are intermediate models between one-fluid and two-fluids ones. It consists of solving the conservation equations for the mixture plus a continuity equation for one phase. This makes it possible to treat the mass transfer term explicitly. ∂ρ m ∂t + ∇. ρ m u m = 0 (2.13) ∂ρ m u m ∂t + ∇. ρ m u m ⊗ u m = -∇(p m ) + ∇.(τ m ) + ρ m F m (2.14) ∂ρ m E m ∂t + ∇. ρ m E m u m = -∇. q m -∇. pu m + ∇. τ m .u m + ρ m F m .u m (2.15) ∂α 1 ρ 1 ∂t + ∇. α 1 ρ 1 u 1 = Γ 1 (2.16) There exists different models according to the modeling of the mass exchange term between the phases. This assumption has the effect of decoupling the mass conservation equation and the momentum conservation equation with the energy conservation equation. In fact, the temperature no longer appears in the first two equations,therefore it has no more influence on the other physical properties. The equations of state Tait law For the case of a slightly compressible flow it is possible to take into account the compressibility of a fluid by the relation : ∆P = c 2 ∆ρ Tait law : ρ ρ re f = [ n] P + P 0 P re f + P 0 where ρ re f and P re f are reference density and pressure. For water, P 0 = 3 × 10 8 and n = 7. It is the formulation used by [START_REF] Venkateswaran | Computation of multiphase mixture flows with compressibility effects[END_REF][START_REF] Pouffary | Simulation numérique d'écoulements 2D/3D cavitants, stationnaires et instationnaires[END_REF] to take into account the compressibility in the pure phases for the modeling of cavitation. The speed of sound c is a given value for each phases. Perfect gas law This state law allows to model a large number of gases with a good approximation: PV = nRT avec R=8.314 J/(K.kg). It is also written in the form: P = ρrT where r = R/M = C p -C v (=287 SI unit for air). According to the internal energy : P ρ, e = (γ -1)ρe where γ = C p C v is the ratio of specific heats. With Joule's law : ∆e = C v ∆T and ∆h = C p ∆T where C v and C p are constants. There is also the semi-perfect gas law, which defines C p (T) and C v (T) no longer to be constant, but by using polynomial laws as a function of temperature. Van der Waals law This law was first introduced by van der Waals in 1873. It contains two constants a and b which are calibrated on the behavior of the fluid at the critical point. It represents one of the first state laws for real gases. P + a v 2 (v -b) = rT where v is the specific volume (2.17) This law produces a negative sound speed (dP/dρ < 0) in the phase transition zone (unstable thermodynamic equilibrium). Stiffened gas law This low is detailed in [START_REF] Rolland | Modélisation et résolution de la propagation de fronts perméables[END_REF]. It is valid for a large number of fluids, and is sometimes used for solids : P ρ, e = (γ -1)ρ(eq) -γp ∞ The term (γ-1)ρ(eq) represents the intermolecular repulsive effect. The term -γp ∞ represents the molecular attraction which is responsible for the cohesion of liquids or solids. This term is null for the perfect gas state law. It is set for each fluid by the constants γ and p ∞ (q=0). In the phase change the parameter q, which refers to the energy of the fluid at a given reference state, is non-zero. The heat capacities are constants in the approximation of stiffened gas law. In the same way as for the perfect gas law, a semi-stiffened gas law makes it possible to define C v and C p by polynomial laws as a function of temperature. Several sets of parameters for cold water have been proposed as shown in Table 2.2 : [START_REF] Saurel | A multiphase godunov method for compressible multifluid and multiphase flows[END_REF] 4.4 6 × 10 8 0 -1625 Barberon et Helluy [START_REF] Barberon | Finite volume simulation of cavitating flows[END_REF] 3 8.533 × 10 8 -0.1148 × 10 7 4200 1569 Paillere et al. [START_REF] Paillere | On the extension of the ausm+ scheme to compressible two-fluid models[END_REF] 2.8 8.5 × 10 8 0 4186 1486 Le Metayer et al. [START_REF] Metayer | Elaborating equations of state of a liquid and its vapor for two-phase flow models[END_REF] 2.35 10 9 -0.1167 × 10 7 4268 1300 Chang et Liou [START_REF] Chang | A robust and accurate approach to computing compressible multiphase flow: stratified flow model and AUSM+-up scheme[END_REF] 1.932 1.1645 × 10 9 0 8095 1487 Table 2.2: Parameters of the stiffened gas law for cold water by different authors Authors γ P ∞ (Pa) q (J/kg) C p (J/K.kg) c (m/s) Saurel et Abgrall Tamman law This law is equivalent to the stiffened gas law : P + P c = ρ L K(T + T c ) The use of parameters P c , K, T c , is another formulation but is equivalent to those of stiffened gas law q, P ∞ and γ. Mie-Grüneisen type law This law is written as : P(ρ, e) = P ∞ (ρ) + Γ(ρ)ρ ee re f (ρ) where Γ = 1 ρ ∂p ∂e ρ is the coefficient of Grüneisen and P ∞ (ρ) is given as a function of the fluid. The stiffened gas law is obtained with the assumption of low density variations from the Mie-Grüneisen law. For isentropic evolutions, it becomes the Tait law. Another particular case : if P ∞ is null, then the perfect gas law is obtained. Benedict-Webb-Rubin law To get as close as possible to the representation of real gases, there are even more complex form of state laws such as the Redlich-Kwong-Soave equation or the Benedict-Webb-Rubin equation [START_REF] Benedict | An empirical equation for thermodynamic properties of light hydrocarbons and their mixtures: methane, ethane, propane and n-butane[END_REF]. The Benedict-Webb-Rubin law is written as : P = RT d + d 2 RT (B + bd) -A + ad -aαd 4 - 1 T 2 C -cd 1 + γd 2 exp -γd 2 With P the pressure, R the perfect gas constant, T the temperature, d the molar density, and a, b, c, A, B, C, α, γ the empirical parameters. This law is for example used to represent refrigerants. It is used to characterize hydrogen in the formulation "condensable fluid" in the code Fine T M /Turbo. Presentation of different models of cavitation In this section, a review of various models available in the literature that describe the phenomena of cavitation with or without the consideration of thermodynamic effect is presented. In cold water, or more generally for a non-thermosensitive fluid, the dynamic and thermal phenomena are decoupled. The energy equation is therefore not necessary. In contrary, in thermosensitive fluid, it is necessary to include the equation of energy. Models with the mixture state law These are models with three equations (or two equations without the energy) for which the phase change is controlled by a state law. There are several types of closure relations to link the two phases in the literature : • Sinusoidal barotropic law [START_REF] Delannoy | Two phase flow approach in unsteady cavitation modelling[END_REF] • Logarithmic barotropic law [START_REF] Schmidt | A fully compressible, two-dimensional model of small, high-speed, cavitating nozzles[END_REF][START_REF] Moreau | A numerical study of cavitation influence on diesel jet atomisation[END_REF][START_REF] Xie | Isentropic one-fluid modelling of unsteady cavitating flow[END_REF] • Saurel's equilibrium law [Saurel et al., 1999] • Tabulated state law [START_REF] Ventikos | A numerical study of the steady and unsteady cavitation phenomenon around hydrofoils[END_REF][START_REF] Clerc | Numerical simulation of the homogeneous equilibrium model for two-phase flows[END_REF] • Equilibrium law based on free enthalpy [START_REF] Edwards | Low-diffusion flux splitting methods for real fluid flows with phase transition[END_REF] • Polynomial law (of degree 5) [START_REF] Song | Current status of cfd for cavitating flows[END_REF] • Barotropic law "Italian" [START_REF] Rapposelli | A barotropic cavitation model with thermodynamic effects[END_REF][START_REF] Sinibaldi | A numerical method for 3D barotropic flows in turbomachinery[END_REF]. • State law based on entropy [START_REF] Barberon | Finite volume simulation of cavitating flows[END_REF] • Mixture of stiffened gas law [START_REF] Goncalves | Numerical simulation of cavitating flows with homogeneous models[END_REF] a/ Sinusoidal barotropic law The barotropic model existing in Fine T M /Turbo was developed by the successive theses of Coutier [START_REF] Coutier-Delgosha | Modelisation des écoulements cavitants: etude des comportements instationnaires et application tridimensionnelle aux turbomachines[END_REF]] and Pouffary [START_REF] Pouffary | Simulation numérique d'écoulements 2D/3D cavitants, stationnaires et instationnaires[END_REF]. It was originally proposed by Delannoy et Kueny [START_REF] Delannoy | Two phase flow approach in unsteady cavitation modelling[END_REF]. This law relates the pressure to the density by a sinusoidal relation : ρ = ρ L + ρ V 2 + ρ L -ρ V 2 sin p -p vap c 2 min 2 ρ L -ρ V (2.18) c min represents the minimum speed of sound in the mixture. This law introduces a small nonequilibrium effect on the pressure. The unequilibrium is controlled by the value of c min . b/ Schmidt's barotropic law From the integration of the Wallis mixture speed of sound which is the propagation velocity of acoustic waves without mass transfer, Schmidt [START_REF] Schmidt | Cavitation in Diesel fuel injector nozzles[END_REF] proposes a barotropic law in the form of : P = p sat + ρ V c 2 V ρ L c 2 L ρ V -ρ L ρ 2 V c 2 V -ρ 2 L c 2 L ln ρ V c 2 V ρ L + α ρ V -ρ L ρ L ρ V c 2 V -α ρ V c 2 V -ρ L c 2 L (2.19) This expression is used in [START_REF] Moreau | A numerical study of cavitation influence on diesel jet atomisation[END_REF][START_REF] Dumont | Modélisation de l'écoulement diphasique dans les injecteurs Diesels[END_REF] to simulate the cavitation of diesel in the injectors of piston engine. A modified version was proposed by [START_REF] Xie | Isentropic one-fluid modelling of unsteady cavitating flow[END_REF] in order to avoid the appearance of negative pressure. c/ Saurel's equilibrium law For compressible flows, Saurel [Saurel et al., 1999] uses the Tait law for the liquid and the perfect gas law for the vapor to calculate the pressure in each phase. The mixture is assumed to be in kinematic and thermodynamic equilibrium. In this way, there is a logarithmic relation to connect P and T in the form of : ln(P/P 0 ) = k a k (T/T 0 ) k (2.20) The densities of each phase are given by polynomial functions of the temperature. The void ratio is defined as : α = ρ -ρ Lsat (T) ρ V sat (T) -ρ Lsat (T) (2.21) d/ Edwards equilibrium law [START_REF] Edwards | Low-diffusion flux splitting methods for real fluid flows with phase transition[END_REF] propose an equilibrium model to simulate twophase octane flows. The pure phases are governed by Sanchez-Lacombe's law. Thermodynamic equilibrium is defined by the equality of free enthalpies (g = h -T s) between phases : g L = g V . The iterative resolution of this equation makes it possible to determine the vapor pressure P vap (T). The void ratio is calculated by : α = ρ-ρ Lsat (T) ρ V sat (T)-ρ Lsat (T) e/ Rapposelli's barotropic law Using thermal analysis on a bubble, a relation between the speed of sound in the two-phase mixture and the temperature can be obtained [START_REF] Rapposelli | A barotropic cavitation model with thermodynamic effects[END_REF]. It is possible to find a law between the density and the temperature by integrating the speed. This law has been used for the calculation of hydrofoil in non-viscous flow. The speed of sound is expressed as the relation : 1 ρ c 2 = 1 ρ ∂ρ ∂p ∼ = 1 -α p 1 -ε L p ρ L c 2 L + ε L g * p c p η + α γ V p (2.22) In this expression, γ V = C p V Cv V and ε L represent the liquid fraction participating in the heat exchanges with the vapor and : ε L = α 1 -α 1 + δ T R 3 -1 (2.23) where δ T R is a controlled parameter obtained from calibration of the model from experimental results. The other parameters are as follows : For cold water : g * = 1.67; η = 0.73; P c = 221.29 10 5 Pa For nitrogen : g * = 1.3; η = 0.69; P c = 3.4 10 6 Pa f/ State law based on entropy [START_REF] Barberon | Finite volume simulation of cavitating flows[END_REF] proposed to calculate the entropy of the mixture to evaluate the pressure and the temperature. The pure phases are both governed by the stiffened gas law. The specific entropy of the mixture is maximal at thermodynamic equilibrium. During the process of maximization the entropy can be determined when equilibrium is reached and then also for the pressure P = T ∂s ∂v , where v is the specific volume. g/ Mixture of stiffened gas law With the assumption of thermal and mechanical equilibrium, an expression for the pressure and the temperature can be deduced as follows [START_REF] Goncalves | Numerical simulation of cavitating flows with homogeneous models[END_REF] : P ρ, e, α = (γ(α) -1)ρ(e -q(α)) -γ(α)P ∞ (α) 1 γ(α) -1 = α γ V -1 + 1 -α γ L -1 and ρ q(α) = αρ V q V + (1 -α)ρ L q L P ∞ (α) = γ(α) -1 γ(α) α γ V P V ∞ γ V -1 + (1 -α) γ L P L ∞ γ L -1 T ρ, h, α = h -q(α) C p(α) with ρC p(α) = αρ V C p V + (1 -α)ρ L C p L The void ratio is computed with saturation values of densities : α = ρ-ρ Lsat ρ V sat -ρ Lsat . An extension version considering thermodynamic effects for thermosensible fluids is proposed in [START_REF] Goncalves | Numerical study of cavitating flows with thermodynamic effect[END_REF] by introducing a linear variation relation of P vap , ρ L and ρ V with the temperature. However, this law failed to obtain reasonable results for Venturi case of 4 degree. Models with four equation, transport-based equation models (TEM) In these models, a conservation equation for one of the phases is added by means of the source term S which models the mass exchange between the phases. There are different formulations for the source term (more or less empirical constants) : • Merkle's model [START_REF] Merkle | Computational modeling of the dynamics of sheet cavitation[END_REF]] • Kunz's model [START_REF] Kunz | A preconditioned navier-stokes method for two-phase flows with application to cavitation prediction[END_REF] • Senocak and Shyy model [START_REF] Senocak | A pressure-based method for turbulent cavitating flow computations[END_REF] • Saito's model [START_REF] Saito | Numerical analysis of unsteady vaporous cavitating flow around a hydrofoil[END_REF] • Vortmann's model [START_REF] Vortmann | Thermodynamic modeling and simulation of cavitating nozzle flow[END_REF] • Utturkar's model [START_REF] Utturkar | Recent progress in modelling of cryogenic cavitation for liquid rocket propulsion[END_REF] • Hosangadi and Ahuja model [START_REF] Hosangadi | Numerical study of cavitation in cryogenic fluids[END_REF] • Goncalvès model [START_REF] Goncalvès | Numerical study of expansion tube problems: Toward the simulation of cavitation[END_REF] • Source term based on the simplified Rayleigh-Plesset equation a/ Merkle's model (1998) The model proposed by [START_REF] Merkle | Computational modeling of the dynamics of sheet cavitation[END_REF]] is one of the first models that uses the mass conservation equation for the vapor phase to simulate the cavitation. The equation solved for the vapor phase is as follows: ∂x V ∂t + u.∇x V = - x V τ V = x L τ L (2.24) where x V and x L are the mass fractions of the vapor and liquid phases respectively (αρ V = x V ρ). The source term is defined as: 1 τ V = 0 when P < P vap 1 kτ re f P-P vap q when P > P vap τ L is defined in the same way for condensation. τ re f = L re f U re f is the reference time scale of the fluid, and k is a constant with the value around 10 -3 . The parameter q is not specified in the article [START_REF] Merkle | Computational modeling of the dynamics of sheet cavitation[END_REF]] but seems to be the reference dynamic pressure q = 0.5ρU 2 re f . b/ Kunz's model (2000) Kunz's model [START_REF] Kunz | A preconditioned navier-stokes method for two-phase flows with application to cavitation prediction[END_REF] is based on an empirical source term split into two contributions for the evaporation and condensation process : ∂α L ∂t + ∇.(α L u) = ṁ+ + ṁ- (2.25) This model is implemented in the IZ code [START_REF] Coutier-Delgosha | Simulation of unsteady cavitation with a two-equation turbulence model including compressibility effects[END_REF][START_REF] Rolland | Modélisation et résolution de la propagation de fronts perméables[END_REF][START_REF] Patella | Numerical model to predict unsteady cavitating flow behavior in inducer blade cascades[END_REF]. The evaporation and condensation source terms are given as following expressions : ṁ-= C dest ρ V α L M in(0, P -P vap ) ρ L (ρ L U 2 re f /2)t ∞ and ṁ+ = C prod ρ V α 2 L (1 -α L ) ρ L t ∞ (2.26) where t ∞ is the relaxation time, C dest and C prod are the constants to be calibrated. The condensation rate is modeled as being proportional to the liquid volume fraction and the amount by which the pressure is below the saturated vapor pressure. For the evaporation rate, a simplified Ginzburg-Landau relationship is used. c/ Senocak and Shyy model (2001) Senocak et Shyy [START_REF] Senocak | A pressure-based method for turbulent cavitating flow computations[END_REF] try to eliminate the empirical constants by adopting from Kunz's model. It is carried out by the idea of introducing the normal interfacial velocity. However there will be a problem of locating the interface arises. This difficulty is overcome by the calculation of the density gradient. In this way, a fictitious interface is obtained because of modeling effort inside it (see Figure 2.1). The mass transfer source terms are as follows : Saito [START_REF] Saito | Numerical analysis of unsteady vaporous cavitating flow around a hydrofoil[END_REF]] uses a mass transfer equation for the vapor phase. The system is closed by the modeling of the source term and a mixture state law. The mixture state law is determined by the weighting of each phase form the Tamman law for the liquid phase and the perfect gas law for the vapor phase respectively. : ṁ-= ρ V α L M in(0, P -P vap ) ρ V (U V ,n -U I,n ) 2 (ρ L -ρ V )t ∞ and ṁ+ = (1 -α L )Max(0, P -P vap ) (U V ,n -U I,n ) 2 (ρ L -ρ V )t ∞ (2. 1 ρ = 1 ρ L (1 -x) + 1 ρ V x or ρ = P P + P c K (1 -x) P(T + T c ) + rx P + P c T (2.28) The vapor pressure is given by an empirical formula as a function of the temperature. The mass transfer source term is proportional to the pressure difference, P vap -P, as well as the inverse of the square root of the saturation temperature. ṁ =          ṁ+ = C e Aα (1 -α) ρ L ρ V P * V ap -P 2πRT S if P < P * vap ṁ-= C c Aα (1 -α) P * vap -P 2πRT S if P ≥ P * vap where T S is the saturation temperature and A = C a α (1 -α) A denotes the interfacial area concentration in the vapor-liquid mixture. The saturation vapor pressure of cold water is given by the empirical formula as : P * vap = 22.13 × 10 6 exp 1 - 647.31 T 7.21379 + 1.152 × 10 -5 -4.787 × 10 -9 T (T -483.16) 2 The parameters C a , C c and C e are empirical constants. e/ Vortmann's model (2003) A rate equation for vapor quality x is formulated by Vortmann [START_REF] Vortmann | Thermodynamic modeling and simulation of cavitating nozzle flow[END_REF] as : ∂x ∂t + u.∇x = (1 -x)K l→v -xK v→l (2.29) The terms K l→v and K v→l mean the probabilities of phase change from liquid to vapor and from vapor to liquid respectively. These terms integrate the Gibbs free energy and involve the relaxation time set to 10 -4 s kg/m 3 . The vapor pressure is supposed to be constant. f/ Utturkar's model (2005) The previous IDM model is adapted by [START_REF] Utturkar | Recent progress in modelling of cryogenic cavitation for liquid rocket propulsion[END_REF] to take the thermodynamic effects into account. The new model is then called Mushy Interfacial Dynamics Model. The original model without thermodynamic effects uses the averaged interface coditions of a liquid-vapor interface to construct the mass transfer source term. This approach is justified by the authors that the sheet cavitation of the cold water contain a significant void ratio. Starting from the analysis of Hord [START_REF] Hord | Cavitation in liquid cryogens, vol 4, combined correlations for venturi, hydrofoil, ogives and pumps[END_REF] for the composition of cryogenic sheet cavitation which describes the vapor zone as the mixture zone with lower void ratio, a model using the averaged CHAPTER 2. REVIEW OF CAVITATION MODELING interface conditions between the liquid and the mixture is formulated. The mass transfer source terms are given below : ṁ-= ρ L α L M in(0, P -P vap ) ρ i (U m,n -U I,n ) 2 (ρ L -ρ V )t ∞ and ṁ+ = ρ L (1 -α L )Max(0, P -P vap ) ρ j (U m,n -U I,n ) 2 (ρ L -ρ V )t ∞ (2.30) if α L ≥ 0.99 ρ i = ρ m and ρ j = ρ m otherwise ρ i = ρ V and ρ j = ρ L This model is valid for the cavitating flows of cold water. As for the sharp IDM model: U m,n = u.n with n = ∇α L |∇α L | For the steady calculation, the normal component of interfacial velocity, U I,n is equal to zero. Numerical simulations are presented for the analysis of the thermodynamic effects for 2D turbulent liquid nitrogen around a warhead. The pressure profiles at the wall in the sheet show a good qualitative behavior of the model, and the void ratio inside the sheet is significantly decreased in comparison with the calculations for cold water. g/ Hosangadi and Ahuja model (2005) Hosangadi et Ahuja [START_REF] Hosangadi | Numerical study of cavitation in cryogenic fluids[END_REF] use the source term based on the one of Merkle [START_REF] Merkle | Computational modeling of the dynamics of sheet cavitation[END_REF]]. The formulation has been implemented within a 3D unstructured code CRUNCH. To our knowledge, this is the first one to calculate the performance drop of a liquid cryogenic inducer (LH2). The transport equation of the vapor phase is shown as below : ∂ρ V α ∂t + ∇.(ρ V αu) = m t (2.31) m t is the mass transfer source term : m t = ṁ-αρ V + ṁ+ (1 -α)ρ L with : ṁ-=   0 P < P vap 1 τ V U re f L re f P-P vap 1 2 ρ L U 2 re f P > P vap   and ṁ+ =   0 P > P vap 1 τ L U re f L re f P-P vap 1 2 ρ L U 2 re f P < P vap   τ V and τ L are the time constants for liquid reconversion and vapor formation respectively which were set at 0.001 s. Fluid thermodynamic properties of each phase are calculated from the NIST Chemistyr WebBook (http://webbook.nist.gov). h/ Goncalvès model (2013) Goncalvès [START_REF] Goncalvès | Numerical study of expansion tube problems: Toward the simulation of cavitation[END_REF] presents the first version of transport equation which has a form that includes two quantities not used before: the speed of sound, c, and the Wallis mixture speed of sound [START_REF] Wallis | One-dimensional two-phase flow[END_REF], c wall is . The Wallis speed of sound is expressed as a weighted harmonic mean of speeds of sound of each phase: 1 ρ c 2 wall is = α ρ v c 2 v + 1 -α ρ l c 2 l (2.32) 2. MODELING OF TWO-PHASE FLOWS The void ratio equation can be expressed as : ∂α ∂t + u ∂α ∂t =   ρ l c 2 l -ρ v c 2 v ρ l c 2 l 1-α + ρ v c 2 v α   =K ∂u ∂x +   c 2 v α + c 2 l 1-α ρ l c 2 l 1-α + ρ v c 2 v α   =1/ρ I the interfacial density ṁ (2.33) where ṁ is the mass transfer between phases and ρ I is the interfacial density. The term K involves the speed of sound of pure phases and it reflects the effects of changes in volume of each phase. By assuming that the mass transfer is proportional to the divergence of the velocity, the mass transfer ṁ is expressed as : ṁ = ρ l ρ v ρ l -ρ v 1 - c 2 c 2 wall is div V (2.34) Computations are performed by Goncalvès and Charrière [START_REF] Goncalvès | Modelling for isothermal cavitation with a four-equation model[END_REF] for several cases including an underwater explosion with cavitation, bubble collapse by a pressure wave and the 8 degree Venturi geometry. I/ Source term based on the simplified Rayleigh-Plesset equation model These models, unlike the bubble tracking approach, do not solve the complete Rayleigh-Plesset equation [START_REF] Brennen | Cavitation and bubble dynamics[END_REF] which is written as : R d 2 R dt 2 + 3 2 dR dt 2 = 1 ρ P vap + P g -P ∞ (t) - 2σ R - 4µ R dR dt (2.35) This equation describes the evolution of a spherical bubble in an infinite domain of liquid. R is the radius of the bubble, P vap the saturation pressure, P g the pressure of dissolved gas, the last two terms represent the surface tensions and the viscous effects respectively. • The CAVKA code (CAVitation KArlsruhe) This code was developed at the University of Karlsruhe by Sauer [START_REF] Sauer | Unsteady cavitating flow -a new cavitation model based on modified front capturing method and bubble dynamics[END_REF][START_REF] Schnerr | Physical and numerical modeling of unsteady cavitation dynamics[END_REF]. It models the cavitation by a void ration equation where the source term S is obtained from a number of nuclei, a characteristic radius of these nuclei and the radius growth rate according to the simplified Rayleigh-Plesset equation. S = n 0 4πR 2 1 + n 0 4 3 πR 3 dR dt (2.36) n 0 represent the bubble number density (10 8 by default), and the radius growth rate is expressed as : dR dt = ± 2 3 P -P vap ρ 1 (2.37) CHAPTER 2. REVIEW OF CAVITATION MODELING The expression for the mass transfer source term is derived from the relation between the void ratio and the bubbles : α = V V V cell = N B 4 3 πR 3 V V + V L = n 0 4 3 πR 3 1 + n 0 4 3 πR 3 (2.38) where N B represents the number of bubbles in the computational cell and V cell is the volume of the computational cell. A comparison of the results between the CAVKA code and another CATUM code for the unsteady non-viscous flow on 2D hydrofoils is given in [START_REF] Schnerr | Shock and wave dynamics of compressible liquid flows with special emphasis on unsteady load on hydrofoils and a cavitation in injection nozzles[END_REF]. A model with thermodynamic effects was proposed by the same authors [START_REF] Sauer | Unsteady cavitating flow -a new cavitation model based on modified front capturing method and bubble dynamics[END_REF]. They propose to associate the thermodynamic properties of each phase, especially for the vapor pressure, with the temperature obtained from the resolution of the mixture energy equation.. Simulation of unsteady cavitating flow of hot water has been carried out in a 2D nozzle. • The codes of Fluent and ACE+ The model of Singhal [START_REF] Singhal | Mathematical basis and validation of the full cavitation model[END_REF][START_REF] Dular | Experimental evaluation of numerical simulation of cavitating flow around hydrofoil[END_REF][START_REF] Zhang | Computational fluid dynamic study on cavitation in liquid nitrogen[END_REF][START_REF] Tani | Development and validation of new cryogenic cavitation model for rocket turbopump inducer[END_REF] where the phase change is managed by the conservation equation of vapor phase : ∂x V ρ m ∂t + ∇.(x V ρ m u) = R e -R c (2.39) x represents the vapor, R e and R c are the evaporation and condensation source terms respectively : R e = Ce k σ ρ L ρ V 2 3 P vap -P ρ L 1 -x V -x g if P < P vap (2.40) R c = Cc k σ ρ L ρ L 2 3 P vap -P ρ L x g otherwise (2.41) Ce and Cc are the empirical constants, k is the local kinetic energy, σ is the surface tension, and x g is the mass fraction of dissolved gases. According to Singhal [START_REF] Singhal | Mathematical basis and validation of the full cavitation model[END_REF], Ce=0.02 and Cc=0.01. An extened version taking into account thermodynamic effects is proposed by [START_REF] Tani | Development and validation of new cryogenic cavitation model for rocket turbopump inducer[END_REF]. The formulation shows the vapor pressure P vap (T) varies with the temperature which is derived from an analytical calculation based on the B-factor theory. • The CFX code The model contains the TEM model [START_REF] Bouziad | Physical modelling and simulation of leading edge cavitation, application to an industrial inducer[END_REF][START_REF] Mejri | Comparison of computational results obtained from a homogeneous cavitation model with experimental investigations of three inducers[END_REF]. The form of the source term S is based on the simplified Rayleigh-Plesset equation : S =        S V = F V N V ρ V 4πR 2 B 2 3 |Pvap-P| ρ L if P < P vap S L = F C N C ρ V 4πR 2 B 2 3 P vap -P ρ L if P > P vap (2.42) MODELING OF TWO-PHASE FLOWS N V and N C represent the number of bubbles for different modeling of physical phenomenon (vaporization or condensation). N V = (1 -α) 3α d 4πR 3 B and N C = 3α 4πR 3 B (2.43) F V and F C are the empirical constants and represent different time scales for the vaporization and condensation processes : F V =50 and F C =0.01. In addition, R B is the initial radius for the bubbles and x d is non-condensable gases which provide nucleation sites for the cavitation process : R B = 10 -6 m and x d = 10 -5 by default. • The Star-CD code This code contains the Rayleigh-Plesset model. However the expressions of the sources terms are unknow, they are not explained in the articles [START_REF] Kimura | Numerical simulation for unsteady cavitating flow in a turbopump inducer[END_REF][START_REF] Ugajin | Numerical simulation of cavitating flow in inducers[END_REF]. Remark: The free, open source software OpenFOAM is a one-fluid RANS solver which is developed under the pressure-based schemes (SIMPLE and PISO). Several TEM models are implemented in OpenFOAM [START_REF] Erney | Verification and validation of single phase and cavitating flows using an open source CFD tool[END_REF], including the Kunz's model, Merkle's model and Sauer and Schnerr model. The validation was carried out in a flow behind a hemisphere and two different hydrofoils (NACA0012 and NACA66). Saurel's model with five equations This model is composed of four conservation equations (two mass balance for each pure phase, one mixture momentum , one mixture energy) plus a non-conservative equation for the void ratio. The inviscid formulations are written as : ∂αρ V ∂t + div αρ V u = ṁ (2.44) ∂(1 -α)ρ L ∂t + div (1 -α)ρ L u = -ṁ (2.45) ∂ρu ∂t + div ρu ⊗ u + grad P = 0 (2.46) ∂ρE ∂t + div u(ρE + P) = 0 (2.47) ∂α ∂t + u.∇α = α(1 -α)(ρ L c 2 L -ρ V c 2 V ) αρ L c 2 L + (1 -α)ρ V c 2 V div (u) (2.48) + α(1 -α) αρ L c 2 L + (1 -α)ρ V c 2 V Γ V α + Γ L 1 -α Q + ρ V c 2 V α + ρ L c 2 L 1-α c 2 V α + c 2 L 1-α ṁ The mass transfer term ṁ and the heat transfer term Q between phases are expressed as : CHAPTER 2. REVIEW OF CAVITATION MODELING ṁ = νρ(g L -g V ) and Q = H(T L -T V ) These terms involve free Gibbs enthalpy and the temperature of pure phases as well as two relaxation coefficients ν and H. The property Γ = 1 ρ ∂p ∂e ρ is the Grüneisen coefficient of pure phases governed by the stiffened gas law. With the assumption of mechanical equilibrium, it is possible to calculate the mixture pressure : p ρ, e, α, ρ V , ρ L = (ρe -αρ V q V -(1 -α)ρ L q L ) -α γ V P V ∞ γ V -1 + (1 -α) γ L P L ∞ γ L -1 α γ V -1 + 1-α γ L -1 (2.49) The relaxation coefficients n and H are unknown and very difficult to determine in practice. To remove this uncertainty, the resolution is carried out in two steps : 1. It is assumed that the thermodynamic equilibrium is reached, that is to say the relaxation coefficients are taken as infinite. This allows to determine the equilibrium void ration α eq . 2. Then the whole system is solved. The mass exchange term between phases is evaluated by : ṁ = νρ(g L -g V ) = ∂αρ V ∂t = (αρ V ) eq -αρ V ∆t . The same goes for the heat transfer term Q. The model has been tested on the problems of 1D expansion tube, 2D supercavitating flow around an obstacle and 2D Venturi flow corresponding to a fuel injector (dodecane). Two-fluid models a/ Grogger and Alajbegovic model (1998) This two-fluid model has been initially applied to the cavitation in cold water in 2D and 3D Venturi [START_REF] Grogger | Calculation of the cavitating flow in venture geometries using two fluid model[END_REF]. It has been applied more recently to a three-phase (liquid, vapor and air) turbulent cavitating flows simulation for the high-pressure swirl injector geometry (3D) [START_REF] Alajbegovic | Three-phase cavitating flows in high-pressure swirl injectors[END_REF]. This model is based on the resolution of mass and momentum conservation equations for each phase (four equations in two phases and six equations in three phases). It has the advantage of modeling the interactions of momentum between phases. The system solved in the two phases is shown as below (k = L, V ) : ∂α k ρ k ∂t + ∇. α k ρ k u k = Γ k (2.50) ∂α k ρ k u k ∂t + ∇. α k ρ k u k ⊗ u k = -α k ∇p + ∇.(α k τ k ) + F d k + u k Γ k (2.51) The mass transfer source term is based on the bubble growth rate from the simplified Rayleigh-Plesset equation : Γ L = ρ L N4πR 2 ∂R ∂t = -Γ V (2.52) The bubble radius R is defined in function of the void ratio α according to the following expres- sion : R = 1 2 6α Nπ 1/3 N represents the bubble number density : -α) for α > 0.5 where N 0 is the initial bubble number density and is set to 10 12 . N = N 0 for α ≤ 0.5 2 N 0 -1 (1 The bubble growth rate is modeled by : ∂R ∂t = ± 2 3 p -p V ρ L The interfacial momentum transfer term is based on the effects of drag force of a sphere and turbulent dispersion forces : F d L = c TD ρ L k L ∇α + c D ρ L |u rel | u rel 2 A = -F d V with u rel = u L -u V (2.53) The term A represents the surface of a spherical bubble. C TD is the turbulent dispersion coefficient and k is the turbulent kinetic energy. The drag coefficient c D is given by the relations : This model is based on the seven-equation model of [START_REF] Baer | A two-phase mixture theory for the deflagration-to-detonation transition (DDT) in reactive granular materials[END_REF], c D = 24/R e(1 + 0.15R e 0 .687) if R e P < which uses the transport equation of the volume fraction α 1 to close the two-fluid model with six equations [START_REF] Saurel | A multiphase godunov method for compressible multifluid and multiphase flows[END_REF]. It was originally proposed without considering the mass transfer. In the thesis of Le Métayer [START_REF] Rolland | Modélisation et résolution de la propagation de fronts perméables[END_REF], a phase change term is introduced to the model. It allows to solve the complex problems such as flows with three components of velocity. The seven equations of Baer-Nunziato without phase change are as follows (k=1,2) : ∂α k ρ k ∂t + ∇. α k ρ k u k = 0 ∂α k ρ k u k ∂t + ∇. α k ρ k u k ⊗ u k = -∇(α k P k ) + ∇.(α k τ k ) + α k ρ k F k + F d k + P I ∇α k ∂α k ρ k E k ∂t + ∇. α k ρ k E k u k = -∇. α k q k -∇. p k u k + ∇. τ k .u k + P I u I .∇α k + µP I P k -P k + α k ρ k F k .u k + F d k .u I + Q kI ∂α 1 ∂t + u I .∇α 1 = -µ (P 1 -P 2 ) CHAPTER 2. REVIEW OF CAVITATION MODELING The index I represents the interface. The term µ (P 1 -P 2 ) which represents the production of the volume fraction α 1 , is equal to the pressure difference between phases multiplied by a coefficient µ that controls the velocity at which pressure equilibrium is reached. This term induces µP I (P k -k ) due to the interfacial pressure work to energy conservation equations. To close the system, the interfacial pressure P I is represented by the most compressible phase pressure and the interfacial velocity u I is represented by the less compressible phase velocity in the original model. Subsequently, Saurel et Abgrall [START_REF] Saurel | A multiphase godunov method for compressible multifluid and multiphase flows[END_REF] estimate the interfacial pressure by the mixture pressure : P I = 2 k=1 α k P k After solving an inert Riemann problem i.e. without phase change, the study carried out by Le Métayer consists of considering the mass transfer term by the Rankine-Hugoniot relations across the shock front. Different models are proposed. The most complex one deals with three velocities, one for each phase plus one for the front. More details about this model are available in the references [START_REF] Saurel | A multiphase godunov method for compressible multifluid and multiphase flows[END_REF][START_REF] Saurel | A multiphase model for compressible flows with interfaces, shocks, detonation waves and cavitation[END_REF][START_REF] Rolland | Modélisation et résolution de la propagation de fronts perméables[END_REF][START_REF] Bibliography Abgrall | Efficient numerical approximation of compressible multi-material flow for unstructured meshes[END_REF]]. These models have been used and validated on a 1D inviscid flow expansion tube problem. c/Saurel and Le Métayer model with ten equations (2001) Saurel et Le Métayer [START_REF] Saurel | A multiphase model for compressible flows with interfaces, shocks, detonation waves and cavitation[END_REF] propose a multiphase model composed of five equations for each phase. It is able to deal with a wide range of applications with the very general formulation (interfaces between compressible materials, homogeneous two-phase flows, the problems of shocks and cavitation). This approach is based on the one proposed by Baer et Nunziato [START_REF] Baer | A two-phase mixture theory for the deflagration-to-detonation transition (DDT) in reactive granular materials[END_REF]. The system of equation for each phase k is written as : ∂α k ρ k ∂t + ∇. α k ρ k u k = m k ∂α k ρ k u k ∂t + ∇. α k ρ k u k ⊗ u k = -∇(α k P k ) + m k u I + F d k + P I ∇α k ∂α k ρ k E k ∂t + ∇. α k ρ k E k u k = -∇. [P k u k ] + P I u I .∇α k + m k E kI -µP I (P k -P k ) + F d k .u I + Q kI ∂α k ∂t + u I .∇α k = µ (P k -P k ) + m k ρ X ∂N k ∂t + ∇. (N k u k ) = Ṅk with the average interface conditions : k m k = 0 k m k u I + F d k + P I ∇α k = 0 k P I u I .∇α k + m k E kI -µP I (P k -P k ) + F d k .u I + Q kI = 0 The conservation law of mass, momentum and energy, appear the terms related volume fraction transport equation α k of phase k, as well as the number density of the individual entity N k composing phase k (ex : number of bubbles for a liquid-gas flow mainly liquid). The source term Ṅk of the equation represents the phenomena of breakup or coalescence. The interfacial velocity u I as well as the interfacial pressure p I of the system are defined as : u I = k α k ρ k u k k α k ρ k and p I = k α k P k + ρ k (u I -u k ) 2 (2.55) The momentum transfer term due to the drag force between phases is modeled by the velocity relaxation term : F d k = λ k u k -u k The mass transfer term is obtained from the interface-averaged equations : m 1 = -Q 1I + Q 2I E 1I -E 2I (2.56) The energy transfer at the interface is provided by empirical correlations : Q kI = h k T kI -T k A ex (2.57) where A ex is the exchange interfacial area. For spherical entities (bubbles, drops) : A ech = N 1 4πR 2 The heat exchange coefficient is defined as : h k = λ k N u 2R The Nusselt number is defined from the Reynolds number and the Prandtl number : N u = 2 + 0.6R e 0.5 P r 0.33 (2.58) The µ (P k -P k ) and µP I (P k -P k ) terms are related to the pressure relaxation process and are controlled by the value of µ. When mass transfer occurs an extra source term, m k ρ X , is present in the transport equation of the volume fraction, where ρ X represents the density of the less compressible phase. The main purpose of this term is to separate the mass transfer and the acoustic propagation during numerical resolution. d/ Model of Saturne code of EDF Mimouni et al. [START_REF] Mimouni | Modelisation et simulation des ecoulements cavitants par une approche diphasique[END_REF] present the simulation of cavitation carried out with the NEPTUNE code which is originally developed by EDF and CEA for the study of two-phase flow with ebullition. This code uses a two-fluid approach and allows to simulate a large number of flow configurations : gaseous or liquid phase with solid particles, liquid phase and vapor , as well as different flow topologies : disperesd phases, flows with a continuous phase... The mass transfer term is obtained from the interface-averaged equations : Γ L = -Γ V = q LI + q V I h V I -h LI A I (2.59) where q kI represents the interfacial heat flux in the phase k, h kI the enthalpy of the phase k at the interface and A I = 6α d the exchange interfacial area (α is the void ration and d the averaged bubble diameter taken equal to 0.1mm). The enthalpies of each phase at the interface are assumed to be saturated. The interfacial heat flux is modeled as follows : q kI = c kI (T sat (P) -T k ) (2.60) With the specific heats as follows : c LI = N u L λ L d and c V I = αρ V C p V ∆t The Nusselt number of liquid is modeled as : The code has also been tested on a case of cavitation behind an orifice (EPOCA test case). N u L = 2 + 0.6R e 1/2 P r 1/3 L with R Summary The vast majority of computer codes dedicated to the simulation of cavitation is based on an averaged approach for both the two-phase flow and the turbulence. With proper averaging, the mean values of fluid motions and properties can be obtained. Within the averaged model family, there are different approaches according to the physic assumptions made on thermodynamics equilibrium and slip condition between phases. This has resulted in the development of various systems ranging from seven (two-fluid) to three (one-fluid) equations only. The two-fluid approach is the most complete and is also known to be a real challenge for numerical simulation due to the complicated characteristics of the equation system and the troublesome non-conservative terms. SUMMARY On the other hand, the one-fluid method, or homogeneous method considers the flows as a mixture of two fluids behaving as one that is similar to the single-phase flow. In this way, only one set of conservation equation is employed to express the fluid interaction for the mixture. Because of the difficulty of modeling nonequilibrium thermodynamics pattern during a phase transition, the existing models have systematic use of mechanical equilibrium assumptions (single pressure model) and thermal (single temperature model). Besides, vaporization and condensation processes are assumed to be instantaneous. Then, this method cannot reproduce strong thermodynamic or kinetic non-equilibrium effects. Considering its simplicity and low computation cost, the homogeneous method is more attractive for numerical simulations of cavitating flows. On the assumptions of velocity equilibrium and pressure equilibrium for each phase from the full models, a reduced model five-equation model is obtained. This model is capable of modeling the mass and energy transfer terms between phases and taking the thermo non-equilibrium into account. By assuming the the velocity, pressure and thermal equilibrium between phases, a four-equation model can be expressed. With an additional transport equation, usually the void ratio, the mass transfer between phases can be modeled. The main problem of this model is the formulation of the source term and the tunable parameters involved for the cavitating processes. With the assumption of complete thermodynamic equilibrium between phases, that is local temperature, pressure and free Gibbs enthalpy are supposed to be in equilibrium, the three-equation models or Homogeneous Equilibrium Models (HEM) are derived. The most difficult part for this approach is to define a proper equation of state (EOS) for the thermodynamic behavior of the mixture to close the system. The one-fluid method (homogeneous method) has received more attention up to now, because of its lower computational cost and easier coupling with turbulence models. Among the various existing models, the main differences count on the relation between the pressure and density fields. This coupling is generally treated through a barotropic equation of state, or developed by the framwork of transport-based equation method. In this study, the HEM based formulations coupled with a barotropic state law [START_REF] Goncalves | Numerical simulation of cavitating flows with homogeneous models[END_REF]] and a four-equation based model completed with a void ratio transport equation [START_REF] Goncalvès | Numerical study of expansion tube problems: Toward the simulation of cavitation[END_REF] Immersed Boundary Method (IBM) have been successfully implemented in NSMB. The details about the solver can be referred to the NSMB Handbook [Vos et al., 2013]. The cavitation models and numerical method that are implemented in this study will be presented in the following sections. Governing Equations The governing equations of the HEM based formulations coupled with a barotropic state law and a four-equation based model completed with a void ratio transport equation are presented below. A three-equation model The movement of fluids is governed by three basic physical conservation equations: mass, momentum and energy conservation equations. The homogeneous mixture approach is used to model two-phase flows. The phases are assumed to be sufficiently well mixed and the disperse particle size are sufficiently small thereby eliminating any significant relative motion. The phases are strongly coupled and moving at the same velocity. In addition, the phases are assumed to be in kinematic and thermodynamic equilibrium: they share the same pressure, temperature and velocity . The evolution of the two-phase flow can be described by the conservation laws that employ the representative flow properties as unknowns just as in a single-phase problem. The void fraction α is introduced to characterize the volume of vapor in each cell: α = 1 means that the cell is completely filled with vapor; inversely, a complete liquid cell is represented by α = 0. The density ρ, the center of mass velocity u and the internal energy e for the mixture are defined 3.1. NSMB by [START_REF] Ishii | Thermo-Fluid Dynamics of Two-Phase Flow[END_REF]: ρ = αρ v + (1 -α)ρ l (3.1) ρu = αρ v u v + (1 -α)ρ l u l (3.2) ρ e = αρ v e v + (1 -α)ρ l e l (3.3) where the subscripts v and l are the vapor and liquid phase respectively. The inviscid compressible Navier-Stokes equations in 3-D Cartesian corrdinates (x, y, z) can be expressed in conservative form as: ∂ ∂t (W) + ∂ ∂x ( f ) + ∂ ∂y (g) + ∂ ∂z (h) = 0 (3.4) where t denotes the time. The state vector W is given by: W =          ρ ρu ρv ρw ρE          (3.5) and the convective fluxes are defined as: f =          ρu ρu 2 + P ρuv ρuw u(ρE + P)          , g =          ρv ρvu ρv 2 + P ρvw v(ρE + P)          , h =          ρw ρwu ρwv ρw 2 + P w(ρE + P)          (3.6) Here u, v and w are the Cartesian velocity components, P is the pressure and E is the total energy. The specific total energy E can be expressed in terms of the specific internal energy e and kinetic energies as: E = e + 1 2 u 2 + v 2 + w 2 (3.7) It is sometimes useful to recast the energy equation in terms of enthalpy. The specific total enthalpy is given by: H = γ γ -1 P ρ + 1 2 u 2 + v 2 + w 2 (3.8) Also, the specific total energy can be written as: E = 1 γ -1 P ρ + 1 2 u 2 + v 2 + w 2 (3.9) From the Equations of 3.8 and 3.9, E = H -P ρ is obtained. To close the system of equations the pressure P must be related to the state vector W. This relation depends on the model used to describe the thermodynamic properties of the gas. The difficulty with this homogeneous approach is to specify an equation of state (EOS) that covers all possible fluid states: pure phases (incompressible region) and two-phase mixture (compressible region). For the pure phases, the convex stiffened gas EOS [START_REF] Metayer | Elaborating equations of state of a liquid and its vapor for two-phase flow models[END_REF]] is used: P(ρ, e) = (γ -1)ρ(e -q) -γP ∞ (3.10) P(ρ, T) = ρ(γ -1)C v T -P ∞ (3.11) T(ρ, h) = h -q C p (3.12) where γ = C p /C v is the polytropic coefficient, C p and C v are thermal capacities,h the enthalpy, q the energy of the fluid at a given reference state and P ∞ is a constant reference pressure. The speed of sound c is given by: c 2 = γ P + P ∞ ρ = (γ -1)C p T (3.13) In terms of computational methods, the application of a compressible formulation to simulate low speed cavitating flows results in poor convergence and erroneous calculations. To achieve this goal, a preconditioned method is necessary. The preconditioning matrix proposed by Turkel [Guillard andViozat, 1999] [Turkel, 1987] is used in this research (see Appendix A). For the two-phase mixture, a sinusoidal barotropic law [START_REF] Delannoy | Two phase flow approach in unsteady cavitation modelling[END_REF]] is applied: P(ρ, α) = P vap + ρ sat l -ρ sat v 2 c 2 min Arcsin(A(1 -2α)) (3.14) T(ρ, h) = h l -q l C p l = h v -q v C p v = h -q(α) C p (α) (3.15) This law is characterized by its maximum slope 1/c 2 min . The quantity c min is an adjustable parameter of the model, which can be interpreted as the minimum speed of sound in the mixture. With this barotropic law, there is no coupling with the temperature and the cavitation phenomenon is assumed to be isothermal. In the original approach, pure phases are considered as incompressible and the speed of sound is infinite in each phase. In order to join compressible pure phases, a constant A, close to 1, is introduced to avoid infinite value of speed of sound. The speed of sound can be computed by: c 2 = ∂P ∂ρ s = ∂P ∂ρ T = Ac 2 min 1 -A 2 (1 -2α) 2 (3.16) NSMB A four-equation model Consider the five-equation Kapila model [START_REF] Kapila | Two-phase modeling of deflagration-to-detonation transition in granular materials: Reduced equations[END_REF] by assuming the liquid is at its saturation state, a four-equation model is obtained. The model consists of three conservation laws for mixture quantities as in Equation (3.4) and an additional equation for the void fraction α. The void fraction equation can be expressed as: ∂α ∂t + u ∂α ∂x + v ∂α ∂ y + w ∂α ∂z = K ∂u ∂x + K ∂v ∂y + K ∂w ∂z (3.17) K =   ρ l c 2 l -ρ v c 2 v ρ l c 2 l 1-α + ρ v c 2 v α   (3.18) The term K involves the speed of sound of pure phases and it reflects the effects of change in volume of each phase. To compute the pressure and the temperature, the convex stiffened gas EOS is used for the pure phases as presented above for the HEM model. For the two-phase mixture, an expression for the pressure and the temperature can be deduced from the thermal and mechanical equilibrium assumption [START_REF] Saurel | Modelling phase transition in metastable liquids: application to cavitating and flashing flows[END_REF] on the basis of the stiffened gas EOS. These expressions are available in all possible fluid states along with the function of the void fraction and the mass fraction of gas Y = αρ v /ρ: P ρ, e, α, Y = γ (α) -1 ρ (e -q (Y )) -γ (α) P ∞ (α) (3.19) 1 γ (α) -1 = α γ v -1 + 1 -α γ l -1 (3.20) q (Y ) = Y q v + (1 -Y ) q l (3.21) P ∞ (α) = γ (α) -1 (α) α γ v γ v -1 P v ∞ + (1 -α) γ l γ l -1 P l ∞ (3.22) By assuming the thermal equilibrium between phases, the mixture temperature is expressed as: T ρ, h, Y = h -q (Y ) C p (Y ) with C p (Y ) = Y C p v + (1 -Y ) C p l (3.23) Without mass transfer, the propagation of acoustic waves follows the Wood or Wallis speed of sound. This speed c wall is is expressed as a weighted harmonic mean of speeds of sound of each phase: 1 ρ c 2 wall is = α ρ v c 2 v + 1 -α ρ l c 2 l (3.24) Numerics The numerical simulations are carried out using an explicit time integration and based on a finite-volume discretization. The convective flux through the cell interface is computed with a HLLC scheme [START_REF] Batten | On the choice of wavespeeds for the hllc riemann solver[END_REF][START_REF] Toro | Restoration of the contact surface in the hll-riemann solver[END_REF]. The method considers two averaged intermediate states U * L and U * R separated by the contact wave of speed S M . The numerical flux Φ i,l at cell interface l can be expressed as: Φ i,l (U L ,U R ) =                  G (U L ) n i,l if S L > 0 G U * L n i,l if S L ≤ 0 ≤ S M G U * R n i,l if S M ≤ 0 ≤ S R G (U R ) n i,l if S R > 0 (3.25) where S L and S R are referred to the speeds of the smallest and largest waves at the cell interface. The normal velocity component V n = V.n. The left (K = L) and right (K = R) states of the variables U * K , and corresponding fluxes G U * K , are defined by: U * K =             ρ * K ρu * K ρv * K ρw * K ρE * K α * K             = 1 S K -S M             ρ K S K -V n K ρu K S K -V n K + (P * -P K ) n i,l ρv K S K -V n K + (P * -P K ) n i,l ρw K S K -V n K + (P * -P K ) n i,l ρE K S K -V n K + P * S M -P K V n K α K S K -V n K             (3.26) G U * K n i,l =             ρ * K S M ρu * K S M + P * n i,l ρv * K S M + P * n i,l ρw * K S M + P * n i,l ρE * K S M + P * S M α * K S M             (3.27) where the pressure P * is given by: P * = P L + ρ L V n L -S L V n L -S M = P R + ρ R V n R -S R V n R -S M (3.28) And the contact-wave speed S M is defined by: S M = P R -P L + ρ L V n L S L -V n L -ρ R V n R S R -V n R ρ L S L -V n L -ρ R S R -V n R (3.29) 3.2. OPENFOAM The HLLC solver requires the estimates of wave speeds S L and S R in the Riemann problem. A direct and simple wave speed estimation is used: S L = Min V n L -c L , V n R -c R ; S R = Max V n L + c L , V n R + c R (3.30) The non-conservative term is discretized following the idea of Daude et al. [START_REF] Daude | Numerical experiments using a hllc-type scheme with ale formulation for compressible twophase flows five-equation models with phase transition[END_REF]. The integral term is approximated with the following relation: C i B (U) divV dS = Bi l∈∂C i ∂C i V.n i,l dl (3.31) where Bi is some average of B on cell C i . Here, Bi = B (U i ) is used. the cell interface value u i,l is expressed as: u i,l (U L ,U R ) =                  V L .n i,l if S L > 0 S L -V n L S L -S M S M if S L ≤ 0 ≤ S M S R -V n R S R -S M S M if S M ≤ 0 ≤ S R V R .n i,l if S R > 0 (3.32) Details on the numerical method implemented in the NSMB solver can be referred to the textbook from Blazek [START_REF] Blazek | Computational fluid dynamics: Principles and applications[END_REF] and the course from Goncalvès [START_REF] Da Silva | Résolution numérique des équations d'euler 1d[END_REF] OpenFOAM Open Unlike many other commercial CFD packages, OpenFOAM does not have a graphical user interface to help the user in preparing a case. In order to run an OpenFoam case, it involves typically the manual preparation of all the required input and mesh files. The minimum required set of files are system, constant and time directories as shown in Table 3.1. • system folder The system directory contains run-time control and solver numerics. The decomposePar file describes how the computational domains to be subdivided for multiple processors. The controlDict file contains general simulation settings such as the time step, duration and data saving interval. The fvSchemes file includes the definition of the numerical schemes implemented to discretize the equations and interpolate the solutions. The fvSolution files defines the parameters used to solve the flow equations and the residual tolerance. • constant folder The constant directory contains physical properties, turbulence modeling properties, advanced physics and so on. The polyhedral mesh information is placed in the polyMesh directory. • 0 folder The 0 directory contains several individual files for every relevant flow quantity including both the initial and boundary conditions. When a new timestep is computed and stored, it is written as a time directory in the case name directory. The time directories contains the solution and derived fields and are created by the solver according to the setting of saving frequency. Turbulence Closures Most cavitation phenomena involve turbulence and the turbulence-cavitation interaction is an under-known and documented phenomenon (due in particular to the difficulty of performing experimental measurements in cavitating flows). Compressibility effects on turbulence and the effects of the dispersed phase are also unknown. The numerical accuracy for turbulent cavitation depends on both cavitation and turbulence modeling. Thus, the choise of a turbulence modeling is an important issue for the simulation of cavitation. Direct numerical simulation (DNS) has the highest capability of resolving all turbulent scales. However, it requires a very fine grid resolution and therefore it is still pretty hard to be applied because of the high consuming of computer performance. Although the Large Eddy Simulation (LES) has already been implemented for the turbulent cavitating flows [START_REF] Wang | Large eddy simulation of a sheet/cloud cavitation on a {NACA0015} hydrofoil[END_REF][START_REF] Huang | Large eddy simulation of turbulent vortex-cavitation interactions in transient sheet/cloud cavitating flows[END_REF][START_REF] Gnanaskandan | Large eddy simulation of the transition from sheet to cloud cavitation over a wedge[END_REF], the usual codes are formulated in a Reynolds-averaged Navier-Stokes (RANS) to tensor turbulent closure model by a transport equation kε (Boussinesq hypothesis) considering the balabce between the computational effort and accuracy. This hypothesis suggests that the turbulent shear or Reynolds stresses could be replaced by the product of the mean velocity gradient and a "turbulent or eddy-viscosity", µ t . In this way, the stress tensor and heat flux vector of the set of transport equations contain additional terms due to the Reynolds stresses. The Reynolds stresses -ρu i u j (i, j = x, y, z) need to be modeled to close the system of equations. The Boussinesq's gradient transport hypothesis for turbulence closure by using the eddy viscosity concept as the following equation: -ρu i u j = µ t ∂u i ∂x j + ∂u j ∂x i - 2 3 δ i j µ t ∂u k ∂x k + ρk (3.33) The eddy viscosity µ t is not a fluid property, but a property that depends on the local turbulence structure. The variable k is the turbulent kinetic energy, defined as k = 1 2 u i u i . The second The Spalart-Allmaras model For a one equation turbulence model, the state vector contains six unknowns. The last variable is either ν or F = k 2 with corresponding source terms S ν or S F , ∂ ∂t (W) + ∂ ∂x ( f -f v ) + ∂ ∂y (g -g v ) + ∂ ∂z (h -h v ) = S (3.34) W = ρ, ρu, ρv, ρw, ρE, ν or F (3.35)          f = ρu, ρu 2 + p, ρuv, ρuw, u(ρE + p), u ν or uF , g = ρv, ρvu, ρv 2 + p, ρvw, v(ρE + p), v ν or vF , h = ρw, ρwu, ρwv, ρw 2 + p, w(ρE + p), w ν or wF . (3.36)            f v = 0, τ xx -ρu x u x , τ x y -ρu x u y , τ xz -ρu x u z , (τU) x -q x , D νx or D F x , g v = 0, τ yx -ρu y u x , τ yy -ρu y u y , τ yz -ρu y u z , (τU) y -q y , D νy or D F y , h v = 0, τ zx -ρu z u x , τ z y -ρu z u y , τ zz -ρu z u z , (τU) z -q z , D νz or D F z . (3.37)    S ν = (0, 0, 0, 0, 0, P ν -Φ ν) , S F = (0, 0, 0, 0, 0, P F -Φ F ) . (3.38) The Reynolds stresses and heat diffusion are calculated using the Boussinesq approximation. D denotes the different diffusion terms and Φ the destruction terms. A transport equation for the turbulent viscosity ν is assembled, using empiricism and arguments of dimensional analysis, Galilean invariance, and selective dependence on the molecular viscosity [START_REF] Spalart | A one-equation turbulence model for aerodynamic flows[END_REF]: D ν Dt convection = c b1 S ν production + 1 σ [∇ • ((ν + ν)∇ ν) + c b2 (∇ ν) 2 ] di f f usion -c w1 f w (r)( ν d ) 2 dissi pation (3.39) The eddy viscosity is defined as: µ t = ρ ν f v1 ≡ ρν t (3.40) 3.3. TURBULENCE CLOSURES To ensure that ν equals K yu τ in the log layer, in the buffer layer and viscous sublayer, the damping function f v1 is defined as: f v1 = χ 3 χ 3 + c v1 3 (3.41) as function of the totally local variable λ: χ ≡ ν ν (3.42) The function S must be modified to maintain its log-layer behavior ( Ŝ = U τ /(K d)) all the way to the wall: S = S 1/2 + ν (kd) 2 f v2 (3.43) which is accomplished with the help of the function f v2 : f v2 = 1 - χ 1 + χ f v1 (3.44) The destruction term should vanish in the outer region of the boundary layer. Spalart-Allmaras proposed the function: f w (r) = g[ 1 + c 6 w3 g 6 + c 6 w3 ] 1/6 (3.45) with the argument r: r = ν (kd) 2 S (3.46) Both r and f w equal 1 in the log layer, and decrease in the outer region. g = r + c w2 (r 6 -r) (3.47) the function g is merely a limiter that prevents large values of f w . The constants of the Spalart-Allmaras model are: c b1 = 0.1355, c b2 = 0.622, c w2 = 0.3, c v1 = 7.1, σ = 2 3 , c w1 = c b1 k 2 + (1 + c b2 ) σ , c w3 = 2 (3.48) The kε model Two-equation turbulence models are widely used, as they offer a good compromise between numerical effort and computational accuracy. The kε model requires the solution of transport equations for the turbulent kinetic energy k and the turbulent dissipation rate ε, the same for kω models. This basis model is typically a "high Reynolds number" model. To take into account for the interaction between turbulence and fluid viscosity, many different low-Reynolds number versions have been implemented. These versions differ in the form of the source terms, in the surface boundary conditions imposed, in the values of closure coefficients and also in the form of the damping functions. They both solve an equation for the isotropic component of the turbulent dissipation. The proposal to use this modified dissipation variable is due to Jones and Launder [START_REF] Jones | The prediction of laminarization with a two-equation model of turbulence[END_REF] who cited decisive computational advantages. The model uses the following transport equations: Turbulent kinetic energy: ∂ρk ∂t + ∂ρU j k ∂x j = ∂ ∂x j (µ + µ t σ k ) ∂k ∂x j + τ i j ∂U j ∂x j -ρε (3.49) Dissipation rate: ∂ρε ∂t + ∂ρU j ε ∂x j = ∂ ∂x j (µ + µ t σ ε ) ∂ε ∂x j + ε k C ε1 τ i j ∂U j ∂x j -C ε2 ρε (3.50) The eddy viscosity is specified as: µ t = ρC µ k 2 ε (3.51) relating the variables k and ε via a dimensionless constant C µ . The constants of the model are: C µ = 0.09, C ε1 = 1.44, C ε2 = 1.92, σ k = 1.0, σ ε = 1.3 (3.52) The Menter kω SST model The k-ω model is also a two-equation turbulence model. Instead of using the turbulent disspiation rate ε as the second transported variable, it uses the specific turbulent dissipation rate ω = ε k . The standard Wilcox [START_REF] Wilcox | Reassessment of the scale-determining equation for advanced turbulence models[END_REF] kω model is extremely accurate and robust in the near-wall region in comparison to the more commonly used k-ε models, but it suffers from high dependency on the chosen inlet freestream turbulence properties. Menter [START_REF] Menter | Influence of freestream values on kω turbulence model predictions[END_REF][START_REF] Menter | Zonal two equation kω turbulence models for aerodynamic flows[END_REF] D ρ k Dt = τ i j ∂U j ∂x j -β * ρ k ω + ∂ ∂x j (µ + σ k µ t ) ∂ k ∂x j (3.53) Specific dissipation rate: D ρ ω Dt +U j = γ ν t τ i j ∂U j ∂x j -β ρ ω2 + ∂ ∂x j (µ + σ ω µ t ) ∂ ω ∂x j + 2(1 -F 1 ) ρσ ω 2 1 ω ∂ k ∂x j ∂ ω ∂x j (3.54) with ν t = k/ ω. The new constants are generated using the relations: φ = F 1 φ 1 + (1 -F 1 )φ 2 (3.55) where φ 1 and φ 2 represent constants in the original Wilcox model and transformed kε respectively. The φ 1 constants (Wilcox kω) are: σ k1 = 0.5, σ ω 1 = 0.5, β 1 = 0.0750 β * = 0.09, κ = 0.41, γ 1 = β 1 /β * -σ ω 1 κ 2 / β * The φ 2 constants (standard Launder-Sharma kε) are: σ k2 = 1, σ ω 2 = 0.856, β 2 = 0.0828 β * = 0.09, κ = 0.41, γ 2 = β 2 /β * -σ ω 2 κ 2 / β * The other definitions are given by where y is the distance to the closest wall and CD kω is the positive part of the cross-diffusion term in the blended specific turbulence dissipation equation: F 1 = CD kω = max 2 ρσ ω 2 1 ω ∂ k ∂x j ∂ ω ∂x j ; 10 -20 (3.58) The SST model is identical to the BSL model except the set of constants φ 1 and the definition of the eddy viscosity. The new constants (SST φ 1 ) are: σ k1 = 0.85, σ ω 1 = 0.5, β 1 = 0.0750, a 1 = 0.31 β * = 0.09, κ = 0.41, γ 1 = β 1 /β * -σ ω 1 κ 2 / β * In this case the eddy viscosity is defined as: ν t = a 1 k max(a 1 ω; |Ω|F 2 (3.59) |Ω| is the norm of the mean vorticity vector but the norm of the rate of strain tensor is now preferred [START_REF] Menter | Ten years of industrial experience with the sst turbulence model[END_REF]. F 2 is given by F 2 = tanh(ar g 2 2 ) ar g 2 = max 2 k 0.009 ωy ; 500ν y 2 ω (3.60) Menter also recommends to limit the production in the turbulent kinetic energy equation to ten times the dissipation to avoid excessive turbulence production, e.g. near stagnation points. However, the standard eddy-viscosity models based on the Boussinesq hypothesis tend to overpredict eddy-viscosity that reduce the effect of re-entrant jet and two-phase structure shedding [START_REF] Sorgüven | Modified kω model for simulation of cavitating flows[END_REF][START_REF] Li | A modified sst kω turbulence model to predict the steady and unsteady sheet cavitation on 2d and 3d hydrofoils[END_REF]. These turbulence models are inadequate to correctly predict the dynamics of cavitation bubbles. Several solutions have been proposed and tested to reduce the eddy viscosity and improve the behavior of turbulence models. Reboud [START_REF] Reboud | Two phase flow structure of cavitation: experiment and modeling of unsteady effects[END_REF]] proposed an arbitrary modification by introducing an eddy viscosity limiter assigned as a function of density, f (ρ), instead of using the mixture density directly in the computation of the turbulent viscosity for the kε turbulence model. µ t = f ρ C µ k 2 ε with f ρ = ρ v + ρ m -ρ v n ρ l -ρ v n-1 (3.61) where n is a parameter set to 10. The density function f (ρ) will be equal to ρ l and ρ v in the regions with pure liquid and vapor, but decreases rapidly in the region with a mixture of liquid and vapor. A filter-based method (FBM) which combines the filter concept and the RANS model was investigated [Wu et al., 2005] [Tseng andWang, 2014] by imposing an independent filter scale, usually the grid size, in the computation of the eddy viscosity. Once the turbulence length scale is larger than the filter size, the eddy viscosity can be reduced by a linear filter function. These corrections have shown some success, but do not take into account the dynamics of small scales [START_REF] Coutier-Delgosha | Numerical simulation of the unsteady behaviour of cavitating flows[END_REF][START_REF] Goncalvès | Unsteady simulation of cavitating flows in venturi[END_REF][START_REF] Goncalvès | Numerical study of unsteady turbulent cavitating flows[END_REF]. The interplay between turbulence and cavitation regarding the unsteadiness and the structure of the flow is complex and not well understood. Moreover, there are less studies about the influence of the turbulence models on cavitating flow. In this study, the Reboud correction is implemented into three different turbulence models and simulated with different cavitation models. The end goal is to provide an insight into the interaction between the turbulence and cavitation models. to solve these test cases. Interface movement in a uniform pressure and velocity flow A discontinuity of volume fraction movement between two fluids in a uniform pressure and velocity flow at 100 m/s is considered. The discontinuity separates two nearly pure fluids from each other and is initially located at x = 0.5 m in a one meter length tube. There is liquid water in the left chamber and air in the right chamber. The uniform pressure is set equal to P = 10 5 Pa. The fluid properties and initial condition for this test are given in Table 4.1. WATER-AIR MIXTURE SHOCK TUBE Water-air mixture shock tube This test case, as proposed in [START_REF] Ansari | Numerical simulation of compressible two-phase flow using a diffuse interface method[END_REF], considers a one meter long shock tube containing two chambers which involves a discontinuity of the volume fraction at the location of x = 0.7 m. Each chamber contains a nearly pure fluid where the left chamber is filled with high-pressure fluids and the right one is filled with low-pressure fluids. The initial velocity is equal to 0 m/s. The fluid properties and initial condition for this test are given in Table 4.2. Computations have been performed with a mesh of 1000 cells and with a time step ∆t = 10 -7 s. in [START_REF] Saurel | Simple and efficient relaxation methods for interfaces separating compressible fluids, cavitating flows and shocks in multiphase mixtures[END_REF]. In addition, the three-equation model was not able to predict well the phenomena. Water-air mixture expansion tube |u| = 2 m/s An expansion tube problem is considered with an initial velocity discontinuity located at the middle of the tube. This test consists in a one meter long tube filled with liquid water at atmospheric pressure and with density ρ l = 1150 kg/m 3 . A weak volume fraction of vapor α = 0.01 is initially added to the liquid. The initial discontinuity of velocity is set at 0.5 m, the left velocity is -2 m/s and the right velocity is 2 m/s. The solution involves two expansion waves. As gas is present, the pressure cannot become negative. To maintain positive pressure, the gas volume fraction increases due to the gas mechanical expansion and create a pocket. Liquid water is expanded until the saturation pressure is reached and then evaporation appears and quite small amount of vapor is created. The parameters of the stiffened gas EOS and saturation values for densities are given in Table 4.3. The quantities have been evaluated with a saturation table at the reference temperature. The vapor pressure P vap = 51000 Pa. The solution obtained with the three-equation model is presented at time t = 3.2 ms in Figure 4.5. Results are compared with the two-fluid solution computed in [START_REF] Zein | Modeling phase transition for compressible two-phase flows applied to metastable liquids[END_REF]. The mesh contains 1000 cells. The time step is set to ∆t = 10 -8 s. The approximate HLLC Riemann solver and the four-equation model were not able to provide a solution and the Jameson-Schmidt-Turkel scheme [START_REF] Jameson | Numerical solution of the Euler equations by finite volume methods using Runge-Kutta time stepping schemes[END_REF] is used instead. Computation results of the void fraction and pressure profiles show large discrepancy with the two-fluid solution. |u| = 100 m/s The same conditions are used except regarding velocities which are set to u = -100 m/s on the left, and u = 100 m/s on the right. This case is stiffer than the previous one because of the high value of the initial velocity and evaporation is much more intense resulting in a large cavitation pocket where the gas volume fraction is close to 1. Computations are performed on a 1000-cell mesh with the time step set to ∆t = 10 -8 s. The result obtained with the three-equation model is presented at time t = 1.5 ms in Figure 4.6. The void fraction profile is in good agreement with the two-fluid solution whereas the pressure simulated by the three-equation model is not able to capture the pressure drop inside the cavitation pocket. Again, the approximate HLLC Riemann solver and the four-equation model failed to solve this case. Water-air shock bubble interaction A cylindrical air bubble with an initial diameter D 0 = 6 mm is immersed in a water pool. Due to the symmetry of the problem the calculations are performed in a half-domain above the axis. The center of the bubble is located at (9, 0) mm in the computational domain of size 24 × 12 mm. The bubble is collapsed by a normal wave moving at Mach 1.72, initially located at abscissa x sh = 4 mm. The schematic diagram of the test case is given in Figure 4.7. The initial and post-shock conditions are shown in Table 4.4. The boundary conditions are the following: the top and bottom boundaries are assumed to be a wall and a symmetry axis, respectively. The left and right sides are assumed to be non-reflecting. Simulations are performed using an uniform mesh composed by 1200 × 600 nodes and a time step ∆t = 10 -10 s. The number of nodes for a bubble diameter is 300. The time evolution of the density gradient modulus (Schlieren-type representation) is plotted in Figure 4.8 from time t = 0.6 µs to t = 2.9 µs. After the water shock wave has collided with the bubble, a strong rarefaction wave is reflected backwards from the interface, and a weak shock wave is transmitted inside of the bubble (time t = 1.1 µs). At time t = 1.7 µs the incident water shock has traversed almost the full cavity width. The interaction between this shock and the expansion waves originating at the bubble surface has resulted in significant weakening and curvature of the incident shock. The shock inside the bubble propagates more slowly. Due to the pressure difference between both sides, the bubble is asymmetrically contracted and spreads laterally in the process. This change in shape is driven by vorticity generated at the edge of the bubble due to the passage of the wave which induces a jet of water along the axis of flow Summary To sum up, the validation of numerical method implemented in the NSMB solver for capturing the phenomenon of cavitation is illustrated in this chapter. There are four different test cases investigated and discussed with both the three-equation and four-equation models coupled with the HLLC scheme. The reuslts obtained from the previous test cases indicate that the implementation of these two cavitation models unfortunately could not be the cure-all and be generalized for all the test cases. In other words, it infers that there is the numerical instability for the implementation in the NSMB solver. Since the NSMB is a huge solver, when the two cavitation models are implemented into the solver, there are about 20 % (one thousand) of subroutines which require to be modified or be coded. Therefore, it remains difficuilities and challenges for the modelling based on the viewpoint of the CFD. However, in order to achieve the academic goal of this study, turbulence and cavitation, another free open source software, OpenFOAM, is adopted. With the built-in solver of the OpenFOAM, interphaseChangeFoam, the phenomenon of cavitation will be studied with the 4°Venturi geometry and will be presented in the next chapter. Venturi 2D Experimental conditions The Venturi type test section of the CREMHyG cavitation small tunnel was sized and designed to Mesh and computational set-up The grid is a H-type topology. It contains 251 nodes in the flow direction and 81 nodes in the orthogonal direction. A special contraction on the mesh is applied in the main flow direction just after the throat to better simulate the two-phase flow area, as illustrated in Figure 5.3. Various computations were performed by varying the cavitation model and the turbulence model. An overview that includes all test cases of the Venturi 2D case is shown in The empirical values of the three cavitation models in OpenFOAM solver are specified in C c = 10 C v = 8000 Merkle U ∞ = 10.8(m/s) t ∞ = 0.023(s) C c = 80 C v = 0.001 SchnerrSauer n = 1.6 × 10 13 (m -3 ) dnuc = 2 × 10 -6 (m) C c = 1 C v = 1 Table 5.3: Empirical values of the cavitation models Results for different turbulence models The calculations are done by using the three cavitation models that are implemented in the solver interPhaseChangeFoam of OpenFOAM with three different turbulence models. The Reboud correction is applied to these turbulence models with the exponent value of n = 10. All numerical values are obtained by a time-averaged statistical treatment over a simulation time of 5 s. Figure 5.9 illustrates the evolution of the void ratio and longitudinal velocity profiles for the numerical and experimental results from station 1 to 5 with Merkle cavitation model. At station 1, close to the throat, the cavity thickness is similar by all simulations and is underestimated. Computations with the KEReboud and KWReboud turbulence models over-estimate the maximum value of the void ratio: the discrepancy with the experimental value is around 7% and 10%. On the other hand, computation with the SAReboud turbulence model under-estimates the maximum value of the void ratio (around 7%). At station 2, all computations capture well the cavity thickness with an increase of it from station 1. However, computations with the KEReboud and SAReboud turbulence models under-estimate the maximum value of the void ratio. The discrepancy with the experimental value is about 10%. For the SAReboud turbulence model, the void ratio value at the wall is extremely low in relation to the experimental value. On the contrary, the computation with the KWReboud turbulence model is over-predict the the maximum value of the void ratio (around 4%). For the velocity profiles, at station 1 and 2, all computations over-estimate the maximum value of the longitudinal velocity. The re-entrant jet phenomena are observed with all turbulence models in station 1 and the KEReboud and SAReboud turbulence models at station 2. From station 3, the re-entrant jet becomes noticeable from the velocity measurement. The KEReboud and SAReboud turbulence models reproduce well the effect of recirculating behavior with a re-entrant jet extending through half the sheet thickness at station 3, 4 and 5. However, simulation with the KWReboud turbulence model does not present the thickness of the recirculating area and there is no re-entrant jet. The computation with the KEReboud turbulence model captures better the intensity of the recirculating zone near the wall at station 3 and 4. At station 5, both the KEReboud and SAReboud turbulence models have the similar intensity prediction. As for the void ration profiles, from station 3 to 5, computations with the KEReboud and SAReboud turbulence models provide a better prediction of the void ratio values and the thickness of the cavitation compared to the KWReboud one. In addition, the wall value of the void ratio given by the SAReboud turbulence model at station 3 and 4 has good agreement with the experimental value. The computation with the KWReboud turbulence model largely overestimates the void ratio value. The dimensionless wall pressure profiles are plotted in Figure 5.10 versus the distance xx inl et . For all computations, the pressure remains at an almost constant value P vap in the cavity. Computations with the KEReboud and SAReboud turbulence models show that the recompression starts from station 4 and re-compress slowly afterward in comparison with the experimental data. On the other hand, the KWReboud turbulence model keeps the pressure equal to the vaporization pressure far downstream. The re-compression is lower for all simulation downstream station 5 in relation to the experimental data. The Root Mean Square (RMS) wall pressure fluctuations are plotted in Figure 5.11 versus the distance xx inl et . All models predict different behavior for the pressure fluctuations. The .13 illustrates the evolution of the void ratio and longitudinal velocity profiles for the numerical and experimental results from station 1 to 5 with SchnerrSauer cavitation model. At station 1, inside the attached cavity sheet, all computations capture the similar cavity thickness but is under-predicted. Computations with the KEREboud and KWReboud turbulence models over-estimate the maximum value of the void ration: the discrepancy with the experimental value is 6% and 10%. At station 2, all simulations present well the cavity thickness with a correct estimation of it. Computations with the KEReboud and SAReboud turbulence models under-estimate the maximum value of the void ratio (around 10%) whereas the KWReboud turbulence model over-estimate it (around 4%). For the velocity profiles, at station 1 and 2, all computations over-predict the maximum value of the longitudinal velocity. Moreover, the re-entrant jet phenomena are obtained with all turbulence models which are not observed in the experimental value and the effect is even stronger with the KWReboud turbulence model. From station 3 to 5, the re-entrant jet becomes noticeable from the velocity measurement. The KEReboud and SAReboud turbulence models capture well the effect of recirculating behavior with a re-entrant jet except that the thickness of the recirculating area is over-estimated. In addition, the intensity of the recirculating zone captured by the KEReboud turbulence model is in good agreement with the experiment whereas it is under-estimated by the SAReboud turbulence model. The KWReboud turbulence model is also capable of predicting the thickness of the recirculating area but largely over-estimate the intensity of the recirculating zone. As for the void ratio profiles, at station 3, all computations provide a better prediction of the thickness of the cavity sheet as compared to the experimental data. The void ratio values are over-predicted by all computations especially for the KWReboud turbulence model. At station 4 and 5, the void ratio values and thickness of the cavitation are over-estimated by all computations. For the SAReboud turbulence model, the void ratio value at the wall is in better agreement with the experimental value. The dimensionless wall pressure profiles are plotted in Figure 5.14 versus the distance xx inl et . For all computations, the pressure remains at an almost constant value P vap in the cavity. Computations with the KEReboud and SAReboud turbulence models show that the re-compression starts from station 4 and re-compress slowly after in comparison with the experimental data. On the other hand, the KWReboud turbulence model keeps the pressure equal to the vaporization pressure far downstream. The re-compression is lower for all simulation downstream station 5 in relation to the experimental data. Figure 5.17 illustrates the evolution of the void ratio and longitudinal velocity profiles for the numerical and experimental results from station 1 to 5 with the kε turbulence model with the Reboud correction. At station 1, the similar cavity thickness is estimated by all models and is under-predicted. Computations with the Merkle and SchnerrSauer cavitation models over-estimate the maximum value of the void ratio (around 6%) wheres the Kunz cavitation model under-estimate it (around 6%). At station 2, all models estimate well the cavity thickness but under-estimate the void ratio composition. At station 3, all models estimate still well the cavity thickness but over-estimate the void ratio composition. At station 4 and 5, all models over-estimate both the cavity thickness and the void ratio composition. For the velocity profiles, at station 1 and 2, all models present the recirculating area which is not observed in the experiment. Further downstream the re-entrant jet phenomenon is well observed in the experiment. From At station 4 and 5, all computations over-predict the cavity thickness and composition. For the velocity profiles, at station 1 and 2, computations with all cavitation models present the reentrant phenomenon which is not observed in the experiment. At station 3, 4 and 5, all cavitation models capture the re-entrant phenomenon but the recirculating area and the intensity of the recirculating zone near the wall are under-estimated. The dimensionless wall pressure profiles are plotted in Figure 5.26 versus the distance x - Venturi 3D Computations of the 3D Venturi geometry are performed with Kunz cavitation model with the KEReboud and KWReboud turbulence models. The 3D mesh was built by extruding the 2D mesh with 81 nodes in the cross direction z. As the test section is a square, the same grid evolution was applied in the y and z directions. A view of the mesh is presented in 5. Comparisons between 2D and 3D simulations concern time-averaged quantities extracted on the mid-span plan. Figure 5.30 presents the evolution of the void ratio and longitudinal velocity profiles for the numerical and experimental results from station 1 to 5 with Kunz cavitation model. At station 1, close to the throat, the cavity thickness obtained by all simulations is underestimated. The results of 2D and 3D predicted by the KWREboud turbulence model are almost the same. At station 2, all computations capture well the cavity thickness with an increase of it from station 1. However, all computations under-estimate the maximum value of the void ratio. With the 3D effect, the void ratio value at the wall obtained by the KWReboud turbulence model is extremely low in relation to the experimental value. For the velocity profiles, at station 1 and 2, all computations over-estimate the maximum value of the longitudinal velocity. In addition, there shows the re-entrant jet phenomena with the KEReboud turbulence model which is not observed in the experiment. For station 3, 4 and 5, the re-entrant jet becomes noticeable from the velocity measurement. Simulation with the KWReboud turbulence model in 2D geometry does not present the thickness of the recirculating area at station 3 and the thickness of the recirculating area is under-estimated. On the contrary, the computation in 3D geometry gives better prediction in it. Compare to the KEReboud turbulence model in 2D geometry, the computation with 3D geometry over-predicts the thickness of the recirculating area. The intensity of the recirculating zone near the wall is under-estimated by all calculations no matter in 2D or 3D. Regarding the void ratio profiles, an over-estimation for both the void ratio values and the thickness of the cavitation is observed by all computations. The result obtained with the KWReboud turbulence model in 3D geometry is in better agreement with the experimental data. The dimensionless wall pressure profiles are plotted in Figure 5.31 versus the distance xx inl et . It showslarge discrepancy between the 2D and the 3D computations. The KWReboud turbulence model on 3D geometry presents an almost constant value of pressure in the cavity, and the re-compression is well captured downstream station 5 compared to the experimental data. The KEReboud turbulence model on 3D geometry re-compresses slowly and has an underestimation of the pressure downstream the cavity. The Root Mean Square (RMS) wall pressure fluctuations are plotted in Figure 5.32 versus the distance xx inl et . For the 2D computations, the peak of fluctuations is obvious although the peak location and amplitude are higher and lower for calculations with the KEReboud and the KWReboud turbulence models in comparison with the experimental data. In addition, the decrease of the fluctuations level in the wake of the cavitation sheet is better captured by the KWReboud turbulence model both on 2D and 3D geometry. This is not the case for the KEReboud turbulence model since the level of pressure fluctuations is largely overestimated in the wake. Figure 5.33 compares time-averaged profiles of the viscosity ratio µ t /µ, at the five station, obtained with Kunz cavitation model. At station 1 and 2 inside the cavity, the effect of 3D geometry is less important. Among the results obtained from the simulation which were compared to the experimental data, it is the Kunz's cavitation model coupled with the kω SST turbulence model that could have a better prediction for the 4°Venturi geometry. Nevertheless, the cavity length of all the models unfortunately was over-predicted by all the simulation. In addition, the 3D effect did not much improve the prediction either according to the obtained numerical results. The main reason for the cause might be that the incompressible solver was not suitable for this kind of internal geometry. The three-equation model was closed with a sinusoidal barotropic EOS. The mixture of stiffened gas EOS was applied to the four-equation model. The proposed models were validated through various inviscid test cases including the interface movement problem, water-air shock tube and expansion tube and shock-bubble interaction. The capability to obtain correct solutions of these test cases has been investigated. The reuslts obtained from the test cases indicate that the implementation of these two cavitation models unfortunately could not be the cure-all and be generalized for all the test cases. Although the validations showed the ability of models to simulate the cavitation development, the two models are still suffered from the issue of numerical instability. The main difference between these two models is that the three-equation model has the assumption of complete thermodynamic equilibrium between phases; therefore, it could explain the discrepancies existed for the test cases above. Since the implementation and validation in the NSMB solver had already taken too much time, in order to achieve the purposes of this study, which are the turbulence and cavitation, another free open source software, OpenFOAM, was adopted to perform the cavitating flows on the Venturi geometry. to reduce the eddy viscosity in order to capture the re-entrant jet dynamics. Results showed that the use of an eddy-viscosity limiter lets the model correctly simulate unsteady behaviors of the sheet, however large discrepancies occur between models and the effect of reduction is not strong enough. Generally the three cavitation models were able to reproduce the re-entrant jet phenomenon but the cavity length was over-predicted. This might be due to the calibration problem of the mass transfer term of the condensation rate and vaporization rate coefficient or the lack of thermodynamic coherence. Also, the impact on the value of the exponent n used in this correction is necessary to be investigated. Besides, the interPhaseChangeFoam is an incompressible solver which is less capable of solving the type of internal geometry. Hence, the speed of sound can be several orders of magnitude higher in the liquid phase than in the two-phase mixture. Thus for low-speed applications, the numerical method must be able to properly and efficiently simulate both incompressible and compressible flow areas. In terms of computational methods, the application of a compressible formulation to simulate low speed cavitating flows results in poor convergence and erroneous calculations. To achieve this goal, a preconditioned method is necessary. It is based on the modification of the derivative term by a premultiplication with a suitable preconditioning matrix. The physical acoustic waves are replaced by pseudo-acoustic modes that are much closer to the advective velocity, reducing the stiffness and enhancing the convergence. Therefore, the preconditioned method can provide both efficiency and accuracy over a wide range of Mach numbers. The preconditioned 1-D Euler equations with the primitive variables W = (P, u, s), where s is the entropy, can be expressed as: P -1 e ∂W ∂t + A e ∂W ∂x = 0 where, P e =     β 2 0 0 0 1 0 0 0 1     , P -1 e =     1 β 2 0 0 0 1 0 0 0 1     , A e =     u ρ c 2 0 1 ρ u 0 0 0 u     1 β 2 P t + uP x + ρ c 2 u x = 0 u t + uu x + 1 ρ P x = 0 s t + us x = 0 The equations can be rewritten in vector form as:     1 β 2 0 0 0 1 0 0 0 1         P u s     t +     u ρ c 2 0 1 ρ u 0 0 0 u         P u s     x = 0 with the eigenvalues of u and λ ± = 1 2 u(1 + β 2 ) ± (β 2 -1) 2 u 2 + 4β 2 c 2 . For the conservative variables w c = (ρ, ρu, ρE), the corresponding form is: P -1 c ∂w c ∂t + A c ∂w c ∂x = 0 where the preconditioning matrix P -1 c = (∂w c /∂W)P -1 e (∂W/∂w c ) = R -1 P -1 e R and A c = ∂F c ∂x = R -1 A e R Differential of the primitive variables with respect to the conservative variables Velocity du = 1 ρ d(ρu) - u ρ dρ therefore,          ∂u ∂ρ = -u ρ ∂u ∂ρu = 1 ρ ∂u ∂ρE = 0 Energy e = E - u 2 2 ⇒ de = dE -udu also, with d(ρE) = ρdE + Edρ and ρudu = ud(ρu) -u 2 dρ, we obtain that de = 1 ρ d(ρE) - E ρ dρ - u ρ d(ρu) + u 2 ρ dρ = 1 ρ d(ρE) - u ρ d(ρu) + u 2 -E ρ dρ therefore,          ∂e ∂ρ = u 2 -E ρ ∂e ∂ρu = -u ρ ∂e ∂ρE = 1 ρ Pressure d ( + hdρ = Adρ + BdP ⇒ dP = ρT B ds + h -A B dρ = ∂P ∂ρ s dρ + ∂P ∂s ρ ds ⇒    ∂P ∂ρ s = c 2 = h-A B ∂P ∂s ρ = ρT B dP = ρT B ds + h -A B dρ = ρT B ds + c 2 dρ ⇒ ds = B ρT dP - Bc 2 ρT dρ with dP = 1 B d(ρE) - u B d(ρu) + u 2 2 -A 1 B dρ therefore, ds = 1 ρT d(ρE) - u ρT d(ρu) + u 2 2 -A 1 ρT - Bc 2 ρT dρ ⇒            ∂s ∂ρ = u 2 2 -A -Bc 2 1 ρT = 1 ρT u 2 2 -h where ∂P ∂ρ s = c 2 = h-A B    =      (γ -1) u 2 2 -h + c 2 γ-1 -u(γ -1) γ -1 -u ρ 1 ρ 0 1 ρT u 2 2 -h -u ρT 1 ρT      R -1 = ∂w c ∂W =         =      1 c 2 0 - (γ-1)ρT c 2 u c 2 ρ - (γ-1)uρT c 2 h+ u 2 2 c 2 ρu - γ-1 c 2 ρT u 2 2 + h -c 2 γ-1      The preconditioning matrix P -1 c = I d + γ -1 c 2 1 β 2 -1      u 2 2 -h + c 2 γ-1 -u 1 u 2 2 -h + c 2 γ-1 u -u 2 u u 2 2 -h + c 2 γ-1 H -uH H      with the definition of total enthalpy H = h+ u 2 2 and c 2 γ-1 = h-A, we obtain that u 2 2 -h+ c 2 γ-1 = u 2 2 -A P -1 c = I d + 1 β 2 -1 1 h -A     u 2 2 -A -u 1 u 2 2 -A u -u 2 u u 2 2 -A H -uH H     B.3 Computational results All cases for the BSCW were run using three meshes, namely the coarse, medium and fine resolution multi-block structured meshes supplied by the workshop organizing committee. Different turbulence models, the Spalart-Allmaras model with Quadratic Constitutive Relation, 2013 version (SA QCR 2013) [START_REF] Mani | Predictions of a supersonic turbulent flow in a square duct[END_REF], the Menter Shear Stress Transport model (kw SST) [START_REF] Menter | Influence of freestream values on kω turbulence model predictions[END_REF] and the Chien k-epsilon model (kec) [START_REF] Chien | Predictions of channel and boundary-layer flows with a low-reynolds-number turbulence model[END_REF], were applied in the simulation. Figure 1 1 Figure 1.1: Phase diagram of water. Figure 1.2: Damage of vane by cavitation. Figure Figure 1.3: Traveling cavitation. Figure Figure 1.4: Sheet cavitation. Figure 1 . 5 : 15 Figure 1.5: Cloud cavitation. Figure 1.6: Vortex cavitation. From the thermodynamic point of view, two state variables are sufficient to represent the thermodynamic state of a fluid. The main relationships existing in the literature are : This assumption leads to a very simplified state law : ρ = ρ 0 and C p = C v = C which are the specific heats at constant pressure and constant volume respectively. This equality leads to the following relation between the internal energy and the CHAPTER 2. REVIEW OF CAVITATION MODELING temperature : de = CdT 27) where U V ,n = u.n with n = ∇α L |∇α L | The normal interfacial velocity, U I,n , is zero in steady calculation. This model is called Sharp Interfacial Dynamics Model (IDM). Figure 2 2 Figure2.1: Representation of a vapourous cavity[Senocak et Shyy, 2004] 1000 and c D = 0.44 otherwise (2.54) R e and R e P are the Reynolds number relative to the flow and the Reynolds number relative to the bubbles respectively. This is one of the first cavitation model to simulate the slip between phases. However, the evolution of this model by solving the energy conservation equations seems not yet to be achieved. b/ Saurel and Le Métayer model with seven equations (2003) e the Reynolds number based on the radius of the bubble : R e = U V -U L d ν L and P r L the Prandtl number of liquid : P r L = ν L a L In these equations, λ L is the thermal conductivity of liquid , C p V the specific heat of vapor at constant pressure, ∆t the iterative time step, ν L the kinematic viscosity of liquid and a L the thermal diffusivity of liquid. The momentum transfer is based on a term due to mass transfer and a drag term. The results are presented to a 3D cold water flow through a diaphragm. The calculations are compared with the test data from the case named SUPER MOBY DICK. This study makes it possible to bring a new modeling of the interfacial transfer for the cavitation based on those of ebullition. are implemented and tested with the interface movement and shock-bubble intercation for the NSMB solver. In addition, the transport equation based method for the void ratio including the source terms for vaporization and condensation in the free, open source software OpenFOAM (Open source Field Operation And Manipulation) are also presented for the Venturi geometry. C H A P T E R 3 NUMERICAL SOLVER T wo numerical codes used for the simulation of this study are briefly described in this chapter. The Navier-Stokes Multi-Block (NSMB) solver is a numerical software developed within an european consortium solving the finite volume Navier-Stokes equations. NSMB is a multi-block structured solver and parallelized able to solve the steady or unsteady Navier-Stokes equations in their compressible or incompressible version. Open source Field Operation And Manipulation (OpenFOAM) is a free, open source software for computational fluid dynamics (CFD). OpenFOAM is a Finite Volume Method (FVM) based numerical solver for solving systems of transient transport equations. Plenty of solvers available for a wide range of domains such as incompressible, compressible, multiphase, combustion, etc . . . 3.1 NSMB Navier-Stokes Multi-blocks (NSMB) was based on a structured multi-block Euler code (EULMB) developed at Swiss Federal Institute of Technology in Lausanne (EPFL, "École Polytechnique Fédérale de Lausanne") in 1989 with the support from the European Centre for Research and Advanced Training in Scientific Computation (CERFACS, "Centre Européen de Recherche et de Formation Avancée en Calcul Scientifique") and the Royal Institute of Technology (KTH). Originally developed by Jan Vos (CFS-Engineering, Lausanne) in 1989, NSMB was developed from 1992 to the end of 2003 in the NSMB consortium, which included several universities (EPFL, Lausanne, Switzerland; SERAM ("Société d'études et de recherches de l'École nationale supérieure d'arts et métiers"), Paris, France; Institute of Fluid Mechanics of Toulouse (IMFT, "Institut de Mécanique des Fluides de Toulouse"), Toulouse, France; KTH, Stockholm, Sweden), a research institution (CERFACS, Toulouse, France), and industrial partners EADS-France (Airbus CHAPTER 3. NUMERICAL SOLVER France and EADS Space Technologies), SAAB Military Aircraft Engineering and CFS. Since 2004, NSMB is further developed by the EPF-Lausanne, ETH-Zrich, ICUBE-Strasbourg, IMFT-Toulouse, Polytechnic University of Munich, the military University of Munich, CFS Engineering and RUAG Aerospace. In addition to these groups, NSMB is still used by Airbus France, EADS-ST and KTH. The NSMB solver is a code which is parallelized in MPI (Message Passing Interface) and solves the steady or unsteady Navier-Stokes equations in their compressible or incompressible version on multi block structured grids by means of finite volume method. It provides a wide range of numerical schemes both for spatial and temporal discretisation. There are for example the central schemes (2nd and 4th order with artificial dissipation) and upwind schemes (Roe, AUSM, Van Leer, Harten . . . ) available for the spatial discretisation. Within this solver turbulence can be treated in various ways: LES (Smagorinski, structure functions Lesieur et al, Ducros et al . . . ), turbulence models from zero equations (Baldwin-Lomax, Granville . . . ), one equation (Spalart-Allmaras and several variants), two equations linear models (kε, kω, SST . . . ), nonlinear models or explicit (EARSM) and the RSM model. For all these models, their RANS-LES (DES, DDES, IDDES, WMLES) hybrid variants have been implemented. The SAS model based on Menter kω SST has also been implemented. Chimera Methodologies (overlapping meshes) and source Field Operation And Manipulation (OpenFOAM) is a free, open source software for computational fluid dynamics (CFD). It is owned by the OpenFOAM Foundation and licenced under the GNU General Public Licence (GPL) that gives users the freedom to modify and redistribute the software and a guarantee of continued free use within the terms of the licence. The codes are written in C++ programming language in an object-oriented manner to solve ordinary differential equations (ODEs) and partial differential equations (PDEs). The correspondence between the implementation and the original equation is clear due to the high level programming. This feature makes users straightforward to modify or mimic the existing solvers. As a result, this provides OpenFOAM with good extensibility qualities. Another distinguishing feature of OpenFOAM is that it can be used in massively parallelism through domain decomposition method, where the computational domain is split into a number of subdomains, one for each processor. Each processor receives a separate distribution of the complied code to be run on each subdomain. For the communication between processors, the Message Passing Interface (MPI) is used. In addition, it provides with plenty of pre-and post-processing utilities for users to perform. OpenFOAM is a Finite Volume Method (FVM) based numerical solver for solving systems of transient transport equations. Regarding the finite-volume discretization, a variety of discretization schemes are implemented for the temporal, convection, diffusion and source terms in the transport equations. Meanwhile, there are plenty of solvers available for a wide range of domains such as incompressible, compressible, multiphases, combustion, etc . . . The above-mentioned advantages give OpenFOAM solvers a great capabilities and extensibility. 3.3. TURBULENCE CLOSURESThe solver interPhaseChangeFoam of OpenFOAM is selected to simulate the cavitation. It is a solver for two incompressible, isothermal immiscible fluids with phase-change (e.g. cavitation) and uses a volume-of-fluid (VOF) phase-fraction based interface capturing approach. The momentum and other fluid properties are of the "mixture" and a single momentum equation is solved. The set of phase-change models provided are designed to simulate cavitation but other mechanisms of phase-change are supported within this solver framework. The solver includes Kunz, Merkle and SchnerrSauer cavitation models. It uses the Pressure-Implicit with Splitting of Operators (PISO) algorithm to solve the Navier Stokes equations by first solving the momentum equations with pressure from the previous time step followed by solving the pressure equation for the new velocity field followed by velocity correction. Details on the numerical principles and specific implementation can be referred to the documentation of OpenFOAM (http://www.openfoam. com). CHAPTER 3. NUMERICAL SOLVER term on the right hand side affects only the normal stresses which equals to twice the turbulent kinetic energy. Three turbulence models, the one-equation Spalart-Allmaras model, the two-equation kε model and the Menter kω SST modle, are chosen for the study and briefly described in the following section. attacked this problem by first transforming the kε model into a kω type formulation before introducing a blending function dependent, among other things, on distance from the nearest wall. Compared to the original k -ω model, the differences are the values of the model constants and the presence of an additional (cross-diffusion) term (Menter discards a small additional diffusion term during the transformation). The two models are then combined by multiplying the original kω model by a function F 1 , the transformed model by (1 -F 1 ), and then adding. The result is a model which keeps the robust and accurate near-wall formulation of the original Wilcox kω model and improves freestream independence through use of the kε model in the outer part of the boundary layer. Futher to this baseline (BSL) model, Menter then added a shear-stress transport correction to form the kω model. The Menter kω SST model is known to be particularly capable of capturing flow fields featuring large separated shear layers. The equations governing the kω SST model are presented below. the validation of numerical method implemented in the NSMB solver for capturing the phenomenon of cavitation. Four test cases including the one-and twodimensional compressible two-phase flows with interface conditions are considered. Both the three-equation and four-equation model coupled with the HLLC scheme have been proposed Figure 4 . 1 : 41 Figure 4.1: Interface movement discontinuity problem. Void fraction and pressure profiles by 3-equation model (symbols) and the exact solution (solid line). Figure 4 4 Figure 4.2: Interface movement discontinuity problem. Void fraction and pressure profiles by 4-equation model (symbols) and the exact solution (solid line). Numerical solutions computed with the 3-equation and 4-equation model at 240 µs are shown in Figure 4.3 and 4.4 respectively. In this test case, strong pressure waves are propagated. The obtained result with the four-equation model are in close agreement with solutions presented Figure 4 4 Figure 4.4: Water-air mixture shock tube problem. Density, pressure, velocity and void fraction profiles by 4-equation model (symbols) and the exact solution (solid line). Figure 4 . 5 : 45 Figure 4.5: Water-air mixture expansion tube problem |u| = 2 m/s. Void fraction and pressure profiles by central scheme with 3-equation model (symbols) and 7-equation model (solid line). Figure 4 4 Figure 4.6: Water-air mixture expansion tube problem |u| = 100 m/s. Void fraction and pressure profiles by central scheme with 3-equation model (symbols) and 7-equation model (solid line). Figure 4 . 7 : 47 Figure 4.7: Initial situation for the shock bubble interaction D 0 = 0.006 m and M sh = 1.72. Figure 4 4 Figure 4.8: Water-air shock bubble interaction. Time evolution of the density gradient. Figure 4 4 Figure 4.9: Water-air shock bubble interaction. Time evolution of the pressure (in bar). Figure 4 . 4 Figure 4.10: Water-air shock bubble interaction. Time evolution of the axial velocity (in m/s). Figure 4 . 4 Figure 4.11: Water-air shock bubble interaction. Time evolution of the vertical velocity (in m/s). is investigated in this chapter for cavitating flow. First, the investigation is conducted on a 2D Venturi geometry with available experimental data tested in the cavitation tunnel of CREMHyG (Centre de Recherche et d'Essais de Machines Hydrauliques de Grenoble). Second, the 3D effect of the same geometry is considered. The simulation of cavitating flow is carried out by the free software OpenFOAM. The built-in solver, interPhaseChangeFoam, is used for the computation. For the Venturi case, three turbulence models, i.e. the Spalart-Allmaras, kε and kω SST models with the Reboud correction are considered together with three cavitation models of interPhaseChangeFoam. Figure 5.1: Schematic view of the Venturi profile. Figure 5 . 2 : 52 Figure 5.2: Photograph of the cavity. Figure 5 . 3 : 53 Figure 5.3: Enlargement of the mesh near the Venturi throat. Figure 5 .Figure 5 . 4 : 554 Figure 5.4 illustrates the different cavities where the time-averaged void ratio is plotted. All the cavitation and turbulence models except for Merkle and SchnerrSauer cavitation models coupled with KWReboud turbulence model show an attached cavity sheet with a large reentrant jet and with the presence of small clouds of mixture in the closure region of the sheet. The configuration of quasi-stable sheet is observed. Among them, only Kunz cavitation model coupled with the KWReboud turbulence and SchnerrSauer cavitation model coupled with the Figure 5 . 5 Figure 5.5 presents the evolution of the void ratio and longitudinal velocity profiles for the numerical and experimental results from station 1 to 5 with Kunz cavitation model. At stations 1 and 2, inside the attached cavity sheet, a relative strong effect of the vaporization phenomenon is clearly represented from the void ratio profile. The void ratio values of the first two stations are 0.9 and 0.95 near the wall according to experiments. At station 1, computations with the KEReboud and SAReboud turbulence model are close and with the discrepancy of about 15% compared to the experimental value. On the contrary, computation with the KWReboud turbulence model over-estimates the maximum value of the void ratio (around 10%). The numerical cavity thickness is under-estimated for all turbulence models. At station 2, the distribution is similar to that obtained for station 1, with an increase in the sheet thickness. Computations with the KEReboud and SAReboud turbulence models largely under-estimate the maximum value of the void ratio (around 20%), as observed for station 1. The computation with the KWReboud turbulence model is in better agreement with the experimental data. For the velocity profiles, at station 1 and 2, all computations over-estimate the maximum value of the longitudinal velocity. In addition, there shows the re-entrant jet phenomena with the KEReboud and SAReboud turbulence models which is not observed in the experiment.Further downstream, for stations 3, 4 and 5, the re-entrant is observed on the velocity measurement. The experiment observation indicates a recirculating behavior with a re-entrant jet extending roughly half the sheet thickness. However, the effect is not very evident especially for the computation with the KWReboud turbulence model. In addition, the thickness of the recirculating area is largely under-estimated by the KWReboud turbulence model at station 3 and 4. The intensity of the recirculating zone near the wall is under-estimated by all calculations.Regarding the void ratio profiles, an over-estimation for both the void ratio values and the thickness of the cavitation is observed by all computations. Computations with the KEReboud and SAREeboud turbulence models are similar. Both models have the similar sheet thickness and the void ratio values. For the KWReboud turbulence model, the void ratio value at the wall is in very good agreement with the experimental value although the maximum value of it and the thickness of the sheet are greatly over-estimated.The dimensionless wall pressure profiles are plotted in Figure5.6 versus the distance x-x inl et .The first five data are located inside the cavity (where the void ratio and velocity profiles are measured). Both the KEReboud and SAReboud turbulence models have a slow re-compression which results in an underestimation of the pressure downstream the cavity. The KWReboud turbulence model presents an almost constant value of pressure in the cavity, but the value keeps the same as the vaporization pressure further downstream compared to the experimental data.The Root Mean Square (RMS) wall pressure fluctuations are plotted in Figure5.7 versus the distance xx inl et . The pressure fluctuation is divided by the time-averaged pressure P av .Experimental data indicate an augmentation of pressure fluctuations at the end of the sheet cavity, with a peak located at station 5. The peak of pressure fluctuation predicted by the Figure 5 . 5 Figure 5.8 compares time-averaged profiles of the viscosity ratio µ t /µ, at the five station, obtained with Kunz cavitation model. The effect of the Reboud correction is not obvious for both the KEReboud and SAReboud turbulence models. The KWReboud turbulence model induces a large reduction of the ratio in the sheet at station 1 to 3. Figure 5 . 5 :Figure 5 . 6 :Figure 5 55565 Figure 5.5: Time-averaged void ratio (left) and velocity (right) profiles from station 1 to -Kunz model comparison Figure 5 .Figure 5 .Figure 5 . 555 Figure 5.9: Time-averaged void ratio (left) and velocity (right) profiles from station 1 to 5 -Merkle model comparison The Root Mean Square (RMS) wall pressure fluctuations are plotted in Figure 5.15 versus the distance xx inl et . Different behaviors for the pressure fluctuations are obtained by the turbulence models. The peak position predicted by the KWReboud turbulence model is present downstream station 5 and the amplitude of the fluctuation peak is underestimated. The pressure fluctuations simulated with both the KEReboud and SAReboud turbulence models are too high. Figure 5 .Figure 5 .Figure 5 .Figure 5 .Figure 5 . 55555 Figure 5.16 compares time-averaged profiles of the viscosity ratio µ t /µ, at the five station, obtained with SchnerrSauer cavitation model. Computational results with the KEReboud turbu- station 3 to 5, computations with the Merkle and SchnerrSauer cavitation models reproduce the recirculating area although over-estimate the thickness of it. The Kunz cavitation model does not capture the recirculating area. The dimensionless wall pressure profiles are plotted in Figure 5.18 versus the distance xx inl et . The Kunz cavitation model has a slow re-compression which results in an underestimation of the pressure downstream the cavity. Computations with the Merkle and SchnerrSauer cavitation models show that the re-compression starts from station 4 and re-compress slowly afterward in comparison with the experimental data. The Root Mean Square (RMS) wall pressure fluctuations are plotted in Figure 5.19 versus the distance xx inl et . The pressure fluctuation simulated with the Kunz cavitation model is too high. The Merkle and SchnerrSauer cavitation models provide fluctuations in better agreement with experimental data inside the cavity but not in the re-compression area. Figure 5 .Figure 5 .Figure 5 .Figure 5 .Figure 5 .Figure 5 . 555555 Figure 5.20 compares time-averaged profiles of the viscosity ratio µ t /µ, at the five station, obtained with the KEReboud turbulence model. With both the Merkle and SchnerrSauer cavitation models, the reduction effect is similar. Figure 5 . 5 Figure 5.24 compares time-averaged profiles of the viscosity ratio µ t /µ, at the five station, obtained with the KWReboud turbulence model. Computational results show that the reduction effect with the SchnerrSauer cavitation model is less evident. Figure 5 .Figure 5 .Figure 5 .Figure 5 .Figure 5 . 55555 Figure 5.21: Time-averaged void ratio (left) and velocity (right) profiles from station 1 to 5 -kω SST model with the Reboud correction comparison x inl et . The Kunz cavitation model has a slow re-compression which results in an underestimation of the pressure downstream the cavity. The Merkle and SchnerrSauer cavitation models give the similar results with an almost constant value of pressure in the cavity. The re-compression starts from station 4 and re-compress slowly afterward in comparison with the experimental data. The Root Mean Square (RMS) wall pressure fluctuations are plotted in Figure 5.27 versus the distance xx inl et . The pressure fluctuation simulated with the Kunz and Merkle cavitation models are too high. The SchnerrSauer cavitation model provide fluctuations in better agreement with experimental data inside the cavity but not in the re-compression area. Figure 5 . 5 Figure 5.28 compares time-averaged profiles of the viscosity ratio µ t /µ, at the five station, obtained with the SAReboud turbulence model. All the cavitation models give the similar results except at station 4 and the computed cavity is nearly steady. Figure 5 .Figure 5 .Figure 5 . 555 Figure 5.25: Time-averaged void ratio (left) and velocity (right) profiles from station 1 to 5 -Spalart-Allmaras model with the Reboud correction comparison Figure 5 . 5 Figure 5.29: View of the 3D mesh composed of 251 nodes in the flow direction and 81 nodes in each transversal direction. Figure 5 .Figure 5 .Figure 5 . 555 Figure 5.30: Time-averaged void ratio (left) and velocity (right) profiles from station 1 to 5 -Kunz model comparison C in this thesis deals with the study and implement of the three-equation and four-equation cavitation models which are developed in LEGI into the NSMB solver. A comparison of various cavitation models coupled with turbulence models by OpenFOAM on 2D and 3D Venturi geometry was proposed. The interPhaseChangeFoam solver was used to simulate the cavitation pocket by the formulation of void ratio transport equation cavitation models including Kunz, Merkle and SchnerrSauer models. Numerical results have been compared with experimental data concerning the time-averaged void ration and longitudinal velocity, wall pressure, RMS wall pressure fluctuations and turbulent eddy viscosity. For the turbulence closure, three models are considered: the one-equation Spalart-Allmaras model, the two-equation CHAPTER 6. CONCLUSIONS AND PERSPECTIVES kε model and the Menter kω SST model. The Reboud eddy-viscosity limiter is introduced Based on the results obtained in this work, suggestions for improvements and future work are proposed as follows. Regarding the four-equation cavitation model implemented in the NSMB solver, the mass transfer rate ṁ in the formulation of the void ration transport equation could be introduced to better modeling the cavitation phenomenon. Then coupled to the turbulence model for turbulent cavitating flow. In the meanwhile, further looking into the code of NSMB is necessary by fresh viewpoints. For OpenFOAM simulation, an investigation of the calibration of the condensation and vaporization constants appearing in the mass transfer formulation to improve the results for the Venturi case. Regarding the options of the turbulence model, the nonlinear and hybrid models could be implemented for better prediction of the structure of characterized by the incompressible region of the pure liquid and compressible region of the two-phase mixture and associated with large variations in the local Mach number, where M < 0.1 in the liquid phase and M > 1 in the mixture zone. is the Jacobian matrix of the convective fluxes.Expressions of the matrices are derived as follow. have the matrices of R and R -1 as follows, Figure B.1: (a) An isometric view of the BSCW (b) Cross-sectional view of the SC(2)-0414 airfoil (c) BSCW model mounted in TDT Figure B. 2 :Figure B. 5 : 25 Figure B.2: Case 1 (Mach 0.7, R e = 1.12 × 10 7 , AoA = 3°): Mean C p for unforced system data at 60% and 95% wing span with the SA QCR 2013 turbulence model. Table 2 2 .1. Table 2 2 .1: Class of models for cavitating flows Table 3 3 .1: Directory structure of an OpenFOAM case case name system controlDict decomposePar fvSchemes fvSolution constant physical properties polyMesh points cells faces boundary 0 BC and initial conditions time directories Table 4 4 4. VALIDATION CASES three-equation model are in good agreement with the previous investigations[START_REF] Saurel | Simple and efficient relaxation methods for interfaces separating compressible fluids, cavitating flows and shocks in multiphase mixtures[END_REF][START_REF] Ansari | Numerical simulation of compressible two-phase flow using a diffuse interface method[END_REF] and there is no oscillation in the solution. However, the volume fraction of gas computed by the four-equation model shows oscillation at the outlet. This problem might be the issue of the boundary condition . .1: Properties of air and water and initial condition for interface movement in a uniform pressure and velocity flow. 0 < x < 0.5 0.5 < x < 1 Air Water Air Water ρ (kg/m 3 ) 10 1000 10 1000 α 10 -6 1 -10 -6 1 -10 -6 10 -6 p ∞ (Pa) 0 6 × 10 8 0 6 × 10 8 γ 1.4 4.4 1.4 4.4 The numerical solution is plotted in Figure 4 .1 and 4.2 at time t = 2.79 ms and is compared to the exact one. A mesh contains 200 uniform cells is used. The results obtained with the CHAPTER Table 4 . 4 2: Properties of air and water and initial condition for the water-air shock tube. 0 < x < 0.7 0.7 < x < 1 Air Water Air Water ρ (kg/m 3 ) 1 1000 1 1000 P (Pa) 10 9 10 9 10 5 10 5 α 10 -6 1 -10 -6 1 -10 -6 10 -6 p ∞ (Pa) 0 6 × 10 8 0 6 × 10 8 γ 1.4 4.4 1.4 4.4 Table 4 . 4 3: Parameters of the stiffened gas EOS for water at T = 355 K. γ P ∞ (Pa) q (J/kg) C p (J/K kg) ρ sat (kg/m 3 ) Liquid 2.35 10 9 -0.1167 × 10 7 4267 1149.9 Vapor 1.43 0 0.2030 × 10 7 1487 0.31 Table 4 . 4 4: Properties of air and water and initial condition for the water-air shock tube. γ ρ (kg/m 3 ) u (m/s) v (m/s) P (Pa) P ∞ (Pa) Water 4.4 1000 0 0 1 ×10 5 6 ×10 8 Air 1.4 1 0 0 1 ×10 5 0 post-shock 4.4 1323.65 681.58 0 1.9 ×10 9 6 ×10 8 Table 5 5 .1. is applied at the upstream inflow and a pressure outlet condition is used at the outlet boundary for the computational set-up. The vaporization pressure P vap is set to 2300 Pa. The time step ∆t and the maximum Courant number are set to 10 -5 s and 1.0 respectively. Table 5 . 5 2: Boundary conditions, flow and turbulence properties of the Venturi tested cases Table 5 5 Cavitation model Kunz U ∞ = 10.8(m/s) t ∞ = 0.023(s) .3. Here, U ∞ is set to the freestream value, t ∞ represents a relaxation time, n is the bubble number density, dnuc is the nucleation site diameter and C c and C v are the condensation rate coefficient and vapourisation rate coefficient respectively. Case 1, BSCW Mean Cp Upper surface at 95% Wing Span B.3. COMPUTATIONAL RESULTS AePW-2 Case 1, BSCW Mean Cp AePW-2 Case 1, BSCW Mean Cp -1.5 Upper surface at 60% Wing Span -1.5 Lower surface at 60% Wing Span Coarse Grid Coarse Grid Medium Grid Medium Grid -1 Fine Grid -1 Fine Grid Experiment Experiment -0.5 -0.5 CP 0 CP 0 0.5 0.5 1 1 -0.2 1.5 0 0.2 0.4 0.6 0.8 1 1.2 -0.2 1.5 0 0.2 0.4 0.6 0.8 1 1.2 X/C X/C AePW-2 Case 1, BSCW Mean Cp AePW-2 -1.5 Upper surface at 95% Wing Span -1.5 Coarse Grid Coarse Grid Medium Grid Medium Grid -1 Fine Grid -1 Fine Grid -0.5 -0.5 CP 0 CP 0 0.5 0.5 1 1 -0.2 1.5 0 0.2 0.4 0.6 0.8 1 1.2 -0.2 1.5 0 0.2 0.4 0.6 0.8 1 1.2 X/C X/C 2.1. MODELING OF TWO-PHASE FLOWS 4.3. WATER-AIR MIXTURE EXPANSION TUBE ACKNOWLEDGEMENTS F irst of all, I would like to express my deepest gratitude and special thanks to my supervisor, Prof. Yannick Hoarau, who took time out to hear, guide and keep me on the correct path symmetry. When this water jet impacts the stationary water at the front of the bubble (time t = 2 µs), an intense blast wave also called water hammer shock [START_REF] Hawker | Interaction of a strong shockwave with a gas bubble in a liquid medium: a numerical study[END_REF] is formed generating a high-pressure zone. The blast front, which expands continuously, is highly asymmetric due to the high-speed water jet (time t = 2.4 µs). The rightward blast wave increases as a spherical wave. Both shocks lose strength as they advance, the rightward wave more so than its leftward twin. The interaction of the blast wave with the bubble fragments lead to high pressure levels (time t = 2.7 µs). Finally, at time t = 2.9 µs, the blast wave continues its expansion and the cavity its shrinkage. These results show a good agreement with previous numerical results [START_REF] Ball | Shock-induced collapse of a cylindrical air cavity in water: a free-lagrange simulation[END_REF][START_REF] Nourgaliev | Adaptive characteristics-based matching for compressible multifluid dynamics[END_REF][START_REF] Ozlem | A numerical study of shock-induced cavity collapse[END_REF]. The pressure evolution during the collapse is illustrated in Figure 4.9. During the impact of the water jet with the stationary water at the front of the bubble, a blast wave is generated leading to the pressure increase (time t = 2 µs). As previously described, the blast fronts are highly asymmetric. The rightward wave increases as a spherical wave and expands continuously in the radial direction ((time t = 2.4 µs and after). The shock intensity decreases during the propagation, especially for the rightward front. At time t = 2.4 µs, the more intense pressure peak is generated by the leftward front on the bubble axis. At time t = 2.7 µs, the interaction of the leftward blast wave with the bubble pieces leads to a very strong pressure peak, which is the most intense reached during the collapse. At time t = 2.9 µs, the low-pressure area inside the vortices core are well illustrated. Both blast wave fronts continue to expand. The time evolution of the axial velocity is plotted in Figure 4.10. The reflected rarefaction wave, resulting from the impact of the incident shock with the upstream bubble interaction, relaxes the pressure, which accelerates the flow and forms a high-speed water jet (time t = 1.1 µs and after). The velocity magnitude is higher than 2000 m/s. At time t = 2 µs, the water jet strikes the downstream bubble interface leading to the blast wave generation. After time t = 2.1 µs, the bubble is cut in half and forms a pair of distinct vortical structures. The developing leftward wave advances relatively slowly due to the high water velocity in the jet fluid. After time t = 2.6 µs, the front of the leftward wave can be observed (abscissa x 0.012 m). The evolution of the vertical velocity is illustrated in Figure 4.11 at the same time as previously. When the water jet impinges with the downstream edge of the bubble, at time t = 2 µs, the bubble forms a pair of distinct vortical structures. At time t = 2.6 µs, caused by the leftward blast wave, secondary jets penetrate into the smaller bubbles and cut the initial bubble into four pieces. CHAPTER 5. RESULTS ON THE VENTURI GEOMETRY CAVITATING FLOW peak position varies among models. With the KWReboud turbulence model, the peak is present downstream station 5, whereas the peak obtained with the SAReboud turbulence model is upstream. The KEReboud turbulence model provides fluctuations in better agreement with experimental data inside the cavity but not in the re-compression area. Figure 5.12 compares time-averaged profiles of the viscosity ratio µ t /µ, at the five station, obtained with Merkle cavitation model. At stations 2 and 3, a drastic decrease of µ t close to the wall for all models due to the Reboud correction can be observed. Therefore, a better prediction of the unsteadiness and the separation can be expected. The KWReboud turbulence model shows a large reduction of the ratio in the sheet at all stations. On the contrary, the SAReboud turbulence is not in the same case. Entropy and B.1 Introduction The aeroelastic prediction workshop aims at the assessment of the state-of-the-art in numerical methods for simulating flow-fields about wings undergoing prescribed motions or static and dynamic aeroelastic deformations at transonic flight conditions. The first AIAA Aeroelastic Prediction Workshop (AePW-1) conducted three configurations: the Rectangular Supercritical Wing (RSW), the Benchmark Supercritical Wing (BSCW) and the High Reynolds Number Aerostructural Dynamics (HIRENASD). These cases focus on the prediction of unsteady pressure distributions resulting from forced motion. The second workshop (AePW-2) extends the benchmarking effort to aeroelastic flutter solutions with flow conditions in transonic regime and focuses on a single configuration. The configuration chosen for the second workshop is the BSCW. The primary analysis condition has been chosen such that the influence of separated flow is considered to be minimal, yet a shock is still present. This is a step back in flow complexity from the BSCW cases for AePW-1. The goal in moving to the lower transonic Mach number is to have analysis teams progress through unforced system analyses, forced oscillation solutions and flutter analyses. Revisiting the AePW-1 analysis condition is included in AePW-2 as an optional case, also extending it to include flutter solutions. B.2 The Benchmark Supercritical Wing The BSCW is chosen for the workshop. It is a rigid, semispan, rectangular supercritical wing with a chord of 16 inches, a span of 32 inches, and a SC(2)0414 supersonic airfoil with design normal force coefficient of 0. Tunnel (TDT). The first experiment in 1991 is a flutter test performed on the pitch and plunge apparatus (PAPA) system which provides two-degree-of-freedom flutter [START_REF] Dansberry | Experimental unsteady pressures at flutter on the supercritical wing benchmark model[END_REF]. The PAPA experimental data consists of unsteady data at flutter points and averaged data on a rigidified apparatus at the flutter condition. The second experiment in 2000 is a forced excitation test performed on the oscillating turntable (OTT), in which the wing was oscillated in pitch about an axis at the 30% chord [START_REF] Heeg | Experimental data from the benchmark supercritical wing wind tunnel test on an oscillating turntable[END_REF] 60% and 95% span stations for the unforced system by considering the effects of grid size for each selected turbulence models. The unforced system computations show good agreement with the experimental data, except that the peak value at the 10% wing span near the leading edge on the upper surface is under-predicted. In addition, form these figures the influence of the grid sizes is minor. B.3.2 Test Case 2 The second test case focuses on flutter prediction at Mach 0.74 and 0°angle of attack. The experimental comparison results obtained from the PAPA test are available for both the inboard span station (60% wing span) and the outboard span station (95% wing span). Comparison of the steady-state solution using different turbulence models among different grid sizes for the mean pressure coefficients at 60% and 95% span stations are shown from Figure B.9 to B.11. For the 60% span station all computational results for the unforced system converged to the same results and are in good agreement to the experimental data for both the upper and lower surface. For the 95% span station the computations show good agreement with the experimental data, except that the peak value at the 10% wing span near the leading edge on the upper surface is over-predicted. From these two sets of comparison, the computational results are in very good agreement with those of experiment. The influence of turbulence models and grid sizes is not evident for test case #2. B.3. COMPUTATIONAL RESULTS B.3.3 Test Case 3 This test case (Mach = 0.85, 5°angle of attack) was also analyzed in AePW-1 and the results from the different analysis teams showed that it is difficult to find a converged steady state solution. This case is chosen as an optional case for AePW-2 in order to continue working on the problem and resolving the discrepancies. For both the unforced system and forced oscillation, experimental data of mean pressure coefficients obtained from the OTT test are available at the inboard span station (60% wing span). B.4 Conclusion Numerical simulations were performed with the NSMB solver for the second Aeroelastic Prediction Workshop (AePW-2). For case#1, at Mach 0.7, 3°angle of attack, the effects of grid sizes and turbulence model are not obvious. The computation results are in very good agreement with the experimental data. There exists only some difference in the suction peak region in the upper surface and in some regions of the lower surface. For forced oscillation simulation, computational results are performed with the medium grid. All turbulence models predict well the pressure values after the shock region. For case #2, at Mach 0.74, 0°angle of attack, the simulations predicted accurately the steady pressure distribution. The influnece of turbulence models and grid size is not obvious. For case #3, at Mach 0.85, 5°angle of attack, the effect of grid sizes is not important for the computational results of the mean pressure distribution. The kω SST turbulence model captures better the upper-and lower-surface shock location, but fails to predict the pressure values at the recovery area behind the shock. For forced oscillation simulation, all turbulence models are not able to predict the mean pressure for the upper surface.

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2013

https://hal.science/hal-01743926/file/2013-Coolsaet-et-al.-Study-for-the-implementation-in-Belgium-of-the-Nagoya-Protocol-on-ABS-to-the-CBD.pdf

Prof Charles-Hubert Born Prof Van Geertrui Prof Delphine Overwalle Prof Missone Cliquet An Isabelle Durant Koen Van Den Bossche Arianna Broggiato Arul Scaria Anne Liesse Heike Rämer Caroline Van Schendel Tom Dedeurwaerdere email: [email protected] LIST OF TABLES Table 1 - LIST OF FIGURES  A phased approach should be adopted for the implementation of the Nagoya Protocol, allowing to benefit from the implementation of the basic principles in a timely manner and to deal with more fine-grained choices at a later stage. Specific recommendations  Alongside the designation of Competent National Authorities (CNAs), a centralized input system to the CNAs should be established.  With regard to compliance measures, sanctions should be provided for cases of non-compliance with PIC and MAT requirements set out by the provider country. When checking content of MAT, a provision in the code of international private law should provide for reference to provider country legislation, with Belgian law as a fallback option.  At this stage of the implementation, the monitoring of the utilization of genetic resources and traditional knowledge by a checkpoint should be done on the basis of the PIC available in the ABS Clearing-House.  With regard to access to Belgian genetic resources, it is recommended to refine the existing legislation relevant for protected areas and protected species, combined with a general notification requirement for access to other genetic resources. Later stages of implementation can then include refinement of additional relevant legislation as well as having ex-situ collections process the other access requests.  At this stage of the implementation, and apart from the general obligation to share benefits, no specific benefit-sharing requirements should be imposed for the Mutually Agreed Terms. A combination of more specific requirements, including the possibility to use standard agreements, can be considered in a later stage of the implementation.  The Royal Belgian Institute of Natural Sciences should be mandated to fulfill the information sharing tasks on Access and Benefit Sharing under the Nagoya Protocol, through the ABS Clearing-House. This study aims to contribute to the ratification and the implementation in Belgium of the Nagoya Protocol on Access and Benefit-sharing (ABS), thereby contributing to the conservation of biological diversity and the sustainable use of its components. This is in support of the overall goal to implement the Convention on Biological Diversity (CBD) since the 2010 Nagoya Protocol on Access to Genetic Resources and the Fair and Equitable Sharing of Benefits Arising from their Utilization is a protocol to the CBD. The CBD is the main international framework for the protection of biodiversity. It has three objectives: (1) the conservation of biological diversity, (2) the sustainable use of its components and (3) the fair and equitable sharing of benefits arising from the utilization of genetic resources. The Nagoya Protocol therefore delineates the means of implementation of the third objective of the CBD. ABS potentially encompasses a large range of issues extending far beyond sole environmental matters, including market regulation and access, international trade, agriculture, health, development cooperation, research & development and innovation. As a consequence, the future implementation of the Nagoya Protocol could be relevant to several departments and several levels of competence in Belgium. Access and Benefit-sharing (ABS) in Belgium Following successive transfers of competences since 1970, the federated entities have the greatest responsibility in ABS-related issues, including environmental policy, agricultural policy, research and development, and economic and industrial policy. However, within these matters, the Federal Government possesses reserved and residual competences, with relevant examples including, among others, the export, import and transit of non-indigenous plant varieties and animal species, industrial and intellectual property, and scientific research that is necessary to the execution of its own competences. The large range of issues also implies an extended administrative distribution of ABSrelated competences within each power level. The implementation of the Nagoya Protocol, as a "double mixed treaty"1 , will thus necessitate competences from both the federal and federate entities and require extensive inter-and intra-departmental coordination. Access to genetic resources, as understood in the Nagoya Protocol, is not as such yet regulated by Belgian public law measures. Nevertheless, existing public and private law provisions already regulate related matters such as property rights, physical access to (genetic material in) protected areas and protected species, or modification and transformation of natural environments. Several of these existing provisions could be used as a basis for the implementation of the Nagoya Protocol in Belgium. In order to fully understand the usefulness of these existing measures, four important preliminary remarks need to be made. First, throughout this study, access and utilization of genetic resources and traditional knowledge are analyzed within the framework of the Nagoya Protocol. The Protocol covers genetic resources and traditional knowledge that are provided by Parties from where such resources originate or by Parties that have acquired them in accordance with the Convention on Biological Diversity. Hence, this report covers:  genetic resources possessed by a country in in-situ conditions and on which that country holds sovereign rights ; and  genetic resources possessed by a country in ex-situ collections and which have been acquired after the entry into force of the Nagoya Protocol and / or in accordance with the obligations of the Convention on Biological Diversity. Second, the CBD distinguishes "genetic material" (i.e. any material of plant, animal, microbial or other origin containing functional units of heredity) from "genetic resources" (i.e. genetic material of actual or potential value). Third, a distinction has to be made between the question of legal ownership of genetic resources in their quality of material goods on the one hand, and the regulation of the access and utilization of genetic resources according to the Nagoya Protocol as an exercise of a sovereign right, on the other. The Belgian State holds sovereign rights over its genetic resources and can thus regulate the utilization of these resources by public law measures, as long as these are justified. However, physical access to and use of genetic material are already regulated by property law and the liability and redress options made available under both civil and criminal procedures related to the enforcement of property rights. Fourth, it is important to remember that while genetic resources can be seen as biophysical entities (e.g. a plant specimen, a microbial strain, an animal, etc.), they also include an "informational component" (i.e. the genetic code). Access to genetic resources therefore relates both to the physical component and/or the informational component. Taking the above into account, currently available national provisions relevant for the legal status of genetic resources in Belgium mainly relate to the question of legal ownership over genetic material. Flowing from the central tenets of the right to property found in the civil code, the conditions and rules surrounding the legal ownership of the genetic material, as a biophysical entity, follow from those governing the ownership of the organism this material can be found in. Property over an organism means that the proprietor possesses the rights to use, perceive the benefits and alienate the specimen. Furthermore, any legal measure regulating access to genetic resources could benefit from building upon existing legislation on physical access to and use of genetic material. The rules regulating physical access and use of genetic material depend upon the type of ownership (movable, immovable or res nullius), the existence of restrictions to the ownership such as specific protection (protected species, protected areas, forests or marine environments) and the location (all four Authorities apply their own rules) of the genetic material. As opposed to its physical components, the informational components regarding the genetic resources may constitute a res communis: "things owned by no one and subject to use by all". While access to such informational components is not covered by subject-specific legislation, the exercise of some use rights can however be limited through intellectual property rights that have been recognized on portions, functions, or uses of biological material resulting from innovations on these materials. These intellectual property rights can take the form of patents, plant variety rights or geographical indications. Alongside these principles surrounding the legal status of genetic resources, a number of rules found in civil, criminal and private international law, offer prospects of liability and redress in cases where an illicit acquisition of genetic resources is established. Their application is different with regard to genetic resources as physical specimens or as informational goods, but also with regard to where the illicit acquisition has taken place. Finally, there are no contemporary legal provisions in Belgium explicitly governing the concepts of "traditional knowledge", "traditional knowledge associated with genetic resources" and "indigenous and local communities". However, concerns over traditional knowledge and the rights of indigenous and local communities have been addressed in some international instruments to which Belgium is a Party, such as the 1957 International Labor Organization (ILO) Convention No. 107 on Indigenous and Tribal Populations, the ILO Convention No. 169 on Indigenous and Tribal Peoples, and the United Nations Declaration on the Rights of Indigenous Peoples. Preliminary recommendations for the options for the implementation of the Nagoya Protocol While the Nagoya Protocol is a recent protocol, it is nonetheless the further implementation of the third objective of the CBD which contains basic principles and ABS related provisions, such as the sovereignty of States over their natural wealth and resources, the fair and equitable sharing of benefits, and the importance of indigenous and local communities and their traditional knowledge. Many Parties to the CBD throughout the world therefore have implemented a series of measures on ABS, which can serve as useful first-hand experience for the implementation of the Nagoya Protocol. Through these experiences, two sets of preliminary recommendations were established in this study, with regard to the available options for the implementation of the Nagoya Protocol in Belgium. The first set of recommendations relates to instruments required for the implementation of the core obligations emanating from the Protocol2 . The second set of recommendations relates to additional measures which are important elements to be taken into account during the implementation of the obligations, but which go beyond the core obligations. With regard to the core obligations, the following is recommended:  Clarify access conditions: By holding sovereign rights over its genetic resources, Belgium can choose whether or not to require users to obtain Prior Informed Consent through the competent authority for access to genetic resources under its jurisdictibon.  Determine the format of the Mutually Agreed Terms: Once the Nagoya Protocol enters into force in Belgium, users operating on its territory will be required to share benefits arising from the utilization of genetic resources. Such sharing shall be based upon MAT. However, the Nagoya Protocol does not impose a specific format for MAT, which can be left to the discretion of stakeholders or flow from guidelines and/or mandatory measures imposed by the State.  Ensure ABS serves conservation and sustainable use of biodiversity: It should be made sure that the implementation of the Nagoya Protocol supports the other two objectives of the CBD: conservation of biodiversity and sustainable use of its components. This can be done, for instance, by linking PIC to mandatory conditions on the sharing of the benefits or by establishing a "benefit-sharing" fund which redirects the benefits towards conservation and sustainable use of biodiversity.  Facilitate access for biodiversity-related research: In order to foster biodiversity-related research and avoiding putting too much burden on non-commercial research utilizing genetic resources, measures could be developed to facilitate access to genetic resources for noncommercial biodiversity-related research.  Establish a Competent National Authority: Each Party has to designate a Competent National Authority that grants access, issues written evidence that access requirements have been met and advises users on applicable procedures and requirements to get access to genetic resources. Given the institutional reality in Belgium, more than one CNA can be established. It should be noted that this task is of the highest priority, as Belgium needs to notify the CBD Secretariat of the contact information of its Competent National Authority/Authorities (and of its national focal point, which is already appointed) no later than the date of entry into force of the Protocol.  Give binding effect to the domestic legislation of provider countries regarding PIC and MAT: As part of the implementation of the Protocol, the basic obligations domestic users have to comply with when utilizing genetic resources in Belgium will have to be laid out. This obligation comes down to giving binding effect to the provider country's PIC and MAT. This could be done by establishing an obligation in the Belgian legislation to comply with the provider country legislation regarding PIC and MAT, or by establishing a self-standing obligation in the Belgian legislation to have PIC and MAT if so required by the provider country.  Designate checkpoint(s) for the monitoring of the utilization of genetic resources: In order to comply with the Nagoya Protocol, at least one institution has to be designated to function as a checkpoint which monitors and enhances transparency about the utilization of GR. This can be a new or existing institution. With regard to additional measures, the following issues are to be taken under consideration: a) specifying benefit-sharing requirements for the MAT; b) establishing a clear and transparent access procedure; c) clarifying additional rights and duties of the Competent National Authorities; d) establishing a monitoring system; e) creating incentives for users to comply; and f) encouraging the development of model clauses, codes of conducts and guidelines. Selected options for the implementation of the Nagoya Protocol In light of the preliminary recommendations for the options for the implementation of the Nagoya Protocol described above, six measures, each including several policy-options, were discussed at the first stakeholder meeting on the 29 th of May 20123 . Based on the results of that meeting, they were selected by the Steering Committee of this study for an in-depth analysis of environmental, social, economic and procedural impacts. Prior to implementing these measures, it should be decided whether to establish both Prior Informed Consent and benefit-sharing as general legal principles in Belgium. While the latter is necessary to comply with the Nagoya Protocol, the former flows from the sovereign rights Belgium holds over its genetic resources and is not necessary for compliance. If Prior Informed Consent is established as a general principle, a procedure needs to be established for access to Belgium's own genetic resources (measure 1). This can be done by modifying existing legislation, by relying upon qualified ex-situ collections, by requiring prior registration or by a combination of these instruments. If benefit-sharing is established as a general principle, the conditions for the specific benefit-sharing requirements through the Mutually Agreed Terms, need to be clarified (measure 2). The specific benefit sharing requirements can be left to the discretion of users and providers (option 1), or be imposed by the state with more or less standardization (options 2 and 3). Measure 2: specifying the benefit-sharing requirements for Mutually Agreed Terms 0. Option 0: No requirement of benefit-sharing for the utilization of genetic resources and traditional knowledge in Belgium; 1. Option 1: No specific benefit-sharing requirements are imposed by the competent authorities for the MAT. Users and providers are free to decide jointly on the content. 2. Option 2: Specific benefit-sharing requirements are imposed, including through standard formats for the MAT for certain uses, which are differentiated depending on the finality of access. b. For unprotected genetic resources: access is provided for through the Belgian ex-situ collections. 2. Option 2 -The baseline fishing net model a. For protected genetic resources: access is made possible through a refinement of existing legislation relevant for protected areas and protected species; b. For unprotected genetic resources: access is accorded upon notification to the competent authority. 3. Option 3 -Modified fishing net model a. For protected genetic resources and genetic resources already covered by specific GR-relevant legislation: access is made possible through a refinement of existing legislation; b. For unprotected genetic resources: access is accorded upon notification to the competent authority. 3. Option 3: Specific benefit-sharing requirements are imposed but without standard formats for the MAT. While taking into account the benefit-sharing requirements, the MAT are tailored on a case-bycase basis by the users and providers. The benefit-sharing requirements are differentiated depending on the finality of access. In order to comply with the Nagoya Protocol, one or several competent national authorities will need to be established (measure 3). Their task is to grant access, to issue written evidence that access requirements have been met and to advise users on applicable procedures and requirements to get access to genetic resources. To fulfill these tasks, the competent national authorities will need to establish entry-points for users of genetic resources. This can be done separately, with each authority having its own entry-point (option 1), or jointly, with a single entry-point for the different authorities (option 2). Once the Nagoya Protocol enters into force in Belgium, it will need to set up compliance measures to make sure that genetic resources and traditional knowledge utilized on its territory have been accessed in accordance with the law of the provider country (measure 4). This can be achieved by referring back to the legislation of the provider country in question and opening review of the content of MAT in accordance with provider country legislation with Belgian law as a fall-back option (option 1), or by setting-up a self-standing obligation under Belgian law (option 2). In the latter option, Belgian legislation would only refer to the specific obligation of requiring PIC and MAT by the provider country without referring to the actual ABS legislation of the provider country. Measure 4: setting-up compliance measures 0. Option 0: No legal provisions on compliance with the Nagoya Protocol are introduced under Belgian law 1. Option 1: A general criminal provision is created that refers back to the legislation regarding PIC and MAT of the provider country. The state enacts a general prohibition to utilize genetic resources and traditional knowledge accessed in violation of the law of the providing country. Review of the content of MAT by judges is subject to provider country legislation, with Belgian law as a fall-back option. 2. Option 2: A provision is created containing an obligation to have PIC and MAT from the provider country for the utilization in Belgium of foreign genetic resources, if this is required by the legislation of the provider country. In order to comply with the Nagoya Protocol, at least one checkpoint needs to be created to monitor the utilization of genetic resources and traditional knowledge in Belgium (measure 5). If Belgium decides to introduce checkpoints, their implementation could take place in several phases. In order to respect the political commitment for a timely ratification of the Nagoya Protocol, the first phase could look at a minimal implementation requiring the establishment of a single checkpoint. Two possible options seem relevant for the first phase, namely monitoring the PIC obtained by users, which is available in the ABS Clearing-House (option 1) and to upgrade the existing patent disclosure obligation (option 2). As options 1 and 2 are not mutually exclusive, a joint implementation could be envisaged. Finally, a Belgian component of/entry-point to the ABS Clearing-House will be created to support exchange of information on specific ABS measures within the framework of the Nagoya Protocol (measure 6). Even if the discussions on the exact modalities of the ABS Clearing-House are still ongoing internationally, three possible candidates have been identified: the Royal Belgian Institute of Natural Sciences (option 1), the Belgian Federal Science Policy Office (option 2), and the Scientific Institute for Public Health (option 3). Impact of the selected options for the implementation of the Nagoya Protocol The evaluation of the possible consequences of the implementation of the above options was conducted through a detailed comparative multi-criteria analysis. This analysis also allowed identifying the possible affected stakeholders. For the operationalization of access to genetic resources (measure 1), the bottleneck option (option 1) and the modified fishing net option (option 3) came out very close. The preference for these options can be explained by the fact they are expected to provide more legal certainty, will have a better environmental impact and correspond better to current practices than the other two options. These two options first require establishing, as a general legal principle, that access to Belgian genetic resources requires Prior Informed Consent. For the specification of benefit-sharing requirements for Mutually Agreed Terms (measure 2) the two options that impose specific benefit-sharing requirements by the Belgian State (options 2 and 3) both ranked better than the option where no specific benefit-sharing requirements are imposed (option 1). This is due to their good economic, environmental and procedural performance (option 2 also has a good social performance). Choosing these options requires adopting benefit-sharing as a general legal principle in Belgium. Alongside the establishment of the Competent National Authorities, a centralized input system clearly came out as the recommended option (option 2 of measure 3). This option scores best on all the criteria and is strictly better on legal certainty and effectiveness for users and providers of genetic resources, at low cost. For the setting up of compliance measures (measure 4), the option to refer back to provider country legislation, with Belgian law as a fallback option, is the recommended option that comes out of this analysis. This can be explained by the closer conformity of this option with existing practices (mainly under the Belgian code of private international law). For the designation of one or more checkpoints (measure 5), the option of monitoring PIC in the ABS Clearing-House stands as the recommended option. It scores at least as well on all criteria and has a better social and procedural performance. Finally, for the sharing of information through the Clearing-House (measure 6), the preference goes to appointing the Royal Belgian Institute of Natural Sciences (RBINS), which has a better performance than other options on most of the analyzed criteria. Recommendations resulting from the impact assessment Two general recommendations result from the impact analysis of the study, along with a set of more specific recommendations for each of the measures. First, the analysis shows that the no policy change baseline (the "0" option for each measure) clearly has the worst performance. This result leads to a first general recommendation, which is to implement both Prior Informed Consent and benefit-sharing as general legal principles in Belgium. Second, the analysis confirmed the validity of a phased approach to the implementation of the Protocol. A phased approach will allow to benefit from the implementation of the basic principles in a timely manner and to deal with more fine grained choices in a later stage. Moreover, the phased approach will be necessary in order to be able to timely ratify the Nagoya Protocol and allow Belgium to participate as a Party to the Nagoya Protocol at the first COP/MOP in October 2014. Finally, the impact assessment has led to a set of specific recommendations on each of the six measures described above: 1. Alongside the designation of Competent National Authorities (CNAs), a centralized input system to the CNAs should be established. 2. With regard to compliance measures, sanctions should be provided for in cases of noncompliance with PIC and MAT requirements set out by the provider country. When checking content of MAT, a provision in the Code of international private law should provide for reference to provider country legislation, with Belgian law as a fallback option. 3. At this stage of the implementation, the monitoring of the utilization of genetic resources and traditional knowledge by a checkpoint should be done on the basis of the PIC available in the ABS Clearing-House. 4. With regard to access to Belgian genetic resources, it is recommended to refine existing legislation relevant for protected areas and protected species, combined with a general notification requirement for access to other genetic resources. Later stages of implementation can then include refinement of additional relevant legislation as well as having ex-situ collections process the other access requests. 5. At this stage of the implementation, and apart from the general obligation to share benefits, no specific benefit-sharing requirements should be imposed for the Mutually Agreed Terms. A combination of more specific requirements, including the possibility to use standard agreements, can be considered in a later stage of the implementation. 6. The Royal Belgian Institute of Natural Sciences should be mandated to fulfill the information sharing tasks on Access and Benefit-sharing under the Nagoya Protocol, through the ABS Clearing-House. Implementation of the recommendations To implement these recommendations, the phased approach could be organized through a three step process: 1. In the first step, a political agreement should be agreed upon by the competent authorities with a clear statement on the general legal principles to be adopted, along with some specification of the actions to be undertaken by the federal and the federated entities to establish these principles and put them into practice. These should include: a. Establishment of benefit sharing as a general legal principle in Belgium. b. Establishment as a general legal principle that access to Belgian genetic resources requires PIC. c. Establishment of the general principle concerning the designation of four Competent National Authorities. d. Commitment that legislative measures will be taken to provide that genetic resources utilized within Belgian jurisdiction have been accessed by PIC and MAT, as required by provider country legislation, and to address situations of non-compliance. e. Designation of the Belgian CBD Clearing-House Mechanism, managed by the Royal Belgian Institute of Natural Sciences, as the Belgian contribution to the ABS Clearing-House, for dealing with the information exchange on ABS under the Nagoya Protocol. The reason for recommending such a political agreement is double. On the one hand, such an agreement provides for a clear political commitment to the core obligations of the Nagoya Protocol, as it specifies the intentions of the competent authorities, within the limits of the decisions already taken at the international and European level at the time of the agreement. On the other hand, it does not prejudge the political decisions to be taken by the different authorities and thus allows for sufficient flexibility to further adjust the implementation process in a later stage. The latter is especially important given the many questions that are still undecided at the present stage, both at the EU and international level, as mentioned and taken into account in this report. 2. In a second step, the specified actions should be subsequently implemented, for example through a cooperation agreement and/or by adding provisions in the relevant legislations such as the environmental codes of the federated entities and the federal government, along with other possible requirements. 3. In a third step, additional actions can be undertaken once there is more clarity from the negotiations on the EU and the international level. 4. RÉSUMÉ ANALYTIQUE Recommandations générales  Tant le consentement préalable donné en connaissance de cause (Prior Informed Consent, PIC) que le partage des avantages (benefit-sharing) devraient être établis comme principe général juridique en Belgique  Une approche par étapes devrait être adoptée pour la mise en oeuvre du Protocole de Nagoya. Celle-ci permettrait de s'appuyer sur l'instauration, dans les temps requis, de principes juridiques de base et de traiter les options plus précises à un stade ultérieur Cette étude a pour objectif de contribuer à la ratification et à la mise en oeuvre en Belgique du Protocole de Nagoya sur l'Accès et le Partage des Avantages (APA), qui à son tour doit contribuer à la conservation de la diversité biologique et à l'utilisation durable de ses éléments. En tant que protocole à la Convention sur la Diversité Biologique (CDB), l'implémentation du Protocole de Nagoya de 2010 sur « l'Accès aux ressources génétiques et le partage juste et équitable des avantages découlant de leur utilisation » participe à l'objectif général de mise en oeuvre de la CDB. La CDB est le principal instrument international pour la protection de la biodiversité. Elle a trois objectifs: (1) la conservation de la diversité biologique, (2) l'utilisation durable de ses éléments et (3) le partage juste et équitable des avantages découlant de l'exploitation des ressources génétiques. Le Protocole de Nagoya dessine les moyens de mise en oeuvre du troisième objectif. L'APA comprend une grande diversité de questions allant bien au-delà des seules matières environnementales, telles que la régulation et l'accès aux marchés, le commerce international, l'agriculture, la santé, le développement et la coopération, la recherche et développement, et l'innovation. Par conséquent, la future mise en oeuvre du Protocole de Nagoya pourrait être pertinente pour plusieurs départements et plusieurs niveaux de compétence en Belgique. L'Accès et le Partage des Avantages (APA) en Belgique Suite aux transferts successifs de compétences depuis 1970, les entités fédérées ont la responsabilité première pour les questions liées à l'Accès et au Partage des Avantages (APA), parmi lesquelles la politique environnementale, la politique agricole, la recherche et le développement, et la politique économique et industrielle. Cependant, le gouvernement fédéral détient dans ces domaines des compétences réservées et résiduelles, s'appliquant entre autres à l'importation, de l'exportation et du transit des espèces végétales et animales non indigènes, à la propriété industrielle et intellectuelle, et à la recherche scientifique nécessaire à l'exercice de ses propres compétences. La grande diversité des questions traitées nécessite aussi une distribution administrative étendue des compétences relatives à l'APA au sein de chaque niveau de pouvoir. La mise en oeuvre du Protocole de Nagoya, en tant que « traité mixte »4 , exigera donc des compétences à la fois de l'Etat fédéral et des entités fédérées, et requerra une coordination inter-et intra-départementale approfondie. L'accès aux ressources génétiques, tel que défini dans le Protocole de Nagoya, n'est pas encore régis en tant que tel par le droit public belge. Néanmoins, des dispositions existantes en droit public et privé réglementent déjà des cas apparentés, tels que les droits de propriété, l'accès physique aux (matériel génétique dans les) régions protégées et aux espèces protégées, ou encore la modification et la transformation des environnements naturels. Plusieurs de ces dispositions existantes pourraient servir de base pour la mise en oeuvre du Protocole de Nagoya en Belgique. Pour comprendre pleinement l'utilité de ces mesures existantes, il y a lieu de faire quatre remarques préliminaires importantes. Premièrement, tout au long de cette étude, l'accès et l'utilisation des ressources génétiques et du savoir traditionnel sont analysés dans le cadre du Protocole de Nagoya. Le Protocole traite des ressources génétiques et du savoir traditionnel qui sont fournis par les Parties qui sont les pays d'origine de ces ressources ou par les Parties qui les ont acquises conformément à la Convention sur la Diversité Biologique. Par conséquent, ce rapport traite:  des ressources génétiques qu'un pays possède dans des conditions in-situ et sur lesquelles il exerce un droit de souveraineté; et  des ressources génétiques qu'un pays possède dans des collections ex-situ et qui ont été acquises après l'entrée en vigueur du Protocole de Nagoya et/ou en accord avec les obligations de la Convention sur la Diversité Biologique. Deuxièmement, la Convention sur la Diversité Biologique distingue "matériel génétique" (c.-à-d. tout matériel végétal, animal, microbien ou de tout autre origine contenant des unités fonctionnelles d'hérédité) des "ressources génétiques" (c.-à-d. matériel génétique de valeur réelle ou potentielle). Troisièmement, il faut distinguer d'une part, la question de la propriété légale de ressources génétiques en leur qualité de biens matériels, et, d'autre part, la réglementation de l'accès et de l'utilisation des ressources génétiques en conformité avec le Protocole de Nagoya en tant qu'exercice d'un droit souverain. L'Etat belge détient des droits souverains sur ses ressources génétiques et peut donc réglementer l'utilisation de ces ressources par des mesures de droit public, pour autant que celles-ci soient justifiées. Cependant, l'accès physique au matériel génétique et leur utilisation sont déjà réglementés par la loi sur la propriété et par les options de responsabilité et de réparation accessibles dans les procédures civiles et pénales relatives au renforcement des droits de propriété. Quatrièmement, il est important de rappeler que si les ressources génétiques peuvent être considérées comme des entités biophysiques (par exemple, un spécimen végétal, une souche microbienne, un animal, etc.), elles comprennent un "composant informationnel" (c.-à-d. le code génétique). L'accès aux ressources génétiques concerne à la fois le composant physique et/ou le composant informationnel. intellectuelle peuvent prendre la forme de brevets, de protection des obtentions végétales ou d'indications géographiques. En parallèle de ces principes directeurs régissant le statut légal des ressources génétiques, le droit civil, pénal, et international privé contiennent des règles et procédures en matière de responsabilité et de réparation relatives à l'acquisition illicite de ressources génétiques. Leur application est différente selon que les ressources génétiques sont des spécimens physiques ou des biens d'information, mais aussi selon l'endroit où l'acquisition illicite a eu lieu. Enfin, il n'y a pas actuellement de dispositions légales en Belgique qui régissent explicitement les concepts de « connaissances traditionnelles », de « connaissances traditionnelles associées à des ressources génétiques » et de « communautés autochtones et locale ». En ce qui concerne les mesures supplémentaires, les questions suivantes doivent être prises en considération: a) spécifier les conditions pour les conditions convenues d'un commun accord (MAT); b) instaurer une procédure claire et transparente pour l'accès aux ressources génétiques; c) clarifier les droits et devoirs supplémentaires de(s) l'autorité(s) nationale(s) compétente(s); d) instaurer un système de surveillance (monitoring); e) créer des incitants à se conformer à l'adresse des utilisateurs ; f) encourager le développement de clauses contractuelles types, de codes de conduite et de lignes directrices. Options sélectionnées pour la mise en oeuvre du Protocole de Nagoya A la lumière des recommandations préliminaires relatives aux options pour la mise en oeuvre du Protocole de Nagoya décrites plus haut, six mesures, chacune comprenant plusieurs options politique, ont été discutées à la première réunion des parties prenantes le 29 mai 2012. Sur base des résultats de cette réunion, elles ont été sélectionnées par le Comité de Pilotage de l'étude pour une analyse en profondeur des impacts environnementaux, sociaux, économiques et procéduraux. Avant de mettre en oeuvre ces mesures, il doit être décidé s'il faut inscrire le consentement préalable donné en connaissance de cause (PIC) et le partage des avantages comme principes juridiques généraux en Belgique. Si ce dernier est indispensable pour se conformer au Protocole de Nagoya, le premier (PIC) découle des droits souverains que la Belgique exerce sur ses ressources génétiques et n'est pas indispensable pour la conformité avec le Protocole. Si le consentement préalable en connaissance de cause est effectivement inscrit comme principe général, il faut instaurer une procédure pour l'accès aux ressources génétiques belges (mesure 1). Cela peut être fait en modifiant la législation existante, en s'appuyant sur les collections ex-situ autorisées, en exigeant une notification préalable ou au travers d'une combinaison de ces dispositifs. Impact des options sélectionnées pour la mise en oeuvre du Protocole de Nagoya L'évaluation des conséquences possibles de l'application des options décrites ci-dessus a été conduite par une analyse comparative détaillée à critères multiples. Cette analyse a également permis d'identifier les parties prenantes qui pourraient être affectées. Pour l'opérationnalisation de l'accès aux ressources génétiques (mesure 1), le modèle « bottleneck » (option 1) et le modèle « fishing net » modifié (option 3) ont des performances très similaires. La préférence pour ces options peut être expliquée par le fait qu'elles sont supposés apporter une plus grande sécurité juridique, qu'elles auront un meilleur impact environnemental, et qu'elles correspondent mieux aux pratiques actuelles que les deux autres options. Ces deux options requièrent d'abord l'instauration du consentement informé préalable (PIC) pour l'accès aux ressources génétiques belges comme principe juridique général. En ce qui concerne la spécification des dispositions pour les conditions convenues d'un commun accord (MAT) (mesure 2), les deux options qui imposent des dispositions spécifiques par l'Etat belge (options 2 et 3) se classent mieux que l'option sans dispositions spécifiques (option 1). Cela s'explique par une meilleure performance économique, environnementale, et procédurale (l'option 2 présente aussi une bonne performance sociale). Choisir ces 2 options impose d'établir le 'partage d'avantage' comme principe juridique général en Belgique. En plus de l'instauration d'autorités nationales compétentes, l'option privilégiant un point d'entrée commun est clairement apparue comme l'option recommandée (option 2 de la mesure 3). Cette option a une bonne performance sur tous les critères, offre un meilleure sécurité juridique et est plus efficace pour les utilisateurs et les fournisseurs de ressources génétiques, à bas coût. Pour l'instauration des mesures de mise en conformité (mesure 4), l'option créant une disposition pénale générale se référant à la législation du pays fournisseur, avec la loi belge comme option de rechange, obtient le meilleur résultat. En effet, cette option présente une meilleure adéquation aux pratiques existantes (dans le Code de droit international privé). Quant à la désignation d'un ou plusieurs points de contrôle (mesure 5), l'option contrôlant le consentement préalable en connaissance de cause (PIC) de l'utilisateur dans le Centre d'Echange pour l'APA, est l'option recommandée. Cette option présente d'aussi bons résultats sur tous les critères que les autres options et présente un meilleur score sur le plan de la performance sociale et procédurale. Enfin, en ce qui concerne le partage de l'information par l'intermédiaire du Centre d'échange pour l'APA (mesure 6), la préférence va à la nomination de l'Institut Royal des Sciences Naturelles de Belgique, qui récolte de meilleurs résultats que les autres options sur la plupart des critères. Recommandations résultant de l'évaluation d'impact Deux recommandations générales résultent de l'analyse d'impact, en même temps qu'un ensemble de recommandations plus spécifiques pour chacune des mesures. D'abord, l'analyse montre que les options n'envisageant pas de changements de politique (les options « 0 » de chaque mesure) obtiennent clairement le résultat le moins bon. Ce score conduit à une première recommandation générale, qui est de mettre en oeuvre à la fois le 'Consentement informé préalable' (PIC) et le 'partage des avantages' (benefit-sharing) comme principes juridiques généraux en Belgique. Ensuite, l'analyse a confirmé la validité d'une approche par étapes pour la mise en oeuvre du Protocole. Une approche par étapes permettra de mettre en place les principes de base dans les temps requis et de traiter les options plus précises à un stade ultérieur. De plus, l'approche par étapes est nécessaire pour être en mesure de ratifier le Protocole de Nagoya dans les temps requis et de permettre à la Belgique de participer comme Partie au Protocole à la première Conférence des Parties (COP/MOP1) en octobre 2014. Enfin, l'évaluation d'impact a conduit à un ensemble de recommandations spécifiques pour chacune des six mesures : Implémentation des recommandations Pour réaliser ces recommandations, l'approche par étapes pourrait être organisée en par un processus en trois étapes : SAMENVATTING Algemene aanbevelingen  Zowel voorafgaande geïnformeerde toestemming (Prior Informed Consent, PIC) als de verdeling van voordelen (benefit-sharing) moeten worden ingevoerd als algemene vereisten in België.  Een gefaseerde aanpak moet worden gevolgd voor de implementatie van het Protocol van Nagoya. Op die manier kan voordeel worden gehaald uit de tijdige invoering van basisprincipes en kunnen specifiekere keuzes in een later stadium worden gemaakt. Specifieke aanbevelingen  Naast de oprichting van de Bevoegde Nationale Instanties, moet ook een gecentraliseerd aanspreekpunt worden gecreëerd voor deze instanties.  Wat de maatregelen inzake naleving van wet-of regelgeving (compliance) betreft, moeten sancties worden voorzien voor situaties van vaststelling van niet-naleving van de PIC en van de Onderling Overeengekomen Voorwaarden (Mutually Agreed Terms, MAT), zoals opgelegd door het oorsprongsland. Voor het controleren van de inhoud van de MAT zou een bepaling in het Wetboek van internationaal privaatrecht moeten verwijzen naar de wetgeving van het oorsprongsland, met de Belgische wetgeving als een eventuele terugvaloptie.  In de eerste uitvoeringsfase zou het controleren van het gebruik van genetische rijkdommen en traditionele kennis moeten gebeuren op basis van de PIC die beschikbaar is via het ABS Clearing-House mechanisme.  Met betrekking tot de toegang tot Belgische genetische rijkdommen, is het aanbevolen de bestaande relevante wetgeving inzake beschermde natuurgebieden en beschermde soorten te verfijnen, in combinatie met een algemene notificatievereiste voor de toegang tot andere genetische rijkdommen. In latere uitvoeringsfasen kan bijkomende relevante wetgeving dan eveneens worden verfijnd, en kan het verwerken van toegangsaanvragen voor andere genetische rijkdommen overgelaten worden aan ex-situ collecties.  In de eerste uitvoeringsfase, en los van de algemene verplichting om de voordelen te verdelen, zouden er geen specifieke vereisten moeten opgelegd worden voor het opstellen van Onderling Overeengekomen Voorwaarden (Mutually Agreed Terms). Een combinatie van meer specifieke vereisten, met de mogelijkheid om standaardakkoorden te gebruiken, kan in een latere uitvoeringsfase worden overwogen.  Het Koninklijk Belgisch Instituut voor Natuurwetenschappen zou moeten gemandateerd worden om de informatieuitwisselingstaken in verband met toegang en verdeling van de voordelen in het kader van het Protocol van Nagoya te vervullen, via het ABS Clearing-House. Het doel van deze studie is bij te dragen tot de ratificatie en implementatie in België van het Protocol van Nagoya inzake toegang en verdeling van voordelen (Access and Benefit-sharing, ABS), welke op haar beurt moet bijdragen tot het behoud van de biologische diversiteit en het duurzame gebruik van bestanddelen daarvan. De implementatie van het Protocol van Nagoya inzake "toegang tot genetische rijkdommen en de eerlijke en billijke verdeling van voordelen voortvloeiende uit hun gebruik" (2010), past in de algemene doelstelling die de implementatie van het Verdrag inzake biologische diversiteit (VBD) beoogt, daar het een protocol is bij het VBD. Het VBD is het voornaamste internationale instrument voor de bescherming van de biodiversiteit. Het heeft drie doelstellingen: (1) het behoud van de biologische diversiteit, (2) het duurzame gebruik van bestanddelen daarvan en (3) de eerlijke en billijke verdeling van de voordelen voortvloeiende uit het gebruik van genetische rijkdommen. Het Protocol van Nagoya bepaalt hoe de derde doelstelling gerealiseerd kan worden. ABS kan een breed scala van gerelateerde aangelegenheden omvatten die veel verder gaan dan louter milieuaangelegenheden, zoals regulering van en toegang tot de markt, internationale handel, landbouw, gezondheid, ontwikkelingssamenwerking, onderzoek & ontwikkeling, en innovatie. Bijgevolg zal de toekomstige implementatie van het Protocol relevant zijn voor verschillende departementen en verschillende beleidsniveaus in België. Toegang en verdeling van voordelen in België Na de opeenvolgende staatshervormingen sinds 1970, ligt de verantwoordelijkheid voor ABSaangelegenheden vooral bij de deelstaten, zoals het milieubeleid, landbouwbeleid, onderzoek en ontwikkeling, en het economisch en industriebeleid. Binnen die domeinen heeft de federale overheid echter gereserveerde en residuaire bevoegdheden. Voorbeelden hiervan zijn o.a. de in-, uit-en doorvoer van inheemse planten-en diersoorten, industriële en intellectuele eigendom, en wetenschappelijk onderzoek dat nodig is voor de uitoefening van haar eigen bevoegdheden. Het brede scala aan gerelateerde aangelegenheden veronderstelt ook een ruime administratieve verdeling van ABS-bevoegdheden binnen elk bevoegdheidsniveau. Voor de implementatie van het Protocol van Nagoya, als een "dubbel gemengd vedrag"6 , spelen de bevoegdheden van zowel de federale overheid als de deelstaten dus een belangrijke rol, en zal een uitgebreide inter-en intradepartementale samenwerking nodig zijn. De toegang tot genetische rijkdommen, zoals die in het Protocol van Nagoya is vastgelegd, is als dusdanig nog niet gereguleerd door Belgische publiekrechtelijke maatregelen. Toch worden gerelateerde aangelegenheden zoals het eigendomsrecht, de toegankelijkheid van (genetisch materiaal in) beschermde natuurgebieden en beschermde soorten, of het wijzigen van vegetatie, al gereguleerd door bestaande publiek-en privaatrechtelijke bepalingen. Deze bestaande bepalingen kunnen als basis worden gebruikt voor de implementatie van het Protocol van Nagoya in België. Om het nut van deze bestaande maatregelen volkomen te begrijpen moeten vier belangrijke, voorafgaande opmerkingen worden gemaakt. Ten eerste wordt in deze studie de toegang tot en het gebruik van genetische rijkdommen en traditionele kennis onderzocht in het kader van het Protocol van Nagoya. Het Protocol betreft genetische rijkdommen en traditionele kennis die worden verschaft door Partijen die het land van oorsprong van deze rijkdommen en/of kennis zijn of door Partijen die genetische rijkdommen in overeenstemming met het VBD hebben verworven. Bijgevolg betreft dit rapport:  genetische rijkdommen die een land bezit onder in-situ omstandigheden en waarop dat land soevereine rechten heeft; en  genetische rijkdommen die een land bezit in ex-situ collecties en die verworven werden na de inwerkingtreding van het Protocol van Nagoya en/of overeenkomstig de verplichtingen uit het Verdrag inzake Biologische Diversiteit. Ten tweede maakt het VBD een onderscheid tussen "genetisch materiaal" (m.a.w. alle materiaal van plantaardige, dierlijke, microbiële of andere oorsprong dat functionele eenheden van de erfelijkheid bevat) en "genetische rijkdommen" (m.a.w. genetisch materiaal van feitelijke of potentiële waarde). Ten derde moet een onderscheid worden gemaakt tussen het juridisch eigendom van genetische rijkdommen als materiële goederen enerzijds, en het reguleren van de toegang tot en het gebruik van genetische rijkdommen overeenkomstig het Protocol van Nagoya in het kader van de uitoefening van een soeverein recht, anderzijds. De Belgische Staat heeft als soevereine staat het recht om het gebruik van haar genetische rijkdommen te reguleren door middel van publiekrechtelijke maatregelen, op voorwaarde dat die maatregelen gerechtvaardigd zijn. De fysieke toegang tot en het gebruik van genetisch materiaal wordt echter al gereguleerd door het eigendomsrecht en door de aansprakelijkheids-en schadeloosstellingsmogelijkheden van de burgerlijke en strafrechtelijke procedures die gebruikt kunnen worden voor het afdwingen van eigendomsrechten. Ten vierde is het belangrijk te onderlijnen dat genetische rijkdommen, ook al kunnen ze als biofysische entiteiten worden beschouwd (e.g. een plantenspecimen, een bacteriële stam, een dier, enz.), ook een "informationele component" bevatten (i.e. hun genetische code). Gelet op het voorgaande zijn de geldende nationale bepalingen met betrekking tot het wettelijke statuut van genetische rijkdommen in België vooral te vinden in het eigendomsrecht van genetisch materiaal. Het juridisch eigendom van genetisch materiaal als biofysische entiteit vloeit voort uit de voorwaarden en regels die de eigendom regelen van het organisme waarin dit materiaal kan worden gevonden, welke vastgelegd zijn door de basisprincipes van het eigendomsrecht in het burgerlijk wetboek. De eigendom op een organisme betekent dat de eigenaar het recht heeft om het organisme te gebruiken, ervan te genieten en er materieel en juridisch over te beschikken. Bovendien zou elke wettelijke maatregel waarin de regulering van de toegang tot genetische rijkdommen wordt overwogen, voordeel kunnen halen uit de bestaande wetgeving die de toegankelijkheid en het gebruik van genetisch materiaal reguleert. Deze wetgeving varieert naargelang het soort eigendom van het materiaal (roerend, onroerend of res nullius), het bestaan van beperkingen op het eigendomsrecht zoals een specifieke bescherming (beschermde soorten, beschermde natuurgebieden, bossen of mariene omgevingen) en de locatie van het genetische materiaal (de vier bevoegde instanties passen elk hun eigen regels toe). In tegenstelling tot de fysieke componenten kunnen de informationele componenten van de genetische rijkdommen aanzien worden als res communis: "zaken die niemands eigendom zijn en door iedereen gebruikt mogen worden". De toegang tot dergelijke informationele componenten valt niet onder een specifieke wetgeving, maar de uitoefening van bepaalde gebruiksrechten kan wel beperkt worden door het intellectuele eigendom dat werd toegestaan op uitvindingen die betrekking hebben op een voortbrengsel dat uit biologisch materiaal bestaat of dit bevat, of op een werkwijze waarmee biologisch materiaal wordt verkregen, bewerkt of gebruikt. Deze intellectuele eigendomsrechten kunnen de vorm aannemen van octrooien, bescherming van kweekproducten of geografische indicaties. Naast deze principes met betrekking tot het wettelijke statuut van genetische rijkdommen bieden enkele burgerrechtelijke, strafrechtelijke en internationale privaatrechtelijke regels ook aansprakelijkheids-en schadeloosstellingsmogelijkheden voor gevallen waarin een illegale verwerving van genetische rijkdommen wordt vastgesteld. Hun toepassing varieert naargelang de aard van het goed (fysieke goederen of informationele goederen), maar ook naargelang de plaats waar de illegale verwerving gebeurt. Tot slot zijn er in België momenteel geen wettelijke bepalingen waarin de concepten "traditionele kennis", "traditionele kennis met betrekking tot genetische rijkdommen" en "inheemse en lokale gemeenschappen" uitdrukkelijk zijn vastgelegd. Traditionele kennis en de rechten van inheemse en lokale gemeenschappen werden echter wel aangekaart in enkele internationale akkoorden waarbij België partij is, zoals het Verdrag nr. 107 van de Internationale Arbeidsorganisatie (IAO) betreffende inheemse en in stamverband levende volken uit 1957, het Verdrag nr. 169 van de IAO betreffende inheemse en in stamverband levende volken, en de VN-verklaring over de rechten van inheemse volken. Voorbereidende aanbevelingen met betrekking tot de opties voor de implementatie van het Protocol van Nagoya Hoewel het Protocol van Nagoya een recent protocol is, is het niettemin de verdere uitvoering van de derde doelstelling van het VBD, welke basisprincipes en ABS aanverwante bepalingen bevat, zoals de soevereine rechten van Staten op hun natuurlijke rijkdommen, de eerlijke en billijke verdeling van voordelen en het belang van inheemse en lokale gemeenschappen en hun traditionele kennis. Verschillende Partijen bij het VBD wereldwijd hebben daarom ABS-maatregelen ingevoerd, welke nuttige ervaringen opleveren voor de implementatie van het Protocol. Op basis van deze ervaringen werden twee groepen voorbereidende aanbevelingen uitgewerkt in deze studie, die betrekking hebben tot de beschikbare opties voor de implementatie van het Protocol in België. De eerste groep aanbevelingen houdt verband met de vereiste instrumenten voor de naleving van de kernverplichtingen die voortvloeien uit het Protocol7 . De tweede groep aanbevelingen houdt verband met belangrijke bijkomende maatregelen waarmee rekening moet worden gehouden bij de naleving van de verplichtingen, maar die verder gaan dan de kernverplichtingen. Voor het implementeren van de kernverplichtingen worden de volgende aanbevelingen gedaan:  De toegangsvoorwaarden verduidelijken: dankzij haar soevereine rechten op de genetische rijkdommen kan België kiezen of gebruikers al dan niet de voorafgaande geïnformeerde toestemming (Prior Informed Consent, PIC) van de bevoegde instantie moeten verkrijgen om toegang te krijgen tot de genetische rijkdommen die onder haar bevoegdheid vallen.  De format van de onderling overeengekomen voorwaarden bepalen: Eenmaal het Protocol van Nagoya in werking treedt in België, moeten gebruikers die op Belgisch grondgebied actief zijn de voordelen die voortvloeien uit het gebruik van genetische rijkdommen verdelen. Die verdeling moet gebeuren op basis van onderling overeengekomen voorwaarden (Mutually Agreed Terms, MAT). Het Protocol van Nagoya legt echter geen specifiek format op voor deze onderling overeengekomen voorwaarden. Deze kunnen worden overgelaten aan het goeddunken van belanghebbenden of voortvloeien uit richtlijnen en/of verplichte maatregelen die door de Staat worden opgelegd.  Ervoor zorgen dat ABS bijdraagt aan behoud en duurzaam gebruik van biodiversiteit: men moet ervoor zorgen dat de implementatie van het Protocol bijdraagt tot de twee andere doelstellingen van het VBD: het behoud van de biologische diversiteit en het duurzame gebruik van bestanddelen daarvan. Dit is bijvoorbeeld mogelijk door aan de PIC verplichte voorwaarden te koppelen voor het verdelen van voordelen of door een "voordelenverdelingsfonds" op te richten waarbij de voordelen voor behoud en duurzaam gebruik van biodiversiteit worden bestemd.  De toegang faciliteren voor biodiversiteit gerelateerd onderzoek: om onderzoek naar biodiversiteit te stimuleren en om niet-commercieel onderzoek met genetische rijkdommen niet te overbelasten, kunnen maatregelen worden uitgewerkt om de toegang tot genetische rijkdommen te faciliteren voor niet-commercieel en biodiversiteit gerelateerd onderzoek.  Een Bevoegde Nationale Instantie oprichten: elke Partij moet een Bevoegde Nationale Instantie (Competent National Authority) aanstellen. Deze instantie is verantwoordelijk voor het verlenen van toegang, of, indien van toepassing, voor de afgifte van schriftelijk bewijs dat voldaan is aan de vereisten voor toegang en voor advisering over de toepasselijke procedures en vereisten voor het toegang krijgen tot genetische rijkdommen. Gelet op de institutionele realiteit in België kan meer dan één Bevoegde Nationale Instantie worden aangesteld. Deze aanstelling is van de hoogste prioriteit, aangezien België uiterlijk op de datum van inwerkingtreding van het Protocol het VBD Secretariaat in kennis moet stellen van de contactgegevens van haar bevoegde nationale instantie of instanties (en van haar nationale contactpunt, dat reeds is aangesteld).  De wetgeving van oorsprongslanden bindend maken: als onderdeel van de implementatie van het Protocol moeten de basisverplichtingen worden vastgelegd waaraan gebruikers moeten voldoen bij het gebruik van genetische rijkdommen in België. Deze verplichting komt neer op het bindend maken van de wetgeving van het oorsprongsland inzake PIC en MAT. Dit zou kunnen gebeuren door in de Belgische wetgeving te verwijzen naar de ABS-wetgeving van het oorsprongsland, of door een op zichzelf staande verplichting vast te leggen in de Belgische wetgeving die PIC en MAT oplegt, indien vereist door het oorsprongsland.  Controlepunt(en) vastleggen om het gebruik van genetische rijkdommen te volgen: om het Protocol van Nagoya na te leven moet minstens één instelling worden aangeduid die als controlepunt zal fungeren om het gebruik van genetische rijkdommen te volgen en de transparantie over het gebruik daarvan te vergroten. Het kan om een nieuwe of bestaande instelling gaan. Wat bijkomende maatregelen betreft, moet het volgende worden overwogen: a) de vereisten voor de MAT verduidelijken; b) een duidelijke en transparante toegangsprocedure uitwerken; c) bijkomende rechten en plichten van de bevoegde nationale autoriteiten verduidelijken; d) een monitoringssysteem invoeren; e) aanmoedigingsmaatregelen voorzien voor de naleving van wet-of regelgeving door gebruikers; en f) de ontwikkeling van contractuele modelbepalingen, gedragscodes en richtlijnen stimuleren. Geselecteerde opties voor de implementatie van het Protocol van Nagoya Gelet op de hierboven beschreven voorbereidende aanbevelingen met betrekking tot de beschikbare opties voor de implementatie van het Protocol, werden zes maatregelen, elk inclusief verschillende beleidsopties, besproken op de eerste vergadering met belanghebbende partijen op 29 mei 20128 . Op basis van deze vergadering selecteerde het Stuurcomité van de studie deze maatregelen voor een grondigere analyse van ecologische, maatschappelijke, economische en procedurele gevolgen van hun implementatie. Alvorens deze maatregelen in te voeren, moet worden besloten of PIC en de verdeling van de voordelen (benefit-sharing) als algemene vereisten moeten gelden in België. Hoewel dit laatste nodig is voor de naleving van het Protocol, vloeit het eerste voort uit de soevereine rechten die België bezit op haar genetische rijkdommen en is het niet nodig voor de naleving van het Protocol. Indien PIC als een algemeen principe wordt beschouwd, moet een procedure worden uitgewerkt voor de toegang tot de eigen genetische rijkdommen van België (maatregel 1). Dit kan door de bestaande wetgeving aan te passen, door op gekwalificeerde ex-situ collecties te vertrouwen, door een voorafgaande notificatie te vereisen of door een combinatie van deze instrumenten. Maatregel 1: de toegang tot genetische rijkdommen operationaliseren Optie 0 -Geen voorafgaande geïnformeerde toestemming Een voorafgaande geïnformeerde toestemming is niet vereist voor het gebruik van genetische rijkdommen en traditionele kennis in België; 5. Optie 1 -Het "bottleneck" model a. Voor beschermde genetische rijkdommen: de toegang wordt mogelijk gemaakt door de bestaande wetgeving relevant voor beschermde natuurgebieden en beschermde soorten te verfijnen; b. Voor niet-beschermde genetische rijkdommen: de toegang wordt mogelijk gemaakt via Belgische ex-situ collecties. 6. Optie 2 -Het "fishing net" model a. Voor beschermde genetische rijkdommen: de toegang wordt mogelijk gemaakt door de bestaande wetgeving relevant voor beschermde natuurgebieden en beschermde soorten te verfijnen; b. Voor niet-beschermde genetische rijkdommen: de toegang wordt toegestaan na notificatie aan de bevoegde instantie. 7. Optie 3 -Het aangepaste "fishing net" model Indien de verdeling van de voordelen als een algemene vereiste wordt beschouwd, moeten de specifieke vereisten voor het opstellen van de Onderling Overeengekomen Voorwaarden (MAT) worden gespecificeerd (maatregel 2). Het bepalen van deze vereisten kan worden overgelaten aan de gebruikers en aanbieders (optie 1), of op een min of meer gestandaardiseerde wijze worden opgelegd door de staat (optie 2 en 3). Maatregel 2: specificeren van de vereisten voor het opstellen van Onderling Overeengekomen Voorwaarden 4. Optie 0: Geen verdeling van voordelen voor het gebruik van genetische rijkdommen en traditionele kennis in België. 5. Optie 1: De bevoegde autoriteiten leggen geen specifieke vereisten op voor het opstellen van de MAT. Het staat gebruikers en aanbieders vrij om gezamenlijk te beslissen over de inhoud. 6. Optie 2: Specifieke vereisten voor het opstellen van MAT worden opgelegd, inclusief door middel van contractuele modelbepalingen die verschillen naargelang van het doel van de toegang. 7. Optie 3: Specifieke vereisten voor het opstellen van MAT worden opgelegd, maar zonder contractuele modelbepalingen. Die specifieke vereisten verschillen naargelang van het doel van de toegang Ze vormen de basis voor onderhandelingen over MAT door de gebruikers en aanbieders van genetische rijkdommen die geval per geval zullen plaatsvinden. Met het oog op de naleving van het Protocol van Nagoya moeten een of meer bevoegde nationale instanties worden aangesteld (maatregel 3). Zij moeten toegang verlenen, schriftelijk bewijs verschaffen dat voldaan is aan de vereisten voor toegang en/of gebruikers adviseren over de toepasselijke procedures en vereisten voor het toegang krijgen tot genetische rijkdommen. Om die taken uit te voeren moeten de bevoegde nationale instanties aanspreekpunten voorzien voor de gebruikers van genetische rijkdommen. Dergelijke aanspreekpunten kunnen afzonderlijk worden voorzien, waarbij elke instantie zijn eigen aanspreekpunt heeft (optie 1), of gezamenlijk, waarbij er één enkel aanspreekpunt is voor de verschillende instanties (optie 2). a. Voor beschermde genetische rijkdommen en genetische rijkdommen die al onder een specifieke relevante wetgeving vallen: de toegang wordt mogelijk gemaakt door de bestaande wetgeving te verfijnen; b. Voor niet-beschermde genetische rijkdommen: de toegang wordt toegestaan na notificatie aan de bevoegde instantie. Maatregel 3: een of meer bevoegde nationale instanties aanstellen Met het oog op de naleving van het Protocol van Nagoya door gebruikers moet minstens één controlepunt worden voorzien voor de monitoring van het gebruik van genetische rijkdommen en traditionele kennis in België (maatregel 5). Indien België besluit om controlepunten in te voeren, kan de invoering daarvan in verschillende fasen gebeuren. Gelet op het politieke engagement voor de tijdige ratificatie van het Protocol van Nagoya, zou in de eerste fase naar een minimale invoering kunnen worden gekeken, met de oprichting van één enkel controlepunt. Voor die eerste fasen lijken twee mogelijke opties relevant, nl. het monitoren van de PIC van de gebruiker, die beschikbaar is via de ABS Clearing-House (optie 1), en het upgraden van de bestaande verplichting van vermelding van de geografische oorsprong in de octrooiaanvragen (optie 2). Aangezien optie 1 en optie 2 elkaar niet uitsluiten, kan een gezamenlijke invoering worden overwogen. Maatregel 5: een of meer controlepunten aanduiden 3. Optie 0: België voorziet geen controlepunten voor de monitoring van het gebruik van genetische rijkdommen en traditionele kennis. 4. Optie 1: het monitoren van de PIC van de gebruiker, die beschikbaar is via de ABS Clearing-House 5. Optie 2: Het octrooibureau wordt als controlepunt gebruikt voor de monitoring van het gebruik van genetische rijkdommen en traditionele kennis. Tot slot zal een Belgische component van of aanspreekpunt voor het ABS Clearing-House worden voorzien, ter ondersteuning van de uitwisseling van informatie over specifieke ABS-maatregelen in het kader van het Protocol van Nagoya (maatregel 6). Hoewel er internationaal nog wordt gediscussieerd over de precieze modaliteiten van het ABS Clearing-House, werden de volgende drie kandidaten reeds geïdentificeerd: het Koninklijk Belgisch Instituut voor Natuurwetenschappen (optie 1), het Federaal Wetenschapsbeleid (optie 2) en het Wetenschappelijk Instituut Volksgezondheid (optie 3). Impact van de geselecteerde opties voor de implementatie van het Protocol van Nagoya De mogelijke gevolgen van de invoering van de bovenvermelde opties werden geëvalueerd door middel van een vergelijkende multicriteria-analyse. Aan de hand van deze analyse konden ook de mogelijk betrokken belanghebbenden worden geïdentificeerd. Wat de operationalisering van de toegang tot genetische rijkdommen betreft (maatregel 1), kwamen het "bottleneck" model (optie 1) en het aangepaste "fishing net" model (optie 3) als beste uit de analyse. De voorkeur voor deze opties kan verklaard worden door het feit dat ze verwacht worden meer rechtszekerheid te zullen bieden, een positiever impact te hebben op het milieu en beter bij de bestaande praktijken te passen dan de andere twee opties. Voor deze opties moet eerst als algemene vereiste worden ingevoerd dat voor de toegang tot Belgische genetische rijkdommen een voorafgaande geïnformeerde toestemming vereist. Wat de specificering van de vereisten voor het opstellen van Onderling Overeengekomen Voorwaarden betreft (maatregel 2), scoorden de twee opties waarbij specifieke vereisten worden bepaald in Belgie (optie 2 en optie 3) beter dan de optie waarbij geen specifieke vereisten worden opgelegd (optie 1). De reden hiervoor zijn hun goede economische, ecologische en procedurele prestaties (optie 2 biedt ook nog goede maatschappelijke prestaties). Om deze opties te kunnen kiezen moet de verdeling van voordelen als een algemene vereiste worden ingevoerd in België. Naast de oprichting van de bevoegde nationale instanties, was ook de oprichting van een gecentraliseerd aanspreekpunt duidelijk de aanbevolen optie (optie 2 van maatregel 3). Wat de uitwerking van nalevingsmaatregelen betreft (maatregel 4), scoort de optie om terug te verwijzen naar de wetgeving van het oorsprongsland (optie 1), met de Belgische wetgeving als terugvaloptie, het best. Dit valt voornamelijk te verklaren door de overeenkomst tussen deze optie en de bestaande praktijken (overeenkomstig het Belgische wetboek van internationaal privaatrecht). Wat de aanduiding van een of meer controlepunten betreft (maatregel 5), is het monitoren, in het ABS Clearing-House, van de door de gebruikers verkregen voorafgaande geïnformeerde toestemming, de aanbevolen optie. Die optie scoort minstens even goed voor alle criteria, en biedt betere maatschappelijke en procedurele prestaties. Tot slot, wat de uitwisseling van informatie via het ABS Clearing-House betreft (maatregel 6), gaat de voorkeur uit naar de aanstelling van het Koninklijk Belgisch Instituut voor Natuurwetenschappen (KBIN), dat voor de meeste onderzochte criteria beter presteert dan de andere opties. Aanbevelingen volgend op de impactanalyse Uit de impactanalyse van deze studie vloeien twee algemene aanbevelingen voort, alsook enkele specifiekere aanbevelingen voor elk van de bovenvermelde maatregelen. Ten eerste blijkt uit de analyse dat de opties die geen beleidsverandering met zich meebrengen (de "0" optie voor elke maatregel) duidelijk de slechtste prestaties bieden. Dat resulteert in een eerste algemene aanbeveling, nl. dat zowel een voorafgaande geïnformeerde toestemming (PIC) als de verdeling van voordelen (benefit-sharing), als algemene vereisten moeten worden ingevoerd in België. Ten tweede bleek uit de analyse de meerwaarde van een gefaseerde aanpak voor de implementatie van het Protocol. Op die manier kan voordeel worden gehaald uit de tijdige invoering van de basisprincipes en kunnen specifiekere keuzes in een later stadium worden gemaakt. Bovendien is een gefaseerde aanpak nodig om het Protocol van Nagoya tijdig te ratificeren en België toe te laten om deel te nemen als een Partij bij het Nagoya Protocol tijdens de eerste bijeenkomst van de Partijen in oktober 2014. Tot slot leverde de impactanalyse enkele specifieke aanbevelingen op voor elk van de zes maatregelen: 1. Naast de oprichting van de Bevoegde Nationale Instanties moet ook een gecentraliseerd aanspreekpunt worden uitgewerkt voor deze instanties. 2. Wat de maatregelen inzake naleving met wet-of regelgeving (compliance) betreft, moeten sancties worden voorzien wanneer de niet-naleving van de PIC en de MAT, zoals opgelegd door het oorsprongsland, wordt vastgesteld. Voor het controleren van de inhoud van de MAT zou een bepaling in het Wetboek van internationaal privaatrecht moeten verwijzen naar de wetgeving van het oorsprongsland, met Belgische wetgeving als een eventuele terugvaloptie. 3. In de eerste uitvoeringsfase zou het controleren van het gebruik van genetische rijkdommen en traditionele kennis moeten gebeuren op basis van de PIC die beschikbaar is via de ABS Clearing-House. 4. Met betrekking tot de toegang tot Belgische genetische rijkdommen is het aanbevolen bestaande relevante wetgeving inzake beschermde natuurgebieden en beschermde soorten te verfijnen, in combinatie met een algemene notificatievereiste voor de toegang tot andere genetische rijkdommen. In latere uitvoeringsfasen kan bijkomende relevante wetgeving dan worden verfijnd, en kan het verwerken van andere toegangsaanvragen overgelaten worden aan ex-situ collecties. 5. In de eerste uitvoeringsfase, en los van de algemene verplichting om de voordelen te verdelen, zouden er geen specifieke vereisten moeten worden opgelegd voor het opstellen van Onderling Overeengekomen Voorwaarden (Mutually Agreed Terms). Een combinatie van meer specifieke vereisten, met de mogelijkheid om standaardakkoorden te gebruiken, kan in een latere uitvoeringsfase worden overwogen. 6. Het Koninklijk Belgisch Instituut voor Natuurwetenschappen zou moeten gemandateerd worden om de informatieuitwisselingstaken in verband met toegang en verdeling van de voordelen in het kader van het Protocol van Nagoya te vervullen, via het ABS Clearing-House. Implementatie van de aanbevelingen Om deze aanbevelingen te implementeren kan voor de gefaseerde aanpak een driestappenproces worden gevolgd: 1. ALs eerste stap kan een politiek akkoord worden afgesloten tussen de bevoegde autoriteiten, die de algemene vereisten uitschrijft en een opsomming maakt van de acties die de federale overheid en de deelstaten moeten ondernemen om deze principes in de praktijk om te zetten. Hiertoe behoren onder andere: a. Het invoeren van de verdeling van voordelen (benefit-sharing) als algemeen vereiste in België. b. Het invoeren van een algemeen principe dat bepaalt dat voor de toegang tot Belgische genetische rijkdommen een PIC nodig is. c. Het bepalen dat van vier Bevoegde Nationale Instanties zullen worden opgericht. d. Het voorzien van wetgevende maatregelen die ervoor zorgen dat het gebruik van genetische rijkdommen onder Belgisch rechtsgebied onderhevig is aan voorafgaande geïnformeerde toestemming (PIC) en onderling overeengekomen voorwaarden (MAT), zoals vereist door de wetgeving van het oorsprongsland. Deze maatregelen moeten er ook in voorzien dat de niet-naleving van deze regels wordt aangepakt. e. Het aanduiden van het Belgisch knooppunt van het VBD Clearing-House Mechanisme, beheerd door het KBIN, als de Belgische deelname aan de ABS Clearing-House in het kader van het Protocol van Nagoya. De reden om een dergelijke politiek akkoord te gebruiken is tweeledig. Enerzijds verschaft het een duidelijk politiek engagement ten opzichte van de kernverplichtingen van het Protocol van Nagoya. Het vermeldt immers de intenties van de bevoegde autoriteiten, binnen de grenzen van de beslissingen die reeds op internationaal en Europees vlak werden genomen op het moment van het akkoord. Anderzijds loopt een dergelijk akkoord niet vooruit op de politieke beslissingen die nog moeten genomen worden door de bevoegde autoriteiten en is het dus voldoende flexibel om het uitvoeringsproces in een later stadium verder aan te passen. Dit laatste is vooral belangrijk gezien de momenteel vele onbeantwoorde vragen, zowel op Europees als op internationaal vlak, die in het evaluatieverslag werden vermeld en behandeld. 2. In de tweede stap zouden de specifieke acties moeten worden geïmplementeerd, bijvoorbeeld aan de hand van een samenwerkingsakkoord en/of door bepalingen toe te voegen aan relevant wetgeving zoals de milieuwetgeving van de deelstaten en de federale overheid, naast andere mogelijke vereisten. 3. Als derde stap kunnen bijkomende acties worden ondernomen eens er meer duidelijkheid is op internationaal en Europees vlak. INTRODUCTION This study aims to contribute to the ratification and the implementation in Belgium of the Nagoya Protocol (NP) on Access and Benefit-sharing (ABS) of the Convention on Biological Diversity 9 . The need for this study was decided by the Interministerial Conference on the Environment of 31 st March 2011 to allow for an early ratification by Belgium of the NP. The objective of the study is to identify and evaluate the possible consequences for the Belgian national legislation and regulation, as well as for Belgian stakeholders, resulting from the implementation of the NP in Belgium. The study involves four phases of work:  Phase 1: Analysis of the regulatory framework of ABS in Belgium  Phase 2: Identification of options and recommendations for possible measures and instruments (legal and non-legal) for the implementation of the NP in Belgium  Phase 3: Impact analysis of the selected options  Phase 4: Conclusions and recommendations Background to ABS and the Nagoya Protocol The Nagoya Protocol on Access to Genetic Resources and the Fair and Equitable Sharing of Benefits Arising from their Utilization is a protocol to the UN Convention on Biological Diversity (CBD) 10 . The objective of the NP is expressed as follows: The objective of this Protocol is the fair and equitable sharing of the benefits arising from the utilization of genetic resources, including by appropriate access to genetic resources and by appropriate transfer of relevant technologies, taking into account all rights over those resources and to technologies, and by appropriate funding, thereby contributing to the conservation of biological diversity and the sustainable use of its components. (Article 1 NP) The CBD is the main international framework for the protection of biodiversity. It has three objectives: (1) the conservation of biological diversity, (2) the sustainable use of its components and (3) the fair and equitable sharing of benefits arising from the utilization of genetic resources (GR), including through access. With currently 193 Parties, the CBD has almost universal membership. Since 1996, Belgium is a Party to the CBD, as is the EU and its other Member States.  44(n) Promote the wide implementation of and continued work on the Bonn Guidelines on Access to Genetic Resources and Fair and Equitable Sharing of Benefits arising out of their Utilization, as an input to assist the Parties when developing and drafting legislative, administrative or policy measures on access and benefit-sharing as well as contract and other arrangements under mutually agreed terms for access and benefit-sharing; and  44(o) Negotiate within the framework of the CBD, bearing in mind the Bonn Guidelines, an international regime to promote and safeguard the fair and equitable sharing of benefits arising out of the utilization of genetic resources 13 . This led to the granting of a detailed negotiating mandate for the ABSWG by the CBD COP7 and negotiations were undertaken at CBD COP8 in March 2006. Guided by the Bonn Roadmap (adopted at COP8), Parties committed themselves to complete negotiations at the earliest possible time before CBD COP10 in October 2010. Formal agreement on the textual basis for the final negotiations was only achieved in July 2010, following numerous negotiation meetings between COP9 and COP10 14 . On 30 th October 2010, the final plenary of CBD COP10 successfully adopted the Nagoya Protocol on "Access to Genetic Resources and the Fair and Equitable Sharing of Benefits Arising from their Utilization". The NP elaborates on and implements the basic principles laid down in the CBD. Of relevance are its Articles 15 and 8(j), in particular: Article 15(1) of the CBD recognizes the sovereign right of States over their natural resources and that the authority to determine access to GR rests with the national governments and is subject to national legislation.  Fair and equitable sharing of benefits arising from GR utilization Pursuant to Article 15(7) of the CBD, the results of research and development and the benefits arising from the commercial and other utilization of GR must be shared in a fair and equitable way with the Contracting Party providing such resources on Mutually Agreed Terms (MAT).  Role and importance of indigenous and local communities (ILCs) and their traditional knowledge (TK) Article 8(j) of the CBD lays down that each contracting Party must, as far as possible and as appropriate and subject to its national legislation, respect, preserve and maintain knowledge, innovations and practices of ILCs embodying traditional lifestyles relevant for the conservation and sustainable use of biological diversity. With the approval and involvement of the holders of such knowledge, innovations and practices, wider application should be promoted and the equitable sharing of the benefits arising from the utilization of such knowledge, innovations and practices should be encouraged.  Adoption and entry into force of the NP The text of the NP was formally adopted on 30 th October 2010 15 and the NP was opened for signature on 2 nd February 2011 till 1 st February 2012 16 . Only Parties to the CBD can sign the NP and only States and Regional Economic Integration Organizations having signed the NP when it was open for signature, can proceed to ratify it 17 . Others will have to accede to the Protocol. Signature in itself does not establish consent to be bound, hence the necessity of an act of ratification 18 or accession 19 . The NP will enter into force on the ninetieth day after the date of deposit of the 50 th instrument of ratification, acceptance, approval or accession by States or REIO that are Parties to the Convention 20 . The Secretary-General of the UN serves as the Depositary of the Protocol 21 . 50 ratifications or equivalent instruments are needed in order for the NP to enter into force. Consequently, there will be one single date of entry into force for the first 50 ratifying Parties, i.e. 90 days after deposit of the 50 th instrument 22 . The ratifying Parties will be bound by treaty obligations upon entry into force. Another date of entry into force will apply for any Party depositing their act of accession after the 15 COP 10 Decision X/1, Access to genetic resources and the fair and equitable sharing of benefits arising from their utilization. Available at: http://www.cbd.int/decision/cop/?id=12267. Nagoya Protocol on Access to Genetic Resources and the Fair and Equitable Sharing of Benefits Arising from Their Utilization to the Convention on Biological Diversity, Nagoya, 29 th October 2010, available at http://www.cbd.int/ cop10/doc/ (accessed 30 th December 2010). 16 Ibid. 91 States and 1 regional economic integration organization (REIO), i.e. the EU, have signed the NP. 17 The NP was open for signature by Parties to the CBD. See NP, op. cit., Article 32. 18 Ratification requires the deposit of a formal instrument following completion of internal procedures, as determined by the constitutional law of each Party. 19 The NP remains open to accession for Parties who have not signed it during the time when it is open for signature. 20 See NP, op. cit., Article 34(1). 21 COP 10 Decision X/1, op. cit. 22 It should be noted that the EU instrument of approval is not to be counted as additional to those ratification instruments deposited by the EU Member States since the NP falls within an area of shared competences. See NP, op. cit., Article 34(3). date of deposit of the 50 th instrument (i.e. 90 days after deposit of their instrument23 ). At the time of writing, 15 States had ratified the NP24 . The entry into force of the NP also determines the date of the 1 st Meeting of the Parties to the Nagoya Protocol (NP COP/MOP) and consequently also the decisionmaking capacity of this organ. COP/MOP1 is expected to be held in 2014, concurrently with CBD COP12. Annex 1 of this report contains an analysis of the legal obligations emanating from the NP that has been provided with the terms of reference of this study, by the four Belgian environmental administrations that commissioned this study. This list serves as the background for this study. Ratification process in the European Union The EU and eleven Member States signed the NP on 23 th June 2011. Eleven more did so during July/September 2011. Five Member States have not signed it (but can still accede to the Protocol) 25 . The ratification procedure is laid down in Article 218 of the Treaty on the Functioning of the European Union (TFEU). The expression of EU consent to be bound requires a Council Decision to "conclude" the NP with the consent of the European Parliament (EP). The procedure is triggered by a Commission proposal for a decision, which is submitted to the Council and the EP. The EP expresses its consent in a legislative Resolution, but not through the ordinary legislative procedure (does not involve extensive readings, the EP can only give or withhold its consent). It is for the Council to formally adopt the decision by means of Qualified Majority Voting. As required by Article 34 of the CBD, a declaration of competence is to be included in the instrument of approval, meaning that the EU must declare the extent of its competences with respect to matters governed by the NP. Negotiations are currently on-going at EU level on the basis of a proposal from the Commission to implement the NP in the Union. The ratification of the NP by the EU is equally being prepared. Structure of the report Chapters 2 to 5 analyze the current state of the art of ABS in Belgium. Chapter 2 takes stock of the current political and administrative distribution of ABS-related competences in Belgium. Chapter 3 analyzes how genetic resources and traditional knowledge are currently addressed in Belgian law, including the legal implications of their ownership, access and use. Chapter 4 describes currently existing policy measures and other initiatives in Belgium which are directly relevant to the implementation of the Nagoya Protocol and chapter 5 discusses the conformity of the current situation with the obligations of the NP. Chapter 6 then goes on by taking stock of existing measures and instruments (legal and non-legal) used for the implementation of ABS throughout the world. This allows, in chapter 7, for the establishment of preliminary sets of legal, institutional and administrative measures which could be implemented in Belgium. The recommended measures are divided into two separate sets: the first one containing actions to be taken in case of minimal implementation of the core obligations stemming from the NP and the second one containing measures in case of additional implementation. The core obligations reflect the obligations identified in the terms of reference of this study as requiring special attention. Chapter 8 presents and describes the different options for the minimal implementation of core measures stemming from the NP. Those options were discussed at the first stakeholder meeting on the 29 th of May 201226 . Based on the results of the stakeholder meeting, the options to be further examined were selected by the Steering Committee of this study and submitted to an in-depth analysis of environmental, social, economic and procedural impacts. Chapter 9 analyzes the implementation modalities of the options described in chapter 8, taking into account the existing legal and institutional situation in Belgium described in chapters 2 to 5. Chapter 10 then analyzes the potential impact and compares the selected options through a multicriteria analysis using the set of evaluation criteria described below. A ranking of the options is also established. Finally, chapter 11 outlines some recommendations for a set of instruments and measures (legal or non-legal) for the implementation of the Protocol in Belgium. Scope of the study In order to realize the objectives of the Convention on Biological Diversity and the Nagoya Protocol, this study aims to contribute to the ratification and implementation of the Nagoya Protocol in Belgium. It is based on the list of legal obligations emanating from the NP (Annex 1) provided with the terms of reference of this study, by the four Belgian environmental administrations that commissioned this study. For this study, access and utilization of GR are analyzed in the context of the scope of the Nagoya Protocol. The Protocol applies to GR that are provided by Contracting Parties that are countries of origin of such resources or by the Parties that have acquired the GR in accordance with the Convention on Biological Diversity (Article 15.3, CBD). Countries of origin are countries that possess those GR in in-situ conditions (Article 2, CBD). In Belgium this means that these GR exist within ecosystems and natural habitats in Belgium, or, in the case of domesticated or cultivated species, in the surroundings in Belgium where they have developed their distinctive properties (Article 2, CBD). The status of the GR in ex-situ conditions that have been acquired before the entry into force of the Nagoya Protocol is still under discussion. Therefore, this report only considers the  GR that a provider country possesses in in-situ conditions and  GR in ex-situ collections acquired after the entry into force of the Nagoya Protocol and/or in accordance with the obligations of the Convention on Biological Diversity. It is further important to highlight the provisional nature of the findings presented in this document, as the on-going discussions around the implementation of the Nagoya Protocol in international and European fora will further influence the application of the results of this study. THE DISTRIBUTION OF ABS-RELATED COMPETENCES IN BELGIUM In Belgium, competences relating to ABS are divided between the federal level, the Regions and the Communities. This distribution results from successive transfers of competences to federated entities through the five state reforms since 197027 , the general contours of the sixth state reform having being enacted in 201128 . As a general principle, federated collectivities possess the full competence for matters that have been attributed to them, while the Federal State possesses those competences that have been reserved on its behalf by the Constitution or legislation enacted with special voting quorums, as well as those residual competences that have not been otherwise attributed to other entities 29 . ABS potentially encompasses a large range of issues extending far beyond sole environmental matters, including market regulation and access, international trade, agriculture, health, development cooperation, research & development and innovation. Consequently, several departments and several levels of competence could be responsible for the future implementation of the NP, at federal, regional and community level30 . It should however be noted that in 1995, the Regions and the Federal Government have concluded a cooperation agreement on international environmental matters. This cooperation agreement provides inter alia for an Intra-Belgian coordination framework supplied by the Coordination Committee of the International Environment Policy31 that is used for preparing the implementation of the Nagoya Protocol in Belgium. The political distribution of ABS-related competences Environmental policy The main principle pertaining to the distribution of competences with regard to environmental policy and nature conservation is laid out in Article 6- §1, II and III of the special law (SL) of institutional reform dated as of 8/8/1980, which provides for the so-called exclusive regional "competence block" in accordance with Article 39 of the Constitution. This Article has been modified numerous times, especially in 1993, where the competences attributed to regions were notably strengthened. Today it is the three Regions (Flemish Region, Walloon Region and Brussels Capital Region) that are competent on overall environmental policy, and thus have the greatest responsibility in biodiversityrelated issues. However, applicable legislation also reserves a number of competences to the Federal State, as an "exception" to the general competence on environmental policy and nature conservation of the Regions. When reading the text through the lens of ABS issues, it becomes clear that the Regions are inter alia responsible for the following environmental matters 32 :  the protection of the environment, notably of the soil, subsoil, water and air against pollution (…);  nature conservation;  the protection and conservation of nature;  green area zones, park zones, green areas;  forests;  fluvial fishing and fish farming;  non-navigable waterways, including verges, and polders. Although environmental matters are in principle a regional competence, the Federal Government has retained some reserved competences on the following ABS-related environmental matters in accordance with the special law 8/8/80, as an exception to the general regional competence on environmental matters:  Article 6 §1, II indent 2 of SL 8/8/80: the establishment, for purposes of environmental protection, of product norms for market access (regional governments need to be consulted when drafting these norms).  Article 6 §1, III, 2° of the SL 8/8/80: the export, import and transit of non-indigenous plant varieties as well as non-indigenous animal species and their cadavers. 32 Article 6 §1, II of the SL 8/8/80, 1° and Article 6 §1, III, 2°, 3°, 4°, 6° and 7° of the SL 8/8/80; See also Geeraerts K, Bursens P, Leroy P(2004) Vlaams milieubeleid steekt de grenzen over. De Vlaamse betrokkenheid bij de totstandkoming van Europees en multilateraal milieubeleid. Steunpunt Milieubeleidswetenschappen As the Belgian territorial sea is not considered a part of the territory of (one of the) Regions, the exercise of environmental and nature conservation competences within the Belgian territorial sea is considered to fall under the residual competence of the Federal Government. Having specific regards to the potential changes in the distribution of competences triggered by the current sixth reform of the State, competences regarding ABS related environmental policy are not expected to significantly change 33 . Agricultural policy and maritime fishery Agricultural policy, including the application of the European CAP measures is a regional competence in accordance with Article 6- §1 V of the SL 8/8/80. However, Regions are not responsible for the standardization and monitoring of the quality of raw and vegetal material and the standardization and monitoring of animal welfare in order to ensure the security of the food chain, as these are reserved federal competences. The agreement of regional governments should be sought with regard to animal welfare measures affecting agricultural policy. It should be noted that animal welfare legislation shall be transferred to Regions in the near future, in accordance with the terms of the 2011 institutional agreement establishing the framework for the sixth State reform. Furthermore, those quality or origin labels that possess a regional or local character (such as geographical indications for instance), are included within the realm of the regional competences (Article 6- §1 VI, alinea 5, 4°, of SL 8/8/80 that excludes these measures from those competences reserved to the federal level). Research and development Before the third 1988 state reform, the Federal Government was responsible for virtually all research and development (R&D) related activities. With these amendments, major research-related competences where transferred to the federated entities. Fundamental research and higher education, as well as the regulation of researchers' funding and the management of research institutions were transferred to the French and the Flemish Communities, as exclusively cultural subject-matters falling under the scope of Article 127 of the Constitution and Article 4 of the special law of 8/8/80. The 1993 state reform confirmed this evolution by making the federated entities the prime responsible authorities in matters of R&D 34 . 33 However, it might be relevant to note that the botanical garden located in Meise is mentioned in the transfers of competences that the reform would operate. This transfer is subject to the ratification of a cooperation agreement, the socalled "Peeters-Demotte" plan enacted in 2008 but that has not yet been adopted. The agreement states that the botanical garden's estate and management would fall within the federated competences (of the Flemish Region), under specific conditions. Indeed, the current collections would remain under federal ownership, as these would be considered as "leased" to the Flemish Region and the Flemish Community for a limited period, and the access to collections would be open and free of charge to "all researchers", while "mainstream collections" would be accessed at the same price for all visitors. 34 Wautrequin J. (2011), Nouveaux Transferts de Compétences en Matière de Politique Scientifique? Critères D'appréciation. Intervention au colloque 'Paroles de chercheurs. Etats des lieux et solutions ', 4 mars 2011 ; Goux C. (1996), La recherche scientifique dans la Belgique fédérale: examen de la répartition des compétences, Série Faculté de droit de Namur Centre de droit régional, La Charte, Bruges. With the insertion of Article 6bis into the special law of 8/8/80, Communities and Regions -and thus not only the federated entities falling under the scope of Article 127 of the Constitution -have become "competent with regard to scientific research within the framework of their respective competences, including research carried out in execution to international agreements or acts". Communities and Regions became thus competent in the field of research related to the exercise of their respective Community competences. As for the Regions, they are notably responsible for R&D activities in the following fields 35 : • economically oriented and industrial research, i.e. research or critical investigation aimed at discovering knowledge and skills to develop new products, processes or services, or a significant improvement of products, processes or services; • support for R&D and innovation; • research for technological development; • knowledge diffusion in the industrial sector; • research related to the exercise of other Regional competences. Finally, the Federal Government, remains nonetheless "competent for scientific research that is necessary to the execution of its own competences, including those carried out in execution of international agreements or acts" (Article 6 bis- §2). In accordance with Article 6bis- §2 the federal level also remains competent with regard to 36 : • the implementation and organization of data exchange networks between scientific institutions on the national and international level; • the scientific and cultural federal institutions, including their research and public service activities; • the programs and actions requiring a homogenous implementation on the national and international level in the fields and according to the modalities set out by the cooperation agreement aimed at in Article 92bis- §1 of the special law; • the holding of a permanent inventory of the scientific potential of the country; • the participation of Belgium to the activities of the international research organizations according the modalities set out by the cooperation agreements aimed at in Article 92bis- §1 of the special law; Moreover, the Federal Government can take initiatives, establish structures and provide financial resources for scientific research for the matters that are of regional or community competence, but are related to national or international agreements to which Belgium is a Party, or are related to actions and programs exceeding the interest of a Region or a Community. In that case, the Federal Authority must first submit a proposal for cooperation to the Regions and Communities. 35 Ibid. 36 Ibid. The sixth state reform contains a number of measures that might influence the vertical distribution of ABS related competences, as both interuniversity and technological attraction poles would respectively be transferred to Communities and Regions. Economic and industrial policy The second state reform of 1980 granted economic and industrial competences to the Regions37 . Viewed in the ABS context, the relevant subject-matters of these exclusive regional competences are listed in Article 6 §1 VI of the SL 8/8/80 and include (without being limited to): • economic policy, (Article6- §1 VI, indent1°); • export policy, without prejudice to federal competences in terms of both the grant of warranties against risks of import, export and investment, and of multilateral trade policy (Article 6 §1 VI, indent3°b). The Federal Authority holds besides a full competence on the control and monitoring of import and export of goods and services; • natural resources (Article6- §1 VI, indent5°). Furthermore, Article 6 §1 VI, alineas 4 and 5 designates some reserved competences of the Federal Government. With specific regards to the regulation of ABS-related economic matters, the Federal Authority is competent for the general rules related to the organization of business (Article 6 §1 VI, alinea 4,3°). It conserves also a full competence for the following matters: • Competition law and trade practices, excluding the assignment of quality labels and designations of origin, regional or local character, which are attributed to the Regions (Article6 §1, VI, alinea 5, 4°); • Industrial and intellectual property (Article6 §1, VI, alinea5, 7°); • contingent and permits, for import and export of industrial and agricultural products(Article6,- §1, VI, alinea 5, 8°); Foreign policy and development cooperation Since the 1993 revision of the Constitution, the regulation of international relations is divided according to the principle 'in foro interno, in foro externo': the Federal Government, the Communities and Regions are all responsible for foreign policy related to their respective material competences 38 . Currently, the development cooperation is a shared competence between the Federal Authority, the Regions and the Communities. In this framework, the Federal Authority holds a general competence, whose scope is thus not limited to the other federal material competences. As for them, the Regions and Communities are only competent for the matters related to their material competences39 . The administrative distribution of ABS-related competences At the federal level The main public service at the federal level competent for the implementation of ABS is the Federal Public Service for Health, Food Chain Safety and Environment (FPS Health, Food Chain Safety and Environment). The Directorate-General for the Environment (DG5) is involved in the negotiation and follow-up of a number of international environmental treaties related to its competences. In order to set up the Belgian position at EU and international level, a coordination process with other federal departments and with the federated entities is established since 1995 through the Belgian Coordination Committee on International Environmental Policy (CCIEP). The DG5 is also responsible for the protection of the North Sea and deals with trade in animals and plants through the Convention on International Trade in Endangered Species of Wild Fauna and Flora (CITES). Two of its civil servants currently serve respectively as the "Belgian Focal Point for Access and Benefit-sharing" to the CBD and as Belgian Focal Point for Genetically Modified Organisms (GMO) issues related to the Cartagena Protocol. The DG Animal, Plant and Food (DG4) of the FPS Health, Food Chain Safety and Environment, is responsible for the protection against plant diseases, the standardization of food and cosmetic products and the part of the regulation of GMOs. ABS measures might however encompass much more than environmental competences. Hence, a lot of other federal services and administrations might need to contribute to the implementation of ABS in Belgium. The Science Policy (BELSPO) of the Federal Public Planning Service is in charge of the scientific aspects of sustainable development at the federal level and of the implementation of the international obligations of the CBD. It manages long-term scientific support schemes for the federal sustainable development policy. It assures the financing of research activities and makes funds available for CBD implementation and overarches several scientific institutions, including the Royal Belgian Institute of Natural Sciences (RBINS) and the Royal Museum for Central Africa (RMCA), which are major players in Belgian scientific expertise in the field of biodiversity. RBINS ensures the function of Belgian National Focal Point (NFP) to the CBD and the Clearing-House Mechanism (CHM). BELSPO also supports the Federal Council for Sustainable Development (CFDD-FRDO). This council advises the Federal Government on its policy on sustainable development. Particular attention is given to the implementation of international obligations, such as those under the Convention on Biological Diversity. The Interministerial Conference on Science Policy serves as the consultative body between Federal Government and federated entities. Another relevant public service is the Federal Public Service for Economy, SMEs, Middle Classes and Energy (FPS Economy), which is responsible for the overall functioning of markets and the commercialization of (biodiversity-related) goods and services as well as for the regulation of their market approval. Its Directorate-General for Market Regulation and Organization (E3) is responsible for the functioning of the markets for goods and services. Its mission is to create a legal and regulatory environment favorable to business, and to promote effective and fair competition between them. Through the "Immaterial Economy" Service, E3 covers the legal and regulatory framework of intellectual property rights. This service is also responsible for the dissemination of information on these rights and on the technical information to be found in patents. The Directorate-General for Economic Potential (E4) is competent for the follow-up and monitoring of key economic sectors which fall under the competences of the Federal Government. It represents and coordinates Belgian efforts in the international economic institution such as the WTO. This DG is also responsible for allowing or denying the right to import and export goods and services. The Federal Public Service for Finances and the Administration of Customs and Excises, is responsible for the collection of excise duties on imported products, for the monitoring of trade in exotic and endangered plant and animal species under CITES and for the monitoring of import of timber under the Forest Law Enforcement Governance and Trade (FLEGT). The Federal Public Service for Foreign Affairs, Foreign Trade and Development Co-operation regroups different sections which are directly related to the CBD and ABS. Foreign Affairs is responsible for the diplomatic aspects of international negotiations such as those at the CBD. It makes sure the Belgian position in different international forums is consistent. Foreign Trade is in charge of the regulation of international trade of goods and commodities. This service also coordinates the Belgian representation for multilateral trade policy (WTO, OECD) and European trade policy. For foreign economic missions, the FPS is assisted by the parastatal Belgian Foreign Trade Agency which coordinates federal and regional efforts in this matter. The Directorate-General for Development Cooperation (DGD) implements Belgian development cooperation projects. Together with associated research institutions it manages and provides funding and structural assistance for research, capacity building and awareness-raising projects under the CBD. The DGD also manages the Belgian financial contribution to the CBD and the Global Environmental Facility (GEF). It is supported by the Belgian Technical Cooperation (BTC) which is exclusively responsible for implementing direct bilateral cooperation set up by DGD, including biodiversity related cooperation. The Federal Service for Justice regroups various missions which includes the preparation of legislation for the Minister of Justice, the supervision of the operational support to the judiciary power and the monitoring of the execution of administrative and judicial decisions. At the regional level Regional environmental administrations, the Bruxelles Environnement/Leefmilieu Brussel (IBGE-BIM), the Departement Leefmilieu, Natuur en Energie (LNE) of the Flemish government and the Direction générale opérationnelle Agriculture, Ressources naturelles et Environnement (DGARNE) of the Service public de Wallonie are the main official authorities for conservation and sustainable use of biodiversity and genetic resources. These administrations are generally flanked by specialized public agencies such as the Flemish Instituut voor natuur -en bosonderzoek (INBO) or the Flemish Agency for Nature and Forest (ANB) (http://www.natuurenbos.be/), or specific internal administrative units like the Walloon Département de l'Etude du milieu naturel et agricole (DEMNA) among many others. But these power levels also have a strong (and heterogeneous) horizontal breakdown when it comes to ABS-related competences. In Flanders, environmental matters are separated from agricultural matters, for which the Department Landbouw en Visserij is responsible. This is not the case in Wallonia, while in Brussels it is managed by the economic department (Bestuur Economie en Werkgelegenheid, BEW/ Administration de l'Economie et de l'Emploi, AEE). The administrations responsible for foreign trade on the one hand, and for innovation and research policy on the other could play a major role in the regulation and monitoring of non-public research activities, as they are responsible for economically oriented, industrial and innovation research. The following administrations are responsible for these competences: • In Flanders, Departement Economie, Wetenschap en Innovatie (EWI); • in Wallonia, Direction générale opérationnelle Economie, Emploi et Recherche (GDO 6); • in Brussels, BEW/AEE. Each region has its own foreign policy administration, as well as agencies in charge of development cooperation: • In Wallonia, Wallonie-Bruxelles International (WBI) ; • in Flanders, Departement Internationaal Vlaanderen and Vlaams Agentschap voor Internationale Samenwerking (VAIS); • in Brussels, foreign policy is taken care of by the Secretariat General of the Region. Partly linked to these foreign policy administrations are the regional bodies promoting foreign trade. The Agence wallonne à l'Exportation et aux Investissements étrangers (AWEX), Flanders Investment and Trade (FIT) and Brussels Export, are responsible for the management of international entrepreneurship of regional companies and the accommodation of international companies in Belgium. No specific administration is assigned with research in the German Community, but if necessary the Ministry of the German Speaking Community could take the required administrative steps to exercise this competence. The inter-and intra-level coordination of the exercise of ABS-related competences The conclusion of international agreements that fall under the competence of the federal and of federate entities is regulated by the coordination agreement for mixed treaties. This agreement considers three types of international treaties in Belgium40 : (1) treaties under the exclusive federal competence, (2) treaties under the exclusive competence of the Regions and/or Communities and which are concluded and ratified by the regional and/or community Governments and (3) "mixed" treaties when the agreement covers both the competence of the federal and federate entities. The first two types of treaties do not necessarily require coordination between federal and regional authorities. The "mixed" treaty however, must be concluded by a special procedure, agreed on by all concerned Governments, and must also be approved by all competent parliaments. The different power levels coordinate their environmental policy in cross-departmental ways through the Belgian Coordination Committee on International Environmental Policy (CCIEP), for which the secretariat is provided by the Federal Public Sevice for Environment. The CCIEP assigns the task to a specific coordination body (e.g. an existing CCIEP Steering Committee, an ad-hoc Steering Committee, or a coordination group) and appointed experts of the different relevant governments for dealing with specific issues41 . Considering the distribution of competences described previously, the CBD and the NP are obviously "mixed" treaties. The federal and federated governments coordinate issues related to the CBD and the NP through the Biodiversity Steering Committee of the CCIEP. For ABS-related matters, a specific ABS contact group was created under the CCIEP Biodiversity Steering Committee. In cases in which no consensus can be reached through the CCIEP, contentious issues or political issues can be transferred to the Interministerial Conference on the Environment. LEGAL STATE OF THE ART REGARDING ABS IN BELGIUM Access and use of genetic resources under national jurisdiction in Belgium For the analysis below, there are important preliminary distinctions to be highlighted. First, a distinction has to be made between the question of legal ownership of genetic resources in their quality of material goods on the one hand, and the regulation of the access and use of genetic resources according to the Nagoya Protocol as an exercise of a sovereign right (pursuant to Article 15.1 CBD) on the other. Second, it is important to recall the definitions included in the text of the CBD. Article 2 clearly distinguishes between "genetic material" that is defined as any material of plant, animal, microbial or other origin containing functional units of heredity on the one hand, and "genetic resources" that are defined as genetic material of actual or potential value on the other. These definitions make it clear that "genetic resources" are a subset of "genetic material". The distinction between the two terms on the basis of whether or not the material is "of actual or potential value" seems to signify that genetic material only becomes a genetic resource when a use can be or is likely to be ascribed to it42 . The Belgian State holds sovereign rights over its genetic resources and as such can regulate the access and use of these resources by public law measures, as long as these are justified (which in this case would be in particular in the context of the objectives of the CBD and the Nagoya Protocol) and are proportionate to those objectives. Access to genetic resources in their generality (as genetic material having actual or potential value) is not as such yet regulated by Belgian public law measures. However, physical access to genetic material is regulated through various private law provisions and through public regulation of access to genetic materials in national parks and protected species. As this might be relevant in enacting public law measures on genetic resources, physical access to genetic material, including the question of legal ownership over genetic materials as biophysical entities, is briefly discussed hereafter, as well as the legal stakes related to the "informational component" of the genetic resources. Legal status of genetic resources under Belgian legislation The major part of the currently available national provisions addressing the status of and access to genetic resources relates to the regulation of physical access to the genetic material itself, as found in property law and the liability and redress options made available under both civil and criminal procedures related to the enforcement of property rights. The conditions and rules surrounding the legal ownership of genetic material follow from those governing the ownership of the organism as a whole. Eventual restrictions of use can be put upon genetic resources' informational component through intellectual property rights. The violations of both property rights and intellectual property rights are sanctioned through criminal and civil liability procedures, mainly directed at theft charges. Unauthorized access to the informational component of genetic resources is as such today not sanctioned by legislation pertaining to property rights, but should rather be sought under the umbrella of concealment or breach of trust proceedings. Physical access to genetic material subject to property law Belgium is a civil law country, with a property regime centered on the exercise of three categories of prerogatives that follow from legal ownership of goods: the right to use the good (usus), to perceive its benefits and fruits (fructus) and to alienate it (abusus). The central tenets of the right to property established by Articles 544 to 546 of the civil code are as follows43 :  The property of soil includes the property above and beneath (Article 552 of the civil code), limited in its concrete application by laws and regulations pertaining for instance to the exploitation of mines (as by the Decree of Walloon Regional Council of 7 th June 1988, M.B., 27 th January 1989)  The property extends to all the fruits and the products generated by the material good (Article 546 of the civil code), except when the production is the result of a third party's activity, in which case the proprietor would have to reimburse the costs of labor and seeds borne by the third party, in accordance with the theory of unjust enrichment of Article 548 of the civil code ("enrichissement sans cause")  The property of soil extends to all that is united and incorporated to it, to everything that constitutes its accessory through mechanisms coined natural or artificial accessions regulated by Article 546 of the civil code44 . Therefore, the conditions and rules surrounding the legal ownership of the genetic material as a biophysical entity (such as a plant specimen, a microbial strain, an animal, etc.) follow from those governing the ownership of the organism as a whole:  If the organism as a whole is "res nullius": then the bona fide possession of the organism or the specimen leads to legal ownership of the genetic material. o Example: bees as governed by Article 14 of the rural code, which states that when a swarm is in liberty, it is res nullius, until it settles on a specific beehive, where it becomes the property of the person who owns the land to which the hive is attached; also fish in rivers, wild animals, etc.  If the organism as a whole is personal property that is by definition movable: then the legal ownership of the genetic material is a consequence of the legal ownership of the organism as a whole. o Example: flowers bought on the market  If the organism as a whole is a real property (immovable) by incorporation or destination falling under the realms of full private property: then the legal ownership of the genetic material is a consequence of the legal ownership of the organism as a whole. The holder of this private property can be the state (if the good is on state land) or a private person (if the good is on private land) o Example: Article 524 of the civil code governing domestic animals in cages; trees; etc. Property over a specimen and/or its genetic code means that the proprietor possesses, in accordance with the central tenets of Belgian national law, the rights to use, perceive the benefits and alienate the specimen. Access to the informational component of genetic resources In today's growingly digitalized world, access to the informational, rather than the biophysical component of genetic resources can be quite easily provided for, yet difficultly controlled. As opposed to genetic resources' physical specimens, these resources' informational components may constitute a res communis viewed as "things (as light, air, the sea, running water) incapable of entire exclusive appropriation, thereby owned by no one and subject to use by all". However, these resources might also be viewed within a property regime parallel to that of the material components of GR for reasons of clarity and legal coherence. As such, access to such informational components is today not covered by subject-specific legislation, as it does not fall under property laws. The exercise of some use rights can however be limited through intellectual property rights that have been recognized on portions, functions, or uses of biological material resulting from innovations on these materials (precluding thus the material or the information as it directly found in nature) Genetic resources subject to intellectual property law In the context of the discussion on the relevant legislation on intellectual property law, it is important to remember the scope of this study. This report only considers the genetic resources that a provider country possesses in in-situ conditions or has acquired in accordance with the obligations of the Convention on Biological Diversity. Moreover, for these resources, it considers possible measures for implementing the Protocol in relation to the exercise of national sovereignty of States over these resources in their generality. Therefore, the discussion on intellectual property rights (IPR) is relevant insofar it is related to the further downstream utilization of genetic resources. This discussion will be particularly useful for evaluating the best available options for the monitoring process, e.g. a patent application might be an indication of commercial interest in the genetic resource and an upgraded patent application could potentially be used as a checkpoint. The competence pertaining to intellectual property rights is reserved to the federal level, as a formal exception to the attributed competence of regions in terms of economic policy (Article 6 §1 VI, indent 4, 7° of SL8/8/80). However, protection tools which constitute designations of origin with a regional or local character fall under regional competence (Article 6 §1 VI, indent 4, 4° of SL8/8/80). In this framework, three categories of IPR protection can be distinguished: patents, plant variety rights and geographical indications. In Belgium, patents are regulated mainly by the law of 28 th March 1984. A patent is an "exclusive and temporary right to exploit any novel invention that also implies an inventive step while being susceptible of industrial application" (Article 2). The law states that "inventions are patentable even when they relate to biological material or contain a process that enables the production, treatment or use of the biological material" (Article 2). Furthermore, "a biological material isolated from its natural environment can be subject to patent protection, even when it pre-existed under its natural state". Patents are for instance quite often granted for molecular markers that are developed to assist plant breeders in the identification of interesting genetic sequences. Recent European case-law has however reduced the possibilities surrounding the patentability of so-called "native traits" and of "conventional breeding techniques"45 . However, a general research exemption to the rights granted by patents is provided by the law. These rights do not extend to "acts accomplished in a private environment and for non-commercial purposes, nor to acts accomplished for scientific purposes on and with the object of the patented invention" (Article 28 §1 (indents 1 and 2) of the 1984 law, as amended by the law of 28 th May 2005). The exact scope of "research on and with" has been defined in the "travaux préparatoires" of the 2005 amendments of the law, indicating that "research on" relates to "acts accomplished for experimental reasons that verify the function, the efficiency or the operational nature of the patented object". "Research with" relates to "acts accomplished for experimental reasons where the patented invention is used to research something else, as a tool or instrument"46 . Scientific purposes should in this regard be understood in a large sense. Following obligations stemming from the CBD (particularly its Articles 8(j), 15 and 16), the patent law has been amended to include a (qualified) origin indication requirement, if the origin of the material is known (Article 15 §1(6)) 47 . In order for the patent application to be admissible, the filing must contain a statement regarding the geographical origin of the biological material that has been used as a basis for the invention, if known 48 . Plant variety right protection is granted to those new, distinct, stable and uniform plant varieties. A variety is defined in Article 2 of the law of 10 th January 2011 49 as "a plant grouping within a single botanical taxon of the lowest known rank, which grouping, irrespective of whether the conditions for the grant of a breeders' rights are fully met, can be:  defined by the expression of the characteristics resulting from a given genotype,  distinguished from any other plant grouping by the expression of at least one of the said characteristics and  considered as a unit with regard to its suitability for being propagated unchanged". As a consequence, the production, reproduction, conditioning for the purpose of propagation, sale, marketing, import, export or stocking of this variety would need the authorization of the breeder (Article 12 of the law of 10 th January 2011), with the exception of certain specific prerogatives granted for research on the material and breeding with the variety, as well as for certain flexibilities recognized towards small farmers (Articles 14 and 15). Plant variety rights also enjoy research and breeding exemptions. The plant variety rights do not extend to "acts accomplished in a private capacity and for non-commercial purposes, acts accomplished in an experimental capacity or acts accomplished in view of creating or discovering and breeding new varieties" (Article 15 of the law of 10 th January 2011on plant variety rights). Plant variety rights were formerly regulated in Belgium by the law of 20 th May 1975, which has been recently abrogated and replaced by the law of 10 th January 2011. The law of 10 th January 2011 has not yet entered into force 50 , but gives nonetheless the necessary general framework so as to put Belgium in conformity with the provisions of the 1991 UPOV Convention (Union for the protection of plant variety rights). Geographical Indications (GI) are names used to describe a specific agricultural product or a foodstuff that is protected due to its regional and local nature, within general agricultural quality common appreciation of the relationship between intellectual property rights and the relevant provisions of the TRIPS Agreement and the Convention on Biological Diversity, in particular on issues relating to technology transfer and conservation and sustainable use of biological diversity and the fair and equitable sharing of benefits arising out of the use of genetic resources, including the protection of knowledge, innovations and practices of indigenous and local communities embodying traditional lifestyles relevant for the conservation and sustainable use of biological diversity. 48 This requirement is much narrower than the first proposed Bill, which stated that non-compliance with CBD provisions would be considered as contrary to the public order and morality, while the Council of State declared that such obligation would deviate from the initial objective of transposition measures and run counter to the objective of achieving effective harmonization throughout the European Union. See Van Overwalle G. (2006), Implementation of the Biotechnology Directive in Belgium and its After-Effects. International Review of IP and Competition Law, 37:8,pp. 889-1008 (especially at pp. 895-897) 49 Loi du 10 janvier 2011 sur la protection des obtentions végétales 50 See. Article72 of the law for the conditions of its entry into force, which render the mandatory force of the text conditional to the adoption of a royal decree, which has to this day not yet been adopted. As long as the required Royal Decree has not been adopted, the relevant legal framework is still the law of 1975. policies. GI's are usually distinguished between protected designation of origin (PDO), protection of geographical indication (PGI) and traditional specialty guaranteed (TSG) in the European Union 51 . GI's may relate to ABS since the product specification includes a description of the product, comprising the raw materials (and if appropriate the principal physical and microbiological characteristics of such material), and might be stacked on later to the bundle of property rights that surround one particular genetic resource if it is used to produce foodstuff protected by a GI. Liability and redress opportunities in cases of illicit acquisition of genetic resources (material and informational components) Alongside the above legal principles surrounding the legal status of genetic resources, there are a number of rules found in civil, criminal and private international law, that are relevant for the regulation of ABS in cases where an illicit acquisition of genetic resources is established. These legal provisions would indeed be of importance when read in concordance with the obligations related to compliance in the Nagoya Protocol. Liability and redress prospects, when referring to GR, should be analyzed both as physical specimens and as informational goods, through a national lens, and in an international context. Liability and redress for illicit acquisition of GR as physical specimen As with the discussion on the existing legislation on physical access to genetic material as biophysical entities, this legislation concerns access to biophysical specimens and therefore is not directly relevant for the regulation of access and utilization of genetic resources under the Nagoya Protocol. Nonetheless, the discussion on this legislation might be useful when assessing possible overlap and/or inconsistency with the measures that would be proposed for implementing the compliance 51 Regulations 510/2006 on the protection of geographical indications and designations of origin for agricultural products and foodstuffs, JOL, 93, 31.3.2006, p. 12-25, and 509/2006 on agricultural products and foodstuffs as traditional specialties guaranteed, JOL 93, 31.3.2006, p. 1-11;  A "designation of origin" refers to the name of a region, a specific place or, in exceptional cases, a country, used to describe an agricultural product or a foodstuff originating in that region, specific place or country, if the quality or characteristics of which are essentially or exclusively due to a particular geographical environment with its inherent natural and human factors, and the production, processing and preparation of which take place in the defined geographical area.  A "geographical indication" refers to the name of a region, a specific place or, in exceptional cases, a country, used to describe an agricultural product or a foodstuff originating in that region, specific place or country, and which possesses a specific quality, reputation or other characteristics attributable to that geographical origin, and the production and/or processing and/or preparation of which take place in the defined geographical area. provisions of the Nagoya Protocol. When assessing which legal principles should address the issues of liability and redress when facing illicit acquisitions of genetic resources as physical entities, it should first be noted that most conflicts will bear an international dimension, thereby precluding any analysis of applicable legal principles to the determination of actually applicable law and competent authorities. This assessment is made in accordance with the principles of private international law that have been favored by the country where litigation is brought. If Belgian law is deemed applicable to the conflict, then liability and redress opportunities will depend on the existence of a contractual relationship or not, in which case extra-contractual liability schemes both in civil and criminal law should be analyzed. A. Contractual breach If a contract has been used between the user and the provider of the genetic material, then any conflict, whether of a national or an international dimension, will be settled in accordance with the clauses set out by the parties with regard to dispute settlement. A number of national and European legislative texts govern the cases where no applicable law has been set by the parties. In Belgian national law, Article 98 of the private international law code refers to Regulation (CE) No. 593/2008 of 17 th June 2008 on the law applicable to contractual obligations (Rome I) (transposing the 1980 Rome Convention), which states that the law of the country of residence of the principal executor of the contract should apply in times of contractual silence. B. Extra-contractual liability and redress (absence of contract) If no contract has been signed by the user and provider of the genetic material, then positive law will come in to fill the void and establish the terms governing dispute settlement if Belgian law is found to be applicable to the conflict in accordance with the principles of either Belgian private international law (if the case is filed in Belgium) or another country's rules on conflicts of laws and the designation of applicable legislation (if the case is filed in another country) 52 . In the absence of a contract, the illicit appropriation of material goods may qualify as a "simple theft" (in accordance with Articles 461 al 1 and 463 of the criminal code), thereby triggering both criminal and civil liability vis-à-vis the perpetrator. The proprietor of the material good can respectively: (1) Seek injunction against a conduct that is judged to be in contradiction with the social order as a violation of property rights, (CRIMINAL PROCEEDINGS) In accordance with Article 461 of the criminal code, an act corresponding to an "unauthorized/ fraudulent removal of the material good that belongs to a third party" shall qualify as a theft, a criminal offense that shall be repressively punished 53 . The concealment of these objects by third parties knowing of their illegal acquisition is also punished through the concealment offense (Article 505 of the criminal code) 54 . Criminal law is regulated by separate provisions which determine under which circumstances Belgian courts have jurisdiction to hear cases over the alleged infringement of Belgian criminal law. The effectiveness of judgments can be complicated in an ABS context by lack of resources and the priorities of criminal prosecution, as well as issues regarding the execution of judgments 55 . (2) and/or seek compensation for the damage caused by the loss of the material good or by the fault of the person having wrongfully appropriated the good (CIVIL PROCEEDINGS). According to the Belgian Court of Cassation, the res nullius character of material goods cannot exempt the perpetrator from repairing the damage resulting from illicit acts 56 . The physical or legal person that is the legal owner of material goods, can therefore also seek civil compensation/damages ("actions en dommages et intérêts") in parallel to the criminal case being prosecuted ("constitution de partie civile", in accordance with Articles 63 and 70 of the criminal instruction code) 57 , or start civil proceedings before criminal jurisdictions if the prosecutor has dropped the case (in accordance with Article 162 of the criminal instruction code). Both intentional and non-intentional torts engage the extra-contractual responsibility of the perpetrator, when the constitutive elements of civil liability are proven; i.e., the fault, damage, and causal link between the fault and the damage.  With fraudulent intention, an illicit appropriation of genetic resources would qualify as an intentional tort or offense ("délit"), triggering delictual liability under Article 1382 of the civil code.  Without fraudulent intention, an illicit appropriation would qualify as a non-intentional tort ("quasi-délit"), a tort/offense committed by imprudence or negligence, and triggering civil liability. This would lead to a civil procedure concerned with the attribution of compensatory damages under Article 1383 of the civil code. C. Specificity of ABS context: an omnipresent international dimension in conflicts The illicit acquisition of material goods, whether with fraudulent intent or not, can have an international dimension. In an ABS context where the actors would most probably be of different nationalities, and where the contentious access or use of genetic resources might occur in a different country than the country where the alleged owner of the resource is established, it is useful to study 54 Concealment will be further analyzed in part 3.1.2.2. of this section. 55 Aside from the complex issues of competence and applicable law dealt with by private international law, criminal proceedings might also be hindered and further complexified due to the international nature of the conflict brought before the courts at the stage of decision implementation. Indeed extradition procedures would in principle need to be initiated in order to execute the judgment against the person convicted for theft55, or that there would need to be control over his property in order to execute the judgment against his property) These procedures would be expedited depending on the international conventions that have been adhered to by the States concerned (CASTIAUX, J., "Extradition en Belgique", in Chome P., Klees O., Lorent A. (eds.), Droit penal et Procédure pénale, Kluwer, Malines, 2011, p. 155). For instance, the Second Protocol to the 1959 European Convention on mutual assistance in criminal matters provides for transboundary observation when there are suspicions of aggravated theft (Article 17). 56 Cass., 28 janvier 2009, Amén., 2009, p.309 (in this case, damage caused by beavers) 57 The State could also directly start civil proceedings before civil courts, however, it would need to wait for the criminal verdict, in accordance with Article 4 of the code of criminal procedure ("le crimineltient le civil en l'état"). extra-contractual liability58 through the lens of private international law, which would apply, "in default of particular rules" adopted by the legislator in this regard. Private international law determines both the rules pertaining to the conflicts of laws and jurisdiction, respectively determining the legal rules that apply to the case, and the judiciary that would be competent to rule on the subject-matter for civil and commercial matters. A number of specific legal provisions of the private international law code59 govern material goods and the case of their theft. It is in this framework that private international law reveals itself relevant for regulating the illicit acquisition and use of foreign genetic material. The international private law legal principles can contribute in particular to uphold the conditions specified in private law access agreements, in situations where the procedures for mutually agreed terms, established by the country of origin include private law contracts. However, even if these principles are a useful contribution, they are certainly insufficient. In particular, in the ABS context, utilization of GR often occurs on the information components (the DNA code, published research results, databases etc.). Moreover, utilization is often based on the use of a copy of the GR (a clone of the entire biological material or a clone/reproduction of a component of it), even when the GR is not situated in Belgium. These frequent cases of research done on/utilization of GR that are not physically in Belgium are not covered by the legal dispositions the private international law code which does not explicitly refer to the use of GR under the Nagoya Protocol in its current scope60 . In addition, compliance with PIC obligations will involve public law requirements and/or administrative acts in the country of origin of the GR, which fall out of the scope of private international law. Therefore, additional measures might be needed to comply with the obligations under Articles 15, 16 and 18.  Conflict of Jurisdictions (Which jurisdiction is competent?) Article 85 of the code of private international law states that the Belgian judiciary is competent to rule on disputes involving a physical access to a material good "if the good is located in Belgium at the time the claim is made". However, the application of this Article to the situations covered by the Nagoya Protocol is quite limited. Indeed, as stated above, utilization often involves the informational component of GR and/or physical components of GR (copies/clones) of which the original GR is not situated in Belgium.  Conflict of Laws (Which laws to apply?)  Property rights related to a material good are governed by the laws of the State where the good is situated at the time the claim is made, in accordance with Article 87- §1 of the code of private international law. The acquisition and loss of property rights are established by the laws of the State where the good was situated at the moment these acts or facts have occurred. o If the good is an integral part of an ensemble of goods affected to a particular use, it is presumed to be situated in the State that has the strongest ties to the patrimony, in accordance with Article 8- §2 of the code of private international law.  Specific provisions exist for stolen material goods, which could be possibly applied in the ABS framework in the case potential users of genetic resources would come to possess resources that have not been obtained through a legal means of property or possession transfer pursuant to Article 92 of the code of private international law o The "native" proprietor has the choice to refer the case to be ruled by  Either the laws of the State where the material good was situated at the moment of its disappearance,  Or the laws of the State where the material good is located at the moment of the claim. However, in the first scenario, if a possessor in good faith is not protected by the internal legal order of the State, he may invoke the protection offered by the laws of the State where the material good is located at the moment of the claim. Liability and redress for illicit acquisition of GR as informational goods A. Contractual breach As is the case with physical specimens of GR, contractual provisions will prevail in terms of liability and redress if such a contract does exist. In the absence of any contractual relationship, torts law and criminal law will apply. B. Extra-contractual liability and redress (absence of contract) Theft of information is not a qualified infraction under Belgian law, and should most probably be fought through provisions related to breach of trust if the informational component is accessed by third parties without the transfer of actual material possession of the specimen. The use of informational components of genetic resources without PIC or MAT will most probably not be covered by those remedies addressing theft. Indeed, if the informational component of genetic resources is viewed as res communes, the usage of which is common to all, such component may not be subject to theft as long as it is not appropriated 61 . Furthermore, theft provisions apply solely to corporeal objects. However, there exists prominent jurisprudence regarding the theft of computer programs, where these have been considered as corporeal because of their economic value and because of them constituting an element of the patrimony of the original software's proprietor 62 . Neither the doctrine nor the jurisprudence is nonetheless unanimous on this issue, as the fraudulent copying of software has been ruled not to constitute a theft or a breach of trust due to its incorporeal nature, precluding the possibility to cede its ownership63 . These controversies have in this instance led to the draft of Article 504quater of the criminal code on informatics fraud. Other possibilities of redress recognized in Belgian criminal law may be exploited besides. Thus, a first option that might be envisaged is the concealment offense, which normally only applies to corporeal objects. "An offense punished through the criminal code's Article 505" concealment punishes the act of a third party to fraudulently conceal a contentious good, knowing that such good has been acquired through a crime or infraction. Concealment therefore implies the preliminary recognition of a crime. It could therefore only be relevant in the ABS context to genetic resources viewed as informational goods if the criminal code is amended to constitute the "use of the informational component of genetic resources in contradiction to PIC and MAT" as a criminal offense. Indeed, concealment proceedings require that the author of the infraction possesses materially or legally the good, knowing of its illicit acquisition; and both the existence of possession and of such knowledge is appreciated by the judiciary64 . Another possible -but non-exclusive -option would be the breach of trust. As an infraction against property rights, the breach of trust is enshrined in Article 491 of the criminal code, which punishes diverts or dispels goods of any kind from the initial usage or determined use that had been convened, with a prison sentence of one month to five years and a fine from 26 to 55 EUR. This provision could for instance be applied in an ABS context with regard to the exceptions that ought to be provided for research purposes (Article 8a NP), but most importantly against uses of genetic resources contrary to MAT or in absence of MAT in countries where the NP has been ratified and MAT has been requested in national legislation. The turning point for the constitution of this infraction is considered as the moment where the user cannot restore the genetic resources, or use them in a manner consistent with the initial destination65 . All of these approaches require an important stretch from currently applicable legislation so as to address specifically the use of informational components of genetic resources without PIC or MAT. However, breach of trust may be adequately used in cases of change of intent in the use of GR. In order to achieve a high level of dissuasion, the opportunity of addressing "information theft" or "genetic resources" theft should be assessed by law-makers, drawing perhaps on experience acquired with regard to software. Civil proceedings drawing on Articles 1382 and 1383 of the civil code might also be envisaged provided that the existence of damage, fault (negligence or imprudence) and causal link is adequately proven. Specificity of ABS context: an omnipresent international dimension in conflicts With regard to the international dimension of ABS conflicts and the determination of competent jurisdictions and applicable law vis-à-vis informational components of GR, since property rights are not recognized as such components, Articles 87 and 92 of the private international law code are not applicable. Answers may be found in the provisions of the aforementioned code on contractual and extra-contractual obligations, especially Articles 103 and 104 dealing with conflicts of jurisdiction and laws with regard to torts and liability deriving from a damaging act. Legal consequences for access to genetic material Under the current legislation in Belgium, the access to genetic resources for their utilization is not subject to a Prior Informed Consent. However, any legal measure that would consider introducing Prior Informed Consent could benefit from building upon existing legislation on physical access to and use of genetic material. That is why legal consequences for physical access to genetic material are investigated in some more detail in this section. Legislation relevant to physical access depends upon the type of ownership (private, public or res nullius), the existence of restrictions to the ownership, such as specific protection (protected species, protected areas, forests or marine environments) and the location (all four authorities apply their own rules) of the genetic material. Private ownership or res nullius In case of private ownership or res nullius (cf. chapter 3.1.1), access to the territory on which the genetic material (i.e. the specimen) is situated requires consent of the legal owner of the territory to get into his territory for the purpose of physically accessing the genetic material (i.e. the specimen). If a disagreement arose ex-post on the consent, the legal property rights would prevail in absence of proof of the consent (for example in the absence of a written contract) As for access to the genetic material (i.e. specimen) itself:  If it is res nullius (e.g. a bee swarm in liberty): then by law no access permits or contracts are needed. Moreover, if you take possession (that is material deeds of controlling the good for exclusive use), then you automatically become the legal owner of the specimen (Article2279 of the Civil Code)  If it is on territory in private ownership of an individual or a non-state organization: then you need a contract with the private owner, except if special restrictions apply to the legal ownership, which is the case of protected species (cf. discussion below)  If it is on territory in state ownership: then there is the need of an access permit (cf. discussion below on protected areas and territory and on territory in the public domain) Protected species Protected species in the Flemish Region In the Flemish Region, protection of species is regulated by the 'Soortenbesluit'66 of the 13 th August 2009. Under this act, it is forbidden to:  deliberately capture specimens of protected animal species, or to collect their eggs (Article 10- §1)  deliberately pick, collect, cut, uproot, destroy or transplant specimens of protected plantspecies or other types of organisms (Article 10- §2)  transport, sell or exchange or offer for sale or exchange specimens of protected animal species, of protected plant-species or other types of organisms (Article 12)  take away nests of protected birds and breeding sites or resting places of protected animals other than birds (Article 14- §1) The act specifies that, if no other satisfactory solutions exist and if it does not affect the conservation of these species, exceptions can be made for purposes related to research or education, repopulation or reintroduction, for the necessary breeding (Article 20- §1) as well as for reasons of economic, social or cultural nature (Article 20- §2). Request for exceptions needs to be addressed and approved by the "Agentschap voor Natuur en Bos" of the Flemish authorities (Article 22). Protected species in the Walloon Region Protection of species in the Walloon Region is regulated by the nature conservation law of 12 th July 197367 , which contains a general prohibition to:  Capture, kill, detain or transport animal species that are protected (Article 2 for birds, with a number of exceptions according to the species; Article 2bis to 2sexies for other animals)  Collect, pick up, cut, uproot, detain or transport specimens or portions of specimens that belong to those plant species that are listed in Annex 6 of the law (Article 3). Management and maintenance activities do not fall under this prohibition. For partially protected species the prohibition is attenuated by Article 3bis, which states that the "aerial parts of the specimens of the plant species listed in Annex 7 can be collected, picked up or cut in small quantities", but they cannot be sold or intentionally destructed. Derogations to the general prohibition can be awarded in accordance with Articles 5 and 5bis of the 1973 law. These are in principal unique (individual, personal and un-transferrable) but annual derogations can be awarded for physical or moral persons conducting research on one or more biological groups on the entire territory of the Walloon Region (with additional conditions in Article 5bis- §3). Derogations with regard to birds can only be awarded if there is no other satisfying condition and if they do not endanger the population concerned (Article 5 §2) and only for reasons of public health and security, research and education, protection of wild animal or plant species, air security, prevention of important damages to cultures, farm animals, forests or water, as well as allowing the capture, detention or other sound exploitation of small quantities of certain birds selectively, in strictly controlled conditions 68 . Similarly, derogations to the general prohibition with regard to mammals, amphibians, reptiles, fish and wild invertebrates, as well as wild plant species (Article 5 §3) 69 can only be awarded if there is no other satisfactory solution and if such derogation does not harm the maintenance of the population's favorable conservation status in their natural repartition area. These derogations can only be obtained for reasons of protection of wild animal or plant species, prevention of important damages to cultures, farm animals, forests or water, research and education, as well as allowing the taking or detention of certain specimens listed in Annex 2 point A selectively and in limited steps. Article 4 of the same law on nature conservation also mandates the government to regulate the modalities of collection and analysis of biological information on wild animal or plant varieties and the natural habitats falling under the scope of the law by the Walloon government. An administrative order was adopted on 24 th July 2003 70 , stating that the agents of the "Centre 71 " and their collaborators are authorized to enter private property, with prior notification of the owner, to proceed to operations that are indispensable to the collection of biological information (Article 4). Protected species in the Brussels-Capital Region The protection of species is regulated in the region of Brussels-Capital by the Ordinance of 1 st March 2012 regarding nature conservation 72 . With regard to animal species, this act awards strict protection to animal species listed in its Annex II.2.1° throughout the Region's territory, and to species cited in Annex II.3 part 1A throughout protected zones established in the Region (Article 6- §1 of the Ordinance). Such protection implies the interdiction, amongst other acts, to hunt or capture specimens, transport, pick up their eggs, sell, or expose in public spaces (Article 68- §1), except if they fall within the scope of management activities foreseen in the protected zone's management plan (Article 68- §3). Exceptions are made for imports, exports or transit of non-indigenous species, which is a federal competence (see chapter 2.1). With regard to plant species, the Ordinance awards strict protection to plant species listed in its Annex II.2.2° throughout the Region's territory, and to species cited in its Annex II.3 B part 1 B and II.3 part 2 throughout protected zones established in the Region (Article 70- §1 of the Ordinance). Such protection implies the interdiction, amongst other acts, to pick up, cut, uproot, unplant or harm the species in their natural repartition zones or within zones where they benefit from active protection and to detain, transport, or sell specimens collected within these active protection zones (Article 70- §2). Exceptions are made for imports, exports or transit of non-indigenous species, which is a federal competence (see chapter 2.1), except if these acts fall within the scope of management activities foreseen in the protected zone's management plan (Article 70 §3). For species presenting a regional or community interest active protection zones can be set out in accordance with Article 72 of the Ordinance. The measures adopted may for instance include prescriptions restricting the access to certain zones, preserving reproduction or resting areas, or regulating the periods, zones or methods of the sampling and exploitation of Annex II.3 specimens outside protected areas (Article 72- §1, 4°). Special dispensations can be awarded to the above interdictions in accordance with Article 83- §1 of the Ordinance, and the rationale include imperative reasons of major public interest (whether of a social or economic nature) and that would entail primordially beneficial consequences for environmental protection, as well as research or educational purposes. The Article also states that derogations might be granted in order to permit the capture and detention of a limited and specified number of specimens determined by competent authorities, in a strictly controlled, selective and limited fashion. The violation of these rules is punished by imprisonment from 10 days to 1 year, and/ or an administrative fine from 150 EUR to 150.000 EUR. The 2012 Ordinance on nature conservation in the Brussels-Capital Region also contains an Article on the sampling and exploitation of specimens in nature as a whole, stating that the Government is habilitated to take the measures necessary to ensure that the sampling and exploitation of species listed in Annex II.5 are compatible with their maintenance in favorable conservations status, including measures pertaining to the interdiction of capture, detention, transport or sale (Article 82). Protected areas and forests Protected areas and forests in the Flemish Region Nature conservation in the Flemish Region is regulated through the "Natuurdecreet" of 21 st October 199773 , through which the Flemish Government can take all necessary measures for nature conservation, regardless of the type of area. This includes regulating access (Article 13- §1, 6°), prohibiting certain activities or subject them to conditions (Article 13- §3, 6°). These conditions and activities may require a permit. A permit is required for the transformation 74 of the vegetation 75 or the modification of all or part of small landscape elements or their vegetation in the following areas: green areas; park areas; buffer areas; forest areas; nature development areas; valley areas; source areas; agricultural areas with ecological importance or value; and agricultural areas of special value or similar areas designated as such in spatial implementation plans (Article 13). However, it is not allowed to change all types of vegetation, nor do all actions producing change require a permit 76 . The prospecting of GR is not included in the actions requiring a permit. If a permit is delivered, the competent authority shall ensure that no avoidable damage to nature may arise by imposing reasonable conditions to prevent damage, to minimize or, if not impossible, to recover (Article 16). Certain areas in the Flemish Region enjoy a "special" status, where different rules apply. In the Flemish Ecological Network (Vlaams Ecologisch Netwerk, VEN) it is forbidden to change vegetation, including perennial crops or small landscape elements. In the nature reserves (natuurreservaten) it is forbidden to deliberately pick, collect, cut, uproot or destroy plants (Article 35). It should be noted that public servants working in relation to matters governed by the "Natuurdecreet" (i.e. nature conservation), may access real property (excluding houses and buildings intended for private or business use) to make measurements and to conduct research (Article 57bis). Forest areas in the Flemish Region are regulated by the "Bosdecreet" of 13 th June 1990. Although it applies to public access for social and educational purposes 77 , forests can only be accessed through the forest roads ('boswegen'). The Flemish Government can however decide to allow access to the forests outside of the roads for other activities (Article 10- §2). Physical access cannot lead to any reduction of the surface covered by the forest (Article 11). It is regulated through an "access regulation" ("toegankelijkheidsregeling") for forest for which a management plan ("beheersplan") is required 78 . Forest for which no management plan is needed do not need "access regulation" (Article 12). Part of these forest areas can be designated by the Flemish Government as protected "forest reserves" ("bosreservaten") because of the ecologic or scientific function these parts fulfil (Article22). In these "forest reserves" it is not allowed to remove plants or parts of plants (Article 30.1) or to extract material from soil or from the substrate (Article 30.2). Violation of this provision is punishable by a fine of 50 to 200 Euros (Article 30). 74 Change of small landscape elements and vegetation are all acts or works that are not understood to include the normal maintenance. Actions to be considered as normal maintenance are described in Annex 1 of Omzendbrief LNW/98/01 betreffende algemene maatregelen inzake natuurbehoud en wat de voorwaarden voor het wijzigen van vegetatie en kleine landschapselementen betreft volgens het besluit van de Vlaamse regering van 23 juli 1998 tot vaststelling van nadere regels ter uitvoering van het decreet van 21 oktober 1997 betreffende het natuurbehoud en het natuurlijk milieu 75 Vegetation has to be understood as the natural and semi-natural vegetation with all spontaneously established herb, bushes and forest covers, and this independently of possible influence of the abiotic environment by humans (Omzendbrief LNW/98/01) 76 This has been regulated by: Besluit van de Vlaamse Regering betreffende de vergoeding van wildschade of van schade door beschermde soorten en tot wijziging van hoofdstuk IV van het besluit van de Vlaamse Regering van 23 juli 1998 tot vaststelling van nadere regels ter uitvoering van het decreet van 21 oktober 1997 betreffende het natuurbehoud en het natuurlijke milieu. 77 The social and educational function of the forest includes the accessibility of the forest to the public for the purpose of recreation or education. 78 A management plan is required for all public forests and for private forests of at least five acres. Further provisions for physical access to both forest and nature reserves in the Flemish Region are provided by a specific Executive Order 79 which applies only to pedestrians, cyclists, horse riders, fishermen, swimmers, skaters, divers, kayakers, sailors, rowers and windsurfers (Article 5- §2) Protected areas and forests in the Walloon Region In the Walloon Region there is a general obligation to request a permit (for land planning) for acts that consist of "clearing the ground or transforming the vegetation of a zone that is judged by the government to be in need of protection, with the exception of the specific management plan of national and aggregated natural reserves", in accordance with Article 84- §1, 12° of the Walloon code for urban and land planning. Furthermore, Article 136 of the same code states that the execution of acts may be "either prohibited or subject to specific conditions for the protection of persons, goods or the environment when those acts relate to national natural reserves, a humid zone of biological interest, an underground cavity of scientific interest, a Natura 2000 site or a forest reserve (Article 452/27)". In natural reserves and national natural reserves physical access is regulated by Article 12 of the 1973 nature conservation law, in accordance to which the ministerial decree of 23 rd October 1975 80 has been enacted. Access to the non-protected material found in these zones is regulated by Article 11 of the nature conservation law, which states that it is forbidden to take out, cut, destroy or harm trees or the vegetative soil as such, or to modify the soil. For national natural reserves, in addition to those acts prohibited by Article 11 of the nature conservation law, it is also forbidden to "take out plants or vegetal parts, notably moss; or to pick up blueberries or cranberries with the help of a hairbrush", in accordance with Article 5 of the ministerial decree. In humid zones of biological interest, in accordance with Articles 2 and 3 of the Walloon Government decree of 8 th June 1989 81 regulating humid zones of biological interest, it is "forbidden at all times to pick up, unplant, harm or destroy all indigenous species of the flora growing in a wild state in the humid zone". For fauna, it is forbidden to hunt, kill, destroy, capture or disturb all indigenous species, except those for which hunting or fishing is authorized and those listed in the Annex of the decree. In underground cavities of scientific interest, in accordance with Article 3 of the Walloon Government Decree of 26 th January 1995 82 , it is the ministerial decrees establishing the specific protected zone that regulates both the physical access and conditions for research or other utilization of GR. In general 83 , the decrees state that access to the site is only authorized for 83 The texts of these ministerial decrees may be found on http://environnement.walloni.e.be/legis/consnat.htm, for an example, see the decree of 18 th September 2001 on the Ivoz-Ramet Vegetation grotto, http://environnement.walloni.e.be/legis/cavites%20souterraines/cavite041.htm management and scientific follow-up operations with the mandate of the managing committee. Scientific and speleological research can be done with the consent of the managing committee, with due respect for the integrity of the cavity and the scientific follow-up measures. In natural parks, regulated by the Decree of 16 th July 1985 84 , the particular terms of access shall be managed by the Managing commission set up in accordance with Articles 11 and 12. In accordance with the interpretation made by the high administrative authority, that is the Council of State, the Walloon code for urban planning defines the acts that are subject to a permit in natural parks as those that are susceptible of having a significant impact on the landscape and the environment 85 . In forest reserves, in accordance with Article 20 of the Walloon forest code of 15 th July 2008 86 , the access of pedestrians is forbidden outside roads and resting areas. However, access can be granted by the agents designated by the Walloon Government (in accordance with Article 92 of the forest code), under the conditions set out by these agents, for medical, pedagogic, scientific, cultural or nature conservation purposes. In accordance with Articles 32 and 34, it is forbidden to cut out, take out or tear down trees, or take out their sap without the authorization of its owner. Furthermore, Article 50 states that no sampling of any product of the forest can be undertaken without the consent of the owner and without respecting the conditions that could be adopted by the government (implying that such conditions may not be adopted). The fine for violation ranges between 25 and 100 Euros (Article 102). What about those acts that do not require permits? The establishment and prescription of protected zones is considered to be a "servitude légale d'utilité publique", restricting the use and affectation of a specific portion of land. The notions of "acts and works" should be understood as those activities characterized by a physical link to the soil or the vegetation, or causing a physical modification of the soil or the vegetation 87 . Therefore, utilization of GR as such may in certain cases not be considered as a modification or transformation of the ecosystemic balance set out by the protected zone. However, if this is the case, this needs to be specified in the general access rules of the protected zone or the permit. Further, within this understanding of passive obligations, those acts that are normally not subject to a permit might, according to doctrinal and jurisprudential thought, still have to respect the destination of the zone, otherwise they would fall under administrative sanctions 88 . Protected areas and forests in the Brussels-Capital Region The access to natural areas (both protected and non-protected) in Brussels is regulated by the Ordinance of 1 st March 2012 regarding nature conservation. In non-protected areas the Government may regulate public access and behavior applicable to the regional parks, gardens, squares, green areas and unoccupied land managed by the Region and publicly available (Article 66- §2). There is no general prohibition/permit requirement on the collection of natural resources in these areas. According to Article 82 of the Ordinance, the Government has to take the necessary measures to make sure the prospecting and use of specimens of species listed under Annex II of the Ordinance is compatible with the conservation of these species. Measures include the prohibition or limitation of their capture, detention, transportation and sale. In protected areas 89 it is forbidden to:  pick, remove, collect, cut, uproot, transplant, damage or destroy native plant species and bryophytes, lichens and macro-fungi, and destroy, damage or transform the vegetation (Article 27- §1, 1°);  leave the roads and paths open to public traffic (Article 27- §1, 10°). If no other satisfactory solutions exist and if it does not affect the conservation of native species, derogations to Article 27 can be made for purposes related to research or education, repopulation or reintroduction, and for the necessary breeding (Article 83). The requests for derogations, including information on the purposes of the request, need to be addressed and approved by the Brussels Institute for Environmental Management (IBGE/BIM), which delivers a permit (Article 84). Non-compliance with Article 27 is punishable by imprisonment from 10 days to 1 year and a fine of 150 EUR to 150 000 EUR (Article 93). Marine environment There are two main legal sources regarding the protection of the marine environment: the so called "MMM" Law of 20 th January 1999 and the "EEZ" Law of 22 nd April 1999 90 . The first one establishes a general regime of protection of animal and plant species. The second one specifies the rights Belgium detains on the exclusive economic zone and the territorial sea. The "MMM" Law The law of 20 th January 1999 defines the legal principles to be respected in order to preserve the Belgian part of the North Sea against marine pollution, and to conserve and develop its natural environment 91 . To this end, the law of 20 th January 1999 integrates within the Belgian legal order the different general principles of the environmental law: prevention principle, precaution principle, «polluter-pays» principle, etc. 89 Applies to all protected areas found in the Brussels Region: The "MMM" Law sets up a general regime of protection of natural resources and marine areas. In this regard, the Federal Government can take all the necessary measures concerning the protection of marine spaces 92 , including -amongst others -the obligations resulting from the CBD (Article 6). The Federal Government can also create protected marine areas (Article 7). The law so organizes and determines the categories and physical borders of the protected zones. It introduces moreover a categorization of the different potentially concerned zones:  In the integral and "directed" marine areas, any activity is forbidden, except for those areas specified in Article 8 93 . However, some specific activities are authorized in exceptional cases for the "directed" marine areas 94 .  A general authorization is given to the special protection zones and special conservation zones even if some activities can be punctually forbidden. Thus, the 2003 federal Masterplan led to delineate five marine zones designed to specifically protect animal species. The access to and use of these zones are submitted to specific conditions determined by the various users of the North Sea for specific periods of the calendar year. The Federal Government establishes a list of protected species in the marine areas 95 , which benefit from a strict prohibition regime forbidding to capture, kill, detain or transport animal species that are protected, and to collect, pick up, cut, uproot, detain, transport or intentionally destruct specimens or portions of specimens that belong to the plant species that are listed as protected (Article 10 §1). Derogations to the general prohibition can be nonetheless awarded for the needs of public health, scientific research, education, restocking or reintroduction of these species (Article 10 §2). Lastly, the deliberate introduction of non-indigenous organisms is forbidden unless otherwise stated by the Government, as is the deliberate introduction of GMO (Article 11). Finally, the law stipulates that any construction activity or industrial, commercial and advertising activity taking place in marine spaces requires a license (Article 25- §1). The granting of this license depends on an environmental impact assessment of the expected activity (Article 28) 96 . However it should be noted that some activities remain excluded from the scope of Article 25- §1, such as professional fishing, or marine scientific research -whose implementation is regulated in the EEZ law hereafter described (Article 25- §3). 92 See. Article 2 §1. The marine spaces are defined as « the territorial see, the exclusive economic zone and the continental shelf aimed par the Law of 13 th June 1969 on the continental shelf of Belgium » 93 The following activities are accepted in the marine areas: (i) surveillance and control; (ii) monitoring and scientific research carried out for or with the consent of the authority;(iii) sailing; (iv) professional fishing, notwithstanding the restrictions or prohibitions imposed by the Government; (v) nature conservation and development activities; (vi) military activities (Article 8) 94 For an example of directed marine area, see : Executive order of 5 th March 2006 créant une réserve marine dirigée dans les espaces marins sous juridiction de la Belgique et modifiant l'Arrêté royal du 14 th October 2005 créant des zones de protection spéciales et des zones de conservation spéciales 95 See Annex I of the Government Executive Order of 21 st December 2001 aiming for the protection for species in the marine areas under the jurisdiction of Belgium (M.B., 14 th February 2002) 96 The specific devices organising the license granting process are defined in the executive order of 7 th September 2003 establishing the granting procedure of the permits and authorizations required for some activities carried out in marine areas, M.B.,17 th September 2003 The "EEZ" Law The law of 22 nd April 1999 97 specifies the legal status of the territorial sea and broadens the sovereign rights of Belgium to a maritime zone located beyond the territorial sea and adjacent to it: the Economic Exclusive Zone (EEZ). The regulation of the economic exclusive zone concerns the exploration and exploitation of the natural resources of the waters in contact with ("surjacent") the marine soils, i.e. the marine soils themselves as well as their subsoil 98 . Belgium has sovereign rights for the exploration and exploitation, conservation and management of natural, biological and non-biological resources found within the EEZ, as well as for other activities tending to the exploration and the exploitation of the zone to economic ends (Article 4- §1). Belgium also has jurisdiction with regard to the settlement and utilization of artificial islands, installations and construction works, to the marine scientific research and to the protection and preservation of marine environment (Article 4- §2). In this framework, any scientific research in territorial sea and in the exclusive economic zone must be submitted to the consent of the Minister of Foreign Affairs, who has then to consult the different involved ministers (Article 40) 99 . Such consent is supposed to be given if Belgium is part of the institutional organization or of the bilateral agreement on the basis of which the scientific research project is developed -unless Belgium objects to it within the two months following the official research request. Finally, the scientific research carried out by foreign ships in the territorial sea and the economic exclusive zone is under the jurisdiction of the Belgian Law related to the protection and conservation of marine environment (Article 42). In the territorial sea, the exclusive economic zone and in high sea, the Federal Government can take the necessary measures to ensure the conservation of biological resources (Article 1- §1, al.1 of the law of 12 th April 1957 entitling the King to prescribe measures in order to conserve the marine biological resources, as modified by Article 6 of the law of 22 nd April 1999). The fishing in the territorial sea and in the exclusive economic zone is forbidden for foreign fishing boats (Articles 10 and 17), except in the exclusive economic zone and in the territorial sea if allowed by the rights deriving from the Treaty of the European Union and the applicable rules of international law. In this framework, the Federal Government can take the necessary measures ensuring the respect of this general prohibition 100 . Finally, Belgium exercises sovereignty on territorial sea and holds sovereign rights on the continental shelf as for the exploration and exploitation of mineral and non-living resources (Article 27). 97 M.B., 20 th July 1999 98 The limits of the economic exclusive zone are fixed through different bilateral agreements : Agreement between the Government of the Kingdom of Belgium and the Government of the Kingdom of England and North Ireland related to the delimitation of the continental shelf between the two countries, signed in Brussels on 29 th May 1991 and approved by the the Law of 17 th February 1993 (M.B., 1 st December 1993) ; Agreement between the Government of the Kingdom of Belgium and the Government of the French Republic related to the delimitation of the continental shelf between the two countries, signed in Brussels on 8 th October 1990 and approved by the law of 17 th February 1993 (M.B., 18 th December 1993) ; Agreement between the Government of the Kingdom of Belgium and the Government of the Netherlands related to the delimitation of the continental shelf between the two countries and Annexes, signed in Brussels on 18 th December 1996 and approved by the law of 10 th August 1998 (M.B., 19 th June 1999). 99 For the general regulation of the matter, see Article40-44 of the law of 22 nd April 1999 100 See Article 10 and foll.; 17 and foll. Access in state owned land outside of protected zones Access to genetic material on state owned land outside protected zones also requires the authorization of the competent state authority, except if the land is explicitly designated as public domain. In the latter case, under the current legislation it is still unclear how access to genetic material is regulated. In general the public domain encompasses "the goods specifically assigned for public use or arranged with the view to realize a public service objective". The specificity of the destination of such goods requires "a specific legal protection and therefore the application of a specific administrative legal regime" 101 . Access to genetic material is not explicitly mentioned in the current legal framework applicable to public domain goods. However, each public entity has its own public domain that it regulates in accordance with the competences attributed or granted by the Belgian legal order. For instance, with regard to the public domain at the municipal level, the regulation of the administrative police of Gesves in the province of Namur, states in its Article 1 that it is forbidden to pick the flowers found on the public domain, as well as to take out grass, soil, rocks or materials belonging to the public domain without prior authorization. In the absence of such specific regulation by the competent authority, access to genetic material in public domain is still a grey legal zone. This question certainly deserves further clarification. 1.1.2.5 101 Willieme C., Boland M., Simon V. ( 2010), Valorization des biens situés dans le domaine public : Quel encadrement juridique ?. Rec. gén. enr. not., 2010, pp. 253-271 (at p. 256). The status of traditional knowledge associated to genetic resources under national legislation in Belgium Traditional knowledge in the context of the CBD is usually understood as "knowledge, innovations and practices of indigenous and local communities" that "embody "traditional lifestyles relevant for the conservation and sustainable use of biological diversity" (Article 8(j) of the CBD). Traditional knowledge is "developed from experience gained over the centuries and adapted to the local culture and environment" and "transmitted orally from generation to generation". Moreover traditional knowledge "tends to be collectively owned and takes the form of stories, songs, folklore, proverbs, cultural values, beliefs, rituals, community laws, local language, and agricultural practices, including the development of plant species and animal breeds" 102 . There are no contemporary legal provisions in Belgium explicitly governing the concepts of "traditional knowledge", "traditional knowledge associated with genetic resources" and "indigenous and local communities". One might argue that some types of knowledge could be qualified as "knowledge, innovations and practices" that "embody traditional lifestyles relevant for the conservation and sustainable use of biological diversity". One example would be knowledge involved in the conservation and use of old seed varieties by farmers. However, this knowledge is not related to specified local communities and their traditional lifestyles as specified in the CBD's understanding of the concept. Nevertheless, concerns over traditional knowledge and the rights of indigenous and local communities have been addressed in some international instruments, especially in the area of development cooperation and sustainable development, to which Belgium is a Party 103 . Three international instruments broach the rights of indigenous and local communities and recognize the importance of traditional knowledge: Verdrag van de Verenigde Naties ter bestrijding van desertificatie in de landen die te kampen hebben met ernstige verdroging en/of woestijnvorming, in het bijzonder in Afrika, BS: 10-12-1997; 17 JUNI 1994. -BIJLAGE II (Bijlage inzake regionale uitvoering van Azië) aan het Verdrag van de Verenigde Naties ter bestrijding van desertificatie in de landen die te kampen hebben met ernstige verdroging en/of desertificatie, in het bijzonder in Afrika, gedaan te Parijs op 17 juni 1994, BS: 10-12-1997; BIJLAGE III (bijlage inzake regionale uitvoering voor Latijns-Amerika en het Caraibisch gebied) aan het Verdrag van de Verenigde Naties ter bestrijding van desertificatie in de landen die te kampen hebben met ernstige verdroging en/of desertificatie, in het bijzonder in Afrika, gedaan te Parijs op 17 juni 1994.   the United Nations Declaration on the Rights of Indigenous Peoples. The UN Declaration on the Rights of Indigenous People might have a practical and a political interest as it is explicitly "noted" in the preamble of the Nagoya Protocol and might therefore provide a framework in the further elaboration of decisions under the Nagoya Protocol relevant to the rights of indigenous and local communities. It nonetheless remains a non-binding instrument, whose provisions do not create any legal obligations. A fourth instrument of relevance is Agenda 21, following the United Nations Conference on Environment and Development (UNCED) held in Rio de Janeiro, Brazil, 3 rd to 14 th June 1992. Its chapter 26 focuses on the role of indigenous people and their communities. It is provided that such communities possess a unique knowledge of their environment and the natural characteristics thereof. Consequently, indigenous people and their communities should acquire the right of selfdetermination, manage their own resources and participate in the decision-making on development programs affecting them104 . This instrument is not legally-binding and merely addresses issues of potential future action. As pointed out in the fourth National Report on the implementation of the Convention on Biological Diversity in and by Belgium (2009), certain policy initiatives have been adopted or identified in order to support actions105 of indigenous and local communities situated in developing countries. Also the ratification of ILO Convention 169 (Indigenous and Tribal Peoples Convention) was put on the agenda. Bilateral official cooperation provides limited direct support to indigenous and local communities, since this is not often taken up as a priority by the partner countries, neither in their national development and poverty reduction policies, nor in their policy dialogue with donor countries. Belgium ratified the ILO Convention No. 107106 but not the ILO Convention No. 169. The 1957 ILO Convention No. 107 on Indigenous and Tribal Populations The 1957 Indigenous and Tribal Populations Convention (No. 107) was a first attempt to codify international obligations of States in respect of indigenous and tribal populations. It was the first international convention on the subject, and was adopted by the International Labor Organization. ILO Convention Relevant provisions of the ILO Convention No. 107 The Convention does not avail itself of the concept of indigenous and local communities, rather it applies to indigenous tribal or semi-tribal populations in independent countries whose social and economic conditions are at a less advanced stage than the stage reached by the other sections of the national community, and whose status is regulated wholly or partially by their own customs or traditions or by special laws or regulations (Article 1). This convention entails certain obligations incumbent on Belgium, but which have not been addressed by it. Three particular provisions are however of particular relevance for the implementation of the NP by Belgium. These concern:  Article 7(1): In defining the rights and duties of the populations concerned regard shall be had to their customary laws.  Article 11: The right of ownership, collective or individual, of the members of the populations concerned over the lands which these populations traditionally occupy shall be recognized.  Article 13: 1. Procedures for the transmission of rights of ownership and use of land which are established by the customs of the populations concerned shall be respected, within the framework of national laws and regulations, in so far as they satisfy the needs of these populations and do not hinder their economic and social development. 2. Arrangements shall be made to prevent persons who are not members of the populations concerned from taking advantage of these customs or of lack of understanding of the laws on the part of the members of these populations to secure the ownership or use of the lands belonging to such members. EXISTING ABS-RELATED POLICY MEASURES AND OTHER INITIATIVES IN BELGIUM Measures resulting from coordination between the three regions and the federal level In 2006, Belgium adopted its National Biodiversity Strategy 2006-2016 107 , which established 15 strategic objectives and 78 operational objectives to reduce and prevent the causes of biodiversity loss. The 6 th strategic objective aims to contribute to an equitable access to and sharing of benefits arising from the use of genetic resources. This objective is projected to be realized mainly through capacity building of national ABS stakeholders and further implementation of the Bonn Guidelines on ABS. In 2006, a study on the awareness of Belgian users of GR concerning the CBD and the level of implementation of ABS dispositions and the Bonn Guidelines in their activities has revealed mixed knowledge within stakeholder groups 108 . The Convention seemed to be better known in upstream activities (e.g. fundamental research) than in downstream activities (e.g. commercial products). Collections and research sectors, both private and public, have a good understanding of the CBD, while other sectors, predominantly composed of private actors, have little or no knowledge. Concerning the implementation of ABS dispositions, the report showed that PIC-related dispositions seem to be relatively widespread, whereas benefit-sharing provisions are nearly inexistent 109 . Other operational objectives of the National Biodiversity Strategy include the enhancement of synergies between actors for addressing ABS, the protection of local communities and their traditional knowledge and the establishment of an international regime on ABS. However, these seem to be general goals the government wants to strive for, rather than specific delineated strategic actions. The strategy has been evaluated at the end of 2011 and is currently under review in order to bring it into line with the new multilateral and European biodiversity objectives (the Biodiversity Strategic Plan 2011-2020 and its Aïchi Targets, the EU biodiversity Strategy and other national and international commitments) and to extend subsequently the reviewed strategy until 2020. As part of the present impact-study, two stakeholder workshops have been organized. The aim of the workshops was to identify the wide range of stakeholders concerned with the implementation of the Protocol in Belgium, to make them aware of the content of the Protocol and its obligations, and to give stakeholders the possibility to exchange views and provide input on the options for and consequences regarding the implementation of the Protocol110 . Federal measures The National Biodiversity Strategy followed the Second Federal Plan for Sustainable Development 2004-2008111 . It calls for a coherent national position on access and benefit-sharing. A third Federal Plan for Sustainable Development, calling for an "equitable distribution of the commercial exploitation of biological resources", was drafted for the period 2009-2012 but never adopted. The second plan was instead extended until 2012. The two plans above are partly concretized by the Federal Plan for the integration of biodiversity in four key sectors, adopted by the Federal Government in 2010. Three of these key sectors are particular relevant for ABS-implementation: the economy, the development cooperation and the scientific policy. For each of these sectors a separate and detailed action plan has been developed for integration of biodiversity, including several ABS-related measures. For the economic sector the plan mainly focuses on awareness-raising and capacity building of the private sector and call for a proactive participation of the Federal Government in the establishment of an international ABS-regime. The plan also calls for an increased participation of the customs administration in biodiversity policy, albeit not directly linked to ABS. This stronger understanding of biodiversity-related issues inside the customs could however be beneficial for and facilitate the implementation of the NP. TEMATEA is a web-based capacity-building utility to support the coherent implementation of international and regional biodiversity related conventions and provides an overview of national obligations regarding ABS, as derived from several international agreements. In the science policy field, the first proposed action of the Federal Plan for the integration of biodiversity in four key sectors is particularly relevant to ABS as it calls for an inventory of the national collection of plant germplasm. This objective will directly benefit from existing projects and initiatives. For instance, the BELSPO, together with the Ghent University, developed straininfo.net115 , a pilot project using bioinformatics tools (web crawlers and search engines) to access and make available data and information stored in 60 biological resource centers worldwide. A standard format to allow for culture collection catalogue information to be exchanged easily has also been developed. PLANTCOL 116 Regional measures Regions each have separate biodiversity policy-plans, mostly as part of a broader environmental strategy, in which ABS measures could be taken up. Although these plans all explicitly refer to the CBD as guidance for biodiversity policy, none of them contain ABS-related provisions. In its recently released Environmental Policy Plan 2011-2015 (MINA-4), as well as in the latest Flemish Strategy for Sustainable Development118 , the Flemish Government also refers to the 10 th COP of the CBD as an important watershed moment, but without identifying or emphasizing the need for ABS-related actions. Research institutions' and private initiatives and policies on ABS In 1997, the Belgian Coordinated Collection of Micro-organisms (BCCM) launched the Microorganisms Sustainable Use and Access Regulation International Code of Conduct (MOSAICC) initiative. MOSAICC is a voluntary code of conduct to facilitate access to microbial genetic resources in line with the CBD, the TRIPS Agreement and other applicable national and international law, and to ensure that the transfer of material takes place under appropriate agreements between partners and is monitored to secure benefit-sharing. It aims, in particular, to develop an integrated conveyance system that has reliable tools to evaluate the economic value of microbiological resources; that disposes of validated model documents with standard provisions to enable tracking via an uncomplicated procedure, widely applied by microbiologists; and, that combines valuation and tracking in one system for trading of microbiological resources, with balanced benefit-sharing for those that are entitled to be rewarded for the services and products they provide to society. BCCM uses a standard BCCM Material Transfer Agreement (MTA) for getting access to the genetic resources of its public collection. If necessary, the MTA can be amended with additional conditions already attached to the biological material. The resources are distributed for a fee covering expenses. The MTA stipulates that anyone seeking to access genetic resources hold by the BCCM has the responsibility to obtain any intellectual property licenses necessary for its use and agrees, in advance of such use, to negotiate in good faith with the intellectual property rights owner(s) to establish the terms of a commercial license. The National Botanic Garden of Belgium (NBGB) is member of International Plant Exchange Network (IPEN), a network of Botanic Gardens that organizes the exchange of living plant specimens. IPEN's members have adopted a Code of Conduct regarding access to genetic resources and benefit-sharing. The NBGB only supplies seed material to other IPEN-members, unless the "Agreement on the supply of living plant material for non-commercial purposes leaving the International Plant Exchange Network" is signed by authorized staff. Although not explicitly linked with ABS, stakeholder conferences and workshops have been organized in 2010 by the Association for Forests in Flanders (Vereniging voor Bos in Vlaanderen) on the importance of preservation of autochthonous genetic bush and tree material. This initiative led to the Plant van Hier project, which included the development of study material119 and the creation of a product label120 encouraging the commercialization of native bushes and trees. Existing ABS-related EU instruments and other initiatives Implementation of the Bonn Guidelines The EU Biodiversity Action Plan (BAP) lays down the political commitment to promote full implementation of the CBD Bonn Guidelines on ABS and other agreements relating to ABS such as the FAO International Treaty on Plant Genetic Resources for Food and Agriculture (ITPGRFA). With regard to the implementation of Article 8(j) of the CBD, the EU BAP put forward the political commitment to apply from 2006 onwards the principle of PIC when commercially using TK relating to biodiversity and encourage the equitable sharing of benefits arising from the use of such knowledge. Therefore, Member States were encouraged to implement the relevant aspects of the Bonn Guidelines in MS when granting access to TK relating to biodiversity. In particular Member States were encouraged to enhance awareness of stakeholders to effectively participate in and contribute to EU preparations for international ABS negotiations and to effectively contribute to on-going negotiations of the Standard Material Transfer Agreement (SMTA) under the International Treaty on Plant Genetic Resources for Food and Agriculture. In order to assess the status of ABS within Europe, the European Commission undertook to calculate the percentage of European patent applications for inventions based on GR. Indicators were to be developed under the lead of the joint Secretariat of the Pan European Biological and Landscape Diversity Strategy (PEBLDS) with the assistance of the European Patent Office and the World Intellectual Property Organization. In 2010, in the context of its reporting obligations to the EU, Belgium qualitatively monitored the implementation of BAP actions and achievement of targets. It was noted that over the period 2006-2009:  Belgium did not provide funding for the ABS Working Group;  no national legislation implementing the CBD Bonn Guidelines on Access and Benefit-sharing existed;  no national activities that raise awareness of the CBD Bonn Guidelines had been implemented;  no national legislation implementing the MTA Agreement of the ITPGRFA existed;  no national activities raising awareness of the MTA of ITPGRFA had been implemented. The EU funds the BIOPOMA project for ABS capacity building in ACP countries (twenty million Euros) in order to enhance existing institutions and networks by building their capacity to strengthen policy and to implement well informed decisions on biodiversity conservation and protected areas management. The project has two components. Firstly, enhancing the effective planning and management of protected areas in ACP countries through the intensive use of scientific and policy information accessible from appropriate database reference systems combined in one information tool and the establishment of a "Centre for Protected Areas and Biodiversity" in each of the 3 regions. The second component aims to contribute to the Access and Benefit-sharing Capacity Development Initiative. This initiative aims to further build the capacities of stakeholders in the access and benefitsharing in each of the 3 ACP regions and is implemented through a trust fund managed by the German Cooperation Agency (GIZ). Proposal for a Regulation on ABS In October 2012, the European Commission proposed a Regulation on Access to Genetic Resources and the Fair and Equitable Sharing of Benefits Arising from their Utilization in the Union 121 . The proposal was based on two previously conducted impact assessments 122 CONFORMITY OF THE EXISTING NATIONAL LEGISLATION AND MEASURES WITH THE OBLIGATIONS OF THE NAGOYA PROTOCOL To the best of our knowledge, no existing national legislation or measures are in contradiction with the obligations under the Protocol. However, existing legislation that addresses physical access to genetic material and instruments regulating benefit-sharing between users and providers of genetic resources need to evolve and be complemented by additional instruments in order to implement the obligations of the Protocol. As indicated above, this analysis is based on the list of legal obligations summarized in annex 1 to this report. Conformity of existing instruments in Belgium that already address obligations of the Protocol Articles 6.1 + 6.3 Under the current legislation in Belgium access to GR is not subject to Prior Informed Consent (PIC) by the Belgian State as a Party to the NP (that is based on a written decision by a Competent National Authority (CNA) on access and benefit-sharing). Even if it is not compulsory, under the Nagoya Protocol, the Belgian State can decide that access is subject to PIC if it so wishes and take the necessary legislative, administrative or policy measures, as appropriate, to provide for access permits by one or more Competent National Authorities and establish the mutually agreed terms for these access permits. Articles 13.1, 13.2 and 13.4 The ABS national focal point already exists. Belgium nominated a civil servant of DG Environment of the FPS Environment that currently ensures the function of national focal point on ABS. However, the obligations related to the CNA still have to be implemented. Articles 15.1 and 16.1 Under the current legislation in Belgium (more specifically the provisions of private international law), the acquisition and the loss of property rights over genetic materials are established by the laws of the State where the good was situated at the moment these acts or facts have occurred (that is at the moment of the acquisition). However, as discussed in chapter 3, even if these principles are a useful contribution to comply with private law contracts over genetic materials, they are certainly insufficient for the Nagoya Protocol, as the compliance with PIC obligations involves public law requirements and compliance with administrative acts of the Country of Origin of the GR, which fall out of the scope of private international law. Furthermore, at present, "use of GR under the Nagoya Protocol" is not explicitly mentioned within the scope of the Belgian code of private international law. In particular, as stated above, utilization of GR often occurs on the information components (the DNA code, published research results, databases etc.) or might be based on the use of a copy of the GR (a clone of the entire biological material or a clone/reproduction of a component of it), even when the GR is not situated in Belgium. These frequent cases of research done on/utilization of GR that isnot physically in Belgium is not covered by the legal dispositions of the private international law code. Therefore, additional measures will be needed to comply with the obligations under Articles 15 and 16. Article 17.1 One measure to monitor use of genetic resources has already been taken, which is the disclosure of the information on the country origin in patent application under Belgian law, whenever this information is available (cf. detailed discussion in 3.1.1.). However, this measure still needs to be completed by other measures in order to comply with Article 17.1 as it is not organized nor designated as a formal checkpoint. Articles 18.2 and 18.3 Regarding the concrete measures linked to the international ABS regime, three main issues would have then to be addressed: (a) determining the jurisdiction that is internationally competent to deal with disputes raised within ABS agreements; (b) determining the applicable law which has to be applied in the case of ABS-related disputes; (c) recognizing and enforcing in another country, party to the NP, judgments' rendered by a jurisdiction in the ABS context. The first two points (a) and (b) are related to Articles 18.1 and 18.2 of the Protocol, of which the provisions seem to state the obvious and have little added value. Most if not all countries in the world with a legal system provide for an opportunity to seek recourse in cases of breach of contract, and have established specific provisions regulating lawsuits involving a "foreign" law element. See chapter 3.1.2 on the existing Belgian private law and international private law provisions regarding contractual breach, amongst which the EC Regulation 44/2001 (Brussels 1) and the Rome Convention on contractual obligations (as well as the Council Regulation "Rome I") 125 . The third point (c) relates to Article 18.3 from a strict reading of which emerges that a Party could demonstrate compliance by proving ratification -or any effort leading to it -of certain international legal arbitration instruments. First, as convincingly put forward by the IUCN Explanatory Guide to the Nagoya Protocol on Access and Benefit-sharing, "it is important to note that it is not for the Parties jointly to take the measures referred to […] it is for "each Party" to enact such measures at the domestic level. Second, the measures shall be taken (only) if it is judged by the Party "appropriate" to do so". 126 125 To broach it more specifically, Article 18.1 does not need to be analyzed under "existing legislation" as it refers to MAT between Parties to NP: let us also note that the Article 18.1 only "encourages" providers and users of genetic resources to include dispute resolution provisions. Article 18.2, however, sets and obligation for each Party at the domestic level to ensure that recourse is available under its legal system if a dispute arises in the framework of a contractual obligation such as the one established by MAT. Moreover Article 18.2 is drafted in such a way that it does not mention whether the opportunity shall also be granted to foreign citizens. It makes clear though that such recourse has to be consistent with applicable jurisdictional requirements of the Party concerned, leaving this issue to national legislation. 126 Article 20.1 Existing measures that deserve to be mentioned are the codes of conduct of IPEN and MOSAICC. These will be further discussed in the action cards below under section 6.2. 127 Extradition procedures would in principle need to be initiated in order to execute the judgment against the person convicted for theft, or that there would need to be control over his property in order to execute the judgment against his property. These procedures would be expedited depending on the international conventions that have been adhered to by the States concerned (Castiaux J. (2011), Extradition en Belgique. In Chome P., Klees O., Lorent A. (red.), Droit pénal et procédure pénale, Mechelen: Kluwer, p. 155). Here, the Second Protocol to the 1959 European Convention on mutual assistance in criminal matters more peculiarly provides for transboundary observation when there are suspicions of aggravated theft (Article 17). 128 This convention, negotiated at the European Union level, requires user countries to take effective measures to ensure that provider countries have recourse to their legal system to obtain redress. It includes an obligation to provide access to administrative or judicial procedures to challenge breaches of national law in a similar way as provided for by Article 18(2) of the Protocol. Obligations of the Nagoya Protocol currently not addressed by legal or non-legal instruments in Belgium To the best of our knowledge, no other obligations of the Nagoya Protocol are explicitly and specifically addressed by existing legal or non-legal instruments in Belgium. Therefore additional instruments will be needed to implement these obligations. These possible legal and non-legal measures will be considered in a systematic manner in the next section. For the purpose of the analysis, a distinction is made however between the Articles that need to be considered most urgently, because of their core nature in the implementation of the Protocol, and additional measures that are important elements during implementation of the obligations, but that are less urgent. The core measures that are considered are the measures specified in the terms of reference of this study ("measures requiring special attention"): 1 REVIEW OF EXISTING MEASURES AND INSTRUMENTS ON ABS IN OTHER COUNTRIES In the next section, a brief overview of measures adopted in other countries is presented. It is based on the review of primary and secondary information related to existing ABS regulations in other countries. In order to provide a clear and structured overview, they are grouped under the following broad themes: access, benefit-sharing, conservation activities and biodiversity research, competent National Authority, and user compliance and monitoring. Under each theme, a number of issues found in the consulted information and which are relevant for the discussion on implementation in Belgium are listed, with a particular focus on the measures listed in the Bonn Guidelines on Access to Genetic Resources and the Fair and Equitable Sharing of the Benefits arising out of their Utilization. Whenever possible, detailed reference is made to how these issues have been solved in other countries in existing legislation or in detailed assessments of possible legislation. Relevant actions for the implementation of the NP in Belgium are summarized for each theme and a distinction is made between actions which are relevant in case of minimal implementation of the core obligations and actions which are relevant in case of additional implementation (beyond the minimal implementation of the core obligations and beyond the core obligations). This overview of existing ABS measures is by no means exhaustive, nor does it imply anything for the implementation at Belgian level. It rather serves as a base for the reflection on the identification of the possible implementation measures for Belgium. Therefore issues which have been identified as being already present in Belgium (e.g. the designation of a NFP) or which have already been discussed previously are not repeated in this section, even if they will be used in the further assessment of the measures. Furthermore, it has to be noted that most of the options identified below are not mutually exclusive. If necessary and/or desirable, a combination of the options also represents a possible outcome. Access First, as part of the core obligations, each Party to the Protocol will have to determine if access to GR will be subject to PIC by the State or not, and, if requiring PIC for access, will have to take legislative, administrative or policy measures, as appropriate, containing minimum requirements for access rules and procedures (Article 6 of the NP). To determine the applicable access rules for GR, legal ownership of GR under national legislation will need to be fleshed out in order to decide which access conditions and procedures can and need to be implemented in relation to the prior informed consent requirements. In most countries that have ABS legislation in place, ownership of GR is derived from ownership of natural resources, which is defined by the Constitution or the civil code, or by common law129 . This ownership applies to the physical component of these resources. In the exceptional case where patents are already attached to genetic components of natural resources at the moment of accessing a natural resource in its insitu environment (because the same genetic sequence exists in the organism that is accessed and in another organism that was accessed earlier in relation to the patent), an additional layer of ownership rights can be claimed on the genetic information, but only in relation to the specific genetic component as used for the specific industrial use claimed in the patent. In all other cases, under the current Belgian legal framework it seems that no legal ownership could be claimed on the informational component due to its nature. (cf. the analysis in chapter 3.1.1). As shown by a study of national legislation in selected countries 130 , two ownership systems are generally in place with regard to natural resources: they can fall under private and/or communal property, they can be property of the state or they can be both. In both situations, it depends on national legislation in place how these property rights relate to the genetic components of these natural resources. In some countries, these directly derive from the ownership of the natural resources. Other countries have decided to explicitly create legal measures to limit the extent of private ownership of natural resources and to place all GR under state ownership. This option is not an obvious one. In particular, according to some scholars, it would require a modification of the property rights system which could infringe on the existing system for regulating private ownership rights 131 . Second, improperly conceived access legislation can be a major cause of legal uncertainty and/or "scare off" potential applicants. The measures to be created should hence establish a predictable and clear situation. As such, the following measures are considered in countries with ABS legislation:  the conditions under which access will be granted: Most countries, having access legislation in place, require both PIC from the providers of GR and the proof of MAT to grant access to prospective users. However, in order to ease the access procedure, some countries have "decoupled" the access requirement and the benefit-sharing requirement. The South African Government amended its 2004 Biodiversity Act with a distinction between the "discovery phase" and the "commercialization phase" of utilization of GR. As such, it acknowledges the unpredictability of the scientific process and allows for benefit-sharing agreements to be made at a later stage in the research process, once results are clearer and potential value is easier to evaluate. The discovery phase only requires a notification to be made to the relevant Minister, while prospective "commercial users" need to apply for a permit, linked to a BS agreement, before entering in the commercialization phase 132 .  the types of utilization requiring an access permit/PIC: Although not always easy to make, some countries have been trying to differentiate between access for commercial and noncommercial reasons, in order to facilitate access to the latter. This has been done through different approaches. In Brazil, the Genetic Patrimony Management Council (CGEN), responsible for granting access to the country's GR, established a list of the types of research and scientific activities exempted for access requirements 133 . In Australia, access for noncommercial purposes such as taxonomy is free, while the permit fee for commercial purposes is AUD $50 134 . In Costa Rica, biodiversity related research conducted in public universities has been left out of the ABS law's scoop, except if it has commercial purposes 135 . Some countries also established differential treatment depending on the type of commercial purpose 136 .  the actors requested to have an access permit: access requirements can be different for domestic and foreign users. Three approaches are being used for this matter. In India, access requirements only apply to foreign individuals, institutions or companies or any Indian organization which has "non-Indian participation in its share capital or management" 137 . In South Africa, foreign nationals can only apply for access jointly with a juristic person registered in terms of South African law 138 . Most countries, however, do not make this distinction. Moreover, when countries do not require access permits for domestic researchers and companies, they still expect these actors to comply with BS 139 .  the access procedure: a transparent and non-arbitrary procedure needs to be set up in order to provide legal certainty to users. The main steps of an access procedure include: (1) the 132 Sections 29, 38 and 39 of the National Environmental Laws Amendment Act, Government Gazette No. 14 of 2009, Republic of South Africa. 4) approval or denial of the application; (5) appeal 140 . Some countries have chosen to enshrine the procedure in a legal act in order to enhance legal certainty. This is the case in Costa Rica for example, where a "General Access Procedure" was developed as a bylaw to the Biodiversity Law 141 . Table 1 - Benefit-sharing An important debate in the literature concerns the benefit-sharing requirements laid down in the mutually agreed terms. This is also clearly mentioned as a core measure in the "Bonn Guidelines on Access to Genetic Resources and the Fair and Equitable Sharing of the Benefits arising out of their Utilization" (Article 41 to 50). In the context of developing straightforward legislation and provide legal clarity, the following set of issues are addressed in the literature:  the format of MAT: Most countries tend to have an ad-hoc approach for MAT, where the content of the MAT is negotiated on a case-by-case basis. However some countries also strived to go further by providing indicative sets of guidelines for the establishment of MAT 142 : the Australian Government has thus published two model agreements on benefitsharing for public and privately owned material 143 . Finally, some countries decide to opt for a more coercive approach. In Australia, the Environmental Protection Diversity Conservation Regulations 2000 (Regulation 8A.10) imposes different substantial and procedural conditions to the MAT. Two model-agreements are provided, one for publicly owned areas and the other for privately owned lands. These models serve as guidelines: parties to a contract are free to set up their own format, based on bilateral negotiations 144 . The third approach imposes a standard model to be used by all the users. In South Africa, the Biodiversity Act lays down the mandatory content of the MAT, composed of a benefit-sharing agreement (BSA) and a material transfer agreement (MTA). A prescribed format is provided by the Competent National Authority for both the BSA and the MTA 145 .  the types of utilization of GR leading to BS: BS could be claimed for all types of access, notwithstanding the prospects of utilization (commercial and non-commercial) flowing from this access. However, in order to avoid to putting too much of a burden on non-commercial research, some countries have limited their benefit-sharing requirements only to those utilization activities with prospects for commercial use 146 . The access application however generally includes a "return clause", obliging researchers to return to the negotiation table and settle benefit-sharing terms if and when they enter into a commercialization phase 147 .  the moment in the procedure at which BS agreements need to be settled: In 2007, Brazil amended its domestic ABS legislation to allow users and providers to set-up a benefit-sharing contract at a later stage than the moment of access. The aim was to make the result of the planned research clearer and allow for an easier evaluation of the value generated by the GR 148 .  the type of benefits to be shared: Some countries set out the types of benefits to be shared. These include participation of domestic institutions, joint ownership of patents, royalties, 142 Burton G (2009) technology transfer, etc. In India, the National Biodiversity Authority (NBA), responsible for benefit-sharing, determines possible benefit-sharing options 149 .  making BS fair and equitable: Whether shared benefits are fair and equitable is up for debate between the stakeholders agreeing on MAT. However, to avoid unequal bargaining power between users and providers, some countries have set-up minimum benefit-sharing criteria, such as the creation of a minimum royalty rate 150 . Other countries created a trust fund which collects all money arising from benefit-sharing agreements 151 . Although a promising solution to guarantee an equitable distribution of benefits between stakeholders, in some cases the fund only serves to channel benefits to the involved stakeholders in accordance with the provisions of the BS agreement 152 . In India, only those types of benefits determined by the National Biodiversity Authority can be considered as being fair and equitable 153 . Conservation activities and biodiversity research Creating conditions to foster biodiversity-related research and making sure ABS serves a conservation purpose and encourages sustainable use of natural resources is a transversal objective. Most of the issues addressed in the literature in relation to this objective do not concern stand-alone measures, but a set of measures listed under various obligations of the NP that contribute to conservation activities and biodiversity research. The following measures are considered in relation to other objectives to make sure they serve the national biodiversity interest:  Ownership: If the ownership of GR is vested in the state and the state collects all benefits arising from their use, it is much easier to make sure resources are accessed in a sustainable way and that benefits are redirected towards conservation activities 154 .  Geographical scope: The management of ABS and of protected areas/natural parks presents interesting synergies which could be promoted. Protected areas play a crucial role in the conservation of biodiversity as they host unique habitats, species and genetic resources. These could be of interest to users. As such, linking protected areas and ABS could be an innovative funding source for biodiversity conservation 155 .  Benefit-sharing: Several initiatives have been taken to redirect benefit-sharing towards conservation and sustainable use. Both Costa Rica and Peru require a fixed percentage (10%) of the value of gross sales, before tax (Peru) or of the research budget (Costa Rica) to be invested in conservation activities or capacity building initiatives for indigenous communities 156 . In South Africa, ABS regulations stipulate that surplus generated by the benefit-sharing fund should be directed towards conservation and capacity building initiatives 157 . An additional measure could be to allow the administrations responsible for the management of nature and/or biodiversity to handle sharing of benefits. This would establish a link between biodiversity conservation activities and the use of benefits. In Costa Rica, the National Commission for the Management of Biodiversity (CONAGEBIO), for example, is responsible for both the development and coordination of the national strategy concerning biodiversity conservation and the management of the utilization process 158 .  Access: Access conditions can be a major leverage for the sustainable use of GR and for the encouragement of biodiversity related research. Firstly, non-commercial biodiversity related research could be exempted from any access requirements, or their access requirement could be simplified, as is explicitly foreseen in Article 8a of the NP. Secondly, the granting of access permits could, for example, be subjected to a mandatory environmental impact th April 1998 157 However, it is unlikely that the fund generates any significant surplus as it is no more than a conduit for money due to stakeholders; See Wynberg R, Taylor M (2009), op. cit. 158 Article 5 of the General Rules for the Access of Genetic and Biochemical Elements and Resources of Biodiversity. Executive Decree No. 31514, Republic of Costa Rica; See also http://www.conagebio.go.cr/quienes/Funciones.html assessment for users, as is currently being done in several countries having ABS legislation in place 159 . In Kenya, for example, the holders of an access permit are required to provide reports on the environmental impacts of the collection of genetic resources or their intangible components 160 . Table 3 - Competent National Authority The CNA is the official institution that grants access, issues written evidence that access requirements have been met and advises users on applicable procedures and requirements to get access to GR 161 . In order to do so, the CNA has to be designated on behalf of the Party and given powers to fulfill its tasks as listed in the NP. The following two issues are addressed in the literature:  Designation: Two approaches are considered in the literature in regard to the establishment of the CNA. The first one consists of the creation of a new institution, which is then designated as the CNA, and possibly also as the authority fulfilling other related tasks, as is the case with CONAGEBio in the national ABS legislation in Costa Rica 162 . The second approach, as used in South Africa, is the designation of an existing institution as CNA, in this case the Ministry of Environmental Affairs and Tourism 163 . According to the used wording in Article 13 of the NP, a Party may also designate more than one CNA. The African Model legislation, for example, includes the possibility for other institutions to take over the role of the CNA 164 .  Empowering: In order to provide users with legal certainty, the role and mandate of the CNA should be clearly defined. Under current ABS measures, the CNA takes on three types of roles. First, it can function as a "one-stop shop", i.e. as a single point of contact for potential users applying for an access permit, granting the ABS permit, but also channeling the applications of other related permits to the competent authorities and the outcome of the procedure back to the user 165 . Secondly, in addition to the roles under the first option, the CNA could exercise the general responsibility on coordinating/facilitating the access procedure, including the coordination of the procedures of ABS and other ABS related permits, such as the granting of related environmental (non-ABS) permits. Third, it could be the responsible authority not only for channeling, coordinating or facilitating the application and permit delivery for ABS related permits, but also be the competent authority for directly granting all ABS and ABS related permits. This latter option might require an in-depth integration with other power-levels and processes. Additionally, it might provide an opportunity to create synergies between the access granting authority and the authority responsible for compliance monitoring 166 . Another important issue to be solved is the clarification of the powers of the CNA in relation to the confidential treatment of data supplied during the access procedure. The Andean Community's model law, for example, describes the conditions under which such confidential data can be treated167 . It is worth recalling that the first upcoming task of the existing NFP and/or of the newly designated CNA will be to comply with the obligation notify the contact information of the NFP and the CNA and, in case of the designation of more than one CNA, of the relevant information on the division of responsibilities between them. Table 4 -Summary of relevant measures for the Competent National Authority Relevant measures for the minimal implementation of core obligations Establish CNA (Article 13)  Option 1: Designate an existing institution as CNA  Option 2: Establish and designate a new institution as CNA  Option 3: Establish more than one CNA Relevant measures for additional implementation Additional legal rights and duties for the CNA  Option 1: CNA responsible for access permit acting as a single stop shop for all ABS related permits, that it channels through to the competent authorities for granting these permits;  Option 2: CNA has full and sole responsibility for the access application (as in option 1), but also coordinates/facilitates the procedure and the granting of ABS related permits;  Option 3: CNA directly grants all ABS and ABS related permits. Compliance Efficient compliance measures, in particular through monitoring the use of GR, are key to a successful implementation of the NP. The following issues are considered in the literature for the design of compliance measures and monitoring systems:  Giving binding effect to domestic legislation of the provider country: A critical step in the regulation of ABS is to lay out the basic obligations domestic users have to comply with when importing and/or utilizing genetic resources. This obligation comes down to give binding effect to the provider country's PIC and MAT. A first approach would be to consider that the prime responsibility for regulating ABS lies with the provider country. In such a case, a Party would only require users under its jurisdiction to act in accordance with the foreign legislation. A second option would be to establish a self-standing obligation in the legislation of the user country. As such, the legislation does not refer to the actual ABS legislation of the provider country, but only to the specific obligation of requiring PIC and MAT for access to its GR 168 , if so requiered by the provider country.  Monitoring the utilization of GR: Very few monitoring systems for ABS are operational yet. The following issues need to be addressed to serve as a basis to implement a monitoring system: o Checkpoints: at least one institution has to be designated by each Party to function as a checkpoint to monitor the use of PIC and MAT during the valorization process of GR. This can be an existing institution, such as the IPR office amongst others, or a newly created institution. The wording of Article 17.1(a) suggests that more than one checkpoint can be designated/created. o Monitoring system: The heterogeneity of utilization activities makes it very difficult to establish a 'one-size-fits-all' monitoring system. Three approaches are generally considered for the implementation of such a system. The less stringent one is the establishment of a 'voluntary monitoring system' where users would be required to report to the checkpoint(s) on a voluntary basis. It requires a strong commitment and understanding of ABS by private users as well as close collaboration between the monitoring authority and these users. In Australia, a "Biodiscovery Industry Panel" was established to foster this type of collaboration 169 . The second option is the socalled "due-diligence" monitoring system 170 . This system is a self-monitoring system requiring that users make sure they are using GR that has been accessed in compliance with the national and/or foreign ABS legislation. This type of system can be particularly relevant when GR is being transferred to third parties during the valorization process. A cost-effective way to support such an approach has been setup in Australia by creating the Genetic Resources Information Data Base (GRID), where all existing ABS agreements are freely viewable online 171 . It creates a transparent system, allowing any prospective investors to verify the legal status of the genetic resources acquired on Australian territory at no cost. A third approach would rely on monitoring by previously established checkpoints at specific stages of the valorization chain. Particularly relevant here is the choice of the time at which the right of use of GR should be controlled. Possible stages are the research fund granting, the patent granting, the market access authorization and the moment of import into the country 172 . This choice would also influence the type and number of checkpoints to be established.  Foster compliance among users: The strength of the motivation of users to comply is likely to be a determinant factor of the regime's effectiveness. Therefore, the state might want to create incentives and motivations for its users to comply. This could be done by offering financial benefits (e.g., tax deductions, rebates, and other rights), opportunities (e.g., special priority for other filings, permits or opportunities, access to special materials or programs that cannot be accessed by others) and positive publicity to complying users 173 . The latter measures are also clearly mentioned in Article 51 of the "Bonn Guidelines on Access to Genetic Resources and the Fair and Equitable Sharing of the Benefits arising out of their Utilization". Table 5 -Summary of relevant measures for compliance Relevant measures for the minimal implementation of core obligations Give binding effect to domestic legislation of provider country (Article 15, 16, 18)  Option 1: Leave responsibility to the provider country  Option 2: Create a self-standing obligation Designate checkpoints (Article 17.a)  Option 1: Designate existing institution as checkpoint  Option 2: Establish and designate new institution as checkpoint  Option 3: Establish more than one checkpoint Relevant measures for additional implementation 171 https://apps5a.ris.environment.gov.au/grid/public/perrep.jsp 172 Kamau EC, Fedder B and Winter G (2010), op. cit. 173 Young T (2006) Covering ABS: Addressing the Need for Sectoral, Geographical, Legal and International Integration in the ABS Regime. IUCN Environmental Policy and Law Paper No. 67/5 RECOMMENDATIONS FOR LEGAL, INSTITUTIONAL AND ADMINISTRATIVE MEASURES IN BELGIUM Based on the preliminary assessment of existing ABS measures (chapter 6) and the legal gap analysis (chapter 5), this section lists a set of recommendations to support identification of policy options for the implementation of the NP in Belgium. Each recommendation is listed as an "action card", including different options for implementation. For each implementation option, a number of advantages and disadvantages are identified, which are the basis of a first selection of the recommended measures for the impact assessment and which can serve as guidelines for a more in depth discussion. Most of these implementation options are not exclusive: they should be combined in order to achieve an efficient implementation of the NP. Each action card also provides a short description of the rationale behind the action to be taken and shortly states some of the existing Belgian measures which are relevant for the action. The action cards have been divided into two groups.  The first set of actions comprises measures related to the core obligations for the implementation of the NP in Belgium, as specified above (cf. chapter 5.2). They form the basis of compliance with the NP and represent a case of 'minimal implementation' for Belgium (addressing the minimal implementation of the core obligations).  A second set of additional measures which are important elements during implementation of the obligations, but that are less urgent (going beyond the minimal implementation of the core obligations or going beyond the core obligations). For each of the action cards a preliminary recommendation is provided, based on the arguments advanced and organized according to the following categories:  recommended measure  preferred measure, potentially interesting and meriting further analysis  more than one of the suggested measures potentially interesting and meriting further analysis  not recommended for a particular reason Recommendations for actions to be taken in case of minimal implementation of the core obligations As indicated above, this section analyzes and evaluates a set of legal and non-legal instruments for the minimal implementation of the core obligations of the Protocol codified under Articles 5 to 9, Article 13 and Articles 15 to 18. This list is by no means exhaustive, but contains a set of recommendations resulting from the analysis in previous chapters and which supported the selection of the options of which the impact was analyzed in this study. Priority of the measures The below-mentionned action cards have been assigned a priority score according to the following scale:  Action card -Determine format of MAT Description: Under the NP, Belgian users are required to share benefits upon MAT. These MAT should hence be given binding effect under Belgian law. The NP, however, does not impose a format for MAT, which can be left to the discretion of stakeholders or flow from (mandatory) measures. Related Article of the NP: 5, 18 Nature of the measure: Legal Priority for Belgium:  Relevant existing measures in Belgium  BCCM's MOSAICC Option 1 -Leave full discretion on how to execute the BS obligation to users and providers of genetic material Possible advantages:  high flexibility for users and providers to agree on specific benefit-sharing  might be less of a burden for large company users, as they will choose to conclude MAT that generate the least cost  might represent a low-cost measure for public authorities as no additional resources are needed concerning MAT. Possible disadvantages:  does not allow the state to control the benefit-sharing procedure  does not allow to make sure benefits are shared in a fair and equitable way  does not allow to make sure that benefits contribute to the conservation of biological diversity and sustainable use of its components Option 2 -Develop mandatory MAT terms and conditions and/or default MAT provisions Possible advantages:  might provide stronger legal clarity to all stakeholders involved  allows the state to control the content of the MAT and can make sure benefits are shared according to principles of fairness and equity  might also smoothen the negotiation process between commercial users and providers Possible disadvantages:  might offer less flexibility to stakeholders EVALUATION The 2 options are potentially interesting and deserve further analysis. Action card -Clarify access conditions Description: Holding sovereign rights over its genetic resources, Belgium can choose whether or not to require bioprospectors to obtain Prior Informed Consent for the competent authority for access to genetic resources under its jurisdiction Related Article of the NP: 6, 7 Nature of the measure: Legal Priority for Belgium:  Examples of relevant existing measures in Belgium  There are no existing legal measures on PIC to access GR in Belgium Option 1 -Require PIC from the Belgian State as a Party to the Protocol Possible advantages:  Contributes to the implementation of the Nagoya Protocol and its objective of promoting the conservation and sustainable use of genetic materials as well as their fair benefit-sharing  Allows to keep track of accessed Belgian GR  Allows for access statistics to be kept  Provides for legal certainty for users, through clarifying the legal state on access in Belgium and through the possibility of providing them with a PIC/international certificate of compliance Possible disadvantages:  Need to develop access rules and procedures  Depending on the implemented procedure, could create some additional administrative burden for users Option 2 -Do not require PIC from the Belgian State as a Party to the Protocol Possible advantages:  No additional legal measures needed, for establishing an access procedure  Lower administrative burden for users at time of access (as not needing to go through an access procedure) Possible disadvantages:  Does not allow to keep track of accessed Belgian GR  No information to base a potential policy review on;  Does not allow for access statistics to be kept  Would still need to take legal measures in order to clarify the current legal status for access to GR in Belgium  Would not provide legal certainty for users of Belgian GRs (e.g. use in third countries, subsequent use, etc.) EVALUATION Option 1: recommended measure ; contributes to the implementation of the Nagoya Protocol by assuring more legal certainty Action card -Ensure ABS serves conservation and sustainable use of biodiversity Description: Alongside the aim to share benefits in a fair and equitable way, the implementation of the Nagoya Protocol should serve the broader goal of the CBD: conservation of biodiversity and sustainable use of its components Related Article of the NP: 9 Nature of the measure: Administrative and/or legal Priority for Belgium:  Examples of relevant existing measures in Belgium  Article 16, Flemish Natuurdecreet: If a permit for access is delivered, the competent authority shall ensure that no avoidable damage to nature may arise by imposing reasonable conditions  Article 20, Flemish Soortenbesluit: Access to protected species can only be allowed if it does not affect the conservation of these species Option 1 -Link access permit to mandatory conditions on the use of benefits Possible advantages:  Could be a way to ensure at least a minimal part of benefits is directly flowing to conservation/sustainable use  Institutions could be reluctant to implement new tasks, as they might not be considered as 'core tasks'  Could be ineffective if the institution(s) does(do) not have sufficient know-how, related experience and resources  One centralized CNA is not in line with the actual division of competences for environmental issues, which are mainly situated at the level of the Regions (cf. chapter 2) Option 2 -Establish and designate a new institution as CNA Possible advantages:  Could establish very efficient procedures as it would be the 'core task' of the new institution.  Possibility to create synergies between CNA and NFP, providing more process certainty for users Possible disadvantages:  Could have a high financial and transaction cost, as new structure needs to be established  Some necessary information might be confidential and/or difficult to access  One centralized CNA is not in line with the actual division of competences for environmental issues, which are mainly situated at the level of the Regions (cf. chapter 2) Option 3 -Designate more than one CNA Possible advantages:  Would better fit the Belgian institutional framework, considering the actual division of competences Possible disadvantages:  Could create uncertainty for potential users on the competent CNA in certain specific cases of mixed competences (however in most cases access will be clearly granted in one of the regions). Coordination mechanisms (such as a web based centralized input system for access requests) might be required then EVALUATION If PIC is required, option 3 seems the most straightforward, as access is mostly clearly granted in one of the 3 regions and as the access requirements to indigenous species in the Regions are part of the regional competences. would only be an addition to existing tasks (but depending on workload, institutional set-up, possible synergies, …) Action card -Give binding effect to domestic legislation of provider country Possible disadvantages:  Institutions could be reluctant to implement checkpoint-tasks, as they are not considered as "core tasks"  Could be ineffective, if these institutions do not have sufficient relevant knowledge, know-how, related experience, etc.  Could represent high administrative burden, if these institutions have to create a whole new "section" for GR monitoring, with few synergies Option 2 -Establish and designate new institution as general checkpoint Possible advantages:  Could establish very efficient monitoring as it would be the "core task" of the new institution.  Possibility to create synergies with CNA and NFP, providing more process certainty for users Possible disadvantages:  Could have a high financial and transaction cost, as new structure needs to be established  Some necessary related information might be confidential and difficult to access by such a "monitoring" institution (compared to existing institutions that already have acquired data, the confidence from stakeholders and users, etc.) Option 3 -Establish more than one checkpoint Possible advantages:  Would better fit a monitoring system to support compliance if transparency is created and monitoring done at specific stages of the valorization chain  Could be better adapted to the institutional reality in Belgium  Could be more cost effective than one centralized institution  Allows to exploit existing institutional capacity to address the monitoring requirements  If based on existing institutions, these could benefit from more confidence of stakeholders and users Possible disadvantages:  Might need more time to install an effective set of checkpoints, and might incur a higher financial, legislative and administrative cost compared to option 1, but lower than option 2  Might increase the complexity of the monitoring process  Might need additional coordination mechanisms amongst the checkpoints EVALUATION Recommended measure: Option 2: the necessary combination of technical, scientific and administrative competences will probably require a new structure to be effective. Could be combined with option 3, if needed. Recommendations for actions to be taken in case of additional implementation This section presents a list of recommendation for measures, which go beyond the minimal implementation of the core obligations and/or beyond the core obligations as explained above. Priority of the measures The below-mentioned action cards have been assigned a priority score according to the following scale:     Action card -Set additional specifications for benefit-sharing upon MAT Description: Alongside the basic mandatory conditions for access (PIC and MAT), Belgium can set additional specifications on BS upon MAT: as a mandatory access condition (in general terms (e.g. established in a legislative instrument/in standard PIC conditions/…) for all uses, for types of uses, or as specific terms (e.g. in the PIC) for the particular use(s) for which access is requested), as default access conditions (in case not provided for otherwise in the terms of the PIC), as a mandatory condition on the use in general terms (e.g. through a legislative instrument) for all uses, for certain uses, or in particular terms for the use envisaged (established in a particular terms for the use, probably through an approval….) Related Article of the NP: 6, 8 Nature of the measure: Legal Priority for Belgium:   Might be difficult to establish a finite list EVALUATION Option 1 is not recommended because of a lack of flexibility; Option 3 is not recommended as it could be illegal. Options 2 and 4 are equally potentially interesting and are recommended for further analysis. Option 5 is also potentially interesting and recommended for further analysis and could be envisioned in combination with option 2 and 4. Examples of relevant existing measures in Action card -Establish clear and transparent access procedure Description: The application and approval procedure should be made clear, including identifying required action to be taken, the consecutive steps of the application, setting time limits for decision-making process and provide a clear record of the final decision. Difficulties can arise when the application process is not clearly defined and/or stated in law or when the law leaves too much discretion to the competent access authority. Related Possible disadvantages:  requires a strong commitment and understanding of ABS by private users  requires close collaboration between the monitoring authority and these users  Could be constraining for non-commercial research  Depending on how it is implemented could still impose a considerable financial and administrative burden on users and administrations Option 3 -Monitoring by checkpoints at specific stages of the valorization chain Possible advantages:  Could be easily combined with the establishment of certain actions (e.g. patenting, commercialization) as triggers for BS, instead of access  Allow a more in-depth monitoring than voluntary based methods Possible disadvantages:  Depending on how it is implemented could be a high(er)-cost option for users and administrations  If existing institutions: institutions could be reluctant to implement new tasks, as they are not considered as 'core tasks'.  If existing institutions: could be ineffective if the institution does not have sufficient knowledge, knowhow EVALUATION These options should be assessed in combination with other action cards. If there is only 1 checkpoint, option 1 and 2, in combination with user incentives, is potentially interesting and deserves further analysis. If the option of more than one checkpoints is considered, option 3 is potentially interesting (seen the potential cost-effectiveness) and deserves further analysis. Action card -Create incentives for users to comply EVALUATION Option 2 is recommended (can build upon existing practices and has proven its effectiveness). Option 3 might impose a higher burden on authorities. Option 1 gives more responsibility to the sectors, but might lack effectiveness. DEFINING THE POLICY OPTIONS AND PRELIMINARY ANALYSIS OF THEIR EXPECTED IMPACTS As for previous chapters, this chapter mainly focuses on the core measures specified in the terms of reference of this study as requiring special attention (see chapter 5.2). Building upon chapter 6 and 7, this chapter presents policy options discussed at the first stakeholder meeting and, based on the discussion with stakeholders, selected by the Steering Committee of this study for the implementation of six core measures that are needed for the minimal implementation of the Nagoya Protocol in Belgium: It is important to remember that at least one of the legal provisions (designation of National Competent Authorities and the National Focal Point, Article 13.4) needs to be implemented no later than the entry into force of the Nagoya Protocol for each Party (that is the ninetieth day after the date of deposit of the 50 th instrument of ratification if the Party ratified until the deposit of the 50 th instrument, or on the ninetieth day after the date of deposit of the instrument of ratification if the Party ratifies after the deposit of the 50 th instrument). Therefore, Article 13 and the core obligations directly related to that Article (such as Article 6 which has a direct impact on the tasks of the Competent National Authority) deserve a special urgent attention. Further, in line with the EU guidelines, the general principle of the impact assessment is to assess the impact of policy options as net changes compared to the "no policy change" baseline. For this purpose, a general description of the "no policy baseline" is given and for each measure the particular expressions of this baseline are specified. For this purpose, a distinction is also made between a general default "no policy change over all the options" and a specific "0" option for each section, which considers a "no policy change" over a specific obligation, in a situation where nevertheless some other measures could have been taken. The description of the options and the preliminary analysis regarding their expected impacts are based on the discussions held and the comments received during the first stakeholder meeting on 29 th May 2012174 . Description and discussion of the general "0" option The general "0" option represents a situation where "no policy change" takes place for any of the items considered below; that is if none of the options discussed below are implemented. This would lead to a non-ratification of the Nagoya Protocol. However, in this situation, Belgium would still have to comply with the international obligations pertaining to GR and TK, mainly the CBD's Articles 8(j) and 15 (for GR and TK associated to GR) and the ILO Convention No.107, Articles 7(1), 11 and 13 (for TK), which Belgium both ratified. In particular, Belgium would still have to take measures to clarify access to GR for their utilization, which may or may not be covered under existing legislation, and to take potential compliance measures with the aim of sharing benefits from the utilization of GR and TK in a fair and equitable way with the countries of origin of these resources. These measures related to existing international obligations would have to be taken in any of the specific "0" option measures discussed below as well. Moreover, after the entry into force of the Nagoya Protocol (in this case, without ratification by Belgium), there would be a need to clarify (reinterpret/amend) the current Belgian legal framework in the light of the adoption of the Nagoya Protocol. For example, clarify whether existing requirements on access also apply on access in the meaning of the Nagoya Protocol and for setting up a framework to enable dealing with transactions with GR from/to countries that have ratified the Nagoya Protocol. An obvious disadvantage of the general "0" option is the failure to create the legal certainty and transparency, both prominent aspects of the implementation of the Nagoya Protocol, thereby potentially increasing transaction and litigation costs for users and providers. Moreover, as stated above, even in case of non-ratification, Belgium needs to take a set of legal measures on access and benefit-sharing. These would nevertheless be different than the set of measures required for implementing the Nagoya Protocol, creating a confusing situation for users and providers (i.e. existence of many different legal regimes for the same issue). Furthermore, non-ratification would lead to complex relations of Belgium as a non-Party with Parties to the Nagoya Protocol. It would also probably lead to a loss of Belgian credibility and trust on the international forum; with a risk of straining multilateral relations and also loss of exchanges (research, development, collections, industry, …) and hence opportunities for Belgian individuals and institutions (e.g. in relation with Parties to the Protocol). Defining the policy options for the core measures and their expected impacts Access to GR and Benefit-sharing As a first preliminary finding of the study on the implementation in Belgium of the Nagoya Protocol, it is recommended to establish Prior Informed Consent (PIC) and Benefit-sharing (BS) as general legal principles in Belgium in order to implement Article 5 (on benefit-sharing) and Article 6 (on access) of the Nagoya Protocol. As a general principle, the operationalization of PIC and BS should be phased, flexible and based on the subsidiarity principle This operationalization of PIC and BS can then be divided in two implementation components that are interrelated: the operationalization of PIC (first component) and the specification of the Mutually Agreed Terms (second component). Description of the options  The specific "0" option: Benefit-sharing-component The specific "0" option on BS would consider taking no measures on benefit-sharing in Belgium (such as establishing benefit-sharing as a general legal principle). This would lead to a nonratification of the Nagoya Protocol, and would still require to take the measures specified in the general "0" option. Moreover, it is unclear if this would not amount to non-compliance by Belgium with Article 15 of the Convention on Biological Diversity.  The specific "0" option: Access-component This specific "0" option on Access would consider no PIC requirement, with or without benefitsharing as a horizontal principle. This would not necessary lead to a non-ratification of the Nagoya Protocol (if benefit-sharing is established as a horizontal principle). If it leads to a nonratification, this "0" option would still require to take the measures specified in the general "0" option above. However in both cases (with or without benefit-sharing as a horizontal principle), this "0" option would create less legal certainty for users of Belgian GR, would not allow to deliver an international certificate of compliance for such users (which serves as evidence that GR, which it covers, has been accessed in accordance with PIC and that MAT have been established) and could lead to a lack of data on the use of Belgian GR for evaluating policy and promoting research and development.  General ABS option 1: No Prior Informed Consent required, but Benefit-sharing as horizontal principle Under option 1, no PIC would be required, but BS would be established as a general legal principle in Belgium in order to implement Article 5 (on benefit-sharing) and Article 6 (on access) of the Nagoya Protocol (which specifies that a Party might determine not to require PIC). However, even if no PIC is established, the current legal framework on access will still need to be clarified, in a way that allows complying with the obligations of the Nagoya Protocol and the options for implementing the core measures discussed in this report.  General ABS option 2: Prior Informed Consent and Benefit-sharing as horizontal principles Under option 2, PIC and BS would be established as general legal principles in Belgium in order to implement Article 5 (on benefit-sharing) and Article 6 (on access) of the Nagoya Protocol. As a general principle, the subsequent operationalization of this general obligation through PIC and MAT should be phased, based on subsidiarity and flexible. Expected impacts  General ABS option 1: No Prior Informed Consent (PIC) required, but benefit-sharing as a horizontal principle This option might seem easily implementable as it would not require any additional legal measures to be taken and could imply a relatively low administrative burden, as the requirements for operationalizing PIC would be avoided (this possible advantage will only be important if the options chosen below imply a heavy administrative burden). However, this option would still require clarifying the current legal framework on access, in a way that would not only take into account the adoption of the Nagoya Protocol (see general baseline), but would also allow complying with the obligations of the Nagoya Protocol and the options for implementing the core measures discussed further in this section. Furthermore, it is unclear how this option could provide legal clarity for users after access, in particular since it does not allow the State to offer users a proof of legal access such as an international certificate. Nor will it allow post-access tracking and/or monitoring of the utilization of genetic resources and the collection of data, which could result in missing out important input of valuable data for research, innovation and conservation policy. In other words, under this option, the Belgian State would not give itself the means to get information on its GR accessed or to monitor/control the use of its own genetic resources. It could lose out on an important incentive to promote conservation and sustainable use of its own GR.  General ABS option 2: Prior Informed Consent (PIC) and Benefit-sharing as horizontal principles The Nagoya Protocol contributed to turn the debates about PIC and BS around. Whereas previously, it was considered to be more interesting for users to access genetic resources in states having the least regulation in place, now users might prefer states with public and transparent access and benefit-sharing legislation in order to optimize the legal certainty for the subsequent utilization of these resources. A major advantage of the option 2 is that it paves the way for the delivery of an internationally recognized certificate of compliance to users by the Belgian State, hence increasing transparency and legal certainty. It could further allow more efficient and effective monitoring and tracking of the use of its GR. Keeping track of access to GR will also give a better view of the available genetic resources, and facilitate data and statistics collection which are useful for biodiversity policy in general and for further implementation of the Nagoya Protocol in particular. To be functional, this option however needs additional legal access rules and a clearly defined access procedure. Depending on the further operationalization, it could create administrative burden, both for users as for public authorities involved in administrating PIC. Further operationalizing general option 2 on PIC If both Prior Informed Consent and Benefit-sharing are established as horizontal principles (general ABS option 2 above) two additional interrelated measures should be implemented: the operationalization of PIC (first component) and the specification(s) of the possible requirement of and conditions for the Mutually Agreed Terms (second component). The first implementation component could consider the operationalization of PIC through a notification/registration/approval requirement 175 to the Competent National Authority or authorities. In the second component, implementation measures related to the content of the mutually agreed terms of the access agreements, including as specified in the notification/registration/approval procedure, should be considered. In line with Articles 4 and 8 of the Nagoya Protocol, these measures should have due regard for the particular features of certain sectors, species or areas and, in line with Articles 1 and 9, they should contribute to the objectives of conservation and sustainable use of biodiversity. Description of the options The options to establish and operationalize PIC are built up in two parts: Starting point: limit administrative burdens by building on existing legislation Two reasons make a preliminary analysis of the existing legislation necessary for the study of the different PIC "sub-options". First, situations should be avoided where different permits from different administrations would have to be obtained for accessing the same material: the superposition of different requirements and procedures for the same material would furthermore complicate the administrative follow-up and increase the administrative burden, in particular if the same data would have to be resubmitted to different, unrelated permit databases. Second, protected areas (PA) and protected species (PS) contain GR which are important for conservation and sustainable use of biodiversity and may be of actual/potential (high) value. The first step in the implementation of the PIC and BS requirements could (a) consider refining existing PA and PS relevant legislation in order to include more specific regulation for the access to GR for utilization, as defined under the Nagoya Protocol. (b) beyond refining PA and PS relevant legislation, potentially include other relevant categories of GR with e.g. actual or potential value, by also considering other existing 175 "Notification" and "registration" refer to an easy and less burdensome permit requirement: the permit is automatically provided/generated if the applicant provides certain data and complies with certain general conditions. "Approval" refers to permit-requirement that demands an individual assessment of each individual application, that apart from general permit conditions, might also imply the imposition of permit specific conditions. legislation relevant for the access to GR to build upon with the view to further operationalize PIC. Default option to complement the starting point (cf. to build on existing legislation) Additionally, for all the GR which are not covered through PA or PS legislation, a default rule could be adopted. This could be done (c) by only allowing such access from/through Belgian collections, or (d) by allowing access from anywhere, providing the user has registered/notified the Competent National Authority (CNA). When combining the above, the assessment of the impacts of the following three options seems to be the most relevant: This option combines an enlarged approach to refining existing legislation relevant for GR, with a default rule that GR, not covered by such modified legislation, can be accessed from anywhere, providing the user has registered/notified the Competent National Authority.  Expected impacts  Option 1: The bottleneck model: only existing PS/PA relevant legislation & measures + only access to GR through ex-situ collections as default rule Possible Advantages: This option would allow the collections to keep a copy of each accessed GR in Belgium whenever this is feasible at a minor cost. The existing scientific and administrative infrastructure of the culture collections could foster ex-post follow up. Existing databases and standard information could be used. The newly encoded information could contribute to biodiversity research such as taxonomic research. Finally, and as a general remark, the cost for access to collections is very high and should be kept as low as possible. To this end, it should take into account the high number of transactions by the collections. In case part of the benefits arising from the utilization of GR would be directed to the conservation activities of the collections, benefit-sharing could generate additional financial support for the collections. This option would not necessarily lead to heavy transaction costs for the collections, as most collections already have standard Material Transfer Agreements (MTA) in place which could be easily adapted, on the condition that these are in line with CBD provisions, including the Bonn Guidelines. Possible Disadvantages: A lot of the relevant GR might be situated outside the collections, such a configuration requiring thus additional resources for the handling of access requests. For these GR two situations can be distinguished. (a) The collection decides to keep a copy of the GR (for example when it is feasible for the collection to keep the GR at a minor cost and whenever it is scientifically relevant). In that case there are no additional resources required for handling the access request, as it is part of the standard procedures of the collections (including encoding in databases, handling of MTAs, etc.). However additional financial resources might be needed to bear the cost of handling the access request and storing information or samples that would not have passed by the collections otherwise (e.g. depending on whether the access concerns physical samples or only information). (b) The collection decides not to keep a copy of the GR (e.g. because it is expensive/beyond the capacity/scientifically not relevant/technically not possible). In that case, if information has to be kept on the access of the genetic resource, it would require the extension of the database infrastructure beyond the ex-situ holdings, to include documentation on access provided to in-situ resources through the collections. However, this might not represent an important additional cost, as it is possible to build upon the existing infrastructure. This second sub-option could also require the handling of MTA for the in-situ resources accessed through the collections, but not kept in the collection. . Furthermore, the relation between the culture collections and the CNA and the specific access-related powers of the collections will need to be clarified, as the CNA is the final authority able to grant access for utilization in the context of the NP. This could lead to an additional step in the access procedure and could create additional administrative burden for users wishing to access GR. However this border is not necessarily higher than under the other options as it would be based on a division of labor in the PIC procedure over the different entities.  Option 2: The fishing net model: only existing PS/PA relevant legislation & measures + access from everywhere, but with registration as default rule Possible Advantages: For the default rule, this option could strongly encourage utilization, as the administrative burden for users would be low. Financial and transaction costs for the State could also be relatively low, as the notification obligation could be easily set up through a standardized system. Moreover the notification/registration obligation would (1) provide data on the type of users of the genetic resource and (2) facilitate possible policy review. Possible Disadvantages: Under this option the default rule could prove to be ineffective or even create a loophole in the basic rule, if cases where species are found only within protected areas prove to be rare, and/or if most species within protected areas can also be found outside of these areas. Furthermore, this option would not allow to obtain as easily a copy of accessed GR in Belgian collections (whenever feasible) and it might be harder to coordinate with the existing databases of the ex-situ collections which already contain information on previous accesses and utilization of Belgian GR. Moreover, the default rule under this option might need to be limited to non-commercial use only.  Option 3: Existing PA/PS relevant legislation & measures + other specific GR relevant legislation/measures + access from everywhere, but with registration as default rule Possible Advantages: It can be expected that this option would mainly give the same positive impacts as under option 2(PA/PS legislation +access from everywhere as default access to GR).It will however have a bigger impact, as it would apply similar requirements as those for PA and PS to a broader set of GR and by integrating the new regulation with a broader set of related legislation. This option thus allows coping with cases where access to genetic resources is not limited to PA and PS. Microorganisms with potential value for research and development, for example, are generally found where natural selection has taken a different path i.e. in extreme environments that do not necessarily coincide with the PA/PS category. This option thus allows to extend the further operationalized PIC requirements to the broadest range of potentially interesting GR and reduces the amount of GR falling under the default category. Possible Disadvantages: Similar disadvantages as for the option 2. Furthermore, the amount of existing legislation relevant to GRs beyond the PS/PA related legislation, but also the amount of areas/material in Belgium beyond PA/PS, that are of particular importance for biodiversity, will determine whether or not this option has any added value beyond option 2. Specification of the Mutually Agreed Terms If PIC would be required in Belgium, it should also be clarified whether MAT is required and under what conditions (e.g. as a condition to obtain PIC). Given that a phased approach would allow to finetune the measures as more feedback is gathered, the initial MAT requirements could be further developed over timer after a rather limited first implementation phase. This section therefore further describes the "sub-options" considered in the case where both Prior Informed Consent and Benefit-sharing are established as horizontal principles (general ABS option 2 above). Description of the options In order to develop an idea of possible impact, 3 types of MAT are proposed for further exploration:  Option 1: No specific BS requirements imposed for the MAT A first type where, in the exercise of its sovereign rights over its GR, the Belgian State decides not to impose any specific benefit-sharing requirements from users in MAT (apart from the general legal obligation to share benefits and the structural benefits occurring from working of the future Belgian ABS system).  Option 2: Specific BS requirements imposed, through standard agreements, depending on finality of access For the second type, specific benefit-sharing requirements are imposed through standard formats for the MAT (e.g. a limited number of standard MAT-agreements), depending on the finality of the access. This could imply that no specific BS requirements are imposed in the MAT if no commercial utilization of the GR is planned, while more specific BS requirements are imposed if commercial purposes are envisaged (e.g. the collection of revenues from that use or the sale of the GR itself). The related MAT for non-commercial utilization would include a re-negotiation requirement in case of change in intent to commercial use.  Option 3: Specific BS requirements imposed, but their implementation is negotiated on a case by case basis, depending on finality of access Under option 3, specific benefit-sharing requirements are developed by the Belgian Authorities for each access request. These requirements can be of a different nature (e.g. a general regulatory obligation, a specific condition as a PIC-conditionality, etc.) and will be differentiated according to the finality of access. Expected impacts  Option 1: No specific BS requirements imposed for the MAT Possible Advantages: This option provides for high flexibility for users and providers to agree on specific benefit-sharing, depending on the specificities of the exchange of GR. It thus probably represents less of a burden for large company users, as they will choose to conclude MAT that generate the least costs, but it might be burdensome for non-commercial and small company users to negotiate individual MAT (e.g. if no standard MAT are available/applied in their sector). For the authorities, this option also represents a low-cost measure as no additional resources are needed concerning MAT. Possible Disadvantages: This option does not allow the Belgian State to control the benefitsharing procedure and to make sure benefits are shared in a fair and equitable way, or that benefits contribute to the conservation of biological diversity and sustainable use of its components. According to paragraph 45 of the Bonn Guidelines, fair and equitable benefitsharing varies "in light of the circumstances" and a third independent stakeholder (i.e. the state) might be needed to identify these circumstances.  Option 2: Standardized formats for BS requirements, depending on finality of access Possible Advantages: This option provides strong legal clarity to all stakeholders involved. It also allows the Belgian State to control the content of the MAT and can make sure benefits are shared according to principles of fairness and equity. It might also smoothen the negotiation process between commercial users and providers, as it could offer standard formats containing guidelines/default rules/requirements to follow, while providing security to providers that changes of intent will need a renegotiation. Possible Disadvantages: This option offers less flexibility to commercial users that already have their own systems or prefer a more flexible approach.  Option 3: Specific BS requirements, depending on finality of access Possible Advantages: This option provides strong legal clarity to all stakeholders involved. It also allows the Belgian State to control the content of the MAT and make sure benefits are shared according to principles of fairness and equity. It furthermore provides much flexibility to fine tune the BS requirements to cover concerns of both users and providers, including the contribution to conservation and sustainable use. Possible Disadvantages: Non-commercial users and small commercial users might suffer from this option, as they might not necessarily possess the needed resources to negotiate and fulfill the specific BS requirements. Establishing one or more Competent National Authorities Description of the options The designation of one or more Competent National Authorities needs to be implemented no later than the entry into force of the Protocol for each Party. Therefore this measure deserves special attention. Based on the options for the operationalization of PIC, the choice of the Competent National Authority would in the first place be based on the relevant competent authorities for the existing legislation and measures concerning in protected areas and/or protected species. This means four Competent National Authorities would be needed: one for each of the three Regions and a federal one, hence flowing from the actual division of competences in Belgium. The difference between the proposed options lies in the way users might have to request access to GR.  Specific "0" option for the CNA The specific "0" option on the Competent National Authority would consider not creating a Competent National Authority under the Nagoya Protocol. This would lead to a nonratification of the Nagoya Protocol, and still require to take the measures specified in the general "0" option.  Option 1: Decentralized input Each authority would have a separate entry-point, and users of genetic resources would need to request access through separate entry-points depending e.g. on the kind of GR or where they are found.  Option 2: Single entry-point Under this option, the four responsible authorities could agree on a centralized input system. Users would request access through a single point of contact, independently of where/which types of GR are accessed. Expected impacts  Option 1: Decentralized input Possible Advantages: Flowing from the actual division of competences in Belgium, this option could provide more liberty to the federated entities to independently organize their biodiversity and/or genetic resources access policy. Possible Disadvantages: Having four different Competent National Authorities might strongly complicate the access procedure, not in the least for foreign users. Additional efforts will be needed in order to clarify the access procedure, e.g. providing users with a clear overview on which of the four Competent National Authorities is responsible for handling access requests, depending on where/which GR are accessed. This might result in a higher administrative burden for both users and administrations. Moreover a decentralized input system for the data generated might lead to additional data coordination and exchange problems.  Option 2: Single entry-point Possible Advantages: A uniform or harmonized process could increase the legal and procedural clarity for users. This might result in less administrative burdens related to the search for information on access procedures and requirements under the Nagoya Protocol in Belgium. Furthermore, some economies of scale could be possible here for the public authorities concerned. Depending on the scope of these economies of scale, it might be decided to opt for more or less coordination through the single entry-point. Possible Disadvantages: This option potentially has a higher initial administrative burden and transaction cost. A common system needs to be established and close coordination between the different authorities needs to be ensured. Setting up compliance measures Description of the options The options for compliance in order to fulfill the obligations of articles 15, 16 and 18 of the Nagoya Protocol are dependent both on the sufficiency of the existing relevant dispositions contained, inter alia, in the existing criminal code, civil procedural code and on implementation of PIC in Belgium. A general criminal provision covering situations where PIC and MAT are required by the provider country is considered. In situations where a civil judge has to consider the contents of MAT, an extension of the field of application of art 15176 of the Code of private international law is envisaged. The granting of PIC on the access to genetic resources within the context of the Nagoya Protocol pertains to the country of origin of the GR applying its sovereign rights. Therefore, compliance with PIC involves public law and administrative acts, which fall outside of the scope of private international law. To contribute to the implementation of Articles 15, 16 and 18, the following options are proposed:  Specific "0" option for the compliance measures The specific "0" option on compliance measures would consider not introducing any legal provision on compliance. This would lead to a non-ratification of the Nagoya Protocol, and still require to take the measures specified in the general "0" option. Moreover, even if these measures were be taken in order to comply with the obligations of the CBD and the ILO Convention No.107, users and providers would not be able to benefit from the clarified legal framework that the compliance measures envisioned under the Nagoya Protocol would entail. This might lead to increased litigation and transaction costs (for clarifying exactly what the compliance to the CBD implies in a situation of no additional measures).  Option 1: Ensuring compliance with provider country legislation regarding PIC and MAT, with Belgian Law as fall-back option Under this option, a general criminal provision is created that refers back to PIC and MAT obligations as specified in the legislation of the provider country while the private international law code would determine that provider country legislation is applicable to disputes regarding compliance with the PIC and MAT. Sanctions would be provided for cases of non-compliance with PIC and MAT requirements set out by the provider country. When checking content of MAT, a provision in the code of international private law would provide for reference to provider country's legislation, with Belgian law as a fallback option. The state would enact a general prohibition to use GR/TK accessed in violation of the law of the providing country, by specifying that the reference to foreign law in the Belgian code of private international law also applies to the use of GR within the context of the Nagoya Protocol177 . The sanctions for violation could in that case be a fine and a confiscation. The state could act ex officio to enforce this criminal provision, which is usually taken up on the basis of complaints by individuals. The fact that a violation of foreign law would be considered as a violation of national, Belgian law, and could be prosecuted and sanctioned as such, would also make it easier for providers to subsequently claim civil law damages. A provision in the private international law code would determine that provider country legislation is applicable to disputes regarding compliance with PIC and MAT. If it is impossible to determine the content of the foreign law in due time, Belgian law should be applied178 . Option 2: Self-standing obligation in the Belgian legislation to have PIC and MAT if so required by the provider country. Under this option, a provision is created containing an obligation to have PIC from the provider country and MAT for the utilization in Belgium of foreign genetic resources, if the legislation of the provider country requires PIC and MAT for access to its GR. As such, Belgian legislation would not refer to the legislation of the provider country regarding PIC and MAT, but only to the specific obligation of requiring PIC and MAT for access to its GR. Expected impacts  Option 1: Ensuring compliance with provider country legislation regarding PIC and MAT, with Belgian-law as fall-back Possible Advantages: This option could serve as a strong measure to support compliance by Belgian users with the entire provider country ABS legislation. Possible Disadvantages: The option relies upon the assumption that the legislation of the country of origin properly implements the Nagoya Protocol provisions and that it is clear enough and acceptable for enforcement. If not, the option might entail legal uncertainty and unpredictability for the users. This disadvantage is attenuated to a certain extent through the fall-back clause in the code of private international law (cf. description of the option above).  Option 2: Self-standing obligation in the Belgian legislation to have PIC and MAT if so required by the provider country. Possible Advantages: It could create less legal complexity for users and enforcement authorities in Belgium Possible Disadvantages: It might be a less stringent measure for acting against potential illegal utilization of GR by Belgian users, although the criminal provision could later be extended to encompass other elements. Designating one or more checkpoints Description of the options Belgium could consider not introducing checkpoints as envisioned under the Nagoya Protocol, within the general "0" option. If Belgium does decide to introduce checkpoints, their implementation could take place in several phases. In order to respect the political commitment to timely ratify the Nagoya Protocol, the first phase could look at a minimal implementation requiring the establishment of a single checkpoint. Two possible options seem relevant for the first phase, namely PIC ("Option 1") and an upgraded patent disclosure ("Option 2"). In subsequent phases, more effective checkpoints might need to be developed in order to monitor the utilization of GR. Possible checkpoints to be explored at a later stage could possibly include public research funding, ex-situ collections or intellectual property related checkpoints other than the patent authorities, such as authorities for assessing applications for geographical indications of origin. Working with different phases could allow for a fast start with limited resources to prepare for an early ratification of the Nagoya Protocol. It also provides time to better identify concrete problems and to learn from the experience of others. However, it might take a longer time to arrive at the most effective and/or relevant checkpoints for the situation on the ground. Therefore caution should be taken not to delay addressing existing and known problem areas.  The specific "0" option on checkpoints This option would consider not introducing checkpoints as envisioned under the Nagoya Protocol (whether through an integrated PIC requirement or upgrading the disclosure requirements in patent applications or through any other means). This would lead to a nonratification of the Nagoya Protocol, but still require to take the measures specified in the general "0" option. Moreover, in the implementation of the CBD and ILO Convention No.107 obligations, it would lead to a lack of monitoring of the requirements under these conventions (and therefore a lack of transparency and data), compared to a situation where the compliance provisions of the Nagoya Protocol would have been implemented. In particular this might create a lack of legal certainty through the lack of checkpoints (where they are supporting compliance)which, if established, would clarify the relevant obligations.  Option 1: Monitoring PIC in the ABS Clearing-House as a checkpoint For this option, PIC might need to comply with more specific information collection and transfer obligations for checkpoints (irrespective of e.g. the obligation to make available permits to the ABS-CH (Article 14.2(c) of the NP), or the obligations linked to the obtaining of an internationally recognized certificate of compliance (Article 17.2-4 of the NP) which may to a certain extent overlap. Using the ABS CH as checkpoint will depend on further policy decisions taken regarding the CNA and the ABS CH.  Option 2: Using the patent office as a checkpoint Legislation is already in place for the disclosure of origin in patent applications (whenever the information is available): a logical step in this first phase could thus be that the patent office would function as a checkpoint. This might be made possible by an upgrade of the disclosure requirement in the patent applications, including information related to both the country of origin (as under the current legislation) and information on PIC from the country of origin. However, as Article 17 of the NP talks about "relevant information related to PIC, to the source of GR, to the establishment of MAT, and/or to the utilization of GR, as appropriate", an upgrade might not even be necessary in order for the patent office to qualify as a checkpoint. Further clarification on the necessity to comply with the obligation to provide for "appropriate, effective and proportionate measures to address situations of noncompliance" is under negotiations in other multilateral fora (WTO). Expected impacts  Option 1: Using the ABS Clearing-House as a checkpoint Possible Advantages: This option could lead to very few additional obligations in the case that general ABS option 2 (PIC as a general legal principle) would be adopted, except for linking the PIC approval to the information obligations to the Clearing-House, and would therefore be sufficient to contribute to respect the political commitment to timely ratify the Nagoya Protocol. Moreover, if appropriately linked to the Clearing-House, the PIC could constitute an internationally recognized certificate of compliance under the Nagoya Protocol and thereby contribute to the objective of increasing overall legal certainty and transparency. Possible Disadvantages: Some extra administrative burden, as the PIC approval would need to be linked to the information obligations under the Clearing-House (which however will probably not be a heavy obligation).  Option 2: Using the patent office as a checkpoint Possible Advantages: This option could lead to very few additional information exchange obligations and hence administrative burden for the patent authorities and users, as the information on the country of origin of the GR has to be provided by the users in the patent application, is available. The microbial ex-situ collections that are recognized as international deposit authorities (IDA) also keep already records of such information in the current situation. Possible Disadvantages: The Belgian patent office currently covers only a very small proportion of the transactions concerned by the Nagoya Protocol. A legal change could be required to upgrade the patent disclosure in order to be able to use it as a checkpoint within the framework of the Nagoya Protocol. In particular, the information on PIC should be included, wherever applicable, and a link with the information obligations under the Clearing-House should be made. Sharing information through the Clearing-House As the discussions on the exact modalities of the ABS CH are still ongoing internationally, it remains unclear if a separate Belgian ABS Clearing-House (ABS CH) component or only a Belgian entry-point will be required. The "0" option would therefore consist in not taking any steps regarding such a component or entry-point nor provide ABS specific information to the central ABS CH. This would lead to a non-ratification of the Nagoya Protocol, but still require to take the measures specified in the general "0" option. In particular, this "0" option would still need to comply with the obligations concerning the Belgian Clearing-House Mechanism to the Convention on Biological Diversity (CBD CHM), which also concerns information exchange on ABS as explained below. Description of the options beyond the "0" option A distinction needs to be made between two separate functions of a Clearing-House component for ABS: 1. Information exchange on ABS, including on the Nagoya Protocol, within the framework of the CBD  This is ongoing and can be further strengthened by integrating more relevant material into the Belgian CBD CHM managed by the Royal Belgian Institute of Natural Sciences (RBINS). This obligation flows from the CBD and is therefore independent of the future ratification of the NP. Support exchange of information on specific ABS measures within the framework of the Nagoya Protocol  Measures are needed to organize the technical information to be provided according to the Nagoya Protocol (for example on PIC, checkpoints and the ABS CH) as well as other information to be decided upon at international level by the NP COP/MOP. The modalities of a separate Belgian ABS Clearing-House (ABS CH) component therefore still depend on the ongoing multilateral negotiations. In this context, it remains unclear whether a Belgian CHM-component or only a Belgian information entry-point will be required. If such a component/entry-point is required, it is clear that the generated information will be useful for Belgian research and development, as well as for the objectives of conservation and sustainable use of biodiversity. Depending on the decision regarding the exact ABS CH modalities, three options could be explored. Three institutions could be potential candidates to support a Belgian component/entry-point of the ABS Clearing-House, if required. The strengths of these different options can be summarized as follows:  Expected impacts These will depend highly on the decisions taken on the exact role and technical specifications of the Clearing-House. However, in general the following points could be expected under these three options:  Option 1: Royal Belgian Institute of Natural Sciences (RBINS) as ABS Clearing-House Possible Advantages: Interesting synergies could be created under this option. The RBINS already hosts the National Focal Point (NFP) to the CBD and ensures the Belgian component of the CBD Clearing-House Mechanism. The RBINS also runs several biodiversity-related research units which could directly benefit from the generated information. Additionally, the RBINS ensures an important awareness building mission towards the broader public through its Museum of Natural Sciences. It operates the Belgian Clearing-House Mechanism for the CBD, with a strong focus on awareness raising, education and communication. It has development projects running (in collaboration with DGD) on establishing CHM in partner countries. Through these capacity building activities with the partner countries, it could play an important role in supporting developing countries with their obligations under the Nagoya Protocol with regard to the ABS CH. Furthermore, the administrative burden for the RBINS could be relatively low if additional information obligations related to the ABS CH could build upon the experience of the RBINS with the general CBD CHM. Possible Disadvantages: Nevertheless, the CHM only has a general communication, information sharing approach and does not handle specific scientific or technical data, contrary to the WIV with the Biosafety Clearing-House (BCH) (see option 3). Its applicability will therefore heavily depend on the level of technical requirements for the ABS CH.  Option 2: Belgian Federal Science Policy Office (Belspo) as ABS Clearing-House Possible Advantages: This option would be ideal for biodiversity research that contributes to sustainable development, as Belspo already hosts the Biodiversity Platform, which has as main task to foster such research. It has several collection databases that could support the working of PIC/checkpoints/ABS-CH. Belspo also hosts several other consultative bodies linking scientific and policy analysis and is involved at international level with digitalization of collection databases. Possible Disadvantages: Compared to the other options, the administrative burden might be heavier, as Belspo does not currently have any information obligations towards the secretariat of the CBD.  Option 3: Scientific Institute for Public Health (WIV-ISP) as ABS Clearing-House Possible Advantages: As it hosts the Belgian component of the CBD's Biosafety Clearing-House (BCH), the WIV-ISP is used to exchanging scientific, technical data with the CBD Secretariat. It also runs several health-related research units which could directly benefit from the generated information. Furthermore, the administrative burden for the WIV-ISP could be relatively low if additional information obligations related to the ABS CH could build upon the experience of the WIV-ISP with the BCH. Possible Disadvantages: The current BCH is very disconnected from the CBD CHM which would be a disadvantage for the ABS CHM where the link between the three objectives is a prime requisite for any implementation option. It also might have little added value regarding awareness raising, capacity building etc. Its relevance will therefore strongly depend on how much the BCH is taken into account at international level as the example to develop the ABS CH. Target Groups and Stakeholders for Which Potential Impact is Assessed For the purpose of the impact assessment of the recommended options in chapter 10, a list with categories of target groups and stakeholders, that could be affected by the proposed measures, is established. Users and providers of genetic resources Land owners  Protected areas: both public and private areas managed for conservation purposes (as providers (not necessarily in the meaning of the NP) of potentially valuable GR).  Other land owners: any public/private land owner might become a provider of GR (not necessarily within the meaning of the NP) with potential interest for R&D. Agriculture sector The agricultural sector includes a variety of public and private organizations, working in the fields of crop and animal selection/improvement, horticulture, fisheries, forestry and biological control. It is an important sector, given the share of Belgium in the world's agricultural products export. Several types of genetic resources are used by the Belgian agricultural sector, including animal genetic resources for food and agriculture (AnGR), fisheries and aquatic genetic resources for food and agriculture (AqGR), forest genetic resources for food and agriculture (FGR), plant genetic resources for food and agriculture (PGR), microbial genetic resources for food and agriculture (MiGR) and genetic resources relevant for biological control and crop protection. Healthcare sector In the context of this study, the healthcare sector includes the pharmaceutical industries, the care and cosmetics industries, so-called 'soft' natural medicines and in vitro diagnostic companies/laboratories. In the healthcare sector, the industries from the private sector in general play a predominant role. This sector is made up of both major multinationals and small family-style firms. Belgium hosts around twenty multinational companies in this sector. The SME sector is much more developed, with almost one hundred companies in Belgium179 . The country is the world's third largest importing country of biopharmaceutical products and the world's one-but-largest exporter180 . The pharmaceutical sector is thus a major player in the Belgian economy. The sector claims to provide the country with more than 30,000 jobs and to account for up to 40% of private R&D funding181 . IMPLEMENTATION OF THE OPTIONS WITHIN THE EXISTING LEGAL SITUATION IN BELGIUM This chapter analyzes the implementation modalities of the policy options described in chapter 8, taking into account the existing legal and institutional situation in Belgium described in chapters 2 to 5. The structure of the chapter is based on the six core measures used in chapter 8. Operationalizing PIC Two different components of these options need to be compared in the assessment of the options for operationalizing PIC:  First, for the GR which are not PS/PA, comparing the bottleneck option to the fishing net option (access through ex-situ collections, compared to access from everywhere).  Second, comparing the "baseline" fishing net model, which envisions the refinement of existing PA/PS relevant legislation to the "modified" fishing net model, where, in addition, other existing GR relevant legislation would be refined. The impacts identified below are an aggregate of the impacts likely to occur along these two components that are present in options 1, 2 and 3. Under the specific "0" option for access, only the situation where BS as a horizontal principle has been adopted is considered, as the situation, where BS is not addressed as a horizontal principle will be assessed under measures for BS as the specific "0" option for MAT (chapter 9.3). As such the specific "0" option for PIC considered here is equivalent to the general ABS option 1 (BS, but no PIC). IMP 1.0 -Implementation of the specific "0" option for operationalizing PIC Under the "0" option, benefit-sharing would still be established as a general legal principle in Belgium, which is not currently the case (cf. chapter 5). In addition, the European Commission's Summary of the selected options for the operationalization of PIC 8. Specific "0" option (access component): the specific "0" option on access would consider no PIC requirement, with benefit-sharing as a horizontal principle 9. Option 1 -The bottleneck model: refining existing PS/PA relevant legislation & measures + only access to GR through ex-situ collections as default rule 10. Option 2 -The baseline fishing net model: refining existing PA/PS relevant legislation & measures + access to GR from everywhere but with registration as default rule 11. Option 3 -Modified fishing net model: potentially enlarged refinement of existing PA/PS relevant legislation & measures + refinement of other specific GR relevant legislation/measures+ access to GR from everywhere but with registration as default rule For a detailed description of the options please refer to chapter 8.2. proposal for a Regulation on ABS182 , which is currently under discussion, encourages benefit-sharing but in its current form, does not establish benefit-sharing as a general legal principle. Seen the division of competences in Belgium, this general legal principle should be firmly anchored in the environmental competences of the Regions and the Federal Government. Indeed, as argued in section 3.2 of chapter 3, any legal measure that would consider introducing Prior Informed Consent could benefit from building upon existing legislation on physical access to and use of genetic material. Under the current regulations, the rules regulating physical access depend upon the type of ownership (private, public or res nullius), the existence of restrictions to the ownership, such as specific protection (protected species, protected areas, forests or marine environments) and the location (all four authorities apply their own rules) of the genetic material. As these regulations currently are part of the environmental competences of the Regions and the Federal Government such anchorage seems the most logical way forward. The implementation and the subsequent operationalization of this general principle would be phased, based on subsidiarity and flexible. Moreover, as for the implementation of other multilateral environmental agreements such as the Kyoto Protocol183 and the Cartagena Protocol184 , considering the need for a minimum level of harmonization of the implementation procedure in Belgium, a cooperation agreement between the Regions and the Federal Government may be necessary. On this basis, the implementation of option "0" could be based on three components: (1) A political agreement from the competent governments to establish benefit-sharing as a general legal principle, to be implemented for example through a cooperation agreement and/or analogous provisions in relevant legislations, such as the basic environmental code of the three Regions and at the federal level. (2) The subsequent or parallel implementation of this general principle through a cooperation agreement and/or analogous provisions in relevant legislations, such as the basic environmental code of the three Regions and at the federal level 185 . (3) The subsequent operationalization of the general principle by the respective governments at the regional (through executive orders) and federal level (through royal orders), establishing rules and procedures for further implementation of the benefit-sharing provision as envisioned in the other options considered below. would be implemented for example through a cooperation agreement and/or analogous provisions in relevant legislations such as the basic environmental code of the three Regions and at the Federal level (which would not be necessarily part of the first implementation step, cf. considerations in chapter 11). This access rule would specify that access to Belgian GR, that are not covered by PA/PS relevant legislation, would need to be sought and processed through qualified Belgian collections (which are equipped for deposit of data and/or samples). (2) Subsequent or parallel implementation of this general principle for example through a cooperation agreement and/or analogous provisions in relevant legislations such as the basic environmental code of the three Regions and at the federal level, which deal with the establishment of the Competent National Authorities and the rules and procedures for processing access requests by these Authorities (cf. IMP 3.1. below). IMP 1.2 -Implementation of option 2 for operationalizing PIC Similarly to option 1, the implementation of option 2 for operationalizing PIC can be broken down in four subsequent steps: The first three implementation steps are identical to the first three implementation steps of option 1 (IMP 1.1.1; IMP 1.1.2; IMP 1.1.3) The fourth implementation step of option 2 (IMP 1.2.4) is similar to the fourth step of option 1 (IMP 1.1.4), with the exception that the default access rule would specify that PIC would require minimally a registration/notification to the Competent National Authority. As also discussed below, a combination of IMP 1.2.4 and IMP 1.1.4, as a general principle in a cooperation agreement, can also be envisioned. However, for the purposes of the assessment under this section, at this stage these options are considered separately. IMP 1.3 -Implementation of option 3 for operationalizing PIC The implementation components of IMP 1.3 are the same as under IMP 1.2, except that it would also include the refinement of other existing legislation relevant to access to GR187 in an analogous way as the refinement of the access provisions under the PA/PS relevant legislation. IMP 1.3.1 -Further refinement of GR legislation (1) Amendment of other existing legislation relevant for access to GR, establishing that any access in that context not only concerns physical access but also access within the meaning of the Nagoya Protocol and that such access also automatically amounts to PIC under the implementation of the principle established under IMP 1.1.2. (through a Decree/Ordinance of the Regions). refinement of the legislation. The matter of conservation varieties do not fall under Annex 1 of the ITPGRFA Treaty and are currently regulated by the following legislations : Ministerieel besluit van 2 juni 2009 tot vaststelling van bepaalde afwijkingen voor de toelating van landrassen en rassen in de landbouw die zich op natuurlijke wijze hebben aangepast aan de lokale en regionale omstandigheden en die door genetische erosie worden bedreigd (Vlaams Gewest); Arrêté du Gouvernement wallon introduisant certaines dérogations pour l'admission des variétés de légumes traditionnellement cultivées dans des régions spécifiques ou sans valeur commerciale (Région Wallonne) ; Arrêté ministériel du 10 décembre 2010 introduisant certaines dérogations pour l'admission des races primitives et variétés de légumes traditionnellement cultivées dans des localités et régions spécifiques et menacées d'érosion génétique (Région Bruxelloise). Specification of MAT Given that a phased approach would allow fine-tuning the measures as more feedback is gathered, the initial MAT requirements analyzed hereunder could be further developed over time after a rather limited first implementation phase. IMP 2.0 -Implementation of the specific "0" option for the specification of MAT The specific "0" option under MAT would lead to a non-ratification of the Nagoya Protocol. However, as discussed in the preliminary assessment above, it is unclear how the specific "0" option would still allow the Belgian State to comply with the BS obligations of the CBD and the obligations under ILO 107, and what implementation steps would result from this alternative scenario. IMP 2.1 -Implementation of option 1 for the specification of MAT The implementation of option 1 would require establishing the general principle of benefit-sharing (cf. IMP 1.0. above). However, as option 1 considers no specific regulation in addition to the general BS principle, no additional implementation steps are needed. IMP 2.2 -Implementation of option 2 for the specification of MAT As under IMP 2.1 the implementation of option 2 is part of the subsequent operationalization of the general principle of benefit-sharing under IMP 1.0., as envisioned in step 3 of IMP 1.0. However, in this case, specific requirements on MAT are considered. Therefore, to assess this option, we will consider the following implementation component:  The subsequent operationalization of the general principle formulated under IMP 1.0. by the respective governments at the regional (through executive orders) and federal level (through royal orders), establishing specific requirements on MAT, including the use of standard agreements, depending on the finality of use. Summary of the selected options for the specification of MAT 0. Specific "0" option: No benefit-sharing 1. Option 1: No specific benefit-sharing requirements imposed for the MAT 2. Option 2: Standard agreements with specific benefit-sharing requirements, depending on finality of access 3. Option 3: Specific benefit-sharing requirements, negotiated on a case by case basis, depending on finality of access For a detailed description of the options please refer to chapter 8.2. IMP 2.3 -Implementation of option 3 for the specification of MAT Idem as IMP 2.2., but the specification of the BS requirements in the general rules does not impose the use of standard agreements. In this option, the implementation of the specific BS requirements would always be negotiated on a case by case basis. Therefore, to assess this option, we will consider the following implementation component:  The subsequent operationalization of the general principle formulated under IMP 1.0. by the respective governments at the regional (through executive orders) and federal level (through royal orders), establishing specific benefit-sharing requirements, negotiated on a case by case basis. IMP 3.2 -Implementation of option 2 on the Competent National Authorities The implementation of option 2 would be very similar to the implementation of option 1 (IMP 3.1). The choice of the Competent National Authority would in the first place be based on the relevant competent authorities and the division of competences (IMP 3.2.1). The main difference is that option 2 would provide for a centralized input system to access requests, which are then referred to one of the 4 CNAs and their respective rules and procedures. This would require the establishment of a single entry-point (such as a webportal) and the specification of rules and procedures for the single entry-point (IMP 3.2.2). Therefore this assessment will consider the following implementation of option 2  A political agreement from the competent governments to establish a single entry-point for access requests (including the specification of its rules and procedures) through a cooperation agreement between the Regions and the federal level  Subsequent or parallel implementation through a cooperation agreement which would include the rules and procedures for requesting access through a single entry-point as this would avoid differences between the Regions and the federal level Setting up compliance measures As highlighted in chapter 8.2, the Belgian national law enacting the code of private international law states in its Article 15 that, if a foreign law needs to be applied to a case that is examined by a Belgian judge, the content of such applicable law should be identified by the judge, according to interpretations received in the "country of origin" (sic). Collaboration can be required if the content cannot be established clearly by the Belgian judge. If it is "impossible to determine the content of foreign law in due time, Belgian law should be applied" (art.15 §2al2)". Therefore, the implementation of option 1 would only entail a minimal amendment to this code by including explicit reference to the use of GR within the context of the Nagoya Protocol as being part of the scope of this code. At the same time, private international law gives to the legislator the possibility to enact "mandatory laws", that rule out the application of the foreign law even though it would have been applicable according to the usual rules of private international law (for instance in cases when the foreign applicable law is inexistent). The "mandatory law" is applied -with a large interpretation -if the Summary of the selected options on compliance 0. Specific "0" option: not introducing any legal provision on compliance 1. Option 1: Ensuring compliance with provider country legislation regarding PIC and MAT, with Belgian law as a fall-back 2. Option 2: Self-standing obligation in the Belgian legislation to have PIC and MAT if so required by the provider country. For a detailed description of the options please refer to chapter 8.2. State reckons that a national application is necessary. The criteria defining a "mandatory law" are not clearly cut by the jurisprudence and the doctrine, and thus provide the legislator with a certain political margin: that is the ground upon which this report envisages the option 2. Finally, and independently of the options chosen, the effectiveness of the ABS compliance regime will largely depend on the effectiveness of both the national focal points and the Clearing-Houses 188 . This is particularly true when referring back to provider country legislation, as these two institutions are responsible for the channelling of information. Some of the assumptions made in the following part could differ in light of the disparity between provider countries. The amount of legal certainty under option 1 for instance, could greatly differ when dealing with a provider country effectively relaying information to the Clearing-House or when dealing with a provider country which is not. IMP 4.0 -Implementation of the specific "0" option on compliance Idem as under IMP 2.0 IMP 4.1 -Implementation of option 1 on compliance The implementation of the compliance provisions of the Nagoya Protocol is explicitly addressed in the EC's proposal for a Regulation on ABS 189 . Therefore, seen the important effort of harmonization at the EU level concerning compliance, and the ongoing discussions on the proposal, a phased approach to the implementation of the compliance obligations is indicated. Moreover, at the present state, it is unclear to what extent the proposed Regulation will be sufficient to implement the core obligations on compliance and/or what additional compliance measure will be needed in case the Regulation is not sufficient. On the basis of these considerations, the assessment of option 1 on compliance will consider the following implementation components: (1) A political agreement from the competent governments to express the commitment that legislative measures will be taken to provide that GR utilized within Belgian jurisdiction have been accessed by PIC and MAT as required by provider country legislation and to address situations of non-compliance. This political agreement would be executed in a later stage of the implementation, as soon as sufficient clarity is provided at the EU level. (2) Implementation of this general principle for MAT through the referring back to the provider country legislation, with Belgian law as a fall-back. As these two elements are currently already part of the Belgian code of International Private Law, such implementation would minimally only entail to amend this code by including explicit reference to the use of GR within the context of the Nagoya Protocol as being part of the scope of this code. (3) Implementation of a criminal provision on complying with provider country legislation regarding PIC and MAT. Due to the ongoing EU negotiations it is premature at this stage to provide for a detailed analysis of criminal sanctions. This will be evaluated once relative certainty on type of behaviors concerned and level of sanctions are available. 188 Tveldt, Fauchald, (2011), op.cit., p.398 189 EC (2012b), op. cit. IMP 4.2 -Implementation of option 2 on compliance Idem as IMP 4.1, except that the implementation of the general principle of compliance would be based on a self-standing obligation, which requires Belgian users to have PIC and MAT from the provider country (as part of Belgian Law), as far as the legislation of the provider country requires PIC and MAT for access to its GR. IMP 5.1 -Implementation of option 1 on checkpoints The implementation of the monitoring obligations under option 1 is closely related to the establishment of the ABS Clearing-House considered below in section 2.6. Indeed, information regarding uses of GR in Belgium, as obtained from the CNAs of the provider countries, will be made available through the Clearing-House Mechanism of the Nagoya Protocol. If PIC is provided within this information, it will be considered as an international certificate of compliance and be acceptable as a checkpoint. The use of PIC as checkpoint therefore could be organized through ensuring that PIC for GRs accessed and or used in Belgium is available in the Belgian node of the Clearing-House Mechanism. No other implementation components therefore are currently required for the timely ratification of the protocol, in addition to the implementation components considered under the establishment of the Clearing-House (see chapter 9.6 below). IMP 5.2 -Implementation of option 2 on checkpoints The Belgian legislation, while implementing recital 27 of the Directive 98/44/EC of 6 th July 1998 on the legal protection of biotechnological inventions, which has due regard to the obligations stemming from the CBD with specific regards to its Articles 8(j), 15 and 16 has included a (qualified) origin indication requirement (if the origin of the material is known) in its Article 15 §1(6). In order for the patent application to be admissible, the filing must contain a statement regarding the geographical origin of the biological material that has been used as a basis for the invention, if known. This provision would need to be amended to allow its use as checkpoint under the Nagoya Protocol, specifying that patent application should contain relevant information related to prior informed consent, to the source of the genetic resource, to the establishment of mutually agreed terms, and/or to the utilization of genetic resources, as appropriate (NP Art 17. For a detailed description of the options please refer to chapter 8.2. The discussions on the exact modalities of the ABS Clearing-House (CH) are still on-going internationally and decisions will only be taken at the NP COP/MOP1 (earliest: October 2014). In the meantime, it remains unclear if a separate Belgian ABS CH component or only a Belgian entry-point will be required. Moreover, the impact of the CH will highly depend on the decisions taken on the exact role and technical specifications of the Clearing-House. Therefore, the impact assessment of this implementation provision is still tentative and will need to be refined in the future. 9.6.1 IMP 6.0 -Implementation of the specific "0" option for the CH Idem as IMP 2.0 IMP 6.1 -Implementation of option 1 for the CH Seen the still ongoing discussions on the international level and the uncertainty regarding the obligations of Belgium under the Nagoya Protocol, the implementation of this option will benefit from a phased approach. As it is likely that the information tasks under the ABS CH will need to be implemented in Belgium in any case, in a first phase, a CH could be established that specifically deals with the information tasks. In a second phase, collaboration between this CH and other institutions/databases could be established if required to implement the more technical tasks of an ABS CH. Given the existing CBD CHM at the RBINS and the strong Belgian preference to ensure coherence between the different Clearing-Houses under the CBD, it seems logical to start this exercise at the RBINS by extending the current ABS part of the CBD CHM. Therefore, two implementation components will be considered in the assessment of this option: (1) Specify in the cooperation agreement that the RBINS will be appointed as the ABS CH, for dealing with the information exchange on ABS under the Nagoya Protocol and indicate that further development of the ABS CH in terms of more technical or specific tasks related to the implementation of the NP will be undertaken after the first COP/MOP of the NP). IMPACT ANALYSIS Methodology of the impact analysis The evaluation of the possible consequences of the implementation of the NP is conducted through a detailed comparative impact analysis (IA) related to the options described in chapters 8 and 9. The IA has three main objectives: 1. Identifying the possible effects of the options 2. Identifying the affected stakeholders 3. Comparing the different options In this framework, the IA is conducted through a multi-criteria analysis (MCA). MCA has been developed as an alternative to the conventional cost-benefit analysis (CBA). CBA assumes value commensurability between the different objectives (i.e. the possibility to measure them through a common monetary metric, which supposes that it makes sense to construct monetized proxies of all criteria and that information is available to do so) and compensability (i.e. the assumption that a loss observed in one attribute or good can be compensated in quantitative terms by a gain in another, which supposes, for example, that one can quantitatively compare through a common metric such as the loss of biodiversity conservation benefits, profits for industry relating to facilitated access to resources or administrative costs). However, there is a wide literature showing that, from an environmental, social and economic perspective, these assumptions are clearly not substantiated for sustainability impact assessments190 . Nevertheless quantitative monetary values are not to be dismissed completely from the evaluation in a MCA: wherever possible, quantification of certain advantages and disadvantages are a crucial input component the MCA, as shown below, even if there is no commensurability or equivalent compensation across all the criteria. But unlike CBA, MCA allows to compare impacts represented both qualitatively and quantitatively. The goal of the IA is thus to identify the existence of qualitative elements in addition to the quantitative elements that can build the basis for a comparison amongst the options for the implementation of the Nagoya Protocol, rather than the calculus of a specific quantitative threshold of aggregated monetary benefits in a common metric, able to justify the expected aggregated costs. The evaluation of the impact is conducted against a set of evaluation criteria described below, which leads to a performance score per criteria for each of the options. Using the performance scores, a dominance analysis and an outranking analysis are performed to compare and rank the alternatives, based on pre-defined weighting (cf. description below). A sensitivity analysis is then conducted to describe the "behavior" of the outcome when changing the weighting (cf. sensitivity analysis paragraph 10.1.3, Step 4). Figure 1 gives an overview of the different steps of the MCA. The formulation of the set of evaluation criteria has been obtained through two overarching questions: 1. To what objectives is the implementation of the Nagoya Protocol seeking to contribute? 2. How would a good option be distinguished from a bad option, given the decision-making context? Although no clear rules exist on the definition of criteria and their number, it is generally considered that it should be kept as low as is operationally desirable (i.e. the model should be as simple as possible). Different economic, social, environmental and procedural criteria were considered, checking them against the preferences of stakeholders and against quality requirements.  Stakeholder-preferences: Analysis of the preferences of stakeholders, expressed both during the first stakeholder workshop as during the interviews, helped to refine a first set of criteria derived from the above questions. Examples of these preferences include flexibility, continuity, knowledge-improvement, legal certainty, non-redundancy and cost-effectiveness in the establishment of the regulatory framework of the Nagoya Protocol.  Quality requirements: the criteria were then checked against a range of qualities such as value relevance (relation with the overall objective), cognitive relevance (shared understanding of concepts), measurability (some form of measurement or judgment, objective or subjective191 ) and non-redundancy (several indicators measuring the same factor). This selection process allowed identifying four criteria to assess the impacts of the proposed options, which are described below. The assessment of the environmental and social impacts is based on two individual sub-criteria (S1 and M1), while the assessment of the economic impact is composed of three sub-criteria (E1, E2, E3). Four procedural (G1 to G4) sub-criteria have also been added to reflect the overall policy process. The different sub-criteria for the economic and the procedural impact have been created for analytical ease, as the assessment would have been too complex if grouped into one single criterion. Having more sub-criteria does not confer more importance to a particular impact. For this report, the social and economic impact are considered to have the same weight (i.e. they are considered of equal importance), while the environmental impact is slightly more important, given the objectives of the Protocol and the CBD to contribute to conservation and sustainable use of biodiversity. The procedural impact has the lowest importance and serves mainly to help refine the preference for an option in case the difference is not clear enough when using the substantive criteria. If the total weight of the impacts represents 100%, the weighting is distributed based on the following basic allocation key: environmental impact (37,5%), social impact (25%), economic impact (25%) and procedural impact (12,5%) (see also sensitivity analysis below). Economic impact E1 Legal certainty and effectiveness for users and providers of GR, at low cost Four indicators are taken into account to evaluate this criterion. Legal certainty refers to the consistency and predictability of the rules and the process in place. Effectiveness of the legal framework refers to a set of indicators including:  Enforceability: the level with which an option allows the ABS regulation to be enforced.  Redundancy relates to existing legislation regulating related obligations.  Proximity with other international agreements. When combining these indicators, an option will be preferred when it increases, at an equivalent cost, legal certainty, allows better enforceability, reduces redundancy and does not conflict with obligations under other existing international treatments. In addition, an option with similar level of legal certainty and effectiveness, compared to another option, will be considered preferable if it leads to less legal costs (such as the cost for drafting new legislation and the cost for asking legal advice.) E2 Maximizing economic innovation and product development (in particular through its contribution to R&D) at reasonable financial and administrative costs Extensive research on private sector return from public and private investment in research infrastructures involving genetic resources shows a clear correlation between improved conditions for R&D and an increase in likelihood of the development of innovative products and services. Options that maximize research and development opportunities for users and providers of GR are therefore considered preferable. These benefits will be assessed while taking into account the changes in research costs that stakeholders incur for the necessary steps they need to take in order to allow for research that complies with the NP to take place. Such costs include, among others, costs involved in the negotiation of the ABS agreement, the acquisition of genetic resources and transaction costs related to the transferring of the GR. E3 Minimizing implementation costs Implementation costs are costs related to obligations flowing from the implementation of the Nagoya Protocol. They include, for example, the administrative costs related to keeping track of the ABS agreements, the financial costs for the creation of new institutions (if needed), the costs for asking for legal advice in the course of the implementation or the cost of monitoring utilization. An option having a lower cost is considered preferable over another with a higher cost for an equivalent level of produced benefits. In addition, an option leading to a one-time expense is preferred over an option which generates recurring expenses. Social impact S Achievement of social objectives Innovation resulting from R&D with GR is expected to contribute to the achievement of important social objectives, be it health, nutrition, food security, or else. Options that maximize opportunities for the users in socially relevant fields are therefore considered preferable over options that create less such opportunities. Options contributing to the transfer of knowledge and technologies to developing countries and to job creation/preservation in the sectors utilizing genetic resources, both in developing and developed countries, are also considered preferable. A particular social aspect is the contribution to the effective protection of the rights of indigenous and local communities over their traditional knowledge associated with GR. Options that effectively protect or advance indigenous rights are preferable over options that do not achieve this aim. Environmental impact M Promotion of conservation and sustainable use of biodiversity Options that enhance conservation and sustainable use of biodiversity, inter alia through improving its knowledge base (e.g. by enhancing taxonomic research), enhance capacity building and technology transfer, through channeling benefit-sharing to conservation and sustainable use, improving protected areas and protected species management and raising awareness are preferable. Procedural impact G1 Flexibility to accommodating sectorial differences The implementation of the Nagoya Protocol will impact different types of actors, using GR under different conditions, in many different ways and at varying moments in the development process. Therefore it seems important that implementing measures offer some flexibility to accommodate for differences between diverse sectors utilizing genetic resources. An option will be considered preferable if it better balances the need for clear and certain rules with flexibility to accommodate for sectorial differences. An inflexible "onesize-fits-all" regime might have negative effects and might contradict the objectives of the CBD192 . G2 Temporal flexibility to allow for future policy and adjustments The boundaries and needs of the utilization of GR evolve continuously, with new resources being discovered every day. The political and socio-economic context of the NP also changes rapidly, as ABS is a relatively new field. This evolving reality creates the need for a flexible and adaptable implementation over time, in particular in light of implementation measures taken by other Parties of the Protocol and in light of future sectorial initiatives. An option is considered as advantageous if it leaves space for adaptation of the implementation and future policy and adjustments over time. In addition, an option providing such a temporal flexibility at lower costs will be considered preferable over another option. G3 Improving knowledge on the exchange of GR and existing ABS agreements for future policy development and evaluation Currently, little data exists on the exchange of GR and existing ABS agreements. Increasing this knowledge is primordial to design efficient rules addressing the needs of the different stakeholders involved. Furthermore, ABS has a clear link with the conservation and sustainable use of biodiversity. Improving the understanding and knowledge of their interlinkage is an important part of the efforts to halt the erosion of biodiversity. An option is preferred when it allows increasing the knowledge in these two fields. G4 Correspondence with existing practices Previous research stresses the importance of relying upon previously established relationships and existing practices of genetic resource use for the success of ABS agreements. For example, Täuber et al. show that strengthening existing research capacities and existing relationships fosters understanding and mutual trust, attracts users and lowers transaction costs 193 . Options building upon existing practices will therefore be generally considered preferable. An option that would require a significant change in practice or which would run against the basic economic model of a practice will be considered less preferable. Data collection for the indicators Due to the scarcity of data and knowledge on the flows of GR and on the current practices of the ABS in Belgium, three types of sources needed to be triangulated. 1. Primary sources such as internal documents, activity reports and policy documents/reports, inasmuch as these were available and shared, have been collected and analyzed. The list of documents can be found in the bibliography of this report. 2. Existing literature on the economics of genetic resource use has been consulted and integrated whenever possible. For a complete list of reference see footnote references and bibliography. 3. In-depth interviews have been conducted to collect information and data specific to the Belgian situation. These are discussed below. It should be noted that relevant data for the measurement of indicators was not always existent, available or shared. Especially quantitative data was very scarce. An evaluation of the most relevant quantitative costs has nevertheless been attempted and applied towards the fine-tuning of choices amongst closely ranked options (cf. also tables in annex 2 and annex 3). For the data gathering, based on these three sources, a list of general indicators was used, as indicated in Table 6. Interviews 29 interviewees, pertaining to groups of potentially impacted stakeholders, were selected in a nonrandom way based on their proven relevant experience and knowledge of the subject. 17 out of 29 accepted the request for an interview, which were conducted between 23 rd July and 20 th August 2012. A complete list of interviewees can be found at the end of this report (annex 5). 12 others declined or did not reply to the requests for an interview. However this did not result in an overall unbalanced representation of certain sectors, as stakeholders from all sectors were interviewed. The decline by some contacted persons could point to a lack of knowledge, understanding and/or interest for the NP by certain persons within the Belgian stakeholder groups. Any form of future implementation will need to address this by setting up targeted capacity-building activities. A full list of the contacted persons has been sent to the accompanying committee of the study. Most interviews were conducted face to face. Interviewees were briefly introduced to the objectives and progression of the study, if needed. Two sets of structured questionnaires were used for the interviews: one for users of genetic resources and one for providers. Some specific additional questions were also prepared for specific profiles of interviewees which were neither users nor providers. These sets addressed both the quantitative and qualitative evaluation of the options through two distinct parts. Questions related to quantitative data aimed at collecting objective figures related to the access, the distribution and the sharing of benefits related to genetic resources in order to try to map the flows of GR in Belgium. Questions included, inter alia, the amount of access made/received and their related costs, the patenting and commercialization rates of acquired GR or the costs of managing collections. Questions related to qualitative data were used to further elucidate stakeholder preferences observed during the first stakeholder workshop and were mostly open-ended and behavior-based questions (e.g. "If you have the choice between options 1 and 2, which one would you chose and why?"). The questionnaires can be found in annexes 2 and 4. As can be seen in the correspondence table between the criteria and the indicators in annex 3, the majority of the indicators are related to the economic and the environmental criteria. As can be seen in the table, the indicators for the environmental criteria refer both to the quantitative aspects (gaps in biodiversity research, incentive for conservation by potential increased use of Belgian GR, etc.) and with more qualitative aspects of the environmental criterion (as these are more difficult to capture in a quantitative indicator). These qualitative aspects, such as increased awareness of biodiversity issues and education for example, were also discussed during the interviews, and the results of the discussion on these qualitative elements have also been included in our discussion below. The same comment applies to the social aspects, such as promotion of indigenous and local communities and social impact through capacity building, which were also discussed both in a quantitative and qualitative manner with the interviewees. Comparing the alternatives The general principle of the impact analysis is to assess the impact of several policy options as net changes compared to a no-policy-change baseline ("0" options) and to compare the impacts of the options amongst each other. The overall goal is to establish a ranking amongst the options. To this purpose, under each section the proposed options were contrasted with each other and with the specific "0" options. In this exercise, it is important to state from the outset that the evaluations do not give any absolute figures/values for each of the criteria, but give a set of values that allow seeing which option would, comparatively, score higher or lower on each of the criteria. As for the comparison with the general "0" option ("no policy change" over PIC, BS, CNA, compliance, etc.), this can be done through an indirect method, based on the aggregated effects of specific "0" options. If all the specific "0" options rank lower than the list of proposed options under the several measures, then the general "0" option (which is the sum of all the specific "0" options, which is no policy change at all) will a fortiori rank lower than the list of the proposed options under these several measures. Therefore this issue is addressed after having assessed the impacts of all the specific "0" options and seen what consequences can be drawn from an aggregation of all the specific decisions not to act on a certain measure. Step 1: Performance of the options Each option is thus analyzed in relation to the others and described in an accompanying text divided per individual criterion. The impact on stakeholders is described for each individual group of stakeholders (land owners, agriculture sector, healthcare sector, biotechnology and processing industry sector, governmental research institutions, collections, university research sector, and other; as described in chapter 8.3, the agriculture, healthcare and biotechnology sectors are evaluated jointly, except when there are major differences in impact that justify to treat them separately). The economic, social and environmental assessments are then represented in a separate impact grid, indicating whether the impact is positive or negative, the likelihood of the occurrence, the magnitude of the impact as well as a general score. The score ranges from [---] (most negative) to [+ + +] (most positive). Neutral and unimportant impacts are indicated with a "0". Table 6 offers an overview of the scoring system. Reading and interpretation of the impact grids is to be done with caution, as some assessments are based on assumptions that are justified in the text. Also, some options have a different subject-matter (see the IA of the establishment of the CNA for example), or represent an aggregate of different possible scenarios (see the IA of the operationalization of PIC for example). The procedural sub-criteria (G1 to G4), which were outside of the Terms of Reference of the study194 , are not represented in an impact grid. They are submitted to an assessment of their contribution to overall quality and effectiveness of the policy process in the following steps of the MCA, instead of a likelihood/magnitude analysis which is less appropriate for these criteria. Step 2: Visual dominance analysis Options and (sub-)criteria are then compared on the basis of a performance chart. The performance chart visually represents the differences between the options and allows for a dominance analysis to be made. The goal here is to identify if there exists an ideal point: the option that dominates all others. An option dominates another if it scores at least as well on all criteria and is strictly better at least on one. However, having an ideal point is rare: only three of our cases present such an ideal point. To allow for this dominance analysis to be based on all criteria, the procedural sub-criteria are included in this visual analysis 195 . Step 3: Ranking the alternatives If no ideal point can be identified, a ranking the alternatives can nonetheless be made based on their performance. As the scores are the result of comparisons between the options within the criteria, and not of comparisons amongst the criteria (cf. introduction), the results for one criterion cannot simply be added up to the results for another. Therefore, the "Preference Ranking Organization Method for Enrichment of Evaluations" (PROMETHEE) was applied, which allows building an outranking relation on the set of alternatives (called "options" in this report). An outranking relation allows building an ordering of the alternatives through a series of pairwise comparisons of these same alternatives 196 . The basic principle of this method is that an option outranks another if that option outweighs all the other options over a larger number of (sub-)criteria than any other option. PROMETHEE uses a preference function for each of the alternatives, which allows identifying the intensity of preference. The intensity of preference represents the importance of the difference between two alternatives when comparing them. The values of the preference function (i.e. the different levels of intensities) lie in an interval from zero to one, within which higher value of the preference function corresponds to a better alternative. In other words, when option 1 outranks option 2 for a certain criterion, the amount of the difference between option 1 and option 2 determines the intensity of the preference of option 1 for that criterion: the higher the difference, the higher the intensity of the preference 197 . A preference index can then be set up for one option over the other. The preference index is the weighted average of preferences on the individual criteria: Where P k (option1, option2) represents the intensity of the preference of option 1 over option 2 for criterion k, and W k represents the weight of criterion k. In this analysis, a Usual preference function is used (Figure 2) for all the alternatives, which is best suited for qualitative criteria with a small number of levels on the criteria scale 198 . With this function, it is considered that values of the intensity of the preference can only be 0 or 1. In other words, the importance of the difference (d in Figure 2) does not matter. Preference is given to the alternative which has a higher value of criterion. Figure 2 -Usual preference function The preference index of each comparison between two alternatives is then summed up to create two indices: the positive outranking flow and the negative outranking flow. The positive outranking flow represents the strength of an alternative when compared to all others (i.e. when it outranks all others). The negative outranking flow represents the weakness of an alternative when compared to all others (i.e. when it outranks all others). These flows are defined as follows:  Positive outranking flow for option 1:  Negative outranking flow for option 1: Positive and negative flows allow calculating the net flow of each alternative, by which a complete pre-order of the alternatives can be established: Option 1 then outranks option n if the net flow of option 1 is higher than the net flow of option n -. For each subset of proposed policy options, the performance of the options will be evaluated and presented in the impact grid along with the explanatory text. A visual dominance analysis is then performed followed by a first ranking of the alternatives. In this first approximation, as indicated earlier, the environmental impact is considered more important than both the social and economic impacts, which in turn are weighted more than the procedural impact (used for fine-tuning the choice amongst closely ranked options). If the total weight of the impacts represents 100%, the weighting is distributed based on the following basic allocation key:  environmental impact: 37,5%  social impact: 25%  economic impact: 25%  procedural impact: 12,5 % Step 4: Sensitivity analysis In addition to the analysis based on the predefined weight distribution of the criteria, a sensitivity analysis has been performed by changing the weighting amongst the criteria and analyzing the impact on the ranking of the options. The sensitivity analysis is used to test the robustness of the outcome of the ranking. It allows assessing how sensitive the outcome is to changes in the problem definition. To perform the sensitivity analysis, two additional weighting scenarios are compared with the basic allocation scenario, to see if there is a reasonable low threshold of change in these criteria that leads to a change in choice amongst the options: 1. The equalized weighting scenario equalizes the importance of the impacts: an equal weighting (25%) is applied to all the four groups of criteria (environmental, social, economic and procedural). 2. The economic weighting scenario puts a stronger focus on the economic impact, which becomes the most important one (37.5%), while social and environmental impacts are considered of equal importance and procedural impact remains unchanged. Wherever a change in ranking occurs, a ranking of the alternatives based on these new weights has been presented in addition to the environmental weighting scenario, in order to be able to compare the weighting choices amongst each other. this option might be problematic in the cases where the GR is not yet accurately known at the point of access, as it will be very hard to control the accuracy of the provided notification and its adequacy for later monitoring if the GR is of uncertain nature. On the other hand, ex-situ culture collections dispose of all the necessary technical and scientific expertise for the appropriate identification (e.g. genetic profiling) of the accessed GR, providing additional information on the GR to the information available with the PIC (compared to a PIC issued for example for an in-situ resource of uncertain nature). Also, if utilized in combination with a postaccess self-monitoring system (e.g. a due diligence system), the bottleneck option will guarantee that only well identified GR enters the value chain of "legally acquired GR", creating strong legal certainty and easing the auto-monitoring by users 199 . The modified fishing net model would lead to some increase in legal certainty, compared to the baseline fishing model, as it considers also to refine legislation pertaining to GR that are outside PA/PS 200 . However, such an additional refinement would imply an additional legislative cost compared to the two other options. As shown in chapter 9.1 (IMP 1.3), while the implementation of all three options implies the same legislative cost for the amendment of PA/PS relevant legislation, option 3 also includes the identification and refinement of other relevant legislation.. The latter additional cost that is specific to option 3 would only be worthwhile if the GR covered by that legislation would be of potential or actual value not found elsewhere. However, in spite of this uncertainty and the cost, the impact of increase in legal certainty for users of GR can still be rated of medium magnitude. and at this stage it seems difficult to go beyond an illustration of what the "refined fishing net" option could entail. The case of the conservation varieties, cited in chapter 9.1 however provides a plausible illustration of such a legislation that is different from the PA/PS legislation and where the current legislation on the "admission to use" could be considered also as a PIC under the Nagoya Protocol, in further refinement of the legislation (cf. references provided in the footnote in that section). By establishing both PIC and BS as legal principles and by refining existing legislation (see chapter 9.1, IMP 1.1, 1.2 and 1.3), it can be argued that options 1, 2 and 3 increase legal certainty, which is likely to consolidate or increase the use of Belgian GR and therefore is expected to lead to more economic innovation and product development, in particular through higher R&D benefits, compared to the situation of the specific "0" option. Conversely, under the specific "0" option, the absence of legal certainty generated by the obligations to share benefits but the absence of any proof of PIC (see chapter 9.1, IMP 1.0) is likely to lead to less use of Belgian GR and could therefore be an obstacle to innovation and development. Giving ex-situ collections a central role in the PIC process, might strongly foster an increase of deposits, as collections are used to deposit a physical copy of GR they work with201 . This could imply an important financial cost for the collection providing those resources that are accessed for utilization outside PA/PS and that are usually not deposited in an ex-situ collection. If it is assumed that both access situations (through fishing net and through bottleneck) lead to an equivalent increase in economic benefits, then the fishing net is to be preferred over the bottleneck under this criterion, as resources are not deposited under the fishing net model. Measuring this cost is difficult, as it strongly differs depending on the type of resource being deposited. The costs of storing GR ranges from a few Euros for herbaria, between 100 and 250€ for plant collections and microbes and up to 40 000€ for animal breeds202 . Moreover, users and providers could decide to deposit only the information, only the physical resource, or both, which would lead to different price tags. These costs can also be nuanced in light of the positive effects the storing of GR can have for other research users (not intended by the users accessing the GR), such as further taxonomic research in the case the deposited GR is of a yet unknown taxonomic nature. The additional cost for organizing the access to materials for research under the bottleneck and the fishing net model, compared to the specific "0" option is likely to be low in both cases. It would be limited to the working hours for administrative requirements such as the establishment of the agreement, including settling the specifications of use of the material, the scope of the agreement and the drafting. Under the bottleneck option, this effort will be shared between users and providers. Under the fishing net, these costs do not take place, as the sole obligation is that users notify the CNA of the access to a GR. Some time investment will nevertheless be required for this notification obligation, but if a centralized notification system is established, whether digitally or physically (cf. E3 below), the working-time is expected to be low. These additional costs are estimated to be ranging from 70 to 140€ per transaction, with the fishing net model having the least additional costs (between 1 and 54€) 203 . The baseline and the modified fishing net model do not show any difference according to this criterion (they both equally promote economic innovation and there is no expected difference in research costs).  Impact on stakeholders: o Coll.: financial impact if increase of deposits under option 1 (but magnitude of impact is unclear). Bear part of the costs for accessing material under option 1. No impact under option 2 and 3. o Gov. Res.: Impact in terms of working hours for complying with the administrative requirements under options 1, 2 and 3. Indirect positive impact from possible additional storage under option 1. o Ag., health and biotech: Impact in terms of working hours for complying with the administrative requirements under options 1, 2 and 3. Indirect positive impact from possible additional storage under option 1. o Univ.: Impact in terms of working hours for complying with the administrative requirements under options 1, 2 and 3. Indirect positive impact from possible additional storage under option 1. o Land: No impact o Other: No impact E3 -Minimizing implementation costs Implementation costs for the access procedure are mostly administrative costs for the later follow-up of the process, such as drafting the PIC notification/registration/approval and handling the ABS agreements, the genetic profiling and the storage of a track-record of the exchange in a centralized database (e.g. the ABS Clearing-House). These costs are shared between users and providers, but they are small (between 1 and 24€ per transaction) and, on the exception of the costs for drafting, occur equally in options 1, 2 and 3 (except for the genetic profiling, not applicable for option 2)204 . Implementation costs for the public administration will occur under all options, related to the structure of notification that will be set up for the PIC. Notification could be done through a digital access portal where these notifications will be made directly by users or a physical access point for input by an administrative agent. Such a structure could also build synergies with existing services in the collections. However, as Belgium counts around 150 different collections205 , the need for operability and transparency could necessitate the centralization of access requests in a few qualified collections 206 . The expected increase of the access requests could then possibly lead to some increase in administrative costs for these collections, even though this could be shared between the collections and the users requesting access (e.g. through a fee). Under option 3, the public administration for PA/PS could also incur some additional costs, as it will have to handle more access requests due to the inclusion of GR in a refined legislation relevant to PA/PS. Overall, in the various structures that could be set up, the additional administrative costs for implementation of the PIC can be considered to be equivalent between the three options, but potentially incurred by different stakeholders. It should also be noted that, as indicated in chapter 9.1, the impact generated by a growing number of access requests will depend upon the relationship with other institutions created for the implementation of the NP, such as the CNA and the ABS CH. The impact of the specific "0" option is unclear as this option still implies to organize BS, which is highly likely to also lead to implementation costs for the users and providers of the GR. However, probably the costs would be lower than under a systematic PIC requirement.  Social impact S -Achievement of social objectives It is likely that the overall contribution to economic innovation and product development of options 1, 2 and 3 (cf. criterion E2 above) will also have (at least indirect) positive effects on socially important sectors such as food security, health and nutrition, albeit with a difference between option 1 and option 2/3 as discussed above. This contribution to the R&D sector is also expected to contribute to job creation in the overall economy, and in public and private research institutions in particular. Requiring PIC for Belgian genetic resources might improve the knowledge base on ABS in Belgium and could therefore contribute to education activities and help to build capacity. However, making a causal link between the requirement of PIC in Belgium and the fulfillment of specific social objectives, especially concerning the impact on transfer of technology or on the ILCs and TK in developing countries, would require more in-depth and long-term data on the effects of the PIC requirement, which will only be available once the implementation is in place. Environmental impact M -Promotion of conservation and sustainable use of biodiversity In spite of the higher costs of the bottleneck option, environmental benefits can be expected to be higher than under the fishing net model. Indeed, by obliging users to come back to the ex-situ collections for each new acquisition, the quality and the accurate documentation of the exchanged resources can be guaranteed. Furthermore, as stressed earlier, giving the ex-situ collections a central role in the PIC process will increase deposits, which will eventually support further biodiversity research and improve the knowledge base of conservation and sustainable use of biodiversity. Even though all the options will lead to an increased awareness on biodiversity conservation, sustainable use and access and benefit-sharing, option 1 is expected to have a larger impact in terms of increased awareness, in particular through increased attention to the use of biodiversity and increased availability of resources for research. If the bottleneck model allows for a more efficient follow-up of the accessed resources (see E1), the benefit-sharing is likely to be more easily monitored and channeled to conservation and sustainable use activities. As argued here, these benefits are likely to be important (generating increase awareness, knowledge and documentation amongst others). However, the importance of expected benefits for biodiversity conservation can be considered quite similar under options 1, 2, and 3, with some advantage of option 1 over options 2 and 3. In any case these benefits are much broader then the specific category of agreed upon monetary benefits from possible returns on commercial profits from the utilization of GR, which would be low over all the options 207 . The modified fishing net model offers another form of environmental benefit over options 1 and 2, in that, alongside the amendments to PA/PS relevant legislation, it also refines other legislation and thus allows covering a broader range of genetic resources (see also chapter 9.1, IMP 1.3). Furthermore, it could remedy situations where the default rule of option 1 proves to be ineffective or even creates a loophole. This could happen in cases where species found exclusively within protected areas prove to be rare, and/or if most species within protected areas can also be found outside of these areas. As the overall contribution to economic innovation and product development is positive, the research benefits for knowledge on biodiversity can also be expected to be positive, but this applies equally under the three options. Procedural impact G1 -Flexibility to accommodating sectorial differences As indicated in chapter 9.1, the assessment only applies to the establishment of the principle of PIC for access to Belgian GR, which is a necessary condition for the implementation of all three options. As this will be done through for example a cooperation agreement or provisions in relevant legislation, flexibility to accommodate sectorial differences will fully be preserved. The choice of the default access procedure (whether through ex-situ collections or through a notification requirement) is not a crucial step for the implementation of the NP and could be taken at a later stage of the implementation. Therefore, all 3 options can be implemented once sectorial specificities have clearly been identified and adapt accordingly. The same goes for the refinement of the existing legislation relevant to PA/PS, which applies to all three options. The specific "0" option does not lead to sector specific differences. G2 -Temporal flexibility to allow for future policy and adjustments As the assessment only applies to the establishment of the general legal principle of PIC for access to Belgian GR (see chapter 9.1), all options allow for temporal flexibility and can be adjusted to integrate future developments, in particular by integrating elements from the other options. G3 -Improving knowledge for future policy development and evaluation The possible increase in deposits under the bottleneck option, if feasible at reasonable cost (cf. criteria E2), will strongly foster overall understanding and knowledge of the type of GR that is being exchanged and valued. Furthermore, ex-situ collections host an important part of the Belgian biodiversity heritage, which is currently underexploited or unknown. Allowing the collections to play a more important role will help to increase knowledge on currently unknown specimen. The bottleneck component is thus preferable under this criterion. The baseline and modified fishing net would both generate information from all type of use sectors and uses (according to the content of the registration), so there is no difference over this criterion. The specific "0" option scores very badly over this criterion. Indeed, in contrast to all the other options, it will not generate systematic data on notification/registration, on user requests for information on PIC when accessing GR or other knowledge generated in the operationalization of the PIC. G4 -Correspondence with existing practices For the reasons discussed under G1, the bottleneck model might require some changes in practices from those sectors that acquire their GR from outside PA/PS and who do not rely on the public culture collections (so acquisition from in situ outside PA/PS and acquisition from informal exchanges with in house collections, often without systematic track record and documentation). However, as explained above, these practices only concern some sectors and some uses. Indeed, in Belgium, as several interviewees have pointed out, most utilized GR come from ex-situ collections, while the use of in-situ GR presents a diminishing trend. Consequently, ex-situ facilities act as important agents in the production chain: studies show that the use of ex-situ collections for new material is larger than both in situ and induced mutation 208 . The requested changes would essentially re-enforce this existing practice and growing trend of relying upon ex-situ collections for the exchange of GR. Moreover, the ex-situ collections already have a practice of documenting GR and dealing with CBD requirements. So overall one can say that the discordance with existing practices is not very likely to occur in the bottleneck model and that for most access situations the correspondence is very high. Under the fishing net component, the GR can be accessed from everywhere, but the introduction of the notification/registration requirement for the GR outside PA/PS would require a change in practices, although with an expected minor impact on these practices as the intent is to have a light notification/registration requirement. Comparing the baseline fishing net to the modified fishing net model, one can conclude that the modified fishing net model has a slight advantage over this criterion, as it relies on additional pieces of existing legislation covering GR. In particular, and although counter-intuitive, there is no evidence that only protected areas contain interesting genetic material. Refining GR legislation beyond the focus on PA and PS only could therefore increase the correspondence with the existing practices of utilization of GR. Visual dominance analysis No ideal point can be identified in the performance chart (Figure 3). 208 Ranking the alternatives A preference can be observed for options 1 and 3. This can be explained by the fact that option 2 is dominated by both option 1 and 3 for legal certainty (E1), for the environmental impact (M) and for correspondence to current practice (G4). A reasonable change in the weighting amongst the options does not allow changing this result. It should be noted that the social impact is not accounted for in this analysis, as the performance of the options is unclear (see description of performances above). Additional data could alter the outcome of the ranking, given that the social criterion is of substantial importance in all three weighting scenarios. As for the difference between option 1 and 3, a further analysis by changing the weighting can refine the analysis, as this difference is not very high. Option 3 scores comparatively better over the economic innovation criterion (E2). Option 1 has advantages over option 3 due to gain in legal certainty (E1), overall environmental benefits (M), and knowledge gathering for future policy making The sharing of benefits for the exchange or the utilization of GR in Belgium is currently self-regulated by the sector, each provider institution proposing its own rules and standard agreements. In this context, option 1 does not impose important legal costs, as it would simply rely on the same model. However, this option does not allow the Belgian State to specify the circumstances of the benefitsharing procedure and to make sure benefits are shared in a fair and equitable way. In addition, both option 1 and the specific "0" option would not allow the users to benefit from the advantages on legal certainty and effectively provided by options 2 and 3. Indeed, from a perspective of legal effectiveness and legal certainty, working with model contractual clauses (option 2) or tailoring the BS agreements to the specificities of each new transaction (option 3) encompass various advantages for users, providers and public authorities. First the development of standard agreements could eliminate variations between ABS regimes, hence providing legal certainty, facilitating transaction initiation, and suppressing information gaps created by extraneity factors 209 . Second, it could give more incentives to respect the rules already in place, insofar as the actors of the private sector currently prefer to trade with informal private collections that do not follow BS standards 210 . Third, using contracts will facilitate the enforcement of contracts between providers and users: "a contract would be binding as long as it is not found to be void, and could, 209 Tauber et al. (2011), op.cit. The term of « extraneity » is used when a legal issue confronts two or more different national legal systems and requires thus to rule a conflicts of laws or jurisdictions. It is envisaged here that the situation where the identity of the physical provider of a genetic resource (e.g. a public collection) differs from the identity of the owner or the original provider of this resource. 210 Laird S., Wynberg R. ( 2012), Bioscience at a crossroads: Implementing the Nagoya Protocol on Access and Benefit-sharing in a Time of Scientific, Technological and Industry Change, CBD. Summary of the selected options for the specification of MAT 0. Specific "0" option: No benefit-sharing 1. Option 1: No specific benefit-sharing requirements imposed for the MAT 2. Option 2: Standard agreements with specific benefit-sharing requirements, depending on finality of access 3. Option 3: Specific benefit-sharing requirements, negotiated on a case by case basis, depending on finality of access For a detailed description of the options please refer to chapter 8.2 and chapter 9. depending on the dispute settlement clause included in the contract, be brought to arbitration" 211 . Fourth, standardizing the negotiation and/or the agreement allows overcoming unbalanced bargaining power resulting from asymmetries in information, knowledge, negotiation, skills and capacity 212 , which is a barrier to fair and equitable benefit-sharing. Fifth -and related to the fourth point -it allows the state to control if benefits arising from potentially high-value resources are being shared accordingly to their value and being used accordingly with the objectives of the Protocol and the Convention. Option 2 might smoothen the negotiation process between users and providers, as it offers guidelines while providing security to providers that changes of intent will be renegotiated. Nonetheless, the legal setup of option 2 and option 3 (i.e. the inclusion of specific BS requirements in the provisions of the environmental code, cf. chapter 9.2) has yet to overcome the difficulty of delineating practically how divergent finalities of access can be distinguished from each other. Failing to specify the nature of different types of utilization, especially commercial utilization, and the correlated adequate BS, is likely to deprive the Belgian State from possible benefit-sharing which might contribute to conservation and sustainable use of biodiversity. However, this report has identified examples of how to deal with this distinction (cf. chapter 6.1 and 6.2). Hence, these practical difficulties do not seem to outweigh the benefits offered by options 2 and 3 through legal certainty and effectiveness. In particular, in the case of option 3, even if the legal costs are likely to be substantially higher, the benefits for effectiveness discussed above could be higher as well. Impact on stakeholders: o Coll.: Benefit from higher legal certainty under options 2 and 3. o Gov. Res.: Benefit from higher legal certainty under options 2 and 3. o Ag., health and biotech: Benefit from higher legal certainty under options 2 and 3. o Univ.: Benefit from higher legal certainty under options 2 and 3. o Land: Benefit from higher legal certainty under options 2 and 3.. o Other: No impact E2 -Maximizing economic innovation and product development (in particular through its contribution to R&D) at reasonable financial and administrative costs Option "0" would also lead to a non-ratification of the Nagoya Protocol as well as to non-compliance with the BS obligations of Belgium in the framework of the CBD. Conversely, the adoption of BS as a horizontal principle envisioned in options 1, 2 and 3 is likely to maximize economic innovation in the future. In general, if BS is adopted, then the increase in legal certainly under options 2 and 3 will spur the utilization of Belgian GR while leading to the lowest level of transaction costs for the different categories of users of GR. Option 1 could also prove to serve economic innovation and product development, as it offers the advantage to agree upon benefits that generate the least costs. But this advantage mainly applies to users, with providers risk to invest a lot of resources to make sure the benefits shared with them reflect a fair and equitable share of the benefits. Under option 3 this flexibility for users (for instance by encompassing an ex-post renegotiation process once the "non-commercial" utilization of the resource finds a commercial application) can be conserved, while offering a basis of negotiation for providers through specific requirements. Option 3 thus has the advantage of both providing a certain level of certainty for providers and small users (through a predefined set of requirements adapted for this sector) and leaving a certain level of flexibility for bigger commercial users (through the case-by-case negotiation). Furthermore, options 2 and 3 will streamline the utilization of GR for both formally and informally organized providers (such as research laboratories that distribute GR that they collected from in situ or official ex-situ collections that work in the context of public-private partnerships). These advantages seem strongest in option 3 as compared to option 2, where both public and private sector research might feel hindered from the lack of flexibility in a fully standardized set of BS requirements. This could rebalance the role of public collections which sometimes lack the resources/bargaining power to impose the appropriate rules.  Impact on stakeholders: o Coll.: As user, might be disadvantaged under option 3, as resources are generally limited to negotiate specific BS requirements. o Gov. Res.: Might be disadvantaged under option 3, as resources could be limited to negotiate specific BS requirements. o Ag., health and biotech: Bigger users might prefer option 1 or 3, due to the flexibility to accommodate to existing functioning. Small commercial users might be disadvantaged under option 2, as resources could be limited to negotiate specific BS requirements. o Univ.: Might be disadvantaged under option 3, as resources are generally limited to negotiate specific BS requirements. o Land: No impact on economic innovation and product development of land owners o Other: Economic innovation more likely if benefit-sharing is adopted as a general principle. E3 -Minimizing implementation costs The impact of the specific "0" option for minimizing implementation costs is unclear as it is unclear how the specific "0" option would still allow the Belgian State to comply with the BS obligations of the CBD (cf. chapter 8) and what implementation costs would result from this alternative scenario. The implementation costs of the options 1, 2 and 3 will be different for stakeholders on the one hand and for the public administration on the other. Stakeholders will be impacted by the negotiation costs they incur to agree upon the sharing of benefits. The level of these costs is inversely proportional to the standardization of (the process to establish) MAT 214 . When BS is not specified, negotiations can include agreeing on the types of benefits to share, the time-frame of the benefitsharing, the distribution of the benefits between the different stakeholders involved, etc. The less the process is standardized and/or is facilitated by pre-existing requirements, the more time stakeholders will spend on the negotiation of terms they both agree upon. This cost is also likely to differ depending on the moment negotiations are taking place. They can take place before the exchange (ex-ante negotiation), most likely at the moment of access, or after the negotiations (expost negotiation) when the agreement specifies that benefit-sharing terms are to be settled at a later stage of the development chain (e.g. the patent stage, the commercialization stage, etc.) or when terms of a project need to be renegotiated. It is assumed that the cost of ex-post negotiation is substantially larger than the costs of agreeing on BS ex-ante, because of the relationship-specific investment related to an already pre-developed product 215 . This is also voiced by some interviewees, who fear that deciding on the amount of benefits to share at a later stage than the moment of access will create higher expectations, resulting in difficult negotiations between users and providers. Taking the above into account, the negotiation cost could range from no costs at all for a fully standardized procedure to more than 1000€ per transaction for ex-post negotiation in a fully flexible context 216 . For the public administration, option 1 leads to the least implementation costs, while option 2 might lead to high set-up and follow-up costs for implementation and option 3 might lead to high implementation costs due to the recurring need to adapt the BS requirements to every new transaction (including legal advice). At the same time, the legislative costs for the drafting of the different options are not necessarily very high, depending on how the standardized mutually agreed terms are specified in the implementation provisions of the Nagoya Protocol in Belgium. In particular, such implementation provisions can draw some lessons from the practices with existing standardized MTA clauses (Material Transfer Agreements) already put into practice by the collections, which shows the benefits from using standardized material acquisition and transfer arrangements. 214 Täuber et al., (2011), op.cit. 215 O.E. Williamson (1985), The Economic Institutions of Capitalism, New York: Free Press; relationship-specific investment are investments whose return depends on the duration on the relationships continuation (See V.P. Crawford (1990), "Relationship-Specific Investment", The Quarterly Journal of Economics, 105(2), pp. 561-574). In other words, the return on the investment made by users of GR in developing a product depends upon the continuation of the relationship (the ABS agreement) they have with the provider. 216 These figures are based on a quantitative evaluation, by the study team, of the negotiation costs related to the MAT under the various options. This evaluation is based on the data generated in the interviews (especially indicators IND 3.1 and 3.2, data collected for the various stakeholder groups). Combining these two contrasted impacts for comparing options 2 and 3, it can be considered that one time set up costs lead to fewer impacts, compared to recurrent costs such as transaction and negotiation costs (cf. E1 and E2). Under options 2 and 3, negotiation and transaction costs are born on a regular basis by the users and providers of GR. However, the setting-up costs of standardized formats for option 2 are costs incurred only once and lead to less recurrent negotiation costs. As a result, it might be said that the overall impact of option 2 on minimizing implementation costs is better than option 3 which leads to higher recurrent costs for all stakeholders. Option 1 is hard to evaluate as it could both be minimalistic or very extensive, even though it has less set-up costs for the State compared to options 2 and 3. Social impact S -Achievement of social objectives It is likely that the overall contribution to economic innovation and product development of the adoption of BS as a horizontal principle might also have (at least indirect) positive effects on socially important sectors such as food security, health and nutrition. The sharing of (monetary and/or nonmonetary) benefits could take the form of management support, educational programs, technology transfer, institutional capacity building, collaboration among companies, etc. which are expected to support social objectives. A concurrent contribution to the R&D sector is also expected to contribute to job creation in the overall economy, and in public and private research institutions in particular. Conversely contribution for socially important sectors under option 0 can be considered negative. However, the different options might have contrasted effects on various sectors of use, impacting their capacity to innovate, create jobs and contribute to social objectives. Option 2, and to a lesser extent option 3, could at least partially contribute to overcome problems of unbalanced bargaining power between the actors. Small commercial users and the non-commercial users might suffer from option 3 if they do not have sufficient capacity to negotiate the case by case agreements, while at the same time this option could potentially better take into account the specificities of the small commercial and non-commercial users and providers. Option 1 is hard to evaluate as it could both be minimalistic or very extensive. Overall, options 3 and 2, compared to option 1, offer a better opportunity for the Belgian authorities to control the types of benefits being shared and monitor whether use made of them by stakeholders serves social objectives. Combining these effects (specific BS requirements and public control over benefits), it seems that option 2, and to a lesser extent option 3, offer an advantage for this criterion over option 1. Procedural impact G1 -Flexibility to accommodating sectorial differences The utilization of genetic resources is very heterogeneous, ranging across different sectors of the biotechnology industry and including different types of users. BS agreements could reflect heterogeneous uses of genetic resources, in different sectors and different production chains. The possibility to accommodate sectorial differences will clearly be highest in option 1 and to a lesser extent in option 3. At first glance, options 1 and 3 will provide some advantage as the BS conditions will be specified upon case by case transaction. In the case of option 1 this will be without having to follow specific BS requirements, in the case of option 3 there will be specific BS requirements but these will leave a large degree of flexibility to accommodate sector specificities. However, option 1 might have some hidden costs of accommodating flexibility, for the public authorities implementing the Nagoya Protocol, as they have to specify the circumstances under which a certain type of benefitsharing can be considered fair and equitable. Therefore, an option such as option 3, leaving sufficient room for flexibility while providing such indications of "appropriate circumstances" by some standardization, might have a lower cost for organizing flexibility. Overall, for the authorities, both option 2 and option 3 represent a low-cost measure for organizing flexibility with sectorial differences, while providing clear and certain rules for benefit-sharing. Comparing the impact on various stakeholders, it can be said that the lack of flexibility regarding sectorial differences in option 2 might be disadvantageous for larger commercial users while it could benefit non-commercial and small commercial users who do not have the same amount of resources for detailed negotiations under option 3 as larger commercial users. The impact of the specific "0" option for accommodating sectorial differences is unclear as it is unclear how the specific "0" option would still allow the Belgian State to comply with the CBD BS obligations (cf. chapter 8 and 9) and how flexibility would be built in this alternative scenario. G2 -Temporal flexibility to allow for future policy and adjustments Overall, the options 1, 2 and 3 allow for some temporal flexibility as they could provide for gradual fine-tuning, or possibly even for a gradual increase of the level of requirements (1->2->3), for example when more knowledge about the use of GR would become available. However, options requiring a substantial investment in order to put into place (such as options 2 and 3 where substantial effort is needed to define the specific requirements) will be more difficult (and there will be more resistance) to change at a later stage. The specific "0" option is likely to have a contrasted effect on temporal flexibility. On one hand, it would lead to non-ratification of the Nagoya Protocol and also to a default on the implementation of the CBD BS obligations. This would still require, in a later stage, to move towards better implementation of the CBD and could lead towards the ratification of the Nagoya Protocol in a second step by adopting option 1, 2 or 3 in a later stage. However, postponing the implementation of the BS obligations to the future is likely to create comparatively higher costs in the future, than the costs that are envisioned now (for example in option 1). Indeed, not ratifying the Nagoya Protocol would still require legal action, as a non-party, to clarify the relationship with Parties to the Protocol and to deal with implementation measures in other countries when distributing GR to these countries (for example if these countries would put a due diligence system in place, requiring clarification of legal provenance for the GR from Belgium). Changing legal actions that would have been taken place outside the system of the Protocol, in a later stage, and revert back to the implementation of the BS sharing under the Nagoya Protocol at a later stage, in a way which is consistent with the legal developments in other countries, would probably lead to additional costs which outweigh the temporal flexibility gained by postponing the implementation. G3 -Improving knowledge for future policy development and evaluation The absence of a horizontal BS requirement (specific "0" option) comparatively would generate less information than the options requiring BS. Indeed these options would generate information on the way actors deal with benefit-sharing obligations (both under options 1, 2 and 3) that could prove relevant for future policy making. G4 -Correspondence with existing practices The exchange of GR in Belgium is currently self-regulated and most current exchanges of GR already include a benefit-sharing clause based on semi-standardized or standardized BS used between user and provider, many of which make a difference between commercial and non-commercial use purposes. Option 1 could therefore easily build upon existing practice, but there would be no decisive difficulty for those providers to adapt themselves to the options 2 or 3, if the specific conditions imposed by the state would be sufficiently flexible. As for the private actors, the small companies could prefer a certain use of standard models given their possible lack of direct legal expertise and/or resources for extensive negotiations. In contrast, the specific "0" option would move away from the existing trends and practices of the Belgian stakeholders. Visual dominance analysis No ideal point can be identified in the performance chart (Figure 5). Ranking the alternatives With our basic allocation key, option 2 stands out as the preferred solution (Figure 6): it performs better or at least as good as other options on all the criteria except on criterion E2. However, the differences with option 3, which comes second, are rather small, as option 3 scores well on economic (E1 and E2), environmental (M) and most procedural criteria. The leading position of option 2 is maintained throughout the sensitivity analysis but is slightly attenuated. In light of this analysis, option 1 and option 0 are not valuable alternatives. Option 2 will generate lower transaction costs in accessing GR, due to a simplified one stop access procedure for users, but the difference in costs with option 1 is unlikely to have a major impact on economic innovation and product development. Moreover, this effect will be stronger for foreign users of GR, and will be more nuanced for Belgian users. Most Belgian actors already function in the strongly decentralized Belgian system. Therefore, option 1 and option 2 cannot clearly be differentiated along this sub-criterion. With its high process as well as legal uncertainty (due to the non-ratification of the Nagoya Protocol), option 0 represents the least favored option for this sub-criterion. E3 -Minimizing implementation costs Under option 1, both users and public administrations will be faced with higher implementation costs. For public administrations, the cost of establishing the entry-point is fourfold higher under option 1, as four separate entry-points will have to be created and manned. As indicated earlier, for users (especially foreign users), identifying the competent entry-point is likely to require more working time than under option 2. Option 2 will have a higher coordination cost, at least in the initial phase of the implementation, as internal mechanisms and procedures will have to be established to deal with the different CNA's in line with their internal legislations and procedures. However, an initial higher set-up cost is to be preferred over continuing higher operating costs. The impact of the specific 0 option along this criterion is unclear as this option would lead to nonratification of the Nagoya Protocol and therefore depend on the alternative measures taken to clarify the access requests.  Impact on stakeholders: public administrations (for setting-up and operating costs), users (for search and input costs Social impact S -Achievement of social objectives Choosing between a single entry-point and four different entry-points to the CNAs is unlikely to have any significant impacts on any social objective. With its high process as well as legal uncertainty (due to the non-ratification of the Nagoya Protocol), option 0 clearly does not benefit any social objective and therefore represents the least favored option for this criterion. Procedural impact G1 -Flexibility to accommodating sectorial differences As indicated in chapter 9.3, the choice of the entry-point, as dealt with in this analysis, is independent from the establishment of the four CNAs (one for each of the regional + federal authorities). The latter applies in any situation, hence this analysis is only concerned with the entrypoints to the CNA. In this context, both options 1 and 2 potentially offer the same level of sectorial flexibility in the implementation of the NP. The impact of this criterion on the specific 0 option is unclear as it would lead to non-ratification of the Nagoya Protocol and depend on the alternative measures taken to clarify the access procedures to Belgian GR. G2 -Temporal flexibility to allow future policy and adjustment The specific 0 option would lead to non-ratification of the Nagoya Protocol. This could lead to keep more flexibility at present, but is likely to lead to less flexibility later on as explained above (cf. analysis under criterion G2 for specifying MAT). The higher set-up costs of a centralized entry-point would partially lead to less flexibility to allow future changes (due to higher resistance to change, lower willingness to renegotiate) or to change modalities that are inherent to the institutional set-up. However, it is important to note that this partial difference in flexibility for change in modalities only concerns the difference in the entry-point to the CNAs and so can be considered asminor (the establishment of the CNA by each relevant authority applies in any situation, see chapter 9.3) G3 -Improving knowledge for future policy development and evaluation Option 0 would strongly inhibit the gathering of information, as no PIC would be granted and no information would be kept about previous access requests in case of the non-establishment of the CNA. It is unclear which of the options 1 or 2 would provide better opportunities to improve the knowledge base, but it might be argued that a centralized system will avoid redundant knowledge acquisition, improve the consistency of the generated data on the various access requests, and hence be more efficient. G4 -Correspondence with existing practices Both option 1 and option 2 build upon existing practices to a certain extent. On the one hand, the decentralized input of access requirements proposed under option 1 clearly corresponds to the current exercise of competences over GR, where there is no coordination between authorities in charge for dealing with access to GR in PA/PS. From this perspective, establishing an increased coordination would introduce a change to the existing practices, albeit at a low cost as it can be implemented through a one-stop digital portal. A notable exception to this is the longstanding coordination within the BCCM consortium for the culture collections, where information on GR and access procedures is available through a common portal which redirects users to the decentralized member collections. On the other hand, there is already an established practice of coordination amongst the federated entities and the Federal Government on matters pertaining to ABS policy. For example, issues related to the CBD and the NP are coordinated through the Biodiversity Steering Committee of the CCIEP, for which the secretariat is provided by the Federal Public Sevice for Environment. The Belgian Clearing-House Mechanism, managed by the focal point to the CBD is a central access point for information and awareness-raising pertaining to the CBD. Hence, establishing a single entry-point for facilitation/channeling of requests/advice (option 2) would also to a certain extent correspond with existing practices. The impact of the specific 0 option under this criterion is unclear, as this option would lead to a nonratification of the Nagoya Protocol and the impact would therefore depend on the alternative measures taken to clarify the access requirements. Visual dominance analysis Option 2 is the dominant alternative: compared to options 0 and 1, it scores at least as well on all criteria and is strictly better on one economic sub-criteria (legal certainty). However, this dominance has little relative value. In light of the preceding analysis of the performance and the impact on the stakeholders, this chart shows that little difference can be observed between the impact of establishing a single entry-point and the impact of establishing four separate entry points. Even if the measures under the general "0" option were taken in order to comply with the obligations of the CBD and the ILO 107, users and providers would not be able to benefit from the clarified legal framework that the compliance measures envisioned under the NP. This would not create a sufficient level playing field for stakeholders. Therefore both options 1 and 2 are preferable over option 0 under this criterion. Option 1 refers back to the legislation of the provider country while the private international law code would determine that provider country legislation is applicable to disputes regarding compliance with the MAT. If it is impossible to determine the content of the foreign law in due time, Belgian law should be applied. This option therefore relies on the assumption that the legislation of the country of origin properly implements the NP provisions and is clear enough and acceptable for enforcement based on the provider country legislation. Instructing courts and authorities to directly apply the terms set by the provider country could create a level of uncertainty for Belgian users and public authorities, "given that access legislation will vary among countries, creating legal uncertainty as to whether and how each country's provider-side law will affect rights and obligations of users" 217 . However, the latter disadvantage is attenuated by the fall-back clause of the Belgian code of private international law, which specifies that if it is "impossible to determine the content of foreign law in due time, Belgian law should be applied" (art.15 §2al2). In addition, now that Belgium has a national code of private 217 Tvedt, Fauchald, (2011), op.cit.p.386. Summary of the selected options on compliance 0. Specific "0" option: not introducing any legal provision on compliance 1. Option 1: Ensuring compliance with provider country legislation regarding PIC and MAT, with Belgian law as a fall-back option 2. Option 2: Self-standing obligation in the Belgian legislation to have PIC and MAT if so required by the provider country. For a detailed description of the options please refer to chapter 8.2 and chapter 9. international law, the application and control of foreign law is quite common218 . There are today more than 300 judgments on the most-used database (Jura) with the keyword "applicable law to contracts in situations of international private law"; and at least 50 cases regarding "the application of foreign law by a Belgian judge". This is therefore absolutely not a new phenomenon, and would not create an additional burden to the judiciary system. Nevertheless, in this perspective, the passing of a self-standing obligation as envisioned under option 2 could instead create less complexity for users, courts and enforcement authorities in Belgium. However, the obvious disadvantage is that it would respect to a lesser degree the political options and the legal requirements of the provider country pertaining to its national ABS legislation. E2 -Maximizing economic innovation and product development (in particular through its contribution to R&D) at reasonable financial and administrative costs The 0 option would lead to non-implementation of the Nagoya Protocol, which would likely lead to an increased difficulty for Belgian users to acquire foreign GR for research and development and result in a barrier for economic innovation. As stated in chapter 3, there are already various substantial (material rules) and formal (private international law) provisions that could be applied to the contractual and extra contractual conflicts related to GR benefit-sharing. However, these are not fully adapted to the NP playing field (e.g. absence of notion of "informational component"). In the field of research and development, collaborations with users in foreign countries which are Parties to the Protocol might be hampered, which will also hinder obtaining internationally recognized standards for proof of good legal provenance of GR. In comparison, options 1 and 2 would allow implementing the Nagoya Protocol and thereby safeguard, or even extend, the level of trust in the research sector. The expected positive impact from the implementation of the Nagoya Protocol will however differ between option 1 and 2. Indeed, under option 1, giving the priority to the law of the provider country within the Belgian legal system could entail significant transaction costs. The extent of these transaction costs will of course largely depend on the effectiveness of the ABS Clearing-House in providing detailed and up to date information. On the other hand, option 2 would be less complex for users to comply with the provider country requirement regarding the existence of PIC and MAT, which would promote the use of GR for economic innovation and product development.  Impact on stakeholders: o Coll.: As users, incurring higher costs under option 1 o Gov. Res.: Incurring higher costs under option 1 o Ag., health and biotech: Incurring higher costs under option 1 o Univ.: Incurring higher costs under option 1 o Land: No impact as providers o Other: No impact E3 -Minimizing implementation costs The impact of the 0 option under this criterion is unclear. The 0 option would lead to non-ratification of the Nagoya Protocol and therefore the impact on implementation costs will depend on the alternative measures that are taken to comply with the obligations of CBD and ILO Convention 107, which are both ratified by Belgium. ABS disputes would relate to disagreement about implementation of provisions included in the MAT. In this context, the absence of related jurisprudence will create a challenge for the initial cases in all the considered options, which might be reduced only by a legislative draft that is as precise as possible. This remark applies equally to options 1 and 2. Social impact S -Achievement of social objectives The envisioned implementation of both option 1 and 2 would contain a firm commitment under Belgian law to the compliance with PIC and MAT of the provider countries, both for GR and TK associated to GR (IMP 4.1 (1)). However, contrary to option 2, in option 1, the actual provisions for the PIC and MAT and the compliance with those provisions, would also be considered by the courts in the context of the relevant legislations of the provider country. If this legislation covers social objectives, and these are included in the MAT, then option 1 could have a better performance. The 0 option is likely to have negative effects on the social objectives. As discussed earlier, option 0 could hinder access, due to lack of trust and level-playing field, and thus the social objectives of possible BS provisions. Furthermore, it would impede R&D in Belgium, which could be a barrier for innovation in the health, nutrition or food security sectors. Environmental impact M -Promotion of conservation and sustainable use of biodiversity, including biodiversity research The implementation of the Nagoya Protocol is expected to have a positive impact on conservation activities and biodiversity research. Therefore, options 1 and 2 are to be preferred over option 0 on this criterion. The difference between option 1 and option 2 are difficult to assess, as the environmental benefits would depend in both cases on the Mutually Agreed Terms specified in the provider country legislations and/or the clauses negotiated on a case by case basis upon the access of the GR. However, contrary to option 2, in option 1, the actual provisions for the PIC and MAT and the compliance with those provisions, would also be subject to revision by the courts in the context of the relevant legislations of the provider country. If this legislation includes provisions on conservation and sustainable use of biodiversity, and these are included within the MAT, then option 1 would better address conservation and sustainable use of biodiversity. Procedural impact G1 -Flexibility to accommodating sectorial differences N/a (the three options are neutral as regards the specificities of various user sectors). All options can be adapted to sectorial and utilization differences. G2 -Temporal flexibility to allow for future policy and adjustments The impact of options 1 and 2 on temporal flexibility is probably quite similar, as all options would still leave room for further adjustments. Shifting from one option to another, or combining the options, still seems possible at further points in time, even though it would always imply a legislative cost. The specific 0 option would lead to the non-implementation of the Nagoya Protocol. This could lead to keep more flexibility at present, but is likely to lead to less flexibility later on as explained above (cf. analysis under criterion G2 for specifying MAT). G3 -Improving knowledge for future policy development and evaluation Both options could generate knowledge through court decisions. Under option 1, this information could also serve provider countries. G4 -Correspondence with existing practices As indicated in the introduction to this section on compliance, currently there is no reference to GR in the scope of the Belgian code on private international law. Both option 1 and 2 would therefore require a change compared to current practices (including the specific 0 option, because of the obligations under CBD and ILO 107 that still need to be implemented). However, option 1 clearly is closer to the existing practices, as it basically extends the existing code of private international law, in order to explicitly address situations of disputes on the content of MAT. The impact of option 0 is unclear as it depends on the way that the obligations under CBD and ILO 107 would be implemented in a situation of non-ratification of the Nagoya Protocol. Visual dominance analysis No ideal point can be identified in the performance chart (Figure 8). Ranking the alternatives With our basic allocation key, option 1 stands out as the preferred solution (Figure 9): it performs better or at least as good as other options on all the criteria except on criterion E1. However, the difference with option 2, which comes second, is so small that the ranking is of very little relative value. As can be observed in the performance chart, the only significant difference between option 1 and 2 is that option 1 clearly corresponds better to existing practices. Option 0 is clearly the least preferred option. chain, option 2 does not incentivize early users (if their utilization never makes it to the patent stage) to acquire GR legally, increasing the legal uncertainty of end users219 . Furthermore, the patent office currently covers only a very small proportion of the transactions concerned by the Nagoya Protocol. In order to be effective to prevent misappropriation of GR, this option will need to be complemented with other checkpoints. But the small amount of transactions covered by option 2 could provide better opportunities for the enforcement procedures for those GR it will possibly cover220 . By linking the monitoring and the patenting process, it could be easy for the authorities to ensure the monitoring of GR likely to have high(er) commercial value. For users, option 2 also makes it possible to combine patenting and ABS obligations, hence limiting the obligation redundancy. As indicated in chapter 9.5, option 2 will require amending the existing patent law, whereas option 1 does not require any extra legal drafting beyond what can be foreseen under the obligations regarding PIC and the ABS Clearing-House. Hence, option 2 is likely to generate higher legal costs than option 1.  Impact on stakeholders: o Coll.: As users, higher legal and process certainty with option 1 and limiting obligation redundancy under option 2 (for users using patents) o Gov. Res.: Higher legal and process certainty with option 1 and limiting obligation redundancy under option 2 (for users using patents) o Ag., health and biotech: Higher legal and process certainty with option 1 and limiting obligation redundancy under option 2 (for users using patents) o Univ.: Higher legal and process certainty with option 1 and limiting obligation redundancy under option 2 (for users using patents) o Land: Option 2 allows more effective enforcement and monitoring of use of genetic resources o Other: No impact E2 -Maximizing economic innovation and product development (in particular through its contribution to R&D) at reasonable financial and administrative costs Option 0 would lead to non-ratification of the Nagoya Protocol, which would likely result in higher distrust with the provider countries and have a negative impact on the acquisition of GR from foreign countries and thereby on the overall capacity of the Belgian GR sector to innovate. Option 1 and 2 on the contrary are expected to increase trust and have a positive overall impact. Under option 2, users acquiring GR from third-parties will face additional financial and administrative costs and efforts to make sure GR have been legally acquired, in order to avoid complications and unforeseen costs at the moment of patenting. However, the exact cost is unclear as it will strongly vary depending on the type of users, their utilization of GR and the moment in the development chain at which they acquire GR and the possible combination with a self-monitoring scheme. On the other hand, for users acquiring GR mostly directly from developing provider countries, option 2 could also prove to be positive for the collection of GR. Being "the option favored by developing countries in the negotiations on the Protocol"221 , this measure could foster trust with partner countries and thus facilitate access to GR in these countries. No additional research costs are expected with options 0 and 1.  Impact on stakeholders: o Coll.: As users, under option 2, higher costs to make sure GR has been acquired legally (for users using patents) o Gov. Res.: Under option 2, higher costs to make sure GR has been acquired legally (for users using patents) o Ag., health and biotech: Under option 2, higher costs to make sure GR has been acquired legally (for users using patents) o Univ.: Under option 2, higher costs to make sure GR has been acquired legally (for users using patents) o Land: No impact o Other: No impact E3 -Minimizing implementation costs The impact of option 0 under this criterion is unclear, as it will depend on the other measures taken by Belgium to comply with CBD and the ILO 107 Convention. Options 1 and 2 are roughly equal in terms of costs related to the establishment of new institutions. Both options require an additional monitoring authority to be created. Although this new service will be hosted in the existing patent office with option 2, option 1 will most probably make use of the ABS CH (cf. chapter 9.5). Exact costs for the monitoring tasks are difficult to determine, as it will also depend on the interpretation of the term "monitoring", possible requirements at EU level and on how other information exchange measures are implemented such as the ABS Clearing-House. If the task of the checkpoint is understood as being limited to the collection and transfer of information as is currently the case (see chapter 9.5), the cost is roughly limited to the cost of storing and handling the information in a database. This cost is likely to be very small under option 1, as the ABS CH will already be used for the collection of information regarding the implementation of the NP, including on Prior Informed Consent. If, on the other hand, the provided information is to be effectively monitored and verified by the entity collecting the information, cost may be substantially higher. While monitoring costs for the patent office are likely to be reasonable 222 , costs related to the monitoring of a very high amount of GR used in the country (full development of option 1) will be much more important. In the absence of specific figures on the utilization of GR in Belgium, it is difficult to assess this financial cost. The latter will also depend on the way the option is implemented in Belgium (cf. discussion on the phased approach above). Implementation costs for users will be differently distributed depending on which option is chosen. Whereas under option 1 the cost of providing the information is incurred by users utilizing GR on Belgian territory, under option 2 this cost is incurred by the users wishing to patent a (semi-)finished product. However, the cost of providing the information will be small for users using legally acquired GR.  Social impact S -Achievement of social objectives Options 1 and 2 have no direct major impact on social circumstances with regard to the access or benefit-sharing of the GR nor on related research and therefore these options are not expected to have a substantial impact on socially relevant objectives. Option 0 could seriously threaten R&D, as access to GR in provider countries will be much more difficult if Belgium's does not ratify the NP. With regard to the protection of traditional knowledge, in light of the current level of details of the options, both options 1 and 2 offer the same possibilities to protect the TK of local communities in third countries. However, option 1 could offer more opportunities: if all relevant information concerning GR utilized in the country could be monitored (not just collected), this would offer a higher protection rate against fraudulent use of TK, including in non-commercial settings. But as indicated earlier, this is difficult and costly to enforce, so this impact will depend on the way option 1 will be implemented if it would be selected. Option 0 does not lead to the protection of TK.  Environmental impact M -Promotion of conservation and sustainable use of biodiversity, including biodiversity research Options 1 and 2 have no direct major impact on conservation or sustainable use of biodiversity. Better monitoring of the use of GR, and in particular of the PIC and MAT obligations under the Nagoya Protocol, is likely to lead to increased benefit-sharing effectively reaching provider countries, which could in turn benefit activities related to the conservation and sustainable use of biodiversity and raise the awareness on the value of biodiversity for research and innovation. Moreover, both options 2 and 1, being in line with demands of provider countries if properly implemented, could facilitate cooperation activities focusing on conservation and sustainable use of biodiversity, while option 0 could create the opposite effect. Option 0 could seriously threaten generation of benefits for biodiversity conservation and sustainable use, as access to GR in provider countries will be much more difficult if Belgium does not ratify the NP. Procedural impact G1 -Flexibility to accommodating for sectorial differences The 0 option would lead to non-implementation of the Nagoya Protocol. The impact of this option on criterion G1 is unclear as it would depend on what alternative measures would be taken to comply with the obligations under the CBD and the Convention ILO 107 and how Belgium would ensure legal consistency between these measures (taken as a non-party) and the measures taken by other countries that would be Party to the Protocol. Even though option 1 and 2 potentially apply equally to all sectors, adopting option 2 instead of option 1 could impose a relative higher duty on users that are more heavily involved in commercial activities that involve patenting activity. Hence, option 2 is less flexible to accommodate sectorial differences. However, as the upgrading of the patent disclosure procedure is not expected to impose any significant costs other than the amendment to the patent law (cf. chapter 9.5), the actual magnitude of this impact can be considered very low. G2 -Temporal flexibility to allow for future policy and adjustments The 0 option would lead to non-implementation of the Nagoya Protocol. As discussed under G2 above (section G2 under MAT), this might lead to some additional flexibility in the short term, but would probably lead to higher adaptation costs at a later point in time. The implementation of the monitoring obligations under option 2 would require an amendment of the federal law transposing the Directive 98/44/EC on the legal protection of biotechnological inventions to include such a new provision, while the option 1 does not require any additional action (cf. chapter 9.5). Therefore, option 2 will be less flexible for future adjustments. G3-Improving knowledge for future policy development and evaluation The 0 option would lead to non-ratification of the Nagoya Protocol and would not allow benefiting from the measures; in particular the information generated from the PIC procedures that would generate knowledge on the flow and the utilization of GR. If efficient, the broad-scale collection/reception of information related to PIC under option 1 could strongly contribute to the knowledge on the utilization of GR in Belgium, as most GR accessed after the entry into force of the NP would theoretically be covered. However, this has to be nuanced, as the implementation of this measure would probably be phased and therefore the contribution to the knowledge base will be incomplete in the first implementation phases, especially if the checkpoints tasks are limited to collect the information only, without verification. Knowledge improvement under option 2 will be weak, as it only covers a relatively small sub-set of GR accessed for R&D and would not bring more knowledge on the transaction of GR compared to the existing disclosure of origin requirement in the patent legislation. However, upgrading of the patent disclosure to a checkpoint recognized under the NP could lead to improved knowledge gathering. Indeed, in its current form the disclosure obligation is reported to be ineffective. At the same time, the patent authority is not competent to verify the correctness of the information provided by the user. Moreover, some stakeholders complain about the difficult implementation of the current disclosure obligation. G4 -Correspondence with existing practices The 0 option would lead to non-ratification of the Nagoya Protocol, which would go against the existing policy and stakeholder efforts to comply with the internationally adopted environmental commitments under the CBD and its Protocol. There are no existing practices on monitoring the use of GR, hence both option 1 and option 2 represent a change compared to existing practices. However, the requirement for the disclosure of origin is already included in the patent law. While it will still need amending the patent law, option 2 thus corresponds more closely than option 1 to existing practices, due to its link with the patent authority, which is already a necessary step for users willing to patent their products. Visual dominance analysis Option 1 is the dominant alternative: compared to options 0 and 2, it scores at least as well on all criteria and is strictly better on the social criterion (note: E3 is undefined). However, this dominance has little relative value. In light of the preceding analysis of the performance and the impact on the stakeholders, what this chart shows is that little difference can be observed between the impact of using the ABS Clearing-House as a checkpoint and using the patent authority as a checkpoint. Furthermore, as mentioned earlier, these two options are not mutually exclusive. In a phased implementation approach, it can be envisioned to implement both options. In this context, the procedural criteria G1, G2 and G3 give an advantage to option 1 compared to option 2 for earlier implementation The 0 option would lead to non-ratification of the Nagoya Protocol and have a major negative impact on legal certainty and effectiveness, as Belgium would not benefit from the transparency and legal clarity advantages of the Protocol. Options 1, 2 and 3 would equally contribute to the objectives of legal certainty and effectiveness, albeit at a different cost, depending on the final requirements of any ABS CH, as indicated above. Indeed, RBINS is expected to be most cost-effective for more general information tasks (as it is in line with it existent practices and expertise), while BELSPO and WIV-ISP are expected to be most costeffective for the coordination of more technical information (as it is in line with it existent practices and expertise). Option 0 would lead to non-ratification of the Nagoya Protocol, which would likely result in higher distrust with the provider countries and have a negative impact on the acquisition of GR from foreign countries and thereby on the overall capacity of users to innovate. Options 1, 2 and 3 on the contrary, as they contribute to the implementation of the Protocol, are expected to increase trust and have a positive overall economic impact. In general the information tasks of the Belgian input point/component of the ABS CH are expected to generate information on exchanges of GR and on-going innovation activities with GR that are useful for R&D in Belgium. The cost-efficient contribution to research will however be higher if the chosen option also generates more information integration. From that perspective options 1 and 2 are preferable over option 3, as they favor integration of the information handled by the CH with more existing biodiversity initiatives (option 1), or within existing powerful database infrastructures (option 2). E3 -Minimizing implementation costs The impact of option 0 under this criterion is positive, as it would lead to no additional costs. However, this option would lead to non-ratification of the Nagoya Protocol and therefore would still lead to information obligations related to the alternative measures taken by Belgium to comply with CBD and the ILO 107 Convention in the absence of the ratification of the Nagoya Protocol. All three options would potentially offer cost-effective solutions to the CH as they all host some expertise that could be relevant for the CH's task. The main criterion for comparing the cost-effective implementation amongst the options is the possible synergies with the existing infrastructures and/or existing tasks under the CBD. For the general information tasks, this is likely to give an advantage of options 1 and 3 over option 2. Indeed, in terms of software, both RBINS and ISP/WIV already have an online portal for their respective tasks with the Clearing-House. Hence, creating an additional ABS CH portal might not be such a big additional cost. The system at RBINS has the advantage to be designed for easy replication as it is used to set up national Clearing-House nodes in partner countries, leading to an additional advantage. For the coordination of the technical information, both BELSPO and ISP/WIV have existing infrastructure for the management of technical data that could support the information on the working of the PIC/checkpoints/ABS CH. Given this analysis, and pending the final outcome of the international negotiations on the ABS CH, it would seem logical to propose a collaboration of all three, as this would create the highest level of synergy with existing infrastructures. Social impact S -Achievement of social objectives The impact of option 0 is likely to be negative on socially relevant objectives. On the contrary, the impact of options 1, 2 is positive while option 3 is neutral. A particular social impact deserves to be highlighted. RBINS is running development projects on establishing CBD CHMs in partner countries. Entrusting RBINS with additional information tasks pertaining to NP issues could promote interesting synergies and additional capacity building in developing countries that are Party to the Protocol for handling NP and CBD requirements in a coherent and efficient way. Environmental impact M -Promotion of conservation and sustainable use of biodiversity, including biodiversity research The impact of the 0 option is likely to have a negative impact on conservation and sustainable use of biodiversity. On the contrary, the impact of options 1, 2 and 3 is respectively positive and neutral. RBINS is running development projects on conservation and sustainable use of biodiversity, including capacity building, biodiversity research and technology transfers, in partner countries. RBINS also organizes educative and communication actions towards stakeholders and broader awarenessraising campaigns on conservation and sustainable use of biodiversity. Reinforcing the role of RBINS in ABS will create synergies between ABS and the conservation/sustainable use initiatives, both nationally and internationally. Option 2 would be ideal for biodiversity research that contributes to sustainable development as BELSPO already hosts the Biodiversity Platform, which has as main task to foster such research. BESLPO also hosts several other consultative bodies linking scientific and policy analysis and is involved at international level with digitalization of collection databases. ISP-WIV has little connection with conservation or sustainable use of biodiversity and would therefore be the least preferable option for this criterion. From an environmental perspective, options 1 and 2 are the preferred alternatives (Figure 12). They both have similar performances on most of the criteria, but option 1 has a better social impact. This outcome is confirmed in the equalized weighting scenario. But, in line with what can be observed from the visual dominance analysis, the difference between the two options is to be nuanced, with the preference for the two options being almost equal. As RBINS and BELSPO are related institutions, a combination of these two alternatives could produce an ideal outcome. Again, as for the checkpoints, it is important to note that both options 1 and 2 rank substantially better than the "0" option and option 3. An important note of caution is in order here. Although this rank might usefully inform decision making on the choice between the options in Belgium, the final evaluation of the most appropriate mechanism will mainly depend on the decisions still to be taken globally on the ABS CH. RECOMMENDATIONS ON INSTRUMENTS AND MEASURES RESULTING FROM THE IMPACT ASSESSMENT As explained in chapter 5 and 8, the impact assessment in this study considers policy options for implementing 6 core measures that are minimally needed to implement the Nagoya Protocol in Belgium:  Operationalizing Prior Informed Consent;  Specification of the Mutually Agreed Terms;  Establishment of the Competent National Authorities;  Setting up compliance measures;  Designation of one or more checkpoints;  Sharing of information through the Clearing-House. The impacts of the selected options for each of these measures were assessed in chapter 10 on a double comparative basis. First, the impacts of the options were compared to the impacts of the "no policy change" base line (0 options). Second, for each measure, the impacts of the options were compared amongst each other. Two general recommendations result from the analysis in chapter 10, along with a set of more specific recommendations for each of the measures. First, the analysis shows that the" no policy change baseline" (the "0" option) for each measure clearly has the worst performance. Amongst other reasons, this is due to the lack of legal clarity that the "no policy change" would entail for users in Belgium and the absence of the environmental benefits that would follow from not implementing the Protocol. This result leads to a first general recommendation, which is to implement both PIC and benefit-sharing as a general legal principle in Belgium. Second, the analysis confirmed the validity of a phased approach to the implementation of the Protocol, which is the second general recommendation. As seen throughout the impact assessment, a phased approach will allow to benefit from the implementation of the basic principles in a timely manner and to deal with more fine-grained choices in a later stage. These more fine-grained choices can then be based on the experience the administrations and/or users will gather on the utilization of Belgian and foreign genetic resources, through the operation of the Competent National Authorities, the checkpoints and the Clearing-House amongst others. Moreover, the phased approach will be necessary in order to be able to ratify the Nagoya Protocol before June 2014 in order to participate as Party to the next COP/MOP in October 2014. The phased approach could be organized through a 3 step process. Such a process could consist of, 5. In the first step, a political agreement in the form of a declaration of intent from the competent governments on the general legal principles, along with some specification of the actions to be undertaken by the federal and the federated entities to establish these principles and put them into practice. 6. In a second step, the specified actions would be subsequently implemented, for example through a cooperation agreement and/or by adding provisions in the relevant legislations such as the environmental codes of the federated entities and the Federal Government, along with other possible requirements. 7. In a third step, additional actions can be undertaken once there is more clarity from the negotiations on the EU and the international level. The impact assessment has led to a set of specific recommendations on each of the 6 measures that have been analyzed. Not all the options have a clear preferred ranking in the basic weighting scenario, in part because of the ongoing discussions under the Nagoya Protocol, in particular regarding the modalities for the ABS Clearing-House. This result did not change by adjusting the weighting scenario through the sensitivity analysis. For these measures the study recommends to combine features of the best options that came out of the assessment. For 3 of the 6 measures a clear first best ranking came out of the impact assessment:  For the establishment of the Competent National Authorities, a centralized input system clearly came out as the recommended option. This option scores best on all the criteria and is strictly better on legal certainty and effectiveness for users and providers of GR, at low cost.  For the setting up of compliance measures, the option to refer back to provider country legislation regarding PIC and MAT, with Belgian law as fallback is the recommended option that comes out of this analysis. This can be explained by the closer conformity of this option with existing practices (under the Belgian code of private international law).  For the designation of one or more checkpoints, the option of using the PIC available in the Access and Benefit-sharing Clearing-House,as a checkpoint stands as the recommended option. It allows timely ratification, while additional checkpoint systems could evolve from there, in particular by adding other checkpoints to further collect or receive, as appropriate, relevant information related to PIC, to the source of the genetic resource, to the establishment of MAT, and/or to the utilization of genetic resources, (such as through an upgraded patent disclosure or monitoring of PIC upon public grants for research). For the other 3 measures, more than one remaining best option came out of the assessment or the remaining best options were very close:  For the operationalization of PIC, the bottleneck option and the refined fishing net option came out very close. These options require establishing as a general legal principle that access to Belgian GR requires PIC. This could be included in a political agreement from the competent governments, expressing the intent to establish such principle while specifying that this would be implemented afterwards for example through a cooperation agreement and/or analogous provisions in relevant legislations such as the basic environmental codes. The two options also have a common component, namely the refinement of the PA/PS relevant legislation. This refinement considers that the access to specimens under PA/PS relevant legislation, 223 , would also be considered as PIC in the context of the Nagoya Protocol by the Belgian federated entities. This general principle would be included in the analogous provisions of the relevant legislation of the three Regions and at the federal level. The actual refinement could then be implemented in the third step (additional actions for further implementation) for example through executing acts, specifying which access provisions exactly are considered as PIC 224 . Therefore, the recommendation that comes out of the analysis of the operationalization of PIC is to proceed with such a refinement of the PA/PS relevant legislation in the third step. As the two first best options rank very close (and in a contrasted way on different criteria), the best way forward when considering GR beyond PA/PS, might be to combine these options in a phased manner. Therefore, the recommendation resulting from the analysis is to implement first the 'fishing net' approach with a general registration/notification requirement to the Competent National Authorities for GR outside PA/PS. In a later stage, the 'bottle neck' approach, through which access requests are processed through qualified Belgian ex-situ collections in conformity with the Nagoya Protocol, could be organized through a set of administrative arrangements between the Competent National Authorities and the collections. In addition, in this later stage the adjustment of other GR relevant legislation can be implemented as envisioned under the refined fishing net model225 .  For the specification of the Mutually Agreed Terms, the two options that impose specific BS requirements by the Belgian State both ranked better than the option where no specific BS requirements are imposed. Nevertheless, as the specification of Mutually Agreed Terms is not a prerequisite for ratification, this can be done in the third step of the implementation. The choice between these two options can therefore be part of a later phase. The recommendation is therefore not to take action on this point before ratification (and therefore by default implement the "no specific benefit-sharing requirements" option) and to consider, in a later stage, a combination of the options that consider introducing specific benefit-sharing requirements to further implement the Protocol. As indicated in chapter 9.2, this further specification would entail specifying rules for the specific BS requirements in relevant legislation for example in the provisions of the environmental code of the three Regions and at the federal level, including rules for the use of standard agreements for some types of uses if needed. It is considered under this option that the implementation of these rules will be done through executive orders of the federated entities.  Finally, for the sharing of information through the ABS Clearing-House, the assessment makes a distinction between the basic information sharing tasks on Access and Benefitsharing by the Clearing-House and the more technical tasks related to the organization of the c. Establishment of the general principle concerning the designation of four Competent National Authorities, which will be implemented for example through a cooperation agreement and/or relevant legislations. This would be implemented by the respective authorities dealing with legislations and measures related to protected areas and protected species at the Regional level and in the respective authority dealing with environmental issues at the federal level 227 (IMP 3.1.1. ( 1)). d. Commitment that legislative measures will be taken to provide that GR utilized within Belgian jurisdiction have been accessed by PIC and MAT as required by provider country legislation and to address situations of non-compliance (IMP 4.1 (1); IMP 5.1). e. The CBD CHM, managed by the RBINS, will be considered as the Belgian contribution to the ABS CH, for dealing with the information exchange on ABS under the Nagoya Protocol and if required, further steps will be taken after the first COP/MOP to develop the correct modalities for the ABS CH. (IMP 6.1 (1)) 2. Subsequent implementation of the principles stated in the political agreement, for example through a cooperation agreement and/or the introduction of analogous provisions relevant legislations such as the environmental codes of the three Regions and at the federal level (cf. footnote 226). (IMP 1.1.1 (2); IMP 1.1.2 (2); IMP 3.1.1. (2)) 3. Subsequent legal and policy measures as soon as more clarity is provided on EU level and on the global level and more practical experience is gained with the implementation of the NP. This will especially apply to the measures on compliance (after conclusion of discussions on compliance at EU level and at COP/MOP1), the subsequent measures on PIC and MAT, and the administrative agreements to further implement the ABS Clearing-House provisions of the Protocol. These subsequent measures might imply, at a later stage, the need for a second cooperation agreement (IMP 1.1.3, possibly with IMP 1.3.1 in addition; IMP 1.1.4/ IMP 1.2.4 combined; IMP 2.2/IMP 2.3 combined; IMP 3.2.2; IMP 4.1 (2); IMP 6.1 (2)) 227 That is, as stipulated above, the "Agentschap voor Natuur en Bos" in the Flemish Region, the "Division de la nature et des forêts" in the Walloon Region, the "Institut Bruxellois pour la gestion de l'environnement" in the Brussels Region and one authority to be established at the Federal level, probably at the Directorate-General Environment of the Federal Public Service "Health, Food Chain Safety and Environment" (for GR that are not under competences of the federated entities, such as Marine GR and ex-situ GR held at federal institutions). CONCLUSIONS This study addresses the implementation in Belgium of the Nagoya Protocol on Access and Benefitsharing to the Convention on Biological Diversity. For an appropriate understanding of the recommendations presented in this study, it is important to recall the various steps and the intermediary conclusions that have led to these recommendations. The report proceeds through four core phases. Chapters 2 to 5 analyze the current state of the art of ABS law and policy in Belgium (phase 1). Chapters 6 to 9 analyze, present and describe the different options for the minimal implementation of core measures stemming from the NP (phase 2). The argument in these chapters served as a basis for the choice of a set of options selected by the Steering Committee and discussed with relevant stakeholders, for further study. Chapters 10 and 11 conducted a multi-criteria impact assessment of the selected options (phase 3) and concluded with a set of recommendations for the implementation of the Nagoya Protocol in Belgium (phase 4). Conclusion of the first phase of the study The main conclusion of the first step is that the existing legislation that addresses physical access to genetic material and the instruments regulating benefit-sharing between users and providers of genetic resources need to evolve and be complemented by additional instruments in order to implement the obligations of the Protocol. In particular, under the current legislation in Belgium, access to GR is not subject to Prior Informed Consent (PIC) by the Belgian State as a Party to the NP (that is based on a written decision by a Competent National Authority (CNA) on access and benefit-sharing). Even if it is not compulsory for compliance with the Nagoya Protocol, the Belgian State can nonetheless decide to subject access to its GR to a PIC-requirement and take the necessary legislative, administrative or policy measures, as appropriate, to provide for access permits by one or more Competent National Authorities. Alongside the requirement for PIC, Belgium needs to require its users to share benefits arising from the utilization of GR and TKaGR, based on mutually agreed terms. Further, for the implementation of the core obligations on compliance, monitoring through checkpoints and the ABS Clearing-House, additional legal measures need to be put into place for the implementation of the Protocol. While the Belgian code of private international law already contains a set of principles that can be directly used for the implementation of the compliance provisions, these principles are insufficient to comply with the Nagoya Protocol. In particular, the "utilization of GR under the Nagoya Protocol" is not explicitly mentioned within the current scope of the Belgian code of private international law. For the monitoring obligations, the Belgian patent law already requires the disclosure of information on the country origin of biological material in patent applications. However, this measure still needs to be completed by other measures in order to comply with Article 17.1 of the NP, as it is not organized nor designated as a formal checkpoint. Finally, a dedicated ABS Clearing-House for information sharing under the Nagoya Protocol will need to be put into place, whether simply as a node of the international ABS Clearing-House or as a separate Belgian ABS Clearing-House. It is important to highlight the provisional nature of these findings, as the on-going discussions on the implementation of the Nagoya Protocol in international and European fora will further influence the results of this analysis. This is particularly relevant for the issue of compliance, some aspects of which will be addressed in the EU regulation on the Implementation of the Nagoya Protocol, and the issue of information sharing through the ABS Clearing-House, as the international mechanism still needs to be clarified. Conclusion of the second phase of the study The main conclusion of the second step is the importance of a phased approach to the implementation, which would first address a set of options for minimal implementation. As such, the analysis lead to distinguish two categories of actions to be undertaken for the implementation: The detailed analysis of the first set of actions, has led to the formulation of a set of options for 6 implementation measures that were the basis of the multi-criteria impact assessment in the third step: 1. Operationalizing Prior Informed Consent 2. Specification of the Mutually Agreed Terms 3. Establishment of the Competent National Authorities 4. Setting up compliance measures 5. Designation of one or more checkpoints 6. Sharing of information through the Clearing-House 5. For the specification of the Mutually Agreed Terms, the two options that impose specific BS requirements by the Belgian State both ranked better than the option where no specific BS requirements are imposed. Nevertheless, as the specification of Mutually Agreed Terms can be done in the third step of the implementation, the choice between these two options can be part of a later phase. 6. Finally, for the sharing of information through the ABS Clearing-House, the assessment makes a distinction between the basic information sharing tasks on Access and Benefitsharing by the Clearing-House and the more technical tasks related to the organization of the technical information to be provided to the ABS Clearing-House mechanism, amongst others. The first task is already ongoing at the Royal Belgian Institute of Natural Sciences (RBINS). The recommendation from the analysis is therefore to further mandate the RBINS to fulfill the information sharing tasks on Access and Benefit-sharing under the Nagoya Protocol. In a second stage, administrative arrangements between this Clearing-House and other relevant institutions could be put into place to extend the tasks, as soon as more clarity is provided by the international negotiations. appropriate, containing minimum requirements for access rules and procedures -OR determine that access is not subject to PIC Applies to GR Minimum-information to be made available to the CHM when notifying permits (read in conjuncture with Article 14.2.c) Permits or equivalents issued in accordance with Article 6.3.e) and made available to CH have to be accepted as internationally recognized certificates of compliance and have to be accepted as evidence that GR have been accessed with PIC and that MAT have been established, as required by provider country. Applies to GR Summary of relevant measures for access ............................................................................. Table 2 -Summary of relevant measures for benefit-sharing ............................................................Table 3 -Summary of relevant measures for conservation activities and biodiversity research ....... Table 4 -Summary of relevant measures for the Competent National Authority .............................. Table 5 -Summary of relevant measures for compliance .................................................................. Table 6 -List of indicators ................................................................................................................... Table 7 -Scoring system of the impact grid ....................................................................................... Table 8 -Economic impact of the options for the operationalization of PIC ...................................... Table 9 -Social impact of the options for the operationalization of PIC ............................................ Table 10 -Environmental impact of the options for the operationalization of PIC ............................ Table 11 -Economic impacts of the options for the specification of MAT ......................................... Table 12 -Social impacts of the options for the specification of MAT ............................................... Table 13 -Environmental impacts of the options for the specification of MAT ................................. Table 14 -Economic impacts of the establishment of the CNA .......................................................... Table 15 -Social impacts of the establishment of the CNA ................................................................ Table 16 -Environmental impacts of the establishment of the CNA .................................................. Table 17 -Economic impacts of the compliance measures ................................................................ Table 18 -Social impacts of the compliance measures ....................................................................... Table 19 -Environmental impacts of the compliance measures ........................................................ Table 20 -Economic impacts of the options for designating checkpoint(s) ....................................... Table 21 -Social impacts of the options for designating checkpoint(s) .............................................. Table 22 -Environmental impacts of the options for designating checkpoint(s) ............................... Table 23 -Economic impacts of the options for the ABS CH .............................................................. Table 24 -Social impacts of the options for the ABS CH ..................................................................... Table 25 -Environmental impacts of the options for the ABS CH ...................................................... Figure 1 - 1 Figure 1 -Steps of the MCA ................................................................................................................. Figure 2 -Usual preference function ................................................................................................... Figure 3 -Performance chart of the options for the operationalization of PIC .................................. Figure 4 -Net flows of the alternatives for operationalizing PIC (basic weighting scenario) ............. Figure 5 -Performance chart of the options for the specification of MAT ......................................... Figure 6 -Net flows of the alternatives for specification of MAT (basic weighting scenario) ............ Figure 7 -Performance chart for the establishment of the CNA ........................................................ Figure 8 -Performance chart for the options setting up compliance measures ................................ Figure 9 -Net flows of the alternatives for setting up compliance measures (basic weighting scenario) ............................................................................................................................................................. Figure 11 -Performance chart for the options designating checkpoints ............................................ Figure 12 -Performance chart of the alternatives for the ABS CH ..................................................... Figure 13 -Net flows of the alternatives for the ABS CH (basic weighting scenario) ......................... Measure 1 : 1 operationalizing access to genetic resources 0. Option 0 -No PIC No requirement of Prior Informed Consent for the utilization of genetic resources and traditional knowledge in Belgium; 1. Option 1 -The bottleneck model a. For protected genetic resources: access is made possible through a refinement of existing legislation relevant for protected areas and protected species; Measure 3 : 3 establishing one or more competent national authorities 0. Option 0: No competent national authority/authorities are established in Belgium; 1. Option 1: Competent authorities are established, with a separate entry-point for each authority; 2. Option 2: Competent authorities are established, with a single entry-point. Measure 5 : 5 designating one or more checkpoints 0. Option 0: no checkpoints are established in Belgium to monitor the utilization of genetic resources and traditional knowledge 1. Option 1: monitoring the PIC obtained by users, which is available in the ABS Clearing-House 2. Option 2: the patent authority is used as a checkpoint to monitor the utilization of genetic resources and traditional knowledge Measure 6: sharing information through the ABS Clearing-House 0. Option 0: not creating a Belgian entry point to/component of the Clearing-House 1. Option 1: appointing Royal Belgian Institute of Natural Sciences (RBINS) as Clearing-House 2. Option 2: appointing Belgian Federal Science Policy Office (BELSPO) as Clearing-House 3. Option 3: appointing Scientific Institute for Public Health (ISP/WIV) as Clearing-House  La création d'autorités compétentes nationales (Competent National Authorities, CNA) devrait être accompagnée d'un système d'input centralisé pour les différentes autorités.  En ce qui concerne les mesures de conformité, des sanctions devraient être prévues en cas de non-respect des exigences du PIC et des conditions convenues d'un commun accord (Mutually Agreed Terms, MAT) fixées par le pays fournisseur. Pour la vérification du contenu des MAT, une disposition dans le Code de droit international privé devrait se référer à la législation du pays fournisseur, avec le droit belge comme option de rechange.  A ce stade de la mise en oeuvre, la surveillance de l'utilisation des ressources génétiques et du savoir traditionnel par un point de contrôle devrait se faire sur base du PIC disponible dans le Centre d'échanges pour l'APA (ABS Clearing-House).  En ce qui concerne l'accès aux ressources génétiques belges, il est recommandé d'une part de préciser la législation en vigueur pertinente pour les zones et les espèces protégées, et d'autre part d'instaurer une obligation générale de notification pour l'accès aux autres ressources génétiques. Les étapes ultérieures de la mise en oeuvre pourront alors introduire des dispositions supplémentaires appropriées et prévoir que le traitement d'autres requêtes d'accès se fasse par les collections ex-situ.  A ce stade de la mise en oeuvre, et indépendamment de l'obligation générale de partager les avantages, aucune disposition spécifique de partage d'avantages ne devrait être imposée pour les conditions convenues d'un commun accord (Mutually Agreed Terms, MAT). Un ensemble de règles plus standardisées, y compris la possibilité d'utiliser des accords types, peut être envisagée à un stade ultérieur de l'implémentation.  l'Institut Royal des Sciences Naturelles de Belgique devrait être mandaté pour remplir les tâches de partage d'information via le Centre d'échange pour l'APA (ABS Clearing-House), comme imposées par le Protocole de Nagoya.  Désigner le(s) point(s) de contrôle pour la surveillance de l'utilisation des ressources génétiques : Pour se conformer au Protocole de Nagoya, au moins un point de contrôle doit être désigné, qui surveille et garantit la transparence quant à l'utilisation des ressources génétiques en Belgique. Il peut s'agir d'une institution existante ou d'une nouvelle instance. Mesure 6 : 6 Partage d'information via le Centre d'échange pour l'APA (ABS Clearing-House) 0. Option 0 : Pas de création de point d'entrée /composant belge du Centre d'échange pour l'APA 1. Option 1 : Nommer l'Institut Royal des Sciences Naturelles de Belgique (IRSNB) comme centre d'échange 2. Option 2 : Nommer la Politique Scientifique Fédérale (BELSPO) comme centre d'échange 3. Option 3 : Nommer L'Institut Scientifique de Santé Publique (ISP) comme centre d'échange Finally, Communities are also competent for research, related to the exercise of other Community competences. The following administrations are specifically responsible for research related competences: In the French community, the General Administration for Education and Scientific Research (Fédération Wallonie-Bruxelles).  In the Flemish community: Administration of Higher Education and Research. They are protected in Belgium through different legislative texts, including:  Federal law of 6 th April 2010 on trade practices and consumer protection, chapter 7 on geographical indications and protected designations of origin  Decree of the Walloon Region of 7 th September 1989 related to the local geographical indication and designated Walloon certificate  Ministerial Decree of the Flemish Government of 19 th October 2007 on the protection of geographical indications 61 See A. Lorant, "Le vol de la chose d'autrui", op.cit. 62 Anvers, 13 dec. 1984, Bruxelles, 5 dec. 1986, See for instance Corr. Bruxelles 24 juin 1993 J.L.M.B. 1994, which states that "Un logiciel -ou programme informatiqueindépendamment même de son support (disquette) ne constitue pas un bien immatériel: il possède une valeur économique propre et est susceptible d'un transfert de possession qui peut être constaté matériellement. Le fait que le propriétaire du logiciel reste, en cas de duplication illicite de celui-ci, en possession des données originaires, n'exclut pas l'application des Article 461 et 505 C. pénal ». Several ABS-related actions are also planned in the context of development cooperation. These include awareness-raising and capacity-building actions with ABS stakeholders in developing countries; inter-university cooperation programs on traditional knowledge associated with genetic resources and on conservation of biodiversity; the monitoring on effective biodiversity efforts in the development cooperation; the creation of toolkits to support implementation of biodiversity conventions; and the support of gene banks and ex-situ conservation techniques for genetic resources. In the development cooperation sector the Federal Plan for the integration of biodiversity in four key sectors makes direct links with existing initiatives established or supported by the Belgian authorities. Both the RBINS and the RMCA have established biodiversity-related capacity-building initiatives in developing countries, although they do not directly focus on ABS. In 2003, the RBINS started supporting ILCs in developing countries in their implementation efforts of the CBD, through a convention with the Federal DGD 112 . The first phase of this convention has been running from 2003 to 2007, but has been renewed in 2008 and runs until 2012. In April 2008, the RMCA, together with the Belgian Technical Cooperation (BTC), has launched the Central African Biodiversity Information Network (CABIN). The aim of this project is to establish a network of databases on biodiversity information, in collaboration with several Central African research institutions 113 . Awareness-raising on ABS could easily be added to such programs. Also, the FPS Environment and the DGD have contributed to the creation of the TEMATEA Project that was managed by the United Nations Environment Program (UNEP) and the International Union for Conservation of Nature (IUCN) 114 until 2011. . Generalo The National Competent Authorities and the National Focal Points (Article 13) o Legal conformity: the conformity with the national legislation of the provider country and the contractual rules (Articles 15, 16, 17 and 18)  Access to genetic resources and traditional knowledge (Articles 6, 7 and 8).  Benefit-sharing (Articles 5 and 9)  Compliance and monitoring o Monitoring of the use of genetic resources and the designation of one or several checkpoints (Article 17) o The compliance with the legislations or the requirements of the provider country (Articles 15 and 16) o The compliance with the Mutually Agreed Terms (MAT) (Article 18) 133 Santili J. (2009), Brazil's Experience in Implementing its ABS Regime -Suggestions for Reform and the Relationship with the International Treaty on Plant Genetic Resources for Food and Agriculture. In Kamau E.C. and Winter G. (Eds.) Genetic Resources, Traditional Knowledge & the Law. Solutions for Access & Benefit-sharing. London: Earthscan ; 134 Burton G. (2009), Access and Benefit-sharing: ABS Law and Administration in Australia. Revista Internacional de Direito e Cidadania, n. 5, p. 93-101, October 2009; Burton G. (2009), Australian ABS Law and Administration -A Model Law and Approach? In Kamau E.C. and Winter G. (Eds.) Genetic Resources, Traditional Knowledge & the Law. Solutions for Access & Benefit-sharing. London: Earthscan ; 135 Article 4 of the Biodiversity Law, No 7788, Legislative Assembly of the Republic of Costa Rica, 30 th April 1998 136 Carrizosa S., Brush B.S., Wright B.D., McGuire P.E. (2004), Accessing Biodiversity and Sharing the Benefits: Lessons from Implementing the Convention on Biological Diversity. IUCN Environmental Policy and Law Paper No. 54 137 Article 3(2) of the Biological Diversity Act 2002. No 18 of 2003, Republic of India. 138 Article 9(c) of the Regulations on Bio-Prospecting, Access and Benefit-sharing. Government Gazette No. 30739, 8 th February 2008, Republic of South Africa 139 Suneetha M.S., Pisupati B. (2009), Benefit-sharing in ABS: Options and Elaborations. UNU-IAS Report submission to the CNA; (2) the review of the application; (3) the negotiation of PIC and possibly MAT; (  Option 1: no PIC required by the State but clarification of national legislation regulating legal ownership of genetic material for access to GR as provided for in the NP  Option 2 : PIC required by the State with a change in national legislation Relevant measures for additional implementation Clarify access requirements  Option 1: "One-size-fits-all" requirement (same access procedure for all applicants and situations)  Option 2: Differentiate access requirements depending on type of projected utilization (for example by allowing stakeholders to agree on some MAT/BS conditions at later stage than moment of access)  Option 3: Differentiate access requirements depending on type of actors (for example foreign / national) Establish clear and transparent access procedure  Option 1: Enshrine procedure in legal act  Option 2: Develop administrative guidance  Option 3: Provide assistance procedure to facilitate transaction between applicant and private stakeholder 140 Young TR (2009), Legal Certainty for Users of Genetic Resources under Existing Access and Benefit-sharing (ABS) Legislation and Policy. In Young T (Ed.) Covering ABS: Addressing the Need for Sectoral, Geographical, Legal and International Integration in the ABS Regime. IUCN Environmental Policy and Law Paper No. 67/5 141 Medaglia JC (2009) The Role of the National Biodiversity Institute in the Use of Biodiversity for Sustainable Development -Forming Bioprospecting Partnerships. In Kamau E.C. and Winter G. (Eds.) Genetic Resources, Traditional Knowledge & the Law. Solutions for Access & Benefit-sharing. London: Earthscan Table 2 - 2 Summary of relevant measures for benefit-sharing Relevant measures for the minimal implementation of core obligations Determine format of MAT  Option 1: Leave full discretion on how to execute the BS obligation to users and provider of genetic material  Option 2: Develop mandatory MAT terms and conditions and/or default MAT provisions  Option 3: Impose standard MAT(s) Relevant measures for additional implementation Clarify benefit-sharing requirements  Option 1: "One-size-fits-all" requirements  Option 2: Differentiate BS requirements depending on type of projected utilization  Option 3: Differentiate BS requirements depending on type of actors  Option 4: Utilize the actual trigger of MAT/BS, instead of access  Option 5: Specify types of benefits to be shared Ensure benefit-sharing is fair and equitable 149 Article 21(2) of the Biological Diversity Act 2002. No 18 of 2003, Republic of India. 150 Ruiz M., Lapeña I., Clark S.E. (2004), The Protection of Traditional Knowledge in Peru: A Comparative Perspective. Washington University Global Studies Law Review, 3(3): 755-97 151 Carrizosa et al. (2004), op. cit. 152 Wynberg R, Taylor M (2009), op. cit. 153 Article 2(g) of the Biological Diversity Act 2002. No 18 of 2003, Republic of India. Summary of relevant measures for conservation activities and biodiversity research Relevant measures for the minimal implementation of core obligations Ensure ABS serves conservation activities/sustainable use (Article 9)  Option 1: Link access permit to mandatory conditions that direct benefits towards conservation activities/sustainable use  Option 2: Require environmental impact assessment prior to access  Option 3: Establish a "benefit-sharing" fund or other mechanism which redirects the benefits Facilitate access for biodiversity-related research (Article 8a)  Option 1: Exempt (non-commercial) biodiversity-related research from any access requirement  Option 2: Facilitate access for biodiversity-related research Relevant measures for additional implementation / 159 Carrizosa et al. (2004), op. cit. 160 Article 15.2(g) of the Conservation of Biological Diversity and Resources, Access to Genetic Resources and Benefitsharing Regulations 2006 of the Environmental Management and Co-ordination Act, Republic of Kenya, 2006 Belgium In Flanders, Article 57bis of Natuurdecreet allows access to real property for research conducted by public servants and related to nature conservation  Besluit van de Vlaamse Regering betreffende de toegankelijkheid van de bossen en de natuurreservaten, 05/12/2008 Option 1 -'One-size-fits-all' requirements for benefit-sharing Possible advantages:  Easy to implement  High legal certainty Possible disadvantages:  Might be inefficient  Might be too constraining and inflexible for certain types of users Option 2 -Differentiate benefit-sharing requirements depending on type of projected utilization, at the moment of access Possible advantages:  Possibility to facilitate access for non-commercial/low-profit research, under the condition of clearly specifying additional conditions in the case of change in intent (from non-commercial to commercial) Possible disadvantages:  Might be difficult to establish efficient and effective conditions on down-stream use at time of access Option 3 -Differentiate benefit-sharing requirements depending on type of actors, at the time of access Possible advantages:  Possibility to foster domestic research  Could facilitate tracing of accessed GR Possible disadvantages:  Might foster reluctance of foreign prospectors  Might conflict with EU-rules, WTO MFN and national treatment  Might not advance the objectives of the CBD and NP  Might create loop holes in the benefit-sharing obligations, distinction domestic / non-domestic difficult to monitor Option 4 -Utilize the actual trigger for establishing MAT/BS conditions, instead of access Possible advantages:  Allows to settle BS agreement based on clearer view of potential value of GR  Lower administrative burden for users at time of access Possible disadvantages:  Requires efficient monitoring of utilization  Requires return clause to make sure users do come back when entering at different/certain phases of utilization (e.g. commercialization phase) Option 5 -Specify types of benefits to be shared Possible advantages:  Could include the (re)direction of (part of) the benefits towards conservation/sustainable use Possible disadvantages:   Higher legislative cost  Will require more time to set up Option 2 -Develop administrative regulations, guidance for access procedure Possible advantages:  Easily modifiable in case of changing circumstances  Could be quickly operational Possible disadvantages:  Would still need a legal basis, containing the essential elements of the procedure and the rights and obligations of individuals  Possibility of less legal certainty EVALUATION A combination of option 1 (legal basis with the essential elements of the procedure and setting out the rights and obligations of individuals) Could create a more efficient process and follow-up  Less administrative burden for users  More process certainty for the user Possible disadvantages:  Could be difficult to implement, given division of ABS-related Requires a strong commitment and understanding of ABS by private users  Requires close collaboration between the monitoring authority and these users  Could be constraining for non-commercial research Option 2 -'Due-diligence' monitoring system Possible advantages:  Relevant when GR is being transferred to third parties during the valorization process  Could have a lower legislative and administrative cost for authorities  Could be more flexible for users  Could build in a different type and level of standards according to users/use of GR  Could build in subsidiarity and responsibility for sectors Option 1: The bottleneck model: only existing PS/PA relevant legislation & measures + only access to GR through ex-situ collections as default rule (a) + (c) This option combines the refinement of existing PA and PS relevant legislation with the default rule for GR which are not in a protected area or which are not protected species that only Belgian collections can provide access to GR.  Option 2: The fishing net model: only existing PA/PS relevant legislation & measures + access to GR from everywhere but with registration as default rule (a) + (d) This option combines the refinement of existing PA and PS relevant legislation, with the default rule that GR can be accessed from anywhere, providing the user has registered/notified the CNA.  Option3: potentially enlarged existing PA/PS relevant legislation & measures + other specific GR relevant legislation/measures + access to GR from everywhere but with registration as default rule (b) + (d) Option 1: Royal Belgian Institute of Natural Sciences (RBINS) as ABS Clearing-House  Option 2: Belgian Federal Science Policy Office (Belspo) as ABS Clearing-House  Option 3: Scientific Institute for Public Health (WIV-ISP) as ABS Clearing-House 1) in the patent applications. The implementation of the monitoring obligations under option 2 would require an amendment of the federal law transposing the Directive 98/44/EC on the legal protection of biotechnological inventions to include such a new provision. It would also imply a change in the tasks of option : not creating a Belgian entry-point to/component of the clearing-house 9. Option 1: Royal Belgian Institute of Natural Sciences as ABS Clearing-House (RBINS) 10. Option 2: Belgian Federal Science Policy Office (BELSPO) as ABS Clearing-House 11. Option 3: Scientific Institute for Public Health (ISP/WIV) as ABS Clearing-House Figure 1 - 1 Figure 1 -Steps of the MCA 197 Belton V., Stewart T.J. (2002), Multiple criteria decision analysis: an integrated approach. Kluwer Academic Publishers; Podvezko V.,Podviezko A. (2010), Use and choice of preference functions for evaluation of characteristics of socioeconomical processes. 6th International Scientific Conference, 13th-14th May 2010, Vilnius, Lithuania; Brans J.P., Marechal B. (2002), Prométhée-Gaia. Une méthodologie d'aide à la décision en présence de critères multiples. Editions de l'Universite Libre de Bruxelles..198 The PROMETHEE-GAIA FAQ "How to choose the right preference function?"; http://www.promethee-gaia.net/faqpro/?action=Article&cat_id=003002&id=4&lang=; Another preference function can however be applied easily if needed. Figure 3 - 3 Figure 3 -Performance chart of the options for the operationalization of PIC Figure 4 - 4 Figure 4 -Net flows of the alternatives for operationalizing PIC (basic weighting scenario) certainty and effectiveness for users and providers of GR, at low cost Figure 5 - 5 Figure 5 -Performance chart of the options for the specification of MAT Figure 6 - 6 Figure 6 -Net flows of the alternatives for specification of MAT (basic weighting scenario) Figure 7 - 7 Figure 7 -Performance chart for the establishment of the CNA Figure 8 - 8 Figure 8 -Performance chart for the options setting up compliance measures Figure 9 - 9 Figure 9 -Net flows of the alternatives for setting up compliance measures (basic weighting scenario) Figure 10 - 10 Figure 10 -Performance chart for the options designating checkpoints  Impact on stakeholders: o Coll.: indirect benefit from increased legal certainty and effectiveness o Gov. Res.: indirect benefit from increased legal certainty and effectiveness o Ag., health and biotech: indirect benefit from increased legal certainty and effectiveness o Univ.: indirect benefit from increased legal certainty and effectiveness o Land: indirect benefit from increased legal certainty and effectiveness o Other: No impact E2 -Maximizing economic innovation and product development (in particular through its contribution to R&D) at reasonable financial and administrative costs  Impact on stakeholders: o Coll.: indirect benefit from increased legal certainty and effectiveness o Gov. Res.: indirect benefit from increased legal certainty and effectiveness o Ag., health and biotech: indirect benefit from increased legal certainty and effectiveness o Univ.: indirect benefit from increased legal certainty and effectiveness o Land: indirect benefit from increased legal certainty and effectiveness o Other: No impact  Impact on stakeholders: o Coll.: No impact o Gov. Res.: No impact o Ag., health and biotech: No impact o Univ.: No impact o Land: No impact o Other: No impact Figure 11 - 11 Figure 11 -Performance chart of the alternatives for the ABS CH Figure 12 - 12 Figure 12 -Net flows of the alternatives for the ABS CH (basic weighting scenario) 1. A first set of actions, which form the basis of compliance with the NP and address the core obligations for the implementation of the NP in Belgium, including :  The establishment of National Competent Authorities and the National Focal Points (Article 13)  Conformity with the national legislation of the provider country and the contractual rules (Articles 15,16,17 and 18)  Access to genetic resources and traditional knowledge (Articles 6, 7 and 8).  Benefit-sharing (Articles 5 and 9)  Monitoring of the use of genetic resources and the designation of one or several checkpoints (Article 17)  Compliance with the legislations or the requirements of the provider country (Articles 15 and 16)  The compliance with the Mutually Agreed Terms (MAT) (Article 18) 2. A second set of additional measures which are important elements during implementation of the obligations, but that are less urgent (going beyond the core obligations). - take measures, as appropriate, with the aim of ensuring that TK is accessed with PIC and MAT of the ILC holding TK Applies to TK Create conditions to promote and encourage biodiversity research, including simplified measures on access for non-commercial research -Pay due regard to cases of present and imminent emergencies that threaten or damage human, animal or plant health -Encourage users and providers to direct benefits towards conservation of biological diversity and sustainable use of its components Applies to GR for and modalities of a global multilateral benefit-sharing mechanism for 1) GR and TK that occur in transboundary situations or 2) for which it is not possible to grant or obtain PIC Applies to GR+TK Article 11 a. subject Each Party b. obligation Endeavour to cooperate in instances: -where the same GR are found in situ within the territory of more than one Party a. subject Parties b. obligation As far as possible cooperate in cases of alleged violation of provider country legislation Applies or policy measures to provide that TK utilized within jurisdiction has been accessed in accordance with PIC and MAT with legislation of country where ILCs are Adoption of measures to address situations of non-compliance with Article 16.Adoption of measures to monitor and enhance transparency about the utilization of GRs, which shall include a) the adoption of one or more checkpoints, b) encouraging the inclusion of provision on the sharing of information on the implementation in MAT, c) encourage the use of cost-effective communication tools and systems Applies ..... INBO Instituut voor natuur -en bosonderzoek IPEN International Plant Exchange Network IPR Intellectual Property Rights ITPGRFA EXECUTIVE SUMMARY International Treaty on Plant Genetic Resources for Food and Agriculture ICE Interministerial Conference on Environment IUCN International Union for Conservation of Nature LNE MAT General recommendations Department Leefmilieu, Natuur en Energie of the Flemish government Mutually Agreed Terms MOP MOSAICC  Both Prior Informed Consent and benefit-sharing should be implemented as general legal principles Meeting of the Parties Micro-organisms Sustainable Use and Access Regulation International Code of Conduct in Belgium. MS Member State MTA Material Transfer Agreement NBGB National Botanic Garden of Belgium NFP National Focal Point NP Nagoya Protocol OECD Organization for Economic Co-operation and Development PDO Protected Designation of Origin PGI Protected Geographical Indication PIC Prior Informed Consent PROMETHEE Preference Ranking Organization Method for Enrichment of Evaluations R&D Research and Development RBINS Royal Belgian Institute of Natural Sciences REIO Regional Economic Integration Organization RMCA Royal Museum for Central Africa SL Special Law SMTA Standard Material Transfer Agreement TFEU Treaty on the Functioning of the European Union TK Traditional Knowledge TKaGR Traditional Knowledge associated with genetic resources TRIPS Trade-Related Aspects of Intellectual Property Rights TSG Traditional Speciality Guaranteed UN United Nations UNCED United Nations Conference on Environment and Development UNCLOS United Nations Convention on the Law of the Sea UNEP United Nations Environment Program VAIS Vlaams Agentschap voor Internationale Samenwerking WBI Wallonie-Bruxelles International WHO World Health Organization WIPO World Intellectual Property Organization WTO World Trade Organization WSSD World Summit on Sustainable Development préliminaires relatives aux options pour la mise en oeuvre du Protocole de Nagoya Toutefois, certaines préoccupations en matière de connaissances traditionnelles et des droits des communautés indigènes et locales ont été traitées dans certains instruments internationaux auxquels la Belgique est partie, telle que la Convention N° 107 de l'Organisation Internationale du Travail (OIT) relative aux populations aborigènes et tribales, la Convention N° 169 de l'OIT relative aux peuples indigènes et tribaux, et la Déclaration des Nations Unies sur les droits des peuples autochtones. Par ses droits souverains sur les ressources génétiques, la Belgique peut choisir si elle exige, ou non, que les utilisateurs obtiennent un consentement préalable donné en connaissance de cause (Prior Informed Consent, PIC) par l'autorité compétente pour accéder aux ressources génétiques dans sa juridiction. Recommandations Même si le Protocole of Nagoya est récent, il n'en est pas moins l'application du troisième objectif de la CDB, qui contient des principes de base et des dispositions apparentées à l'APA, tels que la souveraineté des Etats sur leurs richesses et ressources naturelles, le partage juste et équitable des avantages, et l'importance des communautés locales, des populations autochtones et de leurs connaissances traditionnelles. Beaucoup de Parties à la Convention à travers le monde ont donc mis en oeuvre une série des mesures sur l'APA, qui peuvent servir d'expériences utiles pour l'exécution du Protocole de Nagoya. A l'analyse de ces expériences, deux groupes de recommandations préliminaires ont pu être établies dans cette étude, quant aux options disponibles pour la mise en oeuvre du Protocole de Nagoya en Belgique. Le premier groupe de recommandations concerne les instruments nécessaires pour l'exécution des obligations fondamentales résultant du Protocole 5 . Le second groupe de recommandations concerne des mesures supplémentaires à prendre en compte au cours de la mise en oeuvre des obligations du Protocole, mais qui vont au-delà des obligations fondamentales. Recommandations relatives aux obligations fondamentales:  Clarifier les conditions d'accès :  Déterminer le format des conditions convenues d'un commun accord (Mutually Agreed Terms, MAT) : Une fois que le Protocole de Nagoya entre en vigueur, les utilisateurs oeuvrant sur le territoire belge auront l'obligation de partager les avantages provenant de l'utilisation des ressources génétiques. Un tel partage sera basé des conditions convenues d'un commun accord (Mutually Agreed Terms, MAT). Cependant, le Protocole de Nagoya n'impose pas un format spécifique pour ces MAT qui peuvent être laissés à l'appréciation des parties prenantes ou découler des lignes de directrices et/ou de mesures obligatoires imposées par l'Etat.  Assurer que l' APA contribue à la conservation et l'utilisation durable de la biodiversité: La mise en oeuvre du Protocole de Nagoya devra servir les deux autres objectifs de la CDB: la conservation de la biodiversité et utilisation durable de ses composants. Cela peut être réalisé par exemple en soumettant l'obtention du PIC à des conditions obligatoires sur le partage des avantages ou en instaurant un «fonds de partage des avantages » qui redirige les avantages vers la conservation et l'usage durable.  Faciliter l'accès pour la recherche relative à la biodiversité : pour soutenir et promouvoir la recherche relative à la biodiversité et pour réduire la charge de la réglementation pour recherche non commerciale qui utilise des ressources génétiques, des mesures pourraient être mise en place pour faciliter l'accès aux ressources génétiques pour de la recherche non commerciale liée à la biodiversité.  Instaurer des autorités compétentes nationales (Competent National Authorities, CNA): Chaque partie doit désigner une autorité ou des autorités nationales compétentes qui sont chargées d'accorder l'accès ou, s'il y a lieu, de délivrer une preuve écrite que les conditions d'accès ont été respectées, et de fournir des conseils sur les procédures et les conditions en vigueur pour accéder aux ressources génétiques. Etant donné la réalité institutionnelle en Belgique, plus d'une autorité nationale compétente peut être instaurée. Cette tâche est de la plus haute priorité, puisque la Belgique doit communiquer au Secrétariat de la Convention, au plus tard à la date d'entrée en vigueur du Protocole pour elle, les coordonnées de son correspondant national et de son autorité ou ses autorités nationales compétentes.  Accorder force contraignante à la législation des pays fournisseurs concernant le PIC et les MAT MAT) ou aux consentements préalable donné en connaissance de cause (PIC) du pays fournisseur. Cela pourrait être fait en imposant, dans la législation belge, le respect de la législation du pays fournisseur en ce qui concerne le PIC et les MAT ou en instaurant une règle de police interne dans la législation belge imposant l'obtention d'un consentement préalable et la conclusion de dispositions communes convenues d'un commun accord, si requis par le pays fournisseur. : Parties intégrantes du Protocole, les obligations fondamentales auxquelles les utilisateurs nationaux doivent se conformer lorsqu'ils utilisent des ressources génétiques en Belgique, doivent être énoncées clairement. Cette obligation consiste à accorder force contraignante aux dispositions convenues d'un commun accord ( Mesure 1 : opérationnalisation de l'accès aux ressources génétiques Si le partage des avantages est bien inscrit comme principe général, les dispositions spécifiques de partage des avantages à inclure dans les conditions convenues d'un commun accord (Mutually Agreed Terms, MAT), doivent être spécifiées (mesure2). Ces dispositions spécifiques peuvent être laissées à l'appréciation des utilisateurs (option1), ou être imposées par l'Etat avec plus ou moins de standardisation (options 2 et 3). Pas de dispositions spécifiques imposées par les autorités compétentes pour les conditions convenues d'un commun accord. Les utilisateurs et les fournisseurs sont libres de décider conjointement de leur contenu. 2. Option 2: Imposition de dispositions spécifiques, y compris par des formats standardisés, pour les conditions convenues d'un commun accord pour certains usage, qui seront différenciés selon la finalité de l'accès. Mesure 2 : Spécifier les dispositions pour les conditions convenues d'un commun accord (Mutually Agreed Terms, MAT) 0. Option 0: Pas de partage d'avantage pour l'utilisation des ressources génétiques et du savoir traditionnel en Belgique. 1. Option 1: b. Pour les ressources génétiques non protégées : l'accès est autorisé via les collections ex-situ 2. Option 2 -Modèle « Fishing Net » a. Pour les ressources génétiques protégées : accès possible en affinant la législation existante pertinente pour les zones et les espèces protégées. b. Pour les ressources génétiques non protégées : accès accordé sur notification préalable auprès de l'autorité compétente 3. Option 3 -Modèle « Fishing Net » modifié a. Pour les ressources génétiques protégées et les ressources génétiques déjà régies par une législation pertinente existante: accès possible en affinant la législation existante. 0. Option 0 -Pas de consentement préalable Pas d'exigence de consentement préalable en connaissance de cause (Prior Informed Consent, PIC) pour l'utilisation des ressources génétiques et du savoir traditionnel en Belgique; 1. Option 1 -Modèle « Bottleneck » a. Pour les ressources génétiques protégées : accès possible en affinant la législation existante pertinente pour les zones et les espèces protégées. b. Pour les ressources génétiques non protégées : accès accordé sur notification préalable auprès de l'autorité compétente 3. Option 3: Imposition de dispositions spécifiques mais sans formats standardisés pour les conditions convenues de commun accord. Tout en prenant en compte les conditions exigées de partage d'avantages, les MAT sont définies au cas par cas par les utilisateurs et les fournisseurs. Les dispositions sont différenciées selon la finalité de l'accès. Pour respecter le Protocole de Nagoya, une ou plusieurs autorités nationales devront être établies (mesure 3). Leur tâche sera d'accorder l'accès ou, s'il y a lieu, de délivrer une preuve écrite que les conditions d'accès ont été respectées, et de fournir des conseils sur les procédures et les conditions en vigueur pour accéder aux ressources génétiques. Pour remplir ces tâches les autorités nationales compétentes devront également établir un point d'entrée pour les utilisateurs de ressources génétiques. Cela peut être fait séparément, chaque autorité instaurant son propre point d'entrée (option 1), ou conjointement, un seul point entrée pour les différentes autorités (option 2). Mesure 3: instaurer une ou plusieurs autorités compétentes rechange (option1), ou en établissant une règle de police interne en droit belge (option 2). Dans cette deuxième option, la législation belge se référerait uniquement aux obligations spécifiques de PIC et de MAT, comme fixées par le pays fournisseur, sans se référer à la législation en vigueur dans le pays fournisseur. Option 1 : Une disposition pénale générale est créée qui se réfère à la législation du pays fournisseur concernant le PIC et les MAT. L'Etat édicte une interdiction générale d'utiliser les ressources génétiques et le savoir traditionnel obtenus en violation de la loi du pays fournisseur. Le contrôle du contenu des MAT par un juge se fait sur base de la législation du pays fournisseur, avec le droit belge comme option de rechange 2. Option 2 : Une disposition est créée instaurant l'obligation d'avoir obtenu un PIC et des MAT Mesure 4 : instaurer des mesures de mise en conformité 0. Option 0 : Pas d'introduction de dispositions légales sur la conformité dans le droit belge. 1. 0. Option 0 : pas d'autorité nationale compétente en Belgique 1. Option 1 : Instauration d'autorités nationales compétentes, avec un point d'entrée séparé pour chaque autorité. 2. Option 2 : les autorités nationales compétentes sont instaurées, avec un point d'entrée commun. Une fois le Protocole de Nagoya entré en vigueur en Belgique, il sera indispensable de mettre en place des mesures de mise en conformité pour s'assurer que les ressources génétiques et le savoir traditionnel utilisés sur le territoire belge ont bien été acquis en accord avec le droit du pays fournisseur (mesure 4). Cela peut être réalisé en se référant à la législation du pays fournisseur concerné et en contrôlant le contenu des MAT sur base de cette même législation, avec le droit belge comme option de de la part du pays fournisseur pour l'utilisation en Belgique de ressources génétiques étrangères, s'ils sont requis par la législation du pays fournisseur (d'origine). Pour se conformer au Protocole de Nagoya, au moins un point de contrôle doit être créé pour surveiller l'utilisation des ressources génétiques et du savoir traditionnel (mesure 5). Si la Belgique décide d'introduire des points de contrôle, leur mise en oeuvre pourrait être réalisée en plusieurs étapes. Pour respecter l'engagement politique d'une ratification rapide du Protocole, une première étape pourrait consister en une implémentation minimale requérant la création d'un point de contrôle unique. Deux options possibles semblent pertinentes pour cette première étape, à savoir le contrôle du consentement préalable en connaissance de cause (PIC) des utilisateurs, lequel est disponible via le Centre d'échange pour l'APA (ABS Clearing-House) (option 1) et/ou le renforcement de l'obligation de mention de l'origine géographique de la matière biologique dans les brevets d'invention (option2). Comme les options 1 et 2 ne s'excluent pas mutuellement, une mise oeuvre combinée pourrait être envisagée. Mesure 5 : Désigner un ou plusieurs points de contrôle Même si les discussions sur les modalités exactes du Centre d'échange pour l'APA sont encore en cours au niveau international, trois candidats possibles ont été identifiés : l'Institut Royal de Sciences Naturelles de Belgique (option1), la politique scientifique fédérale (BELSPO) (option 2), et l'Institut Scientifique de Santé Publique (ISP) (option3). 0. Option 0: pas d'instauration de point de contrôle pour surveiller l'utilisation de ressources génétiques et du savoir traditionnel. 1. Option 1: contrôler le consentement préalable en connaissance de cause (PIC) de l'utilisateur, lequel est disponible via le Centre d'Echange APA (ABS Clearing-House). 2. Option 2: L'autorité des brevets est sollicitée comme point de contrôle pour surveiller l'utilisation des ressources génétiques et du savoir traditionnel. Enfin, un composant ou un point d'entrée belge au Centre d'échange pour l'APA (ABS Clearing-House) sera créé pour soutenir l'échange d'information sur les mesures spécifiques d'accès et de partage des avantages dans le cadre du Protocole de Nagoya (mesure 6). 1. La création d'autorités compétentes nationales (Competent National Authorities) devrait être accompagnée d'un système d'input centralisé pour les différentes autorités. 2. En ce qui concerne les mesures de conformité, des sanctions devraient être prévues en cas de non-respect des exigences du PIC et des conditions convenues d'un commun accord (MAT) fixées par le pays fournisseur. Pour la vérification du contenu des MAT, une disposition dans le Code de droit international privé devrait se référer à la législation du pays fournisseur, avec le droit belge comme option de rechange. 3. A ce stade de la mise en oeuvre, la surveillance de l'utilisation des ressources génétiques et du savoir traditionnel par un point de contrôle devrait se faire sur base du PIC disponible dans le Centre d'échanges pour l'APA (ABS Clearing-House). appropriées et prévoir que le traitement d'autres requêtes d'accès se fasse par les collections ex-situ. 5. A ce stade de la mise en oeuvre, et indépendamment de l'obligation générale de partager les avantages, aucune disposition spécifique de partage d'avantages ne devrait être imposée pour les conditions convenues d'un commun accord (MAT). Un ensemble de règles plus standardisées, y compris la possibilité d'utiliser des accords types, peut être envisagée à un stade ultérieur de l'implémentation. 6. l'Institut Royal des Sciences Naturelles de Belgique devrait être mandaté pour remplir les tâches de partage d'information via le Centre d'échange pour l'APA (ABS Clearing-House), comme imposées par le Protocole de Nagoya. 4. En ce qui concerne l'accès aux ressources génétiques belges, il est recommandé d'une part de préciser la législation en vigueur pertinente pour les zones et les espèces protégées, et d'autre part d'instaurer une obligation générale de notification pour l'accès aux autres ressources génétiques. Les étapes ultérieures de la mise en oeuvre pourront alors introduire des dispositions supplémentaires La raison pour laquelle un tel accord politique est recommandé est double. D'une part, il offre un engagement politique clair quant aux obligations fondamentales du Protocole de Nagoya. En effet, il spécifie les intentions des autorités compétentes, dans la limite des décisions déjà prises aux niveaux européen et international au moment de l'accord. les actions spécifiées devront être mises en oeuvre, par exemple à l'aide d'un accord de coopération et/ou en ajoutant des dispositions dans la législation pertinente, comme les Codes de l'environnement des entités fédérées et de l'Etat fédéral, en plus d'autres conditions éventuelles. 1. Dans la première étape, un accord politique devrait être décidé entre les autorités compétentes, comprenant une déclaration claire quant aux principes juridiques généraux à mettre en place, en plus de certaines spécifications sur les actions à entreprendre par l'Etat fédéral et les entités fédérées pour mettre ces principes en application. Cet accord devrait inclure: a. Instauration du partage d'avantages comme principe juridique général en Belgique. b. Instauration d'un principe juridique général selon lequel l'accès aux ressources génétiques belges requiert un Consentement informé préalable (PIC). c. Instauration d'un principe juridique général concernant la création de quatre Autorités Nationales Compétentes. d. Engagement que des mesures législatives seront prises afin de s'assurer que les ressources génétiques utilisées sous la juridiction belge, ont été acquises moyennant un PIC et des MAT, comme fixé par la législation du pays fournisseur, et de répondre aux situations de non-respect. e. Désignation du Centre d'échange d'informations belge de la CDB (Clearing-House Mechanism), géré par l'Institut Royal des Sciences Naturelles, comme Centre d'échange pour l'APA, traitant les échanges d'information sur l'accès et le partages des avantages au titre du Protocole de Nagoya. D'autre part, il ne préjuge pas des décisions politiques qui seront prises par les différentes autorités et offre ainsi une flexibilité suffisante pour ajuster le processus de mise en oeuvre à un stade ultérieur. Ce dernier point est particulièrement important étant donné les nombreuse questions encore en suspens au stade actuel, tant au niveau européen qu'au niveau international, comme indiqué et pris en compte dans ce rapport. 2. Dans une seconde étape, 3. Dans une troisième étape, des actions supplémentaires peuvent être entreprises, une fois qu'il y a plus de clarté aux niveaux européen et international. Deze optie Maatregel 6: informatie uitwisselen via het ABS Clearing-House scoort het best voor alle criteria; strikt genomen scoort ze ook beter op het vlak van rechtszekerheid en efficiëntie voor de gebruikers en aanbieders van genetische rijkdommen, met minder kosten. 4. Optie 0: Er wordt geen Belgisch component van of aanspreekpunt voor het uitwisselingscentrum voorzien. 5. Optie 1: Het Koninklijk Belgisch Instituut voor Natuurwetenschappen (KBIN) wordt aangesteld tot uitwisselingscentrum 6. Optie 2: Het Federaal Wetenschapsbeleid (BELSPO) wordt aangesteld tot uitwisselingscentrum 7. Optie 3: Het Wetenschappelijk Instituut Volksgezondheid (WIV) wordt aangesteld tot uitwisselingscentrum States are sovereign over their natural wealth and resources For a more complete historical account of the latest months prior to the adoption of the NP see Chiarolla C. (2010), Making Sense of the Draft Protocol on Access and Benefit-sharing for COP 10. Idées pour le débat, Institut du Développement Durable et des Relations Internationales (IDDRI) 11 See Decision VI/24. Available at: http://www.cbd.int/decision/cop/?id=7198 12 World Summit on Sustainable Development. Plan of Implementation. Available at: http://www.johannesburgsummit.org/html/documents/summit_docs/2309_planfinal.htm (accessed 26th March 2013) 13 Paragraph 44(o) of the Johannesburg Plan of Implementation. 14 The FPS Economy also hosts the National Institute of Statistics, which is in charge of compiling the Belgian data on biodiversity, as well as the Belgian Office for Intellectual Property (DIE/OPRI). The DIE/OPRI manages the attribution of industrial property titles, informs users regarding intellectual property, advises Belgian governments and represents Belgium at the WIPO. The Office is advised by field-experts and specialists gathered in thematic councils. Relevant councils include the Council for Plant Variety Rights and the Council for Intellectual Property. 79 Besluit van de Vlaamse Regering betreffende de toegankelijkheid van de bossen en de natuurreservaten, 05/12/2008 80 Arrêté ministériel du 23 octobre 1975 établissant le règlement relatif à la surveillance, la police et la circulation dans les réserves naturelles domaniales, en dehors des chemins ouverts à la circulation publique(M.B., 31 décembre 1975) 1.1.2 81 8 juin 1989 -Arrêté de l'Exécutif régional wallon relatif à la protection des zones humides d'intérêt biologique (M.B. 12.09.1989) 1.1.2.1 82 Arrêté du Gouvernement wallon du 26 janvier 1995 organisant la protection des cavités souterraines d'intérêt scientifique, (M.B., 18 mars 1995) Regional Natural Reserves, Certified Natural Reserves, Forest Reserves, Natura 2000 Reserves. 90 Law of 20 th January 1999 aiming to protect the marine environment falling under the jurisdiction of Belgium, M.B. 12 th March 1999 ; Law of 22 nd April 1999 M.B., 20 July 1999 91 Law of 20 th January 1999 aiming to protect the marine environment falling under the jurisdiction of Belgium, M.B. 12 th March 1999 Traditional Knowledge and the Convention on Biological Diversity, available at http://www.cbd.int/doc/publications/8j-brochure-en.pdf103 Overeenkomst tot wijziging van de partnerschapsovereenkomst tussen de leden van de groep van Afrika, het Caribisch gebied en de stille oceaan, enerzijds, en de Europese gemeenschap en haar lidstaten, anderzijds, ondertekend te Cotonou op 23 juni 2000, BS: 30-04-2008; Overeenkomst inzake politieke dialoog en samenwerking tussen de Europese Gemeenschap en haar Lidstaten, enerzijds, en de Andesgemeenschap en haar Lidstaten (Bolivia, Colombia, Ecuador, Peru en Venezuela), anderzijds, en met de Bijlage, gedaan te Rome op 15 december 2003, BS: 03-06-2008; Internationaal Verdrag inzake plantgenetische hulpbronnen voor voeding en landbouw, gedaan te Rome op 6 juni2002, BS: 21-12-2007; Populations;  the ILO Convention No. 169 on Indigenous and Tribal Peoples; and 102 CBD Secretariat, the 1957 International Labor Organization (ILO) Convention No. 107 on Indigenous and Tribal No. 107 is a broad development instrument, covering a wide range of issues such as land; recruitment and conditions of employment; vocational training, handicrafts and rural industries; social security and health; and education and means of communication. Particularly the provisions of Convention No. 107 with regard to land, territories and resources have a wide coverage and are similar to those of Convention No. 169. Convention No. 107 was ratified by 27 countries. It was revised during 1988-1989, through the adoption of Convention No. 169. Although since the adoption of Convention No. 169, Convention No.107 is no longer open for ratification, it is still in force for 28 States, including Belgium, a number of which have significant populations of indigenous peoples, and remains a useful instrument in these cases as it covers many areas that are key for indigenous and local communities. 107 CCIEP (2006) Belgium's National Biodiversity Strategy 2006-2016. Belgian Coordination Committee for International Environment Policy, Directorate-General for the Environment. The process of drafting the National Biodiversity Strategy was initiated by the Interministerial Conference for the Environment in June 2000. The Strategy was elaborated by a team representing the major actors in the field of biodiversity in Belgium. It acted as a contact group under the "Biodiversity Convention" Steering Committee. This Steering Committee was established under the Belgian Coordination Committee for International Environment Policy (CCIEP) under the auspices of the Interministerial Conference for the Environment, which endorsed the strategy the 26 th October 2006. 108 C. Frison and T. Dedeurwaerdere (2006) Infrastructures publiques et régulations sur l'accès aux ressources génétiques et le partage des avantages qui découlent de leur utilisation pour l'innovation de la recherche des sciences de la vi.e. Accès, conservation et utilisation de la diversité biologique dans l'intérêt général. Enquête Fédérale Belge. Centre de Philosophie du Droit, Université Catholique de Louvain. 109 Ibid. is another similar Belgian initiative, taken by the Association of Botanical Gardens and Arboreta. It has developed a navigation system for sharing plant information from different databases in a common format. It is also worth noting that a Belgian Biodiversity Platform 117 was created by BELSPO in 2003, in the context of the Second Multi-annual Scientific Support Plan for a Sustainable Development Policy. The Platform functions as an interface between providers and users of biodiversity information. Other proposed ABS-related actions in this field closely relate to those in the development cooperation field, including capacity-building initiatives in Central Africa and the promotion of ex-situ conservation.In accordance with COP Decision V/26 of the CBD, a civil servant of the DG Environment of the FPS Environment currently ensures the function of national focal point on ABS. At federal level, a "long term strategic vision for sustainable development to 2050" is currently under development. ABS concerns should be included. . The proposal was discussed during the first Environment Council of the European Union under the Irish Presidency, on 21 st March 2013 123 , as well as during a workshop on Access to Genetic Resources and Fair and Equitable Sharing of Benefits held on the 19 th March 2012 in the European Parliament. Negotiations on the Regulation are still ongoing in the Council's Working Party on Environment. The European Parliament committee vote is scheduled for July 2013. Preceding the impact assessments, from October 2011 to December 2011, the European Commission also held a public consultation on the implementation and ratification of the Nagoya Protocol, with the aim of exploring the possible effects of the Protocol and to gather concrete proposals on the practical challenges of the implementation. Results of this public consultation are publicly available Proposal for a Regulation of the European Parliament and of the Council on Access to Genetic Resources and the Fair and Equitable Sharing of Benefits Arising from their Utilization in the Union. COM(2012) 576 final 122 EC (2012), Impact Assessment accompanying the document "Proposal for a Regulation of the European Parliament and of the Council on Access to Genetic Resources and the Fair and Equitable Sharing of Benefits Arising from their Utilization in the Union, European Commission Staff Working Document, COM(2012) 576 final; IEEP, Ecologic and GHK (2012), Study to analyze legal and economic aspects of implementing the Nagoya Protocol on ABS in the European Union, Final report for the European Commission, DG Environment. Institute for European Environmental Policy, Brussels and London, April 2012 123 Council of the European Union (2013). Press Release of the 3233rd Council meeting Environment. Brussels, 21 st March 2013 124 http://ec.europa.eu/environment/consultations/abs_en.htm on the European Commission website 124 . 121 EC (2012b), , taking a more comprehensive understanding of Article 18.3, the recognition and enforcement of decisions on civil and commercial matters are ruled by the EC Regulation 44/2001 (Brussels 1) as well as the 2007 Convention of Lugano on jurisdiction and the recognition and enforcement of judgments in civil and commercial matters. The 2005 Convention on Choice of Court Agreements adopted in the framework of the Hague Conference on Private International Law is also a useful tool in this regard, as it sets rules for when a court must take jurisdiction or refuse to do so, where commercial parties have entered into an exclusive choice of court agreement. The Convention also provides for the recognition and enforcement of resulting judgments, with an option for States Parties to agree on a reciprocal basis to recognize judgments based on a choice of court agreement that was not exclusive.Moreover, various conventions could act as "effective measures regarding access to justice" (Article 18.3.a). Regarding the investigation procedure, Belgium did not ratify the 1970 Hague Convention on the taking of evidence abroad in civil or commercial matters. This convention is mainly referring to "commissions rogatoires", through which a judge delegates his investigation powers through a limited mandate allowing another judge or judicial officer to execute an investigation act on his behalf in another jurisdiction. Nonetheless Belgium ratified, amongst other applicable conventions, the Second Protocol to the 1959 European Convention on mutual assistance in criminal matters 127 and the 1965 Hague Convention on notification and communication abroad of judicial and extrajudicial acts in civil or commercial matters. Taking an extensive definition of "access to justice", it is relevant to mention that Belgium ratified also the Aarhus Convention on Access to Information, Public Participation in Decision-making and Access to Justice in Environmental Matters 128 . Greiber T., Peña Moreno S., Åhrén M., Carrasco J.N., Kamau E.C., Cabrera Medaglia J., Julia Oliva M., Perron-Welch F. in cooperation with Ali N. and Williams C. ( 2012 ), An Explanatory Guide to the Nagoya Protocol on Access and Benefit-sharing. IUCN, Gland, Switzerland. However , op. cit.143 See : http://www.environment.gov.au/biodiversity/science/access/model-agreements/index/html144 Burton G (2009), op. cit. 145 Wynberg R., Taylor M. (2009), op. cit. 146 Santili J. (2009), op. cit. 147 Burton G. (2009), op. cit. 148 Santili J. (2009), op. cit. Ley que Establece el Régimen de Protección de los Conocimiento Colectivos de los Pueblos Indígenas vinculados a los Recursos Biológicos, 2002. Ley No 27811, Comisión Permanente del Congreso de la República del Perú; Article 76 of the Biodiversity Law, No 7788, Legislative Assembly of the Republic of Costa Rica, 30 154 Wynberg R, Taylor M (2009), op. cit. 155 UNEP, Natural Justice and IUCN (2011) Report of the International Experts Meeting on Access & Benefit-sharing and Protected Areas, Gland, Switzerland, 6 th -8 th July 2011 156 Article 8 of ) of the African Model Legislation for the Protection of the Rights of Local Communities, Farmers and Breeders, and for the Regulation of Access to Biological Resources, 2000165 For example in Kenya, the National Environment Management Authority (NEMA) collects all the necessary permits, issued by other authorities, before granting access permit. SeeKamau E.C., Winter G. (2009), Streamlining Access Procedures and Standards. In Kamau E.C. and Winter G. (Eds.) Genetic Resources, Traditional Knowledge & the Law. Solutions for Access & Benefit-sharing. London: Earthscan 166 Young TR (2009), op.cit. 161 Article 13(2) of the Nagoya Protocol 162 Article 14 of the Biodiversity Law, No 7788, Legislative Assembly of the Republic of Costa Rica, 30 163 Article 6 of the Regulations on Bio-Prospecting, Access and Benefit-sharing. Government Gazette No. 30739, 8 th April 1998 th February 2008, Republic of South Africa 164 Article 7(1 benefit-sharing" fund or other mechanism which redirects the benefits Possible advantages:  Allows for in-depth monitoring of distribution of benefits Possible disadvantages:  Institutional burden Option 4 -Integrate ABS in biodiversity policies Possible advantages:  Links the 3 objectives of the CBD together  ABS could benefit from more political attention through biodiversity policy  Could generate synergies between policies/actions/actors/administrations Possible disadvantages:  ? EVALUATION Option 1 and 4 recommended. The other options are only recommended if they do not lead to a disproportionately high institutional burden. Description: In order to foster biodiversity-related research, Belgium could develop additional measures to facilitate access to GR Related Article of the 8 NP: Nature of the measure: Administrative and/or legal Priority for Belgium:  Examples of relevant  For example, in Flanders, Article 57bis of the Natuurdecreet existing measures in allows access to real property for research conducted by public Belgium servants and related to nature conservation Possible disadvantages:  Difficult to impose on private sector legal owners Option 2 -Require environmental impact assessment of collection, prior to access Possible advantages:  Has already legal basis for certain types of access Possible disadvantages:  High administrative and financial burden/cost for users  May be ineffective, as collecting a sample probably does not have a major environmental impact Option 3 -Establish a " 7.1.4 Action card -Facilitate access for biodiversity-related research Option 1 -Exempt biodiversity-related research by certain actors from any access requirements Possible advantages  requires return clause to make sure users do come back when entering commercialization phase EVALUATION Option 1 and 2: both are recommended measures if they would lead to simplify the access procedure (the relevance of such an option would depend thus on the complexity of the proposed default procedures). They contribute to a core objective of the Nagoya Protocol and can build upon existing legal measures 7.1.5 Action card -Establish CNA Description: Each Party has to designate a CNA that grants access, issues written evidence that access requirements have been met and advises users on applicable procedures and requirements to get access to GR Related Article of the Article 13 NP: Nature of the measure: Institutional Priority for Belgium:  Examples of relevant existing measures in Belgium :  Has already legal basis in (part of) Belgium (cf. example of existing measure) Possible disadvantages:  Does not allow for post-access monitoring Option 2 -Facilitated access measures for non-commercial biodiversity related research (with a retun clause before entering in a commercial phase) Possible advantages:  Allows to settle BS for the commercial phase at a later stage, based on clearer view of potential value of GR  Lowers administrative burden for non-commercial research at time of access Possible disadvantages:  Requires efficient monitoring of utilization  Article 22, Flemish Soortenbesluit: access to protected species needs to be approved by the "Agentschap voor Natuur en Bos" of the Flemish government Option 1 -Designate one existing institution as CNA Possible advantages:  Low institutional cost, as institution(s) already exist  Low financial cost, as tasks would only be an addition to existing tasks Possible disadvantages: Set up financial incentives (tax reductions, rebates, …) for complying users Possible advantages Could favor important users who can more easily share benefits Description: Incentives might be efficient complementary tools to enforcement mechanism. Related Article of the 15 NP: Nature of the measure: Administrative Priority for Belgium:  Option 1 - :  Could foster greater compliance motivation among (private) users Possible disadvantages:  Would have to transfer part of the extra cost arising out of BS to the state  Option 2 -Set up structural incentives (e.g., special priority for other filings, permits or opportunities, (facilitated) access to special materials, programs, funds, …) for complying users Possible advantages :  Could foster greater compliance motivation among (private) users  Lower financial cost than financial incentives Possible disadvantages:  Could favor important users who can more easily share benefits Option 3 -Set up positive publicity measures (e.g. label) for complying user Possible advantages:  Could foster greater compliance motivation among (private) users Possible disadvantages:  Labels need to be established and monitored  Could favor important users who can more easily share benefits EVALUATION The 3 options are potentially interesting and deserve further analysis. 7.2. 6 Action card -Encourage the development of model clauses, codes of conducts and guidelines sector specific aspects Possible disadvantages:  Effectiveness might be doubtful  Does not allow to address differences in bargaining power between stakeholders Option 2 -Develop ABS guidelines Possible advantages:  Could build on existing measures (MOSAICC )  Part of stakeholders already use it Possible disadvantages:  Difficult to impose on private sector users Option 3 -Develop model contractual clauses or mandatory code of conduct Possible advantages:  Increases control by the state on the content of ABS agreements  Could build on existing measures (IPEN, ECCO core MTA)  Could combine mandatory and non-mandatory provisions Possible disadvantages: Description: Encourage the development of model clauses, codes of conducts and guidelines, to help stakeholders develop appropriate agreements when exchanging GR  Guidelines: Non-mandatory provisions aiming to facilitate the exchange of GR and generalize best practices  Code of conduct: Set of rules outlining the responsibilities of stakeholders when exchanging GR (e.g. IPEN)  Model contractual clauses: Specific clauses to be included in an ABS contract (e.g. ECCO core MTA) Related Article of the 19 and 20 NP: Nature of the measure: Administrative Priority for Belgium:  Examples of relevant  BCCM's MOSAICC existing measures in  ECCO core MTA Belgium  IPEN Code of conduct Option 1 -Rely upon current ABS practices of stakeholders Possible advantages:  Low administrative burden  Gives responsibility to the sectors, taking into consideration  Provides less flexibility for users  Difficult to establish one model that fits all types of utilization  Could conflict with models of contracting Party  Higher administrative burden for authority Table 6 - 6 List of indicators Criteria Selected indicators for assessment E1 Legal certainty and effectiveness for users and Legal certainty providers of GR, at low cost  IE1: consistency and predictability of the rules and the process in place. Effectiveness of the legal framework  IE2: enforceability (the level with which an option allows the ABS regulation to be enforced)  IE3: limiting redundancy (if existing legislations regulate related obligations).  IE4: proximity with other international agreements. E2 Maximizing economic innovation and product  IE5: maximize research and development opportunities for users development (in particular through its and providers of GR contribution to R&D) at reasonable financial and  IE6: allow economic and research stakeholders to compliance administrative costs with the NP at reasonable costs (negotiation costs, costs related to acquisition/transfer of GR, etc.) E3 Minimizing implementation costs  IE7: minimize administrative costs related to keeping track of the ABS agreements (including monitoring costs)  IE8: minimize financial costs for the creation of / changes in institutions (including costs for asking for legal advice) S Achievement of social objectives  IS1: job creation/preservation in the sectors utilizing genetic resources (including through management support, collaboration programs amongst companies, educational programs, etc.)  IS2: maximize research and innovation opportunities in socially relevant fields such as health, nutrition and food security  IS3: support to small and medium enterprises  IS4: transfer of knowledge and technologies to developing countries  IS5: effective protection of the rights of indigenous and local communities over their traditional knowledge associated with GR M Promotion of conservation and sustainable use  IM1: helping ensure fair and equitable benefit-sharing of biodiversity, including biodiversity research  IM2: more predictable conditions for access (including through creating greater legal certainty for users/providers of GR)  IM3: encouraging advancement of research on GR and biodiversity  IM4: creating incentives to conservation and sustainable use of GR (for ex. through recognizing their value and through benefit- sharing, through capacity building and technology transfer)  IM5: enhancing the contribution of biodiversity to development Table 7 -Scoring system of the impact grid 7 Likelihood Magnitude If positive effect If negative effect High Strong + + + --- Medium High Strong Medium + + -- Medium Medium High Weak + - Low Strong Medium Weak Low Medium 0 0 Low Weak economic innovation and product development (in particular through its contribution to R&D) at reasonable financial and administrative costs 199 EC (2012) Impact Assessment accompanying the document "Proposal for a Regulation of the European Parliament and of the Council on Access to Genetic Resources and the Fair and Equitable Sharing of Benefits Arising from their Utilization in the Union, European Commission Staff Working Document, COM(2012) 576 final200 As mentioned under IMP 1.3, the refined fishing net would consider any other legislation where notification/registration/permit exist and specify that such notification/registration/permit is also considered as a PIC under the Nagoya Protocol. This option is part of the later implementation steps (step 3 of the implementation, cf. chapter 11)  Impact on stakeholders: o Coll.: impacted under option 0, 1, 2 and 3, depending on level of legal certainty o Gov. Res.: impacted under option 0, 1, 2 and 3, depending on level of legal certainty o Ag., health and Biotech : impacted under option 0, 1, 2 and 3, depending on level of legal certainty o Univ.: impacted under option 0, 1, 2 and 3, depending on level of legal certainty o Land: impacted under option 0, 1, 2 and 3, depending on level of legal certainty o Other: None E2 -Maximizing Impact on stakeholders: o Coll.: Limited administrative costs per transaction under option 1 (as providers) and under options 2 and 3 (as users). If centralization of access in qualified collections (without prejudice to CNAs), possible increase of costs. o Gov. Res.: Limited administrative costs per transaction under option 1, 2 and 3. o Ag., health and biotech: Limited administrative costs per transaction under option 1, 2 and 3 o Univ.: Limited administrative costs per transaction under option 1, 2 and 3 o Land: Limited administrative costs per transaction under option 1, 2 and 3 (as providers). o Other: No impact Table 8 -Economic impact of the options for the operationalization of PIC Selection criteria Option Pos/Neg Likelihood of occurrence Effect magnitude Score 8 Option 0 Negative High Strong --- E1 Option 1 Option 2 Positive Positive High Medium Strong Medium +++ + Option 3 Positive High Medium ++ Option 0 Negative Medium Medium - E2 Option 1 Option 2 / Positive Medium Medium Weak (*) Medium 0 + Option 3 Positive Medium Medium + Option 0 Negative High Weak - E3 Option 1 Option 2 Negative Negative High High Medium Medium ---- Option 3 Negative High Medium -- (*)The main reason of ranking « weak » instead of « medium » for this option is the possible financial cost of storage (cf. discussion under E2)  Impact on stakeholders: o Coll.: Increasing R&D could trigger job creation o Gov. Res.: Increasing R&D could trigger job creation o Ag., health and biotech: Increasing R&D could trigger job creation o Univ.: Increasing R&D could trigger job creation o Land: Could help to build institutional capacity with small land owners, in particular on ABS issues related to use of biodiversity for R&D o Other: Indirect positive effects for society as a whole through innovation and new available products in the field of food security, health and nutrition Table 9 -Social impact of the options for the operationalization of PIC Selection criteria Option Pos/Neg Likelihood of occurrence Effect magnitude Score 9 Option 0 Negative Unclear S Option 1 Option 2 Positive Positive Unclear Unclear Option 3 Positive Unclear Table 10 -Environmental impact of the options for the operationalization of PIC Selection criteria Option Pos/Neg Likelihood of occurrence Effect magnitude Score 10 Economic Valuation of Biodiversity, Final Report for the MOSAICS Project. Korea Institute for International Economic Policy; Rausser G.C., Small A. (2000), Valuing Research Leads: Bioprospecting and the Conservation of Genetic Resources. Journal of Political Economy. 108(1), pp. 173-206; Biber-Klemm S., Martinez S.I., Jacob A. (2010), Access to Genetic Resources and Sharing of Benefits -ABS Program 2003-2010, Swiss Academy of Sciences, Bern, Switzerland; Martin B.R. and Tang P. (2007), The benefits from publicly funded research. SPRU Working Paper: 161.  Impact on stakeholders: o Coll.: More opportunities for own (taxonomic) research due to increased deposits under option 1 o Gov. Res.: No impact o Ag., health and biotech: No impact o Univ.: No impact o Land: Could increase awareness on ABS related to use of biodiversity for R&D o Other: Indirect positive effects for society through increase of knowledge base and closer monitoring of benefits under option 1 and through refined PA/PS under option 3 M Option 0 Negative Medium Medium - 207 Yun M. (2005), Stromberg P., Dedeurwaerdere T., Pascual U. (2007), An empirical analysis of ex-situ conservation of microbial diversity. Presentation at the 9th International BIOECON Conference, Kings College Cambridge, 19 th -20 th September 2007; Täuber et al., (2011), op.cit. The resulting legal uncertainty (see also E1) is likely to lead to less utilization of Belgian GR in research and development and thus potentially hamper economic innovation and product development substantially213 . This expected impact of this hypothetical situation for the Belgium providers and users can be illustrated with the historical example of the legal vacuum, between 1992 and 1994, of the international network of the CGIAR collections 211 M.W. Tvedt, O.K. Fauchald, O. K. (2011), Implementing the Nagoya Protocol on ABS: A Hypothetical Case Study on Enforcing Benefit-sharing in Norway. The Journal of World Intellectual Property, 14: p. 392 212 Tauber et al., (2011), op.cit. 213 (Consortium for International Agricultural Research). As documented in the literature, this legal vacuum led to a temporary, but spectacular, decrease by over 50% of the use of the GR of these collections, see Byerlee, D. and Dubin, J. 2010 . Crop improvement in the CGIAR as a global success story of open access and international collaboration. International Journal of the Commons (4) 1. Table 11 -Economic impacts of the options for the specification of MAT Selection criteria Option Pos/Neg Likelihood of occurrence Effect magnitude Score 11  Impact on stakeholders: o Coll.: Incurring less implementation costs under option 2 and, to a lesser extent, option 3, both as users and providers. o Gov. Res.: Incurring less implementation costs under option 2 and, to a lesser extent, option 3. o Ag., health and biotech: Incurring less implementation costs under option 2 and, to a lesser extent, option 3. More flexibility under option 3 o Univ.: Incurring less implementation costs under option 2 and, to a lesser extent, option 3. o Land: Incurring less implementation costs under option 2 and, to a lesser extent, option 3 o Other: No impact Option 0 Negative High Strong --- E1 Option 1 Option 2 / Positive Medium Medium Weak Medium 0 + Option 3 Positive Medium Medium + Option 0 Negative High Medium - E2 Option 1 Option 2 Positive Positive Medium Medium Medium Medium + + Option 3 Positive High Medium + + Option 0 Unclear E3 Option 1 Option 2 Positive Positive Medium Medium Weak Medium 0 + Option 3 / Medium Weak 0  Impact on stakeholders: o Coll.: As provider, more opportunity to contribute to social objectives with BS as horizontal principle. o Gov. Res.: Contribution to the R&D can lead to job creation and contribute to social objectives o Ag., health and biotech: Contribution to the R&D can lead to job creation and contribute to social objectives o Univ.: Contribution to the R&D can lead to job creation and contribute to social objectives o Land: As provider, more opportunity to contribute to social objectives with BS as horizontal principle o Other: the adoption of BS as a horizontal principle might also have (at least indirect) positive effects on socially important sectors Table 12 -Social impacts of the options for the specification of MAT Selection criteria Option Pos/Neg Likelihood of occurrence Effect magnitude Score 12 Option 0 Negative Medium Medium - S Option 1 Option 2 / Positive Medium High Weak Medium 0 ++ Option 3 Positive High Weak + Environmental impact M - Promotion of conservation and sustainable use of biodiversity It is likely that BS, as a horizontal principle would have positive effects on conservation of sustainable use of biodiversity, in particular when non-monetary and monetary benefits, as included in the MAT, are directed towards the objectives of conservation and sustainable use of biodiversity. Conversely, contribution of the option 0 can be considered negative.Like for social objectives, options 3 and 2 offer a better opportunity for the Belgian authorities to control the types of benefits being shared and monitor whether they contribute to conservation and sustainable use of biodiversity.  Impact on stakeholders: o Coll.: No impact o Gov. Res.: No impact o Ag., health and biotech: No impact o Univ.: No impact o Land: Option 1 offers less possibility to channel benefits towards conservation and sustainable use o Other: BS as a horizontal principle is expected to offer positive environmental effects for society as a whole Table 13 -Environmental impacts of the options for the specification of MAT Selection criteria Option Pos/Neg Likelihood of occurrence Effect magnitude Score 13 Option 0 Negative Medium Medium - M Option 1 Option 2 Positive Positive Medium Medium Weak Medium 0 + Option 3 Positive Medium Medium + Table 14 -Economic impacts of the establishment of the CNA Selection criteria Option Pos/Neg Likelihood of occurrence Effect magnitude Score 14 ). o Coll.: No impact o Gov. Res.: No impact o Ag., health and biotech: Foreign users can benefit from lower implementation costs under option 2 o Univ.: Foreign universities can benefit from lower implementation costs under option 2 o Land: No impact o Other: No impact Option 0 Negative High Strong --- E1 Option 1 / Medium Weak 0 Option 2 Positive Medium Medium + Option 0 Negative High Strong --- E2 Option 1 / Low Medium 0 Option 2 / Low Medium 0 Option 0 Unclear E3 Option 1 Negative High Medium -- Option 2 Negative High Medium -- Table 15 -Social impacts of the establishment of the CNA Selection criteria Option Pos/Neg Likelihood of occurrence Effect magnitude Score 15  Impact on stakeholders: o Coll.: No impact o Gov. Res.: No impact o Ag., health and biotech: No impact o Univ.: No impact o Land: No impact o Other: No impact M - Promotion of conservation and sustainable use of biodiversity, including biodiversity research Option 0 would not allow keeping track of the access requests and analyzing the ways in which the resources are accessed, missing a major opportunity to enhance knowledge which could be used for the improvement of conservation and sustainable use. Options 1 and 2 do not lead to a different impact on the environment.  Impact on stakeholders: o Coll.: No impact o Gov. Res.: No impact o Ag., health and biotech: No impact o Univ.: No impact o Land: No impact o Other: No impact Table 16 -Environmental impacts of the establishment of the CNA Selection criteria Option Pos/Neg Likelihood of occurrence Effect magnitude Score 16 Option 0 Negative High Strong --- M Option 1 / Low Weak 0 Option 2 / Low Weak 0 Table 17 -Economic impacts of the compliance measures Selection criteria Option Pos/Neg Likelihood of occurrence Effect magnitude Score 17 Option 0 Negative High Strong --- E1 Option 1 Positive Medium Medium + Option 2 Positive Medium Medium + Option 0 Negative High Medium -- E2 Option 1 Positive Medium Weak 0 Option 2 Positive Low Weak 0 Option 0 Unclear / / / E3 Option 1 Unclear / / / Option 2 Unclear / / /  Impact on stakeholders: o Coll.: As users, possible additional costs o Gov. Res.: Possible additional costs o Ag., health and biotech: Possible additional costs o Univ.: Possible additional costs o Land: No impact as providers o Other: No impact Table 18 -Social impacts of the compliance measures Selection criteria Option Pos/Neg Likelihood of occurrence Effect magnitude Score 18 Option 0 Negative High Medium -- S Option 1 Positive Medium Medium + Option 2 Positive Medium Medium +  Impact on stakeholders: o Coll.: Capacity for social innovation is limited. o Gov. Res.: Capacity for social innovation is limited. o Ag., health and biotech: Capacity for social innovation is limited. o Univ.: Capacity for social innovation is limited. o Land: No impact. o Other: Option 1 and 2 both offer opportunities for increasing protection of ILCs and TK in provider countries Table 19 -Environmental impacts of the compliance measures 19  Impact on stakeholders: o Coll.: No impact o Gov. Res.: No impact o Ag., health and biotech: No impact o Univ.: No impact o Land: Better level playing field, within an effective regime, would benefit awareness raising on biodiversity issues more generally, and in in situ environments in particular. o Other: capacity building and technology transfer to third countries for conservation and sustainable use easier if NP is implemented Selection criteria Option Pos/Neg Likelihood of occurrence Effect magnitude Score Option 0 Negative High Medium -- M Option 1 Positive Medium Medium + Option 2 Positive Medium Medium + Table 20 -Economic impacts of the options for designating checkpoint(s) Selection criteria Option Pos/Neg Likelihood of occurrence Effect magnitude Score 20 Impact on stakeholders: o Coll.: As users, impact dependent upon type of utilization and moment of acquiring GR o Gov. Res.: impact dependent upon type of utilization and moment of acquiring GR o Ag., health and biotech: impact dependent upon type of utilization and moment of acquiring GR o Univ.: impact dependent upon type of utilization and moment of acquiring GR o Land: no impact. o Other: No impact Option 0 Negative High Strong --- E1 Option 1 Positive Low Medium + Option 2 Positive Low Medium + Option 0 Negative Medium Medium - E2 Option 1 / Medium Weak 0 Option 2 / Medium Weak 0 Option 0 Unclear E3 Option 1 Unclear Option 2 Unclear Impact on stakeholders: o Coll.: No impact o Gov. Res.: No impact o Ag., health and biotech: No impact o Univ.: No impact o Land: No impact o Other: Higher level of protection of TK of communities in third countries under option 1 Table 21 -Social impacts of the options for designating checkpoint(s) Selection criteria Option Pos/Neg Likelihood of occurrence Effect magnitude Score 21 Option 0 Negative Medium Medium - S Option 1 Positive Medium Medium + Option 2 / Medium Weak 0 Table 22 -Environmental impacts of the options for designating checkpoint(s) Selection criteria Option Pos/Neg Likelihood of occurrence Effect magnitude Score 22 Coll.: biodiversity research could be hindered under option 0 o Gov. Res.: biodiversity research could be hindered under option 0 o Ag., health and biotech: No impact o Univ.: biodiversity research could be hindered under option 0 o Land: Both option 1 and 2 will better contribute to awareness raising, which can contribute to biodiversity conservation activities in general, and in in situ environments in particular o Other: Generation of benefits for biodiversity conservation and sustainable use in provider countries hindered under option 0 Option 0 Negative Medium Medium - M1 Option 1 Positive Medium Medium + Option 2 Positive Medium Medium +  Impact on stakeholders: o  Impact on stakeholders: o Coll.: No impact o Gov. Res.: No impact o Ag., health and biotech: No impact o Univ.: No impact o Land: No impact o Other: No impact Table 23 -Economic impacts of the options for the ABS CH 23 Selection criteria Option Pos/Neg Likelihood of occurrence Effect magnitude Score Option 0 Negative High Strong --- E1 Option 1 Option 2 Positive Positive Medium Medium Medium Medium + + Option 3 / Medium Weak 0 Option 0 Negative High Strong --- E2 Option 1 Option 2 Positive Positive Medium Medium Medium Medium + + Option 3 / Medium Weak 0 Option 0 Positive High Medium ++ E3 Option 1 Option 2 Positive Positive Medium Medium Medium Medium + + Option 3 Positive Medium Medium + Table 24 -Social impacts of the options for the ABS CH Selection criteria Option Pos/Neg Likelihood of occurrence Effect magnitude Score 24 Option 0 Negative High Strong --- S Option 1 Option 2 Positive Positive High Medium Medium Medium ++ + Option 3 / Medium Weak 0 The Nagoya Protocol has been declared a "double mixed treaty" by the Working Group on Mixed Treaties on 22/11/2010. This means that the federal State, the Regions and the Communities need to give their consent in order for Belgium to be able to ratify. The core obligations are the obligations specified in the terms of reference of this study as requiring special attention: Access to genetic resources and traditional knowledge; Benefit-sharing; the National Competent Authorities and the National Focal Points; Conformity with the national legislation of the provider country and the contractual rules; and compliance and monitoring. Report of the stakeholder meeting is available here: http://www.biodiv.be/implementation/cross-cuttingissues/abs/workshop-np-20120529/20120529-nagoya-stakeholder-workshopreport-final.pdf Le Protocole de Nagoya a été déclaré « traité doublement mixte » par le Groupe de travail Traités Mixtes de la Conférence interministérielle de la Politique étrangère le 22/11/2010. L'Etat fédéral, les Régions et les Communautés doivent donner leur consentement pour que la Belgique puisse ratifier le Protocol. Au regard des remarques qui précèdent, les dispositions nationales actuellement disponibles régissant le statut légal des ressources génétiques en Belgique concernent principalement la question de la propriété légale du matériel génétique. Il résulte des principes fondamentaux sur le droit de propriété que l'on trouve dans le code civil, que les conditions et règles relatives à la propriété légale du matériel génétique, en tant qu'entité biophysique, découlent de celles qui régissent la propriété de l'organisme dans lequel ce matériel peut être trouvé. La propriété sur un organisme signifie que le propriétaire possède les droits de l'utiliser, d'en jouir et d'en disposer juridiquement et matériellement. De plus, toute mesure légale qui envisagerait de réglementer l'accès aux ressources génétiques pourrait se baser sur la législation existante sur l'accès physique et sur l'usage de matériel génétique. Les lois réglementant l'accès physique et l'usage du matériel génétique dépendent du type de propriété (mobilière, immobilière, ou res nullius), de l'existence de restrictions à la propriété comme une protection spécifique (espèces protégées, zones protégées, forêts ou environnements marins) et de la situation géographique du matériel génétique (les quatre autorités appliquent leurs propres règles).Contrairement à ses composants physiques, les composants informationnels des ressources génétiques peuvent constituer une res communis : "chose qui qui n'appartient à personne mais est sujet à l'usage par tous". Tandis que l'accès à de tels composants informationnels n'est pas couvert par une législation spécifique, l'exercice de certains droits d'utilisation peut cependant être limité par des droits de propriété intellectuelle qui touchent à des parties, des fonctions ou des utilisations de matériel biologique résultant d'innovations faites sur ces matériaux. Ces droits de propriété Les obligations fondamentales sont les obligations spécifiées dans les termes de référence de la présente étude comme requérant une attention spéciale : accès aux ressources génétiques et au savoir traditionnel ; partage des avantages ; l'Autorité Nationale Compétente et les correspondants (coordinateurs) nationaux ; conformité avec la législation nationale du pays d'origine (fournisseur) et règles contractuelles ; conformité et monitoring. Het Protocol van Nagoya werd dubbel gemengd verklaard door de Werkgroep Gemengde Verdragen (WGV) van de Interministeriële Conferentie voor Buitenlands Beleid op 22/11/2010. De instemming van de federale Staat, de Gewesten en de Gemeenschappen is vereist voor de instemming met het Protocol. De kernverplichtingen zijn die verplichtingen die volgens de referentievoorwaarden van deze studie bijzondere aandacht verdienen: toegang tot genetische rijkdommen en traditionele kennis; batenverdeling; de Nationale Bevoegde Autoriteiten en de Nationale Contactpunten; naleving van de nationale wetgeving van het oorsprongsland en de contractuele regels; en naleving en monitoring. Het verslag van deze vergadering is beschikbaar op het volgend adres: http://www.biodiv.be/implementation/crosscutting-issues/abs/workshop-np-20120529/20120529-nagoya-stakeholder-workshopreport-final.pdf The following 'acts' express the consent of a State to be bound by a treaty: ratification, accession, approval and acceptance. The legal implications, i.e. the binding nature of ratification, accession, approval, and acceptance are the same. Status of Signature, and ratification, acceptance, approval or accession, available at http://www.cbd.int/abs/nagoyaprotocol/signatories/default.shtml(accessed 26 March 2013) 25 Estonia, Latvia, Malta, Slovakia and Slovenia. Report of the stakeholder meeting is available here: http://www.biodiv.be/implementation/cross-cuttingissues/abs/workshop-np-20120529/20120529-nagoya-stakeholder-workshopreport-final.pdf Belgian State reforms were performed in 1970, 1980, 1988, 1993 and 2001. The main provisions pertaining to these reforms are to be found in the "special law" dated 8 th August 1980 related to the general institutional reforms, and the special law of 12 th January 1989 pertaining to the institutions of the Brussels Region. The sixth reform will follow on the footsteps of the institutional agreement adopted on 11 th October 2011 and operate additional transfers of competences towards federated entities, especially the Regions. However, this reform, which has not officially been transcribed into applicable legislative texts, shall not highly affect the distribution of competences that may be linked with ABS. This principle applies notwithstanding the future entry into force of Article 35 of the Constitution. The Nagoya Protocol has been declared a double mixed treaty by the Working Group on Mixed Treaties on 22/11/2010. A double mixed treaty indicates that the competent entities for its implementation are the Governments of the Regions (Flemish, Walloon and Brussels-Capital Region), the Governments of the Communities (Flemish, French and German Community) and the Federal Government (see also the analysis in chapter 2 on the distribution of ABS related competences in Belgium). Accord de coopération du 5 avril 1995 entre l'Etat fédéral, la Région flamande, la Région wallonne et la Région de Bruxelles-Capitale relatif à la politique internationale de l'environnement / Samenwerkingsakkoord van 5 april 1995 tussen de Federale Staat, het Vlaamse Gewest, het Waalse Gewest en het Brussels HoofdstedelijkGewest met betrekking tot het internationaalmilieubeleid. Article 6,- § 1er, VI de la loi spéciale du 8 août 1980 de réformes institutionelles / Artikel 6,- § 1st, VI van de wet van 8 augustus 1980 tot hervorming der instellingen [START_REF] Duran | Een vergelijkend onderzoek naar en bestedingsanalyze van het buitenlands beleid en de diplomatieke representatie van regio's met wetgevende bevoegdheid en kleine staten[END_REF], Een vergelijkend onderzoek naar en bestedingsanalyze van het buitenlands beleid en de diplomatieke representatie van regio's met wetgevende bevoegdheid en kleine staten. Rapport, Antwerpen: Steunpunt Buitenlands Beleid, 418 p As for the development cooperation field, the State Reform of 2001 intended to further clarify the distribution of competence between the federal and federated entities through the new Article 6ter of the SL8/8/80 that reads: "certain fragments of development cooperation will be transferred on 1 st January 2004, to the extent which they concern competences attributed to Communities and Regions (Inserted by Article 41 of the special law of 13 th July 2001 (M.B., 3 rd August 2001), which has entered into force on 1st January 2002). A specific working group is constituted to propose a list of subject-matters concerning Community and Regional competences at the latest on 31 st December 2002. Such a working group was created in 2004 to solve the issue but has not yet led to any conclusion. [START_REF] Paquin | Paradiplomatie identitaire et diplomatie en Belgique fédérale : le cas de la Flandre[END_REF], Paradiplomatie identitaire et diplomatie en Belgique fédérale : le cas de la Flandre. Canadian Journal of Political Science/Revue canadienne de science politique(2003), 36 : pp 621-642 [START_REF] Geeraerts | De Vlaamse betrokkenheid bij de totstandkoming van Europees en multilateraal milieubeleid[END_REF], Vlaams milieubeleid steekt de grenzen over. De Vlaamse betrokkenheid bij de totstandkoming van Europees en multilateraal milieubeleid, Steunpunt Milieubeleidswetenschappen, Antwerp, UA. Glowka, L., Burhenne-Guilmin, F., Synge, H., in collaboration with McNeely, J., A. and Gündling, L. (1994), A guide to the convention on biological diversity, Environmental Policy and Law Paper No. 30, IUCN Environmental Law Center [START_REF] Van Den Haselkamp-Hansenne | L'étendue de la propriété immobilière[END_REF], L'étendue de la propriété immobilière. In X., Guide de droit immobilier, 2011, liv. 64, Kluwer, Waterloo, sections I.5.-1 toI.5.3.-4 [START_REF] Hansenne | L'accession immobilière[END_REF], L'accession immobilière, in X., Guide de droit immobilier, 2011, liv. 64, Kluwer, Waterloo, sections I.8.-1 to I.8.4.-1. Indeed, according to the European Patent Office, a process for plant production that contains steps of crossing the entire genome of plants followed by the selection of obtained plants is not patentable. These steps should be seen as "essentially biological", as mentioned in Article 53 (b) of the 1973 Convention on the European patent; See DEN HARTOG, J., (2011), "Interpretatie van Article 53(b) EOV; werkwijzen van wezenlijke biologische aard", BIE, pp. 20-23. Projet de loi modifiant la loi du 28 Mars 1984 sur les brevets d'invention, en ce qui concerne la brevetabilité des inventions biotechnologiques, Rapport fait au nom de la Commission des Finances et Affaires Economiques par Mme Zrihen, Doc.Senat, sess. 2004-2005, no.3-1088/3, p.3. See also Van Overwalle G. (2004), Van groene muizen met rode oortjes: de EU-Biotechnologierichtlijn en het Belgisch wetsontwerp van 21st September 2004.IRDI, This clause is a transposition of Directive 98/44/EC of 6 th July 1998 on the legal protection of biotechnological inventions, which takes Articles 8(j) and 15 of the CBD into consideration. Its preamble notes that in case an invention is based on biological material of plant or animal origin or if such material is used, the patent application should, where appropriate, include information on the geographical origin of such material, if known. The Directive furthermore stresses that Member States must give particular weight to Article 8(j) of the CBD when bringing into force the laws, regulations and administrative provisions necessary to comply with this Directive. The Directive is motivated by the need to develop a The determination of applicable law and juridical competence will be studied with greater detail in part C of this section devoted to the implementation of the route taken by Belgium with regard to private international law. There are three constitutive elements to "theft": « soustraction, chose d'autrui et intention frauduleuse » Indeed, by virtue of the principle of "pacta sunt servanda" ("principe de la convention-loi", Article 1134 al.1 of the civil code), the procedure following breaches of contract and the compensation for the violation shall be determined by the contractual clauses themselves. Law of 16 th July 2004 related to the code of private international law, M.B., 27 th July 2004, pp. 57344 Concerns can also be raised for the lack of reference in these legal dispositions of important issues of "access to justice" addressed in the Nagoya Protocol, such as the legal standing of ILCs before Belgium courts. Liège, 25 avr. 1991, Rev. dr. pén., 1991, p. 1013. Cass. (2e ch.) RG P.98.0082.N, 5 octobre 1999 (Indestege) Cass. RG 2941, 9 avril 1991 (Marchand / Strubbe) One can for instance foresee the starting point of breach of trust at the change of nature of the recipient institution, turning for instance from a public non-profit organisation into a commercial structure. Besluit van 15 mei 2009 van de Vlaamse Regering met betrekking tot soortenbescherming en soortenbeheer, (B.S.,13 augustus 2009). Loi du 12 juillet 1973 sur la conservation de la nature: Région wallonne (M.B., 11 septembre 1973) Decreet betreffende het natuurbehoud en het natuurlijk milieu See: Belgian Senate, parl. sess. 1995[START_REF] Goux | La recherche scientifique dans la Belgique fédérale: examen de la répartition des compétences[END_REF], Bulletin 1-2 à, 18 th June 1996, vraagn°43 (Mrs. V. Dua), 26 th March 1996. See: Federal Plan for SustainableDevelopment (2000Development ( -2004) ) Convention concerning the Protection and Integration of Indigenous and Other Tribal and Semi-Tribal Populations in Independent Countries , 26 th June 1957, Genève, ILO, ratified by Belgium on 19 th November 1958, (M.B., 6 th December 1958), entered into force on 2 nd June 1959. Report of the first stakeholder meeting is available here: http://www.biodiv.be/implementation/cross-cuttingissues/abs/workshop-np-20120529/20120529-nagoya-stakeholder-workshopreport-final.pdf 111 CIDD/ICDO (2008) Federaal plan inzake duurzame ontwikkeling 2004-2008/Plan Fédéral de DéveloppementDurable 2004-2008. Interdepartmental Commission for Sustainable Development 112 More information on http://www.biodiv.be/ info0405/activities/ 113 More information on http://www.africamuseum.be/museum/about-us/cooperation/index_html More information on http://www.tematea.org More information on http://straininfo.net More information on http://www.plantcol.be/ More information on http://www.biodiversity.be Samen Grenzen Ver-Leggen. Vlaamse strategie duurzame ontwikkeling, Vlaamse Regering, 2011. Available at http://do.vlaanderen.be/sites/default/files/VSDO2_3.pdf VBV (2011) Plant van Hier. Praktisch vademecum met oog op het behoud en de promotie van autochtone planten. Available on http://www.vbv.be/projecten/plantvanhier/Vademecum_PlantVanHier_web.pdf http://www.plantvanhier.be/ [START_REF] Cbd | Report on the Legal Status of Genetic Resources in National Law. Including Property Law, where applicable, in a Selection of Countries[END_REF] Report on the Legal Status of Genetic Resources in National Law. Including Property Law, where applicable, in a Selection of Countries.UNEP/CBD/WG-ABS/5/1. Ibid. Wynberg R, Taylor M (2009) Finding a Path Through the ABS Maze -Challenges of Regulating Access and Benefit-sharing in South Africa. In Kamau EC and Winter G (Eds.) Genetic Resources, Traditional Knowledge & the Law. Solutions for Access & Benefit-sharing. London: Earthscan Article 19 of Decision 391 on the Common Regime on Access to Genetic Resources. Cartagena Agreement Official Gazette No. 213 of 17 th July 1996 Report of the stakeholder meeting is available here: http://www.biodiv.be/implementation/cross-cuttingissues/abs/workshop-np-20120529/20120529-nagoya-stakeholder-workshopreport-final.pdf The Belgian national law enacting the code of private international law states in its Article 15 that, if a foreign law needs to be applied to a case that is examined by a Belgian judge, the content of such applicable law should be identified by the judge, according to interpretations received in the "country of origin" (sic). Collaboration can be required if the content cannot be established clearly by the Belgian judge. If it is "impossible to determine the content of foreign law in due time, Belgian law should be applied" (art.15 §2al2)" The Belgian national law enacting the code of private international law states in its Article 15 that, if a foreign law needs to be applied to a case that is examined by a Belgian judge, the content of such applicable law should be identified by the judge, according to interpretations received in the "country of origin" (sic). Collaboration can be required if the content cannot be established clearly by the Belgian judge. If it is "impossible to determine the content of foreign law in due time, Belgian law should be applied" (art.15 §2al2)" cf. supra, previous footnote. Frison C., Dedeurwaerdere T. (2006), op. cit. UN Comtrade (2010) Medicinal and pharmaceutical products (SITC 54) Available at http://comtrade.un.org/. Figures from Pharma.be, http://www.pharma.be/newsitem.aspx?nid=2174 EC (2012b), op. cit. 19 th February 2007 -Accord de coopération entre l'Autorité fédérale, la Région flamande, la Région wallonne et la Région de Bruxelles-Capitale relatif à la mise en oeuvre de certaines dispositions du Protocole de Kyoto 25th April 1997 -Accord de coopération entre l'Etat fédéral et les Régions relatif à la coordination administrative et scientifique en matière de biosécurité (M.B. 14.07.1998) The provisions IMP 1.0 (2); IMP 1.1.1 (2); IMP 2.2 ; IMP 2.3 would require a Federal Law and Decrees of the Federated Entities, to amend the basic environmental codes of the Regions and the Federal State : Natuurdecreet, 21 st of October 1997 (Vlaams Gewest) ; Loi sur la Conservation de la Nature, 12 th of July 1973 (Région Wallonne) ; Ordonnance sur la conservation de la nature, 1 st of March 2012 (Région Bruxelloise) ; Law on the protection of the Marine Environment, 20th January 1999. For a detailed description of these laws, cf. above section 3.1 of the study on the "Access and use of genetic resources under national jurisdiction in Belgium". The refined fishing net would consider any other legislation where notification/registration/permit exist and specify that such notification/registration/permit is also considered as a PIC under the Nagoya Protocol. The case of the conservation varieties, cited for illustration only, shows such a legislation that is different from the PA/PS legislation and where the current legislation on the "admission to use" could be considered also as a PIC under the Nagoya Protocol, in further For an overview see[START_REF] Vatn | Institutions and the Environment[END_REF], Institutions and the Environment. Edward Elgar Pub Due to the scarcity of quantitative data, most performances of criteria in this report reflect a subjective assessment and evaluation based on the various available input and data. Nevertheless, quantitative figures have been included and detailed whenever possible and reliable. See C. Correa (2000), Implications of national access legislation for germplasm flows, In Strengthening partnerships in agricultural research for development in the context of globalization, proceedings of the GFAR Conference, 21-23 May 2000, Dresden, Germany. FAO/Global forum on agricultural research; K.D. Prathapan, R. Dharma, T.C. Priyadarsanan and Narendran, C.A. Viraktamath, N.A. Aravind, J. Poorani, J. (2008) Death sentence on taxonomy in India. Current Science, 94 (2). pp. 170-171. S. Täuber, K. Holm-Müller, T. Jacob, U. Feit (2011), An Economic Analysis of new Instruments for Access and Benefitsharing under the CBD -Standardisation options for ABS Transactions. Research project of the Federal Agency for Nature Conservation, Germany. Terms of reference No. DG5/AMSZ/11008 Procedural sub-criteria have been dichotomized through a Boolean expression, as these criteria were not attributed performance in the previous particle. Options positively addressing the issues raised by a general criterion were assigned a true (1) status, whereas others were considered as false (0). This Boolean expression has been broadened a little in the course of the analysis, as it quickly became apparent that some options could be both true and false. Therefore, a third category was created (0,5) in some cases. In order to allow for software-based comparison scores have been quantified on a scale from 1 to 7: the "---" score corresponds to value 1 and the "+++" to value 7. The software used for the PROMETHEE calculus is Visual Promethee, version 1.0.10.1. For some sectors, the rate of deposits of material used for research outside of the collections is very low. According to one interviewee, for the microbial GR, for instance, less than 1% of the GR serving for research outside of the collections is currently being deposited in an ex-situ collection. M.M.[START_REF] Watanabe | Innovative Roles of Biological Resource Centers[END_REF], Innovative Roles of Biological Resource Centers. Proceedings of the Tenth International Congress for Culture Collections, Japan Society for Culture Collections ; World Federation for the Culture Collections p. 435-438; K B. Koo, P.G. Pardey, B.D. Wright and others (2004), Saving Seeds; The economics of conserving crop genetic resources ex situ in the future harvest centre of CGIAR, CABI Publishing, p.45.; Interviews. These figures are based on a quantitative evaluation, by the study team, of the additional costs for accessing materials under the options for the operationalization of PIC. This evaluation is based on the data generated in the interviews (especially indicators IND 3.1 and 3.2, data collected for the various stakeholder groups, and IND 8.1 to 8.5 for the collections) and existing models from the literature for assessing the implementation costs (in particularTäuber et al. (2007);[START_REF] Eaton | A Moving Target: Genetic Resources and Options for Tracking and Monitoring their International Flows[END_REF]; CBD Bonn Guidelines (2002); Visser B,[START_REF] Visser | Transaction costs of germplasm exchange under bilateral agreements[END_REF], op. cit.). Ibid. Data from address database from Belgian users of GR, acquired for the 2006 awareness study on access and benefitsharing[START_REF] Frison | Infrastructures publiques et régulations sur l'Accès aux ressources génétiques et le Partage des Avantages qui découlent de leur utilisation pour l'innovation dans la recherche des sciences de la vie : Accès, conservation et utilisation de la diversité biologique dans l'intérêt général[END_REF]. 206 This does not imply that these key collections acquire the authority to decide whether or not to grant access, as this task is reserved to the CNA. [START_REF] Marchal | Le droit comparé dans la Jurisprudence de la Cour de Cassation[END_REF] Le droit comparé dans la Jurisprudence de la Cour de Cassation. Revue de Droit International etde Droit Comparé, liv.2-3, 2008 -p. 418. [START_REF] Ieep | Study to analyze legal and economic aspects of implementing the Nagoya Protocol on ABS in the European Union, Final report for the European Commission[END_REF], op.cit. Even if the efficiency depends upon additional training of the patent office, or close contribution of a more technically skilled monitoring office IEEP, Ecologic and GHK (2012), op. cit., p. 139 Eaton D., Visser B. (2007), op. cit. cf. The detailed analysis in section 3.1 on "Access and use of genetic resources under national jurisdiction in Belgium". Non-exhaustive list of examples of legislation covering PA/PS which do not expressly cover access of GM or GR for utilization as defined under the NP: Ordonnance du 1 mars 2012 relative à la conservation de la nature; Besluit van de Vlaamse Regering van 15 mei 2009 met betrekking tot soortenbescherming en soortenbeheer; Besluit van de Vlaamse Regering van 5 december 2008 betreffende de toegankelijkheid van de bossen en de natuurreservaten; Décret du 15 juillet 2008 relatif au Code forestier; Decreet van 21 oktober 1997 betreffende het natuurbehoud en het natuurlijk milieu; Arrêté du Gouvernement wallon du 26 janvier 1995 organisant la protection des cavités souterraines d'intérêt scientifique; Decreet van 13 juni 1990 Bosdecreet; Arrêté de l'Exécutif régional wallon du 8 juin 1989 relatif à la protection des zones humides d'intérêt biologique, modifié par l'arrêté du 10 juillet 1997; Loi du 12 juillet 1973 sur la conservation de la nature For examples of other relevant legislation please refer to footnote 187 The provisions under (1) a, b and c would require a Federal Law and Decrees of the Federated Entities, to amend the basic environmental codes of the Regions and the Federal State : Natuurdecreet, 21 st October 1997 (Vlaams Gewest) ; Loi sur la Conservation de la Nature, 12 th July 1973 (Région Wallonne) ; Ordonnance sur la conservation de la nature, 1st of March 2012 (Région Bruxelloise) ; Law on the protection of the Marine Environment, 20th January 1999. For a detailed description of these laws, cf. above section 3.1 of the study on the "Access and use of genetic resources under national jurisdiction in Belgium". The interpretation of this analysis, especially the results of the outranking flow calculus of the PROMETHEE method, is to be done with care and in light of both the context described in the evaluation of the performance of the options (step 1 of the IA, as described above) and the analysis of the relationship between the options as presented in the visual dominance analysis (step 2 of the IA). Access and Benefit-sharing ABS CH ABS Clearing-House ABSWG Ad Hoc Open-ended Working Group on Access and Benefit-sharing ANB Flemish Agency for Nature and Forest AWEX Agence wallonne à l'Exportation et aux Investissements étrangers BAP EU Biodiversity Action Plan BCCM Belgian Co-ordinated Collections of Micro-organisms BCH Biosafety Clearing-House BELSPO Belgian Federal Science Policy Office BEW/AEE Economy and employment administration of the Brussels-Capital Region BS Benefit-sharing BTC Belgian Technical Cooperation CAP Common Agricultural Policy CBD Convention on Biological Diversity CCIEP Coordinating Committee for International Environment Policy CFDD Conseil Fédéral du Développement Durable CHM Clearing-House Mechanism to the CBD CITES Convention on International Trade in Endangered Species of Wild Fauna and Flora CNA Competent National Authority COP Conference of the Parties COP/MOP Conference of the Parties serving as the Meeting of the Parties DIE-OPRI Dienst voor Intellectueel Eigendom/Office belge de la Propriété intellectuelle DG Directorate-General DG4 DG Animal, Plant and Food of the FPS Health, Food Chain Safety and Environment DG5 DG Environment of the FPS Health, Food Chain Safety and Environment DGARNE Wallonia's Operational Directorate-General for Agriculture, Natural Resources and the Environment DGD/DGOS DG Development Cooperation of the FPS Foreign Affairs, Foreign Trade and Development Co-operation DGO6 Wallonia's Operational Directorate-General for Economy, Employment and Research E3 DG Market Regulation and Organization of the FPS Economy, SMEs, Middle Classes and Energy E4 DG Economic Potential of the FPS Economy, SMEs, Middle Classes and Energy EC European Commission EP European Parliament EU European Union EWI Department Economie, Wetenschap en Innovatie of the Flemish government FLEGT Forest Law Enforcement Governance and Trade FIT Flanders Investment and Trade FPS Federal Public Service GEF Global Environment Facility GMO Genetically Modified Organism GI Geographical Indications GR Genetic Resources IA Impact Assessment ILCs Indigenous and Local Communities ILO International Labour Organization Establish monitoring system  Option 1: Voluntary monitoring system  Option 2: "Due-diligence" monitoring system  Option 3: Monitoring by checkpoints at specific stages of the valorization chain Create incentives for users to comply  Option 1: Set up financial incentives (tax reductions, rebates, …)  Option 2: Set up structural incentives (e.g., special priority for other filings, permits or opportunities, access to special materials or programs that cannot be accessed by others)  Option 3: Set up positive publicity measures (e.g. label) Option 2 -Self-standing obligation in the Belgian legislation to have PIC and MAT, if so required by the provider country. Possible advantages:  could create less legal complexity for users and enforcement authorities in Belgium Possible disadvantages:  might be a less stringent measure for acting against potential illegal utilization of GR by Belgian users EVALUATION The 2 options are potentially interesting and deserve further analysis. Action card -Designate checkpoints Description: At least one institution has to be designated by Belgium to function as a checkpoint to monitor and enhance transparency about the utilization of GR Related Article of the NP: competences in Belgium Option 2 -CNA is single point of contact for user on all ABS related permits, but serves as coordination/facilitation body between NFP, other ABS or non-ABS related access granting authorities Possible advantages:  More suited for Belgium, given shared competences Possible disadvantages:  Could lead to lower process certainty for user  Probably longer application process  Would still need a degree of harmonization and integration of the different permits EVALUATION Option 2 is potentially interesting (especially for coordination between various permits/contracts if multiple permits/contracts are requested) and deserves further analysis. Action card -Establish monitoring system Description: Measures should be taken to ensure an efficient monitoring Related Article of the NP: Biotechnologies and processing industry sector The sector is made up of the remaining stakeholders active in the field of biotechnologies, but not active in the healthcare sector. It covers, amongst other, the following fields: energy, materials, biocatalysts, and chemical industries. The processing industry sector in Belgium is mainly focused on food industries and animal feed industry. Ex-situ collections of genetic resources The ex-situ collections sector in Belgium includes over 300 organizations and covers botanical gardens, zoos, aquariums, museums, herbaria, gene banks, collections of micro-organisms/cells, collections of dead material, both in public and private collections. Governmental research institutions Researchers in governmental institutions are accessing genetic resources, as well as traditional knowledge associated with genetic resources, on a regular basis for research purposes. In this category we only consider the specificities of public and academic research, while private research is dealt with under the other stakeholder categories. Key players may include:  The Royal Belgian Institute of Natural Sciences (RBINS);  The Royal Museum for Central Africa (RMCA)  The Walloon Agricultural Research Center (http://cra.walloni.e.be)  Veterinary and agrochemical Research Center (http://www.coda-cerva.be) University research sector Key players targeted are the departments of Belgian universities that deal in particular with life science, engineering, as well as chemical, agricultural, environmental, health research, etc. Other possible stakeholders 8.3.2.1 Civil society Civil society organizations (advocacy NGOs, interest groups, etc.) do not seem to be directly impacted by the Nagoya Protocol as such. However, they might be consulted if they have gathered relevant information on the provision and/or use of genetic resources that can contribute to the impact assessment (if they developed a certain expertise, or have a privileged contact to information from a main user/provider of GR for example). Citizens and consumers Same comment as for civil society. IMP 1.1 -Implementation of option 1 for operationalizing PIC The implementation of option 1 for operationalizing PIC can be broken down in four subsequent components: IMP 1.1.1 -The establishment of BS as a general legal principle Option 1 also includes (as option 2 and 3) the establishment of BS as a general legal principle in Belgium. For this implementation part, the same components are considered as for IMP1.0: (1) A political agreement from the competent governments to establish benefit-sharing as a general legal principle to be implemented for example through a cooperation agreement and/or analogous provisions in relevant legislations such as the basic environmental code of the three Regions and at the federal level. (2) Subsequent or parallel implementation of this general principle through a cooperation agreement and/or analogous provisions in relevant legislation such as the basic environmental code of the three Regions and at the federal level. (3) Subsequent operationalization of the general principle by the respective governments at the regional (through executive orders) and federal level (through royal orders), establishing rules and procedures for further implementation of the benefit-sharing provision as envisioned in the other options considered below. IMP 1.1.2 -Establishing a general legal principle to require PIC for access to Belgian GR In addition, option 1 would require establishing as a general legal principle that access to Belgian GR requires PIC. For the implementation of this principle, the same considerations as those considered for IMP 1.0 apply. Therefore, the same three phased components are considered as for IMP1.0: (1) A political agreement from the competent governments to establish PIC as a general legal principle for access to Belgian GR, with the specifications that this would be implemented for example through a cooperation agreement and/or analogous provisions in relevant legislations such as the basic environmental code of the three Regions and at the Federal level. (2) Subsequent or parallel implementation of this general principle through a cooperation agreement and/or analogous provisions in relevant legislations such as the basic environmental code of the three Regions and at the federal level. (3) Subsequent operationalization of the general principle by the respective governments at the regional (through executive orders) and federal level (through royal orders), establishing rules and procedures for further implementation of the general PIC provision as envisioned in the other options considered below. IMP 1.1.3 -Refinement of relevant legislation for Protected Areas (PA) and Protected Species (PS) Option 1 could be implemented by refining existing PA/PS relevant legislation to establish that access provisions to PA/PS not only concern physical access but also access within the meaning of the Nagoya Protocol and that such access would also amount to prior informed consent from the Belgian State. Once more information becomes available over time regarding experience with the implementation of that general provision and taking into account ongoing discussions and/or practices at international and Party level, the modalities for executing this general principle/provision could be further refined. In addition, IMP 1.1.3 is a further operationalization of IMP 1.1.2, which in itself already provides a sufficient legal basis for establishing PIC as a general principle in the context of the ratification of the Nagoya Protocol. Therefore, any further specification can be made in a later stage after IMP 1.1.1 and IMP 1.1.2, as soon as more experience is available, and for the implementation of this aspect of option 1, the assessment only considers the following component: (1) Amendment of existing legislation relevant for PA/PS to establish that access provisions to PA/PS not only concern physical access but also access within the meaning of the Nagoya Protocol and that such access also automatically amounts to PIC under the implementation of the principle established under IMP 1.1.2. (through a Decree/Ordinance of the Regions) 186 IMP 1.1.4 -Establishing the default access rule (from qualified Belgian collections only) Option 1 would specify that, for GR outside PA/PS, access to Belgian GR would need to be sought and processed as much as possible through qualified Belgian collections (which are equipped for deposit of data and/or samples). Once IMP 1.1.2 is established, IMP 1.1.4 is a further operationalization that is part of the specification of the procedures for processing access requests by the Competent National Authorities, including the designation of the qualified collections by the Regions and the Federal Government. Therefore, under IMP 1.1.4, only the establishment of the general principle is considered, while the detailed operationalization will be considered under IMP 3.1 below. (1) A political agreement from the competent governments to establish a default access rule from qualified Belgian collections between the Regions and the Federal Government which 186 As explained in the detailed analysis in section 3.2., the current access provisions are regulated by various legal measures, depending on the nature of the material, the region and the environmental competence. Therefore, it would be probably most effective (and efficient) to implement this first through a general provision in the basic environmental code and second make specific amendements in the other applicable codes as discussed in ch 3. th January 1999. The second step could build upon several specific existing access regulations (for example whenever access is given for research), see explanations on existing PA/PS relevant legislation (chapters 3.2.2 -3.2.5). Establishing one or more Competent National Authorities The choice of the Competent National Authority would in the first place be based on the relevant competent authorities for the existing legislation and measures related to GR (that is PA/PS and possibly other existing legislation on GR). This means four Competent National Authorities would be needed: one for each of the three regions and a federal one, flowing from the actual division of competences in Belgium. These CNAs would thereby build upon existing institutions and be responsible for granting the access permits. Given this institutional context, the options do not reflect the amount of CNAs to be established but rather the ways in which users can request access (i.e. directly through one of the CNAs vs. through a centralized entry-point). In particular, under a centralized access system, the CNAs would coordinate through channeling, facilitating and/or advising the access requests. This has consequences for the level of comparability of the proposed options. Whereas options 1 and 2 focus on different scenarios to organize the ways in which a user requests access, option 0 focuses on the non-establishment of the CNA. The latter would lead to a non-implementation of the Nagoya Protocol, but still require from the Belgian State to clarify the access procedures to Belgian GR, as discussed in chapter 8. IMP 3.0 -Implementation of the specific "0" option on the Competent National Authorities Idem to IMP 2.0 IMP 3.1 -Implementation of option 1 on the Competent National Authorities The implementation of option 1 on the CNA implies two distinct steps: Summary of the selected options on the Competent National Authority 6. Specific 0 option: non-establishment of the CNAs 7. Option 1: Decentralized input to the CNAs 8. Option 2: Single entry-point to the CNAs For a detailed description of the options please refer to chapter 8.2. IMP 3.1.1 -Establishing the four Competent National Authorities The choice of the Competent National Authorities should take into consideration the division of competences in Belgium on environmental issues, and the objective of the Nagoya Protocol to contribute to conservation of biological diversity and the sustainable use of its components. The choice of the four authorities competent for the existing legislations and measures related to protected areas and protected species, or for other existing legislation on access to GR, would seem logical. Therefore, option 1 and option 2 consider the logical situation where the CNAs would be established in the respective authorities (cf. section 3 of this report), that is the "Agentschap voor Natuur en Bos" in the Flemish Region, the "Division de la nature et des forêts" in the Walloon Region, the "Institut Bruxellois pour la gestion de l'environnement" in the Brussels-Capital Region and one authority to be established at the federal level, probably at the Directorate-General for the Environment of the Federal Public Service "Health, Food Chain Safety and Environment" (for GR that are not under competences of the federated entities, such as Marine GR and ex-situ GR held at federal institutions). Considering that both option 1 and 2 would benefit from the additional legal clarity that will be provided through a timely ratification of the Nagoya Protocol (in particular through the decisions at the first COP/MOP to the NP), two phased implementation components for this option are considered in this assessment: (1) A political agreement from the competent governments to establish four Competent National Authorities to be implemented for example through a cooperation agreement and/or analogous provisions in relevant legislations such as the environmental codes of the three Regions and at the Federal level. (2) Subsequent or parallel implementation for example through a cooperation agreement and/or the analogous provisions of relevant legislations such as the basic environmental code of the three Regions and at the federal level. The specification of the rules and procedures for processing access requests by these Authorities would be done to the maximum possible extent through executive orders of the governments of the Federated Entities. (3) Administrative arrangements could be established between designated ex-situ collections and the CNAs for processing access requests (under option 1 for PIC)or for the management of the notification/registration procedures by the four CNAs (as envisioned under option 2 and 3 for PIC). Such administrative arrangements would not require any additional legal measures (legislative or executive), but could be supported by policy guidance (advice, provision of technical information). IMP 3.1.2 -Decentralized input system A decentralized input would not require any additional implementation measures to the measures under IMP 3.1.1. Designating one or more checkpoints Summary of the selected options on checkpoints 0. Specific "0"Option : No checkpoints would be introduced as envisioned under the Nagoya Protocol 1. Option 1: Monitor PIC in the ABS Clearing-House 2. Option 2: Using the patent office as a checkpoint For a detailed description of the options please refer to chapter 8.2. The provision of Article 17 of the NP is of binding nature. Therefore, as discussed in chapter 8, option 0 (introducing no checkpoints) would lead to a non-ratification of the Nagoya Protocol. Option 1 envisions using the ABS Clearing-House, which monitors the PIC established in the implementation of the Nagoya Protocol, as checkpoint. However, the choice of the ABS CH as checkpoint depends among others on the options chosen for the operationalization of PIC and the ABS CH (cf. discussion on operationalizing PIC above and the ABS CH below). Under option 2, it is the patent office that would function as such a checkpoint. In any case, the tasks of the institution handling the checkpoint should also be further specified, so that it can address, as appropriate, the monitoring of GR and TKaGR used in Belgium, both for Belgian GR and GR or and TKaGR acquired from other countries. The further specification of these tasks can be done gradually, as part of the phased implementation of the Nagoya Protocol, but would include in the first stance the collection and transfer of relevant information on prior informed consent. "PIC as checkpoint" thus can be understood as the collection/reception (by a yet to be defined authority) of the proof of PIC from the provider country (whether that is Belgium or not) as a condition for the utilization of GR in Belgium. Options 1 and 2 are not mutually exclusive and, in a phased implementation approach, it can be envisioned to implement both options together. Checkpoints are monitoring services collecting or receiving information at different stages of the development chain: before utilization (activities such as collecting, identifying and storing GR), during utilization (basic and applied research, and research for product development) or after utilization stages of the development chain (e.g. commercial sale). Options 1 and 2 respectively represent early and later stages of this development chain. The difference is therefore less regarding what is monitored, than when the monitoring takes place. With its focus on the early steps of the development chain, it is assumed that monitoring under option 1 will cover the broadest possible amount of GR and its users in order to be effective. Option 2, on the contrary, would only focus on the specific situations where the utilization of GR is part of a patent application procedure. IMP 5.0 -Implementation of the specific "0" option on checkpoints Idem as under IMP 2.0 (2) Subsequent implementation by establishing cooperation between this CH and other institutions, through appropriate administrative arrangements between all the players involved. Importantly, in this assessment, for comparative purposes, we consider option 1, 2 and 3 separately. However, in practice it is likely that, based on the assessment of the respective strengths and weaknesses of the players, the first step will only involve the RBINS while a combination of the options might be considered for the full implementation of the CH obligations. IMP 6.2 -Implementation of option 2 for the CH Idem as under IMP 6.1 except that BELSPO would be appointed in the first phase for contributing to the information tasks of the CH. IMP 6.3 -Implementation of option 3 for the CH Idem as under IMP 6.1 except that WIV-ISP would be appointed in the first phase for contributing to the information tasks of the CH. Operationalizing PIC Performance of the options Economic impact E1 -Legal certainty and effectiveness for users and providers of GR, at low cost Option 0 is the least preferred option for this criterion: it does not provide any legal certainty to the user, due to the fact that it does not establish proof of legal access; nor is it enforceable as it does not allow any post-access tracking and monitoring to take place; and it makes responsibility so diffuse that no Party can be held accountable. Moreover, it would not allow issuing an internationally recognized certificate of compliance, which is one of the main contributions of the Nagoya Protocol for increasing legal certainty and transparency of exchanges of GR. Under the fishing net model, PIC is operationalized through a simple notification obligation upon the point of access. Therefore, this model provides users with a high level of process certainty and legal certainty, at an early stage of the ABS application process (before identification and storage procedures in the public ex-situ collections or in research laboratories). The simplified nature of the model also allows for a good overview regarding this process (compared to the sometimes lengthy and complex laboratory operations required before a GR can enter an ex-situ collection). However, Summary of the selected options for the operationalization of PIC 0. Specific "0" option (access component): the specific "0" option on access would consider no PIC requirement, with benefit-sharing as a horizontal principle 1. Option 2 has, compared to option 1, a set of advantages pertaining to legal certainty and effectiveness. Indeed, this option is likely to increase legal clarity, as it would lead to a more standardized input system of access requests, and reduce the redundancy in information provision on access procedures. In addition, it can reduce costs related to the search of the adequate information for users, as only one input system will be in place. The set-up costs of a single entrypoint to the CNAs is to be born only once (and might benefit from some economies of scale), while the operating costs are likely to remain low once implemented (for example through a single digital portal as entry-point to the CNAs). These effects would probably be different for foreign users and for Belgian users. If accessed resources are to be used mainly by Belgian users, the impact on users has to be nuanced, as Belgian users are accustomed to the decentralized Belgian system. In contrast, for foreign users, the choice between 4 different entry-points could indeed create confusion and thereby lead to higher legal uncertainty and/or ineffectiveness. Option 0 clearly represents the least favored option as the non-implementation will create high process as well as legal uncertainty for users. It will also make it impossible to keep track of legally accessed resources and hence prevent the public authority to enforce any obligations at a later stage of the utilization. Performance of the options Economic impact E1 -Legal certainty and effectiveness for users and providers of GR, at low cost Option 0 clearly represents the least favored option as the non-ratification will create high process and legal uncertainty for users. Options 1 and 2 will allow ratifying the Nagoya Protocol and thereby allow users and providers to benefit from the higher legal certainty and transparency created by the Protocol. In addition, the monitoring measures put into place under options 1 and 2 are envisioned as an important contribution to promoting transparency and compliance. In a phased implementation of these measures, it is expected that this additional contribution would only be minor in a first phase (as they would cover a sub-set of GR and/or GR on which information on PIC is already available), but this impact is expected to increase in the later implementation stages. However, creating a common level playing field would provide substantial benefit from the outset. Under ideal conditions, option 1 would be looking at the available information at the beginning of the development chain, thereby providing users and providers with the possibility to review whether all genetic resources utilized in Belgium have been acquired in compliance with the PIC provisions of the provider country. However, much will depend upon the effectiveness of the ABS Clearing-House(s) (internationally, but also in both the provider country and Belgium, cf. chapter 9.5). In case of ineffective transfer of information between the provider country and Belgium, users may face situations of uncertainty. Furthermore, the enforceability of the option is very doubtful, as it will prove hard to systematically control the high quantity of GR being utilized in Belgium from a high variety of sources and by very different users. However, a phased implementation might be a possible answer to these concerns. Option 2 provides both providers and users with less possibility to monitor the correct use of the GR in Belgium than option 1. The patent stage is an advanced stage in the development process. Collecting proof of compliance at this late stage could generate uncertainty for users using GR that have transited through third-parties. By putting the burden of proof at the end of the development Procedural impact G1 -Flexibility to accommodating sectorial differences The general information exchange tasks and the organization of the technical information do not make any differences amongst the sectors. Therefore, the options can be considered neutral in regards to this criterion. G2 -Temporal flexibility to allow for future policy and adjustments The 0 option would lead to non-implementation of the Nagoya Protocol. As discussed under G2 above (section G2 under MAT), this might lead to some additional flexibility in the short term, but would probably lead to higher adaptation costs at a later point in time. The temporal flexibility of the other 3 options will highly depend on the initial set-up costs of the various obligations. If these are high, then it might lead to less flexibility to change the options later on. In addition, if high level of technical expertise was required this might lead to less temporal flexibility in the change of the options, as it would necessitate to acquire again the same expertise by other actors. Both these arguments lead to favor options that build upon existing practices and expertise, over options that less do so. This applies equally to all three options. G3 -Improving knowledge for future policy developments and evaluation The option 0 would lead to non-implementation of the Nagoya Protocol and would not allow the necessary generation of PIC/checkpoints etc. that is useful for informing future policy developments. In contrast, the other options would allow improving and systematizing this knowledge base. As in the case of criterion G2, solutions that build upon existing practices and expertise would probably give a better guarantee of knowledge quality and integration, compared to solutions that less do so. G4 -Correspondence with existing practices The impact of option 0 over this criterion is unclear. Indeed, it would depend on the other measures taken by Belgium to comply with CBD and Convention ILO 107. For options 1, 2 and 3, all three options build upon existing practices for information coordination on ABS issues in Belgium and/or biodiversity policy matters. The RBINS already hosts the Belgian component of the CBD Clearing-House Mechanism (CHM) and the NFP to the CBD. RBINS is, together with BELSPO, part of the Belgian Biodiversity Platform (BBP), while the ISP/WIV hosts the Belgian Biosafety Clearing-House (BCH). Visual dominance analysis No ideal point can be identified in the performance chart. technical information to be provided to the central ABS Clearing-House, amongst others. The first task is already ongoing at the Royal Belgian Institute of Natural Sciences (RBINS) within the framework of the CBD CHM. The recommendation from the analysis is therefore to further strengthen the RBINS to fulfill the information sharing tasks on Access and Benefitsharing under the Nagoya Protocol. In a second stage, based on the modalities to be determined at COP/MOP1, administrative arrangements between this Clearing-House and other relevant institutions might be necessary to extend the tasks. Before summarizing the final outlook of the phased approach based on these recommendations, it is worthwhile to clarify that the three step approach to the implementation presented here is based on a concern for maximum legal clarity for all parties concerned and compliance as a Party with the core obligations of the NP, while at the same time allowing a timely ratification. The proposed approach is therefore to start with a political agreement which would include, in general terms, the principles on which the federated entities and the Federal State will take subsequent actions. The reason for recommending such a political agreement with a specification of the actions that will have to be taken to implement the Protocol in Belgium is double. On the one hand, such an agreement provides for a clear political commitment to the core obligations of the NP as it specifies the intentions of the competent authorities, within the limits of the decisions already taken at the international and European level at the time of the agreement. On the other hand, it does not prejudge the political decisions to be taken by the different authorities and thus allows for sufficient flexibility to further adjust the implementation process in a later stage. The latter is especially important given the many questions that are still undecided at the present stage, both at the EU and international level, as mentioned and taken into account in the assessment report. Based on the above considerations, the recommended phased approach for implementation of the Protocol that results from this study can be summarized as follows. 1. A political agreement by the competent authorities with a clear statement on the general legal principles for a minimal implementation of the Nagoya Protocol in Belgium that came out of the study : a. Establishment of benefit-sharing as a general legal principle in Belgium, which will be implemented for example through a cooperation agreement and/or analogous provisions in relevant legislations such as the environmental codes of the three Regions and at the federal level 226 (IMP 1.1.1 (1)) b. Establishing as a general legal principle that access to Belgian GR requires PIC, which will be implemented for example through a cooperation agreement and/or analogous provisions in relevant legislations such as the environmental codes of the three Regions and at the federal level (IMP 1.1.2 (1)) Conclusion of the third and fourth phase of the study The main conclusions of the third step have been presented in detail in the chapters 10 and 11 of the report and resulted in two general recommendations, along with a set of more specific recommendations for each of the 6 implementation measures. First, the analysis shows that the "no policy change" baseline for each measure clearly has the worst performance. This result has led to a first general recommendation, which is to implement both Prior Informed Consent (PIC) and benefit-sharing as general legal principles in Belgium. Second, the analysis confirmed the validity of a phased approach to the implementation of the Protocol, which is the second general recommendation and could be organized through a 3 step implementation process: 1. In the first implementation step, through a political agreement by the competent authorities which would include a clear statement on the general legal principles, along with the specification of the actions to be undertaken by the federal and the federated entities to put these principles into practice. 2. In a second implementation step, the specified actions would be subsequently implemented for example through a cooperation agreement and/or analogous provisions in the relevant legislations such as the environmental codes of the federated entities and the Federal Government, along with other possible requirements. 3. In a third implementation step, additional actions can be undertaken once there is more clarity on the EU and the international level. Finally, a set of specific recommendations on each of the 6 measures arise from third step of this study: 1. For the establishment of the Competent National Authorities, a centralized input system clearly came out as the recommended option. 2. For the setting up of compliance measures, the option to refer back to provider country legislation, with Belgian law as fallback option, is the recommended option that comes out of this analysis. 3. For the designation of one or more checkpoints, the option of using the PIC of users available in the international ABS Clearing-House (and therefore also through the Belgian node/or the Belgian ABS CH), in the first step of the implementation, stands as the recommended option. 4. For the operationalization of PIC, the bottleneck option and the refined fishing net option came out very close. a. First, both these options require establishing as a general legal principle that access to Belgian GR requires PIC. This could be implemented for example through a cooperation agreement and/or analogous provisions in relevant legislations such as the environmental code of the three Regions and at the federal level b. Second, additional measures should be envisioned afterwards, the most important of which are the refinement to existing PA/PS relevant legislation and the general registration/notification requirements to the Competent National Authorities for GR outside PA/PS. ANNEX 1 -OVERVIEW OF ARTICLES OF THE NAGOYA PROTOCOL THAT CONTAIN LEGAL OBLIGATIONS FOR A PARTY/PARTIES This list contains an analysis of the legal obligations emanating from the NP that has been provided with the terms of reference of this study, by the four Belgian environmental administrations that commissioned this study. This list serves as the background for this study.

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Dr Phillip Lee M P Chair Foreword W e are in tough economic times and there is no doubt that the overall British approach and our short-term expectations with regards to environmental policy will have to be revised. At the same time, as the impact of climate change becomes apparent, we do then have to address the fundamental problems and act swiftly to protect the environment. Britain has a dual approach to European and international climate change policy: to work with key countries, partners and sectors to demonstrate the potential of low carbon growth, and at the same time to make progress with a legally-binding global deal curbing emissions to tackle climate change. Our focus will now have to be on building a global roadmap to an agreement by 2015 and we also need a strategic and viable approach to renewable energy. I am delighted that, under the UN Framework, both developed and developing countries were able to join together and agree on the strategy at the recent Durban Climate Change Conference -European leadership played a large part in this. In addition, a second commitment of the Kyoto Protocol was agreed and I am proud of the role that the UK played in galvanising support for both agreements. Europe is an essential global player in shaping overall targets and leading the way on environmental policy. It is a hugely complex area and so I am pleased to present this report. I hope that you will find it useful in guiding you through the maze of proposals that cover the wide range of issues that need to be tackled. These issues are certainly very significant, but it is clear that we have the opportunity to make real progress if we act now. Executive summary EU environmental policy is facing a new and challenging context. The current economic and financial preoccupations in Europe are unlikely to fade away quickly. It is difficult to forecast when instabilities in financial markets, uncertainties over economic and job prospects and pressure to maintain austerity regimes will end. The crisis in the Eurozone has led to bigger questions concerning the role of regulation and aspects of the EU project itself; particularly but not exclusively in the UK where political tensions have been brought to the fore in recent months. Details of a new inter-governmental agreement on the economic governance of the Eurozone are currently being negotiated. Most existing EU policies, including those concerning the environment, are not likely to be affected by this agreement. However, the political repercussions and dynamics of the new economic governance structure are yet to unfold and may spread beyond the arenas of fiscal and budgetary policy. For these reasons, conditions for the further development of a proactive EU environmental policy may not look favourable. Nonetheless, several environmental challenges call for a response as a matter of urgency, both within Europe and on a global scale. Many of these issues need to be addressed at a European level and there is a clear link to the single market as well as the ecological integrity of the continent. The current economic situation also offers a number of opportunities for promoting the environmental policy agenda, particularly in view of fostering an efficiency revolution. It has given an impetus to concepts such as the green economy, green growth, resource efficiency etc., which are increasingly reflected in mainstream political discourse both in the EU and domestically. Thus, even in a period of economic recession and political upheaval, the environmental perspective should remain a cornerstone of strategies for the future economy. Whilst far from perfect, environmental policy is certainly one of the success stories of the EU and is an area in which one can clearly see the benefits of the Union, both on the ground and internationally. Over the past four decades, a range of key pressures on the environment have been reduced and several aspects of Europe's environment have improved. Major progress has been made, including reductions in overall air and water pollution, improvements in the preservation of the natural environment and efforts in relation to waste and resource use. EU policy has played a very significant role in achieving these results. A particular strength of EU policy is that it addresses the environmental agenda rather systematically and is less affected by short-term political and budgetary shocks than most national governments. This has provided the conditions for a longer-term view which is particularly valuable in environmental policy. In the next two to three years, safeguarding jobs and stimulating growth are likely to remain an over-riding political priority. Thus securing support for new environmental measures will mean convincing leaders of the costs of inaction and the cost-effectiveness of action. Strong arguments and solid evidence will be at a premium. At the same time, policy is moving in new directions. The rise of emerging economies is dramatically changing the international landscape and the role of the EU therein. Moreover, the nature of contemporary environmental challenges is such that many cannot be addressed by environmental policy alone. Rather, they require wider economic and social changes, with implications for a suite of policies, ranging from trade and international relations to industrial policy, research and development, and fisheries. Although relatively comprehensive already, the body of EU environmental policy remains dynamic and is constantly being updated, subjected to scrutiny and potential modifications or roll-back. It is now at a critical point, with a number of important policy processes and strategic discussions taking place in various areas. Many of the key areas of policy development are reviewed in Chapter 4 of this report, while Chapter 5 provides an overview of the wider strategic context. The Annexes of the report provide an overview of forthcoming strategic events, EU targets and legislative proposals awaiting adoption. The following are amongst some of the key issues and policy processes that will be prominent on the agenda in the next two to three years: • Climate change concerns have infiltrated the main political discourse and there are currently several issues on the agenda. A shift to a 30 per cent EU greenhouse gas emission reduction target remains possible, as well as desirable, despite the reduced impetus from global negotiations. There are also specific proposals to promote energy efficiency more effectively. Other issues on the agenda include addressing emissions from the transport sector and the decarbonisation of transport fuels, as well as securing finance for climate-related investments within the EU and externally • A new approach to resource use in Europe is signalling efforts in the next eight years to improve resource efficiency and, more tentatively, to reduce resource use, while linking to strategies to promote green growth. This emerging agenda needs to be converted into concrete actions at EU and national level starting with the development of concrete targets and indicators for reducing resource consumption. • With regard to the natural environment, the valuation of natural capital and ecosystem services is increasingly recognised. However, it needs to be translated into concrete measures to protect biodiversity in practice, including adequate funding for Natura 2000 and a revised approach in the CAP. • Comprehensive reviews of existing legislation in a number of important areas of EU environmental policy are underway. A 'Blueprint to safeguard Europe's waters' is expected to be presented in November 2012, addressing the broad scope of EU water policy and making recommendations for improvements. These might include legislative changes and initiatives to improve implementation, which has been slow in the case of the Water Framework Directive for example. A review of the EU's approach to regulating the production and use of chemicals through the REACH Regulation is also expected in 2012. A review of EU air quality policy should conclude in 2013, with the presentation of a new clean air package, updating existing policies and directives. • In addition to specific legislative developments, there are also a host of strategies and roadmaps which set out where the EU is heading on the economy, energy policy, climate, innovation and the environment itself. Particularly significant in 2012 will be the emergence of a proposal for the 7th Environment Action Programme which is expected to set the direction for EU environmental policy for the coming years. • Funding for the environment will be a frontline issue with the EU budget for 2014-2020 in principle being agreed during the year. Reforms of the Common Agriculture, Cohesion and Common Fisheries Policies, all with large environmental components, will run through 2012 and beyond. The investment needed to achieve EU environmental objectives and to support the transition to a low-carbon, resource efficient economy is substantial. Rising public debts in several Member States and flailing capital markets have dented the ability to invest in the critical infrastructure and innovative technologies and services. Therefore, securing adequate financing to support environmental commitments in the main EU funds will be an important test of the commitment to environmental progress. The outcomes of these processes will have an important influence on the context and scope of EU environmental policy to 2020 and beyond. In broad terms, the main environmental challenges ahead include reducing the intensity of natural resources used for economic activity, decreasing the negative environmental impacts associated with the use of natural resources, preserving and restoring natural capital and ecosystem services, and improving human well-being and quality of life. The inter-linkages and trade-offs between different thematic areas such as climate change, biodiversity, and natural resources, as well as between environmental policy and sectoral policies such as agriculture, energy or transport will need to be addressed more vigorously. Improving the implementation of existing environmental policy has been and remains a key challenge. It requires a more honest alignment of aspirations, regulatory means and implementation capacity with the political realities of a Union of 27 Member States. Maintaining sufficient institutional and administrative capacities for good governance and regulatory foresight in the face of fiscal austerity and pressures for budgetary cuts will be an important challenge for national authorities as well as the EU institutions. However, recent history suggests that well designed and effectively implemented environmental policy can provide some of the foundations for long term prosperity, as well as steering us towards a more sustainable society. Introduction The European Union (EU) is currently embroiled in significant turmoil. The crisis in the Eurozone has not only had serious economic, political and social ramifications, but has also led to bigger questions concerning the EU project itself. It has exposed some of the weaknesses of the Union within a rapidly changing global context, and has highlighted the overwhelming need for reform in the Eurozone, with possible implications beyond fiscal policy. Growing scepticism about the EU has been voiced in a number of Member States, including the UK where a skirmish over a referendum on the UK's membership of the EU in the autumn brought some of the underlying political tensions to the fore. A veto exercised by the British Prime Minister at a December Council meeting of EU leaders created further furore, both among other Member States and domestically, and may have marked a significant moment in the history of the UK's relationship with its European partners. The details of the subsequent inter-governmental agreement are currently being finalised. Although many existing EU policies, including those concerning the environment, are not likely to be affected by the agreement; the new economic governance structure can be expected to create a momentum of its own and the political repercussions and dynamics of this are yet to unfold. Given this rather volatile climate, some are likely to question the value of having a discussion on EU environmental policy at all. However, it would be a mistake to underestimate the significance of environmental policy on account of the current situation. Whilst far from perfect, environmental policy is certainly one of the success stories of the EU and is an area in which one can clearly see the benefits of the Union, both on the ground and internationally. In the short-to medium-term, the EU is likely to remain the primary source of environmental policy within Europe, driving most national decisions in the same way it does today. Many environmental issues need to be addressed at a European level and there is a clear link to the single market as well as the ecological integrity of the continent. Moreover, while there is a preoccupation with economic performance at present, this needs to be linked to a green agenda, both in the EU and domestically. Concepts such as the green economy, green growth, resource efficiency etc., are increasingly reflected in mainstream political discourse. A relevant example is the Europe 2020 Strategy which has an explicit environmental dimension to it. Thus, even in a period of economic recession and political upheaval, the environmental perspective should remain a key component of strategies to stimulate growth. Despite progress in a number of areas, the overall state of Europe's environment and the EU's impacts on the environment in other parts of the world still require further action [START_REF] Eea | EU Environmental Policy Handbook: A Critical Analysis of EU Environmental Legislation, Making it accessible to environmentalists and decision makers[END_REF]. Inter-linkages between thematic areas such as climate change, biodiversity, natural resources and environment and health, and their links with sectoral policies such as agriculture, energy or transport are far from being adequately addressed. As in the UK, the EU's environment and climate change agenda is now covered by different departments within the European Commission, thus further increasing the need for coherence and coordination. Improving implementation of the substantial body of law already agreed has been, and still is, a key challenge. There are also a number of remaining gaps in the coverage of EU environmental policy, such as tackling climate change adaptation, addressing water scarcity, issues related to nanotechnologies and the cocktail effects of chemicals, where new measures may still be developed. Markets and consumers continue to get distorted price signals that do not fully account for the cost of environmental damage and significant knowledge gaps exist for several environmental problem areas. Addressing these and other remaining challenges will be a priority in the coming years. Although relatively comprehensive already, the body of EU environmental policy remains dynamic and is constantly being updated, subjected to scrutiny and potential modifications or roll-back. It is now at a critical point, with a number of important policy processes and strategic discussions underway. They include the adoption of new strategic plans including in relation to future biodiversity policy and comprehensive reviews of existing legislation in important areas of EU environmental policy such as water, air and chemicals. Substantial discussions on the future EU budget are currently taking place, as are efforts to take forward the EU's new economic strategy -the so-called Europe 2020 Strategy and related Flagship Initiatives. Alongside this, there are also a number of reform processes in sectors with a large environmental impact such as agriculture, transport and energy. These different processes are taking place against the backdrop of a difficult economic and political climate and will have an important influence on the context and scope of EU environmental policy to 2020 and beyond. This report provides an update to the 2009 IEEP report for the All-party Parliamentary Environment Group on the 'The Future of EU Environment Policy: Challenges and Opportunities'. It begins with a brief review of the development of EU environment policy and the changing nature of environmental issues addressed. It then goes on to provide a brief review of the key environmental challenges facing the EU in a number of different thematic areas. This is followed by an overview of the main policy and strategic discussions currently underway. The report concludes with a discussion on some of the prospects and key challenges for the future. Setting the scene Over the past 40 years, the EU has set up a relatively comprehensive and dense body of environmental legislation (the environmental acquis) which is accredited with reducing several pressures on the environment and improving environmental standards in the majority of its Member States, including the UK. The number of items of EU environmental legislation has increased rapidly over the years (see Figure 1) and in the UK it is estimated that over 80 per cent of environmental legislation originates in the EU. The same is probably true of most other EU Member States. Major progress has been made, including in relation to reductions in overall air and water pollution, improvements in the preservation of the natural environment and efforts in relation to waste and resource use. A number of factors have contributed to these achievements including strong networks of environmental actors, political will, and the effective use of 'windows of opportunity'. While there has been a good pace in the adoption of EU policies relating to the environment, progress has not been linear and has tended to be sensitive to wider economic and political cycles (Hey, 2005). The focus of EU environmental policy has shifted from an initial emphasis on addressing issues within Europe to growing consideration of the international dimension. The approach has evolved from an emphasis on controlling pollution from point sources, mainly through 'end of pipe' legislation towards a more holistic, integrated approach aimed at tackling the underlying causes of environmental damage, particularly in key economic sectors such as agriculture, transport and energy. Actor constellations have changed accordingly. The formulation and implementation of EU environmental policy now takes place in a complex system of multi-level governance involving not only environment Ministries and the Commission's Directorate-General (DG) for the Environment, but also sectoral Ministries and other DGs within the Commission, different levels of government, and a growing number of non-state actors. The European Parliament has also played an increasingly active and assertive role in relation to the development of EU environmental policy over the years. Today's environmental challenges are no longer distinct, independent and straightforward. The increasing complexity of inter-linkages between policies on climate change, biodiversity, natural resources, and environment and health has become ever more apparent in recent years. For example, the link between water and climate change is increasingly evident as large parts of Southern Europe are affected by water scarcities while other parts of Europe suffer from a rise in the frequency of major floods and related damage. Over the past few years, the focus of environmental policy has also shifted towards resource inputs in the economy and their environmental impacts. This agenda is taken forward not only by focussed environmental initiatives but also by economic policy drivers and their offshoots, including the Europe 2020 Strategy and related Flagship Initiatives. In particular, the 2011 resource efficiency Flagship Initiative has spawned a number of relevant strategies including the low carbon Roadmap and the resource efficiency Roadmap and has led to the re-conceptualisation of a number of environmental issues to relate them to the resource efficiency agenda. The web of different policies and strategies and the links between them is becoming more complex and sometimes less transparent. For example, feedback mechanisms between policies can lead to unintended negative side effects from well-intended measures, such as the case of indirect land use change impacts of biofuels (Bowyer 2010). Moreover, in a highly interdependent world, many key drivers of environmental pressures operate on a global scale and are likely to unfold over decades. Changes in one region can trigger a cascade of impacts that also affect other regions. The World Economic Forum has warned that a comprehensive set of interlinked global risks (see Figure 2) is evolving and that domestic governance systems lack the capacity to deal with them effectively (WEF 2011). Against this challenging backdrop, the political difficulty of agreeing different response options has increased markedly. In the current context of an extended economic crisis, many of the enabling conditions which have helped advance the development of EU environmental policy in the past are no longer fully present. How to ensure sustainability and promote environmental objectives in times of austerity is thus a critical question as we move forward. Policy is moving in new directions and a major expansion of the EU environmental acquis no longer seems likely in the coming decade. Moreover, the nature of environmental challenges faced today is such that they cannot be addressed by environmental policy alone. Rather, they require wider economic and social changes and relate to a suite of policies from trade and international relations to industrial policy, research and development and fisheries. Over the past four decades, a range of key pressures on the environment have been reduced and several aspects of Europe's environment have improved. EU policy has played a very significant role in achieving these results. However, a number of serious challenges remain; some in areas with a long history of policy efforts (e.g. managing waste, biodiversity) and those where efforts have been more limited to date (e.g. aviation, marine environment, transport). The EU's impact on the environment in other parts of the world, our so-called 'footprint', continues to grow. Further action is still required. The 2010 review by the European Environment Agency on the state of the European environment and outlook (EEA 2010c) identified a number of future priorities. These included the need to improve implementation and the management of natural capital and ecosystem services, to further integrate environmental considerations in sectoral policy domains, such as the CAP, and to achieve the transformation to a green economy. These will be amongst the principal challenges in the years ahead. However, it is less clear how we are going to address them. Given the current economic and political climate, there is a growing sense that the EU cannot proceed by regulation alone. There are active debates underway on the pros and cons of different types of intervention, their effectiveness, the associated administrative burden, costs entailed, etc. Issues of competitiveness and growth have been brought to the forefront of political priorities, creating tensions with environmental objectives. These questions are not purely theoretical; they are feeding into contemporary decisions over the next generation of policies. In the following sections we offer an overview of how these issues are being addressed in a range of themes of particular environmental significance. They include a summary of the current EU policy response and future options being explored. It is not intended as a comprehensive account of all areas of EU environmental policy, but is deliberately selective. Climate change and energy Despite being among some of the largest emitters of greenhouse gases (GHGs) in the world, EU Member States are also among the most active in seeking to address the issue. At the international level, the EU negotiates as a bloc within the UN Framework Convention on Climate Change (UNFCCC) and its Kyoto Protocol. The Kyoto Hungary, Latvia, Lithuania, Poland, Romania, Slovakia Sweden and the UK) (EEA 2011). Some, including Austria, Italy and Spain, are significantly off track. Moreover, these figures do not cover Europe's emissions effectively imported through the substantial trade of goods and services with third countries. Net emission transfers from third countries, particularly from newly advanced economic countries such as China, have increased continuously since 1990 and in total may offset Europe's emission reductions [START_REF] Peters | Growth in emission transfers via international trade from 1990 to 2008[END_REF]. Progress towards the achievement of the other 20-20-20 targets is mixed. Despite progress under several energy saving policies, estimates suggest that the projected impact of these policies would need to triple in order to meet the target of saving 20 per cent of energy use by 2020 (Ecofys and Fraunhofer Institute 2010). By contrast, the share of renewable energy sources continues to increase steadily (Figure 4 overleaf). Several Member States (Austria, Bulgaria, Czech Republic, Denmark, Germany, Greece, Spain, France, Lithuania, Malta, Netherlands, Slovenia and Sweden) are forecast to surpass their own targets, putting the EU on track to meet (or even exceed) its target of increasing the share of renewable energies in the overall energy mix to 20 per cent by 2020 (EC 2011d). The UK has a challenging target under the renewable energy Directive which it aims to meet mainly by a sizeable increase in wind power. A number of studies have demonstrated that more ambitious climate mitigation policies are needed in Europe and are technically and economically feasible. They point to the economic benefits of an ambitious climate policy which could function as a motor for modernising the EU economy and its infrastructure, create jobs and enhance Protocol commits the EU-15 to reducing average GHG emissions by 8 per cent below 1990 levels between 2008 and 2012. The EU has developed an array of internal policies to implement its international commitments and also to achieve the more ambitious overarching objective of limiting global warming to two degrees Celsius above average pre-industrial temperature levels. In 2009, the EU adopted a package of climate and energy measures to implement the so-called '20-20-20' targets agreed by EU leaders in 2007 as the centrepiece of EU climate policy. The targets are to reduce GHG emissions by 20 per cent, increase the share of renewable energy by 20 per cent, and reduce energy consumption by 20 per cent, all by 2020. Although a number of Member States, including the UK, continue to support a move to strengthen EU emission reduction targets, from 20 to 30 per cent, political efforts have yet to bear fruit in this regard. While overall figures offer some encouragement (see Box 1 above), major differences remain between Member States (see Figure 3). competitiveness in fast growing global markets for low-carbon goods and services. Such a proactive approach is supported by many Member States including the UK, and a substantial group of commercial as well as environmental stakeholders are arguing for the adoption of a 30 per cent emissions reduction target as soon as possible (see Box 2 overleaf). However, there continue to be some Member States which are reluctant to take this step and the debate over moving to a 30 per cent target remains central to EU climate policy. Whilst it is important to resolve this, the eight year period to 2020 is only the near horizon. Looking beyond 2020, current information indicates that existing and planned measures on their own are not likely to be sufficient to bring the EU on a pathway to achieve its long-term emission reduction objective of 80-95 per cent by 2050 compared to 1990 levels (EEA 2011). In the near future, new climate measures will be coming into force and others are in the pipeline. These include progressive reinforcement of the EU Emissions Trading System (ETS) which is entering its third phase in 2013 and has a pivotal role in mitigation policy, although some are questioning this in the light of the low carbon price at present. The ETS has been extended to include additional GHG gasses and to include aviation from January 2012 (see next section on transport). Energy efficiency has been less prominent in EU policy but is the topic of considerable action at present. The background is provided by the Commission's energy efficiency Plan presented in March 2011 which aims to ensure the EU delivers on existing policy commitments and goes beyond this to achieve a 25 per cent overall GHG emission reduction by 2020 (EC 2011e). The public sector is allocated a key role in driving change, in particular through the market power of public spending. Complementary to this are the November 2010 energy 2020 strategy and the December 2011 energy Roadmap 2050 which explores different scenarios for the decarbonisation of the energy system (EC 2011h). The Roadmap concludes that the decarbonisation of the energy system is technically and economically feasible and that the overall costs of transforming the energy system are similar in all scenarios. The Roadmap maintains that while energy prices will rise until 2030 or so, new energy systems can lead to lower prices after that. A major programme of new investments will pay off in terms of growth, employment, greater energy security and lower fuel costs. The Roadmap is to be followed by initiatives in specific energy policy areas starting with proposals on the internal market, renewable energy and nuclear safety in 2012. The EU has been prominent in international climate change negotiations, playing a leading role in the establishment of the Kyoto Protocol and in maintaining a future for it after 2012. To some extent it has been willing to accept greater emission reduction commitments than most other major players, such as the USA and Canada, although these have not been very demanding. However, it is now signalling that further European action depends on the willingness of others to commit. The latest round of UNFCCC negotiations in Durban, South Africa led to an agreement among all parties to draft a new protocol, legal instrument or an agreed outcome with 'legal force' by 2015. This was a key condition of the EU to commit to new binding emission reduction targets under a second commitment period of the Kyoto Protocol from 2013. Several countries, including Norway, Iceland and Switzerland, will also be part of this second commitment period; although others including Canada, Russia, and Japan will be notably absent. The targets and time span of this new scheme will be discussed and finalised next year. The EU remains pivotal to the future of the Kyoto Protocol and after a disappointing performance at the 2009 conference in Copenhagen seems to have somewhat re-established its leadership on climate change on the global stage. This is to be welcomed and has provided a much needed boost to EU morale in this area. However, a significant amount of work lies ahead to deliver on the agreed deal and to mobilise the resources that have been committed to developing countries to help them respond to the climate challenge. This will be a priority for the EU and for the more active national governments, such as the UK. Globally or domestically, it will only be possible to adequately address the climate agenda by building a low carbon economy. This is a very substantial undertaking but the EU has started to move in this direction. Climate change concerns have begun to infiltrate the economic discourse and are now reflected in the EU's core economic strategy and in its spending priorities to 2020. Despite the difficult economic context, efforts are being made to incorporate climate change concerns in relevant policies such as energy, transport and regional policy, and are leading to the gradual acceptance of a new approach. Key challenges for the EU in the next decade include the following: • Consolidating, reinforcing and strengthening the existing agenda (e.g. extending action on energy efficiency through mandatory and other measures) so that it is in line with the trajectory set out in the EU's 2050 low carbon roadmap. • Building climate concerns more tightly into economic policy, including measures aimed at innovation and research and at key sectors such as energy supply and transport. • Making more progress with securing adequate finance for climate related investments both within the EU and externally. • Improving coherence between climate change and other environment policy areas so as to ensure an integrated approach in which climate change considerations are sufficiently embedded in other policies and to address any potential negative environmental aspects of climate measures, as exemplified by the biofuels debate. • Securing an appropriate global framework for addressing climate issues more urgently. Transport The transport sector continues to be a source of significant environmental pressure in the EU. Emissions from transport are a major source of the EU's GHG emissions (Figure 5 below). In 2010 transport was responsible for more than a fifth of GHG emissions from the EU (EEA 2011c). The increasing demand for transport has offset potential gains from improvements in the energy efficiency of new vehicles. Transport emissions also exacerbate problems with poor air quality and noise, particularly in urban areas. Additionally, transport infrastructure and its users constitute one of the main drivers of pressure on Europe's ecosystems and biodiversity, particularly with regard to the fragmentation of landscapes and ecosystems, and account for the use of large quantities of raw materials. Given its contribution to EU GHG emissions, the transport sector has become an increasingly important target of the EU's climate change and energy policy. Under the 2009 renewable energy Directive, there is a target that a minimum of 10 per cent of the final energy consumption used by transport is to come from renewable sources by 2020 in each Member State. A 2009 amendment to the fuel quality Directive included a target for the reduction of lifecycle GHG emissions from liquid transport fuels of at least 6 per cent by the end of 2020. In March 2011, for the first time, the Commission proposed a specific GHG emission reduction target for the European transport sector. In the White Paper on transport, the target is to reduce GHG emissions from transport by 60 per cent by 2050, in addition there are a number of goals for a competitive and resource efficient transport system (EC 2011i). Policy on vehicle emissions is now an important means of pursuing progress in the sector, albeit relatively slowly. Fuel efficiency requirements for new passenger cars are established in the passenger car CO 2 Regulation. This sets an average target of 130gCO 2 /km to be met by manufacturers by 2015 and an average target for 2020 of 95gCO 2 /km. Another Regulation aims to reduce average CO 2 emissions from new vans to 175gCO 2 /km by 2017 and 147gCO 2 /km by 2020 [START_REF] Skinner | Carbon impact of HS2: Overview of relevant policy issues and advice on modelling assumptions[END_REF]. A report on progress towards the targets set out in the passenger car CO2 Regulation suggest that in 2010, the average new car in the EU-27 had CO 2 emissions of 140.3g/km. This is an improvement of 3.7 per cent on the 2009 figure. In the UK, average CO 2 emissions of new cars declined from 150g/km to 144g/km (Transport and Environment 2011). These figures provide some initial indications of the effectiveness of the regulatory measures introduced following the failure to deliver significant emission reductions under an earlier voluntary agreement between the Commission and car manufacturers. A revised euro-vignette Directive on road charging of heavy good vehicles (HGVs) agreed in 2011 allows Member States to levy additional charges on HGVs to cover the cost of the air and noise pollution they create in addition to existing infrastructure charges. Although Member States are currently not obliged to introduce such charges, the measure paves the way for a more intelligent approach both to taking account of external costs in charges and achieving greater European harmonisation, and could ultimately enhance incentives for the use of cleaner vehicles. Concrete measures have also been introduced in relation to aviation. In 2008, a new Directive was adopted to expand the scope of the EU ETS to include the aviation sector. However the Directive, which requires all flights landing and taking off from EU airports to be covered by the ETS from 2012, therefore levying costs on airlines, has been condemned by a number of developed and emerging economies which do not have equivalent measures themselves. Just how controversial the measure has been is evident in the actions taken over the past few months which include the approval by the US House of Representatives in October 2011 of a draft law that if passed would ban US airlines from participating in the ETS; threats of legal action by the Chinese Air Transport Association and the Indian government; and a legal challenge brought before the High Court of Justice of England and Wales by a group of US airlines. Aside from the question of emissions, the transport sector is the principal consumer of oil-based fuels on which it is almost wholly dependent. Currently there is a major push to encourage the use of biofuels and to accelerate the process of electrifying road vehicles. There are, however, many challenges to ensuring that these alternative fuels and energy sources are sustainable and in fact low carbon. Efforts to decarbonise the transport sector are inevitably linked to efforts to decarbonise energy supply (European Expert Group 2011) as the use of low carbon electricity in the transport sector requires the decarbonisation of the electricity supply industries. Moreover, GHG benefits from biofuels varies according to the feedstock used, direct land use change and associated emissions from planting feedstock, emissions from processing, transport and the use of the by-products, as well as the potential impact of indirect land use change (Bowyer 2010). In addition, EU and other public spending will need to be shifted to the creation of new low-carbon infrastructure (such as high-speed rail, new bus and rapid urban transit systems in cities) and charging facilities for electric vehicles, as well as initiatives to reduce the need to travel. In October 2011, the Commission set out its proposal for the funding mechanism for EU infrastructure priorities in the transport, energy and telecommunications sectors in the 2014-2020 period -the Connecting Europe Facility (CEF) (EC 2011j). A proposed budget of €31.7 billion is to be invested in transport infrastructure. However the total amount of financing available may be significantly larger with the use of the new EU financial instrument -the Project Bond Initiative -which will be used to attract additional private finance to EU priority projects [START_REF] Withana | Mobilising private investment for climate change action in the EU: The role of new financial instruments[END_REF]. These are potentially powerful levers for change if they can be directed into the most appropriate areas of investment. The transport sector is one in which GHG emissions continue to rise; policy interventions remain quite controversial and in some cases are relatively recent. There are a number of important issues on the horizon, including the following: • The adoption of mandatory targets for vehicle emissions has proved quite successful in reducing emissions in the last year, although there remains some way to go. A proposal to amend legislation on CO 2 from cars and vans is due in 2012 and may lead to the development of targets beyond 2020. Similar legislation for CO 2 emission standards for vehicles in other modes could be developed in the coming years, as called for in the 2011 White Paper on transport. • We are now part way through the process of extending provisions in the EU ETS to the aviation sector. Although this expansion has been agreed by the EU, it has proved to be particularly controversial among international players and is under heavy fire from the US in particular. It is nonetheless important if rising emissions from the sector are to be tackled effectively and addressing related concerns will be a key challenge in the coming year. • The issue of shipping emissions is still at an early stage in development and the Commission is expected to put forward proposals in this regard in 2012 in the absence of international action. • Another key challenge relates to the decarbonisation of transport fuels and ensuring the sustainability of alternative fuels and energy sources as well as addressing issues arising from the use of 'unconventional' sources such as oil sands which have the potential to increase the carbon content of transport fuel. Here the fuel quality Directive is pivotal [START_REF] Skinner | Carbon impact of HS2: Overview of relevant policy issues and advice on modelling assumptions[END_REF]). • At the same time, the need for new approaches to transport infrastructure is increasingly recognised, at least in principle; including the weight of investment in rail, the provision of charging points for electric cars etc. Whether this will be translated into actual results on the ground remains to the seen. Water Over the years, the discharge of pollutants to fresh and coastal waters has fallen in much of Europe, leading to improvements in freshwater quality, for example in relation to inland bathing waters. However, pollution levels remain significant in several European rivers which show a mix of different pollutants, including nutrients, biocides, industrial and household chemicals or pharmaceuticals. The quality of coastal waters is also affected (EEA 2010c). The concentration of nitrates in rivers and ground waters remains a persistent challenge. Diffuse pollution from agriculture is the main source of nitrate pollution, contributing to eutrophication in coastal and marine waters and pollution of drinking water, particularly where ground waters are contaminated. There are also growing problems in relation to water quantity. Although water abstraction rates have fallen in the majority of EU Member States, particularly in the eastern Member States, overexploitation remains a challenge in many parts of Europe. While water scarcity is a growing problem, in particular in the south of Europe, (see Figure 6), other parts of Europe, including many urban areas in England are suffering from a rise in the frequency of major floods (see Figure 7) and related flood damage. The costs of floods have increased markedly as a consequence (EEA 2010c). European water bodies have also been altered through physical modifications, leading to changes in water flows, habitat fragmentation and obstructions to species migration. The water framework Directive (WFD), adopted in 2000, provides the overall policy framework for preserving and restoring the quality of European water bodies. It creates a timeframe for policy action that should bring all water bodies in the EU to good status by 2027. The WFD is widely appraised as a good example of integrated approaches to environmental policy-making, highlighting the ecological assessment of ecosystems and the focus on river-basin management, full cost-recovery and water pricing. It is complemented by daughter directives on groundwater and on environmental quality standards; and is linked to the emission-oriented approach to water protection set out in the urban wastewater treatment Directive and the nitrates Directive. Due to its broad character, the WFD lacks clarity in detail and leaves a lot of room for diverging interpretation by Member States of the actions required. Implementation remains a major challenge with many Member States making slow progress towards their obligations. The recent publication of Member States' river basin management plans (RBMPs), which are required by the Directive, has made clear that a variety of emissions of hazardous substances continue to pose a threat to the quality of Europe's surface water (EEA 2011g). For these reasons the achievement of the targets under the Directive are uncertain. Further action is needed, particularly with regard to using water in agriculture and buildings more efficiently. Progress by Member States in introducing economic instruments such as water pricing needs to be accelerated, while the principle of cost-recovery remains controversial. The scope of EU policy was expanded to flood risk management by the 2007 floods Directive. The Directive aims to establish a framework for the assessment and management of flood risks and is strongly linked to the WFD implementation process. The Directive requires Member States to assess flood risks for each of their river basins and associated costal zones, develop good hazard maps, and produce flood risk management plans. By the end of 2011, Member States are to undertake a preliminary assessment to identify the river basins and associated coastal areas at risk of flooding; flood risk maps are to be developed by 2013. The mapping phase will provide a major improvement in information available. By 2015, Member States are to establish flood risk management plans focused on prevention, protection and preparedness. This requires a particular effort by the legally competent authorities within Member States to get the new system in place. Drafting these plans would benefit from a stronger link to issues of land management, including agriculture. The approach taken so far has tended to be rather reactive, in terms of better preparation for floods, rather than seeking to mitigate their causes. EU water policy provides a comprehensive legislative framework that aims to address issues related to water quality as well as water demand and availability. However a number of challenges remain, including the following: • EU Member States have considerable autonomy and flexibility with regard to meeting the objectives of the WFD, for example in relation to adequate pricing of water use. Many are proceeding slowly, so implementation continues to be a significant challenge. • Economic instruments focusing on efficiency in water supply are not widely used in Europe, while the principle of cost-recovery and water pricing remain controversial. • An effective approach to better integrating water concerns in key sectoral policies is missing, particularly with regard to increasing the efficiency of water use in agriculture and buildings. For example the introduction of efficiency standards for water use in building offers significant potential for future savings. • Despite some progress in addressing the potential of water savings in different sectors, the widespread and potentially growing challenges of water scarcity and droughts are largely outside the current policy framework. There is currently no consensus on whether future regulatory action on droughts is needed although there is widespread agreement on the need for increased policy coordination in the area. • There is also a need to improve the quality and availability of information and data on water issues, together with related institutional capacity, even more so to help improve understanding and respond to climate change in future (Deloitte and IEEP, 2011). The year 2012 will be important for EU water policy. The Commission is currently undertaking a 'fitness check' of EU water law, covering the WFD, groundwater, environmental quality standards for water, urban waste water treatment, nitrates and floods Directives and the Communication on water scarcity and droughts. The aim of such 'fitness checks' (which are part of the EU's better/smart regulation agenda) is to identify excessive burdens, overlaps, gaps, inconsistencies and/or obsolete measures which may have appeared over time. The conclusions of the fitness check, together with a report on implementation of the WFD, an assessment on the RBMPs and the vulnerability of water resources to climate change, will feed into a 'Blueprint to safeguard Europe's waters'. This is expected to be presented in November 2012. The Blueprint is due to address the broad scope of core EU water policy, making recommendations for improvements, which might include legislative changes. Air Forty years of EU policy on air pollution have resulted in an absolute decoupling of direct air emissions from economic growth (ETC/SCP 2011). In the period 1990-2009, levels of several air pollutants in Europe dropped significantly, in particular of sulphur dioxide (SO 2 ), nitrogen oxides (NO x ) and lead (Pb) (EEA 2011d). However, a reduction of emissions does not equate to a reduction of ambient concentrations, in particular for ground-level ozone (O 3 ) and particulate matter (PM), concentrations of which have not decreased despite a reduction in respective emissions. It is estimated that 17 per cent of the EU urban population lives in areas where the EU ozone target value set by the air quality framework Directive is not met (Figure 8 opposite). The WHO has estimated that large majorities of the European urban population breathe air that largely exceeds their recommendations for PM10 (WHO/JRC, 2011), causing a decrease in life expectancy and a rise in respiratory and cardiovascular problems among others. Past emission reductions have not always produced a corresponding drop in atmospheric concentrations because of complex linkages between the two, making it more of a challenge for policy-makers to move trends downwards. Poor air quality is also linked to the increase in transport volumes in the EU (see section on transport). Moreover, while action taken to improve air quality is expected to yield benefits for climate change and vice versa, in certain cases the use of some new technologies to reduce CO 2 emissions could be counter-productive to efforts to improve air quality. For example, a 2011 report by the EEA points to the risk of increasing air pollutant emissions if carbon capture and storage (CCS) technology is to be applied widely in power and industrial plants in the EU (EEA 2011e). The last strategic review of the EU air policy framework resulted in the Thematic Strategy on air pollution in 2005 which set objectives for air quality for the period up to 2020 related to impacts on, and risks to, human health (e.g. a 47 per cent reduction in loss of life expectancy) and the environment (e.g. a 43 per cent reduction in areas or ecosystems exposed to eutrophication). To reach these objectives, sector specific priorities for EU action were identified and revisions of the three main air pollution directives called for. These were the air quality framework Directive, the national emission ceilings (NEC) Directive and the integrated pollution prevention and control (IPPC) Directive. The air quality framework Directive was subsequently revised with some modernisation in assessment and monitoring and changes to air limit values, including the introduction of new air quality limit values for fine particulate matter (PM2.5). The IPPC Directive was recast as the industrial emissions Directive, incorporating other industry-related legislation, including the large combustion plants Directive. This should lead to tighter controls on industrial air pollution emissions, both in terms of conditions set in permits and enforcement of those conditions. However, no proposal to revise the NEC Directive has emerged, despite repeated rumours that it would be forthcoming. Various reasons for this delay have been cited over the years, including first the need to develop GHG emission targets, then the need to take account of the revision of the IPPC Directive and, now, the need to take account of the economic crisis. Furthermore, measures taken to reduce GHG emissions, such as those contained in the climate and energy package, are also expected to lead to important reductions in pollutant emissions. The implementation of EU air quality policy continues to be a major challenge. Issues relating to compliance and enforcement in the Member States were identified as problems associated with the previous IPPC Directive. The new industrial emissions Directive was designed to address some of these issues and progress can be expected over time. However, given that implementation problems were significant with the preceding legislation it will be necessary to pay close attention to implementation of the new Directive on the ground. This will be an important factor in the functioning and performance of the measure, particularly where limit values are more stringent or where installations are included for the first time. Moreover, the value of the air quality Directive depends on the performance and functioning of measures introduced at the European level to reduce emissions at source and on the implementation of national, regional and local measures to ensure air quality limit values are met. Several Member States, including the UK, have requested and been granted derogations from meeting their obligations under the air quality framework Directive. A 2011 report from the UK Environmental Audit Committee on air quality noted that the UK is still failing to meet European targets for safe air pollution limits across many parts of the country. The report found that 30,000 deaths in the UK were linked to air pollution in 2008, with 4,000 in London alone, and that poor air quality is shortening the life expectancy of people in the UK by an average of seven to eight months; costing society up to £20 billion per year (EAC 2011). Given the scale of the challenge and the on-going threat to health, especially in urban areas; the continued problems of non-compliance, in particular for particulate matter and ozone need to be addressed and the legislative framework brought up to speed. Although economic concerns need to be taken into account, they should not derail the process entirely. Moreover, interactions with other policy developments (such as on agriculture, transport, biodiversity etc.) will need to be taken into account more fully. Such issues could be among those taken up in the review of EU air quality policy currently underway. This is expected to conclude in 2013 with the presentation of an EU clean air package, updating existing policies and directives including the NEC Directive. The review is expected to propose stricter emission ceilings for 2020 and potentially see the introduction of a ceiling for fine particulate matter. To support this, a broad consultation process has been launched, which includes an online public consultation, the establishment of a stakeholder group, the organisation of dedicated workshops and events and dialogue with international organisations (such as the WHO, UNECE) (EC 2011). Chemicals The production, use and disposal of chemicals have been linked to a range of environmental and health related problems. Human exposure to chemicals takes place through multiple sources like water, air, food, consumer products and indoor dust. Of particular concern are persistent and bio-accumulative compounds, endocrinedisrupting chemicals and heavy metals used in plastics, textiles, cosmetics, dyestuffs, pesticides, electronic goods and food packaging (EC 2010b). Chemicals in consumer goods may also be of concern when products become waste and chemicals migrate to the environment and can be found in wildlife, ambient air, indoor dust, wastewater and sludge. There is also growing attention to the possible combined effects of exposure to a mixture of chemicals found at low levels in the environment or in consumer goods, especially among vulnerable young children [START_REF] Eea | EU Environmental Policy Handbook: A Critical Analysis of EU Environmental Legislation, Making it accessible to environmentalists and decision makers[END_REF]. The cornerstone of the EU's approach to regulating the production and use of chemical substances is the Regulation on the registration, evaluation, authorisation and restriction of chemicals (REACH). The Regulation entered into force in 2007 following a major lobbying offensive by industry groups, consumer, health and environmental organisations. Under REACH, all chemical substances manufactured or imported in quantities of 1 tonne or more must be registered by the manufacturer/importer with the European Chemicals Agency (ECHA). The registration contains a dossier with information to enable the substance to be used safely. The ECHA can evaluate dossiers and substances. Downstream users are to contribute to the dossier. Substances of very high concern are not to be used unless authorised. Companies will be required to make efforts to find safer substitutes as part of the authorisation procedure; and the manufacture, marketing and use of substances can be restricted. The Regulation is based on the principle that it is for manufacturers, importers and downstream users to ensure that they manufacture, place on the market or use only those substances that do not adversely affect human health or the environment. REACH is an important piece of legislation, not least since it combines traditional regulation with other approaches, notably enhanced producer responsibility. It is also one of the more ambitious and complex environmental regulations, involving a lengthy implementation process. As part of the REACH procedure, substances of very high concern (SVHC) are identified as part of a process to phase out the use of the most hazardous substances. So far 53 SVHC substances of very high concern have been identified and included in the candidate list. From this candidate list priority substances are recommended by the ECHA and their inclusion into the so called Annex XIV list is decided through comitology. Once a substance is added to this list any manufacturer, importer or downstream user of that substance must apply for an authorisation from the Commission or they will not be permitted to use it after a certain deadline (the sunset date). In September 2010, the first six SVHCs were added to the authorisation list. The substance specific sunset dates range from 2014 to 2015. In December 2011, ECHA added twenty SVHCs to the candidate list. Nineteen of these substances are classified as carcinogenic and/or toxic for reproduction. In addition, for the first time, one substance has been identified as an SVHC because of its endocrine disrupting properties which give rise to an 'equivalent level of concern' due to likely serious effects on the environment (ECHA 2011). Specific legislation on the authorisation and use of pesticides has had a considerable impact in recent years. A new Regulation concerning the placing of plant protection products on the market and a Directive establishing a framework for EU action working towards the sustainable use of pesticides were adopted in 2009. The new rules on pesticides are based on hazard-based criteria for granting authorisations and apply tougher controls or a ban on several SVHC. The Regulation aims to harmonise rules for placing pesticides on the market while also addressing agricultural practices. The Directive establishes a framework by promoting the use of integrated pest management (IPM) and alternative approaches or techniques, such as non-chemical alternatives to pesticides. Under the Directive, Member States are required to adopt national action plans to set up quantitative objectives, targets, measures and timetables to reduce risks and the impacts of pesticide use on human health and the environment, and to encourage the development and introduction of IPM and alternative approaches to reduce dependency on pesticides in farming. Member States are also required to ensure that the use of pesticides is minimised or prohibited in certain specific areas. Implementation of the REACH Regulation remains a critical challenge, as is evident in the slow pace of application to date. Implementation is likely to be further complicated by the fact that many binding dates under REACH lie quite far in the future and by the complex procedures of authorisation and restrictions. Moreover, REACH places administrative and procedural burdens, not only on public authorities, but also on the ECHA to compel industry to discharge its responsibilities. Thus resource constraints and the need for prioritisation, has meant that there is still a long way to go before appropriate risk management measures are actually taken for many 'phase-in' substances. Implementation of new pesticides rules will also prove challenging. There remains a need to improve knowledge on the environment and health impacts of chemicals, particularly with regard to the effects of low doses and multiple exposures, methods for risk assessment of endocrine disruptors, cumulative risk assessment, effects of chemical cocktails etc. Planned reviews of relevant legislation to be presented in 2012 may help to address some of these challenges. The forthcoming review of the REACH Regulation is expected to assess the operation of the Regulation to date, identify lessons learnt in particular with regard to the costs and administrative burden, review the scope and potential overlaps of REACH with other EU legislation on chemicals, and review the ECHA. Although the review is more likely to focus on enforcing existing rules rather than a major overhaul of the legislation, one can expect sensitivities concerning the competitiveness of the European chemical industry to be brought to the fore yet again. A revision of the EU strategy on endocrine disruptors is also expected from the Commission in late 2012. Waste In 2011, the EU economy generated around six tons of waste per person every year. Although Europe has become more efficient in managing material resources, in absolute terms, the consumption of materials continues to increase and a consistent trend to reduce waste has not (yet) been achieved. The EU has a long policy tradition and track record in waste management, with the emphasis shifting over time from disposal to recycling and prevention. Reduction of waste generation however, has not proved easy to achieve on the ground and there remain substantial differences in results achieved between Member States (EEA 2010). On the positive side, waste policy has contributed to higher recycling rates, now amounting to up to 60 per cent for packaging waste and 39 per cent for municipal waste (ETC/SCP, 2011). However, half of the total waste generated is still sent to landfills, with large differences between Member States (see Figure 9). Waste legislation continues to suffer from sub-optimal implementation and enforcement in many Member States which has meant less change on the ground than implied by the legislation in place. The philosophy behind current EU waste policy was set out in 2005 in the Thematic Strategy on waste prevention and recycling. This advocated a lifecycle approach and shift towards a materials-based approach to recycling, a new focus on the prevention of waste and a transition towards more flexible mechanisms of policy making and standard setting at the EU level. In 2008, the revised waste framework Directive reset the baselines for much of EU waste management, redefined key terms and concepts, reinforced the waste hierarchy and set the EU's first ever sector-wide targets for re-use and recycling, complementing existing product-based action. However, progress to date has been disappointing. A 2009 Commission report on the implementation of EU waste legislation from 2004-2006 underlined a series of weaknesses including a lack of waste treatment infrastructure, separate waste collections, recycling and recovery targets in many countries (EC 2009). While 90 per cent of hazardous waste is estimated to be treated in the EU-15, the monitoring of illegal shipments of waste still needs improvement. Volumes of electronic waste have grown rapidly and exports need to be better regulated to avoid the potential environmental burdens arising from such waste (which contains hazardous substances) being treated in third countries with less stringent environmental standards than those in the EU. This requires a recast of the WEEE Directive which was first proposed in 2008, but divided the European Parliament and the Council until a recent agreement in December. The 2011 review of the Thematic Strategy on waste (EC 2011a) found that significant progress has been made in the improvement and simplification of legislation, the establishment and diffusion of key concepts such as the waste hierarchy and lifecycle thinking, an increasing focus on waste prevention, efforts to improve knowledge, and new European collection and recycling targets. In terms of waste management performance, recycling rates have improved, the amount of waste going to landfill has decreased, the use of hazardous substances in some waste streams has been reduced, and the relative environmental impacts per tonne of waste treated have decreased. These achievements are however offset by the negative environmental impacts caused by the increase in overall waste generation. The review concluded that the Thematic Strategy has played an important role in guiding policy development, but that the EU is still some way from a 'recycling' society. More impetus is now needed in a number of areas to inter alia: • Properly implement and enforce existing EU waste legislation. In this regard, the Commission suggests the development of a 'proactive verification procedure' and early warning system on compliance with key EU targets, based on national waste management plans. • Define new and more ambitious (material-specific) prevention and recycling targets, • Improve the knowledge base on waste and resources, • Support national actions on waste prevention, • Increase coordination of national inspection activities, • Promote combinations of economic and legal instruments for waste management, • Improve the competitiveness of EU recycling industries and develop markets for secondary raw materials, • Improve measures to prevent illegal waste exports, • Improve stakeholder participation and raise public awareness on waste, and • Promote lifecycle thinking (e.g. through more consistency between waste and product design policies). The Commission is expected to set out proposals to address some of the above mentioned issues in 2012. Resource use The overall environmental impacts of EU natural resource use in and beyond Europe are growing. Europe is struggling to achieve absolute decoupling of resource use from economic growth, despite a range of efforts to improve resource efficiency over the years. Growth in resource productivity has been significantly lower than growth in labour productivity. While many products are gradually becoming more energy-efficient, efficiency gains are often off-set by changing consumption patterns. Some forms of material use are expanding significantly and the ecological footprint of the average European citizen exceeds 4.5 global ha per capita (ETC/SCP, 2011). A new approach to resource use in Europe is increasingly seen as a central plank of the EU environmental agenda for the next decade. There is growing recognition that Europe cannot continue to consume more than its share of global resources and an understanding that the problem goes well beyond the issue of industrial raw materials. This is not a new theme in EU policy; there has been a history of efforts to achieve sustainable consumption and production (SCP) for example, as well as more sectoral initiatives concerned with water, wastes etc. However, there is now a much stronger link being made to the EU's economic strategy (see section 5). Improving resource efficiency and, more tentatively, reducing resource use, is being seen as part of a strategy for green growth. The economic benefits and scope for win-wins for business and also for consumers are being emphasised. In January 2011, the Commission published a Flagship Initiative on resource efficiency (EC 2011b) as part of the Europe 2020 Strategy. It provides a long-term framework for actions in several areas to support the shift towards a resource-efficient, low-carbon economy aiming at sustainable growth. Policy is to be developed in a set of roadmaps, namely on the decarbonisation of Europe's economy, improving resource efficiency and initiating long-term energy transition, as well as more sectoral plans such as the Blueprint for water policy and the 2020 biodiversity Strategy. The Flagship Initiative lacks ambition in tackling the ever increasing trends in European consumption, which are one of the main roots of the problem. Greater efficiency alone will not lead to a decrease in absolute resource use, and some voices in the EU are now calling for a 'resource-intelligent' Europe which refers to the combination of a less materialistic approach to well-being and new economic models (e.g. a goods-as-services based economy). The Commission's thinking is developed considerably further in the Roadmap to a resource-efficient Europe published in September 2011 (EC 2011c). Natural resources, ecosystem services and natural capital concerns are linked with resource use which is understood in a broad sense to include biodiversity. The Roadmap postulates an ambitious long-term policy vision, including no net land-take in 2050 and a sustainable use of resources within planetary boundaries. These are coupled with more detailed milestones of varying levels of ambition, including the phasing out of environmentally harmful subsidies by 2020. However, there are no overarching targets, such as those that apply in the area of climate change policy. Agreement on concrete and detailed indicators to guide action by policy-makers, business and investors is deferred to 2013. Although this may delay critical progress, the Roadmap does refer to the principal areas of European resource consumption, namely buildings, transport and food, and while it is cautious in approaching the topic of absolute reductions in consumption, the implications of inaction are not disguised. The main question now is how far the Roadmap will be converted into concrete proposals at a national as well as EU level over the next eight years. A potentially large range of measures could be introduced over the next decade or so if the Roadmap is to be converted into a concrete action plan, although it would require a step change in the level of determination, starting with the adoption of targets and indicators. Future measures could include: • The review and tightening of existing standards and targets, not least to increase recycling rates for metals and other materials, e.g. current targets for collection rates under the batteries Directive are for 25 per cent in 2012 and 45 per cent in 2016. • Stimulating changes in the design and use of certain products, for example by extending the scope of the ecodesign Directive. • More focussed efforts to apply existing measures, such as realistic pricing for water under the WFD and respect for the principle of Maximum Sustainable Yield for commercial fish stocks under the CFP. • The wider application of economic instruments, including appropriate incentives for recycling and refund systems, the selective use of levies and taxes (generating funds that can be applied to complementary uses), the withdrawal of environmentally harmful subsidies, greater use of green public procurement etc. • An initiative to bring a land-use dimension to EU policy, globally as well as domestically. • Investment in innovation, research and development, education and more innovative approaches to addressing consumption levels. The resource efficiency agenda extends well beyond the sensitive issue of reducing dependence on raw materials, such as rare earths and must embrace the health of ecosystems more broadly. The Roadmap is a good foundation for building a new generation of policies that grasp the opportunity to make resource efficiency central to the creation of a greener economy. Soil Soil has received much less attention than other environmental media in EU policy, even though soil degradation is accelerating in many parts of Europe. European soils continue to face multiple threats such as erosion, organic matter decline, contamination, compaction, salinisation, landslides, contamination, sealing and biodiversity decline (EEA 2010c). Although a number of sectoral EU policies have an impact on soil management practices, including measures taken under industrial emissions policy, as well as water, waste and agricultural policy; they provide only a patchy level of protection. The development of specific EU legislation addressed primarily to issues of soil protection and prevention of land degradation has been limited by concerns among many Member States that soil protection is as an area of national competence and potentially costly, particularly with respect to contaminated land restoration. The last serious EU-level initiative in this area was the Commission's 2006 Thematic Strategy on soil protection intended to pave the way for future policy. The Thematic Strategy aimed to develop a new approach to the management and protection of Europe's soils and was built around four pillars for action: the integration of soil protection into national and Community policies; closing recognised knowledge gaps; increasing public awareness; and the development of framework legislation aimed at protection and sustainable use of soils. The Commission's 2006 proposal for a soil framework Directive was meant to have a significant impact on soil protection and the retention of soil functions in Europe. In its current form, it would require the identification of soils at the greatest risk of degradation and actions to address this, with the precise obligations on Member States being a matter of controversy. The UK, Germany, France, Austria and the Netherlands have established a blocking minority, resisting the adoption of the proposal in its current form for the reasons cited above. Many other Member States support the proposal quite strongly. Some progress has however been made under the other pillars of the Thematic Strategy. Under the first pillar, efforts have been made to integrate aspects of soil protection into relevant EU policies, e.g. the requirements of the IPPC Directive to ensure the protection of soil when an industrial operation is discontinued were too vague to enforce changes in actual practices and they have been clarified in the new industrial emissions Directive. Under the second pillar, a number of studies have considerably improved the existing body of knowledge in the area of soil protection. However, a continuing concern relates to the lack of harmonised information at EU level on soil conditions. Under the third pillar, the adoption of the Thematic Strategy led to several EU-wide stakeholder conferences on soil related issues, attended by scientists, Member State representatives, civil society and other stakeholders. This rising level of awareness has been one of the factors in deepening stakeholder engagement in the debate on future policy in an area which has received much less attention than others such as air and water. There has also been a growing level of awareness of soil issues linked to the climate change debate (e.g. carbon sequestration) and the role of soils in delivering ecosystem services. There is now greater understanding of soil interactions with other priorities such as the need to sequester carbon, manage land in a way that enables adaption to climate change and ensure the protection of water both, in terms of quality and quantity. It is clear that soil conservation needs more priority in both agricultural and environmental policy. However, it is not yet clear whether the adoption of a new framework Directive will be the way forward in this regard. There are a number of issues on the horizon which could contribute to soil protection. They include a Commission technical document on soil sealing expected to be published in early 2012. Also relevant will be the outcome of deliberations on CO 2 emissions associated with land-use change, discussions on carbon credits for the protection of forests (and other terrestrial carbon sinks), the evolution of criteria for the sustainable production of biofuels, cross compliance in future EU agriculture policy, or the EU climate adaptation strategy which is expected to appear in early 2013. Soil management certainly should attract more attention within the CAP in the future. Biodiversity Despite a fairly wide ranging regulatory framework in place and considerable efforts to establish a European network of protected areas (Natura 2000), the loss of Europe's biodiversity remains a persistent problem. On a global scale, biodiversity is still threatened by increases in the five principal pressures: habitat change, overexploitation, pollution, invasive alien species and climate change (CBD 2010). In Europe, species decline is particularly marked in agricultural and grassland ecosystems, mainly due to intensified farming and unsustainable land-management. The fate of wildlife rich habitats varies. Some face very considerable pressures; whilst others, such as European forest coverage grow despite being affected by acidification, eutrophication, forest fires and other regional pressures (Forest Europe, UNECE and FAO 2011). In 2001, EU leaders committed to halting the decline of biodiversity in the EU by 2010 and to restoring habitats and natural systems. Despite some progress in various areas such as the extension of the Natura 2000 network, the EU failed to achieve this target. This is due to continuing negative trends in key pressures, such as changes in agricultural systems, pollution of freshwater, land abandonment, and habitat fragmentation. There have been particular problems related to the implementation of the nature Directives and the biodiversity action plan, including slow or incomplete identification and designation of Natura 2000 sites (especially in the marine environment), inadequate management of habitats and species within Natura sites and especially in the wider environment, as well as problems related to relevant sectoral policies. Over the last two years there has been increasing recognition of the economic value of biodiversity and ecosystem services (such as healthy soils, clean water, carbon sequestration) in the policy process. This has been driven in part by developments in the knowledge base, including inter alia the TEEB (The Economics of Ecosystems and Biodiversity) initiative. This has helped to raise the political profile of biodiversity issues in recent years. A new biodiversity target was agreed by the Council in March 2010 to halt the loss of biodiversity and the degradation of ecosystem services in the EU by 2020 and restore them in so far as feasible; while stepping up the EU contribution to averting global biodiversity loss. The explicit addition of ecosystem services to the target reflects the increased recognition of the value of biodiversity to society and the need to broaden concern for biodiversity across society and sectoral interests (see Box 4). Box 4. Business and biodiversity -Making the case for a lasting solution A 2010 report by the United Nations Environment Programme (UNEP) set out a case for incorporating biodiversity in business models. The report looks at a broad spectrum of businesses across different sectors and for each provides examples of companies that have taken actions to reduce their impacts on biodiversity. The report outlines the case for businesses to examine their impact on biodiversity including expanding market opportunities, brand advantage, opportunities for new business ideas, and the potential of new green technologies. The report argues that a failure to address biodiversity issues could affect the supply of resources, access to markets, reputation, licenses to operate and access to finance. In addition, companies could face a consumer backlash, as an increasing number of customers demand sustainably produced products and services. Source: UNEP-WCMC and UNEP-DTIE, 2010 A new EU biodiversity Strategy to 2020 was subsequently produced in May 2011 which sets out six main targets, 20 actions and 36 measures (EC 2011k). The targets relate to full implementation of the birds and habitats Directives, maintaining and restoring ecosystems and their services, increasing the contribution of agriculture and forestry to maintaining and enhancing biodiversity, ensuring the sustainable use of fisheries resources, combating invasive alien species and helping to avert global biodiversity loss. Biodiversity remains a contested area and despite being one of the oldest areas of EU environmental policy, issues of implementation remain a major challenge, including in several older Member States, such as the UK. The EU's 2010 biodiversity target was very difficult to achieve, particularly given the scale of the task and insufficient political support and action by Member States in this regard. It is not clear whether there will be sufficient commitment to the action now required and whether necessary mechanisms are now in place to deliver the new EU biodiversity objectives for 2020. Some key challenges going forward include how to secure Member States commitment to full implementation of existing measures, the development of new measures to address gaps in the coverage of EU policies (e.g. on invasive alien species, on green infrastructure), how to 'mainstream' biodiversity policy in other relevant policy areas and secure adequate financing for implementation of biodiversity commitments including expansion of the Natura 2000 network. Marine environment and fisheries The European marine environment and the ecosystem services it provides are under considerable pressure. The majority of pollutants in freshwater bodies (described in the section on water above), is ultimately discharged to coastal waters. Runoff from fertilisers and pesticides from land-based sources has led to oxygen depletion and to ecosystem collapses (e.g. Black and Baltic Seas) (EEA 2010c). While some marine protected areas have been established under the Natura 2000 network, marine sites currently only account for around 6 per cent of Sites of Community Importance (SCIs) and 10 per cent of Special Protection Areas (SPAs) (EEA 2010a). Other concerns relate to the threat of invasive species, marine plastic litter and the future impact of climate change. The fisheries sector has a major impact on the overall state of the marine environment. Despite changes made during and since the 2002 reform of the Common Fisheries Policy (CFP), overexploitation of marine fisheries remains a major problem and has led to a situation where 26 per cent of fish stocks are below safe biological limits [START_REF] Sissenwine | An overview of the state of stocks[END_REF]. Despite an apparent improvement in the current state of stocks, there is also pressure to reduce levels of bycatch, eliminate discards of non-target fishing species, and avoid damage to habitats from several types of fishing gear [START_REF] Lutchman | Towards a Reform of the Common Fisheries Policy in 2012 -A CFP Health Check[END_REF]. In 2008, the EU adopted the marine strategy framework Directive (MSFD) under which Member States are required to take measures to achieve or maintain good environmental status in the marine environment by 2020. To this end, marine strategies are to be developed and implemented to protect and preserve the marine environment, prevent its deterioration or, where practicable, restore marine ecosystems and prevent and reduce inputs in the marine environment. Working groups have been established to support the interpretation and practical application of parts of the MSFD. In some countries the structure and responsibilities for implementation of the MSFD are clear. However, some Member States are still discussing how the Directive should be implemented; while others are still working on a process for the identification of potential programmes and/or parameters for good environmental status. When fully implemented, the Directive can be expected to make a significant contribution to improving the state of the marine environment. However, there are some limitations to its use and a number of issues it cannot address which will have to be confronted through other instruments such as the CFP. The MSFD is the environmental pillar of the EU's integrated maritime policy (IMP) which aims to provide a framework for the development of policies affecting maritime areas. The IMP has resulted in, or influenced, a number of subsequent policy documents covering substantive issues (e.g. on maritime spatial planning and integrated maritime surveillance) as well as on sectoral policies and regional policies (e.g. on offshore wind energy and maritime transport). A 2009 report on implementation of the IMP (EC 2009b) highlights a number of positive developments at Member State level in integrating maritime governance, including the UK's Marine Bill. A key challenge in the years ahead will be the issue of integration not only in terms of making marine spatial planning work, but also integrating it with the separate domain of fisheries policy. It will become increasingly necessary to address conflicting policy objectives which will inevitably arise. The pressure to manage fisheries sustainably and responsibly is growing and the current reform of the common fisheries policy (CFP) has highlighted the shortcomings of the current approach and the need for critical changes. The reform is to be completed by 2012 and is likely to include major changes to the principle CFP Regulation including a ban on discards, the introduction of transferable quotas, decentralising some decision-making powers to the regions, measures to move towards multi-species fisheries management, and the introduction of marketbased quota management. The most significant proposed change to the general objectives of the CFP is the aim of reaching maximum sustainable yield of commercial fish stocks by 2015. Another new objective is that the CFP shall integrate the requirements of EU environmental legislation. The Commission's proposals for fisheries funding under the future EU budget also appear to be moving in the direction of sustainability not just of fisheries but of the broader marine environment, with the introduction of a new European Maritime Fisheries Fund (EMFF) (replacing the current European Fisheries Fund). As proposed it should support fishing which is more selective, producing no discards, doing less damage to marine ecosystems and relating to the science that supports these activities. The extent to which these provisions are taken up in final legislation will depend on the outcome of on-going negotiations between the European Parliament and the Council. The 2010 BP Deepwater Horizon oil spill in the Gulf of Mexico and the observed shortcomings of response strategies promoted a review of EU practices and provisions covering off-shore oil and gas exploitation. Developments in technologies and in oil and gas exploration techniques have also rendered current legislation obsolete or ineffective. An assessment of the EU approach concluded that there is insufficient coverage of environmental protection, disaster prevention and response and that the industry's liability for environmental degradation is not clearly defined. Moreover, only segments of the EU's environmental liability Directive, habitats Directive and birds Directive directly apply to off-shore petroleum activities (EC 2010). The Commission thus proposed new rules for the safety of off-shore oil and gas prospection, exploration and production activities in October 2011. The proposal extends the environmental liability Directive to cover all EU marine waters within 370 kilometres from coastal areas and sets rules that cover the lifecycle of all exploration and production activities. Given the sensitivity of the proposal in terms of its encroachment on an area of traditional national competence, discussions on its finalisation are likely to be contentious. Whatever the outcome of current efforts to address the Eurozone crisis, the current economic and financial preoccupations in Europe are unlikely to fade away rapidly. Instabilities in financial markets, uncertainties over growth and job prospects and pressure to maintain austerity regimes could continue in some form for several years. Safeguarding jobs and stimulating growth is thus likely to remain an overriding political priority. The added value of EU policies will frequently be measured against this yardstick. Similarly, political support for specific environmental measures is likely to only be achieved by convincing leaders of the costs of inaction and the costeffectiveness of action. At the same time, the current economic situation also offers a number of opportunities for promoting the environmental policy agenda, particularly in view of fostering an efficiency revolution. There are currently a number of strategic and sectoral processes underway in the EU which will affect the general context and scope for environmental policy action to 2020 and beyond. This includes taking forward the EU's 10year economic strategy known as the Europe 2020 Strategy, promoting eco-innovation, discussions on the EU budget for the 2014-2020 period and on the future strategic framework for EU environmental policy under the 7th Environment Action Programme (7th EAP). Improving implementation of the substantial body of environmental law already agreed has been, and still is, a key challenge. This section provides a brief overview of some of these wider strategic issues and the challenges and opportunities they offer for future EU environmental policy. The Europe 2020 strategy: promoting smart, sustainable and inclusive growth In 2010, the EU adopted a new medium-term growth strategy known as the 'Europe 2020 Strategy' (EC 2010). The Strategy aims to turn the EU into a smart (based on knowledge and innovation), sustainable (promoting resource efficient, greener and more competitive growth) and inclusive (high employment, delivering economic, social and territorial cohesion) economy. These priorities are linked to five headline targets for employment, social inclusion, education, innovation, climate change and energy, which are to be reached by 2020 (see Table 1). These targets are common goals to be achieved through a mix of European and national action. The targets have been translated by Member States into corresponding national objectives and measures reflective of their own geographic, socio-economic and political situation . The Europe 2020 Strategy also identified 'Flagship Initiatives' in seven areas within which EU and national authorities should coordinate their efforts (see Box 5). The 'Flagship Initiatives' were presented by the Commission in 2010/2011 and have led to the adoption of a series of subsequent strategies, roadmaps and measures. Of particular relevance to the environmental agenda is the resource efficiency Flagship Initiative which has spawned a number of strategies and roadmaps (see section on environmental challenges), and has led to the re-conceptualisation of a number of environmental issues so as to relate them to the resource efficiency agenda. Progress in implementing the Europe 2020 Strategy at both EU and Member State level is pursued via the EU's new cycle of economic and fiscal policy coordination (the 'European Semester') and is to be closely followed by EU leaders. The six-month cycle includes the preparation of an annual growth survey by the Commission, the assessment of Member States' stability and convergence reform programmes and national reform programmes, and the adoption of country-specific recommendations. The first cycle of the European Semester ended in June 2011 and it is still too early to judge the results. Some initial assessments suggest that although the topics of energy efficiency, addressing environmentally harmful subsidies and the reduction of GHG emissions in key sectors are highlighted in Member State reports, the treatment of climate change concerns and the overall performance of the Strategy have been relatively weak. A report for the Greens/EFA in the European Parliament concluded that the priorities of the 2011 Annual Growth Survey do not cover all the agreed headline targets, with particular gaps for those concerned with climate change. National recommendations do not appear to be based on Member State progress towards respective goals, but rather focus on fiscal consolidation needs [START_REF] Derruine | The first European Semester and its contribution to the EU 2020 strategy[END_REF]. The Europe 2020 Strategy aims to be central to economic policy in the EU and is strongly supported by the Commission and its President. It has thus become a key strategic document directing EU action across the spectrum, including EU spending (see section below). Most recent Commission policy documents have been linked to the priorities of the Europe 2020 Strategy and framed accordingly. The Strategy underlines the need to combat climate change and increase Europe's resource efficiency, thus placing these objectives high on the overall EU policy agenda. However, given its overarching priority to stimulate economic growth, its primary focus is on 'win-win' environmental options (i.e. those that can bring financial gains, improve competitive advantage, and in some cases reduce dependence on foreign resources). This narrow focus ignores other key policy objectives that are nonetheless firmly embedded in the EU environmental acquis. For example issues such as biodiversity and the broader notion of ecosystems and their services, which are of central relevance to human well-being and economic performance, are side-lined. Too much of a focus on 'win-wins' also creates the risk of developing policies that do not take into account the inter-linkages and trade-offs between different areas. Promoting eco-innovation Environmental challenges and resource constraints have led to growing demand for greener technologies, products and services and have facilitated the emergence of new types of manufacturing and services. Their development creates huge market opportunities as well as new challenges and pressures on companies. The global market for eco-industries is estimated to stand at roughly €600 billion a year, with over one third of this stemming from the EU. The US and Japan account for a large part of the remaining global turnover of eco-industries (ECORYS 2009). The EU's comparative advantage and niche markets are in the areas of renewable power generation technologies (with over 40 per cent of global market share) and waste management and recycling technologies (with 50 per cent of global market share). Although an established market player in certain segments; the EU's eco-industry sector is facing increasing competition from Japanese, US, Taiwanese and Chinese players (ECORYS 2009). There has been some investigation of the factors affecting the level of innovation in the EU. While environmental regulation can help to promote eco-innovation, a number of negative factors persist including under-investment in the knowledge base (where other countries like the US and Japan are out-investing the EU and China is rapidly catching up); unsatisfactory framework conditions, such as poor access to finance, high costs of intellectual property rights, ineffective use of public procurement; and fragmentation and duplication of efforts (EC 2010c). Eco-innovation represents a key area of synergy between environmental and economic objectives and is an important part of delivering smart, sustainable and inclusive growth. The Commission estimates that European eco-industries currently have an annual turnover of €319 billion, or about 2.5 per cent of EU GDP and have recently been growing by 8 per cent each year. The main sub-sectors deal with waste management (30 per cent), water supply (21 per cent), wastewater management (13 per cent) and recycled materials (13 per cent). The sector directly employs 3.4 million people, with around 600 000 additional jobs created between 2004 and 2008 (EC 2011d). The annual growth rate in employment in all subsectors between 2000 and 2008 was roughly 7 per cent (see Figure 11). Germany (24 per cent), France (20 per cent) and the UK (17 per cent) have the highest number of eco-industry jobs (WIFO 2006). When taking into account those people directly employed in jobs related to the environment, including jobs in sectors that depend on a good quality environment as an input, such as organic agriculture, sustainable forestry, and tourism, the number of people employed in the sector in 2008 is estimated at 5.6 million (ECORYS 2009). The EU can help to accelerate eco-innovation through well-targeted policies and actions such as regulatory initiatives, voluntary agreements, incentives, private and public procurement, standards and performance targets all of which can help to create stronger and more stable markets for eco-innovation. The EU can also help to mobilise additional funding for investment in eco-innovation and policy measures to lower and manage risks for entrepreneurs and private investors (EC 2011d). Over the years, the EU has introduced various measures that seek to promote further eco-innovation. Recent developments have been closely related to the Europe 2020 Strategy. In October 2010, the Commission presented a Flagship Initiative on the Innovation Union with the aim of improving conditions and access to finance for research and innovation, ensuring that innovative ideas can be turned into products and services that create growth and jobs (EC 2010c). This was followed in December 2011 with a new Eco-innovation Action Plan (Eco-AP) (EC 2011d). The programme is the successor to the EU's Environmental Technologies Action Plan (ETAP) launched in 2004. The new plan has a broader remit than its predecessor and includes a variety of measures intended to overcome the barriers preventing the development and spread of eco-technologies, particularly among SMEs. Proposed actions include the use of environmental policy and legislation as a driver to promote eco-innovation, supporting demonstration projects and partnering to bring promising technologies to the market, developing new standards which will boost eco-innovation, mobilising financial instruments and support services for SMEs, and supporting the development of emerging skills, jobs and training programmes. Of the total budget of €87bn of the programme, the Commission proposes that at least 60 per cent support sustainable development objectives, out of which around 35 per cent should be climate change related. More specifically it is proposed that €4.7bn will be used to secure sufficient supplies of safe and high quality food and other bio-based products by developing productive and resource-efficient primary production systems; €6.5bn will be allocated to the transition to a reliable, sustainable and competitive energy system; €7.7bn will be allocated to a resource efficient, environmentally-friendly and safe transport system; and €3.5bn will support the objective of achieving a resource efficient and climate change resilient economy, protected ecosystems, biodiversity and sustainable supply of raw materials (EC 2011e). The Horizon 2020 package is expected to be adopted by the end of 2013 with a view to enter into force on 1 January 2014. The Commission has also included a strong innovation component in its proposals for the 2014-2020 Cohesion Policy (EC 2011d). The EU budget: financing environmental policy in times of austerity The growth focussed agenda of the Europe 2020 Strategy is met by an increasingly vigilant push for financial austerity in many Member States. This tension underlies the on-going discussions on Europe's next long-term budget (the so-called multi-annual financial framework (MFF)), that will set out EU spending priorities for the 2014-2020 period. From an environmental perspective there are however a number of opportunities to re-focus significant elements of EU spending so as to support the transition to a low-carbon, resource-efficient economy. Raising further revenues from environmental fiscal reform may also be possible. In both ways a greener budget could contribute positively to the wider political objectives of promoting economic recovery and creating jobs. Reforming EU spending for environmental purposes Public expenditure through the EU budget, albeit relatively small in size, remains an important source of financing for the environment and can exert a strong influence on patterns of investment and related policies in Member States. Since the launch of the review of EU spending and resources in September 2007, there has been growing recognition of the need for reform of the EU budget to reflect new and emerging challenges, such as climate change. The Commission Communication on the EU budget review (EC 2010b) presented in October 2010 argued that the future budget should be closely aligned to the Europe 2020 Strategy and play a key role in its delivery. The need to address climate change, resource efficiency and energy security is highlighted and the case for ensuring the necessary investments in green technologies, services and jobs is clearly made. In June 2011, the Commission formally tabled its proposals for the 2014-2020 MFF under the title 'A Budget for Europe 2020' (EC 2011). The Commission proposes an overall increase for the period to €1,025 billion (1.05 per cent of GNI, which is in fact a slight decrease from the current budget which represents 1.12 per cent of GNI). The CAP (€372 billion) remains a sizeable element of the overall budget but is now to account for a fractionally smaller share than Cohesion Policy (€376 billion). A key function of the budget is to provide a means of responding to persistent and emerging challenges that require a common, pan-European approach such as environmental protection and climate change. With a relatively small sum (€3.2 million) allocated to the future environment funding instrument (LIFE); 'mainstreaming' is put forward as the principal mechanism for financing environment and climate change priorities. Most notable is a requirement that at least 20 per cent of the EU budget is allocated to climate change financing. If mainstreaming is applied rigorously it could help to shift investment patterns in several sectors, aiding energy conservation, the growth in renewables, a greener transport infrastructure etc. To make mainstreaming effective, it requires policy shifts in the main spending areas embodied in EU regulations and willingness in the Member States where the money is spent to take such considerations into account in their planning and decision-making. issues of resource efficiency and biodiversity etc. feature less prominently. More balance is required to take into account the wider suite of environmental issues. Several governments are now arguing for an effective freezing of the future budget (including the UK, Denmark and the Netherlands), while others such as France and most central and eastern European countries aim to defend the traditional spending blocs on agriculture and cohesion [START_REF] Medarova | When financial needs meet political realities. Implications for Climate Change in the Post-2013 EU Budget[END_REF]. This positioning contrasts with that of the European Parliament which supports inter alia a 5 per cent increase in the overall EU budget and the abolition of all rebates and correction mechanisms. Whilst there is no environmentally optimal size of budget, one that is squeezed down may well lose key environmental elements as almost occurred earlier in 2011 with an attack on rural development spending. Despite its small size, the EU budget can have significant multiplier effects in important policy areas such as energy and transport and build institutional capacity at a European scale. However, Member State positions indicate that traditional issues (e.g. the total size of the MFF, the share of CAP and Cohesion Policy, national rebates and new sources of revenues) could dominate the debate. The risk in doing so is that the 'greener' elements of the proposals will be watered down or lost [START_REF] Medarova | When financial needs meet political realities. Implications for Climate Change in the Post-2013 EU Budget[END_REF]. Exploring new revenue sources The difference between national contributions to the EU budget and national receipts is a matter of significant contention and underlies the position of many Member States on the EU budget. The UK has negotiated a sizeable national 'rebate' to reduce the size of its national net contribution and defends it as far as it can during the decisionmaking process. Some other Member States have smaller rebates. The Commission's proposals for the 2014-2020 MFF aim to simplify Member State contributions, introduce a new system of own resources, and reform 'correction mechanisms' (including a review of the UK rebate). With its proposals, the Commission aims to move towards a system in which revenue flows directly to the EU budget, thus reducing reliance on national contributions. The most controversial Commission proposals in this respect concern the introduction of a new EU financial transaction tax (FTT) and new EU VAT resource. The FTT in particular has been criticised by several Member States including the UK. Given the need for unanimity in the Council for the adoption of any fiscal measures the future of these proposals is uncertain. Given continued constraints on public budgets, the Commission is also proposing to increase the use of 'innovative financial instruments' as a means of attracting additional public and private financing to projects of EU interest. Most of the proposals extend, with some modifications or extensions, existing financial instruments. These include risk-sharing instruments (e.g. the Risk-Sharing Finance Facility for investments in research, development and innovation (RSFF)), financial engineering and technical assistance under Cohesion Policy, guarantees and venture capital for SMEs under the Competitiveness and Innovation framework Programme (CIP), and equity instruments such as the Marguerite Fund. One new proposal under the Connecting Europe Facility is the EU project bond initiative, which focuses on securing investment for strategic infrastructure projects in the energy, transport and ICT sectors [START_REF] Withana | Mobilising private investment for climate change action in the EU: The role of new financial instruments[END_REF]. The current focus on fiscal issues in the EU raises questions about whether there should be a shift towards environmental fiscal reform in the years ahead. In principle, shifting part of the current national tax bases from labour to environmentally damaging activities, though environmental tax reform (ETR) could bring about an improvement in both the environment (by properly pricing externalities) and the economy as a whole (e.g. by making the cost of labour cheaper and therefore encouraging employment). The reform and/or phasing out of environmentally harmful subsidies (EHS) could also help to release additional financial resources, including for the environment. Following a significant increase in the use of environmental taxes in the 1990s among EU countries, these levels have remained stable and in some countries have decreased over the past decade. Moreover, despite various EU and international commitments to reforming EHS, progress has been slow. Fiscal issues have always been a particularly sensitive area as can be seen in the on-going discussion on the Commission's proposal to revise the energy taxation Directive to introduce a carbon element which would reflect the environmental impact of various types of fuels. This proposal has been met by resistance from several Member States, including the UK, and the European Parliament. Nevertheless, in the current economic and financial crisis, reforms in this area could play a significant role in the restructuring of EU finances and could also contribute to achieving wider environmental and climate change objectives. Developing the future strategic framework for EU environmental policy: The 7th Environment Action Programme Since 1973, the Commission has periodically issued Environment Action Programmes (EAPs) setting out forthcoming initiatives, legislative proposals, broader approaches and principles for EU environmental policy. In July 2002 the sixth Environment Action Programme (6th EAP) was adopted. The Programme establishes a tenyear framework for EU action on the environment, focusing on four thematic areas: climate change, nature and biodiversity, environment and health, and natural resources and waste. It also outlines governance mechanisms to improve the environmental policy-making process in the EU. More detailed measures to meet the objectives of the Programme were set out in seven Thematic Strategies covering soil protection, marine environment, pesticides, air pollution, urban environment, natural resources and waste. As the Programme nears its last phase, there has been much discussion on its achievements and shortcomings as well as its successor. In August 2011, the Commission presented its final assessment of the 6th EAP (EC 2011a) which concluded that on balance, the Programme has been helpful in providing an overarching framework for EU environmental policy. The 6th EAP acted as an important reference point for Member States, local and regional authorities and other stakeholders. In some areas, the 6th EAP helped to build political will for action (e.g. on marine, soil, urban, resources), while in others it focused on revising existing measures and addressing specific gaps (air, pesticides, waste prevention). However, a number of shortcomings of the 6th EAP were also recognised. The large number of actions (156 in total) and the absence of a longer-term vision were seen to have compromised its capacity to deliver a clear, coherent message. Inadequate implementation and enforcement of EU environmental legislation was another concern, although this problem is not attributable to the EAP. There has been considerable debate and uncertainty about the need for and political added value of a successor EAP. The added value of such a Programme given the current plethora of strategic documents such as the resource efficiency Roadmap was one question on the table. This debate has now been resolved and the Commission has formally announced its intention to present a proposal for a seventh Environment Action Programme (7th EAP) in October 2012. The 7th EAP is expected to set out strategic orientations for EU environmental policy for the shortto-medium term and a longer-term vision, bringing together action to protect natural capital and ecosystems, encourage resource efficiency and improve implementation. The 7th EAP is also expected to build on proposals in the resource efficiency Roadmap and deal with a number of challenging issues including changing consumer behaviour, improving policy coherence, examining environmental determinants for improving public health, the international aspect of environmental policy, and securing better financing (EC 2011b). The development of the 7th EAP is still in its early stages and will be subject to consultation in 2012, thus providing an opportunity to feed strategic thinking into discussions on the future framework for environmental policy in Europe. There are implementation gaps across most of the main topics of environmental law and in almost all Member States. At the end of 2009, Spain had the highest number of on-going infringements cases (40), most of these relate to nature legislation ( 14) and water legislation (10). Italy and Ireland had more than 30 open infringements each and the Czech Republic, France and the UK had 26 each. In the UK, the majority of infringements related to water, air and environmental impact assessments (see Figure 13). Although a number of Member States have received significant fines for poor compliance, the problem persists. The implementation challenge A recent study for the Commission [START_REF] Cowi | The costs of not implementing the environmental acquis[END_REF] attempted to estimate the cost of not implementing the EU environmental acquis. The report notes that the lack of full implementation of the acquis could have negative effects on eco-industries, as uncertainty about environmental measures may hamper investments in new environmental technologies. Uneven implementation can also distort competition across Member States and lead to higher administrative costs when standards vary across countries. Overall, the study suggests that the current cost of not fully implementing key EU environmental legislation, in the fields of water, air, nature and biodiversity, waste, chemicals and noise, may represent about €50 billion per year. Furthermore, it suggests that missing future environmental targets could cost up to €250 billion per year. Although the study is only a first order-of-magnitude estimate, it shows that the lack of full implementation can have real economic impacts. Implementation of the environmental acquis is in the first place the responsibility of EU Member States. However, action at EU level can also be helpful to improve the situation. The Commission has made various efforts over the years to guide Member States in implementing EU law, such as issuing guidance documents interpreting specific matters of EU law, sharing good practices, setting up early 'package meetings' to discuss transposition difficulties with national administrations, etc. Current EU efforts build strongly on a preventative approach, seeking close cooperation with Member States before taking enforcement action via the court. In 2008, an 'EU Pilot project' was launched which aims to correct infringements of EU law at an early stage without recourse to infringement proceedings through closer collaboration between the Commission and Member States. At the end of 2010, the initiative covered 18 Member States, including the UK, and has reportedly contributed to a reduction in the number of infringement proceedings among participating Member States (EC 2011c). Different factors explain implementation failures, including an unwillingness to accept costs, insufficient administrative capacities, lack of political priority for environmental inspections and associated limited resources for inspection authorities at Member State level. A critical barrier remains the lack of political will for real action. Efforts to improve implementation are also not helped by the lengthy nature of litigation procedures, whichdepending on the procedure -can take several years. It can also prevent the Commission from taking action in cases where damage has already occurred and cannot be repaired (IEEP 2011). Full implementation of environmental legislation is not only an issue of credibility but also has economic and social implications. Improving implementation of EU environmental law is a key priority of the current environment Commissioner, Janez Potočnik, and is expected to form an important part of the upcoming 7th Environment Action Programme (see previous section While the difficult economic conditions clearly require attention, strategic environmental priorities are in danger of being neglected or watered down in the face of concerns about streamlining legislation and reducing administrative burdens. Often this is based more on ingrained assumptions than a clear appraisal of the evidence. This more cautious approach has led to a narrower focus on those environmental initiatives that provide win-win solutions and are backed by broader economic interests. Rising public debts in several Member States and flailing capital markets have dented the ability to invest in critical infrastructure and innovative technologies and services necessary for the transition to a low-carbon, resource efficient economy. Moreover, the rise of emerging economies is dramatically changing the international landscape and the role of the EU therein. The crisis also provides EU environmental policy with a number of new opportunities. Addressing the inter-linkages and trade-offs between different thematic areas such as climate change, biodiversity, natural resources and environment and health, as well as between environmental policy and sectoral policies such as agriculture, energy or transport will also be important. The current economic downturn has made it cheaper and thus easier to achieve certain policy objectives, such as the EU's 2020 climate change objectives. Climate change concerns have infiltrated the main political discourse and are now increasingly reflected in the EU's Europe 2020 Strategy, as well as in spending priorities under the future EU budget. The EU has started to develop a policy agenda on resource efficiency that may in the future lead to concrete targets and indicators for reducing resource consumption, thus addressing the key driver behind many of the environmental challenges faced today. Discussions on greater economic convergence among some Member States could provide the conditions for extended efforts on green fiscal reform. Preconceptions are also gradually changing, with growing recognition among policy-makers and business actors that the current model of economic growth is inherently unsustainable and cannot be pursued indefinitely. The inter-connections and inter-dependencies between different economic, political, social, cultural, technological and environmental systems are being re-appraised. Developments in other regions are increasingly affecting Europe and vice-versa. Consequently, there is a need for a more holistic, integrated perspective that looks at the coherence and trade-offs of different policies, and points to a new focus on a green economy. In broad terms, the challenges ahead include reducing the intensity of resources used for economic activity (resource decoupling), reducing the negative environmental impacts from the use of natural resources (impact decoupling), preserving and restoring natural capital, and improving human well-being and quality of life. Improving the implementation of EU environmental policy will remain a key challenge and requires a more honest alignment of aspirations, regulatory means and implementation capacities with the political realities of a Union of 27 Member States. In the face of fiscal austerity and pressures for budgetary cuts across the board, there is a need to defend administrative capacities for good governance and regulatory foresight. The investment needs for achieving EU environmental objectives and to support the transition to a low-carbon, resource efficient economy are substantial and securing adequate financing to support environmental commitments will be another key challenge. Additional financial resources will need to be mobilised through new approaches complementing traditional grant funding, while proper take-up of the proposed mainstreaming approach in the future EU budget could have major implications for investment patterns in Member States. In the current context of economic austerity, there is a tendency to see environmental regulation as a brake on growth without necessarily considering the evidence at hand. Recent history suggests that well designed and implemented environmental policy, regulatory or not, can provide some of the foundations for long term prosperity as well as steering us towards a more sustainable society. Thus, even in a period of economic recession and political upheaval, EU environmental policy is likely to remain dynamic and relevant, offering a number of opportunities and avenues to help move forward from the current stasis. CLIMATE AND ENERGY Energy taxation COM(2011)169 Proposal to revise the existing energy taxation Directive 2003/96/EC under which taxation would be split into two components: a minimum tax rate of €20 per tonne of CO 2 and minimum rates for energy based on the energy content of a fuel rather than volumes. The proposed CO 2 tax rate will apply to all sectors not subject to the EU ETS, namely transport, households, agriculture and small industries. Provisions for some derogations are included. Energy efficiency COM(2011)370 Proposal for the establishment of a common framework for promoting energy efficiency in the EU to ensure the target of 20 per cent primary energy savings by 2020 is met. Placing on the market and use of biocidal products COM(2009)267 The proposal will repeal and replace Directive 98/8/EC on the placing of biocidal products on the market. The Proposal aims to address weaknesses identified in the implementation report on the Directive, such as the costs of compiling a dossier in support of the inclusion of active substance. For the first time the Proposal identifies which active substances may not be used in biocidal products. Control of majoraccident hazards involving dangerous substances COM(2010)781 Proposal to revise Directive on major accident hazards. The main proposed changes are: to align Annex I of the Directive to changes to the EU system of classification of dangerous substances; to include mechanisms to adapt Annex I in the future to deal with changing situations; to strengthen the provisions relating to public access to safety information, participation and access to justice; and to introduce stricter standards for inspections. SOIL Soil protection COM(2006)232 Proposal for the establishment of a framework for the protection of soil defining seven key functions of soil and introducing EU rules on soil condition monitoring, soil erosion, decline in organic matter, and contamination. The Directive would oblige sellers and buyers to provide a soil status report for any transaction of land where a potentially contaminating activity has taken, or is taking, place. Annex II: key EU legislative proposals awaiting adoption the all-party parliamentary environment group One of the larger all-party groups in Parliament, the All-Party Parliamentary Environment Group was set up twelve years ago to strengthen the influence of Parliamentarians on public policy and public debate on the environment. The Group also aims to assist Parliamentarians by improving their access to specialist information through regular group meetings and contact with senior environmental managers and directors from industry and NGOs, written briefings and special reports such as this one. The Group has over 150 Members of Parliament and the House of Lords, and some 180 associate member companies and organisations. It holds regular meetings and receptions at the House of Commons, with talks by leading British and International politicians and captains of industry on key environmental issues. A newsletter and briefing sheet is produced after each meeting. Over the years the Group has played host to quite a number of different British Ministers including David Miliband, Margaret Beckett and Michael Meacher, the Dutch, German and Danish Environment Ministers, senior Brussels officials including Margot Wallstrom, EU Commissioner, and many others from government, business and the campaign groups both in the UK and abroad. The Group meets 5 or 6 times a year at the Houses of Parliament and membership is by invitation. If you would be interested in joining the Group as an associate member, please contact the membership office shown opposite with details of your company or organisation. The Institute for European Environmental Policy The Institute for European Environmental Policy (IEEP) is an independent research organisation working on policies affecting the environment in Europe and beyond. Our aim is to analyse and present policy options and to disseminate knowledge about Europe and the environment. Our research work involves both pressing short-term policy issues and long-term strategic studies, drawing on more than thirty years of experience. Our project portfolio varies from year to year, but we are committed to being at the forefront of thinking about the environmental aspects of EU policies and keeping an open dialogue with policymakers and stakeholders. We have research programmes in several different fields and produce the Manual of European Environmental Policy (http://www.europeanenvironmentalpolicy.eu/). We work closely with the full range of policy actors, from international agencies and the EU institutions to national government departments, NGOs and academics. IEEP has an interdisciplinary staff with experience in several European countries and a wider network of partners throughout the EU. We work closely with universities, specialist institutes and consultancy organisations. The London office of IEEP was founded in 1980 and the Brussels office in 2001. For further information, please see our website: www.ieep.eu Figure 1 : 1 Figure 1: Number of items of EU environmental legislation adopted each year, 1962-2008 Source: IEEP 2011 Figure 2 : 2 Figure 2: Risk interconnections -complexities, interactions and synergies Source: WEF 2011 Figure 3 : 3 Figure 3: Gap between average 2008-2010 emissions and Kyoto targets in sectors not covered by the EU ETS Source: EEA 2011 Box 3 . 3 Adaptation to climate change Adaptation has been another issue on the EU policy map, although less prominently and with less specific measures. The Commission White Paper on adaptation in 2009 (EC 2009a) proposed more than 30 concrete actions in a number of areas, such as the development of a knowledge base, and the integration of adaptation into other EU policies. It called on the EU and Member States inter alia to explore the possibility of making climate impact assessment a condition for public and private investment and to develop indicators to better monitor the impact of climate change, (including vulnerability impacts), and the progress on adaptation. A follow-up EU adaptation Strategy is expected in 2013. Figure 5 : 5 Figure 5: Total GHG emissions from transport, 1990-2008 Source: Annual European Community GHG inventory 1990-2008 and inventory report 2010, Submission to the UNFCCC Secretariat, EEA Technical Report No 6/2010, EEA Figure 6 : 6 Figure 6: Main drought events in Europe (2000-2009) Source: EEA, 2010c Figure 7 : 7 Figure 7: Occurrence of floods in Europe (1998-2009) Source: EEA, 2010c Figure 8 : 8 Figure 8: Percentage of the EU urban population potentially exposed to air pollution exceeding acceptable EU air quality standards Source: EEA, 2011d Figure 9 : 9 Figure 9: Waste management performance/waste treatment in EEA countries in 2010 Source: Derived from information on Eurostat waste data centre 2010, http://epp.eurostat.ec.europa.eu/portal/page/portal/waste/data/sectors/municipal_waste Figure 10 : 10 Figure 10: Use of material resources and material productivity for the EU-15 and EU-12 Note: Domestic material consumption (DMC) is an aggregate of materials (excluding water and air) which are actually consumed by a national economy, calculated based on domestic extraction and physical imports (mass weight of imported goods) minus exports (mass weight of exported goods). Source: EEA, 2010c cent of 20-64 year olds to be employed Innovation • 3 per cent of the EU's GDP (public and private combined) to be invested in R&D/innovation Climate change and energy • Reduction in EU GHG emissions to at least 20 per cent below 1990 levels (reduction of 30 per cent if conditions are right) • 20 per cent of EU energy consumption to come from renewable resources • 20 per cent increase in energy efficiency Education • Reduce school dropout rates below 10 per cent • At least 40 per cent of 30-34 year olds should complete the third level of education Poverty and social exclusion • Reduce the number of people in or at risk of poverty and social exclusion by at least 20 million policy for the globalisation era • An agenda for new skills and jobs • European platform against poverty Source: European Commission 2010 Figure 11 : 11 Figure 11: Employment in various sub-sectors of the EU eco-industry sector Source: ECORYS 2009 The EU also has several sources of funding to support the research and deployment of environmental technologies. As part of the 2007-2013 multi-annual financial framework, the Commission supports research, development and demonstration of eco-innovative technologies and their market penetration within the 7th Framework Programme for Research and Technological Development (FP7) under the 'environmental and climate change' (€1.8 billion) and the 'energy' (€2.3 billion) themes; the Competitiveness and Innovation Framework Programme (CIP) which has a budget of €3.6 billion for the same time period, as well as the Eco-innovation First Application and Market Replication Projects, the European Eco-innovation Platform, and the environmental pillar of the LIFE+ Programme. Moreover, Member States and regions can also draw on funding under Cohesion Policy for the further deployment and replication of eco-innovation (EC 2011d).For the 2014-2020 period, the Commission has proposed a new Horizon 2020 Framework Programme for Research and Innovation which brings together all existing EU research and innovation funding. Horizon 2020 is expected to strengthen the role of eco-innovation and provide financing for the implementation of the Eco-AP. of funds is currently spent on two EU policies: the Common Agricultural Policy (CAP) and 'Cohesion Policy' which is devoted to regional development, social priorities, infrastructure and aid to poorer parts of the EU (see Figure12). A relatively minor fund is dedicated to environmental objectives (LIFE). However environmental spending also takes place through a number of other funds including the Structural and Cohesion Funds and the Rural Development element of the CAP. Figure 12 : 12 Figure 12: EU budget composition, 2007-2013 MFF Source: CEPS 2009 Figure 13 : 13 Figure 13: Infringements of EU environmental legislation by Member State and by sector (as of the end 2009) Source: EC 2010 proposal aims to simplify the existing legal framework, decreasing the share of emissions from two or three-wheel vehicles to overall transport emissions and improving aspects of vehicle functional safety. to add new provisions to the WEEE Directive (2002/96/EC), including new measures on registration and reporting requirements, broaden its scope, new collection and recycling targets, minimum inspection rules and make producers financially responsible for household collection.CHEMICALSExport and import of dangerous chemicals COM(2011)245 Proposal to recast the Regulation on the export and import of dangerous chemical, changing and clarifying some definitions and aspects of the consent procedure and transferring certain tasks to ECHA. the all-party parliamentary environment group chair Phillip Lee MP vice chairs Jack Dromey MP Therese Coffey MP Martin Horwood MP Mike Weir MP Baroness Young The All-Party Parliamentary Environment Group was set up to strengthen the influ- ence of Parliamentarians on public policy and public debate on the environment. The Group also aims to assist Parliamentarians by improving their access to specialist information. It currently has 126 Members of Parliament, 27 Members of the House of Lords and some 180 affiliated member companies, environmental groups and organisations. For more details, please contact: secretariat 45 Weymouth Street London W1N 3LD tel 0207 935 1689 fax 0207 486 3455 email [email protected] membership & accounts 4 Vine Place Brighton BN1 3HE tel 01273 720305 fax 0845 299 1092 email [email protected] Box 1. EU progress in reducing GHG emissions Since 2008, progress towards the EU's 20-20-20 targets has been aided by the economic downturn and the burgeoning development of renewable energies. In 2009, the EU's total GHG emissions decreased by 7 per cent. However, the combination of economic growth in some countries and a cold winter led to a rise in emissions in 2010 by 2.4 per cent. Overall emissions in the EU-15 had fallen by 10.7 per cent by the end of 2010 as a result of domestic emission cuts and activities in the sphere of land use, land use change and forestry (LULUCF), assisted by the use of flexible mechanisms. Hence, the EU-15 is expected to go beyond its very modest Kyoto targets. Although it has no specific target under the Kyoto Protocol, the EU-27 countries as a block follow a similar trend and by the end of 2010 had reduced emissions by 15.5 per cent (not taking LULUCF into account). Kyoto target, since the Protocol was ratified before 12 other countries joined the EU. Most Member States that joined the EU since 2004 have the same 8 per cent reduction target, with the exception of Hungary and Poland which have a reduction target of 6 per cent. Cyprus is a non-Annex-I Party to the Convention and thus does not have a target. Source: EEA 2010b; EEA 2011; EEA 2011b On an individual basis, only 16 of those EU Member States with a Kyoto target are currently on track to meet their individual objectives (Bulgaria, Czech Republic, Estonia, Finland, France, Germany, Greece, 1 The EU-27 does not have a for moving to a low-carbon economy by 2050', as The Plan was followed by a proposal for an energy efficiency Directive, currently under scrutiny by the European Parliament and the Council (EC 2011f). This sets out a number of measures for energy using sectors, and for the European energy supply sector and other proposals to remove barriers and overcome market failures that impede efficiency. As it stands, the draft Directive requires the Commission to assess in 2014 whether the EU can achieve the current EU energy savings target and, if appropriate, to propose binding legislation with mandatory national targets for 2020 which do not exist at present. Looking further ahead, several strategic documents relating to future climate and energy policy have appeared over the past year. In March 2011, as well as the energy efficiency plan, the Commission presented the 'Roadmap part of the Europe 2020 resource efficiency Flagship (EC 2011g). The overarching objective is to reduce GHG emissions by 80-95 per cent by 2050. The Roadmap also sets out the percentage reductions that would have to be achieved in key sectors (power, transport, the built environment, industry, agriculture and forestry) by 2030 and 2050 respectively. Investment needs are estimated to be, on average, around €270 billion annually over the next 40 years. Substantial investment needs have also been identified in Box 2. European companies support more ambitious EU climate policy In June 2011, 72 leading European and global companies signed a declaration calling on the EU to increase its current target of reducing GHG emissions to 30 per cent by 2020. Together the signatories account for more than 3.8 million employees with an annual turnover of more than €1 trillion. The companies call for an ambitious EU policy framework that can spur innovation and investment, notably in renewables and energy efficiency, lead to the creation of new jobs and enable Europe to maintain its leadership position in a global low carbon economy. Source: WWF 2011 Figure 4: Primary energy production in the EU, by fuel, EU-27 (Mtoe) relation to sustainable energy and transport infrastructure, low carbon technologies, research and development and adaptation (see Medarova et al, 2011). The question of securing financing for climate change related action is reflected quite prominently in the Commission's proposals for the future EU budget, although the sums available directly from this source are limited (see section 5). Source: Eurostat May 2011 The first step is in the political spotlight at present. New draft regulations for the CAP, Cohesion Policy and Research Policy among others are being negotiated in 2012-13. At present the main environmental concerns addressed in the Commission's draft regulations relate to climate change and energy, while Member States' record of implementing a large part of EU environmental legislation remains poor. At the end of 2010, environmental infringement procedures accounted for approximately one fifth of all open cases for noncommunication, non-conformity or bad application of EU law in the 27 Member States. A large number of cases relate to waste, nature, and water matters (see Figure13). The number of judgements of the European Court of Justice (ECJ) in environmental matters has increased continuously over the years as has the number of cases of noncompliance by Member States (EU-15) with ECJ judgements (IEEP 2011). Deadlines set in EU environmental legislation are regularly missed by a large number of Member States, with transposition of the environmental liability Directive for example proving a particularly problematic case. These issues are however normally resolved after the launch of infringement proceedings, and protracted delays only occur in a minority of Member States (EC 2011c). ). A new communication on implementing EU environmental law and policy is also under development by the Commission. The Communication is expected to explore practical avenues to improve current gaps in implementation, examining issues of improving coherence, enhancing compliance, strengthening inspections and enhancing the role of national judges in supporting implementation of EU legislation (EC 2010).EU environmental policy is facing a new and challenging context. Political attention is currently focused on the EU's economic and financial crisis, leaving little appetite for major new legislative initiatives in other areas. The crisis in the Eurozone has led to bigger questions concerning the EU project itself and growing scepticism about the EU has been voiced in a number of Member States, including the UK where political tensions have been brought to the fore in recent months. Details of a new inter-governmental agreement on the economic governance of the Eurozone are currently being negotiated. Many existing EU policies, including those concerning the environment, are not likely to be affected by this agreement. However, the political repercussions and dynamics of the new economic governance structure are yet to unfold. 6 Conclusions: priorities for the future development of EU environmental policy Provisions for end-use sectors, for the energy supply sector, and other measures are included. The proposal requires the Commission to assess in 2014 whether the EU can achieve the current energy savings target and, if appropriate, to propose legislation with mandatory national targets for 2020. The Connecting Europe Facility (CEF) is a new integrated instrument for investing in EU infrastructure priorities in the transport, energy and telecommunications sectors. The proposed budget is €50 billion of which €31.7 billion will be invested in transport infrastructure, €9.1 billion in energy infrastructure and €9.2 billion in broadband networks and digital services. The proposed Europe 2020 Project Bond Initiative will be one of a number of risksharing instruments upon which the CEF may draw to attract private finance. The proposal aims to harmonise recreational craft and personal watercraft with stricter emission limits for NOx, hydrocarbons and particulate matter. TRANSPORT Connecting Europe Facility COM(2011)456 AIR QUALITY Recreational craft COM(2007)851 Transport within the EU is heavily dependent on imported oil and oil products which account for more than 96 per cent of the sector's energy needs -EC (2011) Roadmap to a Single European Transport Area, Facts and Figures, http://ec.europa.eu/transport/strategies/facts-and-figures/putting-sustainability-at-the-heart-of-transport/index_en.htm The future for EU environmental policy 41 MARINE ENVIRONMENT AND FISHERIES Detergents COM( 2010)597 The proposal aims to extend the scope of Regulation 648/2004 to introduce a limitation on the content of phosphates and others phosphorous compounds in household laundry detergents. The proposal will set a phosphorous content limit of 0.5 per cent of the total weight of the product in all laundry detergents on the EU market Safety of offshore oil and gas operations COM( 2011)688 Proposal for a Regulation on the safety of offshore oil and gas prospection, exploration and production activities which establishes minimum requirements for industry and national authorities involved in offshore oil and gas operations. The Regulation aims to reduce the risks of a major accident in EU waters, and to limit the consequences should such an accident occur. Oil pollution COM (2000)802 amended by COM(2002)313 Proposal for a Regulation concerning oil pollution in European water. The proposal, part of the Erika package, is to establish a fund (COPE) to compensate for oil pollution damage in European waters and to complement the existing international two-tier regime on liability and compensation for oil pollution damage by oil tankers. Sulphur content of marine fuel COM(2011)439 Proposal to amend Directive 1999/32/EC as regards the sulphur content of marine fuels. If adopted, the Directive would transpose into EU law global limits on the sulphur content of marine fuels adopted in 2008 by the modification of the MARPOL Agreement of the IMO. By 2020 the limit for sulphur in marine fuels would be lowered from 4.5 per cent to 0.5 per cent. Alternative compliance methods are introduced, such as exhaust gas cleaning systems. The strengthening of the EU monitoring and enforcement regime is also proposed. For example, the Commission would be allowed to specify the frequency of sampling, the sampling methods and the definition of a sample representative of the fuel examined.

01743995

en

2024/03/05 22:32:07

2018

https://inria.hal.science/hal-01743995/file/RR-9162.pdf

Davide Frey Marc X Makkes Pierre-Louis Roman François Taïani Spyros Voulgaris Dietcoin: shortcutting the Bitcoin verification process for your smartphone Keywords: blockchain, sharding, UTXO, distributed ledger, cryptocurrency, mobile computing Blockchains have a storage scalability issue. Their size is not bounded and they grow indefinitely as time passes. As of August 2017, the Bitcoin blockchain is about 120 GiB big while it was only 75 GiB in August 2016. To benefit from Bitcoin full security model, a bootstrapping node has to download and verify the entirety of the 120 GiB. This poses a challenge for low-resource devices such as smartphones. Thankfully, an alternative exists for such devices which consists of downloading and verifying just the header of each block. This partial block verification enables devices to reduce their bandwidth requirements from 120 GiB to 35 MiB. However, this drastic decrease comes with a safety cost implied by a partial block verification. In this work, we enable low-resource devices to fully verify subchains of blocks without having to pay the onerous price of a full chain download and verification; a few additional MiB of bandwidth suffice. To do so, we propose the design of diet nodes that can securely query full nodes for shards of the UTXO set, which is needed to perform full block verification and can otherwise only be built by sequentially parsing the chain. Dietcoin: court-circuiter la vérification dans Bitcoin pour les téléphones mobiles Résumé : Les blockchains telles que Bitcoin passent mal à l'échelle, notamment du fait de leurs besoins important de stockage. Les besoins de stockage d'une blockchain typique ne sont pas limités et croissent indéfiniment. Par exemple, en août 2017, les données contenues dans la blockchain Bitcoin représentaient environ 120 GiB, contre 75 GiB un an auparavant, en août 2016. Pour bénéficier des garanties complètes de sécurité apportées par Bitcoin, un noeud qui rejoint le réseau doit télécharger et vérifier l'intégralité des 120 GiB de données. Cette nécessité pose un défi pour les appareils à faibles ressources tels que les smartphones. Heureusement, une alternative existe pour de tels dispositifs qui consiste à télécharger et à vérifier seulement l'en-tête de chaque bloc de la blockchain. Cette vérification partielle des blocs permet aux appareils de réduire leurs besoins en bande passante de 120 GiB à 35 MiB, mais cette diminution drastique ne permet qu'une vérification partielle des blocs, et diminue grandement les garanties de sécurité offertes aux noeuds qui l'utilisent. Dans ce travail, nous proposons une approche qui permet aux appareils à faibles ressources de vérifier entièrement des sous-chaînes de blocs sans avoir à payer le prix onéreux d'un téléchargement et d'une vérification complète de la chaîne ; quelques MiB supplémentaires de bande passante suffisent. Pour ce faire, nous proposons d'introduire des noeuds Dietcoin qui sont capables en toute sécurité d'interroger des noeuds exécutant le protocole complet pour obtenir des fragments d'un ensemble appelé UTXO, nécessaire à la vérification complète des blocs. Mots-clés : blockchain, partitionnement, UTXO, registre distribué, monnaie cryptographique, informatique mobile 1 Trustless Bitcoin Within a decade, blockchains have become extremely popular, and have been used to implement several widely-used crytocurrencies [START_REF] Nakamoto | Bitcoin: A peer-to-peer electronic cash system[END_REF], and smart-contract services [START_REF] Dickerson | Adding concurrency to smart contracts[END_REF]. A blockchain implements a tamper-proof distributed ledger in which public transactions can be recorded in a close-toirrevocable manner. Recorded transactions are stored into blocks, which are then incrementally linked (or chained ) in order to form an append-only list. The irrevocability of these chaining mechanisms exploits cryptographic mechanisms and peer-to-peer exchanges. This combination makes it in principle inconceivably hard for individual participants to revoke past transactions (due to the computational cost involved), while it remains possible for any participant to verify the validity of a blockchain's entire history. Verifying a blockchain remains, however, a particularly costly process. The verifying node must first download the entire blockchain, which in many cases has reached a size beyond the communication capabilities of many mobile devices. The Bitcoin blockchain, for instance, had grown to 120 GiB as of August 2017 (Figure 1), and follows an exponential growth, implying the problem can only become more acute. Once the blockchain has been downloaded, the verifying node must then check its consistency block by block, a lengthy process that can take hours on high-end machines. The exorbitant price of a full chain verification makes it unrealistic for low-resource devices to fully implement a blockchain protocol. Some blockchain systems, such as Bitcoin, therefore enable nodes to perform varying degrees of verification: full nodes verify everything while lightweight nodes only verify a small fraction of the data. In the case of Bitcoin, this lightweight verification is known as Simplified Payment Verification (SPV for short). SPV nodes only download and verify a much reduced version of the Bitcoin blockchain, comprised only of its block headers, which today only weights 35 MiB (a reduction by three orders of magnitude). This summary version however only contains the chaining information making up the blockchain, not the recorded transactions. This information is sufficient for SPV nodes to verify that the chain's structure is valid (and hence very unlikely to have been created by malicious nodes), but not that a past transaction does exist in the chain. As a result, SPV nodes are vulnerable to attacks in which an attacker leads an SPV node to believe a transaction t has occurred, while t is later on rejected by the system because the funds transferred by t have in fact already been spent (known as a double-spend attack ). To protect themselves against double-spend attacks, full nodes keep track of unspent funds in a structure known as the set of Unspent TransaCTion Outputs (UTXO set). The UTXO set is unfortunately costly to construct (as this construction requires the entire blockchain), to exchange (currently weighing 1.9 GiB, see Figure 1), and to maintain, which explains why SPV nodes do not use it. In this report, we propose to bridge the gap between full nodes and SPV nodes by introducing diet nodes, and their associated protocol, Dietcoin. Dietcoin strengthens the security guarantees of SPV nodes by bringing them close to those of full nodes. Dietcoin enables low-resource nodes to verify the transactions contained in a block without constructing a full-fledged UTXO set. In our protocol, diet nodes download from full nodes only the parts of the UTXO set they need in order to verify a transaction of interest. This selective download mechanism must, however, be realized with care. Diet nodes must be able to detect any tampering of the UTXO set itself, at a cost that remains affordable for low-resource devices, both in terms of communication and computing overhead. The rest of this report is structured as follows. We first present the Bitcoin protocol in more detail (Section 2), and explain the workings of full and SPV nodes. We then detail the design of Dietcoin and diet nodes and discuss the security guarantees they provide (Section 3). Finally, we present related work (Section 4), and conclude (Section 5). The Bitcoin system A blockchain is a decentralized ledger composed of blocks containing transactions. The transactions, the blocks, and the resulting chain obey a few core rules that ensure the system remains tamper-proof. Great care is required when modifying these rules, as even minor changes might break the blockchain's properties and its security guarantees. In the following, we first detail the default workings of the Bitcoin blockchain and its rationale 1 . We then build upon these explanations to introduce and justify the changes we are proposing. Overview In a blockchain system such as Bitcoin, the blockchain proper ((B k ) k∈Z ≥0 , label 1 in Figure 2) is maintained by a peer-to-peer network of miners. Each block B k links to the previous block B k-1 by including in its header a cryptographic hash that is (i) easy to verify, but (ii) particularly costly to create (this second point is one of the central element of blockchains with open membership, which we will discuss in detail just below). The leftmost block B 0 is known as the Genesis Block : it is the first and oldest block in the blockchain, and it is the only block with no predecessor. Recording a new transaction To transfer 8 bitcoins from herself to Bob, the user Alice must first create a valid transaction (label 2 ) that contains information proving she actually owns the 8 bitcoins (with a cryptographic signature using asymmetric keys), and encode the resulting transaction output with Bob's public key (such that, in turn, only Bob will be able to demonstrate ownership of the transaction's output). Alice then broadcasts this new transaction to the network of miners 3 , in order for it to be included in the blockchain. Before adding Alice's transaction into the blockchain, Miner A first verifies that the transaction is valid (label 4 in Figure 3, details on the transaction verification process will follow in Section 2.2.1-(BV2)). Miner A then includes Alice's transaction together with other transactions received in parallel into a new block (B 3 , 5 ), and attempts to link it to the current tip of the blockchain. This linkage operation requires Miner A to solve a probabilistically difficult cryptopuzzle 6 that regulates the frequency at which blocks are created (or mined ) by the whole network. (In Bitcoin, this periodicity is set to one block every 10 minutes.) If Miner A succeeds, the new block B 3 is now 1 Blockchains with closed membership or different consensus protocols are not discussed in this section RR n°9162 The new block ultimately reaches Bob 9 , who can check that the transaction has been properly recorded (and can then, for example, sell some goods to Alice). ... þ þ þ B 0 B 1 B 2 B 3 B 0 B 1 B 2 B 3 Bob 9 Irrevocability of deep blocks Because blocks are produced at a limited rate such that all the miners receive block B k before they can successfully mine a concurrent block B k , honest miners are highly likely to extend the chain when producing a new block, ensuring a consistent system state with high probability. The views of individual miners may however diverge in problematic cases, causing branches to appear. When a branch occurs, miners resolve the divergence by choosing as valid branch the one that was the most difficult to create (details on block difficulty will follow in Section 2.2.1-(BV2)). Blocks that are left out of the chain are said to be orphan. The risk of being made orphan decreases exponentially as a block lies deeper in a chain, ensuring the practical irrevocability of deep blocks and the transactions they contain. This is illustrated in Figure 4: consider an attacker who wishes to revoke a block B n-k (targeted block), that lies k blocks away from the chain's tip B n . For this attack to succeed, this attacker must produce an alternative subchain (B n-k , .., B n , B n+1 ) that is more difficult to create than the current chain. Producing this subchain is however extremely costly, and takes time which increases the odds that the legitimate chain grows (with a block B n+1 , thus requesting an even more difficult attack subchain) before the attacker succeeds. When the computing power of the attacker is less than half of that of the rest of the network, his probability of success drops exponentially with k. Inria B n-k-1 B' n-k B n-k B n Current blockchain ... ... ... B' n B' n+1 Hijacking a3empt Targeted block Transactions, Blocks, and UTXO set To benefit from the full security of Bitcoin, Bob should verify the validity of the new block that contains Alice's payment to him (label 9 in Figure 3) in addition to verifying the validity of Alice's transaction. This is because Alice could collude with a miner (or launch herself a miner), and produce an invalid block that she would advertise to Bob. Bitcoin relies on a number of builtin validity checks on blocks and transactions to conduct this verification. However, whereas full nodes exploit all of these checks, Simple Payment Verification nodes (SPV nodes) only perform a limited verification. In the following, we describe the details of these validity checks, we discuss the role of an intermediary set known as the set of Unspent TransaCTion Outputs (UTXO set), and the shortcomings of SPV nodes ensued by the limited verification they perform. Checking block validity A block is valid if and only if it meets the following two conditions. • (BV1) Its header respects the blockchain's Proof-of-Work predicate. • (BV2) It only contains valid transactions (which we discuss further below). BV1: The Proof-of-Work predicate makes it very difficult for malicious actors to alter the blockchain in an attempt to edit the ledger. The Proof-of-Work predicate is used as a lock-in mechanism to anchor blocks in the chain. It is enforced on each block header, whose simplified structure is shown in Figure 5. The header of each block B k points both to the header of the previous block B k-1 (using a hash function, 1 ), and to the transactions contained in the current block B k 2 . To fulfill the Proof-of-Work predicate 4 , a header must contain a nonce 3 such that the hash of the header is less than a difficulty target. The difficulty target is set so that a new block is created every ten minutes by the miners as a whole, regardless of the computation power (the difficulty target is regularly adjusted to cope with changes in their computation power). Finding a nonce respecting the difficulty target is computationally very expensive, as every miner competes to create blocks. This computing cost prevents attackers from easily tampering the chain as they have to recompute fresh nonces for the blocks they wish to replace. To establish a secure and verifiable link between the header of B k and the corresponding block B k , the pointer to B k 's transactions 2 consists of the root of a Merkle tree. A Merkle tree is a hierarchical hashing mechanism for sets that enables a verifier to efficiently test whether an item (here a transaction) belongs to the set by reconstructing the root of the Merkle tree. Each leaf node in a Merkle tree consists of the hash of an item, while each internal node (including A the root) consists of the hash of its children. This makes it possible to reconstruct the root, and thus verify set membership, using only a logarithmic number of intermediate hashes. In Figure 6 for example, a node can verify the presence of transaction A in the set by (i) downloading the root from a secured communication channel (e.g., the blockchain), and (ii) downloading A and the three intermediate hashes shown in red: H B , H CD and H EF GH , and reconstructing the root from the downloaded hashes. The reconstructed root should match the downloaded one. BV2: In addition to the Proof-of-Work predicate (BV1), all the transactions included in a block must also be valid for the overall block to be valid. Figure 7 shows the validity mechanisms included in a typical Bitcoin transaction. In this example, Alice uses 3 coins she owns (the transaction's inputs 1 ) to pay 7 Bitcoins to Bob, and 4 to Tux (the transaction's outputs 2 ). Only coins created in earlier transactions may be spent: each of Alice's inputs therefore points back to the output of an earlier transaction 3 . To ensure that only the recipients (Bob and Tux) are able to spend the output, each new coin contains an ownership challenge (a hashed public key), that must be solved to spend this coin 4 . H ABCD H B H C H D H A H AB H EFGH H F H G H H H E H EF H CD H GH H ABCDEFGH B C D E F G H Reconstructed Downloaded Unneeded Alice's transaction is only valid if the following three conditions are met: • (TV1) The inputs do exist, and Alice owns them. She can prove her ownership of the inputs by providing a public key matching their ownership challenges 5 , and by signing the new transaction with the corresponding private key2 6 ; • (TV2) No money is created in the transaction. In effect, the total value of the transaction's inputs must be greater than or equal to that of its outputs: in∈inputs(t) value(in) ≥ out∈outputs(t) value(out). The difference value(in) -value(out) is given as a fee to the miner of the block containing the transaction; • (TV3) The transaction's inputs (tx_ID i , index j ) have not been spent yet (i.e., they do not appear as inputs of any earlier transaction, an attack known as a double spend ). The set of Unspent TransaCTion Outputs (UTXO set) While the validity of a block's header (BV1) only requires access to the current block B k , and to the header of its predecessor B k-1 , verifying transactions (BV2) requires a lot more information. Verifying the ownership challenges of input coins (TV1), and their amount (TV2) requires access to the transactions recorded in earlier blocks. Worse, verifying that inputs coins have not yet been spent (TV3) potentially requires parsing and verifying the entire blockchain. To avoid performing such a costly operation for each new block, nodes that verify transactions maintain an intermediary set known as the set of Unspent TransaCTion Outputs (UTXO set). The UTXO set contains all the coins that have been created in the chain but not spent in later transactions, it thus contains all the spendable coins. A node verifying transactions can prevent double spends (TV3) by simply ensuring that all the inputs of a transaction appears in its UTXO set. The UTXO set evolves as new correct blocks are added to the chain: transaction outputs are removed from the set when they are spent, and outputs of new transaction are added to the set. The limitations of SPV nodes In spite of its benefits, constructing a local UTXO set is costly: in order to obtain the set, a node must first download the entire chain (120 GiB as of August 2017, see Figure 1) and validate it (a lengthy process that can take hours on high-end machines), even if only the latest block is relevant to its interest. Because of this cost, Bitcoin supports several levels of verification. Miners and users running full nodes construct the UTXO set and check both block headers (BV1) and transactions (TV1,2,3 and as a result BV2). By performing all the possible checks, full nodes benefit from the maximum security that the Bitcoin system has to offer. By contrast, Simple Payment Verification nodes (SPV nodes) do not construct the UTXO set. Instead, they only download the chain's block headers (rather than full blocks), and verify that these headers are valid (BV1). With the headers only, SPV nodes are able to verify the well-formedness of the blockchain including the crucial Proof-of-Work predicate that seals the links of the chain. However, because this verification is only partial, SPV nodes are unable to detect if a new block contains an invalid transaction. This scenario, however probabilistically difficult to accomplish for an attacker, is a vulnerability of SPV nodes. To circumvent this vulnerability, SPV nodes typically wait until miners have created subsequent blocks extending the chain containing a block of interest, which implies that these miners have performed a full verification on it and consider this block as valid. The need for SPV nodes to wait makes it particularly problematic to use Bitcoin on limited devices (i.e. mobile phones) for everyday transactions. SPV nodes are not even able to check whether the inputs used in a transaction do exist. It also limits the ability of SPV nodes to detect faulty transactions as early as possible, which is an important usability feature of modern payment systems. In this work, we propose to overcome the inherent limitations of SPV nodes with Dietcoin. Dietcoin enables nodes with limited resources (diet nodes) to benefit from a security level that is close to that of full nodes, at a fraction of the cost required to run full security checks. The Dietcoin system To address the vulnerabilities of SPV nodes and to improve the confidence mobile users can have in recent transactions, we propose Dietcoin, an extension to Bitcoin-like blockchains. Although our proposal can be applied to most existing Proof-of-Work blockchains using the UTXO model for coins, we describe Dietcoin in the context of the Bitcoin system as presented in Section 2. The core of Dietcoin consists of a novel class of nodes, called diet nodes, which provide lowpower devices with the ability to perform full block verification with minimal bandwidth and storage requirements. Instead of having to download and process the entire blockchain to build their own copy of the UTXO set, diet nodes query the UTXO set of full nodes and use it to verify the legitimacy of the transactions they are interested in (as described in Section 2.2.1-(BV2)) and the correctness of the blocks that contain them. This gives diet nodes security properties that sit in between those of full nodes, and those of Bitcoin's SPV nodes. Consider a user wishing to verify a transaction for the sale of some goods. The user's diet node will initially proceed like a standard SPV node. It will contact a full node to obtain the header of the block that supposedly contains its transaction as well as the corresponding branch of the transaction Merkle tree, to verify that the transaction indeed is included in the block. But while an SPV node would stop at this inclusion check, the diet node continues by verifying both the inclusion and the correctness of all the transactions in the block. To make this possible we introduce the possibility for diet nodes to access the state of the UTXO set of full nodes corresponding to the instant right before the block they want to verify. Since downloading the entire UTXO set would result in prohibitive bandwidth overhead, as shown in Figure 1, Dietcoin-enabled full nodes split their UTXO set into small shards, enabling diet nodes to download only the shards that are relevant to the transactions in the block. Inria To prevent diet nodes from trusting maliciously forged shards, the shard hashes are used as leaves of a Merkle tree, which root is stored in each block. Having the Merkle root stored in blocks enables the UTXO shards to benefit from the same Proof-of-Work protection as transactions. In addition to the full verification done on the block of their interest B k , diet nodes can increase their trust in B k by fully verifying its previous blocks. By doing so, diet nodes are ensured that none of the l verified blocks contain illegal transactions nor erroneous UTXO Merkle root. To make a diet node trust a forged transaction in block B k , an attacker has to counterfeit the subchain of l + 1 blocks (B k-l , ..., B k ). Thanks to the Proof-of-Work protection, the cost of this attack increases exponentially as l increases linearly. In the following, we detail the operation of Dietcoin by first describing how full nodes can provide diet nodes with verifiable UTXO shards. Secondly we discuss how miners link blocks to the state of their UTXO set. We then explain how diet nodes can extend the verification process from one block to a subchain of any length. Finally, we detail the operation of diet nodes when verifying transactions. Sharding the UTXO set To enable the operation of diet nodes, Dietcoin-enabled full nodes need (i) to provide diet nodes with shards of the UTXO set, while (ii) enabling them to verify that these shards are authentic. To satisfy (i), Dietcoin-enabled full nodes store the UTXO set resulting from the application of the transactions in each block in the form of shards with a predefined maximum size of 1 KiB (on average, across all shards). The use of shards enables diet nodes to download only the relevant parts of the UTXO set and also limits the storage requirements at full nodes, which only need to store the modified shards for each block to let diet nodes query older versions of the shards. Similarly, the limit of 1 KiB for the size of each shard limits the bandwidth employed by diet nodes in the verification process. To satisfy (ii), full nodes also maintain a Merkle tree that indexes all the shards of the UTXO set. Using shards also proves advantageous with respect to this Merkle tree. If nodes were to index UTXO entries directly, the continuous changes in the UTXO set would cause the Merkle tree to become quickly unbalanced, leading to performance problems or requiring a potentially costly self-balancing tree. The use of shards, combined with the right sharding strategy, gives the UTXO Merkle tree a relatively constant structure, enabling shards to be updated in place most of the time. Moreover, it makes it possible to predict the size of the UTXO Merkle tree and its incurred overhead, which enables us to better control and balance the storage overhead for full nodes and the bandwidth requirements of diet nodes. A number of sharding strategies satisfy the requirement of a fixed number of shards. In this work, we use the simplest approach consisting of indexing UTXO entries by their first k bits. This strategy resembles a random approach since the first bits of an UTXO are the transaction hash it references, which value is expected to be random due to the uniformity property of the SHA-256 hash function. This strategy comes with the added advantages of obtaining shards of homogeneous size and a full binary Merkle tree with 2 k leaves. Keeping in mind the size cap of 1 KiB per shard, k can be adapted locally by each node to cope with the growth of the UTXO set. When the average shard size breaches the cap of 1 KiB, k is incremented by one resulting in (i) halving the average shard size, and (ii) adding one layer to the Merkle tree, doubling its storage footprint in the process. Linking blocks with the UTXO set With reference to Figure 8, consider a Dietcoin-enabled miner that is mining block B k , and let UTXO k be the state of the UTXO set after applying all the transactions in B k . The miner stores the root of the UTXO Merkle tree associated with UTXO k as an unspendable output in the first transaction of block B k before trying to solve the Proof-of-Work as shown in Figure 8. Storing the root of the UTXO Merkle tree in the first transaction of the block does not require any modification to the structure of Bitcoin's blocks. Thus it is possible for Dietcoin-enabled full nodes and miners to co-exist with their legacy Bitcoin counterparts. Dietcoin-enabled full nodes verify the value of the UTXO Merkle root against their own local copy when they verify a block, while legacy Bitcoin nodes simply ignore it. The UTXO Merkle tree provides a computationally efficient way for diet nodes to verify whether the shards they download during the verification process are legitimate and correspond to the current state of the ledger. Still referring to Figure 8, let us consider a diet node d that wishes to verify a transaction in block B k . Node d needs to obtain: block B k , the UTXO Merkle root in block B k-1 , the shards of the UTXO set between the two blocks, and the elements of the associated Merkle tree that are required to verify their legitimacy. It can then verify the shards using the root stored in block B k-1 's first transaction, and use them to verify the correctness of the transactions in B k . We observe that storing the Merkle-root referring to the state after the block into the block itself forces diet nodes to download two blocks to verify transactions. This makes it inherently harder for an attacker to provide a diet node with a fake block B k because it would need to forge not only block B k but also block B k-1 . Extended verification Diet nodes have the ability to extend their confidence in a block by iterating the verification process towards its previous blocks. By doing so, diet nodes ensure the correctness of the UTXO Merkle root present in block B k-1 used to verify the correctness of block B k . The extended verification can be performed on a subchain of any length l provided that the verifying diet node can query UTXO shards of any age. A diet node fully verifying the subchain (B k-l+1 , ..., B k ) can only be tricked into trusting a malicious transaction in block B k if the attacker manages to counterfeit the l + 1 successive blocks starting from B k-l , which becomes exponentially more costly as l increases linearly. Inria The block B k-l contains the UTXO Merkle root that serves as a basis for the verification of the consequent blocks. Since the block B k-l is not verified, it is thus trusted by the diet node. A comparison can be made with full nodes where the first block of the chain, the genesis block, is hard-coded and is therefore trusted. By shifting the trust from the genesis block to the block B k-l for diet nodes, Dietcoin effectively shortcuts the verification process. Picking the value of l exhibits a trade-off between security and verification costs. On one hand choosing a great l draws the behavior of the diet node closer to that of a full node, while on the other hand choosing a small l draws it closer to that of an SPV node. The user can make her decision based on which block depth l is great enough in her opinion such that all blocks prior to B k-l are unlikely to be counterfeited. For instance, a diet node user can choose l such that the trusted block has a block depth of 6 or greater since it is the de facto standard in Bitcoin to consider blocks of depth 6 or greater as secured 3 . In the case depicted in Figure 8, assuming l = 2, diet node d downloads block B k-1 to verify the transactions and the UTXO Merkle root in B k-1 to further increase its confidence in B k . To verify B k-1 , the diet node uses the UTXO Merkle root in B k-2 . Detailed Operation Equipped with the knowledge of Dietcoin's basic mechanisms, we can now detail the verification process carried out by diet nodes. Algorithm 1 depicts the actions taken by a diet node when its user starts the application using Dietcoin and compares it with those taken by legacy SPV clients. Black dotted lines [•] are specific to diet nodes, while hollow dotted lines [•] are common to both diet and SPV nodes. The algorithm begins when the application using Dietcoin starts and updates its view of the blockchain. In this first part of the algorithm, a diet node behaves exactly in the same manner as an SPV node. First, it issues a query containing the latest known block hash and an obfuscated representation of its own public keys in the form of a bloom filter (lines 4-5). A full node responds to this query by sending a list of all the block headers that are still unknown to the SPV/diet node, together with the transactions matching the SPV/diet node's bloom filter, and the information from the transaction Merkle tree that is needed to confirm their presence in their blocks. Using this information, the SPV/diet node verifies the received headers (including Proof-of-Work verification) and updates its view of the blockchain (line 6). Since the response from the full node might contain false positives due to the use of a bloom filter for public key obfuscation, the next verification step consists in ensuring that the received transactions match one of the user's public keys (lines 8-9). Once false positives are discarded, the SPV/diet node verifies that the received transactions are in the blocks (line 10-12). At this point, a standard SPV node simply returns the received transactions to the application (line 14). A diet node, on the other hand, continues the verification process. To this end, the diet node first computes which blocks to fully verify to ensure that (i) no block is fully verified twice (line 17), (ii) old blocks considered by the user as secured enough are not fully verified (parameter maxDepth p , line 17) and (iii) only a subchain of limited length l is fully verified (parameter maxLength p , line 19). If no block is selected for full verification, the diet node falls back to SPV mode (lines 20-21). To bootstrap the full verification process, the diet node must first download the UTXO Merkle root present in the block prior to the first block to verify (line 23). For each of the selected blocks B k , the diet node downloads from Dietcoin-enabled full nodes (i) the block B k itself (line 25), (ii) the state, before B k , of the UTXO shards associated with both the block's transactions' inputs and outputs (line 26), and (iii) the partial UTXO Merkle tree required to prove the integrity of the downloaded shards (line 26). Once it has all the data, the diet node proceeds with the verification process. For each block B k , it first verifies that the downloaded UTXO shards match the UTXO Merkle root from the previous block B k-1 (lines 27-29). Then it verifies that the transactions in each block use only available inputs from these shards (lines 32-33), and computes the new state of these shards based on the transactions in B k (lines 34, 36). Finally it verifies that the updated shards lead to the UTXO Merkle root contained in B k (lines 37-39). Once the verification process has terminated, the diet node returns the transactions associated with the local user to the application (line 14). Related work Making the UTXO set available for queries between nodes has been discussed several times in the Bitcoin community over the past few years. Bryan Bishop published a comprehensive list of such proposals [START_REF] Bryan | bitcoin-dev] Protocol-Level Pruning[END_REF] that share some of the following goals: (i) enabling faster node bootstrap, (ii) strengthening the security guarantees of lightweight nodes, and (iii) scaling the UTXO set to reduce its storage cost. The primary goal of Dietcoin is to strengthen the security of lightweight nodes. Dietcoin's strongest feature is the ability for diet nodes to efficiently perform subchain verification, as described in Section 3.3, which offers stronger security guarantees than the referenced proposals made by the community. Moreover, even though we focus in this report on the security of lightweight nodes, it is also possible to bootstrap full nodes faster with Dietcoin. Sharing similar goals with Dietcoin, Andrew Miller [START_REF] Miller | Storing UTXOs in a Balanced Merkle Tree (zero-trust nodes with O(1)-storage)[END_REF] suggests to store in blocks the root of a self-balancing Merkle tree built on top of the UTXO set. In such a system, lightweight nodes only download the UTXOs they need, which results in a lower bandwidth consumption than with shards as we propose it, but at the cost of a greater storage overhead since the stored Merkle tree is larger. Moreover, Dietcoin combines a full, and thus always balanced, Merkle tree built on top of 2 k shards that can each be updated in place as blocks are appended to the chain. This combination of tree stability and updatable shards enables efficient subchain verification, as described in Section 3.3, and adapting this feature to a system using a self-balancing Merkle tree does not seem trivial. Vault [START_REF] Leung | Vault: Fast bootstrapping for cryptocurrencies[END_REF] also proposes to use Merkle trees to securely record the state of the distributed ledger in recent blocks, and shards this state across nodes, to reduce storage costs. Contrarily to Dietcoin, however, Vault targets balance-based schemes, such as introduced by Ethereum, in blockchains relying on Proof-of-Stake consensus. Vault further stores individual accounts in the Merkle trees, rather than UTXO shards as we do. This represents a different and to some extent orthogonal trade-off to that of Dietcoin, in that Vault chooses to increase the size of Merkle tree witnesses that must be included in transactions, but removes the need for lightweight nodes to download UTXO shards. Whereas we focus on sharding the resulting state of the blockchain, other systems propose to shard the verification process. Both Elastico [START_REF] Luu | A Secure Sharding Protocol For Open Blockchains[END_REF] and OmniLedger [START_REF] Kokoris-Kogias | OmniLedger: A Secure, Scale-Out, Decentralized Ledger via Sharding[END_REF] proposes a permissionless distributed ledger using multiple classical PBFT consensus protocols each executing within a subset (shard) of nodes. Elastico limits the number of shards a malicious nodes may join (under different identifies) by tying the shard of a node to the result of a Proof-of-Work puzzle. OmniLedger [START_REF] Kokoris-Kogias | OmniLedger: A Secure, Scale-Out, Decentralized Ledger via Sharding[END_REF] extends the ideas proposed by Elastico [START_REF] Luu | A Secure Sharding Protocol For Open Blockchains[END_REF] to increase the size of the shards (and thus reduce the probability of failures), and allow for cross-shards transactions thanks to a Byzantine shard-atomic commit protocol called Atomix. Both Elastico and OmniLedger use sharding to increase the transaction processing power of a distributed ledger, rather than to improve access and verification of the Inria UTXO set, as we do. Chainiac [START_REF] Nikitin | CHAINIAC: Proactive Software-Update Transparency via Collectively Signed Skipchains and Verified Builds[END_REF] combines the ideas of skiplists and blockchains to realize skipchains, an authenticated log with both back and forward long-distance links to implement a distributed authenticated software-release ledger. Chainiac relies on digital collective signatures to implement forward links, which are not available in permissionless Proof-of-Work chains such as Bitcoin. Long distance links are particularly well adapted to navigate a well-identified subset of a blockchain (such as a package's releases). They were not however directly designed to handle the kind of dependencies captured by the UTXO model. An entirely different approach to scaling blockchains for lightweight nodes is the use of Non-Interactive Proofs of Proof-of-Work (NIPoPoWs) [START_REF] Kiayias | Non-interactive proofs of proof-of-work[END_REF] that enable constant size queries. NIPoPoWs strive for minimal cost of proof of inclusion of a transaction in a chain, thus reducing to a minimum the bandwidth requirements of lightweight nodes. NIPoPoWs however do not aim at offering improved security for lightweight nodes as we do with Dietcoin. Conclusion In this report, we have presented the design of Dietcoin, that proposes a new form of Bitcoin nodes that strengthens the security guarantees of lightweight SPV nodes by bringing them closer to those of full Bitcoin nodes. The Dietcoin protocol enables low-resource nodes to verify the transactions contained in blocks without constructing a full-fledged UTXO set. In our protocol, diet nodes download from full nodes parts of the UTXO set they need in order to verify a block, or a subchain of blocks, of interest. Diet nodes are able to detect any tampering of the UTXO set itself, at a cost that remains affordable for low-resource devices, both in terms of communication and computing overhead. In our approach, Dietcoin-enabled full nodes split their UTXO set into small shards, and enable diet nodes to download only the shards that are relevant to the transactions in the block, while verifying that these shards do indeed corresponds to the state of the UTXO set for the block they are verifying. 2 . 3 9 3 239 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.1 Recording a new transaction . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.2 Irrevocability of deep blocks . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Transactions, Blocks, and UTXO set . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2.1 Checking block validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2.2 The set of Unspent TransaCTion Outputs (UTXO set) . . . . . . . . . . . 9 The limitations of SPV nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Dietcoin system 10 3.1 Sharding the UTXO set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 Linking blocks with the UTXO set . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.3 Extended verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.4 Detailed Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Figure 1 : 1 Figure 1: Both the Bitcoin blockchain and the UTXO set have almost tripled in size in the past two years. Inria 2 B 0 B 1 B 2 BobFigure 2 : 20122 Figure 2: A blockchain is formed of a sequence of blocks containing transactions. The current state of the blockchain (here (B 0 , B 1 , B 2 )) is stored by each individual miner. Figure 3 : 3 Figure 3: To add a new transaction to the current blockchain, a miner first verifies the validity of the transaction. It must then solve a costly cryptopuzzle to encapsulate this transaction in a new block (here B 3 ), before disseminating this block to other miners. Figure 4 : 4 Figure 4: Revoking the content of a block B n-k deep in the chain requires constructing a better alternative subchain, which becomes exponentially harder as the block lies deeper. hash(B k-1 .header) merkle_h(B k .transac1ons) tx 2 BFigure 5 : 25 Figure 5: Content of a block header (simplified). Figure 6 : 6 Figure 6: Example of a Merkle tree root reconstruction that only needs log(n) hashes. Figure 7 : 7 Figure 7: Structure of a transaction. Figure 8 : 8 Figure8: The UTXO set is updated every time a block is validated. For a counterfeited block to be validated by diet nodes, a malicious node has to forge at least two consecutive blocks: the first block B k-1 containing a fake Merkle root of UTXO k-1 , and the second block B k spending fake coins validated by the fake UTXO k-1 . RR n°9162 Bitcoin uses a scripting language to encode challenges and proofs of ownership, enabling for more complex schemes, but for ease of exposition we limit ourselves to the typical case.Inria Acknowledgments This work has been partially funded by the Region of Brittany, France, by the Doctoral school of the University of Brittany Loire (UBL), by the French National Research Agency (ANR) project SocioPlug under contract ANR-13-INFR-0003 (http://socioplug.univ-nantes.fr) and by the SIDN Fonds contract 172027. bloomFilter(keys) Compute a bloom filter from keys buildMRoot(hashes) Compute the Merkle root from hashes getShardKey(txId) Apply the sharding algorithm to txId height(header) Index of header starting from the genesis block updateMTreeInPlace(MTree, dataset) Update the hashes of MTree with the new value of dataset verifyHeaders(headers) Add headers to the chain, return the hash of the new chain tip filter ← bloomFilter(pubKeys p ) •5: ({header, txMTree, txs}) ← send queryMerkleBlocks(tipId p , filter) •6: tipId p ← verifyHeaders((header)) •7: for all {header, txMTree, txs} ∈ ({header, txMTree, txs}) do if ∀k ∈ pubKeys p : k / ∈ {tx.inputs ∪ tx.outputs} then continue Ignore bloom filter false positives •10: assert(∀tx ∈ txs : HASH(tx) ∈ txMTree) •11: builtTxMRoot ← buildMRoot(txMTree) •12: assert(builtTxMRoot = header.txMRoot) •13: verifyBlocksUpTo(height(header)) •14: callback(txs, header) Callback to app •15: procedure verifyBlocksUpTo(last) •16: Do not verify blocks twice or below maxDepth p •17: first ← max(highestVerified p , height(tipId p ) -maxDepth p ) •18: Verify up to maxLength p blocks •19: first ← max(first, last -maxLength p ) •20: if first ≥ last then The first UTXO Merkle root is not verified utxoMRoot ← send queryUtxoMRoot(HASH(headerStorep[first])) •24: for all blockId of height ∈ [first + 1, last] do •25: block ← send queryBlock(blockId) •26: {shards, utxoMTree} ← send queryUtxos(blockId) •27: assert(∀shard ∈ shards : HASH(shard) ∈ utxoMTree) •28: builtUtxoMRoot ← buildMRoot(utxoMTree) •29: assert(builtUtxoMRoot = utxoMRoot) •30: for all tx ∈ block.transactions do for all i ∈ tx.inputs do •32: shard ← shards[getShardKey(i)] •33: assert(i ∈ shard ∧ valid proof of ownership of i) •34: shard.remove(i) utxoMTree ← updateMTreeInPlace(utxoMTree, shards) •38: utxoMRoot ← buildMRoot(utxoMTree) •39: assert(utxoMRoot = block.utxoMRoot)

01744091

en

2024/03/05 22:32:07

2016

https://hal.science/hal-01744091/file/PLM016_18_ECNLMS_ProReg_VF.pdf

Farouk Belkadi Ravi Kumar Gupta Ekaterini Vlachou Alain Bernard Dimitris Mourtis Linking modular product structure to suppliers' selection through PLM approach: A Frugal innovation perspective Keywords: PLM, co-evolution, Modular, Supplier selection, frugal innovation To maintain market share rates, frugal innovation is a main solution for competitive enterprises to meet the customer's needs in different regional markets. The co-evolution of product and production network aims to manage local production sites of the OEM and several collaborative relations between OEM and supplier companies for better management of the project resources in the regional market. Supplier selection and evaluation are among the main factors to be resolved at the earlier stage to guarantee successful results from any OEM-Supplier collaboration. This paper discusses the potential of using a modular-based approach as a kernel methodology to support the co-evolution of product structure and production network definition, especially in the case of supplier selection for frugal innovation perspective. The application of PLM approach to manage interconnected data describing the co-evolution of the product structure and production network is also discussed. Introduction In context of hard competitiveness and economic pressures, companies need to reach new markets (i.e. both emerging and mature market) by further sharpening their strategic focus on what customers really need. For this, frugal innovation theory is introduced to explain new market trends and to propose new solutions supporting these evolutions [START_REF] Zeschky | Frugal Innovation in Emerging markets: The case of Mettler Toledo[END_REF]. For Tiwari and Herstatt [START_REF] Tiwari | Frugal Innovation: A Global Networks' Perspective[END_REF], frugal innovation refers to innovative products and services that "seek to minimize the use of material and financial resources in the complete value chain (from development to disposal) with the objective of reducing the cost of ownership while fulfilling or even exceeding certain pre-defined criteria of acceptable quality standards". This theory results new category of products named "frugal" as an aggregation of the following attributes: Functional to answer the exact customer need by focusing on key product features; Robust with integration of recent technologies and facilities of maintenance high life duration; User-friendly through a simple and easy to use functions and interfaces; Growing through large production volumes enabling economies of scale; Affordable by offering to customer good "value for his money" through adaptable prices according to their socio-economic context; and Local to propose products mainly tailored to local requirements but also built using some production facilities (i.e. suppliers) and components from the targeted market [START_REF] Berger | Frugal Products: Study Results[END_REF]. To develop such kind of products, companies should adopt a customer-driven design process and an optimal co-evolution of the production strategy to reduce manufacturing and logistic costs, taking in consideration the capabilities, constraints and resources available in the targeted market. The design strategy respecting frugal attributes can be conducted through a set of following design actions: • Design new specific modules or modifying the features of existing ones, • Reuse existing solutions developed by the company in previous projects, • Use standard modules developed by external suppliers for several products. The co-evolution of the production network to support the adaptation of an existing product to a new market implies new changes on the production process or on suppliers to cope with new modification in the product structure (it can be the change of one module or modification of some module features). In parallel, the modification of the product structure can be a consequence of using a new module (or technology) proposed by a supplier in the local market which results in the co-evolution of the product structure with the production strategy [START_REF] Arndt | Customer-driven Planning and Control of Global Production Networks -Balancing Standardisation and Regionalisation[END_REF]. The big challenge is then to have an efficient collaboration between the OEM company and all its suppliers. The co-evolution of the production system should be extended to the production network level that aims to create several collaborative relations between the OEM (Original Equipment Manufacturer) and suppliers' companies for better management of their distinctive skills and resources in the whole production process [START_REF] Hochdörffer | Evaluation of global manufacturing networks -a matter of perspective[END_REF]. Thus, Supplier selection and evaluation are among the main factors to be resolved at the earlier stage to guarantee successful results from any OEM-Supplier collaboration [START_REF] Cheraghi | Critical Success Factors for Supplier Selection: An Update[END_REF]. As per our study, an emerging market's needs can be solved by adapting an existing product and production which are well stablished in westerned and European markets to fulfil regional customer's requirements (new market) especially when the customer belongs to emerging markets (e.g. India, Africa and China). This adaptation of the existing product to emerging market is researched using frugal innovation. Few companies adopted the frugal innovation for the above aspects, but took long time to launch their product in the markets. The research in ProRegio [7] is to fill these gaps. One part in this research is suppliers' selection respecting the requirements from regional markets and also company's policies. In this context, this paper discusses the potential of using a modular-based approach as a kernel methodology to support the co-evolution of product structure and production network definition, especially in the case of supplier selection in context of frugal innovation. This approach is mainly suitable for design strategies based on the use of standard modules or the reuse of existing solutions (with possible adaptation). A smart algorithm is used to generate and evaluate the alternative supplier networks. 2 PLM approach for supplier selection Frugal innovation and suppliers' selection With respect to the frugal innovation principle, the global modular-based approach should handle the easier interpretation of customer requirements and the identification of only concerned modules to be considered for the customization process [START_REF] Du | Product Families for Mass Customization: Understanding the Architecture[END_REF][START_REF] Mourtzis | The evolution of manufacturing systems: From craftsmanship to the era of customization. Design and Management of Lean Production Systems[END_REF]. The concept of module represents a physical or conceptual grouping of product components to form a consistent unit that can be easily identified and replaced in one product architecture in order to increase product variety and flexible adaptability [START_REF] Jiao | Product Family Design and Platform-based Product Development: State-of-the-art Review[END_REF]. In the proposed frugal innovation process (Figure 1), the designer can propose several alternatives of product modules with specific features to cope with a set of customer requirements. These features concern technical characteristics used for the engineering perspective as well as useful inputs for building the related production network. Each module is identified with all possible production capabilities or suppliers able to provide it with the desired characteristics. The selection of the best alternatives of production systems or suppliers is fulfilled by taking in consideration of different facilities and constraints in the local market to build the global production network. Then the selection of the best module solutions can be obtained as a consequence of selecting the related production systems or suppliers. Fig 1. Co-definition of product structure and production network Production system refers to technological elements (machines and tools), organizational behavior and managing resources within OEM whereas supplier refers to external supplier of product modules and related supports. A supplier can be local or international based on local requirements and product modules characteristics. By fixing the different production systems and suppliers (for final assembly and for the production of modules), the structure of the production network is defined as a combination of the selected items. The expected behavior of the network is obtained by the definition of the global planning and all collaborative processes supporting information and material exchange among these production systems and suppliers. Thus, the assembly process of the whole product structure is obtained according to the global production planning at network level. Several collaborative relations between the OEM and suppliers are identified in the literature based on the level of integration of the supplier in the final project of the OEM [START_REF] Calvi | How to manage early supplier involvement (ESI) into the new product development process (NPDP): Several lessons from a French study[END_REF]. At the low levels, suppliers are assimilated to simple executors of detailed specifications from the OEM. In more collaboration level, the supplier is more involved in the development project and participates to the definition of the product architecture (1 st rank suppliers). This nature of collaboration can provide serious advantages for the deployment of frugal innovation strategy in new markets. In this case the OEM will exploit some interesting and innovative solutions proposed by suppliers to design new frugal product or to adapt an existing one to one specific market. By this, the development process will follow a concurrent path in which the selection of the best product modules can be obtained from the identification of the best suppliers respecting the frugal requirements. Supplier selection strategy contributes principally to the improvement of "Robust", ""Affordable" and "Local" attributes since it gives the possibility to the company to: use new solution that can enhance the product quality respecting the target market standard, moderate the cost by selling from competitive suppliers, especially when those ones are coming from the target market. The concept of product module can be used to connect the product structure to different outputs of the suppliers involved in the related production network. Thus, using product module features combined with additional information from the final assembly process planning and logistic constraints in the local market can give interesting requirements for the evaluation of KPIs (Key performance indicators) useful to get an impartial assessment of potential suppliers' capacities before considering them in the production network. The mapping of product module features (from product configuration view) to KPIs (suppliers selection view) is presented in Figure 2. The product modules are the products of the suppliers. The decision making for supplier selection and product module selection is based on the mapping between requirements (product module features) and real values of KPIs for the suppliers and their products. Global PLM approach for supplier selection The role of mapping is to classify all available suppliers according to the matching level between their KPI values (representing real assessment) and product features (representing requirements). Some time, when there are limited possibilities of suppliers to cover the requested module, the mapping table can be used to adapt, if necessary, some product features according to the capacities of the related supplier. The PLM approach can support the concurrent design process of frugal products and the supplier network through a smart management of product modules and related suppliers alternatives for several product configurations addressed to various markets. This is ensured through an optimal integration, storage and connection of numerous data coming from several applications, including suppliers' selection tools. Indeed, the implementation of PLM approach results from the integration between heterogeneous IT systems such as ERP (Enterprise Resource Planning), PDM (Product Data Management), SCM (Supply Chain management), etc. [START_REF] Bosch-Mauchand | Knowledge based assessment of manufacturing process performance: integration of product lifecycle management and value chain simulation approaches[END_REF]. In the proposal, the supplier selection process is achieved based on several data stored on different business tools. This data is identified and managed in the PLM system as specific features connected to different product modules alternatives. Each module is analyzed to generate supplying requirements. The final decision is the identification of optimal "Modules-Suppliers" combination as fragment co-definition of best product structure and supplier network with respect to frugal attributes. Product features for supplier selection To illustrate the proposed principle of solution for the use of modularity for supplier selection problem, an example of concurrent design process of frugal product and identification of possible suppliers for each module is presented in Figure 3. This scenario deals with the case of adapting existing product architecture to a new market based on the capabilities of existing suppliers in the targeted market. Supplier selection strategy based on global modular product design approach The process starts by the analysis of new market requirements (including specific customer requirements) and then search for existing product structures covering these needs. The analysis of modules functions and features as requirements is used to generate list of possible suppliers. The list of suppliers' alternatives will provide list of possible module alternatives matching with the same functions of the originally design product. Then several alternatives of product structures from the original one (developed in older projects) can be obtained as combination of the product modules proposed by the selected suppliers. The performance assessment of each alternative will help to fix the best product structure and consequently the best product modules replacing the original ones. In final the list of supplier is fixed as those providing the selected modules. To perform these decision making processes, additional categories of information should be embedded in the product module concept. The definition of product module through features is to connect different views of the product design and development. Product module features are defined to translate the regional customer requirements to product design and connect the product design to production planning and supplier network design. The product modules' features are defined to address three categories objectives as inputs for frugal decision making problems regarding the objectives of the global modular product design approach: • Analyzing customer/market requirements and linking product modules to requested functions in specific product architecture. • Defining modules' parameters and its interfaces to support the design of new product (and related alternatives) as a consistent combination of product modules. • Responding to production strategy (production system and network definition) through a consistent connection of modules to optimal production capabilities. These three categories of features are used to identify requirements for supplying properties and production planning as well as to refine the selection of product module. From these categories of features, the following Table 1 includes the list of relevant features that should be used as requirements for supplier selection process. Table 1. List of modules features for supplier selection process Feature Description Performance Acceptable standardization and tolerance level for product module performance values, regarding the global product performance. Interfacing Capacity to one module to be interfaced with other ones Interchangeability Capacity for one module to be replaced by one or more other modules from other suppliers but providing similar functions. Customization Requested possibility to change some proprieties of the module and level of options proposed in the supplied module. Process position Connection of the module to different steps of the final assembly process and the level of dependency with the connected modules Criticality The importance of the related function regarding its added value to the final product structure. Supplying tolerance Level of request on the supplier in terms of cost, delivery time and confidence level as a consequence of the previous modules' features. Figure 4 illustrates an example of using product module features (green color rectangle) to analyze product life cycle and generate additional Supplying Tolerance Features (inner blue color rectangle), to be considered as requirements for decision criteria for the selection of best suppliers for frugal innovation issue. These supplying decision criteria and/or related KIPs can be used for production network design. Through the application of the concurrent design of product structure and suppliers' network, conflict can appear from contradictions on results based on analysis of product module features on one side and supplying requirements on the other side, for example a module can have high performance but real risk on delivery time regarding the supplier capacities, especially if the module is requested earlier in the production process. This will imply important delays on the final assembly process of the product. Affordable and growing attributes are seriously affected. Product module features as requirements for supplying decision criteria Figure 5 illustrates such kind of conflicts between product modules "as designed" (inner blue color rectangle) and real modules "as produced by suppliers" (inner green color rectangle). It is shown that the product module "PM1" has product module feature "Criticality" value as "High". PM1.1 is the "Best" implementation and PM1.2 is the "Worst" one according to the related suppliers KPIs. However, PM1.1 presents a low level of interfacing with the PM 2.1 that is the only possible implementation of PM2. The conflict is that if the best product module with high criticality (PM1.1) is selected then supplying of other product module (PM2) is not possible. There are two possibilities to resolve this type of situations either (i) Select the modules to be supplied as per the requested module features and redesign (or reselect new supplier) the product modules which are not possible to be supplied by any existing supplier, or (ii) Relax features' values for the conflicted product module in the requested product. Thus, product module features and KPIs for production network design should be connected together through the PLM approach to evaluate all possible combinations of product-suppliers in the production network. Conflicts possibilities on the co-definition of product structure and supplier network This connection provides real positive impacts on the frugal attributes. For instance, deciding optimal supplier based not only on its own properties but also on the criticality of the related modules allows managers adapting the balance between product cost and quality by favoring reputable suppliers for the critical modules (high function criticality requesting high quality) even the price is high, and obtaining the less critical modules (less added value for the target market) from low cost suppliers. Supplier selection and network design method The supplier selection and network design method should be capable of generating alternative combinations of product-suppliers and evaluate their performance based on multiple and conflicting criteria based on the constrains-suggestions provided by the design during the generation of the different product structures. One of the main challenges during the supplier's selection and the supplier networks design is to select the optimum supplier based on their suitability and their availability. Therefore, the proposed work proposes a smart search algorithm capable of subset of total number of alternative manufacturing network configurations by utilizing three adjustable parameters [START_REF] Chryssolouris | Manufacturing Systems: Theory and Practice[END_REF]. The three control parameters that are utilized are the maximum number of alternatives (MNA) which controls the breadth of the search, the decision horizon (DH) which controls the depth of the search, and the sampling rate (SR) which guides the search towards the high quality branches of the tree of alternatives [START_REF] Mourtzis | A Multi-Criteria Evaluation of Centralized and Decentralized Production Networks in a Highly Customer-Driven Environment[END_REF]. The optimum values of these factors will be obtained through a statistical Design of Experiments (SDoE) [START_REF] Phadke | Quality Engineering Using Robust Design[END_REF]. The main inputs of the proposed algorithm are the various product features generated by the product configuration and the pre-filtered list of suppliers based on their compatibility and their suitability, as well the bill of processes to produce the product modules and in general the final product. The decision-making procedure following by the supplier selection and networks design algorithm is based on resource-task assignment decisions. Considering that, in order to produce a part/module of the product a task should be performed by a resource and the resource belongs to a plant or to a supplier. In case of suppliers the task is directly connected with the supplier. The main steps of the proposed algorithm the formalization of the alternatives, the criteria determination to satisfy the objectives, the definition of the criteria weights, the calculation of the criteria values and last is the selection of the optimum alternative based on their performance (Figure . 6) [START_REF] Mourtzis | A Multi-Criteria Evaluation of Centralized and Decentralized Production Networks in a Highly Customer-Driven Environment[END_REF][START_REF] Mourtzis | Design and Operation of Manufacturing Networks for Mass Customisation[END_REF]. Multiple and conflicting criteria are considered and are calculated by the smart search algorithm and during the decision making procedure. Production and Transportation Cost, Quality, Lead time, total Energy Consumption, CO2 emissions are among the main criteria considered [START_REF] Mourtzis | F A Toolbox for the Design, Planning and Operation of Manufacturing Networks in a Mass Customisation Environment[END_REF]. To address the need of frugal innovation in the context of the supplier's selection method also the locality of the suppliers is considered as main criterion. Locality shows how close to the targeted market is the supplier. This will contribute to enhance the affordable and local attributes by reducing logistics costs and using existing standards in the target market since the providers are coming from the same market. Moreover, through the consideration of Locality, as criterion, local suppliers can be considered, increasing the market share of local markets. The total network performance is calculated by measuring defined KPIs. In that stage, also the suggested KPIs, from the product configuration stage, are considered. The final results of the selected network are sending back to the PLM tool in order to fix the different modules of the product and finalize the design of the product. constraints and resources. Global modular approach has been used to support the design and development of frugal product. The application of module concept for frugal innovation perspectives showed the need to re-think the definition of the product model and its integration in future generation of PLM systems. New features should be considered to represent requirements of production from frugal product side and to connect product design to production and supplying strategies Considering product features as decision criteria for supplier selection strategy will contribute to enhance the product frugal attributes, especially affordable, robust and local attributes. This is given through an optimal balance between cost, quality and delivery time requests, assessed separately for each product module according to its criticality in the target market, impact on the global performance and implication on the final assembly process. The logic is that for critical modules, strong trust on suppliers' performance is required. Less important modules or with high interchangeability allow more flexibility on the selection strategy for cost reduction. The proposed methodology can provide a great advantage to companies that are moving towards frugal innovation concept by providing local, low-cost and affordable products taking into account local customer's requirements. Moreover, through the consideration of Locality, as criterion, during the proposed supplier's selection and networks design tool, local suppliers can be considered, increasing the market share of local markets. This paper focuses on the conceptual solution instead of presenting the final implementation for industrial use case which is under development. The developed methodology is well appreciated by industrial partners as well as academic partners. Fig 2.Global PLM approach for supplier selection Fig 3.Supplier selection strategy based on global modular product design approach Fig 4.Product module features as requirements for supplying decision criteria Fig 5.Conflicts possibilities on the co-definition of product structure and supplier network Fig 6 . 6 Fig 6.Supplier selection and network design Algorithm-Smart search algorithm Acknowledgments. The presented results were conducted within the project "ProRegio" entitled "customer-driven design of product-services and production networks to adapt to regional market requirements". This project has received funding from the European Union's Horizon 2020 program under grant agreement no. 636966. The authors would like thank the industrial partners involved in this research.

01744104

en

2024/03/05 22:32:07

2017

https://theses.hal.science/tel-01744104/file/TH2017HAOWENJUN.pdf

come Atomic layer deposition of boron nitride

01744138

en

2024/03/05 22:32:07

2018

https://hal.science/hal-01744138/file/2018BlogCybergeo-Tannier_en.pdf

Cécile Tannier About fractal models in urban geography and planning: refuting the aesthetics and the universal norm Fractal models have been used in urban geography for forty years. Their main applications were for analyzing urban forms but they were also used to simulate urban growth. Research in the field has significantly contributed to better characterise the local and global shape of cities and to better understand their evolutions. Yet concomitantly, one can deplore the circulation of some myths about the interpretation of fractal analysis results and about the possible usage of fractal models for urban analysis and planning. I propose here to undermine some of these myths on the basis on some scientific publications in the field1. Cities are not fractal. Indeed, measures of fractal dimensions vary in space: the fractal dimension estimated for a whole city differs from the fractal dimension estimated for its neighbourhoods, each having its own dimension (see e.g. [START_REF] Thomas | Comparing the fractality of European urban neighbourhoods: do national contexts matter[END_REF]). Moreover, considering a given built pattern, the slope of the curve that represents the number of counted elements with respect to the size of the counting window may exhibit local variations (Frankhauser 1998[START_REF] Frankhauser | Comparing the morphology of urban patterns in Europe. A fractal approach[END_REF][START_REF] Tannier | Fractals in Urban Geography: A Theoretical Outline and an Empirical Example[END_REF][START_REF] Thomas | Clustering patterns of urban built-up areas with curves of fractal scaling behaviour[END_REF]. Very early on, pioneer works in geography have insisted on the fact that the fractal dimension is not expected to be constant in reality [START_REF] Goodchild | Fractals and the Accuracy of Geographical Measures[END_REF]; most often, it is constant over a limited range of scales but varies somewhat over successive ranges of scales [START_REF] Lam | On the Issues of Scale, Resolution, and Fractal Analysis in the Mapping Sciences[END_REF][START_REF] White | Urban Systems Dynamics and Cellular Automata: Fractal Structures between Order and Chaos[END_REF]. Consequently, properties of scale-invariance and statistical self-similarity are locally specific but not universal, and concern limited scale ranges. Thus, by definition, cities are not fractal. Nevertheless, measures of fractal dimension are interesting for geographers as they enable the characterisation of spatial distributions being highly heterogeneous. Indeed, as fractal dimension is determined via counting the elements of a spatial distribution according to several nested spatial resolutions, it informs us about the systematic variations of a given geographical fact through scales. Such variations are not proportional, not linear, and would not be detected when we use other spatial concentration indexes such as density [START_REF] François | Villes, densité et fractalité. Nouvelles représentations de la répartition de la population[END_REF]. In practice with fractal analysis, the starting point is to set the hypothesis that a spatial distribution is scale-invariant (or statistically self-similar). Then deviations to scale-invariance are studied. Such deviations may appear for some scale ranges but not others. Deviations may also vary in space, which allows the identification of spatial differentiations. 2. Evolution of the shape of cities does not comply with a unique model that would finally end in a fractal order state. For P. [START_REF] Frankhauser | Comparing the morphology of urban patterns in Europe. A fractal approach[END_REF] and N. [START_REF] Salingaros | A universal rule for the distribution of sizes[END_REF], the emergence of fractal urban patterns originates in the combination of two types of processes:  bottom-up processes resulting from individual actions (for instance, the choice of households to become owner of an individual house in a suburban area) or from collective actions (for instance, actions of local pressure groups according to the Nimby (Not In My Backyard) logic);  top-down processes (i. e. urban and regional planning). For F. [START_REF] Schweitzer | Analysis and Computer Simulation of Urban Cluster Distributions[END_REF], fractal urban patterns emerge from the interaction of contradictory actions at an individual level only, i. e. residential location choices that minimize the distance to both the city centre and the urban boundary (countryside). Underlying those assumptions is the idea of self-organisation. In a self-organised system, a meso-or macroscopic order emerges from interactions at a microscopic level and constraints in return the future evolutions of the system2. Accordingly, F. [START_REF] Schweitzer | Analysis and Computer Simulation of Urban Cluster Distributions[END_REF] have set the hypothesis that the rank-size distribution of urban built clusters changes in the course of time toward becoming a Pareto distribution. If this hypothesis is confirmed, the conclusion would be that a Pareto exponent indicates the development stage of a city and that the deviations from a Pareto distribution indicates potentials for future urban developments. Yet some empirical research results have infirmed this hypothesis. In particular, L. [START_REF] Benguigui | The Dynamics of the Tel Aviv Morphology[END_REF] have shown that the rank-size distribution of built clusters of the city of Tel Aviv (Israel) followed a Zipf's law from 1935 to 1964 then gradually deviated from it between 1974 and 2000. Several scholars have also studied the evolution of fractal dimensions of cities in the course of time. Although the comparison of values of fractal dimension obtained for different studies is not possible since the data used and the calculation methods involved are different, we note similar general tendencies. P. [START_REF] Frankhauser | Aspects fractals des structures urbaines[END_REF] calculated an increase of fractal dimension of Berlin (Germany) in the course of time: 1.43 in 1875, 1.54 in 1920, 1.69 in 1945[START_REF] Shen | Fractal Dimension and Fractal Growth of Urbanized Areas[END_REF] calculated an increase of fractal dimension of Baltimore (USA), from 1.015 in 1822 to 1.722 in 1992. An increase of fractal dimension of the built surface has also been shown for the metropolitan area of Basel (Switzerland, France, Germany) between 1882 and 1994 [START_REF] Tannier | Fractals in Urban Geography: A Theoretical Outline and an Empirical Example[END_REF] as well as for the metropolitan area of Lisbon (Portugal) between 1960 and 2004 [START_REF] Encarnação | Fractal cartography of urban areas[END_REF]). Yet if the evolution of city shapes evolves according to a fractal growth process, their fractal dimension should not change in the course of time, which studies quoted above contradict. An objection can be that these studies consider each city within a spatial extent that is fixed in the course of time and that comprises on the one hand the urban area itself which expands gradually, and on the other hand, its periphery. That's why other studies have considered cities within a study area that expands according to the urbanisation process. For instance, L. Benguigui, D. Czamanski, M. Marinov et J. [START_REF] Benguigui | When and Where Is a City Fractal? Environment and Planning B: Planning and Design[END_REF] have analysed the evolution of Tel Aviv's metropolitan area from 1935 to 1991 taking into account three nested study areas. They have shown that the growth differed for each part of the metropolitan area: the fractal dimension increased at different speed in each part of the metropolitan area; some parts have become "fractal" (i. e. statistically self-similar for a given scale range) earlier than others; and the whole metropolitan area has become "fractal" until the mid 80s. The fact is that a city does not evolve for centuries according to a fractal growth process that would be unique and that would go on until the achievement of a final "maturity" stage of the urban form. First, rules that determine the location and the shape of new urban developments change in the course of time. Second, the urban sprawl process results in the gradual integration of peripheral built areas (villages, hamlets, diffuse suburban settlements/buildings) within the inner city. Third, an urban built pattern can be deeply modified through destruction and reconstruction (for instance, the re-shaping of Paris during the 19th century, the reconstruction of cities after massive destructions resulting from bombing or natural disasters, or the massive destructions of old built neighbourhoods and the construction of large buildings and skyscrapers in contemporary Chinese cities). Last, we can observe creations ex nihilo of new towns in the periphery of large cities. 3. Nothing proves that fractal urban forms are optimal by nature. Starting from the statement that a fractal order can emerge at a meso-or macroscopic level from self-organising processes, fractality is sometimes seen as a desirable equilibrium state. « Multifractality represents optimal structure of human geographical systems because a fractal object can occupy its space in the most efficient way. Using the ideas from multifractals to design or plan urban and rural terrain systems, we can make the best of human geographical space » (Chen 2016)3. Thus, if self-organised fractal forms are satisfying (even optimal), urban planning becomes useless (and even annoying) because top-down constraints may engender deviations from fractality (Genre-Grandpierre 2017). Subsequently, deviations from fractality can be seen as signs of dysfunction. For Y. [START_REF] Chen | Multifractal Characterization of Urban Form and Growth: The Case of Beijing[END_REF], for instance, deviations from a multifractal structure of Beijing's urban area denote its decline and the degeneration of the inner city. Yet we have previously seen that shapes of cities and urban built patterns are not fractal. Nonetheless they are not all in decline neither are they all degenerated. Adopting an organicist point of view, other scholars support the idea that urban planning and design should aim at the creation of fractal forms because such forms exist in nature (in which they spontaneously emerge) and thus are "by nature" virtuous and optimal. Fractals are then raised to a universal aesthetic principle, see e. g. [START_REF] Jiang | A New Kind of Beauty Out of the Underlying Scaling of Geographic Space[END_REF]. Besides the fact that such a principle falls under belief more than science, its adoption leads most often to give more importance to emerging forms than to their generative mechanisms. Resulting models are essentially structural: they take into account neither the real behaviours of individuals (as a result of a combination of aspirations, constraints and available means) nor the strong emergence that characterizes social systems4. The fact is that functional advantages of realistic fractal urban developments (starting from an existing urban pattern) with respect to non fractal developments are still poorly known because they have rarely been studied until now. However, a shape is not intrinsically optimal: it is optimal only with regards to the processes (i. e. behaviours, practices) that this shape allows to optimise. Moreover, the absolute does not exist in the case of spatial distributions of human settlements because of the diversity of contexts (social, political, economic, natural, etc.) from which results a high diversity of human spatial behaviours and practices. The "good" fractal dimension for urban planning does not exist. A same value of fractal dimension can characterise very different urban shapes and can result from generative processes being qualitatively and quantitatively very different [START_REF] Pumain | Commentaire sur le chapitre 3 -Les fractales doivent-elles guider l'aménagement urbain ? In G. Dupuy (dir) Villes, réseaux et transport. Le défi fractal[END_REF]. As a Most ideas exposed in this blog post are taken from the dissertation entitled "Analysis and simulation of the concentration and the dispersion of human settlements from local to regional scale. Multi-scale and trans-scale models" [In French], chap. 3 "Variation de la concentration et de la dispersion des implantations humaines à travers les échelles : modèles mono-et multi-fractals", pp. 114-175, C.[START_REF] Tannier | Analyse et simulation de la concentration et de la dispersion des implantations humaines -Modèles multi-échelles et trans-échelles[END_REF]. https://tel.archivesouvertes.fr/tel-01668615v1 Additionally, scholars commonly introduce in self-organising models a limiting factor that often corresponds to a maximum city size, which can not be overcome[START_REF] Schweitzer | Analysis and Computer Simulation of Urban Cluster Distributions[END_REF], or to a maximum urbanisation rate[START_REF] Chen | Defining urban and rural regions by multifractal spectrums of urbanization[END_REF]. Y.[START_REF] Chen | Defining urban and rural regions by multifractal spectrums of urbanization[END_REF] goes further as he suggests that the urbanisation process should ideally stop at the stage at which the urbanisation rate L equals 0.618 and the urban-rural ratio (called "golden ratio" in the quoted paper) equals 1.618. It is possible to distinguish weak emergence, where the macroscopic structures resulting from microscopic behaviours can be observed by an external observer identifying a particular regularity in the observed process, from strong emergence, where the microscopic entities observe themselves the macroscopic structures they have produced and adapt their behaviours accordingly[START_REF] Livet | Ontology, as a mediator for agent-based modeling in social science[END_REF]. Social systems are characterised by a strong emergence: through their membership to a social group or to a place, individuals participate in the creation of collective references that refer to this group or place (weak emergence). In return, the adoption (or not) of these collective references by the group or the individual influences its behaviour (strong emergence).

01316563

en

2024/03/05 22:32:07

2015

https://shs.hal.science/halshs-01316563/file/The%20politics%20of%20post-suburban%20densification.pdf

Eric Charmes Roger Keil THE POLITICS OF POST-SUBURBAN DENSIFICATION IN CANADA AND FRANCE This debate specifically focuses on densification as a particular dimension of (post-) suburbanization. In the introduction, we discuss densification, along with 'compactness' and 'intensification', conceptual terms that have become buzzwords within urban planning. Objectives associated with these tend to be presented in the literature within a normative framework, structured by a critique of the negative effects attributed to sprawl. The perspective here is different. It is not normative but critical, and articulated around the analysis of political and social issues, related to the transformation of wider metropolitan space. Three main themes are developed: (1) the politics of densification (the environmental arguments favouring densification are highly plastic, and are thus often used to defend projects or initiatives which are actually determined by other agendas); (2) why morphology matters (a similar number of houses or square metres can be established in many different ways, and those different ways have political and social meaning); (3) the diversity of suburban densification regimes (it is not only the landscapes of the suburbs that are diverse, but also the local bodies governing them--between the small residential municipalities of the Paris periurbs and the large inner suburbs of Toronto lies a broad spectrum). Introduction 1 While newly developed areas of North America and Western Europe continue to be filled with detached single-family dwellings (see Figure 1), many older suburbs are changing [START_REF] Harris | Meaningful types in a world of suburbs[END_REF]Hamel and Keil, 2015). This process has been described as post-suburbanization (Phelps and Wu, 2011a). It does not affect all suburbs, and resistance to change is strong (Filion, 2015, this issue), but the changes are significant in many places. It is important to state that we are not talking here about a distinct typology--suburbs versus post-suburbs--but rather a historical change in direction: a process of dedensification (classical suburbanization) is partly converted, inverted or subverted into a process that involves densification, complexification and diversification of the suburbanization process (see Figure 2). Los Angeles, for example, long viewed as the ultimate suburban city, is really one of the densest metropolitan areas in the United States, an 'inverted city' where suburbanization has begun to fold in on itself as areas traditionally considered sprawling, mono-cultural and mono-functional have increasingly become denser, more multicultural and mixed in use. In the European case, postsuburbanization involves a slight shift in focus from the discourse on the traditional (dense, centralized, politically integrated) European city--as argued masterfully by Patrick Le [START_REF] Galès | European Cities: Social Conflicts and Governance[END_REF]--to a model that acknowledges the dissolution as posited in the literature on the 'in-between city' or zwischenstadt (Sieverts, 2003). In the Canadian case, we have witnessed the turn away from the classical North American model, with its clear separation of classical nineteenth-century industrial inner city and twentieth-century suburbanization, towards 'in-between' or 'post-suburban' forms of peripheral urbanization: in this process, the post-second world war 'inner suburbs' with their particular assemblage of issues--decaying high-rise housing stock, concentrations of poverty, mobility imbalances, often racialized social segregation, a dramatic lack of services, etc.--have become a focus of attention among researchers [START_REF] Hulchanski | The Three Cities Within Toronto. Income Polarization Among Toronto's Neighbourhoods, 1970-2005[END_REF][START_REF] Keil | Post-Suburbia and City-region Politics[END_REF][START_REF] Keil | In-Between Mobility in Toronto's New (Sub)urban Neighbourhoods[END_REF]Poppe and Young, 2015, this issue). intra-muros 12, 74x100 cm, 2008, www.jeanpierreattal.com) As the growing literature on post-suburbs shows, such processes are by no means ignored. We can consult the most comprehensive and influential recent contribution to the debate on postsuburbanization in the introduction to a recent collection of studies on the subject by Phelps and Wu (2011a). In this view, post-suburbanization refers primarily to an era involving a double re-definition of classical suburbia: a 'maturation' of the suburbs and a host of new influences changing the nature of those areas. Some interventions, like the extensive literature on Los Angeles since the 1980s, have gone so far as to claim an epochal shift in urbanization patterns. More broadly, however, post-suburbia entails the notion of a reversal of the linearity of historical processes, as traditional geographical typologies of ordered concentric segmentation have given way to a more splintered or fragmented urbanism (as encountered under broader processes of neoliberalization). Post-suburbanization does not refer to a complete and featureless dissolution but to a reconsolidation of the urban fabric, even a balancing, and a rejection of classical functional or conceptual dichotomies such as live-work. This is particularly the case in the technoburbs that have been associated with the process. Postsuburbanization also entails a profound re-scaling of the relations and modes of governance that have traditionally regulated the relationships between centre and periphery in the suburban model [START_REF] Phelps | The New Post-suburban Politics?[END_REF]Phelps and Wu, 2011b;Hamel and Keil, 2015). In this contribution to Debates & Developments in IJURR, we specifically focus on densification as a particular dimension of (post-)suburbanization. This introduction and the four essays that follow make no larger claims about suburbanization in France and Canada or even their comparative trajectories. Yet we do think that these essays have significance in engaging critically with the strategic and normative preference for density and compactness in a sustainability paradigm that often remains unchallenged on both sides of the Atlantic. In this context the Canadian and French examples, and their comparison through the work presented here, is valuable. Low-density morphological patterns have not only been at the heart of the original programme of suburban utopia with its setting of pastoral life [START_REF] Fishman | Bourgeois Utopias: The Rise and Fall of Suburbia[END_REF], but they are also key to understanding the recent transformation of suburbs. Yet in both national contexts under review here, Canada and France, higher-density peripheral development has also been part of the development regime since as far back as the 1960s. More recently (and most relevant to the debate presented here), densification, along with 'compactness' and 'intensification', have become buzzwords in urban planning. Objectives associated with densification tend to be presented in the literature within a normative framework, structured by a critique of the negative effects attributed to sprawl. There is a wealth of propositions under the banners of New Urbanism or Smart Growth in North America and Sustainable Cities in Europe, around the 'battle against urban sprawl' and the need to increase the density of cities to make them more resilient and sustainable. Our perspective here is different. It is not normative but critical, and articulated around the analysis of political and social issues, related to the transformation of wider metropolitan space. The aim of this debate is to document with empirical surveys the issues at stake with densification policies. These transformations serve certain interests yet neglect many others. They often lead to displacement of less well-off populations. More broadly, their geography is uneven, predominantly targeting the low-income and working-class suburbs. The success of the themes of densification and of the battle against urban sprawl should also be related to the fact that those themes converged with the interests of urban planners (under the buzzwords of growth control), politicians in core urban areas (who welcome new residents and activities) and developers (who can exploit the rent gap, new opportunities, etc.). Besides, the need for dense cities is a convenient argument in overcoming the strong local resistance to urban development (see below). Such critical perspective has of course already been developed in the literature. Among others, John [START_REF] Logan | Urban Fortunes: The Political Economy of Place[END_REF]Harvey Molotch (2007 [1987]: xx) noted (in the second edition of their book Urban Fortunes) that densification not only serves environmental interests but also helps defend the 'same old growth machine'. Emergence of consensual discourse on sustainable development is often accompanied by de-politicization of the issues at stake [START_REF] Béal | Le développement durable changera-t-il la ville ? Le regard des sciences sociales, Saint-Etienne[END_REF]. The gravitas gained by sustainable development ideology contributes to silencing debates on political and social issues. Sustainability itself becomes the stand-in for better (sub)urbanization and is not usually exposed to critical scrutiny [START_REF] Keil | Cities and the Politics of Sustainability[END_REF]. For example, the website of Richard Rogers' planning and architecture practice (one of the leading proponents of compactness) proclaims that 'Compact polycentric cities are the only sustainable form of development' (Rogers Stirk Harbour + Partners, n.d.). Since the survival of humanity is at stake, there is no point debating the opportunity to increase density (which is key for compactness) of cities. With this debate, however, we propose to discuss this normative ideal. Three main themes are being developed. The first is the politics of densification. All the essays address some of the political, social and economic stakes of densification. Within this introduction, we will stress the plasticity of the environmental arguments favouring densification. In fact, the idea that the dense city is more sustainable than the low-density city can be contested on environmental grounds. This plasticity of the environmental discourse makes it all the more obvious to consider densification as a political process favouring some interests while disadvantaging others. In any case, smart growth and new urbanist models are eagerly supported by many land developers, builders and local political elites favouring growth. The second dominant theme of this debate is how and why morphology matters. For reasons that will be developed in this introduction and in the essays, discussing density will not suffice to qualify changes related to densification. Density is a poor predictor of urban forms; a development consisting of terraced houses may, for example, have the same density as a modernist estate of tower blocks. The third and last dominant theme is the diversity of suburban densification regimes. It is not only the landscapes of the suburbs that are diverse, but also the local bodies governing them. Between the small residential municipalities of the Paris periurbs and the large inner suburbs of Toronto lies a broad spectrum. The debate presented here develops these considerations from four case studies, two in Canada and two in France. This is fortuitous, the result of encounters between participants in two research projects (a French project on urban change in low-density residential areas and the Global Suburbanisms project, a globally scaled Canadian study)2 . Moreover, by focusing on Canada and France respectively, we are not arguing that such a comparison is entirely new, nor are we ignoring the wealth of literature that exists in each country on processes of suburban diversification and change. Yet the comparison of France and Canada is highly relevant, especially in a context where US case studies have largely dominated the Anglophone (and indeed other) literatures. What the US tells us about the suburbs and post-suburbs is relevant for other countries, including Canada and France. Moreover, global dynamics exist and we can in fact talk about the phenomenon of 'global sprawl' [START_REF] Keil | Editorial, Global sprawl: "Urban form after Fordism?[END_REF]). Yet each national context gives a specific flavour to suburbanization. And since the focus of this debate is on the politics of densification specifically, Canada and France are two interesting cases to discuss and to compare with the US case, as the societal conditions differ significantly and the variegations of (neoliberal) social formations matter in terms of (post-)suburban outcomes. Indeed, we can detect a Wacquantian landscape of difference: 'In this sense, Canada/Toronto is located in the mid-range of a scale in which France/Paris and USA/ Chicago are extremes. This has to do as much with the traditionally mixed social and capitalist economy in Canada as with the particular nature of neoliberalization in that country' [START_REF] Young | Rebuilding the Modern City After Modernism in Toronto and Berlin[END_REF]Keil, 2014: 1594). These differing political histories translate into different representations of the suburbs. Low-density residential suburbs are numerous in France for example, and combating sprawl is a central focus of public policies and debates. Yet the dominant image of the suburb is not associated with individual detached houses. In France, the word banlieues (which translates to suburbs in English) evokes images of apartment towers and barred windows (see Figures 3 and4) rather than a grid-like alignment of detached single-family dwellings, and images of marginalized immigrant populations rather than a white middle-class community fully integrated into the economy. As explained by Max Rousseau (2015, this issue) in the first part of his essay, this image can be explained by the particular role played by the French state in the production of the city. In Canada too the suburbs are not homogeneous neighbourhoods of single-family homes, instead largely comprising higherdensity morphological forms of comprehensive socio-economic and ethno-cultural diversity (see below). The contributors to this debate are building on a robust literature in Canada that has specifically explained the country's suburbanization through its historical-geographical diversity (see e.g. [START_REF] Harris | Unplanned Suburbs. Toronto's American Tragedy[END_REF]2004;[START_REF] Walks | The Causes of City-Suburban Political Polarization? A Canadian Case Study[END_REF]2007;[START_REF] Walks | Urban Form, Everyday Life, and Ideology: Support for Privatization in Three Toronto Neighbourhoods[END_REF]Addie et al., 2015;Keil et al., 2015). Finally, the four essays to follow in this debate focus on transformations in residential space. This choice is, above all, practical: by limiting the diversity of cases, we facilitate significant comparisons (although the terrains under investigation were limited to two countries). That said, this choice is not meant to reduce the suburbs to their residential status. As is well documented, the suburbs do not consist of housing alone. For quite some time, they have brought together employment, commerce, cultural organizations, infrastructural and logistical facilities, ecological spaces (parks and greenbelts) and large-scale institutions such as hospitals and universities. These changes are at the very heart of the move from suburbanization to post-suburbanization. The politics of densification: towards sustainable cities or the new guise of growth coalitions? The debate presented here focuses on densification. Densification is by no means the only morphological change that affects suburbs and post-suburbs, but it is certainly among the most discussed strategies within the planning community. The densification of residential suburbs is commonly considered a key objective. It is evidenced in France by the so-called 'Grenelle 1 et 2 de l'environnement' legislation, enacted in 2009 and 2010. Grenelle 2, for example, permits a minimum level of density, especially in areas close to public transport links. Such dispositions are almost unopposed, either from the left or from the right (at least at national level). In the next section of this introduction, we present the politics behind this consensus in favour of densification. As we will see, much recent research points towards a questioning of the relationship between density and sustainability. A re-politicization of the anti-sprawl discourse within urban planning can be expected from this turn in the debate. The density turn within the environmentalist discourse In the 1970s, an ecologist was often someone who escaped the city and its pollution; in the 1980s, however, a reversal occurred with respect to the environmental discourse and the city. This can be symbolized by the success of the work of Peter Newman and Jeffrey Kenworthy (1999), who popularized the simple equation that a dense city was a sustainable city (Charmes, 2010a). Living in a lower-density environment, attractive for its greenness, usually means being far from the concentrated resources of a city and depending on the car for even the shortest trips. More energy is consumed and this lifestyle therefore has a negative effect on the environment. This critique of car dependency reinforced previously expressed critiques. In both France and Canada, leapfrog development has since the 1970s been perceived as a threat to agriculture. A related but different criticism focuses on excessive land consumption driven by individual housing. Lastly, a more economistic critique (yet one often integrated with ecological discourse) highlights the cost of sprawl, since the more spread out the city is, the longer its networks and infrastructures need to be (see [START_REF] Sewell | The Shape of the Suburbs: Understanding Toronto's Sprawl[END_REF] for a typical summary of those arguments in the Canadian case; Jaglin, 2010 in respect of France). Fostered by these arguments, the war against sprawl mobilizes most urbanists and planners (in both France and Canada) and most of them now ideologically favour dense and compact cities (planning history in the twentieth century shows that this was not always the case--see Touati, 2010). Critics of low-density residential spaces foster the diffusion of now well-established planning norms like: densification of residential neighbourhoods (see Figure 5); infill on brownfield sites (see Figures 6 and7); functional diversification (with the development of businesses and employment within a polycentric pattern); and concentration of urban development around train stations and public transport nodes. In any case, the anti-sprawl discourse questions the traditional suburban pastoral ideal of subdivisions and detached single-family dwellings. Living in a low-rise residential space is no longer perceived as being about getting closer to the countryside, rather it is deemed unfriendly to the environment. Yet the arguments behind this reasoning are debatable. There is insufficient space here for a comprehensive discussion, but recent research has shown that many of the arguments presenting the dense city as more sustainable are at best questionable (see e.g. [START_REF] Echenique | Growing cities systainably[END_REF]. In their assessment of energy consumption in transportation, Newman and Kenworthy (1999) only take daily trips into consideration, ignoring longdistance trips (for pleasure or business). Yet the latter increase with higher densities, due among other things to the need for respite from the noise and stress of dense city centres [START_REF] Holden | Three Challenges for the Compact City as a Sustainable Urban Form: Household Consumption of Energy and Transport in Eight Residential Areas in the Greater Oslo Region[END_REF]Nessi, 2012). More research needs to be undertaken to evaluate the impact of such compensatory trips, but it is significant and severely reduces the presumed advantages of density regarding energy consumption in transportation. Transportation is not the only source of energy consumption in cities. Buildings are also a major one. Yet if the energy consumption of a group of buildings tends to decrease with density, pre-existing detached houses are much more adaptable than high-rise buildings, and it is easier to reduce energy consumption of the former (by installing heat pumps, wells or solar panels). Regarding land consumption, countries like France and Canada do not lack space for urbanization. In France, if every household were to live in a detached house on a 1,000 m2 lot, only about 10% of the nation's land mass would be urbanized [START_REF] Charmes | L'artificialisation est-elle vraiment un problème quantitatif ?[END_REF]. For the preservation of natural land and agriculture, the issue is less about limiting sprawl, and more about organizing it and controlling so-called leapfrog development. We could go much further. By raising these points of discussion, our intent is certainly not to advocate sprawl, as for example Robert [START_REF] Bruegmann | Sprawl: A Compact History[END_REF] did. Nor do we fall in line with the conventional suburb-boosting arguments, emanating usually from American libertarian scholars and pundits (Cox, Kotkin and Richardson for example) who identify classical suburbanization with the promises of the 'American dream'. Density has many other advantages beyond environmental ones, especially for urbanity, serendipity, creativity and so on. Our intent is rather to point to the normativity of the environmental discourse in favour of density. In a context of energy scarcity and global climate change, density appears to be a non-disputable issue, a 'given' beyond political debate. Yet it is not, and density should be re-politicized. Density = sustainability: the new motto of growth coalitions? These scientific uncertainties about the real environmental benefits of density show how important it is for research on the transformation of the suburbs to distance itself from the planning discourses on sprawl. This is all the more important at the metropolitan level, where environmental interests of densification and limitation of sprawl are most disputable. Even if the dense city proves to have some climate change virtues, these may only be apparent at the international level. Worse, the inhabitants of the densified areas will suffer from increased exposure to local pollution [START_REF] Echenique | Growing cities systainably[END_REF]. At the metropolitan scale, other than an uncertain improvement of the local environment the stakes of urban redevelopment reside, for example, in: the mobilization of scarce land resources for new construction; improving the image of socially distressed neighbourhoods (an image that looms over the city itself ); and the growth of the metropolis. Indeed, the density turn within environmental discourse is especially convenient for promoting projects of growth coalitions or (more fundamentally) urban growth, and often serves to override local resistance. In many suburbs, residents expect that their representatives seek to protect the environment, preserve quality of life and limit growth, rather than attract new jobs or homes. Of course, everything depends on the context, but the general trend is one of decline in positions favourable to growth. In the light of that, the density turn helps to counter the main arguments used by movements opposed to urban growth, especially in recent cases of new-build gentrification. It helps to dismiss any desire to preserve the original programme of suburban utopias, such as low-density built-up areas, as local expressions of selfishness. And it helps to discredit and delegitimize local mobilizations by designating them as nimbyism [START_REF] Wolsink | Invalid theory impedes our understanding: a critique on the persistence of the language of NIMBY[END_REF]. More broadly, the density turn helps to weaken local opposition to growth from its own environmental perspective. By referring to the future of the planet, one can justify the densification of suburbs in the name of sustainable development and denounce opposition to local intensification projects as self-serving. Such disqualification of local opposition to growth is questionable. In fact, behind the motivations associated with sustainable development, and more specifically behind the equation that density equals a sustainable city, often hides the old conflict between exchange value and use value [START_REF] Logan | Urban Fortunes: The Political Economy of Place[END_REF][START_REF] Logan | Urban Fortunes: The Political Economy of Place[END_REF]). The defensive politics of suburbanites vis-à-vis continuous development is not just a defensive stance of private interests; 'such a politics also recognizes the constraints of the jumbled, anaesthetic environments [of post-suburban in-between cities] as the true playgrounds of a new and potentially productive politics of the urban region' (Keil and Young, 2011: 77). Within that politics, citizens have learned to suspect other special interests behind the positions put forward. Thus, the idea that growth is good for a local community is often obliterated by the notion that this growth serves the interests of developers (growth is greed, to put it bluntly). From this point of view, defending one's suburban 'castle' against higher density is perceived by observers to be a defence by 'the little guy' against the greed of developers and those financing their projects. This local perspective appears all the more legitimate since there is a growing interest in local policymaking processes, in conjunction with the rise of participatory forms of democracy. Last but not least, local opposition to growth is successful precisely because it is not reduced to expressions of local selfishness, and is able to make its views resonate with issues that go beyond the local [START_REF] Keil | Going up the country: Internationalization and urbanization on Frankfurt's northern fringe[END_REF]. For that matter, such opposition is often associated with the use of environmental arguments: new construction is not only bad for house values, it is also bad for the environment. Without necessarily adhering to those ideologies, many citizens leverage environmental concerns to serve their local interests. All this confirms that environmental arguments are particularly plastic. And two of the central terms of this debate below, sustainability and density, are themselves 'chaotic terms'. They can be used to support growth as much as to justify stopping or limiting growth. They can justify the fight against urban sprawl, while justifying the purchase of a house with a large garden on the outskirts of a city. This plasticity, coupled with the ability of environmental arguments to generate consensus, creates legitimizations that are often used to defend projects or initiatives, which are actually determined by other agendas. It is therefore necessary to analyse and critique these issues. In short, the morphological changes of suburbs must be analysed as much as those faced by urban centres. These transformations bring with them political and social issues. Morphology matters: deconstructing densification If suburbs are ever-less frequently called suburbs, but zwischenstadt, in-between cities or post-suburbs instead, it is primarily because many so-called suburbs do not look like dormitory towns with their agglomeration of detached houses. Yet those morphological changes are rarely considered in and of themselves, but instead as signs or symptoms of something else, like structural modifications of daily mobility, changes in suburban politics, redefinition of metropolitan centrality and so on (Phelps and Wu, 2011b). Within this framework, a new multi-storey building in a low-density suburb manifests among other things the evolution of the position of that suburb within a metropolitan system. More generally, it manifests that some quintessential attributes of centrality are now to be found within suburbs. Yet density matters not only as a signifier or as a symbol, but as an important component of the production of the city. Through the various forms it may take, density reveals power relations. It also mediates between different interests, favouring some and disadvantaging others. As Figure 8 shows, density can take many different forms. A comparable number of houses or square metres can take many different forms. And those different forms have political and social meaning. Thus (and this is what we are debating here), it remains to be understood why a multi-storey building emerges in one particular suburb and not in another one; why the densification process should take the form of a multi-storey building and not of infill semi-detached or terraced houses. These questions are discussed below by Anastasia Touati (2015, this issue), who contrasts hard and soft densification (see also [START_REF] Touati | Economie Politique de la densification des espaces à dominante pavillonnaire: l'avènement de stratégies post-suburbaines différenciées[END_REF]. She makes the illuminating statement that soft densification can be a compromise between exchange value and use value. In a low-density residential neighbourhood, the addition of individual houses through infill is a way of reconciling economic interests emergent from urban growth with the interests of the inhabitants, since the residential image of the neighbourhood is preserved. Soft densification can indeed overcome resistance from inhabitants, while hard densification may trigger strong opposition. Yet, soft densification may not be sufficient to sustain a strategy seeking to establish a suburb as a metropolitan sub-centre. In this sense, the type of densification is revealing of power relations, particularly between local and metropolitan interests. Other interesting questions addressed in this debate are: which are the social groups that densification policies aim to attract, and who are those coming to inhabit the new building. Indeed, as shown by Max Rousseau (this issue), densification may (depending upon local context) be part of a process of social downgrading as well as a process of upgrading. In an upscale residential suburb, the construction of a multi-storey building is perceived as a threat, both from the perspective of landscape conservation and from the perspective of social engineering. The same building on a derelict modernist-era housing estate (the so-called grands ensembles) is on the contrary perceived as a way of rehabilitating the place and attracting middle-class households. In this debate, two out of the four essays--those of Will Poppe and Douglas Young and of Max Rousseau--focus on large modernist housing estates. As stated above, in France the image of the suburb (banlieue) is less associated with the single-family home (pavillon), and more with the grands ensembles, and thus towards verticality and concrete rather than horizontality and greenery. This image of the suburbs contrasts sharply with the dominant one in North America, even if Canadian cities are renowned for high-rise neighbourhoods that are often found at the urban periphery. Particularly relevant to the debate presented here, the French grands ensembles are the object of a policy focusing on morphology. Between 1954 and 1973, millions of new homes were built in grands ensembles. For various reasons we cannot present in detail here (see [START_REF] Subra | Histoire des discours politiques sur la densité » In E. Charmes (Ed.) « La densification en débat[END_REF], the grands ensembles are today the places where France's urban crisis is concentrated, places like Clichy-sous-Bois and Montfermeil, municipalities in the Paris suburbs that were at the epicentre of the 2005 riots [START_REF] Dikec | Badlands of the republic. Space, politics and urban policy[END_REF] Figure 3). These incidents drive public intervention, and significant public funds have been mobilized. This intervention is largely morphological: it targets towers and housing blocks that 'disfigure' the landscape, focusing on the image of 'concrete neighbourhoods', and it reconstructs a habitat of more traditional (low-rise) urban forms, which is supposed to appeal to the middle classes. It is also said that the grands ensembles should be 'de-densified'. Indeed, the French grands ensembles are often associated with hyper-density, and their repugnant image is regularly mobilized by those opposing densification (many French grand ensembles technically have the same density as a town core made up of terraced houses with small gardens, but what matters is the size of individual buildings). This focus on morphology may be related to the French state's poor capacity to trust the people, especially those from immigrant backgrounds (Bacqué and Sintomer, 2004;[START_REF] Baudin | Faut-il vraiment démolir les grands ensembles ?[END_REF]. The comparison with Canada is telling. While Canada shares with France a built environment of peripheral high-rise housing estates that have become concentrations of poor, immigrant, non-white tenant populations, the (Anglo-)Canadian urban experience also includes communitarian recognition, through the institutions of a multicultural society which provides safeguards against some of the problems associated with the exclusionary tendencies embedded in the republican tradition. While no federal urban policy exists, the Canadian (local) state has institutionalized integrationist measures, some of them place-based, allowing for targeted interventions through a variety of mechanisms: schools, community welfare, culture, etc.3 In the case of Toronto, renewal of residential areas in the inner suburbs has focused in recent years on what has been termed tower renewal (see Figure 4). Discussed at greater length in the essay below by Will Poppe and Douglas Young, this process has been seen as an attempt not only to apply architectural and energy retrofits to existing concrete towers, but also to re-engineer entire tower neighbourhoods. This has been part of a general place-based strategy targeting 13 so-called 'priority neighbourhoods', selected (for both socio-demographic and built-form reasons) as concentration points for social policy interventions. While the discourse around concrete towers and the communities inhabiting them resembles the negative practices found in France, there has also been considerable movement by urban specialists and residents alike to rehabilitate rather than demonize these high-rise suburbs. Many initiatives and projects specifically seek to empower residents of these neighbourhoods and overcome the negative images associated with their environment. In any case, morphological changes do not happen easily in the suburbs. They can be hindered by many factors. Some have already been mentioned, but a fact often overlooked outside the inner circle of urban designers and planners is that one of the biggest obstacles to change in suburban housing estates (be they high-rise buildings or detached houses) is their functional specialization. Suburbs were built as dormitory areas. As documented in Pierre Filion's essay below, functional specialization is one of the main causes of inertia in suburban landscapes, however dated that may seem. What remains prevalent is the morphological structure of the suburban street and road network that became predominant in the second half of the twentieth century. Throughout that period, all the planning manuals recommended the environmental area model, to preserve low-density residential areas from the nuisance of through traffic. According to the creator of the concept, Colin Buchanan (1963), environmental areas should be designed so as to have no extraneous traffic, no drifting through of traffic without business in the area, which should be accessible from arterial or distributor roads at one intersection only (see Figure 9). These environmental areas are highly favourable to functional specialization: there is only one type of user, the resident. Those who access the space do so either as residents or as visitors of the residents. Of course, functional specialization is not inherent to environmental areas: mono-functional zoning is a planning decision that can be made relatively independently from the design of the road system. Yet, a comparison of how residential suburbs structured along a gridded street system evolve, and how suburbs structured along pods develop, shows that the former are much more open to functional mixing than the latter. In fact, the dense and multifunctional city is facilitated where there are flows of pedestrians, cars, etc. [START_REF] Mangin | Influence du contexte urbain et du rapport au cadre de vie sur la mobilité en Ile-de-France et à Rome[END_REF]Charmes, 2010b). The organization of the urban fabric within a collection of enclaves where all traffic is excluded prevents this activation of land by circulatory flows. Since the road and street networks, as well as the underlying infrastructures such as water and sewerage, exert very strong inertia in the cities, circulation flows in many suburbs will remain separated from urban life for generations to come. And the transformation of the suburban landscape must happen within that framework, which means that often densification happens and will happen without functional diversity or, more accurately, with limited functional diversity. The low-rise office park perhaps changes into a high-rise office park, not into a dense multifunctional urban centre like those of Lyon, Toronto or Paris. This is one of the major reasons why the densification processes occurring in many suburbs produce a landscape that is very different from that found in older urban centres. This is why many suburbs are turning into post-suburbs and not into urban centres. Of course, suburbs include older towns and cities which can become denser through functional diversity, but those towns and cities do not constitute the major part of suburban landscapes. Densification regimes Factors hindering or promoting densification are present in almost all circumstances. However, they exist in varying degrees. Variations occur from one country to another depending on the particular urban history, attitudes towards nature and density, and systems of local government. For example, the idea that density or, more broadly, that the dense city is collectively desirable may historically have been a more accepted notion in France than in Canada (yet it is also obvious that there are convergences here: Canadians have learned to accept the diktats of a climate-change-driven push towards greater compactness; and French suburbanites have learned to escape the grands ensembles, gravitating towards the lotissements of pavillons on the outskirts of not only large conurbations but also many smaller towns and villages, generating a pervasive pattern of leapfrog development similar to that found in many North American cities). Variations are also observed within a single city. This is illustrated well in the essay by Max Rousseau (this issue): the capacity of households living in lowincome areas to protect their quality of life is not the same as that of affluent households (primarily because low-income households are less able to mobilize legal or other means to challenge development projects). In addition, as stated above, municipalities with poor populations are often motivated to transform the urban landscape, whereas richer municipalities do not have this same desire. These are only a few cases from a broad variety. The diversity of reactions to the dynamics of change in suburban morphology reflects the diversification of governmental regimes in suburbs and postsuburbs, with each being more or less favourable to certain coalitions and particular morphological changes [START_REF] Phelps | The New Post-suburban Politics?[END_REF]. These suburban regimes are formed among actors with varying levels of status and intervention capacity. We propose an overview of that diversity in Tables 1 and2, which should help readers make the fullest sense of the four essays comprising this debate (especially those based on case studies). The tables are based on the knowledge of the authors, acquired both through fieldwork and from the literature. Due to space constraints, our accompanying commentary is brief. The suburban political game is played out through the central modalities of suburban governance on the terrain of local communities, where the state (at various scales), capital accumulation (land development) and authoritarian governmentalities (articulated through politicized class interests) interact [START_REF] Ekers | Governing Suburbia: Modalities and Mechanisms of Suburban Governance[END_REF]. In France, the public actor predominates (see Table 1). This corresponds with the prevailing image of France abroad. Yet contrary to that image, the state has lost many of its prerogatives. Remnants of its former power may be glimpsed in places like Défense or in new towns around Paris, but today the state remains largely in the background (working through national regulations or project funding). The image of a highly centralized country that remains prevalent around the world does not reflect the highly fragmented nature of France's public actors. Municipalities play much more of a key role in suburban regimes than the national state does. Moreover, municipal authorities are extremely fragmented, especially in periurban (or exurban) areas. Inter-municipal cooperation has developed since the turn of the millennium but, as Max Rousseau's essay (this issue) demonstrates, power remains largely in the hand of the municipalities themselves. This results in a strong emphasis on local (sometimes very local) perspectives, focused on the defence of quality of life and residential interests more generally [START_REF] Charmes | La ville émiettée. Essai sur la clubbisation de la vie urbaine[END_REF]. Everywhere, residents mobilize to preserve the landscape and maintain certain social qualities of their local communities. Such resistance has become very important, both in North America and in Europe, as evidenced by the literature on nimbyism or no-growth coalitions [START_REF] Subra | Histoire des discours politiques sur la densité » In E. Charmes (Ed.) « La densification en débat[END_REF]. This resistance represents an extension from the home to the local environment of the domain over which people consider that they have property rights. Householders do not only buy a home, but also the local environment that comes with it. Through that process, the relationship to the neighbourhood becomes more and more similar to one of co-ownership, a process we described as 'clubbisation' [START_REF] Charmes | On the Clubbisation of the French periurban municipalities[END_REF]. This process is one of the main driving forces powering not only residents' movements, but also the development of private residential neighbourhoods and gated communities. And in the case of France this process has a significant effect on municipalities. The fragmentation of the French municipal fabric gives residents' movements a singular ability to influence planning regulations. A metropolitan region like Lyon is composed of 514 municipalities with a population of 2.1 million, within which are about 380 periurban municipalities each with an average population of 1,560 inhabitants. In such municipalities, extremely local issues prevail in residents' concerns. At the same time, the prerogatives of those municipalities are far reaching, and include planning (which allows suburbanites to control types of construction, lot size, etc.). The resultant conservationist agenda acts as a significant barrier to redevelopment projects. This type of planning is often exclusive too, because it usually limits (or even halts) urbanization, which not only prevents population increase but also raises prices and thus restricts access to those who can afford to obtain entry (see Figure 10). In Canada, suburban municipalities also play an important role, but they are much larger (see Table 2). And they (e.g. Surrey, BC; Mississauga, Ontario; Markham, Ontario; Laval, Quebec; Brossard, Quebec) challenge, rival and sometimes supersede the political centrality of the core city as they attempt to redefine their (sub)urban future in the context of more general calls for more sustainable (and now increasingly resilient) forms of development. In any case, suburban politics is not only about residential qualities of places. The suburbs have historically often lacked employment. Gradually, however, the work commute has been reconfigured: the suburbs to centre transportation flow has lessened as more people commute from suburb to suburb, or even from centre to suburb, following employment opportunities in Fordist factories and post-Fordist manufacturing, logistics and office locations, as well as commercial and entertainment enterprises (see Figures 7,11 and 12). In parallel, during the decades following the second world war, governments sought to organize suburban growth by creating new settlements and satellite cities. While this policy has had relatively limited impact in the US, the creation of the Municipality of Metropolitan Toronto in the 1950s was specifically linked to the siting of multi-density housing developments away from the centre [START_REF] Young | Rebuilding the Modern City After Modernism in Toronto and Berlin[END_REF]. In France, an ambitious policy of new town construction was launched in the late 1960s, and the five new towns created around Paris at that time now constitute major sub-centres (see Figure 10). Likewise, edge cities have developed around highway interchanges serving shopping malls in the US and Canada [START_REF] Garreau | Edge City: Life on the New Frontier[END_REF]. Large cities also include neighbouring smaller cities within their orbit, with all their shops, jobs, equipments and services. Finally, apart from edge cities, one notices a dissemination of employment and commerce to the outskirts, to what have been named 'edgeless cities' [START_REF] Lang | Edgeless Cities: Exploring the Elusive Metropolis[END_REF]. Within that context, in both France and Canada (albeit in different guises) suburban regimes that have historically been formed in contradistinction to the central city have recently been freeing themselves from the inside-outside duality traditionally characterizing their political frame. The increasing recognition of regional and 'in-between' issues, particularly in policy sectors such as transportation, welfare, ecology and housing, has led to an increase of rhetoric (if not action) regarding cooperation between suburbs and the central city on one hand, and coordination in a competitive regional environment among suburban municipalities in decentralizing regions on the other [START_REF] Lehrer | Producing global metropolitanism in the periphery: Toronto and Frankfurt[END_REF]. These logics articulate themselves in variable configurations depending on context. In France, for example, while inner-and middle-ring suburbs clearly have an urban identity, and are often integrated into metropolitan communities (communautés urbaines) in which the core municipality cooperates with its neighbouring municipalities, periurban areas retain a more rural identity and tend to adopt a defensive stance against the city. Again, space constraints prevent us from exploring in detail all the regimes presented in Tables 1 and2. But the examples above, and the cases discussed in the essays, show the importance of considering densification and, more broadly, morphological transformations of suburbs in their respective contexts--considering different scales that interact. Suburbs are also extremely diverse, to a degree that makes it difficult to talk about suburbs or post-suburbs in general. From that perspective, the comparison between cities and between countries is very helpful. It helps to disentangle the contingent from the structural. It also helps to identify the various factors determining a suburban regime. The politics of post-suburban densification There are differences, but also similarities, in the formulaic morphology of post-second world war metropolitan landscapes, and in the crisis besetting those landscapes. The archetypes of the concrete residential tower and the single family pavillon are just the external markers of that dialectics of difference and convergence. Beyond that dialectics, in the debate we present here, we can note certain convergences. Both French and Canadian metropolitan suburbs and periurban areas are caught up in the frantic pace of modernization, although collective actors in both settings continue to mobilize around conservation of present scales, forms and communities. On both sides of the Atlantic, we detect strong conflicts between local and metropolitan perspectives, private and public interests, community and corporate actors. And we have noted the multifaceted nature of the suburban theatre of collective action, which cannot be dismissed as mere nimbyism. The suburban political realm is not static, nor is it dormant. There are strong and growing mobilizations around issues of everyday suburbanism, but also around long-term planning and policy in regional matters. There are expectations in both Canada and France that there will be (decisive) state action when needed. While Canadian jurisdiction lies mostly with provincial government (of which local communities are mere creatures), in France the post-1981 decentralization has led to a strong dialectic of central and local political action. These essays speak of multiple rationalized projects (the social, the environmental, the economic), around which actors play a political game in which they are conscious and deliberate participants that don't just react, but also set new boundaries and rules. In this sense, all the essays pay tribute to [START_REF] Logan | Urban Fortunes: The Political Economy of Place[END_REF][START_REF] Logan | Urban Fortunes: The Political Economy of Place[END_REF]) initial formulation, that sees suburbanization as a process of complex interrelationships of individual decisions in firm structures, located in a political universe of use and exchange value decisions by individual and institutional actors. They speak of multiple scales and recognize the metropolitan significance of change in place (most notably in the case of Rousseau, this issue) and take seriously the actors that are involved. Suburbs are not just treated as objects of external planning and policy, but also as places subject to endogenous, maybe even autonomous, agency. All the essays presented here discuss different suburban forms (tower blocks, houses, pavillons, grands ensembles) and demonstrate an ability to discern the differences between those morphologies, the consequences they have for the political process and their engagement with the modalities of suburban governance. And finally, they all elaborate upon both the ideological and material aspects of the relationships of urban form and social structures, an important part of the new debate on suburbs and suburbanization to which we hope to contribute with this debate. The French essays (more than the Canadian essays) display a comparative perspective that involves an excellent recognition of the current state of debate on post-suburban politics. North American-European differences are critically acknowledged and productive solutions are found. The Canadian essays present two very different views of the changeability of suburban form (tower renewal versus stasis of the suburban morphology). They also engage different scales of suburbanization, one local and place-based, the other regional and trans-jurisdictional. The concepts we suggest readers of this debate might contemplate may be summarized in the terms 'soft' and 'hard', which are most forcefully introduced by Anastasia Touati who discusses densification in rather different suburban environments. Poppe and Young demonstrate that Toronto's peripheral concrete tower neighbourhoods, in respect of which reformers seek renewal, operate on the plane of both the hard material retrofit and the soft social engineering of inventing a new contextual fit for communities in a changing post-suburban landscape. Rousseau discusses the hard and soft edges of the urban region depending on alternative socioeconomic and political structures. And finally Filion's essay questions whether the suburban morphologies of Canadian urban regions are hard or soft, and likely to resist pressure for change. We can conclude that post-suburbia has well and truly arrived, and we may propose that we need to accept that post-suburbia is now ubiquitous. No new frontiers are part of this particular set of case studies; their view is directed towards the inside. In all of this, there is some clear transatlantic convergence but also lots of diversity, both internally and between the French and Canadian cases. The debate also highlights the fact that comparative studies of this nature are now more important than ever in order to create productive conversations about what needs to be done. The choice of Canada and France for such a comparison was productive as it allowed the authors of the individual essays (as well as us as editors and editorializers of the case studies) to nod politely to the 'classical' US case, but then to break free of it, liberating innovative and new modes of thinking that engage with the shared realities and divergent idiosyncrasies of both cases. The cases and the comparison leave us with the insight that all measures are inevitably socio-ecological and socio-economic, as well as politically negotiated. Despite clear path dependencies (in morphology, institutions, ideology and political process), political choices and options remain available in our postsuburban futures. Figure 1 : 1 Figure 1: An artist's view of an archetypical outer-suburban development in France (Jean-Pierre Attal,intra-muros 12, 74x100 cm, 2008, www.jeanpierreattal.com) Figure 2 : 2 Figure 2: In-between city Toronto: York University campus at the northwestern edge of Toronto looking south (photo by Roger Keil) Figure 3 : 3 Figure 3: A barre in Montfermeil: the building is a condominium and a process of acquisition by individual residents is underway (photo by Eric Charmes) Figure 4 : 4 Figure 4: Toronto tower neighbourhood: Thorncliffe Park (photo by Roger Keil) Figure 5 : 5 Figure 5: New urbanist development in Markham, Ontario (photo by Roger Keil) Figure 6 : 6 Figure 6: Infill at former industrial site in eastern Toronto inner suburb of Scarborough (photo by Roger Keil) Figure 7 : 7 Figure 7: Redevelopment of former industrial land in Plaine Saint-Denis, close to Paris city limits (photo by Eric Charmes) Figure 8 : 8 Figure 8: Morphological modulations of density. In all those three cases, the density of construction is the same (source: Institut d'aménagement et d'urbanisme de l'Île-de-France, 2005; Appréhender la densité, Note Rapide, 383) Figure 9 : 9 Figure 9: The 'environmental area' principle, as conceived by Colin Buchanan(1963: 69) Figure 10 : 10 Figure 10: Driving out of a small periurban commune (a territory governed by a municipality) in the first periurban ring of Paris (photo by Eric Charmes) Figure 11 : 11 Figure 11: Industrial-residential mix: industrial plant in eastern Toronto suburb with encroaching highrise and single-family-home residential development (photo by Roger Keil) Figure 12 : 12 Figure 12: The Carré Sénart, south of Paris, in Île-de-France: this 'shopping parc' was designed by the planners of Sénart new town to be the centre of the whole development (photo by Eric Charmes) Table 1 : 1 Diversity of post-suburban regimes in France Suburbs type Population type Dominant politics Main actors Morphological (inhabitants) change Inner and Upper class Exclusionary zoning; Municipality (10,000s Stability with local middle ring occasional local up to 100,000) sporadic changes suburbs redevelopment (the farther projects from the core Diverse with a Redevelopment Municipality (10,000s Densification city the less domination of projects of various up to 100,000); (mostly soft); brown intense the middle classes sizes metropolitan field changes) community (including redevelopment; new core city)*; offices; developers sporadic transformations Diverse with a Redevelopment Municipality (10,000s Densification (soft to domination of through gentrification up to 100,000); hard), brown field lower and lower (including new build metropolitan redevelopment; new middle classes gentrification) community (including commercial core city)*; infrastructure; new developers offices Poor (with Urban renewal Municipality From modernist to many through partial (10,000s); neotraditional immigrants) demolition metropolitan community (including core city) *; national State ; large developers Edge cities: new towns Diverse without Extensive growth National State; Extensive growth (mostly around upper middle from the 1960s; several municipalities ending or slowing Paris and Lyon) nor upper redevelopment from (100,000s); down; classes the 1990s large developers renewal in some neighbourhoods old urban Preserving their Municipality Renewal in some centers influence over their (10,000s); developers neighbourhoods; surroundings extension of the urbanized area through subdivisions; business parks; shopping strips Periurbs: residential Middle to upper No growth; Municipality Stability with municipality middle classes exclusionary zoning; (typically 1,500) occasional (first periurban with a marked clubbisation redevelopment of rings) homogeneity at old village core the municipal level Towns Diverse (within Extensive growth Municipality Subdivisions; new (about 10 % of middle classes) and/or (typically between business parks; new the periurban redevelopment of 3,000 and 10,000); suburban shopping municipalities) town center (the developers strips; densification closer from the core of town center city the less extensive Table 2 : 2 Diversity of post-suburban regimes in Canada Suburbs type Population type Dominant politics Main actors Morphological change Inner and Elite uses in Institutional Universities; state Rapid and large middle ring educational redevelopment projects; institutions; scale change suburbs (In- institutions like capital expenditure into hospitals between universities; little prestige infrastructures; cities; third residential use by highways; in Toronto city) upper classes specifically: conservative upper class appeal through populist politics to lower and middle classes that feel excluded from inner city political power perceived as elitist; end to the 'war on the car' Middle class Some infill; some Developers; private Densification continuing larger scale households; transit (mostly soft), suburbanization through agencies, park and brownfield single family homes, sports redevelopment; townhouses, high rise organizations new offices and condominiums; some warehouses; gentrification along conversions of emerging transit lines industrial spaces (subway extension in into places of Toronto) worship; conversions of places of worship into condominiums; Sporadic transformations Lower to lower Redevelopment of Local state; Retrofits middle class tower neighbourhoods planning and (environmental, and strip malls through architecture aesthetic and state action; hesitant professionals; structural); some gentrification effects public housing new commercial agencies (major infrastructure, new landlords in tower offices; community neighbourhoods); centres in priority school boards neighbourhoods (agents aiming for See http://www.yorku.ca/suburbs. Canada and France have an unfortunate shared experience in terms of their marginalized suburban populations. We are extremely grateful to Imelda Nurwisah for translating the original draft of this introduction from French into English. The Canadian part of the research presented here was supported by the Social Sciences and Humanities Research Council of Canada through funding from the Major Collaborative Research Initiative 'Global Suburbanisms: Governance, Land and Infrastructure in the 21st Century ' (2010-17). The French part of the research was supported by the French National Research Agency (ANR) within the framework of the 'Sustainable City' research programme.

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2018

https://inria.hal.science/hal-01744252/file/RR-9164.pdf

Teddy Furon The illusion of group testing Keywords: Group testing, hypothesis testing, identification, information theory Test par groupe, test d'hypothèse, identification, théorie de l'information This report challenges the assumptions usually made in non-adaptive group testing. The test is usually modelled as a probabilistic mechanism prone to false positive and / or false negative errors. However, the models are still too optimistic because the performances of these non ideal tests are assumed to be independent of the size of the groups. Without this condition, the report shows that the promises of group test (a number of tests and a decoding complexity scaling as c log N ) do not hold. Introduction Group testing has recently received a surge of research works mainly due to its connection to binary compressed sensing [START_REF] Lam | Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms[END_REF][START_REF] Atia | Boolean compressed sensing and noisy group testing[END_REF][START_REF] Scarlett | Converse bounds for noisy group testing with arbitrary measurement matrices[END_REF][START_REF] Scarlett | Phase transitions in group testing[END_REF]or to traitor tracing [START_REF] Meerwald | Group testing meets traitor tracing[END_REF][START_REF] Laarhoven | Asymptotics of fingerprinting and group testing: Tight bounds from channel capacities[END_REF]. The usual setup is often described in terms of clinical screening as it was the first application of group testing [START_REF] Dorfman | The detection of defective members of large populations[END_REF]. Among a population of N individuals, there are c infected people, with c much smaller than N . Screening the whole population by individual blood test is too costly. However, it is possible to mix blood samples from several persons and to perform a test. Ideally, the test is negative if none of these persons are infected, and positive if at least one of them is infected. The application of group testing are nowadays DNA screening [START_REF] Ngo | A survey on combinatorial group testing algorithms with applications to DNA library screening[END_REF], signal processing [START_REF] Gilbert | Recovering simple signals[END_REF], machine learning [START_REF] Zhou | Parallel feature selection inspired by group testing[END_REF]. Indeed, group testing may be a solution to any 'needles in haystack' problem, i.e. aiming at identifying among a large collection the few 'items' sharing a peculiar property detectable by a test, provided that this test can be performed on groups of several items. In this paper, we use the terminology of items and defective items. The dominant strategy nowadays is called non-adaptive group testing [START_REF] Atia | Boolean compressed sensing and noisy group testing[END_REF][START_REF] Lam | Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms[END_REF]. A first stage pools items into groups and performs the tests. A second stage, so-called decoding, analyses the result of these tests to identify the defective items. Tests and decoding are sequential. If the number of tests M is sufficiently big, the decoding stage has enough information to identify the defective items. In a nutshell, the groups are overlapping in the sense that one item is involved in several tests. Decoding amounts at finding the smallest subset of items which would trigger the observed positive tests. The promises of group testing are extremely appealing. First, the theoretical number of tests M asymptotically scales as O(c log N ) as N goes to infinity [START_REF] Atia | Boolean compressed sensing and noisy group testing[END_REF][START_REF] Laarhoven | Asymptotics of fingerprinting and group testing: Tight bounds from channel capacities[END_REF][START_REF] Scarlett | Converse bounds for noisy group testing with arbitrary measurement matrices[END_REF]. This result holds even if c increases with N , but at a lower rate [START_REF] Scarlett | Converse bounds for noisy group testing with arbitrary measurement matrices[END_REF][START_REF] Scarlett | Phase transitions in group testing[END_REF]. Second, recent papers propose practical schemes non only achieving this efficiency (or almost, i.e. O(c log c log N )) but also within a decoding complexity of O(c log N ) (or almost, i.e. O(c log c log N )) [START_REF] Cai | Grotesque: Noisy group testing (quick and efficient)[END_REF][START_REF] Lee | SAFFRON: A fast, efficient, and robust framework for group testing based on sparse-graph codes[END_REF]. This paper takes a complete opposite point of view. The number c of defective items is fixed, and we don't propose more efficient design. On contrary, we show that these promises hold only for some specific probabilistic models. These models are well known in the literature of group testing. They do take into account some imperfection in the test process, however, they are somehow optimistic. As group testing becomes popular, people applying this technique to their 'needles in haystack' problems might be disappointed. The promises of group testing (a number of tests in O(c log N ) together with a computational complexity of O(c log N )) fade away for applications not compliant with these models. The goal of this paper is to investigate what is specific in these models and to better understand the conditions necessary for achieving the promises of group testing. This paper has the following structure. Section 2 describes the recent approaches achieving both the minimum asymptotic number of tests and the minimal decoding complexity. The usual models are introduced together with an information theoretic justification that these approaches are sound. Section 3 introduces some more general models and shows that the total number of tests no longer scales as O(c log N ) in most cases. Previous works A typical paper about non-adaptive group testing proposes a scheme, which is composed of a pooling design and a decoding algorithm. The pooling is the way M groups are composed from a collection of N items. The decoding receives the M binary test results (positive or negative) to infer the defective items. Under a definition of a successful decoding and some probabilistic models, the paper then shows how the necessary number of tests asymptotically scales. For instance, if the decoding aims at identifying all the defective items, the authors show how M should scale as N → ∞ to make the probability of success converge to one. The best asymptotical scaling has been proven to be in O(c log N ) in theoretical analysis [START_REF] Atia | Boolean compressed sensing and noisy group testing[END_REF][START_REF] Laarhoven | Asymptotics of fingerprinting and group testing: Tight bounds from channel capacities[END_REF]. Notations and models The assumptions of the proof of a typical group testing paper concern the distribution of the defective items in the collection and the model of the test. Denote x a binary vector of dimension N encoding which items are defective: x i = 1 if the i-th item is defective, 0 otherwise. X is the random variable associated to this indicator vector. We assume that there are a fixed number c of defective s.t. P[X = x] = 1/ N c if |x| = c, 0 otherwise. As for the test, the models define the probabilistic behavior of its output. Suppose a group G i of n items, and let 0 ≤ K i ≤ max(n, c) be the random variable encoding the number of defectives in this group. Denote first by Z i a binary r.v. s.t. Z i = 1 if K i > 0, 0 otherwise. Now denote by Y i ∈ {0, 1} the r.v. Y i = Z i ⊕ N i with N i independent and identically distributed as Bernouilli B( ) and ⊕ the XOR operator. Dilution: Y i = ∨ j∈Gi [X j ∧ W i,j ], where ∧ and ∨ are the AND and OR operators and W i,j a binary r.v. modeling the detectability of the j-th item in the i-th group. These random variables are independent (both along i and j) and identically distributed: W i,j ∼ B(1-υ). For a given defective and test, the probability of being diluted (i.e. not detectable) is υ. Threshold: Y i = 0 if K i ≤ κ L and Y i = 1 if K i ≥ κ U . There are plenty variants describing what happens for κ L < K i < κ U [START_REF] Cheraghchi | Improved constructions for non-adaptive threshold group testing[END_REF]. Note that some models can be 'concatenated': we can witness a dilution phenomenon of parameter υ followed by a noise channel of parameter . Another way to model a test is through the c + 1 parameters (θ 0 , • • • , θ c ) defined as the following probabilities: θ k := P[Y i = 1|K i = k]. (1) Parameter θ 0 is thus the probability of a false positive, whereas 1 -θ k for 0 < k ≤ c are the probabilities of false negative when k defectives pertain to the test group. For the models above mentioned, we have the equivalent formulation: 1. Noiseless test: θ 0 = 0 and θ k = 1 for 0 < k ≤ c. 2. Noisy test: θ 0 = and θ k = 1 -for 0 < k ≤ c. Dilution: θ k = 1 -υ k with the convention that x 0 = 1, ∀x ∈ R + . 4. Threshold: θ k = 0 if 0 ≤ k ≤ κ L , θ k = 1 if κ U ≤ k ≤ c. Inria Probabilistic group testing The next step is to create the binary design matrix A ∈ {0, 1} M ×N . This matrix indicates which items belong to which groups: A i,j = 1 if item j is involved in test i, and 0 if not. There are constructions which are deterministic (up to a permutation over the N items) such as those relying on disjunct matrices [START_REF] Indyk | Efficiently Decodable Non-adaptive Group Testing[END_REF][START_REF] Ngo | A survey on combinatorial group testing algorithms with applications to DNA library screening[END_REF]. Another popular method is the probabilistic construction where A i,j is set to one depending on a coin flip: P[A i,j = 1] = p. These coin flips are independent w.r.t. indices i (groups) and j (items). The sequence (A 1,j , • • • , A M,j ) is often called the codeword of item j. We shall focus on this last construction. Theoretical studies [START_REF] Scarlett | Phase transitions in group testing[END_REF][START_REF] Scarlett | Converse bounds for noisy group testing with arbitrary measurement matrices[END_REF][START_REF] Atia | Boolean compressed sensing and noisy group testing[END_REF] shows that there is a phase transition: it is possible to identify all defectives with an asymptotically vanishing probability of error (as N → ∞) if M ≥ max ∈{1,••• ,c} log 2 N I(Y ; A G dif |A Geq ) (1 + η); (2) whereas the error probability converges to one for any decoding scheme if M ≤ max ∈{1,••• ,c} log 2 N I(Y ; A G dif |A Geq ) (1 -η). (3) The sets of items (G dif , G eq ) compose a partition of the set of defective items such that |G dif | = and |G eq | = c -, and A G dif (resp. A Geq ) denote the codewords of the items in G dif (resp. A Geq ). These theoretical results are extremely powerful since they still hold when c is not fixed but slowly increasing with N . They are somehow weakly related to practical decoding schemes. For instance, equation (2) comes from a genie aided setup: a genie reveals to the decoder some defective items G eq , and the number of tests needed to identify the remaining ones, i.e. G dif , is evaluated. This is done for different sizes of G eq , from 0 to c -1. A decoder without any genie needs more than the supremum of all these quantities. In the sequel, we consider simpler expressions of the total number of tests but related to practical (or almost) decoders. Joint decoder The joint decoder computes a score per tuple of c items. It spots the tuple of defective items (identifying all of them) with a probability at least 1 -α J ; and it incorrectly points a tuple of non defective items with probability β J . Denote γ J := log(α J )/ log(β J /N c ). T. Laarhoven [START_REF] Laarhoven | Asymptotics of fingerprinting and group testing: Tight bounds from channel capacities[END_REF] showed that a sufficient and necessary number of tests is at least: M J = c log 2 N max p∈(0,1) I J (p) (1 + O( √ γ J )), (4) where I J (p) = I(Y i , (A i,j1 , • • • , A i,jc )|p) is the mutual information between the output of the test and the codeword symbols of the defectives {j 1 , • • • , j c }. In other words, this corresponds to the case where the genie reveals no information: G eq = ∅ [START_REF] Scarlett | Phase transitions in group testing[END_REF]. Since lim N →∞ γ J = 0 for fixed (α J , β J ), this allows to state that M J scales as M J ≈ c log 2 N/I J (p J ) with p J = arg max p∈(0,1) I J (p). For the equivalent model (θ 0 , • • • , θ c ), this amounts to find the maximizer of the following function: I J (p) := h (P (p)) - c k=0 π k h(θ k ), (5) with P (p) := P[Y i = 1|p] = c k=0 π k θ k , (6) π k := c k p k (1 -p) c-k , ∀0 ≤ k ≤ c, (7) and h(x) is the entropy in bits of a binary r.v. distribution as B(1, x). Laarhoven gives the expressions of p J for large c and for the usual models [START_REF] Laarhoven | Asymptotics of fingerprinting and group testing: Tight bounds from channel capacities[END_REF]. The maximizer and the maximum are functions of c and of the parameters of the test model (for example, or υ for the noisy or dilution model). The drawback is that the decoding is exhaustive: it scans the N c possible subsets of size c from a set of N items. Therefore its complexity is in O(N c ). This is called a joint decoder as it jointly considers a subset of c items. The joint decoder is mainly of theoretical interest since its complexity is hardly tractable. Some schemes propose approximations of a joint decoder with manageable complexity resorting to Markov Chain Monte Carlo [START_REF] Knill | Interpretation of pooling experiments using the Markov chain Monte Carlo method[END_REF], Belief Propagation [START_REF] Sejdinovic | Note on noisy group testing: asymptotic bounds and belief propagation reconstruction[END_REF] or iterative joint decoders [START_REF] Meerwald | Group testing meets traitor tracing[END_REF]. Single decoder The single decoder analyses the likelihood that a single item is defective. It correctly identifies a defective item with probability 1 -α S while incorrectly suspecting a non defective item with probability less than β S . Denote γ S = log(β S )/ log(α S /N ). Laarhoven [START_REF] Laarhoven | Asymptotics of fingerprinting and group testing: Tight bounds from channel capacities[END_REF] showed that a sufficient and necessary number of tests is at least: M S = log 2 N max p∈(0,1) I S (p) (1 + O(γ S )) (8) where I S (p) = I(Y i , A i,j1 |p) is the mutual information between the output of the test and the symbol of the codeword of one defective, say j 1 . Again, since lim N →∞ γ S = 0 for fixed (α S , β S ), this allows to state that M S scales as M S ≈ log 2 N/I S (p S ) with p S = arg max p∈(0,1) I S (p). For the equivalent model (θ 0 , • • • , θ c ), this amounts to find the maximizer of the following function: I S (p) := h (P (p)) -ph(P 1 (p)) -(1 -p)h(P 0 (p)) (9) with P 1 (p) := P[Y i = 1|A i,j1 = 1, p] = c k=1 c -1 k -1 p k-1 (1 -p) c-k θ k (10) P 0 (p) := P[Y i = 1|A i,j1 = 0, p] = c-1 k=0 c -1 k p k (1 -p) c-1-k θ k (11) Laarhoven gives the expressions of p S for large c and for the usual models [START_REF] Laarhoven | Asymptotics of fingerprinting and group testing: Tight bounds from channel capacities[END_REF]. It always holds that I J (p) ≥ cI S (p), for any p ∈ [0, 1]. This yields M S inherently bigger than M J [START_REF] Laarhoven | Asymptotics of fingerprinting and group testing: Tight bounds from channel capacities[END_REF]: Both total numbers of tests scale as c log N , but with a bigger multiplicative constant for M S . The simple decoder computes a score for each item. Therefore its complexity is linear in O(N ). Divide and Conquer Papers [START_REF] Cai | Grotesque: Noisy group testing (quick and efficient)[END_REF][START_REF] Lee | SAFFRON: A fast, efficient, and robust framework for group testing based on sparse-graph codes[END_REF] have recently proposed schemes meeting the promises of group testing as listed in the introduction: optimal scaling both in the total number of tests and decoding complexity. Inria Both of them are deploying a 'Divide and Conquer' approach. Identifying c defectives among a collection of N items is too complex. Their strategy splits this problem into S simpler problems. The collection is randomly split into S subsets. S is chosen such that any subset likely contains at most one defective. Indeed, their proof selects S big enough s.t., with high probability, each defective belongs at least to one subset where it is the only defective. Assume that it is possible to detect whether a subset has no, one or more defectives. Then, a group testing approach is applied on each subset containing a single defective (so called 'singleton' subset in [START_REF] Lee | SAFFRON: A fast, efficient, and robust framework for group testing based on sparse-graph codes[END_REF]): The decoding identifies this defective thanks to the result of tests performed on groups composed of items of that subset. It turns out that identifying defectives in a collection is much simpler when knowing there is only one. In a non-adaptive framework, all group tests are performed in the first stage, but the decoding is only run on subsets deemed as 'singleton'. We detail here our own view of this 'Divide and Conquer' approach. Papers [START_REF] Cai | Grotesque: Noisy group testing (quick and efficient)[END_REF][START_REF] Lee | SAFFRON: A fast, efficient, and robust framework for group testing based on sparse-graph codes[END_REF] slightly differ in the way subsets are created. More formally, each subset S k ,1 ≤ k ≤ S, is composed independently by randomly picking N S items in the collection of N items. Denote by π the probability that an item belongs to a given subset: π = N S /N. Subset S k is not useful for identifying a given defective if: • it doesn't belong to subset S k with probability 1 -π, • else, if it is not the only defective in this subset with probability 1 -H(0; N, c, N S ), where H(k; N, c, N S ) is the hypergeometric distribution, • else, if the decoding over this subset misses its identification with probability denoted by α. Over all, this event happens with probability g(N S ) := (1 -π) + π ((1 -H(0; N, c, N S ) + H(0; N, c, N S )α) (12) which is minimized by selecting N S = N +1 /c+1 because g(N S ) -g(N S -1) ≤ 0 iff N S ≤ N +1 /c+1 (we assume that c + 1 divides N + 1). The probability that S k is useless for identifying a given defective simplifies in: g(N S ) = 1 -N S (N -c -1)! N ! (N -N S )! (N -N S -c)! (1 -α), (13) = 1 - c c + 1 c . 1 -α c + 1 .(1 + O(1/N )), (14) where we use the fact that Γ(N +a) /Γ(N+b) = N a-b (1 + (a + b -1) (a-b) /2N + O( 1 /N 2 )) [16]. Suppose that the goal of the decoding is to identify on expectation a fraction (1 -α S ) of the defectives. The probability of missing a given defective because none of the subset is useful for identifying it equals α S : P[Not identifying a given defective] = g(N S ) S = α S . (15) Since ( c /c+1) c ≥ 1 /e, it is safe to choose S = (c + 1)e(-log α S )/(1 -α) . Suppose now that the goal is to identify all the defectives with probability 1 -α J . The probability of identifying them all is given by: P[identifying all of them] = 1 -g(N S ) S c = 1 -α J . ( 16 ) This can be achieved with S ≥ e(c + 1) log(c/α J )/(1 -α) . RR n°9164 The point of this 'Divide and conquer' approach is that m = Θ(log 2 N S ) tests are needed for identifying a unique defective item in a subset of size N S and with a fixed probability of error α (see Sec. 2.4). Since the sizes of the subsets are all equal to N /c, the total number of tests scales as M DC = O(c log c log N /c) to identify all defectives with high probability, which is almost the optimal scaling. In [START_REF] Lee | SAFFRON: A fast, efficient, and robust framework for group testing based on sparse-graph codes[END_REF], the authors show that the decoding can also exploit subsets containing two defectives (so-called 'doubleton') which reduces S = O(c) for identifying all the defectives. To discover a fraction of the defectives the total number of tests scales as M DC = O(c log 2 N /c). This ends up in the optimal scaling achieved by GROTESQUE [START_REF] Cai | Grotesque: Noisy group testing (quick and efficient)[END_REF] and the 'singleton' only version of SAFFRON [START_REF] Lee | SAFFRON: A fast, efficient, and robust framework for group testing based on sparse-graph codes[END_REF]. These schemes have also the following advantages: • The decoding complexity scales like O(cm) = O(c log 2 N /c) if a deterministic construction is used as in [START_REF] Cai | Grotesque: Noisy group testing (quick and efficient)[END_REF] and [START_REF] Lee | SAFFRON: A fast, efficient, and robust framework for group testing based on sparse-graph codes[END_REF]. We decode O(c) 'singleton' subsets in total. In the noiseless setup, decoding a 'singleton' amounts to read the outputs of the tests because it exactly corresponds to the codeword of the unique defective of that subset. If the setup is not noiseless, the outputs are a noisy version of this codeword. An error correcting code whose decoding is in O(m) gets rid of these wrong outputs. For instance, the authors of [START_REF] Lee | SAFFRON: A fast, efficient, and robust framework for group testing based on sparse-graph codes[END_REF] uses a spatially-coupled LDPC error correcting code. • The decoding complexity scales like O(cmN S ) = O(N log 2 N /c) if a probabilistic construction is used per subset. We decode O(c) 'singleton' subsets. Decoding a singleton amounts to compute the likelihood scores for N S items and identifying the defective as the items with the biggest score. The likelihood is a weighted sum of the m test outputs. The next section shows that finding the optimal parameter p of the probabilistic construction is also simple. The main drawback of the 'Divide and Conquer' strategy is that it doesn't apply when θ 1 = θ 0 . This typically corresponds to the 'threshold' model where one unique defective is not enough to trigger the output of a test. Likewise, if θ 1 ≈ θ 0 , any efficient error correcting decoder will fail and the only option is the exhaustive maximum likelihood decoder. At that point, a probabilistic construction is preferable. Identifying the unique defective in a 'singleton' subset This 'Divide and conquer' approach greatly simplifies the model [START_REF] Atia | Boolean compressed sensing and noisy group testing[END_REF]. Since there is a single defective, we only need parameters θ 0 and θ 1 . By the same token, there is no need of joint decoding since the defective is unique. The mutual information in (8) takes a simple expression: I DC (p) = H(Y i |p) -H(Y i |A i,j , p) = h(θ 0 + p(θ 1 -θ 0 )) -(1 -p)h(θ 0 ) -ph(θ 1 ), (17) which is strictly positive on (0, 1) if θ 1 = θ 0 and whose maximisation is simpler than for the single and joint decoders. This concave function has a null derivative for: p DC = 1 θ 1 -θ 0 • 1 2 h(θ 1 )-h(θ 0 ) θ 1 -θ 0 + 1 -θ 0 . ( 18 ) This gives the following application to the usual models: 1. Noiseless test: θ 0 = 1 -θ 1 = 0 so that p DC = 1 /2 and I DC (p ) = 1. 2. Noisy test: θ 0 = 1 -θ 1 = so that p DC = 1 /2 and I DC (p ) = 1 -h( ). Inria 3. Dilution: θ 0 = 0 and θ 1 = 1 -υ so that p DC (υ) = 1 1 -υ • 1 2 h(υ)/(1-υ) + 1 , (19) I DC (p DC (υ)) = h ((1 -υ)p DC (υ)) -p DC (υ)h(υ). (20) Denote f (υ) = 1 /p DC (υ). We have: f (υ) = -1 -2 h(υ) 1-υ 1 + ln υ 1 -υ for υ ∈ [0, 1]. ( 21 ) Since ln(υ) ≤ υ -1 -(1 -υ) 2 /2 , we have on one hand 2 h(υ) 1-υ ≥ 2 υ /(1 -υ) and on the other hand (1 + ln υ 1-υ )/(1 -υ) ≤ -1 /2. This shows that f (υ) ≥ 0. Function f is increasing, therefore p DC is a decreasing function of the dilution factor υ. As υ → 1, h(υ)/(1 -υ) = 1/ ln(2) -log 2 (1 -υ) + O(1 -υ), s.t. p DC → 1 /e ≈ 0.37. The number of tests for a subset is given by ( 8) which multiplied by the number of subsets gives M DC ≈ ce(-log α S ) I DC (p ) log 2 ( N /c) (22) for identifying a fraction 1 -α S of defectives on expectation. Less optimistic models We would like to warn the reader that the promises of group testing are due to the simplicity of the models described in Sec. 2.1. These models are not naive since they do encompass the imperfection of the test over a group. The output of the test is modeled as a random variable. Yet the statistics of the test only depend on the number of defective items inside the group, but not on the size of the group. Consider the noisy setup with parameter modelling the imperfection of the test. The optimal setting is p DC = 1 /2 for the 'Divide and Conquer' scheme. This means that if N = 200, then the size of the groups is around 100, if N = 2 • 10 9 , the groups are composed of a billion of items and, still, the reliability of the test is not degraded. There are some chemical applications where tests can detect the presence of one single particular molecule among billions. But this is certainly not the case of all 'needles in haystack' problems. Our proposed model We believe there are many applications where the reliability of the test degrades as the size n of the group increases. Indeed, when the size of the group grows to infinity, the test might become purely random. For the noisy setup, should be denoted as n s.t. lim n→+∞ n = 1 /2 if the test gets asymptotically random. For the dilution model, υ should be denoted as υ n s.t. lim n→+∞ υ n = 1. This captures the fact that the defectives get completely diluted in groups whose size grows to infinity. Instead of coping with the noisy or dilution setups, we prefer to consider the equivalent model where probabilities (θ 0,n , • • • , θ c,n ) now depend on the size of the group. We make the following assumptions: • For all n, θ 0,n ≤ • • • ≤ θ c,n . Having more defective items in a group increases the probability that the test is positive. • θ 0,n is a non decreasing function of n. Parameter θ 0,n is the probability of a false positive (the test is positive whereas there is no defective in the group). Increasing the size of the group will not help decreasing the probability of this kind of error. • For 0 < k ≤ c, θ k,n is a non increasing function. Again, 1 -θ k,n is the probability of a false negative (the test is negative whereas there k defective items in the group). Increasing the size of the group will not help decreasing the probability of this kind of error. • These probabilities are bounded monotonic functions, therefore they admit a limit as n → +∞, denoted as θk := lim n→+∞ θ k,n . A test is deemed as asymptotically random if θ0 = • • • = θc , whatever the value of this common limit. Application to group testing designs We consider the three schemes above-mentioned: single, joint and 'Divide and conquer'. As described in Sec. 2.5, the 'Divide and conquer' design builds S subsets by randomly picking N S items out of N . The optimal size N S of a subset grows linearly with N . The three schemes then compose random groups from a population whose size N, be it N = N (single and joint schemes) or N = N S ('Divide and conquer' design), grows to infinity. We denote the size of the test groups by n(N) to investigate different choices as N goes to infinity. Note that n(N) ≤ N. When we pick up at random n(N) items the probability p that this group contains a given defective is p = n(N)/N. We analyze two choices concerning the asymptotical size of the test groups: either lim N→∞ n(N) = +∞ or lim N→∞ n(N) < +∞. We derive the mutual information at stake for the three schemes ('Divide and Conquer', simple, and joint) and we deduce the asymptotical scaling of the total number of tests. We introduce the non increasing functions δ k,n := θ k,n -θ 0,n ≥ 0. For the 'Divide and conquer' scheme, the decoding is performed on singleton subset. Therefore δ 1,n shows the speed at which the test gets closer to a random test. For the simple and joint decoder, there are up to c defective items in a group (we suppose that c < n) and δ c,n shows the speed at which the test gets closer to a random test. The most important factor is the limit δk := lim n→∞ δ k,n . For the 'Divide and conquer' scheme, δ1 > 0 means that the two probabilities θ 0,n and θ 1,n have distinct limits, and Sect. 2.6 gives the optimum choice replacing θ 0 and θ 1 by their limits θ0 and θ1 . When there is a single defective in a subset, Eq. [START_REF] Knill | Interpretation of pooling experiments using the Markov chain Monte Carlo method[END_REF] shows that the number of tests to identify it is Θ(log 2 N S ), which in turn is Θ(log 2 N /c). Because the number of subsets S used by the 'Divide and Conquer' strategy in Sect. 2.5 asymptotically gets independent of N , the total number of tests scales as Θ(c log 2 N /c) (identification of a fraction of defective items) or Θ(c log c log 2 N ) (identification of all the items). For the single and joint decoders, δc > 0 means that the test is not asymptotically random. Packing more and more items in groups always provides informative tests. The strategy of selecting n(N) s.t. lim N→∞ n(N)/N = p will deliver a non null mutual information. Parameter p is derived from Sec. 2 replacing (θ 0 , • • • , θ c ) by their limits ( θ0 , • • • , θc ). There might be other way giving a lower total number of tests, but at least this strategy delivers the promises of group testing with a total number of tests scaling as Θ(c log 2 N ). The next sections investigate our main concern : the case where the test is asymptotically random. Inria 4 First strategy: lim N→∞ n(N) = n The first strategy makes the size of the test groups converging to the finite value n := lim N→∞ n(N) for which the test is not random: suppose θ 0,n < θ 1,n . On the other hand, the probability of an item being in a given test group vanishes as p = n/N. 4.1 The case where θ 0,n = 0 Assume first that θ 0,n = 0, Taylor series give the following asymptotics (See Appendices A.1, B.1, and C.1): I J (p) ≈ c n N ∆h 0,1 , (23) I S (p) ≈ n N ∆h 0,1 , (24) I DC (p) ≈ n N S ∆h 0,1 , (25) with ∆h 0,1 := [(θ 1,n -θ 0,n )h (θ 0,n ) + h(θ 0,n ) -h(θ 1,n )]. These three mutual informations only depend on the first two parameters of the model which is unusual for the joint and the single schemes. As the probability p vanishes, the tests are positive for a unique reason: there is a single defective in the groups. More formally, thanks to L'Hôspital's rule, the probability that there is a single defective knowing that there at least one converges to zero: lim p→0 π 1 1 -π 0 = lim p→0 1 -(c -1) p 1 -p = 1 (26) It is therefore quite normal that I DC (p) and I S (p) coincide (except that N S is replaced by N ). However, the 'Divide and Conquer' scheme runs a group testing procedure per subset s.t. it asymptotically needs e(-log α S ) more tests than the single decoder (comparison of ( 22) with ( 8)). I J (p) is exactly c times bigger than I S (p), which in the end offers the same scaling for the total number of tests: M J ≈ M S (comparison of ( 4) with ( 8)). This signifies that the joint decoding doesn't perform better than the single decoding. Indeed, the score computed for a tuple of c items by the joint decoder becomes asymptotically equal to the sum of the scores of the c items as computed by the single decoder. The three schemes need a total number of tests scaling as O(N log N ). It is surprising that it doesn't depend on c, but the most important point is that this is much less appealing than the promise in O(c log N ). 4.2 The case where θ 0,n = 0 When θ 0,n = 0, the expressions above are no longer correct because lim x→∞ h (x) = ∞. New Taylor series give the following asymptotics (See Appendices A.1, B.1, and C.1): I J (p) ≈ cθ 1,n n N log 2 N n , (27) I S (p) ≈ θ 1,n n N log 2 N n , (28) I DC (p) ≈ θ 1,n n N S log 2 N S n . ( 29 ) The same comments as above hold except that this time the schemes provide a better total number of tests scaling as O(N ). The explanation is that a test s.t. θ 0,n = 0 has the advantage of being positive if and only if there is at least one defective in the group. Indeed, there is a single defective in a positive group exactly in the 'Divide and Conquer' scheme and asymptotically for the joint and single decoders. This certainty eases a lot the decoding. These mutual informations show that the multiplicative factor of this scaling is 1/(nθ 1,n ) for the single and joint decoders. We thus need to select n s.t. nθ 1,n > 1 in order to be, asymptotically at least, preferable to an exhaustive search testing items separately. This raises an even more stringent condition for the 'Divide and Conquer' scheme because we need nθ 1,n > e(-log α S ). 5 Second strategy: lim N→∞ n(N) = ∞ The second strategy makes the size of the test groups increasing as N → ∞. Therefore, the rate at which the test becomes random matters. This is reflected by the speed at which δ 1,n ('Divide and Conquer' ) or δ c,n (joint and single) converge to zero. Once again, we make the distinction between tests s.t. θ0 > 0 and those for which θ0 = 0. The case where θ0 = 0 Assume first that θ0 = 0, Taylor series give the following asymptotics (See Appendices A.2.1, B.2.1 and C.2.1): I J (p) ≈ - 1 2 h ( θ0 )Var[θ K,n ], (30) I S (p) ≈ - 1 2 h ( θ0 ) 1 c 2 p(1 -p) Cov(K, θ K,n ) 2 , (31) I DC (p) ≈ - 1 2 h ( θ0 )p(1 -p)δ 2 1,n , (32) with K ∼ B(c, p). Since Cov(K, θ K,n ) 2 ≤ Var[K]Var[θ K,n ] and Var[K] = cp(1 -p), these series comply with the rule that I S (p) ≤ I J (p)/c. We can also check that if c = 1, these three series are equal. Another remark: If θ K,n = Kδ 1,n + θ 0,n ∀0 ≤ K ≤ c, then I DC (p) = I S (p) = I J (p)/c and the three schemes provide the same scaling of the total number of tests. Note however that the probability p of belonging to a group equals n/N in the first two expressions, whereas it equals n/N S in the last expression. Application to the noisy group testing: In this setup, θ 0,n = 1 -θ k,n = n → 1 /2. This simplifies the expressions above as follows: I J (p) ≈ 2 ln 2 (1 -p) c (1 -(1 -p) c )δ 2 1,n (33) I S (p) ≈ 1 ln 2 p(1 -p) 2c-1 δ 2 1,n (34) I DC (p) ≈ 2 ln 2 p(1 -p)δ 2 1,n . (35) If we suppose that δ 1,n = O(n -a ) with a > 0 and we increase the size of the group s. t. n ∝ N b with 0 ≤ b ≤ 1 (or n ∝ N b S for the 'Divide and Conquer' scheme), then the three schemes offer a mutual information of the same order: Inria • If 0 < a ≤ 1 /2, the best option is to choose b = 1, i.e. to fix the value of p, to achieve I = O(N -2a ), which ends up in a total number of test scaling as Ω(N 2a log N ). • If a > 1 /2, the best option is to set b = 0, i.e. to fix the size of the group, to achieve I = O(N -1 ), which ends up in a total number of tests scaling as Ω(N log N ). We rediscover here the results of Sect. 4. These scalings are much bigger than the promised Θ(c log N ). Yet, if the test smoothly becomes random as n increases, i.e. when a < 1 /2, the situation is actually not that bad since the scale of the total number of tests is slower than Θ(N ), i.e. the scaling of the exhaustive screening (yet, we need a setup where the Ω becomes a Θ). The appendices gives upper bounds of the mutual informations of the single and joint decoders in the general case: I J (p) - 1 2 h ( θ0 )(1 -(1 -p) c )δ 2 c,n , (36) I S (p) - 1 2 h ( θ0 ) p 1 -p δ 2 c,n (37) These two upper bounds share the same decrease in O(N -2a ) if δ 1,n = O(n -a ) with 0 < a ≤ 1 /2. Now to get M = O((log N ) d ) we need to have I = Ω((log N ) 1-d ) and therefore, for a fixed p, δ c,n (or δ 1,n for the 'Divide and Conquer' scheme) being Ω((log n) 1-d /2 ). The point of this chapter is to consider that δ c,n converges to zero, therefore d > 1. We are getting closer to the promise of group testing for tests becoming random at a very low speed. The case where θ0 = 0 The appendices A.2.2, B.2.2 and C.2.2 show that: I J (p) ≤ (-θ c,n log 2 θ c,n )(1 -π 0 -π c ) + θ c,n (-(1 -π 0 ) log 2 (1 -π 0 )) + o(θ c,n ) (38) I S (p) ≤ (-θ c,n log 2 θ c,n )(1 -π 0 -π c ) + θ c,n (-(1 -π 0 ) log 2 (1 -π 0 ) + (c -1)p c log 2 p) + o(θ c,n ) (39) I DC (p) = θ 1,n (-p log 2 p) + o(θ 1,n ), (40) with π 0 = (1 -p) c and π c = p c . For a fixed p, the mutual informations of the joint and single decoders are dominated by -θ c,n log 2 θ c,n . If θ c,n = O(n -a ), a > 0, the total number of tests scales as Ω(N a ). This does not hold for the 'Divide and Conquer' scheme: if θ 1,n = O(n -a ), the total number of tests scales as Ω(N a log N ). If n ∝ N b , 0 ≤ b ≤ 1 so that p ∝ N b-1 , then the mutual informations of the joint and single decoders are O(N b(1-a)-1 log N ). If the test is slowly converging to a random test, i.e. a < 1, then we should set b = 1 and we are back to the option of freezing p. Otherwise, it is better to set b = 0 so that the total number of tests scales as Ω(N ), and we find back the first strategy fixing n. The same comment holds for the the 'Divide and Conquer' scheme. Again, this case is preferable to the case θ0 = 0: M = Ω(N a log N ) and not Ω(N 2a log N ), and for a longer range 0 < a ≤ 1 (and not 0 < a ≤ 1 /2). Last but not least, for the 'Divide and Conquer' scheme, to get M = O((log N ) d ) we need to have θ 1,n = Ω((log N ) 1-d ) with d > 1 to make θ 1,n vanishing as n = pN increases (p is fixed). With the same setup, M = O( (log N ) d /log log N ) for the single and joint decoders. Conclusion The point of this chapter is not to find the best choice concerning the asymptotical size of the test groups. We just show that whatever this choice, group testing fails delivering the promise of a total number of tests scaling as O(c log N ). The condition of utmost importance for such an appealing scaling is to have a test which doesn't become purely random as the size of the group grows to infinity. However, group testing almost keeps its promise, i.e. a total number of tests scaling as a power of log N , for setups where the test converges to randomness very slowly, i.e. at a rate in Ω( 1 /log g n) with g > 0. For this kind of setups, it is better to fix p, which means that the size of the groups are proportional to N . However, if the test becomes random too rapidly, i.e. as fast as O(n -a ) with a ≥ 1 /2, it is useful to switch from a fixed p strategy to a fixed n strategy. Setups where there is no false positive (θ 0,n = 0) or no false negative (θ k,n = 1 for k > 0) lead to better performances: the total number of tests is lower and the transition from fixed p to fixed n occurs at a higher rate, i.e. for a = 1. A 'Divide and conquer' A.1 First strategy: lim N→∞ n(N) = n The first strategy makes the size of the test groups converging to the finite value n := lim N S →∞ n(N S ) for which the test is not random, i.e. θ 0,n < θ 1,n . On the other hand, the probability of an item being in a given test group vanishes as p = n/N S . Assume that θ 0,n = 0, a Taylor series of [START_REF] Zhou | Parallel feature selection inspired by group testing[END_REF] gives the following asymptotic: I DC (p) = n N S [(θ 1,n -θ 0,n )h (θ 0,n ) + h(θ 0,n ) -h(θ 1,n )] + o(N -1 S ). (41) If θ 0,n = 0, the result above does not hold because lim x→0 h (x) = +∞. A new Taylor series gives the following asymptotic: I DC (p) = θ 1,n n N S log 2 N S n + o 1 N S log(N S ) . (42) A.2 Second strategy: lim N→∞ n(N) = ∞ A.2.1 When θ0 ∈]0, 1[ The assumption here is that θ1 = θ0 which lies in ]0, 1[. We denote η n := θ 0,n -θ0 and δ 1,n := θ 1,n -θ 0,n . With these notations, we have I DC (p) = h( θ0 + η n + pδ 1,n ) -(1 -p)h( θ0 + η n ) -ph( θ0 + η n + δ 1,n ), (43) Note that η n ≤ η n + pδ 1,n ≤ η n + δ 1,n because 0 ≤ p ≤ 1, which implies that |η n + p n δ 1,n | ≤ max(|η n |, |η n + δ 1,n |). (44) Both |η n | and |δ 1,n | converges to 0 so that, for > 0, there exist n 0 big enough s.t. ∀n ≥ n 0 , max(|η n |, |η n + δ 1,n |) ≤ . We then apply the following Taylor development for θ0 ∈]0, 1[: h( θ0 + ) = h( θ0 ) + h ( θ0 ) + 2 h ( θ0 )/2 + o( 2 ), (45) Inria on the three terms of (43) to simplify it to: I DC (p) = - 1 2 δ 2 1,n h ( θ0 )p(1 -p) + o( 2 ). (46) Since we assume in the text that θ 0,n is non decreasing, it has to converge to θ0 from below s.t. η n is non positive. In the same way, θ 1,n converges to θ0 from above s.t. η n +δ 1,n is non negative. This shows that ≤ δ 1,n ≤ 2 and therefore δ 1,n = Θ( ). This allows to replace o( 2 ) by o(δ 2 1,n ) in (46). A.2.2 When θ0 ∈ {0, 1} We detail the case for θ1 = θ0 = 0. With the same notations as in App. A.2.1, this case implies that η 0,n = 0 because θ 0,n is non decreasing and non negative. The mutual information in this context equals: I DC (p) = h(pδ 1,n ) -ph(δ 1,n ). (47) For > 0, there exist n big enough for which δ 1,n = and where h( ) = -log 2 ( ) + / ln 2 + o( ). Applying this development, we obtain: I DC (p) = δ 1,n (-p log 2 p) + o(δ 1,n ). (48) B Joint decoder We assume that the size of a group is always larger than c. Therefore, (c + 1) parameters define the test (θ 0,n , • • • , θ c,n ). B.1 First strategy: lim N→∞ n(N) = n The mutual information for the joint decoder (5) has the following Taylor series when p = n/N → 0, for θ 0,n > 0: I J (p) = c n N [(θ 1,n -θ 0,n )h (θ 0,n ) + h(θ 0,n ) -h(θ 1,n )] + o(1/N ), (49) and, for θ 0,n = 0 and θ 1,n > 0: I J (p) = cθ 1,n n N log (N ) + o(N -1 log N ). (50) We have supposed that n is chosen s.t. δ 1,n > 0. It is possible to relax this constraint and the first non nul parameter δ k,n will appear in the above equations. Yet, we also get a decay in N -k instead of N -1 , whence choosing n s.t. δ 1,n = 0 should be avoided if possible. This is a real issue for the threshold group testing model where θ 1,n = θ 0,n (see Sect. For 0 < θ0 < 1, we apply the development (45) to h(P (p)) and h(θ k,n ): I J (p) = - 1 2 h ( θ0 ) c k=0 π k θ 2 k,n -P (p) 2 + o( 2 ) (51) = - 1 2 h ( θ0 )Var[θ K,n ] + o( 2 ). (52 C Single decoder We assume that the size of a group is always larger than c. Therefore, (c + 1) parameters (θ 0,n , • • • , θ c,n ) define the test. C.1 Asymptotical analysis for lim N S →∞ n(N S ) < ∞ Assume that θ 0,n = 0, a Taylor series of (9) gives the following asymptotic: I S (p) = n N [(θ 1,n -θ 0,n )h (θ 0,n ) + h(θ 0,n ) -h(θ 1,n )] + o(1/N ). ( 56 ) If θ 0,n = 0, a Taylor series gives the following asymptotic: (60) From ( 10) and [START_REF] Meerwald | Group testing meets traitor tracing[END_REF], it is clear that θ 0,n ≤ P 0 (p) and P 1 (p) ≤ θ c,n . We also have p(1 -p)P (p) = E[(K -cp)θ K,n ] = E[Kθ K,n ] -E[K]E[θ K,n ] = Cov(K, θ K,n ). ( 61 ) with K ∼ B(c, p) because E[K] = cp. We introduce K ∼ B(c, p) independent of K. Then, on one hand: Cov(K -K , θ K,n -θ K ,n ) = Cov(K, θ K,n ) + Cov(K , θ K ,n ) = 2Cov(K, θ K,n ), (62) while, on the other hand, Cov(K -K , θ K,n -θ K ,n ) = E[(K -K )(θ K,n -θ K ,n )] -E[K -K ]E[θ K,n -θ K ,n ] = 0≤k,k ≤c π k π k (k -k )(θ k,n -θ k ,n ). (63) 1 [ 1 2.1). B.2 Second strategy: lim N→∞ n(N) = ∞ Now suppose that the parameters of the model vary with n s.t. θ 0,n ≤ θ 1,n ≤ • • • ≤ θ c,n , and that δ c,n = θ c,n -θ 0,n vanishes to 0 as n increases. The analysis is made for a fixed 0 < p < 1. We remind that π k = c k p k (1 -p) c-k , ∀0 ≤ k ≤ c, the distribution of the binomial B(c, p). T. Furon B.2.1 When θ0 ∈]0, As in the previous section, η n = θ 0,n -θ0 and δ k,n = θ k,n -θ 0,n converge to zero. For any > 0, there exists n large enough for which max(|η n |, η n + δ c,n ) = . This implies that |η n + δ k,n |, ∀0 ≤ k ≤ c, and |η n + P (p) -θ 0,n | are smaller than . ) Note that Var[θ K,n ] ≤ E[(θ K,n -θ 0,n ) 2 ] ≤ δ 2 c,n (1 -(1 -p) c ) < 4 2 .On the other hand,Var[θ K,n ] ≥ π 0 (θ 0,n -P (p)) 2 + π c (θ c,n -P (p)) 2 ≥ π 0 π c π 0 + π c δ 2 c,n ≥ 2 . (53)This shows thatVar[θ K,n ] = Θ( 2 ) so that we can replace o( 2 ) by o(Var[θ K,n ]). B.2.2 When θ0 ∈ {0, 1}We start by applying the development h(x) = -x log 2 (x) + x /ln 2 + o(x) on h(P (p)) and h(θ k,n ):I J (p) = (-P (p) log 2 (P (p))) -c k=0 π k (-θ k,n log 2 θ θ k,n ) + o(θ c,n ).(54)The function x → -x log 2 (x) is increasing over [0, 1 /e) and P (p) ≤ θ c,n (1-π 0 ). On the other hand c k=0 π k (-θ k,n log 2 θ θ k,n ) ≥ π c (-θ c,n log 2 θ c,n ). This inequality follows from these arguments: I J (p) ≤ (-θ c,n log 2 θ c,n )(1 -π 0 -π c ) + θ c,n (-(1 -π 0 ) log 2 (1 -π 0 )) + o(θ c,n ). (55) I S (p) = θ 1,n n N log 2 Nπ 2 n + o (log(N )/N ) . (57) If θ 1,n = θ 0,n , the Taylor series wil show the role of the first non nul coefficient δ k,n but fraction n/N must be replaced by ( n/N ) k . This should be avoided as it Inria C.2 Second strategy: lim N→∞ n(N) = ∞We first present some relations between P (p), P 1 (p) and P 0 (p):P 1 (p) = P (p) + (1 -p)P (p)/c,(58)P 0 (p) = P (p) -pP (p)/c, k θ k,n (k -cp). Since θ k,n is increasing with k, the summands in the last equation are all non negative. We can also lower bound this sum by only keeping the terms |k -k | = c. This shows that0 ≤ δ c,n cp c-1 (1 -p) c-1 ≤ P (p) = 1 p(1 -p) Cov(K, θ K,n ). (64)An upper bound is given by noting that c k=0 π k θ k,n k ≤ cpθ c,n and c k=0 π k θ k,n ≥ θ 0,n , s.t. p(1 -p)P (p) ≤ δ c,n cp. This proves that P (p) = Θ(δ c,n ).Since P (p) ≥ 0, we haveθ 0,n ≤ P 0 (p) ≤ P (p) ≤ P 1 (p) ≤ θ c,n .(65)These five probabilities converge to θ0 as n → ∞.C.2.1 When θ0 ∈]0, 1[ For > 0, there exists n large enough s.t. max(|η n |, δ c,n + η n ) = .The Taylor series of (9) leads to:(1 -p) Cov(K, θ K,n ) 2 + o( 2 ).(67) Now, δ c,n = δ c,n + η n -η n s.t. ≤ δ c,n ≤ 2 and P (p) = Θ( ). This allows to replace o( 2 ) by o(P (p) 2 ) or o(Cov(K, θ K,n ) 2 ) in the equations above. The upper bound on P (p) is used to bound the mutual information: I S (p) ≤ - modeling the output of the test performed on this group. There are four well known models:1. Noiseless test: Y i = Z i . The test is positive if and only if there is at least one defective in a group, 2. Noisy test: RR n°9164 Publisher Inria Domaine de Voluceau -Rocquencourt BP 105 -78153 Le Chesnay Cedex inria.fr ISSN 0249-6399 T. Furon C.2.2 When θ0 ∈ {0, 1} We have: Following the same rationale as in Sect. B.2.2, we get:

01744266

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2024/03/05 22:32:07

2018

https://hal.sorbonne-universite.fr/hal-01744266/file/JABER_Maguy.pdf

Pollyana Trigueiro Silvia Pedetti Baptiste Rigaud Sebastien Balme Jean-Marc Janot Ieda M G Dos Santos Régis Gougeon Maria G Fonseca Thomas Georgelin Maguy Jaber email: [email protected] Going through the wine fining: intimate dialogue between organics and clays Keywords: adsorption, resveratrol, BSA, NMR, Fluorescence spectroscopy, wine, clay Wine chemistry inspires and challenges with its complexity and intriguing composition. In this context, the composites based on the use of a model protein, a polyphenol of interest and montmorillonite in a model hydroalcoholic solution have been studied. A set of experimental characterization techniques highlighted the interactions between the organic and the inorganic parts in the the composite. The amount of the organic part was determined by ultraviolet-visible (UV-VIS) and thermal analysis. X-ray diffraction (XRD) and transmission electronic microscopy (TEM) informed about the stacking/exfoliation of the layers in the the composites. Vibrational and nuclear magnetic resonance spectroscopies methods stressed on the formation of a complex between the protein and the polyphenol before adsorption on the clay mineral. The mobility/rigidity of the organic parts were determined by fluorescence time resolved spectroscopy. Changes in the secondary structure of the protein occured upon complexation with polyphenol on clay mineral due to strong interactions. Although not representating faithfully enological conditions, these results highlight the range and nature of mechanisms possibly involved in wine fining. Introduction Proteins are present in wine in low concentrations, contributing weakly to their nutritive characteristics. Nevertheless, their presence in high proportion may affect the quality of the product, being responsible for the turbidity and time-instability of white wines. The proteins of wine are generally tolerant to the pH of the wine (pH = 3-3.6) [START_REF] Jaeckels | Influence of bentonite fining on protein composition in wine[END_REF][START_REF] Van Sluyter | Wine protein haze: Mechanisms of formation and advances in prevention[END_REF][START_REF] Moreno-Arribas | Analytical methods for the characterization of proteins and peptides in wines[END_REF]. Naturally occurring proteins of grapes, and in particular pathogenesis related (PR) proteins, have been shown to cause turbidity or haze formation in white wines, where the instability can be related to factors such as pH of the medium, ethanol content, ionic strength, temperature or concentration of the organic acids, tannins and polyphenolic compounds [START_REF] Dordoni | Effect of bentonite characteristics on wine proteins, polyphenols, and metals under conditions of different pH[END_REF][START_REF] Vincenzi | Study of combined effect of proteins and bentonite fining on the wine aroma loss[END_REF][START_REF] Esteruelas | Phenolic compounds present in natural haze protein of Sauvignon white wine[END_REF]. Therefore, it is necessary in the fining treatment to eliminate the risk of protein precipitation, which content can vary from 10 to more than 260 mg.L -1 [START_REF] Cilindre | It's time to pop a cork on champagne's proteome![END_REF][START_REF] Wigand | Analysis of Protein Composition of Red Wine in Comparison with Rosé and White Wines by Electrophoresis and High-Pressure Liquid Chromatography-Mass Spectrometry (HPLC-MS)[END_REF]. It must be noted though, that in Champagne, proteins can have a positive role on the foamability. Polyphenols are responsible for the differences between white and red wine, especially in the determination of the color, body, flavor and structure of red wine [START_REF] Pérez-Magariño | Polyphenols and colour variability of red wines made from grapes harvested at different ripeness grade[END_REF]. Among the diversity of polyphenolic compounds, stilbenes and derivatives have been largely studied,in particular resveratrol (RESV), has antioxidant, bactericides, anti-inflamatory and vitamin properties, which are related to its protective effect against cardiovascular diseases [START_REF] Cao | Non-covalent interaction between dietary stilbenoids and human serum albumin: Structure-affinity relationship, and its influence on the stability, free radical scavenging activity and cell uptake of stilbenoids[END_REF].Even though it can also be found in food products and beverages such as peanut butter, mulberries and grape juice, Red wine is believed to be the main source of resveratrol in the human diet [START_REF] Lu | Resveratrol, a natural product derived from grape, exhibits antiestrogenic activity and inhibits the growth of human breast cancer cells[END_REF][START_REF] Wenzel | Metabolism and bioavailability of trans-resveratrol[END_REF]. The phenolic compounds found in wines may vary according to their structure as, tannins, non-flavonoids and flavonoids. RESV (Figure 1 SI, in Supplementary information) is a flavonoid, natural pigment, which can be present in high concentration in grape skin but not in grape flesh, so red wine contains significant amounts of resveratrol compared to white wine, with concentration ranging from 0.1 to 3.0 mg.L -1 , where its amount depends on the grape variety and the vinification process [START_REF] Minussi | Phenolic compounds and total antioxidant potential of commercial wines[END_REF][START_REF] Lucas-Abellán | Cyclodextrins as resveratrol carrier system[END_REF]. It has been reported in the literature that the complexation of resveratrol with macromolecules can modulate the expression of its bioviability and stability as well as its antioxidant effect [START_REF] Lucas-Abellán | Cyclodextrins as resveratrol carrier system[END_REF][START_REF] Richard | Recognition characters in peptide-polyphenol complex formation[END_REF]. Different fining agents in winemaking are reported in the literature, but clay minerals rich in montmorillonite such as bentonite, remain heavily used for white wines. Bentonite behaves as stabilizer by adsorbing proteins in wine wich typically have isoelectric point (IP) between 3 and 9 and a MW between 20-70 KDa [START_REF] Achaerandio | Protein Adsorption by Bentonite in a white wine model solution: Effect of Protein Molecualr Weight and Ethanol Concentration[END_REF]. Montmorillonite displays a layered structure of type 2:1, ideally formed by two sheets of tetrahedrally coordinated silicon linked through a sheet of octahedrally coordinated aluminum. Al 3+ and Mg 2+ generally occupy octahedral sites, whereas Hydrated exchangeable Na + or Ca 2+ cations are present in the interlayers to balance the negative layer charge [START_REF] Jaber | Synthesis, characterization and applications of 2:1 phyllosilicates and organophyllosilicates: Contribution of fluoride to study the octahedral sheet[END_REF][START_REF] Brigatti | Chapter 2 Structures and Mineralogy of Clay Minerals[END_REF][START_REF] Jaber | Mercaptopropyl Al-Mg phyllosilicate: synthesis and characterization by XRD, IR, and NMR[END_REF]. Adsorption of proteins onto clay minerals has been widely reported in the literature [START_REF] Assifaoui | Structural Studies of Adsorbed Protein (Betalactoglobulin) on Natural Clay (Montmorillonite)[END_REF][START_REF] Yu | Adsorption of proteins and nucleic acids on clay minerals and their interactions: A review[END_REF][START_REF] Servagent-Noinville | Conformational Changes of Bovine Serum Albumin Induced by Adsorption on Different Clay Surfaces: FTIR Analysis[END_REF]. It may involve several types of physical and chemical interactions such as cation exchange, electrostatic forces or hydrogen bonding, on the interlayer surface of clay minerals, on the edges or both. The positive charge of the protein when the pH is below the IP allows the adsorption on the negatively charged surface of montmorillonite due to electrostatic forces. Various possible mechanisms can describe the interaction between proteins and clays: intercalation, exfoliation or both [START_REF] Assifaoui | Structural Studies of Adsorbed Protein (Betalactoglobulin) on Natural Clay (Montmorillonite)[END_REF]. Extensive penetration of protein chains into the interlayer space of clays can lead to exfoliation or delamination of the silicate layers [START_REF] Kumar | Effect of Type and Content of Modified Montmorillonite on the Structure and Properties of Bio-Nanocomposite Films Based on Soy Protein Isolate and Montmorillonite[END_REF][START_REF] Gopakumar | Influence of clay exfoliation on the physical properties of montmorillonite / polyethylene composites[END_REF][START_REF] Cai | Adsorption of DNA on clay minerals and various colloidal particles from an Alfisol[END_REF]. The use of clay minerals for the fine treatment of wine may induce not only adsorption of the protein but also of other molecules. As reported in the literature, there may be a decrease in the amount of resveratrol in the wine due to interactions with fining agents [START_REF] Donovan | Effects of Small-Scale Fining on the Phenolic Composition and Antioxidant Activity of Merlot Wine[END_REF][START_REF] Threlfall | Effects of fining agents on transresveratrol concentration in wine[END_REF] or an interaction of resveratrol with the protein due to electrostatic forces, hydrophobic interactions and/or hydrogen bond between these two compounds [START_REF] Liang | Interaction of b-lactoglobulin with resveratrol and its biological implications[END_REF][START_REF] N′ Soukpoé-Kossi | Resveratrol Binding to Human Serum Albumin[END_REF]. Bovine serum albumin (BSA, Figure 2 SI in Supplementary information) with MW of 66.5 KD a is used as model protein for this study. It is characterized by globular structure and is the most abundant soluble plasma protein with a typical circulating concentration of 0.6 mmol.L -1 . In addition, it is extensively used in biochemical studies due to its wide availability and high structural resemblance with human serum albumin [START_REF] Bourassa | Binding sites of resveratrol, genistein, and curcumin with milk α-And β-caseins[END_REF][START_REF] Carter | Structure of serum albumins[END_REF]. Resveratrol has a strong interaction with albumin. Indeed, several molecules can be bound to the hydrophobic pocket of albumin with an affinity constant around 2.10 4 M -1 or 5.10 4 M -1 . This binding occurs via both hydrophillic and hydrophobic interactions and/or hydrogen bonding [START_REF] Bourassa | Binding sites of resveratrol, genistein, and curcumin with milk α-And β-caseins[END_REF][START_REF] Nair | Biology Spectroscopic study on the interaction of resveratrol and pterostilbene with human serum albumin[END_REF][START_REF] Jiang | Study of the interaction between trans-resveratrol and BSA by the multi-spectroscopic method[END_REF]. The aim of the present work is to investigate the co-adsorption of a model protein, the BSA, and a polyphenol of interest, resveratrol, onto a synthetic montmorillonite (Mt)in an hydroalcoholic solution, in order to investigate, via a multi-technical approach, the presence and the nature of interactions between the clay mineral, the protein and the polyphenolic compound present in wine. 2. Experimental part Materials Bovine Serum Albumin (>99%) and resveratrol (>99%) were purchased from Sigma-Aldrich. Potassium phosphate monobasic (>99%) buffer 0.1M and ethanol (solvent system) were used to solubilize the protein and polyphenol. Anapproximation of amodel wine solution was made with the mixture of 25% distilled water, 50% phosphate buffer and 25% ethanol (ratio v/v) at pH = 4.5 (solvent system used in BSA solution and RESV solution). For the synthesis of the montmorillonite,the following reagents were used: Aerosil 130 (Evonik Industries) as source of silica, Bohemite (AlOOH) 74% Al2O3 (Sasol Germany), Magnesium acetate tetrahydrate (Sigma-Aldrich), Sodium acetate (Sigma-Aldrich) and hydrofluoric acid. Synthesis of montmorillonite For the synthesis of sodium montmorillonite, the reagents were mixed in the following order: deionized water, hydrofluoric acid and the sources of interlamellar cation: sodium acetate, magnesium acetate, alumina and silica. The hydrogels were aged under stirring at room temperature for 2h and then were autoclaved for reaction at 220 °C for 72h. The autoclaves were cooled to room temperature and the products were washed thoroughly with distilled water and centrifuged. The solids were then dried at 50 °C for 24 h [START_REF] Jaber | Selectivities in Adsorption and Peptidic Condensation in the (Arginine and Glutaminic Acid)/Montmorillonite Clay System[END_REF][START_REF] Tangaraj | Adsorption and photophysical properties of fluorescent dyes over montmorillonite and saponite modified by surfactant[END_REF]. Adsorption of proteins Adsorption experiments were carried out by adding dropwise BSA solution, dissolved in the above described solvent system, in an aqueous suspension of the clay mineral (1.2mg.mL -1 ). The initial concentration of BSA (1.0 mg.mL -1 ) was determined before addition of montmorillonite. The mixture was stirred at room temperature for 4 hours. After that, the suspension was centrifuged at 2700rpm for 8 minutes. Both supernatants and solids were collected and analyzed. Solids were washed with aqueous solution to remove weakly adsorbed BSA and dried at room temperature. The amount of protein remaining in the supernatant was determined by UV-Vis spectroscopy considering the absorption maximum wavelength at 281 nm. All measurements were performed in triplicate. The concentration of adsorbed protein on montmorillonite was determined by: Г t = V ( C 0 -C t ) m ( 1 ) where V is the total volume of the protein solution, C0 is the initial concentration of protein, Ct is the concentration of protein in supernatant at time t, and m is the mass of the clay mineral. With V (mL), C0 and Ct (mg.mL -1 ), the interfacial concentration Гt of protein is given in mg of protein per mg of clay mineral (mg.mgMt -1 ). Adsorption of resveratrol onto sodium clay The polyphenol adsorption experiments were performed as described for the protein. The initial concentration of RESV (0.5 mg.mL -1 ) was determined before addition of montmorillonite. The amount of resveratrol remaining in the supernatant was determined by UV-Vis spectroscopy with absorption at 306 nm. Simultaneous adsorption of polyphenol and BSA onto sodium clay BSA (1.0 mg.mL -1 ) was dissolved in solution of the above described solvent system. Separately, resveratrol with a concentration of (0.5 mg.mL -1 ) was dissolved in a solution of the above described solvent system. Then, the solution containing RESV was added dropwise to the BSA solution that was then kept under constant stirring for 2 hours. Resulting solution was added dropwise to the aqueous suspension of the clay mineral with a of concentration 1.2 mg.mL - 1 under constant stirring for 2 hours. The resulting mixture was stirred for 4 hours at room temperature. The resulting slurry was washed and centrifuged for removal of excess of protein and polyphenol. The resulting solid was separated by centrifugation and dried at room temperature. UV-Visible Ultraviolet-visible (UV Vis) absorption spectra were recorded using an Ocean View Optics spectrometer DH-2000-BAL. The light source was a Deuterium and Halogen lamp, equipped with 400μm diameters optic fibers, coupled with CUV 1cm cuvette holder. Spectra were acquired by Ocean View Spectroscope Software. The absorbance wavelength range was from 250 to 900 nm. X-ray diffraction (XRD) Powder X-ray diffractograms were recorded using D8 Advance Bruker-AXS Powder X-ray diffractometer with CuKα radiation (λ= 1.5405 Å). The diffractions patterns were measured between 5-70° (2θ) with a scan rate of 0.5 deg.min -1 . The active area of the detector was limited as much as possible in order to reduce the background scattering at low angle between 1-10° (2θ). Samples were kept 72 h before measurements under controlled humidity. Transmission electron microscopy (TEM) TEM study of the samples was performed on a JEOL 2010 microscope, 200kV LaB6 coupled Orius camera, from Gatan Company. Samples in the form of bulk powders were suspended in ethanol and then deposited on 400 mesh copper grids covered with an ultrathin carbon membrane of 2-3 nm thickness. 2.9.Attenuated Total Reflectance (ATR) Infrared spectroscopy Fourier transform infrared spectroscopy (FTIR) of the solid samples was performed on Agilent Cary 630 FTIR spectrometer using Agilent diamond attenuated total reflectance ATR technique. Spectra were acquired on the 4000 and 650 cm -1 range and processed with the Microlab FTIR Software (Agilent Technologies). Thermal analysis Thermal analyses were carried out using a Q600-1694 SDT Q600 TA Instrument and TA Universal Analysis 2000, the heating rate was of 5 °C.min -1 from 25 °C to 1000 °C, with flow rate of 10 mL.min -1 , in air atmosphere. Solid State Cross-Polarization Magic Angle Spinning Carbon 13 Nuclear Magnetic Resonance ( 13 C CP-MAS Solid state nuclear magnetic resonance) [START_REF] Minussi | Phenolic compounds and total antioxidant potential of commercial wines[END_REF] CCP-MAS NMR spectra were obtained on a Bruker Advance 500 spectrometer operating at ΩL=500MHz ( 1 H ) and125 MHz ( 13 C ) with a 4 mm H-X MAS probe. Chemical shifts were calibrated using the CH2signal of adamantane (38.52 ppm) as an external standard. The CP spectra were acquired with a MAS rate of 14 kHz, an acquisition time of 40 ms, a ramp-CP contact time of 1ms, a 1 s recycle delay and with a 1 H spinal 64decoupling sequence. The number of scans to obtain the spectra depending on the S/N obtained for each sample. Spectra were processed with a zero filling factor of 2 and with an exponential decay corresponding to 25 Hz line broadening. Only spectra with the same line broadening were directly compared. The decomposition of the spectra was performed using Dmfit-2015 software. Decay analysis was performed using a Levenberg-Marquardt algorithm. For the analysis, the fluorescence decay law at the magic angle IM(t) was assumed as a sum of exponentials. We assumed a Poisson distribution of counts in the calculation of the χ2 criterion; residual profiles and autocorrelation function as well as Durbin-Watson and skewness factor were used in order to estimate the quality of the adjustment. The number of exponentials used for the fit was increased until all the statistical criterions were improved. All details about calculation of both lifetime are given elsewhere [START_REF] Balme | Highly efficient fluorescent label unquenched by protein interaction to probe the avidin rotational motion[END_REF]. 2.12.Fluorescence spectroscopy Steady Results and Discussions BSA and RESV adsorption onto sodium clay The UV-Vis spectra of the BSA and RESV before and after adsorption on montmorillonite are shown in Figure 1. The intensity of the band maxima of the BSA in Figure 1(a) at 281 nm decreases after adsorption on montmorillonite. The RESV showed band maxima at 306 nm as we can see in Figure 1(b) and follows the same trend as BSA. The influence of the contact duration was also investigated. The adsorption kinetics (C0 = 1.0 mg.mL -1 ) curves are reported in Figure 3 SI in Supplementary information and similar behavior was observed for both protein and resveratrol. In the present study, the Langmuir isotherm model adequately describes the adsorption data, that are as widely reported for protein adsorption on clay minerals [START_REF] Bajpai | Study on the adsorption of hemoglobin onto bentonite clay surfaces[END_REF]. The Langmuir isotherm of pseudo-second-order model can be linearized in agreement to the equation: 𝑡𝑡 Γ 𝑡𝑡 = 1 𝑘𝑘 2 Γ 𝑒𝑒 2 + 1 Γ 𝑒𝑒 𝑡𝑡 (2) where Γ 𝑡𝑡 and Γ 𝑒𝑒 is the interfacial concentration (mg.mgMt) at time t and at the equilibrium respectively, k2 is the equilibrium rate constant of pseudo-second-order adsorption (mg mgMt - 1 .min -1 ). Pseudo-second order equation were used to research the adsorption kinetic behavior for protein and polyphenol on montmorillonite, reported in Figure 4 SI in Supplementary files. The slope and intercept of the plot of t/Γt versus t were used to calculate the K2 rate constant. Values of the Γ 𝑒𝑒 (interfacial concentration maximum) and K2 (adsorption kinetic constant) are reported in Table I. K2 values indicates affinity of the protein and polyphenol for the inorganic support and in the present case the value of K2=0.16 obtained demonstrates the high affinity of the BSA with the surface of the montmorillonite [START_REF] Assifaoui | Structural Studies of Adsorbed Protein (Betalactoglobulin) on Natural Clay (Montmorillonite)[END_REF][START_REF] Lepoitevin | BSA and lysozyme adsorption on homoionic montmorillonite: Influence of the interlayer cation[END_REF]. Table I. Kinect parameters obtained by Langmuir isotherm for BSA and resveratrol adsorption on montmorillonite. Resveratrol BSA Γ 𝑒𝑒 (mg.mgMt -1 ) 0.32 0.87 K2 (mg mgMt -1 .min -1 ) 1.90 0.16 Localization of the organic molecules in the clay mineral XRD patterns of the montmorillonite and all hybrids composites are reported in Figure 2. The X-Ray pattern of the raw montmorillonite shows typical reflections characteristics of the dioctahedral smectite (d060 = 0.149 nm) [START_REF] Lepoitevin | BSA and lysozyme adsorption on homoionic montmorillonite: Influence of the interlayer cation[END_REF][START_REF] Reinholdt | Synthesis and characterization of montmorillonite-type phyllosilicates in a fluoride medium[END_REF]. The d001 value is about 1.26 nm before adsorption corresponding to the thickness of the layer and the presence of hydrated sodium in the interlayer space. Upon adsorption of the protein, the d001 peak disappears while the other characteristic reflection peaks of montmorillonite remained intact, hence the hypothesis of a possible partial exfoliation or delamination can be considered [START_REF] Assifaoui | Structural Studies of Adsorbed Protein (Betalactoglobulin) on Natural Clay (Montmorillonite)[END_REF]. After adsorption of resveratrol, the interbasal spacing of the clay mineral cannot be detected, making difficult to evaluate the degree of intercalation of the polyphenol which may possibly be non-uniformly distributed between the montmorillonite layers. After adsorption of the BSA and RESV together, it is not possible to measure the inter-basal spacing d001, which may suggest partial or total exfoliation of the montmorillonite layers upon inclusion of the protein and the polyphenol. These studies will be complemented with TEM analysis. Presence of the crystalline phase of resveratrol is common, characteristic peaks of the drug at 2θ=16. Transmission electron microscopy (TEM) Transmission electron microscopy (TEM) micrographs show layered structures with alternate dark and bright fringes with a repeat length of 1.26 nm for the raw clay, synthetic clay without modification. In Figure 3 BSA-RESV-Mt (e) and (f). Experimental conditions: BSA (1.0 mg.mL -1 ), RESV (0.5 mg.mL -1 ), Mt (1.2 mg.mL -1 ) in a buffer phosphate solution at pH 4.5 and ethanolic solution in ratio of 25 %. Spectroscopic characterization Study of BSA and RESV conformation before and after adsorption onto clay mineral was performed using ATR-Infrared spectroscopy (Figure 4). The presence of the physisorbed water of montmorillonite is observed by the bending vibration at 1634 cm -1 [START_REF] Georgelin | Inorganic phosphate and nucleotides on silica surface: Condensation, dismutation, and phosphorylation[END_REF] while the characteristic peak at 1015 cm -1 is due to stretching vibrations of the Si-O groups [START_REF] Madejová | FTIR techniques in clay mineral studies[END_REF]. According to the literature [START_REF] Bourassa | Resveratrol, genistein, and curcumin bind bovine serum albumin[END_REF][START_REF] Roy | Spectroscopic and docking studies of the binding of two stereoisomeric antioxidant catechins to serum albumins[END_REF][START_REF] Liu | Molecular Modeling and Spectroscopic Studies on the Interaction of Transresveratrol with Bovine Serum Albumin[END_REF] the spectral ranges for the most intense peaks for proteins occur in the region 1700-1600 cm -1 and 1600-1500 cm -1 that are assigned to amide I and amide II vibrations, respectively. Infrared absorption of resveratrol showed three characteristics intense bands (Figure 4b) at 1606 cm -1 corresponding to C-C aromatic double band stretching, 1585 cm -1 band assigned to C-C olefinic stretching and the band observed at 1381 cm -1 corresponding to a ring C-C stretching [START_REF] Popova | Preparation of resveratrol-loaded nanoporous silica materials with different structures[END_REF][START_REF] Billes | Vibrational spectroscopy of resveratrol[END_REF]. Conformational changes of the structure of the BSA upon confinement onto montmorillonite were pointed. A shift in amide I and amide II bands from 1640 cm -1 to 1643 cm -1 and from 1521 cm -1 to 1525cm -1 in Figure 4(a) were observed. This is consistent with the change in the secondary structure of the protein [START_REF] Servagent-Noinville | Conformational Changes of Bovine Serum Albumin Induced by Adsorption on Different Clay Surfaces: FTIR Analysis[END_REF][START_REF] Della Porta | RSC Advances Conformational analysis of bovine serum albumin adsorbed on halloysite nanotubes and kaolinite : a Fourier transform infrared spectroscopy study[END_REF]. Upon adsorption of RESV onto Mt, we observed a spectral shift of characteristics bands of the RESV in Figure4(b) that move from 1606 to 1608cm -1 ,from 1585 to 1589cm -1 and from 1381 to 1386 cm -1 . The significant shift of this last band corresponding to the ring C-C stretching suggests the interaction of RESV with the Mt surface [START_REF] Popova | Preparation of resveratrol-loaded nanoporous silica materials with different structures[END_REF]. The absorption bands characteristic of the BSA-RESV-Mt composite are showed in Figure 4(c). Shifts from 1640 to 1648 cm -1 and from 1521to 1524cm -1 as well as increase in the intensity of the amide I and amide II bands of protein, respectively were observed. Changes from 1585 to 1588cm -1 and from 1381 to 1388 cm -1 of the characteristics bands of resveratrol after interaction with BSA and montmorillonite were observed. This could be attributed to the presence of interactions between the protein and the polyphenol with montmorillonite or it may suggest the co-adsorption of resveratrol in the protein before interaction with the clay mineral. This information can be complemented with the fluorescence analysis. 3.5.Thermogravimetric Analyses TG curve and their corresponding derivative of the montmorillonite are reported in Figure 5 SI in Supplementary files. The curve relative to montmorillonite alone is characterized by three steps of mass loss associated to endothermal events. The first one, at 82 o C with weight loss of 8.6 %, corresponds to the release of water physically adsorbed on the surface, the weight loss of 1.4 % at 184 o C is due to the departure of the interlayer water molecules. Finally, two peaks at 428 o C and 645 o C are due to dehydroxylation with weight loss of about 3.5 % [START_REF] Xie | Thermal characterization of organically modified montmorillonite[END_REF]. Thermogravimetric analyses of BSA and RESV alone and of BSA-RESV-Mt composite were studied and reported in Figure 6 SI and Figure 7 SI in Supplementary information. DTG curve of free BSA exhibits a broad mass loss at 61 o C related to dehydratation while two peaks observed at 275 o C and 316 o C are likely due to the polypeptide chain thermal decomposition of proteins that corresponds to 55% mass loss. In addition, a second thermal decomposition step of the organic matter is observed between 450-650 o C (mass loss around of 40%) with a maximum at 493 o C and 617 o C, includes both the decomposition of the hard residues of the proteins [START_REF] Duce | Loading of halloysite nanotubes with BSA, α-Lac and β-Lg: A Fourier transform infrared spectroscopic and thermogravimetric study[END_REF]. The free RESV is thermally stable under air flow up to about260°C and its degradation shows two maximum at 310 o C and 549°C, with a total mass loss around 90%. After adsorption of the protein on Mt, an increase of the decomposition temperature for the BSA was observed, which occurs at 308°C and 537°C (mass loss 22%). This is likely due to the changes in the conformational structure of the proteins upon interactions with the clay mineral. After adsorption of RESV on Mt, an increase in thermal stability of the polyphenol was observed with a maximum decomposition temperature of polyphenol at 428 o C, with a mass loss of 14%. This may suggest intercalation or interactions of resveratrol in the external and/or internal surface of clay. The peak observed at 632°C with a mass loss of 2.6 % is attributed to matrix dehydroxylation. Thermal analysis BSA-RESV-Mt composite showed mass loss peaks shifted with respect to single Mt composites, observed in Figure 7 SI in Supplementary files. The differential peaks at 46°C with mass loss of 18 % and 219°C with mass loss of 3 % correspond to the removal of physisorbed water and to the dehydratation of interlayer water, respectively. The progressive decomposition of the organic matter, with mass loss 37%, is indicated by the peaks maximum at 316°Cand 489°C. The significant enhancement in thermal stability of the composites can be attributed to the exfoliation of silicate layers with intercalation of the protein and polyphenol complex preventing the fast decomposition of the products. The results obtained are similar to those observed in the adsorption tests. onto Mt, peaks are shifted to 178.5; 175.8 and 173.0 ppm. For the BSA-RESV-Mt composite the shift to 178.6; 175.7 and 173.7 ppm was observed. It must be noted also that resveratrol is significantly co-adsorbed with BSA, as illustrated by the corresponding intense peaks for the spectrum of the BSA-RESV-Mt composite. Together, these results indicate that an interaction between the inorganic surface and the two organic molecules probably takes place leading to a modification of the secondary structure of the protein. 3.7.Fluorescence spectroscopy The fluorescence is an interesting method to characterize the interaction between resveratrol and protein [START_REF] N′ Soukpoé-Kossi | Resveratrol Binding to Human Serum Albumin[END_REF][START_REF] Nair | Biology Spectroscopic study on the interaction of resveratrol and pterostilbene with human serum albumin[END_REF][START_REF] Jiang | Study of the interaction between trans-resveratrol and BSA by the multi-spectroscopic method[END_REF][START_REF] Balme | Structure, orientation and stability of lysozyme confined in layered materials[END_REF]. The BSA contains 2 Trp residues. Trp212 is located within a hydrophobic binding pocket of the protein and Trp134 is located on the surface of the molecule [START_REF] Bourassa | Resveratrol, genistein, and curcumin bind bovine serum albumin[END_REF][START_REF] Poklar | Interactions of different polyphenols with bovine serum albumin using fluorescence quenching and molecular docking[END_REF]. BSA and resveratrol thus exhibit different photophysical properties. The resveratrol absorbance spectrum covers the emission of the BSA. This induces a fluorescence quenching of the BSA in presence of resveratrol due to a non-radiation energy transfer [START_REF] Xiao | Probing the interaction of trans-resveratrol with bovine serum albumin: A fluorescence quenching study with tachiya model[END_REF][START_REF] Cao | Interaction between trans-resveratrol and serum albumin in aqueous solution[END_REF]. The adsorption mechanism is not totally elucidated since two assumptions have been formulated: (i) The BSA and resveratrol are adsorbed independently or (ii) the BSA binds the resveratrol. The fluorescence emission spectra of BSA-RESV-Mt composite exhibits a similar emission spectra that RESV-Mt under excitation at 290 nm. At the maximum of emission wavelength its intrinsic fluorescence and also confirm that a complex between resveratrol and BSA occurred [START_REF] Liu | Molecular Modeling and Spectroscopic Studies on the Interaction of Transresveratrol with Bovine Serum Albumin[END_REF]. Thus the most likely scenario is a binding of resveratrol by BSA before the adsorption onto montmorillonite. Conclusion The use of montmorillonite as a fining agent in winemaking adsorbed protein thus decreasing -state and time-resolved fluorescence spectra were obtained by the time-correlated single-photon counting technique. The excitation wavelength was achieved using a SuperK EXTREME laser (NKT Photonics, model EXR-15) as a continuum pulsed source combined with SuperK EXTEND-UV super continuum (NKT Photonics, model DUV); the wavelength was selected by coupling to a monochromator (Jobin-Yvon H10). The repetition rate was set to 19.4 MHz; the excitation pulse duration on this device is around 6 ps (full-width-at-halfmaximum, FWHM). The emission of fluorescence is detected, after passing through a polarizer oriented at the magic angle (54.73°) to the polarization of the excitation, through a double monochromator Jobin-Yvon DH10 on a hybrid PMT detector HPM-100-40 (Becker & Hickl).The instrumental response function of the equipment was measured by using a dilute suspension of polystyrene nanospheres in water (70 nm of diameter) as a scattering solution; it was typically about 130-160 ps FWHM. Decays were collected at a maximum counting rate of 17 kHz into 4096 channels using an acquisition card SPC-730 (Becker & Hickl). This limiting count rate was achieved by dilution in water of the sample and after sedimentation of the suspension in order to minimize as much as possible the scattering of the particles. The time per channel was set around 6 ps ch-1 in order to fit a full decay in the experimental time window. Figure 1 . 1 Figure 1. UV-Vis absorption spectra of supernatant solution during adsorption reaction of BSA (a) and RESV (b) on montmorillonite. Experimental conditions: BSA (1.0 mg.mL -1 ), RESV (1.0 mg.mL -1 ), Mt (1.0 mg.mL -1 ) in a buffer phosphate solution at pH 4.5 and ethanolic solution in ratio of 25%. 5, 22.6, 23.8 and 30.1° were observed in RESV-Mt sample and at 16.4, 22.5 and 28.5° (2θ) in the BSA-RESV-Mt. Shifts in the d(001) reflexion in the RESV-Mt sample is probably due to a different hydration state of the interlayer space. In the pattern of BSA-Mt sample, the (001) reflexion is not visible due probably to an heterogeneity in the layer stacking or a delamination of the layers[START_REF] Bertacche | Host-guest interaction study of resveratrol with natural and modified cyclodextrins[END_REF][START_REF] Popova | Preparation of resveratrol-loaded nanoporous silica materials with different structures[END_REF]. These results will be confirmed by TEM. Figure 2 . 2 Figure 2. XRD patterns diffractograms and insert at low angle 1-10 o (2θ) of the montmorillonite (black), BSA absorbed onto clay (violet), RESV absorbed onto clay (pink) and BSA-RESV-Mt composite (green). (a) and 3(b), after BSA adsorption on Mt, the measured d001 value reaches3.2 nm indicating the intercalation and partial delamination of the protein in the interlayer space of the Mt. After resveratrol adsorption, the d001value varies from 1.5 to 1.85 nm, attesting of the heterogeneous incorporation of resveratrol, in Figure 3(c) and 3(d), in agreement with the XRD results. The sample containing both resveratrol and BSA presented different populations of layers: some are exfoliated and others are intercalated with a d spacing of BSA-RESV-Mt composite varying between 2.96-3.9 nm. From these results the incorporation of the BSA and RESV together is supposed to cause exfoliation of the clay mineral shown in Figure 3(e) and 3(f). Figure 3 . 3 Figure 3. Transmission electron micrographs (TEM) of the BSA-Mt (a) and (b); RESV-Mt (c) and (d); Figure 4 . 4 Figure 4. ATR-IR spectra of (a) free BSA and upon adsorption on Mt; (b) free RESV and upon adsorption onto Mt and (c) free BSA, free RESV and BSA-RESV-Mt composite. 3. 6 . 13 CFigure 5 6135 Figure 5(a) exhibits spectra of the protein, the polyphenol and the composites prepared with the Figure 5 ( 5 Figure 5(a). 13 C CP-MAS NMR spectra with a MAS frequency of 14kHz. Free RESV and free BSA and BSA-Mt, RESV-Mt and BSA-RESV-Mt composites. Figure 5 ( 5 Figure 5(b). Deconvoluted spectra for BSA alone and upon adsorption on montmorillonite and BSA-RESV-Mt hybrid as illustrated by components attributed to carbonyl peaks. Figure 5 ( 5 Figure 5(b) exhibits the decomposition of the spectra for the BSA, BSA-Mt and BSA-RESV- Figure 6 ( 6 Figure 6(a). Fluorescence emission spectra of free BSA(black) and RESV-Mt (green), BSA-Mt(red) and BSA-RESV-Mt (blue) hybrids. Figure 6 6 Figure 6(b). Fluorescent decay of BSA-Mt (red), RESV-Mt (green), BSA-RESV-Mt (blue and violet) hybrids. ( 390 nm), the average fluorescence lifetime 2.254 ns is longer than the one of RESV-Mt (under excitation 340 nm). The emission band of BSA has disappeared meaning that the Trp fluorescence is quenched by the polyphenol. In addition, we can observe a decrease of fluorescence lifetime to 1.302 ns measured at 340 nm. The fluorescence quenching of BSA is explained by a non-radiation energy transfer. The lifetime values of BSA-RESV-Mt τ1=2.587 ns, τ2=0.680 ns, τ3=0.03 ns are different from those of BSA-Mt, suggesting that the adsorbed BSA conformation is different in presence of RESV. Both fluorescence quenching and the different fluorescence lifetime of BSA indicates that RESV can interact with protein and quench turbidity or haze formation and can modulate the polyphenol concentration of wine. The results suggest a non denaturated intercalation of the protein in the interlayer space of montmorillonite by electrostatic forces and hydrogen bonding causing delamination and/or partial exfoliation of the layers of clay mineral observed in XRD and TEM. The direct interaction of resveratrol with montmorillonite can also be observed. It can be located on the surface and the edges of the clay mineral. Extensive penetration of the protein and resveratrol together causes exfoliation of the lamellar structure of the montmorillonite. Time resolved fluorescence experiments highlight the strong interaction between resveratrol and BSA and changes in the environment of amino acid residues due to energy transfer from Trp to resveratrol. This is the first example of coadsorption of a polyphenol and protein in presence of montmorillonite. Current studies are under progress to transpose this work on a real wine solution. Table II . II Time resolved fluorescence results for free BSA and BSA-Mt, RESV-Mt and BSA-RESV-Mt hybrids. λEx(nm) λEm (nm) τ1 (Y1 %) τ2 (ns) (Y2 %) τ3 (ns) (Y3 %) τav(ns) χ 2 (ns) BSA 290 340 6.287 (88.8) 2.359 (10.7) 0.189 (0.6) 5.833 1.01 BSA-Mt 290 340 4.556 (71.8) 1.553 (24.6) 0.427 (3.7) 3.667 1.18 RESV-Mt 340 390 1.517 (33.4) 0.808 (42.4) 0.123 (24.2) 1.193 .93 Acknowledgements This is work was supported by Capes/Cofecub Project (N 835/15) as well as with participation of the Archeology Molecular and Structural Laboratory and National School of Chemistry of Montpellier, in France and Fuel and Material Laboratory and Fast Solidification Laboratory, in Brazil. Supplementary information :

01744328

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2024/03/05 22:32:07

2017

https://hal.science/hal-01744328/file/ertek_chi_zhang_2017_RFID.pdf

A Framework for Mining RFID Data From Schedule-Based Systems Gürdal Ertek, Xu Chi, Member, IEEE, and Allan N. Zhang Member, IEEE Abstract-A schedule-based system is a system that operates on or contains within a schedule of events and breaks at particular time intervals. Given RFID data from a schedule-based system, what set of actions and computations, and what type of data mining methods can be applied so that one can obtain actionable insights regarding the system and domain? The research goal of this paper is to answer this posed research question through the development of a framework that systematically produces actionable insights for a given schedule-based system. We show that through integrating appropriate data analysis methodologies as a unified framework, one can obtain many insights from even a very simple RFID dataset, which contains only very few fields. The developed framework is general, and is applicable to any schedule-based system, as long as it operates under a few basic assumptions. The types of insights are also general, and are formulated in the most abstract possible way. The applicability of the developed framework is illustrated through a case study, where real world data from a schedule-based system is analyzed using the introduced framework. Insights obtained include the profiling of entities and events, the interactions between entity and events, and the relations between events. Index Terms-Data mining, Decision support systems, Information systems. I. INTRODUCTION T HE topic of this paper is the mining of data collected through RFID from schedule-based systems. A schedulebased system is a system that operates on (or contains within) a schedule of events and breaks at particular time intervals [START_REF] Schonberger | Applications of single-card and dual-card kanban[END_REF], [START_REF] Benton | Push and Pull Production Systems[END_REF]. Figure 1 illustrates a schedule-based system, which is characterized by a set of entities (or resources) I entering and exiting a particular set of locations that have events J 1 taking place in them according to a schedule. An entity is a distinct, independent, or self-contained being. An event is something that occurs in a certain place/location during a particular interval of time. The events may take place successively or may be separated by breaks, in other words, time intervals of no events. The set of breaks is denoted by J 0 . Events and breaks constitute the set of time intervals J . The schedulebased systems that we are particularly interested in are systems where the entry and exits of entities to location(s) are recorded through a data collection system, typically barcode, RFID, GPS (Global Positioning System), or sensors. Since RFID systems are gaining increasing importance in industry, we have illustrated a schedule-based system with RFID. G. Ertek is with Rochester Institute of Technology -Dubai, Dubai Silicon Oasis, Dubai, UAE, e-mail: [email protected]. X. Chi and A.N. Zhang are with Singapore Institute of Manufacturing Technology, 71 Nanyang Dr, 638075, Singapore, e-mail: [email protected], [email protected]. Manuscript received June 19, 2014; revised December 9, 2014. Schedule-based systems are extensively encountered in a variety of domains, ranging from manufacturing to social event management. However, the basic elements of the system are the same. The basic elements are shown in bold in Figure 1. Table I lists some of the domains where schedule-based systems are present, and maps the key elements of a schedulebased system to domain-specific terminology. An RFID (Radio Frequency Identification) system consists of tags (a.k.a. transponders) and readers (a.k.a. interrogators), typically also linked to an information system [START_REF] Zhu | A review of RFID technology and its managerial applications in different industries[END_REF], [START_REF] Oztekin | An RFID network design methodology for asset tracking in healthcare[END_REF]. In passive RFID, the information on the chip of the tag is read by the reader through radio waves, and the tag cannot transmit radio waves by itself. In active RFID, the tag has its own internal power source and the capability of actively transmitting information to the reader. Passive tags have the advantage of being significantly cheaper, whereas active tags possess larger memory capacity and can be used in more sophisticated scenarios. [START_REF] Zhu | A review of RFID technology and its managerial applications in different industries[END_REF] provides an extensive review of RFID technology and its application in various industries, including logistics, retailing, travel and tourism, library science, food services and health care. A recent study reveals that only 3 percent of the companies in Europe have adopted RFID technology [START_REF] Zhu | A review of RFID technology and its managerial applications in different industries[END_REF]. Thus, only a small percentage of companies have adopted RFID technology in their operations so far. However, the commitment of leading institutions (such as the US Department of Defense) and companies (such as Walmart, JC Penney and PG) is expected to eventually spread the use of RFID, just as the barcode technology has gained acceptance over time. [START_REF] Zhu | A review of RFID technology and its managerial applications in different industries[END_REF], [START_REF] Liao | Contributions to Radio Frequency Identification (RFID) research: An assessment of SCI-, SSCI-indexed papers from 2004 to 2008[END_REF], and [START_REF] Han | Warehousing and mining massive RFID data sets[END_REF] provide a detailed discussion of RFID application domains, as well as a detailed literature review of RFID. [START_REF] Cinicioglu | Use of radio frequency identification for targeted advertising: a collaborative filtering approach using Bayesian networks[END_REF] provides a highly useful list of potential benefits of RFID systems on operations management activities, in a multitude of domains. These benefits include preventing theft and shrinkage, identifying causes of spoilage, and evaluating employees. RFID systems are used to basically produce data that can be mined through data mining methods for knowledge discovery and obtaining actionable insights. Data mining is the growing field of computer science where the goal is to uncover hidden information in -typically large and complex-piles of data [START_REF] Han | Data Mining: Concepts and Techniques[END_REF]. There exist a multitude of data mining methods that can be applied depending on the size and structure of the data at hand. Data mining can thus be considered as a field which encompasses a collection of interrelated and interacting tools, including clustering, classification, association mining, network analysis, data visualization, as well as others. A significant challenge then is the selection of the appropriate set of methodologies and the way they are applied in analyzing a particular dataset. The research question to be answered in this paper is the following: "Given RFID data from a schedule-based system in any domain (such as social event management, manufacturing, healthcare, etc.) what set of actions (including the data cleaning steps) and computations, and what type of data analysis and data mining methods can be applied, so that one can obtain actionable insights regarding the system and the domain?" The research goal to answer the above research question is the development of a framework, that takes RFID data and basic event schedule data and information, and produces actionable insights regarding the system and entities within the system. Our first main motivation was to show that, through appropriate data analysis methodologies, one can obtain many insights from even a very simple RFID dataset, which contains only very few fields. Our second main motivation was that such a framework would be applicable in a wide range of domains. Our third motivation was observing from our survey of the literature that there is a significant gap regarding this type of research. The contributions of our study are multifold: First, we introduce an analysis framework, including its mathematical representation, for mining RFID data coming from a schedulebased system. The framework developed is general, and is applicable to any schedule-based system that operates as described. While the framework is developed assuming a single location, it can also be extended to the case of multiple locations by introducing a set of locations L and a new dimension in the relevant sets and parameters. Second, we enumerate the different types of insights that can be obtained through the introduced framework. These insights are also general, and are formulated in the most abstract way possible. Third, we develop and present the corresponding algorithms that are needed in the analysis framework. The framework depends on these algorithms to do the required data processing, database augmentation, and other computations. Finally, we demonstrate the applicability of the developed framework through a case study, where real world data from a schedulebased system is analyzed using the introduced framework. The case study illustrates how the framework can be applied in the real world for a given domain. The novelty of the research is the introduction of a data mining framework for the first time for this type of a system. The existing research in schedule-based systems mainly focuses on obtaining good, and if possible optimal, schedules, or event processing. However, the interaction of the entities in the system, given the obtained schedule, has not been analyzed in depth in earlier research. The importance of the research lies in its general applicability in a wide range of domains. Table I lists some of the application areas of the developed framework, with a mapping to the domain-specific terminology. Thus the developed framework is applicable in its current form in all the listed domains, because the fundamental aspects of the model are the same across domains. The remainder of the paper is organized as follows: Section II provides a brief review of some relevant literature as the background. Section III discusses the framework developed and proposed. Section IV is devoted to the results and analysis of the case study, where new insights are obtained. Finally, Section VI presents some conclusive remarks. II. LITERATURE A. Schedule-based Systems The primary line of existing research regarding schedulebased systems involves the derivation of good, and if possible optimal, schedules. The primary modeling approach for this line of research is optimization, and typically mixed-integer programming. [START_REF] Pinedo | Scheduling: Theory, Algorithms, and Systems[END_REF] is the classic reference for scheduling theory, and [START_REF] Pinedo | Planning and Scheduling in Manufacturing and Services[END_REF] contains a detailed discussion of practice and application of scheduling, in addition to theory and algorithms. The scheduling research focuses on whether problems are polynomially solvable and optimal under certain conditions [START_REF] Yin | Single-Machine Scheduling With Job-Position-Dependent Learning and Time-Dependent Deterioration[END_REF]. Typical contribution in such research also includes optimization or approximation algorithms and analysis of worst case error bound. Scheduling can be at any resolution, ranging from single-machine machine scheduling [START_REF] Yin | Single-Machine Scheduling With Job-Position-Dependent Learning and Time-Dependent Deterioration[END_REF] to the scheduling of supply chains [START_REF] Lau | Agent-Based Modeling of Supply Chains for Distributed Scheduling[END_REF]. One line of scheduling research develops or applies machine learning and data mining methods and algorithms for generating the schedules [START_REF] Qiu | An AIS-based hybrid algorithm for static job shop scheduling problem[END_REF]. Some of these studies also analyze generated schedules using data mining techniques for coming up with new schedules [START_REF] Balasundaram | Discovering dispatching rules for job shop scheduling using data mining[END_REF]- [START_REF] Wang | Mining scheduling knowledge for job shop scheduling problem[END_REF]. However, while very extensive research exists on scheduling, the interaction of the entities in the system, given the obtained schedule, has not been analyzed from a data mining perspective in earlier research. In our research, we provide the possible practical benefits of such a perspective in Section V. One final stream of research regarding schedulebased systems is regarding the processing of the events data [START_REF] Helaoui | Recognizing interleaved and concurrent activities using qualitative and quantitative temporal relationships[END_REF]. B. Mining RFID Data There exists a large body of literature on the mining of RFID data. However, an extensive survey performed during our study revealed that none of the existing research studies have developed a comprehensive framework for mining RFID data coming from a schedule-based system. One approach could be modifying Knowledge Discovery and Data Mining (KDDM) process models [START_REF] Mariscal | A survey of data mining and knowledge discovery process models and methodologies[END_REF], [START_REF] Sharma | Evaluation of an integrated Knowledge Discovery and Data Mining process model[END_REF] for this particular domain. The most time consuming step in data mining is typically data cleaning. [START_REF] Ku | A Bayesian Inference-Based Framework for RFID Data Cleansing[END_REF] develops a framework for RFID data cleaning. [START_REF] Baba | Spatiotemporal data cleansing for indoor RFID tracking data[END_REF] presents a data cleaning methodology for indoor RFID data, eliminating temporal redundancy and spatial ambiguity, by building a distance-aware graph. The authors test and illustrate the methodology with real data from the baggage handling system of an airport. The success of a data mining process is highly dependent on the underlying data structure. To this end, [START_REF] Han | Warehousing and mining massive RFID data sets[END_REF] develops a data mining infrastructure that allows the efficient data mining of RFID data. Specifically, the authors introduce two new data models, namely path cube and workflow cube. They explain and illustrate their approach using examples and data from supply chain management. Based on our literature review, the domains where one can find the mining of RFID data are supply chain management and logistics, as well as retail. [22] presents a data processing and mining framework for logistics using RFID data. [START_REF] Ilic | Increasing supply-chain visibility with rule-based RFID data analysis[END_REF] performs a rule-based analysis and GIS-based visualization of RFID data for managing items in a supply chain. For example, consistency of velocity and waiting time has to be ensured for an item throughout the supply chain, and any anomalies have to be detected. In a similar study, [START_REF] Shuping | Geotime Visualization of RFID[END_REF] applies 3-dimensional visualization for tracking and understanding object movements through time, again enabling the discovery of irregularities. The following three studies are examples of data mining for retail RFID data: [START_REF] Miyazaki | Analysis of Residence Time in Shopping Using RFID Data -An Application of the Kernel Density Estimation to RFID[END_REF] develops a framework for the analysis of residence time in shopping, based on the mining of RFID data. [START_REF] Cinicioglu | Use of radio frequency identification for targeted advertising: a collaborative filtering approach using Bayesian networks[END_REF] and [START_REF] Fang | A novel mobile recommender system for indoor shopping[END_REF] use RFID data for targeted advertising inside a retail store. [START_REF] Sakurai | Application of the RFID Data Mining to an Apparel Field[END_REF] uses RFID data for predicting retail store sales. Studies on the mining of RFID data for other domains include the following: [START_REF] Lyu | Integrating RFID with quality assurance system -Framework and applications[END_REF] presents a framework for quality assurance, as well as two industry applications. [START_REF] Lee | A RFID-based Resource Allocation System for garment manufacturing[END_REF] mines RFID data through the integration of fuzzy logic for resource allocation in garment manufacturing, and illustrates the applicability of this approach at a company. [START_REF] Wen | An intelligent traffic management expert system with RFID technology[END_REF] presents a framework that uses RFID data for intelligent traffic management. [START_REF] Tsai | Generating touring path suggestions using time-interval sequential pattern mining[END_REF] performs sequential pattern mining of RFID data for generating tourist path suggestions. [START_REF] Sakurai | Application of the RFID Data Mining to an Apparel Field[END_REF] recommends routes for theme park visitors using real time RFID information and historical tourist behavior data. [START_REF] Meiller | Adaptive knowledgebased system for health care applications with RFID-generated information[END_REF] presents a knowledgebased system framework for healthcare using RFID data. [START_REF] Lapalu | Unsupervised Mining of Activities for Smart Home Prediction[END_REF] mines RFID data for smart home prediction. A multitude of studies investigate the outlier detection problem with RFID data. [START_REF] Hsu | RFID-Based Personalized Behavior Modeling[END_REF] carries out behavior modeling using RFID data, using clustering to detect abnormal events. [START_REF] Hsu | RFID-based human behavior modeling and anomaly detection for elderly care[END_REF] also performs behavior modeling using RFID data, detecting abnormal events in elderly care. [START_REF] Delgado | Correct behavior identification system in a Tagged World[END_REF] uses RFID data for behavior identification and anomaly detection. [START_REF] Liu | Mining frequent trajectory patterns for activity monitoring using radio frequency tag arrays[END_REF] mines frequent trajectory patterns and detects abnormal trajectories. [START_REF] Masciari | A Framework for Outlier Mining in RFID data[END_REF] presents an data mining framework that detects outlier observations in RFID data. [START_REF] Cattuto | Dynamics of person-to-person interactions from distributed RFID sensor networks[END_REF] analyzes the dynamics of person-to-person interaction networks using RFID data. Other related papers do not necessarily use data collected through RFID, but illustrate methods and case studies that can be adopted to the analysis of RFID based data. For example, [START_REF] Gao | Data Analysis on Location-Based Social Netwoks[END_REF] presents a very detailed analysis of data on location-based social networks. Some of the research questions investigated in [START_REF] Gao | Data Analysis on Location-Based Social Netwoks[END_REF] include how social connection is affected by geographical distance, how users can be clustered based on their activities, how user mobility is influenced by various factors, and how home locations of users can be predicted. [START_REF] Chen | Mining User Movement Behavior Patterns in a Mobile Service Environment[END_REF] mines matching behavioral patterns based on joining various kinds of entity characteristics in mobile communication. One final related line of research builds social recommender systems with various benefits, such as supporting the creation of new social relations [START_REF] Kazienko | Multidimensional Social Network in the Social Recommender System[END_REF]. C. Mining RFID Data from Social Events RFID technology has a great potential for facilitating and enhancing the management of social events, where humans interact with each other over time and across different locations. The case study in our paper presents the application of RFID in the context of a social event, specifically a scientific conference. [START_REF] Szomszor | Providing Enhanced Social Interaction Services for Industry Exhibitors at Large Medical Conferences[END_REF]- [START_REF] Bravo | Visualization Services in a Conference Context: An Approach by RFID Technology[END_REF] provide information system architectures for collecting data in a conference through RFID. [START_REF] Bravo | Visualization Services in a Conference Context: An Approach by RFID Technology[END_REF] also describes how this data can be used in real time for informing conference attendees and illustrates, through a detailed scenario, how the system operates. [START_REF] Hsu | Extending UML to model Web 2.0-based context-aware applications[END_REF] describes how UML (Unified Modeling Language) can be extended to model Web 2.0-based context-aware applications. The UML profile explained in [START_REF] Hsu | Extending UML to model Web 2.0-based context-aware applications[END_REF] can be used in developing the browseraccessed online services and mobile applications that can be deployed for conference management. RFID systems, when used in social events, generate timestamped location data for each of the attendees/participants of the event. This data, when combined with other data regarding the attributes of the attendees, locations and the event schedule, can generate significant insights regarding the attendees, the structure and the nature of the social network, and the event. Furthermore, the methods employed for mining social network data [START_REF] Atzmueller | Mining social media: key players, sentiments, and communities[END_REF], [START_REF] Huffaker | Group Membership and Diffusion in Virtual Worlds[END_REF] can be fused to obtain hybrid data analysis frameworks. These insights and the information systems designed around them can be used to improve the social event in better serving its intended goals [START_REF] Reinhardt | Awareness-support in scientific events with SETapp[END_REF]. Improved conference management information systems and managerial practices can enable the attendees find sessions and other people that they would be interested in, minimize schedule conflicts, increase participation in the sessions, and improve the overall quality of the event. [50] integrates RFID data with online data from social networks (e.g. Facebook, Twitter) and offline data from earlier conferences, and develops an ubiquitous conference management system. The system generates context-aware recommendations to conference attendees, significantly increasing attendees' satisfaction with the event. [43], [START_REF] Chin | Using proximity and homophily to connect conference attendees in a mobile social network[END_REF], and [START_REF] Atzmueller | Face-toface contacts during a conference: Communities, roles, and key players[END_REF] are the most related studies in the literature to our case study, because these papers carry out posterior visualization and analysis of RFID enriched event data, and furthermore give examples of insight-generating questions whose answers can be obtained through querying the data. [43] develops an infrastructure and a scalable information system for tracking and analyzing human face-to-face (f2f) contact networks, such as people in a scientific conference. The authors employ RFID technology and data reporting and analysis methods for enhancing social interactions between event attendees and industry exhibitors. [44] develops an RFID based system for connecting conference attendees based on their locations, the sessions that they have attended, and the attendees they have interacted with. Posterior analysis of attendee behavior suggested that earlier physical encounter during the conference (proximity), as well as commonality of attributes (homophily) were the most important factors affecting the selection of new contacts. [51] also presents an information system for conference management and detailed analysis of the obtained f2f data, as in [START_REF] Szomszor | Providing Enhanced Social Interaction Services for Industry Exhibitors at Large Medical Conferences[END_REF] and [START_REF] Chin | Using proximity and homophily to connect conference attendees in a mobile social network[END_REF]. The main contribution of [START_REF] Atzmueller | Face-toface contacts during a conference: Communities, roles, and key players[END_REF] is the comprehensive evaluation of the behavioral patterns in a conference setting, developing analysis techniques for revealing roles of the attendees and attendee communities. Explicit and organizing roles are discovered through the analysis of classic centrality measures used in graph theory, such as degree, strength, betweenness, closeness, and eigenvalue centrality. While [START_REF] Szomszor | Providing Enhanced Social Interaction Services for Industry Exhibitors at Large Medical Conferences[END_REF], [START_REF] Chin | Using proximity and homophily to connect conference attendees in a mobile social network[END_REF], and [START_REF] Atzmueller | Face-toface contacts during a conference: Communities, roles, and key players[END_REF] bring fresh perspectives to the mining of RFID data, our work has several additional aspects in comparison to these studies: First, we develop a complete framework that exhaustively explores and exhibits all the possible types of insights, rather than a set of selected few insights. Second, our framework requires a very basic data, with very few attributes, collected by almost every RFID system by default. Third, our framework is described not only conceptually, but also through rigorous mathematical formalism. The algorithms used for data processing are also included in the work. Fourth, rather than discussing a single domain, we generalize the analysis to schedule-based systems, which can include a very rich collection of application domains. Fifth, we discuss the practical implications of our research for not only a single domain (ex: social event management), but for a multitude of domains. III. FRAMEWORK In this section, we describe the framework that we introduce for mining RFID data from schedule-based systems. First we outline the research steps followed in the study. Then we list our assumptions regarding the analyzed system. Third, we introduce the mathematical notation and the database structures in the various stages of the analysis framework. Fourth, we describe the computational algorithms for augmenting the RFID data obtained from a schedule-based system. Finally, we present the novel analysis framework that we have developed, and list the types of insights that can be obtained through this framework. A. Research Steps Our study consists of the steps listed below, and resulted in the framework and case study presented in this paper. We thus suggest the application of similar steps in analyzing the RFID data coming from a system with particular characteristics. 1) Understanding of the data mining research goal, as well as the research question and the domain. 2) Development of a mathematical notation (example: sets, parameters,...) 3) Description of the RFID data and the domain-related data in terms of the developed mathematical notation (example: entities, entity entrance times to events) 4) Identification of the metrics to be computed (example: whether an attendee has attended a particular session or not, as well as the time s/he spent in each session), and the database structures needed. 5) Development of formulas for obtaining the desired performance metrics and insights. 6) Identification some of the possible types of data analysis that can be implemented, as well as some of the possible types of insights that can be obtained through each type of data analysis. 7) Survey of the literature for related studies and recording the types of analysis they present, which can be adopted. 8) Execution of the data analysis process, and the discovery of various types of insights. 9) Elicitation of the obtained results and insights, and the subsequent filtering of the most essential and actionable insights among those obtained. The importance and actionability of insights were decided upon through discussion sessions with conference organizers from academia. 10) Integration of the executed data mining processes in a single unified framework, and proposing it as a general methodology for the analysis of RFID data from schedule-based systems, that can be applied to systems other than schedule-based ones. B. Assumptions Our assumptions regarding the RFID data collection are as follows: 1) The gateway where the RFID reader is located is an in-out-gateway [START_REF] Gonzalez | Modeling massive RFID data sets: a gateway-based movement graph approach[END_REF]. 2) RFID tags are read throughout the event schedule, not missing any of the events, nor people passing through the doors. 3) All passes (entries and exits) made with an RFID tag are read, with the RFID receiver not missing any passes. 4) RFID readings and the final data are accurate. 5) Every entity wears RFID during passes, except when the RFID tag is left in the location, never to be worn again. 6) All events happen in one location. These assumptions (except the last) are required so that the data is accurate and complete. The last assumption is assumed so that the concepts and the developed framework can be easily demonstrated. C. Mathematical Notation We now introduce the mathematical notation that will be used throughout the description of the framework. While the indices are always provided in the notation, for convenience, sometimes the indices are dropped (for example, u) . In that case, the symbol refers to the symbol with the default indices that were specified when the notation was initially introduced (for example, u refers to u ir , because that is how it is defined initially). The database structures and algorithms will also be introduced in this subsection. Sets R : set of unique record IDs ; r : 1 • • • R I : set of entities ; i : 1 • • • I J : set of time intervals ; j : 0, 1 • • • J (The time intervals correspond to actual events and the breaks between these events); J = J 0 ∪ J 1 . J 0 : set of breaks J 1 : set of events Given Data u ir : entry time of entity i in record r U ir : exit time of entity i in record r D 0 : the database of RFID logs; D 0 = d 0 : r, i, u ir , U ir Event Schedule Data s j : start time of time interval j f j : finish time of time interval j d j : duration of time interval j; d j = f j -s j D : the database of time intervals; D = {d : j, intervalT ype(j), s j , f j , d j } where intervalT ype(j) is a lookup function (defined next) that returns whether the time interval corresponds to an event or a break. Lookup Functions intervalT ype (j) =    break, if j ∈ J 0 event, if j ∈ J 1 null, o/w intervalOf (t) = {j ∈ J : s j ≤ t ≤ f j } Intermediary Data u = u ir : entry time of an entity in a record U = U ir : exit time of an entity in a record e = e irj : entry time of an entity to an event in a record x = x irj : exit time of an entity from an event in a record T = T irj = x -e: time spent by entity at an event (in a single record) p = p irj : start time of an entity present at the location for an event in a record (The entity may wait for the event.) q = q irj : end time of an entity present at the location for an event in a record (The entity may be spending additional time at the location after the event is completed.) Computed Metrics p ij : earliest start time of an entity present at the location for an event; p ij = min r∈R p irj . q ij : latest end time of an entity present at the location for an event; q ij = max r∈R q irj . earliness = earliness irj : how early entity i entered event j in a given record r; takes positive value if entity entered early, and takes negative value if entity entered late; earliness irj = s j -p irj . lateness = lateness irj : how late entity i exited from event j in a given record r; takes positive value if entity exited late, and takes negative value if entity exited early; lateness irj = q irj -f j . earliness = earliness ij : how early entity i entered event j; takes positive value if entity entered early, and takes negative value if entity entered late; earliness ij = max r∈R earliness irj . lateness = lateness ij : how late entity i exited event j; takes positive value if entity exited late, and takes negative value if entity exited early; lateness ij = max r∈R lateness irj . entryStatus =        N oEntry, if earliness = null and not an entry from previous event EarlyEntry, if earliness ≥ 0 and not an entry from previous event LateEntry, if earliness < 0 and not an entry from previous event EntryF romP reviousEvent, if entry from previous event exitStatus =        N oExit, if lateness = null and not an exit into next event EarlyExit, if lateness ≥ 0 and not an exit into next event LateExit, if lateness < 0 and not an exit into next event ExitIntoN extEvent, if exit into next event Z ij = 1, if attendee i attended event j, i ∈ I, j ∈ J 0, o/w n ij : number of times that entity i has entered and/or exited event j T ij : total time (stay duration) that entity i spent in event j; T ij = r∈R T irj . Databases The databases whose structures are given here are shown as cylinders in Figure 2. For example, cylinder with the label 0 refers to D 0 and the cylinder with the label 1 refers to D 1 . The database structures for the Raw RFID Database and the Joined Database 1 are D 0 = d 0 : r, i, u ir , U ir D 1 = d 1 : r, i, u ir , U ir , j 1 , j 2 , ε 1 , ε 2 , s j1 , s j2 , f j1 , f j2 The Augmented Database is D 2 ={d 2 : i, j, r, u, U, e, x, t, p, q, earliness , lateness , entryStatus, exitStatus , j ∈ J 1 } Entity-Event Profile database and the databases derived from that database are D 3 = d 3 : i, j, earliness , j ∈ J 1 D 4 = d 4 : i, j, Γ 1 (earliness) , j ∈ J 1 where Γ 1 (earliness) =    N oEntry, if earliness = null EarlyEntry, if earliness ≥ 0 LateEntry, if earliness < 0 D 5 = d 5 : i, j, Γ 2 (earliness) , j ∈ J 1 where Γ 2 (earliness) = N oEntry, if earliness = null Entry, o/w Entity Profile database is D 6 ={d 6 : i, avg(t), avg j (earliness), min j (earliness), max j (earliness), stdev j (earliness), avg j (lateness), min j (lateness), max j (lateness), stdev j (lateness), count j (i) , j ∈ J 1 } where the computations for D 6 are done using D 2 . The remaining databases are D 7 = d 7 : i 1 , i 2 , i 1 , i 2 ∈ I D 8 = d 8 : i 1 , i 2 , count(i 1 , i 2 ) , i 1 , i 2 ∈ I D 9 = d 6 ∪ metrics(i), i ∈ I , d 6 ∈ D 6 where I is the set of entities in D 8 which have a support count greater than the minimum support count threshold, and metrics(i) is a function that returns the array of computed graph metrics for an item i. D. Computational Algorithms for Data Augmentation The computational algorithms for augmenting the RFID data are given in Appendix A. The first of these algorithms takes the raw RFID data D 0 and the schedule data, and joins these two tables to form a new data table, namely D 1 . The second algorithm is more complicated, and is focused only in what is happening with respect to events (rather than breaks). This second algorithm transforms the data which is in the form of entry/exit records into a database D 2 that contains information only on entities and events. The augmented database D 2 contains the entry and exit times of entities to events, as well as their earliness (positive value if early entry), lateness (positive value if late exit), as well as other data. D 2 is critical, because it is used in later stages of the analysis framework to extract new databases and to obtain insights. E. Analysis Framework The developed analysis framework is given as a flowchart in Figure 2. The framework starts with raw data coming from RFID system, as well as data regarding the schedule of events in the system. The data is then brought to a richness so that it can be analyzed to obtain insights. The analysis centers around three lines; shown with the numbers 3, 6, and 7 in the figure. The insights are obtained through analyzing entities, events, entity-event interactions, and entity-entity interactions. Figure 2 shows that the analysis begins by joining the raw RFID data D 0 (shown with the cylinder with the label 0) with the event schedule data to form D 1 , and then augmenting D 1 to generate D 2 . Next, three basic types of data are obtained: D 3 , Entity-Event Profile Data is obtained through pivoting on D 2 , and shows the earliness of each entity for each event. Some of the values are missing, indicating that the entity did not enter the system at all during a particular event. D 6 , Entity Profile Data shows the metric statistics for each entity as computed over all the sessions. D 7 , Entity-Entity Interaction Data lists the entity pairs that have entered or exited the system simultaneously. The data is obtained through running a pattern matching algorithm (Appendix B). Having obtained these three basic types of data, further data transformations and/or algorithms are applied to obtain Insight No Question Answered Example in Behavior Analytics 1 Based on the behavioral patterns of the entities with respect to specific events... 1 • Which entities are positioned close to each other? Figure 3 2 • Which groups of entities behave most similar? 3 • Which entities can be clustered into which clusters? 4 • What are the profiles of these entity clusters? Event Analytics Based on the behavioral patterns of the entities with respect to specific events... 5 • Which events are similar to each other? 6 • Which groups of events are most similar? Figure 4 7 • Which events can be clustered into which clusters? 8 • What are the profiles of these event clusters? 9 • What is the correlation between different events? Table III 10 • Which earlier events affect a particular event, and how? Figure 5 11 • Can the entry of specific entities to an event be predicted? Table IV Behavior Analytics 2 Based on the general behavioral patterns of the entities... 12 • Which entities are positioned close to each other? Figure 6 13 • Which groups of entities behave most similar? Figure 7 14 • Which entities can be clustered into which clusters? 15 • What are the profiles of these entity clusters? Figure 8 Relationship Network Analysis Based on the joint actions of the entities... 16 • Which entity pairs enter/exit many events together? Table V 17 • How are the entities related to each other? Figure 9 18 • Which entities are influencers and which are followers? Table VI 19 • How can the behavioral attributes be analyzed Figure 10 together with the relationship network? 20 • How can the behavioral attributes be analyzed together with the graph metrics? TABLE II INSIGHTS THAT CAN BE OBTAINED THROUGH THE INTRODUCED ANALYSIS FRAMEWORK insights into the system. These insight types are numbered from 1 to 20 in Figure 2, inside the circles. These 20 insight types are then listed in Table II. In Table II, the insights that can be obtained using our proposed framework have been classified into categories of behavior analytics, event analytics, and relationship network analysis. The behavior analytics category has been further labeled as 1 or 2 based on the analytics quest and the data source. Table II also lists (in its last column) the figure and/or tables which illustrate the insight type in the case study. For example, consider the line corresponding to Insight 2 in Table II. This insight aims at answering the question "Which entities are positioned close to each other?". The same line in the table tells that an example of the analysis that leads to this type of insight is illustrated in Figure 3. Due to space limitations in the paper, we are able to provide examples for only some of the insights, hence the empty cells under the last column of the table. IV. CASE STUDY In this section we firstly describe the data used in the case study, and then illustrate the various insights that can be obtained through the presented framework. The insights that will be presented are listed in Table II, along with the figure and table numbers. The data mining processes applied are described in Appendix C, and the software tools used are described in Appendix D. A. Data The data used in the study belongs to the domain of social event management, and comes from a four-day medical conference. Each attendee of the conference was provided with a unique RFID tag and their entry and exit times to the single conference hall were recorded. The raw RFID data D 0 consists of 9624 rows (entry-exit combinations), and four columns, where the columns are the record ID, attendee name (masked), entry date and time, and exit date and time. The total number of attendees (total number of entities in the schedule-based system) is 272. The schedule consists of 17 events and 11 breaks (including the time interval before the first event and the time interval after the last event) giving a total of 28 time intervals. B. Behavior Analytics 1: Based on Entity-Event Data The first illustration is for Insight 1, and is given in Figure 3. This insight answers the question "Which entities are positioned close to each other?". The analysis here is based on temporal proximity [START_REF] Wang | Efficient mining of group patterns from user movement data[END_REF] of entities. The data mining method used for this purpose is Multi-Dimensional Scaling (MDS) (as read from the box that corresponds to Insight 1 circle in Figure 2), which maps multi-dimensional data onto two dimensions, based on how close the data points are [START_REF] Borg | Applied Multidimensional Scaling[END_REF]. Figure 3 shows the mapping of attendees (entities) on a two-dimensional plane. The most significant associations are shown with lines between the points. Since Insight 1 is using the database D 4 (EarlyEntry/LateEntry/NoEntry data), the closeness of the points, as well as the links between them, are based on the Hamming distance in between. Hamming distance is a distance measure that computes the number of bits two strings are different from each other [START_REF] Bose | Information Theory, Coding and Cryptography[END_REF]. As an example, if two entities entered all events early, but differed only in their behavior with respect to one event (for example one entered early, and the other entered late into the last event), the Hamming distance between them would be 1. The Hamming distance measure has been selected, rather than other distance measures, because it is a very popular distance measure when the data points are binary vectors. One can observe a highly dense region in Figure 3, to the right of the figure, as well as a less dense region in the middle of the figure, and some sparse links. This shows that the entities in the dense cluster are very much close to each other, whereas there are other closely positioned entities among the remaining entities. Furthermore, given a particular entity, one can find the other entities closely positioned to this entity from the figure. C. Event Analytics Based on Entity-Event Data The next illustration is for Insight 6, and is given in Figure 4. This insight answers the question "Which groups of events are most similar?". The data mining method used for this purpose (as read from Figure 2) is hierarchical clustering, which hierarchically builds clusters from data, starting from individual points ( [56], p44). An interesting point here is that the data points are not the entities, but rather the events (sessions in the case study). So the goal is to see which events are similar to each other, based on the entry-exit patterns of the entities. This analysis also uses database D 4 (EarlyEntry/LateEntry/NoEntry data), however, carries hierarchical clustering of events (rather than the entities), based on the Hamming distances between the events. The dendrogram in Figure 4 shows that events {S2, S4, S6} are similar to each other, based on the entityevent profile data. Similarly, {S18, S24, S25, S26} are very similar, since they form a cluster together. Other groups of similar events are {S20, S21, S22}, {S8, S10}, {S12, S13}, and {S15, S16}. The next illustration is for Insight 9, and is given in Table III. This insight answers the question "What is the correlation between different events?". The data mining / statistics method used for this purpose is correlation analysis, which computes the linear association between pairs of observations [START_REF] Rodgers | Thirteen ways to look at the correlation coefficient[END_REF]. Again, the observations are events (rather than entities). Table III shows the correlation values above 0.50 and below -0.50. High correlation between successive events indicates that the entities which entered the former of those successive events also mostly entered the latter, or those who did not enter the former did not enter the latter. This is the case for session pairs (S25, S26), (S20, S21), (S2, S4), (S26, S27), and (S21, S22). This high correlation can indicate that the former event encouraged entry to the latter, that the two events catered to the same set of entities, or both of these. Negative correlation between two successive events is also an important observation, and may be due to one or combination of several reasons: First, it may be that the former event was (not) successful, en(dis)couraging entry to the latter. Second, the two events may be catering to different set of entities. Third, there may be another reason, such as the latter event being the last event of the day, and entities exiting the system early. The successive event pairs with high negative correlation are (S8, S10), (S15, S16), and (S24, S25). The following illustration is for Insight 10, and is given in Figure 5. This insight answers the question "Which earlier events affect a particular event, and how?". The data mining method used for this purpose is classification tree analysis [START_REF] Rokach | Data mining with decision trees: theory and applications[END_REF]. Classification trees summarize rule-based information about classification as trees. In classification tree models, each node is split (branched) according to a criterion. Then, a tree is constructed with a depth until all the rules are displayed on the graph under a stopping criterion. At each level, the attribute that creates the most increase compared with the previous level is observed. The algorithms for classification tree analysis are explained in [START_REF] Rokach | Data mining with decision trees: theory and applications[END_REF]. In the implementation that we utilized in our analysis, selecting the attributes for the splits is based on information gain. In classification trees, identifying the nodes that differ noticeably from the root node are important, because the path that leads to those nodes tells us how significant changes are observed in the sub-sample compared with the complete data. By observing the shares of slices and comparing with the parent and root nodes, one can discover interesting classification rules and insights. Figure 5 shows the classification tree where the Entry/NoEntry into event S27 (last session in the case study) is the predicted attribute. The very first split, based on the value of S26 provides the most information. In the complete data, 80.5% of the entities did not participate in event S27 (light shaded slice). However, among those entities that did not enter S26 at all (NoEntry), this percentage is 96.2%. On the other hand, among the entities that did enter S26, the percentage of Entry into S27 is higher (55.6%). So, approximately half (55.6%) of those entities that entered S26 entered S27, whereas almost all (96.2%) of the entities that skipped S26 also skipped S27. This connectedness between S26 and S27 could also be hypothesized based on Table III, which shows a correlation of 0.60 between these sessions. However, the classification tree analysis provides us with specific percentages of Entry/NoEntry for S27 based on the values of S26. The following illustration is for Insight 11, and is given in Table IV. This insight answers the question "Can the entry of specific entities to an event be predicted?". The data mining method used for this purpose is classification analysis. In classification analysis, the dataset is divided into two groups, namely, learning dataset and test dataset. Classification algorithms, also called classifiers (or learners), use the learning dataset to learn from data and predict the class attributes in the test dataset ( [59], p17). The prediction success of each learner is measured through classification accuracy (CA) [START_REF] Brodley | Identifying Mislabeled Training Data[END_REF], the percentage of correct predictions among all, as well as receiver operating characteristic (ROC) curves [START_REF] Fawcett | ROC Graphs: Notes and Practical Considerations for Researchers[END_REF]. Classifiers which result in higher CA and a greater area under the ROC curve (AUC) correspond to better predictive models. The following classification algorithms are among the bestknown classifiers in the machine learning field, and have been used in our analysis: CN2, k-Nearest Neighbor (kNN), Classification Tree, Support Vector Machines (SVM), Naive Bayes, and Neural Networks [START_REF] Alpaydin | Introduction to Machine Learning[END_REF]. Firstly, the entries of entities into event S27 are predicted with a very small learning dataset of 50% (around 130 observations), with 100 experimental repeats (using percentage split of the full dataset into learning and testing datasets). The CA and AUC values are displayed in Table IV, showing that if the behavior of half of the entities for S27 are known, the remaining entry or no entries can be predicted with a very high accuracy, up to 83.15%, with neural network classifier. Besides the black box neural networks technique, which does not tell the reasoning behind classification, CN2 and classification tree might be considered, since they provide the classification rules openly. D. Behavior Analytics 2: Based on Entity Profile Data The next illustration is for Insight 12, and is given in Figure 6. This insight answers the question "Which entities are positioned close to each other?". While the question Fig. 6. MDS results for entity analysis based on entity profile data Fig. 7. Hierarchical clustering results for entity analysis based on entity profile data answered is the same as that of Insight 1, the way it is answered is different. In Insight 1, the answer was computed based on entity-event data, whereas this time it is computed based on entity profile data. The data mining method used for this purpose is again Multi-Dimensional Scaling (MDS). Figure 6 shows the mapping of attendees (entities) on a two-dimensional plane. The most significant associations are shown with lines between the points. Insight 12 is using the database D 6 , which contains only numerical values. Hence, the closeness of the points, as well as the links between them, are based on the Euclidean distance in between. One can observe a highly dense region in Figure 6, to the middle of the figure, as well as a less dense region to the left of the figure, and some sparse links. This means that the entities in the dense cluster are very much close to each other, whereas there are other closely positioned entities among the remaining entities. The results of Insight 12 are different than that of Insight 1, since both the values in the database and the distance measure used are different. This illustrates that one should use the appropriate dataset (and the associated distance measure) that is aligned with the goals of the analysis. The next illustration is for Insight 13, and is given in Figure 7. This insight answers the question "Which groups of entities behave most similar?". The data mining method Fig. 8. Cluster profiles for entity analysis based on entity profile data used for this purpose is hierarchical clustering, just as in Insight 6. The data points this time are entities, rather than events. So the goal is to see which entities are similar to each other, based on their overall behavior patterns, particularly their entry and exit timings. This analysis uses database D 6 , and carries hierarchical clustering of entities based on the Euclidean distance between them. The dendrogram in Figure 7 shows that entity groups {A132, A188}, {A030, A177}, {A043, A122, A169}, {A179, A262} are similar to each other, based on the entity profile data. When partitional clustering is carried out, the entities are partitioned into distinct clusters. One of the analysis to be done given these clusters is to profile the clusters using exploratory data visualization. This cluster profiling constitutes Insight 15, and an illustration of this insight is given in Figure 8. This insight answers the question "What are the profiles of these entity clusters?". Here, the clusters are again based on numerical data coming from the database D 6 . Figure 8 profiles the clusters based on three attributes, namely average stay duration, average earliness, and average lateness, and also provides the number of entities in each cluster. The 45 entities in the first cluster C1 have the highest average stay duration (56.31 minutes), and have the enter and exit events (sessions in the case study) almost with a perfect timing, neither early nor late. The 23 entities in the next cluster, C8, also stay in the events for a long time (average of 45.97 minutes), and exit with almost no earliness or lateness, but arrive an average of 15.45 minutes late. Each cluster has a profile, that can be similarly read from the figure. For example, at the other extreme, the last cluster, C2, consists of 16 entities who stay the least in the events (average of 29.94 minutes) and enter the events very late (average of 29.59 minutes). A particularly interesting cluster is C10. The 9 entities in cluster C110 stay for a long time in the events, and arrive almost on time, but they stay for a long time (average of 28.71 minutes) after the event is over. In the case study, this can be referring to the after-session discussions participated by these entities. E. Relationship Network Analysis Based on Entity-Entity Interaction Data The next illustration is for Insight 16, and is given in Table V. This insight answers the question "Which entity pairs enter/exit many events together?". The data mining method used for this purpose is association mining [START_REF] Agrawal | Mining association rules between sets of items in large databases[END_REF], [START_REF] Agrawal | Fast algorithms for mining association rules[END_REF]. The (where an entity pair appears as many times as they are seen together). Then, association mining is carried out to compute the entity pairs that appear together frequently, and this information is populated into database D 8 . Association mining provides us with the frequent itemsets, namely itemsets that appear together frequently. Only the itemsets that appear at least "minimum support (count) threshold" times are mined and listed. Table V gives a snapshot of D 8 for our case study, where the minimum support threshold is given in terms of support count (absolute threshold) as 6. So, only the entity pairs that appear together at least 6 times are selected in the association mining analysis and for further analysis. From Table V we can observe that entities {A150, A161} have entered and/or exited together 39 times, which is more than the number of events. This means that they entered and exited together many times during the events, as well, revealing a social connection between these two entities. Other entity pairs with the "strongest" social connection include {A164, A150} and {A009, A150}. It should be noticed here that A150 appears frequently with A161, A164, and A009. So A150 is among the most influential entities. The analysis of "influencer" entities will be extended later in the illustration of Insight 18. The next illustration is for Insight 17, and is given in Figure 9. This insight answers the question "How are the entities related to each other?". The identified relationship networks can be used for personalization and generating recommendations for human entities [START_REF] Atzmueller | Face-toface contacts during a conference: Communities, roles, and key players[END_REF]. The data mining methods used for this purpose are network visualization and analysis [START_REF] Herman | Graph visualization and navigation in information visualization: A survey[END_REF]- [START_REF] Opsahl | Node centrality in weighted networks: Generalizing degree and shortest paths[END_REF]. Figure 9 provides a grid visualization of the 163 entities that appear in D 8 . Each circular node represents an entity; the area of each circle represents the support count (number of times the entity is observed in D 8 ); arcs between nodes represent an association between two entities. The visualization is constructed so as to minimize arc crossings. The entities in the upper region of the visualization are those that appear in many interactions. Among those that appear in many interactions, those with smaller area are even more interesting, since we can be inclined in thinking that their interactions were not due to frequent entry-exits, but rather due to interactions with other entities. Furthermore, by visual querying, it is possible to observe how each entity is related to each other entity. In Figure 9, a particular node (entity) is selected and all the associations that it has are highlighted. Attendee1 Attendee2 Support Count A150 A161 39 A164 A150 31 A009 A150 29 • • • • • • • • • A055 A230 6 A249 A230 6 A249 A126 6 Another way of characterizing the nodes (entities) is to compute their graph metrics. One of these metrics is degree, which denotes the number of connections for each node. It is an integer value, and it is the summation of in degrees and out degrees of the node. such metric is betweenness centrality, which represents the total number of shortest paths for each pair of nodes, if the node is on that path (it can take values between 0 and 1). Detailed information on this and other graph metrics can be found in [START_REF] Christensen | Using graph concepts to understand the organizatýon of complex systems[END_REF] and [START_REF] Opsahl | Node centrality in weighted networks: Generalizing degree and shortest paths[END_REF]. The next illustration is for Insight 18, and is given in Table VI. This insight answers the question "Which entities are influencers and which are followers?". The data mining method used for this purpose is network analysis, and specifically network characterization through computing node metrics [START_REF] Christensen | Using graph concepts to understand the organizatýon of complex systems[END_REF], [START_REF] Opsahl | Node centrality in weighted networks: Generalizing degree and shortest paths[END_REF]. Table VI lists the most active and influential entities (A161 through A119), as well as some of the least associated ones (which appear only 6 times with other entities). The most influential entities have high value for betweenness centrality, indicating that they are at the "crossroads" of social networks. The next illustration is for Insight 19, and is given in Figure 10. This insight answers the question "How can the behavioral Fig. 10. Results for association mining analysis with Harel-Koren layout algorithm attributes be analyzed together with the social network?". This analysis is specifically aimed at the scenarios where the entities are humans. The data mining method used for this purpose is network visualization, where some behavioral attributes are mapped onto the nodes. The network in Figure 10 is exactly the same as that in Figure 9 with respect to node-link structure. However, the selected layout algorithm is different (Harel-Koren algorithm; [START_REF] Harel | Graph drawing by high-dimensional embedding[END_REF]), and nodes are colored according to average stay duration (a behavioral attribute). Lighter colors denote longer average stay durations. Size again denotes the support count of the node. The visualization in Figure 10 is constructed using D 9 , which is obtained by joining the two different databases of D 6 (entity profile data) and D 8 (association graph). The nodes in the center are influencers and the ones on the outside are followers. However, we are now also able to see which influencers stay in the events for long, and which stay less. Thus, we are able to see the influencers that enhance our desired goals (longer stay durations in sessions with less frequent entry-exits, in our case study) versus the influencers that disrupt the system (by staying very short in sessions and entering and exiting many times). So we are not only able to identify the social network and the influence the entities have on the network, but also the direction of the effects (positive or negative) that they have. V. PRACTICAL IMPLICATIONS In this section, we discuss how the insights and information obtained through our analysis can be used in a multitude of ways, for improving the system and achieving various goals. First, information regarding similar-behaving entities in a schedule-based system can be used in several ways: • In the context of social event management, ubiquitous information systems can use this information to suggest new people for professional social networks. For example, in a conference, when two attendees are identified as entering and exiting similar events, the conference mobile application can recommend them each other to add into LinkedIn and other professional social networks. Another use of the information is the suggestion of events to social event attendees in which similar-behaving attendees have already entered. • In the context of manufacturing, if one of the entities has entered the production system, a-priori planning can be done for similar-behaving entities. Furthermore, the production system can be set up to accommodate not only the already entered entity, but also those that may potentially enter, in a way to reduce total setup time. • In the context of warehousing, an example scenario where the information on similar-behaving entities can be used is the following: Consider pallets of similarbehaving products entering the warehouse. There is a high chance that they will also exit together. Therefore, the warehouse management system (WMS) software can be programmed so as to allocate neighboring locations for these two pallets, so that they can be put away and picked on the same route, saving time and cost. • In the context of healthcare, similar-behaving entities can be equipment used for surgical operations. In this scenario, these equipment can be stored in the same storage room when not in use. This way, they can be accessed in the least possible time when an urgent surgical operation is to be conducted. • In the context of education, similar-behaving students can be identified automatically based on their entries and exits to classrooms. Then this information is populated into the school's information system databases. In case a student can't be reached by phone, the school management or the instructors can try to reach him through contacting his friend. • Finally, in the context of tourism, visitors in a museum can be offered special places of interest in the museum (visited by similar-behaving visitors) through the smart mobile devices that are guiding them. Information regarding groups of similar events in a schedule-based system can also be used in a multitude of ways for improving the system and achieving various goals. One example use case from social event management is the joint design and improvement of similar sessions in the social event. The session managers can come together and discuss possible opportunities of improving the similar sessions in future conferences. Information regarding the correlation between events can be used for improving schedules. For example, in the context of manufacturing, successive production periods with negative correlation may experience great changes in the product mix entering the production. These can also be sources of long setup times and costs. Therefore, schedule can be adjusted based on the results of data mining. Information regarding earlier events affecting later events, as well the predictability of entry of specific entities to an event, can also be used for benefiting the system. Consider a warehousing scenario where the entities are the various types of products loaded on pallets. Based on the past behaviors of these products, one can estimate for each product the probability of entering a particular warehouse zone during a particular time interval (event). This can be used to predict whether capacity will be exceeded in that zone of the warehouse during that time interval. Then, if necessary, additional capacity can be created, for example, through establishing temporary additions to that zone using pallet stacking frames. Finally, the insights regarding influencers in the system can be utilized in many ways. For example, in social event management, once these influencers are determined, they can be consulted for help in promoting newly established sessions or for increasing membership to the organizing society. VI. CONCLUSIONS AND FUTURE RESEARCH The importance of RFID systems for data collection and processing is ever increasing. RFID systems find applications in a very wide range of domains, including in schedulebased systems, which operate based on (or contain within) a schedule of events. In this paper, we have presented a comprehensive framework for mining of RFID data coming from schedule-based systems, for the first time in the literature. Our framework is generic, and can be applied to any schedulebased system that operates as described. There exists two very fundamental future research avenues for extending the current work: • RFID tags can collect and/or carry not only location and time information, but other information, as well. Such information typically includes entity type, entity affliation, physical attributes, and assigned attributes. In a logistics context, examples of these attributes are product type, manufacturer, weight, and price [START_REF] Gonzalez | Modeling massive RFID data sets: a gateway-based movement graph approach[END_REF]. The additional information may also be collected through various sensors (e.g. temperature, GPS) integrated within or mounted on the tags. While the framework that we have presented here considers only time data in relation to schedule data, it can be highly enriched with the incorporation of analysis of these additional attributes. For example, scheduling and plans are important in manufacturing context, and are very much dependent on quality level achieved in the production process. Quality-related attributes can be analyzed together with product attributes and schedule data to improve the production process in the dimensions of quality, time, and cost. To this end, the re-mining framework of [START_REF] Demiriz | Re-mining item associations: Methodology and a case study in apparel retailing[END_REF] can be integrated with the framework here to augment the data and to discover further insights. • The other fundamental research avenue is extending the framework from the temporal domain to the spatiotemporal domain, by extending it to handle multiple locations. • While the analysis of serial events is fundamental, consideration can be made in future research for concurrent events (events that can independently take place at the same time) in the system. The consideration for concurrent events would require significant changes in the augmentation algorithm, as it would require complex event processing [START_REF] Helaoui | Recognizing interleaved and concurrent activities using qualitative and quantitative temporal relationships[END_REF]. However, the applicability of the current framework, as well as the types of analysis and the insights obtained, would still be relevant and useful. • One of the important challenges in industrial applications is the challenge of big data [START_REF] Mayer-Schönberger | Big Data: A Revolution That Will Transform How We Live, Work, and Think[END_REF]. A possible future research can involve the development of the framework to accommodate for big data applications. To support the large volumes of input data, when the proposed framework is implemented, the data processing of this framework should be split into independent tasks to support parallel processing systems such as MapReduce. As indicated by Figure 2 in the manuscript, our framework has very few interactions between different branches of data flows and thus splitting the overall data processing into multiple tasks is possible and can be highly feasible. The methods for MapReduce implementation of the individual data mining and data visualization algorithms used in this manuscript, such as hierarchical clustering, can be found in the literature [START_REF] Sun | An efficient hierarchical clustering method for large datasets with Map-Reduce[END_REF]. Hence, the proposed framework can support the big data environment if its implementation is properly designed. Other possible future research avenues include the following: • The concepts and methods used for the analysis of behavior in electronic games and virtual worlds [START_REF] Huffaker | Group Membership and Diffusion in Virtual Worlds[END_REF] [71] [72] can be used in the analysis of RFID data, and vice versa. • The methods used for analyzing animal societies based on RFID data [START_REF] Cabanes | Mining RFID behavior data using unsupervised learning[END_REF] can be adopted to analyzing the movement of entities in schedule-based systems in general. • Data from RFID (and other types of sensors) have been used in some literature [START_REF] Oztekin | An RFID network design methodology for asset tracking in healthcare[END_REF], [START_REF] Chang | A stack-based prospective spatio-temporal data analysis approach[END_REF], [START_REF] Cheung | A methodological approach to optimizing RFID deployment[END_REF] to (optimally) allocate the RFID readers. Data mining frameworks can be integrated with such methods to come up with better allocation of reader within an environment. • The study can be extended such that it encompasses more of the available data mining algorithms and techniques. For example, besides using k-Means Clustering in the unsupervised learning process, one can use k-Means++ [START_REF] Arthur | k-means++: The advantages of careful seeding[END_REF], to reduce both clustering errors and running times. • Last but not least, mining of RFID data can be used in the general context of ambient intelligence applications, which are surveyed and discussed in [START_REF] Ramos | Ambient intelligence-The next step for artificial intelligence[END_REF]- [START_REF] Augusto | Ambient intelligence and smart environments: A state of the art[END_REF]. APPENDIX A. AUGMENTATION ALGORITHMS The first augmentation algorithm is the following: Input: D 0 , D Output: D 1 foreach d 0 ∈ D 0 = r, i, u ir , U ir do d 1 = r, i, u ir , U ir , j 1 = intervalOf (u ir ) , j 2 = intervalOf (U ir ) , ε 1 = intervalT ype (j 1 ) , ε 2 = intervalT ype (j 2 ) , s j1 , s j2 , f j1 , f j2 ; D 1 d 1 ; end Algorithm 1: Generate D 1 By using the schedule data D , this algorithm augments each record of the RFID database D 0 with the information of the intervals covering the entry time and exit time. This information includes the type of interval, start time, and finish time. The augmented records forms a new database D 1 for further analysis. Algorithm 1 includes a single loop that requires the initial construction of the lookup tables for the lookup functions. Each lookup has to scan through the J intervals for each of the R records. Running time of this initialization stage is O(RJ). After this, each record is augmented, taking O(R) time. So the running time of Algorithm 1 is O(RJ). The second augmentation algorithm is given below. For each record of database D 1 , this algorithm first identifies the types of the sequence of intervals that partially or completely falls between the entry time u and exit time U of that particular record. If an interval j is an event, its entry time scenario is analyzed to derive e and p. This is followed by the analysis of exit time scenario to determine x and q. Finally, the time T spent on that event, the earliness and the lateness are computed for that event based on e, p, x, and q. The first four fields in this record are then augmented with intermediary data e, p, x, q as well as the computed t, earliness and lateness . The augmented new record is added to a new database D 2 . This procedure is repeated for all records in database D 1 . Algorithm 2 has two interleaved loops, and runs for each record and for each interval. So the running time of Algorithm 2 is O(RJ). In the below algorithm, by noting that two successive break intervals are impossible and at least part of the event falls within the time span bounded by u and U , the following possible entry time scenarios for the event are considered: 1) Entry time u falls within the event 2) Entry time u is before the start time of the event a) The interval j -1 preceding the event and in the sequence is a break i) The second interval j -2 preceding the event and in the sequence is an event ii) The second interval j -2 preceding the event and in the sequence does not exist (Type of interval is null) b) The interval j -1 preceding the event and in the sequence is an event APPENDIX D. SOFTWARE TOOLS USED There exist a multitude of data analysis and data mining software tools, and we have used different tools for different purposes. Matlab1 was used for coding the developed and presented algorithms. Orange2 data mining software [START_REF] Demšar | Orange: data mining toolbox in Python[END_REF] was used for clustering, classification, and classification tree analysis. RapidMiner3 [START_REF] Hofmann | RapidMiner: Data Mining Use Cases and Business Analytics Applications[END_REF] was used to compute the correlation matrix for the sessions. Borgelt's implementation of the apriori algorithm 4 [83]- [START_REF] Borgelt | Recursion Pruning for the Apriori Algorithm[END_REF] was used to compute frequent itemsets (attendees frequently appearing together). Finally, NodeXL5 [START_REF] Hansen | Analyzing social media networks with NodeXL: Insights from a connected world[END_REF] was used to visualize association mining results and to compute graph metrics, enabling association-based social network analysis. Fig. 1 . 1 Fig. 1. A schedule-based system where entities entering and exiting the system are tracked with RFID Fig. 2 . 2 Fig. 2. The developed analysis framework for mining RFID data from a schedule-based system Fig. 3 . 3 Fig. 3. MDS results for entity analysis based on entity-event profile data Fig. 4 . 4 Fig. 4. Attribute (session) clustering results for Case 2 Fig. 5 . 5 Fig. 5. The classification tree for the case where class attribute is entry into session S27 Fig. 9 . 9 Fig. 9. Results for association mining analysis with grid visualization Fig. 11 .Fig. 12 . 1112 Fig. 11. The unsupervised data mining process conducted in the study TABLE I APPLICATION I AREAS OF THE DEVELOPED METHODOLOGY, WITH A MAPPING OF THE VARIOUS ASPECTS OF THE MODEL. TABLE IV THE IV CLASSIFICATION RESULTS FOR PREDICTING ATTENDEES TO SESSION S27 (LAST EVENT IN THE SCHEDULE) TABLE V THE V LIST OF SELECTED ENTITY PAIRS AND THEIR SUPPORT COUNT (ABSOLUTE SUPPORT) VALUES TABLE VI THE VI LIST OF TOP 10 "INFLUENCERS" AND 5 OF THE "FOLLOWERS" http://www.mathworks.com http://orange.biolab.si/ http://rapidminer.com/ http://www.borgelt.net/apriori.html http://nodexl.codeplex.com/ ACKNOWLEDGEMENTS The authors thank Akın Altunbaş and Ahmet Erdem Altunbaş from Borda Technology and Enes Eryarsoy from İstanbul Şehir University for introducing the problem to the research group and providing the data for the case study. The authors also thank Utku Kaymaz, Berkay Dönmez, and Çagrı Başel from Sabancı University for their assistance in the editing of the paper. if intervalT ype(j + 2) = Event then q = fj + (fj+1 -sj+1) /2; end if intervalT ype(j + 2) = null then q = U ; end end if intervalT ype (j + 1) = Event then exitStatus = ExitIntoN extEvent; q = fj ; end end / * Compute the metrics * / T = x -e; earliness = sj -p; lateness = q -fj ; end d ← i, j, r, u, U, e, x, T, p, q, earliness , lateness , entryStatus, exitStatus ; D 2 d; end end Algorithm 2: Generate D 2 Similarly, the possible exit time scenarios under consideration are 1) Exit time U falls within the event 2) Exit time U is after the finish time of the event a) The interval j + 1 following the event and in the sequence is a break i) The second interval j + 2 following the event and in the sequence is an event ii) The second interval j + 2 following the event and in the sequence does not exist (Type of interval is null) b) The interval j + 1 following the event and in the sequence is an event Different scenarios of entry time and exit time have different expressions for e, x, p, q respectively as presented in the algorithm. The computation for the metrics, however, is unified in the sense that it is independent of the types of scenarios. APPENDIX B. PATTERN MATCHING ALGORITHM The pattern matching algorithm is given below: Algorithm 3: Generate D 7a and D 7b whose union forms D 7 This algorithm first generates all possible combinations of two arbitrary records from D 1 that have different entities. For each combination, if the entry time difference between the two records are less than or equal to a predefined length T RD , the transaction ID for entry time is first updated. Then two new records with updated transaction ID, entity, entry time and record ID are generated and added to a database D 7a . Similarly, if the exit time difference is less than or equal to T RD , the transaction ID for exit time is updated and two new records are inserted into a database D 7b . The union of D 7a and D 7b form a new database D 7 containing all detected pairs showing the close relationship between entry or exit time of two entities for a particular event. Algorithm 3 has two interleaved loops, each executed for up to R records. So the running time of Algorithm 3 is O(R 2 ). APPENDIX C. DATA MINING PROCESSES Figures 11 and12 display the data mining processes carried out. The first process in Figure 11 shows an unsupervised machine learning model, whereas the second process shown in Figure 12 shows a supervised machine learning model. The unsupervised data mining process (Figure 11) starts with reading data from file (File block), verifying that the data is read correctly (Data Table 1 block), and handling any missing values (Impute block). Next, data is again verified, this time visually, using a scatter plot (Scatterplot 1 block). The attributes are selected and specified (Select Attributes) and the unsupervised learning is initiated. The first type of analysis uses entity-entity distances (Example Distance) and conducts MDS (MDS), as well as Hierarchical Clustering, and detects any Outliers. The next analysis is k-Means Clustering, whose results are visually inspected (Scatterplot 2) and exported into a data table (Data Table 2). The final analysis is the computation of distances between events (Attribute Distance) and the conduct of hierarchical clustering (Hierarchical Clustering 2). The supervised data mining process (Figure 12) also starts with the same steps. However, the attribute selection is different, because -unlike the previous process-one categorical attribute (S27, in our case study) has to be selected as the class attribute to be predicted. Next, multiple classifiers are

01744359

en

2024/03/05 22:32:07

2018

https://hal.science/hal-01744359/file/dL-extended.pdf

Étienne Miquey A Classical Sequent Calculus with Dependent Types Dependent types are a key feature of the proof assistants based on the Curry-Howard isomorphism. It is well-known that this correspondence can be extended to classical logic by enriching the language of proofs with control operators. However, they are known to misbehave in the presence of dependent types, unless dependencies are restricted to values. Moreover, while sequent calculi naturally support continuation-passing style interpretations, there is no such presentation of a language with dependent types. The main achievement of this paper is to give a sequent calculus presentation of a call-by-value language with a control operator and dependent types, and to justify its soundness through a continuation-passing style translation. We start from the call-by-value version of the λµ μ-calculus. We design a minimal language with a value restriction and a type system that includes a list of explicit dependencies to maintains type safety. We then show how to relax the value restriction and introduce delimited continuations to directly prove the consistency by means of a continuation-passing-style translation. Finally, we relate our calculus to a similar system by Lepigre, and present a methodology to transfer properties from this system to our own. INTRODUCTION Control operators and dependent types Originally created to deepen the connection between programming and logic, dependent types are now a key feature of numerous functional programming languages. From the point of view of programming, dependent types provide more precise types-and thus more precise specifications-to existing programs. From a logical perspective, they permit definitions of proof terms for statements like the full axiom of choice. Dependent types are provided by Coq or Agda, two of the most actively developed proof assistants. They both rely on constructive type theories: the calculus of inductive constructions for Coq [START_REF] Coquand | Inductively defined types[END_REF], and Martin-Löf's type theory for Agda [START_REF] Martin-Löf | Constructive mathematics and computer programming[END_REF]. Yet, both systems lack support for classical logic and more generally for side effects, which make them impractical as programming languages. In practice, effectful languages give the programmer a more explicit access to low-level control (that is: to the way the program is executed on the available hardware), and make some algorithms easier to implement. Common effects, such as the explicit manipulation of the memory, the generation of random numbers and input/output facilities are available in most practical programming languages (e.g., OCaml, C++, Python, Java). In 1990, Griffin discovered that the control operator call/cc (short for call with current continuation) could be typed by Peirce's law ((A→ B) →A) →A) [START_REF] Griffin | A formulae-as-type notion of control[END_REF], thus extending the formulas-as-types interpretation. Indeed, Peirce's law is known to imply, in an intuitionistic framework, all the other forms of classical reasoning (excluded middle, reductio ad absurdum, double negation elimination, etc.). This discovery opened the way for a direct computational interpretation of classical proofs, using control operators and their ability to backtrack. Several calculi were born from this idea, for example Parigot's λµ-calculus [START_REF] Parigot | Proofs of strong normalisation for second order classical natural deduction[END_REF], Barbanera and Berardi's symmetric λ-calculus [START_REF] Barbanera | A symmetric lambda calculus for classical program extraction[END_REF], Krivine's λ c -calculus [START_REF] Krivine | Realizability in classical logic. In interactive models of computation and program behaviour[END_REF], Curien and Herbelin's λµ μ-calculus [START_REF] Curien | The duality of computation[END_REF]. c b n a This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 Unported License. 1:2 Étienne Miquey Nevertheless, dependent types are known to misbehave in the presence of control operators, and lead to logical inconsistencies [START_REF] Herbelin | On the degeneracy of sigma-types in presence of computational classical logic[END_REF]. Since the same problem arises with a wider class of effects, it seems that we are facing the following dilemma: either we choose an effectful language (allowing us to write more programs) while accepting the lack of dependent types, or we choose a dependently typed language (allowing us to write finer specifications) and give up effects. Many works have tried to fill the gap between effectful programming languages and logic, by accommodating weaker forms of dependent types with computational effects (e.g., divergence, I/O, local references, exceptions). Amongst other works, we can cite the recent works by Ahman et al. [START_REF] Danel Ahman | Dependent Types and Fibred Computational Effects[END_REF], by Vákár [START_REF] Vákár | A framework for dependent types and effects[END_REF][START_REF] Vákár | In Search of Effectful Dependent Types[END_REF] or by Pédrot and Tabareau who proposed a systematical way to add effects to type theory [START_REF] Pédrot | An effectful way to eliminate addiction to dependence[END_REF]. Side effects-that are impure computations in functional programming-are often interpreted by means of monads. Interestingly, control operators can be interpreted similarly through the continuation monad, but the continuation monad generally lacks the properties necessary to fit these frameworks. Although dependent types and classical logic have been deeply studied separately, the problem of accommodating both features 1 in one and the same system has not found a completely satisfying answer yet. Recent works from Herbelin [START_REF] Herbelin | A constructive proof of dependent choice, compatible with classical logic[END_REF] and Lepigre [START_REF] Lepigre | A classical realizability model for a semantical value restriction[END_REF] proposed some restrictions on dependent types to make them compatible with a classical proof system, while Blot [START_REF] Blot | Hybrid realizability for intuitionistic and classical choice[END_REF] designed a hybrid realizability model where dependent types are restricted to an intuitionistic fragment. Call-by-value and value restriction In languages enjoying the Church-Rosser property (like the λ-calculus or Coq), the order of evaluation is irrelevant, and any reduction path will ultimately lead to the same value. In particular, the call-by-name and call-by-value evaluation strategies will always give the same result. However, this is no longer the case in presence of side effects. Indeed, consider the simple case of a function applied to a term producing some side effects (for instance increasing a reference). In call-by-name, the computation of the argument is delayed to the time of its effective use, while in call-by-value the argument is reduced to a value before performing the application. If, for instance, the function never uses its argument, the call-by-name evaluation will not generate any side effect, and if it uses it twice, the side effect will occur twice (and the reference will have its value increased by two). On the contrary, in both cases the call-by-value evaluation generates the side effect exactly once (and the reference has its value increased by one). In this paper, we present a language following the call-by-value reduction strategy, which is as much a design choice as a goal in itself. Indeed, when considering a language with control operators (or other kinds of side effects), soundness often turns out to be subtle to preserve in call-by-value. The first issues in call-by-value in the presence of side effects were related to references [START_REF] Wright | Simple imperative polymorphism[END_REF] and polymorphism [START_REF] Harper | Polymorphic type assignment and CPS conversion[END_REF]. In both cases, a simple solution (but often unnecessarily restrictive in practice [START_REF] Garrigue | Relaxing the value restriction[END_REF][START_REF] Lepigre | A classical realizability model for a semantical value restriction[END_REF]) to solve the inconsistencies consists in the introduction of a value restriction for the problematic cases, restoring then a sound type system. Recently, Lepigre presented a proof system providing dependent types and a control operator [START_REF] Lepigre | A classical realizability model for a semantical value restriction[END_REF], whose consistency is preserved by means of a semantical value restriction defined for terms that behave as values up to observational equivalence. In the present work, we will rather use a syntactic restriction to a fragment of proofs that allows slightly more than values. As we will see, the restriction that arises naturally coincides with the negative-elimination-free fragment of Herbelin's dPAω system [START_REF] Herbelin | A constructive proof of dependent choice, compatible with classical logic[END_REF]. A sequent calculus presentation The main achievement of this paper is to give a sequent calculus presentation2 of a call-by-value language with classical control and dependent types, and to justify its soundness through a continuation-passing style translation. Our calculus is an extension of the λµ μ-calculus [START_REF] Curien | The duality of computation[END_REF] with dependent types. Amongst other motivations, such a calculus is close to an abstract machine, which makes it particularly suitable to define CPS translations or to be an intermediate language for compilation [START_REF] Downen | Sequent calculus as a compiler intermediate language[END_REF]. As a matter of fact, the original motivation for this work was the design of a program translation for Herbelin's dPAω system (that already encompasses control operators and dependent types) to justify its soundness. However, this calculus was presented in a natural deduction style, making such a translation hard to obtain. We thus developed the framework presented in this paper to have an intermediate language more suitable for a continuation-passing style translation at our disposal. Additionally, while we consider in this paper the specific case of a calculus with classical logic, the sequent calculus presentation itself is responsible for another difficulty. As we will see, the usual call-by-value strategy of the λµ μ-calculus causes subject reduction to fail, which would already happen in an intuitionistic type theory. We claim that the solutions we give in this paper also works in the intuitionistic case. In particular, the system we develop might be a first step towards the adaption of the well-understood continuation-passing style translations for ML to design a (dependently) typed compilation of a system with dependent types such as Coq. Delimited continuations and CPS translation The main challenge in designing a sequent calculus with dependent types lies in the fact that the natural relation of reduction one would expect in such a framework is not safe with respect to types. As we will discuss in Section 2.6, the problem can be understood as a desynchronization of the type system with respect to the reduction. A simple solution, presented in Section 2, consists in the addition of an explicit list of dependencies in typing derivations. This has the advantage of leaving the computational part of the original calculus unchanged. However, it is not suitable for obtaining a continuation-passing style translation. We thus present a second way to solve this issue by introducing delimited continuations [START_REF] Ariola | A type-theoretic foundation of delimited continuations[END_REF], which are used to force the purity needed for dependent types in an otherwise non purely functional language. It also justifies the relaxation of the value restriction and leads to the definition of the negative-elimination-free fragment (Section 3). In addition, it allows for the design, in Section 4, of a continuation-passing style translation that preserves dependent types and permits us to prove the soundness of our system. Finally, it also provides us with a way to embed our calculus into Lepigre's calculus [START_REF] Lepigre | A classical realizability model for a semantical value restriction[END_REF], as we shall see in Section 5. This embedding has in particular the benefit of furnishing us with a realizability interpretation for free. Contributions of the paper Our main contributions in this paper can be listed as follows: • We soundly combine dependent types and control operators by means of a syntactic restriction to the negative-elimination-free fragment; • We give a sequent calculus presentation and solve the type-soundness issues it raises in two different ways; • Our first solution simply relies on a list of dependencies that is added to the type system • Our second solution uses delimited continuations to ensure consistency with dependent types and provides us with a CPS translation (carrying dependent types) to a calculus without control operator; • We relate our system to Lepigre's calculus, which gives us a realizability interpretation for free and offers an additional way of proving the consistency of our system. This paper is an extended and revised version of the article presented at ESOP 2017 [START_REF] Miquey | A classical sequent calculus with dependent types[END_REF]. A MINIMAL CLASSICAL LANGUAGE WITH DEPENDENT TYPES 2.1 A short primer to the λµ μ-calculus We recall here the spirit of the λµ μ-calculus, for further details and references please refer to the original article [START_REF] Curien | The duality of computation[END_REF]. The syntax and reduction rules (parameterized over a subset of proofs V and a subset of evaluation contexts E) are given in Figure 1, where μa.c can be read as a context let a = [ ] in c. A command ⟨p||e⟩ can be understood as a state of an abstract machine, representing the evaluation of a proof p (the program) against a co-proof e (the stack) that we call context. The µ operator comes from Parigot's λµ-calculus [START_REF] Parigot | Proofs of strong normalisation for second order classical natural deduction[END_REF], µα binds a context to a context variable α in the same way that μa binds a proof to some proof variable a. The λµ μ-calculus can be seen as a proof-as-program correspondence between sequent calculus and abstract machines. Right introduction rules correspond to typing rules for proofs, while left introduction are seen as typing rules for evaluation contexts. In contrast with Gentzen's original presentation of sequent calculus, the type system of the λµ μ-calculus explicitly identifies at any time which formula is being worked on. In a nutshell, this presentation distinguishes between three kinds of sequents: (1) sequents of the form Γ ⊢ p : A | ∆ for typing proofs, where the focus is put on the (right) formula A; (2) sequents of the form Γ | e : A ⊢ ∆ for typing contexts, where the focus is put on the (left) formula A; (3) sequents of the form c : (Γ ⊢ ∆) for typing commands, where no focus is set. In a right (resp. left) sequent Γ ⊢ p : A | ∆, the singled out formula3 A reads as the conclusion "where the proof shall continue" (resp. hypothesis "where it happened before"). For example, the left introduction rule of implication can be seen as a typing rule for pushing an element q on a stack e leading to the new stack q • e: Γ ⊢ q : A | ∆ Γ | e : B ⊢ ∆ Γ | q • e : A → B ⊢ ∆ → l As for the reduction rules, we can see that there is a critical pair if V and E are not restricted enough: c[α := μx .c ′ ] ←-⟨µα .c || μx .c ′ ⟩ -→ c ′ [x := µα .c]. The difference between call-by-name and call-by-value can be characterized by how this critical pair 4is solved, by defining V and E such that the two rules do not overlap. Defining the subcategories Remark 2.1 (Application). The reader unfamiliar with the λµ μ-calculus might be puzzled by the absence of a syntactic construction for the application of proof terms. Intuitively, the usual application p q of the λ-calculus is replaced by the application of the proof p to a stack of the shape q • e as in an abstract machine 5 . The usual application can thus be recovered through the following shorthand: Γ ⊢ t :A | ∆ Γ | e : A ⊢ ∆ ⟨t ||e⟩ : (Γ ⊢ ∆) (Cut) (a : A) ∈ Γ Γ ⊢ a : A | ∆ (Ax r ) Γ, a : A ⊢ p : B | ∆ Γ ⊢ λa.p : A → B | ∆ (→ r ) c : (Γ ⊢ ∆, α : A) Γ ⊢ µα .c : A | ∆ (µ) (α : A) ∈ ∆ Γ | α : A ⊢ ∆ (Ax l ) Γ ⊢ p : A | ∆ Γ | e : B ⊢ ∆ Γ | p • e : A → B ⊢ ∆ (→ l ) c : (Γ, a : A ⊢ ∆) Γ | μa.c : A ⊢ ∆ ( μ) (c) Typing rules p q ≜ µα .⟨p||q • α⟩ Finally, it is worth noting that the µ binder is a control operator, since it allows for catching evaluation contexts and backtracking further in the execution. This is the key ingredient that makes the λµ μ-calculus a proof system for classical logic. To illustrate this, let us draw the analogy with the call/cc operator of Krivine's λ c -calculus [START_REF] Krivine | Realizability in classical logic. In interactive models of computation and program behaviour[END_REF]. Let us define the following proof terms: call/cc ≜ λa.µα .⟨a||k α • α⟩ k e ≜ λa ′ .µβ .⟨a ′ ||e⟩ The proof k e can be understood as a proof term where the context e has been encapsulated. As expected, call/cc is a proof for Peirce's law (see Figure 2), which is known to imply other forms of classical reasoning (e.g., the law of excluded middle, the double negation elimination). Let us observe the behavior of call/cc (in call-by-name evaluation strategy, as in Krivine λ c -calculus): in front of a context of the shape q • e with e of type A, it will catch the context e a : (A → B) → A ⊢ a : (A → B) → A | • (Ax r ) •, a ′ : A ⊢ a ′ : A | • (Ax r ) • | α : A ⊢ α : A, • (Ax l ) ⟨a ′ ||α⟩ : (•, a ′ : A ⊢ α : A, β : B) (Cut) •, a ′ : A ⊢ µβ .⟨a ′ ||α⟩ : B | α : A (µ) • ⊢ λa ′ .µβ .⟨a ′ ||α⟩ : A → B | α : A (→ r ) | α : A ⊢ α : A (Ax l ) • | λa ′ .µβ .⟨a ′ ||α⟩ • α : (A → B) → A ⊢ α : A (→ l ) ⟨a||λa ′ .µβ .⟨a ′ ||α⟩ • α⟩ : (a : (A → B) → A ⊢ α : A) (Cut) a : (A → B) → A ⊢ µα .⟨a||λa ′ .µβ .⟨a ′ ||α⟩ • α⟩ : A | (µ) ⊢ λa.µα .⟨a||λa ′ .µβ .⟨a ′ ||α⟩ • α⟩ : ((A → B) → A) → A | (→ r ) (where • is used to shorten useless parts of typing contexts.) Fig. 2. Proof term for Peirce's law thanks to the µα binder and reduce as follows: ⟨λa.µα .⟨a||k α • α⟩||q • e⟩ → ⟨q|| μa.⟨µα .⟨a||k α • α⟩||e⟩⟩ → ⟨µα .⟨q||k α • α⟩||e⟩ → ⟨q||k e • e⟩ We notice that the proof term k e = λa ′ .µβ .⟨a ′ ||e⟩ on top of the stack (which, if e was of type A, is of type A → B, see Figure 2) contains a second binder µβ. In front of a stack q ′ • e ′ , this binder will now catch the context e ′ and replace it by the former context e: ⟨λa ′ .µβ .⟨a ′ ||e⟩||q ′ • e ′ ⟩ → ⟨q ′ || μa ′ .⟨µβ .⟨a ′ ||e⟩||e ′ ⟩⟩ → ⟨µβ .⟨q ′ ||e⟩||e ′ ⟩ → ⟨q ′ ||e⟩ This computational behavior corresponds exactly to the usual reduction rule for call/cc in the Krivine machine [START_REF] Krivine | Realizability in classical logic. In interactive models of computation and program behaviour[END_REF]: call/cc ⋆ t • π ≻ t ⋆ k π • π k π ⋆ t • π ′ ≻ t ⋆ π Inconsistency of classical logic with dependent types The simultaneous presence of classical logic (i.e. of a control operator) and dependent types is known to cause a degeneracy of the domain of discourse. Let us shortly recap the argument of Herbelin highlighting this phenomenon [START_REF] Herbelin | On the degeneracy of sigma-types in presence of computational classical logic[END_REF]. Let us adopt here a stratified presentation of dependent types, by syntactically distinguishing terms-that represent mathematical objects-from proof terms-that represent mathematical proofs. In other words, we syntactically separate the categories corresponding to witnesses and proofs in dependent sum types. Consider a minimal logic of strong existentials and equality, whose formulas, terms (only representing natural number) and proofs are defined as follows: Formulas A, B ::= t = u | ∃x N .A Terms t, u ::= n | wit p Proofs p, q ::= refl | subst p q | (t, p) | prf p (n ∈ N) Let us explain the different proof terms by presenting their typing rules. First of all, the pair (t, p) is a proof for an existential formula ∃x N .A where t is a witness for x and p is a certificate for A[t/x]. This implies that both formulas and proofs are dependent on terms, which is usual in mathematics. What is less usual in mathematics is that, as in Martin-Löf's type theory, dependent types also allow for terms (and thus for formulas) to be dependent on proofs, by means of the constructors wit p and prf p. Typing rules are given with separate typing judgments for terms, which can only be of type N: Γ ⊢ p : A(t) Γ ⊢ t : N Γ ⊢ (t, p) : ∃x N .A (∃ I ) Γ ⊢ (t, p) : ∃x N .A Γ ⊢ prf p : A[wit p/x] (prf ) Γ ⊢ t : ∃x N .A Γ ⊢ wit t : N (wit) n ∈ N Γ ⊢ n : N Then, refl is a proof term for equality, and subst p q allows us to use a proof of an equality t = u to convert a formula A(t) into A(u): t → u Γ ⊢ refl : t = u (refl) Γ ⊢ p : t = u Γ ⊢ q : B[t] Γ ⊢ subst p q : B[u] (subst) The reduction rules for this language, which are safe with respect to typing, are then: wit (t, p) → t prf (t, p) → p subst refl p → p Starting from this (sound) minimal language, Herbelin showed that its classical extension with the control operators call/cc k and throw k (that are similar to those presented in the previous section) permits to derive a proof of 0 = 1 [START_REF] Herbelin | On the degeneracy of sigma-types in presence of computational classical logic[END_REF]. The call/cc k operator, which is a binder for the variable k, is intended to catch its surrounding evaluation context. On the contrary, throw k discards the current context and restores the context captured by call/cc k . The addition to the type system of the typing rules for these operators: Γ, k : ¬A ⊢ p : A Γ ⊢ call/cc k p : A Γ, k : ¬A ⊢ p : A Γ, k : ¬A ⊢ throw k p : B allows the definition of the following proof: p 0 ≜ call/cc k (0, throw k (1, refl)) : ∃x N .x = 1 Intuitively such a proof catches the context, gives 0 as witness (which is incorrect), and a certificate that will backtrack and give 1 as witness (which is correct) with a proof of the equality. If, besides, the following reduction rules6 are added: wit (call/cc k p) → call/cc k (wit (p[k(wit { })/k])) call/cc k t → t (k FV (t)) then we can formally derive a proof of 1 = 0. Indeed, the term wit p 0 will reduce to call/cc k 0, which itself reduces to 0. The proof term refl is thus a proof of wit p 0 = 0, and we obtain the following proof of 1 = 0: ⊢ p 0 : ∃x N .x = 1 ⊢ prf p 0 : wit p 0 = 1 (prf ) wit p 0 → 0 ⊢ refl : wit p 0 = 0 (refl) ⊢ subst (prf p 0 ) refl : 1 = 0 (subst) The bottom line of this example is that the same proof p 0 is behaving differently in different contexts thanks to control operators, causing inconsistencies between the witness and its certificate. The easiest and usual approach (in natural deduction) to prevent this is to impose a restriction to values (which are already reduced) for proofs appearing inside dependent types and within the operators wit and prf , together with a call-by-value discipline. In the present example, this would prevent us from writing wit p 0 and prf p 0 . 1:8 Étienne Miquey A minimal language with value restriction In this section, we will focus on value restriction in a similar framework, and show that the obtained proof system is coherent. We will then see, in Section 3, how to relax this constraint. We follow here the stratified presentation7 from the previous section. We place ourselves in the framework of the λµ μ-calculus to which we add: • a language of terms which contain an encoding 8 of the natural numbers, • proof terms (t, p) to inhabit the strong existential ∃x N .A together with the first and second projections, called respectively wit (for terms) and prf (for proofs), • a proof term refl for the equality of terms and a proof term subst for the convertibility of types over equal terms. For simplicity reasons, we will only consider terms of type N throughout this paper. We address the question of extending the domain of terms in Section 6.2. The syntax of the corresponding system, that we call dL, is given by: Terms t ::= x | n | wit V Proof terms p ::= V | µα .c | (t, p) | prf V | subst p q Proof values V ::= a | λa.p | λx .p | (t, V ) | refl Contexts e ::= α | p • e | t • e | μa.c Commands c ::= ⟨p||e⟩ (n ∈ N) The formulas are defined by: Formulas A, B ::= ⊤ | ⊥ | t = u | ∀x N .A | ∃x N .A | Π a:A B. Note that we included a dependent product Π a:A B at the level of proof terms, but that in the case where a FV (B) this amounts to the usual implication A → B. Reduction rules As explained in Section 2.2, a backtracking proof might give place to different witnesses and proofs according to the context of reduction, leading to inconsistencies [START_REF] Herbelin | On the degeneracy of sigma-types in presence of computational classical logic[END_REF]. The substitution at different places of a proof which can backtrack, as the call-by-name evaluation strategy does, is thus an unsafe operation. On the contrary, the call-by-value evaluation strategy forces a proof to reduce first to a value (thus furnishing a witness) and to share this value amongst all the commands. In particular, this maintains the value restriction along reduction, since only values are substituted. The reduction rules, defined in Figure 3 (where t → t ′ denotes the reduction of terms and c ⇝ c ′ the reduction of commands), follow the call-by-value evaluation principle. In particular one can see that whenever a command is of the shape ⟨C[p]||e⟩ where C[p] is a proof built on top of p which is not a value, it reduces to ⟨p|| μa.⟨C[a]||e⟩⟩, opening the construction to evaluate p 9 . Additionally, we denote by A ≡ B the transitive-symmetric closure of the relation A ▷ B, defined as a congruence over term reduction (i.e. if t → t ′ then A[t] ▷ A[t ′ ]) and by the rules: 0 = 0 ▷ ⊤ 0 = S(u) ▷ ⊥ S(t) = 0 ▷ ⊥ S(t) = S(u) ▷ t = u ⟨µα .c ||e⟩ ⇝ c[e/ Typing rules As we explained before, in this section we limit ourselves to the simple case where dependent types are restricted to values, to make them compatible with classical logic. But even with this restriction, defining the type system in the most naive way leads to a system in which subject reduction will fail. Having a look at the β-reduction rule gives us an insight of what happens. Let us imagine that the type system of the λµ μ-calculus has been extended to allow dependent products instead of implications. and consider a proof λa.p : Π a:A B in front of a context q • e : Π a:A B. A typing derivation of the corresponding command would be of the form: Π p Γ, a : A ⊢ p : B | ∆ Γ ⊢ λa.p : Π a:A B | ∆ (→ r ) Π q Γ ⊢ q : A | ∆ Π e Γ | e : B[q/a] ⊢ ∆ Γ | q • e : Π a:A B ⊢ ∆ (→ l ) ⟨λa.p||q • e⟩ : Γ ⊢ ∆ (Cut) while this command would reduce as follows: ⟨λa.p||q • e⟩ ⇝ ⟨q|| μa.⟨p||e⟩⟩. On the right-hand side, we see that p, whose type is B[a], is now cut with e whose type is B[q]. Consequently, we are not able to derive a typing judgment 10 for this command anymore: Π q Γ ⊢ q : A | ∆ Γ, a : A ⊢ p : ¨B[a] | ∆ Γ, a : A | e : ¨B[q] ⊢ ∆ ⟨p||e⟩ : Γ, a : A ⊢ ∆ Mismatch Γ | μa.⟨p||e⟩ : A ⊢ ∆ ( μ) ⟨q|| μa.⟨p||e⟩⟩ : Γ ⊢ ∆ (Cut) The intuition is that in the full command, a has been linked to q at a previous level of the typing judgment. However, the command is still safe, since the head-reduction imposes that the command ⟨p||e⟩ will not be executed before the substitution of a by q11 is performed, and by then the problem would be solved. This phenomenon can be seen as a desynchronization of the typing process with respect to computation. The synchronization can be re-established by making explicit a list of dependencies σ in the typing rules, which links μ variables (here a) to the associated proof term on the left-hand side of the command (here q). We can now obtain the following typing derivation: Γ ⊢ p : A | ∆; σ Γ | e : B ⊢ ∆; σ {•|p} B ∈ A σ ⟨p||e⟩ : Γ ⊢ ∆; σ (Cut) (a : A) ∈ Γ Γ ⊢ a : A | ∆; σ (Ax r ) (α : A) ∈ ∆ Γ | α : A ⊢ ∆; σ {•|p} (Ax l ) c : (Γ ⊢ ∆, α : A; σ ) Γ ⊢ µα .c : A | ∆; σ Π q Γ ⊢ q : A | ∆ Π p Γ, a : A ⊢ p : B[a] | ∆ Π e Γ, a : A | e : B[q] ⊢ ∆; σ {a|q}{•|p} ⟨p||e⟩ : Γ, a : A ⊢ ∆; σ {a|q} (Cut) Γ | μa.⟨p||e⟩ : A ⊢ ∆; σ {.|q} ( μ) ⟨q|| μa.⟨p||e⟩⟩ : Γ ⊢ ∆; σ (Cut) Formally, we denote by D the set of proofs we authorize in dependent types, and define it for the moment as the set of values: D ≜ V . We define a list of dependencies σ as a list binding pairs of proof terms 12 : σ ::= ε | σ {p|q}, and we define A σ as the set of types that can be obtained from A by replacing all (or none) occurrences of p by q for each binding {p|q} in σ such that q ∈ D: A ε ≜ {A} A σ {p |q } ≜ A σ ∪ (A[q/p]) σ if q ∈ D A σ otherwise. The list of dependencies is filled while going up in the typing tree, and it can be used when typing a command ⟨p||e⟩ to resolve a potential inconsistency between their types: Γ ⊢ p : A | ∆; σ Γ | e : B ⊢ ∆; σ {•|p} B ∈ A σ ⟨p||e⟩ : Γ ⊢ ∆; σ (Cut) Remark 2.2. The reader familiar with explicit substitutions [START_REF] Fridlender | Pure type systems with explicit substitutions[END_REF] can think of the list of dependencies as a fragment of the substitution that is available when a command c is reduced. Another remark is that the design choice for the (Cut) rule is arbitrary, in the sense that we chose to check whether B is in A σ . We could equivalently have checked whether the condition σ (A) = σ (B) holds, where σ (A) refers to the type A where for each binding {p|q} ∈ σ with q ∈ D, all the occurrences of p have been replaced by q. Furthermore, when typing a stack with the (→ l ) and (∀ l ) rules, we need to drop the open binding in the list of dependencies 13 . We introduce the notation Γ | e : A ⊢ ∆; σ {•| †} to denote that the dependency to be produced is irrelevant and can be dropped. This trick spares us from defining a second type of sequents Γ | e : A ⊢ ∆; σ to type contexts when dropping the (open) binding {•|p}. Alternatively, one can think of † as any proof term not in D, which is the same with respect to the list of dependencies. The resulting set of typing rules is given in Figure 4, where we assume that every variable bound in the typing context is bound only once (proofs and contexts are considered up to α-conversion). Note that we work with two-sided sequents here to stay as close as possible to the original presentation of the λµ μ-calculus [START_REF] Curien | The duality of computation[END_REF]. In particular this means that a type in ∆ might depend on a variable previously introduced in Γ and vice versa, so that the split into two contexts makes us lose track of the order of introduction of the hypotheses. In the sequel, to be able to properly define a typed CPS translation, we consider that we can unify both contexts into a single one that is coherent with respect to the order in which the hypotheses have been introduced. Example 2.3. The proof p 1 ≜ subst (prf p 0 ) refl which was of type 1 = 0 in Section 2.2 is now incorrect since the backtracking proof p 0 , defined by µα .(0, µ_.⟨(1, refl)||α⟩) in our framework, is not a value in D. The proof p 1 should rather be defined by 14 µα .⟨p 0 || μa.⟨subst (prf a) refl||α⟩⟩ which can only be given the type 1 = 1. Subject reduction We start by giving a few technical lemmas that will be used for proving subject reduction. First, we will show that typing derivations allow weakening on the lists of dependencies. For this purpose, we introduce the notation σ ⇛ σ ′ to denote that whenever a judgment is derivable with σ as list of dependencies, then it is derivable using σ ′ : σ ⇛ σ ′ ≜ ∀c ∀Γ ∀∆.(c : (Γ ⊢ ∆; σ ) ⇒ c : (Γ ⊢ ∆; σ ′ )). This clearly implies that the same property holds when typing evaluation contexts, i.e. if σ ⇛ σ ′ then σ can be replaced by σ ′ in any typing derivation for any context e. Lemma 2.4 (Dependencies weakening). For any list of dependencies σ we have: 1. ∀V .(σ {V |V } ⇛ σ ) 2. ∀σ ′ .(σ ⇛ σσ ′ ) Proof. The first statement is obvious. The proof of the second one is straightforward from the fact that for any p and q, by definition A σ ⊂ A σ {p |q } . □ As a corollary, we get that † can indeed be replaced by any proof term when typing a context. Corollary 2.5. If σ ⇛ σ ′ , then for any p, e, Γ, ∆: Γ | e : A ⊢ ∆; σ {•| †} ⇒ Γ | e : A ⊢ ∆; σ ′ {•|p}. Proof. Assume that e is of the form μa.c (other cases are trivial), then we have c : Γ ⊢ ∆; σ {a| †}. By definition of † and from the hypothesis, we get that σ {a| †} ⇛ σ ′ , i.e. that c : Γ ⊢ ∆; σ ′ is derivable. By applying the previous Lemma, we get that c : Γ ⊢ ∆; σ ′ {a|p} is derivable for any proof p, whence the result. □ We first state the usual lemmas that guarantee the safety of terms (resp. values, contexts) substitution. Lemma 2.6 (Safe term substitution ). If Γ ⊢ t : N | ∆; ε then: (1) c : (Γ, x : N, Γ ′ ⊢ ∆; σ ) ⇒ c[t/x] : (Γ, Γ ′ [t/x] ⊢ ∆[t/x]; σ [t/x]), (2) Γ, x : N, Γ ′ ⊢ q : B | ∆; σ ⇒ Γ, Γ ′ [t/x] ⊢ q[t/x] : B[t/x] | ∆[t/x]; σ [t/x], (3) Γ, x : N, Γ ′ | e : B ⊢ ∆; σ ⇒ Γ, Γ ′ [t/x] | e[t/x] : B[t/x] ⊢ ∆[t/x]; σ [t/x], (4) Γ, x : N, Γ ′ ⊢ u : N | ∆; σ ⇒ Γ, Γ ′ [t/x] ⊢ u[t/x] : N | ∆[t/x]; σ [t/x]. Lemma 2.7 (Safe value substitution). If Γ ⊢ V : A | ∆; ε then: (1) c : (Γ, a : A, Γ ′ ⊢ ∆; σ ) ⇒ c[V /a] : (Γ, Γ ′ [V /a] ⊢ ∆[V /a]; σ [V /a]), (2) Γ, a : A, Γ ′ ⊢ q : B | ∆; σ ⇒ Γ, Γ ′ [V /a] ⊢ q[V /a] : B[V /a] | ∆[V /a]; σ [t/x], (3) Γ, a : A, Γ ′ | e : B ⊢ ∆; σ ⇒ Γ, Γ ′ [V /a] | e[V /a] : B[V /a] ⊢ ∆[V /a]; σ [V /a], (4) Γ, a : A, Γ ′ ⊢ u : N | ∆; σ ⇒ Γ, Γ ′ [V /a] ⊢ u[V /a] : N | ∆[V /a]; σ [V /a]. Lemma 2.8 (Safe context substitution). If Γ | e : A ⊢ ∆; ε then: (1) c : (Γ ⊢ ∆, α : A, ∆ ′ ; σ ) ⇒ c[e/α] : (Γ ⊢ ∆, ∆ ′ ; σ ), ( 2 ) Γ ⊢ q : B | ∆, α : A, ∆ ′ ; σ ⇒ Γ ⊢ q[e/α] : B | ∆, ∆ ′ ; σ , ( 3 ) Γ | e : B ⊢ ∆, α : A, ∆ ′ ; σ ⇒ Γ | e[e/α] : B ⊢ ∆, ∆ ′ ; σ , ( 4 ) Γ ⊢ u : N | ∆, α : A, ∆ ′ ; σ ⇒ Γ ⊢ u : N | ∆, ∆ ′ ; σ ]. Proof. The proofs are done by induction on typing derivations. □ We can now prove the preservation of typing through reduction, using the previous lemmas for rules which perform a substitution, and the list of dependencies to resolve local desynchronizations for dependent types. Theorem 2.9 (Subject reduction). If c, c ′ are two commands of dL such that c : (Γ ⊢ ∆; ε) and c ⇝ c ′ , then c ′ : (Γ ⊢ ∆; ε). Proof. The proof is done by induction on the typing derivation of c : (Γ ⊢ ∆; ε), assuming that for each typing proof, the conversion rules are always pushed down and right as much as possible. To save some space, we sometimes omit the list of dependencies when empty, writing c : Γ ⊢ ∆ instead of c : Γ ⊢ ∆; ε, and we denote the composition of consecutive rules (≡ l ) as: Γ | e : B ⊢ ∆; σ Γ | e : A ⊢ ∆; σ Π q Γ ⊢ q : A ′ | ∆ Γ ⊢ q : A | ∆ (≡ l ) Π p Γ, a : A ⊢ p : B | ∆ Γ, a : A ⊢ p : B ′ | ∆ (≡ r ) Π e Γ, a : A | e : B ′ q ⊢ ∆; {a|q}{•|p} B ′ q ∈ B ′ {a |q } ⟨p||e⟩ : Γ, a : A ⊢ ∆; {a|q} (Cut) Γ | μa.⟨p||e⟩ : A ⊢ ∆; {.|q} ( μ) ⟨q|| μa.⟨p||e⟩⟩ : Γ ⊢ ∆ (Cut) using Corollary 2.5 to weaken the dependencies in Π e . • Case ⟨µα .c ||e⟩ ⇝ c[e/α]. A typing proof for the command on the left-hand side is of the form: Π c c : Γ ⊢ ∆, α : A Γ ⊢ µα .c : A | ∆ (µ) Π e Γ | e : A ⊢ ∆; {•|µα .c} ⟨µα .c ||e⟩ : Γ ⊢ ∆ (Cut) We get a proof that c[e/α] : Γ ⊢ ∆ is valid by Lemma 2.8. • Case ⟨V || μa.c⟩ ⇝ c[V /a]. A typing proof for the command on the left-hand side is of the form: Π V Γ ⊢ V : A | ∆ Π c c : Γ, a : A ′ ⊢ ∆; {a|V } Γ | μa.c : A ′ ⊢ ∆; {•|V } ( μ) Γ | μa.c : A ⊢ ∆; {•|V } (≡ l ) ⟨V || μa.c⟩ : Γ ⊢ ∆ (Cut) We first observe that we can derive the following proof: Π V Γ ⊢ V : A | ∆ Γ ⊢ V : A ′ | ∆ (≡ l ) and we get a proof for c[V /a] : Γ ⊢ ∆; {V |V } by Lemma 2.7. We finally get a proof for c[V /a] : Γ ⊢ ∆ by Lemma 2.4. • Case ⟨(t, p)||e⟩ ⇝ ⟨p|| μa.⟨(t, a)||e⟩⟩, with p V . A proof of the command on the left-hand side is of the form: Π t Γ ⊢ t : N | ∆ Π p Γ ⊢ p : A[t/x] | ∆ Γ ⊢ (t, p) : ∃x N .A | ∆ (∃ r ) Π e Γ | e : ∃x N .A ⊢ ∆; {•|(t, p)} ⟨(t, p)||e⟩ : Γ ⊢ ∆ (Cut) We can build the following derivation: Π p Γ ⊢ p : A[t/x] | ∆ Π (t,a) Γ, a : A[t/x] ⊢ (t, a) : ∃x N .A | ∆ (∃ I ) Π e Γ | e : ∃x N .A ⊢ ∆; {a|p}{•|(t, a)} ⟨(t, a)||e⟩ : Γ, a : A[t/x] ⊢ ∆; {a|p} (Cut) Γ | μa.⟨(t, a)||e⟩ : A[t/x] ⊢ ∆; {•|p} ( μ) ⟨p|| μa.⟨(t, a)||e⟩⟩ : Γ ⊢ ∆ (Cut) where Π (t,a) is as expected, observing that since p D, the binding {•|(t, p)} is the same as {•| †}, and we can apply Corollary 2.5 to weaken dependencies in Π e . • Case ⟨prf (t, V )||e⟩ ⇝ ⟨V ||e⟩. This case is easy, observing that a derivation of the command on the left-hand side is of the form: Π t Π V Γ ⊢ V : A(t) | ∆ Γ ⊢ (t, V ) : ∃x N .A(x) | ∆ (∃ r ) Γ ⊢ prf (t, V ) : A(wit (t, V )) | ∆ (prf ) Π e Γ | e : A(wit (t, V )) ⊢ ∆; {•| †} ⟨prf (t, V )||e⟩ : Γ ⊢ ∆ (Cut) Since by definition we have A(wit (t, V )) ≡ A(t), we can derive: Π V Γ ⊢ V : A(t) | ∆ Π e Γ | e : A(wit (t, V )) ⊢ ∆; {•|V } Γ | e : A(t) ⊢ ∆; {•|V } (≡ l ) ⟨prf (t, V )||e⟩ : Γ ⊢ ∆ (Cut) • Case ⟨subst refl q||e⟩ ⇝ ⟨q||e⟩. This case is straightforward, observing that for any terms t, u, if we have refl : t = u, then A[t] ≡ A[u] for any A. • Case ⟨subst p q||e⟩ ⇝ ⟨p|| μa.⟨subst a q||e⟩⟩. This case is similar to the case ⟨(t, p)||e⟩. • Case c[t] ⇝ c[t ′ ] with t → t ′ . Immediate by observing that by definition of the relation ≡, we have A[t] ≡ A[t ′ ] for any A. □ Soundness We here give a proof of the soundness of dL with a value restriction. The proof is based on an embedding into the λµ μ-calculus extended with pairs, whose syntax and rules are given in Figure 5. A more interesting proof through a continuation-passing translation is presented in Section 4. We first show that typed commands of dL normalize by translation to the simply-typed λµ μcalculus with pairs (i.e. extended with proofs of the form (p 1 , p 2 ) and contexts of the form μ(a 1 , a 2 ).c). We do not consider here a particular reduction strategy, and take ↣ to be the contextual closure of the rules given in Figure 5. The translation essentially consists in erasing the dependencies in types 15 , turning the dependent products into arrows and the dependent sum into a pair. The erasure procedure is defined by: (∀x N .A) * ≜ N → A * ⊤ * ≜ N → N (∃x N .A) * ≜ N ∧ A * ⊥ * ≜ N → N (Π a:A B) * ≜ A * → B * (t = u) * ≜ N → N and the corresponding translation for terms, proofs, contexts and commands is given by: 1:16 Étienne Miquey Proofs p ::= V | µα .c | (p 1 , p 2 ) Values V ::= a | λa.p | (V 1 , V 2 ) Contexts e ::= α | p • e | μa.c | μ(a 1 , a 2 ).c Commands c ::= ⟨p||e⟩ Γ ⊢ p 1 : A 1 | ∆ Γ ⊢ p 2 : A 2 | ∆ Γ ⊢ (p 1 , p 2 ) : A 1 ∧ A 2 | ∆ (∧ r ) c : Γ, a 1 : A 1 , a 2 : A 2 ⊢ ∆ Γ | μ(a 1 , a 2 ).c : A 1 ∧ A 2 ⊢ ∆ (∧ l ) (a) Syntax (b) Typing rules ⟨µα .c ||e⟩ ↣ c[e/α] ⟨λa.p||q • e⟩ ↣ ⟨q|| μa.⟨p||e⟩⟩ ⟨p|| μa.c⟩ ↣ c[p/a] ⟨(p 1 , p 2 )|| μ(a 1 , a 2 ).c⟩ ↣ c[p 1 /a 1 ][p 2 /a 2 ] µα .⟨p||α⟩ ↣ p μa.⟨a||e⟩ ↣ e (c) Reduction rules ⟨p||e⟩ * ≜ ⟨p * ||e * ⟩ α * ≜ α (t • e) * ≜ t * • e * (q • e) * ≜ q * • e * ( μa.c) * ≜ μa.c * x * ≜ x n * ≜ n (wit p) * ≜ π 1 (p * ) a * ≜ a refl * ≜ λx .x (λa.p) * ≜ λa.p * (λx .p) * ≜ λx .p * (µα .c) * ≜ µα .c * (prf p) * ≜ π 2 (p * ) (t, p) * ≜ µα .⟨p * || μa.⟨(t * , a)||α⟩⟩ (subst V q) * ≜ µα .⟨q * ||α⟩ (subst p q) * ≜ µα .⟨p * || μ_ .⟨µα .⟨q * ||α⟩||α⟩⟩ (p V ) where π i (p) ≜ µα .⟨p|| μ(a 1 , a 2 ).⟨a 1 ||α⟩⟩. The term n is defined as any encoding of the natural number n with its type N * , the encoding being irrelevant here as long as n ∈ V . Note that we translate differently subst V q and subst p q to simplify the proof of Proposition 2.12. We first show that the erasure procedure is adequate with respect to the previous translation. Lemma 2.10. The following holds for any types A and B: (1) For any terms t and u, (A[t/u]) * = A * . (2) For any proofs p and q, (A[p/q]) * = A * . ( ) If A ≡ B then A * = B * . (4) For any list of dependencies σ , if A ∈ B σ , then A * = B * . 3 Proof. Straightforward: (1) and ( 2) are direct consequences of the erasure of terms (and thus proofs) from types. (3) follows from (1),( 2) and the fact that (t = u) * = ⊤ * = ⊥ * . (4) follows from [START_REF] Ariola | A type-theoretic foundation of delimited continuations[END_REF]. □ We can extend the erasure procedure to typing contexts, and show that it is adequate with respect to the translation of proofs. Proposition 2.11. The following holds for any contexts Γ, ∆ and any type A: (1) For any command c, if c : Γ ⊢ ∆; σ , then c * : Γ * ⊢ ∆ * . (2) For any proof p, if Γ ⊢ p : A | ∆; σ , then Γ * ⊢ p * : A * | ∆ * . (3) For any context e, if Γ | e : A ⊢ ∆; σ , then Γ * | e * : A * ⊢ ∆ * . Proof. By induction on typing derivations. The fourth item of the previous lemma shows that the list of dependencies becomes useless: since A ∈ B σ implies A * = B * , it is no longer needed Γ * | μa.⟨(t * , a)||α⟩ : A * ⊢ ∆ * , α : N∧A * ( μ) ⟨p * || μa.⟨(t * , a)||α⟩⟩ : Γ * ⊢ ∆ * , α : N∧A * (Cut) Γ * ⊢ µα .⟨p * || μa.⟨(t * , a)||α⟩⟩ : N ∧ A * | ∆ * (µ) □ We can then deduce the normalization of dL from the normalization of the λµ μ-calculus [START_REF] Polonovski | Strong normalization of lambda-bar-mu-mu-tilde-calculus with explicit substitutions[END_REF], by showing that the translation preserves the normalization in the sense that if c does not normalize, then neither does c * . Proposition 2.12. If c is a command such that c * normalizes, then c normalizes. Proof. We prove this by contraposition, by showing that if c does not normalize (i.e. if it admits an infinite reduction path), then c * does not normalize either. We will actually prove a slightly more precise statement, namely that each step of reduction is reflected into at least one step through the translation: ∀c 1 , c 2 , (c 1 1 ⇝ c 2 ⇒ ∃n ≥ 1, (c 1 ) * n ↣ (c 2 ) * ). Assuming this holds, we get from any infinite reduction path (for ⇝) starting from c another infinite reduction path (for ↣) from c * . Thus, the normalization of c * implies the one of c. We shall now prove the previous statement by case analysis of the reduction c 1 ⇝ c 2 . • Case wit (t, V ) → t: Proof. Proof by contradiction: if c does not normalize, then by Proposition 2.12 neither does c * . However, by Proposition 2.11 we have that c * : Γ * ⊢ ∆ * . This is absurd since any well-typed command of the λµ μ-calculus normalizes [START_REF] Polonovski | Strong normalization of lambda-bar-mu-mu-tilde-calculus with explicit substitutions[END_REF]. (wit (t, V )) * = π 1 (µα .⟨V * || μa.⟨(t * , a)||α⟩⟩) ↣ π 1 (µα .⟨(t * , V * )||α⟩) ↣ π 1 (t * , V * ) = µα .⟨(t * , t * )|| μ(a 1 , a 2 ).⟨a 1 ||α⟩⟩ ↣ µα .⟨t * ||α⟩ ↣ t * • Case ⟨µα .c ||e⟩ ⇝ c[e/α]: (⟨µα .c ||e⟩) * = ⟨µα .c * ||e * ⟩ ↣ c * [e * /α] = c[e/α] * • Case ⟨V || μa.c⟩ ⇝ c[V /a]: (⟨V || μa.c⟩) * = ⟨V * || μa.c * ⟩ ↣ c * [V * /a] = c[V /a] (⟨prf (t, V )||e⟩) * = ⟨π 2 (µα .⟨V * || μa.⟨(t * , a)||α⟩⟩)||e * ⟩ ↣ ⟨π 2 (µα .⟨(t * , V * )||α⟩)||e * ⟩ ↣ ⟨π 2 (t * , V * )||e * ⟩ = ⟨µα .⟨(t * , V * )|| μ(a 1 , a 2 ).⟨a 2 ||α⟩⟩||e * ⟩ = ⟨(t * , V * )|| μ(a 1 , a 2 ). □ Using the normalization, we can finally prove the soundness of the system. Theorem 2.14 (Soundness). For any p ∈ dL, we have ⊬ p : ⊥ . Proof. We actually start by proving by contradiction that a command c ∈ dL cannot be welltyped with empty contexts. Indeed, let us assume that there exists such a command c : (⊢). By normalization, we can reduce it to c ′ = ⟨p ′ ||e ′ ⟩ in normal form and for which we have c ′ : (⊢) by subject reduction. Since c ′ cannot reduce and is well-typed, p ′ is necessarily a value and cannot be a free variable. Thus, e ′ cannot be of the shape μa.c ′′ and every other possibility is either ill-typed or admits a reduction, which are both absurd. We can now prove the soundness by contradiction. Assuming that there is a proof p such that ⊢ p : ⊥, we can form the well-typed command ⟨p||⋆⟩ : (⊢ ⋆ : ⊥) where ⋆ is any fresh α-variable. The previous result shows that p cannot drop the context ⋆ when reducing, since it would give rise to the command c : (⊢). We can still reduce ⟨p||⋆⟩ to a command c in normal form, and see that c has to be of the shape ⟨V ||⋆⟩ (by the same kind of reasoning, using the fact that c cannot reduce and that c : (⊢ ⋆ : ⊥) by subject reduction). Therefore, V is a value of type ⊥. Since there is no typing rule that can give the type ⊥ to a value, this is absurd. □ Toward a continuation-passing style translation The difficulties we encountered while defining our system mostly came from the interaction between classical control and dependent types. Removing one of these two ingredients leaves us with a sound system in both cases. Without dependent types, our calculus amounts to the usual λµ μ-calculus. And without classical control, we would obtain an intuitionistic dependent type theory that we could easily prove sound. To prove the correctness of our system, we might be tempted to define a translation to a subsystem without dependent types, or without classical control. We will discuss later in Section 5 a solution to handle the dependencies. We will focus here on the possibility of removing the classical part from dL, that is to define a translation that gets rid of the classical control. The use of continuationpassing style translations to address this issue is very common, and it was already studied for the simply-typed λµ μ-calculus [START_REF] Curien | The duality of computation[END_REF]. However, as it is defined to this point, dL is not suitable for the design of a CPS translation. Indeed, in order to fix the problem of desynchronization of typing with respect to the execution, we have added an explicit list of dependencies to the type system of dL. Interestingly, if this solved the problem inside the type system, the very same phenomenon happens when trying to define a CPS translation carrying the type dependencies. Let us consider, as discussed in Section 2.5, the case of a command ⟨q|| μa.⟨p||e⟩⟩ with p : B[a] and e : B[q]. Its translation is very likely to look like: q μa.⟨p||e⟩ = q (λa.( p e )), where p has type (B[a] → ⊥) → ⊥ and e type B[q] → ⊥, hence the sub-term p e will be ill-typed. Therefore, the fix at the level of typing rules is not satisfactory, and we need to tackle the problem already within the reduction rules. We follow the idea that the correctness is guaranteed by the head-reduction strategy, preventing ⟨p||e⟩ from reducing before the substitution of a was made. We would like to ensure that the same thing happens in the target language (that will also be equipped with a head-reduction strategy), namely that p cannot be applied to e before q has furnished a value to substitute for a. This would correspond informally to the term 16 : ( q (λa. p )) e . Assuming that q eventually produces a value V , the previous term would indeed reduce as follows: ( q (λa. p )) e → ((λa. p ) V ) e → p [ V /a] e Since p [ V /a] now has a type convertible to (B[q] → ⊥) → ⊥, the term that is produced in the end is well-typed. The first observation is that if q, instead of producing a value, was a classical proof throwing the current continuation away (for instance µα .c where α FV (c)), this would lead to the unsafe reduction: (λα . c (λa. p )) e → c e . Indeed, through such a translation, µα would only be able to catch the local continuation, and the term would end in c e instead of c . We thus need to restrict ourselves at least to proof terms that could not throw the current continuation. The second observation is that such a term suggests the use of delimited continuations 17 to temporarily encapsulate the evaluation of q when reducing such a command: ⟨λa.p||q • e⟩ ⇝ ⟨µ t p.⟨q|| μa.⟨p|| t p⟩⟩||e⟩. Under the guarantee that q will not throw away the continuation 18 μa.⟨p|| t p⟩, this command is safe and will mimic the aforedescribed reduction: ⟨µ t p.⟨q|| μa.⟨p|| t p⟩⟩||e⟩ ⇝ ⟨µ t p.⟨V || μa.⟨p|| t p⟩⟩||e⟩ ⇝ ⟨µ t p.⟨p[V /a]|| t p⟩||e⟩ ⇝ ⟨p[V /a]||e⟩. This will also allow us to restrict the use of the list of dependencies to the derivation of judgments involving a delimited continuation, and to fully absorb the potential inconsistency in the type of t p. In Section 3, we will extend the language according to this intuition, and see how to design a continuation-passing style translation in Section 4. EXTENSION OF THE SYSTEM Limits of the value restriction In the previous section, we strictly restricted the use of dependent types to proof terms that are values. In particular, even though a proof term might be computationally equivalent to some value (say µα .⟨V ||α⟩ and V for instance), we cannot use it to eliminate a dependent product, which is unsatisfactory. We will thus relax this restriction to allow more proof terms within dependent types. 16 We will see in Section 4.4 that such a term could be typed by turning the type A → ⊥ of the continuation that q is waiting for into a (dependent) type Π a:A R[a] parameterized by R. This way we could have q : ∀R .(Π a:A R[a] → R[q]) instead of q : ((A → ⊥) → ⊥). For R[a] := (B(a) → ⊥) → ⊥, the whole term is well-typed. Readers familiar with realizability will also note that such a term is realizable, since it eventually terminates on a correct term p[q/a] e . 17 We stick here to the presentations of delimited continuations in [START_REF] Ariola | A type-theoretic foundation of delimited continuations[END_REF][START_REF] Herbelin | An approach to call-by-name delimited continuations[END_REF], where t p is used to denote the top-level delimiter. 18 Otherwise, this could lead to an ill-formed command ⟨µ t p.c ||e ⟩ where c does not contain t p. Proofs p ::= ::= V | (t, p N ) | µ⋆.c N fragment | prf p N | subst p N q N c N ::= ⟨p N ||e N ⟩ e N ::= ⋆ | μa.c N (a) Language ⟨µα .c ||e⟩ ⇝ c[e/α] ⟨λa.p||q • e⟩ q ∈nef ⇝ ⟨µ t p.⟨q|| μa.⟨p|| t p⟩⟩||e⟩ ⟨λa.p||q • e⟩ ⇝ ⟨q|| μa.⟨p||e⟩⟩ ⟨λx .p||V t • e⟩ ⇝ ⟨p[V t /x]||e⟩ ⟨V p || μa.c⟩ ⇝ c[V p /a] ⟨(V t , p)||e⟩ p V ⇝ ⟨p|| μa.⟨(V t , a)||e⟩⟩ ⟨prf (V t , V p )||e⟩ ⇝ ⟨V p ||e⟩ ⟨prf p||e⟩ ⇝ ⟨µ t p.⟨p|| μa.⟨prf a|| t p⟩⟩||e⟩ ⟨subst p q||e⟩ p V ⇝ ⟨p|| μa.⟨subst a q||e⟩⟩ ⟨subst refl q||e⟩ ⇝ ⟨q||e⟩ ⟨µ t p.⟨p|| t p⟩||e⟩ ⇝ ⟨p||e⟩ c → c ′ ⇒ ⟨µ t p.c ||e⟩ ⇝ ⟨µ t p.c ′ ||e⟩ wit p → t ⇐ ∀α, ⟨p||α⟩ ⇝ ⟨(t, p ′ )||α⟩ t → t ′ ⇒ c[t] ⇝ c[t ′ ] where: We can follow several intuitions. First, we saw at the end of the previous section that we could actually allow any proof term as long as its CPS translation uses its continuation and uses it only once. We do not have such a translation yet, but syntactically, these are the proof terms that can be expressed (up to α-conversion) in the λµ μ-calculus with only one continuation variable (that we write ⋆ in Figure 6), and which do not contain application 19 . We insist on the fact that this defines a syntactic subset of proofs. Indeed, ⋆ is only a notation and any proof defined with only one continuation variable is α-convertible to denote this continuation variable with ⋆. For instance, µα .⟨µβ ⟨V ||β⟩||α⟩ belongs to this category since: µα .⟨µβ .⟨V ||β⟩||α⟩ = α µ⋆.⟨µ⋆.⟨V ||⋆⟩||⋆⟩ Interestingly, this corresponds exactly to the so-called negative-elimination-free (nef) proofs of Herbelin [START_REF] Herbelin | A constructive proof of dependent choice, compatible with classical logic[END_REF]. To interpret the axiom of dependent choice, he designed a classical proof system with dependent types in natural deduction, in which the dependent types allow the use of nef proofs. V t ::= x | n V p ::= a | λa.p | λx .p | (V t , V p ) | refl c[t] ::= ⟨(t, p)||e⟩ | ⟨λx .p||t • e⟩ (b) Reduction rules Second, Lepigre defined in recent work [START_REF] Lepigre | A classical realizability model for a semantical value restriction[END_REF] a classical proof system with dependent types, where the dependencies are restricted to values. However, the type system allows derivations of judgments up to an observational equivalence, and thus any proof computationally equivalent to a value can be used. In particular, any proof in the nef fragment is observationally equivalent to a value, and hence is compatible with the dependencies of Lepigre's calculus. From now on, we consider the system dL of Section 2 extended with delimited continuations, which we call dL t p , and we define the fragment of negative-elimination-free proof terms (nef). The syntax of both categories is given by Figure 6, the proofs in the nef fragment are considered up 19 Indeed, λa .p is a value for any p, hence proofs like µα . ⟨λa .p ||q • α ⟩ can drop the continuation in the end once p becomes the proof in active position. to α-conversion for the context variables 20 . The reduction rules, given in Figure 6, are slightly different from the rules in Section 2. In the case ⟨λa.p||q • e⟩ with q ∈ nef (resp. ⟨prf p||e⟩), a delimited continuation is now produced during the reduction of the proof term q (resp. p) that is involved in the list of dependencies. As terms can now contain proofs which are not values, we enforce the call-by-value reduction by requiring that proof values only contain term values. We elude the problem of reducing terms, by defining meta-rules for them 21 . We add standard rules for delimited continuations [START_REF] Ariola | A type-theoretic foundation of delimited continuations[END_REF][START_REF] Herbelin | An approach to call-by-name delimited continuations[END_REF], expressing the fact that when a proof µ t p.c is in active position, the current context is temporarily frozen until c is fully reduced. Delimiting the scope of dependencies Regarding the typing rules, which are given in Figure7, we extend the set D to be the nef fragment: D ≜ nef and we now distinguish two modes. The regular mode corresponds to a derivation without dependency issues whose typing rules are the same as in Figure 4 without the list of dependencies; plus the new rule ( t p I ) for the introduction of delimited continuations. The dependent mode is used to type commands and contexts involving t p, and we use the symbol ⊢ d to denote these sequents. There are three rules: one to type t p, which is the only one where we use the dependencies to unify dependencies; one to type context of the form μa.c (the rule is the same as the former rule for μa.c in Section 2); and a last one to type commands ⟨p||e⟩, where we observe that the premise for p is typed in regular mode. Additionally, we need to extend the congruence to make it compatible with the reduction of nef proof terms (that can now appear in types), we thus add the rules: A[p] ▷ A[q] if ∀α (⟨p||α⟩ ⇝ ⟨q||α⟩) A[⟨q|| μa.⟨p||⋆⟩⟩] ▷ A[⟨p[q/a]||⋆⟩] with p, q ∈ nef Due to the presence of nef proof terms (which contain a delimited form of control) within types and lists of dependencies, we need the following technical lemma to prove subject reduction. Lemma 3.1. For any context Γ, ∆, any type A and any e, µ⋆.c: ⟨µ⋆.c ||e⟩ : Γ ⊢ d ∆, t p : B; ε ⇒ c[e/⋆] : Γ ⊢ d ∆, t p : B; ε. Proof. By definition of the nef proof terms, µ⋆.c is of the general form µ⋆.c = µ⋆.⟨p 1 || μa 1 .⟨p 2 || μa 2 .⟨. . .|| μa n-1 .⟨p n ||⋆⟩⟩⟩⟩. For simplicity reasons, we will only give the proof for the case n = 2, so that a derivation for the hypothesis is of the form (we assume the Thus, we have to show that we can turn Π e into a derivation Π ′ e of Γ | e : A ⊢ d ∆ t p ; {a 1 |p 1 }{•|p 2 } with ∆ t p ≜ ∆, t p : B, since this would allow us to build the following derivation: Π 1 Γ ⊢ p 1 : A 1 | ∆ Π 2 Γ, a 1 : A 1 ⊢ p 2 : A | ∆ Π ′ e • • • | e : A ⊢ d ∆ t p ; {a 1 |p 1 }{•|p 2 } ⟨p 2 ||⋆⟩ : Γ, a 1 : A 1 ⊢ ∆ t p ; {a 1 |p 1 } (Cut) Γ | μa 1 .⟨p 2 ||e⟩ : A 1 ⊢ d ∆ t p ; {•|p 1 } ( μ) ⟨p 1 || μa 1 .⟨p 2 ||e⟩⟩ : Γ ⊢ d ∆ t p ; ε (Cut) It suffices to prove that if the list of dependencies is used in Π e to type t p, we can still give a derivation with the new one. In practice, it corresponds to showing that for any variable a and any list of dependencies σ : {a|µ⋆.c}σ ⇛ {a 1 |p 1 }{a|p 2 }σ . For any A ∈ B σ , by definition we have: A[µ⋆.⟨p 1 || μa 1 .⟨p 2 ||⋆⟩⟩/b] ≡ A[µ⋆.⟨p 2 [p 1 /a 1 ]||⋆⟩/b] ≡ A[p 2 [p 1 /a 1 ]/b] = A[p 2 /b][p 1 /a 1 ]. Hence for any A ∈ B {a |µ⋆.c }σ , there exists A ′ ∈ B {a 1 |p 1 } {a |p 2 }σ such that A ≡ A ′ , and we can derive: A ′ ∈ B {a 1 |p 1 } {a |p 2 }σ Γ | t p : A ′ ⊢ d ∆, t p : B; {a 1 |p 1 }{b|p 2 }σ A ≡ A ′ Γ | t p : A ⊢ d ∆, t p : B; {a 1 |p 1 }{b|p 2 }σ (≡ l ) □ We can now prove subject reduction for dL t p . Theorem 3.2 (Subject reduction). If c, c ′ are two commands of dL t p such that c : (Γ ⊢ ∆) and c ⇝ c ′ , then c ′ : (Γ ⊢ ∆). Proof. Actually, the proof is slightly easier than for Theorem 2.9, because most of the rules do not involve dependencies. We only give some key cases. • Case ⟨λa.p||q • e⟩ ⇝ ⟨µ t p.⟨q|| μa.⟨p|| t p⟩⟩||e⟩ with q ∈ nef. A typing derivation for the command on the left is of the form: Π p Γ, a : A ⊢ p : B | ∆ Γ ⊢ λa.p : Π a:A B | ∆ (→ l ) Π q Γ ⊢ q : A | ∆ Π e Γ | e : B[q/a] ⊢ ∆ Γ | q • e : Π a:A B ⊢ ∆ (→ l ) ⟨λa.p||q • e⟩ : Γ ⊢ ∆ (Cut) • Case ⟨µ t p.c ||e⟩ ⇝ ⟨µ t p.c ′ ||e⟩ with c ⇝ c ′ . This case corresponds exactly to Theorem 2.9, except for the rule ⟨µα .c ||e⟩ ⇝ c[e/α], since µα .c is a nef proof term (remember we are inside a delimited continuation), but this corresponds precisely to Lemma 3.1. □ Remark 3.3. Interestingly, we could have already taken D ≜ nef in dL and still be able to prove the subject reduction property. The only difference would have been for the case ⟨µα .c ||e⟩ ⇝ c[e/α] when µα .c is nef. Indeed, we would have had to prove that such a reduction step is compatible with the list of dependencies, as in the proof for dL t p , which essentially amounts to Lemma 3.1. This shows that the relaxation to the nef fragment is valid even without delimited continuations. To sum up, the restriction to nef is sufficient to obtain a sound type system, but is not enough to obtain a calculus suitable for a continuation-passing style translation. As we will now see, delimited continuations are crucial for the soundness of the CPS translation. Observe that they also provide us with a type system in which the scope of dependencies is more delimited. A CONTINUATION-PASSING STYLE TRANSLATION We shall now see how to define a continuation-passing style translation from dL t p to an intuitionistic type theory, and use this translation to prove the soundness of dL t p . Continuation-passing style translations are indeed very useful to embed languages with classical control into purely functional ones [START_REF] Curien | The duality of computation[END_REF][START_REF] Griffin | A formulae-as-type notion of control[END_REF]. From a logical point of view, they generally amount to negative translations that allow us to embed classical logic into intuitionistic logic [START_REF] Ferreira | On various negative translations[END_REF]. Yet, we know that removing classical control (i.e. classical logic) from our language leaves us with a sound intuitionistic type theory. We will now see how to design a CPS translation for our language which will allow us to prove its soundness. Target language We choose the target language to be an intuitionistic theory in natural deduction that has exactly the same elements as dL t p , except the classical control. The language distinguishes between terms (of type N) and proofs, it also includes dependent sums and products for types referring to terms, as well as a dependent product at the level of proofs. As is common for CPS translations, the evaluation follows a head-reduction strategy. The syntax of the language and its reduction rules are given by Figure 8. The type system, also presented in Figure 8, is defined as expected, with the addition of a secondorder quantification that we will use in the sequel to refine the type of translations of terms and nef proofs. As in dL t p , the type system has a conversion rule, where the relation A ≡ B is the symmetric-transitive closure of A ▷ B, defined once again as the congruence over the reduction -→ and by the rules: 0 = 0 ▷ ⊤ 0 = S(u) ▷ ⊥ S(t) = 0 ▷ ⊥ S(t) = S(u) ▷ t = u. Translation of proofs and terms We can now define the continuation-passing style translation of terms, proofs, contexts and commands. The translation is given in Figure 9, in which we tag some lambdas with a bullet λ • for technical reasons. The translation of delimited continuations follows the intuition we presented in Section 2.8, and the definition for stacks t • e and q • e (with q nef) inlines the reduction producing a command with a delimited continuation. All the other rules are natural in the sense that they Γ ⊢ q : t = u Γ ⊢ q : A[t] Γ ⊢ subst p q : A[u] (subst) Γ ⊢ p : A A ≡ B Γ ⊢ p : B (CONV) (c) Type system reflect the reduction rule ⇝, except for the translation of pairs (t, p): (t, p) p ≜ λk. p p ( t t (λxa.k (x, a))) The natural definition would have been λk. t t (λu. p p λq.k (u, q)), however such a term would have been ill-typed (while the former definition is correct, as we will see in the proof of Lemma 4.9). Indeed, the type of p p depends on t, while the continuation (λq.k (u, q)) depends on u, but both become compatible once u is substituted by the value return by t t . This somewhat strange definition corresponds to the intuition that we reduce t t within a delimited continuation 22 , in order to guarantee that we will not reduce p p before t t has returned a value to substitute for u. The complete translation is given in Figure 9. Before defining the translation of types, we first state a lemma expressing the fact that the translations of terms and nef proof terms use the continuations they are given once and only once. In particular, it makes them compatible with delimited continuations and a parametric return type. This will allow us to refine the type of their translation. Lemma 4.1. The translation satisfies the following properties: (1) For any term t in dL t p , there exists a term t + such that for any k, we have t t k → * β k t + . (2) For any nef proof p N , there exists a proof p + N such that for any k, we have p N p k → * β k p + N . wit p t ≜ λk. p p (λ • q.k (wit q)) V t V t ≜ λk.k V t a V ≜ a λa.p V ≜ λ • a. p p (V t , V p ) V ≜ ( V t V t , V V ) V p ≜ λk.k V V µα .c p ≜ λ • α . c c prf p p ≜ λ • k.( p p (λ • qλk ′ .k ′ (prf q))) k (t, p) p ≜ λ • k. p p ( t t (λxλ • a.k (x, a))) subst V q p ≜ λk. q p (λ • q ′ .k (subst V V q ′ ))) subst p q p ≜ λk. p p (λ • p ′ . q p (λ • q ′ .k (subst p ′ q ′ ))) (p V ) α e ≜ α t • e e ≜ λp.( t t (λ • v.p v)) e e q N • e e ≜ λp.( q N p (λ • v.p v)) e e (q N ∈ nef) q • e e ≜ λ • p. q p (λ • v.p v e e ) ( q n V t ≜ n x V t ≜ x refl V ≜ refl λx .p V ≜ λ • x . p p µ t p.c p ≜ λk. c t p k μa.c e ≜ λ • a. c c ⟨p|| t p⟩ t p ≜ p p μa.c e t p ≜ λ • a. c t p Fig. 9. Continuation-passing style translation In particular, we have : t t λx .x → * β t + and p N p λa.a → * β p + N Proof. Straightforward mutual induction on the structure of terms and nef proofs, adding similar induction hypothesis for nef contexts and commands. The terms t + and proofs p + are given in Figure 10. We detail the case (t, p) with p ∈ nef to give an insight of the proof. Proof. Simple proof by induction on the reduction rules for ⇝, using Lemma 4.1 for cases involving a term t. □ (t, p) p k → β p p ( t t (λxa.k (x, a))) → β ( t t (λxa.k (x, a))) p + → β (λxa.k (x, a)) t + p + → β (λa.k (t + , a)) p + → β k (t + , p + ) ( by x + ≜ x n + ≜ n (wit p) + ≜ wit p + a + ≜ a refl + ≜ refl (λa.p) + ≜ λa. p p (λx .p) + ≜ λx . p p (t, p) + ≜ (t + , p + ) (prf p) + ≜ prf p + (subst p q) + ≜ subst p + q + (µ⋆.c) + ≜ c + (µ t p.c) + ≜ c + (⟨p||⋆⟩) + ≜ p + (⟨p|| t p⟩) + ≜ p + (⟨p|| μa.c t p ⟩) + ≜ c + [p + /a] Normalization of dL t p We can in fact prove a finer result to show that normalization is preserved through the translation. Namely, we want to prove that any infinite reduction sequence in dL t p is responsible for an infinite reduction sequence through the translation. Using the preservation of typing (Proposition 4.10) together with the normalization of the target language, this will give us a proof of the normalization of dL t p for typed proof terms. To this purpose, we roughly proceed as follows: (1) we identify a set of reduction steps in dL t p which are directly reflected into a strictly positive number of reduction steps through the CPS; (2) we show that the other steps alone can not form an infinite sequence of reductions; (3) we deduce that every infinite sequence of reductions in dL t p gives rise to an infinite sequence through the translation. The first point corresponds thereafter to Proposition 4.5, the second one to the Proposition 4.6. As a matter of fact, the most difficult part is somehow anterior to these points. It consists in understanding how a reduction step can be reflected through the translation in a way that is sufficient to ensure the preservation of normalization (that is the third point). Instead of stating the result directly and giving a long and tedious proof of its correctness, we will rather sketch its main steps. First of all, we split the reduction rule → β into two different kinds of reduction steps: • administrative reductions, that we denote by -→ a , which correspond to continuationpassing and computationally irrelevant (w.r.t. to dL t p ) reduction steps. These are defined as the β-reduction steps of non-annotated λs. • distinguished reductions, that we denote by -→ • , which correspond to the image of a reduction step through the translation. These are defined as every other rules, that is to say the β-reduction steps of annotated λ • 's plus the rules corresponding to redexes formed with wit, prf and subst . In other words, we define two deterministic reductions -→ • and -→ a , such that the usual weakhead reduction → β is equal to the union -→ • ∪ -→ a . Our goal will be to prove that every infinite reduction sequence in dL t p will be reflected in the existence of an infinite reduction sequence for -→ • . Second, let us assume for a while that we can show that for any reduction c ⇝ c ′ , through the translation we have: c c t 0 t 1 t 2 c ′ c β * a * • 1 β * 1:30 Étienne Miquey Then by induction, it implies that if a command c 0 produces an infinite reduction sequence c 0 ⇝ c 1 ⇝ c 2 ⇝ . . ., it is reflected through the translation by the following reduction scheme: Using the fact that all reductions are deterministic, and that the arrow from c 1 c to t 02 (and c 2 c to t 12 and so on) can only contain steps of the reduction -→ a , the previous scheme in fact ensures us that we have: c c 0 c t 00 t 01 t 02 t 10 t 11 t 12 t 20 t 21 c 1 c c 2 c β * a * a * • 1 β * β * • 1 β * β * • 1 This directly implies that c 0 c produces an infinite reduction sequence and thus is not normalizing. This would be the ideal situation, and if the aforementioned steps were provable as such, the proof would be over. Yet, our situation is more subtle, and we need to refine our analysis to tackle the problem. We shall briefly explain now why we can actually consider a slightly more general reduction scheme, while trying to remain concise on the justification. Keep in mind that our goal is to preserve the existence of an infinite sequence of distinguished steps. The first generalization consists in allowing distinguished reductions for redexes that are not in head positions. The safety of this generalization follows from this proposition: Proof. By induction on the structure of t, a very similar proof can be found in [START_REF] Joachimski | Short proofs of normalization for the simply-typed λ-calculus, permutative conversions and gödel's t[END_REF]. □ Following this idea, we define a new arrow ? -→ • by: u -→ • u ′ ⇒ t[u] ? -→ • t[u ′ ] where t[] :: = [] | t ′ (t[]) | λx .t[], expressing the fact that a distinguished step can be performed somewhere in the term. We denote by -→ β + the extended reduction relation defined as the union -→ β ∪ ? -→ • , which is not deterministic. Coming back to the thread scheme we described above, we can now generalize it with this arrow. Indeed, as we are only interested in getting an infinite reduction sequence from c 0 c , the previous proposition ensures us that if t 02 (t 12 , etc.) does not normalize, it is enough to have an arrow t 01 * -→ β + t 02 (t 11 * -→ β + t 12 , etc.) to deduce that t 01 does not normalize either. Hence, it is enough to prove that we have the following thread scheme, where we took advantage of this observation: c 0 c t 00 t 01 t 02 t 10 t 11 t 12 t 20 t 21 c 1 c c 2 c β * β * β * a * a * • 1 β + * • 1 β + * • 1 In the same spirit, if we define = a to be the congruence over terms induced by administrative reductions -→ a , we can show that if a term has a redex for the distinguished relation in head position, then so does any (administratively) congruent term. -→ • u and t = a t ′ , then there exists u ′ such that t ′ 1 -→ • u ′ and u = a u ′ . Proof. By induction on t, observing that an administrative reduction can neither delete nor create redexes for -→ • . □ In other words, as we are only interested in the distinguished reduction steps, we can take the liberty to reason modulo the congruence = a . Notably, we can generalize one last time our reduction scheme, replacing the left (administrative) arrow from c i c by this congruence: c 0 c t 00 t 01 t 02 t 10 t 11 t 12 t 20 t 21 c 1 c c 2 c β * β * β * a a • 1 β + * • 1 β + * • 1 For all the reasons explained above, such a reduction scheme ensures that there is an infinite reduction sequence from c 0 c . Because of this guarantee, by induction, it is enough to show that for any reduction step c 0 ⇝ c 1 , we have: c 0 c t 0 t 1 t 2 c 1 c β * • 1 β + * a (1) In fact, as explained in the preamble of this section, not all reduction steps can be reflected this way through the translation. There are indeed 4 reduction rules, that we identify hereafter, that might only be reflected into administrative reductions, and produce a scheme of this shape (which subsumes the former): c 0 c * -→ β + t = a c 1 c (2) This allows us to give a more precise statement about the preservation of reduction through the CPS translation. Proposition 4.5 (Preservation of reduction). Let c 0 , c 1 be two commands of dL t p . If c 0 ⇝ c 1 , then it is reflected through the translation into a reduction scheme (1), except for the rules: ⟨subst p q||e⟩ p V ⇝ ⟨p|| μa.⟨subst a q||e⟩⟩ ⟨subst refl q||e⟩ ⇝ ⟨q||e⟩ ⟨µ t p.⟨p|| t p⟩||e⟩ ⇝ ⟨p||e⟩ c[t] ⇝ c[t ′ ] which are reflected into the reduction scheme (2). Proof. The proof is done by induction on the reduction ⇝ (see Figure 6). To ease the notations, we will often write λ • v.(λ • x . p p ) v -→ • λ • x . p p where we perform α-conversion to identify λ • v. p p [v/x] and λ • x . p p . Additionally, to facilitate the comprehension of the steps corresponding to the congruence = a , we use an arrow ? -→ a to denote the possibility of performing an administrative reduction not in head position, defined by: u -→ a u ′ ⇒ t[u] ? -→ a t[u ′ ] We write -→ a + the union -→ a ∪ ? -→ a . • Case ⟨µα .c ||e⟩ ⇝ c[e/α]: We have: ⟨µα .c ||e⟩ c = (λ • α . c c ) e e -→ • c c [ e e /α] = c[e/α] c • Case ⟨λa.p||q • e⟩ ⇝ ⟨q|| μa.⟨p||e⟩⟩: We have: ⟨λa.p||q • e⟩ c = (λk.k (λ • a. p p )) λ • p. q p (λ • v.p v e e ) -→ a (λ • p. q p (λ • v.p v e e )) λ • a. p p -→ • q p (λ • v.(λ • a. p p ) v e e ) ? -→ • q p (λ • a. p p e e ) = ⟨q|| μa.⟨p||e⟩⟩ c • Case ⟨λa.p||q N • e⟩ q N ∈nef ⇝ ⟨µ t p.⟨q N || μa.⟨p|| t p⟩⟩||e⟩: We know by Lemma 4.1 that q N being nef, it will use, and use only once, the continuation it is applied to. Thus, we know that if k -→ • k ′ , we have that: q N p k * -→ β k q + N -→ • k ′ q + N β ←-q N p k ′ and we can legitimately write q N p k -→ • q N p k ′ in the sense that it corresponds to performing now a reduction that would have been performed in the future. Using this remark, we have: ⟨λa.p||q N • e⟩ c = (λk.k (λ • a. p p )) λp.( q N p (λ • v.p v)) e e 2 -→ a ( q N p (λ • v.(λ • a. p p ) v)) e e -→ • ( q N p (λ • a. p p )) e e a ←-(λk.( q N p (λ • a. p p )) k) e e = ⟨µ t p.⟨q N || μa.⟨p|| t p⟩⟩||e⟩ c • Case ⟨λx .p||V t • e⟩ ⇝ ⟨p[V t /x]||e⟩: Since V t is a value (i.e. x or n), we have V t t = λk.k V t V t . In particular, it is easy to deduce that p[V t /x] p = p p [ V t V t /x], and then we have: ⟨λx .p||V t • e⟩ c = (λk.k (λ • x . p p ))λp.( V t t (λ • v.p v)) e e 2 -→ a ( V t t (λ • v.(λ • x . p p ) v)) e e -→ a ((λ • v.(λ • x . p p ) v) V t V t ) e e -→ • ((λ • x . p p ) V t V t ) e e -→ • ( p p [ V t V t /x]) e e = p[V t /x] p e e = ⟨p[V t /x]||e⟩ • Case ⟨V || μa.c⟩ ⇝ c[V /a]: Similarly to the previous case, we have V p = λk.k V V and thus c[V /x] c = p p [ V V /a]. ⟨V p || μa.c⟩ c = (λk.k V V )λ • a. c c -→ a (λ • a. c c ) V V -→ • c c [ V V /a] = c[V /a] c • Case ⟨(V t , p)||e⟩ p V ⇝ ⟨p|| μa.⟨(V t , a)||e⟩⟩: We have : ⟨(V t , p)||e⟩ c = (λ • k. p p ( V t t (λxλ • a.k (x, a))) e e -→ • p p ( V t t (λxλ • a. e e (x, a))) -→ a + p p ((λxλ • a. e e (x, a)) V t V t ) -→ a + p p (λ • a. e e ( V t V t , a)) a +←-p p (λ • a. (V t , a) p e e ) a +←-(λk p p (λ • a. (V t , a) p k)) e e = ⟨p|| μa.⟨(V t , a)||e⟩⟩ c • Case ⟨prf p||e⟩ ⇝ ⟨µ t p.⟨p|| μa.⟨prf a|| t p⟩⟩||e⟩: We have: ⟨prf p)||e⟩ c = λ • k.( p p (λ • aλk ′ .k ′ (prf a))) k) e e -→ • ( p p (λ • a.λk ′ .k ′ (prf a))) e e a ←-(λk.( p p (λ • a.λk ′ .k ′ (prf a))) k) e e = ⟨µ t p.⟨p|| μa.⟨prf a|| t p⟩⟩||e⟩ c • Case ⟨prf (V t , V p )||e⟩ ⇝ ⟨V p ||e⟩: We have: ⟨prf (V t , V p )||e⟩ c = λ • k.((λk.k ( V t V , V p V )) (λ • qλk ′ .k ′ (prf q))) k) e e -→ • ((λk.k ( V t V , V p V )) (λ • qλk ′ .k ′ (prf q))) e e -→ a ((λ • qλk ′ .k ′ (prf q))( V t V , V p V )) e e -→ • (λk ′ .k ′ (prf ( V t V , V p V ))) e e -→ a e e (prf ( V t V , V p V ))) ? -→ • e e V p V a ←-⟨V p ||e⟩ c • Case ⟨subst p q||e⟩ p V ⇝ ⟨p|| μa.⟨subst a q||e⟩⟩: We have: ⟨subst p q||e⟩ c = (λk. p p (λ • a. q p (λ • q ′ .k (subst a q ′ )))) e e -→ a p p (λ • a. q p (λ • q ′ . e e (subst a q ′ ))) ? a ←-p p (λ • a.(λk. q p (λ • q ′ .k (subst a q ′ ))) e e ) = ⟨p|| μa.⟨subst a q||e⟩⟩ c • Case ⟨subst refl q||e⟩ ⇝ ⟨q||e⟩: We have: ⟨subst refl q||e⟩ c = (λk. q p (λ • q ′ .k (subst refl q ′ ))) e e -→ a q p (λ • q ′ . e e (subst refl q ′ )) ? -→ • q p (λ • q ′ . e e q ′ ) ? -→ • q p e e = ⟨q||e⟩ c • Case ⟨µ t p.⟨p|| t p⟩||e⟩ ⇝ ⟨p||e⟩: We have: .c ′ ||e⟩ • Case t → t ′ ⇒ c[t] ⇝ c[t ′ ]: As such, the translation does not allow an analysis of this case, mainly because we did not give an explicit small-step semantics for terms, and defined terms reduction through a big-step semantics: ∀α, ⟨p||α⟩ * ⇝ ⟨(t, q)||α⟩ ⇒ wit p → t However, we claim that we could have extended the language of dL t p with commands for terms: c t ::= ⟨t ⟨V t || μx .c t ⟩ ⇝ c t [V t /x] ⟨wit (V t , V p )||e t ⟩ ⇝ ⟨V t ||e t ⟩ ⟨V p || μ ť p.⟨ ť p||e⟩⟩ ⇝ ⟨V p ||e⟩ c ⇝ c ′ ⇒ ⟨p|| μ ť p.c⟩ ⇝ ⟨p|| μ ť p.c ′ ⟩ It is worth noting that these rules simulate the big-step definitions we had before while preserving the global call-by-value strategy. Defining the translation for terms in the extended syntax: wit V t t ≜ λk.k (wit V t V t ) wit p t ≜ λk. p p (λ • q.k (wit q)) μ ť p.c t t ≜ c t t μx .c t ≜ λ • x . c c ⟨t ||e t ⟩ t ≜ t t e t t ť p p ≜ λ • k.k We can then prove that each reduction rule satisfies the expected scheme. Case ⟨λx .p||t • e⟩ ⇝ ⟨µ t p.⟨t || μx .⟨p|| t p⟩⟩||e⟩: We have: We have: ⟨λx .p||t • e⟩ = (λ • k.k λ • x . p p ) ( wit (V t , V p ) t e t t = (λk.k (wit ( V t V t , V p V ))) e t t -→ a e t t (wit ( V t V t , V p V )) -→ • e t t V t V t a ←-(λk.k V t V t ) e t t = V t t e t Case ⟨V t || μx .c t ⟩ ⇝ c t [V t /x]: We have: V t t μx .c t = (λk.k V t V t ) λ • x . c c -→ a (λ • x . c c ) V t V t -→ • c c [ V t V t /x] = c[V t /x] c Case ⟨V || μ t p.⟨ t p||e⟩⟩ ⇝ ⟨V ||e⟩: We have: V p μ t p.⟨ t p||e⟩ e = (λk.k V V ) ((λk.k) e e ) -→ a ((λk.k) e e ) V V -→ a e e V V a ←-(λk.k V V ) e e = ⟨V ||e⟩ c Case c ⇝ c ′ ⇒ ⟨V || μ t p.c⟩ ⇝ ⟨V || μ t p.c ′ ⟩: This case is similar to the case for delimited continuations proved before, we only need to use the induction hypothesis for c c to get: V p μ t p.c e = (λk.k V V ) c c -→ a c c V V * -→ β + t V V = a c ′ c V V a +←-(λk.k V V ) c ′ c = V p μ t p.c ′ e □ Proposition 4.6. There is no infinite sequence only made of reductions: (1) ⟨subst p q||e⟩ p V ⇝ ⟨p|| μa.⟨subst a q||e⟩⟩ (2) ⟨subst refl q||e⟩ ⇝ ⟨q||e⟩ (3) ⟨µ t p.⟨p|| t p⟩||e⟩ ⇝ ⟨p||e⟩ (4) c[t] ⇝ c[t ′ ] Proof. It is sufficient to observe that if we define the following quantities: (1) the quantity of subst p q with p not a value within a command, (2) the quantity of subst within a command, (3) the quantity of t p within a command, (4) the quantity of wit terms within a command. then the rule (1) makes quantity (1) decrease while preserving the others. Likewise, (2) decreases quantity (2) preserves the other, and so on. All in all, we have a bound on the maximal number of steps for the reduction restricted to these four rules. □ Proposition 4.7 (Preservation of normalization). If c c normalizes, then c is also normalizing. Proof. Reasoning by contraposition, let us assume that c is not normalizing. Then in any infinite reduction sequence from c, according to the previous proposition, there are infinitely many steps that are reflected through the CPS into at least one distinguished step (Proposition 4.5). Thus, there is an infinite reduction sequence from c c too. □ Proof. Using the preservation of typing that we shall prove in the next section (Proposition 4.10), we know that if c is typed in dL t p , then its image c c is also typed. Using the fact that typed terms of the target language are normalizing, we can finally apply the previous proposition to deduce that c normalizes. □ Translation of types We can now define the translation of types in order to show further that the translation p p of a proof p of type A is of type A * . The type A * is the double-negation of a type A + that depends on the structure of A. Thanks to the restriction of dependent types to nef proof terms, we can interpret a dependency in p (resp. t) in dL t p by a dependency in p + (resp. t + ) in the target language. Lemma 4.1 indeed guarantees that the translation of a nef proof p will eventually return p + to the continuation it is applied to. The translation is defined by: A * ≜ ( A + → ⊥) → ⊥ t = u + ≜ t + = u + ∀x N .A + ≜ ∀x N . A * ⊤ + ≜ ⊤ ∃x N .A + ≜ ∃x N . A + ⊥ + ≜ ⊥ Π a:A B + ≜ Π a: A + B * N + ≜ N Observe that types depending on a term of type T are translated to types depending on a term of the same type T , because terms can only be of type N. As we shall discuss in Section 6.2, this will no longer be the case when extending the domain of terms. To extend the translation for types to the translation of contexts, we consider that we can unify left and right contexts into a single one that is coherent with respect to the order in which the hypotheses have been introduced. We denote this context by Γ ∪ ∆, where the assumptions of Γ remain unchanged, while the former assumptions (α : A) in ∆ are denoted by (α : A ⊥ ⊥ ). The translation of unified contexts is given by: Γ, a : A ≜ Γ + , a : A + Γ, x : N ≜ Γ + , x : N Γ, α : A ⊥ ⊥ ≜ Γ + , α : A + → ⊥. As explained informally in Section 2.8 and stated by Lemma 4.1, the translation of a nef proof term p of type A uses its continuation linearly. In particular, this allows us to refine its type to make it parametric in the return type of the continuation. From a logical point of view, it amounts to replacing the double-negation (A → ⊥) → ⊥ by Friedman's translation [START_REF] Friedman | Classically and intuitionistically provably recursive functions[END_REF]: ∀R.(A → R) → R. It is worth noticing the correspondences with the continuation monad [START_REF] Filinski | Representing monads[END_REF]. Also, we make plain use here of the fact that the nef fragment is intuitionistic, so to speak. Indeed, it would be impossible to attribute this type 23 to the translation of a (really) classical proof. Moreover, we can even make the return type of the continuation dependent on its argument (that is a type of the shape Π a:A R(a)), so that the type of p p will correspond to the elimination rule: ∀R.(Π a:A R(a) → R(p + )). This refinement will make the translation of nef proofs compatible with the translation of delimited continuations. Lemma 4.9 (Typing translation for nef proofs). The following holds: (1) For any term t, if Γ ⊢ t : N | ∆ then Γ ∪ ∆ ⊢ t t : ∀X .(∀x N .X (x) → X (t + )). (2) For any nef proof p, if Γ ⊢ p : A | ∆ then Γ ∪ ∆ ⊢ p p : ∀X .(Π a: A + X (a) → X (p + ))). (3) For any nef command c, if c : (Γ ⊢ ∆, ⋆ : B) then Γ ∪ ∆ , ⋆ : Π b:B + X (b) ⊢ c c : X (c + )). Proof. The proof is done by induction on typing derivations. We only give the key cases of the proof. • Case (µ). For this case, we could actually conclude directly using the induction hypothesis for c. Rather than that, we do the full proof for the particular case µ⋆.⟨p|| μa.⟨q||⋆⟩⟩, which condensates the proofs for µ⋆.c and the two possible cases ⟨p N ||e N ⟩ and ⟨p N ||⋆⟩ of nef commands. This case corresponds to the following typing derivation in dL t p : Π p Γ ⊢ p : A | ∆ Π q Γ, a : A ⊢ q : B | ∆ • • • | ⋆ : B ⊢ ∆, ⋆ : B ⟨q||⋆⟩ : Γ, a : A ⊢ ∆, ⋆ : B the reduction rules for the fragment of Lepigre's calculus we use, for the type system we refer the reader to [START_REF] Lepigre | A classical realizability model for a semantical value restriction[END_REF]: Values Terms Stacks Processes Formulas v, w ::= x | λx .t | {l 1 = v 1 , l 2 = v 2 } t, u ::= a | v | t u | µα .t | p | v.l i π , ρ ::= α | v • π | [t]π p, q ::= t * π A, B ::= X n (t 1 , . . . , t n ) | A → B | ∀a.A | ∃a.A | ∀X n .A | {l 1 : A 1 , l 2 : A 2 } | t ∈ A The reduction ≻ is defined as the smallest relation satisfying: t u * π ≻ u * [t]π v * [t]π ≻ t * v • π λx .t * v • π ≻ t[x := v] * π µα .t * π ≻ t[α := π ] * π p * π ≻ p (v 1 , v 2 ).l i ≻ v i It is worth noting that the call-by-value strategy is obtained via the construction [t]π which allows to evaluate the argument of t to a value before pushing it onto the stack. Even though records are only defined for values, we can define pairs and projections as syntactic sugar: ( t 1 , t 2 ) ≜ (λv 1 v 2 .{l 1 = v 1 , l 2 = v 2 }) t 1 t 2 fst(t) ≜ (λx .(x .l 1 )) t snd(t) ≜ (λx .(x .l 2 )) t A 1 ∧ A 2 ≜ {l 1 : A 1 , l 2 : A 2 } Similarly, only values can be pushed on stacks, but we can define processes 25 with stacks of the shape t • π as syntactic sugar: t * u • π ≜ tu * π We first define the translation for types (extended for typing contexts) where the predicate Nat(x) is defined 26 as usual in second-order logic: Nat(x) ≜ ∀X .(X (0) → ∀y.(X (y) → X (S(y))) → X (x)) and t t is the translation of the term t given in Figure 11. (∀x N .A) * ≜ ∀x .(Nat(x) → A * ) (∃x N .A) * ≜ ∃x .(Nat(x) ∧ A * ) (t = u) * ≜ ∀X .(X ( t t ) → X ( u t )) ⊤ * ≜ ∀X .(X → X ) ⊥ * ≜ ∀XY .(X → Y ) (Π a:A B) * ≜ ∀a.((a ∈ A * ) → B * ) (Γ, x : N) * ≜ Γ * , x : Nat(x) (Γ, a : A) * ≜ Γ * , a : A * (Γ, α : A ⊥ ⊥ ) * ≜ Γ * , α : ¬A * Note that the equality is mapped to Leibniz equality, and that the definitions of ⊥ * and ⊤ * respectively correspond to (0 = 1) * and (0 = 0) * in order to make the conversion rule admissible through the translation. The translation for terms, proofs, contexts and commands of dL t p , given in Figure 11 is almost straightforward. We only want to draw the reader's attention on a few points: • the equality being translated as Leibniz equality, refl is translated as the identity λa.a, which also matches with ⊤ (α). By hypothesis, σ realizes Γ, α : A ⊥ ⊥ from which we obtain ασ = σ (α) ∈ A ⊥ σ . ( * ). We need to show that tσ * πσ ∈ B ⊥⊥ σ , so we take ρ ∈ B ⊥ σ and show that (tσ * πσ ) * ρ ∈ ⊥ ⊥. By anti-reduction, it is enough to show that (tσ * πσ ) ∈ ⊥ ⊥. This is true by induction hypothesis, since tσ ∈ A ⊥⊥ σ and πσ ∈ A ⊥ σ . (µ). The proof is the very same as in [START_REF] Lepigre | A classical realizability model for a semantical value restriction[END_REF]Theorem 6]. (∀ l ). By induction hypothesis, we have that πσ ∈ A[x := t] ⊥ σ . We need to show the inclusion A[x := t] ⊥ σ ⊆ ∀x .A ⊥ σ , which follows from ∀x .A σ = t ∈Λ A[x := t] σ ⊆ A[x := t] σ . (⇒ l ). If t is a value v, by induction hypothesis, we have that vσ ∈ A σ and πσ ∈ B ⊥ σ , and we need to show that vσ • πσ ∈ A ⇒ B ⊥ σ . The proof is already done in the case (⇒ e ) (see [START_REF] Lepigre | A classical realizability model for a semantical value restriction[END_REF]Theorem 6]). Otherwise, by induction hypothesis, we have that tσ ∈ A ⊥⊥ σ and πσ ∈ B ⊥ σ , and we need to show that tσ • πσ ∈ A ⇒ B ⊥ σ . So we consider λx .u ∈ A ⇒ B σ , and show that λx .u * tσ • πσ ∈ ⊥ ⊥. We can take a reduction step, and prove instead that tσ * [λx .u]πσ ∈ ⊥ ⊥. This amounts to showing that [λx .u]π ∈ A ⊥ σ , which is already proven in the case (⇒ e ). (let). We need to show that for all v ∈ A σ , v * [tσ ]πσ ∈ ⊥ ⊥. Taking a step of reduction, it is enough to have tσ * v •πσ ∈ ⊥ ⊥. This is true since by induction hypothesis, we have tσ ∈ A ⇒ B ⊥⊥ Proof. The proof is an easy induction on the typing derivation Γ ⊢ p : A | ∆. Note that in a way, the translation of a delimited continuation decompiles it to simulate in a natural deduction fashion the reduction of the applications of functions to stacks (that could have generated the same delimited continuations in dL t p ), while maintaining the frozen context (at top-level) outside of the active command (just like a delimited continuation would do). This trick allows us to avoid the problem of dependencies conflict in the typing derivation. For instance, assuming that q 1 p (resp. q 2 p ) reduces to a value V 1 (resp. V 2 ) . p p [V 1 /a 1 ]] e e ≻ * p p [ V 1 p /a 1 ][ V 2 p /a 2 ] * e e * ≺ q 2 p * [λa 2 . p p [V 1 /a 1 ]] e e * ≺ q 1 p * [λa 1 a 2 . p p )] q 2 p • e e * ≺ (λa 1 a 2 . p p ) * q 1 p • q 2 p • e e = ⟨λa 1 λa 2 .p||q 1 • q 2 • e⟩ c where we observe that e e is always kept outside of the computations, and where each command ⟨q i || μa i .c t p ⟩ is decompiled into (µα . q i p * [λa i . c t p t p ].α) * e e , simulating the (natural deduction style) reduction of λa i . c t p t p * q i p • e e . These terms correspond somehow to the translations of former commands typable without types dependencies. □ As a corollary we get a proof of the adequacy of dL t p typing rules with respect to Lepigre's realizability model. This immediately implies the soundness of dL t p : Theorem 5.4 (Soundness). For any proof p in dL t p , we have: ⊬ p : ⊥. Proof. By contradiction, if we had a closed proof p of type ⊥, it would be translated as a realizer of ⊤ → ⊥. Therefore, p p λx .x would be a realizer of ⊥, which is impossible. □ Furthermore, the translation clearly preserves normalization (in the sense that for any c, if c does not normalize then neither does c c ), and thus the normalization of dL t p is a consequence of adequacy. It is worth noting that without delimited continuations, we would not have been able to define an adequate translation, since we would have encountered the same problem 27 than with a naive CPS translation (see Section 2.8). FURTHER EXTENSIONS As we explained in the preamble of Section 2, we defined dL and dL t p as small languages containing all the potential sources of inconsistency we wanted to mix: classical control, dependent types, and a sequent calculus presentation. It had the benefit to focus our attention on the difficulties inherent to the issue, but on the other hand, the language we obtain is far from being as expressive as other usual proof systems. We claimed our system to be extensible, thus we shall now discuss this matter. Intuitionistic sequent calculus There is not much to say on this topic, but it is worth mentioning that dL and dL t p could be easily restricted to obtain an intuitionistic framework. Indeed, just like for the passage from LK to LJ, it is enough to restrict the syntax of proofs to allow only one continuation variable (that is one conclusion on the right-hand side of sequent) to obtain an intuitionistic calculus. In particular, in such a setting, all proofs will be nef, and every result we obtained will still hold. Extending the domain of terms Throughout the paper, we only worked with terms of a unique type N, hence it is natural to wonder whether it is possible to extend the domain of terms in dL t p , for instance with terms in the simply-typed λ-calculus. A good way to understand the situation is to observe what happens through the CPS translation. We saw that a term t of type T = N is translated into a proof t * which is roughly of type T * = ¬¬T + = ¬¬N, from which we can extract a term t + of type N. However, if T was for instance the function type N → N (resp. T → U ), we would only be able to extract a proof of type T + = N → ¬¬N (resp. T + → U * ). There is no hope in general to extract a function f : N → N from such a term, since such a proof could be of the form λx .p, where p might backtrack to a former position, for instance before it was extracted, and furnish another proof. Such a proof is no longer a witness in the usual sense, but rather a realizer of f ∈ N → N in the sense of Krivine classical realizability. This accounts for a well-know phenomenon in classical logic, where witness extraction is limited to formulas in the Σ 1 0 -fragment [START_REF] Miquel | Existential witness extraction in classical realizability and via a negative translation[END_REF]. It also corresponds to the type we obtain for the image of a dependent product Π a:A B, that is translated to a type ¬¬Π a:A + B * where the dependence is in a proof of type A + . This phenomenon is not surprising and was already observed for other CPS translations for type theories with dependent types [START_REF] Barthe | CPS translations and applications: The cube and beyond[END_REF]. Nevertheless, if the extraction is not possible in the general case, our situation is more specific. Indeed, we only need to consider proofs that are obtained as translation of terms, which can only contains nef proofs in dL t p . In particular, such proofs cannot drop continuations (remember that this was the whole point of the restriction to the nef fragment). Therefore, we could again refine the translation of types, similarly to what we did in Lemma 4.9. Once more, this refinement would also coincide with a computational property similar to Lemma 4.1, expressing the fact that the extraction can be done simply by passing the identity as a continuation 28 . This witnesses the fact that for any function t in the source language, there exists a term t + in the target language which represents the same function, even though the translation of t is a proof t . To sum up, this means that we can extend the domain of terms in dL t p (in particular, it should affect neither the subject reduction property nor the soundness), but the stratification between terms and proofs is to be lost through a CPS translation. If the target language is a non-stratified type theory (most of the presentations of type theory correspond to this case), then it becomes possible to force the extraction of terms through the translation. Another solution would consist in the definition of a separate translation for terms. Indeed, as it was reflected by Lemma 4.1, since neither terms nor nef proofs may contain continuations, they can be directly translated. The corresponding translation is actually an embedding which maps every pure term (without wit p) to itself, and which performs the reduction of nef proofs p to proofs p + so as to eliminate every µ binder. Such a translation would intuitively reflect an abstract machine where the reduction of terms (and the nef proofs inside) is performed in an external machine. If this solution is arguably a bit ad hoc, it is nonetheless correct and it is maybe a good way to take advantage of the stratified presentation. Adding expressiveness From the point of view of the proof language (that is of the tools we have to build proofs), dL t p only enjoys the presence of a dependent sum and a dependent product over terms, as well as a dependent product at the level of proofs (which subsumes the non-dependent implication). If this is obviously enough to encode the usual constructors for pairs (p 1 , p 2 ) (of type A 1 ∧ A 2 ), injections ι i (p) (of type A 1 ∨ A 2 ), etc..., it seems reasonable to wonder whether such constructors can be directly defined in the language of proofs. In fact, this is the case, and we claim that is possible to define the constructors for proofs (for instance (p 1 , p 2 )) together with their destructors in the contexts (in that case μ(a 1 , a 2 ).c), with the appropriate typing rules. In practice, it is enough to: • extend the definitions of the nef fragment according to the chosen extension, • extend the call-by-value reduction system, opening if needed the constructors to reduce them to a value, • in the dependent typing mode, make some pattern-matching within the list of dependencies for the destructors. The soundness of such extensions can be justified either by extending the CPS translation, or by defining a translation to Lepigre's calculus (which already allows records and pattern-matching over general constructors) and proving the adequacy of the translation with respect to the realizability model. For instance, for the case of the pairs, we can extend the syntax with: We then need to add the corresponding typing rules (plus a third rule to type μ(a 1 , a 2 ).c in regular mode): Γ ⊢ p 1 : A 1 | ∆ Γ ⊢ p 2 : A 2 | ∆ Γ ⊢ ( )||e⟩⟩⟩ ⟨(V 1 , V 2 )|| μ(a 1 , a 2 ).c⟩ ⇝ c[V 1 /a 1 , V 2 /a 2 ] We let the reader check that these rules preserve subject reduction, and suggest the following CPS translations: (p 1 , p 2 ) p ≜ λ • k. p 1 p (λ • a 1 . p 2 p (λ • a 2 .k (a 1 , a 2 ))) (V 1 , V 2 ) V ≜ λ • k.k ( V 1 V , V 2 V ) μ(a 1 , a 2 ) .c e ≜ λp. split p as (a 1 , a 2 ) in c c which allow us to prove that the calculus remains correct with these extensions. We claim that this methodology furnishes a good approach to handle the question "Can I extend the language with ... ?". In particular, it should be enough to get closer to a realistic programming language and extend the language with inductive fixed point operators 29 . CONCLUSION Several directions remain to be explored. We plan to investigate possible extensions of the syntactic restriction we defined, and its connections with notions such as Fürhmann's thunkability [START_REF] Führmann | Direct models for the computational lambda calculus[END_REF] or Munch-Maccagnoni's linearity [START_REF] Munch-Maccagnoni | Models of a Non-associative Composition[END_REF]. Moreover, it might be of interest to check whether this restriction could make dependent types compatible with other side effects, in presence of classical logic or not. More generally, we would like to better understand the possible connections between our calculus and the categorical models for dependently typed theory. On a different perspective, the continuation-passing style translation we defined is at the best of our knowledge a novel contribution, even without considering the classical part. In particular, our translation allows us to use computations (as in the call-by-push value terminology) within dependent types with a call-by-value evaluation strategy, and without any thunking construction. It might be the case that this translation could be adapted to justify extensions of other dependently typed calculi, or provide typed translations between them. Last but not least, we plan to present an application of dL t p to solve the problem that was our original motivation to design such a calculus. In [START_REF] Miquey | Classical realizability and side-effects[END_REF]Chapter 8], we present a sequent calculus equivalent to Herbelin's dPA ω [START_REF] Herbelin | A constructive proof of dependent choice, compatible with classical logic[END_REF], whose presentation is inspired from dL t p . This leads to the definition of a realizability model inspired from Lepigre's construction and from another technique developed with Herbelin [START_REF] Miquey | Realizability interpretation and normalization of typed call-by-need λ-calculus with control[END_REF] to give a realizability interpretation to calculi with laziness and memory sharing (two features of dPA ω ). As a consequence, we deduce the normalization and the soundness of our system. Proofs p ::= a | λa.p | µα .c ⟨p|| μa.c⟩ → c[a := p] p ∈ V Contexts e ::= α | p • e | μa.c ⟨µα .c ||e⟩ → c[α := e] e ∈ E Commands c ::= ⟨p||e⟩ ⟨λa.p||u • e⟩ → ⟨u || μa.⟨p||e⟩⟩ (a) Syntax (b) Reduction rules Fig. 1 . 1 Fig. 1. The λµ μ-calculus Fig. 4 . 4 Fig. 4. Typing rules of dL Fig. 5 . 5 Fig. 5. λµ μ-calculus with pairs Theorem 2 . 13 . 213 ⟨a 2 ||e * ⟩⟩ ↣ ⟨V * ||e * ⟩ = (⟨V ||e⟩) * • Case ⟨subst refl q||e⟩ ⇝ ⟨q||e⟩: (⟨subst refl q||e⟩) * = ⟨µα .⟨q * ||α⟩||e * ⟩ ↣ ⟨q * ||e * ⟩ = (⟨q||e⟩) * • Case ⟨subst p q||e⟩ ⇝ ⟨p|| μa.⟨subst a q||e⟩⟩ (with p V ): (⟨subst p q||e⟩) * = ⟨µα .⟨p * || μ_ .⟨µα .⟨q * ||α⟩||α⟩⟩||e * ⟩ ↣ ⟨p * || μ_ .⟨µα .⟨q * ||α⟩||e * ⟩⟩ ↣ ⟨µα .⟨q * ||α⟩||e * ⟩ = (⟨subst a q||e⟩) * □ If c : (Γ ⊢ ∆; ε), then c normalizes. Fig. 6 . 6 Fig. 6. dL t p : extension of dL with delimited continuations Fig. 8 . 8 Fig. 8. Target language definition) (by induction) (by induction) □ Moreover, we can verify that the translation preserves the reduction: Proposition 4.2. If c, c ′ are two commands of dL t p such that c ⇝ c ′ , then c c = β c ′ c Fig. 10 . 10 Fig. 10. Linearity of the translation for nef proofs Proposition 4 . 3 . 43 If u -→ • u ′ and t[u ′ ] does not normalize, then neither does t[u]. Proposition 4 . 4 . 44 If t 1 • ⟨µ t p.⟨p|| t p⟩||e⟩ c = (λk. p p k) e e -→ a p p e e = ⟨p||e⟩ c Case c ⇝ c ′ ⇒ ⟨µ t p.c ||e⟩ ⇝ ⟨µ t p.c ′ ||e⟩: By induction hypothesis, we get that c c * -→ β + t = a c ′ c for some term t. Therefore, we have: ⟨µ t p.c ||e⟩ = (λk. c c k) e e -→ a c c e e * -→ β + t e e = a c ′ c e e a ←-(λk. c ′ c k) e e = ⟨µ t p Theorem 4 . 8 ( 48 Normalization). If c : Γ ⊢ ∆, then c normalizes. σ and πσ ∈ B ⊥ σ , thus v • πσ ∈ A ⇒ B ⊥ σ . □ It only remains to show that the translation we defined in Figure 11 preserves typing to conclude the proof of Proposition 5.3. Lemma 5.2. If Γ ⊢ p : A | ∆ (in dL t p ), then (Γ ∪ ∆) * ⊢ p p : A * (in Lepigre's extended system). The same holds for contexts, and if c : Γ ⊢ ∆ then (Γ ∪ ∆) * ⊢ c c : ⊥. Proposition 5 . 3 ( 53 Adeqacy). If Γ ⊢ p : A | ∆ and σ is a substitution realizing (Γ ∪ ∆) * , then p p σ ∈ A * ⊥⊥ σ . p ::= • • • | (p 1 , p 2 ) e ::= • • • | μ(a 1 , a 2 ).c • • • | µ t p.c t p Delimited c t p ::= ⟨p N ||e t p ⟩ | ⟨p|| t p⟩ continuations e t p ::= μa.c t p nef p N ||e t ⟩ e t ::= μx .c[t] c[] ::= ⟨([], p)||e⟩ | ⟨λx .p||[] • e⟩ and adding dual operators ť p/ μ ť p for (co-)delimited continuations to allow for a small-step definition of terms reduction: ⟨λx .p||t • e⟩ ⇝ ⟨µ t p.⟨t || μx .⟨p|| t p⟩⟩||e⟩ ⟨wit p||e t ⟩ ⇝ ⟨p|| μa.⟨wit a||e t ⟩⟩ ⟨(t, p)||e⟩ ⇝ ⟨p|| μ ť p.⟨t || μx .⟨ ť p|| μa.⟨(x, a)||e⟩⟩⟩⟩ λp.( t t (λ • v.p v)) e e ) -→ • (λp.( t t (λ • v.p v)) e e ) λ • x . p p -→ a ( t t (λ • v.(λ • x . p p ) v))e e +←-λk.(( t t (λ • x . p )) k) e e = ⟨µ t p.⟨t || μx .⟨p|| t p⟩⟩||e⟩ c Case ⟨(t, p)||e⟩ ⇝ ⟨p|| μ ť p.⟨t || μx .⟨ ť p|| μa.⟨(x, a)||e⟩⟩⟩⟩: We have: ⟨(t, p)||e⟩ = (λ • k. p p ( t t (λx .λ • a.k (x, a)))) e e -→ • p p ( t t (λx .λ • a. e e (x, a))) a +←-p p ( t t (λx .(λk.k)λ • a. e e (x, a))) a +←-p p ( t t (λx .(λk.k)λ • a.(λk.k (x, a)) e e )) = ⟨p|| μ ť p.⟨t || μx .⟨ ť p|| μa.⟨(x, a)||e⟩⟩⟩⟩ c Case ⟨wit p||e t ⟩ ⇝ ⟨p|| μa.⟨wit a||e t ⟩⟩: We have: wit p t e t t = (λk. p p (λ • a.k (wit a))) e t t -→ a p p (λ • a. e t t (wit a))) a +←-p p (λ • a.(λk.k (wit a)) e t t ) = ⟨p|| μa.⟨wit a||e t ⟩⟩ c Case ⟨wit (V t , V p )||e t ⟩ ⇝ ⟨V t ||e t ⟩: ? -→ • ( t t (λ • x . p )) e e a • e e ≜ q p • e e t • e e ≜ t t • e e μa.c e ≜ [λa. c c ]• (•). By definition, we have ⊥ σ = ∀X .X σ = ∅, thus for any stack π , we have π ∈ ⊥ ⊥ σ = Π. In particular, • ∈ ⊥ ⊥ σ . x t ≜ x (t, p) p ≜ ( t t , p p ) q n t ≜ λzs.s n (z) µα .c p ≜ µα . c c wit p t ≜ π 1 ( p p ) prf p p ≜ π 2 ( p p ) a p ≜ a refl p ≜ λa.a λa.p p ≜ λa. p p subst p q p ≜ p p q p λx .p p ≜ λx . p p α e ≜ α * , we have: ⟨µ t p.⟨q 1 || μa 1 .⟨q 2 || μa 2 .⟨p|| t p⟩⟩⟩||e⟩ c = µα .(µα .( q 1 p * [λa 1 . ⟨q 2 || μa 2 .⟨p|| t p⟩⟩ t p ]α) * α) * e e ≻ µα .( q 1 p * [λa 1 . ⟨q 2 || μa 2 .⟨p|| t p⟩⟩ t p ]α) * e e ≻ q 1 p * [λa 1 . ⟨q 2 || μa 2 .⟨p|| t p⟩⟩ t p ] e e ≻ * q 2 p * [λa 2 p 1 , p 2 ) : (A 1 ∧ A 2 ) | ∆ ∧ r c : Γ, a 1 : A 1 , a 2 : A 2 ⊢ d ∆, t p : B; σ {(a 1 , a 2 )|p} Γ | μ(a 1 , a 2 ).c : (A 1 ∧ A 2 ) ⊢ d ∆, t p : B; σ {•|p} ∧ land the reduction rules:⟨(p 1 , p 2 )||e⟩ ⇝ ⟨p 1 || μa 1 .⟨p 2 || μa 2 .⟨(a 1 , a 2 Aside from strictly logical considerations as in[START_REF] Herbelin | A constructive proof of dependent choice, compatible with classical logic[END_REF], there are motivating examples of programs that could only be written and specified in such a setting. Consider for instance the infinite tape lemma that states that from any infinite sequence of natural numbers, one can extract either an infinite sequence of odd numbers, or an infinite sequence of even numbers. Its proof deeply relies on classical logic, and the corresponding program (which, given as input a stream of integers, returns a stream that consists either only of odd integers or only of even ones) can only be written in a classical setting and requires dependent types to be specified. See[START_REF] Lepigre | Semantics and Implementation of an Extension of ML for Proving Programs[END_REF] Section 7.8] for more details. In the sense of a formulas-as-types interpretation of a sequent calculus à la Hilbert (as Curien-Herbelin's λµ μ-calculus[START_REF] Curien | The duality of computation[END_REF] or Munch-Maccagnoni's system L[START_REF] Munch-Maccagnoni | Focalisation and Classical Realisability[END_REF]), as opposed to traditional type systems given in a natural deduction style. This formula is often referred to as the formula in the stoup, a terminology due to Girard. Observe that this critical pair can be also interpreted in terms of non-determinism. Indeed, we can define a fork instruction by ⋔≜ λab .µα . ⟨µ_⟨a ||α ⟩ || μ_. ⟨b ||α ⟩ ⟩, which verifies indeed that ⟨⋔||p 0 • p 1 • e ⟩ → ⟨p 0 ||e ⟩ and ⟨⋔||p 0 • p 1 • e ⟩ → ⟨p 1 ||e ⟩. To pursue the analogy with the λ-calculus, the rest of the stack e can be viewed as a context C e [ ] surrounding the application p q, the command ⟨p ||q • e ⟩ thus being identified with the term C e [p q]. Similarly, the whole stack can be seen as the context C q•e [ ] = C e [[ ]q], whence the terminology. Technically this requires to extend the language to authorize the construction of terms call/cc k t and of proofs throw t . The first rule expresses that call/cc k captures the context wit { } and replaces every occurrence of throw k t with throw k (wit t ). The second one just expresses the fact that call/cc k can be dropped when applied to a term t which does not contain the variable k . This design choice is usually a matter of taste and might seem unusual for some readers. However, it has the advantage of exhibiting the different treatments for terms and proofs through the CPS in the next sections. The nature of the representation is irrelevant here as we will not compute over it. We can for instance add one constant for each natural number. The reader might recognize the rule (ς ) of Wadler's sequent calculus[START_REF] Wadler | Call-by-value is dual to call-by-name[END_REF]. Observe that the problem here arises independently of the value restriction (that is whether we consider that q is a value or not), and is peculiar to the sequent calculus presentation. Note that even if we were not restricting ourselves to values, this would still hold: if at some point the command ⟨p ||e ⟩ is executed, it is necessarily the case that q has produced a value to substitute for a. (µ) c : (Γ, a : A ⊢ ∆; σ {a|p}) Γ | μa.c : A ⊢ ∆; σ {•|p} ( μ) Γ, a : A ⊢ p : B | ∆; σ Γ ⊢ λa.p : Π a:A B | ∆; σ (→ r ) Γ ⊢ q : A | ∆; σ Γ | e : B[q/a] ⊢ ∆; σ {•| †} q D → a FV (B) Γ | q • e : Π a:A B ⊢ ∆; σ {•|p} (→ l ) Γ, x : N ⊢ p : A | ∆; σ Γ ⊢ λx .p : ∀x N .A | ∆; σ (∀ r ) Γ ⊢ t : N ⊢ ∆; σ Γ | e : A[t/x] ⊢ ∆; σ {•| †} Γ | t • e : ∀x N .A ⊢ ∆; σ {•|p} (∀ l ) Γ ⊢ t : N | ∆; σ Γ ⊢ p : A(t) | ∆; σ Γ ⊢ (t, p) : ∃x N .A(x) | ∆; σ (∃ r ) Γ ⊢ p : ∃x N .A(x) | ∆; σ p ∈ D Γ ⊢ prf p : A(wit p) | ∆; σ prf Γ ⊢ p : A | ∆; σ A ≡ B Γ ⊢ p : B | ∆; σ (≡ r ) Γ | e : A ⊢ ∆; σ A ≡ B Γ | e : B ⊢ ∆; σ (≡ l ) Γ ⊢ p : t = u | ∆; σ Γ ⊢ q : B[t/x] | ∆; σ Γ ⊢ subst p q : B[u/x] | ∆; σ (subst) Γ ⊢ t : N | ∆; σ Γ ⊢ refl : t = t | ∆; σ (refl) Γ, x : N ⊢ x : N | ∆; σ (Ax t ) n ∈ N Γ ⊢ n : N | ∆; σ(Ax n ) Γ ⊢ p : ∃x .A(x) | ∆; σ p ∈ D Γ ⊢ wit p : N | ∆; σ (wit) In practice we will only bind a variable with a proof term, but it is convenient for proofs to consider this slightly more general definition. It is easy to convince ourselves that when typing a command ⟨p ||q • μa .c ⟩ with {• |p }, the "correct" dependency within c should be {a |µα ⟨p ||q • α ⟩ }, where the right proof is not a value. Furthermore, this dependency is irrelevant since there is no way to produce such a command where a type adjustment with respect to a needs to be made in c. That is to say let a = p 0 in subst (prf a) refl in natural deduction. (≡ l )where the hypothesis A ≡ B is implicit.• Case ⟨λx .p||t • e⟩ ⇝ ⟨p[t/x]||e⟩.A typing proof for the command on the left-hand side is of the form:Π p Γ, x : N ⊢ p : A | ∆ Γ ⊢ λx .p : ∀x N .A | ∆ (∀ r ) Π t Γ ⊢ t : N | ∆ Π e Γ | e : B[t/x] ⊢ ∆; {•| †} Γ | t • e : ∀x N .B ⊢ ∆; {•|λx .p} (∀ l ) Γ | t • e : ∀x N .A ⊢ ∆; {•|λx .p} (≡ l ) ⟨λx .p||t • e⟩ : Γ ⊢ ∆ (Cut)We first deduce A[t/x] ≡ B[t/x] from the hypothesis ∀x N .A ≡ ∀x N .B. Then, using the fact that Γ, x : N ⊢ p : A | ∆ and Γ ⊢ t : N | ∆, by Lemma 2.6 and the fact that∆[t/x] = ∆, we get a proof Π ′ p of Γ ⊢ p[t/x] : A[t/x] | ∆.We can thus build the following derivation:Π ′ p Γ ⊢ p[t/x] : A[t/x] | ∆ Π e Γ | e : B[t/x] ⊢ ∆; {•|p[t/x]} Γ | e : A[t/x] ⊢ ∆; {•|p[t/x]} (≡ l ) ⟨p[t/x]||e⟩ : Γ ⊢ ∆ (Cut)using Corollary 2.5 to weaken the binding to p[t/x] in Π e .• Case ⟨λa.p||q • e⟩ ⇝ ⟨q|| μa.⟨p||e⟩⟩.A typing proof for the command on the left-hand side is of the form:Π p Γ, a : A ⊢ p : B | ∆ Γ ⊢ λa.p : Π a:A B | ∆ (→ r ) Π q Γ ⊢ q : A ′ | ∆ Π e Γ | e : B ′ [q/a] ⊢ ∆; {•| †} Γ | q • e : Π a:A ′ B ′ ⊢ ∆; {•|λa.p} Γ | q • e : Π a:A B ⊢ ∆; {•|λa.p} (≡ l ) ⟨λa.p||q • e⟩ : Γ ⊢ ∆ (Cut)If q D, we define B ′ q ≜ B ′ which is the only type in B ′ {a |q } . Otherwise, we define B ′ q ≜ B ′ [q/a] which is a type in B ′ {a |q } . In both cases, we can build the following derivation: for the (cut)-rule. Consequently, it can also be dropped for all the other cases. The case of the conversion rule is a direct consequence of the third case. For refl, we have by definition that refl * = λx .x : N * → N * .The only non-direct cases are subst p q, with p not a value, and (t, p). To prove the former with p V , we have to show that if:Γ ⊢ p : t = u | ∆; σ Γ ⊢ q : B[t/x] | ∆; σ Γ ⊢ subst p q : B[u/x] | ∆; σ (subst)then subst p q * = µα .⟨p * || μ_ .⟨µα .⟨q * ||α⟩||α⟩⟩ : B[u/x] * . According to Lemma 2.10, we have that B[u/x] * = B[t/x] * = B * . By induction hypothesis, we have proofs of Γ * ⊢ p * : N * → N * | ∆ * and of Γ * ⊢ q * : B | ∆ * . Using the notation η q * ≜ µα .⟨q * ||α⟩, we can derive:Γ * ⊢ p * : N * → N * | ∆ * Γ * ⊢ q * : B * | ∆ * Γ * ⊢ η q * : B * | ∆ * α : B * ⊢ α : B * ⟨η q * ||α⟩ : Γ ⊢ ∆ * , α : B * (Cut) Γ * | μ_ .⟨η q * ||α⟩ : B * ⊢ ∆ * , α : B * ( μ) ⟨p * || μ_ .⟨η q * ||α⟩⟩ : Γ * ⊢ ∆ * , α : B * (Cut) Γ * ⊢ µα .⟨p * || μ_ .⟨η q * ||α⟩⟩ : B * | ∆ * (µ)The case subst V q is easy since (subst V q) * = q p has type B * by induction. Similarly, the proof for the case (t, p) corresponds to the following derivation: Γ * ⊢ p * :A * | ∆ * Γ * ⊢ t * : N | ∆ * a : A * ⊢ a : A * Γ * , a : A * ⊢ (t * , a) : N∧ A * | ∆ * (∧ r ) α : N∧ A * ⊢ α : N∧ A * ⟨(t * , a)||α⟩ : Γ, a : A * ⊢ ∆ * , α : N∧A * (Cut) We actually even consider α -conversion for delimited continuations t p, to be able to insert such terms inside a type. Even though this might seem strange at first sight, this will make sense when proving subject reduction. Everything works as if when reaching a state where the reduction of a term is needed, we had an extra abstract machine to reduce it. Note that this abstract machine could possibly need another machine itself for reducing proofs embedded in terms, etc. We could actually solve this by making the reduction of terms explicit, introducing for instance commands and contexts for terms with the appropriate typing rules. However, this is not necessary from a logical point of view and it would significantly increase the complexity of the proofs, therefore we rather chose to stick to the actual presentation. (µ) c : (Γ, a : A ⊢ ∆) Γ | μa.c : A ⊢ ∆ ( μ) Γ, a : A ⊢ p : B | ∆ Γ ⊢ λa.p : Π a:A B | ∆ (→ r ) Γ ⊢ q : A | ∆ Γ | e : B[q/a] ⊢ ∆ q D ⇒ a FV (B) Γ | q • e : Π a:A B ⊢ ∆ (→ l ) Γ, x : N ⊢ p : A | ∆ Γ ⊢ λx .p : ∀x N .A | ∆ (∀ r ) Γ ⊢ t : N ⊢ ∆ Γ | e : A[t/x] ⊢ ∆ Γ | t • e : ∀x N .A ⊢ ∆ (∀ l ) Γ ⊢ t : N | ∆ Γ ⊢ p : A(t) | ∆ Γ ⊢ (t, p) : ∃x N .A(x) | ∆ (∃ r ) Γ ⊢ p : ∃x N .A(x) | ∆ p ∈ D Γ ⊢ prf p : A(wit p) | ∆ prf Γ ⊢ p : A | ∆ A ≡ B Γ ⊢ p : B | ∆ (≡ r ) Γ | e : A ⊢ ∆ A ≡ B Γ | e : B ⊢ ∆ (≡ l ) Γ ⊢ p : t = u | ∆ Γ ⊢ q : B[t/x] | ∆ (→ I ) Γ ⊢ p : Π a:A B Γ ⊢ q : A Γ ⊢ p q : B[q/a] (→ E ) Γ, x : N ⊢ p : A Γ ⊢ λx .p : ∀x N A (∀ 1 I ) Γ ⊢ p : ∀x N .A Γ ⊢ t : N Γ ⊢ p t : A[t/x] (∀ 1 E ) Γ ⊢ p : A X FV (Γ) Γ ⊢ p : ∀X .A (∀ 2 I ) Γ ⊢ p : ∀X .A Γ ⊢ p : A[P/X ] (∀ 2 E ) Γ ⊢ t : N Γ ⊢ p : A[u/x] Γ ⊢ (t, p) : ∃x N A (∃ I ) Γ ⊢ p : ∃x N A Γ ⊢ prf p : A(wit p) (prf ) Γ ⊢ p : ∃x N A Γ ⊢ wit p : N (wit) Γ ⊢ refl : x = x (refl) In fact, we could define it formally, which would require a kind of co-delimited continuation. A classical proof might backtrack, thus it translation might use a former continuation. The return type of continuations thus need to be uniform (usually ⊥) and can not be parametrized by ∀R. (Cut) Γ | μa.⟨q||⋆⟩ : A ⊢ ∆, ⋆ : B ( μ) ⟨p|| μa.⟨q||⋆⟩⟩ : Γ | ∆, ⋆ : B (Cut) Γ ⊢ µ⋆.⟨p|| μa.⟨q||⋆⟩ | ∆⟩ : B (µ)We want to show that for any X we can derive:Γ ∪ ∆ ⊢ λk. p p (λa. q p k) : Π b:B X (b) → X (q + [p + /a]).By induction, we have:Γ ∪ ∆ ⊢ p p : ∀Y .(Π a:A + Y (a) → Y (p + )) Γ ∪ ∆ , a : A + ⊢ q t : ∀Z .(Π b:B + Z (b) → Z (q + )),so that by choosing Z (b) ≜ X (b) and Y (a) ≜ X (q + ), we get the expected derivation:Γ ∪ ∆ ⊢ p p : . . . Γ ∪ ∆ , a : A + ⊢ q p : . . . k : Π b:B X (b) ⊢ k : k : Π b:B X (b) Γ ∪ ∆ , k : Π b:B X (b), a : A + ⊢ q p k : X (q + ) (→ E ) In particular, Lepigre's semantical restriction is so permissive that it is not decidable, while it is easy to decide whether a proof term of dL t p is in nef. This will allow us to ease the definition of the translation to handle separately proofs and contexts. Otherwise, we would need formally to define ⟨p ||q • e ⟩ c all together by p p q p * e e . Where 0 is defined as λzs .z and S (t ) as (λzs .s(t zs)), i.e. as the translation of the corresponding 0 and successor from dL t p . ⟨p||e⟩ c ≜ p p * e e µ t p.c p ≜ µα . c t p ⟨p|| t p⟩ t p ≜ p p ⟨p|| μa.c⟩ t p ≜ (µα . p p * [λa. c t p ]α) * α Fig. 11. Translation of proof terms into Lepigre's calculusΓ ⊢ t : A Γ ⊢ π : A ⊥ ⊥ Γ ⊢ t * π : B * Γ ⊢ • : ⊥ ⊥ ⊥ • Γ, α : A ⊥ ⊥ ⊢ α : A ⊥ ⊥ α Γ, α : A ⊥ ⊥ ⊢ t : A Γ ⊢ µα .t : A µ Γ ⊢ π : (A[x := t]) ⊥ ⊥ That is, the translation q p * [λa . p p * e e ]• of a command ⟨q || μa . ⟨p ||e ⟩ ⟩ (where e is of type B[q] and p of type B[a]) would have been ill-typed (because p p * e e is). To be precise, for each arrow in the type, a double-negation (or its refinement) would be inserted. For instance, to recover a function of type N → N from a term t : ¬¬(N → ¬¬N) (where ¬¬A is in fact more precise, at least ∀R .(A → R) → R), the continuation needs to be forced at each level: λx .t I x I : N → N. We do not want to enter into to much details on this here, as it would lead us to much more than a paragraph to define the objects formally, but we claim that we could reproduce the results obtained for terms of type N in a language with terms representing arithmetic functions in finite types. The interested reader could see for instance[START_REF] Miquey | Classical realizability and side-effects[END_REF] Chapter 8] where a similar language with pairs, patter-matching, inductive and coinductive fixed points is defined. Acknowledgments. The author wishes to thank Pierre-Marie Pédrot for a discussion that led to the idea of using delimited continuations, Gabriel Scherer for his accurate observations and the constant interest he showed for this work, Hugo Herbelin who provided valuable help all along the writing of this paper, Rodolphe Lepigre for the example of the infinite tape lemma, as well as Alexandre Miquel and anonymous referees of this paper for their constructive remarks. Regular mode: We can thus build the following derivation for the command on the right: • Case ⟨prf p||e⟩ ⇝ ⟨µ t p.⟨p|| μa.⟨prf a|| t p⟩⟩||e⟩. We prove it in the most general case, that is when this reduction occurs under a delimited continuation. A typing derivation for the command on the left to be of the form: The proof p being nef, so is µ t p.⟨p|| μa.⟨prf a|| t p⟩⟩, and by definition of the reduction for types, we have for any type A that: so that we can prove that for any b: σ {b|prf p} ⇛ σ {b|µ t p.⟨p|| μa.⟨prf a|| t p⟩⟩}. Thus, we can turn Π e into Π ′ e a derivation of the same sequent except for the list of dependencies that is changed to σ {•|µ t p.⟨p|| μa.⟨prf a|| t p⟩⟩}. We conclude the proof of this case by giving the following derivation: • Case ⟨µ t p.⟨p|| t p⟩||e⟩ ⇝ ⟨p||e⟩. This case is trivial, because in a typing derivation for the command on the left, t p is typed with an empty list of dependencies, thus the type of p, e and t p coincides. prf (t, p) → β p subst refl q → β q (a) Language and formulas (b) Reduction rules • Case (wit). In dL t p the typing rule for wit p is the following: We want to show that: By induction hypothesis, we have: hence, it amounts to showing that for any X we can build the following derivation: • Case (∃ I ). In dL t p the typing rule for (t, p) is the following: Hence, we obtain by induction: and we want to show that for any Z : So we need to prove that: We let the reader check that such a type is derivable by using X (x) ≜ Π a:A(x ) Z (x, a) in the type of t p , and using Y (a) ≜ Z (t + , a) in the type of p p : □ Using the previous Lemma, we can now prove that the CPS translation is well-typed in the general case. Proposition 4.10 (Preservation of typing). The translation is well-typed, i.e. the following holds: (1 Proof. The proof is done by induction on the typing derivation, distinguishing cases according to the typing rule used in the conclusion. It is clear that for the nef cases, Lemma 4.9 implies the result by taking X (a) = ⊥. The rest of the cases are straightforward, except for delimited continuations that we detail hereafter. We consider a command ⟨µ t p.⟨q|| μa.⟨p|| t p⟩⟩||e⟩ produced by the reduction of the command ⟨λa.p||q • e⟩ with q ∈ nef. Both commands are translated by a proof reducing to ( q p (λa. p p )) e e . The corresponding typing derivation in dL t p is of the form: A Classical Sequent Calculus with Dependent Types 1:39 By induction hypothesis for e and p we obtain: Applying Lemma 4.9 for q ∈ nef we can derive: We can thus derive that: Γ ∪ ∆ ⊢ q p (λa. p p ) : B[q + ] * , and finally conclude that: Γ ∪ ∆ ⊢ ( q p (λa. p p )) e e : ⊥ . □ We can finally deduce the correctness of dL t p through the translation: Theorem 4.11 (Soundness). For any p ∈ dL t p , we have: ⊬ p : ⊥. Proof. Any closed proof term of type ⊥ would be translated in a closed proof of (⊥ → ⊥) → ⊥. The correctness of the target language guarantees that such a proof cannot exist. □ EMBEDDING INTO LEPIGRE'S CALCULUS In a recent paper [START_REF] Lepigre | A classical realizability model for a semantical value restriction[END_REF], Lepigre presented a classical system allowing the use of dependent types with a semantic value restriction. In practice, the type system of his calculus does not contain a dependent product Π a:A B strictly speaking, but it contains a predicate a ∈ A allowing the decomposition of the dependent product into ∀a.((a ∈ A) → B) as it is usual in Krivine's classical realizability [START_REF] Krivine | Realizability in classical logic. In interactive models of computation and program behaviour[END_REF]. In his system, the relativization a ∈ A is restricted to values, so that we can only type V : V ∈ A: However, typing judgments are defined up to observational equivalence, so that if t is observationally equivalent to V , one can derive the judgment t : t ∈ A. Interestingly, as highlighted through the CPS translation by Lemma 4.1, any nef proof p : A is observationally equivalent to some value p + , so that we could derive p : (p ∈ A) from p + : (p + ∈ A). The nef fragment is thus compatible with the semantical value restriction. The converse is obviously false, observational equivalence allowing us to type realizers that would be untyped otherwise 24 . We shall now detail an embedding of dL t p into Lepigre's calculus, and explain how to transfer normalization and correctness properties along this translation. Additionally, this has the benefits of providing us with a realizability interpretation for our calculus. While we do not use it in the current paper, we take advantage of this interpretation (and in particular of the interpretation of dependent types) in [START_REF] Miquey | Classical realizability and side-effects[END_REF]Chapter 8] to prove the normalization of dLPA ω , the sequent calculus which originally motivated this work and whose construction relies on dL t p . Actually, his language is more expressive than ours, since it contains records and patternmatching (we will only use pairs, i.e. records with two fields), but it is not stratified: no distinction is made between a language of terms and a language of proofs. We only recall here the syntax and Γ ⊢ π : • the strong existential is encoded as a pair, hence wit (resp. prf ) is mapped to the projection π 1 (resp. π 2 ). In [START_REF] Lepigre | A classical realizability model for a semantical value restriction[END_REF], the coherence of the system is justified by a realizability model, and the type system does not allow us to type stacks. Thus, we cannot formally prove that the translation preserves typing, unless we extend the type system in which case this would imply the adequacy. We might also directly prove the adequacy of the realizability model (through the translation) with respect to the typing rules of dL t p . We will detail here a proof of adequacy using the former method. We then need to extend Lepigre's system to be able to type stacks. In fact, his proof of adequacy [START_REF] Lepigre | A classical realizability model for a semantical value restriction[END_REF]Theorem 6] suggests a way to do so, since any typing rule for typing stacks is valid as long as it is adequate with the realizability model. We denote by A ⊥ ⊥ the type A when typing a stack, in the same fashion we used to go from a type A in a left rule of two-sided sequent to the type A ⊥ ⊥ in a one-sided sequent (see the remark at the end of Section 2.5). We also add a distinguished bottom stack • to the syntax, which is given the most general type ⊥ ⊥ ⊥ . Finally, we change the rule ( * ) of the original type system in [START_REF] Lepigre | A classical realizability model for a semantical value restriction[END_REF] and add rules for stacks, whose definitions are guided by the proof of the adequacy [START_REF] Lepigre | A classical realizability model for a semantical value restriction[END_REF]Theorem 6] in particular by the (⇒ e )-case. These rules are given in Figure 12. We shall now show that these rules are adequate with respect to the realizability model defined in [START_REF] Lepigre | A classical realizability model for a semantical value restriction[END_REF]Section 2]. Proposition 5.1 (Adeqacy). Let Γ be a (valid) context, A be a formula with FV (A) ⊂ dom(Γ) and σ be a substitution realizing Γ. The following statements hold: Proof. The proof is done by induction on typing derivations, we only need to do the proof for the rules we defined above (all the other cases correspond to the proof of [START_REF] Lepigre | A classical realizability model for a semantical value restriction[END_REF]Theorem 6]). 1:46 Étienne Miquey A fully sequent-style dependent calculus While the aim of this paper was to design a sequent-style calculus embedding dependent types, we only presented the Π-type in sequent-style. Indeed, we wanted to be sure above all that it was possible to define a sound sequent-calculus with the key ingredients of dependent types (i.e. dependent pairs and dependently-typed functions). In particular, rather than having left-rules (as in sequent calculi) for every syntactic constructors, we presented the existential type and the equality type with the following elimination rules (as in natural deduction): However, it is now easy to replace both elimination rules (and thus the corresponding destructors) by equivalent left-rules (and thus syntactic constructors for contexts). For instance, we could rather have contexts of the shape μ(x, a).c (to be dual to proofs (t, p)) and μ=.c (dual to refl). We could then define the following typing rules: and define prf p and subst p q as syntactic sugar: prf p ≜ µ t p.⟨p|| μ(x, a).⟨a|| t p⟩⟩ subst p q ≜ µα .⟨p|| μ=.⟨q||α⟩⟩. Observe that prf p is now only definable if p is a nef proof term. Since for any p ∈ nef and any variables a, α, the formula A(wit p) belongs to A(wit (x, a)) {(x,a)|p } , this allows us to derive the admissibility of the former (prf )-rule: . As for the reduction rules, we can define the following (call-by-value) reductions: and check that they advantageously 30

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F Pacheco-Torgal S Jalali J C Morel J E Aubert Y Millogo E Hamard A Fabbri Some observations about the paper "Earth construction: Lessons from the past for future eco-efficient construction" by As the starting point of this discussion, we would like to congratulate the authors for this interesting review [START_REF] Pacheco-Torgal | Earth construction: Lessons from the past for future ecoefficient construction[END_REF], which defends the use of earth as a building material. Indeed, while this is one of the oldest building materials in the world, it is also one of the less studied by the scientific community, and thus, one of the less understood. However, as stated by the two authors of this review, the number of scientific studies on this subject has increased dramatically in recent years. There searches on earth as a building material are mainly motivated by the growing demand of masons and construction companies for scientific data and evidence to evaluate and improve the wealth, the hygrothermal comfort and the seismic resistance of earth construction. First of all, we share the approach proposed by the author that consists of connecting the past and present (and even the future). This point is well illustrated by the first paragraph of the paper and its attractive title. Indeed, we think that the comparisons between the characteristics of modern earthen material sand existing ones, which have proven their effectiveness over the decades, is a major key to improve our understanding of this multi-scale composite material. The authors wish here to compare their views with those of Pacheco-Torgal and Jalali and highlight the following three main points of disagreement that justify this discussion. The present and the future of unstabilised earth constructions Throughout the article, the authors seem to postulate that the stabilisation technique (e.g., addition of hydraulic binders) is a compulsory step for earth construction. This leads to quite surprising conclusions about the cost and environmental impacts and their assumed direct link with the nature and amount of the binder used. These conclusions become even stranger if we consider cement stabilisation, which could be irrelevant from environmental, economic and technical perspectives. Indeed, if this stabilisation is efficient in the case of kaolinic clay materials containing appreciable amount of sand [START_REF] Millogo | Microstructural characterization and mechanical properties of cement stabilised adobes[END_REF], the same is not necessarily the case for raw clay materials rich in montmorillonite [START_REF] Molard | Study of the extrusion and stabilization with cement of monomineralogic clay[END_REF][START_REF] Amor | Cold stabilization of montmorillonitebased materials using Portland cement[END_REF][START_REF] Temimi | Making building products by extrusion and cement stabilization: limits of the process with montmorillonite clay[END_REF]. From this partial point of view on the stabilisation, we can strongly question the consistency displayed by the authors to link traditional earth constructions to the modern use of soil as a building material. Indeed, this former is mostly constituted by structures made of unstabilised earth, even for areas subject to heavy rains (Northern Europe). As a consequence, while it is true that in some countries, the temptation to accelerate the strengthening of the material by the addition of hydraulic binders can be justified for industrial production rates [START_REF] Ciancio | Experimental investigation on the compressive strength of cored and molded cement-stabilized rammed earth samples[END_REF] or for maintenance purposes [START_REF] Millogo | Microstructural characterization and mechanical properties of cement stabilised adobes[END_REF][START_REF] Reddy | Cement stabilised rammed earth. Part B: Compressive strength and stress-strain characteristics[END_REF][START_REF] Reddy | Structural behavior of story-high cementstabilized rammed-earth walls under compression[END_REF], an understanding review should not overshadow the research that is ongoing on un-stabilised earth constructions. An illustration of the significant importance of taking into consideration both stabilised and unstabilised materials is given in Germany, which is regularly used as a reference by the authors in their review. Indeed, after updating their professional rules, the Dachverband Lehm wrote a draft standard on earth-based bricks considering only un-stabilised bricks (except plant fibres that can be considered as a stabiliser in some cases). Based on this premise that un-stabilised earth constructions were only useful in the past, the majority of this review loses its relevance and contradicts the title that suggests that we can build the future using knowledge from the past. This contradiction becomes particularly annoying during the discussions on economic and environmental impacts. The unstabilised earth is solely able to be returned to its initial state (as a soil) without any "waste" of energy, by simply wetting. Moreover, it is possible to reuse the material with the same embodied energy to build again. This is the only material with drystone masonry to be able to do that. Using cement or lime stabilisation increases the embodied energy of the material. The authors are not at all against the stabilisation, particularly if it is done with the real three dimensions of sustainable building, when for example it enables to use local materials and develop local skills and employment but they are just aware that using earth is not sufficient to be sustainable. Material properties The presentation of the material properties of the soil used for earth construction is interesting but definitely lacks a discussion on the compressive strength. This feature has been extensively studied by various researchers since the 1980s, see for examples . Indeed, this characteristic is currently a feature required by all parties involved in construction as a proof of durability. However, there is a paradox between this parameter and the observation of existing earth constructions that have long shown sufficient durability. Thus, despite years of research, there is still no consensus on how to measure this characteristic [START_REF] Morel | Compressive strength testing of compressed earth blocks[END_REF]. As an example, similarly to what it is observed for other building materials, such as concrete, Anglo-Saxon culture advocates the measurement of the "confined" resistance, whereas in French culture, we continue to be attached to the "unconfined" measurements. Many discussions about the extent and relevance of this feature continue to animate the scientific debate within the community working on this subject. Moreover, as is rightly stated by the authors, the compressive strength will depend on the sample shape (and this is where the main problem with the compression test lies). However, to echo the previous discussion, it is important to underline here that the stabilisation will also significantly affect the fracture behaviour of the test sample. Actually, stabilisation created by rigidifying the material will induce a commonly observed behaviour in brittle materials as concrete or stone. In contrast, theun-stabilised material is likely to be closer to conventional behaviour of soils. In this case, the soil behaviour elastoplastic models are a priori better suited to earthen materials [START_REF] Nowamooz | Finite element modelling of a rammed earth wall[END_REF][START_REF] Jaquin | The strength of unstabilised rammed earth materials[END_REF]. Thus, any comparison between the compressive strengths of stabilised and unstabilised earth samples should be made with care. Hygrothermal properties Finally, a similar discussion can be undertaken on the hygrothermal properties of earthen materials and their impact on comfort and indoor air quality. One of the main assets that is used to promote earth constructions is their role in controlling moisture and indoor air quality. To our knowledge, there are also few studies that demonstrate what the influence of stabilisation on the hygroscopic behaviour might be. It is well known since [START_REF] Olivier | Le matériau terre: Essai de compactage statique pour la fabrication de briques de terres compressées[END_REF] that, for materials of the same soils manufactured at the optimum water content, stabilisation increases the volume fraction porosity of the material. The consequence is that the sorptivity of the cement stabilised samples is higher than the unstabilised sample [START_REF] Hall | Moisture Ingress in Rammed Earth: Part 3 -The Sorptivity and the Surface Inflow Velocity[END_REF]. But recent studies in the moisture buffering in buildings, clearly stated that the vapour transfer were reduced by the stabilisation with lime or cement [START_REF] Mcgregor | The effect of stabilisation on humidity buffering of earth walls[END_REF][START_REF] Eckermann | Auswirkung von Lehmbaustoffen auf die Raumluftfeuchte[END_REF]. However, the prediction of this phenomenon through the integration of the measured physical properties of moisture and the heat transfer coupled models is extremely rare [START_REF] Allinson | Hygrothermal analysis of a stabilised rammed earth test building in the UK[END_REF][START_REF] Hall | Chapter 2: Hygrothermal behaviour and thermal comfort in modern earth buildings[END_REF][START_REF] Hall | Analysis of the Hygrothermal Functional Properties of Stabilised Rammed Earth Materials[END_REF] and should also be completed in further studies. Conclusions We believe that it is necessary to continue discussions among scientists on the use of this material in modern "green buildings". Moreover, it is quite relevant, as suggested by Pacheco-Torgal and Jalali, to study existing earth constructionsfor, at least, the transmission of cultural know-how. However, the existence of these structures is, by itself, evidence of the durability of these types of constructions, which have remained intact for decades. It will be necessary to increase our knowledge of this material to renovate it properly (including from the energy point of view, for example, in the countries of Northern Europe). Finally, the question remains open on stabilisation. While it is entirely appropriate for some applications (low-cost buildings in India subjected to the monsoon to avoid having to rebuild every year for example), its routine use in industrialised countries can be questioned. Some local soils are known to exhibit sufficient mechanical characteristics without amendment.

01744386

en

2024/03/05 22:32:07

2016

https://hal.univ-grenoble-alpes.fr/hal-01744386/file/Poster_AlgaEurope.pdf

CONTACT : EXTRACTION BY SPARK GENERATED SHOCKWAVES Contact: Jean-Maxime Roux | [email protected] Authors : H. Lamotte 1 , J-M. Roux 1 , J. Lupette 2 , E. Marechal 2 , J-L. Achard 3 1 CEA LETI MINATEC Campus, Univ. Grenoble Alpes, F-38054 Grenoble, France 2 Laboratoire Physiologie Cellulaire & Végétale (CEA, CNRS, INRA, Univ. Grenoble Alpes), Grenoble, France 3 Laboratoire des Écoulements Géophysiques et Industriels (GrenobleINP, Univ. Grenoble Alpes, CNRS), Grenoble, France The production of lipids using oleaginous microorganisms is widely studied to develop the next generation of renewable fuels. Now, the extraction of intracellular lipids has been identified as one of the crucial elements that have the largest impact on the cost of microalgal biorefinery process. In this context, a new electric field based technique is studied to disrupt cell membranes and to extract their content. High voltage electrical pulses are used without any transducer to create underwater spark discharges that produce a high-pressure plasma/vapor bubble whose expansion, collapse and rebounds generate shockwaves. The high induced variations of pressure produce a cloud cavitation above the electrode; bubbles collapses are known to destroy mechanically cells membranes. Moreover, the pressure rise induced by the process can be used to renew the solution containing destroyed cells at each pulse. Underwater spark discharges can extract chlorophyll and lipids from Phaeodactylum tricornutum and Nannochloropsis gaditana and also DNA from bacteria such as Bacillus subtilis. The present communication presents results of a parametric study of the process. It is shown that efficiencies depends on the microorganisms' concentration and the water height above the electrodes. Perspective : recover the same amount as chlorophyll Cloud cavitation The pressure rise and the bubble growth can expel water from the treatment unit. Picture of the cavitation bubbles taken 150 µs after the electric discharge. Shockwaves induce pressure changes that lead to the formation of cloud cavitation. Phaeodactylum tricornutum % extraction % extraction Underwater discharges produce heat and free radicals that combine to create very reactive molecules such a hydrogen peroxyde. But Phaeodactylum tricornutum does not release pigments when heated up to 90°C. Also 1 mM of hydrogen peroxyde does not seem to make the algae release pigment. Release of intracellular molecules seem to be therefore due to the mechanical effect of cloud cavitation. • About 25-35% of chlorophyll are released by 200 electric pulses • A water height of about 13,5mm above the electrodes seems to be optimal • Dilution seems to improve the treatment efficiency Phaeodactylum tricornutum % extraction Following Extraction of about 6%

01744426

en

2024/03/05 22:32:07

2018

https://inria.hal.science/hal-01744426/file/TACO_HAL.pdf

Oleksandr Zinenko Stéphane Huot Cédric Bastoul Keywords: CCS Concepts:, Human-centered computing → Human computer interaction (HCI), •So ware and its engineering → Compilers, Polyhedral model, direct manipulation ACM Reference format: Parallelism is one of the key performance sources in modern computer systems. When heuristics-based automatic parallelization fails to improve performance, a cumbersome and error-prone manual transformation is o en required. As a solution, we propose an interactive visual approach building on the polyhedral model that visualizes exact dependences and parallelism; decomposes and replays a complex automatically-computed transformation step by step; and allows for directly manipulating the visual representation as a means of transforming the program with immediate feedback. User studies suggest that our visualization is understood by experts and non-experts alike, and that it may favor an exploratory approach. Visual Program Manipulation in the Polyhedral Model OLEKSANDR ZINENKO, Inria and University Paris-Saclay ST ÉPHANE HUOT, Inria C ÉDRIC BASTOUL, University of Strasbourg and Inria INTRODUCTION Large-scale adoption of heterogeneous parallel architectures requires e cient solutions to exploit the available parallelism from applications. Despite signi cant e ort in simplifying parallel programming through new languages, high-level language extensions, frameworks and libraries, manual parallelization may still be required although o en ruled out as time consuming and errorprone. us, programmers mostly rely on automatic optimization tools, such as those based on the polyhedral model, to improve program performance. e polyhedral model [START_REF] Feautrier | Polyhedron Model[END_REF] has been the cornerstone of loop-level program transformation in the last two decades [START_REF] Bastoul | Code Generation in the Polyhedral Model Is Easier an You ink[END_REF][START_REF] Bondhugula | e Pluto+ Algorithm: A Practical Approach for Parallelization and Locality Optimization of A ne Loop Nests[END_REF][START_REF] Feautrier | Some E cient Solutions to the A ne Scheduling Problem. Part II. Multidimensional Time[END_REF]. It features exact iteration-wise dependence analysis and optimization for both parallelism and locality. However, automatic polyhedral compilation is based on imprecise heuristics [START_REF] Bondhugula | e Pluto+ Algorithm: A Practical Approach for Parallelization and Locality Optimization of A ne Loop Nests[END_REF][START_REF] Bondhugula | A Practical Automatic Polyhedral Parallelizer and Locality Optimizer[END_REF]. Polyhedral compilers give user some (limited) control over the optimization process, which requires understanding their internal operation anyway. Furthermore, they are applicable globally and do not allow for nergrain control, e.g., a ecting only one loop nest. Visual interfaces for con guring the polyhedral compiler [START_REF] Papenhausen | PUMA-V: An Interactive Visual Tool for Code Optimization and Parallelization Based on the Polyhedral Model[END_REF] partially mitigate these issues by making polyhedral compiler blocks discoverable, but still require a deep understanding of internal operation of a compiler. Semi-automatic approaches provide the user with a set of prede ned program transformations, typically exposed as compiler directives [START_REF] Girbal | Semi-Automatic Composition of Loop Transformations for Deep Parallelism and Memory Hierarchies[END_REF][START_REF] Kelly | A Unifying Framework for Iteration Reordering Transformations[END_REF][START_REF] Yuki | Alphaz: A System for Design Space Exploration in the Polyhedral Model[END_REF]. ey shi the expertise requirements from heuristics to loop-level code transformations. ey also require program transformation to be performed from scratch (as polyhedrally-transformed code is barely readable) while o ering li le for (i = 0; i < N ; i ++) for (j = 0; j < N ; j ++) S : z[i+j] += x[i ] * y [ j ]; (a) Original #pragma omp parallel for private ( t2 ) for ( t1 = 0; t1 <= 2* N -2; t1 ++) for ( t2 = max (0 , t1 -N +1); t2 <= min ( t1 , N -1); t2 ++) S : z where arr corresponds to the name of accessed array and a 1 corresponds to its only dimension. Program Transformation and Schedules. Changing the relative execution order of statement instances transforms the program. We can de ne a scheduling relation to map iteration domain points to logical execution dates. If these dates are multidimensional, statement instances are executed following the lexicographical order of their dates. Scheduling relations are expressive enough to encode a complex composition of program transformations including, e.g., loop interchange, fusion, ssion, skewing, tiling, index-set-spli ing, etc. [START_REF] Girbal | Semi-Automatic Composition of Loop Transformations for Deep Parallelism and Memory Hierarchies[END_REF]. For example, loop tiling [START_REF] Irigoin | Supernode Partitioning[END_REF] for the polynomial multiply can be expressed by the schedule θ S (N ) = {(i, j) T → (t 1 , t 2 , t 3 , t 4 ) T | (32t 1 ≤ t 3 ≤ 32t 1 +31) ∧ (32t 2 ≤ t 4 ≤ 32t 2 +31) ∧ t 3 = i ∧ t 4 = j}, where 32 is the tile size. Note that t 3 and t 4 are de ned explicitly by equalities while t 1 and t 2 are de ned implicitly by bounding inequalities, which correspond to integer division. Schedule relations can be constructed manually or using high-level frameworks [START_REF] Bagnères | Opening Polyhedral Compiler's Black Box[END_REF][START_REF] Girbal | Semi-Automatic Composition of Loop Transformations for Deep Parallelism and Memory Hierarchies[END_REF][START_REF] Kelly | A Unifying Framework for Iteration Reordering Transformations[END_REF]. Automatic optimizers directly construct a scheduling relation with certain properties, including minimal reuse distances, tilability and inner/outer parallelism [START_REF] Bondhugula | e Pluto+ Algorithm: A Practical Approach for Parallelization and Locality Optimization of A ne Loop Nests[END_REF][START_REF] Bondhugula | Automatic Transformations for Communication-Minimized Parallelization and Locality Optimization in the Polyhedral Model[END_REF]. However, they may fail to improve performance when achieving di erent properties requires contradictory transformations, for example exploiting spatial locality may be detrimental for parallelism [START_REF] Shirako | Oil and Water Can Mix: An Integration of Polyhedral and AST-Based Transformations[END_REF]. 2.1.3 Encoding Lexical Order. roughout this paper, we use the so called (2d + 1) structure of scheduling relations. It introduces (d + 1) auxiliary dimensions to the scheduling relation [START_REF] Kelly | A Unifying Framework for Iteration Reordering Transformations[END_REF] to represent lexical order. ey are referred to as β-dimensions [START_REF] Girbal | Semi-Automatic Composition of Loop Transformations for Deep Parallelism and Memory Hierarchies[END_REF], as opposed to α dimensions that represent the execution order of the d loops. Zero-based contiguous constant values of β i enforce the relative order between di erent objects (loops or statements) at depth i. ey express code motion transformations such as loop fusion and ssion. For example, the (2d + 1) form of the identity scheduling relation for polynomial multiply is θ S (N ) = {(i, j) T → (β 1 , α 1 , β 2 , α 2 , β 3 ) T | β 1 = 0 ∧ α 1 = i ∧ β 2 = 0 ∧ α 2 = j ∧ β 3 = 0}. Given that β-dimensions are constant, they can be concisely rewri en as a vector ì β = (0, 0, 0) T . β-vectors uniquely identify statements since no two statements can have the same lexical position. Pre xes of β-vectors (β-pre xes) uniquely identify loops, with their length corresponding to the nesting depth. Statements that share d loops, have identical β-pre xes of length d. 2.1.4 Program Analysis and Parallelism. ey key power of the polyhedral model is its ability to compute exact instance-wise dependences [START_REF] Feautrier | Data ow Analysis of Array and Scalar References[END_REF]. Two statement instances are dependent if they access the same array element and at least one of them writes to it. For a program transformation to preserve original program semantics, it is su cient that pairs of dependent instances are executed in the same order as before the transformation [START_REF] Kennedy | Optimizing Compilers for Modern Architectures: A Dependence-Based Approach[END_REF]. A dependence relation maps statement instances (dependence sources) to the instances that must be executed a er them (dependence sinks). If a transformation inverses the execution order of dependent instances or assigns them the same logical execution time, the dependence becomes violated and the transformation is illegal. e polyhedral model provides means to verify the legality of a scheduling relation [START_REF] Bastoul | Mapping Deviation: A Technique to Adapt or to Guard Loop Transformation Intuitions for Legality[END_REF][START_REF] Feautrier | Data ow Analysis of Array and Scalar References[END_REF][START_REF] Pugh | e Omega Test: A Fast and Practical Integer Programming Algorithm for Dependence Analysis[END_REF][START_REF] Vasilache | Violated Dependence Analysis[END_REF]. Groups of instances, including loops, that do not transitively depend on each other may be executed in an arbitrary order, including in parallel. Loop-level parallelism is expressed by a aching a "parallel" mark to an α dimension, which requires code generator to issue a parallel loop. Code Generation. A er a scheduling relation is de ned, code generation is a ma er of building a program that scans the iteration domain with respect to the schedule [START_REF] Ancourt | Scanning Polyhedra with DO Loops[END_REF]. Modern code generators rely on generalized change of basis that combines the iteration domain and the scheduling relation and puts scheduling dimensions in the foremost positions before creating loops from all dimensions. Several e cient algorithms and tools exist for that purpose including CLooG [START_REF] Bastoul | Code Generation in the Polyhedral Model Is Easier an You ink[END_REF], CodeGen+ [START_REF] Chen | Polyhedra Scanning Revisited[END_REF] and ppcg [START_REF] Grosser | Polyhedral AST Generation Is More an Scanning Polyhedra[END_REF]. For example, given the schedule T S = {(i, j) T → (t1, t2) T | t1 = i + j ∧ t2 = j} that implements loop skewing for the polynomial multiply kernel and a parallel mark for dimension t1, CLooG may generate the code in Fig. 1b. Transformation Directives Even though polyhedral and syntactic approaches can be combined in an automatic tool [START_REF] Shirako | Oil and Water Can Mix: An Integration of Polyhedral and AST-Based Transformations[END_REF], the polyhedral optimizer does not operate in syntactic terms and provides only li le control over its parameters through compiler ags. Recently, Bagnères et.al. proposed the Clay transformation set that expresses a large number of syntactic loop transformations as structured changes to scheduling relations and rely on β-pre xes to identify targets [START_REF] Bagnères | Opening Polyhedral Compiler's Black Box[END_REF]. ey also proposed the Chlore algorithm that identi es a sequence of Clay primitives that would transform any given scheduling relation into another scheduling relation. For example, the aforementioned loop skewing transformation is expressed as a dimension substitution: S ( ì ρ, i, k): ∀θ S : ì β S,1.. dim ì ρ = ì ρ, α dim ì ρ → α dim ì ρ + k • α i . Any occurrence of the output dimension α dim ì ρ is replaced by a linear combination of itself with another output dimension α i . us, the schedule T S from the previous section is obtained from the identity schedule by S ((β 1 ) T = (0) T , i = 2, k = 1) where β 1 identi es the outer i loop. Loop R is similar to S except that it uses a linear combination of the input rather than output dimensions. Transformations of the lexical order are encoded as modi cations of β-vectors. For example, fusing two subsequent loops is expressed as F N ( ì ρ): ∀θ S : ì β S,1.. dim ì ρ-1 = ì ρ 1.. dim ì ρ-1 ∧ ì β S,dim ì ρ = ì ρ dim ì ρ + 1, ì β S,1.. dim ì ρ-1 ← ì ρ S,1.. dim ì ρ , ì β S,dim ì ρ ← ì β S,dim ì ρ + max T : ì β T = ì ρ ì β T ,dim ì ρ , where dim ρ encodes fusion depth. is transformations assigns equal β values up to given depth, which corresponds to fusion, and updates the remaining ones to maintain uniqueness and contiguity. Clay transformations are applicable to unions of scheduling relations such that the entire union (but not necessarily individual relations) is le -total and injective. Internally, Clay operates on a matrix representation of systems of linear inequalities and supports arbitrarily complex transformations as long as the properties of an union are preserved. Chlore algorithm builds on matrix decompositions to identify sequences of Clay primitives that transform one set of matrices into another set. More information and full speci cation of transformations is available in [START_REF] Bagnères | Opening Polyhedral Compiler's Black Box[END_REF]. Although Clay and Chlore enable interaction with a polyhedral engine using syntactic terms, they face several challenges in application. (1) Target selection-β-pre xes are required for each transformation, yet they are not easily accessible in the source code. (2) Target consistency-the generated code may have a di erent structure than the original code, for example due to loop separation [START_REF] Bastoul | Code Generation in the Polyhedral Model Is Easier an You ink[END_REF], resulting in a mismatch between β-vectors and loop nesting. (3) E ect separationeven if Chlore produces a sequence of primitive transformations, it is di cult to evaluate (potentially negative) e ects of individual transformation by reading the polyhedrally transformed code. We address these challenges with Clint, a new interactive tool based on a graphical representation of SCoPs which: simpli es target selection to directly choosing a visualization of a transformation target; maintains target consistency by matching the visualization to the original (o en simpler) code; and replays primitive "steps" of transformation to separate their e ects, supporting further interactive modi cation. DIRECTLY MANIPULATING POLYHEDRAL VISUALIZATIONS To reduce the burden of code editing and transformation primitive application, we propose Clint, an interactive loop-level transformation assistant based on the polyhedral model. It leverages the geometric nature of the model by presenting SCoPs in a directly manipulable [START_REF] Shneiderman | Direct Manipulation: A Step Beyond Programming Languages[END_REF] visualization that combines sca er plots of iteration domains and node-link diagrams of instance-wise dependences. is approach is similar to the one commonly used in the polyhedral compilation community to illustrate iteration domains. Clint goes beyond these static views by allowing program transformation to be initiated directly from the visualization, and provides an animation-based visual explanation of an automatically computed program transformation. Animated transitions correspond to program transformations that, when applied, would change the program to obtain the nal visualization. e user can replicate the action by directly manipulating the visualization similarly to the transition or in a more elaborate way. e set of interactive manipulations builds on the geometry-related vocabulary of classical loop transformations, such as skewing or shi ing, which is expected to give the user supplementary intuition on the transformation e ects and to support exploration and learning. e design of Clint is motivated by the need for (1) a single and consistent visual interface to bridge the gap between dependence analysis and subsequent program transformation; (2) an e cient way to explore multiple alternative loop transformations without rewriting the code; (3) explaining the code modi cations yielded by an automatic optimization. Although built around the complete Clay transformation set [START_REF] Bagnères | Opening Polyhedral Compiler's Black Box[END_REF], it can be extended to support di erent transformations as long as e ects of any transformation can be undone by (a sequence of) other transformations. Clint seamlessly combines loop transformations to support reasoning about execution order and dependences rather than loop bounds and branch conditions. e interactive visual approach reduces parallelism extraction to visual pa ern recognition [START_REF] Ware | Information Visualization: Perception for Design[END_REF] and code transformation to geometrical manipulations, giving even non-expert programmers a way to manage the complexity of the underlying model [START_REF] Norman | Living with Complexity[END_REF]. Finally, it brings insight into the code-level e ects of the polyhedral optimization by decomposing a complex program transformation into primitive steps and providing a step-by-step visual replay, independent of how an automatic optimizer operates internally. Structure of the Visualization Clint visualizes scheduled iteration domains, e.g., statement instances mapped by the scheduling relation to the new coordinates in logical time space, see Fig. 2, for an example of a simple code and its corresponding visualization. e main graphical elements are as follows. Points and Polygons. Our visualization consists of polygons containing points on the integer la ice. Each point represents a statement instance, positioned using values of α dimensions. Points are linked together by arrows that depict instance-wise data ow between them. e polygon delimits the loop bounds in the iteration space and is computed as a convex hull of the points it includes. e space itself is displayed as a coordinate system where axes correspond to loop iteration variables. Color Coding. Statements are color coded to ensure matching between code and visual representations. A transformation, such as peeling or index-set-spli ing, may result in sets of instances of the same statement being executed in di erent loop nests. We refer to this case as multiple occurrences of the statement. Di erent occurrences of the statement share the same color coding. for (i = 0; i < 2*N-1; ++i) for (j = max(0,i-N-1); j < min(i+1,N); ++j) z[i] += x[i-j] * y[j]; Coordinate Systems. Each coordinate system is at most two-dimensional. e horizontal axis represents the outer loop, and the vertical axis represents the inner loop. Statement occurrences enclosed in both loops are displayed in the same coordinate system, with optional slight displacement to discern them (see Fig. 4). Statement occurrences that share only the outer loop are placed into di erent coordinate systems, vertically aligned so that they visually share the horizontal axis. We refer to this structure as pile (see Fig. 8b). Finally, statement occurrences not sharing loops are displayed as a sequence of piles (see Fig. 8a), arranged to follow the lexical order. We use β-vectors internally to arrange polygons and coordinate systems. Statements with identical β-pre xes of length d share a coordinate system if d is the depth of the inner loop, and a pile if d is the depth of the outer loop. Consequently, coordinate systems and piles are uniquely identi ed by a β-pre x. Execution Order. Statement instances are executed bo om to top, then le to right, crossing the bounds of coordinate systems in both cases. Multiple instances sharing a loop iteration are executed in the order of increasing displacement. Arrows point at the instance executed second. Tiling. Tiled domains are displayed as polygons with wide lines inside to delimit tile shapes. All dimensions that are implicitly de ned (see Section 2.1.2) are considered as tile loops and serve to build the tile shapes creating sca erplots with nested axes [START_REF] Rey Heer | A Tour rough the Visualization Zoo[END_REF]. Tiling makes execution order two-level: entire tiles are executed following the previously described order; instances inside each tile are executed bo om to top, then le to right without crossing tile boundaries. Multiple Projections. e overall visualization is a set of two-dimensional projections, where loops that are not matched to the axes are ignored. As the goal of Clint is program transformation, we only display projections on the schedule α-dimensions, which coincide with iteration domain dimensions before transformation. For a single statement occurrence, they may be ordered in a sca erplot matrix as in Fig. 5a. e points are displayed with di erent intensity of shade depending on how many multidimensional instances were projected on this point. We motivate the choice of 2D projections vs 3D visualization by easier direct manipulation with a standard 2D input device (e.g., mouse) [START_REF] Beaudouin-Lafon | Designing Interaction, Not Interfaces[END_REF][START_REF] Cockburn | 3D or Not 3D?: Evaluating the E ect of the ird Dimension in a Document Management System[END_REF] and consistency of the visualization for even higher dimensionality. Dependences and Parallelism. Dependences between points in the same coordinate system are shown as arrows pointing from source to sink. By default, only direct (i.e., non transitively-covered) dependences are shown. When hovering a point, all its dependences are visualized. Dependences between vertically or horizontally adjacent coordinate systems are aggregated into large dots (Fig. 7b). Finally, dependences between points in distant coordinate systems are only visualized when either their source or sink is being manipulated to avoid visual clu ering. Arrows and dots turn red if the dependence is violated. Transformation legality check is performed parametrically. If legality violation exists for values of parameters other than currently selected, the polygon contour turns red instead of arrows. Generally, parallel dependence arrows imply some parallelism is present in the loops -e.g., if they are orthogonal to an axis, the loop corresponding to an axis features D A parallelism. Clint highlights "parallel" axes in green to simplify parallelism identi cation (see Fig. 2). Parametric Domains. Domains whose bounds involve parametric expressions are visualized for a xed value of the parameters. By default, all parameters are assigned identical values computed as follows. Clint computes the dependence distance sets from dependence relations by subtracting the relation's range from its domain. It then takes the maximum non-parametric absolute value across all dimensions. Finally, it takes a minimum of this value and a prede ned constant. We selected this constant as 6 from our preliminary studies, observing that it is su cient to represent the majority of dependence pa erns in our test suite. e user can dynamically modify values of individual parameters and the visualization will be automatically updated. Directly Manipulable Visual Objects Since program transformations in the polyhedral model correspond to changes of the statement instance order, they can be performed on the visual representation of that order. In Clint, the execution dates are mapped to point positions. erefore, moving points corresponds to program transformations. Visual marks such as points and polygons a ord direct manipulation, i.e., they can be dragged and dropped directly to the desired position. Because many of the visual elements are mapped from the underlying SCoP properties, manipulation should be structured so as to maintain those properties. For example, point coordinates should remain integer to properly map to counted for loops. Furthermore, the polyhedral model represents parametric iteration domains-having constant yet unknown sizes-making it technically impossible to schedule each instance separately. erefore, we only enable structured point manipulation that can be mapped to similarly structured program transformations as expressed in, e.g., Clay framework. Visually, we use polygons and coordinate systems as manipulation substrates [START_REF] Klokmose | Webstrates: Shareable Dynamic Media[END_REF] that mediate interaction with groups of points while ensuring structure preservation. We refer to polygons and coordinate system as point containers. ey can be seen as persistent selection of the points manipulable together and sharing a common property: representing instances of the same statement or being enclosed in the same loops. Polygons and coordinate systems also allow to reify the conventional target selection and make it a rst-class interactive object [START_REF] Beaudouin-Lafon | Rei cation, Polymorphism and Reuse: ree Principles for Designing Visual Interfaces[END_REF]. e user no longer needs an explicit (and sometimes cumbersome) selection step, by either clicking or lassoing the objects with cursor, before starting the manipulation. Mapping Interactions to Loop Transformations As motivated above, we center the manipulation around polygons. We augment the polygon with handles at its corners and borders, similarly to a conventional graphical editor. ey appear when the polygon is hovered and support many transformations without using any instruments or modes. We rely on structured scheduling relation modi cations of Clay framework, most of which were inspired by well-known "classical" loop transformations [START_REF] Joseph | High Performance Compilers for Parallel Computing[END_REF]. Some of them map directly to Clint visualization (e.g., S ), while others do not (e.g., I ) or, even worse, can be mapped in a misleading way (S ). erefore, instead of trying to map Clay transformations, we rather follow an interaction-centered approach by mapping the possible graphical actions to sequences of Clay transformations. Fig. 3 lists the graphical actions and the corresponding program transformations. e action parameters correspond to the a ributes of the object being manipulated or properties of the manipulation. Drag polygon between CS ì β, x, , ì ρ 1. R (β 1.. , put last), β ← max S β S -1 2. D (β 1.. ), β -1.. = (β -1 + 1, 0) T 3. repeat 1,2 until dimension x 4. R (β 1..x , put a er ρ x ), β x ← ρ x + 1 5. F N (β 1..x ), β x ..x +1 ← (β x -1, max S β S x +1 +1) 6. repeat 4,5 until dimension 7. R (β 1.. put last) Drag corners from center ì β, x, , dx, d , sx, s 1. R ( ì β, , x, dx/s ) 2. R ( ì β, x, , d /sx ) use skew when possible Drag corners towards center ì β, x, dx, sx ( axis used if d > dx) 1. I ( ì β, x, ) if dx/sx mod 2 = 1 2. R (β 1..x ) if 1 ≤ dx/sx mod 4 ≤ 2 3. R (β 1.. ) if dx/sx mod 4 ≥ 2 Drag border x, dx, sx 1. D ( ì β) 2. R ( ì β) if dx < 0 3. G ( ì β, dx/sx ) Click on rect- angular selec- tion of points ì β, x, , tx, t 1. I (β 1.. +2 , , +1) if implicitly de ned 2. L (β 1.. +1 ) if + 1 implicitly de ned 3. L (β 1..x ) if x implicitly de ned 4. S M (β 1..x , tx) 5. S M (β 1.. +1 , t ) 6. I (β 1.. 2 , , + 1) Select points and move ì β, ì ρ, selection shape { f i (x, ) ≥ 0} 1. ∀i, I S S ( ì β, f i ) if ì ρ = ∅ C ( ì ρ) otherwise. Fig. 3. Mapping between interactive polygon manipulations and Clay transformations. ì β identifies the statement occurrence corresponding to the polygon; ì ρ identifies the β-prefix of the coordinate system; x and are loop depths corresponding to the horizontal and vertical axes, respectively; dx and d are cursor o sets from its position when the manipulation started; sx and s are sizes of the polygon; tx and t are sizes of the selection. O sets and sizes are expressed in coordinate system units, i.e., iterations. For example, dragging a polygon along one of the axes directly corresponds to the S transformation. However, dragging it to a di erent coordinate system corresponds to a complex sequence of Clay directives that perform code motion (see "Drag polygon between CS" in Fig. 3). Transformations that result in an identical schedule are omi ed, for example, no R is applied before D if the statement occurrence is already the last in the loop. Polymorphic Actions. e coordinate system can be automatically extended to t the polygon being dragged. We leverage the equivalence property of transformation to stop automatic extension. Shi ing past the largest bound does not change the relative execution order. In such cases, the polygon goes outside the coordinate system, which is shrunk to t only the remaining polygons. Parametric Transformations. Transforming a parametrically-bounded domain may result in parametric transformations. In particular, we look for a parametric bound closest to the mouse cursor at the end of manipulation. For example, the amount of S is computed with respect to the closest bound of the polygon other than the one being shi ed. Alternatively, the conditions for I S S are ( rst) computed as a ne expressions of the closest bound. If there is no such expression, they are computed without using parameters. Skew and Reshape. By default, the graphical action of skewing corresponds to the R transformation, and not the S transformation. e la er transforms the loop with respect to the current expression for the other loop rather than to the original iterator. is makes S transformation combine badly: if the x loop is skewed by to become (x + ), it becomes impossible to skew by x as it does not appear independently of anymore. e graphical intuition behind loop skewing does not hold for combinations of skews. However, when a R is identical to S , Clint will perform a S since it is one of the well-known classical transformations 1 . Targeting Individual Statements. Many Clay transformations operate on β-pre xes, that is loops rather than statements. We circumvent this by distributing away the target statement, applying the desired transformation to a loop nest with only this statement, and then fusing everything back. Manipulating Multiple Statements. If multiple polygons are selected within a coordinate system, transformations are applied to all of them in inverse lexicographical order of their respective β-vectors. Inversion prevents transformations from modifying β-vectors used to target subsequent transformations. If a user manipulates a pile (or a coordinate system), the action is propagated to all the polygons it contains, making the pile an implicit selector for the polygons it contains. Manipulating Groups of Points. Individual points or groups thereof can be manipulated by turning them into a polygon rst. Selecting a group of points and dragging it away from existing polygon separates it into two parts, mapping to the I S S transformation. It creates a new statement occurrence that can be manipulated separately. Dropping this polygon on top of another polygon that represents a di erent occurrence of the same statement is mapped to the C transformation. In cases of selections that are not adjacent to borders and/or not convex, multiple I S S transformations are performed. Each of the two resulting parts may correspond to multiple occurrences of the statement, but is visualized and manipulated as a whole. Cross-Projection Selections. When multiple projections are used, the selection of statement instance points is combined from di erent projections. e overall multidimensional selection is an intersection of constraints imposed by each separate two-dimensional selection. Empty selection in a projection is thus equivalent to selecting everything. Decoupling Visualization from Code. In Clint, we keep the visualization consistent with the original program structure unless the user manually modi es the code. is allows for manipulating multiple statement occurrences together, for example in case of shi ing one statement with respect to another inside the loop, which may result in loop separation as in Fig. 4. Fig. 4. Manipulation for Transformation: the darker polygon is dragged right so that dependence arrows become vertical without spanning between di erent iterations on i. The visualization is then decoupled from the code structure, and both statements can still be manipulated as if they were not split between two loops. Transformation Legality Feed-Forward. Clint graphical interactions are structured so that it is possible to identify the transformation before it is completed. For example, dragging a corner of a polygon away from its center corresponds to a R , the dragging direction and distance de ne transformation parameters. Since they are typically expressed in units of iteration steps through division, we can use ceil instead of normal rounding to obtain the parameters earlier. Hence Clint can perform a transformation before the end of corresponding user interaction. is allows to provide feed-forward about the transformation, i.e., its e ects (in particular dependence violation) are visualized during the interaction, guiding the user in their choice. In addition, this approach allows Clint to hint the user about the state of the visualization if they nish manipulation immediately using a grayed-out preview shape (see Fig. 8). Mapping Loop Transformations to Animated Transitions Clint visualization enables the illustration of step-by-step execution of a Clay transformation script, either constructed manually or translated from a compiler-computed schedule using Chlore [START_REF] Bagnères | Opening Polyhedral Compiler's Black Box[END_REF]. Instead of providing a one-to-one mapping between individual transformations and animated transitions, we take a generalized approach based on the structure of transformations. ey can be divided based on the scheduling relation dimensions they a ect: (1) only α, (2) only β or (3) both α and β. e rst group contains all transformations except F N , D and R , which belong to the second group, and S M , L , I S S and C , which belong to the third group. is classi cation allows us to limit the animation scope. Transformations that do not modify β-dimensions may only a ect points inside one container while points cannot be moved between containers. Furthermore, only the projections on iterators involved in the transformation should be updated. Transformations that only modify β-dimensions a ect entire containers without modifying the point positioning inside them. Within-Container Transformations. Transformations of the rst group are animated by simultaneously moving individual points to their new positions. During the transition, polygonal shapes are updated to match the convex hull of the respective points. us S transformation moves all points simultaneously in one direction and corresponds to visual displacement, while R transformation moves rows (or columns) of points at di erent lengths and results in shape skewing. Multiple Projections. Several transformations operate on two dimensions, for example R and I . For these cases, we consider the projection on both of these dimensions as the main one, and the projections on one of the dimensions as auxiliary ones. In the main projection, the one-to-one point transition remains applicable. On the other hand, in the auxiliary ones, points may be created or deleted. For example, an auxiliary projection retains the rectangular shape a er a R but becomes larger as some points are projected onto new coordinates (see Fig. 5). Clint handles this by introducing a temporary third axis, orthogonal to the screen plane. is axis corresponds to the dimension present in the transformation, but not in the projection. Points and arrows are then re-projected on three dimensions. Extra objects become visible only during the animated transition and create a pseudo-3D e ect. A er the transition, the third axis is deleted while the projected points remain in place (see Fig. 5b). is technique is analogous to Sca erDice [START_REF] Elmqvist | Rolling the Dice: Multidimensional Visual Exploration Using Sca erplot Matrix Navigation[END_REF], but without axis switching. Between-Container Transformations. As transformations of the second group a ect entire polygons only, we can translate them into motion of polygons. If all polygons of a container are moved, the entire container is moved instead. Target containers are identi ed using β-pre xes. Container Creation and Deletion. Transformations of the third group may result in containers being created or deleted. However, without points, a polygon would correspond to statement occurrence that has no instances and thus is never executed. erefore, it must be impossible to create empty containers. e only way to create a container in Clint is by spli ing an existing container into multiple parts. is exactly corresponds to the I S S transformation if the container is a polygon. It also maps to the D transformation when the container is a coordinate system or a pile. Conversely, C and F N transformations correspond to visually joining two containers. Clint Interface Clint combines three editable and synchronized representations (see Fig. 5a): (1) the interactive visualization; (2) a navigable and editable transformation history view based on Clay scripts; and (3) the source code editor. A consistent color scheme is used between the views to match code statements to the visualization. Transformation directives corresponding to graphical actions are immediately appended to the history view. e user can then navigate through the history by selecting an entry, which will update the visualization to the corresponding previous state, or edit it directly using Clay syntax. As the target code tends to become complex and unreadable a er several manipulations, the user has the option to keep the original code visible instead of the transformed one. Finally, when the code is edited, the visualization is updated, thus making Clint a dynamic visualizer for polyhedral code. USE SCENARIOS Clint can be used as a stand-alone program transformation tool or in conjunction with an automatic optimizer. In the rst case, the user must decide on the transformation to perform. In the second case, Clint proposes a sequence of primitive transformations equivalent to the automatically computed one, le ing the user complement or modify it independently from the optimizer. In both cases, the user may reason in terms of an instance-wise dependence graph rather than in terms of loop transformations or parameters of the optimization algorithm. Our approach does not impose a particular transformation heuristic. Instead, we suggest to build intuition by visualizing (optimized) programs that perform well and identifying visual pa erns. For an optimization expert, these pa erns may eventually lead to a novel heuristic. We provide two end-to-end illustrative examples, in which we a empt to make dependence arrows short to improve reuse and orthogonal to axes to exploit parallelism. Assisted Semi-Automatic Transformation Clint can be used as a tool for applying loop-level transformations that provides instant legality feedback and generates transformed code automatically. Let us continue with the polynomial multiplication kernel example, see Fig. 1a, to demonstrate how a long sequence of transformations can be applied. Default representation of the kernel, with parameters set to 4, is shown in Fig. 6a. e loop j features parallelism and is marked accordingly. Inner parallelism is o en less desirable as it would incur barrier synchronization cost on every iteration of the outer loops. erefore, observing that dependence arrows are diagonal, the user may decide to make them orthogonal to the i loop to make it parallel. ey can do so by dragging the top right handle of the polygon right, Fig. 6a. However, such transformation is illegal as indicated by the red arrows that point in the direction opposite to the j access. is dependence violation can be removed by switching the direction of arrows, which is achieved by dragging the top right handle le to rotate the polygon around its center, Fig. 6b. e combined transformation sequence is now legal yet potentially ine cient: di erent iterations of parallel loop i execute di erent numbers of statement instances. Observing the symmetry of the polygon, the user selects a triangular-shaped group of points on the right, Fig. 6c, and drags it to the empty space on the le , Fig. 6d, until the balanced, rectangular shape is reconstructed, Fig. 6e. e nal transformation corresponds to loop skewing, followed by two loop reversals and shi s, then by index-set spli ing, and nally by shi ing. However, at no time during transformation, the user must be aware of particular loop transformations, their legality or the transformed code. ey can operate on an instantiation of the instance-wise dependence graph as opposed to directive-based approaches where, even with visualization, they would have to nd the transformation directive that would result in a desired visual shape. Understanding, Improving and Rectifying Automatic Transformation Manual program transformation, even with e cient support tools, may require su cient e ort from the programmer. Fully automated program optimizers are designed to yield decent performance in most cases. However, they are based on imprecise heuristics, which may fail to improve performance or even degrade it. Polyhedral optimizers are essentially source-to-source black boxes o ering li le control over the optimization process. Clint relies on Chlore [START_REF] Bagnères | Opening Polyhedral Compiler's Black Box[END_REF] to nd a sequence of primitive directives equivalent to the automatically computed optimization and let the user replay and modify it, independently of the optimization algorithm. e user does not have to know or understand the internal operation of the optimizer and its con guration. Consider the Multi-Resolution Analysis Kernel code, available in doitgen benchmark of the PolyBench/C 4.2 suite [START_REF] Pouchet | PolyBench/C 4.2. Polyhedral Benchmark Suite[END_REF] and presented in Fig. 7a. A sequential version of this kernel runs in 0.83s on our test machine. 2 We applied Pluto3 polyhedral compiler [START_REF] Bondhugula | A Practical Automatic Polyhedral Parallelizer and Locality Optimizer[END_REF] to extract parallelism from this code. We also requested Pluto to tile the transformed code, which is likely to improve performance thanks to data locality and expose wavefront parallelism. A simpli ed version of the resulting code is presented in Fig. 7c. It indeed contains tiled and parallelized loops. Yet this code executes in 0.91s, a 10% slowdown compared to the sequential version (untiled parallel version executes in 50.1s, a 62× slowdown). Without any further suggestion from Pluto, the user may either stick with a non-transformed sequential version or with a non-e cient parallel one. e code was transformed so aggressively that the user is unlikely to a empt code modi cations or even understanding the transformation that was applied. Comparing Clint visualizations before, Fig. 7b, and a er, Fig. 8a, transformation suggests loop ssion took place, which can also be inferred from the generated code. Step-by-step replay con rms this and also demonstrates loop tiling followed by skewing. It also shows that inner loops were parallelized, which is known to result in large barrier synchronization overheads. A fat dot between coordinate systems indicates there is some reuse between loops, but it is unclear whether Pluto performed ssion to ensure legality of skewing and tiling or because of its fusion heuristic. To discover that, the user may undo the ssion by fusing the loops back together, Fig. 8a. While they drag the polygon, legality feedforward appears in a shape of gray arrows that indicate that transformation would be legal and would preserve parallelism. Motivated by the success and observing the remaining reuse, the user may decide to fuse the remaining loop as well. is transformation would be illegal as indicated by red arrows appearing as the polygon is being dragged. e users can still nish the manipulation, and then use a conventional "undo" command. Particular use cases of the previous section illustrate well the potential bene ts of the tool in speci c cases, but they do not help evaluating and understanding its overall usability in more general cases and with di erent users. erefore, as it is commonly done in Human-Computer Interaction, we conducted a series of user studies considering more abstract tasks that assess the usability of Clint. Understanding the Visualization Although similar visualizations have been already used for descriptive or pedagogical purposes, there is no empirical evidence of their appropriateness for conveying program structures. We designed an experiment to assess the suitability of our visual representation. In particular, we test whether both experts in the polyhedral model and non-expert programmers can establish a bidirectional mapping between Clint visualization and code. Participants. We recruited 16 participants (aged 18-53) from our organizations. All of them had experience in programming using imperative languages with C-like syntax and basic understanding of the polyhedral model and its limitations. Six participants reported to have manually constructed similar visualizations from scratch and were therefore considered Experts. Because participants were asked to construct visualizations following given rules, previous exposure to these rules is a more relevant criterion of expertise than familiarity with the polyhedral model. Procedure. Our experiment is a [3 × 2] mixed design having two factors: • T : mapping direction (between participants) -Visualization to Code (VC) -writing a code snippet corresponding to a given visualization using a C-like language featuring loops and branches with a ne conditions; -Code to Visualization (CV ) -drawing an iteration domain visualization given the corresponding code. • D : problems may be (within participants) -Simple -two-dimensional with constant bounds; -Medium -multi-dimensional with constant bounds; -Hard -two-dimensional with mutually-dependent bounds and branches. We divided participants in two groups with equal number of experts. Group 1 performed the VC task, group 2 performed the CV task. is between participant factor allowed us to present the same problems to all participants while avoiding learning e ect. Both tasks were performed on paper with squared graph support for the CV task. Participants were instructed about visualization and performed two practice tasks before the session. ey were asked to work as accurately as possible without time limit and were allowed to withdraw from a task. Expected solutions were shown at the end of the experiment. Each session lasted about 20 minutes. Data Collection. For each trial, we measured Completion Time, Error and Abandon rates. e errors were split in two categories: Parameter Errors, when the shape of the resulting polyhedron was drawn correctly, but linear sizes or position were wrong; Shape Errors, when the shape of the polyhedron was incorrect. Codes describing the same iteration domain were considered equivalent (e.g., i <= 4 and i < 5). Upon completion, participants lled out a demographics questionnaire. Data Processing and Analysis. We performed log-transformation of the Completion Time to compensate for the positive skew of its distribution, resulting in asymmetric con dence intervals. Due to concerns over the limits of null hypothesis signi cance testing in various research elds [START_REF] Cumming | e New Statistics Why and How[END_REF][START_REF] Dragicevic | Fair Statistical Communication in HCI[END_REF], our analyses are based on estimation [START_REF] Cumming | Inference by Eye: Con dence Intervals and How to Read Pictures of Data[END_REF]. We report symmetric e ect sizes on means -es = 2(m 1 -m 2 )/(m 1 + m 2 ) where m 1 , m 2 are means-and 95% con dence intervals (CIs). Results . We did not observe signi cant order e ect on the Error Rate or Completion Time, meaning that there were neither learning nor fatigue e ect along the experiment. Completion Time. We discarded 7 trials in which participants produced erroneous code. T did not strongly a ect the Completion Time: VC took 182s (95%CI = [127s, 262s]) on average while CV took 215s (95%CI = [156s, 296s]) on average, resulting in an e ect size of 16.3% (95%CI = [-39.2, 50.9]). Despite Experts being familiar with similar representations, we observed no interaction between expertise and T . Experts performed 56.7% (95%CI = [26.8, 98.8]) faster than Non-Experts for Hard tasks. Both performed similarly on Easy and Medium tasks. In general, Completion Time is more consistent across Non-Expert participants than across Expert participants (Fig. 9a). ese results suggest that our representation is suitable for both Experts and Non-Experts if the complexity of the task remains limited. ey also con rm our assessment of task di culty. Errors and Abandons. Participants performed the tasks with very low error rates, 8.3% (95%CI = [-3.6%, 20.3%]) for VC tasks and 4.2% (95%CI = [-4.5%, 12.8%]) for CV. Non-Experts proposed wrong code for Hard VC tasks, equally split between Parameter and Shape Errors. Experts made Parameter Errors for some Medium tasks. We observed only two withdrawals during a trial, both from nonexperts on a Hard task, one in VC and CV, and a er more than 500s (Fig. 9b). Overall, such low error rates make it di cult to conclude on the causes of the errors, but suggest that both experts and non-experts users can reliably map Clint visual representation to the code and vice versa. Interactive Manipulation A er assessing the visualization approach, we focused on interactive program transformation with Clint. We conducted a preliminary usability study with users already familiar with the visualization. In order to separate the e ect of direct manipulation from individual di erences in expertise, participants were not allowed to use any automatic parallelizing compiler that would help experts to achieve be er performance. We also decided not to use Clay syntax directly as it is li le-known and was designed as an intermediate representation for graphical manipulation. Noone a empted to use other diretive-based tools. Protocol. Participants. We recruited 8 participants (aged 23-47) by direct email to the participants of the previous study. Since they all were familiar with Clint, our expertise criterion does not apply. Apparatus. e study was conducted with a prototype of Clint running on a 15" MacBook Pro. Participants were interacting with the laptop keyboard and a standard Apple mouse. Procedure. e task consisted in transforming a program part so that the maximum number of loops becomes parallelizable. Participants had to transform the program, but not to include parallelism-speci c constructs, e.g., OpenMP pragmas, in order to avoid bias from individual expertise di erences. e experiment has a [3 × 3] within-subject design with two factors: • T used in the trial: Code -writing code in an editor of user's choice, no visualization available; Viz -direct manipulation, no code visible; Choice -full interface, with direct manipulation and source code editing. • D of the task: Easy -two-dimensional case with at most two transformations; Medium -two-or three-dimensional case with rectangular bounds and at most three transformations; Hard -two-or three-dimensional case with mutually-dependent bounds and at least two transformations. Trials were grouped in three blocks by T . e Code and Viz blocks were presented rst. eir order was counterbalanced across participants. Choice was always presented last in order to assess participants' preference in using code editing or direct manipulation a er having used both. In each block, participants were presented with one task of each di culty level in random order. Tasks were randomly picked into di erent blocks across participants. ey were drawn from real-world program examples and polyhedral benchmarks. Trials were not limited in time and participants were asked to explicitly end the trial by pushing an on-screen bu on. Prior to the experiment, participants were instructed about source code transformations and the corresponding direct manipulation techniques. ey also practiced 4 trials of medium di culty for each technique before the experiment and were allowed to perform two "recall" practice trials before each T block. Each session lasted about 60 minutes. e study was completed by a demographics questionnaire. Data Collection. For each trial, we measured: • the overall trial Completion Time; • First Change Time, the amount of time from the start to the rst change in the program structure (code edited or visualization manipulated); • Success Rate, the ratio between the number of loops made parallel by transformation and the total number of possibly parallel loops. We recorded both the nal state and all intermediary transformations to the program. During the analysis, we performed a log-transform of the Completion Time and First Change Time. Results and Discussion . Because this experiment was conducted with a small sample, we mostly report results graphically in order to illustrate general trends. We did not observe any ordering e ect of T or D on Completion Time and Success Rate. Accuracy and E ciency. Fig. 10a suggests, despite large variability, that participants were in general more successful in transforming the program with direct manipulation than with code editing. E ect sizes reach 40% and 44% for Easy and Medium tasks. However, for Hard tasks, the success rates are identical. is suggests that nding a multi-step transformation is a key di culty. Fig 10b suggests that, for successful trials, participants performed the transformation consistently faster in Viz condition. e di erence in variability between Code and Viz suggests that direct manipulation compensates for individual expertise di erences. Similar Completion Times for failed trials can be explained, a er analyzing the transformations, by participants "abandoning" the trial if their rst a empt did not expose parallelism and submi ing a non-parallelizable version. Strategy and Exploration. Participants at least tried to perform a transformation in 76% cases with Code and 94% with Viz, suggesting that visualization engages participants by changing the perception of task di culty. We computed the ratio First Change Time/Completion Time as a measure of "engagement" (Fig. 10c). It increases with di culty for Code, but drastically decreases for Viz, suggesting that participants were more likely to adopt an exploratory trial-and-error strategy supported by the interactive visualization as opposed to code. In Choice condition, the ratio remains stable, as participants spent time choosing which representation to use. Choice between Code Editing and Direct Manipulation. In the Choice condition, only 3 participants interacted with the code. ey made edits during the rst 30s and then switched to the visualization. A er the experiment, they explained to have modi ed the code for the sake of analysis, e.g., to see whether a dependence was triggered by a particular access they temporarily removed. We observed that most participants were examining the code, but not selecting it. is observation suggests that, although they see the limitations of code representation, participants may need it to relate to the conventional program editing that be er corresponds to their expertise. Preference for Code or Visualization Our last experiment investigates the use of textual and visual representations for SCoPs. We relied on eye tracking technology in order to precisely measure visual a ention between code and visualization when both were available. We expect that, given su cient training, users will prefer visualization to code analysis if there is a meaningful task-relevant mapping between the two. is experiment required a pair of small program analysis tasks such that either code or visualization support each of them be er, but never both. Participants had to answer a binary question, with positive or negative formulation to avoid bias. e study was structured as the previous one. Protocol. Participants. We recruited 12 participants (aged 21-34, mean=27) through mailing lists. ey did not participate in previous studies and had a self-reported experience in programming of 5 to 15 years. All had normal uncorrected vision. Apparatus. e experimental setup consisted of a 15" MacBook Pro with 2880 × 1800 screen at 220 ppi connected to the SMI-ETG v1 eye-tracking system 4 . e participant was seated 70 cm away from the screen, which resulted in gaze position accuracy of 27.7px in screen space. e tracking system outputs a 30 FPS video stream from its frontal camera. We placed bright-colored tokens on the screen corners to locate it in the video and compensate for perspective distortion. ese tokens were tracked by a custom OpenCV-based script that generated gaze position in screen coordinates through linear interpolation with perspective correction. We ensured that the sizes of both representations are identical across conditions, with the content centered in each of them. Unused space was lled with neutral gray to avoid distraction. When visible, multiple representations were 60 px away (2× resolution) to identify gaze into one of them. Procedure. e study is a [3 × 3 × 2] within-participants experiment with 4 repetitions per participant and the following factors: • R used in the trial, one of visual representation (Viz), source code (Code) or both simultaneously (Choice); • D of the question, one of Easy, a loop nest with constant conditions, Medium, a loop nest with at least 3 non-constant conditions, or Hard, a loop nest with a branch inside and at least 5 non-constant conditions; • a binary asked to the user, either concerns the textual form of loop bounds (Bounds) or a statement instance being executed or not inside a loop (Execution). Bounds questions were targeted at Code, where the answer is immediately visible, while Execution questions were targeted at Viz. We refer to these conditions as matching questions, and to other conditions as mismatching questions. In total, we collected data for 12 • 3 • 3 • 2 • 4 = 864 trials. Trials were rst blocked by R and then by repetition. R blocks are ordered identically to the previous study. Each of them comprises 4 repetition blocks, each of which has 6 trials with di erent and D in a randomized order. R blocks were preceded by a practice session with 4 trials of Medium di culty. A er each trial in Choice condition, participants were asked about their preferred representation for this question. Blocks featuring only Code or Viz were conducted without eye tracking. Participants were wearing the eye-tracking glasses for the third block, a er we performed a 3-point calibration with 30px tokens and checked if the glasses did not a ect their vision by performing a read-aloud test. Participants started the trial by clicking the "start" bu on and ended it by clicking the answer bu on. ey could abandon the trial a er at least 15s to avoid immediate abandons for Hard tasks with mismatching questions. So ware provided the correct answer a er each trial. One session lasted 50 minutes on average and was complemented by a demographic questionnaire. Data Collection and Processing. We collected the following data: • Completion Time of the trial; • Correctness of the answer; • Preference between R for the last block; • Gaze from the eye-tracking glasses for the last block. Given gaze position in screen coordinates, we identi ed the widget in the focus of a ention as one out of three: Code Widget, Viz Widget or estion Widget. Outside any of the widget areas, the gaze was considered O Screen. We randomly sampled 10 frames from each video and veri ed manually that the script provides exact classi cation. Completion Time was log-transformed to compensate the positive skew of its distribution. Results and Discussion . Ordering e ects. We observed a slight decrease in Completion Time between rst blocks, e ect size -13.6% (95%CI = [-37.7, 6.1]), but large variability does not allow to conclude on the presence of a learning e ect. Correctness did not vary substantially between blocks. Completion Time. Mismatching questions required substantially more time to complete the trial than matching questions, except for Easy tasks as shown in Fig. 11a . ese results suggest that, given two representations, participants are likely to chose the matching one. Although they do not spend more time on average, the variability is larger for Choice condition. It suggests that participants only e ectively use one representation, but consider both. We illustrate this later with eye tracking data. Correctness. e participants succeeded to answer the majority of the questions with 93% (95%CI = [90, 95]) of correct results on average as shown in Fig. 11b. Abandoned trials were considered as incorrect answers. Overall trends are similar to Completion Time. Given Choice, participants had a high success rate overall, except Easy Execution questions with mean Success Rate 89.5% (95%CI = [79%, 100%]). is may be explained by choosing the mismatching Code representation due to visible task simplicity. Due to low error rates, we did not perform any further analyses. Only 4 trials were abandoned, all featuring mismatching questions, 3 of which with Code. Abandons took place a er 91s on average whereas the mean trial duration is 13.7s. Representation Choice. Our analyses are built on the following metrics, de ned prior to the study. Visual Preference, VP -total duration of gaze on the Viz Widget divided by the total duration of gaze on Viz or Code Widget. Values close to 1 indicate participant looking more at the visualization. Representation Uncertainty, RU -the measure of a ention distribution computed as RU = 2 • abs(VP -0.5). High values mean a ention was distributed evenly between representations, low values -that only one representation was used. We expect Completion Time to increase with Representation Uncertainty as the participant uses two representations where one would su ce. At the same time, it may increase even more for lower values of Representation Uncertainty and high Visual Preference for the unadapted representation. Fig. 12a shows the Visual Preference for di erent conditions, the center line corresponding to the equal distribution of visual a ention. For Medium and Hard tasks, participants spent more time on matching representations, e ect sizes reach 66.6% (95%CI = [4.8, 128.5]) and 81.2% (95%CI = [16.7, 145.7]), respectively. For Easy tasks they relied on the Code independent of . e reported Preference, depicted on Fig. 12b, shows the same tendency. e preference for Code drops from 56% in Easy Execution tasks to 6% in Medium and Hard Execution tasks. Since we asked which representation they found "most useful", the di erence between reported Preference and Visual Preference suggests that participants tend to look at both representations even though they do not nd one of them useful. Nevertheless, we observed a positive correlation between reported Preference and Visual Preference, r =0.41 (95%CI = [0.20, 0.57]), suggesting that participants tend to use more the representation they nd useful. Overall, we observed a correlation between Representation Uncertainty rate and Completion Time, r =0.41 (95%CI = [0.19, 0.58]) as well as a negative correlation between Representation Uncertainty rate and Correctness, r =-0.27 (95%CI = [-0.47, -0.04]): the more participants' a ention was distributed between representations, the less correct answers they gave. Although the correlation does not imply causality, the connection between the simultaneous use of di erent representations and the total trial duration suggests that one matching representation should be preferred to two. RELATED WORK Interactive Program Parallelization. Program editors supporting interactive program parallelization date back to wide adoption of parallelism for scienti c programming. We review those speci cally targeting loop-level optimizations. e ParaScope editor [START_REF] Kennedy | Interactive Parallel Programming Using the ParaScope Editor[END_REF] provided dependence analysis and interactive loop transformation for High-Performance F (HPF). It reported the dependence analysis results and allowed the user to perform various loop transformations, including parallelization. e D Editor interacted with an distributed HPF compiler to report optimization choices regarding data distribution and parallelization [START_REF] Hiranandani | e D Editor: A New Interactive Parallel Programming Tool[END_REF]. SUIF Explorer took a di erent approach, collecting dynamic execution and dependence data to suggest loops (or parts thereof thanks to program slicing [START_REF] Weiser | Program Slicing[END_REF]) for parallelization [START_REF] Liao | SUIF Explorer: An Interactive and Interprocedural Parallelizer[END_REF]. Similarly, DECO records traces of the memory accesses along with cache hit information and uses pa ern recognition algorithms to suggest memory optimizations [START_REF] Tao | An Interactive Graphical Environment for Code Optimization[END_REF]. NaraView provides a navigable 3D visualization of loop-level access pa erns [START_REF] Sasakura | NaraView: An Interactive 3D Visualization System for Parallelization of Programs[END_REF]. Contrary to these tools, Clint uses the polyhedral model with its instancewise dependence analysis and static guarantees of loop transformation legality. It also allows for transforming the program using its visualization. Chlore-based transformation replay is not tied to particular compiler transformations. Semi-Automatic Polyhedral Transformations. User-assisting tools based on the polyhedral model emerged as a means to express "classical" loop transformations [START_REF] Joseph | High Performance Compilers for Parallel Computing[END_REF] in the model, the Uni ed Transformation Framework (UTF) stemming from the rst approach [START_REF] Kelly | A Unifying Framework for Iteration Reordering Transformations[END_REF]. URUK was proposed to improve loop transformation composability and enable automated traversal of a transformation search space [START_REF] Girbal | Semi-Automatic Composition of Loop Transformations for Deep Parallelism and Memory Hierarchies[END_REF], delaying the legality analysis until code generation. Loop Transformation Recipes combine loop transformations, mapping to accelerators and code generation directives from CHiLL [START_REF] Chen | CHiLL: A for Composing High-Level Loop Transformation[END_REF] with the POET [START_REF] Yi | POET: Parameterized Optimizations for Empirical Tuning[END_REF] language for auto-tuning speci cation. AlphaZ focuses on equational programming and enables complex memory mapping and management [START_REF] Yuki | Alphaz: A System for Design Space Exploration in the Polyhedral Model[END_REF]. Clay is arguably the rst complete set of directives for polyhedral program transformations [START_REF] Bagnères | Opening Polyhedral Compiler's Black Box[END_REF]. Clint uses visualization and direct manipulation to address the challenges of directive-based approaches, such as identifying a promising transformation, targeting it at a program entity or evaluating its e ects. Visualizations for the Polyhedral Model. e literature on the polyhedral model heavily relies on sca erplot-like visualizations of iteration domains. Polyhedral libraries include components for visualization, including VisualPolylib [START_REF] Loechner | PolyLib: A library for manipulating parameterized polyhedra[END_REF] for Polylib and islplot [START_REF] Grosser | islplot: Library to Plot Sets and Maps[END_REF] for isl [START_REF] Verdoolaege | Isl: An Integer Set Library for the Polyhedral Model[END_REF]. LooPo was arguably the rst tool to visualize the polyhedral dependence analysis information during program transformation [START_REF] Griebl | e Loop Parallelizer LooPo-Announcement[END_REF]. Tulipse integrates polyhedral visualization into Eclipse IDE [START_REF] Wong | Tulipse: A Visualization Framework for User-Guided Parallelization[END_REF]. Clint goes beyond static visualization by enabling direct manipulation to transform the program. 3D iteration space visualizer lets the user interactively request loop parallelization through a visual representation [START_REF] Yu | Loop Parallelization Using the 3D Iteration Space Visualizer[END_REF]. Polyhedral Playground [START_REF] Grosser | PollyLabs Polyhedral Playground[END_REF] augments a web-based polyhedral calculator with domain and dependence visualizations. PUMA-V provides a set of visualizations that expose internal operation of the R-Stream compiler [START_REF] Papenhausen | PUMA-V: An Interactive Visual Tool for Code Optimization and Parallelization Based on the Polyhedral Model[END_REF][START_REF] Papenhausen | Polyhedral User Mapping and Assistant Visualizer Tool for the R-Stream Auto-Parallelizing Compiler[END_REF]. It allows the user to control the optimizationrelated compiler options from the visualization. Clint builds on Clay as intermediate abstraction and does not require the user to control or even understand the operation of a compiler. CONCLUSION Clint addresses the issues of directive-based approaches in the polyhedral model: target identication is made direct without exposing polyhedral-speci c concepts; transformation legality and e ects are visible immediately during manipulation; reading polyhedrally-transformed code is no longer necessary. It makes loop optimization accessible, interactive and independent of a particular algorithm. Our approach enables human-machine partnership where an automatic framework performs heuristic-driven transformation and provides feedback on demand while a user brings in domain knowledge to tweak the transformation without modifying the heuristics. Such domain knowledge may be unavailable to framework designers and di er between use cases. Experiments suggest that visualizations lower the expertise necessary to perform aggressive program restructuring and decrease the time necessary for program analysis. Semi-automatic transformation decreases the time of program transformation. In our studies, visual semi-automatic approach to program transformation doubled the success rate and decreased the required time by a factor of 5 for some program structures. We also contribute to the discussion on visualization acceptance, suggesting its perceived utility increases with the relative complexity of the task. Limitations. As Clint was designed using a set of polyhedral test cases with small number of statements nested in shallow loops, it may be subject to clu ering for larger program parts. Long blocks of interdependent statements may result in a profusion of dependence arrows. Visual replay may become distracting when multiple projections are rendered for deep loops. However, program parts amenable to the polyhedral model are typically small yet require aggressive transformation. Future Work. Drawing from the eye-tracking study conclusions and existing limitations, the visual approach seems promising yet restricted for di cult cases. We plan to address those by interleaving visual representations and code fragments and by proposing a zoomable interface with di erent levels of detail. At the same time, the visualization may be bene cial for learning, which can be supported with a smooth transition between code and visual representation. Visual clu ering can be addressed by only displaying salient parts. ey can be identi ed directly by the users, or inferred from their behavior. On the other hand, a polyhedral compiler may provide additional feedback on, e.g., dependences that prevent parallel execution. Finally, Clint visualizations may be used conjointly with performance models and runtime evaluators, and integrated into a larger development environment in order to account for program parallelization all along the development process. Fig. 1 . 1 Fig. 1. Polynomial Multiply computation kernel. = 0; i < N; ++i) for (j = 0; j < N; ++j) z[i+j] += x[i] * y[j]; Fig. 2 . 2 Fig. 2. Performing a skew transformation to parallelize polynomial multiplication loop by deforming the polygon. The code is automatically transformed from its original form (le ) to the skewed one (right). for (i = 0; i < N; ++i) for (j = 0; j < N; ++j) { A[i+1][j+1] += 0.5 * A[i+1][j]; B[i+1][j+1] += A[i][j]; } for (j = 0; j < N; ++j) B[1][j+1] += A[0][j]; #pragma omp parallel for private(j) for (i = 0; i < N; ++i) for (j = 0; j < N; ++j) { A[i+1][j+1] += 0.5 * A[i+1][j]; B[i+1][j+1] += A[i][j]; } for (j = 0; j < N; ++j) A[N][j+1] += 0.5 * A[N][j]; (a) Clint interface includes: (1) interactive visualization with multiple projections, (2) editable history view of transformations, and (3) source code editor; all coordinated with each other. (b) When main projection is manipulated, auxiliary projections are updated simultaneously. Fig. 5 . 5 Fig. 5. Clint displays multiple projections for deep loop nests. Fig. 6 . 6 Fig. 6. Users can directly manipulate the visual representation of SCoPs and have the transformed program generated automatically. Dragging the corner from the center performs loop skewing, to the center-reversal; selecting the points and dragging them performs index-set spli ing followed by loop shi ing. Dependence arrows orthogonal to axes enable parallel execution. Fig. 8 . 8 Fig. 8. Using visual representation to re-adjust automatically computed transformation with immediate feed-forward on semantics preservation. Dependences violated by the intended transformation turn red, within shapes depict tiles. Shaded shapes are positions before manipulation. Fig. 9 . 9 Fig. 9. (a) Completion Times increase with task di iculty but less so for Experts. Results are similar between Experts and Non-Experts. Error bars are 95% confidence intervals. (b) overall Error Rate is low. Experts are more successful but fail at simpler tasks; Non-Experts may abandon. Fig. 10 . 10 Fig. 10. (a) Success Rate is higher with Viz, except for Hard tasks. (b) Completion Time is lower with Viz, especially in successful trials. (c) Ratio First Change Time / Completion Time; the change in trend between Code and Viz may be due to users adopting an exploratory strategy. Error bars are 95% CIs. Fig. 11 . 11 Fig. 11. (a) mismatching questions required up to 4× more time. (b) Medium and Hard questions with mismatching representation result in more incorrect answers. Completion Times and Correctness Ratios for C are close to those for matching representation. Dots are means, error bars are 95% CIs, vertical density plots show underlying distributions. . With Code, participants spent 14% (95%CI = [-22, 40]) more time on Easy Execution questions, and respectively 132% (95%CI = [108, 146]) and 134% (95%CI = [111, 147]) more time on Medium and Hard Execution questions than on the Bounds questions of the same di culty. Similarly, with Viz representation, they answered Execution questions 9% (95%CI = [-20, 49]), 40% (95%CI = [1, 102]) , and 57% (95%CI = [10, 129]) faster for increasing D . is result supports the de nition of mismatching question suggesting that a representation not adapted for the question slows participants down. e smaller increase of Completion Time with Viz compared to Code suggests that Viz representations allows to reason about mismatching questions easier than Code. Choice condition shows Completion Times close to those for matching representation. For Bounds questions, it took on average 6% (95%CI = [-27, 56]), -3% (95%CI = [-40, 56]), and 7% (95%CI = [-33, 70]) more time compared to Code for increasing D . For Execution questions, it took 5% (95%CI = [-27, 53]), -14% (95%CI = [-54, 54]) and -21% (95%CI = [-63, 58]) more time than Viz for increasing D Fig. 12 . 12 Fig. 12. (a) matching representations are more used for Medium and Hard tasks, but Code for Easy tasks. (b) reported preference demonstrates similar trend. Dots are means, error bars are 95% CIs. ACM Transactions on Architecture and Code Optimization, Vol. 15, No. 1, Article 16. Publication date: March 2018. In fact, we created R transformation in Clay to address the skew combination problem. It was the last missing transformation that enabled completeness of the set. ACM Transactions on Architecture and Code Optimization, Vol. 15, No. 1, Article 16. Publication date: March 4× Intel Xeon E5-2630 (Sandy Bridge, 6 cores, 15MB L3 cache), 64 GB RAM, running CentOS Linux 7.2.1511, compiled with GCC 4.9.3 with -O3 -march=native ags, benchmark size LARGE, NQ= 140, NR= 150, NP= 160. Average of 12 runs is reported, kernel execution time only, using high-resolution CPU timers. Pluto 0.11.4 with --parallel --tile, as available on h ps://github.com/bondhugula/pluto/releases/tag/0.11.4 ACM Transactions on Architecture and Code Optimization, Vol. 15, No. 1, Article 16. Publication date: March 2018. h p://www.eyetracking-glasses.com/ ACM Transactions on Architecture and Code Optimization, Vol. 15, No. 1, Article 16. Publication date: March 2018. for (r = 0; r < NR ; r ++) for (q = 0; q < NQ ; q ++) { for (p = 0; p < NP p ++) { sum [p] = 0.0; for (s = 0; s < NP ; s ++) sum e nal manually retouched version runs in 0.67s with a (modest) 25% speedup. Without step-by-step replay and direct manipulation, it would be hard to experiment with di erent fusion structures using a general trial-and-error strategy. Although loop fusion is o en implemented as a separate optimization problem in polyhedral optimizers, it is no easier to control externally. Clint allows users to understand and directly modify the fusion/ ssion structure, instead of reasoning about how a particular heuristic would behave.

01744451

en

2024/03/05 22:32:07

2018

https://hal.science/hal-01744451/file/Kwok%20Diana_JWB%20PDW2.pdf

Keywords: mergers and acquisitions, interpersonal trust, boundary spanning, emerging economies, Malaysia How does trust develop in new subordinate-leader relationships during post-acquisition integration? Does boundary spanning facilitate or hamper trust? Cultural differences between the acquirer and acquired firm add to integration uncertainty, occurring at multiple, interconnected levels (national, organizational, functional cultures). Multicultural societies add another layer of complexity. This study compares a domestic M&A with a South African cross-border acquisition in the Malaysian financial industry. The analysis reveals that boundary spanning by acquirer and acquired-firm managers facilitated subordinate-leader trust development. I thus posit that boundary spanning mitigates uncertainty and cultural differences during integration. Further and paradoxically, integrating domestic rather than cross-border acquisitions can be more complex when intra-national culture differences are accounted for. This paper offers insights for advancing Western-developed theories and for more successful integration. INTRODUCTION Mergers and acquisitions (M&As) are burdened by high failure rates. At best only half succeed [START_REF] Cartwright | The role of culture compatibility in successful organizational marriage[END_REF][START_REF] Haspeslagh | Managing acquisitions: Creating value through corporate renewal[END_REF]. i Post-acquisition integration is a vital process characterized by complexity, uncertainty and unpredictability [START_REF] Cording | Reducing causal ambiguity in acquisition integration: Intermediate goals as mediators ofintegration decisions and acquisition performance[END_REF]Graebner et al., 2017). Birkinshaw et al.'s (2000) seminal paper argues that successful human integration facilitates the effectiveness of task integration, although both processes determine acquisition success. Cultural differences between the acquirer and acquired firm add to integration uncertainty and complexity, and have long been blamed for unsuccessful domestic and crossborder M&As [START_REF] Buono | When cultures collide: The anatomy of a merger[END_REF][START_REF] Cartwright | The role of culture compatibility in successful organizational marriage[END_REF][START_REF] Chatterjee | Cultural differences and shareholder value in related mergers: linking equity and human capital[END_REF]Nahavandi & Malekzadeh, 1988). Culture is a multifaceted construct incorporating shared assumptions, practices, values, artifacts, and rituals [START_REF] Taras | Half a century of measuring culture: Review of approaches, challenges, and limitations based on the analysis of 121 instruments for quantifying culture[END_REF]2016). Cultural differences often lead to complex trust dynamics and relationships between acquirer and acquired-firm personnel [START_REF] Buono | The human side of mergers and acquisitions[END_REF][START_REF] Stahl | Do cultural differences matter in mergers and acquisitions? A tentative model and examination[END_REF][START_REF] Teerikangas | The culture-performance relationship in M&A: From yes/no to how[END_REF]. Cultural challenges occur at multiple, interconnected levels including national, industrial, organizational, functional and professional cultures in M&As [START_REF] Teerikangas | The culture-performance relationship in M&A: From yes/no to how[END_REF]. Scholars have yet to address this interconnectivity adequately (ibid.), despite pinpointing national and organizational culture differences as the main integration challenges [START_REF] Brannen | Merging without alienating: Interventions promoting cross-cultural organizational integration and their limitations[END_REF][START_REF] Chatterjee | Cultural differences and shareholder value in related mergers: linking equity and human capital[END_REF][START_REF] Shimizu | Theoretical foundations of cross-border mergers and acquisitions: A review of current research and recommendations for the future[END_REF]. Moreover, the M&A cultural 'baggage' [START_REF] Teerikangas | The culture-performance relationship in M&A: From yes/no to how[END_REF]) can be weighed down further by within-country cultural diversity. Recent studies highlight the complexities of multifaith and multilingual societies in cross-border M&As [START_REF] Cuypers | The effects of linguistic distance and lingua franca proficiency on the stake taken by acquirers in cross-border acquisitions[END_REF][START_REF] Dow | The effects of within-country linguistic and religious diversity on foreign acquisitions[END_REF][START_REF] Kroon | Explaining employees' reactions towards a cross-border merger: The role of English language fluency[END_REF][START_REF] Kwok | CEOs we trust: Religious homophily and crossborder acquisitions in multifaith Asian emerging economies[END_REF]. In particular, [START_REF] Dow | The effects of within-country linguistic and religious diversity on foreign acquisitions[END_REF] found that multifaith and multilingual societies can add complexity to behavioral uncertainty and information asymmetry between foreign acquirers and local acquired firms. Trust is a complex, multilevel [START_REF] Currall | A multilevel approach to trust in joint ventures[END_REF] and multifaceted construct [START_REF] Mayer | An integrative model of organizational trust[END_REF][START_REF] Rousseau | Not so different after all: A cross-discipline view of trust[END_REF]. It refers to positive expectations when the trustor is vulnerable to the trustee's actions (ibid.). Trust is essential in M&As [START_REF] Graebner | Caveat venditor: Trust asymmetries in acquisitions of entrepreneurial firms[END_REF][START_REF] Lander | Boarding the aircraft: Trust development amongst negotiators of a complex merger[END_REF][START_REF] Maguire | Citibankers' at Citigroup: a study of the loss of institutional trust after a merger[END_REF][START_REF] Stahl | Trust dynamics in acquisitions: A case survey[END_REF]2012). During postacquisition integration, trust facilitates cooperation, job performance, resource sharing, and knowledge transfer [START_REF] Bauer | Speed of acquisition integration: Separating the role of human and task integration[END_REF][START_REF] Stahl | Trust in mergers and acquisitions[END_REF]2010). However, unanswered questions remain about how trust develops during integration, especially in subordinateleader relationships (Graebner et al., 2017). Inspired by calls for a multilayered, multifaceted and contextual view of culture in multiple domains (e.g., [START_REF] Dietz | Unravelling the complexities of trust and culture[END_REF][START_REF] Leung | Culture and international business: Recent advances and their implications for future research[END_REF]Tung, 2008), this study addresses the following research question: How does trust develop in new subordinateleader relationships amidst the uncertainty and cultural differences of post-acquisition integration? The research is contextualized in Malaysia, a multicultural, middle-ranking emerging economy in Southeast Asia for cross-border M&As (UNCTAD, 2017). Emergingeconomy firms actively engage in both domestic and cross-border M&As [START_REF] Aybar | Cross-border acquisitions and firm value: An analysis of emerging-market multinationals[END_REF][START_REF] Bandeira-De-Mello | Theoretical and empirical implications for research on South-South and South-North expansion strategies[END_REF][START_REF] Lebedev | Mergers and acquisitions in and out of emerging economies[END_REF]. Interestingly, Malaysia's M&A volume and value increased in 2017 despite the decline in global deals [START_REF] Duff | Transaction Trail: Annual Issue[END_REF]. Equally, this paper addresses another gap in M&A scholarship, on the difference between integrating domestic vs. international acquisitions. Most researchers contend that integrating cross-border M&As is riskier and more complicated than domestic deals [START_REF] Angwin | Strategic perspectives on European cross-border acquisitions: A view from top European executives[END_REF][START_REF] Barkema | Foreign entry, cultural barriers, and learning[END_REF][START_REF] Olie | Shades of culture and institutions in international mergers[END_REF]. Yet, some demonstrate that domestic M&A integration can be more challenging: despite the merging firms' shared national culture, organizational culture differences can hinder the integration process [START_REF] Véry | A cross-national assessment of acculturative stress in recent European mergers[END_REF]1997). In fact, the differences between domestic and cross-border M&As are still poorly understood [START_REF] Bris | The value of investor protection: Firm evidence from crossborder mergers[END_REF][START_REF] Gregory | Do cross border and domestic acquisitions differ? Evidence from the acquisition of UK targets[END_REF][START_REF] Reynolds | The international experience in domestic mergers-Are purely domestic M&A a myth[END_REF]. I apply a comparative two-case research design in the Malaysian financial services industry, with a domestic M&A and a South African cross-border acquisition. South Africa and Malaysia are newly developed mid-range emerging economies [START_REF] Hoskisson | Emerging multinationals from mid-range economies: The influence of institutions and factor markets[END_REF]. The domestic M&A combines potentially conflicting cultures: two government-linked corporations (GLCs) with majority ethnic Malay personnel, and two entrepreneurial firms with mainly ethnic Chinese personnel. ii Using an abductive approach [START_REF] Timmermans | Theory construction in qualitative research: From grounded theory to abductive analysis[END_REF][START_REF] Welch | Theorising from case studies: Towards a pluralist future for international business research[END_REF], my analysis reveals how boundary spanning by both acquirer and acquired-firm managers facilitated subordinate-leader trust development. I had found puzzling the apparent mismatch between theory-led expectations (complexity of integrating domestic vs. cross-border acquisitions) and interview narratives. Post-acquisition integration had been relatively smooth in both cases except the domestic M&A's first year, when it had been "absolute chaos" and "very stressed and a challenging environment for all the people". How did integration uncertainty and cultural differences dissipate? Moreover, other similar mergers in Malaysia had involved "a lot of infighting… intensified lobbying [and] backstabbing". Abduction led to "a re-description or re-contextualization of the phenomenon" (Welch et al., 2011, p. 748). The systematic analysis revealed boundary spanning behavior by acquirer and acquired-firm senior and middle managers. This indicates the widening locus of M&A leadership. The manager's relating, scouting, persuading and empowering behaviors [START_REF] Druskat | Managing from the boundary: The effective leadership of self-managing work teams[END_REF]Wheeler, 2003) spanned horizontal, vertical, stakeholder, demographic andgeographical boundaries (Palus et al., 2013), and encompassed cultural differences at national (between-country), intra-national (within-country), organizational and functional levels [START_REF] Teerikangas | The culture-performance relationship in M&A: From yes/no to how[END_REF]. Boundary spanning overcame the 'us and them' mentality between acquirer/acquired-firm subordinates and acquired-firm/acquirer leaders by reducing uncertainty and cultural differences, thus enabling the development of trust. Thus, I posit that boundary spanning mitigates uncertainty and cultural differences during post-acquisition integration. This paper makes two other contributions to M&A scholarship and offers managerial insights for more successful post-acquisition integration. First, it extends the role of boundary spanning in M&As, beyond the pre-M&A negotiation phase [START_REF] Lander | Boarding the aircraft: Trust development amongst negotiators of a complex merger[END_REF] to post-acquisition integration. This complements the process perspective [START_REF] Jemison | Corporate acquisitions: A process perspective[END_REF] and elucidates on the temporality of integrative actions [START_REF] Birkinshaw | Managing the post-acquisition integration process: How the human integration and task integration processes interact to foster value creation[END_REF][START_REF] Froese | Integration management of Western acquisitions in Japan[END_REF][START_REF] Teerikangas | Structure first! Temporal dynamics of structural and cultural integration in cross-border acquisitions[END_REF] from the perspective of South-South M&As. Second, it contributes to the literature on the complexities of multilingual societies in M&As [START_REF] Cuypers | The effects of linguistic distance and lingua franca proficiency on the stake taken by acquirers in cross-border acquisitions[END_REF][START_REF] Dow | The effects of within-country linguistic and religious diversity on foreign acquisitions[END_REF][START_REF] Kroon | Explaining employees' reactions towards a cross-border merger: The role of English language fluency[END_REF]. The domestic M&A was unable to establish a lingua franca for all personnel despite the possibility of speaking in two languages (English and the national language) at meetings. This demonstrates that when intra-national culture differences are taken into consideration, integrating domestic rather than cross-border M&As can be more challenging [START_REF] Véry | A cross-national assessment of acculturative stress in recent European mergers[END_REF]1997). LITERATURE REVIEW Subordinate-leader trust and M&As The paper adopts the widely quoted definition of trust from [START_REF] Rousseau | Not so different after all: A cross-discipline view of trust[END_REF], as "a psychological state comprising the intention to accept vulnerability based upon positive expectations of the intentions or behavior of another " (p. 395). This definition captures two essential concepts: positive expectations of trustworthiness, including beliefs and perceptions of being able to rely on the trustee; and, willingness to accept vulnerability or a suspension of uncertainty [START_REF] Colquitt | Trust, trustworthiness, and trust propensity: a meta-analytic test of their unique relationships with risk taking and job performance[END_REF][START_REF] Ferrin | Can I trust you to trust me? A theory of trust, monitoring, and cooperation in interpersonal and intergroup relationships[END_REF]. In M&As individual trust can be directed at: a leader, subordinate, or between peers; a group including top management team (TMT) and headquarters (HQ); and, an organization, e.g., during M&A negotiations [START_REF] Currall | A multilevel approach to trust in joint ventures[END_REF]. Interpersonal trust refers to trust between individuals. In one of the most influential trust models [START_REF] Burke | Trust in leadership: A multilevel review and integration[END_REF][START_REF] Lewicki | Models of interpersonal trust development: Theoretical approaches, empirical evidence, and future directions[END_REF], [START_REF] Mayer | An integrative model of organizational trust[END_REF] proposes three main characteristics for trusting work relationships. Ability refers to the trustee's skills and domain-specific competence. Benevolence represents the trustor's belief that the trustee wants to "do good to the trustor" (p. 718). Integrity refers to the perception that the trustee "adheres to a set of principles that the trustor finds acceptable" (p. 719). These trust antecedents juxtapose with the trust constructs introduced by McAllister (1995): cognition-based trust based on rational information on the trustee's competence, reliability and credibility; and, affect-based trust which refers to an emotional attachment or concern for the other party's interests and welfare. Cognition-based trust precedes affectbased trust in organizations [START_REF] Mcallister | Affect-and cognition-based trust as foundations for interpersonal cooperation in organizations[END_REF][START_REF] Rousseau | Not so different after all: A cross-discipline view of trust[END_REF], and increases job satisfaction and performance [START_REF] Yang | Examining the effects of trust in leaders: A basesand-foci approach[END_REF]. Studies of subordinate-leader trust in the context of M&As emphasize subordinate trust in leaders and emerging economies are still relatively rare [START_REF] Kwok | CEOs we trust: Religious homophily and crossborder acquisitions in multifaith Asian emerging economies[END_REF][START_REF] Stahl | Does national context affect target firm employees' trust in acquisitions? A policy-capturing study[END_REF]. Trust between subordinates and leaders is not necessarily mutual or reciprocal [START_REF] Brower | A model of relational leadership: The integration of trust and leader-member exchange[END_REF][START_REF] Korsgaard | It Isn't Always Mutual: A Critical Review of Dyadic Trust[END_REF][START_REF] Schoorman | An integrative model of organizational trust: Past, present, and future[END_REF]. Their antecedents differ. Leaders emphasize the subordinate's receptivity, availability, and discreteness; whereas, subordinates emphasize the leader's availability, competence, discreteness, integrity, and openness [START_REF] Brower | A closer look at trust between managers and subordinates: Understanding the effects of both trusting and being trusted on subordinate outcomes[END_REF]. This asymmetry may lead to longer-term implications on trust dynamics and post-acquisition integration. Boundaries, cultural differences, boundary spanning and M&As Boundaries separate an organization's internal and external environments. Boundaries exist within organizations [START_REF] Palus | Boundary-Spanning Leadership in an Interdependent World[END_REF][START_REF] Schotter | Boundary spanning in global organizations[END_REF]. Managers often face five types of boundaries: horizontal, vertical, stakeholder, demographic, and geographic [START_REF] Palus | Boundary-Spanning Leadership in an Interdependent World[END_REF]. Horizontal boundaries separate functions and units by specialized expertise, and include functional, occupational and professional culture differences. During post-acquisition integration, horizontal boundaries often stem from organizational culture differences (pre-M&A legacies). Vertical boundaries are found across hierarchical levels and define title, rank and power. Stakeholder boundaries refer to shareholders, Boards of Directors, customers, partners, governments, and other communities. Stakeholder boundaries can overlap with organizational culture differences in M&As, and with national cultural differences in crossborder M&As. Demographic boundaries separate people by gender, age, ethnicity, religion, political ideology, etc., and include within-country cultural differences. Geographic boundaries are defined by physical location including countries, regions and East/West, and include national and regional culture differences. Boundaries are linked to identities (i.e., who we are and how we define ourselves; ibid.). Identity has attracted the attention of M&A researchers [START_REF] Clark | Transitional identity as a facilitator of organizational identity change during a merger[END_REF][START_REF] Drori | One out of many? Boundary negotiation and identity formation in postmerger integration[END_REF][START_REF] Maguire | Citibankers' at Citigroup: a study of the loss of institutional trust after a merger[END_REF]. Aldrich and Herker (1977) describe boundary spanning in organizations in terms of information processing -to selectively interpret, filter, summarize, and facilitate the transmission of information; in other words "uncertainty absorption" (p. 219). More recently, [START_REF] Schotter | Boundary spanning in global organizations[END_REF] offers a multidimensional definition of boundary spanning, as "a set of communication and coordination activities performed by individuals within an organization and between organizations to integrate activities across multiple cultural, institutional and organizational contexts" (p. 404). In bridging and mediating across one or multiple contexts, boundary spanners overcome the 'us and them' mentality, thus allowing trust to develop [START_REF] Schotter | Boundary spanning in global organizations[END_REF][START_REF] Yip | The nexus effect: when leaders span group boundaries[END_REF]. Boundary spanners can perform a specific function or responsibility such as public relations, union representation, acquiring/disposing resources and relationship management (Aldrich & Herker, 1977), or improvise his/her actions to get the job done [START_REF] Yagi | Boundary work: An interpretive ethnographic perspective on negotiating and leveraging cross-cultural identity[END_REF]. Boundary spanning contexts include expatriates (Au & Fukuda, 2002), bicultural managers [START_REF] Yagi | Boundary work: An interpretive ethnographic perspective on negotiating and leveraging cross-cultural identity[END_REF], and global vendor-client partnerships [START_REF] Søderberg | Boundary Spanners in Global Partnerships: A Case Study of an Indian Vendor's Collaboration with Western Clients[END_REF]. In M&As, [START_REF] Lander | Boarding the aircraft: Trust development amongst negotiators of a complex merger[END_REF] studied the Air France-KLM merger where chief negotiators acted as "boundary role persons" [START_REF] Currall | Measuring trust between organizational boundary role persons[END_REF], enabling interorganizational collaboration and trust. This contrasts with [START_REF] Drori | One out of many? Boundary negotiation and identity formation in postmerger integration[END_REF] analysis into the process of boundary creation/re-creation in shaping the post-merger identity of an organization, and allowing managers and personnel to maintain key aspects of their previous identities. Anecdotal evidence suggests that leaders engage in boundary spanning during post-acquisition integration, for example, in the Lenovo-IBM merger [START_REF] Stahl | Lenovo-IBM: Bridging Cultures, Languages, and Time Zones-Integration Challenges (B)[END_REF][START_REF] Yip | The nexus effect: when leaders span group boundaries[END_REF]. Outstanding boundary spanning leaders constantly engage in internal (team) and external (organization) oriented initiatives for their teams [START_REF] Druskat | Managing from the boundary: The effective leadership of self-managing work teams[END_REF]. From studying the interactions between external leaders and their team members and managers in a large manufacturing firm, Druskat and Wheeler inductively categorized eleven boundary spanning behavior into four functional clusters. Table 1 summarizes this typology. - ------------------------------------------INSERT TABLE 1 ABOUT HERE ------------------------------------------- RESEARCH METHODS AND DATA Empirical design and context This research applies comparative case-study research and focuses on trust development in new subordinate-leader relationships arising from M&As in multicultural emerging economies. Case study research is particularly suited for in-depth probing into complex concepts such as trust and culture, which quantitative approaches cannot easily reveal [START_REF] Yin | Case study research: Design and methods[END_REF]. I compare a domestic and a cross-border acquisition from Malaysia's financial services industry. Malaysia is multiethnic, multifaith, and multilingual. There are three main ethnic groups (Bumiputra-comprising Malay and other indigenous groups-63%, Chinese 25%, and Indian 7%), and four main religions (Muslim 61%, Buddhist 20%, Christian 9%, Hindu 6%; Malaysian Department of Statistics, 2010). Generally the Malays are Muslims, the Chinese are Buddhists or Christians, and the Indians are Hindus, Muslims or Christians. Most Malaysians are bilingual in Bahasa Malaysia and English, the national and business languages, but many are trilingual (in Chinese or Indian languages) or quadrilingual [START_REF] Fontaine | Cross-cultural research in Malaysia[END_REF]. The financial services industry was selected to control for environmental variation and a transparently observable phenomenon of interest [START_REF] Eisenhardt | Building theories from case study research[END_REF]. The Malaysian financial industry comprises banking intermediaries, insurance firms and capital market intermediaries (IMF, 2014). As Malaysia's seventh largest sector, it accounts for 4.5% of nominal GDP (IHS, 2016) and has undergone substantial consolidation and rationalization subsequent to the 1997-1998 Asian financial crisis (IMF, 2014). In 2015, the financial industry ranked fourth in terms of emerging economies' M&A activity (Thomson Reuters, 2016) and accounted for 19% of global M&A volume (Bloomberg, 2016). Further, I had worked in the Malaysian financial services industry and am familiar with its characteristics. Case selection and description This paper compares a domestic M&A "DoeMez" versus a South African acquisition of a Malaysian firm "Cross". Purposive sampling was applied during case selection and care was taken to match them on contextual appropriateness [START_REF] Poulis | The role of context in case study selection: An international business perspective[END_REF][START_REF] Yin | Case study research: Design and methods[END_REF]. Given South Africa's cultural diversity and history of apartheid, the two cases provide an intriguing comparison of cross-cultural trust between acquired-firm and acquirer personnel. In Malaysia, the Bumiputra have enjoyed special rights since 1971 including in employment, resulting in sensitive although peaceful ethnic relations [START_REF] Bhopal | Ethnicity as a management issue and resource: Examples from Malaysia[END_REF][START_REF] Haque | The role of the state in managing ethnic tensions in Malaysia: A critical discourse[END_REF]IHS, 2016). South Africa introduced a comparable majority-favoring regime in 1998 to eliminate post-apartheid employment discrimination against the 91% black population [START_REF] Thomas | Employment equity in South Africa: Lessons from the global school[END_REF][START_REF] Lee | Affirmative Action in Malaysia and South Africa: Contrasting Structures, Continuing Pursuits[END_REF]. Bumiputra and blacks are increasingly represented in high-level occupations, mainly in the public sector [START_REF] Lee | Affirmative Action in Malaysia and South Africa: Contrasting Structures, Continuing Pursuits[END_REF]. Cross is a Malaysian firm acquired by "SAF", a South African multinational corporation with operations in approximately 40 emerging economies. DoeMez is a domestic acquisition of two sister firms (Mez1 and Mez2) which were then merged with two subsidiaries of the acquirer Doe to create DoeMez1 and DoeMez2. The acquirer and its subsidiaries are majority controlled by GLCs (IMF, 2014) and employ 80% Malay personnel, whereas the acquired firms were founded by an ethnic Chinese entrepreneur, had "Chinaman entrepreneurial" culture, and employed 80% ethnic Chinese personnel. iii Table 2 summarizes the main features of the cases. - ----------------------------------------- - INSERT TABLE 2 ABOUT HERE ------------------------------------------- Both acquisitions were friendly deals, had similar motives (business expansion and scaling up), involved experienced acquirers, and were completed approximately two years prior to data collection. To gain access to the firms, personalized invitation letters were sent to the CEOs of the acquired firms and their acquirers/parent firms, together with an offer to share my findings with them. The participating firms and informants were promised complete confidentiality and anonymity. Data collection The analysis was based mainly on 35 semi-structured interviews with TMT, senior and middle managers, and supplemented by 2 conference calls with dealmakers, participatory observations, and archival data. To capture new acquired-firm/acquirer subordinate-acquirer/acquired-firm leader relationships arising from the M&A, I sought individuals who had reported to different managers pre-and post-M&A. The interviewees were selected by their firms-23 from DoeMez and 12 from Cross-and originated from the acquired firm, acquirer, or were 'neutrals' recruited during the focal M&As, as detailed in Table 3. The interviews lasted an hour on average, transpired face-to-face except for one (by conference call), and were recorded with permission and transcribed. All communication was in English except for local expressions sprinkled in some interviews. - ----------------------------------------- ------------------------------------------The interview questions covered: organizational conditions around the time of the acquisition including the pre-acquisition mood at the individual's firm and the impact of the acquisition on the participant; post-acquisition working relationships with his/her new direct manager and subordinates; and, interactions with the new TMT. An interview protocol was developed beforehand and refined through pilot interviews with 7 Malaysians who were either M&A legal advisors or professionals with M&A experience. - INSERT TABLE 3 ABOUT HERE - Due to practical constraints, the interviews were conducted in a tandem arrangement: DoeMez in mid-November 2016, then Cross in late November/mid-December 2016. At DoeMez especially, 23 interviews in 5 consecutive days left little time for reflection. However, since the interviews were held at the firms' premises, I was able to engage in informal conversations and observe the work environment and personnel interactions. Further, I participated in one of Cross's day-long workshops in December 2016 when the acquirer's values and culture were introduced. The workshop was facilitated by an external consultant and attended by approximately 100 Cross personnel, its TMT, and 2 senior executives from the acquirer's HQ. Data analysis My analysis applied abduction which is partly deductive (theory-driven) and partly inductive (data-driven) [START_REF] Timmermans | Theory construction in qualitative research: From grounded theory to abductive analysis[END_REF][START_REF] Welch | Theorising from case studies: Towards a pluralist future for international business research[END_REF]. Abduction facilitates the discovery of new variables and relationships by moving back-and-forth between framework, data sources and analysis [START_REF] Timmermans | Theory construction in qualitative research: From grounded theory to abductive analysis[END_REF]. Abduction has been used in studies on boundary spanning [START_REF] Søderberg | Boundary Spanners in Global Partnerships: A Case Study of an Indian Vendor's Collaboration with Western Clients[END_REF][START_REF] Yagi | Boundary work: An interpretive ethnographic perspective on negotiating and leveraging cross-cultural identity[END_REF], M&As [START_REF] Monin | Giving sense to and making sense of justice in postmerger integration[END_REF][START_REF] Reynolds | The international experience in domestic mergers-Are purely domestic M&A a myth[END_REF], and international business [START_REF] Barron | Exploring the performance of government affairs subsidiaries: A study of organisation design and the social capital of European government affairs managers at Toyota Motor Europe and Hyundai Motor Company in Brussels[END_REF][START_REF] Maitland | Managerial cognition and internationalization[END_REF]. Initially, I built individual case descriptions by triangulating the multiple sources of evidence [START_REF] Graebner | Caveat venditor: Trust asymmetries in acquisitions of entrepreneurial firms[END_REF]. Next, the interview transcripts were analyzed iteratively in groups: acquirer, acquired-firm and neutral managers; TMT and department heads vs. managers; and, subordinate-leader dyads where possible. For DoeMez's transcripts, withinand between-division analysis was also conducted since its integration involved three 'standards': one division followed Doe's practices, a second division followed Mez's practices, while the third division adopted a mix of both. Multiple examples of TMT and manager boundary spanning were identified. Finally, I categorized the examples according to [START_REF] Druskat | Managing from the boundary: The effective leadership of self-managing work teams[END_REF] typology and identified the boundaries spanned-to reveal the intent or purpose behind each boundary spanning behavior, for clearer linkage to trust development. FINDINGS The post-acquisition integration approaches of DoeMez and Cross is described first, followed by the boundary spanning actions of the acquirer and acquired-firm managers and my interpretation of how this behavior facilitated subordinate-leader trust development. Integration approaches Cross' integration by SAF corresponds with the light-touch integration approach [START_REF] Liu | Light-Touch Integration of Chinese Cross-Border M&A: The Influences of Culture and Absorptive Capacity[END_REF] of Asian acquirers [START_REF] Kale | Don't integrate your acquisitions, partner with them[END_REF][START_REF] Liu | Light-Touch Integration of Chinese Cross-Border M&A: The Influences of Culture and Absorptive Capacity[END_REF][START_REF] Marchand | Do All Emerging-Market Firms Partner with Their Acquisitions in Advanced Economies? A Comparative Study of 25 Emerging Multinationals' Acquisitions in France[END_REF]. In DoeMez's case, the integration approach resembles symbiosis [START_REF] Haspeslagh | Managing acquisitions: Creating value through corporate renewal[END_REF]. Table 4 compares their integration approaches. - ----------------------------------------- - INSERT TABLE 4 ABOUT HERE ------------------------------------------- Boundary spanning behaviors My analysis identified boundary spanning in the domestic and the cross-border acquisition cases, by TMT and managers of both the acquirers and acquired firms. The boundary spanning behaviors are described below according to [START_REF] Druskat | Managing from the boundary: The effective leadership of self-managing work teams[END_REF] clusters. Feedback on the behaviors is provided where possible. Horizontal, vertical, stakeholder, demographic, and geographic boundaries [START_REF] Palus | Boundary-Spanning Leadership in an Interdependent World[END_REF] were spanned, representing national, intra-national, organizational and functional culture differences. Tables 5 and6 show boundary spanning examples from Cross and DoeMez, the boundaries and cultural differences spanned, and trust attributes. - --------------------------------------------------- --------------------------------------------------- Cross managers' social and political awareness spanned all five boundaries and differences in national, organizational, functional and personal cultures (see Table 5). A senior manager enthused: "[SAF are] totally different from shareholders who are American or British. They're very respectful of local cultures and very keen on sharing their knowledge… They don't dictate that they know better than we do, which is a total contrast to where I come from." DoeMez manager's social and political awareness spanned all boundaries, organizational culture differences and especially intra-national culture differences. A Mez leader said, "I go the extra mile to make sure the [Doe] colleagues feel very welcome." Particular attention was paid to cater for halal food preferences at work, on business trips, annual dinners, and a company trip overseas. Chinese colleagues were reminded eat nonhalal food outside the workplace. Personnel were instructed to speak in English and the national languages only during meetings, for everyone to understand. One senior manager commented, "I heard some departments had [a language-related] problem, [like] immediate separation between [Doe and Mez people who] tend to not mingle [and] seldom talk... There's quite a lot of staff in [X] Department who were from Chinese schools. Their main language… is Cantonese and Mandarin... Sometimes in meetings… it's an issue." Another manager, whose team spanned two locations, introduced a regular weekly conference call and persevered against Asian reticence to seek team members' negative feedback. Other examples are provided in Table 6. In developing subordinate trust in leaders, Cross leaders showed transparency and willingness to span all boundaries, and national, intra-national and organizational culture differences. Group TMT visited Cross' offices four times in the first eighteen months from the acquisition announcement. Personnel were informed of SAF's intentions and future plans, which dispersed a lot of uncertainty. Managers received regular CEO and CFO performance updates to disseminate in their departments/units. In his first townhall with Cross personnel, the CEO presented his short-term goal and personal interest in being assigned to Malaysia. The CEO's trust-building approach is detailed in Table 5. DoeMez's leaders developed subordinate trust across intra-national and organizational culture differences. Like Cross, this was initiated from early post-acquisition integration and a personal approach was used sometimes (see Table 6). Cross and DoeMez managers demonstrated their care for subordinates in numerous examples with noteworthy examples in Tables 5 and6. Other caring behavior at DoeMez included a department head covering for subordinates who had exceeded their transaction limits, and a senior manager's insistence that the whole team leaves work by 6:30 pm. Subordinate-leader bonding at DoeMez and Cross transpired over coffee, drinks and personal chats. Two DoeMez senior managers also organized weekly/daily lunches and outings for within-team bonding. Cluster 2: Scouting Scouting involves seeking information within the organization to clarify the manager's understanding or team and organizational needs, and to solve problems. This second function of managerial boundary spanning includes a set of three behaviors: seeking information from managers, peers and specialists, diagnosing subordinate behavior, and investigating problems systematically. Seeking information helped to span national, intra-national and organizational culture differences at both Cross and DoeMez, as evidenced from the examples pertaining to incoming expatriates and the expatriate CEO's efforts to develop sensitivity to Malaysian cultural nuances (Table 5), and solving a religion-related problem at DoeMez (Table 6). Subordinate behavior diagnosis spanned mainly vertical boundaries and, organizational culture differences at Cross and intra-national cultural differences at DoeMez (Tables 5 and6). Cross and DoeMez managers investigated problems involving horizontal and vertical boundaries. At Cross, this was related to 'silos' where subordinates performed their functions with very little understanding of what happened in other areas of the business. Table 5 provides an example where this where intra-national and functional culture differences were involved. A DoeMez department manager recounted a similar problem when personnel who withhold information, while another manager mediated such a situation involving intranational culture differences (Table 6). Cluster 3: Persuading Persuading involves managers seeking resources from within the organization to support their teams' needs, and influencing subordinates' priorities to support organizational objectives. Successful persuasion enables a manager to align his/her team's objectives with organizational objectives. The data reveals that DoeMez leaders influenced their subordinates more than Cross leaders. In seeking organizational-level resources for their teams, Cross managers spanned mainly stakeholder boundaries with its two shareholders and Board of Directors. The CEO had monthly engagements and weekly video conferences with various people at HQ, while department managers had a dedicated support person at HQ (e.g., IT coordinator for the IT head, HR coordinator for the HR head). Cross managers were formally and informally linked with Group TMT, e.g., unit heads who also reported to the group's department heads, or the COO and his department manager who liaised regularly with the Group COO. The CEO also recognized the potential of Cross' minority shareholder and its deep local market knowledge (see Table 5). DoeMez managers spanned vertical and horizontal boundaries in seeking their leaders to intervene on critical occasions. Table 6 presents two examples including one with organizational and intra-national culture differences in the attitudes of certain subordinates. In terms of influencing subordinates, Cross managers did so less than DoeMez managers, perhaps reflecting its light-touch integration approach (cf. DoeMez's symbiotic approach). One Cross senior manager said, "[The TMT] don't bulldoze things. They will [talk to everybody] and see people's acceptance. When a few fellows don't accept, they will try to convince the persons." The TMT encouraged subordinates to span organizational and national culture differences by debating and differing in opinion from their leaders. For example: "I am used to seriously being challenged at [HQ]… whereas here when I make statements, I must say 'You are allowed to disagree. Please disagree if you feel so, talk to me, tell me this is a wrong assumption or this is a wrong statement or whatever.'" The TMT concurred that their people were respectful and uncomfortable with discord, reflecting their Asian culture. Nevertheless, two expatriates noted that subordinates, from watching the TMT challenge one another, began doing so themselves. DoeMez managers influenced their subordinates at organizational-and team-levels through the organization's four core values which were enacted from the outset of the M&A. First, the core values shaped changes in culture and thought processes across the organization. As one dealmaker/leader explained, "We initiated [culture change] by the new brand. We launched an idea behind that brand, these disciplines, this other culture we want to emulate... We started building new thought processes." Second, the core values guided managers to focus/refocus their teams. For example, one leader reminded his team, "No, no, our core value is to collaborate. Guys, we need to collaborate…" and "We need to be entrepreneurial. That means we need to think how to solve the problem." The core values spanned organizational and intra-national culture differences as seen in the same manager's comment: "Without the core values, everybody will define their own… I think without that, this merger would be a nightmare. I honestly tell you that." Department heads influenced their subordinates to accept incoming members from the acquirer/acquired firm and fuse horizontal boundaries. One entire team was instructed to treat the newcomers well regardless of the circumstances and to refer back to the manager if any issue arose. Another manager instructed his unit heads similarly. Managers also emphasized the amount of time the two sides would spend together and the need for a pleasant work atmosphere. One subordinate remarked that human integration was "very well-driven… We're all no longer ex-Mez or ex-Doe people, we're all DoeMez." Cluster 4: Empowering Empowering involves delegating authority to subordinates and coaching to improve subordinates' capabilities. This final function of managerial boundary spanning includes a set of three behaviors that span horizontal, vertical and demographic boundaries: delegating authority, flexible on team decisions, and coaching subordinates. My data reveals that the most frequent empowering behavior was coaching subordinates, with some coaching actions implemented through the HR departments. Delegating was observed in DoeMez and Cross. One DoeMez department head spanned organizational and intra-national culture differences through delegation: "When I first took [the Doe people] in, they weren't comfortable doing a lot of things... They weren't given a lot of freedom in the old days… I found that a lot of them are quite capable... They did make a lot of mistakes and I do hold them accountable for it, but [no] penalties...'" For one Cross expatriate, delegating authority overlaps with being flexible on team decisions behaviors: "If [you] have to change approaches, then we change if it makes sense... [The initial approach is] not necessarily right. It's only right if it makes sense to you also." The leader explained, "They'll say something works like this. Then I'll ask 'Why?'… 'Do you think it's right?'… I'm starting to ask questions [but] not telling them what to do." I did not observe any flexible on team decisions behavior in DoeMez. Subordinate coaching, the most prominent behavior in this cluster, was implemented both through the HR departments and independently. First, HR mobilized and trained managers to be relays: Cross managers were coached to coach their subordinates and thus support them more effectively, while DoeMez senior managers were trained to counsel their teams on how to deal with a different organizational culture, new processes and new workflows. If each HR department had provided support to personnel on an organizational level, more resources would have been required (e.g., more HR personnel, outsourcing). One Cross expatriate introduced a 360-degree feedback system through the HR department, where an individual's leader, peers, and direct subordinates provide confidential and anonymous feedback on "This is what I'd like to see more of you, in…" Second, as a personal initiative, a Cross expatriate coached department managers who had been working in silos on the impact of this behavior on team outcome, and how to deliver as a team. At DoeMez, several managers gave indirect answers to subordinates' questions, for example, a department head: "I say 'I want to hear from you first. Come back to me, say, in 2-3 hours' time and see what you have.'… I put pressure to them to think through properly… Otherwise they will never learn. Always remember, don't give the man fish all the time. You must teach him how to fish." DISCUSSION The objective of this study was to explore how trust develops in new subordinate-leader relationships in the M&A nucleus, amidst the uncertainty and cultural differences of integration. Three bodies of literature are connected to understand the trust development process: M&A, boundary spanning, and trust. The paper's main contribution is to posit that boundary spanning mitigates the uncertainty and cultural differences arising from postacquisition integration. Managerial boundary spanning overcame the 'us and them' mentality between acquirer/acquired-firm subordinates and acquired-firm/acquirer leaders, reduced uncertainty and cultural differences, and thus catalyzed subordinate-leader trust development. Focusing on a domestic and a cross-border acquisition of Malaysian firms, my systematic analysis reveals that both acquirer and acquired-firm managers engaged in boundary spanning behavior. The managers ranged from TMT, integration managers [START_REF] Teerikangas | Integration managers' value-capturing roles and acquisition performance[END_REF] and middle managers in diverse functions. This indicates the widening locus of M&A leadership (Graebner, 2004;[START_REF] Junni | The role of leadership in mergers and acquisitions: A review of recent empirical studies[END_REF] and suggests more widespread managerial influence for smoother and more successful integration. Second, this research extends the ambit of managerial boundary spanning beyond the pre-M&A negotiation phase [START_REF] Lander | Boarding the aircraft: Trust development amongst negotiators of a complex merger[END_REF] to post-acquisition integration, and adds evidence that effective leaders engage in boundary spanning behavior [START_REF] Druskat | Managing from the boundary: The effective leadership of self-managing work teams[END_REF]. This paper also complements the process perspective of M&As [START_REF] Jemison | Corporate acquisitions: A process perspective[END_REF] and elucidates on the temporality of integrative actions. Birkinshaw et al.'s (2000) influential study argues for a two-phased implementation of effective integration. First, human integration is emphasized while the acquirer and acquired units achieve acceptable performance, and then revisiting task integration so that the existing success of human integration facilitates task integration across units. Other researchers suggest closer interrelation between these elements of integration. [START_REF] Froese | Integration management of Western acquisitions in Japan[END_REF] find that human integration and organizational integration occur simultaneously, from the beginning of post-acquisition integration. [START_REF] Teerikangas | Structure first! Temporal dynamics of structural and cultural integration in cross-border acquisitions[END_REF] observe that cultural integration begins once structural integration is underway, as long as both are implemented in complementarity. In this study, DoeMez implemented extensive organizational integration and human integration rapidly, similar to the Renault-Nissan acquisition [START_REF] Froese | Integration management of Western acquisitions in Japan[END_REF]. In contrast, SAF selectively implemented Cross' organizational integration over time and to a lesser extent, with human integration from the third year post-acquisition. Both deals are considered successful, as are SAF's other similarly integrated South-South acquisitions. Third, the study's focus on South-South acquisitions extends M&A scholarship which is heavily focused on acquisitions to and from developed economies [START_REF] Lebedev | Mergers and acquisitions in and out of emerging economies[END_REF], and contributes to the literature on the complexities of multilingual societies in M&As [START_REF] Cuypers | The effects of linguistic distance and lingua franca proficiency on the stake taken by acquirers in cross-border acquisitions[END_REF][START_REF] Dow | The effects of within-country linguistic and religious diversity on foreign acquisitions[END_REF][START_REF] Kroon | Explaining employees' reactions towards a cross-border merger: The role of English language fluency[END_REF]. DoeMez exposes that domestic rather than cross-border M&As can be more challenging to integrate [START_REF] Véry | A cross-national assessment of acculturative stress in recent European mergers[END_REF]1997), especially when the intra-national culture differences of a multicultural emerging economy are considered. The language diversity among Malaysian personnel was more problematic for DoeMez than Cross and there was no lingua franca for all DoeMez personnel. Malaysian personnel's proficiency in English varies significantly, particularly in the younger generation [START_REF] Darmi | English language in the Malaysian education system: Its existence and implications[END_REF][START_REF] Yin | Case study research: Design and methods[END_REF]. Some ethnic Chinese personnel were not sufficiently fluent in English or Bahasa, despite the possibility of using both languages in meetings. Malaysian Chinese can be differentiated by their schooling: those with Chinese-school education relate strongly to traditional Chinese values, while those with non-Chineseschool education identify weakly with such values [START_REF] Fontaine | Cross-cultural research in Malaysia[END_REF][START_REF] Ong | Chinese ethnicity: Its relationship to some selected aspects of consumer behaviour[END_REF]. This paradoxical finding draws attention to the intricacy of cultural challenges in M&As [START_REF] Teerikangas | The culture-performance relationship in M&A: From yes/no to how[END_REF], while extending our knowledge of the differences between integrating domestic and cross-border acquisitions [START_REF] Bris | The value of investor protection: Firm evidence from crossborder mergers[END_REF][START_REF] Gregory | Do cross border and domestic acquisitions differ? Evidence from the acquisition of UK targets[END_REF][START_REF] Reynolds | The international experience in domestic mergers-Are purely domestic M&A a myth[END_REF]. More broadly, my research adds to the literature on multilingual organizations including multinational corporations [START_REF] Bordia | Employees' willingness to adopt a foreign functional language in multilingual organizations: The role of linguistic identity[END_REF][START_REF] Brannen | The multifaceted role of language in international business: Unpacking the forms, functions and features of a critical challenge in MNC theory and performance[END_REF][START_REF] Feely | Language management in multinational companies[END_REF], and the conversation that "context matters". The Malaysian context diversifies the geographic focus of emerging market studies away from China [START_REF] Jormanainen | International activities of emerging market firms[END_REF] and offers tentative insights to advance Western-developed theories in strategy and management [START_REF] Jormanainen | International activities of emerging market firms[END_REF][START_REF] Wright | Strategy research in emerging economies: Challenging the conventional wisdom -Introduction[END_REF][START_REF] Xu | Linking theory and context: 'Strategy research in emerging economies' after Wright et al[END_REF]. The findings support the argument that national cultures are more heterogeneous than homogeneous [START_REF] Shenkar | Cultural distance revisited: Towards a more rigorous conceptualization and measurement of cultural differences[END_REF][START_REF] Taras | Does country equate with culture? Beyond geography in the search for cultural boundaries[END_REF][START_REF] Tung | The cross-cultural research imperative: The need to balance crossnational and intra-national diversity[END_REF][START_REF] Tung | Beyond Hofstede and GLOBE: Improving the quality of cross-cultural research[END_REF]. Fourth, the paper contributes to trust scholarship by diversifying from its reliance (Fulmer & Gelfand, 2012) on WEIRD samples (Western, educated, industrialized, rich, and democratic;[START_REF] Henrich | Most people are not WEIRD[END_REF]. While Mayer et al.'s (1995) model has become one of the most well-known and influential trust models, it remains unclear whether each component of trustworthiness (ability, benevolence, integrity) is equally as important in trust outcomes [START_REF] Burke | Trust in leadership: A multilevel review and integration[END_REF]. My analysis of the linkage between boundary spanning and subordinate-leader trust shows that benevolence and integrity are more prominent than ability, for facilitating Malaysian subordinate-leader trust in M&As. This complements [START_REF] Poon | Effects of benevolence, integrity, and ability on trust-in-supervisor[END_REF] findings on the significance of benevolence for Malaysian personnel to trust their leaders. The precedence of affect-based trust over cognition-based trust is consistent with other emerging economies where personal relationships are important, e.g., China [START_REF] Chua | Guanxi vs networking: Distinctive configurations of affect-and cognition-based trust in the networks of Chinese vs American managers[END_REF][START_REF] Jiang | Effects of cultural ethnicity, firm size, and firm age on senior executives' trust in their overseas business partners: Evidence from China[END_REF], Singapore and Turkey [START_REF] Tan | Understanding interpersonal trust in a Confucian-influenced society: An exploratory study[END_REF][START_REF] Wasti | Cross-cultural measurement of supervisor trustworthiness: An assessment of measurement invariance across three cultures[END_REF]. The importance of building personal relationships was evident in DoeMez and Cross where informal social activities facilitated bonding between subordinates, peers and leaders. Managerial implications This research shows that a wider leadership can influence and smoothen M&A processes, outcomes and cultural differences. Regardless of their hierarchical positions, managers from both the acquirer and acquired firm can contribute to reducing post-acquisition uncertainty among personnel, thereby facilitating subordinate-leader trust during integration. The managers should be empowered to play a more central role in human integration from early post-acquisition integration. For example, through awareness-building workshops on where boundaries and cultural differences may lie, with illustrative boundary spanning behaviors. Moreover, since boundary spanning behaviors are often improvised by managers in doing their jobs [START_REF] Yagi | Boundary work: An interpretive ethnographic perspective on negotiating and leveraging cross-cultural identity[END_REF], integration and HR managers should develop ways to formalize and share such experiences with other managers in the organization so that larger scale benefits can be drawn. This research highlights how language diversity can be an issue not only cross-border acquisitions but in domestic acquisitions in a multicultural society. Boundary spanning behaviors to address this divide includes seeking colleagues to translate or summarize the essential points, and cascading information downwards through department/unit managers so that subordinates have a more accessible point of reference for clarification. In the longer term, personnel should have training and opportunities to overcome their "language anxiety" [START_REF] Darmi | English language in the Malaysian education system: Its existence and implications[END_REF][START_REF] Yin | Case study research: Design and methods[END_REF]. Limitations and future research directions The study's findings should be interpreted caution. Despite exposing the positive effects of managerial boundary spanning in post-acquisition integration, further research is required to ascertain the conditions under which boundary spanning influences successful integration. First, the analysis compares two M&A case studies within a single industry in Malaysia. Malaysia falls under [START_REF] Schwartz | A theory of cultural value orientations: Explication and applications[END_REF] South Asia cluster of cultural value orientations, which characterize hierarchical and collective societies. Future studies should examine M&As in the same cluster (e.g., Indonesia, India), in other multicultural emerging economies (e.g., Turkey, Nigeria, Chile), as well as vertical acquisitions and acquisitions of financially-troubled firms. Moreover, the integration approach adopted may have a bearing on the extent of boundary spanning. It would be useful to study not only other M&As applying symbiosis and light-touch integration like DoeMez and Cross, but also preservation and absorption [START_REF] Haspeslagh | Managing acquisitions: Creating value through corporate renewal[END_REF]. Second, the managerial boundary spanning behaviors may be related to the individual's job function, level of experience, seniority in the organization, and previous acquisition experience. To analyze this requires a larger sample. Third, the study relied mostly on interview data to analyze trust development retrospectively. Despite my efforts to reduce social desirability bias, some interviewees may not have been entirely forthcoming since they were selected by their firms. Ethnographic research or a longitudinal study would be more robust. Looking beyond, it would be fruitful to enquire into how boundary spanning affects trust between peers during post-acquisition integration, and trust between the acquired firm and HQ personnel. The linkage between boundary spanning, interpersonal trust and identity formation is another interesting research area. I look forward to more active scholarly pursuit along this path. -INSERT TABLES 5 AND 6 ABOUT HERE - - Cluster 1: RelatingRelating involves building relationships within the team and organization while showing social and political awareness. This first function of managerial boundary spanning includes a set of three behaviors: social and political awareness, building subordinate trust in leaders, and caring about subordinates. TABLE 1 : Typology of boundary spanning actions 1 (adapted from Druskat & Wheeler, 2003) Function Behavior Examples Relating Social and political Building rapport by: understanding the awareness concerns and interests of specific groups, appreciating power relationships, politicking Build subordinate trust in Showing that the leader is reliable, fair and leaders honest and focused on the team's best interests Care about subordinates Caring actions towards a team or individual Scouting Seek information from Making the effort to go beyond the boundary managers, peers and spanner's knowledge or understanding by specialists at organizational- referring to someone in the organization level Diagnose subordinate Analyzing a subordinate or team's verbal or behavior nonverbal behavior to understand their needs Investigate problems Solving a problem by breaking it down, systematically systematically tracing the cause of a problem Persuading Obtain organizational Persuading external groups so that they assist support for their teams or support the needs of the leader's teams Influence subordinates Encouraging subordinates to make choices that support team or organizational goals Empowering Delegate authority Giving responsibility, control or authority to subordinates Flexible on team decisions When the leader is open-minded about how subordinates fulfil a role, assignment, etc. Coach subordinates Developing subordinates' knowledge and skills TABLE 2 : The research cases 2 Cross DoeMez TABLE 3 : Dealmakers and interviewees 3 Position / Firm-of-origin Cross DoeMez Dealmakers Acquirer 2 2 3 2 a Acquired firm - 1 Interviewees 12 23 TMT Acquirer 4 b 2 c Acquired firm - 3 Neutral - 1 Department heads Acquirer - 3 Acquired firm 4 2 Neutral 3 1 Other managers Acquirer - 5 Acquired firm 1 4 Neutral a Includes a TMT. - 2 b Non-Malaysians. c Includes a non-Malaysian. TABLE 4 : The integration approaches 4 Cross DoeMez Integration Financial controls; HR, operations Integration of business activities, and and processes were improved or operations, policies, procedures and controls cleaned up processes. 3 offices were converged into 2 Corporate Former shareholder's name was Launched "DoeMez" (new name, logo, rebranding removed; Cross endorsed as "A identity, image); new corporate HQ member of the SAF Group" TMT 4 expatriates from SAF Acquirer, acquired firm and neutral TMT Cultural Initiated during third year post- None integration acquisition Integration approach; Speed Light-touch integration; Slow Symbiosis; Fast TABLE 5 : 5 Cross boundary spanning examples(HOD = head of department; SM = senior manager) Boundary Cultural difference Trust attribute Selective quotes Relating behaviors Social and political Vertical, National, Benevolence, [P]art of my job is… to cushion off all [issues and]… complement awareness horizontal, organizational integrity, my staff… so everyone works in harmony [with the boss] or SAF… stakeholder ability I'll do a lot of [consulting], very frequent advice to my staff that this is how [the key people] work. (HOD) Vertical, National, Ability, [The CFO] is trying to [develop more informal interaction with HQ] stakeholder, organizational integrity, through conf calls, updates… We always have this difficulty to align geographic benevolence [their] understanding of certain topics because their environment, perspective, legal and operating environments are different. (SM) Horizontal, Personal, Benevolence, Introverts… tend to be more comfortable on email. Very often [after vertical functional integrity, a] meeting,… I'll email [a summary]… Extroverts are normally on ability their feet and more direct [but] you have IT people, [actuaries and accountants] who are normally introverts. (Expatriate) Build subordinate trust Vertical, Intra-national, Benevolence, You will build trust easier with your senior people than the rest of horizontal, national integrity, the company [because of more frequent face-to-face interaction]. demographic ability The rest of the company will be my communications… On [festive occasions] I send a nice message… You must do those things repeatedly, then people will see 'He genuinely means he embraces diversity, he doesn't favor any specific groups.' When we have a Divali celebration, I dress like an Indian and those types of things. People like that. They start thinking 'Yeah, [he] can walk in my shoes, I can trust him.' (CEO) Care about subordinates Vertical Personal Benevolence, People must come and tell me [when problems arise]… then try to integrity fix it themselves… I concentrate on the fixing, not on the blame. (Expatriate) I would draft a speech and say to my secretary-she's a Muslim-and my HR person [who] is Chinese, 'Here is my speech. Please look at it. Am I using the wrong words? Have I said the wrong thing?'... TABLE 6 : 6 DoeMez boundary spanning examples (HOD = head of department; SM = senior manager) People no longer should have that notion of 'I used to manage client A, it's my client'. Today, client A is the firm's clients and is being serviced by a team of people from Doe and Mez... By doing that we avoided a fight, people trying to fight for accounts, for clients, because naturally people want the bigger accounts. (HOD) lot of time trying to tell my acquiree employees that… certain aspects of [their] perception [of our corporate culture]… may not be the reality… Obviously we are a government-linked company [with] Malay mentality and… mindset… What can I dispel? … I can certainly dispel that the management in Doe does not have interference of any sort from the government shareholding that we have, that we are not a racial-biased entity, that we are predominantly merit-based. (Leader) to explain… and remind [my team] what we're going through… [Doe people] have to learn [Mez's] system and now 'I have to lead you.' (DoeMez HOD) The big boss]… was really pushing for [the seller] to accept Doe's bid… because he knows that [if another bidder] were to acquire us, then most of the staff will be jobless. (DoeMez HOD) Muslim [senior management] colleagues together and we brainstormed [on the religion-related problem]… I said 'These are the issues that we have. How should we manage it?' … [The Head of Doe] said, 'Let me talk to [the Muslim staff].' … He helped ease the whole process. (Leader) Some of my people], they are very hierarchical 'I'm the boss, you guys take instructions'… [Other people] are waiting for instructions [or] order-takers. [Another] set of people [are] cookie-cutters,… only can do one thing... [and no more]. (HOD) You have pockets of people who… hold [the] organization to ransom. They know but… don't volunteer the information. For example, 'do we need to do this?' They will say yes. But they never tell you the consequence… I said [to my team], 'You need to probe them further to make sure we don't [miss any details].' (HOD) Two subordinates doing the same job and withholding information from each other: The Chinese guy [from Mez] asked the Malay guy [from Doe] 'Have you done this...?' This Malay guy [felt] 'Why do I have to report to you?'... Then same thing happened [the other way around]. (Manager) [the Group COO] and give her heads-up how to fix the problem… She gave me a lot of air-cover. (HOD) [the] different [corporate] culture [and personal attitude] that I have a problem [with in some of my team]… I discussed it with [my boss]… I think he did talk to the team. Things are better now. (SM)We initiated [culture change] by the new brand. We launched an idea behind that brand, these disciplines, this other culture we want Boundary Cultural difference Trust attribute Illustrative quotes demographic to emulate... We started building new thought processes. (Leader) You have me telling the team that 'you better treat [the Mez] people nicely; I don't care what they do to you, you just treat them nicely. Any [issues], you just come to me.' (HOD) When I first took [them] in, they weren't comfortable doing a lot of things... They weren't given a lot of freedom in the old days… I found that a lot of them are quite capable... They did make a lot of mistakes and I do hold them accountable for it, but [no] penalties...' (HOD) I want to hear from you first. Come back to me, say, in 2-3 hours' time and see what you have.'… I put pressure to them to think through properly… Otherwise they will never learn. Always remember, don't give the man fish all the time. You must teach him how to fish. (HOD) HR Department… arranged training on mergers and acquisitions, post-mergers, and things like that for the senior management team. Then we will go back to our own department and advise or counsel [our teams] on that different culture… the new flow, new process, new way of doing things. (SM) Boundary Cultural difference Trust attribute Illustrative quotes Relating behaviors Social and political Horizontal, Organizational, Benevolence, It's not for the acquiring company to reach out to us. Because I'm awareness vertical intra-national integrity the acquirer, I felt it was important for me to reach out to them. (Leader) Vertical, Organizational, Benevolence, stakeholder intra-national integrity Build subordinate trust I take pains Care about subordinates Vertical, horizontal, demographic, stakeholder Organizational, intra-national Benevolence, integrity, ability I spent a Vertical, horizontal, demographic Organizational, intra-national Benevolence, integrity, ability Vertical Personal Benevolence, integrity [Scouting behaviors ENDNOTES i The terms "M&As" and "acquisitions" are used interchangeably in this paper, as in extant literature. ii Malaysian GLCs are usually Malay-owned, controlled and managed, with substantial ethnic Malay managers and personnel (Lee, 2016). GLC shareholding in Malaysia's financial industry is high: four of eight banking groups have direct and indirect government shareholding of 40%-60% (IMF, 2014). iii This paper refers to Malaysians of Chinese ancestry as ethnic Chinese/Malaysian Chinese to distinguish them from Chinese who are nationals of China; idem for Malaysians of Indian descent.

00174446

en

2024/03/05 22:32:07

2007

https://hal.science/hal-00174446/file/nedelec_revised.pdf

C Mansuy Dr J M Nedelec C Dujardin R Mahiou Concentration effect on the scintillation properties of Sol-Gel derived LuBO 3 doped with Eu 3+ and Tb 3+ Keywords: 78.55.Hx, 81.20. Fw, 29.40.Mc, 73.61.Tm scintillators, borate, sol-gel, medical imaging, x-ray conversion Lu 1-x Eu x BO 3 and Lu 1-x Tb x BO 3 powders have been prepared by a sol-gel process with 0 < x < 0.15 for Eu 3+ and 0 < x < 0.05 for Tb 3+ . The purity of powders has been verified by X-Ray diffraction and the results confirm that all the materials have the vaterite type even if the calcination has been performed at 800°C. Furthermore, the solid solution for LuBO 3 vaterite is observed up to x=0.15 and x=0.05 for europium and terbium ions respectively. So doping with Eu 3+ or Tb 3+ ions does not affect the structure. These materials have also been analyzed by Fourier Transform Infra Red Spectroscopy. The morphology of the powders has been studied by Scanning Electron Microscopy and shows a very nice morphology with small spherical particles with narrow size distribution. Optical properties have then been studied to confirm the effective substitution of Eu 3+ or Tb 3+ for Lu 3+ ions and to determine the materials scintillation performances. The optima, in term of scintillation yield, are obtained for Eu 3+ and Tb 3+ concentration of x=0.05 in both cases. The afterglows have also been measured and confirm the potentiality of these materials as scintillators. Introduction Nowadays, research directed towards scintillating materials is in constant development. These materials, that convert high energy radiation into UV-Visible light, are used in various applications: medical imaging, high energy physics, airport security and industrial control. The use of these scintillating materials in medical equipment (X-rays, γrays, positron emission, ... ) requires improvement of their properties, in particular their conversion yield. Soft chemistry routes and in particular the sol-gel process offer an attractive alternative solution for the production of efficient scintillators with a control of the optical properties. Rare earth activated lutetium orthoborates, which present a high density due to lutetium, appear to be good scintillators [1,[START_REF] Moses | Procceding of the International Conference on Inorganic Scintillators and Their Applications[END_REF]. We consequently decided to prepare LuBO 3 :Eu 3+ and LuBO 3 :Tb 3+ powders by an original sol-gel route. Indeed, the major advantage of this soft chemistry process is that it allows the control of the morphology and the texture of the materials and also the preparation of the material as thin films [START_REF] Schmidt | [END_REF][START_REF] Nedelec | [END_REF]5]. Moreover, the sol-gel route allows the elaboration of materials of different composition and doped easily with different ions, in various concentration and the sol-gel derived materials are synthesized at lower temperature than the ones elaborated by classical solid state synthesis. The powders have been characterized using various techniques. The purity and the morphology have been respectively analysed by X-Ray Diffraction (XRD) and Scanning Electron Microscopy (SEM). The scintillation properties of LuBO 3 :Eu 3+ and LuBO 3 :Tb 3+ powders have also been studied. Experimental section Powders preparation LuBO 3 :Eu 3+ and LuBO 3 :Tb 3+ powders have been prepared by a sol-gel process described elsewhere [6]. In a first step, lutetium and Ln (Ln: Eu or Tb) chlorides are dissolved in required amounts in isopropanol during 2 hours. Rare earth chlorides and solvents used were anhydrous and all the experiments were carried out in an argon inert atmosphere to prevent any influence of air moisture. Potassium isopropoxide was prepared by reacting metallic potassium (Aldrich) with anhydrous 2-propanol (Accros). After dissolution of chlorides, the solution of potassium isopropoxide is added and then Lu/Ln chlorides mixture is reacted with the potassium alcoholate which substitutes for chloride leading to the formation of the rare earth alkoxides and the immediate precipitation of KCl according to: LnCl 3 + 3 K + + 3 i OPr -→ Ln( i OPr) 3 + 3 KCl↓ This mixture was then refluxed for 2h at 85°C, in order to complete the formation of the rare earth alkoxides. Secondly, an alcoholic solution of boron tri-isopropoxide is added. A homogeneous solution is obtained after 4h-reflux at 85°C. After cooling down at room temperature, centrifugation is performed in order to separate KCl from the sol. The hydrolysis of the sol with distilled water yields a gel, which is dried at 80°C to obtain a white xerogel. This xerogel is then fired at 800°C for 18h in order to obtain Lu (1-x) Ln x BO 3 (Ln :Eu or Tb) crystalline powders. Samples with 0.005 < x < 0.15 and 0.002 < x < 0.05 have been prepared respectively for LuBO 3 :Eu 3+ and LuBO 3 :Tb 3+ . Characterization All powders have been checked by X-Ray diffraction on a Siemens D501 diffractometer working in the Bragg-Brentano configuration with Cu-K α radiation (λ = 1.5406 Ǻ). Infrared spectra were recorded on a Perkin Elmer 2000 FTIR spectrometer using the KBr pellet technique. Thermogravimetric analysis was performed using a Metler Toledo 851 apparatus. Samples were heated in air with a rate of 1 °C.min -1 . Micrographs were recorded using a Cambridge StereoScan 360 SEM operating at 20 kV. Samples were prepared by depositing a small quantity of powder on adhesive carbon film before coating the surface with gold. The excitation spectra of all the powders, doped with Eu 3+ or Tb 3+ , were recorded at room temperature using a Xenon lamp as continuous excitation source and a Triax 320 monochromator coupled with a CCD detector. The scintillation spectra were recorded with a Jobin-Yvon Triax 320 monochromator coupled with a CCD camera after excitation of the samples with a tungsten X-ray tube working at 35 kV and 15 mA. The signal was collected near the sample with an optical fiber. For relative conversion yield estimation, the samples were placed in a quartz tube with a fixed position throughout the measurements. Commercial polycristalline Gd 2 O 2 S:Tb 3+ powder supplied by Riedel de Haën was used as a standard for the scintillation yields measurements. The setup was kept constant between the measurements (excitation and detection), simply changing the sample tube. Equivalent masses of samples were used for measurements and all samples were milled in similar conditions in order to keep the granulometry as constant as possible. Reproducibility was tested on the gadox sample yielding values within a 5 % deviation range. The afterglow measurements were performed at room temperature on the samples corresponding to concentration optima. The excitation was performed during 10 s with a Xray source working at 40 kV and 35 mA. Gd 2 O 2 S:Tb 3+ powder was used as a reference. presents two crystalline types depending on the thermal treatment. As mentioned in [7] the vaterite form is obtained at 800°C instead of the calcite form when the material is prepared by solid state reaction. However, all the recorded diffractograms are identical and show exclusively the vaterite form of LuBO 3 with no evidence for LuBO 3 calcite, EuBO 3 or TbBO 3 phases. Refinement of powder X-ray diffraction patterns allows deriving the cell parameters for the vaterite phase of LuBO 3 . Plot of the cell volume as a function of Eu 3+ content as shown in Figure 2, clearly shows a linear evolution as expected from Vegard's Law for a solid solution. So rare earth ions (Eu 3+ , Tb 3+ ) substitute for Lu 3+ cation in LuBO 3 vaterite structure and the solid solution is observed up to 15% and 5% for respectively europium and terbium ions. Solid solution limits have not been determined and higher concentration might be possible while keeping the vaterite structure and a monophasic material. Results and discussion Characterisation X-Ray diffraction FTIR Spectroscopy Fourier transform Infrared spectroscopy has been carried out on the different samples doped with Eu 3+ or Tb 3+ ions. All recorded spectra, displayed in Figure 3 All the bands observed in the range 500-1200 cm -1 correspond to B-O group vibration modes and the positions agree well with published results [START_REF] Nedelec | [END_REF]8]. Thermal analysis The evolution of LuBO 3 from the amorphous to crystalline form has been studied by thermal analysis. This study gives informations on the cristallisation of the material. A thermogravimetric analysis has been carried out on LuBO 3 powder elaborated by sol-gel process and the resulting thermogramm is presented in Figure 4. The first derivative is also shown in order to clearly identify the temperatures associated with the different weight losses. A first weight loss is observed around 100°C and can be allotted to the elimination of adsorbed species such as alcohol or water molecules. A second significant weight loss, observed at about 150°C, corresponds to the condensation of the material. There is a condensation of the alkoxy and hydroxy groups with subsequent alcohol or water elimination. Some residual organic compounds can also be directly pyrolyzed. At this temperature, the inorganic skeleton is formed. A last weight loss is observed at 700°C and corresponds to the crystallisation temperature. From this temperature, the crystalline growth occurs and no more weight loss is observed. This last loss is characteristic of the mineral network reorganization and of the final elimination of residual OH groupments. Total loss of weight is approximately 17%. Thermal behavior appear very consistent with former results concerning sol-gel derived oxides. Scanning Electron Microscopy To complete the study, Scanning Electron Microscopy has been performed on vaterite LuBO 3 powders synthesized by sol-gel process and treated at 800°C for 18h. The micrograph recorded at 30 000 x magnification, given in Figure 5, indicate that LuBO 3 powders are homogeneous and constituted of small spherical particles of about 200nm. The size distribution of these particles is uniform, which is a usual consequence of using the solgel process. Excitation and emission spectra Excitation spectra The excitation spectra of LuBO 3 :Eu 3+ and LuBO 3 :Tb 3+ vaterite powders were recorded for different concentrations of Eu 3+ (0.05 < x < 0.15) and Tb 3+ (0.002 < x < 0.05). Figure 6 shows the excitation spectra recorded at room temperature by fixing the emission wavelength at respectively 591nm and 541nm for the optima (Lu 0.95 Eu 0.05 BO 3 and Lu 0.95 Tb 0.0 5BO 3 ). The excitation spectrum recorded for LuBO 3: Eu 3+ powder is constituted of lines corresponding to 4f-4f transitions. The line observed at about 470 nm is attributed to 7 F 0 →5 D 2 transition and the ones situated in the range between 300-430 nm correspond to 7 F 0 → 5 F 2 , 5 H J , 5 D 4 , 5 G J , 5 L 8 , 5 L 6 , 5 D 3 transitions. The excitation band located below 250 nm is assigned to the charge transfer absorption [9]. The excitation lines observed for LuBO 3 :Tb 3+ powder, in the range 300-500 nm, are characteristic of 4f-4f transitions. They correspond to 7 F 6 → 5 H 6 , 5 H 7 , 5 L 8 , 5 L 9 , 5 D 2 , 5 G 5 , 5 L 10 , Emission spectra The emission spectra recorded at room temperature under X-ray excitation for Eu 3+ and Tb 3+ doped LuBO 3 vaterite, with different concentrations of Eu 3+ and Tb 3+ , are presented in Figure 7. Gadox (Gd 2 O 2 S:Tb) emission spectrum has also been recorded in order to calculate the scintillation yields of Eu and Tb doped materials. In the case of LuBO 3 :Eu 3+ (Fig. 7a), the spectrum is constituted of lines corresponding to 5 D 0 → 7 F J (J = 0-4) transitions of Eu 3+ ions. The spectral distribution of the Eu 3+ doped materials results in a global orange-red emission. LuBO 3 :Tb 3+ emission spectrum (Fig. 7b) exhibits, in the range between 475-650 nm, several lines characteristic of 5 D 4 → 7 F J (J = 3-6) transitions of Tb 3+ ions. 5 D 4 → 7 F 5 transition is the most intense and confers to the materials an overall green emission. Scintillation yields Scintillation yields have been calculated for all the powders by comparing the integrating areas of the emission spectra of the sample and Gadox. Scintillation yields under γ-rays excitation is 78000 photons/Mev [11] for Gadox. The yields of our materials were calculated from reference values, which are obtained under γ-ray excitation. Our measurements have been performed under X-ray excitation, so the results given for the scintillation yields under γ-ray excitation might be under-estimated. Yields have been calculated for concentrations of Eu 3+ and Tb 3+ ions with respectively 0.005 < x < 0.15 and 0.002 < x < 0.05. The scintillation yields for all the samples and their evolution as a fonction of the doping ion concentration is presented in Figure 8. For Eu 3+ doped LuBO 3 , the optimum is obtained for a Eu concentration of 5% with a scintillation yield of about 8923 photons/MeV. This scintillation yield is 11% of the Gadox one, which is a good value. In the case of Tb 3+ doped LuBO 3 powders, LuBO 3 :Tb 3+ (5%) present the higher scintillation yield since this one is equal to about 4398 photons/MeV. Afterglow Precise knowledge of the afterglow is required for practical applications. Afterglow measurements were recorded at room temperature, using an X-Ray source, which operated at 40kV with an intensity of 25mA. The material was excited for 10s and the signal was collected using a photomultiplier. The afterglow behaviours for Eu 3+ and Tb 3+ doped LuBO 3 powders are presented in Figure 9. The afterglow of Gadox (Gd 2 O 2 S:Tb 3+ ) was also measured as a reference. The afterglow values, mesured 1 s after X-ray turn-off, are 1%, 0.2% and 0.007% respectively for Lu 0.95 Eu 0.05 BO 3 , Lu 0.95 Tb 0.05 BO 3 and Gadox. Conclusion The sol-gel process has been proven to be a good technique for the preparation of scintillating materials doped with different rare earth ions both as powders and thin films. The advantages of this method compared to traditional syntheses are, the lower temperature of treatment, the good crystallinity and purity of the samples, the morphology control and a homogeneous distribution of the particles size. The scintillation properties of Eu 3+ and Tb 3+ doped LuBO 3 powders were studied for different concentrations of doping ion and good scintillation yields were obtained. Sol-gel derived LuBO 3 :Eu 3+ and LuBO 3 :Tb 3+ appear to be promising scintillators. Figures captions Figures captions Figure 1 1 Figure 1 presents the X-Ray diffraction patterns recorded for LuBO 3 powders heated at 800°C for 18h and doped with Eu 3+ (Fig. 1 a) or Tb 3+ (Fig. 1 b) ions. Orthoborate LuBO 3 , are similar. No significant change is observed upon doping with Eu 3+ (Figure 3(a)) or Tb 3+ (Figure 3(b)) ions. Figure 1 :Figure 2 : 12 Figure 1: X-Ray diffraction patterns recorded for (a) Lu 0.85 Eu 0.15 BO 3 and (b) Lu 0.95 Tb 0.05 BO 3 Figure 3 :Figure 4 :Figure 5 :Figure 6 : 3456 Figure 3: IRTF spectra of (a) LuBO 3 :Eu 3+ and (b) LuBO 3 :Tb 3+ powders of vaterite form Figure 7 : 7 Figure 7: Emission spectra recorded, at room temperature, under X-ray excitation on (a) LuBO 3 :Eu 3+ and (b) LuBO 3 :Tb 3+ of vaterite form, with different concentrations of Eu 3+ and Tb 3+ ions. Figure 8 :Figure 9 : 89 Figure 8: Relative scintillation yields of (a) LuBO 3 :Eu 3+ and (b) LuBO 3 :Tb 3+ powders Figure 1 : 1 Figure 1: X-Ray diffraction patterns recorded for (a) Lu 0.85 Eu 0.15 BO 3 and (b) Lu 0.95 Tb 0.05 BO 3 with the corresponding ASTM reference patterns (dotted lines) 1 )Figure 3 :Figure 4 :Figure 5 :Figure 6 :Figure 7 : 134567 Figure 3: IRTF spectra of (a) LuBO 3 :Eu 3+ and (b) LuBO 3 :Tb 3+ powders of vaterite form heated at Figure 8 : 8 Figure 8: Relative scintillation yields of (a) LuBO 3 :Eu 3+ and (b) LuBO 3 :Tb 3+ powders. Figure 9 : 9 Figure 9: Afterglow measurement on Lu 0.95 Eu 0.05 BO 3 , Lu 0.95 Tb 0.05 BO 3 and Gadox under X-ray G , 5 D 3 and 5 D 4 transitions[10]. Acknowledgements The authors would like to thanks the French FRT for financial support under project LuminiX (RNTS-01B262).

00174449

en

2024/03/05 22:32:07

2007

https://hal.science/hal-00174449/file/NIMB_2007_1.pdf

J Lao J M Nedelec Ph Moretto Edouard Jallot email: [email protected] Biological activity of a SiO 2 -CaO-P 2 O 5 sol-gel glass highlighted by PIXE-RBS methods Keywords: 68.08.-p, 81.05.Kf, 81.20.Fw, 82.80.Ej, 82.80.Yc, 87.64.Gb, 87.68.+z PIXE-RBS methods, biomaterials, bioactive glass, sol-gel It is proposed in this study to observe the influence of P 2 O 5 on the formation of the apatite-like layer in a bioactive glass via a complete PIXE characterization. A glass in the SiO 2 -CaO-P 2 O 5 ternary system was elaborated by sol-gel processing. Glass samples were soaked in biological fluids for periods up to 10 days. The surface changes were characterized using Particle Induced X-ray Emission (PIXE) associated to Rutherford Backscattering -2 -Spectroscopy (RBS), which are efficient methods for multielemental analysis. Elemental maps of major and trace elements were obtained at a micrometer scale and revealed the bone bonding ability of the material. The formation of a calcium phosphate-rich layer containing magnesium occurs after a few days of interaction. We demonstrate that the presence of phosphorus in the material has an impact on the development and the formation rate of the bone-like apatite layer. Indeed, the Ca/P atomic ratio at the glass/biological fluids interface is closer to the nominal value of pure apatite compared to P 2 O 5 -free glasses. It would permit, in vivo, an improved chemical bond between the biomaterials and bone. Introduction With the development of biologically active materials, new field of applications arise in surgical therapeutics. Clinical operations on bone defects and fractures may call for a filling material that also presents the ability to contribute to the healing process. For this purpose, bioactive glasses are of huge interest. In contact with living tissues, bioactive glasses establish an enduring interface consisting of a calcium phosphate-rich layer that shows a bone-like apatite structure. The bioactivity mechanisms and growth of the layer at the interface deeply depend on the composition of the glass. The biological activity of bioactive glasses is linked to their capability, in aqueous solution, to leach ions from their surface; a porous silica-gel layer is then formed, which will play the part of support for bone-like apatite crystals growth [1]. The development mechanisms of that calcium phosphate-layer lie in the diffusion of calcium and phosphorus ions from the glass and from the aqueous medium to the material surface [2]. As a result, controlling the surface reactions rates and kinetics is of major importance. Optimizing determining parameters such as the material chemical composition and textural properties is in keeping with the concern that the material will be used as an efficient implant, capable of forming a strong interfacial bond with host tissues and stimulating bone-cell proliferation [3]. For this purpose, a SiO 2 -CaO-P 2 O 5 bioactive glass was elaborated using the sol-gel method, which permits the synthesis of materials with higher purity and homogeneity at low processing temperature [4]. Samples of gel-glass powder and glass compacted discs were immersed in biological fluids for varying periods. As a network former, phosphorus was expected to influence the in vitro bioactivity via a modification of the dissolution kinetics [5]. Analyses of major, minor and trace elements present at the biomaterial/biological fluids interface were performed by particle-induced X-ray emission (PIXE) associated to Rutherford backscattering spectroscopy (RBS). Obtaining PIXE elemental maps at a micrometer scale permits the complete follow-up of the bone-like layer formation along with major and trace element quantification. It allows important evaluation for the in vivo bioactivity of such a bone-forming material. Materials and methods Preparation of the bioactive glass samples Gel-glass powders containing 67.5wt% SiO 2 -25wt% CaO-7.5 wt% P 2 O 5 were prepared using the sol-gel process. Tetraethylorthosilicate (TEOS; Si(OC 2 H 5 ) 4 ), triethylphosphate (PO(OC 2 H 5 ) 3 ) and calcium nitrate Ca(NO 3 ) 2 . 4 H 2 O were mixed in a solution of ethanol in presence of water. The prepared sol was then transferred to an oven at 60°C for gelification and aging. Four hours later, the obtained gel was dried at 125°C for 24 hours, then finally reduced to powder and heated at 700°C for 24 hours. The final surface area of the glass was found to be ??? m2/g by nitrogen sorption analysis. Part of the dry gel powder was then compacted into discs of 13 mm diameter and 2 mm height. In vitro assays The glass discs were immersed at 37°C for 1, 6 h and 1, 2, 5, 10 days in 45 mL of a standard Dulbecco's Modified Eagle Medium (DMEM, Biochrom AG, Germany), which composition is almost equal to human plasma. 10 mg of gel-glass powder samples were soaked at 37°C for 1, 6 h and 1, 2, 3, 4 d in DMEM, with a surface area to DMEM volume ratio fixed at 500 cm -1 . After interaction, the samples were removed from the fluid, air dried and embedded in resin (AGAR, Essex, England). Before characterization, the glass discs were cut into thin sections of 30 micrometers nominal thickness using a Leica RM 2145 microtome. 1000 nm thin sections of the glass powder samples were prepared by mean of a Leica EM UC6 Ultramicrotome, and laid out on 50 mesh copper grids. Then, the sections and grids are placed on a mylar film with a hole of 3 mm in the centre. PIXE-RBS analysis Analyses of the biomaterial/biological fluids interface were carried out using nuclear microprobes at CENBG (Centre d'Études Nucléaires de Bordeaux-Gradignan, France). For PIXE analyses, we chose proton scanning micro-beam of 1.5 MeV energy and 100 pA in intensity. The beam diameter was nearly 2 µm. An 80 mm 2 Si(Li) detector was used for X-ray detection, orientated at 135° with respect to the incident beam axis and equipped with a beryllium window 12 µm thick. PIXE spectra are treated with the software package GUPIX [6] [6]. Relating to RBS, a silicon particle detector placed 135° from the incident beam axis provided us with the number of protons that interacted with the sample. Data were treated with the SIMNRA code [7]. Results and discussion Glass powder samples Elemental maps for each immersion time in DMEM were recorded. Figure 1 represents the elemental distribution of a powder grain after 1 h of interaction with biological fluids. The grain is still homogeneous and dissolution has not begun. Its composition is in the order of the primary SiO 2 -CaO-P 2 O 5 synthesized glass. Nevertheless, some grains (not shown) present a gradient of Ca and P concentration from the centre to the periphery of the material, indicating that ionic exchanges are imminent. After 6 h soaking, we note that calcium and phosphorus started to diffuse from the glass (data not shown). Ion exchange between the grains and the solution has occurred and traces of magnesium are detected at the periphery of the material. However silicon is still uniformly distributed through the grains. The breakdown of the silicate network occurs within 24 h of interaction, and a calcium phosphate-rich layer is formed on particular nucleation sites, located at the periphery of the grains. Calcium and phosphorus ions continue to diffuse from the glass; those are added to the calcium ions and phosphates coming from biological fluids, forming an amorphous calcium phosphate layer on the glass surface. That is illustrated in Figure 2, which shows the multielemental maps of powder grains after 2 days soaking. As visible on the biggest grain, a homogeneous calcium phosphate-rich layer containing magnesium surrounds the material. The core of the grain is composed of the silicate network enduring dissolution. The smallest grains (in the picture corners) already changed into calcium phosphates. Glass compacted discs Glass pastilles react more slowly than powder samples, since their massive powder compacted shape does not grant the same porous-gel open structure as single grains. However, their retarded behavior is similar to that of powder grain samples. Figure 3 shows the multi-elemental maps across the periphery of a glass pastille after 1 h soaking. Measuring the elemental concentrations in the material reveals no changes in the material composition. On 6 h immersed samples, we observe the presence of thin Ca-P enriched areas disseminated on the surface of the discs. Growth of those areas is supplied by constant ionic exchanges between the material and biological fluids. It results in the formation of a large calcium phosphate layer after a few days of interaction (Figure 4). We have measured the Ca/P atomic ratio at the periphery of the glass: it is equal to 1.89 after 10 d soaking. That is an essential indication for the formation of bone-like apatite, which Ca/P nominal value is equal to 1.67. Conclusion Thanks to micro-PIXE associated to RBS, we are able to specify the role of major and trace elements in physico-chemical reactions occurring at the periphery of the glass. In contact with body fluids, bioactive glasses induce a specific biological response at their surface. The initial SiO 2 -CaO-P 2 O 5 glass network is quickly enduring dissolution. Then, following the different stages of the bioactivity process, a bone-like layer is quickly formed at the material periphery. The calcium phosphate-rich layer formation and evolution of the glass network are highlighted. Magnesium is proved to be blended into the material: that is new information of capital importance since magnesium can play an important role during spontaneous formation of in vivo calcium phosphates and bone bonding [8,9]. The specific preparation protocol developed permits the characterization of highly porous powders with grains of a few micrometers. We demonstrate that the presence of phosphorus in the material composition has an impact on the development and the formation rate of the bone-like apatite layer. In a previous work on P 2 O 5 -free glasses, we found that the Ca/P atomic ratio at the material periphery was equal to 2.05 after 10 d of interaction [10]. The Ca/P atomic ratio at the SiO 2 -CaO-P 2 O 5 glass/biological fluids interface is equal to 1.89 after 10 d soaking, which is closer to the 1.67 nominal value of pure apatite. Furthermore, phosphorus-based glass compacted discs present slower dissolution kinetics compared to P 2 O 5 -free glasses (peut on donner une info + quantitative ?). It might permit, in vivo, an improved bonding ability with host tissues. Biological studies are now being performed to confirm this point. Figure captions Figure 1 : 1 Figure 1: Elemental maps of a SiO 2 -CaO-P 2 O 5 powder grain after 1 h of interaction with biological fluids (53 × 53 µm 2 ). Figure 2 : 2 Figure 2: Elemental maps of SiO 2 -CaO-P 2 O 5 powder grains after 2 days of interaction with biological fluids (101 × 101 µm 2 ). Figure 3 : 3 Figure 3: Elemental maps at the periphery of a SiO 2 -CaO-P 2 O 5 glass disc after 1 h of interaction with biological fluids (53 × 53 µm 2 ). Figure 4 : 4 Figure 4: Elemental maps at the periphery of a SiO 2 -CaO-P 2 O 5 glass disc after 10 d of interaction with biological fluids (179 × 179 µm 2 ). Figures Figure 2 Figure 3

01744529

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2024/03/05 22:32:07

2018

https://insu.hal.science/insu-01744529/file/1-s2.0-S1352231018301857-main-1.pdf

Dandan Li Likun Xue email: [email protected] Wen Liang Xinfeng Wang Tianshu Chen Abdelwahid Mellouki email: [email protected] Jianmin Chen Wenxing Wang Characteristics and sources of nitrous acid in an urban atmosphere of northern China: Results from 1-yr continuous observations Keywords: Nitrous acid, Seasonal variation, Heterogeneous conversion, Atmospheric oxidizing capacity, North China Plain Nitrous acid (HONO) is a key reservoir of the hydroxyl radical (OH) and plays a central role in the atmospheric chemistry. To understand the sources and impact of HONO in the polluted atmosphere of northern China, continuous measurements of HONO and related parameters were conducted from September 2015 to August 2016 at an urban site in Ji'nan, the capital city of Shandong province. HONO showed well-defined seasonal and diurnal variation patterns with clear wintertime and nighttime concentration peaks. Elevated HONO concentrations (e.g., over 5 ppbv) were frequently observed with a maximum value of 8.36 ppbv. The HONO/NO X ratios of direct vehicle emissions varied in the range of 0.29%-0.87%, with a mean value of 0.53%. An average NO 2 -to-HONO nighttime conversion frequency (k het ) was derived to be 0.0068±0.0045 h -1 from 107 HONO formation cases. A detailed HONO budget analysis suggests an unexplained daytime missing source of 2.95 ppb h -1 in summer, which is about seven times larger than the homogeneous reaction of NO with OH. The effect of HONO on OH production was also quantified. HONO photolysis was the uppermost source of local OH radical throughout the daytime. This study provides the year-round Introduction Nitrous acid (HONO) is a key precursor of the hydroxyl radical (OH), one of the main tropospheric oxidants in the gas phase. Numerous field and modeling studies have shown that HONO photolysis contributes significantly to the OH sources not only in the early morning but also during the rest of the daytime (Acker et al., 2006b;[START_REF] Kleffmann | Daytime formation of nitrous acid: A major source of OH radicals in a forest[END_REF][START_REF] Xue | Oxidative capacity and radical chemistry in the polluted atmosphere of Hong Kong and Pearl River Delta region: analysis of a severe photochemical smog episode[END_REF]. This is mainly ascribed to the unexpectedly high concentrations of HONO during daytime which would have been kept at lower levels due to its rapid photolysis (R1). Therefore, the knowledge of characteristics and sources of HONO is critical for a better understanding of the tropospheric oxidation chemistry processes. HONO + hυ → OH + NO (320 nm < λ < 400 nm) (R1) So far, field observations of HONO have been carried out at remote, rural and urban areas. The reported ambient concentrations rang from several pptv up to 15 ppbv (e.g., [START_REF] Beine | Surprisingly small HONO emissions from snow surfaces at Browning Pass, Antarctica[END_REF][START_REF] Elshorbany | Oxidation capacity of the city air of Santiago, Chile[END_REF][START_REF] Zhou | Snowpack photochemical production of HONO: A major source of OH in the Arctic boundary layer in springtime[END_REF]). However, the potential sources that could explain the observed elevated daytime HONO are still under controversial discussion. The well accepted HONO sources include direct emissions from vehicle exhaust [START_REF] Kurtenbach | Investigations of emissions and heterogeneous formation of HONO in a road traffic tunnel[END_REF] and homogeneous gas phase reaction of NO with OH (R2) [START_REF] Pagsberg | Kinetics of the gas phase reaction OH + NO(+M) →HONO(+M) and the determination of the UV absorption cross sections of HONO[END_REF]. NO + OH +M → HONO + M (R2) Heterogeneous reactions of NO 2 occurring on wet surfaces (R3) have been also proposed as an important source of HONO according to both laboratory studies and field observations (e.g., [START_REF] Finlayson-Pitts | The heterogeneous hydrolysis of NO 2 in laboratory systems and in outdoor and indoor atmospheres: An integrated mechanism[END_REF]. Nonetheless, the source strength of reaction (R3) has not been accurately quantified and relies on the NO 2 concentrations, surface area density and water content [START_REF] Finlayson-Pitts | The heterogeneous hydrolysis of NO 2 in laboratory systems and in outdoor and indoor atmospheres: An integrated mechanism[END_REF]. These reactions could occur on various types of M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT 3 surfaces including ground, buildings, vegetation, and aerosol surfaces [START_REF] Liu | Evidence of aerosols as a media for rapid daytime HONO production over China[END_REF][START_REF] Vandenboer | Understanding the role of the ground surface in HONO vertical structure: High resolution vertical profiles during NACHTT-11[END_REF]. Up to now, the contribution of the ground surfaces to the overall production of HONO is still under discussion and subject to intensive research activity [START_REF] Wong | Vertical profiles of nitrous acid in the nocturnal urban atmosphere of Houston, TX[END_REF][START_REF] Zhang | Potential Sources of Nitrous Acid (HONO) and Their Impacts on Ozone: A WRF-Chem study in a Polluted Subtropical Region[END_REF]. 2NO 2 + H 2 O → HONO + HNO 3(ads) (R3) In addition, the heterogeneous reduction of NO 2 on soot particles, mineral dust, and surfaces containing organic substrates was also proposed as a source of HONO (R4) [START_REF] Ammann | Heterogeneous production of nitrous acid on soot in polluted air masses[END_REF]2005;[START_REF] Ma | SO 2 initiates the efficient conversion of NO 2 to HONO on MgO surface[END_REF], and these processes can be further photo-enhanced during the daytime [START_REF] George | Photoenhanced uptake of gaseous NO 2 on solid organic compounds: a photochemical source of HONO?[END_REF][START_REF] Monge | Light changes the atmospheric reactivity of soot[END_REF][START_REF] Ndour | Photoenhanced uptake of NO 2 on mineral dust: Laboratory experiments and model simulations[END_REF][START_REF] Stemmler | Photosensitized reduction of nitrogen dioxide on humic acid as a source of nitrous acid[END_REF]. Although the heterogeneous NO 2 conversion on soot surfaces has high potential to produce HONO, it decreases rapidly with aging and is usually regarded to be less important for ambient HONO formation [START_REF] Han | Heterogeneous photochemical aging of soot by NO 2 under simulated sunlight[END_REF]. NO 2 + HC red → HONO +HC ox (R4) Besides, some other HONO sources have also been proposed, including soil emissions [START_REF] Su | Soil nitrite as a source of atmospheric HONO and OH radicals[END_REF], photolysis of adsorbed nitric acid (HNO 3 ) and nitrate (NO 3 -) at UV wavelengths of 300 nm [START_REF] Zhou | Snowpack photochemical production of HONO: A major source of OH in the Arctic boundary layer in springtime[END_REF], and homogeneous reaction of NO 2 with HO 2 •H 2 O [START_REF] Li | Missing gas-phase source of HONO inferred from Zeppelin measurements in the troposphere[END_REF]. Despite the abovementioned significant progress, the 'missing' daytime source(s) of atmospheric HONO is still under exploration. In comparison with sources, the sink pathways of HONO are relatively well established. The chemical losses of HONO include the photolysis (R1) and reactions with OH radicals (R5). Moreover, HONO can be also removed through dry deposition on ground surfaces. Budget analysis of HONO sources and sinks has been proved to be a robust method to examine the unknown sources and quantify their source strength [START_REF] Sörgel | Quantification of the unknown HONO daytime source and its relation to NO 2[END_REF]Su et al., 2008b). HONO + OH → H 2 O + NO 2 (R5) A number of field studies have been conducted to measure ambient HONO in the polluted urban and rural atmospheres of China during the last decade. High concentration M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT 4 levels and strong potential missing source(s) of HONO have been reported in some metropolises (e.g., Beijing, Shanghai and Guangzhou) and surrounding regions (e.g., [START_REF] Bernard | Measurements of nitrous acid (HONO) in urban area of Shanghai, China[END_REF][START_REF] Qin | An observational study of the HONO-NO 2 coupling at an urban site in Guangzhou City, South China[END_REF][START_REF] Tong | Exploring the nitrous acid (HONO) formation mechanism in winter Beijing: direct emissions and heterogeneous production in urban and suburban areas[END_REF]. However, most of these studies were mainly based on short-term intensive observations. While long-period measurements are necessary to support a holistic investigation of characteristics and sources of HONO, they are very scarce [START_REF] Hendrick | Four years of ground-based MAX-DOAS observations of HONO and NO 2 in the Beijing area[END_REF]. In the present study, we have carried out 1-yr continuous observations of HONO and related parameters at an urban site of Ji'nan city, which is located almost in the center of the North China Plain (NCP), the most polluted region of China with dense population and industries. A large amount of observational data and HONO formation cases provided an opportunity of a thorough examination of temporal variations, sources and impacts of HONO in this polluted urban atmosphere of northern China. In the following sections, we will first show the seasonal and diurnal variations of HONO and related species. Then, several sources of HONO will be explored, including vehicle emission, nighttime heterogeneous formation and potential unknown daytime sources. We will finally evaluate the impacts of HONO photolysis on the primary OH sources and hence atmospheric oxidizing capacity. Experimental Site description The measurements were conducted from September 1 st 2015 to August 31 st 2016 at an urban site of Ji'nan, the capital city of Shandong Province, with approximately 7 million A detailed description of the study site can be found elsewhere [START_REF] Wang | HONO and its potential source particulate nitrite at an urban site in North China during the cold season[END_REF]. Measurement techniques HONO was measured by a commercial instrument of LOPAP (long path absorption photometer, QUMA GmbH, Germany). The LOPAP is a wet chemistry based real-time measurement device, with which HONO is sampled in an external sampling unit as a stable diazonium salt and is subsequently detected photo-metrically after conversion into an azodye in a long-path absorption tube of 2.4 m Teflon AF. The LOPAP is conceived as a 2-channel system to correct for the potential interferences. In channel 1, HONO as well as possible interfering gases are determined, while in channel 2 only the interfering gases are quantified. The difference of both channels yields the HONO concentrations. A detailed description of the LOPAP instrument has been described in detail by [START_REF] Heland | A new instrument to measure gaseous nitrous acid (HONO) in the atmosphere[END_REF]. In the present study, the sampling gas flow and the peristaltic pump velocity were set to 1 L min -1 and 20 r min -1 during the whole measurement period. With these settings, the HONO collection efficiency was ensured above 99.99%. Zero air calibration by ultrapure nitrogen (purity of 99.999%) was performed for 30 min automatically at a time interval of 12 h 30 min. An experimental cycle of 8 days was calibrated twice manually by using a known concentration of nitrite (NO 2 -) standard solution. The detection limit of our measurements was 3 ppt at a time resolution of 30 s, with an accuracy of 10% and a precision of 1%. We note that although the LOPAP instrument may collect data in 30 s (or 1 min) intervals, the physical time resolution of the instrument is relatively longer, ca. 3 min. The [START_REF] Xue | Source of surface ozone and reactive nitrogen speciation at Mount Waliguan in western China: new insights from the 2006 summer study[END_REF] . Results and discussion Data overview Figure 2 shows an overview of the measured HONO, NO, NO 2 , PM 2.5 , PM 1.0 , J HONO , and meteorological parameters in the present study. During the 1-yr measurement period, the prevailing winds were from the east and southwest sectors, indicating the general influence of industrial emissions on the study site (see Fig. 1). The air temperature ranged from -15 ℃ to 39 ℃ with a mean value (±standard deviation) of 16±11 , ℃ and the relative humidity showed a clear seasonal variation pattern with higher levels in winter and summer. Markedly poor air quality was observed as expected. Throughout the 1-yr period, 137 haze episodes occurred with daily mass concentration of PM 2.5 exceeding the National Ambient Air Quality Standard (Class II: 75 µg m -3 ), including 9 severe polluted haze episodes with daily average PM 2.5 concentrations above 250 µg m -3 . In addition, elevated levels of NO X , i.e., up to 350 ppbv of NO and 108 ppbv of NO 2 , were also frequently recorded, possibly as a result of intensive vehicle emissions nearby the study site. Overall, these observations highlight the nature of our measurement station as a typical polluted urban environment in North China. M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT Table 1 documents the measured levels of HONO, NOx, HONO/NOx ratios and the comparison with the results obtained previously elsewhere. The measured HONO mixing ratios in Ji'nan ranged from 17 pptv to 8.36 ppbv with a mean (±SD) value of 1.15±1.07 ppbv. Elevated HONO concentrations were frequently observed during the measurement period, with the daily maximum values exceeding 2 ppbv and 4 ppbv on 156 and 50 days, respectively (see Fig. 2). The maximum hourly value of 7.39 ppbv was recorded in the early morning of 6 December 2015. Such high levels of ambient HONO indicate the intense sources of HONO and potentially strong atmospheric oxidizing capacity in urban Ji'nan. The nighttime average (18:00-06:00, LT) HONO concentration was 1.28±1.16 ppbv, compared to the daytime average value (6:00-18:00, LT) of 0.99±0.95 ppbv. In particular, the mean HONO mixing ratio around noontime (11:00-13:00, LT) was even as high as 0.76±0.61 ppbv, which is nearly the highest levels ever recorded in the urban atmospheres, and about 27% of the noontime HONO data were above 1.00 ppbv during the measurement period. This implies the existence of strong daytime sources of HONO in the atmosphere of Ji'nan, which will be further discussed in Section 3.4. The seasonal variations of ambient HONO and related parameters are depicted in Fig. 3. The highest concentrations of HONO occurred in winter (i.e., December-January), followed by spring (i.e., April-May), summer (especially August) and autumn, with seasonal mean (±SD) values of 1.71±1.62, 1.16±0.90, 1.12±0.93 and 0.78±0.60 ppbv, respectively. Overall, the seasonal variation of HONO coincided with that of NO 2 , an important precursor of HONO. Such measured seasonal pattern of HONO is different from those measured in Hong Kong [START_REF] Xu | Nitrous acid (HONO) in a polluted subtropical atmosphere: Seasonal variability, direct vehicle emissions and heterogeneous production at ground surface[END_REF] and Beijing [START_REF] Hendrick | Four years of ground-based MAX-DOAS observations of HONO and NO 2 in the Beijing area[END_REF], where the highest levels were found in the autumn season. The wintertime peak of ambient HONO in Ji'nan should be the result of the lower boundary layer height, weaker photolysis, and enhanced heterogeneous production of HONO given the more abundant NO 2 . The relatively higher springtime HONO mixing ratios might be related to some degree to the more intense heterogeneous reactions of NO 2 on the surface of mineral particles [START_REF] Nie | Asian dust storm observed at a rural mountain site in southern China: chemical evolution and heterogeneous photochemistry[END_REF], as indicated by the coincident higher concentrations of NO 2 and PM 2.5 (note that PM 10 was not measured in the present study). Indeed, the air quality of Ji'nan in the spring of 2016 was characterized by high levels The diurnal profiles of HONO and related supporting parameters are shown in Figure 4. Overall, the diurnal variations of HONO in different seasons were similar, which dropped rapidly after sunrise and reached a minimum at around 15:00 LT, and then increased and peaked during the morning rush hours (an exception is the winter case that showed a concentration peak at midnight). The diurnal variation trend of HONO was similar to that of NO, owing to a variety of chemical and physical processes, and the similar nighttime profiles suggest that vehicle emissions may pose a significant effect on the measured HONO levels. Such nighttime pattern was also found at Tung Chung, Hong Kong [START_REF] Xu | Nitrous acid (HONO) in a polluted subtropical atmosphere: Seasonal variability, direct vehicle emissions and heterogeneous production at ground surface[END_REF], a roadside site in Houston, U.S. [START_REF] Rappenglück | Radical precursors and related species from traffic as observed and modeled at an urban highway junction[END_REF] and in a tunnel in Wuppertal, Germany [START_REF] Kurtenbach | Investigations of emissions and heterogeneous formation of HONO in a road traffic tunnel[END_REF]. The average diurnal profiles of HONO/NO 2 ratio are also shown in Fig. 4f. The HONO/NO 2 ratio generally decreased after sunrise due to the increase of HONO photolysis, and then increased during the nighttime. An interesting finding was the second peak of HONO/NO 2 at around noontime in spring, summer and winter seasons. If the HONO sources during nighttime were the same as those at daytime, the minimum HONO/NO 2 ratios should be found at noon due to the strong photolysis of HONO. Thus, the higher ratios at noontime indicated the existence of additional daytime sources of HONO. Moreover, the HONO/NO 2 ratios increased with solar radiation (e.g., J HONO ), implying that the additional sources may be related to the solar radiation intensity. We will further discuss the potential daytime sources of HONO in Section 3.4. Contribution of vehicle emissions As our study site is close to several major roads of large traffic fleet, it is necessary to evaluate the contribution of vehicle emissions to the measured HONO concentrations. The HONO/NO X ratio was usually chosen to derive the emission factor of HONO in the freshly emitted plumes [START_REF] Kurtenbach | Investigations of emissions and heterogeneous formation of HONO in a road traffic tunnel[END_REF]. In order to ensure the fresh air masses, the Table 2 summarizes the estimated emission factors of HONO/NOx for the 12 vehicular emission plumes. The average ∆NO/∆NO X ratio of the selected plumes was 94%, indicating that the air masses were indeed freshly emitted. The correlation coefficients (r 2 ) of HONO with NOx varied case by case and were in the range of 0.58-0.96, which may be due to the inevitable mixing of vehicle plumes with other air masses and/or heterogeneous conversion of NO 2 on soot particles and ground surface. The derived ∆HONO/∆NO X ratios varied in the range of 0.19%-0.87%, with an average value (±SD) of 0.53%±0.20%. This is comparable to the emission factors obtained in Santiago, Chile (0.8%; [START_REF] Elshorbany | Oxidation capacity of the city air of Santiago, Chile[END_REF] and Wuppertal, Germany (0.3-0.8%; [START_REF] Kurtenbach | Investigations of emissions and heterogeneous formation of HONO in a road traffic tunnel[END_REF], but is substantially lower than those derived in Guangzhou (with a minimum value of 1.4%; [START_REF] Qin | An observational study of the HONO-NO 2 coupling at an urban site in Guangzhou City, South China[END_REF] and Houston, U.S. (1.7%; Rappengluck et al., 2013). The emission factors should be dependent on the types of vehicle engines, fuels and catalytic converters [START_REF] Kurtenbach | Investigations of emissions and heterogeneous formation of HONO in a road traffic tunnel[END_REF]. The variance in the HONO/NOx ratios derived from different metropolitan areas highlights the necessity of examining the vehicular emission factors of HONO in the target city in the future studies. In the present study, the average HONO/NOx value of 0.53% was adopted as the emission factor in urban Ji'nan, and was used to estimate the contributions of traffic 1) with a rough assumption that the observed NOx at our site was mainly emitted from vehicles. HONO = NO × 0.0053 (E1) Where, HONO emis is the HONO concentration arising from the direct vehicle emissions. The calculated HONO emis levels contributed on average 12%, 15%, 18% and 21% of the whole measured nighttime HONO concentrations in urban Ji'nan in spring, summer, autumn and winter, respectively. (3) the meteorological conditions, especially surface winds, should be stable. Figure 5 presents an example of the heterogeneous HONO formation case occurring on 6-7 September, 2015. In this case, the HONO mixing ratios increased rapidly after sunset from 0.08 ppbv to 0.78 ppbv. Since the HONO concentrations and HONO/NO 2 almost increased linearly throughout the night, the slope fitted by the least linear regression for HONO/NO 2 ratios against time can be taken as the conversion frequency of NO 2 -to-HONO (k het ; also referred to as C HONO in other studies). During the 1-yr period, a total of 107 cases were finally selected. Such a large set of cases facilitated a more robust statistical analysis of the heterogeneous formation of HONO. Heterogeneous conversion of NO As our study site is close to the traffic roads, it is necessary to subtract the contribution from direct vehicle emissions. The emission ratio of HONO/NO X derived in Section 3.2 was M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT then used to adjust the HONO concentrations by Eq. ( 2). The NO 2 -to-HONO conversion frequency can be computed by Eq. ( 3), by assuming that the increase of HONO/NO 2 ratio was caused by the heterogeneous conversion (Su et al., 2008a;[START_REF] Xu | Nitrous acid (HONO) in a polluted subtropical atmosphere: Seasonal variability, direct vehicle emissions and heterogeneous production at ground surface[END_REF]. HONO = HONO -NO × 0.0053 (E2) k = [ ! "#$$ ](& ' ) [! ' ](& ' ) ( [ ! "#$$ ](& ' ) [! ' ](& ) ) ( ' ( ) ) (E3) The k het values derived from the 107 cases showed a large variability, from 0.0013 h -1 to 0.0194 h -1 , with a mean value of 0.0068±0.0045 h -1 . These results are well within the range of k het obtained previously from other urban areas. For example, the k het in Ji'nan is comparable to that derived at an urban site of Shanghai (0.007 h -1 ; [START_REF] Wang | Long-term observation of atmospheric nitrous acid (HONO) and its implication to local NO 2 levels in Shanghai, China[END_REF], and less than those in Guangzhou (0.016 h -1 ; [START_REF] Qin | An observational study of the HONO-NO 2 coupling at an urban site in Guangzhou City, South China[END_REF], Milan (0.012 h -1 ; [START_REF] Alicke | Impact of nitrous acid photolysis on the total hydroxyl radical budget during the Limitation of Oxidant Production/Pianura Padana Produzione di Ozono study in Milan[END_REF] and Kathmandu (0.014 h -1 ; [START_REF] Yu | Observations of high rates of NO 2 -HONO conversion in the nocturnal atmospheric boundary layer in Kathmandu, Nepal[END_REF]. Figure 6 provides the seasonal variation of the NO 2 -to-HONO conversion rate in Ji'nan. Clearly, the largest average k het was found in winter with a value of 0.0073±0.0044 h -1 . This should be ascribed to the higher S/V surface density within the shallower boundary layer in the wintertime. Moreover, weak correlation (R = 0.07) between k het and aerosol surface density was also found, which suggests that the efficient heterogeneous formation of HONO may be independent on the aerosol surface. The uptake coefficient of NO 2 on various surfaces to yield HONO (* +, ' →.,+, ) is a key parameter with large uncertainty in the air quality models to simulate HONO and OH radicals [START_REF] Zhang | Potential Sources of Nitrous Acid (HONO) and Their Impacts on Ozone: A WRF-Chem study in a Polluted Subtropical Region[END_REF]. The overall * +, ' →.,+, on the bulk surface of ground and particles can be estimated from Eq. ( 4). Where, > +, ' is the mean molecular velocity of NO 2 (370 m s -1 ); S/? @ and S/? A are the surface area to volume ratio (m -1 ) for both aerosol and ground, respectively. Considering the land use of the study site, the ground was treated as an uneven surface, and a factor of 2.2 per unit ground surface measured by [START_REF] Voogt | Complete urban surface temperatures[END_REF] was adopted to calculate the total M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT 12 active surface. Hence, S/? A can be calculated by Eq. ( 5), where H is the mixing layer height and was obtained from the European Centre for Medium-Range Weather Forecasts (ECMRWF, ERA-Interim; http://apps.ecmwf.int/datasets/data/interim-full-daily/levtype=sfc/). The calculated uptake coefficients for the 107 cases varied in a wide range from 6.1×10 -8 to 1.7×10 -5 , whilst the majority (5%-95% percentiles) fell in a narrower range of 1.1×10 -7 to 4.5×10 -6 . The mean γ ΝΟ2 value was 1.4±2.4×10 -6 . Current laboratory studies have reported the range of γ 01 ' →2101 from 10 -6 to 10 -5 on the ground surface [START_REF] Kurtenbach | Investigations of emissions and heterogeneous formation of HONO in a road traffic tunnel[END_REF][START_REF] Vandenboer | Understanding the role of the ground surface in HONO vertical structure: High resolution vertical profiles during NACHTT-11[END_REF] and from 10 -7 to 10 -5 on the aerosol surface [START_REF] Ndour | Photoenhanced uptake of NO 2 on mineral dust: Laboratory experiments and model simulations[END_REF][START_REF] Wong | Vertical profiles of nitrous acid in the nocturnal urban atmosphere of Houston, TX[END_REF]. Obviously, the uptake coefficient in different orders of magnitude would definitely lead to different assessment of the importance of heterogeneous HONO sources [START_REF] Li | Impacts of HONO sources on the photochemistry in Mexico City during the MCMA-2006/MILAGO Campaign[END_REF]. The average uptake coefficient obtained from such a large set of samples could serve as a reference for modeling studies to simulate ambient HONO and atmospheric oxidation processes in the urban atmospheres of North China. Furthermore, the total area of ground surface is much larger than the reactive surface provided by aerosols, suggesting that the heterogeneous reactions of NO 2 on ground surface may play a dominant role. It should be noted that the exact uptake coefficients of NO 2 on ground and aerosol surfaces are variable and should be different, and the present analysis simplified this process by treating the ground and aerosol surfaces the same. Our derived uptake coefficients can be regarded as an equivalent γ NO2 on the bulk surface of ground and particles. Daytime HONO budget analysis In this section, we examine the potential unknown source(s) of daytime HONO by a detailed budget analysis. Equation ( 6) summarizes the main factors affecting the ambient concentrations of HONO. Even though taking a mean noontime [HONO] level of 1 ppbv, a value of 6×10 -5 ppb s -1 was derived [START_REF] Dillon | Chemical evolution of the Sacramento urban plume: Transport and oxidation[END_REF][START_REF] Sörgel | Quantification of the unknown HONO daytime source and its relation to NO 2[END_REF], which is much smaller compared to L phot (1×10 -3 ppb s -1 ). The noontime data (11:00-14:00 LT) with the strongest solar radiation were chosen to calculate the unknown HONO source strength based on Eq. ( 7). Here the d[HONO]/dt was approximated by ∆HONO/∆t, which is the difference of the measured HONO concentrations every 10 minutes [START_REF] Sörgel | Quantification of the unknown HONO daytime source and its relation to NO 2[END_REF]. Where, l .,+, and l +, ' are the photolysis frequencies of HONO and NO 2 (s -1 ), respectively. P EFGF HF = L J + L 12D2101 + L B J + ∆[2101] ∆ -P 12D01 -P (E7) = [HONO] \J 2101 + K 12D2101 [OH] + ^ ! ;$#_`a 2 b + ∆[2101] ∆ -K 12D01 [OH][NO] - ∆[01 c ]×d.e5% ∆ [OH] = a(J 1 ) h ) i (J 01 ' ) j k01 ' D4 01 ' ' DB01 ' D4 ( Direct measurements of l .,+, and l +, ' were made in this study except for the period from December 2015 to May 2016. For the reaction of HONO with OH, a rate constant m ,.D.,+, of 6.0×10 -12 cm 3 molecules s -1 was taken from [START_REF] Atkinson | Evaluated kinetic and photochemical data for atmospheric chemistry: Volume I -gas phase reactions of Ox, HOx, NOx and SOx species[END_REF]. The OH mixing ratios were expressed by the NO 2 concentrations and photolysis frequencies of O 3 and NO 2 , as shown in Eq. ( 8) [START_REF] Alicke | Impact of nitrous acid photolysis on the total hydroxyl radical budget during the Limitation of Oxidant Production/Pianura Padana Produzione di Ozono study in Milan[END_REF]. In the present study, the calculated daily peak OH concentrations were in the range of 0.3-2×10 7 molecules cm -3 , which are comparable to those M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT 14 measured in the polluted atmospheres of northern China [START_REF] Lu | Missing OH source in a suburban environment near Beijing: observed and modelled OH and HO 2 concentrations in summer 2006[END_REF]. Nonetheless, it should be noted that the OH calculation from such empirical equation may be subject to some uncertainty. V ZMW was calculated by assuming a HONO daytime dry deposition velocity of 2 cm s -1 and an effective mixing height of 200 m. Due to the rapid photolysis of HONO at daytime, most of HONO cannot reach the height above 200 m [START_REF] Alicke | Impact of nitrous acid photolysis on the total hydroxyl radical budget during the Limitation of Oxidant Production/Pianura Padana Produzione di Ozono study in Milan[END_REF]. m ,.D+, is the rate constant for the reaction of OH with NO, using a value of 9.8×10 -12 cm 3 molecules -1 s -1 from [START_REF] Atkinson | Evaluated kinetic and photochemical data for atmospheric chemistry: Volume I -gas phase reactions of Ox, HOx, NOx and SOx species[END_REF]. The emission source strength was estimated from the HONO/NO X emission ratio of 0.53% as determined in Sec. 3.2. Figure 7 shows the average contributions of all the source and sink terms to the HONO budget in August 2016, when accurate J values observations were available and elevated daytime HONO were observed. An unknown source L QRSRTUR was clearly the dominant part, accounting for over 80% of the HONO production. An average P unknown value of 2.95 ppb h -1 was derived, which is more than 7 times greater than that of the homogeneous formation rate (L ,.D+, , 0.40 ppb h -1 ). The major loss pathway of HONO was the photolysis with a mean V WXTY value of 2.80 ppb h -1 , followed by dry deposition (V ZMW , 0.49 ppb h -1 ), and V ,.D.,+, was very small and almost less than 3% of V WXTY . The unknown source strength of daytime HONO in Ji'nan is higher than those derived in Santiago, Chile (1.69 ppb h -1 ; [START_REF] Elshorbany | Oxidation capacity of the city air of Santiago, Chile[END_REF], Beijing (1.83 ppb h -1 ; [START_REF] Hou | Comparison of atmospheric nitrous acid during severe haze and clean periods in Beijing, China[END_REF], and Houston, US (0.61 ppb h -1 ; Wong et al., 2012). Some studies have reported much lower L QRSRTUR obtained from a rural site in Guangzhou, China (0.76 ppb h -1 ; [START_REF] Li | Exploring the atmospheric chemistry of nitrous acid (HONO) at a rural site in Southern China[END_REF], a mountain site in Hohenpeissenberg, Germany (0.40 ppb h -1 ; Acker et al., 2006), and a forest site in Julich, Germany (0.50 ppb h -1 ; [START_REF] Kleffmann | Daytime formation of nitrous acid: A major source of OH radicals in a forest[END_REF]. We further explored the potential unknown daytime source(s) of HONO based on our measurement data. According to the laboratory studies, heterogeneous reactions of NO 2 on wet surfaces should be an important contributor to the ambient HONO concentrations, and the reaction rate is first order in NO 2 . It has been also proposed that these heterogeneous reactions can be photo-enhanced [START_REF] Stemmler | Photosensitized reduction of nitrogen dioxide on humic acid as a source of nitrous acid[END_REF]. Thus, the strength of the unknown HONO source (P unknown ) can be expressed by equation (E9), if the heterogeneous reactions were the major HONO sources. M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT L QRSRTUR ∝ l +, ' * [pq = ] * r s t u (E9) Correlation analysis between P unknown and related parameters has been widely adopted to diagnose the potential HONO sources (e.g., Su et al., 2008b). The NO 2 concentration ([NO 2 ]) is usually used as an indicator of the heterogeneous reactions on the ground surface since the ground surface to volume ratio (S/V g ) can be assumed to be constant for the well-mixed boundary layer at noontime, whilst [NO 2 ]*[S/V a ] can be taken as a proxy for the HONO formation on the aerosol surface. JNO 2 *[NO 2 ] and JNO 2 *[NO 2 ]*[S/V a ] can be used to infer the photo-enhanced heterogeneous reactions on ground and aerosol surfaces. Figure 8 shows the scatter plots of the calculated P unknown versus the abovementioned four indicators for the summer case (i.e., August 2016) when accurate J value measurements were available. P unknown showed a moderate correlation with [NO 2 ] with a correlation coefficient (R) of 0.55, and it was significantly improved after J NO2 *[NO 2 ] was considered (R=0.76). When aerosol surface density was taken into account, however, the correlation became even weaker with R values of 0.40 and 0.43. This suggests that the photo-enhanced heterogeneous reactions of NO 2 on the ground surface played a major role in the daytime HONO formation in Ji'nan in summer. Impact on the primary OH sources Photolysis of HONO presents an important primary source of OH in the atmosphere. The from O 3 photolysis can be calculated by Equation ( 12) (Su et al., 2008b). P 12 (O w ) = 2J 1 ) h × [O w ] / (1 + k w [M]/k = [H = O]) (E12) Figure 9 shows the daytime profiles of OH production rates from photolysis of HONO and O 3 in the summer (i.e., August 2016) period when accurate J value measurements were available. Clearly, photolysis of HONO dominated the daytime OH production in urban Ji'nan. The mean L ,. (vqpq) was 1.88 ppb h -1 , almost 3 times higher than L ,. (q w ). Furthermore, in contrast to most of the earlier studies which suggested that the contribution of HONO photolysis was mainly concentrated in the early morning and neglected at noon, photolysis of HONO presents the dominant OH contributor throughout the daytime at our study site. Even though at noontime in summer, high contributions of HONO photolysis were still found. These results demonstrate the significant role of HONO in the atmospheric oxidizing capacity in the urban atmosphere of Ji'nan. Summary Highly time-resolved continuous field observations of HONO, related air pollutants and meteorological parameters were performed at an urban site of Ji'nan in North China, for a year from September 2015 to August 2016. The measured mean concentration of HONO was 1.15 ppbv with a maximum level of 8.36 ppbv. The ambient HONO concentrations presented a seasonal variation with the highest level in winter as well as elevated concentrations in spring (April-May) and summer (August). Well-defined diurnal cycles of HONO with concentration peaks in the early morning and valleys in the afternoon were found for all the four seasons. Direct emissions from vehicle exhaust posed a large contribution to the ambient HONO, with an average emission ratio ∆HONO/∆NO X of 0.53%. During the nighttime, the heterogeneous conversion of NO 2 on the ground surface is an important source of HONO. The average conversion frequency of NO 2 to HONO was derived as 0.0068 h -1 from over hundred cases. At daytime, a missing HONO source with an average strength of 2.95 ppb h -1 was derived in summer, which was about seven times larger than the gas phase reactions. Our One-year continuous measurements of HONO were made at a typical urban site in northern China. Seasonal and diurnal variations, vehicle emission factors, heterogeneous formation, and daytime sources were examined based on a large observational data set. A strong missing daytime source was needed to explain the measured HONO concentrations. HONO photolysis is the dominant OH source throughout the daytime. population and 1.6 million automobiles. The site is located in the central campus of Shandong University (36º40'N, 117º03'E), a typical urban area surrounded by massive buildings and condensed population and close to several main traffic roads (Fig.1). Large-scale industries, including steel plants, thermal power plants, cement plants, oil refineries and chemical plants in suburban areas are the major industrial emission sources of local air pollution in Ji'nan, and are mainly distributed in the northeast and southwest of the site. By observing the spikes of sulfur dioxide (SO 2 ) concentrations under northeasterly and/or southwesterly winds, we know that the site was affected by the local industrial emission sources. All the measurements were carried out on the rooftop of a six-floor teaching building, around 22 m above the influenced by several dust storms and urban dust (http://www.sdein.gov.cn /dtxx/hbyw/201605/t20160512_294758.html). Besides, the elevated HONO levels in August under the condition of intense solar radiation suggests the presence of strong HONO sources as well as the important contributions of HONO as a potential source of OH radicals to the atmospheric oxidation chemistry. were adopted to select the cases: (a) only data in the morning rush hours (6:00-8:30 LT) in winter (e.g., November-February) were used; (b) NO/NO X>0.7; (c) good correlation between HONO and NO X ; (d) short duration of the plumes (<2 hours). Rush hours are the prominent period with strong traffic emission and thus greatest contribution of vehicle exhaust to HONO concentrations. Furthermore, during the winter early morning rush hours when the solar radiation is weak and boundary layer height is relatively stable, the derived HONO/NOx is less interfered by atmospheric photochemical reactions and mixing with the air masses aloft. Criteria (b) was used as an indicator for identifying the freshly emitted plumes. Criteria (c) and (d) further confirmed that the increase of HONO was mainly attributed to direct emissions instead of heterogeneous reactions of NO 2 . The slopes of the scatter plot of HONO versus NO X can be considered as the emission ratios. With such strict selection criteria, a total of twelve cases were screened out to estimate the vehicle emission factors of HONO in urban Ji'nan. ambient nocturnal HONO levels (Equation 2 to HONO during nighttimeIt has been widely accepted that the heterogeneous reactions of NO 2 on wet surfaces present an important formation pathway of ambient HONO, and generally make a dominant contribution at nighttime (e.g.,[START_REF] Finlayson-Pitts | The heterogeneous hydrolysis of NO 2 in laboratory systems and in outdoor and indoor atmospheres: An integrated mechanism[END_REF]. To investigate the heterogeneous production of HONO in Ji'nan, a number of nighttime HONO formation cases were identified to estimate the NO 2 -to-HONO conversion frequency. The selected cases should meet the following criteria: (1) only the nighttime data in the absence of sunlight (i.e., 20:00-05:59 LT in autumn and winter and 20:00-04:59 LT in spring and summer) were used considering the fast HONO loss via photolysis and potential existence of unknown HONO sources at daytime; (2) both HONO concentrations and HONO/NO 2 ratios increased steadily during the target case; photolysis and gas phase reactions at noontime. The average daytime wind speed in Ji'nan in the present study was 1.7 m s -1 , so that at a HONO lifetime of about 15 min at noontime, the horizontal transport had to occur within 1.6 km (in which no large-scale pollution sources) to reach the site.[START_REF] Dillon | Chemical evolution of the Sacramento urban plume: Transport and oxidation[END_REF] proposed a parameterization for the dilution by background air to estimate the magnitude of vertical transport (T V = k (dilution) ([HONO]-[HONO] background ). E8) (α = 0.83, β = 0.19, a = 4.1×10 9 , b = 140, c = 0.41, and d = 1.7) photolysis (O 1 D+H 2 O), another important OH source, based on the concurrent observations of HONO, O 3 , J HONO and J O1D in summer. The other primary OH sources, such as photolysis of peroxides and ozonolysis reactions of alkenes, are generally not very important in urban areas, especially at daytime, and were not considered in the present study. We also don't consider the primary sources of HO 2 and RO 2 radicals (such as photolysis of OVOCs) due to the lack of measurement data for these radical precursors. The net OH production rate from HONO photolysis (L ,. (vqpq) RMY ) was calculated by the source strength subtracting the sink terms due to reactions (R2) and (R5)(Equations 10 and 11). The OH production rate HONO) F = P 12 (HONO) -k 01D12 [NO][OH] -k 2101D12 [HONO][OH] (E11) the photo-enhanced heterogeneous reaction of NO 2 on the ground surface may be a major source of daytime HONO in summer. Photolysis of HONO presents the predominant OH contributor not only in the early morning but also throughout the daytime in urban Ji'nan, and hence plays a vital role in the atmospheric oxidation and ozone formation in the polluted urban atmosphere of northern China. (18:00-06:00, LT); D: daytime (06:00-18:00, LT) 1: Elshorbany et al. (2009); 2: Acker et al. (2006a); 3: Yu et al. (2009); 4: Bernard et al. (2016); 5: Qin et al. (2009); 6: Tong et al. (2015); 7: Su et al. (2008a); 8: Alicke et al. (2002); 9: Li et al. (2012); 10: this study. Figure 1 . 1 Figure 1. Locations of Ji'nan and the sampling site. The left map is color-coded by the anthropogenic NOx emissions (Zhang et al., 2009), while the right is color-coded by the geographical height. The large industrial sources are labeled with different colors, including steel plants (light blue), thermal power plants (pink), cement plants (red), oil refineries (grey) and chemical plants (green). Figure 2 .Figure 3 .Figure 4 .Figure 5 . 2345 Figure 2. Time series of HONO, NO, NO 2 , PM 2.5 , PM 1.0 , J HONO , temperature (T), relative humidity (RH) and surface wind in Ji'nan from September 2015 to August 2016. The data gap is mainly due to the maintenance of the instruments. Figure 6 .Figure 7 . 67 Figure 6. Seasonal variation of the NO 2 -to-HONO conversion frequency (k het ) in Ji'nan. Figure 8 .Figure 9 . 89 Figure 8. Scatter plots of the unknown daytime HONO source strength (P unknown ) with (a) NO 2 , (b) NO 2 *(S/V) a , (c) NO 2 *J NO2 , and (d) NO 2 *J NO2 *(S/V) a during August 2016. Table 1 . 1 Overview of the measured HONO and NOx levels in urban Ji'nan and comparison with other studies. Location Time HONO(ppb) N D NO 2 (ppb) N D NO X (ppb) N D HONO/NO 2 N D HONO/NO X N D Ref. Santiago, Chile (urban) Mar-Jun 2005 3.00 1.50 30.0 20.0 200.0 40.0 0.100 0.075 0.015 0.038 1 Rome, Italy (urban) May-Jun 2001 1.00 0.15 27.2 4.0 51.2 4.2 0.037 0.038 0.020 0.024 2 Kathmandu, Nepal (urban) Jan-Feb 2003 1.74 0.35 17.9 8.6 20.1 13.0 0.097 0.041 0.087 0.027 3 Shanghai, China (urban) Oct 2009 1.50 1.00 41.9 30.0 / / 0.038 0.032 / / 4 Guangzhou, China (urban) Jun 2006 3.5 2.00 20.0 30.0 / / 0.175 0.067 / / 5 Beijing, China (urban) Oct-Nov 2014 1.75 0.93 37.6 35.3.0 94.5 53.4 0.047 0.026 0.019 0.017 6 Xinken, China (suburban) Oct-Nov 2004 1.30 0.80 34.8 30.0 37.8 40.0 0.037 0.027 0.034 0.020 7 Milan, Italy (suburban) May-Jun 1998 0.92 0.14 33.2 18.3 117.5 23.4 0.028 0.008 0.008 0.006 8 Backgarden, China (rural) Jul 2006 0.95 0.24 16.5 4.5 20.9 5.5 0.057 0.053 0.045 0.043 9 Sep 2015-Aug 2016 1.28 0.99 31.0 25.8 46.4 40.6 0.079 0.056 0.040 0.035 10 Sep-Nov 2015 (autumn) 0.87 0.66 25.4 23.2 38. 37.5 0.049 0.034 0.034 0.022 10 Ji'nan, China (urban) Dec 2015-Feb 2016 (winter) 2.15 1.35 41.1 34.6 78.5 64.8 0.056 0.047 0.034 0.031 10 Mar-May 2016 (spring) 1.24 1.04 35.8 25.8 47.3 36.0 0.046 0.052 0.035 0.041 10 Jun-Aug 2016 (summer) 1.20 1.01 22.5 19.0 29.1 25.8 0.106 0.079 0.060 0.049 10 N: nighttime Table 2 . 2 The emission ratios ∆HONO/∆NO X of fresh vehicle plumes. Date Local Time ∆NO/∆NO X R 2 ∆HONO/∆NO X (%) 11/03/2015 06:08-07:45 0.89 0.91 0.29 11/05/2015 06:00-07:30 0.92 0.84 0.63 11/17/2015 06:00-07:30 1 0.95 0.75 11/20/2015 06:00-07:15 0.92 0.72 0.59 12/06/2015 06:00-07:48 0.83 0.75 0.87 12/08/2015 06:02-07:30 0.95 0.61 0.58 12/25/2015 06:00-07:30 0.94 0.72 0.30 12/31/2015 06:00-07:30 0.96 0.94 0.47 01/02/2016 06:00-07:30 1 0.61 0.46 01/20/2016 06:44-07:48 0.92 0.96 0.71 01/21/2016 06:26-07:56 0.94 0.58 0.54 01/27/2016 07:00-08:12 0.89 0.77 0.19 Acknowledgments We thank Chuan Yu, Ruihan Zong, Dr. Zheng Xu and Dr. Long Jia for their contributions to the filed study. We are grateful to the European Centre for Medium-Range Weather Forecasts for sharing the boundary layer height data and to the National Center Atmospheric Research for providing the TUV model. This work was funded by the National Natural Science Foundation of China (No. 41505111 and 91544213), the National Key Research and Development Programme of the Ministry of Science and Technology of China (No. 2016YFC0200500), the Natural Science Foundation of Shandong Province (ZR2014BQ031), the Qilu Youth Talent Program of Shandong University, and the Jiangsu Collaborative Innovation Center for Climate Change.

01744547

en

2024/03/05 22:32:07

1994

https://hal.science/tel-01744547/file/1994_PIERRON_Fabrice.pdf

Fabrice Pierron may come The iosipescu in-plane shear test : modelization and experimental procedure for composites

00174455

en

2024/03/05 22:32:07

2007

https://hal.science/hal-00174455/file/Nedelec_et_al.pdf

Keywords: thermoporosimetry, confinement, crystallisation, nanoporous materials The thermal behaviour of carbon tetrachloride confined in silica gels of different porosity was studied by differential Scanning Calorimetry. Both the melting and the phase transition at low temperature were measured and found to be inextricably dependant upon the degree of confinement. The amount of solvent was varied through two sets of experiments, sequential addition and original progressive evaporation allowing the measurement of the DSC signals for the various transitions as a function of the amount of CCl 4 . These experiments allowed the determination of transition enthalpies in the confined state which in turn allowed the determination of the exact quantities of solvent undergoing the transitions. A clear correlation was found between the amounts of solvent undergoing the two transitions (both free and confined) demonstrating that the formation of the adsorbed layer t does not interfere with the second transition. The thickness of this layer and the porous volumes of the two silica samples were measured and found to be in very close agreement with the values determined by gas sorption. Crystallization of carbon tetrachloride in confined geometries Adil Meziane 1 , Jean-Pierre E. Grolier 2 , Mohamed Baba 2 and Jean-Marie Nedelec Introduction The peculiar behaviour of liquids in confined geometry has attracted a lot of interest in particular during the past ten years. A comprehensive review has been published in 2001 [1]. The case of water [2,3] is particularly relevant because of the numerous works dealing with water and also because of obvious practical applications. The revival in the interest in transitions in confined geometry undoubtedly comes from the considerable progress in the preparation of nanoporous materials with controlled pore size and with spatially controlled pore distribution and connectivity. In this context discovery of MCM type materials [4] has played an important role. The use of organized molecular systems (surfactants) to limit spatially the condensation of alkoxide precursors is now common and has been extended to various systems and various pore organizations. The availability of such porous materials with controlled porosity, and to some extent with tuneable porosity, has lead to an increased interest for the study of crystallisation in confined geometry. Practical interest of liquids in porous materials is also very widespread and the case of oil recovery is a major example. The chemistry of water in clouds is also greatly affected by confinement effects. More importantly, the research devoted to the preparation of nanocrystals with a good control of both crystal size and size distribution has been incredibly expanding in the last twenty years [5]. In particular semi-conducting nanocrystals or quantum dots have been the subject of many research papers [6,7] due to the possible observation in these materials of a direct quantum effect correlated to the size of the crystals. Porous materials appeared to be ideal candidates for the preparation of such nanocrystals, utilizing the pores as nanoreactors where the crystallization of the desired material could be confined. In particular abundant examples concerning MCM-41 and SBA-15 mesoporous silicas templates can be found in the literature, see [8,9] for instance. Another very interesting example of crystallization in confined geometries is Biomineralization [10,11]. Biomineralization is a complex process in which the solution conditions, organic template, and crystal confinement coordinate to yield nanostructured composite materials with controlled morphology and mechanical and structural properties. Over the past few decades, research has examined various aspects of this mineralization process both by characterizing those found in nature and by creating synthetic composites. Another field in which crystallization in confined geometries play a major role is polymer science. Numerous examples demonstrate how the confinement can modify the kinetics of crystallization of polymers and also the morphology of the crystals [12]. All these selected examples demonstrate how crucial it is to get information concerning crystallization in confined geometries. In particular the energetic of crystallization in confining media is not well documented. The well known modification of the freezing point temperature of liquids in confined geometry has led to the development of characterization techniques for the measurement of porosity in solids. Such techniques are based upon the Gibbs-Thomson equation [START_REF] Gibbs | Collected works[END_REF][START_REF] Thomson | [END_REF] which relates the shift ∆T of the crystallization temperature to the pore size of the confining material according to [15]: p m p s m SL p R H k R H T Cos T T T ∆ ≈ ∆ = - = ∆ ρ θ σ 0 0 . 2 (Equation 1) where T p is the melting temperature of a liquid confined in a pore of radius R p , T 0 is the normal melting temperature of the liquid, σ SL is the surface energy of the solid/liquid interface, θ the contact angle, ∆H m is the melting enthalphy, ρ S the density of the solid and k a constant. The measurement of ∆T by calorimetry or NMR technique leads to thermoporosimetry [16] and NMR cryoporometry [17] respectively. The advantages of both techniques have been discussed extensively [18]. As first proposed by Kuhn [19] in the 1950's, thermoporosimetry can also be of great value for soft networks characterization like polymeric gels [20]. In this case, the confinement is created by the meshes defining the 3-dimensional polymer network. The study of polymer architecture modification by thermoporosimetry requires knowledge of the behaviour of liquids able to swell these organic materials. We recently developed reference porous materials for calibration of thermoporosimetry with various solvents [21,22]. In our systematic work, we observed that some solvents presenting a low temperature phase transition in the solid state offered even more interest [23]. Indeed, this transition is also affected by the confinement and is an interesting alternative to the use of liquid to solid transition since it is usually much more energetic. From a practical point of view the use of these transitions does not change the procedure requiring the calibration of the technique with samples of known porosity. But from a fundamental point of view, this observation raises some questions about the underlying thermodynamics. The objective of this paper is to discuss the transitions of carbon tetrachloride in confined geometry because CCl 4 is an effective solvent for polymer swelling and also presents this solid state phase transition as observed before. [24,25]. Theoretical considerations According to Equation (1), the shift of the transition temperature of a confined liquid ∆T is inversely proportional to the radius of the pore in which it is confined. In fact it is well known that not all the solvent takes part in the transition and that a significant part of it remains adsorbed on the surface of the pore. The state of this adsorbed layer has been discussed extensively in the case of water. Consequently, the radius measured by application of the Gibbs-Thomson equation should be written R=R p -t where t is the thickness of the adsorbed layer leading to a reformulation [7] of Equation 1as 2) t T H k R m p + ∆ ∆ = (Equation The value of t can be determined by the calibration procedure using materials of various pore sizes and this is the traditionally adopted procedure. The problem in doing so is that the underlying hypothesis is that the thickness of the adsorbed layer t does not vary with pore size. For small pores, the error on t can lead to large errors on the measurement of R p . We proposed an alternative method to measure t by adding sequentially various amounts of liquid in the porous material [26]. As stated before this layer t represents the part of the solvent which does not crystallize. For solvents like CCl 4 which exhibit a further transition at low temperature the behaviour of this adsorbed layer is an open question. Does this solvent participate in the second transition? Is a new adsorbed layer created on the top of the first one? To get further insight into these questions we studied the behaviour of CCl 4 in mesoporous silica gels in this paper as described in the following section. Experimental section Mesoporous silica gels Mesoporous monolithic silica gels (2.5 mm × 5.6 mm diameter cylinders) were prepared by the acid catalysed hydrolysis and condensation of a silicon alkoxide, following procedures reviewed elsewhere [START_REF] Hench | Sol-gel silica : processing, properties and technology transfer[END_REF]. Careful control of the aging time performed at 900°C allowed the production of samples with controlled textural properties. In this study two samples (A and B) with different textural properties (Specific Surface Area (SSA), total pore volume (V p ) and pore size distribution (PSD)) were used. The textural characteristics of the samples were determined by N 2 sorption. Gas sorption measurements Textural data of the silica gels were determined on a Quantachrome Autosorb 1 apparatus. The instrument permits a volumetric determination of the isotherms by a discontinuous static method at 77.4 K. The adsorptive gas was nitrogen with a purity of 99.999%. The cross sectional area of the adsorbate was taken to be 0.162 nm 2 for SSA calculations purposes. Prior to N 2 sorption, all samples were degassed at 100°C for 12 h under reduced pressure. The masses of the degassed samples were used in order to estimate the SSA. The BET [START_REF] Brunauer | [END_REF] SSA was determined by taking at least 4 points in the 0.05<P/P 0 <0.3 relative pressure range. The pore volume was obtained from the amount of nitrogen adsorbed on the samples up to a partial pressure taken in the range 0.994<P/P 0 <0.999. Pore size distributions were calculated from the desorption isotherm by the BJH method [29]. The mean pore radius R av was calculated according to BET p av S V R 2 = (Equation 3) corresponding to a cylindrical shape for the pores which is also the underlying hypothesis in equation (1). Textural data for the two samples are displayed in Table 1. In this table, the modal pore diameter R p is also shown. This value fairly matches the R av derived from S BET measurement with cylindrical shape assumption thus confirming the validity of the hypothesis on the pore shape. Sample SSA (m 2 /g) Vp (cm 3 /g) R av (nm) R p (nm) A 183 1, DSC measurements A Mettler-Toledo DSC821 instrument calibrated (both for temperature and enthalpy) with metallic standards (In, Pb, Zn) and with n-heptane was used to record the thermal curves. It was equipped with an intracooler set allowing a scanning range of temperature between -70 and 600 °C. About 10 or 20 mg of the studied material was introduced into an aluminium DSC pan to undergo an appropriate temperature program. To allow the system to be in an equilibrium state, a slow freezing rate is required [30]. A rate of -0.7 °C/min was chosen. Other slower cooling rates were tested which did not show any significant discrepancy. CCl 4 (Aldrich) of HPLC quality was used without any supplementary purification. Results and discussion Thermal behavior of free CCl 4 Bulk CCl 4 was studied before and its thermal phase transitions were well characterized [24,25]. It exhibits a complex thermal transitions system as shown in Figure 1. - (R) (M) (Liquid) (Liquid) (Liquid) (M) (M) (R) (R) (FCC) (FCC) Heat Flow (mW.g -1 ) T (°C) As it is cooled down, liquid CCl 4 crystallizes into Face-Centered-Cubic phase (FCC) which follows a phase transition upon further cooling to a Rhombohedral one (R) which, in turn, transforms to Monoclinic crystalline structure (M) around -48°C. Heating the (M) phase leads to (R) in a reversible way but upon heating (R) melts directly without transforming into the (FCC) phase. Observing the transition heat values (Figure 1), it can be pointed out that the R to-liquid transition releases an enthalpy (13.6 J/g) equivalent to the total heat liberated by the liquid-to-FCC (9.6 J/g) together with the FCC-to-R (3.8 J/g). Takei et al. [24] showed that both solid-to-solid and liquid-to-solid transitions of CCl 4 were strongly dependent on the average pore size of the material in which the liquid is confined. In particular, they demonstrated that the FCC-to-R transition is no longer observed when the pore radius is smaller than 16.5 nm, which is the case for our silica samples (see Table 1). Because of the complex behaviour of CCl 4 upon cooling, we chose to use the heating of the solvent to limit the study to the M-to-R and R-to liquid transitions. The two transitions were studied for CCl 4 confined in the two porous samples A and B. Thermal behaviour of CCl 4 confined in sample A The objective is to get quantitative information on the solvent undergoing both transitions (both confined and free solvent). In order to do so, we performed sequential addition of precise quantities of CCl 4 in the sample as described in [26]. Briefly, a known mass of silica gel (about 20 mg) is set in the DSC pan which is sealed. A small hole is drilled in the cover allowing further injection of known masses of carbon tetrachloride. This procedure allows a precise control of the added mass of solvent. After each thermal cycle, a new injection is performed. For the first time to our knowledge, we also performed some experiments in the reverse way, by progressively evaporating the solvent starting from a large excess. This was performed by inert gas flushing in the DSC pan at 25 °C. The subsequent evaporation of the solvent is controlled by the flushing time. Obviously in this case we do not know the remaining mass of CCl 4 , but we can calculate it from the measured enthalpies. The thermograms recorded for various quantities of CCl 4 added to sample A are shown in Figure 2. In this figure, 4 peaks can be observed which are labelled from 1 to 4 starting from low temperature to room temperature. The assignment of all peaks is presented in Table 2. In Figure 2, it can be seen that for small quantities of added CCl 4 , no transition is observed. This first step corresponds to the creation of the adsorbed layer t onto the surface of the porous silica gel. For higher amounts of CCl 4 , peak 3 appears at a temperature shifted with respect to the normal melting temperature of solid CCl 4 . At about the same time, peak 1 also appears corresponding to the M-to-R transition for the confined solvent. The intensities of these two peaks increase upon further addition of solvent until they remain constant coinciding with the appearance of peaks 2 and 4 corresponding to excess free solvent. A plot of the heats corresponding to peaks 3 and 4 (H3 and H4) as a function of the mass of CCl 4 added (m CCl4 ) is presented in Figure 4. The different steps are clearly observable. The point where H3 is different from zero corresponds to the end of the creation of the adsorbed layer allowing the determination of the quantity of solvent (m t ) participating in the formation of this layer. At a given point, H3 remains constant and this point corresponds to the total filling of the pores (H3 Max ) thus allowing the determination of the porous volume of the sample (see section 3.4). To measure the amounts of CCl 4 involved in each transition precisely, we need to know the transition enthalpy at the given temperature. These values are known for free solvent transiting at regular temperatures but not for confined solvent which undergoes transitions at lower temperature. From Figure 4, we can measure H3 Max at the point where all pores are filled in the constant part of the curve. In this case the enthalpy corresponds to a mass of solvent equal to the total mass added (m vp ) minus the mass required for the creation of the adsorbed layer (m t ) namely m=m vp -m t . We can then deduce the enthalpy of melting per gram for the confined solvent ∆H3 = 13.67 J.g -1 . For the M-to-R transition, the situation is different. Because of the overlapping of peaks 1 and 2 we can only use the sum H1+H2. If we plot the evolution of H3 and H4 as a function of (H1+H2) we obtained the curves presented in Figure 5. It is worth noting that the points corresponding to desorption experiments complete nicely the points corresponding to sequential addition of solvent (empty and full symbols respectively). The point where H4 differs from zero corresponds to the H1 Max value corresponding to the totality of solvent undergoing the transition (in this case H2=0). The enthalpy of transition ∆H1 can then be deduced ∆H1= 27.22 J.g -1 . Knowing ∆H1 and ∆H3, we can now calculate the masses of solvent which undergo the various transitions for all points. Figure 6 presents the correlation between these masses M3 and M1 (the indexes correspond to the different peaks). Progressive filling (•) and desorption (○) experiments. The line y=x is also plotted. A clear correlation is observed between the confined solvent which undergoes the R-tliquid and the M-to-R transitions. This correlation is observed both for addition and evaporation experiments. This clearly confirms that all solvent undergoing the first transition also undergoes the second one. This observation is further confirmed by the plot of Figure 7 showing the correlation between M2 and M4, the masses of free solvent which undergo the transitions 2 and 4. M2 is determined through the following equation: ( ) 2 1 2 1 2 H H H H M Max ∆ - + = Equation (4) where H1 Max is the enthalpy required for the M-to-R transition of the liquid totally filling the pores (see Figure 5) and ∆H2=46.6 J.g -1 the specific enthalpy for the M-to-R transition of free CCl 4 . Once again a clear correlation is observed between the two quantities confirming that all the solvent which has crystallised outside the pores undergoes the second transition at a regular temperature (no confinement). These conclusions also demonstrate that the layer t remains adsorbed and does not participate in the low temperature transition. Thermal behaviour of CCl 4 confined in sample B The same experiments and calculations were applied to sample B which presents smaller pores i.e. higher confinement. The thermograms recorded for sample B filled with CCl 4 upon progressive evaporation are displayed in Figure 8. Because of the higher degree of confinement, the two peaks 1 and 2 are well resolved and can be discriminated. -60 -50 -40 -30 -20 Heat Flow (a.u.) T (°C) Following the same procedure, we can plot the evolution of H3 and H4 as a function of m CCl4 as performed in Figure 9. The plot of H1 and H2 as a function of m CCl4 (not shown here) can also be performed. From these curves, ∆H1 and ∆H3 for sample B can be derived (∆H1=22.19 J.g -1 and ∆H3=10.13 J.g -1 ). Together with the known values of ∆H2 (46.6 J.g -1 ) and ∆H4 (25.07 J.g -1 ) they allow the calculation of the quantities of solvent which undergo the different transitions for the various experiments. Calculation of porous volumes and thicknesses of adsorbed layers Considering the curves of Figure 4 and 9, we can measure the mass of solvent corresponding to total filling of the pore m vp . The porous volume of the gel can then be calculated according to: ρ being the density of CCl 4 . We took the value at -20°C ( ρ=10.85 kmol.m -3 ) [31]. From the same figures, we can also measure the mass of the adsorbed layer m t , the thickness of this layer can be calculated according to: SSA being the specific surface area of the silica sample given in Table 1. The results are summarized in Table 3. Sample V p (cm 3 .g -1 ) V N2 (cm The calculated value of V p are in very good agreement with the value measured by nitrogen sorption, the error is less than 2%. The values of t determined for samples A and B are also in good agreement with average value given in [25] after calibration procedure with samples of various pore size. All calculations were performed with a constant value of ρ CCl4 measured at -20°C. Obviously no information can be found in the literature for densities of carbon tetrachloride at lower temperatures since it is usually solid at these temperatures. Nevertheless using the value at -20°C, the error must be small. Furthermore, with the validity of such an approach demonstrated, we can now consider the exact porous volume to calculate the exact density of the confined solvent at various low temperatures. Conclusions The thermal behavior of carbon tetrachloride confined in two mesoporous silica gels of different porosity was studied. The two transitions (solid to liquid and Monoclinic to Rhombohedral) were measured and are affected by the confinement. The enthalpies of these two transitions were determined for the first time at the temperatures corresponding to confined solvent. Using these enthalpies, a clear correlation has been shown between the solvent undergoing the first and the second transitions. Consequently, the adsorbed layer which is created during the intrusion of CCl 4 inside the porosity of the silica gels is kept constant and does not participate in the two transitions. The thickness of this layer was measured for both samples and is found to be slightly dependant on the pore radius. Finally the porous volumes of the silica gels have been measured and the values agree very closely with those derived from nitrogen sorption isotherm. It has been demonstrated that using porous samples of known porosity (measured by mercury intrusion porosimetry or gas sorption analysis) could allow the measurement of thermodynamical data of confined liquids (Enthalpy of transition, density,….). Figure 1 : 1 Figure 1: DSC thermogram of pure CCl 4 showing the different transitions. 2 : 2 Labelling of the different peaks observed in the DSC curves. Figure 3 3 Figure 3 presents the DSC curves recorded upon desorbing the CCl 4 by gas flushing. As can Figure 2 : 2 Figure 2: Thermograms recorded for various amount of CCl 4 added to sample A. Figure 3 : 3 Figure 3: Thermograms recorded for various flushing times for sample A filled with CCl 4 . Figure 4 : 4 Figure 4: Evolution of H3 (circles) and H4 (squares) as a function of the mass of CCl 4 Figure 5 : 5 Figure 5: Evolution of H3 (circles) and H4 (squares) for progressive filling (full symbols) and Figure 6 : 6 Figure 6: Evolution of the mass of confined solvent undergoing transition M-to-R (M1) as a Figure 7 : 7 Figure 7: Correlation between the masses of free solvent undergoing the R-to-liquid (M2) Figure 8 : 8 Figure 8: Thermograms recorded for various flushing times for sample B filled with CCl 4 . Figure 9 : 9 Figure 9: Evolution of H3 (circles) and H4 (squares) as a function of the mass of CCl 4 Figure 10 :Scheme 1 : 101 Figure 10: Correlation between M3 and M1 (circles) and M2 and M4 (squares) for 3 * 3 1 Laboratoire de Photochimie Moléculaire et Macromoléculaire, UMR CNRS 6005 2 Laboratoire de Thermodynamique des Solutions et des Polymères, UMR CNRS 6003 3 Laboratoire des Matériaux Inorganiques, UMR CNRS 6002 TransChiMiC Ecole Nationale Supérieure de Chimie de Clermont Ferrand & Université Blaise Pascal, 24 Avenue des landais 63177 Aubière Cedex, FRANCE. Table 1 : 1 Porous characteristics of the silica gels samples. 327 14,5 14,25 B 166 0,991 11,9 8,7 Acknowledgements Financial support from the French ANR under project Nanothermomécanique (ACI Nanosciences N°108) is gratefully acknowledged. The authors would like to thank A. Gordon and Pr S. Turrell for careful reading of the paper. Contact author for correspondence and return of proofs: e-mail : [email protected]

01744623

en

2024/03/05 22:32:07

2018

https://hal.sorbonne-universite.fr/hal-01744623/file/AAA_Deuve%202018%20version%20ultime.pdf

Thierry Deuve email: <[email protected]> What is the epipleurite? A contribution to the subcoxal theory as applied to the insect abdomen Keywords: Arthropoda, Hexapoda, Insecta, morphology, morphogenesis, segment, limb, pleurite, sternite, subcoxa, precoxa, ectodermal genitalia, wings has shown that they are instead eupleural (i.e. appendicular) and correspond to a dorsal part of the subcoxa. Their presence in the abdominal segments of insects illustrates the fundamental importance of the subcoxa in segmental structure, with a function of anchoring and supporting the appendage when the latter is present. However, the epipleurites are normally separated and functionally dissociated from the coxosternum, which integrates the ventral component of the subcoxa. In females, the epipleurite of segment IX of the abdomen corresponds to the gonangulum, as already pointed out by Deuve in 1994 and 2001, and it is involved in gonopod articulation. At segments VIII and IX of both males and females of holometabolans, the formation process of the genital ducts leads to an internalisation of the whole subcoxosternum (i.e. the coxosternum with the exception of the coxal and telopodal territories), and it is the two flanking epipleurites that ventrally close the abdomen in relation to the rearward displacement of the gonopore. This model may be generalised, in its broad lines, to a large part of the hemimetabolans. The body plan of the insect abdomen underlines the morphological and functional importance of the subcoxa in its fundamental structure, but the study of the Hexapoda in general also indicates the presence of a more proximal segment, the precoxa, which would belong to the groundplan but is more cryptic because it is often closely associated with the subcoxa and/or the paranotal lobe. Its location, which is sometimes on the ventral flank of the paranotal lobe, is in line with the hypothesis of a dual origin of the pterygote wing. Résumé. Qu'est-ce qu'un épipleurite ? Une contribution à la théorie subcoxale appliquée à l'abdomen des Insectes. Les épipleurites ont été d'abord décrits par Hopkins en 1909 sur l'imago et la larve d'un Coléoptère. Puis ce terme a été largement utilisé en morphologie de l'Insecte pour désigner des sclérites de la région pleurale, surtout pour les larves. Ils ont récemment été interprétés comme tergopleuraux (c'est-à-dire pleuraux mais non strictement appendiculaires) par Deuve en 2001, mais une étude du développement embryonnaire par Kobayashi et al. en 2013 a montré qu'ils étaient en réalité eupleuraux (c'est-à-dire appendiculaires) et correspondaient à la partie dorsale de la subcoxa. Leur présence aux segments abdominaux des Insectes illustre l'importance fondamentale de la subcoxa dans l'architecture segmentaire, avec une fonction d'ancrage et de support de l'appendice lorsque celui-ci est présent. Cependant, les épipleurites sont habituellement séparés et fonctionnellement dissociés du coxosternum, lequel intègre la composante ventrale de la subcoxa. Chez les femelles, l'épipleurite du segment IX de l'abdomen correspond au gonangulum, comme déjà indiqué par Deuve en 1994 et 2001, et participe à l'articulation du gonopode. Aux segments VIII et IX des mâles et des femelles d'Insectes Holométaboles, la morphogenèse des conduits génitaux provoque une internalisation de tout le subcoxosternum (c'est-à-dire le coxosternum moins les territoires coxaux et télopodiaux), et ce sont les deux épipleurites adjacents qui ferment ventralement l'abdomen en accompagnement du déplacement vers l'arrière du gonopore. Ce modèle peut être étendu, dans ses grandes lignes, à une large partie des Insectes hémimétaboles. Le plan structural de l'abdomen des Insectes souligne l'importance morphologique et fonctionnelle de la subcoxa dans l'architecture fondamentale, mais l'étude de tous les Hexapodes indique aussi la présence d'un article plus proximal de l'appendice, la precoxa, qui appartiendrait au plan de base mais serait plus discret car souvent étroitement associé à la subcoxa et/ou au lobe paranotal. Sa position, souvent au flanc ventral du lobe paranotal, s'accorde avec l'hypothèse d'une origine duale de l'aile des Ptérygotes. Introduction For more than a century, the question of structure of the arthropod segment along its dorsoventral axis has been subject of intensive debate. In theory, this structure corresponds to a simple model: a dorsal tergum is separated from a ventral sternum by lateral pleura [START_REF] Audouin | Recherches anatomiques sur le thorax des animaux articulés et celui des insectes hexapodes en particulier[END_REF][START_REF] Milne-Edwards | Introduction à la zoologie générale, ou, considérations sur les tendances de la[END_REF]. In practice, however, it is often difficult to delimit these respective territories if we consider the morphological diversity of arthropods and the adaptive differentiations that respond to mechanical and functional constraints. Sclerified areas and membranous areas delineate visible territories that do not always correspond to the fundamental morphological fields. Sclerites of different origins may merge, or, conversely, a given sclerite may be reduced or fragmented and partially replaced by a membranous area with blurred or indefinable limits. Invaginations of some skeletal territories to form internal ducts or apophyses for muscle attachment have also been reported. Several approaches make it possible to search for the identity and limits of the original territories: the classical methods of comparative anatomy, including study of muscles, whose inserts can act as landmarks, those of descriptive and comparative embryology, and the more recent ones of developmental genetics using molecular markers of anatomical territories. Over the past few decades, the combined use of these methods has illustrated the complexity of the issues involved, but has also clarified our understanding of the fundamental organisation of the skeleton. Pleural regions present the majority of problems, because their boundaries are ipso facto those of the tergal and sternal regions. In addition, they include the limbs and a subsequent question is whether the pleura are entirely appendicular in nature or whether some parts of the pleura would instead be 'non-appendicular' or 'formerly appendicular'. Hypothetical homologies with a prearthropodan ancestor may even be sought. Added to this is the complex structure of the arthropodal limb itself, of a pleural nature, whose fundamental organisation needs to be elucidated in order to uncover the true pattern of its diversification within the different arthropod lineages and, for a given organism, according to the metamere considered. The pleura are lateral areas, but the insertion of the arthropodal limb has shifted to a more lateroventral location. This arrangement and functional necessities lead to a differentiation of the most proximal segments of the appendage (precoxa and coxa in crustaceans, precoxa and subcoxa in hexapods): on the ventral side, these segments tend to merge with the sternum or even to replace it [START_REF] Börner | Die Gliedmassen der Arthropoden[END_REF][START_REF] Weber | Die Gliederung der Sternopleuralregion des Lepidopterenthorax. Eine vergleichende morphologische Studie zur Subcoxaltheorie[END_REF][START_REF] Ferris | Some general considerations[END_REF]Ferris , 1940a)), on the dorsal side, they tend to be embedded into the body wall, or even to merge with it [START_REF] Börner | Die Gliedmassen der Arthropoden[END_REF]. In the dorsal region of the pleuron, there is indeed a peculiar area situated between the apparent base of the appendage and the tergum, the interpretation of which is problematic. Respiratory organs-the spiracles in Hexapoda and Myriapoda-are located in this laterodorsal region. [START_REF] Crampton | Notes on the thoracic sclerites of winged insects[END_REF] proposed the name eupleurites for the pleural sclerites belonging to the limb, and [START_REF] Prell | Das Chitinskelett von Eosentomon[END_REF] introduced the term tergopleurites for those which, being 'not appendicular', are located between the base of the limb and the genuine tergum (the paranotal lobes could probably be included among these). In the Hexapoda, Snodgrass first named some laterodorsal sclerites "tergopleurites" (1927), then "paratergites" (1931) and finally "laterotergites" (1935a), because, from his final point of view, they belonged not to the limb but to the tergum. He then drew a boundary, called the "dorsopleural line" (1931, 1935a, 1958), which would separate the pleural and tergal regions. This "pleural line" was the so-called "pleural suture" of [START_REF] Hopkins | Contributions toward a monograph of the scolytid beetles. I. The genus Dendroctonus[END_REF] with respect to the abdomen. It should be noted that in Snodgrass' diagrams (ibid.), the spiracles are more dorsal than this boundary and therefore located in the tergal or, more precisely, 'laterotergal' area. This led me to rename the "laterotergites" of Snhodgrass (1935a) "epipleurites", using the terminology of [START_REF] Hopkins | Contributions toward a monograph of the scolytid beetles. I. The genus Dendroctonus[END_REF] created for both imago and larva of Coleoptera, because they seemed to me to be pleural rather than tergal in nature [START_REF] Deuve | Sur la présence d'un "épipleurite" dans le plan de base du segment des Hexapodes[END_REF](Deuve , 2001a)). In my view, and following [START_REF] Hopkins | Contributions toward a monograph of the scolytid beetles. I. The genus Dendroctonus[END_REF], they were tergopleurites in the sense of [START_REF] Prell | Das Chitinskelett von Eosentomon[END_REF]. It can also be noted that Snodgrass (1935a[START_REF] Snodgrass | Evolution of arthropod mechanisms[END_REF] later identified as "epipleurites" or "epipleural sclerites" the subalar and basalar sclerites of the pterygote thorax, lying between the limb-base and the notum. In a recent work on the embryonic development of a carabid beetle, [START_REF] Kobayashi | Embryonic development of Carabus insulicola (Insecta, Coleoptera, Carabidae) with special reference to external morphology and tangible evidence for the subcoxal theory[END_REF] correctly note that I misplaced the dorsopleural or longipleural fold separating the epipleurite from the subcoxa [START_REF] Deuve | Les sternites VIII et IX de l'abdomen sont-ils visibles chez les imagos des Coléoptères et des autres Insectes Holométaboles ?[END_REF][START_REF] Deuve | Sur la présence d'un "épipleurite" dans le plan de base du segment des Hexapodes[END_REF](Deuve , 2001a)), confusing it in the thorax of a Carabus larva with the paracoxal furrow separating the catepimerite and the anepimerite. Consequently, anepimerite and epipleurite were confused in the metathoracic segment, the latter losing its subcoxal identity. However, I had only reproduced the successive diagrams of Snodgrass that drew the "dorsopleural line" on larvae of Silphidae [START_REF] Snodgrass | Morphology of the insect abdomen. Part I. General structure of the abdomen and its appendages[END_REF], "Fig. 3B") and Carabidae (Snodgrass 1927, "Fig. 25";1935a, "Fig. 139B";1958, "Fig. 8G"). Indeed, Snodgrass (1935a) interpreted this larval thoracic anepimerite as a laterotergite or, later (1958), as a pleurite more dorsal than the anapleurite. It must be noted in this respect that Snodgrass himself reproduced an error of Hopkins (1909, "Fig. 3"), who, on an imago of Coleoptera, confused under the unique name "pleural suture" the true thoracic pleural furrow and the abdominal dorsopleural furrow. Snodgrass (1931, p. 11) subsequently corrected this error of homonomy for the imago, but he did not correct it for the larva as can be seen from his figures cited above. This misinterpretation of larval thoracic segments led to a consecutive error in abdominal segments by serial comparison. The understanding of this repeated error now makes it possible to correct the general interpretation of the abdominal epipleurites (abdominal laterotergites sensu Snodgrass 1935a) by giving them an identity more coherent with the hexapodan segment model according to the general subcoxal theory. In his fundamental work on the skeletal morphology of a scolytid beetle, [START_REF] Hopkins | Contributions toward a monograph of the scolytid beetles. I. The genus Dendroctonus[END_REF] placed great importance on the longitudinal fold which, in the abdominal pleural region, separates a dorsal epipleurite and a ventral hypopleurite. As mentioned above, [START_REF] Snodgrass | Morphology and mechanism of the insect thorax[END_REF][START_REF] Snodgrass | Morphology of the insect abdomen. Part I. General structure of the abdomen and its appendages[END_REF]Snodgrass ( , 1935a) ) named this fold the "dorsopleural line" and for this reason considered the sclerites located above this line as tergal and those below it as pleural (subcoxal). Thus the "epipleurites" of [START_REF] Hopkins | Contributions toward a monograph of the scolytid beetles. I. The genus Dendroctonus[END_REF] and [START_REF] Böving | An illustrated synopsis of the principal larval forms of the order Coleoptera[END_REF] became "paratergites" or "laterotergites". Returning to the conception of Hopkins, I adopted the term epipleurite and named the dorsopleural line of Snodgrass the "longipleural furrow" [START_REF] Deuve | Sur la présence d'un "épipleurite" dans le plan de base du segment des Hexapodes[END_REF]). It should be noted, however, that the epipleurite of Hopkins described for the imago of Dendroctonus (Scolytidae) is located around the spiracle and not below it, as in the campodeiform larva of Carabus (Carabidae) that served as my model. The two territories are not exactly homologous and [START_REF] Böving | On the abdominal structure of certain beetle larvae of the campodeiform type. A study of the relation between the structure of the integument and the muscles[END_REF] made a useful distinction between the abdominal "pleural suture" in the sense of Hopkins and an "antipleural suture" located above it. This antipleural furrow was named "tergopleural suture" by [START_REF] Craighead | The determination of the abdominal and thoracic areas of the cerambycid larvae as based on a study of the muscles[END_REF], and "dorsolateral suture" by [START_REF] Böving | An illustrated synopsis of the principal larval forms of the order Coleoptera[END_REF], adding to the confusion. I actually used the term epipleurite for all beetle larvae in the classical sense generalised by [START_REF] Böving | An illustrated synopsis of the principal larval forms of the order Coleoptera[END_REF], as well as by [START_REF] Snodgrass | Morphology of the insect abdomen. Part I. General structure of the abdomen and its appendages[END_REF]Snodgrass ( , 1935a) ) under the name "paratergite" or "laterotergite". The use of the term epipleurite continues today in most works on larval morphology, but not without maintaining some confusion. As will be seen below, the meaning of this term may refer, depending on the case, to subcoxal or to subcoxal + precoxal territories. In my previous paper (Deuve 2001a), presented at a symposium on the origin of the Hexapoda held in Paris in January 1999, I hesitated whether to interpret the "epipleural field" as a precoxal (appendicular) or tergopleural (not appendicular) territory, finally opting for the second hypothesis, but only after expressing my doubts. In a clumsy way, I gave too much importance to this uncertain choice, including in the title, which had the effect of misleading the reader and diverting attention from the best-supported part of my demonstration, i.e. the presence of some peculiar sclerites in the groundplan of the pterygote abdominal segments and their key involvement in the functioning of the external female genitalia after subcoxosternal plate internalisation. The denomination and interpretation of these sclerites ("epipleurites, hypopleurites, tergopleurites, laterocoxites, subcoxae, precoxae") are secondary but not minor issues. The results of [START_REF] Kobayashi | Embryonic development of Carabus insulicola (Insecta, Coleoptera, Carabidae) with special reference to external morphology and tangible evidence for the subcoxal theory[END_REF] show that the epipleurites have a subcoxal identity and therefore a readjustment must be made. But this does not change the broad lines of my previous study, as will be discussed below. If it is considered that there also exists a longitudinal field in the fundamental organisation of the arthropod metamere that was formerly appendicular but not strictly part of the present arthropodal appendage, and then became located between the primary tergum and the base of the limb and possibly including the paranotal lobe, [START_REF] Prell | Das Chitinskelett von Eosentomon[END_REF] terminology could be used and the term "tergopleural field" should be applied instead of the inappropriate "epipleural field". There are also, especially in certain Chilopoda, some sclerites named "dorsal sclerites" by [START_REF] Bäcker | A forgotten homology supporting the monophyly of Tracheata: the subcoxa of insects and myriapods re-visited[END_REF], located above-and not belonging to-the limb, which I marginally included in my former "epipleural field" concept. They must not now be confused with the epipleurites and, as far as we know, they would be paranotal and/or tergopleural in nature. However, these sclerites deserve special attention in reference to their possible relationship with precoxal extensions (we have in mind the origin of insect wings). In addition, Bäcker et al. point out that species showing these dorsal sclerites are often burrowing forms or species of the edaphon, a milieu that requires functional adaptations of the skeleton with losses (i.e. transformations) of the projecting paranotal lobes. The subcoxal theory The idea that a proximal segment of the appendage, named subcoxa by [START_REF] Heymons | Beiträge zur Morphologie und Entwicklungsgeschichte der Rhynchoten[END_REF], would make up most, if not all, of the pleural and ventral regions, is called the "subcoxal theory". Following the embryological observations of Heymons on Heteroptera, [START_REF] Börner | Die Gliedmassen der Arthropoden[END_REF] promoted and extended this idea to many arthropods, distinguishing (p. 690) between subcoxae that are free, pleural, or integrated into a subcoxosternum. He wrote very explicitly: "It is not uncommon to observe a more or less deep fusion of the subcoxa with the sternum or tergum, as is the case with many crustaceans and hexapods, whereas a fusion with the coxa seems to have occurred only rarely" (p. 656, translated from German). [START_REF] Snodgrass | Morphology and mechanism of the insect thorax[END_REF] and [START_REF] Weber | Die Gliederung der Sternopleuralregion des Lepidopterenthorax. Eine vergleichende morphologische Studie zur Subcoxaltheorie[END_REF] later discussed, employed and developed this theory. In insects, the subcoxal theory could only be really applied to the thoracic segments, where the limbs are particularly developed as legs. Snodgrass described in pterygotes a pleural wall, of a subcoxal nature, split dorsoventrally by an oblique fold called pleural furrow or pleural sulcus. This fold serves for muscular attachments, reinforced in the thoracic tagma whose locomotory function is predominant. The presence of meso-and metathoracic wings further reinforces the mechanical importance of these parietal pleura. However, if the subcoxa is a proximal segment of the limb, it should surround the base of the more distal coxa and therefore, if embedded, it would also have some anterior, posterior and ventral components. [START_REF] Börner | Die Gliedmassen der Arthropoden[END_REF], [START_REF] Weber | Die Gliederung der Sternopleuralregion des Lepidopterenthorax. Eine vergleichende morphologische Studie zur Subcoxaltheorie[END_REF][START_REF] Weber | Morphologie, Histologie und Entwicklungsgeschichte der Articulaten[END_REF] and [START_REF] Ferris | Some general considerations[END_REF]Ferris ( , 1940a) ) have shown that the subcoxa is ventrally associated or integrated with the sternum in formations whose interpretation is complex and sometimes rather speculative. Recent observations of embryonic development have confirmed the preponderant role of some ventral subcoxal formations (particularly the basisternum) in the formation of the secondary sternum, which is in fact a subcoxosternum (e.g. [START_REF] Uchifune | Embryonic development of Galloisiana yuasai Asahina, with special reference to external morphology (Insecta: Grylloblattodea)[END_REF]. In addition, developmental genetic techniques have definitively confirmed the existence of the subcoxa as a proximal segment of the appendage of the Hexapoda, not only in the thorax, but also in the cephalic segments [START_REF] Coulcher | Molecular developmental evidence for a subcoxal origin of pleurites in insects and identity of the subcoxa in the gnathal appendages[END_REF]. Certainly, this theory may be extended to more or less all segments, including those of the abdomen. Despite the bitter opposition of [START_REF] Hansen | Studies on Arthropoda II[END_REF], the subcoxal theory prevailed in the following decades, before [START_REF] Snodgrass | A textbook of arthropod anatomy[END_REF][START_REF] Snodgrass | Evolution of arthropod mechanisms[END_REF][START_REF] Snodgrass | A contribution toward an encyclopedia of insect anatomy[END_REF] changed his own mind, and [START_REF] Bekker | Evolution of the leg in Tracheata. Part 1. Subcoxal theory and a critique of it[END_REF], [START_REF] Sharov | Basic arthropodan stock[END_REF] and [START_REF] Manton | The evolution of arthropodan locomotory mechanisms. Part 10. Locomotory habits, morphology and evolution of the hexapod classes[END_REF] criticised it in quite different ways. [START_REF] Bekker | Evolution of the leg in Tracheata. Part 1. Subcoxal theory and a critique of it[END_REF] criticised Heymons' observations and Börner's old interpretations of the articulation of the limb on the pleuron, but he seems to have been unaware of the fundamental works of Snodgrass and Weber, not citing any bibliographical reference after 1913. For [START_REF] Sharov | Basic arthropodan stock[END_REF], the pleuron of the pterygotes was essentially precoxal in nature ("pleuron" or "basal joint corresponding to the crustacean precoxopodite", p. 186-187, see also the present Figure 2), the true subcoxa being reduced to the trochantin (the latter would then be a genuine segment-the subcoxa-and not a fragment of the coxa as Snodgrass claimed in his subcoxal theory). [START_REF] Bäcker | A forgotten homology supporting the monophyly of Tracheata: the subcoxa of insects and myriapods re-visited[END_REF] adopted the broad lines of this interpretation. In his final model, [START_REF] Snodgrass | A textbook of arthropod anatomy[END_REF][START_REF] Snodgrass | Evolution of arthropod mechanisms[END_REF][START_REF] Snodgrass | A contribution toward an encyclopedia of insect anatomy[END_REF] considered the whole pleural region of arthropods as resulting from differential sclerotisation of the body-wall or from fragmentation of the coxa. In doing so, he totally abandoned his former subcoxal theory. [START_REF] Manton | The evolution of arthropodan locomotory mechanisms. Part 10. Locomotory habits, morphology and evolution of the hexapod classes[END_REF], who gave more importance to functional aspects than to problems of identity and homology, followed Snodgrass' later conceptions and considered the pleurites of arthropods as so many adaptations to different modes of life. For the latter author, there is no subcoxa and it is vain to try to establish such homologies. Today, however, there seems to be a consensus in favour of the subcoxal theory. It implies the existence of a segment more proximal than the coxa, named the subcoxa. Dorsally, located between the coxa and the tergum, the subcoxa is divided in pterygote insects by the pleural furrow into an anterior episternite and a posterior epimerite. The subcoxa should not be confused with the trochantin, a basal sclerite of the appendage known for long time, which according to Snodgrass, as well as [START_REF] Carpentier | Quelques remarques concernant la morphologie thoracique des Collemboles (Aptérygotes)[END_REF], would be genuinely coxal in nature (with serious arguments, Sharov 1966, as well as [START_REF] Bäcker | A forgotten homology supporting the monophyly of Tracheata: the subcoxa of insects and myriapods re-visited[END_REF], did not accept this interpretation). Some collembolans, such as Tetrodontophora and Orchesiella species, are known to have two well-defined segments, more proximal than the coxa, interpreted as precoxa (or "pretrochantin") and subcoxa ("trochantin") [START_REF] Hansen | Studies on Arthropoda II[END_REF], also sometimes referred to respectively as "subcoxa 1" and "subcoxa 2" [START_REF] Denis | Sous-classe des Aptérygotes[END_REF][START_REF] Deharveng | Morphologie évolutive des Collemboles Neanurinae en particulier de la lignée néanurienne[END_REF]. These two segments have also been observed as buds during the embryonic development [START_REF] Bretfeld | Zur anatomie und Embryologie der Rumpfmuskulatur und der abdominalen Anhänge der Collembolen[END_REF]. More generally, these are two arciform sclerites that can be observed in some collembolans and that [START_REF] Willem | Recherches sur les Collemboles et les Thysanoures[END_REF] has considered as true "precoxal segments", i.e. true segments more proximal than the coxa. [START_REF] Hansen | Studies on Arthropoda II[END_REF] clearly described these two segments in Nicoletia, an exobasal zygentom (note that only an ancestor or a node in phylogenetical tree may be said 'basal'; a basally branched extant clade is 'exobasal'). This observation was taken up by [START_REF] Sharov | Basic arthropodan stock[END_REF], who observed these two segments in both Nicoletia and Tricholepidon. As [START_REF] Barlet | Le thorax des Japygides[END_REF] reported, these two segments were also observed in the Diplura as supracoxal rings, while in the Protura they have also been described as "subcoxa 1" and "subcoxa 2" by [START_REF] Denis | Sous-classe des Aptérygotes[END_REF]. These supracoxal arches, named anapleurite (subcoxa 1) and catapleurite (subcoxa 2) by Barlet and Carpentier, are separated from each other by the "paracoxal fold" [START_REF] Matsuda | Morphology and evolution of the insect thorax[END_REF]. [START_REF] Bäcker | A forgotten homology supporting the monophyly of Tracheata: the subcoxa of insects and myriapods re-visited[END_REF] interpreted their presence as forming part of the hexapodan groundplan and named them respectively "eupleurite" and "trochantinopleurite". We could also use Hansen's terminology ("pretrochantin" and "trochantin") or, to make matters even easier and more definitive, the terms precoxa and subcoxa. This problem of terminology may seem a secondary question, but it has crucial theoretical implications. Although Bäcker et al.'s argumentation against the terminology employed in the works of Barlet and Carpentier, as well as those of [START_REF] Snodgrass | Evolution of arthropod mechanisms[END_REF][START_REF] Snodgrass | A contribution toward an encyclopedia of insect anatomy[END_REF], is relevant considering their interpretation, it should be noted that the terms anepimeron, anepisternum, catepimeron and catepisternum had been proposed long before. They were introduced by [START_REF] Crampton | A contribution to the comparative morphology of the thoracic sclerites of insects[END_REF] for the pterygote thorax. Later, the terms anepisternite, anepimerite, catepisternite and catepimerite were widely used by Ferris and his school-of which Matsuda was a young member-to describe the subdivisions of the so-called subcoxa of the pterygotes (e.g. [START_REF] Rees | The morphology of Tipula reesi Alexander (Diptera: Tipulidae)[END_REF]Ferris 1940b) and it is this model that was later adopted and developed by Matsuda. The anapleurite and catapleurite were meticulously studied by Carpentier and Barlet, who clearly observed them in Collembola, Archaeognatha, Zygentoma and Diplura [START_REF] Carpentier | Sur la valeur morphologique des pleurites du thorax des Machilides (Thysanoures)[END_REF][START_REF] Carpentier | Quelques remarques concernant la morphologie thoracique des Collemboles (Aptérygotes)[END_REF][START_REF] Barlet | Remarques sur la musculature thoracique des Machilides (Insectes Thysanoures)[END_REF][START_REF] Barlet | La question des pièces pleurales du thorax des Machilides (Thysanoures)[END_REF][START_REF] Carpentier | Les sclérites pleuraux du thorax de Campodea (Insectes, Aptérygotes)[END_REF][START_REF] Barlet | Le thorax des Japygides[END_REF]). In addition, [START_REF] François | Le squelette thoracique des Protoures[END_REF][START_REF] François | Squelette et musculature thoraciques des Protoures[END_REF] described them accurately in the Protura (Figure 1). This arrangement has also been reported in most orders of pterygote insects by many authors. In this latter clade the pleural fold distinctly separates the anepisternite from the anepimerite and the catepisternite from the catepimerite. [START_REF] Matsuda | Morphology and evolution of the insect thorax[END_REF][START_REF] Matsuda | Morphology and evolution of the insect abdomen[END_REF] gave an excellent theoretical synthesis of the main works on the fundamental pattern of the thorax of hexapods, following the subcoxal theory. In a previous paper, I retained homologies between the anapleuron and the precoxa, and between the catapleuron and the subcoxa (Deuve 2001a). In this model, two segments more proximal than the coxa are then present at the base of the hexapodan appendage. More recently, [START_REF] Kobayashi | Formation of subcoxae-1 and 2 in insect embryos: the subcoxal theory revisited[END_REF] showed that the subdivision of the subcoxa into two segments can be observed during the embryonic development of Coleoptera, Megaloptera, Neuropterida and Trichoptera. In hemimetabolan orders, the differentiation of the paracoxal fold is more difficult to detect (e.g. [START_REF] Mashimo | Embryological evidence substantiates the subcoxal theory on the origin of pleuron in insects[END_REF]), but it has been observed at the imaginal stage in several orders [START_REF] Duporte | The lateral and ventral sclerites of the insect thorax[END_REF] and confirmed for the embryonic stage by Y. Kobayashi (personal communication). However, partly taking into account the remarks of [START_REF] Mashimo | Embryological evidence substantiates the subcoxal theory on the origin of pleuron in insects[END_REF], Y. Kobayashi (personal communication) considers that both the anapleural and catapleural rings are subdivisions of the "subcoxa 2", and thus correspond to the true subcoxa, whereas the "subcoxa 1", as observed in several insect embryos, is a true proximalmost segment of the limb, located at the border with the tergum or paranotum. This model of three basal segments of the hexapodan appendage (i.e. from most proximal to most distal: precoxa, subcoxa and coxa) fits perfectly with the theoretical model of the interpretation of the crustacean protopodite, which would also be three-segmented (from proximal to distal: precoxa, coxa and basis), bearing in mind that the crustacean coxa is homologous with the hexapodan subcoxa and the basis is homologous with the hexapodan coxa [START_REF] Hansen | Zur Morphologie der Gliedmassen und Mundtheile bei Crustaceen und Insekten[END_REF] (Figure 2). Hansen (1893, p. 198) was the first author to describe in Argulus a trisegmented protopodite for the biramous swimming limb of Maxillopoda. In addition, the precoxa seems to belong to both the crustacean groundplan [START_REF] Hansen | Studies on Arthropoda II[END_REF][START_REF] Boxshall | The evolution of arthropod limbs[END_REF]) and the hexapodan groundplan [START_REF] Sharov | Basic arthropodan stock[END_REF][START_REF] Boxshall | The evolution of arthropod limbs[END_REF]. [START_REF] Boxshall | Comparative limb morphology in major crustacean groups: the coxa-basis joint in postmandibular limbs[END_REF] reported it in Remipedia and Maxillopoda. It was long considered to be present in Malacostraca (e.g. [START_REF] Hansen | Studies on Arthropoda II[END_REF]) and in Stomatopoda [START_REF] Balss | Stomatopoda[END_REF]. [START_REF] Vandel | Embranchement des Arthropodes (Arthropoda, Siebold et Stannius 1845). Généralités. Composition de l'Embranchement[END_REF] pointed out that this trisegmentation is not primitive, but derived from secondary divisions of a primordial coxa. Recent studies of embryonic development confirm this view, but do not call into question the fundamentally trisegmented nature of the protopodite, probably common to all arthropods, although this hypothesis is still disputable [START_REF] Bitsch | The hexapod appendage: basic structure, development and origin[END_REF]. In a previous work (Deuve 1994, p. 203-204), I was already speculating that the paranotal lobes of the Arthropoda were pleural in nature and I compared those of the campodeiform larvae of certain Coleoptera to those of amphipods or isopods such as Saduria entomon. In the same vein, Y. Kobayashi (personal communication) recently drew my attention to the coxal plates of some amphipods and isopods, whose embryonic development has been accurately studied. In the late embryo of Orchestia (Pericarida), these plates are flap-like and mobile along the margin of the tergum, and they seem to form part of the limb [START_REF] Ungerer | External morphology of limb development in the amphipod Orchestia cavimana (Crustacea, Malacostraca, Peracarida)[END_REF]. In Porcellio (Oniscidea), they are still distinct but merge with the tergum and retain a shape like a 'paranotal lobe' [START_REF] Wolff | The embryonic development of the malacostracan crustacean Porcellio scaber (Isopoda, Oniscidea)[END_REF]. In the light of these different morphologies observed in crustaceans, the narrow anatomical relations between the precoxa and the paranotal lobe will have to be more accurately analysed in Hexapoda in order to understand the tergopleural territories and their respective links with the base of the limb and the true tergum. The pleural regions of Myriapoda and Hexapoda have often been compared (e.g. Füller 1963a,b) and they offer evident similarities, related to terrestrial life, that might be homologies. In particular, two concentric sclerite rings at the base of the locomotory limb seem to belong to the myriapodal groundplan and would correspond to the two proximal true-segments, which [START_REF] Bäcker | A forgotten homology supporting the monophyly of Tracheata: the subcoxa of insects and myriapods re-visited[END_REF] named eupleurite and trochantinopleurite, partly following [START_REF] Crampton | Notes on the thoracic sclerites of winged insects[END_REF] terminology and [START_REF] Weber | Lehrbuch der Entomologie[END_REF] view. In the same vein, [START_REF] Wesener | Sternites and spiracles -The unclear homology of ventral sclerites in the basal millipede order Glomeridesmida (Myriapoda, Diplopoda)[END_REF] pointed out that in Diplopoda the spiracle-bearing plates previously interpreted as sternites were not of a sternal nature, but were instead subcoxal sclerites associated with the apparent limb base. [START_REF] Bäcker | A forgotten homology supporting the monophyly of Tracheata: the subcoxa of insects and myriapods re-visited[END_REF] gave a reasoned description of a possible evolution of pleural structures in Myriapoda and Hexapoda. The subcoxal theory applied to abdominal segments Because the abdomen of the Hexapoda does not bear multisegmented locomotory limbs, it is more difficult to interpret its skeletal pattern. Few studies refer to the presence of the subcoxa and precoxa on this tagma. However, ventral sclerites (= ventrites) are classically interpreted as 'coxosternites', referring to the integration of some eupleural elements merged with the genuine sternum, and some abdominal pleurites have often been described, more or less associated with the spiracles (see e.g. the pleurites in Figure 11, named "epipleurites"). Bitsch (1994, p. 120) rightly interpreted the pleural sclerites of the abdomen as "arising from a basal appendicular segment (subcoxa) secondarily incorporated into the body wall". In this interpretation, we can stress a fundamental dissociation between the ventral component of the subcoxa, which is part of the coxosternum, and the more isolated dorsal component, which is embedded into the more laterodorsal body-wall and is often named 'pleurite'. A distinction is also made between the pregenital abdomen (segments I-VII), the appendages of which are rudimentary or absent, and the genital/post-genital abdomen (segments VIII-XI), some segments of which may carry visible appendages in the form of complex external genitalia (segments VIII and IX of female, segment X of male) and cerci [START_REF] Bitsch | Morphologie abdominale des Insectes[END_REF]. In particular, the genital segments often carry a pair of gonopods, which are sometimes connected to the tergum by an articular pleurite. The interpretation of these structures is easier in females, whose ovipositor apparatus has sometimes retained its original metameric arrangement and its original connections with the tergum, which is not the case in males. Recently, embryological studies have made it possible to identify the first buds of limbs and hence determine homonomies between thoracic and abdominal segments by serial comparison. Thus, [START_REF] Komatsu | Embryonic development of a whirligig beetle, Dineutus mellyi, with special reference to external morphology (Insecta: Coleoptera, Gyrinidae)[END_REF] and [START_REF] Kobayashi | Embryonic development of Carabus insulicola (Insecta, Coleoptera, Carabidae) with special reference to external morphology and tangible evidence for the subcoxal theory[END_REF] have identified in Coleoptera the "subcoxa 1" and "subcoxa 2", which can respectively be interpreted as the precoxa and subcoxa, more dorsal than the coxa, on the thoracic segments and abdominal segments I-VIII (Figure 3). These authors correctly presented their results as a contribution to the subcoxal theory (see also [START_REF] Kobayashi | Formation of subcoxae-1 and 2 in insect embryos: the subcoxal theory revisited[END_REF]. [START_REF] Kobayashi | Embryonic development of Carabus insulicola (Insecta, Coleoptera, Carabidae) with special reference to external morphology and tangible evidence for the subcoxal theory[END_REF] demonstrated that the epipleurite of the campodeiform Carabus larva corresponds to the subcoxa (subcoxa 2), whereas the juxta-or peri-spiracular area corresponds to the precoxa (subcoxa 1). In so doing, they showed that the 'epipleural field' I had identified and illustrated (Deuve 2001a) is actually a subcoxal territory (or, in some cases, subcoxal + precoxal). I fully agree with this interpretation, which makes my scheme of the hexapodan abdominal segments more coherent. Just as the dorsal part of both subcoxa and precoxa are integrated into the lateral wall of the thoracic segments, with a function of fixation of the appendage to the body, the epipleurite is also integrated into the wall of the abdominal segments, but it remains distinct from the coxosternum (with which it may, however, merge in some cases) on the pregenital segments, and it has the same function of fixation of the appendage on the genital segments. By homonomy, these territories have the same identity, as well as the same functional specialisation. In general, the skeletal organisation of the thoracic and abdominal segments is thus homogenous. A study of the cephalic segments will probably show a similar pattern, with dorsal precoxal and subcoxal areas embedded into the body-wall. Precoxa and subcoxa tend to lose their appendicular morphology, but remain functionally associated with the protruding part of the limb, for which they provide anchoring, support and articulation. In addition, we must keep in mind that if epipleural areas are subcoxal in their nature, they are truly appendicular and must therefore have a corresponding ventral part that is associated with the formation of the secondary sternum. This is precisely what [START_REF] Kobayashi | Embryonic development of Carabus insulicola (Insecta, Coleoptera, Carabidae) with special reference to external morphology and tangible evidence for the subcoxal theory[END_REF] observed in the embryo. In my 2001 paper, I showed that the epipleural areas participate in specialisation of the genital segments by serving as a substitute for the internalised subcoxosternum (formation of the vaginal ducts) through a medio-ventrad shift. This process of ventral closure of the genital segments, starting from the primitive seripleural type, leads to the formation of a sympleural abdomen (the epipleurites, of a subcoxal nature, join and merge ventrally) (Figure 4). In a way, this is the formation of a 'tertiary sternum'. In the 'sympleural-I' type the eighth epipleurites merge and participate in the formation of a 'subgenital plate', while gonopods VIII move rearwards to form a functional ovipositor in association with gonopods IX. In the 'sympleural-II' type, the eighth epipleurites form the first subgenital plate, but the ninth epipleurites also merge together to form a second subgenital plate, while the gonopods are no longer visible. The resulting secondary gonopore is then displaced to the rear of segment IX. The primitive (plesiomorphic) seripleural type is present in Archaeognatha, Zygentoma, Dictyoptera and Dermaptera, as well as some exceptional coleopterans belonging to the genus Eustra, which have retained a neotenic abdominal structure [START_REF] Deuve | L'abdomen et les genitalia des femelles de Coléoptères Adephaga[END_REF](Deuve , 2001b) ) (Figure 5). The sympleural-I type is present in Odonata, orthopteroids, Hymenoptera, Neuropterida, Coleoptera and Diptera; the sympleural-II type occurs in Trichoptera, Lepidoptera and Mecoptera (Deuve 2001a). However, it should be noted that the condition in Mecoptera is peculiar. The most plesiomorphic type of external genitalia occurs in Nannochoristidae, apparently corresponding to the sympleural type II, but the female gonopore is located at the rear of segment VIII rather than at the rear of segment IX [START_REF] Mickoleit | Die Genital-und Postgenital Segmente der Mecoptera-Weibchen. I. Das Exoskelet[END_REF]). In the other families, an apparent second subgenital plate is produced by elements that would belong to the eighth segment ("coxosternites VIII") but are apparently located on segment IX, while the true epipleurites IX ("coxosternites IX") regress and finally disappear [START_REF] Mickoleit | Die Genital-und Postgenital Segmente der Mecoptera-Weibchen. I. Das Exoskelet[END_REF]. [START_REF] Bitsch | Morphologie abdominale des Insectes[END_REF], however, did not adopt this interpretation. In addition, a "genital chamber" is formed at the level of the segments VIII and IX, which contains vestiges of the gonopods on its inner surface [START_REF] Grell | Der Genitalapparat von Panorpa communis L. Zoologische Jahrbücher[END_REF], as discussed and critised by [START_REF] Mickoleit | Die Genital-und Postgenitalsegmente der Mecoptera-Weibchen (Insecta, Holometabola). II. Das Dach der Genitalkammer[END_REF]. The best argument in favour of this seripleural and sympleural-type model is that there are no combined structures: the seripleural type, with retained metamerism, precludes the presence of a genuine subgenital plate when the vaginal duct is formed; the sympleural-II type, with a second subgenital plate, is incompatible with the presence of a true ovipositor. However, a problematic aspect of my interpretation was the structure of the genital segments in some females of Mecoptera. [START_REF] Mickoleit | Die Genital-und Postgenital Segmente der Mecoptera-Weibchen. I. Das Exoskelet[END_REF] recognised the absence of the genuine sternites VIII and IX as a result of the morphogenesis of the genital chamber. He also described the presence of a small "laterotergite" lying near the spiracle of the pregenital segments, but also on segment VIII, where it is adjacent to the subgenital plate. [START_REF] Kristensen | Heterobathmia valvifer n. sp.: a moth with large apparent 'ovipositor valves' (Lepidoptera: Heterobathmiidae)[END_REF] considered this to be an argument against my interpretation of the epipleural nature of the ventral sclerites VIII. The same difficulty arises with Embioptera, which also have "laterotergites" in addition to the epipleurites [START_REF] Klass | The female genitalic region and gonoducts of Embioptera (Insecta), with general discussions on female genitalia in insects[END_REF]. However, as I have previously explained (Deuve 2001a), the 'laterotergite' of some Mecoptera is located beside the spiracle and not below it, as is usually the case for an epipleurite. In fact, if we consider that the epipleurite represents the dorsal part of the subcoxa, it is likely that the small socalled 'laterotergite' of segment VIII in some Mecoptera would be precoxal, ocupying same juxtaspiracular location as this sclerite in the larva of Carabus [START_REF] Kobayashi | Embryonic development of Carabus insulicola (Insecta, Coleoptera, Carabidae) with special reference to external morphology and tangible evidence for the subcoxal theory[END_REF]. A similar situation is found in Neuropterida. [START_REF] Liu | Homology of the genital sclerites of Megaloptera (Insecta: Neuropterida) and their phylogenetic relevance[END_REF] admitted that "the predominant sternite-like sclerite of the female abdominal segment VIII represents the fused gonocoxites VIII" (it is actually formed from the merged epipleurites VIII). However, this subgenital plate is separated from the tergite by a fairly large membranous area containing the spiracle (Liu et al. 2016, "Fig. 12"). This area would represent a large part of the dorsal precoxal territory. The same situation can be observed in Coleoptera, but the dorsal part of the subcoxa (epipleurite) and of the precoxa may merge together, or merge with the tergum and even with the coxosternum or with the subcoxosternum such as in segment VIII of certain Scarabaeoidea [START_REF] Ritcher | Spiracles of adult Scarabaeoidea (Coleoptera) and their phylogenetical significance. 1. The abdominal spiracles[END_REF], or on the contrary become subdivided to allow increased mobility of the pregenital abdomen, as in Staphylinidae [START_REF] Naomi | The lateral sclerites of the pregenital abdominal segments in Coleoptera (Arthropoda: Hexapoda)[END_REF]). It should be noted that a sclerification does not necessarily follow the limits of the different anatomical fields. For example, among beetles, the epipleurite is located below the spiracle in Carabidae larvae, but the spiracle lies in the centre of a large sclerite (epipleurite + precoxite?) in certain larvae of Lampyridae (Deuve 2001a, "Fig. 6 and 7"). It is also well known that spiracles may move, some may disappear and yet others may appear, in the evolution of hexapods, especially in the most exobasal orders, such as the Diplura (see discussion in [START_REF] Kristensen | The groundplan and basal diversification of the hexapods[END_REF]in Klass &[START_REF] Klass | The ground plan and affinities of hexapods: recent progress and open problems[END_REF]). Their location is certainly less variable in the pterygotes, where the location of the spiracular line can reasonably be used as landmark. [START_REF] Ritcher | Spiracles of adult Scarabaeoidea (Coleoptera) and their phylogenetical significance. 1. The abdominal spiracles[END_REF] documented the migration of abdominal spiracles in some Scarabaeoidea (Coleoptera), but it is mainly the areas of sclerification of the abdominal wall that differ, depending on whether they are covered and protected by elytra or, in contrast, exposed to the environment. As Snodgrass (1935b, "Fig. 2") clearly illustrated, the location of the spiracular line can appear to be variable in different orthopteran families, but it is actually the areas of sclerification of the abdominal segments that have moved: in Tettigidea and Rhipipteryx the spiracle lies in the pleural membrane, whereas in Melanoplus it is located on the lateral margin of the so-called 'tergite', which apparently includes tergum + merged precoxae. Recently, [START_REF] Mashimo | Embryological evidence substantiates the subcoxal theory on the origin of pleuron in insects[END_REF] showed that in Gryllus bimaculatus the subcoxa remains undivided during embryonic development and the spiracle appears on the lateral margin of the so-called tergum. Thus, the precoxa cannot be seen-there is only a 'paratergal furrow', which is a simple, not very evident linear depression that delimits the paranotal lobe bearing the spiracle. It is questionable whether the differences between the observations of [START_REF] Kobayashi | Embryonic development of Carabus insulicola (Insecta, Coleoptera, Carabidae) with special reference to external morphology and tangible evidence for the subcoxal theory[END_REF] of the Carabus embryo and those of [START_REF] Mashimo | Embryological evidence substantiates the subcoxal theory on the origin of pleuron in insects[END_REF] in a Gryllus embryo are due to artefacts produced by the techniques used, or whether they are instead real differences between the organisms. In fact, the buds and territories observed on a given embryo are already partly dependent on the future morphology of the imago, being precursors of the latter. That is a very general rule in any development process. In many orthopterans, spiracles are located on the apparent tergite (e.g. see above concerning the abdominal segments in Melanoplus). In this case, it is not unexpected that the same structure can be observed at an earlier embryonic stage of development. Alternatively, one might try to retain the hypothesis of [START_REF] Mashimo | Embryological evidence substantiates the subcoxal theory on the origin of pleuron in insects[END_REF] and consider a tergal (or, more exactly, tergopleural) origin of the spiracle-bearing territory. In this case, the "subcoxa 1" described by [START_REF] Kobayashi | Embryonic development of Carabus insulicola (Insecta, Coleoptera, Carabidae) with special reference to external morphology and tangible evidence for the subcoxal theory[END_REF] in Carabus, apparently homologous with the 'tergal' territory observed in a Gryllus embryo, would also be of a tergopleural nature and could not be interpreted as the precoxa. The problem with this hypothesis is that it contradicts the formation of this "subcoxa 1" through a division of the primordial subcoxa observed by [START_REF] Kobayashi | Embryonic development of Carabus insulicola (Insecta, Coleoptera, Carabidae) with special reference to external morphology and tangible evidence for the subcoxal theory[END_REF] during the embryonic development. It must be recognised that the problem of anatomical relationships between the spiracle and the lateral margins of the 'tergum' in hexapods is not yet resolved. For example, the anatomical studies of [START_REF] François | Squelette et musculature thoraciques des Protoures[END_REF] on proturans clearly indicate the presence of spiracle much more dorsal than the anapleurite and located on the lateral margin of the so-called tergum (Figure 1). Another example amongst many others is that of the embryo of Pedetontus (Archaeognatha), in which the spiracles appear to be located on the margins of the so-called tergum [START_REF] Machida | External features of embryonic development of a jumping bristletail, Pedetontus unimaculatus Machida (Insecta, Thysanura, Machilidae)[END_REF]. The informative work of [START_REF] Niwa | Evolutionary origin of the insect wing via integration of two developmental modules[END_REF] on the latter illustrates the complex genetic relationships controlling the morphogenesis of both paratergal and limb territories. In papers on Hemiptera, Sweet noted the presence of a double row of abdominal sclerites, which he named "laterotergites" [START_REF] Sweet | The external morphology of the pre-genital abdomen and its evolutionary significance in the order Hemiptera (Insecta)[END_REF], more or less following the terminology of [START_REF] Dupuis | Données nouvelles sur la morphologie abdominale des Hémiptères Hétéroptères et en particulier des Pentatomoides[END_REF][START_REF] Dupuis | Les Rhopalidae de la faune française (Hemiptera, Heteroptera). Caractères généraux -Tableaux de détermination -Données monographiques sommaires[END_REF]. Later, he interpreted them as genuine pleurites, which he called hypopleurites and epipleurites [START_REF] Sweet | Comparative external morphology of the pregenital abdomen of the Hemiptera[END_REF], using the terminology of Hopkins (1909) also adopted by [START_REF] Deuve | Sur la présence d'un "épipleurite" dans le plan de base du segment des Hexapodes[END_REF]. According to Sweet, these pleurites would form part of the fundamental organisation of the insect abdomen. It should be noted that in Heteroptera, it is the 'hypopleurite' in Sweet's sense, not the epipleurite, that bears the spiracle. This in turn raises new questions. The fundamental pattern of insect abdomen has been illustrated by diagrams that explain the specialisation of the female genital segments with internalisation of the subcoxosternal plates (Deuve 2001a). These diagrams are reproduced here with slight modifications. Figure 4 illustrates the three abdominal types occurring in female insects: seripleural, sympleural-I and sympleural-II. A precoxite (i.e. precoxal sclerite) is figured next to the spiracle. In my first attempt to introduce a particular sclerite, the "epipleurite", in the groundplan of the hexapodan segment, I mentioned the following homology: "to be cited as an example among other structures, the 'gonangulum' of [START_REF] Scudder | Reinterpretation of some basal structures in the insect ovipositor[END_REF], which would correspond to the epipleurite of the IXth abdominal segment" (Deuve 1994, p. 203, translated from French). Indeed, the interpretative model proposed there makes it possible to understand the nature of the gonangulum, which corresponds to the epipleurite IX [START_REF] Deuve | Sur la présence d'un "épipleurite" dans le plan de base du segment des Hexapodes[END_REF]2001a, p. 210-211). In sympleural-I type, the gonangulum is a more or less triangular sclerite that has retained its articular function, connecting the ovipositor (gonopods VIII and IX) to the ninth tergite. In sympleural-II type, the gonangulum does not have this articular function because the epipleurite IX is a component of the second subgenital plate. This explains why [START_REF] Scudder | Comparative morphology of insect genitalia[END_REF] pointed out the so-called 'absence' of a gonangulum in Mecoptera, Diptera, Trichoptera, Lepidoptera and Coleoptera. In the case of Coleoptera, the abdomen of females indeed belongs to the sympleural-I type, but gonopods VIII are reduced and closely associated with gonopods IX, so that there is no typical ovipositor and the epipleurites IX (gonangula) are, in contrast, strongly developed and do not look at all like 'small triangular articular sclerites'. The gonangulum has sometimes been confused by coleopterists with the lateral extremities of the so-called "tergum IX" expansions [START_REF] Bils | Das abdomenende weiblicher terrestrische lebender Adephaga une seine Bedeutung für die Phylogenie[END_REF][START_REF] Burmeister | Der Ovipositor der Hydradephaga (Coleoptera) und seine phylogenetische Bedeutung unter besondere Berücksichtigung der Dytiscidae[END_REF], following the interpretation of [START_REF] Mickoleit | Ueber Ovipositor der Neuropteroidea und Coleoptera und seine phylogenetische Bedeutung[END_REF]. More recently, Hünefeld et al. (2012) interpreted the holometabolan groundplan as lacking the gonangulum, but I do not share this view. Simply, this epipleurite IX no longer has the morphological and functional appearance of a gonangulum as described by Scudder. In a recent and meticulous work on the gonangulum, [START_REF] Klass | The gonangulum: a reassessment of its morphology, homology, and phylogenetic significance[END_REF] have presented an interpretation that is the same in its broad lines as the one I had previously developed [START_REF] Deuve | Sur la présence d'un "épipleurite" dans le plan de base du segment des Hexapodes[END_REF](Deuve , 2001a)), but using the term laterocoxite (Bitsch 1973b[START_REF] Bitsch | Morphologie abdominale des Machilides (Thysanura) -II. Squelette et musculature des segments génitaux femelles[END_REF]) instead of epipleurite. In a study of the blattarian ovipositor, Klass (1998) had rightly compared the structure observed with that known in Archaeognatha, which led him to compare the gonangula, visible in the eighth and ninth segments, with the laterocoxites described by Bitsch in machilids. Later, in successive works on the abdomen of other exobasal hemimetabolan orders, notably Dermaptera, [START_REF] Klass | The female abdomen of the viviparous earwig Hemimerus vosseleri (Insecta : Dermaptera : Hemimeridae), with a discussion of the postgenital abdomen of Insecta[END_REF][START_REF] Klass | The female genitalic region in basal earwigs (Insecta: Dermaptera: Pygidicranidae (s.l.)[END_REF] was progressively led to generalise the term laterocoxite. Bitsch has named "laterocoxite" a small angular scleritesometimes visible in the genital segments of certain Archaeognatha, more rarely in the pregenital segments, and sometimes merged with the coxa-, that was already known as the "subcoxa" [START_REF] Bekker | K stroyeniyu i proiskhozhdeniyu naruzhnykh polovykh pridatkov Thysanura I Hymenoptera [Structure and origin of the external genital appendages of Thysanura and Hymenoptera[END_REF], the "laterosternite" [START_REF] Gustafson | The origin and evolution of the geniatlia of the Insecta[END_REF] or "laterotergite IX" [START_REF] Livingstone | On the morphology and bionomics of Tingis. Duddleidae Drake (Heteroptera: Tingidae). Part III -Functional morphology of the abdomen, male and female genitalia and abdominal scent glands[END_REF], and which [START_REF] Smith | Evolutionary morphology of external insect genitalia. 1. Origin and relationships to other appendages[END_REF] had correctly interpreted as the subcoxa, independently of the older study by [START_REF] Bekker | K stroyeniyu i proiskhozhdeniyu naruzhnykh polovykh pridatkov Thysanura I Hymenoptera [Structure and origin of the external genital appendages of Thysanura and Hymenoptera[END_REF]. Bitsch proposed an identity of this sclerite with the piece named gonangulum by [START_REF] Scudder | Reinterpretation of some basal structures in the insect ovipositor[END_REF]Scudder ( , 1961aScudder ( ,b, 1964[START_REF] Scudder | Comparative morphology of insect genitalia[END_REF]) and, at the same time, he suggested that it has probably a subcoxal nature. I also wrote about the laterocoxite as described in female machilids: "If this 'laterocoxite' corresponds to Scudder's gonangulum, as Bitsch suggested, it would be an epipleurite" (Deuve 2001a, p. 219). Thus, the link is established and we reach a consensus: [START_REF] Gustafson | The origin and evolution of the geniatlia of the Insecta[END_REF] laterosternite (described on the genital segments of female machilids, but misinterpreted because it is non-sternal), [START_REF] Scudder | Reinterpretation of some basal structures in the insect ovipositor[END_REF] gonangulum (described only in segment IX of the abdomen of some female insects), Bitsch's (1973b) laterocoxite (described on the genital segments of female machilids and correctly interpreted) and Hopkins' (1909) epipleurite (described on the abdominal segments of imagos and larvae of both sexes in coleopteran), are one and the same fundamental sclerite of the hexapodan segment, having a subcoxal nature. In this context, it should be noted that [START_REF] Bekker | K stroyeniyu i proiskhozhdeniyu naruzhnykh polovykh pridatkov Thysanura I Hymenoptera [Structure and origin of the external genital appendages of Thysanura and Hymenoptera[END_REF] was the first author to correctly interpret epipleurite IX as the subcoxa in machilids (Archaeognatha), lepismatids and nicoletiids (Zygentoma), as well as in Gryllidea (Bekker 1932a,b). This consensus might be extended further: [START_REF] Kukalová-Peck | Origin of the insect wing articulation from the arthropodan leg[END_REF][START_REF] Kukalová-Peck | New Carboniferous Diplura, Monura, and Thysanura, the hexapod ground plan, and the role of thoracic side lobes in the origin of wings of Insecta[END_REF], 1992, 1997, 2008) proposed an original model of the structure of the insect segment and limb based on study of some Carboniferous fossil hexapods. I published (Deuve 2001a, p. 220-221) a severe critique of this model because, in my opinion, Kukalová-Peck confused the epipleurite with the subcoxa. However, if we now consider that the epipleurite does indeed have a subcoxal nature, then Kukalová-Peck's model would regain part of its relevance. For this author, the hexapodan limb is composed of several segments integrated into the lateral wall, these being (from the most dorsal to the most distal): epicoxa (archipleuron), subcoxa, coxa and trochanter, followed by the multisegmented telopodite. Kukalová-Peck's model, based on compression fossils, the study of which is very delicate, can still be criticised. Shortcomings worth noting are: the subalar and basalar thoracic sclerites are claimed to be subcoxal (1983, p. 1652), the segment ("epicoxa") described as the most proximal is confused with the paranotal lobe (for this reason, the "epicoxa" of Kukalová-Peck, located above the spiracle, is not really synonymous with the precoxa), the subcoxa described by Kukalová-Peck on the abdomen lies near the spiracle and resembles the precoxa, the trochanter is dissociated from the telopodite. In fact, it is clear that in most of her illustrations-notably the diagrammatic representation of the typical segment of a protoinsect (Kukalová-Peck 1983, "Fig. 4") or the representation of a dipluran ("Fig. 7") or a paleodictyopteroid ("Fig. 1")-this author displaces all respective segments of the appendage. This results in the "epicoxa" at the location of the paranotal lobe, the "subcoxa" at the location of the precoxa, the "coxa" at the location of the subcoxa, and the "trochanter" at the location of the coxa. It should also be noted that the term "epicoxite" can hardly be adopted for the precoxa or paranotal lobe, because it has already been used by [START_REF] Becker | Zur morphologischen Bedeutung der Pleuren bei Ateloceraten[END_REF] and Füller (1963a) in a completely different sense: it designates a sclerite of a subcoxal nature in Myriapoda. The more precise study by [START_REF] Kobayashi | Embryonic development of Carabus insulicola (Insecta, Coleoptera, Carabidae) with special reference to external morphology and tangible evidence for the subcoxal theory[END_REF] on a campodeiform Carabus larva demonstrates the location of the precoxa ("subcoxa 1") on the ventral flank of the paranotal lobe next to the spiracle (Figure 6a). This close association of precoxa and paranotal lobe evokes [START_REF] Rasnitsyn | A modified paranotal theory of insect wing origin[END_REF] model of a dual origin-paranotal and appendicular-of the wing of pterygotes (see also [START_REF] Niwa | Evolutionary origin of the insect wing via integration of two developmental modules[END_REF][START_REF] Prokop | Paleozoic nymphal wing pads support dual model of insect wing origins[END_REF][START_REF] Elias-Neto | Tergal and pleural structures contribute to the formation of ectopic prothoracic wings in cockroaches[END_REF][START_REF] Mashimo | Embryological evidence substantiates the subcoxal theory on the origin of pleuron in insects[END_REF]. Kukalová-Peck (1978), inspired by the paper of [START_REF] Wigglesworth | Evolution of insect wings and flight[END_REF], considered the wing as an exite of her "epicoxa". This idea of a wing originating from an exite (of the precoxa?) was taken up in developmental genetic studies (Averov & Cohen 1997). In a recent study of the postabdomen of Odonata, [START_REF] Klass | The female abdomen of ovipositor-bearing Odonata (Insecta:Pterygota)[END_REF] pointed out the presence of a gonangulum (epipleurite IX) divided into two distinct sclerites, which he named "antelaterocoxa" and "postlaterocoxa". He homologised the former with [START_REF] Bitsch | Morphologie abdominale des Machilides (Thysanura) -II. Squelette et musculature des segments génitaux femelles[END_REF] "precoxite" and the latter with Bitsch's "laterocoxite". It should be noted in this respect that Bitsch (1974, p. 102, footnote) had taken care to specify that his term "precoxite" was purely descriptive and in no way referred to a presumption of precoxal nature. [START_REF] Emeljanov | The evolutionary role and fate of the primary ovipositor in insects[END_REF] used Klass' observation to give the gonangulum a fundamentally dual nature, with "sternal" and "pleural or pleuroparatergal" components. The question of the identity of the antelaterocoxa in Odonata still deserves scrutiny, but Klass' ( 2008) interpretation of a gonangulum with either one-piece or bipartite condition sounds as corresponding to the nature of epipleurite in either subcoxal or subcoxal + precoxal identity. While awaiting precise analyses, it is important to assume that an epipleurite, in its current broad sense, may include a precoxal component, often merged with the subcoxal one. This is consistent with the delineation of the epipleurite as initially given by [START_REF] Hopkins | Contributions toward a monograph of the scolytid beetles. I. The genus Dendroctonus[END_REF]. The present interpretation of the limb with a protopodite composed of three segments (precoxa, subcoxa and coxa) seems coherent. A subsidiary problem is presented by the coxa of the female gonopods, which sometimes appears to be bi-segmented. In Coleoptera, the gonopod IX is often (primitively?) composed of two segments, followed by the so-called stylus that is presumably of a telopodal nature [START_REF] Deuve | L'abdomen et les genitalia des femelles de Coléoptères Adephaga[END_REF] (Figure 6b,c). Following [START_REF] Mickoleit | Ueber Ovipositor der Neuropteroidea und Coleoptera und seine phylogenetische Bedeutung[END_REF] and [START_REF] Bils | Das abdomenende weiblicher terrestrische lebender Adephaga une seine Bedeutung für die Phylogenie[END_REF], Hünefeld et al. (2012) reported the existence of these two joints as "a cranial element of gonocoxite IX articulating with the laterotergite IX [what I named epipleurite IX], and a caudal element bearing a stylus". In his most recent studies on adephagans, Ball names these joints "gonocoxite 1" and "gonocoxite 2", i.e. coxa 1 and coxa 2 respectively (e.g. [START_REF] Ball | Taxonomic review of the Tribe Melaenini (Coleoptera: Carabidae), with observations on morphological, ecological and chorological evolution[END_REF]. The question arises as to whether coxa 2 might be the trochanter, which has become well developed and adapted for oviposition in soil. In this case, the more distal so-called stylus in coleopterans would be the second segment of the telopodite, not the first. However, this idea is still speculative and lacks support. One might think that, quite simply, the coxal segment is dimeric in some coleopterans, showing a kind of annulation found in limb structure of many arthropods [START_REF] Boxshall | The evolution of arthropod limbs[END_REF]), but it is noticeable that some muscles connect the basal joint to the distal one [START_REF] Bils | Das abdomenende weiblicher terrestrische lebender Adephaga une seine Bedeutung für die Phylogenie[END_REF]). In addition, this bi-segmented structure is known in Hymenoptera and, according to [START_REF] Mickoleit | Ueber Ovipositor der Neuropteroidea und Coleoptera und seine phylogenetische Bedeutung[END_REF], it would belong to the groundplan of the Holometabola. It should also be noted that the gonopods VIII of certain Pygidicranidae (Dermaptera) have been described as bi-jointed (Deuve 2001a, "Fig. 28";Klass 2003, "Fig. 68-69"), but they would be gonapophyses. Rearward displacement of the gonopore Epitopy is the apparent rearward displacement of the gonopore as a result of the internalisation of subcoxosternal areas involved in formation of the ectodermal genital ducts. This term means that it is not, strictly speaking, a rearward migration of the gonopore, but rather an 'epitopic location' of it, i.e. peripheral in relation to the whole invaginated area forming the genital pouch, the protovagina and finally the vaginal duct. In females, a location of the secondary gonopore at the rear of segment VIII, or even at the rear of segment IX, is observed in most pterygotes. In contrast, gonopore orthotopy corresponds to the primitive condition, in which the primary female gonopore lies at the posterior margin of segment VII, corresponding to its original metameric position. True orthotopy has been reported in Archaeognatha, Zygentoma and, very exceptionally, in partially neotenic Coleoptera belonging to the genus Eustra [START_REF] Deuve | L'abdomen et les genitalia des femelles de Coléoptères Adephaga[END_REF](Deuve , 2001a)). I have mentioned elsewhere (Deuve 2001b) that a subterranean life may lead to partial neoteny, in relation to environmental stability [START_REF] Gould | Ontogeny and phylogeny[END_REF], and this may reveal some primitive patterns by prematurely stopping the morphogenetic process of the organ concerned. In the case of the genus Eustra, it is the internalisation process of the subcoxosternal plates VIII and IX that has been prematurely halted during its development. The apparent morphogenetic shift of the female gonopore during pre-imaginal development has been studied by several authors, especially Singh-Pruthi (1924), [START_REF] Heberdey | Zur Entwicklungsgeschichte vergleichende Anatomie und Physiologie der weiblichen Geschletsausführwege der Insekten[END_REF] and Metcalfe (1932a,b). [START_REF] Snodgrass | Morphology of the insect abdomen, Part II. The genital ducts and the ovipositor[END_REF], [START_REF] Weber | Lehrbuch der Entomologie[END_REF][START_REF] Weber | Grundriss der Insektenkunde[END_REF] and [START_REF] Vandel | Embranchement des Arthropodes (Arthropoda, Siebold et Stannius 1845). Généralités. Composition de l'Embranchement[END_REF] published useful overview, but they used the diagrams of Heberdey, which are misleading. Indeed, there is no internal connection between three genital pouches, which are located, respectively, at the rear of sterna VII, VIII and IX, but there is a concomitant invagination of the subcoxosternal areas VIII and IX. [START_REF] Styš | Reinterpretation of the theory on the origin of the pterygote ovipositor and notes on the terminology of the female ectodermal genitalia of insects[END_REF] well understood this formation of a "gynatrium" by "invagination of the ventral portions of the VIIIth and IXth urites". Figure 7 illustrates the location of the gonopore ("gon.") and its epitopy as a result of the ventral connection of the epipleurites VIII and epipleurites IX in sympleural types I and II, respectively. It should be noted that under these conditions the invaginated 'sternal' areas correspond to the secondary sternum and, necessarily, include some ventral components of the subcoxa and precoxa (but not the coxal ones, which correspond to the gonopods). These are therefore subcoxosternal areas (and even, as it should be written, precoxal-subcoxosternal). In the case of the sympleural-II type, the gonopods themselves have regressed and could have part of their territory included in this invagination. Thus, the secondary genital ducts (vaginal ducts) of insects are not only sternal, but subcoxosternal or sometimes coxosternal in nature. These considerations need to be taken into account in any study of the musculature supporting the ectodermal genitalia. On the other hand, the anatomical delimitation of these internalised territories is hardly possible in the current state of our knowledge (Figure 8). However, on the vaginal duct or bursa copulatrix of some Adephaga (Coleoptera), [START_REF] Bils | Das abdomenende weiblicher terrestrische lebender Adephaga une seine Bedeutung für die Phylogenie[END_REF] was able to distinguish territories connected by muscles to segment VIII and others connected by muscles to segment IX: the former, ventral, are connected to epipleurites VIII, the latter, dorsal, lateral or caudal, are connected to epipleurites IX or, more rarely, to tergite IX or coxae of the ninth segment. It is interesting to note that formation of the arthropod hypopharynx, which is ectodermal, follows a similar process, with participation of several cephalic segments and invagination of their respective subcoxosternal or sternal areas. The formation of the stomodeum, which is possibly comparable, has been interpreted in a variety of different ways (see review in Bitsch 1973a). In the same vein, the internalisation of the primary sternum in thoracic segments was clearly described and illustrated by [START_REF] Weber | Die Gliederung der Sternopleuralregion des Lepidopterenthorax. Eine vergleichende morphologische Studie zur Subcoxaltheorie[END_REF] in context of the nascent "subcoxal theory". It can also be noted that, in the same way as at the thorax the coxa maintains an articular function, remains associated with the prominent appendage but dissociates itself from the subcoxal territories which are embedded into the body, at the level of the genital segments the coxa remains associated with the gonopod of which it is a part, but dissociates itself from the so-called coxosternal plate, in reality subcoxosternal. This subcoxosternal plate includes only the ventral component of the subcoxa, not the dorsal component corresponding to the epipleurite, which is not internalised. In the case of the insect abdomen, a crucial question is whether the whole subcoxosternum is internalised, or whether lateral margins of it could instead remain in their external position and participate partly in formation of the final subgenital plate in some cases. Just as thoracic subcoxosternal territories are more internalised in holometabolans than in hemimetabolans [START_REF] Weber | Die Gliederung der Sternopleuralregion des Lepidopterenthorax. Eine vergleichende morphologische Studie zur Subcoxaltheorie[END_REF][START_REF] Ferris | Some general considerations[END_REF], it can be supposed that the same is true for the subcoxosternal plates of the genital segments, the two processes probably being, in genetic terms, parallel. For example, [START_REF] Klass | The female abdomen of ovipositor-bearing Odonata (Insecta:Pterygota)[END_REF] considers the ventrite VIII of the Odonata to be of a 'coxosternal' nature, but with inclusion of the epipleurites (which he names "laterocoxites"). This would therefore be, more exactly, an epipleurosubcoxosternum, because the coxae are excluded. The same author indicates that the embiopteran subgenital plate is atypical, being complex in nature and subject to interpretation in various hypothetical ways [START_REF] Klass | The female genitalic region and gonoducts of Embioptera (Insecta), with general discussions on female genitalia in insects[END_REF]. In Coleoptera and certainly also in Neuropterida, which have very similar abdominal structures [START_REF] Mickoleit | Ueber Ovipositor der Neuropteroidea und Coleoptera und seine phylogenetische Bedeutung[END_REF][START_REF] Liu | Homology of the genital sclerites of Megaloptera (Insecta: Neuropterida) and their phylogenetic relevance[END_REF], it is clear that the totality of the subcoxosternal area is internalised because the two well delimited epipleurites are connected medio-ventrally to each other, are juxtaposed, and often even merge to form a kind of 'tertiary sternum'. In segment VIII, this merging of the epipleurites is sometimes so perfect that the resulting subgenital plate (of a strictly sympleural nature), named ventrite VIII, is still often confused by most coleopterists with 'sternite VIII' or 'coxosternite VIII', which is incorrect. The reality of the connection of the epipleurites is particularly well illustrated in the imaginal stage of certain Paussidae (Caraboidea), in which all intermediate steps can be observed between an orthotopic state-with the subcoxosternum still present between the epipleurites and the gonopore located at the rear of segment VII-and an epitopic state-with complete suturing of the epipleurites (Figure 9). Another argument is the presence in the genus Luperca (Caraboidea Siagonidae) of a membranous septum that connects the internal margins of the two epipleurites VIII to the gonopods VIII, which are associated with the ninth segment [START_REF] Deuve | Les sternites VIII et IX de l'abdomen sont-ils visibles chez les imagos des Coléoptères et des autres Insectes Holométaboles ?[END_REF], 1993, p. 145, "Fig. 219"). The plesiomorphic presence of the gonopods VIII in their primitive location-between the two epipleurites VIII-is also observed in some Hydradephaga (Deuve 2001a, "Fig. 14"), whereas in other beetles they have become closely associated with gonopods IX to form an ovipositor and then regress. Concomitantly, the previously lateral epipleurites VIII become connected to each other ventromedially. [The following errors in Deuve (2001a) need to be noted: the references to figures "18" and "19" in the text (p. 207-208) should be corrected to 20 and 21, and those to figures "21" and "22" in the text (p. 208), should be corrected to 18 and 19]. In many beetles, the merging of epipleurites VIII is perfect, such that the resulting ventral sclerite is identical in appearance to a coxosternite. In particular, strong longitudinal muscles are observed that connect ventrite VII to ventrite VIII. This point deserves special attention because it has often been used as an argument to interpret the ventrite VIII as being homonomous with the coxosternite VII. While it is true that muscle insertions can often be used as landmarks to delimit anatomical territories, this principle should not be applied dogmatically. For example, it was the presence of muscles directly connecting the tergum to the coxa that led [START_REF] Snodgrass | A textbook of arthropod anatomy[END_REF][START_REF] Snodgrass | Evolution of arthropod mechanisms[END_REF] to reject the subcoxal theory. In his study on the abdomen of female machilids, Bitsch (1973b) refers to the theoretical possibility of "a secondary displacement of the anterior attachment of the muscle 71" (translated from French). It is known that muscle insertions can actively migrate during development, especially during metamorphosis [START_REF] Williams | Active muscle migration during insect metamorphosis[END_REF]. Also, neoformations of muscles adapted to a new specialised function are frequent in insects. For example, Hünefeld et al. (2012) report the neoformation of a transverse muscle between appendages VIII of the Antliophora. The postabdominal musculature of the Adephaga (Coleoptera) was carefully studied by [START_REF] Bils | Das abdomenende weiblicher terrestrische lebender Adephaga une seine Bedeutung für die Phylogenie[END_REF] and [START_REF] Burmeister | Der Ovipositor der Hydradephaga (Coleoptera) und seine phylogenetische Bedeutung unter besondere Berücksichtigung der Dytiscidae[END_REF]. In Hygrobia, there are muscles connecting epipleurite VIII to gonopod IX and other muscles connecting gonopod VIII to gonopod IX. These are functional specialisations related to the complex movements of the ovipositor. Also the muscles connecting epipleurites VIII to coxosternite VII in beetles are similar in shape and arrangement to the strong, ventral, antagonistic, longitudinal muscles that connect coxosternite VII to coxosternite VI, or coxosternite VI to coxosternite V. These are functional necessities related to the mobility of the whole abdomen and its ability to retract. The analogy of musculatures cannot be used to assert that ventrite VIII is the homonom of ventrite VII. A study of the abdominal muscles of female Paussidae (Caraboidea) beetles would have to be undertaken in the future to show, step by step, the muscular homologies and rearrangements between partially neotenic species of the genus Eustra which have an orthotopic gonopore and seripleural abdomen, and those belonging to other genera of the tribe Ozaenini, in which epipleurites VIII move closer to each other until they become connected and finally merge together to accompany the gonopore epitopy and to close the abdominal venter (Figure 9). In Coleoptera, I showed that ventrites VIII and IX were formed by the suture or merging of the epipleurites in males as well [START_REF] Deuve | Les sternites VIII et IX de l'abdomen sont-ils visibles chez les imagos des Coléoptères et des autres Insectes Holométaboles ?[END_REF]). The differences are that gonopods IX of males are not visible on the ninth ventrite and that the aedeagus is mainly formed from segment X [START_REF] Dupuis | Origine et dévelopement des organes génitaux externes des mâles d'Insectes[END_REF]. The study of a gynandromorph of Cetoniidae (Coleoptera) showed the co-existence in the same specimen of the female (segment IX) and male (segment X) ectodermal genitalia, the latter located behind the former [START_REF] Deuve | Origine segmentaire des genitalia ectodermiques mâles et femelles des Insectes. Données nouvelles apportées par un gynandromorphe de Coléoptère[END_REF], thus providing evidence that they are not homologous. Ventrite IX, which has the misleading shape of an ordinary coxosternite in males (Figure 10f), is 'feminised' in this gynandromorph and shows some subdivisions that have the appearance of epipleurites and rudimentary gonopods (Figure 10e). While it seems clear that ventrite VIII of Coleoptera (see [START_REF] Dupuis | L'abdomen et les genitalia des femelles de Coléoptères Scarabaeoidea (Insecta, Coleoptera)[END_REF], for Scarabaeoidea) and Neuropterida has arisen through the juxtaposition or merging of just the two epipleurites to form the first subgenital plate, without any subcoxosternal component, it is crucial to consider whether the same pattern could be generalised to all insects showing an abdomen of the sympleural type. Indeed, it is conceivable that in this type only a medial part of the subcoxosternum has been invaginated to form the genital ducts and that some lateral parts of it remain closely associated with the epipleurites. However, while the bipartite composition of ventrite VIII has often been documented in various orders, an obvious tripartite arrangement (i.e. with a median relictual subcoxosternal area) has never been reported. In pterygote insects, it appears that the coxosternum of each abdominal segment forms an undivided ventral plate (= ventrite), with only few examples of fragmentation being known. It can therefore be hypothesised that the entire genital subcoxosternal plate (i.e. in segments VIII and IX) is internalised when the vaginal ducts are formed. In contrast, the epipleurites are often well separated from the coxosternum. In many beetles, such as Stictotarsus or Systelosoma (Figure 11), the epipleurites are contiguous with the lateral margins of the coxosternite, but are not merged with it. Such a merging occurs in many pterygotes and we can refer to a resulting epipleuro-coxosternum occupying the ventral side of all pregenital segments of the abdomen. It appears important to insist on this anatomical and functional dissociation between the epipleurite on one side, and coxosternite on the other, on all abdominal segments. On the genital segments, the gonopods are dissociated from the subcoxosternum and therefore the epipleurites must a fortiori be dissociated. This dissociation is fundamental in the pterygote groundplan. In female Coleoptera, [START_REF] Verhoeff | Zur vergleichenden Morphologie des Abdomens der Coleopteren und über die phylogenetische Bedeutung desselben, zugleich ein zusammenfassender kritischer Rückblick und neuer Beitrag[END_REF] had already pointed out the presence of a "bipartite sternite 8" and Tanner (1927) had written: "in some cases it [the VIIIth sternite] is shield-shaped and divided into two parts on the middle by a small strip of membrane". In female Mecoptera, the symmetrical bipartite nature of the ventrite VIII was noted by [START_REF] Mickoleit | Die Genital-und Postgenital Segmente der Mecoptera-Weibchen. I. Das Exoskelet[END_REF], who admits the absence of the true sternum. Indeed, it is certainly in Mecoptera that this bipartite nature of ventrite VIII is most clearly visible. This author generalised this observation, writing: "Furthermore, we can assume that a larger sternal plate is lacking in the groundplan of the genital segments of the pterygotes" (Mickoleit 1975, p. 102, translated from German). However, in Mickoleit's model, which differs significantly from mine on this point, only the genuine primary sternum is internalised and the two remaining components of ventrite VIII retain the name "coxosternite VIII". Some students working with him, such as [START_REF] Bils | Das abdomenende weiblicher terrestrische lebender Adephaga une seine Bedeutung für die Phylogenie[END_REF] and [START_REF] Burmeister | Der Ovipositor der Hydradephaga (Coleoptera) und seine phylogenetische Bedeutung unter besondere Berücksichtigung der Dytiscidae[END_REF], also continued to use "coxosternite VIII" to designate the ventral sclerite of the adephagan beetles, whereas the homonomous ventrites IX are named by them, as by Mickoleit, "tergites IX" or "laterotergites IX". Mickoleit (ibid.) stated precisely: "Therefore, a true sternal plate would not exist at all at the venter of the 8th segment of the Pterygota. Whether and to what extent the subgenital plate of the pterygote groundplan is involved in the formation of the sclerites known as gonocoxosternites is not determined" (translated from German). After studying the postabdominal musculature in a nannochoristid (Mecoptera), [START_REF] Hünefeld | The female postabdomen of the enigmatic Nannochoristidae (Insecta: Mecopterida) and its phylogenetic significance[END_REF] have finally followed Mickoleit and concluded: "The ventral sclerotised parts of segments VIII and IX of mecopterid females (and endopterygote females in general) are derivatives of the genital appendages of these segments and definitively not true sternal sclerotisations". These so-called appendicular formations are subcoxal in nature and correspond to the epipleurites VIII and IX. In addition, this model can most likely be generalised to endopterygote males, as shown for coleopterans [START_REF] Deuve | Les sternites VIII et IX de l'abdomen sont-ils visibles chez les imagos des Coléoptères et des autres Insectes Holométaboles ?[END_REF]. But in my model, not only the primary sternum but the whole subcoxosternum is internalised and therefore absent from the external skeleton of the genital segments in both males and females of the endopterygotes. Although the subgenital plate has often been interpreted as resulting from a rearward expansion of segment VII (see review in [START_REF] Bitsch | Morphologie abdominale des Insectes[END_REF], [START_REF] Kristensen | Heterobathmia valvifer n. sp.: a moth with large apparent 'ovipositor valves' (Lepidoptera: Heterobathmiidae)[END_REF] wrote concerning the subgenital plate of an heterobathmiid (Lepidoptera): "While a derivation from the venter VII territory remains a possibility, it could also be a fusion product of the segment VIII appendage bases, as hypothesized for similar formations in the Mecoptera and Trichoptera [START_REF] Mickoleit | Die Genital-und Postgenital Segmente der Mecoptera-Weibchen. I. Das Exoskelet[END_REF][START_REF] Nielsen | A comparative study of the genital segments and the genital chamber in female Trichoptera. Biologiske Skrifter[END_REF]". Although by 1999 the interpretation of these sclerites as being formed from only appendicular components was beginning to be accepted, [START_REF] Kristensen | Heterobathmia valvifer n. sp.: a moth with large apparent 'ovipositor valves' (Lepidoptera: Heterobathmiidae)[END_REF] referred to Mickoleit's model, with internalisation of the only primary sternum. It is also important to clearly distinguish between two different types of 'subgenital plates', which are not homologous and should not be confused. Posterior expansions of coxosternite VII to protect or support the genital segments are present in many insects and have sometimes been referred to collectively as the 'subgenital plate'. In Lepismatida, [START_REF] Rousset | Squelette et musculature des régions génitales et postgénitales de la femelle de Thermobia domestica (Packard). Comparaison avec la région génitale de Nicoletia sp. (Insecta: Apterygota: Lepismatida)[END_REF] described a "languette" (small tongue-like lobe) which is a prominent membrane fold belonging to segment VII. In Coleoptera, ventrite VII is usually dilated rearwards to support the retracted postabdomen. This is not the typical subgenital plate of the sympleural-type abdomen, which consists of the two joined epipleurites VIII. Among hemimetabolan insects, the Dermaptera have a seripleural type abdomen (Deuve 2001a;[START_REF] Klass | The female abdomen of the viviparous earwig Hemimerus vosseleri (Insecta : Dermaptera : Hemimeridae), with a discussion of the postgenital abdomen of Insecta[END_REF][START_REF] Klass | The female genitalic region in basal earwigs (Insecta: Dermaptera: Pygidicranidae (s.l.)[END_REF], whereas the sympleural-I type is seen in orthopteroids. Their subgenital plate has been the subject of several studies without clear conclusions, probably because both kinds of 'subgenital plates' (expansion of coxosternum VII versus ventral junction of epipleurites VIII) have often been confused under this name. [START_REF] Qadri | On the development of the genitalia and their ducts of orthopteroid insects[END_REF], however, interprets the subgenital plate of Tettigonioidea as resulting from the fusion of lateroventral formations of the "sternite VIII" (read "ventrite VIII") which might be epipleurites. Snodgrass (1935b) indicates that the vaginal ducts of Locustana are formed from a medial groove along the entire length of the eighth ventrite during organogenesis. These observations are consistent, at least in part, with the hypothesis of a subgenital plate formed by the secondary juxtaposition of epipleurites VIII. The abdomen of female Odonata has been well studied by [START_REF] Klass | The female abdomen of ovipositor-bearing Odonata (Insecta:Pterygota)[END_REF] in a work that takes into account musculature and innervation. This author essentially agrees with my interpretation of the female genital abdominal segments (Deuve 2001a) by recognising in the same way the presence of epipleurite IX, which he names "laterocoxite IX", but with an anterior and posterior component. Regarding segment VIII, Klass (2008, p. 137) admits that the ventral plate VIII resembles the coxosternum of the pregenital segments, but in contrast to these the coxae VIII are not included. In such cases, it is reasonable to assume that the subcoxosternum VIII is also not included. However, in the same work Klass (2008, p. 117) rejects my interpretation of the ventral plate VIII as resulting from the medial merging of epipleurites VIII (his "laterocoxites VIII") for the sole reason that I named "coxosternum VII" the ventral sclerite of the preceding segments, which in my model would be "exclusively non-homonomous". That was not my opinion: in many pterygotes the epipleurites of the pregenital segments are indeed merged with the lateral margins of the coxosternum to form a single ventral sclerite. This is certainly the case in Mecoptera, judging by the musculature described by [START_REF] Hünefeld | The female postabdomen of the enigmatic Nannochoristidae (Insecta: Mecopterida) and its phylogenetic significance[END_REF] for the pregenital and genital segments, and probably also in Odonata. I agree that, in this precise case of Zygoptera, it would have been more accurate to use the full term 'epipleuro-coxosternum VII' rather than 'coxosternum VII'. Instead, I opted for a simpler term, as was usual at that time. In addition, rearrangements of the musculature of the genital segments during metamorphosis are significant. A study by [START_REF] Matushkina | Skeletomuscular development of genital segments in the dragonfly Anax imperator (Odonata, Aeshnidae) during metamorphosis and its implications for the evolutionary morphology of the insect ovipositor[END_REF] on the transformation of the abdominal musculature of an aeshnid during metamorphosis shows the absence of several muscles of segment VIII in the imago, mainly "intersegmental sternal" and "intersegmental pleuro-sternal" muscles, in relation to the development of the internal and external genitalia. Conversely, these muscles remain present in the pregenital segments. Therefore, there is a considerable rearrangement of the ventral musculature of segment VIII, and even more so for segment IX. In a subsequent paper, [START_REF] Klass | The female genitalic region and gonoducts of Embioptera (Insecta), with general discussions on female genitalia in insects[END_REF] explicitly admitted a process of internalisation of the primary sternum of the genital segments in the introduction of their study of Embioptera: "the ventral plates of both segments 8 and 9 probably lack a sternal component, yet we call them coxosternites; and in cases of absence of discrete laterocoxites it is often disputable whether laterocoxal sclerotisations are absent or included in the coxosternite, and how large this portion is". However, I cannot follow these authors when they write a few lines earlier: "the laterotergites and perhaps also the pleurites fall into the category of epipleural sclerotisations as defined by Deuve (e.g. 2001)" (Klass & Ulbricht 2009, p. 120). In fact, it is rather the "laterocoxite", as described in detail by Klass since his work on Dermaptera [START_REF] Klass | The female abdomen of the viviparous earwig Hemimerus vosseleri (Insecta : Dermaptera : Hemimeridae), with a discussion of the postgenital abdomen of Insecta[END_REF], that most precisely corresponds to the epipleurite as I characterised and illustrated it in the fundamental structure of the insect abdomen in both sexes [START_REF] Deuve | Sur la présence d'un "épipleurite" dans le plan de base du segment des Hexapodes[END_REF], 2001a, "Fig. 10 to 31"), its subcoxal identity being another question. The "laterotergites and pleurites" described by [START_REF] Klass | The female genitalic region and gonoducts of Embioptera (Insecta), with general discussions on female genitalia in insects[END_REF] in female embiopterans probably correspond to tergopleural and/or precoxal formations. Moreover, it is precisely this fundamental distinction between the epipleurite and a small sclerite (precoxite?) lying next to the spiracle that I pointed out (Deuve 2001a) in response to an objection from [START_REF] Kristensen | Heterobathmia valvifer n. sp.: a moth with large apparent 'ovipositor valves' (Lepidoptera: Heterobathmiidae)[END_REF] in their study of the subgenital plate of a heterobathmiid (Lepidoptera) (see above). That said, additional studies are still needed to assess the importance and limitations of the internalisation of all, or part, of the eighth subcoxosternum in Odonata and in other orders of hemimetabolan insects having a sympleural postabdomen. The question is whether the eighth ventrite consists only of the two connected epipleurites VIII-as I presume-or whether it may also retain some traces of the former subcoxosternum and, if so, in what proportion. However, if coxae VIII are not included in the eighth ventral plate of the Odonata [START_REF] Klass | The female abdomen of ovipositor-bearing Odonata (Insecta:Pterygota)[END_REF]) and if a sternal component is also lacking in the Embioptera [START_REF] Klass | The female genitalic region and gonoducts of Embioptera (Insecta), with general discussions on female genitalia in insects[END_REF], as in all pterygote insects [START_REF] Mickoleit | Die Genital-und Postgenital Segmente der Mecoptera-Weibchen. I. Das Exoskelet[END_REF], almost only the epipleurites VIII remain 'available' to form the subgenital plate. Coxal territories being absent (gonopods), the lateral margins of a subcoxosternite would theoretically be of a strictly subcoxal nature (ventral parts of the subcoxae). Therefore, at most, we could imagine some subcoxal remnants of the coxosternum which would then merge with the epipleurites to form a resulting tripartite or quadripartite subgenital plate that is totally subcoxal in nature. Yet nothing like this has ever been observed and it is not the simplest hypothesis. Except for a few details and the semantic replacement of the term epipleurite by laterocoxite, I finally see no fundamental difference between Klass's model and mine. Conclusions We have arrived at a fairly consensual model to describe the fundamental structure of the insect skeleton. In particular, the importance of the subcoxal area-not only on the thorax, but also on the abdomen-no longer seems to be in doubt. Named epipleurite [START_REF] Hopkins | Contributions toward a monograph of the scolytid beetles. I. The genus Dendroctonus[END_REF][START_REF] Böving | An illustrated synopsis of the principal larval forms of the order Coleoptera[END_REF][START_REF] Deuve | Sur la présence d'un "épipleurite" dans le plan de base du segment des Hexapodes[END_REF][START_REF] Sweet | Comparative external morphology of the pregenital abdomen of the Hemiptera[END_REF]), subcoxa [START_REF] Bekker | K stroyeniyu i proiskhozhdeniyu naruzhnykh polovykh pridatkov Thysanura I Hymenoptera [Structure and origin of the external genital appendages of Thysanura and Hymenoptera[END_REF](Bekker , 1932a,b;,b;[START_REF] Smith | Evolutionary morphology of external insect genitalia. 1. Origin and relationships to other appendages[END_REF]) (in reality, only the dorsal part of the subcoxa, the ventral part being integrated into the secondary sternum), laterosternite [START_REF] Gustafson | The origin and evolution of the geniatlia of the Insecta[END_REF][START_REF] Matsuda | Comparative morphology of the abdomen of a machilid and a rhaphidiid[END_REF], gonangulum [START_REF] Scudder | Reinterpretation of some basal structures in the insect ovipositor[END_REF][START_REF] Scudder | Comparative morphology of insect genitalia[END_REF] or laterocoxite (Bitsch 1973b[START_REF] Bitsch | Morphologie abdominale des Machilides (Thysanura) -II. Squelette et musculature des segments génitaux femelles[END_REF][START_REF] Klass | The female abdomen of the viviparous earwig Hemimerus vosseleri (Insecta : Dermaptera : Hemimeridae), with a discussion of the postgenital abdomen of Insecta[END_REF], a sclerite of a subcoxal nature has been identified at all segments of the abdomen, and in both sexes. The same applies to the cephalic segments, for which developmental genetic techniques have recently revealed subcoxal territories at the base of gnathal appendages [START_REF] Coulcher | Molecular developmental evidence for a subcoxal origin of pleurites in insects and identity of the subcoxa in the gnathal appendages[END_REF]. Also, I have stressed the dissociation, in anatomical and probably also genetic terms, between the dorsal part of the subcoxa, which corresponds to the epipleurite, and the ventral part, which is more closely associated with the primary sternum to form a coxosternum or a subcoxosternum. It is noteworthy how much the existence of a subcoxa has been discussed for more than a century and that it has only recently been recognised in the mandible or on all segments of the abdomen. It might be concluded from this that this segment is cryptic, but this is not the case. On the contrary, it actually occupies a large area and has important functions in the patterning of the hexapodan skeleton. In the thorax, the subcoxa plays a major role in the formation of the secondary sternum and the lateral wall lying between the coxa and the tergum. In the abdomen, the subcoxa plays a similar role in the formation of a secondary sternum (coxosternum), but it also plays this role in the pleural regions below the spiracle and, on the genital segments, in the support and articulation of the gonopods and/or in the formation of a new ventrite (subgenital plate) that is a sort of 'tertiary sternum'. Whereas the coxa has acquired a predominant articular function during evolution, the subcoxa (and probably also the precoxa that originates from it) has become specialised for the functions of fixing the limb to the body-wall and carrying the appendage, while maintaining its relative mobility. This can be observed on the abdominal and thoracic segments, whose general architecture is finally quite similar. There is not only structural but also functional homonomy and identity. The presence or absence of a precoxa at the base of the hexapodan appendage still remains a subject of discussion. There are good arguments to support the existence of this proximalmost segment resulting from a subdivision of the embryonic subcoxal bud, in agreement with the tri-segmented protopodite model of the crustaceans. The hypothesis of a precoxal segment is obviously crucial, particularly for understanding the anatomical origin of the wings of the Pterygota. Some points still remain to be clarified. Whereas the hypothesis of the internalisation of the whole subcoxosternum in the genital segments of Holometabola seems to be firmly established, the possibility in hemimetabolan insects that some marginal subcoxosternal elements might remain external and, if so, in what proportions, still needs to be evaluated. Also, we have yet to identify the nature of certain small enigmatic sclerites, such as the "tergopleurites" described by [START_REF] Prell | Das Chitinskelett von Eosentomon[END_REF] and the "laterotergites" described by [START_REF] François | Le squelette thoracique des Protoures[END_REF][START_REF] François | Squelette et musculature thoraciques des Protoures[END_REF] in Protura and by [START_REF] Bäcker | A forgotten homology supporting the monophyly of Tracheata: the subcoxa of insects and myriapods re-visited[END_REF] in some Chilopoda. More importantly, the problem of the origin of wings in pterygotes needs to be clarified, in relation to the identity of the axillary sclerites. In a recent paper, [START_REF] Mashimo | Embryological evidence substantiates the subcoxal theory on the origin of pleuron in insects[END_REF] stressed the difficulty in delineating the crucial anatomical boundary between the tergum and the arthropod appendage. The question of the fundamental location of the spiracle on the lateral margins of the socalled tergum or on the very base of the appendage remains wide open. These wing and spiracle problems are reflected in the abundant literature dealing with the pleura of arthropods for over a century. It may be questioned whether the intrinsic difficulty in locating this tergal/pleural boundary is related to an earlier appendicular nature of the lateral margins of the so-called tergum itself. As has already been pointed out [START_REF] Deuve | Sur la présence d'un "épipleurite" dans le plan de base du segment des Hexapodes[END_REF], the dorsum of Euarthropoda is fundamentally trilobed [START_REF] Lauterbach | Die Muskulatur der Pleurotergite im Grundplan der Euarthropoda[END_REF] and it is consistent to suppose that paranotal lobes are tergopleural in nature, rather than being strictly tergal formations. Given that the ancestor of Euarthropoda was probably a type of marine worm, it seems likely that the ventral part of its pleura became specialised for locomotion on the seabed, whereas the dorsal part became specialised for respiration and gill protection. This same pleural origin of the base of the arthropodal limb and of the paranotal lobe would make it easier to understand their close functional (and genetic?) association, as well as the 'dual' origin of the wings of pterygote insects. Hopefully this speculative idea will be read with indulgence, but, after more than a century of debate and some probably excessive attempts at schematisation, it must be kept in mind that there is still much work to be done to fully clarify our understanding of the arthropod pleura. Glossary of the main terms used: Appendage: Eupleural set corresponding to a specialised organ and consisting of several successive segments; in hexapods these are: precoxa, subcoxa, coxa, and telopodites. (Synonym: limb) Coxosternum: Ventral sclerite corresponding to the merging of the true sternum, the coxae and some ventral components of the subcoxae. In theory, ventral components of the precoxae would also be included. The coxosternum, often termed "secondary sternum" in the context of the subcoxal theory, or just ventrite, is clearly visible as the only ventral sclerite of the pterygote pregenital abdominal segments. (Plural: coxosterna) Dorsum: The dorsal face of the body (Audouin 1820). In theory, a sclerite of the dorsum would be a "dorsite", but this term is rarely used. Epipleurite: [START_REF] Hopkins | Contributions toward a monograph of the scolytid beetles. I. The genus Dendroctonus[END_REF]). Separated sclerite well visible in the skeleton of many arthropods, especially in pterygotes. It corresponds to the dorsal part of the subcoxa, which dissociates itself from the ventral part to remain in pleural position between the base of the functional appendage and some more dorsal sclerites. It may also have a precoxal component. Conversely, the ventral part of the subcoxa tends to associate with the sternum to form a subcoxosternum or a coxosternum. However, the epipleurite can in some cases merge with the coxosternum. The epipleurite has often been referred to as "laterotergite" by Snodgrass, especially in his famous book "Principles of insect morphology" (1935a). The gonangulum of [START_REF] Scudder | Reinterpretation of some basal structures in the insect ovipositor[END_REF] is the abdominal epipleurite IX [START_REF] Deuve | Sur la présence d'un "épipleurite" dans le plan de base du segment des Hexapodes[END_REF](Deuve , 2001a)). The laterocoxite (Bitsch 1973b[START_REF] Bitsch | Morphologie abdominale des Machilides (Thysanura) -II. Squelette et musculature des segments génitaux femelles[END_REF]) might be a synonym of epipleurite. Epipleuro-coxosternum: Ventral sclerite resulting from the merging of the epipleurites and the coxosternum. It is often observable on the pregenital abdominal segments of insects. Eupleurite: [START_REF] Crampton | Notes on the thoracic sclerites of winged insects[END_REF]). Pleural sclerite forming part of the arthropodal appendage. Exobasal: (Deuve 2013, p. 5, footnote). In a phylogenetical tree, only an ancestor or a node (i.e. a cladogenetical event), or, at the limit, an entire lineage, may be said to be 'basal'; an extant clade branched basally is exobasal, but its most recent common ancestor is basal in the phylogeny of the group. Evidently, the sister-group of an exobasal clade is also exobasal. For example, Archaeognatha is an exobasal order of Hexapoda, which is not the case of Lepidoptera. Paranotal lobe: Often considered as lateral lobes of the tergum, the paranotal lobes may be tergopleural formations [START_REF] Deuve | Sur la présence d'un "épipleurite" dans le plan de base du segment des Hexapodes[END_REF]). In the theory of the dual origin of the pterygote wing [START_REF] Rasnitsyn | A modified paranotal theory of insect wing origin[END_REF], they have become associated themselves with basal elements of the appendage (eupleural elements) to form the functional wing. Pleuron: Lateral area located between the tergum and the sternum. A sclerite of the pleuron is a pleurite. (Plural: pleura) Precoxa: The basalmost segment of the arthropodal appendage. Its existence is still under debate. It has been observed in crustaceans, myriapods and hexapods. It would correspond to the 'subcoxa 1'. A sclerite of the precoxa is a precoxite. (Plural: precoxae) Sternum: Area located ventrally between the pleura, i.e. between the appendages. A sclerite of the sternum is a sternite. (Plural: sterna) Subcoxa: Basal segment of the appendage (homologue of the coxa of crustaceans) located between the precoxa and the coxa. It corresponds to the 'subcoxa 2'. Often cryptic because embedded into the body-wall of arthropods (subcoxal theory), with a function of anchoring and supporting the appendage. The epipleurite corresponds to its dorsal component, which retains a pleural location and remains more or less isolated from the coxosternal plate (coxosternum). A sclerite of the subcoxa is a subcoxite. (Plural: subcoxae) Subcoxosternum: Ventral plate resulting from the merging of the true sternum and some ventral components of both the subcoxae and precoxae. The coxae are not integrated in it. It should be noted that the 'secondary sternum' of the pterygote thoracic segments is a subcoxosternum rather than a coxosternum, the two coxae being separated from it and functioning as articular segments of the appendages. Although they are often confused by authors, the distinction between subcoxosternum and coxosternum is essential, especially at the level of the abdomen, in order to fully understand the structure of sclerites of the pregenital segments on the one hand (with a coxosternum) and that of the genital segments on the other (with internalisation of the subcoxosternum). (Plural: subcoxosterna) Tergopleurite: [START_REF] Prell | Das Chitinskelett von Eosentomon[END_REF]). Pleural sclerite not strictly belonging to the arthropodal appendage. Located between the base of the arthropodal appendage and the true tergum. Tergum: Area located dorsally between the pleura. A sclerite of the tergum is a tergite. (Plural: terga) Venter: The ventral face of the body. A sclerite of the venter is a ventrite. (It should be noted that Snodgrass (1935a[START_REF] Snodgrass | A contribution toward an encyclopedia of insect anatomy[END_REF] considered the venter as synonymous with sternum and [START_REF] Styš | Reinterpretation of the theory on the origin of the pterygote ovipositor and notes on the terminology of the female ectodermal genitalia of insects[END_REF] the ventrite as synonymous with the epipleuro-coxosternum, which he named "zygosternum".) Abbreviations used in the illustrations: (after Sharov 1966). The precoxa of the machilid was named "pleurite" by Sharov. It is noteworthy that the hexapodan subcoxa corresponds to the coxa of the crustaceans and that the hexapodan coxa corresponds to the basis of the crustaceans. In the female, the exodermal genital ducts are well shaped, with oviduct, bursa copulatrix, spermatheca and spermathecal gland. b. In the gynandromorph, the female components are vestigial, with poorly shaped bursa copulatrix but without oviduct or spermatheca; the male components are fully formed, with a complete aedeagus, but its rotation is incomplete (90° instead of 180°). c. In the male, the aedeagus is complete, with 180° rotation. d. In female, the external genitalia show lateral epipleurites IX (subcoxae) and developed gonopods VIII (coxae). e. In the gynandromorph, the ventrite IX is decomposed into distinct epipleural and coxal elements, but the VIII-IX costal area shows a 'genital ring' and a normally shaped spiculum gastrale. f. In the male, the ventrite IX (synsclerite) would be formed from totally merged epipleurites IX. Figure 1 . 1 Figure 1. Mesothoracic segment of proturans (after François 1964, modified). a. Lateral view of the external morphology in Acerentomon. There is a so-called laterotergite more dorsal than the anapleurite and located on the lateral margin of the tergum. It might be a tergopleurite. b. Section of the appendage and of more dorsal areas in Eosentomon. Note the position of the spiracle above the anapleurite on the lateral margin of the tergum, in a probably tergopleural area. Figure 2 . 2 Figure 2. Homologies of the respective segments of the appendage of an Anaspides sp. (Crustacea, Syncarida) (a), a Palaeozoic monuran (b) and a machilid (Archaeognatha) (c)(after Sharov 1966). The precoxa of the machilid was named "pleurite" by Sharov. It is noteworthy that the hexapodan subcoxa corresponds to the coxa of the crustaceans and that the hexapodan coxa corresponds to the basis of the crustaceans. Figure 3 . 3 Figure 3. Carabus insulicola(Coleoptera, Carabidae). Embryo fixed at 60% DT (percentage of total developmental time, from oviposition to hatching), showing thoracic segments and four abdominal segments (afterKobayashi et al. 2013, modified). Different colours refer to the apparent longitudinal fields, blue: precoxalred: subcoxalyellow: coxal and telopodal. The discoid organ lying in the coxal area of the first abdominal segment is the pleuropodium. Figure 4 . 4 Figure 4. Diagrams of the three theoretical abdominal types in female ectognathan hexapods: seripleural, sympleural-I and sympleural-II types. Note that the borders of the coloured regions for the coxosternal plates are arbitrary, being intended only to indicate existence of coxal, subcoxal and sternal components; for simplicity, no precoxal component are represented.In the seripleural type, the original metameric arrangement is maintained; the primary gonopore lies behind segment VII. In sympleural-I type, gonopods VIII join gonopods IX to form an ovipositor; epipleurites IX correspond to gonangula with an articular function; epipleurites VIII join ventrally to form the first subgenital plate that closes the abdomen; the secondary gonopore is apparently located at the rear of segment VIII. In the sympleural-II type, the gonopods are regressed; epipleurites IX join ventrally to form a second subgenital plate; the secondary gonopore is apparently located at the rear of segment IX. Colours refer to the different limb segments, blue: precoxalred: subcoxalyellow: coxal and telopodal. Figure 5 . 5 Figure 5. Female ectodermal genitalia of Eustra leclerci (Coleoptera, Paussidae), external views (a, c) and internal views (b, d) of the ventral side. a and b. Segment IX: gonopods and subcoxosternal plate.The protovagina is shaped, with formation of the spermatheca, accessory gland and vaginal apophysis, but the whole is still separated from the oviductal pouch. c and d. Segments VIII and IX. There is a large subcoxosternal plate VIII between the epipleurites VIII. The oviductal pouch is located anterior to this plate, in the form of a membranous funnel into which the common oviduct opens. Figure 6 . 6 Figure 6. Carabus sp. (Coleoptera, Carabidae). a. Abdominal segment of a larva, lateral view. Note the presence of a precoxal territory on the ventral flank of the paranotal lobe, near the spiracle. b. Distal extremity of the female abdomen. Note the distinctly bipartite morphology of the ventrite VIII, formed by juxtaposition of the two epipleurites. Gonopod IX is dimeric. c. Idem, scanning electron micrograph. Different colours indicate the longitudinal fields, blue: precoxalred: subcoxalyellow: coxal and telopodial. Note that the borders of the coloured regions for the coxosternal plates are arbitrary, being intended only to indicate existence of coxal, subcoxal and sternal components; for simplicity, no precoxal component are represented. Figure 7 . 7 Figure 7. Diagrams illustrating the rearward displacement of the gonopore in a female insect. Note that the colours of the coxosternal plates are here arbitrary, intended only to remind the existence of coxal, subcoxal and sternal components; for simplicity, no precoxal component is represented. In the orthotopic stage (a. seripleural type), the metameric organisation is maintained and the gonopore is located at the rear of segment VII. Note the separation of the primary gonopore (oviductal pouch) and the protovagina. The dashed line indicates the area that will be internalised to form the epitopic gonopore, located at the rear of segment VIII (b. sympleural-I type) or at the rear of segment IX (c. sympleural-II type). Note the anatomical and functional dissociation of the epipleurites and the coxosternal plates. Figure 8 . 8 Figure 8. Female ectodermal genitalia of Pachyteles granulatus (Coleoptera, Paussidae). The dashed line separates the oviductal components (segment VII) from the vaginal components (segments VIII and IX). Figure 9 . 9 Figure 9. Illustration of the abdominal closure by juxtaposition of the epipleurites VIII in various female Paussidae (Coleoptera). a. Eustra lebretoni, in which the epipleurites VIII are in lateral position, still separated from each other by the presence of the subcoxosternal plate VIII. Some elements of subcoxosternum IX are still visible between the gonopods. The oviductal pouch is separated from the protovaginal formations. b. Sphaerostylus punctatostriatus, in which epipleurites VIII are still in lateral position, but the subcoxosternal areas VIII and IX are internalised and the oviduct opens into the vaginal duct. c. Tachypeles pascoei, in which epipleurites VIII and IX joined and are juxtaposed to form a new ventral plate or 'tertiary sternum'. A ligula basalis can be observed, perhaps representing a vestige of the primary gonopore. Figure 10 . 10 Figure 10. Internal (Figures a-c) and external (Figuresd-f) genitalia in Cotinis mutabilis (Coleoptera, Cetoniidae) (after Deuve 1992): a and d, female; b and e, gynandromorph; c and f male. a. In the female, the exodermal genital ducts are well shaped, with oviduct, bursa copulatrix, spermatheca and spermathecal gland. b. In the gynandromorph, the female components are vestigial, with poorly shaped bursa copulatrix but without oviduct or spermatheca; the male components are fully formed, with a complete aedeagus, but its rotation is incomplete (90° instead of 180°). c. In the male, the aedeagus is complete, with 180° rotation. d. In female, the external genitalia show lateral epipleurites IX (subcoxae) and developed gonopods VIII (coxae). e. In the gynandromorph, the ventrite IX is decomposed into distinct epipleural and coxal elements, but the VIII-IX costal area shows a 'genital ring' and a normally shaped spiculum gastrale. f. In the male, the ventrite IX (synsclerite) would be formed from totally merged epipleurites IX. Figure 11 . 11 Figure 11. Distal extremity of the abdomen in beetles, illustrating the dissociation of epipleurites and coxosternites. a. Stictotarsus duodecimpustulatus (Dytiscidae), lateral view (after Burmeister 1976, modified). It is noteworthy that epipleurite VIII acquires mobility following its involvement in the ventral closure of segment VIII which follows the internalisation of coxosterna VIII and IX. b. Systolosoma breve (Trachypachidae). Ventral and dorsal views, the tergites having been cut along the midline. The alignment of the epipleurites is clearly visible. Subcoxosternal plates VIII and IX are entirely internalised. Note that the borders of the coloured regions for the coxosternal plates are arbitrary, being intended only to indicate existence of coxal, subcoxal and sternal components; for simplicity, no precoxal components are represented. c. Systolosoma breve, lateral view. scx: subcoxa scxst: subcoxosternum sp.: spiracle sp.gastr.: spiculum gastrale spm.: spermatheca sty: stylus subgen. pl.: subgenital A1, A2… An: abdominal segments 1 to N oss: subapical setose organ (including the so- acc.gl.: accessory gland called stylus) aed.: aedeagus ov : oviduct ana: anapleurite pcx: precoxa b.c.: bursa copulatrix pl: pleurite ba: basis pp: periproct ca: carpus pr.: proctodeum cata: catapleurite pro: propodus cx: coxa prtvg: protovagina cxst: coxosternum ptar: pretarsus d: dactylus def. gl.: defensive gland ej.duct: ejaculatory duct epl.: epipleurite eppd: epipodite exo: exite fm: femur plate fun.: funnel synscl.: synsclerite gen.ring: genital ring T1, T2, T3: thoracic segments 1-3 gon.: gonopore tar: tarsus gpd.: gonopod tb: tibia is: ischium tg: tergum or tergite lig.bas.: ligula basalis tlpd: telopodite lpnt: paranotal lobe tr: trochanter ltg: laterotergite v.: ventrite mb: membrane vg: vaginal duct mer: merus vg. ap.: vaginal apophysis Acknowledgements I am very grateful to the two reviewers, whose corrections and constructive comments were helpful for improving the manuscript, to Mark Judson (MNHN), who kindly corrected the English text, and to Jocelyne Guglielmi (MNHN), who found for me the little-known articles of Ernest Becker (or "Bekker").

01741085

en

2024/03/05 22:32:07

2018

https://hal.sorbonne-universite.fr/hal-01741085/file/MATCHEMPHYS-D-17-02747R1%281%29_sans%20marque.pdf

Sana Ben Moussa Afef Mehri Michel Gruselle Patricia Beaunier Sana Ben Moussa Guylène Costentin Béchir Badraoui Combined effect of magnesium and amino glutamic acid on the structure of hydroxyapatite prepared by hydrothermal method Keywords: hydrothermal method, apatite surface, glutamic acid, hybrid compound come Combined effect of magnesium and amino glutamic acid on the structure of hydroxyapatite prepared by hydrothermal method Introduction Among the minerals having an interest from an economic point of view, apatites, mostly hydroxy-(CaHAp) and fluoro-apatites (FAp), are of considerable interest in numerous research areas [START_REF] Elliot | Structure and Chemistry of the Apatites and Other Calcium Orthophosphates[END_REF][START_REF] Govindaraj | Carbon nanotubes/pectin/minerals substituted apatite nanocomposite depositions on anodized titanium for hard tissue implant: In vivo biological performance[END_REF][START_REF] Gomez-Morales | Progress on the preparation of nanocrystalline apatites and surface characterization : overview of fundamental and applied aspects[END_REF]. Apatites are used in several applications such as sorbents, catalysts and biomaterials [START_REF] Ben Moussa | Calcium, Barium and Strontium Apatites: A New Generation of Catalysts in the Biginelli Reaction[END_REF][START_REF] Gruselle | Apatites: a new family of catalysts in organic synthesis[END_REF]. Apatites are component of bones and teeth [START_REF] Wang | FT-IR and XRD study of bovine bone mineral and carbonated apatites with different carbonate levels[END_REF][START_REF] Kanwal | In-vitro apatite formation capacity of a bioactive glass -containing toothpaste[END_REF]. Apatites belong to the phosphate family of compounds, and are of general formula M 10 (PO 4 ) 6 Y 2 where M is a divalent cation: Ca, Sr, Ba, Pb…, and Y a hydroxyl (OH) or halide (F, Cl) [START_REF] Bulina | Fast synthesis of Lasubstituted apatite by the dry mechanochemical method and analysis of its structure[END_REF][START_REF] Lim | Enhancing osteoconductivity and biocompatibility of silver-substituted apatite in vivo through silicon cosubstitution[END_REF]. Apatites may contain several different substituents in their structure. This ability to trap substituents in their structures leads to the formation of total or *Manuscript Click here to view linked References partial solid solutions. The substitution processes are controlled by the crystallographic rules related mainly to: ionic radius, charge, electronegativity and polarisability [START_REF] Hamad | Synthèse et étude physico-chimique de fluoroapatites mixtes à cations bivalents Pb-Cd, Pb-Sr et Sr-Cd[END_REF][START_REF] Ben Moussa | Synthesis, Rietveld refinements and electrical conductivity of news fluorobritholite based on lead Ca 7x Pb x La 3 (PO 4 ) 3 (SiO 4 ) 3 F 2 (0  x  2)[END_REF]. Previous works [START_REF] Kaur | Lanthanide (= Ce, Pr, Nd and Tb) Ions Substitution at Calcium Sites of Hydroxyl Apatite Nanoparticles as Fluorescent Bio Probes: Experimental and Density Functional Theory Study[END_REF][START_REF] Badraoui | Synthesis and characterization of Sr (10-x) Cd x (PO 4 ) 6 Y 2 (Y = OH and F): A comparison of apatites containing two divalent cations[END_REF][START_REF] Aissa | Synthesis, X-ray Structural Analysis and Spectroscopic Investigations (IR and 31 P MAS NMR) of Mixed Barium/Strontium Fluoroapatites[END_REF] have shown that steric hindrance related to cations bigger in size than calcium ions, plays an important role in the limitation of the cationic substitution processes. Magnesium is undoubtedly one of the most important bivalent ions associated to biological apatites [START_REF] Geng | Synthesis, characterization and the formation mechanism of magnesium-and strontium-substituted hydroxyapatite[END_REF][START_REF] Geng | Synthesis, characterization and biological evaluation of strontium/magnesium-co-substituted hydroxyapatite[END_REF]. It has been verified that in calcified tissues, the amount of magnesium associated to the apatitic phase is higher at the beginning of the calcification process and decreases on increasing calcification [START_REF] Gu | Influences of doping mesoporous magnesium silicate on water absorption, drug release, degradability, apatitemineralization and primary cells responses to calcium sulfate based bone cements[END_REF][START_REF] Bigi | Structural and chemical characterization of inorganic deposits in calcified human mitral valve[END_REF][START_REF] Burnell | Normal maturational changes in bone matrix, mineral, and crystal size in the rat[END_REF]. This interest is increasing taking into consideration the ability of such mixed apatites to lead to hybrid organic-inorganic materials by reaction with amino-acids [START_REF] Li | Dopamine Modified Organic -Inorganic Hybrid Coating for Antimicrobial and Osteogenesis[END_REF][START_REF] Yu | Synergistic antibacterial activity of multi components in lysozyme/chitosan/silver/hydroxyapatite hybrid coating[END_REF][START_REF] Hong | Fabrication, biological effects, and medical applications of calcium phosphate nanoceramics[END_REF][START_REF] Padilla | High Specific Surface Area in Nanometric Carbonated Hydroxyapatite[END_REF][START_REF] Sanchez-Salcedo | In vitro structural changes in porous HA/β-TCP scaffolds in simulated body fluid[END_REF][START_REF] Zhou | Nanoscale hydroxyapatite particles for bone tissue engineering[END_REF][START_REF] Carrodeguas | α-Tricalcium phosphate: synthesis, properties and biomedical applications[END_REF]. The expected benefit from the introduction of an amino-acid such as a glutamic acid in magnesium modified apatites is correlated with the ability of magnesium to coordinate amino-acids more strongly that calcium ions do and consequently to bind a greater proportion and more firmly proteins on their surfaces. In this aim, we have carried out a structural, morphology and chemical investigation of the combined effect of magnesium and amino glutamic acid on hydroxyapatite structure. We have also studied the interaction between glutamic acid and the apatite surface. The results show that the acid forms a complex on the apatite surface. As glutamic acid functionalization should potentially modify the electrostatic interaction, the corresponding change in the surface charge of the powder was monitored by zeta-potential measurements and [H + ] consumed as a function of the pH value. Experimental and methods Synthesis The mixed Mg/CaHAp of general formula: Ca (10-x) Mg x (PO 4 ) 6 (OH) 2 (x = 0, 0.5 and 1.0), named Ca (10-x) Mg x HAp, have been synthesized using the hydrothermal method [START_REF] Bigi | Magnesium influence on hydroxyapatite crystallization[END_REF]. A demineralised water solution (14 mL, 0.75 M) of a mixture of the two nitrates Ca(NO 3 ) 2 .4H 2 O and Mg(NO 3 ) 2 .6H 2 O in the desired proportions is added to a (NH 4 ) 2 HPO 4 water solution (25 mL, 0.25 M). The pH of the final solution is adjusted to 10 by adding a NH 4 OH solution (d = 0.89, Purity = 28%). The final solution is transferred to an autoclave. The mixture is maintained at 120°C for 12 hours. After filtration and washing using hot demineralised water, the mineral is dried at 120°C overnight. The hybrid materials were prepared according to the same experimental protocol, with addition of a quantity of organic reagent glutamic acid (GA) to the phosphate solution before pH adjustment [START_REF] Bigi | Microstructural investigation of hydroxyapatite-polyelectrolyte composites[END_REF]. The samples will be named as Ca (10-x) Mg x HAp-GA(n), where n is the value of the glutamic acid/(CaHAp) molar ratio (n= 10 and 20). Powder characterization N 2 adsorption-desorption isotherms were performed at 77 K using a Micromeritics ASAP 2000 instrument. The Brunauer-Emmett-Teller equation was used to calculate the specific surface area (S BET ). X-ray diffraction (XRD) analysis were carried out by means of a X'Pert Pro Panalytical X-pert diffractometer using Cu-Kα radiation (λ=1.5418 Å, with θ-θ geometry, equipped with an X'Celerator solid detector and a Ni filter). The 2θ range was from 20 to 70° with a step size Δ2θ=0.0167°. The experimental patterns were compared to standards compiled by the Joint Committee on Powder Diffraction and Standards (JCPDS cards) using the X'Pert High-Score Plus software [29]. The infrared (IR) adsorption analysis of the samples were obtained using a Spectrum Two 104462 IR spectrophotometer equipped with a diamond ATR setup in the range 4000-400 cm -1 . Nitrogen sorption isotherms for dried powders were recorded at 77K using a sorptometer EMS-53 and KELVIN 1040/1042 (Costech International). Points of Zero Charge (PZC) and Iso-Electric Point (IPE) of the samples are determined by zeta potential measurements, using a Malvern Nano ZS. Suspensions were prepared using NaCl (0.1M) as a background electrolyte with each powder using aqueous solutions, starting in an alkaline medium and stopping at pH = 4 under N 2 at 25°C [START_REF] Wu | Surface complexation of calcium minerals in aqueous solution[END_REF]. The titrations were carried out on suspensions of different samples of apatite obtained by adding 0.15 g of apatite to 30 mL of electrolyte (NaCl) and then 1.5 mL of 0.1M NaOH. The titrant used is 0.1 M hydrochloric acid prepared from 1M HCl, at the same ionic strength as the electrolyte by the addition of NaCl. The phosphorus and calcium contents were obtained by ICP-OES on a Horiba Jobin Yvon modele activa. The thermal analysis of the carbon was carried out using a SETARAM SETSYS 1750. Heating was performed in a platinum crucible in air flow at a rate of 10°C /min up to 800°C. For transmission electron microscopy (TEM) investigations, samples were prepared by dispersing the powders in a slurry of dry ethanol, deposited on a copper grid covered with a carbon thin film. High-resolution transmission electron microscopy (HRTEM) observations were performed on a JEOL JEM 2010 transmission electron microscope equipped with a LaB 6 filament and operating at 200 kV. The images were collected with a 4008 X 2672 pixels CCD camera (Gatan Orius SC1000). Circular dichroism (CD) experiments were performed at solid state using 5 mg of powder dispersed in nujol between NaCl pellets [START_REF] Castiglioni | Experimental Aspects of Solid State Circular Dichroism[END_REF]. The measurements were performed by a TASCO J-815 spectropolarimeter. The scans were recorded from 190 to 300 nm wavelength with the following parameters: 0.5 data pitch, 2 nm bandwidth, 100 nm/min scanning speed, and are the result of 3 accumulations. Results and discussion Elemental analysis The results of chemical analysis for mixed CaMgHAp with glutamic acid are reported in Table 1. The CaHAp sample shows a (Ca/P) molar ratio very close to the targeted stoichiometric value of 1.67. For the apatite series, the (Ca+Mg/P) ratio decreased from the starting CaHAp (1.68) to Ca 9 Mg 1 HAp-GA [START_REF] Li | Dopamine Modified Organic -Inorganic Hybrid Coating for Antimicrobial and Osteogenesis[END_REF] (1.55). The presence of the organic anion in the precipitated material is attested and quantified by the total carbon analysis. We can note the increase in carbon amount with the concentration of Mg, this great affinity of glutamic acid to mixed CaMgHAp could be explained by the high electronegativity of magnesium (χ Ca = 1, χ Mg = 1,31) [START_REF] Pauling | The Nature of the Chemical Bond[END_REF], in agreement with the results previously reported for CaCuHAp modified by polyaspartic acid [START_REF] Othmani | Surface modification of calcium-copper hydroxyapatites using polyaspartic acid[END_REF] and CaZnHAp modified by tartric acid [START_REF] Turki | Surface modification of zinccontaining hydroxyapatite by tartaric acid[END_REF]. The larger absorption of carbon in the CaMgHAp-GA would explain their loss of stoichiometry. This indicates that our samples are indeed hydroxyapatite-glutamic acid composites. For the samples CaHAp, Ca 9.5 Mg 0.5 HAp and Ca 9 Mg 1 HAp the increase in carbon amount with the concentration of Mg is explained by the disorder induced by the magnesium substitution promoting the incorporation of carbonates. Thermal analysis The (TG) curves of Ca 9 Mg 1 HAp, Ca 9 Mg 1 HAp-GA [START_REF] Hamad | Synthèse et étude physico-chimique de fluoroapatites mixtes à cations bivalents Pb-Cd, Pb-Sr et Sr-Cd[END_REF] and Ca 9 Mg 1 HAp-GA [START_REF] Li | Dopamine Modified Organic -Inorganic Hybrid Coating for Antimicrobial and Osteogenesis[END_REF] samples are reported in figure 1. The thermal decomposition shows a first weight loss between 50°C and 200°C, assigned to the removal of physisorbed water. The second one, between 200°C and 500°C, corresponds to the elimination of the organic matter of glutamic acid. The weight loss associated with this second process allows evaluation of the relative amount of glutamic acid in the composite hybrids. The values obtained for carbon expressed as wt% of the solid product are reported in Table 2. The relative amount of glutamic acid increases with its increasing concentration in the reaction with CaHAp. This result is in agreement with the increase of the percentage of carbon determined by chemical analysis. Figure 2 reports the differential thermal analysis curves (DTA) of the samples. These curves display an unexpected endothermic effect associated with water desorption. This effect was already reported for Mg modified hydroxyapatites [START_REF] Yasukawa | Preparation and characterization of magnesiumcalcium hydroxyapatites[END_REF][START_REF] Diallo-Garcia | Influence of Magnesium Substitution on the Basic Properties of Hydroxyapatites[END_REF]. It was assigned to the fact that heating in the presence of physisorbed water first induce a structuring effect. The latter may be related both to the surface relaxation at solid-water interface upon water release [START_REF] Ben Osman | Control of calcium accessibility over hydroxyapatite by post-precipitation steps: influence on the catalytic reactivity toward alcohols[END_REF] and to 6 the polarization process of OH groups from the columns known to occur at 200°C that initiates the proton mobility inside the columns [START_REF] Nakamura | Proton Transport Polarization and Depolarization of Hydroxyapatite Ceramics[END_REF]. An exothermal effect is observed in the temperature range 200-500°C with a peak top at 300°C for Ca9Mg1HAp-GA [START_REF] Hamad | Synthèse et étude physico-chimique de fluoroapatites mixtes à cations bivalents Pb-Cd, Pb-Sr et Sr-Cd[END_REF] and Ca9Mg1HAp-GA(20) samples. This peak, which is absent in the DTA plot of non-modified Ca9Mg1HAp, corresponds to the combustion of the organic material. Furthermore, the intensity of these peaks increases with increasing grafted amount content. In fact, their presence confirms that the prepared samples correspond to hydroxyapatite-glutamic acid composites, similar to that previously found for hydroxyapatite modified with glycine and sarcosine acids [START_REF] Ben Moussa | Hybrid organic-inorganic materials based on hydroxyapatite structure[END_REF]. Table 2 Results of TG analysis of ungrafted and grafted apatites. Samples Infrared investigation The IR spectra recorded with or without glutamic acid onto the apatite surface are illustrated in figure 4. The vibrations in the range 1800-1200 cm -1 are summarized in table 3. In particular, the bands observed at 1598, 1445 and 1261cm -1 (Fig. 4b,4c), which are not present in the spectrum of Ca 9 Mg 1 HAp (Fig. 4a) can be attributed to sorbed glutamic acid [START_REF] Navarrete | Vibrational study of aspartic acid and glutamic acid dipeptides[END_REF]. The vibration located at 1635 cm - 1 is attributed to adsorbed water [START_REF] Garcia-Ramos | The adsorption of acidic amino acids and homopolypeptides on hydroxyapatite[END_REF]. The higher water content in the presence of glutamic acid is consistent with the decrease in the ratio (Ca + Mg) / P. In order, to eliminate the hypothesis of a simple mechanical mixture between glutamic acid and apatite, further data have been recorded from such 20) sample. The FT-IR spectrum of this mixture, which is reported in Fig. 4d, displays a number of bands due to the two discrete components of the mixture, and indicates the absence of specific interaction between the carboxylic groups of the glutamic acid and the calcium ions of hydroxyapatite. The comparison with the spectra reported in Fig. 4b and4c, which show also a small shift of the carboxylic stretching band to lower wave numbers, in agreement with an increase of the C-O bond length can be attributed to those of the organic moieties grafted on calcium or magnesium atoms on the surface of the hydroxyapatite, in agreement with the results previously reported for CaHAp modified by amino-acids [START_REF] Ben Moussa | Hybrid organic-inorganic materials based on hydroxyapatite structure[END_REF][START_REF] Bachoua | Preparation and characterization of functionalized hybrid hydroxyapatite from phosphorite and its potential application to Pb 2+ remediation[END_REF]. Table 3 FTIR Spectral data (±5) of Ca 9 Mg 1 HAp-GA(10), Ca 9 Mg 1 HAp-GA [START_REF] Li | Dopamine Modified Organic -Inorganic Hybrid Coating for Antimicrobial and Osteogenesis[END_REF], and pure glutamic acid. Tentative assignments Wave number (cm -1 ) Glutamic acid [START_REF] Navarrete | Vibrational study of aspartic acid and glutamic acid dipeptides[END_REF] Asym.: Asymmetrical, str.: stretching X-ray analysis The X-ray powder diffractograms for CaHAp and Ca 9 Mg 1 HAp synthesized with or without the presence of glutamic acid are shown in figure 5. In table 4, we reported the size of the apatite crystallites induced by magnesium and glutamic acid for the reflections (002) and (310). For all samples, we observe a unique apatitic phase belonging to the P6 3 /m space group (n° 9-432-ICDD-PDF). We could not prepare the grafted apatites Ca 8.5 Mg 1.5 HAp-GA(n), since all our preparations failed. Several authors have shown that the particular behavior of the Ca 8.5 Mg 1.5 HAp compound can therefore be explained by the fact that sample Ca 8.5 Mg 1.5 HAp is a pure non stoichiometric CaMgHAp which would turn into whitlockite under the effect of temperature and already made up of a mixture of phases CaMgHAp crystalline and amorphous whitlockite [Ca 3y Mg y (HPO 4 ) z (PO 4 ) 2-2z/3 ], which is crystallized under the effect of heat treatment at 900 °C [START_REF] Chaudhry | Synthesis and characterization of magnesium substituted calcium phosphate bioceramic nanoparticles made via continuous hydrothermal flow synthesis[END_REF][START_REF] Correia | Wet synthesis and characterization of modified hydroxyapatite powders[END_REF]. Broadening of the diffraction lines increases with the concentration of magnesium and glutamic acid. The crystallite sizes were calculated from the broadening of the (0 0 2) and (3 1 0) using the Scherrer equation [START_REF] Bigi | Strontium-substituted hydroxyapatite nanocrystals[END_REF]: β 1/2 cosθ Kλ D hkl  Where  is the diffraction angle,  the wavelength and K a constant depending on the crystal (chosen as 0.9 for apatite crystallites) and β ½ is the line width at full width at half maximum (FWHM), of a given reflection. The line broadening of the (0 0 2) and (3 1 0) reflections was used to evaluate the crystallite size along the c-axis and along a direction perpendicular to it. The crystallinity (Xc) is defined as the fraction of the crystalline apatite phase in the investigated volume of powdered sample. An empirical relation between X c and β ½ was deduced, according to the following equation [START_REF] Ren | Characterization and structural analysis of zinc-substituted hydroxyapatites[END_REF]: X c = [K A / 1/2 ] 3 . Where K A is a constant set at 0.24 and β ½ is the FWHM of the (0 0 2) reflection, the 4. The crystallite size and the crystallinity decrease with increasing magnesium and amino acid concentration. It can also be deduced that the crystallites are of nanometric sizes and the decrease is more important in the (3 1 0) than in the (0 0 2) direction. Such observation was earlier reported for other organic moieties grafted onto apatite surfaces and can be explained by a better interaction of the glutamic acid with faces parallel to the c axis [START_REF] Boanini | Nanocomposites of hydroxyapatite with aspartic acid and glutamic acid and their interaction with osteoblast-like cells[END_REF][START_REF] Othmani | Surface modification of calcium-copper hydroxyapatites using polyaspartic acid[END_REF]. The individual effect of magnesium and glutamic acid on the crystallite size can be observed in figure 6. The crystallite sizes D (002) and D (310) decrease slowly with the concentration of magnesium whereas the addition of glutamic acid induces a bigger change in the crystallinity. It would appear that the glutamic acid is the component most responsible for the loss of crystallinity which could be explained by the presence of the groups COO -on the surface of materials. We do not observe a change in position of the peaks between CaHAp and Ca 9 Mg 1 HAp diffractograms, here the main phenomenon is the broadning of the peaks, which has already been observed during the substitution of calcium by another divalent ion. When glutamic acid is present, we observe no difference between Ca 9 Mg 1 HAp-GA [START_REF] Hamad | Synthèse et étude physico-chimique de fluoroapatites mixtes à cations bivalents Pb-Cd, Pb-Sr et Sr-Cd[END_REF] and Ca 9 Mg 1 HAp-GA(20) diffractograms. In contrast, peak shifts are observed compared to the CaHAp and Ca 9 Mg 1 HAp one, in particular for the (211) and ( 222) peaks. This phenomenon that we have already described was attributed to a better interaction of the glutamic acid with the faces parallel to the axis c during the process of crystalline growth [START_REF] Ben Moussa | Hybrid organic-inorganic materials based on hydroxyapatite structure[END_REF]. TEM observations Transmission electron microscopy (TEM) analysis micrographs of the samples are illustrated in figure 7A. From the photomicrographs, it can be seen that the size of precipitated apatite particles, prepared with and without glutamic acid, is on the nanometer scale. CaHAp is constituted of well dispersed plate-shaped crystals with an average size of about 40-150 nm long and about 30 nm wide. A small addition of magnesium induces a decrease of the size: 30-60 nm long and about 15-20 nm wide for the Ca 9.5 Mg 0.5 HAp. The presence of magnesium and glutamic acid in the start solution completely modifies the aspect. We obtain large bundles of CaMgHAp-GA fibers (300 nm length /80 nm width). The HRTEM images (Fig. 7B) reveal that theses fibers are thin (15 nm wide) and stacked together to each other. The analysis of the FFT patterns show that the growth of the particles always occurs in the (002) direction, indicating that they are growing along the c axis direction such as the [[D (310) ] Mgx, GA(n) / [D (310) ] CaHAp ]% 0 0,2 0,4 0,6 0,8 1 Functionalization of CaHAp with glutamic acid Speciation of the interface apatite-solution The influence of the glutamic acid amount on the textural properties of CaHAp was examined (Table 5). We obtained 43 m 2 /g for Ca 9 Mg 1 HAp and after functionalization 30 m 2 /g and 24 m 2 /g for Ca 9 Mg 1 HAp-GA [START_REF] Hamad | Synthèse et étude physico-chimique de fluoroapatites mixtes à cations bivalents Pb-Cd, Pb-Sr et Sr-Cd[END_REF] and Ca 9 Mg 1 HAp-GA [START_REF] Li | Dopamine Modified Organic -Inorganic Hybrid Coating for Antimicrobial and Osteogenesis[END_REF], respectively. The decrease in the specific surface area of the apatite samples is related to the structural arrangement of the 2-aminopentanedioic acid on the surface of the solid. These results are in good agreement with those obtained by Oberto Da Silva et al. [START_REF] Silva | Hydroxyapatite organofunctionalized with salivating agents to heavy cation removal[END_REF]. The initial pH i value measured in aqueous solutions increase with the glutamic acid ratio. We recorded 8.9 and 9.3 for Ca 9 Mg 1 HAp-GA [START_REF] Hamad | Synthèse et étude physico-chimique de fluoroapatites mixtes à cations bivalents Pb-Cd, Pb-Sr et Sr-Cd[END_REF] and Ca 9 Mg 1 HAp-GA(20) respectively against only 8.4 for Ca 9 Mg 1 HAp. The Zeta potential of different apatite samples is presented in figure 8. The values of Point of Zero Charge (PZC) and iso-electric point (IEP) were determined (Table 5). In a basic medium, the Ca 9 Mg 1 HAp sample is characterized by a highly negative zeta potential indicating a deficit of positive surface charge, a value lower than that of the apatites Ca 9 Mg 1 HAp-GA [START_REF] Hamad | Synthèse et étude physico-chimique de fluoroapatites mixtes à cations bivalents Pb-Cd, Pb-Sr et Sr-Cd[END_REF] and Ca 9 Mg 1 HAp-GA [START_REF] Li | Dopamine Modified Organic -Inorganic Hybrid Coating for Antimicrobial and Osteogenesis[END_REF]. This result indicates an increase in positive surface charge sites, on amino-acid grafted apatites [START_REF] Kollath | A Modular Approach To Study Protein Adsorption on Surface Modified Hydroxyapatite[END_REF]. Whereas the value of pH IEP is 7. The total number of protons consumed during titrations of different samples is determined using this equation:                                    OH ) pH ) Ke ( Log ( 10 H pH 10 V blanc ) b V b C a V a C ( susp ) b V b C a V a C exp S 8(B) ). This quantity of protons (H + ) is deduced from the difference between the amount of (H + ) added to the suspension and the amount of free (H + ) in solution. This is calculated directly from the measured pH. The amount of the added protons is corrected by the amount of hydroxyl initially added to the suspension. The proton quantity consumed by apatite samples [H + ] shows the same evolution: in an acid medium, [H + ] is between 9 and 12 μmol/m² (pH = 5) for Ca 9 Mg 1 HAp, Ca 9 Mg 1 HAp-GA [START_REF] Hamad | Synthèse et étude physico-chimique de fluoroapatites mixtes à cations bivalents Pb-Cd, Pb-Sr et Sr-Cd[END_REF] and Ca 9 Mg 1 HAp-GA [START_REF] Li | Dopamine Modified Organic -Inorganic Hybrid Coating for Antimicrobial and Osteogenesis[END_REF], respectively. Then, it decreases regularly and becomes negative in a basic medium with the protons consumed in the order of -8 and -10 μmol/m² at pH = 11. Proposed glutamic acid sorption mechanism The IR spectra provide data about the ionization state of the carboxylate groups grafted onto the apatite surface. Indeed, the presence of bands characteristic of -COO -groups and the absence of bands attributed to COOH groups (1700 cm -1 ) indicate the carboxylate form. This shows that the interaction is mainly due to the electrostatic interaction between -COO -groups of the glutamic acid and the calcium Ca 2+ /Mg 2+ ions of the hydroxyapatite. We cannot exclude also that interactions between COO -and surface POH groups are possible. The fixation is due to the simultaneous presence of -COO-/Ca 2+ / Mg 2+ electrostatic interactions and H-bonds between NH 3 + protons and surface oxygen atoms of the PO 4 group [START_REF] Almora-Barrios | Density functional theory study of the binding of glycine, proline, and hydroxyproline to the Hydroxyapatite (001) and (010) surfaces[END_REF][START_REF] Rimola | Ab initiomodelling of protein/biomaterial interactions: glycine adsorption at hydroxyapatite surfaces[END_REF]. The nature of interactions between the apatite surface and a glutamic acid depends on the pH value of the medium. Taking into consideration that the reaction is carried out at a pH above 9.5, we can consider that in aqueous solution glutamic acid exists as a carboxylate ion, the amino group being neutral [57]. Under the same conditions, the apatite surface is considered as negatively charged, therefore some authors consider that the electrostatic interactions between the surface and the aminoacid are very weak [START_REF] Palazzo | Amino acid synergetic effect on structure, morphology and surface properties of biomimetic apatite nanocrystals[END_REF][START_REF] Brown | An analysis of hydroxyapatite surface layer formation[END_REF]. In the present study, glutamic molecules are present in the form of carboxylate ions, which can lead to calcium complexes that participate in the formation of the crystalline edifice. The observation of the IR spectra, showing characteristic vibrations of carboxylate salts, lead us to conclude that the amino-acid in this carboxylate ionic form exchanges an hydroxyl ion. Different model grafting mechanisms proposed based on the results obtained are shown in scheme 1. Scheme 1. Formation of Ca carboxylate salt leading to the grafting of glutamic on the CaHAp surface. Conclusion In conclusion, we have successfully synthesized hydroxyapatite-glutamic composites of different glutamic acid content using the hydrothermal method. The presence of the glutamic acid and/or magnesium in the reaction solution does not change the apatite structure, but reduces the crystallinity and the crystallite sizes. According to IR spectroscopy, the new vibrations after adsorption can be attributed to those of the organic moieties grafted on calcium or magnesium atoms onto the surface of FirstFig. 2 . 2 Fig. 2. DTA plots; (a) Ca 9 Mg 1 HAp, (b) Ca 9 Mg 1 HAp-GA(10) and (c) Ca 9 Mg 1 HAp-GA(20). Fig. 1 . 1 Fig. 1. TG plots; (a) Ca 9 Mg 1 HAp, (b) Ca 9 Mg 1 HAp -GA(10) and (c) Ca 9 Mg 1 HAp-GA(20). Fig. 3 . 3 Fig. 3. (Left) Circular dichroism spectrum and (right) UV absorption curves of (a) glutamic acid and (b) Ca 9 Mg 1 HAp-GA(20). 9 9 Mg 1 HAp and glutamic acid in the same relative amounts as those contained in the Ca 9 Mg 1 HAp-GA( Fig. 4 . 4 Fig. 4. FT-IR spectra of: (a) Ca 9 Mg 1 HAp, (b) Ca 9 Mg 1 HAp-GA(10), (c) Ca 9 Mg 1 HAp-GA(20), (d) Mixed Ca 9 Mg 1 HAp-glutamic acid and (e) glutamic acid. Ca 9 Mg 1 HAp-GA(10) Ca 9 Mg 1 HAp-GA(20) COO-asym.str. CaHAp Ca 9 9 Mg 1 HAp Ca 9 Mg 1 HAp-GA[START_REF] Hamad | Synthèse et étude physico-chimique de fluoroapatites mixtes à cations bivalents Pb-Cd, Pb-Sr et Sr-Cd[END_REF] Ca 9 Mg 1 HAp-GA[START_REF] Li | Dopamine Modified Organic -Inorganic Hybrid Coating for Antimicrobial and Osteogenesis[END_REF] Fig. 5 . 5 Fig. 5. X-ray diffractograms for CaHAp and Ca 9 Mg 1 HAp ungrafted and grafted. Fig. 6 . 6 Fig. 6. Effect of the Mg and GA concentration on the crystallite size D (002) (A) and D (310) (B) relative to the CaHAp values taken as references. Fig. 7B . 7B Fig. 7A. TEM images (Scale bars=50 nm) of CaHAp, Ca 9.5 Mg 0.5 HAp, Ca 9 Mg 1 HAp-GA(10) and Ca 9 Mg 1 HAp-GA(20). specific surface area (m 2 /g) exposed in the suspension, V is total volume of solution; C a and C b are the HCl and NaOH concentration used for titration. K e is the dissociation constant of water. ( H+ ) and ( OH-) are the coefficient of the dissociation activity of H + and OH -calculated by Debye-strength of solution (mol/L), A=0.507, B=0.328.10 -8 and  i 0 is the effective diameter (Fig. Fig. 8 . 8 Fig. 8. (A) Zeta potential IEP (in mV) as a function of pH spectra of the as-prepared Ca 9 Mg 1 HAp-GA(n) (n=10 and 20) and sample Ca 9 Mg 1 HAp. (B) Number of protons consumed by the surface for different samples reported in µmol/m 2 during their 2 hours immersion in a solution containing 0.1 M NaCl + HCl or NaOH as a function of pH of the as-prepared Ca 9 Mg 1 HAp-GA(n) (n=10 and 20) and sample Ca 9 Mg 1 HAp. the apatite. TEM images confirm the reduction of crystallite sizes and indicate the change in its morphology. Table 1 1 Chemical composition (% weight ±0.02) of grafted mixed CaMgHAp. Samples %Ca %Mg %P %C (Ca + Mg)/P CaHAp 39.31 - 18.09 0.12 1.68 Ca 9.5 Mg 0.5 HA 38.14 0.98 18.66 0.15 1.65 p Ca 9 Mg 1 HAp 35.28 1.82 18.01 0.18 1.64 CaHAp-GA(10) 39.05 - 18.15 0.72 1.66 Ca 9.5 Mg 0.5 HAp- 37.94 1.08 18.63 0.87 1.64 GA(10) Ca 9 Mg 1 HAp- 34.05 1.89 17.98 1.12 1.59 GA(10) CaHAp-GA(20) 38.97 - 18.23 1.25 1.65 Ca 9.5 Mg 0.5 HAp- 37.23 1.13 18.58 1.49 1.63 GA(20) Ca 9 Mg 1 HAp- 32.51 2.03 17.88 2.58 1.55 GA(20) Table 4 : 4 Evolution size of the apatite crystallites with magnesium and glutamic acid for the reflexion (002) and (310). Samples β 1/2 (002) D (002) (Å) β 1/2 (310) D (310) (Å) Crystallinity (X C ) CaHAp 0.191(1) 427 0.477(9) 177 1.981 Ca 9.5 Mg 0.5 HAp 0.207(2) 394 0.640(1) 132 1.554 Ca 9 Mg 1 HAp 0.264(2) 309 0.712(1) 119 0.749 CaHAp-GA(10) 0.230(4) 354 0.710(4) 119 1.042 Ca 9.5 Mg 0.5 HAp-GA(10) 0.283(4) 288 0.931(3) 91 0.607 Ca 9 Mg 1 HAp-GA(10) 0.364(5) 224 1.286(3) 66 0.285 CaHAp-GA(20) 0.233(1) 350 0.761(2) 111 1.029 Ca 9.5 Mg 0.5 HAp-GA(20) 0.322(3) 253 1.084(1) 75 0.413 Ca 9 Mg 1 HAp-GA(20) 0.389(2) 210 1.385(1) 61 0.234 Table 5 5 Surface area and pH value of different apatite samples (c) (d) 002 100 Samples S BET (m 2 /g) IEP PZC pH i in aqueous solution Ca 9 Mg 1 HAp 43 7.1 8.2 8.4 Ca 9 Mg 1 Hap-GA(10) 30 8.2 8.6 8.9 Ca 9 Mg 1 Hap-GA(20) 24 8.6 8.8 9.3 PZC: point zero charge is determined by an acid-basic titration equilibrated at 16h IEP: Iso-electric point (mV) is determined by zeta-metry 1, the PZC for the Ca 9 Mg 1 HAp sample is around 8.2. The same value is found in the literature[START_REF] Bell | The point of zero charge of hydroxyapatite and fluorapatite in aqueous solutions[END_REF][START_REF] Attia | The equilibrium composition of hydroxyapatite and fluoroapatitewater interfaces[END_REF][START_REF] Saleeb | Surface properties of alkaline earth apatites[END_REF][START_REF] Wu | Surface complexation of calcium minerals in aqueous solution[END_REF] (Figure8(B)). This result can be explained by the presence of initial charge and specific adsorption at the surface of apatite. After glutamic acid functionalization, the Zeta-potential curve is shifted towards alkaline pH values, with pH IEP of 8.2 and 8.6 for Ca 9 Mg 1 HAp-GA[START_REF] Hamad | Synthèse et étude physico-chimique de fluoroapatites mixtes à cations bivalents Pb-Cd, Pb-Sr et Sr-Cd[END_REF] and Ca 9 Mg 1 HAp-GA[START_REF] Li | Dopamine Modified Organic -Inorganic Hybrid Coating for Antimicrobial and Osteogenesis[END_REF], respectively. This result verified by measurement of PZC (from 8.6 to 8.8), confirms the functionalization of the surface of Ca 9 Mg 1 HAp by the glutamic acid. Acknowledgments This research was carried out with the financial support of the University of Monastir, (Tunisia); the University Pierre and Marie Curie and CNRS (France).

01744738

en

2024/03/05 22:32:07

2018

https://hal.science/hal-01744738/file/EXME2018a-ccsd.pdf

Benoit Voillot email: [email protected] Jean-Lou Lebrun René Billardon François Hild J.-L Lebrun F Hild Lmt Validation of registration techniques applied to XRD signals for stress evaluations in titanium alloys Keywords: DIC, in-situ test, integrated methods, stress analyses, XRD To estimate stresses near specimen surfaces, X-ray diffraction (XRD) is applied to titanium alloys. Some of these alloys are difficult to study since they are composed of various phases of different proportions, shapes and scales. For millimetric probed volumes, such multi-phase microstructures induce shallow and noisy diffraction signals. Two peak registration techniques are introduced and validated thanks to tensile tests performed on two titanium alloy samples. Introduction High performance titanium alloys are used in aeronautical industries and especially large forgings (e.g., landing gears). These critical parts of an aircraft are submitted to various heat and mechanical treatments [START_REF] Hill | Engineering residual stress in aerospace forgings[END_REF]. One of the consequences are changes in microstructure, roughness and residual stresses [START_REF] Cox | The effect of finish milling on the surface integrity and surface microstructure in Ti-5Al-5[END_REF][START_REF] Aeby-Gautier | Isothermal α formation in β metastable titanium alloys[END_REF]. Each of these parameters has an impact on the global in-service mechanical behavior of the whole structure and in particular in fatigue [START_REF] Lütjering | Titanium[END_REF][START_REF] Guillemot | Prediction of the endurance limit taking account of the microgeometry after finishing milling[END_REF][START_REF] Souto-Lebel | Characterization and influence of defect size distribution induced by ball-end finishing milling on fatigue life[END_REF]. It is of high interest to be able to describe the effect of these treatments by several ways (e.g., microstructure studies, roughness measurements). Another way to quantify these treatments is via residual stress analyses. Various methods exist to estimate residual stresses. Most of them are destructive (e.g., incremental hole [START_REF]ASTM E[END_REF], contour methods [START_REF] Prime | Cross-sectional mapping of residual stresses by measuring the surface contour after a cut[END_REF]). To carry out analyses on in-service structures, nondestructive methods need to be performed. One of the most popular nondestructive stress analysis techniques is X-ray diffraction (XRD) [START_REF] Reed Reed | The influence of surface residual stress on fatigue limit of titanium[END_REF][START_REF] Fitzpatrick | Determination of residual stresses by x-ray diffraction -issue 2[END_REF][START_REF] Freour | Determination of the macroscopic elastic constants of a phase embedded in a multiphase polycrystal -application to the beta-phase of Ti17 titanium based alloy[END_REF][START_REF]Non-destructive Testing, Test Method for Residual Stress analysis by X-ray Diffraction[END_REF][START_REF] Cullity | Elements of x-ray diffraction[END_REF][START_REF] Hauk | Structural and Residual Stress Analysis by Non Destructive Methods: Evaluation, Application, Assessment[END_REF]. Various post-processing algorithms are used to evaluate stresses from XRD measurements [START_REF] Pfeiffer | Evaluation of thickness and residual stress of shallow surface regions from diffraction profiles[END_REF][START_REF]Non-destructive Testing, Test Method for Residual Stress analysis by X-ray Diffraction[END_REF]. Among them the centered barycenter is common [START_REF]Non-destructive Testing, Test Method for Residual Stress analysis by X-ray Diffraction[END_REF][START_REF] Cullity | Elements of x-ray diffraction[END_REF][START_REF] Hauk | Structural and Residual Stress Analysis by Non Destructive Methods: Evaluation, Application, Assessment[END_REF]. It consists in determining positions of the diffraction peak by the sliding center of gravity of all the points in its neighborhood. This method is very powerful for well-defined peaks. However it is known that stress analyses are challenging for titanium alloys, especially multi-phase grades [START_REF] Lefebvre | External reference samples for residual stress analysis by x-ray diffraction[END_REF][START_REF] Suominen | Residual Stress Measurement of Ti-Metal Samples by Means of XRD with Ti and Cu Radiation[END_REF]. This difficulty comes from high levels of fluorescence and high noise to signal ratios. As a result, it is difficult to accurately locate diffraction peaks [START_REF] Withers | The precision of diffraction peak location[END_REF]. Other post-processing methods are currently used to register diffraction peaks. Peak to peak registrations are also utilized in some cases [START_REF]Non-destructive Testing, Test Method for Residual Stress analysis by X-ray Diffraction[END_REF]. A prerequisite is good similarity between peaks, which is not necessarily very suitable for two-phase materials or materials whose fluorescence varies with the angle of measurement. The latter phenomenon is observed in two-phase titanium alloys [START_REF] Cullity | Elements of x-ray diffraction[END_REF][START_REF] Lütjering | Titanium[END_REF]. All these methods are implemented in most of commercial codes (e.g., Stressdiff [START_REF]Stressdiff[END_REF] or Leptos by Bruker [START_REF] By | [END_REF]). The last method used in existing commercial codes consists in modelling peaks with a mathematical function (e.g., Lorentz, Pearson VII, pseudo-Voigt or Gauss distributions [START_REF]Non-destructive Testing, Test Method for Residual Stress analysis by X-ray Diffraction[END_REF]). This post-processing method (i.e., model with a known function) will be used to benchmark the integrated approach proposed herein. Stress analyses are most of the time benchmarked with stress-free configurations (i.e., powder of the material [START_REF]Non-destructive Testing, Test Method for Residual Stress analysis by X-ray Diffraction[END_REF][START_REF] Noyan | Residual Stresses: Measurements by Diffraction and Interpretation[END_REF]) or standard coupon [START_REF] Lefebvre | External reference samples for residual stress analysis by x-ray diffraction[END_REF]. One of the issues with the former is finding a powder whose lattice structure is close to those of the material of interest. In some cases such as multi-phase materials, it is hard or impossible to get powder of the real material. An alternative route is to monitor a mechanical test with XRD means. Tools used for evaluating residual (i.e., ex-situ) stresses can also determine applied (i.e., in-situ) stresses [START_REF] Geandier | Elastic strain distribution in metallic filmpolymer substrate composites[END_REF] during mechanical tests. Either the testing machine is put within the goniometer [START_REF] Geandier | Development of a synchrotron biaxial tensile device for in-situ characterization of thin films mechanical response[END_REF] or the goniometer is mounted inside the testing machine [START_REF] Rekik | Dispositif de mesure du comportement magnéto-mécanique d'un alliage de fer-silicium sous chargement mécanique multiaxial[END_REF]. This methodology will be used to validate the stress estimates reported herein. Further, it will also be combined with 2D digital image correlation [START_REF] Sutton | Image correlation for shape, motion and deformation measurements: Basic Concepts, Theory and Applications[END_REF] to monitor surface displacements. Such combination has been used to validate XRD measurements and determine stress/strain curves in biaxial experiments on thin films [START_REF] Djaziri | Combined synchrotron x-ray and image-correlation analyses of biaxially deformed w/cu nanocomposite thin films on kapton[END_REF][START_REF] Djaziri | Investigation of the elastic-plastic transition of nanostructured thin film under controlled biaxial deformation[END_REF]. To address the issue of high noise to signal ratio associated with titanium alloys, it is proposed to require the peak shifts to be expressed in terms of the quantities of interest, namely, the sought elastic strains (or stresses). When a peak registration procedure is used it corresponds to an integrated approach as used in digital image correlation [START_REF] Hild | Digital image correlation: From measurement to identification of elastic propertiesa review[END_REF][START_REF] Roux | Stress intensity factor measurements from digital image correlation: post-processing and integrated approaches[END_REF][START_REF] Leclerc | Integrated digital image correlation for the identification of mechanical properties[END_REF][START_REF] Réthoré | A fully integrated noise robust strategy for the identification of constitutive laws from digital images[END_REF], stereocorrelation [START_REF] Réthoré | Robust identification of elasto-plastic constitutive law parameters from digital images using {3D} kinematics[END_REF][START_REF] Beaubier | CAD-based calibration of a 3D-DIC system: Principle and application on test and industrial parts[END_REF][START_REF] Dufour | CAD-based displacement measurements. Principle and first validations[END_REF][START_REF] Dufour | Shape, Displacement and Mechanical Properties from Isogeometric Multiview Stereocorrelation[END_REF] and digital volume correlation [START_REF] Hild | Toward 4d mechanical correlation[END_REF]. The outline of the paper is as follows. First, the studied alloys and the experimental setup are presented. Then the two registration techniques used herein are introduced. Last, the results obtained on the two titanium alloys are analyzed and validated. Studied alloys Two two-phase titanium alloys are studied herein. First, Ti64 alloy is selected for benchmark purposes because it is well-known and its microstructure is favorable for XRD measurements [START_REF] Lefebvre | External reference samples for residual stress analysis by x-ray diffraction[END_REF][START_REF] Suominen | Residual Stress Measurement of Ti-Metal Samples by Means of XRD with Ti and Cu Radiation[END_REF]. Second, Ti5553 is a two-phase material used by Safran Landing Systems for landing gears [START_REF] Boyer | The use of beta titanium alloys in the aerospace industry[END_REF]. The composition of Ti64 is given in Table 1. It is composed of 6 wt% of Aluminum and 4 wt% of Vanadium. It is a quasi pure α-phase (i.e., 95 wt% of α-phase and only 5 wt% of β-phase). The α-phase is hexagonal close packed (HCP) and the β-phase is body centered cubic (BCC). Table 1: Chemical composition of Ti64 Alloying elements of Ti5553 are mostly composed of 5 wt% of Aluminum, 5 wt% of Vanadium, 5 wt% of Molybdenum, and 3 wt% of Chromium (Table 2). It is a quasi β-metastable material. It contains 60 wt% of α-phase and 40 wt% of β-phase. Such metastable alloys are of interest for aeronautical applications thanks to their high specific strength. Only the α-phase will be considered in XRD measurements. The same lattice family (i.e., {213}-planes) will be studied. The proportion of β-phase in Ti64 can be neglected. This is not the case for Ti5553 alloys for which the stress evaluation will be incomplete. Figure 1 shows micrographs of both alloys. For Ti64 the α-grains are observed as well as the former β-grains. For Ti5553 the matrix of the primary β-phase appears in gray and contains primary and secondary α-nodules (a few µm in diameter) and lamellae (at sub-µm scale) in black [START_REF]Standard Test Method for Determining Average Grain Size[END_REF]. The coexistence of these two phases is an issue for XRD analyses [START_REF] Lu | Handbook of measurement of residual stresses[END_REF][START_REF] Voillot | Evaluation of residual stresses due to mechanical treatment of Ti5553 alloy via XRD[END_REF]. Figure 3 shows that the size of β-grains is millimetric. This dimension is comparable to the extent of the volume probed by the X-ray beam (i.e., 1 mm in diameter for a depth of 5 µm, see Section 3). For one XRD analysis, only few grains of the β-phase are analyzed. It results that the β-phase lattices of the probed volume are not always in Bragg's conditions. Consequently, the β-phase is not adequate for stress analyses. This EBSD picture also reveals that there is no significant texture. The orientations of β-grains are not linked to one another. The α-phase made of nodules and lamellae do not show any preferential orientation with the β-grain in which they are nested [START_REF] Duval | Mechanical Properties and Strain Mechanisms Analysis in Ti5553 Titanium Alloy[END_REF]. This is of prime importance for XRD measurements since a random orientation of α-grains in the interaction volume ensures that Bragg's condition will be satisfied for the α-phase in any probed direction. A complete stress estimation is possible with a lab goniometer [START_REF]Non-destructive Testing, Test Method for Residual Stress analysis by X-ray Diffraction[END_REF][START_REF] Hauk | Structural and Residual Stress Analysis by Non Destructive Methods: Evaluation, Application, Assessment[END_REF] for this phase. 3 and4). Consequently, it is expected that the peak positions are more difficult to determine with Ti5553 than Ti64 alloys, which will presumably have an impact on stress evaluations [START_REF] Withers | The precision of diffraction peak location[END_REF][START_REF] Voillot | Evaluation des contraintes residuelles induites par traitements mecaniques dans un alliage de titane bi-phase par trois methodes de depouillement differentes[END_REF]. The aim of the following analyses is to assess the feasibility of XRD analyses under such challenging conditions. 3 Experimental setup The experimental setup consists of a biaxial testing machine [START_REF] Bertin | Integrated digital image correlation applied to elasto-plastic identification in a biaxial experiment[END_REF] mounted in the X-ray goniometer (Figure 6). The advantage of such testing machine, which can also be used in an SEM chamber, is that two actuators are used to load the sample. In the chosen control mode, the center of the sample is motionless. It was checked (a posteriori) with DIC analyses that the maximum motion was less than 10 µm, which is negligible with respect to the probed surface (i.e., 1 mm 2 ). The testing machine is moved and relocated into the goniometer at each step of loading in order to acquire images for DIC purposes. The goniometer used in the present work follows the χ-method [START_REF]Non-destructive Testing, Test Method for Residual Stress analysis by X-ray Diffraction[END_REF], see Figure 7. The radius of the goniometer is 150 mm. It is equipped with a mobile head composed of the X-Ray source (X) and a single linear position sensor (LPS) that turns about a motionless coupon to be analyzed. The LPS sensor gives a 1D measurement of diffraction intensity. S1, S2, S3 are the basis axes corresponding to the analyzed sample S. Here as the coupon is motionless during all the measurements, the basis corresponds to the goniometer reference axes. 2θ is the diffraction angle between the incident and diffracted beams. For diffraction in titanium alloys, a copper source was selected. The use of a collimator gives a probed volume of the order of 1 mm 2 × 5 µm. A Nickel (K β ) filter is added in front of the LPS sensor to only analyze the K α rays of the copper source. The duration of X-Ray exposure (i.e., 3 min per diffractogram), intensity and voltage (i.e., 20 mA and 40 kV) are a compromise between duration of measurement and diffraction peak quality in the current conditions (Figure 5). To have access to various zones on the diffraction sphere, rotations are made possible in the Eulerian cradle of the goniometer as illustrated in Figure 8. Along the S3 axis (i.e., normal to the sample surface and the vertical axis of the goniometer) various angles φ are reached. They correspond to the main direction of measurement. The rotation along the S1 axis allows various angles χ or ψ to be probed. For one single stress analysis, thirteen diffractograms at various angular positions ψ ranging from -50 • and 50 • are selected [START_REF]Non-destructive Testing, Test Method for Residual Stress analysis by X-ray Diffraction[END_REF]. It is worth noting that the goniometer does not allow for oscillations during measurements. When the moving head is turning within the goniometer, the center of the surface area of the volume probed by the X-ray beam is moving by less than 0.1 mm. This observation ensures that the same volume of material is always impacted by X-rays at each angle of measurement. To check the errors due to the use of the goniometer, an analysis on a stress-free powder made of pure titanium was carried out before and after each measurement campaign. All evaluated stress components varied about 0 MPa with a standard deviation of 10 MPa. Further, a height change of 50 µm of the coupon led to an error in stress of about 4 MPa. This stability of the beam interaction with the probed surface and the verification of stress-free evaluations with a Ti powder are indications of good working conditions for stress analyses. In order to validate the stress analyses performed on titanium alloys, in particular the new integrated method, dog-bone samples (Figure 9) were machined. The coupon has been designed to ensure a controllable and homogenous stress and strain area in the central area of the coupon where X-ray analyses are performed. The aim is to carry out in-situ tensile tests. It is an alternative way to benchmark stress analyses via XRD. Usual validations are carried out with standard coupons [START_REF] Lefebvre | External reference samples for residual stress analysis by x-ray diffraction[END_REF] or powders assumed to be stress-free. For pure titanium, powders exist and were proven to yield good results for the goniometer used herein [START_REF]Non-destructive Testing, Test Method for Residual Stress analysis by X-ray Diffraction[END_REF]. However, the lattice parameters of pure titanium are slightly different for Ti5553 or Ti64 alloys. The position and quality of peaks also vary (Figure 5). Consequently, having reliable results on titanium powders does not necessarily ensure trustworthy estimates for titanium alloys. The Ti64 grade will be the reference material since it is more suitable to XRD analyses. The Ti5553 alloy will be subsequently studied once the registration procedures have been validated. In-situ uniaxial tensile tests are carried out under a load control mode. The direction φ is chosen to coincide with the longitudinal direction S1 of the coupon (Figures 7 and9). To measure the total strains on the sample surface, 2D DIC will be used. The sample surface is coated with black and white paints in order to create a speckle pattern (Figure 9(b)). To avoid any bias, the analyzed XRD zone is not covered. When the targeted load level is reached, the sample and testing machine are moved out of the goniometer to acquire pictures. A telecentric lens is mounted on the digital camera to minimize as much as possible the spurious effects associated with out of plane motions. The actuators of the testing machine induce stress variations less than 16 MPa (standard deviation) for the investigated stress range. An isostatic set-up ensures the testing machine to be positioned very precisely in the goniometer. The angular standard uncertainty in relocation is less than 2 • , which corresponds to a small error in final stress estimations. The spatial repositioning error is less than 0.1 mm, a value ten times smaller than the probed surface diameter. Stress extraction When a material is loaded or has residual stresses, crystal lattices deform. XRD procedures measure the variations of inter-reticular distances by analyzing diffraction peaks in (poly)crystalline materials [START_REF] Hauk | Structural and Residual Stress Analysis by Non Destructive Methods: Evaluation, Application, Assessment[END_REF][START_REF] Noyan | Residual Stresses: Measurements by Diffraction and Interpretation[END_REF][START_REF] Cullity | Elements of x-ray diffraction[END_REF][START_REF] Lu | Handbook of measurement of residual stresses[END_REF]]. In the case of titanium alloys analyzed by a copper source, the peak commonly used to carry out stress analyses diffracts at an angle ≈ 140 • [10] and corresponds to the {213} family of the α-phase, which can be found in both Ti64 and Ti5553 alloys [START_REF] Lefebvre | External reference samples for residual stress analysis by x-ray diffraction[END_REF][START_REF] Boyer | The use of beta titanium alloys in the aerospace industry[END_REF][START_REF] Aeby-Gautier | Isothermal α formation in β metastable titanium alloys[END_REF]. The inter-reticular distance for this angle is given by the dimension of the HCP crystal structure where the edges of the hexagon are a = 0.295 nm and the height of the lattice is c = 0.468 nm [START_REF] Leyens | Titanium and titanium alloys: fundamentals and applications[END_REF]. The link between diffraction peak angle and inter-reticular distances is given by Bragg's law 2d {213} sin θ = λ KαCu (1) where d {213} is the inter-reticular distance for {213} planes of the α-phase, θ the diffraction angle, and λ KαCu the wavelength of the X-ray beam (e.g., copper source with λ KαCu = 0.154 nm). Peak registrations presented in the following do not dissociate K α1 and K α2 rays. Elastic stresses near the sample surface in α-phases are evaluated via XRD by estimating peak shifts [START_REF]Non-destructive Testing, Test Method for Residual Stress analysis by X-ray Diffraction[END_REF][START_REF] Hauk | Structural and Residual Stress Analysis by Non Destructive Methods: Evaluation, Application, Assessment[END_REF]. From channels to stresses Figure 10 describes the whole procedure used from the acquisition of diffractograms by the goniometer equipped with a linear position sensor (LPS) that gives an information expressed in channels x to stress estimations. and angular position depends on the geometry of the goniometer (Figure 11) 2θ(x) = 2θ ref + tan -1 L x -x ref x max (2) with x ref = x max + 1 2 (3) where 2θ ref is the angle corresponding to the central channel of the linear sensor (this angle is equal to 140 • thanks to the use of Ti powder). The ratio /L is a geometric parameter of the goniometer (i.e., it depends on the length of the LPS sensor, = 50 mm, and the distance between the analyzed surface and the sensor, L = 150 mm). The angles φ and ψ determine the location of the goniometer during one single measurement with φ the main measurement direction and ψ the incidence angle of the X-ray source and sensor (see Figures 7 and8). the interreticular distance for the undeformed configuration, and 2θ 0 the corresponding diffraction angle. To get an estimate of the stress tensor in one direction, several measurements (i.e., angular positions ψ) are needed for each point of analysis [START_REF]Non-destructive Testing, Test Method for Residual Stress analysis by X-ray Diffraction[END_REF]. Inter-reticular distances d φψ and corresponding angles θ φψ depend on the main measurement direction φ but also on the incidence angle of the X-ray source ψ. As measurements are carried out very close to the sample surface, the normal stress component perpendicular to the surface (along the axis S3, see Figure 7) will be assumed to vanish (σ 33 = 0), in order to be consistent with a traction-free surface. The relationship between stress and elastic strain components is then given by ˜ φψ = - 1 2 S {213} 2 σ φ sin 2 ψ - 1 2 S {213} 2 τ φ sin 2ψ + ˜ ψ=0 (5) where ˜ φψ = ln(sin θ φψ ), S are invariant for any probed angle for the α-phase. It is worth noting that with the present setting, the knowledge of θ 0 is not needed when using ˜ φψ instead of φψ . However, to probe the consistency of the results, it will be checked a posteriori when analyzing tensile tests (in particular, the value of θ 0 ). In the present case, the X-Ray elasticity parameters S 1 = - ν α E α and 1 2 S 2 = 1 + ν α E α depend on the Poisson's ratio ν α and Young's modulus E α of the α-phase (i.e., 1 2 S 2 = 11.9 × 10 -6 MPa -1 and S 1 = -2.64 × 10 -6 MPa -1 [START_REF] Fréour | Influence of a two-phase microstructure on XEC and XRD stress analysis[END_REF][START_REF] Bruno | Surface and Bulk Residual Stress in Ti6Al4V Welded Aerospace Tanks[END_REF]. These values correspond to E α = 109 GPa and ν α = 0.3. The stress tensor components σ φ and τ φ are determined by least squares minimization min σ φ ,τ φ ,˜ ψ=0 ψ [˜ φψ + 1 2 S 2 σ φ sin 2 ψ + 1 2 S 2 τ φ sin 2ψ -˜ ψ=0 ] 2 (6) in addition to the composite strain ˜ ψ=0 . Registration of diffraction peaks The previous section has shown that the sought stresses can be related to the strains (Equation ( 5)) or equivalently to the channel position of the diffraction peak (Equations ( 2)-( 4)). In both cases, the location of the peaks θ φψ has to be determined. One of the standard approaches consists in registering the measured diffractogram with an a priori chosen function 0 ≤ g(x) ≤ 1 whose maximum location is known (i.e., g(x = x 0 ) = 1). The registration procedure consists in finding the location x 0 of the peak in addition to other parameters needed to model the measured signal to minimize the sum of squared differences min x 0 (ψ),ξ 0 ,∆I,I b ,D x η 2 (x; x 0 (ψ), ξ 0 , ∆I, I b , D) with η(x; x 0 (ψ), ξ 0 , ∆I, I b , D) = f (x; ψ) -I b -Dx ∆I -g x -x 0 (ψ) ξ 0 ( 8 ) where f is raw signal measured by XRD for the angular position ψ, x the channel position on the LPS sensor, ξ 0 the width of the function that is proportional to the full width at half maximum (FWHM), and ∆I = I max -I bgl the peak intensity above the background line (bgl). In the present case, the background intensity is modelled with a linear function I bgl (x) = I b + Dx. Consequently, there are five unknowns to be determined, namely, x 0 , ξ 0 and I max that characterize the diffraction peak, in addition to I b and D that describe the background line (Figure 12). Each diffractogram is analyzed independently (i.e., for each considered angle ψ) with a registration technique that can be referred to as diffraction signal correlation or DSC (i.e., it is the one dimensional version of (2D) digital image correlation or (3D) digital volume correlation [START_REF] Sutton | Image correlation for shape, motion and deformation measurements: Basic Concepts, Theory and Applications[END_REF][START_REF] Hild | Digital Image Correlation[END_REF]. A Gauss-Newton algorithm is implemented to minimize the sum of squared differences [START_REF]ASTM E[END_REF] for each considered angle ψ. The covariance matrix associated with the evaluated parameters, which are gathered in the column vector {p} = {x 0 , ξ 0 , ∆I, I b , D} † , reads [START_REF] Hild | Digital Image Correlation[END_REF] [ Cov p ] = γ 2 ∆I 2 [M] -1 ( 9 ) where γ is the standard deviation of the acquisition noise that is assumed to white and Gaussian, and [M] the Hessian used in the minimization scheme [M] = [m] † [m] (10) with [m] =     ∂η ∂x 0 (x) ∂η ∂ξ 0 (x) ∂η ∂∆I (x) ∂η ∂I b (x) ∂η ∂D (x) . . . . . . . . . . . . . . .     (11) This covariance matrix allows the resolution 1 of the registration technique to be assessed [START_REF] Hild | Digital Image Correlation[END_REF]. The standard resolution of the peak position is then equal to the diagonal term of [Cov p ] corresponding to x 0 , provided the other parameters do not change. The influence of the peak position uncertainty on stress resolutions is easily computed since the stress extraction is a linear least squares problem in terms of ˜ φψ (see Equation ( 6)). The covariance matrix of the stress extraction technique reads [Cov σ ] = [C] -1 [c] † [Cov ][c][C] -1 (12) with [C] = [c] † [c] (13) and [c] =     1 2 S 2 sin 2 ψ 1 2 S 2 sin 2ψ 1 . . . . . . . . .     (14) 1 The resolution of a measuring system is the "smallest change in a quantity being measured that causes a perceptible change in the corresponding indication" [START_REF]International Vocabulary of Metrology -Basic and General Concepts and Associated Terms, VIM. International Organization for Standardization[END_REF]. Commercial softwares (e.g., Stressdiff [START_REF]Stressdiff[END_REF] or Leptos [START_REF] By | [END_REF]) use peak positions issued from such minimization to end up with so-called sin 2 ψ methods [START_REF]Non-destructive Testing, Test Method for Residual Stress analysis by X-ray Diffraction[END_REF][START_REF] Cullity | Elements of x-ray diffraction[END_REF][START_REF] Lütjering | Titanium[END_REF]. This procedure will serve as benchmark for the validation of the following integrated procedure. Given the fact that the peak positions are parameterized in terms of the sought stresses and composite strain (i.e., x 0 = x 0 (σ φ , τ φ , ˜ ψ=0 ), see Equations ( 2) and ( 4)), the previous minimization can be performed over the whole set of diffractograms min σ φ ,τ φ ,˜ ψ=0 ,ξ 0 ,∆ I ,I b ,D ψ x η 2 I (x, ψ; σ φ , τ φ , ˜ ψ=0 , ξ 0 , ∆ I , I b , D) (15) with η I (x, ψ; σ φ , τ φ , ˜ ψ=0 , ξ 0 , ∆ I , I b , D) = f (x, ψ) -I b -Dx ∆I -g x -x 0 (σ φ , τ φ , ˜ ψ=0 ) ξ 0 (16) This approach corresponds to integrated DSC for which the sought quantities are directly determined from the registration procedure and do not need an additional minimization step. A Gauss-Newton scheme is also implemented and the initial guess of the sought parameters comes from a first non integrated analysis. The covariance matrix associated with the evaluated parameters, which are gathered in the column vector {p I } = {σ φ , τ φ , ˜ ψ=0 , ξ 0 , ∆ I , I b , D} † , reads [Cov p I ] = γ 2 ∆I 2 [M I ] -1 (17) where [M I ] is the approximate Hessian used in the global minimization scheme [M I ] = [m I ] † [m I ] (18) with [m I ] =     ∂η I ∂σ φ (x, ψ) ∂η I ∂τ φ (x, ψ) ∂η I ∂˜ ψ=0 (x, ψ) ∂η I ∂ξ 0 (x, ψ) ∂η I ∂∆I (x, ψ) ∂η I ∂I b (x, ψ) ∂η I ∂D (x, ψ) . . . . . . . . . . . . . . . . . . . . .     (19) Tests have been made by selecting various distributions (i.e., Gauss, Pearson VII, Lorentz). They give the same types of results without any significant improvement for any of them in terms of registration residuals. In the following, a Gaussian distribution was selected g ≡ g G (x) = exp - x -x 0 ξ 0 2 (20) A preliminary step is performed to calibrate the LPS sensor in terms of local offset corrections. The offset of the detector is determined by measuring the intensity during diffraction on a material that does not diffract at an angle close to that of {213} planes of the α-phase. Amorphous glass was chosen. The diffractogram on glass is obtained by a measurement lasting at least half an hour in order to get enough signal. After rescaling the offset is subtracted to each diffractogram used in stress analyses. Figure 13 shows that most of the spacial variations are erased and that the diffraction peak appears more clearly. Fig. 13: Subtraction of detector offset For the sake of convenience, the residuals that will be reported hereafter will refer to the raw acquisitions, namely, ρ(x; ψ) = f (x; ψ) -∆Ig x -x 0 (ψ) ξ 0 + I b + Dx (21) and are therefore expressed in counts. Any deviation from random noise will be an indication of model error (i.e., associated with the choice of g, and elasticity). The final outputs of the two registration algorithms are σ φ , τ φ , and ˜ ψ=0 = ln(sin θ 0 ) -S 1 tr(σ). 5 Results of stress analyses Validation of the registration techniques Six loading steps were applied (Figure 14) on the Ti64 sample ranging from 0 to 660 MPa. For such type of alloy, the maximum stress level is lower than the yield stress ≈ 900 MPa [START_REF] Lütjering | Titanium[END_REF][START_REF] Duval | Mechanical Properties and Strain Mechanisms Analysis in Ti5553 Titanium Alloy[END_REF]. The applied stress corresponds to the applied load divided by the cross-sectional area of the sample ligament. Interestingly, the integrated DSC residuals (i.e., 39 counts), which are expected to be higher since less degrees of freedom are available, remain close to the non integrated approach. The similarity of residual levels between integrated and non integrated approaches validates the assumption of elasticity in the probed volume. These two results also show that the registration was successful via DSC and integrated DSC. Stress analyses can also be performed via integrated DIC [START_REF] Leclerc | Integrated digital image correlation for the identification of mechanical properties[END_REF]. The displacement field is first measured with FE-based DIC in which the kinematic field is parameterized with nodal displacements associated with an unstructured mesh (see Figure 16) made of three-noded triangles (with linear interpolation of the shape functions). Even though the central part of the sample was not speckled, the DIC code could converge (i.e., helped by the convergence in the speckled area) and yields results within the XRD measured area. From these measurements, only those corresponding to the two extreme transverse rows of elements are considered and they are prescribed as Dirichlet boundary conditions to an elastic FE analysis under plane stress hypothesis, which allows the stress field to be evaluated everywhere. Macroscopic elasticity constants used are the Young's modulus which is equal to 109 GPa and the Poisson ratio which is equal to 0.3. Figure 16 shows the longitudinal stress field for each loading step. In all the zone probed by XRD, the stress is virtually uniform and the corresponding standard deviation is less than 10 MPa for the highest load level. From this analysis, the mean longitudinal stress is reported and will be compared to the other stress evaluations. Fig. 16: Longitudinal stress (expressed in MPa) field evaluated via integrated DIC at the last loading step. Online version: history of the six analyzed steps (Figure 14) In Figure 17, the comparison between various stress analyses is shown. In the present plot, the reference (i.e., horizontal axis) is the applied stress. Integrated DIC and DIC give the same stress estimation and correspond to the mean longitudinal stress averaged over the XRD zone. These levels are very close to the applied stress. For DIC, the measured strains are assumed to be elastic and the corresponding stresses are evaluated, as would be performed by XRD analyses (i.e., without any equilibrium requirement), by using isotropic elasticity under plane stress The two DSC approaches yield results that are in very good agreement with each other and with the evaluations based on integrated DIC and load measurements. Interestingly, the stress estimations based on both DIC results for which the strains are averaged over the zone of interest of XRD analyses are consistent with each other. This observation proves that the assumption of linear elasticity is satisfied. The fact that DSC results are consistent with IDSC, integrated DIC evaluations and applied stress estimates also validates the choice of X-Ray elasticity constants for the XRD measurements in Ti64 alloy. Figure 18 shows the error quantifications for DSC and integrated DSC with respect to the applied stress. The stress resolutions of both DSC methods are also shown. They are based on the effect of acquisition noise only (see Equations ( 12) and ( 12)). The standard resolution of the two DSC techniques is very low and of the order of 3 MPa (i.e., 2.7 MPa for integrated DSC and 3.3 MPa for regular DSC). Thanks to integration, the standard resolution is decreased by 20 %. The root mean square error between the applied stress and DSC or IDSC is respectively equal to 37 MPa and 41 MPa. This level corresponds to an upper bound given the fact that load fluctuations occur (i.e., inducing stress variations of the order of 16 MPa in the present case). The standard applied stress uncertainty is equal to 16 MPa The last output of the DSC codes is the composite strain ˜ ψ=0 = ln(sin θ 0 ) -S 1 tr(σ). Given the fact that the trace of the stress tensor is equal to the longitudinal stress in a uniaxial tensile test, it is possible to evaluate 2θ 0 , which is the diffraction angle for zero stress of {213} planes for the α-phase. The mean value is 2θ 0 = 140.16 • , which is close to the expected value for pure titanium (i.e., 140 • ), see Figure 20. The corresponding standard deviation is equal to 0.019 • for DSC and 0.018 • for IDSC, which is 5 % lower than DSC. With Bragg's law, it is concluded that this angular uncertainty divided by tan(θ 0 ) is an estimate of the longitudinal strain uncertainty (i.e., ≈ 1.2 × 10 -4 for both methods), and a longitudinal stress uncertainty of ≈ 13 MPa. This level is of the same order as the stress fluctuations induced by the tensile stage. Thanks to the reported consistencies of stress estimations both DSC and IDSC algorithms are now considered as validated and will be used to study the Ti5553 grade. Application to Ti5553 alloy For the specimen made of Ti5553 alloy, 9 loading steps and one globally unloaded step were applied and analyzed via DIC and DSC (Figure 21). The lowest stress level is 0 MPa, and the highest is 1100 MPa (the yield stress for this alloy is of the order of 1250 MPa [START_REF] Martin | Simulation numerique multi-echelles du comportement mecanique des alliages de titane betametastable Ti5553 et Ti17[END_REF]). However during electropolishing, the geometry of the sample was roughened and the thickness of the coupon has been reduced from 0.5 mm to 0.45 mm and some imperfections on the edges were created. As a result, plasticity may occur during the test especially at high load levels. Further, the estimation of the applied stress is more delicate and thus will not be reported. The so-called DIC stresses are preferred. The occurrence of plasticity is confirmed in Figure 22 in which the stresses are evaluated with the DIC strain fields, which are assumed to be elastic. For high load levels, very high (and non physical) stress levels are observed in one area of the sample, which was not probed via XRD. It is worth noting that the last step corresponds to total unloading, which does not result in vanishing strains and stresses, thereby confirming the presence of plastic strains in the lower part of the coupon. However, as will be illustrated in Figure 25, in the area impacted by X-rays it is believed the local stress state remains elastic and homogeneous during the whole experiment. The standard stress uncertainty corresponding to the probed area is less than 30 MPa via DIC estimations. Fig. 22: Longitudinal stress (expressed in MPa) field evaluated via DIC (assuming elasticity) at the last loading step (Figure 21). Online version: history of the ten analyzed steps Figure 23 shows root mean square residuals for both DSC approached. In the present case, the two residuals are virtually coincident for all investigated stress levels. This remarkable agreement between the two DSC techniques enables the hypothesis of elasticity to be fully validated in the present case. This observation is consistent with the fact that the α-phase is finely distributed within the probed volume (i.e., of the order of 1 mm 2 × 5 µm, see Figure 2). When compared to Ti64, the mean residuals of DSC and IDSC are lower (e.g., 25 counts instead of 39 counts for IDSC in Ti64). However, the signal levels were also significantly lower (Figure 5). Due to the coexistence of two phases in the studied alloy (Figures 3 and4), the results will first focus on elastic strains. Figure 24 displays the estimations of elastic strain by various techniques, namely integrated DIC in addition to DSC and integrated DSC. In the present case, the abscissa corresponds to the mean longitudinal strain in the XRD zone measured by DIC. The fact that DIC and integrated DIC results coincide proves that the hypothesis of elasticity was fulfilled during all the experiment and that plasticity was confined in an area that has not impacted the zone probed by XRD (as expected from Figure 22). This effect cannot be attributed to plasticity since linear relationships between DIC and integrated DIC results would not be observed. The explanation comes from the fact that DIC analyses are performed at the macroscale (i.e., the two-phase alloy) whereas DSC analyses only consider the α-phase. In the present case, it is shown that the mean elastic strains evaluated in the α-phase are about 73 % those of the alloy (i.e., at the macroscopic level). This difference is due to the presence of two phases in Ti5553 whose elastic properties are different [START_REF] Hauk | Structural and Residual Stress Analysis by Non Destructive Methods: Evaluation, Application, Assessment[END_REF][START_REF] Fréour | Influence of a two-phase microstructure on XEC and XRD stress analysis[END_REF][START_REF] Sylvain Fréour | Determining ti-17 β-phase singlecrystal elasticity constants through x-ray diffraction and inverse scale transition model[END_REF][START_REF] Martin | Simulation numerique multi-echelles du comportement mecanique des alliages de titane betametastable Ti5553 et Ti17[END_REF][START_REF] Herbig | D short fatigue crack investigation in beta titanium alloys using phase and diffraction contrast tomography[END_REF][START_REF] Duval | Mechanical Properties and Strain Mechanisms Analysis in Ti5553 Titanium Alloy[END_REF]. Contrary to Ti64, which is mostly composed of α-phase, the presence of 40 wt% β-phase induces a deviation for the determination of macroscopic stresses using X-Ray elasticity constants of pure titanium. The deviation is estimated at 27 % in the present case. Figure 25 shows the stress estimates for Ti5553. In the present case the DIC stresses (i.e., the macroscopic stresses) are the reference. Independent tensile tests on the same alloy provided a Young's modulus of 115 GPa, and Poisson's ratio equal to 0.35. DIC and integrated DIC stresses are close, which is consistent with the observations of Figure 24. The longitudinal stresses obtained with both DSC approaches are also in good agreement. By using the same elastic parameters for the α-phase as those considered for Ti64, the mean stress in the α-phase amounts to about 69 % of the macroscopic stress. This stress ratio is again due to the difference in elastic properties of α-and β-phases in this alloy [START_REF] Hauk | Structural and Residual Stress Analysis by Non Destructive Methods: Evaluation, Application, Assessment[END_REF][START_REF] Fréour | Influence of a two-phase microstructure on XEC and XRD stress analysis[END_REF][START_REF] Sylvain Fréour | Determining ti-17 β-phase singlecrystal elasticity constants through x-ray diffraction and inverse scale transition model[END_REF][START_REF] Martin | Simulation numerique multi-echelles du comportement mecanique des alliages de titane betametastable Ti5553 et Ti17[END_REF][START_REF] Herbig | D short fatigue crack investigation in beta titanium alloys using phase and diffraction contrast tomography[END_REF][START_REF] Duval | Mechanical Properties and Strain Mechanisms Analysis in Ti5553 Titanium Alloy[END_REF]. The standard DIC stress uncertainty is equal to 30 MPa The shear stresses are again very small (Figure 26). The mean level is 13.5 MPa for IDSC, and 15 MPa for DSC. Therefore, there is a small bias in the present case. It may be due to small misalignments of the goniometer (whose standard uncertainty was estimated to be 2 • ). From this information, the stress state is subsequently determined. The second route consists in merging both steps into a single analysis. It is referred to as integrated diffraction signal correlation (i.e., IDSC). It is shown that the latter leads to lower stress resolutions when compared to the former. For Ti64 a very good agreement is observed between the stresses estimated with both DSC techniques, integrated DIC and regular DIC. For Ti5553, it is shown that the elastic strains in the α-phase are about 73 % of the macroscopic strain, which is due to the difference in elastic properties of the two phases. Further, the fact that the two DSC techniques yield virtually identical results validates the hypothesis of elasticity at the level of the α-phase. For both materials, it is observed that the diffraction angle for the unstressed state is very close to 140 • (i.e., the reference level for pure titanium). Stress resolutions and uncertainties have been systematically analyzed for both alloys. In particular, it is shown that the uncertainty associated with the knowledge of the unstressed configuration has a limited impact on the overall levels. With the implemented registration techniques there is only a 20 % increase of overall stress uncertainties for Ti5553 with respect to Ti64 even though the peak height has been decreased by a factor of 4. This result validates the two registration techniques and shows their robustness even for Ti5553. Having validated the present registration techniques, they can now be used in other configurations to study, for instance, the surface integrity [START_REF] Lütjering | Titanium[END_REF][START_REF] Guillemot | Prediction of the endurance limit taking account of the microgeometry after finishing milling[END_REF][START_REF] Souto-Lebel | Characterization and influence of defect size distribution induced by ball-end finishing milling on fatigue life[END_REF] associated with different milling conditions of Ti5553 [START_REF] Cox | The effect of finish milling on the surface integrity and surface microstructure in Ti-5Al-5[END_REF]. They may also be used in benchmarks with other materials and procedures [START_REF] Lefebvre | External reference samples for residual stress analysis by x-ray diffraction[END_REF][START_REF] Suominen | Residual Stress Measurement of Ti-Metal Samples by Means of XRD with Ti and Cu Radiation[END_REF]. Fig. 1 :Figure 2 12 Fig. 1: Micrographs of Ti64 (a) and Ti5553 (b) alloys Fig. 2 : 2 Fig. 2: (a) EBSD orientation map of Ti64 sample (α-indexation). The black box is enlarged in sub-figure (b) Fig. 3 : 4 In Figure 4 , 344 Fig. 3: (a) EBSD orientation map of Ti5553 sample (β-indexation). The black rectangle is enlarged in (b). The boxed detail is shown in Figure 4 Fig. 4 :Figure 5 45 Fig. 4: EBSD orientation map of Ti5553 sample (zoom of Figure 3(b)). (a) α-indexation, and (b) β-indexation Fig. 5 : 5 Fig. 5: Diffractograms of the two studied alloys and of pure titanium powder (the same dynamic range is used for all plots) Fig. 6 : 6 Fig. 6: In-situ tensile test enabling for XRD stress analyses Fig. 7 : 7 Fig. 7: Sketch of the linear position sensor (LPS), X-ray source (X), analyzed sample S in the goniometer (after Ref. [12]) Fig. 8 : 8 Fig. 8: Sketch of a rotation of ψ = -50 • around χ axis and locations reached thanks to the Eulerian cradle of χ-method goniometer (after Ref. [12]) Fig. 9 : 9 Fig. 9: Coupon for in-situ tests. (a) Dog-bone geometry (ligament width: 6 mm, thickness: 0.5 mm, radius of the hour-glass shape: 65 mm). (b) Speckle pattern used for DIC procedures with an unpainted area to allow for XRD measurements. The area probed by the X-ray beam is depicted in orange. Fig. 10 : 10 Fig. 10: General procedure used in stress analyses Fig. 11 : 11 Fig. 11: Linear position sensor (LPS) and goniometer geometry 2 2 are the X-Ray elasticity constants depending on the diffracting planes, and ˜ ψ=0 = ln(sin θ 0 ) -S {213} 1 tr(σ) will be referred to as composite strain. The normal and shear stresses are denoted by σ φ ≡ σ φφ and τ φ ≡ σ φ3 , respectively. In the present case elastic isotropy and homogeneity are assumed since anisotropy in {213} planes of α lattice is relatively small, so that S Fig. 12 : 12 Fig. 12: Peak model and notations Fig. 14 : 14 Fig. 14: Loading history during the in-situ tensile test on Ti64 alloy. The standard stress uncertainty is equal to 16 MPa Fig. 15 : 15 Fig. 15: Root mean square residual for Ti64 alloy for the two DSC analyses Fig. 17 : 17 Fig. 17: Longitudinal stress σ φ in Ti64 alloy coupon measured with four different evaluation techniques. The error bars depict the root mean square difference between DSC or IDSC estimates and the applied stress. The standard applied stress uncertainty is equal to 16 MPa Fig. 18 :Fig. 19 : 1819 Fig. 18: Stress resolutions and errors for Ti64 alloy coupon. The standard applied stress uncertainty is equal to 16 MPa Fig. 20 : 20 Fig. 20: Estimation of 2θ 0 for Ti64 alloy from the analysis of the composite strain . The standard applied stress uncertainty is equal to 16 MPa Fig. 21 : 21 Fig. 21: Loading history during the in-situ tensile test on Ti5553 alloy. The standard DIC stress uncertainty is equal to 30 MPa Fig. 23 : 23 Fig. 23: Root mean square residual for Ti5553 alloy for the two DSC analyses. The standard DIC stress uncertainty is equal to 30 MPa Fig. 24 : 24 Fig. 24: Longitudinal elastic strains for the 10 loading steps with the four techniques Fig. 25 : 25 Fig. 25: Longitudinal stresses σ φ in Ti5553 alloy evaluated via different techniques. The error bars depict the room mean square difference between DSC or IDSC estimates and the DIC stresses. Fig. 26 :Fig. 27 :Fig. 28 : 262728 Fig. 26: Shear stress τ φ in Ti5553 alloy coupon evaluated with DSC and IDSC. The error bars depict the room mean square difference between DSC or IDSC estimates and a vanishing shear stress. The standard DIC stress uncertainty is equal to 30 MPa Table 2 : 2 Chemical composition of Ti5553 Acknowledgements This work was funded by Safran Landing Systems. The authors acknowledge Pierre Mella for providing Ti64 alloy and useful discussions on XRD analyses. The authors also thank Adam Cox for providing Ti5553 alloy samples and Thierry Bergey for electropolishing them.

01744748

en

2024/03/05 22:32:07

2018

https://amu.hal.science/hal-01744748/file/garcia.pdf

Ana Paula García-Nieto Ilse R Geijzendorffer Francesc Baró Philip K Roche Alberte Bondeau Wolfgang Cramer Impacts of urbanization around Mediterranean cities: changes in ecosystem service supply Keywords: Land cover, population, urban, rural, spatial analysis, trend, nature's contributions to people Urbanization is an important driver of changes in land cover in the Mediterranean Basin and it is likely to impact the supply and demand of ecosystem services (ES). The most significant land cover changes occur in the periurban zone, but little is known about how these changes affect the ES supply. For eight European and four North African cities, we have quantified changes in peri-urban land cover, for periods of sixteen years (1990)(1991)(1992)(1993)(1994)(1995)(1996)(1997)(1998)(1999)(2000)(2001)(2002)(2003)(2004)(2005)(2006) in the Northern African, and twenty-two years in the European cities, respectively. Using an expertbased method, we derived quantitative estimates of the dynamics in the supply of twenty-seven ES. The nature of land cover changes slightly differed between European and North African Mediterranean cities, but overall it increased in urban areas and decreased in agricultural land. The capacity of the peri-urban areas of Mediterranean cities to supply ES generally reduced over the last 20-30 years. For nine ES the potential supply actually increased for all four North African cities and three out of the eight European cities. Across all cities, the ES timber, wood fuel and religious and spiritual experience increased. Given the expected increase of urban population in the Mediterranean Basin and the current knowledge of ES deficits in urban areas, the overall decrease in ES supply capacity of peri-urban areas is a risk for human wellbeing in the Mediterranean and poses a serious challenge for the Sustainable Development Goals in the Mediterranean basin. Introduction Approximately two thirds of the world's population (i.e., 6,4 billion people for the median projection) and 84% in Europe will be living in urban areas by 2050. In 2014 already more than half of the global population was urban, while in Europe this was 70% [START_REF] Kabisch | Diversifying European agglomerations: evidence of urban population trends for the 21st century[END_REF]United Nations, 2015a, 2014). The increase in total population entails a corresponding increase in demand for natural resources [START_REF] Ma | MA) Millennium Ecosystem Assessment, 2005. Ecosystems and Human Well-being: Synthesis[END_REF], particularly for energy and water. The demand for water is expected to increase with 55% between 2000 and 2050 (United Nations World Water Assessment Programme, 2014). The effect of urban population growth on peri-urban landscapes is expected to be particularly prominent since urban land cover increases even faster than could be expected from demographic pressure, resulting in substantial land use conversions [START_REF] Angel | The dimensions of global urban expansion: Estimates and projections for all countries, 2000-2050[END_REF][START_REF] Seto | A Global Outlook on Urbanization[END_REF][START_REF] Seto | Global forecasts of urban expansion to 2030 and direct impacts on biodiversity and carbon pools[END_REF][START_REF] Seto | A Meta-Analysis of Global Urban Land Expansion[END_REF]. Urban populations in countries around the Mediterranean Sea increased from 152 million to 315 million between 1970 and 2010 (an average rate of 1.9 % per year) (UNEP/MAP, 2012). By 2030, the Mediterranean Basin will be the global biodiversity hotspot with the highest percentage of urban land (5%) [START_REF] Elmqvist | History of Urbanization and the Missing Ecology[END_REF]. Urbanization rates have been accelerated by environmental change; for example, intense drought conditions contributed to a rural exodus in Morocco between 1980and 1990, and in Algeria and Tunisia in 1999(FAO, 2001;[START_REF] Hervieu | Rethinking rural development in the Mediterranean[END_REF]. Tourism and housing development have led to the development of infrastructure close to coastal areas and near culturally important cities [START_REF] Eea | Biogeographical regions in Europe. The Mediterranean biogeographical regionlong influence from cultivation, high pressure from tourists, species rich, warm and drying[END_REF][START_REF] Houimli | The factors of resistance and fragility of the littoral agriculture in front of the urbanization: the case of the region of North Sousse in Tunisia[END_REF]. Mediterranean cities are considered attractive places to settle for retirees from northern Europe [START_REF] Membrado-Tena | Costa Blanca: Urban Evolution of a Mediterranean Region through GIS Data[END_REF], and for return migrants to the Maghreb countries [START_REF] Cassarino | Return migrants to the Maghreb Countries: Reintegration and development challenges[END_REF]. The growth of urban areas often takes place at the expense of agricultural land and this can potentially lead to environmental degradation and socio-economic challenges [START_REF] Orgiazzi | Global soil biodiversity atlas[END_REF]. Although probably the most studied, the direct conversion of agricultural land into urban land is only one of the many impacts of urbanization on the structure and function of ecosystems and their services [START_REF] Mcdonnell | The use of gradient analysis studies in advancing our understanding of the ecology of urbanizing landscapes: current status and future directions[END_REF][START_REF] Modica | Spatio-temporal analysis of the urban-rural gradient structure: an application in a Mediterranean mountainous landscape[END_REF]. Examples of other impacts of urbanization include changes in demand patterns [START_REF] Bennett | Linking biodiversity, ecosystem services, and human well-being: three challenges for designing research for sustainability[END_REF][START_REF] García-Nieto | Mapping forest ecosystem services: From providing units to beneficiaries[END_REF][START_REF] Schulp | Uncertainties in Ecosystem Service Maps: A Comparison on the European Scale[END_REF], or the infrastructure construction for water distribution facilities, energy plants and internet connection [START_REF] Kasanko | Are European cities becoming dispersed?: A comparative analysis of 15 European urban areas[END_REF][START_REF] Seto | Global forecasts of urban expansion to 2030 and direct impacts on biodiversity and carbon pools[END_REF], agricultural land abandonment [START_REF] Hasse | Land resource impact indicators of urban sprawl[END_REF][START_REF] Hervieu | Rethinking rural development in the Mediterranean[END_REF] or the protection of traditional landscapes with the aim to maintain the aesthetic quality [START_REF] Baró | Mapping ecosystem service capacity, flow and demand for landscape and urban planning: a case study in the Barcelona metropolitan region[END_REF]. Urbanization may also affect the diversity of the landscape, with agricultural land being managed by hobby farmers rather than for commercial production [START_REF] Jarosz | The city in the country: Growing alternative food networks in Metropolitan areas[END_REF][START_REF] Zasada | Multifunctional peri-urban agriculture-A review of societal demands and the provision of goods and services by farming[END_REF]. As these examples show, the influence of urban areas on ecosystems extends well beyond the urban boundaries [START_REF] Lead | Drivers of change in ecosystem condition and services[END_REF], but it is unclear how the changes in the peri-urban landscapes affect human well-being. A growing number of studies of human well-being and quality of life in urban areas focus on the benefits provided by natural elements within cities, so called urban ecosystem services [START_REF] Bolund | Ecosystem services in urban areas[END_REF][START_REF] Kremer | Key insights for the future of urban ecosystem services research[END_REF]. Many of these studies have found that the natural elements currently present in cities often do not seem to provide ecosystem services (ES) in sufficient quantities in comparison to the demand for these services [START_REF] Folke | Ecosystem Appropriation by Cities[END_REF][START_REF] Jansson | Reaching for a sustainable, resilient urban future using the lens of ecosystem services[END_REF]. [START_REF] Baró | Mismatches between ecosystem services supply and demand in urban areas: A quantitative assessment in five European cities[END_REF] have recently shown mismatches between ES in supply and demand for five European cities. These mismatches may depend on many factors, e.g. differences in spatial distribution of goods and needs or access restrictions to resources for particular groups, such as women [START_REF] Geijzendorffer | Improving the identification of mismatches in ecosystem services assessments[END_REF]. Some of these factors can be addressed through governance or land use management [START_REF] Jansson | Reaching for a sustainable, resilient urban future using the lens of ecosystem services[END_REF]. Potentially, land cover changes around cities affect ES supply, and these changes may therefore potentially reduce or enhance deficits for ES within cities. The objective of this study is to assess how growing urban areas in the Mediterranean Basin modify the periurban landscapes and, consequently, ES supply. With the current urban population being expected to increase to 385 million people by 2025 (UNEP/MAP, 2012) and the objective of improved human well-being of the Sustainable Development Goals (United Nations, 2015b), an increase of ES is required for this growing urban population. Therefore, there is a particular need to assess the recent dynamics in ES supply, both within cities and in their peri-urban areas. The European Mediterranean areas are estimated to be particularly vulnerable with respect to ES supply, mostly due to climate and land use change [START_REF] Schröter | Ecosystem Service Supply and Vulnerability to Global Change in Europe[END_REF]. Although similar studies for the north-African Mediterranean countries are missing [START_REF] Nieto-Romero | Exploring the knowledge landscape of ecosystem services assessments in Mediterranean agroecosystems: insights for future research[END_REF], it is highly likely that these countries are subject to similar, if not higher, anthropogenic pressures, experience more rapid population increases and are undergoing significant landscape changes. The importance of this knowledge gap goes beyond its implications for regional assessments, by additionally increasing the uncertainty of supra-regional assessments of sustainable futures. The need to evaluate land use and land cover changes and their impacts for future conditions is increasingly being recognized [START_REF] Eea | Biogeographical regions in Europe. The Mediterranean biogeographical regionlong influence from cultivation, high pressure from tourists, species rich, warm and drying[END_REF][START_REF] Fichera | GIS and Remote Sensing to Study Urban-Rural Transformation During a Fifty-Year Period[END_REF]. To inform land management improvements, land use -land cover assessments should take into account spatial and temporal patterns along urban-rural gradients [START_REF] Kroll | Rural-urban gradient analysis of ecosystem services supply and demand dynamics[END_REF]. Previous land use -land cover assessments in the Mediterranean have focused on areas where spatial data was available and, as a consequence, the north-African region has received little attention [START_REF] Haase | A quantitative review of urban ecosystem service assessments: concepts, models, and implementation[END_REF][START_REF] Luederitz | A review of urban ecosystem services: six key challenges for future research[END_REF]. Also, most studies to date focus on single city case studies and are limited to the dense urban fabric. A multi-city analysis therefore fills an important knowledge gap by allowing for a comparison of the impacts of urbanization in peri-urban land and its consequences for ES supply in the Mediterranean Basin. For this study we selected both European and north-African Mediterranean cities: eight Mediterranean European cities (Lisbon, Madrid, Barcelona, Marseille, Florence, Rome, Athens, Thessaloniki) and four Northern African cities (Nabeul, Sfax, Tunis, Rabat). Material and methods For this study, we analyzed: 1) whether land cover changes around cities differed significantly from trends at national level; 2) whether different specific conversions of land cover are common for groups of cities occurred over time and finally 3) whether the spatio-temporal supply patterns of ES over the period 1990 -2012 were shared among these Mediterranean cities, depending on data availability. The assessment was carried out in six steps. First, we selected twelve major Mediterranean cities as case studies. We used a systematic approach to define the peri-urban area for each city. Based on time series of available land cover maps (Fig. 1), we assessed land cover changes in each peri-urban area and we compared them with national dynamics per country. In addition, we identified the main patterns in land cover changes across all periurban areas. Finally, we identified changes in land cover with expert based estimates of ES supply [START_REF] Stoll | Assessment of ecosystem integrity and service gradients across Europe using the LTER Europe network[END_REF] and searched for specific or general dynamics in ES supply. Data were available for the period 1990-2006 for Northern African cities, and for 1990-2012 for European cities (Table 1). These periods allowed for the analysis of important dynamics and they correspond to the used expertbased ES estimates (see below). -Insert Fig. 1 around here - Selection of Mediterranean cities In the selection of cities we aimed to achieve a geographical distribution in the Mediterranean biogeographical region [START_REF] Olson | Terrestrial Ecoregions of the World: A New Map of Life on Earth A new global map of terrestrial ecoregions provides an innovative tool for conserving biodiversity[END_REF] (Fig. 2), with a special attention to include both cities on northern and southern Mediterranean shores. An additional search criterion was that land cover data should be available on at least two moments in time. These criteria allowed for the selection of twelve cities in total, four in Northern Africa (Nabeul, Sfax, Tunis, Rabat) and eight in Southern Europe (Lisbon, Madrid, Barcelona, Marseille, Florence, Rome, Athens, Thessaloniki). Spatial land cover data is available for the entire Mediterranean basin, but the categories, spatial resolution and time series differ (Fig. 1). CORINE Land Cover (CLC) [START_REF] Feranec | European landscape dynamics: CORINE land cover data[END_REF] is a spatial database with a resolution of 100 m and it is available for all European countries for the years 1990, 2000, 2006 and 2012 (Table 1). For the North African countries, CLC is available only for 1990. We used GlobCorine Land Cover (GLC) [START_REF] Bontemps | GlobCorine-A joint EEA-ESA project for operational land dynamics monitoring at pan-European scale[END_REF] to include another point in time ( 2006) for these countries. The GLC land cover map was developed by the European Environmental Agency and European Space Agency attempting to ensure compatibility with CLC (Appendix 1). -Insert Fig. 2 around here - Defining the peri-urban areas There are many different approaches to define the urban and peri-urban areas of a city [START_REF] Orgiazzi | Global soil biodiversity atlas[END_REF]. For our study we searched for a simple, yet objective delineation of the urban areas that could be adapted to include peri-urban areas. We defined the peri-urban area as the rural area located in proximity around the urban area. In addition, the delineation method should be able to deal with the differences in data resolutions between the European countries and the north-African Mediterranean countries. The approach published by [START_REF] Kasanko | Are European cities becoming dispersed?: A comparative analysis of 15 European urban areas[END_REF] assumes a fixed relationship to estimate the boundary of the urban area, separating it into the urban core area (A) and the adjacent urban area (W u ) ( ). For our study, we used this method to additionally define the peri-urban areas (Wp). To parameterize the equation of [START_REF] Kasanko | Are European cities becoming dispersed?: A comparative analysis of 15 European urban areas[END_REF] for the boundary of the peri-urban area (W p ), we used the peri-urban estimate published by [START_REF] Kroll | Rural-urban gradient analysis of ecosystem services supply and demand dynamics[END_REF], obtaining a general equation for peri-urban areas: (see Fig. 3). By using these criteria, the width of the adjacent urban area and the peri-urban area are assumed to depend only on the area of the urban core in each city. Table 2 shows the resulting urban core areas and the corresponding W u and W p areas for 2006. The urban core area (A) was computed for the Mediterranean cities using the urban land cover determined by CLC and GLC using 2006 as reference time period. In the case of CLC, from the 44 land cover classes (Appendix 2), we selected the polygons belonging to continuous and discontinuous urban fabric (categories 111 and 112) whose centroid was inside the administrative boundary of the city. From the fourteen land cover classes of GLC (Appendix 1), class 10 (urban and associated areas) was used to determine the urban core area. In a second step, we repeated this process including those polygons whose centroid was within a radius of 1 km from the selected urban polygons, to calculate the final size and location of urban core area (A). -Insert Fig. 3 around here - Identification of land cover changes Focusing the analysis on the peri-urban area (W p ), we identified land cover changes from 1990 to 2012 for the European cities and from 1990 to 2006 for Northern African cities. Spatial land cover data was extracted for the different time periods in each city and its belonging country, considering the Mediterranean biogeographical region boundaries [START_REF] Olson | Terrestrial Ecoregions of the World: A New Map of Life on Earth A new global map of terrestrial ecoregions provides an innovative tool for conserving biodiversity[END_REF]. The land cover data (area by land cover category) for each year in each W p area was normalized using the size of each peri-urban area, to be comparable across the different case studies. The same normalization was made for each country based on the proportion of that country classified as Mediterranean. In the remainder of this text we will refer to this area as the "national area". The spatial information was analyzed using ArcGIS 10.2.2 (ESRI, 2013). To allow for the comparison between land cover categories from CLC (1990) and GLC (2006) in North Africa, we developed weighing factors using Andalusia and Sicily as most closely representative sampling sites for the Northern African Mediterranean setting. The spatial information of CLC and GLC in 2006 was intersected and extracted to measure the contributions of each CLC category in each GLC class. To transform the spatial GLC information of North African cities into information on each of the 44 CLC classes, we defined X x as the area of a specific GLC category and computed weighing factors (W x ) based on how the GLC categories in Andalusia and Sicily were composed of the different CLC categories. We applied these weighing factors in all calculations on North African areas to transform all GLC data into CLC data to allow for multiplication with the Stoll capacity matrix. This means that the surface of each CLC category (Y x ) is equal to a multiplication of the surface (km 2 ) for each GLC category (X x ) by weighing factors (W x ) (Equation 1). Equation 1: For the statistical analysis, the CLC categories were summarized in 7 different CLC groups (Table 3). To obtain the urbanization trends in Mediterranean cities, we estimated the total standardized surface by CLC group coming from W p and compared it with the total standardized land cover from the respective Mediterranean parts of countries (Portugal, Spain, France, Italy and Greece; Tunisia and Morocco) over the different periods. Assessments for Europe and North Africa follow the same methods, but were applied separately because the uncertainties and applied methods for the input data are different. Conversion of land cover changes into ecosystem services supply dynamics Following an approach developed by [START_REF] Burkhard | Landscapes' capacities to provide ecosystem services-a concept for land-cover based assessments[END_REF][START_REF] Burkhard | Mapping ecosystem service supply, demand and budgets[END_REF], we related land cover data to expert-based values of the capacity for ES supply. In a recent study, [START_REF] Stoll | Assessment of ecosystem integrity and service gradients across Europe using the LTER Europe network[END_REF] developed an ES supply capacity matrix based on CLC types combined with expert-based estimates of the supply capacity for thirty-one ES for European countries. Capacity estimates for ES supply range from 0 (no relevant capacity of the land cover type to provide this particular ecosystem service) to 5 (very high capacity). To assess how land cover changes around the case study cities influence ecosystem service supply over time, we translated our land cover changes into ES dynamics using the ES supply matrix published by [START_REF] Stoll | Assessment of ecosystem integrity and service gradients across Europe using the LTER Europe network[END_REF]. Estimates for each ES (ES x ) at different time periods in every peri-urban area were calculated by multiplying the area of each CLC class (X x ) by the corresponding ES value from ES matrix (ESstoll xn ) (Equation 2). The resulting ES assessment included twenty-seven ES supplied by peri-urban landscapes. Equation 2: Statistical analysis The chosen statistics in this study responded to type of variables, sampling distribution and scientific objectives: 1) analysis of land cover changes around cities and comparison with trends at national level; 2) identification of common specific conversions of land cover among cities over time; and 3) assessment of spatio-temporal supply patterns of ecosystem services over the period 1990 -2012 in Mediterranean peri-urban areas. For each objective, a group of statistical analysis was conducted. To assess whether land cover change patterns around cities differ from trends at the national level, the standardized total surface of land cover groups was statistically compared over time. For this purpose, we selected the non-parametric Wilcoxon test and the parametric Two-sample t-test. According to sampling distribution of CLC and paired groups, non-parametric Wilcoxon test was suitable for the case of permanent crops, complex cultivation patterns and shrub and/or herbaceous. Based on the assumptions of normal sampling distribution and paired groups, the parametric Twosample t-test was conducted for urban, non-irrigated and irrigated arable land, and forest land cover groups. Appendix 4). The data for the European cities was assessed using Within-class Correspondence Analysis (WCA) [START_REF] Benzécri | Analyse de l'inertie intraclasse par l'analyse d'un tableau de correspondance[END_REF][START_REF] Chessel | Méthodes K-tableaux[END_REF] through the "within.coa function" in the R package ade4 [START_REF] Rstudio | RStudio: Integrated development environment for R (Version 0.96.122[END_REF]. Data for the European and North-African cities was analyzed separately due to the differences in the input data (as discussed in section 2.3). Changes on ES supply in peri-urban areas over time were estimated conducting a Within-class Principal Component Analysis (WCP) [START_REF] Benzécri | Analyse de l'inertie intraclasse par l'analyse d'un tableau de correspondance[END_REF][START_REF] Chessel | Méthodes K-tableaux[END_REF], evaluating separately ES estimates for European and North African peri-urban areas (see Appendix 5 and 6). WCA and WCP are similar to standard Correspondence Analysis with a single constraining factor to remove [START_REF] Chessel | Méthodes K-tableaux[END_REF]. As strong differences between cities may mask patterns over time, we used this analysis to compare spatio-temporal variations of land cover distributions and ES supply removing alternatively the effect of the city and the year variables as constraining factors. Results Land cover change under urbanization Common patterns emerge for all selected Mediterranean cities when we compared land cover patterns over time between the selected Mediterranean cities and the trends in the respective countries (Fig. 4A and4B). Overall, changes were more pronounced in the peri-urban areas than at national level. As expected, all peri-urban areas demonstrated a significant increase of urban fabric. Around the European cities this took place mostly at the expense of complex cultivation patterns, non-irrigated and irrigated arable land, and shrub and/or herbaceous and pastures from 1990 to 2012. The parametric two-sample t-test revealed significant differences over the time between peri-urban areas from selected cities and countries in Europe in the case of urban, non-irrigated and irrigated arable land, and forest (p-value = 0.05) (Fig. 4A). In the North African peri-urban areas, the increase of urban area from 1990 to 2006 occurred in parallel with an increase in irrigated arable land, permanent crops, complex cultivation patterns and shrublands and/or herbaceous and pastures, at the expense of non-irrigated arable land and forest, both around in peri-urban areas as well as at the national level (Fig. 4B). -Insert Fig. 4 around here - Mediterranean peri-urban areas, spatio-temporal dynamics in land cover Within-class Correspondence Analyses (WCA) were performed separately to assess how land cover changes differed over time in peri-urban areas of Mediterranean European cities (from 1990 to 2012) and in peri-urban areas of Northern African cities (from 1990 to 2006) (Appendix 7). Results from European cities (where city as a variable was removed) showed that 4.56% of the variation was due to time patterns in land use; whereas when WCA was based on the time period or year (enhancing the differences between cities) 97.87% of the variation was due to the difference between cities and peri-urban land uses. This means that the differences of surrounding land use between cities are a dominant pattern that masks common trends if it is not removed beforehand. In the case of European cities (Fig. 5A Results from WCA in North African cities (where city as a variable was removed) showed that 62.86% of the variation was due to temporal patterns in land use; whereas when removing year variable 82.18% of variation is due to differences between cities and land uses. North African peri-urban areas (Fig. 5B) in 1990 were characterized by a clear pattern from non-irrigated arable land and forest patterns (in negative scores of F1) to an increase in 2006 of urban land, shrubs and herbaceous associated vegetation and pastures (in positive scores of F2). Peri-urban areas of Sfax showed transitions of permanent crops (in positive scores of F1) to urban land, shrubs and herbaceous associated vegetation and pastures (in positive scores of F2) also, with an increase of irrigated arable land. -Insert Fig. 5 around here - Ecosystem services supply: spatio-temporal patterns The supply of ESs (Appendix 5 and 6) was estimated through multivariate within-class Principal Component Analysis (WCP), considering European and North African cities over the indicated periods respectively. Trends in ES supply differed between EU cities (Fig. 6). -Insert Fig. 7 around here -All Mediterranean peri-urban areas show increases in the supply of air quality regulation, timber, wood fuel and religious and spiritual experience (Fig. 8). The European peri-urban areas show small or negative trends for the other ES. North-African peri-urban areas show much larger changes than those found in the European peri-urban areas, but this is maybe caused by the differences between CLC and GLC data. In general, the North-African peri-urban area showed stronger increases of ES supply capacities than the European peri-urban areas. In addition to the previously mentioned ES, the supply capacity of pollination, livestock, religious and cultural heritage increased in North-African peri-urban areas. Peri-urban areas do not show the same patterns for ES supply over time (Fig. 9). In some peri-urban areas the supply of regulating and cultural ES was more important than for the provisioning ES (Lisbon, Barcelona, Marseille, Florence, Athens and Sfax) or the inverse in the case of Thessaloniki. In other cases, the supply of regulating ES was more important than provisioning and cultural ES (Madrid and Rome). Cultural ES increased around Nabeul, Rabat and Tunis. -Insert Fig. 8 around here - Discussion Changes in land cover Land cover changes in European Mediterranean peri-urban areas showed an expansion of urban and forested areas at the expense of agriculture land similar to described by d [START_REF] Amour | Future urban land expansion and implications for global croplands[END_REF] and [START_REF] Depietri | The urban political ecology of ecosystem services: The case of Barcelona[END_REF]. Especially irrigated (Marseille and Florence) and non-irrigated arable land (Madrid and Thessaloniki) were reduced. Around Barcelona, Lisbon and Athens, complex cultivation patterns, shrublands and pastures were more abandoned. In Rome's peri-urban areas where the urban area increased less, general land cover change patterns are also less pronounced. Peri-urban areas of all North African cities, showed the same general pattern of increases in urban land, but instead this coincided with increases of agricultural land, herbaceous associated vegetation and pastures while irrigated agriculture and forest areas were reduced for Rabat, Nabeul and Tunis. In the peri-urban area of Sfax the area of irrigated arable land increased, while complex cultivation systems reduced in area. This expansion of both urban areas as well as agricultural areas in North Africa had been observed earlier [START_REF] Bouraoui | L'agriculture, nouvel instrument de la construction urbaine? Étude de deux modèles agri-urbains d'aménagement du territoire: le plateau de Saclay, à Paris, et la plaine de Sijoumi[END_REF]. Previous European focused studies demonstrated an expansion of woodlands in abandoned and marginal agricultural land [START_REF] Eea | Biogeographical regions in Europe. The Mediterranean biogeographical regionlong influence from cultivation, high pressure from tourists, species rich, warm and drying[END_REF][START_REF] Zanchi | Afforestation in Europe final version 26/01/07[END_REF]. Indeed, our results showed an increase in forest and shrublands for Europe at both peri-urban and national level. For the North African cities and countries, however, it is particularly shrublands which increased while the forest area showed a relatively slight decline. -Insert Fig. 9 around here - Implications of ES trends in peri-urban areas The Mediterranean basin has some serious challenges to advance on global Sustainable Development Goals (UNEP/MAP, 2016), to which the ES supplied by peri-urban areas could positively contribute. In general, the ES supply capacity of peri-urban Mediterranean areas decreased over time, in particular for the supply of provisioning and regulating ES. If we consider the ES that are of most immediate concern for ensuring human well-being, i.e. supply of food, water and protection from hazards, then European and north-African peri-urban regions showed decreasing supply trends for the supply of food and protection from hazards. Supply of freshwater in Europe remains constant, but north-African peri-urban areas showed an increased supply of freshwater. This is linked to a growing surface of continental water bodies, since several dams and reservoirs have been built to address issues related to water scarcity and strong interannual variability of precipitation [START_REF] Tramblay | Future water availability in North African dams simulated by high-resolution regional climate models[END_REF]. However, as the ES of water purification decreased (Fig 8), we may have to be cautious as to the use of this water for all purposes. Total area for crop production decreased over time in peri-urban areas, but this does not necessarily mean that the total food production also decreased. Changes in farm management may have increased the productivity of remaining agricultural land. Also, since these Mediterranean cities already rely heavily on food imports [START_REF] Lead | Drivers of change in ecosystem condition and services[END_REF][START_REF] Soulard | Peri-urban agro-ecosystems in the Mediterranean: diversity, dynamics, and drivers[END_REF], a reduction in locally produced food is likely not to lead immediately to a food deficit. However, an increased dependence on global food market prices does render countries more vulnerable to potential food crises. Urbanization often implies the expansion of impermeable surfaces leading to an increase of surface water runoff, and consequently an increased risk of flooding [START_REF] Gómez-Baggethun | Urban Ecosystem Services[END_REF]. Our results show that the regulation of natural hazards has been decreasing over the years in Mediterranean peri-urban areas which pose a particular threat to people living around urban and peri-urban areas. Depending on whether the supply of ES needs to be local in order to provide benefits, the peri-urban area can supply ES in the urban area. Recent studies on ES mismatches between supply and demand in urban areas have predominantly indicated deficits in local climate regulation (carbon sequestration, urban cooling), air quality regulation, and recreation and nature tourism [START_REF] Baró | Mapping ecosystem service capacity, flow and demand for landscape and urban planning: a case study in the Barcelona metropolitan region[END_REF][START_REF] Baró | Mismatches between ecosystem services supply and demand in urban areas: A quantitative assessment in five European cities[END_REF]. Of these deficits, peri-urban land could provide air quality regulation to reducing the deficit in the future. Although urban areas represent relatively small areas at a global scale, their increase could negatively impact local climate [START_REF] Foley | Global Consequences of Land Use[END_REF][START_REF] Verburg | Challenges in using land use and land cover data for global change studies[END_REF]. Our results indicate that Mediterranean peri-urban capacity to supply climate regulation has been decreasing. The steady trend or the increase of potential supply of some cultural ES by peri-urban areas (religious and spiritual experience) was not considered in previous assessment on ES around the Mediterranean, [START_REF] Nieto-Romero | Exploring the knowledge landscape of ecosystem services assessments in Mediterranean agroecosystems: insights for future research[END_REF][START_REF] Runting | Incorporating climate change into ecosystem service assessments and decisions: a review[END_REF] and this study offers therefore, a first reflection on its trends. There are ES supplied by the peri-urban areas which may not actually reach inhabitants of urban areas, for instance trees only provide shade locally. We can assume however, that for other ES for which distances are less relevant (e.g. global climate regulation) or for which people are likely to travel short distances (e.g. recreation and nature tourism), ES supply by peri-urban areas can be considered relevant for urban areas. For instance, the increased potential of cultural ES in peri-urban areas that we found (due to predominantly an increase in the nonirrigated agricultural and forest areas) is increasingly relevant for people seeking to spend their leisure time outside of urban areas. As the urban population grows, it could be possible that the demand for cultural services, notably in the nearby surroundings of cities, will increase. To determine whether or not the identified increase in supply will be able to meet this presumed increase in demand would need to a more detailed study. Our use of the capacity matrix developed by [START_REF] Stoll | Assessment of ecosystem integrity and service gradients across Europe using the LTER Europe network[END_REF] has several limitations, namely, 1) the capacity matrix was based on expert estimates only reflected the potential ES supply which can be different from the actual supply; 2) land cover information does not incorporate the type of management on arable lands and forest, prohibiting estimations of effects due to changes in land use intensity; 3) the matrix essentially represents a European perspective on land use and potential ES supply, so a capacity matrix adapted to the Mediterranean biome should be required to obtain more accurate estimates. Despite those limitations, we consider that our approach allows assessing land cover and ES changing patterns across Mediterranean peri-urban areas based on openly available data, identifying potential influence on ES supply. The multi-city approach used for this study allowed addressing the complexity of landscapes and management around the Mediterranean basin reflected by the different evolution of supplied ES over time. European peri-urban areas evolved from a bundle of ES provided mainly by agro-ecosystems to forest and natural vegetation ecosystems. Meanwhile, North African peri-urban areas supplied a bundle of ES from agroecosystems and natural vegetation ecosystems that change tending to rangelands over the years. A first step to improve the estimates presented in this paper would be to take into account land management and its diversity, which is likely to entail a larger diversity in ES supply trends that can be obtained by focusing on land cover information only. In a second future step, the identified trends in ES supply should be confronted with the trends in demand for ES, to evaluate in a quantified manner how forecasted population increases in cities around the Mediterranean will affect ES deficits. Conclusion Mediterranean peri-urban areas can play an important role contributing to the supply of some ES to near urban areas (air quality regulation, timber, wood fuel and religious and spiritual experience). However, general trends indicated a decrease of ES supply due to land cover changes in Mediterranean peri-urban areas, induced by nearby urbanization. 111,112,121,122,123,124,131,132,133,141,142 Non Land cover database Multivariate analyses were conducted to identify land cover changes over time and to detect spatio-temporal trends in ES supply around Mediterranean cities. Variables included land cover data for European cities (for theyears 1990years , 2000years , 2006years and 2012, see Appendix 3) , see Appendix 3) and data for North African cities(for 1990 and 2006, see ), different dynamics of land cover change over time occurred in the periurban areas. A clear pattern of change from non-irrigated arable land in 1990 (negative scores of F1) to urban in 2012 (positive scores of F1) was identified in Madrid and Thessaloniki. Barcelona, Lisbon and Athens showed transitions of complex cultivation patterns, shrublands and pastures (negative scores of F2) to forest (in positive scores of F2) in 1990, and to urban in 2012 (in positive scores of F1). Peri-urban areas of Marseille and Florence were characterized by a transition from irrigated arable land (negative scores of F1) towards urban. Land cover change patterns in the peri-urban area of Rome are less pronounced, but mostly correspond to permanent crops, shrublands and pastures in 1990 transforming into urban land in 2012. In conclusion, for European cities the temporal gradient was dominated by an increase of urban areas and marginal changes in agricultural land uses. Figure 1 . 1 Figure 1. Land cover information. A -Spatial information for 1990 (Corine Land Covergreen colour); B -Spatial information for 2006 (Globcorineblue colour); spatial information used in 2000, 2006 and 2012 covers only Europe (Corine Land Covergreen colour). Figure 2 . 2 Figure 2. Mediterranean biogeographical region (Olson et al. 2001) and selected study sites. Figure 3 .Figure 4 .Figure 5 . 345 Figure 3. Boundaries definition concept and applied example. Figure 6 .Figure 7 .Figure 8 .Figure 9 . 6789 Figure 6. Biplots of Within-class Principal Component Analysis (WCP) for the most statistical significant ecosystem services and their relationship with the European peri-urban areas (i.e., Lisbon, Madrid, Barcelona, Marseille, Florence, Rome, Athens, Thessaloniki). A: variables, B: observations. Table 1 . 1 Land cover data. Year Spatial resolution Cities LC categories 1990 Corine Land Cover 2000 2006 100 m Lisbon (Portugal), Madrid (Spain), Barcelona (Spain), Marseille (France), Florence (Italy), Rome (Italy), Athens (Greece), Thessaloniki (Greece) 44 2012 1990 250 m Rabat (Morocco), Tunis (Tunisia), Sfax (Tunisia), GlobCorine 2006 300 m Nabeul (Tunisia) 14 Table 2 . 2 Urban and surrounding areas (km 2 ) CLC groups Table 3 . 3 CLC categories summarized into CLC groups. LISBON MADRID BARCELONA MARSEILLE FLORENCE ROME ATHENS THESSALONIKI RABAT TUNIS SFAX NABEUL Urban core (km 2 ) 103,39 175,20 74,07 104,01 43,50 296,05 163,07 44,77 251,89 427,33 149,85 23,72 Adjacent urban area Wu (km 2 ) 348,61 431,82 219,95 441,96 145,01 1483,68 452,86 135,46 410,34 790,00 209,41 43,37 Peri-urban area Wp (km 2 ) 1412 2409,41 1000,45 1849,99 657,24 5031,76 2107,68 640,91 3451,95 5715,11 1884,03 350,25 Appendix 4. Land cover standardized surface in peri-urban North African cities from 1990 to 2006. Appendix 5. Ecosystem service capacity provision estimated for European peri-urban areas from 1990 to 2012. Appendix 6. Ecosystem service capacity provision estimated for North African periurban areas from 1990 to 2006. 2012 15,289 0,521 0,000 3,061 13,310 12,063 15,027 40,729 THESSALONIKI 1990 7,124 29,634 7,175 0,443 14,032 7,117 14,420 20,055 2000 7,678 29,709 7,088 0,000 13,938 5,893 15,634 20,060 EESS PROVISI ON -EU (km 2 * estimatio n) EESS PROVISI ON -North Africa (km 2 * estimatio n) Year Year Global climate regulation Global climate regulatio n Local climate regulatio n Local climate regulatio n Air quality regulatio n Air quality regulatio n Water flow regulatio n Water flow regulatio n 2006 2012 Water purificati on Water purificati on 15,018 15,045 Nutrient regulatio n Nutrient regulatio n Erosion regulatio n Erosion regulatio n 19,820 19,839 Natural hazard protectio n Natural hazard protectio n Pollinatio n Pollinatio n Pest and disease control Pest and disease control 7,177 7,094 Regulatio n of waste Regulatio n of waste Crops Crops 0,000 0,000 Energy (Biomass) Fodder Energy (Biomass ) Fodder Livestock Livestock 16,724 16,765 Fibre Timber Fibre Timber Wood fuel Wood fuel 5,446 5,472 Wild Wild food, semi-domestic livestock and ornament al s s resource food, semi-domestic livestock and ornament resource al Biochemi cals and medicine Biochemi cals and medicine 15,971 15,942 Freshwat er Recreati on and tourism Recreati Freshwat on and er tourism Landsca pe aesthetic , amenity and inspiratio n Landsca pe aesthetic , amenity and n inspiratio 19,845 19,843 Knowled ge systems Religious and spiritual experien ce Religious Knowled and ge spiritual systems experien ce Cultural heritage and cultural diversity Cultural heritage and cultural diversity Natural heritage and natural diversity Natural heritage and natural diversity NABEUL 312,489 242,464 17,653 52,478 286,019 288,058 51,679 150,971 66,351 240,647 233,519 163,389 137,769 133,886 44,313 89,420 20,001 37,991 250,279 204,304 0,457 308,984 311,816 331,790 32,051 234,060 271,570 110,537 172,486 59,746 147,179 146,541 141,885 124,268 135,472 103,582 164,796 207,246 143,518 111,302 63,028 66,528 62,382 82,525 105,668 173,900 105,718 90,949 342,044 305,668 297,387 115,534 251,342 264,913 SFAX 299,783 220,164 43,534 61,011 279,125 278,395 62,622 125,707 50,959 200,164 184,150 212,441 111,120 86,580 8,308 39,803 136,523 136,534 213,487 159,586 2,043 395,276 348,311 355,137 86,595 272,984 328,299 RABAT STANDARISED SURFACE % (land cover groups) 87,839 164,908 31,465 142,043 326,601 302,231 108,664 86,251 Year URBAN 121,429 110,244 316,500 313,650 106,321 122,225 NON-IRRIGATED ARABLE LAND 135,262 89,403 184,568 125,280 IRRIGATED ARABLE LAND 170,484 184,249 133,374 254,858 250,813 165,262 PERMANENT CROPS 93,498 59,496 162,117 128,870 COMPLEX CULTIVATION PATTERNS 64,994 68,992 36,825 52,307 11,538 70,644 107,736 114,476 FORESTS 153,494 308,826 SHRUB AND/OR 84,466 310,582 HERBACEOUS VEGETATION 90,600 282,710 ASOC AND PASTURES 246,184 6,067 321,737 322,791 OTHERS 269,165 66,221 CATEGORIES 331,931 67,908 207,049 227,962 247,250 292,171 NABEUL 0,429 0,43 12,23 2,68 5,68 15,91 11,982 51,018 121,146 182,818 71,328 147,142 148,046 144,343 137,294 133,701 120,929 164,786 198,491 172,874 124,335 71,461 72,245 70,545 101,461 128,755 174,431 116,848 75,826 334,781 304,327 302,295 113,677 258,980 268,210 7,412 7,41 8,09 3,13 11,03 11,43 14,094 39,552 TUNIS 216,298 224,977 37,844 88,996 177,533 176,861 76,849 135,210 98,911 181,707 174,019 273,091 189,307 136,028 42,442 114,581 43,040 59,951 189,404 158,934 4,822 251,206 248,653 285,513 42,793 235,044 196,021 125,511 SFAX 190,205 75,822 139,873 132,022 0,219 130,222 137,189 0,22 122,203 129,075 156,309 5,48 172,357 232,339 154,723 0,00 90,106 83,330 96,172 34,76 116,172 142,650 10,29 158,694 119,711 1,256 50,553 311,060 285,848 47,984 294,556 108,999 268,805 248,358 4,818 4,82 10,84 2,81 4,38 12,55 9,062 53,864 RABAT 0,267 0,27 28,22 0,03 1,18 5,60 2,950 41,476 3,754 3,75 11,11 2,44 14,08 13,35 16,647 32,238 TUNIS 0,622 0,62 36,66 2,29 9,17 10,31 10,323 27,819 2,691 2,69 16,20 4,94 17,78 15,17 16,383 20,464 ROME 128,936 179,164 92,054 95,175 105,090 97,349 122,211 97,375 118,235 151,830 136,329 215,767 131,043 101,45 9 23,823 86,372 114,023 122,393 137,078 108,127 13,292 212,913 189,015 202,580 78,021 189,790 151,108 119,253 ATHENS 173,531 91,193 94,273 93,847 8,237 86,800 121,234 1,344 93,358 118,361 147,843 0,000 132,092 216,578 3,223 125,779 101,79 0 22,568 19,286 90,082 114,381 120,867 11,588 129,743 102,619 20,912 13,082 201,318 181,481 35,410 194,963 78,664 186,099 142,797 126,792 177,599 90,815 94,060 101,553 10,246 94,491 120,832 1,066 96,304 118,125 152,158 0,000 136,533 215,792 3,027 127,021 103,01 3 22,659 18,005 89,697 113,834 120,506 9,117 135,717 106,952 23,048 13,102 207,970 187,891 35,491 201,081 78,921 189,145 148,904 126,792 177,599 90,815 94,060 101,553 15,122 94,491 120,832 0,574 96,304 118,125 152,158 0,000 136,533 215,792 3,070 127,021 103,01 3 22,659 13,413 89,697 113,834 120,506 14,310 135,717 106,952 16,822 13,102 207,970 187,891 36,689 201,081 78,921 189,145 148,904 Acknowledgements This work has received support from European Union FP7 projects OPERAs (Contract No. 308393, to APGN and WC), EU BON (Contract No. 308454, to IG and WC) and ECOPOTENTIAL project (Contract No. 641762, to IG). The authors acknowledge Labex OT-Med (ANR-11-LABX-0061) funded by the French Government Investissements d'Avenir program of the French National Research Agency (ANR) through the A*MIDEX project (ANR-11-IDEX-0001-02). Special thanks for valuable advice go to Berta Martín-López, Violeta Hevia-Martín, Claude Napoleone, Marina Cantabrana and Benjamin Mary. We also thank three anonymous reviewers for their constructive comments. OTHERS CATEGORIES LISBON 10,739 4,419 0,830 0,129 27,795 7,248 11,590 37,250 18,272 2,955 0,513 0,889 23,866 5,928 10,585 36,992 19,499 2,857 0,548 0,865 23,133 5,798 10,263 37,037 19,998 2,649 1,682 0,624 21,883 6,040 10,005 37,119 MADRID 9,969 41,753 4,952 2,057 7,805 7,353 25,769 0,342 16,950 36,069 4,250 1,910 7,135 7,456 25,801 0,430 21,689 33,182 3,873 1,699 6,335 7,247 25,522 0,454 24,829 31,340 3,432 1,755 5,637 13,383 18,864 0,760 BARCELONA 19,749 8,023 4,481 1,261 4,204 15,084 8,750 38,447 22,411 7,078 3,820 1,232 3,768 14,851 8,356 38,483 25,470 6,233 3,082 1,180 3,411 14,914 7,378 38,332 25,990 4,807 2,566 1,294 1,545 17,633 7,888 38,278 MARSEILLE 10,020 2,773 0,000 2,252 9,017 17,166 14,777 43,994 11,291 2,566 0,000 2,144 8,871 17,120 15,056 42,953 11,541 2,530 0,000 2,135 8,738 16,552 15,328 43,176 12,172 2,268 0,000 2,136 8,441 16,364 15,975 42,644 FLORENCE 8,216 15,634 0,000 24,596 15,808 32,859 2,401 0,485 9,658 14,518 0,000 24,552 15,282 33,217 2,166 0,608 10,138 14,168 0,000 24,516 15,210 33,317 2,136 0,516 11,205 12,872 0,000 25,276 14,493 33,694 1,963 0,497 ROME 4,206 21,181 0,000 11,299 16,693 13,739 3,386 29,495 4,378 21,506 0,000 10,200 17,522 13,750 3,052 29,593 4,569 21,341 0,000 10,223 17,476 13,656 3,115 29,620 4,569 21,341 0,000 10,223 17,476 13,656 3,115 29,620 LISBON 104,945 146,094 52,712 76,263 93,706 83,958 104,976 91,634 97,049 165,774 154,361 136,308 93,783 99,078 48,168 90,359 48,780 69,098 113,234 73,984 9,905 186,355 186,888 189,761 71,990 174,361 140,182 81,677 118,387 46,108 66,024 63,712 53,603 91,949 63,890 90,640 126,148 114,791 120,533 81,791 86,971 46,470 75,527 43,413 62,703 88,290 56,307 9,135 165,358 153,388 153,729 88,236 153,579 105,086 91,473 123,161 45,036 64,557 83,133 74,155 89,405 79,142 89,591 143,037 135,594 118,531 89,452 88,420 44,721 73,541 41,897 60,720 108,176 62,028 9,083 189,302 178,143 174,508 91,149 164,917 120,896 79,364 115,634 47,099 65,324 62,740 53,385 92,397 61,996 88,735 121,708 111,654 116,718 84,229 83,894 50,448 73,522 44,639 62,422 91,044 54,868 9,705 162,858 151,809 150,777 92,415 152,091 102,510 MADRID 137,797 199,918 56,863 108,995 111,356 121,370 132,650 110,338 142,564 135,914 148,838 274,137 195,064 135,37 5 90,761 135,202 50,567 79,411 162,637 127,094 4,652 201,209 225,817 261,502 89,716 250,232 148,864 130,689 186,171 57,627 103,281 111,629 121,891 131,774 103,995 140,643 127,803 141,433 242,262 178,412 124,30 3 85,941 118,698 49,981 79,663 158,859 121,162 4,575 204,671 228,131 253,178 101,874 244,149 147,914 123,451 174,879 56,463 98,327 105,735 114,796 128,026 98,353 136,908 122,045 137,988 223,268 169,395 119,79 8 85,268 108,876 48,865 78,175 153,020 113,923 4,896 202,429 221,640 242,784 105,976 236,262 141,380 131,432 188,827 82,577 102,726 118,983 126,120 148,076 106,672 145,994 129,522 144,785 208,887 169,671 127,08 9 78,080 93,980 82,616 101,141 160,224 129,146 5,733 221,928 228,805 232,985 113,354 238,321 152,210 BARCEL ONA 104,597 139,139 88,391 65,725 107,207 106,103 105,470 84,387 109,993 114,706 115,719 90,618 93,764 46,019 43,271 52,729 68,616 85,006 111,394 85,945 3,829 170,836 189,225 175,857 117,118 176,564 136,369 90,790 126,253 86,867 62,620 94,903 93,991 102,600 75,901 107,380 103,571 104,711 81,508 86,375 40,758 38,859 46,318 67,230 82,956 100,715 76,276 3,512 160,800 178,281 163,086 119,090 168,926 124,043 95,075 125,564 86,586 59,491 100,586 99,067 98,178 76,513 105,049 104,425 105,097 72,774 82,363 39,385 32,015 39,924 66,829 81,916 103,691 78,496 3,458 163,040 179,719 161,798 117,983 165,671 125,924 93,834 130,124 100,189 66,236 106,400 105,671 117,187 80,173 115,280 101,974 103,284 58,125 76,210 37,244 28,351 26,481 77,559 92,762 101,322 83,260 1,991 165,372 183,837 153,062 120,179 164,404 133,256 MARSEI LLE 111,411 157,355 106,099 82,843 131,348 131,976 133,360 108,164 135,612 133,547 131,249 67,212 72,312 45,323 23,159 32,200 73,884 107,622 120,187 114,603 1,850 189,120 215,081 184,777 113,303 172,073 183,176 106,295 151,825 107,475 82,517 126,013 127,594 133,514 107,041 138,397 131,674 128,779 65,940 70,985 44,775 21,833 32,220 73,202 108,238 118,007 116,612 1,643 186,608 215,665 182,889 117,600 172,633 180,830 105,661 149,557 104,906 81,011 124,507 126,091 130,315 105,637 136,935 130,942 128,995 65,374 70,968 44,759 22,872 32,533 70,935 106,469 118,177 114,907 1,635 185,909 214,075 182,918 117,106 171,136 178,250 103,419 146,747 104,621 80,854 122,703 124,202 129,828 104,603 138,177 129,288 128,160 63,504 70,686 43,747 24,031 31,800 70,143 106,866 117,562 113,602 1,643 186,138 214,294 182,850 120,036 171,013 177,062 FLOREN CE 193,919 258,570 195,260 128,613 196,728 192,867 215,729 138,292 177,815 215,029 198,635 241,838 150,413 113,10 9 15,457 75,417 236,077 256,816 189,705 173,841 5,022 339,559 321,765 305,446 182,269 289,536 266,320 193,160 256,651 196,341 127,204 197,859 194,424 216,145 137,630 177,685 213,766 197,500 234,689 147,249 110,60 6 13,717 72,073 236,730 256,974 189,730 173,045 4,846 340,556 322,355 303,497 184,748 288,627 266,316 193,097 256,172 196,780 126,939 198,312 194,873 216,556 137,464 177,697 213,674 197,462 232,615 146,659 109,85 4 13,635 71,143 237,083 257,356 189,882 172,910 4,874 340,651 322,917 303,481 185,496 288,566 266,537 193,283 254,591 199,142 126,494 200,625 197,478 218,644 137,548 178,392 212,681 195,734 226,819 145,902 107,36 4 12,309 66,968 237,373 257,985 190,031 173,269 4,195 345,555 324,019 301,760 187,129 288,851 266,849 ATHENS 113,068 146,677 85,671 82,287 114,243 110,934 129,314 92,356 134,158 148,577 139,569 96,617 78,479 63,284 46,926 65,878 60,262 101,040 120,086 105,884 5,750 181,755 208,424 201,081 18,556 61,988 39,373 92,633 126,313 75,064 77,396 92,753 89,640 117,404 80,931 129,405 132,619 127,237 89,870 74,853 58,148 51,204 63,888 49,300 94,418 105,749 92,215 5,618 164,652 191,603 188,511 18,317 58,793 34,432 110,525 142,303 94,250 77,242 121,486 118,819 129,790 91,032 132,630 139,484 131,413 74,723 74,514 57,621 36,942 44,376 67,672 106,038 121,562 109,146 4,423 190,175 211,711 189,463 19,716 59,197 38,499 88,793 125,752 81,091 74,093 101,116 98,565 120,289 81,288 121,935 124,459 111,326 74,026 68,138 55,001 38,090 42,742 58,631 93,117 105,010 96,477 4,395 180,025 197,139 172,515 19,749 57,520 35,183 THESSA LONIKI 117,510 178,528 55,660 100,109 91,191 94,486 100,697 101,271 116,576 144,519 131,552 240,847 167,882 125,32 8 52,279 129,664 42,002 63,560 134,987 107,776 5,141 162,875 186,291 213,034 318,720 597,785 561,152 100,224 165,622 49,362 95,119 69,034 70,128 91,844 92,801 114,360 133,498 125,396 238,606 165,581 121,46 2 61,864 134,154 37,186 61,732 115,665 97,525 4,977 146,246 169,680 201,209 311,819 555,889 495,746 102,450 155,985 49,606 91,497 81,958 84,854 92,892 90,083 110,168 138,786 123,525 201,948 146,025 113,64 6 51,704 119,170 35,295 61,316 121,027 96,828 4,753 155,449 185,303 203,661 383,423 574,958 543,387 88,737 147,750 49,704 91,528 68,256 71,084 93,205 84,663 110,227 130,477 115,218 201,783 143,016 110,94 6 51,495 118,924 35,482 61,417 110,027 88,646 4,798 144,593 174,359 192,603 349,169 530,562 499,209

01718234

en

2024/03/05 22:32:07

2018

https://inria.hal.science/hal-01718234v2/file/main.pdf

Alejandro Gómez-Boix email: [email protected] Pierre Laperdrix email: [email protected] Benoit Baudry email: [email protected] Hiding in the Crowd: an Analysis of the Effectiveness of Browser Fingerprinting at Large Scale Keywords: browser fingerprinting, privacy, software diversity Browser fingerprinting is a stateless technique, which consists in collecting a wide range of data about a device through browser APIs. Past studies have demonstrated that modern devices present so much diversity that fingerprints can be exploited to identify and track users online. With this work, we want to evaluate if browser fingerprinting is still effective at uniquely identifying a large group of users when analyzing millions of fingerprints over a few months. We analyze 2,067,942 browser fingerprints collected from one of the top 15 French websites. The observations made on this novel dataset shed a new light on the ever-growing browser fingerprinting domain. The key insight is that the percentage of unique fingerprints in this dataset is much lower than what was reported in the past: only 33.6% of fingerprints are unique by opposition to over 80% in previous studies. We show that non-unique fingerprints tend to be fragile. If some features of the fingerprint change, it is very probable that the fingerprint will become unique. We also confirm that the current evolution of web technologies is benefiting users' privacy significantly as the removal of plugins brings down substantively the rate of unique desktop machines. INTRODUCTION Web browsers share device-specific information with servers to improve online user experience. When a web browser requests a webpage from a server, by knowing the platform or the screen resolution, the server can adapt its response to take full advantage of the capabilities of each device. In 2010, through the data collected by the Panopticlick website, Eckersley showed that this information is so diverse and stable that it can be used to build what is called a browser fingerprint to track users online [START_REF] Eckersley | How Unique is Your Web Browser?[END_REF]. By collecting information from HTTP headers, JavaScript and installed plugins, he was able to uniquely identify most of the browsers. With the gathered data, Eckersley not only showed that there exists an incredible diversity of devices around the world but he highlighted that this very same diversity could be used as an identification mechanism on the web. Since this study, researchers have looked at new ways to collect even more information [13, 14, 18, 24-26, 32, 34, 36], measure the adoption of these techniques on the Internet [START_REF] Acar | The Web Never Forgets: Persistent Tracking Mechanisms in the Wild[END_REF][START_REF] Acar | FPDetective: dusting the web for fingerprinters[END_REF][START_REF] Englehardt | Online Tracking: A 1-million-site Measurement and Analysis[END_REF][START_REF] Nikiforakis | Cookieless Monster: Exploring the Ecosystem of Web-Based Device Fingerprinting[END_REF], propose defense mechanisms [START_REF] Baumann | Disguised Chromium Browser: Robust Browser, Flash and Canvas Fingerprinting Protection[END_REF]17,[START_REF] Fiore | Countering Browser Fingerprinting Techniques: Constructing a Fake Profile with Google Chrome[END_REF][START_REF] Laperdrix | FPRandom: Randomizing core browser objects to break advanced device fingerprinting techniques[END_REF][START_REF] Laperdrix | Mitigating browser fingerprint tracking: multi-level reconfiguration and diversification[END_REF][START_REF] Nikiforakis | PriVaricator: Deceiving Fingerprinters with Little White Lies[END_REF], and track devices over long periods of time [START_REF] Vastel | FP-STALKER: Tracking Browser Fingerprint Evolutions[END_REF]. In 2016, a study conducted by Laperdrix et al. [START_REF] Laperdrix | Beauty and the Beast: Diverting modern web browsers to build unique browser fingerprints[END_REF] with the AmIUnique website confirmed Eckersley's findings. The authors noted a shift in the most discriminating attributes with the addition of new APIs like Canvas and the progressive removal of browser plugins. They also demonstrated that fingerprinting mobile devices is possible, but with a lower degree of success. Tracking users with fingerprinting is a reality. If a device presents the slightest difference compared to other ones, it can be identified and followed on different websites. While Panopticlick and AmIUnique proved that tracking is possible, one problem arises when looking at both datasets: their bias. First, both websites are dedicated to fingerprinting, people who visited them are interested in the topic of online tracking. It limits the scope of their studies. Then, looking at the general statistics page of the AmIUnique website from July 2017, we can clearly see a bias as 57% of visitors are on Windows, 15% on Linux, 13% on Mac, 5% on Android and 4% on iOS. The latest statistics from StatCounter for the month of July 2017 reveal that the OS market share is dominated by Android with a percentage around 40%, followed by Windows at 36%, iOS at 13%, Mac at 5% and Linux under 1% [START_REF]Operating System Market Share Worldwide -StatCounter[END_REF]. One can then ponder about the impact of such a big difference on the effectiveness of browser fingerprinting. In this paper, we investigate whether tracking can be extended to websites that target a broad audience. We analyze 2,067,942 fingerprints collected from one of the top 15 French websites, and we investigate whether browser fingerprinting techniques are still effective in identifying users by collecting the same attributes reported in the literature. Our first two research questions are related to this issue: RQ 1. How uniquely identifiable are the fingerprints in our data? RQ 2. Can non-unique fingerprints become unique if some value changes? The other questions are related to the characteristics of the dataset and the possible impact of the evolution of web technologies: RQ 3. Can the circumstances under which fingerprints are collected affect the obtained results? RQ 4. Does the evolution of web technologies limit the effectiveness of browser fingerprinting? Where previous studies reported having above 80% of unique fingerprints, we obtained a surprising number: 33.6% of unique fingerprints. This gap can be explained by the targeted audience as our study looks at fingerprints collected from the global population and not necessarily biased towards users interested in online privacy. The difference is even more noticeable when looking at the 251,166 fingerprints coming from mobile devices. [START_REF] Fifield | Fingerprinting web users through font metrics[END_REF].5% of them are unique which is in direct contradiction with the 81% that has been observed by Laperdrix et al. [START_REF] Laperdrix | Beauty and the Beast: Diverting modern web browsers to build unique browser fingerprints[END_REF]. These results show another aspect of browser fingerprinting and its tracking capabilities with the current evolution of web technologies. Here, we extend the analyses carried out by Eckersley [START_REF] Eckersley | How Unique is Your Web Browser?[END_REF] and Laperdrix et al. [START_REF] Laperdrix | Beauty and the Beast: Diverting modern web browsers to build unique browser fingerprints[END_REF] by putting the browser fingerprinting domain under a different light. Our key contributions are: • We explore the current state of browser fingerprinting with the analysis of 2,067,942 fingerprints composed of 17 different attributes. We also provide the first large-scale study of JavaScript font probing and we measure its real-life effectiveness. • We show that by collecting these attributes and targeting a much broader audience, browser fingerprinting is not as effective as it was reported in the literature. While previous studies reported having above 80% of unique fingerprints, we obtained 33.6%. • We compare our dataset with the ones from Panopticlick and AmIUnique and we explain in details the numerous differences that can be observed. • We provide a discussion on the future of browser fingerprinting and what these results mean for the domain and for future applications of this technique. The paper is organized as follows. Section 2 introduces our new dataset along with the ones from Panopticlick and AmIUnique. Section 3 analyzes the diversity of browser fingerprints in our data and compares the three datasets by providing detailed statistics to help explain the differences. Section 4 discusses the impact of our results on the domain and we simulate possible technical evolutions to have an insight on future applications of this technique. Finally, Section 5 concludes this paper. DATASET This section introduces the three different datasets that form the basis of the comparison in the next section. First, we give a short description of the two available sets of browser fingerprint statistics conducted on a large scale. Then, we describe the attributes collected to form the browser fingerprints analyzed here. Previous studies 2.1.1 Panopticlick. In 2010, Peter Eckersley launched the Panopticlick website with the goal of collecting device-specific information via a script that runs in the browser [START_REF] Eckersley | How Unique is Your Web Browser?[END_REF]. The script collected values for 10 different web browser features and its execution platform. Features were collected from three different sources: HTTP protocol, JavaScript and Flash API. Eckersley collected 470,161 fingerprints from January 27th to February 15th, 2010. Data obtained by Panopticlick is "representative of the population of Internet users who pay enough attention to privacy" [START_REF] Eckersley | How Unique is Your Web Browser?[END_REF], so in this sense the data is quite biased. In the study performed by Eckersley, the list of fonts (collected through the Flash API) and the list of plugins (collected via JavaScript) were the most distinguishable attributes. 2.1.2 AmIUnique. With the aim of performing an in-depth analysis of web browser fingerprints, the AmIUnique website was launched in November 2014. Collected fingerprints are composed of 17 features (among them, those proposed by Eckersley [START_REF] Eckersley | How Unique is Your Web Browser?[END_REF]). These fingerprints include recent technologies, such as the HTML5 canvas element and the WebGL API. In the study conducted by Laperdrix et al. [START_REF] Laperdrix | Beauty and the Beast: Diverting modern web browsers to build unique browser fingerprints[END_REF], 118,934 fingerprints collected between November 2014 and February 2015 were analyzed. The authors validated Eckersley's findings with Panopticlick and provided the first extensive analysis of fingerprints collected from mobile devices. Data collected on this website is biased towards users who care about privacy and their digital footprint. The dataset The fingerprints used in this study have been collected through a script deployed in collaboration with the b<>com Institute of Research and Technology (IRT) on one of the top 15 French websites (according to the Alexa traffic rank) on two specific web pages: a weather forecast page and a political news page. The script ran for a six month period, from December 7th, 2016 to June 7th, 2017. To be compliant with the European directives 2002/58/CE and 2009/136/CE, and with the French data protection authority (CNIL), only visitors who consented to the use of cookies, and thus the use of fingerprinting techniques, were fingerprinted. When users first connect to one of these two pages, we set up a 6-months long cookie in their browser. This supports the identification of returning visitors. Compared to the other two detailed studies, the website used to collect this dataset covers a wide range of topics and it is not dedicated to browser fingerprinting. According to the Hawthorne effect [START_REF] Mccarney | The Hawthorne Effect: a randomised, controlled trial[END_REF], if individuals are aware that they are being studied, a type of reaction occurs, in which individuals modify an aspect of their behavior in response to their awareness of being observed. In our case, this means that the fingerprints in this dataset are more representative of those found in the wild, since users are not enticed to play with their browsers to change their configuration and produce different fingerprints. Fingerprinted attributes. In order to compare ourselves with previous studies, we rely on the same attributes found in the study conducted by Laperdrix et al. in 2016 [START_REF] Laperdrix | Beauty and the Beast: Diverting modern web browsers to build unique browser fingerprints[END_REF]. The complete list of attributes is given in the 'Attribute' column of Table 2. However, to reflect recent technological trends, we made the following modifications to our script: List of fonts. Fonts are usually collected through the Flash plugin. With a few lines of code, one can get access to the entire list of fonts installed on the user's system. However, because of security and stability reasons, plugins are being deprecated in modern browsers in favor of a feature-rich HTML5 environment [START_REF] Schuh | Saying Goodbye to Our Old Friend NPAPI[END_REF]. Flash is expected to disappear definitely as Adobe announced the end-of-life of its solution for 2020 [START_REF]Flash & The Future of Interactive Content -Adobe[END_REF]. All major web browsers like Chrome, Firefox, Edge and Safari already block Flash content or have removed support for it. This means that fingerprinting scripts must turn to another mechanism to get access to the list of fonts. Nikiforakis et al. revealed that it is possible to probe for the existence of fonts through JavaScript [START_REF] Nikiforakis | Cookieless Monster: Exploring the Ecosystem of Web-Based Device Fingerprinting[END_REF]. A script can ask to render a string with a specific font in a div element. If the font is present on the device, the browser will use it. If not, the browser will use what is called a fallback font. By measuring the dimensions of the div element, one can know if the demanded font is used or if the fallback font took its place. The biggest difference between these two gathering methods is that fonts through JavaScript must be checked individually whereas Flash gives all the installed fonts in a single instruction. This means that testing a large number of fonts is time consuming and can delay the loading of a web page. For this reason, we chose to test 66 different fonts, some among the most popular 'web-safe fonts' which are found in most operating systems and other less common ones. Appendix A reports on the complete list of fonts we tested in our script. Before deploying our script in production, we identified a limitation in how JavaScript font probing operates. We found out that some fonts can have the exact same dimensions as the ones from the fallback font. Figure 1 illustrates this problem. In the example, the two tested fonts are metrically comparable and have the exact same width and height. However, they are not identical as it can be seen in the shapes of some of the letters (especially "e", "a" and "w"). This means that font probing here will report incorrect results if one were to ask Times New Roman on a system with the Tinos font installed (or vice versa). To fix this problem, we measured the dimensions of a div against three font style variants. There are different typefaces that can be used by a web browser with the most popular ones being serif, sans-serif, monospace, cursive and fantasy. We chose the first three and we tested each font against the three of them, resulting in 66 * 3 = 198 different tests. This way, we avoid reporting false negatives as the three fallback fonts have different dimensions. Canvas. The Canvas API allows for scriptable rendering of 2D shapes and texts in the browser. Discovered by Mowery et al. [START_REF] Mowery | Pixel Perfect: Fingerprinting Canvas in HTML5[END_REF], investigated by Acar et al. [START_REF] Acar | The Web Never Forgets: Persistent Tracking Mechanisms in the Wild[END_REF], and then collected on a large scale by Laperdrix et al. [START_REF] Laperdrix | Beauty and the Beast: Diverting modern web browsers to build unique browser fingerprints[END_REF], canvas fingerprinting can be used to differentiate devices with pixel precision by rendering a specific picture following a set of instructions. In order to see how far we can go with this technique, we took as a basis the canvas test performed by Laperdrix et al. [START_REF] Laperdrix | Beauty and the Beast: Diverting modern web browsers to build unique browser fingerprints[END_REF] and we made a more complex canvas element by combining new elements of different natures. First, the script asks the browser to render the two following strings: "Yxskaftbud, ge vår WC-zonmö IQ-hjälp" and "Gud hjälpe Zorns mö qvickt få byxa". Both strings are pangrams (a string with all the letters of the alphabet) of the Swedish alphabet. For the first string, we force the browser to use one of its fallback fonts by asking for a font with a fake name. Depending on the OS and the fonts installed on the device, fallback fonts may differ from one user to another. For the second line, the browser is asked to use the Arial font that is common in many operating systems. Then, we ask for additional strings with symbols and emojis. All strings, with the addition of a rectangle are drawn with a specific rotation. A second set of elements is rendered with four mathematical functions: a sine, a cosine and two linear functions. These functions are plotted on a specific interval and using the PI value of the JavaScript Math library as a parameter. The third set of elements consists in drawing a set of ellipsis. These figures are drawn with different colors and with different levels of transparency. Since filters for opacity change among browsers, it creates differences between them. The last element is a centered shadow that overlaps the canvas element. Figure 2 displays an example of a canvas rendering following the instructions of our script. Cookies. Since we only have fingerprints from users who accepted the use of cookies, all fingerprints have the exact same value for this attribute. Descriptive statistics. We distinguish two different kinds of fingerprints: those belonging to mobile devices and those belonging to desktop and laptop machines (we will refer to desktop and laptop machines as personal computers). To prevent collecting multiple copies of the same fingerprint from the same user, we store a cookie on the user's device with a unique ID for six months. Among the 2,067,942 fingerprints, the distinction is as follows: 1,816,764 come from personal computers (87.9% of the data), and the rest, 251,190 fingerprints come from mobile devices (12.1% of the data). 1 reports on the distribution of operating systems in both our dataset and the one from the AmIUnique website. Statistics gathered from StatCounter for the month of July 2017 have also been added to give an idea how close they are from the global population. First, by looking at the differences between our newly collected data and AmIUnique, we can see that there is a significant difference in terms of distribution. Notably, we can see a clear bias in the demographic that AmIUnique attracted since the percentage of Linux desktop machines is much higher than the reported by StatCounter. Then, if we compare our numbers with the ones from StatCounter, we can see that we provide a closer representation of the global population as the percentages for both distributions are close to each other. Table 2 summarizes the essential descriptive statistics of our dataset. The 'Distinct values' column provides the number of different values that we observed for each attribute, while the 'Unique values' column provides the number of values that occurred a single time in our dataset. For example, the Use of local/session storage attribute has no unique values since it is limited to "yes" and "no". Moreover, in our data, all users accepted the use of cookies, so all the fingerprints have "yes" for this attribute. Other attributes can take a high number of values. For example, we observed 6,618 unique values for the list of fonts. In fact, we also know the higher bound for the number of distinct values for this attribute. We perform in total 66 * 3 tests and each one can take the value 'true' or 'false'. These results in 2 66 * 3 possible combinations even if, in practice, many of them will not be found. ANALYSIS AND COMPARISON In this section, we first analyze how diverse browser fingerprints are in our dataset. Then, we analyze the level of identifying information of each attribute that makes up the fingerprint. Finally, we compare our dataset with the two available sets of fingerprint statistics, provided by Eckersley in 2010 [START_REF] Eckersley | How Unique is Your Web Browser?[END_REF] and Laperdrix et al. in 2016 [START_REF] Laperdrix | Beauty and the Beast: Diverting modern web browsers to build unique browser fingerprints[END_REF]. Browser fingerprint diversity Our data was collected on a much larger scale than previous studies and targeting a much broader audience, which leads to the RQ 1. How uniquely identifiable are fingerprints in our data? This question aims at determining how diverse the browser fingerprints are in this novel dataset. Using attributes from Table 2, we succeeded in uniquely identifying 33.6% of fingerprints in our dataset. On personal computers, 35.7% of fingerprints are unique while this number is lower on mobile devices with 18.5%. On personal computers, the threat is less important than reported in other studies. On mobile devices, the number is much smaller but the threat comes from elsewhere: closed platforms with integrated tracking applications. Figure 3 represents the distribution of the anonymity sets. A set represents a group of fingerprints with identical values for all the collected attributes. If a fingerprint is in a set of size 1, it means that this fingerprint is unique and it can be identified. On mobile devices, the percentages of fingerprints belonging to sets of size larger than 50 is around 59%, while on personal computers this percentage is around 8%. It means that the number of devices sharing equal fingerprints on mobile devices is larger than on personal computers. This can be explained by the fact that the software and hardware environments of these devices are much more constrained than on desktop and laptop machines. Users buy very specific models of smartphones that are shared by many. The largest set of mobile devices contains 13,241 fingerprints, while for personal computers it contains 1,394 fingerprints. Low rates of success at uniquely identifying browser fingerprints in our data reveal that, by collecting more than two million browser fingerprints on a commercial website, it is very unlikely that a fingerprint is unique and hence exploitable for tracking. Possibilities for a fingerprint to be unique are three times lower than in the previous datasets collected for research purposes (Panopticlick and AmIUnique). Unique fingerprints. There are 46,459 unique fingerprints on mobile devices and 647,741 on personal computers. A fingerprint is unique due to one of the following reasons: • It has an attribute whose value is only present once in the whole dataset. • The combination of all its attributes is unique in the whole dataset. On mobile devices, 73 % of fingerprints are unique because they contain a unique value, while this percentage is around 35% for personal computers. While mobile fingerprints tend to be unique because of their unique values, laptop/desktop fingerprints tend to have combinations of values so diverse that they create unique fingerprints. The most distinctive attributes are canvas on mobile devices and plugins on personal computers. Fingerprints with unique canvas values represent 62% of unique fingerprints on mobile devices, while on personal computers, fingerprints with unique combinations of plugins represent 30% of unique fingerprints. Investigating changes on browser fingerprints. Over the course of its lifetime, a device exhibits different fingerprints. This comes from the fact web technologies are constantly evolving and thus, web browser components are continually updated. From the operating system to the browser and its components, one single update can change the exhibited browser fingerprint. For instance, a new browser version is directly reflected by a change in the user-agent. A plugin update is noticeable by a change in the list of plugins. When web browsers evolve naturally, changes happen automatically without any user intervention and this affects all users. Natural evolution of web technologies is not the only reason why fingerprints evolve. There are some parameters that usually are the choice of the users, such as the use of cookies, the presence of the "Do Not Track" header, or the activation of specific plugins. Users are allowed to change these values at anytime. Besides, some attributes such as timezone or fonts are indirectly impacted by a Let us take as an example of a user with a non-unique fingerprint. Running Chrome 55 on Windows 10, the browser displays the following value for the Content language header: fr-FR,fr;q=0.8,en-US;q=0.6,en;q=0.4 For some reason, the user decides to add the Spanish language. The browser then displays the following value: fr-FR,fr;q=0.8,en-US;q=0.6,en;q=0.4,es;q=0.2 By changing the language settings, does the fingerprint become unique? In order to answer this type of question and to study how resilient non-unique fingerprints are in the face of evolution, we conducted an experiment. We looked at analyzing the impact made by the user's choice on the uniqueness of their fingerprints. There is a set of attributes whose values cannot be changed such as attributes related to the hardware and software environment on which the browser is running. The Platform attribute is linked to the operating system, while WebGLVendor and WebGLRenderer reveal information about the GPU. Attributes such as User-agent, List of HTTP headers or Content encoding are beyond the control of the user because they are related to the HTTP protocol. However, attributes such as Cookies enabled, Do Not Track, Content language and List of plugins are a direct reflection of the user's choice. Nevertheless, Cookies enabled, Do Not Track, Use of local/session storage are limited to "yes" and "no", so they do not offer a very discriminant information. This leaves the Content language, List of plugins, Available fonts and Timezone under the scope of our analysis. For the experiment, we chose fingerprints belonging to sets larger than 50 fingerprints. New values were chosen randomly from nonunique fingerprints that had the same operating system and web browser (including versions). This was made to ensure that the new values are consistent with fingerprints that can be found in the wild. This way, we avoid choosing values that are not characteristic of the fingerprint environment. For example, two browsers can have the same language configuration, but the encoding is different depending on the web browser. Example: Windows 8.1, Chrome, fr-FR,fr;q=0.8,en-US;q=0.6,en;q=0.4 Windows 8.1, Firefox, fr-FR,fr;q=0.8,en-US;q=0.5,en;q=0.3 Both browsers are running on the same operating system and have the same language configuration: French/France[fr-FR], French[fr], English/United States[en-US] and English [en]. But, depending on the web browser, the final language headers are different. Results. The experiment was repeated ten times and results were averaged. Figure 4 represents the distribution of the anonymity sets resulting of randomly changed values for the Content language, List of plugins, Available fonts and Timezone on mobile devices and desktop/laptop machines. First, we can clearly notice an important difference between devices. For desktop/laptop machines, more than 85% of fingerprints turned into unique fingerprints. This is due to the fact that combinations of values tend to be so diverse on personal computers that they make up unique fingerprints. On mobile devices, when changing the Available fonts and the List of plugins, over 80% of fingerprints remained in large sets. These results are explained by the absence of diversity in these attributes on mobile devices. Results are very different for Content language and Timezone: over 60% of fingerprints turned into unique fingerprints. This can be explained by the lack of diversity in these attributes. As most users share the same timezone and languages, a single change on one of these two attributes dramatically increases the likelihood of the fingerprint to become unique. By looking at the results of the experiment, we can conclude that if one single feature of a fingerprint changes, it is very probable that this fingerprint becomes unique. In the end, desktop/laptop fingerprints tend to be much more fragile than their mobile counterparts. Comparison of attributes Mathematical treatment. We used entropy to quantify the level of identifying information in a fingerprint. The higher the entropy is, the more unique and identifiable a fingerprint will be. Let H be the entropy, X a discrete random variable with possible values x 1 ; ...; x n and P(X ) a probability mass function. The entropy follows this equation: H (X ) = - n i=0 P(x i )loд b P(x i ) (1) We use the entropy of Shannon where b = 2 and the result is expressed in bits. One bit of entropy reduces by half the probability of an event occurring. In order to compare all three datasets which are of different sizes, we applied the Normalized Shannon's entropy: H (X ) H M (2) H M represents the worst case scenario where the entropy is maximum and all values of an attribute are unique (H M = loд 2 (N ) with N being the number of fingerprints in our dataset). The advantage of this measure is that it does not depend on the size of the anonymity set but on the distribution of probabilities. We are quantifying the quality of our dataset with respect to an attribute uniqueness independently from the number of fingerprints in our database. This way, we can qualitatively compare the datasets despite their different sizes. Table 3 lists the Shannon's entropy for all attributes from both the Panopticlick and AmIUnique studies, and our dataset. Column 'Entropy' shows the bits of entropy and column 'Norm. ' shows the normalized Shannon's entropy. The last two rows of Table 3 show the worst case scenario where the entropy is maximum (i.e. all the values are unique) and the total number of fingerprints. In the novel dataset analyzed here, the most distinctive attributes are the List of plugins, the Canvas, the User-agent and the Available fonts. Due to differences in software and hardware architecture between mobile devices and personal computers, we computed entropy values separately. By comparing entropy values between mobile devices and personal computers, we observed three attributes where the difference is significant. The largest difference is for the List of plugins with a difference of 0.485 for the normalized entropy. It can be explained by the lack of plugins on mobile devices as web browsers on mobile devices take full advantage of functionalities offered by HTML5 and JavaScript. On personal computers, the List of plugins is the most discriminant attribute while it is almost insignificant for mobile devices. We can observe in Table 2 that among 251,166 fingerprints coming from mobile devices, there are only 81 distinct values for plugins. The second significant difference is 0.214 for the Available fonts. Installing fonts on mobile devices is much more restrained than on personal computers. Even if we test a very limited set of fonts through JavaScript compared to what could be collected through Flash, we can see that there is clearly more diversity on personal computers. The last significant difference is for the User-agent attribute, with a difference of 0.182. On mobile devices, the user-agent has the highest entropy value. This is because phone manufacturers include the model of their phone and even sometimes the version of firmware directly in the user-agent as revealed by Laperdrix et al. [START_REF] Laperdrix | Beauty and the Beast: Diverting modern web browsers to build unique browser fingerprints[END_REF]. Attributes Use of an ad blocker and Use of local/session storage have very low entropy values because their values are either "yes" or "no". We also tested the impact of compressing a canvas rendering to the JPEG format. It should be noted that the JPEG compression comes directly from the Canvas API and is not applied after collection. Due to the lossy compression, it should come as no surprise that the entropy from JPEG images is lower than the PNG one usually used by canvas fingerprinting tests (from 0.407 to 0.391). In the study realized by Eckersley [START_REF] Eckersley | How Unique is Your Web Browser?[END_REF], the analysis of browser fingerprints was performed without differentiating between mobile and desktop fingerprints. Later, some researchers conducted studies about browser tracking mechanisms either on desktop machines [START_REF] Acar | The Web Never Forgets: Persistent Tracking Mechanisms in the Wild[END_REF][START_REF] Boda | User Tracking on the Web via Cross-Browser Fingerprinting[END_REF] or on mobile devices [START_REF] Spooren | Mobile Device Fingerprinting Considered Harmful for Risk-based Authentication[END_REF][START_REF] Wu | Efficient Fingerprinting-Based Android Device Identification With Zero-Permission Identifiers[END_REF] but not on both. In 2016, Laperdrix et al. [START_REF] Laperdrix | Beauty and the Beast: Diverting modern web browsers to build unique browser fingerprints[END_REF] provided the first extensive study about browser fingerprinting on mobile devices, they proved that both kind of devices presented different discriminating attributes. If the analysis of browser fingerprinting is not carried out by differentiating mobile devices from personal computers, results obtained will not be representative of both kinds of devices. In our data, 12.1% of fingerprints belong to mobile devices so mobile fingerprints represent a small part of the entire data. If we take a look at Table 3, entropy values for attributes like List of plugins or Available fonts are largely influenced by the group that contains the majority of fingerprints which, in our case, is the one with personal computers. For future work, it is strongly recommended to differentiate mobile devices from personal computers (laptops and desktop machines) to obtain more accurate results. Comparison with Panopticlick and AmIunique In the data collected by Panopticlick, Eckersley observed that 83% of visitors had instantaneously recognizable fingerprints. This number reached 94% for devices with Flash or Java installed. With the AmIUnique website, Laperdrix and colleagues observed that 89.4% of fingerprints from their dataset were unique. Thanks to the high percentages of unique browser fingerprints, browser fingerprinting established itself as an effective stateless tracking technique on the web. However, with our study, we provide an additional layer of understanding in the fingerprinting domain. By having 33.6% of unique fingerprints compared to the 80+% of the other two studies, we show that browser fingerprinting may not be effective at a very large scale and that the targeted audience plays an important role in its effectiveness. Comparing data size. When analyzing the percentages of unique fingerprints, the amount of fingerprints is an important element that influences the results. As discussed by Eckersley in [START_REF] Eckersley | How Unique is Your Web Browser?[END_REF], the probability of any fingerprint to be unique in a sample of size N is 1/N . It is clear that probabilities of being unique in our dataset are much lower than the probabilities of being unique in the AmIUnique one. With the aim of establishing a more equitable comparison, we took some samples with the same number of fingerprints as the AmIUnique data and we then calculated the percentage of unique fingerprints. We perform a comparison with the AmIUnique data because the amount of fingerprints is four times smaller than the one collected by Panopticlick. Because our dataset spans a six month period, we divided the data into six parts, each part containing data for one month. We kept the same proportion between mobile devices and desktop machines as the AmIUnique data, so we randomly took 105,829 desktop/laptop fingerprints and 13,105 mobile fingerprints from each month. Results were averaged. On average, 56% of personal computers are unique, while 29% of mobile devices are unique. These percentages show that low ratios of unique fingerprints are influenced by the number of fingerprints. Even so, results obtained on the sample are significantly distant from those obtained by Laperdrix et al. [START_REF] Laperdrix | Beauty and the Beast: Diverting modern web browsers to build unique browser fingerprints[END_REF]. These results show that performing tracking with fingerprinting is possible, yet difficult. Comparing entropy values. Comparison with Panopticlick can be established by taking into account only six attributes. We observe that entropy values for our dataset and Panopticlick differ significantly for all attributes, except for the Screen resolution. Entropy values for the Screen resolution attribute hardly change for the three datasets. Regarding Timezone and Cookies enabled, drops occur in entropy values due to the characteristics of our dataset. As we explained in Section 2, we analyze fingerprints from users who accepted cookies, and most of them live in the same geographic region. The difference in the entropy value for Content language is due to the fact that most users are located in the same geographic region, which implies that most of them share the same language. In fact, 98% of users present the same value for timezone, which corresponds to Central European Time Zone UTC+01:00 and as a direct consequence of this, 97.7% of fingerprints present French as their first language. The noticeable drop in the entropy values for the Timezone and Content language affects the fingerprint diversity. To a great extent, the lack of diversity in any attribute has a direct impact on the fingerprint diversity. By decreasing the amount of values that an attributes can take, the identifying value of the attribute is reduced and therefore the identifying value of the browser fingerprint decreases. It means that the diversity surface is reduced, which reduces the diversity among the browser fingerprints giving as result less identifiable browser fingerprints. For the List of plugins, it is still the most discriminating attribute but a gradual decrease can be observed. From Panopticlick to AmI-Unique, a difference of 0.24 is present. From AmIUnique to our dataset, the difference is 0.126 resulting in a decrease of 0.365 from Panopticlick to our data. This gradual decrease in the entropy value for the List of plugins is explained by the absence of plugins on mobile devices and by the removal of plugins from modern browsers. Over time, features have been added in HTML5 to replace plugins as they were considered a source of many security problems. Chrome stopped supporting the old NPAPI plugin architecture on Chrome in 2015 (topic discussed in [START_REF] Laperdrix | Beauty and the Beast: Diverting modern web browsers to build unique browser fingerprints[END_REF]). Mozilla dropped support in version 52 of the Firefox browser released in March 2017. Safari has never supported plugins, Flash is long discontinued for Android, and MS Edge for Windows 10 does not support most plugins. Anything else reliant on the Netscape Plugin API (NPAPI) is now dropped which means Silverlight, Java and Acrobat are gone [START_REF]Mozilla Developer Network and individual contributors[END_REF]. The difference in the entropy value for Available fonts between Panopticlick and AmIUnique is explained by Laperdrix et al. [START_REF] Laperdrix | Beauty and the Beast: Diverting modern web browsers to build unique browser fingerprints[END_REF]. Half of the fingerprints in the AmIUnique dataset were collected on browsers that do not have the Flash plugin installed or activated. Between AmIUnique and our data, the difference for the entropy value of fonts is 0.117. Even if collecting fonts through JavaScript is not as effective as with Flash, we observe that the entropy of fonts is still high, keeping its place as one of the top distinctive attributes. For the other attributes, we observe that the entropy values for both our dataset and AmIUnique are similar. DISCUSSION In this section, we discuss our results along with the potential implications on the browser fingerprinting domain. The impact of different demographics In Section 3, we compared our dataset with the two available sets of fingerprint statistics. There are two key elements that can influence the results of this analysis: the targeted audience and the evolution of web technologies. Previous datasets were collected through websites dedicated to browser fingerprint collection. Both websites amiunique.org and panopticlick.eff.org inform users about online fingerprint tracking, so users who visit these websites are aware of online privacy, interested in the topic or might be more cautious than the average web user. Our dataset is much different as it was collected by targeting a general audience through a commercial website. We believe that this difference in the fingerprint collection process is key to explain the differences between datasets, giving rise to RQ 3. Can the circumstances under which fingerprints are collected affect the obtained results? Web technologies affect browser fingerprinting. The fact that some technologies are no longer used leads to the evolution of fingerprinting techniques. In some cases, it leads to a decrease in the identifying value of certain features, as we noticed in the attributes List of plugins and Available fonts. As a result of the progressive disappearance of plugins, the List of plugins is rapidly losing its identifying value. In addition to the effects produced by the evolution of technologies, there are some issues resulting from the collection process. Some of them are caused by targeting a specific demographic group. For instance, the market share distribution across the planet is not uniform. According to StatCounter [START_REF]Operating System Market Share Worldwide -StatCounter[END_REF] in 2017, the European mobile market was led by Apple and Samsung, with similar participation percentages above 30%. Although the mobile market in North America is also led by Apple and Samsung, Apple represents about 50% of the market, while Samsung about 24%. If we collect a sample of mobile fingerprints from North America, there is a good chance that the sample will have a greater presence of Apple devices. So, the distribution of some features like the Platform, WebGL Vendor or User-agent will be more representative of Apple devices. In the end, depending on the website, the use case or the targeted demographic, the results can greatly vary and the effectiveness of browser fingerprinting can change. Moreover, if we were to perform a similar study targeted at different countries, can we expect the same diversity of fingerprints? Do more developed countries have access to a wider range of devices and, as a consequence, present a larger set of fingerprints? Do more educated users have a tendency to specialize and configure more their devices which, as a result, would make their fingerprints more unique? From data that we gathered, it is impossible to answer these questions as we do not collect information beyond what is presented by the user's device. Yet, considering these different facets may be the key to understand the extent to which browser fingerprinting can work for tracking and identification. Its actual effectiveness is much more nuanced that what was reported in the past and it is far from being an answer to a simple yes or no question. Towards a potential privacy-aware fingerprinting An arms race is currently developing between users and thirdparties. As people are getting educated on the questions of tracking and privacy on the web, more and more users are installing browser extensions to protect their daily browsing activities. At the end of 2016, 11% of the global Internet population is blocking ads on the web [START_REF]The state of the blocked web -2017 Global Adblock[END_REF]. This represents 615 million devices with an observed 30% growth in a single year. With regards to browser fingerprinting, several browsers already include protection to defend against it. Pale Moon [START_REF]Pale Moon browser -Version 25[END_REF], Brave [START_REF]Fingerprinting Protection Mode -Brave browser[END_REF] and the Tor Browser [START_REF]The Design and Implementation of the Tor Browser [DRAFT] "Cross-Origin Fingerprinting Unlinkability[END_REF] were the very first ones to add barriers against techniques like Canvas or WebGL fingerprinting. Mozilla is also currently adding its own fingerprinting protection in Firefox [START_REF]Fingerprinting protection in Firefox as part of the Tor Uplift Project -Mozilla Wiki[END_REF] as part of the Tor Uplift program [START_REF]Tor Uplift Project -Mozilla Wiki[END_REF]. With our study, we show that we do not know yet the full extent of what is possible with browser fingerprinting and as so, modern browsers are getting equipped with mitigation techniques that require a lot of development to integrate and maintain. But, Does the evolution of web technologies limit the effectiveness of browser fingerprinting? In order to answer the RQ 4, we follow the idea proposed by Laperdrix et al. [START_REF] Laperdrix | Beauty and the Beast: Diverting modern web browsers to build unique browser fingerprints[END_REF]. The authors simulated the effectiveness of browser fingerprinting against possible technical evolutions. We recreated some of their scenarios on our dataset. Scenario n°1 -The end of browser plugins. Web browsers are evolving to an architecture not based on plugins. Despite the progressive disappearance of plugins, the list of plugins is still the most distinctive attribute for personal computers in our data. A glimpse of the impact of this scenario is observed on mobile devices, although some plugins still remain. To estimate the impact of the disappearance of plugins, we simulate the fact that they are all the same in our dataset, but only on personal computers and thus taking mobile devices as reference. The improvement is significant with a decrease of exactly 19.2% from 35.7% to 16.5%, taking slightly lower value than on mobile devices, which is 18.5%. Disappearance of plugins for personal computers reduces significantly the effectiveness of browser fingerprinting at uniquely identifying users, as we observed first on mobile web browsers. Scenario n°2 -Adherence to the standard HTTP headers. Laperdrix et al. [START_REF] Laperdrix | Beauty and the Beast: Diverting modern web browsers to build unique browser fingerprints[END_REF] simulated this scenario assuming that the HTTP header fields had the same value for all fingerprints. We followed this idea, and in addition to that, we reduced the identifying information of the user-agent, just by keeping the name and version of the operating system and the web browser. On personal computers, the improvement is moderate with a decrease of 4.7% from 35.7% to 31% in overall uniqueness. However, on mobile fingerprints, we can observe a drop less significant of 2.3% from 18.5% to 16.2%. In the simulation of this scenario, Laperdrix et al. [START_REF] Laperdrix | Beauty and the Beast: Diverting modern web browsers to build unique browser fingerprints[END_REF] obtained significant results compared with our results. The small drop is due to the low entropy value of the HTTP headers in our data of 0.085 compared with the value obtained by [START_REF] Laperdrix | Beauty and the Beast: Diverting modern web browsers to build unique browser fingerprints[END_REF] of 0.249. Another element to consider is that we included a piece of information contained in the user-agent, illustrating that the combination of operating system and web browser still includes some diversity. Scenario n°3 -The end of JavaScript. By using only features collected through JavaScript (equivalent to remove HTTP features), it is possible to uniquely identify 28.3% of personal computers and 14.3% of mobile devices. By removing all features collected through JavaScript, fingerprint uniqueness drastically drops. On mobile devices, the percentage drops by 14.2% from 18.5% to 4.3%. On personal computers, the drop is abrupt from 35.7% to 0.7%. The improvement in privacy by removing JavaScript is highly visible, but the cost to the ease and comfort of using web services could be overly high. These findings show that the evolution of web technologies can benefit privacy with a limited impact. While some of them are becoming a reality, others are more improbable. Yet, it is possible to envision a future where a "privacy-aware" form of fingerprinting is possible, i.e. one that does not enable identification but that can still provide the security benefits touched upon in the literature. First, the W3C has put privacy at the forefront of discussions when designing new APIs. In 2015, Olejnik et al. performed a privacy analysis of the Battery Status API [30]. They found out that the level of charge of the battery could be used as a short-term identifier across websites. Because of this study, this API has been removed from browsers several years after its inclusion [START_REF] Olejnik | Battery Status Not Included: Assessing Privacy in Web Standards[END_REF] and it changed the way new APIs are making their way inside our browsers. A W3C draft has even been written on how to mitigate browser fingerprinting directly in web specifications [START_REF]Mitigating Browser Fingerprinting in Web Specifications -W3C Draft[END_REF]. This shows how important privacy is going forward and we can expect in the future that new APIs will not reveal any identifying information on the user's device. Then, looking at our own dataset, a privacy-aware fingerprinting seems achievable thanks to the low percentages of unique fingerprints we present in this study. CONCLUSION In this work, we analyzed 2,067,942 browser fingerprints collected through a script that was launched on one of the top 15 French websites. Our work focuses on determining if fingerprinting is still possible at a large scale. Our findings show that current fingerprinting techniques do not provide effective mechanisms to uniquely identify users belonging to a specific demographic region as 33.6% of collected fingerprints were unique in our dataset. Compared to other large scale studies on browser fingerprinting, this number is two to three times lower. This difference is even larger when only considering mobile devices as 18.5% of mobile fingerprints are unique compared to the 81% from [START_REF] Laperdrix | Beauty and the Beast: Diverting modern web browsers to build unique browser fingerprints[END_REF]. The other key elements from our study are as follows. Personal computers and mobile devices have unique fingerprints that are composed differently. While desktop/laptop fingerprints are unique mostly because of their unique combinations of attributes, mobile devices present attributes that have unique values across our whole dataset. We show that by changing some features of the fingerprint, such as Content language or Timezone, it is very probable that the fingerprint will become unique. We also show that User-agent and HTML5 canvas fingerprinting play an essential role in identifying browsers on mobile devices, meanwhile the List of plugins is the most distinctive elements on personal computers, followed by the HTML5 canvas element. Furthermore, in the absence of the Flash plugin to provide the list of fonts, we used an alternative for collecting fonts through JavaScript. Even if the list of tested fonts is much smaller compared to what could be captured through Flash, collecting fonts through JavaScript still presents some good results to distinguish two devices from each other. We also discussed some of the elements that can change the effectiveness of browser fingerprinting, such as the targeted demographic and the existing web technologies. Finally, we analyze the impact of current trends in web technologies. We show that the latest changes in fingerprinting techniques have benefited users' privacy significantly, i.e. the end of browser plugins is bringing down substantively the rate of uniqueness among desktop/laptop fingerprints. Figure 1 : 1 Figure 1: Difference between Tinos (top) and Times New Roman (bottom). Figure 2 : 2 Figure 2: Example of a rendered picture following the canvas fingerprinting test instructions. Figure 3 : 3 Figure 3: Comparison of anonymity set sizes between mobile devices and desktop/laptop machines. Figure 4 : 4 Figure 4: Anonymity sets resulting of changing values randomly in sets larger than 50 fingerprints on mobile devices (a) and Personal computers (b). Table 1 : 1 OS market share distribution. OS Our data AmIUnique StatCounter Nov'14-Jul'17 [22] Jul'17 [6] Windows 93.5% 63.7% 84% MacOS 5.5% 14.9% 11% Linux 0.9% 16.9% 1.8% Android 72% 55.6% 70% iOS 18.8% 42.3% 22% Windows Phone 7.6% <1% 1% Table Table 2 : 2 Browser measurements for the data. change in the environment, such as to a different timezone or adding fonts (fonts can be added intentionally or come as a side effect of installing new software on a device). Which gives rise to the RQ 2. Can non-unique fingerprints become unique if some value changes? and specifically, do non-unique fingerprints become unique if only one value changes? Dataset Mobile devices Personal computers Attribute Distinct Unique Distinct Unique Distinct Unique values values values values values values User-agent 19,775 8,702 10,949 5,424 8,826 3,278 Header-accept 24 9 9 2 19 8 Content encoding 30 8 19 5 25 4 Content language 2,739 1,313 961 529 2,128 958 List of plugins 288,740 196,898 81 33 288,715 196,882 Cookies enabled 1 0 1 0 1 0 Use of local/session storage 2 0 2 0 2 0 Timezone 60 16 39 1 58 18 Screen resolution and color depth 2,971 1,015 434 159 2675 897 Available fonts 17,372 6,618 94 36 17,326 6,603 List of HTTP headers 610 229 158 78 491 164 Platform 32 5 21 2 26 3 Do Not Track 3 0 3 0 3 0 Canvas 78,037 65,787 30,884 28,768 47,492 37,194 WebGL Vendor 27 1 20 2 26 3 WebGL Renderer 3,691 657 95 10 3,656 661 Use of an ad blocker 2 0 2 0 2 0 Table 3 : 3 Shannon's entropy for all attributes from Panopticlick, AmIUnique and our data. Panopticlick AmIUnique Dataset Mobile devices Desktop/laptop machines Attribute Entropy Norm. Entropy Norm. Entropy Norm. Entropy Norm. Entropy Norm. Platform - - 2.310 0.137 1.200 0.057 2.274 0.127 0.489 0.024 Do Not Track - - 0.944 0.056 1.919 0.091 1.102 0.061 1.922 0.092 Timezone 3.040 0.161 3.338 0.198 0.164 0.008 0.551 0.031 0.096 0.005 List of plugins 15.400 0.817 11.060 0.656 9.485 0.452 0.206 0.011 10.281 0.494 Use of local/session storage - - 0.405 0.024 0.043 0.002 0.056 0.003 0.042 0.002 Use of an ad blocker - - 0.995 0.059 0.045 0.002 0.067 0.004 0.042 0.002 WebGL Vendor - - 2.141 0.127 2.282 0.109 2.423 0.135 1.820 0.088 WebGL Renderer - - 3.406 0.202 5.541 0.264 4.172 0.233 5.278 0.254 Available fonts 13.900 0.738 8.379 0.497 6.904 0.329 2.192 0.122 6.967 0.335 Canvas - - 8.278 0.491 8.546 0.407 7.930 0.442 8.043 0.387 Header Accept - - 1.383 0.082 0.729 0.035 0.111 0.006 0.776 0.037 Content encoding - - 1.534 0.091 0.382 0.018 1.168 0.065 0.153 0.007 Content language - - 5.918 0.351 2.716 0.129 2.291 0.128 2.559 0.123 User-agent 10.000 0.531 9.779 0.580 7.150 0.341 8.740 0.487 6.323 0.304 Screen resolution 4.830 0.256 4.889 0.290 4.847 0.231 3.603 0.201 4.437 0.213 List of HTTP headers - - 4.198 0.249 1.783 0.085 1.941 0.108 1.521 0.073 Cookies enabled 0.353 0.019 0.253 0.015 0.000 0.000 0.000 0.000 0.000 0.000 H M (worst scenario) 18.843 16.860 20.980 17.938 20.793 Number of FPs 470,161 118,934 2,067,942 251,166 1,816,776 ACKNOWLEDGMENT We thank the b<>com Institute of Research and Technology (IRT) for their support and we are particularly grateful to Alexandre Garel for his collaboration in setting up the script on the commercial website and collecting the data. This work is partially supported by the CominLabs-PROFILE project and by the Wallenberg Autonomous Systems Program (WASP). APPENDIX A. LIST OF TESTED FONTS Andale Mono, AppleGothic, Arial, Arial Black, Arial Hebrew, Arial MT,Arial Narrow, Arial Rounded MT Bold, Arial Unicode MS,

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2018

https://hal.science/hal-01728605v2/file/conference_071817.pdf

Nicolas Cristal Benoît Cristal Optimisation de tournée de véhicule dans un environnement portuaire 1 st Danloup Keywords: Optimisation, Cplex, AGV Dans un contexte portuaire, le but du projet de recherche est de prouver l'efficacité d'un système complétement automatisé avec guidage sans infrastructure pour le transfert de conteneurs dans les terminaux portuaires. Le moyen de transfert utilisé est une flotte d'AGV (pour Automotated Guided Vehicles). Le but est l'automatisation des ports, qui est devenue une priorité pour les plus grands acteurs du secteur portuaire contraints de s'adapter à la croissance du volume de containers transportés liée à la globalisation de l'économie mondiale et aux capacités des nouvelles générations de bateaux. La concurrence entre terminaux et la nécessité de réduire les délais de chargement et déchargement des bateaux nécessitent de la part des opérateurs portuaires une augmentation de la productivité globale, une réduction des co ûts d'exploitation tout en garantissant une sécurité maximale. Cet article concerne le verrou scientifique de la gestion automatique et optimisée de la flotte d'AGV. Il s'agit de proposer un routage de la flotte d'AGVs sans blocages qui permette de réaliser des missions soumises régulièrement. I. INTRODUCTION Durant les dernières décennies, le commerce à distance s'est très fortement développé. La réduction des coûts de transport, liée à la massification des biens transportés, a engendré une forte augmentation des volumes transportés. En effet, c'est un moyen de transfert adapté aux matières de poids transportées sur de longues distances par des gros navires. Le transport maritime est un moyen de transport à moindre frais ; il est trente fois moins cher que le transport terrestre. L'augmentation du volume à trasnsporter à vue l'augmentation de la taille des navires et, en conséquence, des capacités d'accueil des lieux portuaires pour l'amarrage des navires mais aussi en terme de chargement et/ou déchargement. L'idée actuelle est de proposer des solutions automatisées de chargement/déchargement des cargos. Ce qui est motivé par le développement de véhicule automatisé sur le marché : les AGV (pour Automotated Intelligent Vehicles) et des conteneurs de taille standard (équivalents 20 ou 40 pieds). Le but est de minimiser le temps opérationnel et de garantir une efficacité de fonctionnement. Ainsi, l'automatisation vise à permettre l'accélération du transfert des conteneurs du navire vers les clients et inversement. L'article est organisé suivant trois parties. Nous commencerons par décrire le problème étudié. Dans la partie suivante, le problème est formalisé vers une abstraction mathématique. Il en découle les premiers résultats dans la quatrième section. La partie finale est dédiée à la conclusion de cet étude. II. DESCRIPTION DU PROBL ÈME Les problèmes de gestion des opérations dans les terminaux à conteneurs est une thématique très importante dans la littérature scientifique en recherche opérationnelle et logistique. Il existe des synthèses de résultats dans les articles suivants : [START_REF] Steenken | Container terminal operation and operations research -a classification and literature review[END_REF]- [START_REF] Vis | Transshipment of containers at a container terminal : An overview[END_REF]. Il existe de nombreux problèmes au sein des activités portuaires. On trouve, par exemple, des études d'allocation de quai aux bateaux, d'ordonnancement des grues de quai, celui de l'allocation de stockage aux conteneurs. . . Notre étude porte sur la problématique de routage des AGV dans une zone portuaire. L'envirronement est constitué de bateaux à décharger, les grues de quai sont en place. L'opérateur d'exploitation du port (Terminal Operating System) donne des ordes de déchargement au fil de l'eau. L'exécution est confiée à un prestataire qui est chargé d'envoyer un véhicule pour transporter les conteneurs de la grue de quai vers un emplacement de stockage et inversement. Nous devons donc gérer une flotte d'AGV pour honorer les ordres rec ¸us. Plus précisement, il faut affecter un AGV à chaque mission et lui trouver une route qui permette l'exécution de la mission dans le temps imparti. III. FORMALISATION DU PROBL ÈME A. Moteur d'affectation Cette couche se base sur un algorithme d'affectation de type glouton, c'est à dire que l'on procède par itérations en faisant des choix localement optimaux, ce qui ne garantit toutefois pas l'optimalité globale de la solution. Elle fait l'hypothèse que le nombre d'AGVs disponibles en circulation est supérieur ou égal au nombre de missions à réaliser. Si ce n'est pas le cas, la couche agrégation devra limiter le nombre de missions à traiter par la couche macro. Son principe est le suivant. Soit M l'ensemble des missions à affecter. B. Moteur de routage Cette section décrit le modèle mathématique proposé pour traiter la problématique de routage des AGV vers les emplacements correspondants aux missions auxquelles ils ont été affectés. 1) Topologie: Les AGV peuvent se déplacer en suivant un réseau de transport virtuel, inspiré de l'organisation ferroviaire, défini par des cantons et des carrefours, qui permettent de représenter un graphe dans lequel les AGV évolueront. a) Cantons (blocks b i , b j ).: Les AGV qui ne sont affectés à aucune mission ne font pas partie de cet ensemble. Leurs positions (dans des cantons longs, puisqu'ils sont stationnés) seront toutefois transmises par le moteur d'affectation au moteur de routage, afin que ce dernier puisse interdire l'utilisation des cantons correspondants. On dénote par B f (forbidden) 3 l'ensemble des cantons qui ne peuvent être utilisés dans la résolution courante. On dénote par B = {b 1 , b 2 , . . . , b B } = {b i } i∈[[ Chaque mission est donc associée à un lieu de départ correspondant à la position courante de l'AGV au moment où cette mission est transmise. Chaque mission est également associée à un lieu d'arrivée, en fonction de son type (grues de quai ou grues de stock pour les missions de type 1 et 2, cantons pour les missions de type 3). Ces lieux doivent nécessairement être des cantons longs. Dans la topologie générale proposée dans la figure ??, comme nous ne considérons pas les missions de type 3 dans cette étude, il s'agira donc des cantons représentés sur fond vert. Pour chaque AGV a m ∈ A, on dénotera par s m ∈ B l son canton de départ (source) et par t m ∈ B l son canton d'arrivée. 2) Variables de décision: De manière à pouvoir proposer une formulation linéaire du problème considéré, on introduit un certain nombre de constantes et variables en plus des termes introduits dans la formalisation précédente. • R est une constante suffisamment grande. • ω est une constante suffisamment petite. • X i,j m ∈ B identifie le passage de l'AGV a m du canton b i au canton b j dans cet ordre. X i,j = δ(a m passe de b i à b j ) où la fonction δ(C) est l'indicateur δ(C) = 1 si la condition C est vérifiée, 0 sinon. • Y i m ∈ B identifie le passage de l'AGV a m par le canton b i . Y i m = δ(a m passe par b i ). • Z p m ∈ B identifie le passage de l'AGV a m par le carrefour c p . Z p m = δ(a m passe par c p ). • α i m ∈ N dénote la date d'arrivée de l'avant de l'AGV a m dans le canton b i . • β i m ∈ N dénote la date de sortie de l'avant de l'AGV a m du canton b i . • γ i m ∈ N dénote la date de sortie de l'arrière de l'AGV a m du canton b i . • φ p m ∈ N dénote la date d'arrivée de l'avant de l'AGV a m dans le carrefour c p . • ψ p m ∈ N dénote la date de sortie de l'avant de l'AGV a m du carrefour c p . • δ p m ∈ N dénote la date de sortie de l'arrière de l'AGV a m du carrefour c p . • ε i m ∈ N dénote la durée de stationnement de l'AGV a m dans le canton long b i ∈ B l . • B i m,n ∈ B caractérise l'ordre chronologique de deux AGV s'ils utilisent le même canton b i ∈ B. B i m,n = δ(a m entre dans b i avant a n ). • C p m,n ∈ B caractérise l'ordre chronologique de deux AGV s'ils utilisent le même carrefour c p ∈ C. C p m,n = δ(a m entre dans c p avant a n ). a) Notes: Utiliser une seule variable pour dénoter la présence d'un AGV dans un canton, ainsi que l'intervalle horaire de sa présence dans celui-ci, implique implicitement qu'un AGV ne pourra pas passer deux fois par le même canton sur son chemin entre son origine et sa destination. Cette particularité est classique pour les types de problèmes apparentés au voyageur de commerce. Puisque nous utilisons également une seule variable pour dénoter la présence d'un AGV dans un carrefour, ainsi que l'intervalle horaire de sa présence dans celui-ci, un AGV ne pourra pas non plus effectuer de demi-tour pour s'écarter par exemple de la trajectoire d'un autre AGV dont il barrerait la route avant de reprendre son chemin. 3) Contraintes de routage: Cette section détaille les contraintes régissant le déplacement des AGV de leur origine à leur destination. Elle ne permet pas de garantir les contraintes de sécurité qui seront présentées dans la section suivante. a) Pré-calculs.: Les variables de décision X i,j m doivent être nulles si les cantons i et j ne sont pas accessibles entre eux. Les variables X i,i sont trivialement nulles également. ∀a m ∈ A, ∀b i ∈ B, ∀b j ∈ A B i , X i,j m = 0 (1) ∀a m ∈ A, ∀b i ∈ B, X i,i m = 0 (2) b) Origine et Destination: Chaque AGV doit sortir de son canton de départ. On contraint donc que parmi les variables X i,j m dénotant la sortie de l'AGV a m du canton s m , une et une seule soit non nulle. ∀a m ∈ A, bj ∈ A Bs m X sm,j m = 1 (3) Par ailleurs, l'AGV considéré ne doit pas revenir dans son canton d'origine. ∀a m ∈ A, bi∈B X i,sm m = 0 (4) De manière réciproque, chaque AGV doit entrer dans son canton de destination, et ne plus en ressortir. ∀a m ∈ A, bi∈B X i,tm m = 1 (5) ∀a m ∈ A, bj ∈ A Bt m X tm,j m = 0 (6) c) Note: On pourrait utiliser dans l'équation (5) l'ensemble ← -A B j proposé dans la note en page 2. d) Respect de la topologie: Depuis un canton b i , un AGV a m ne peut se déplacer que vers un seul autre canton b j , à condition que ce dernier soit accessible depuis b i . ∀a m ∈ A, ∀b i ∈ B, bj ∈ A Bi X i,j m ≤ 1 (7) L'opérateur d'inégalité permet d'autoriser la somme à être nulle. En effet, si l'AGV a m n'entre pas dans le canton b i considéré, toutes les variables X i,j m devront être nulles, ce que permet l'inégalité. Au contraire, si la somme est égale à 1, c'est que l'AGV a m quitte le canton b i pour se rendre vers un seul autre canton b j . e) Connexité des parcours des AGVs: Si un AGV a m entre dans un canton b j (sauf le canton de destination t m ), il doit en sortir. Inversement, si cet AGV sort du canton b i (sauf le canton de départ s m ), il doit y être entré. ∀a m ∈ A, ∀b j ∈ B -{s m , t m }, bi∈B X i,j m = b k ∈B X j,k m (8) La première somme permet de caractériser si l'AGV a m entre dans le canton b j . En effet, lorsque cette somme vaut 1, c'est qu'il existe un canton b i pour lequel X i,j m = 1 : l'AGV passe de b i à b j . La seconde somme indique que l'AGV a m sort de b j . La contrainte d'égalité permet d'imposer le comportement souhaité. Au contraire, si l'une des sommes vaut 0, c'est que l'AGV ne passe pas par le canton b j : il ne doit ni y entrer, ni en sortir. f) Interdiction des demi-tours: Un AGV ne doit pas pouvoir effectuer de demi-tour. ∀a m ∈ A, ∀b i ∈ B, ∀b j ∈ A B i , X i,j m + X j,i m ≤ 1 (9) g) Note: Cette contrainte agit comme une coupe, et n'est pas nécessaire pour assurer la correction des solutions renvoyées par le solveur dans la mesure où les contraintes de temps interdisent déjà ce phénomène. Cependant, cette équation est susceptible de guider le solveur dans sa résolution de manière plus efficace. Des jeux d'essais devront être menés pour en mesurer l'impact. ∀a m ∈ A, ∀b i ∈ B -{t m }, bj ∈ A Bi X i,j m = Y i m ( 12 ) ∀a m ∈ A, ∀b j ∈ B -{s m }, bi∈B X i,j m = Y j m (13) d) Note: On pourrait utiliser dans l'équation (13) l'ensemble ← -A B j proposé dans la note en page 2. e) Parcours des carrefours: Lorsqu'un AGV passe du canton b i au canton b j , il utilise l'unique carrefour i c j qui les relie. ∀a m ∈ A, ∀b i ∈ B, ∀b j ∈ A B i , X i,j m ≤ Z i c j m (14) L'inégalité est importante : elle permet d'imposer la valeur de la variable Z ∀a m ∈ A, ∀b i ∈ B, ∀b j ∈ A B j -{s m }, X i,j m = 1 ⇒ β j m = α j m + ε j m + d b ij (15) X i,j m = 1 ⇒ γ j m = β j m + t b ij (16) Pour linéariser ces relations, nous utilisons cette fois un double encadrement. ∀a m ∈ A, ∀b i ∈ B, ∀b j ∈ A B j -{s m }, β j m -α j m -ε j m -d b ij ≤ R • 1 -X ij m α j m + ε j m + d b ij -β j m ≤ R • 1 -X ij m ( 17 ) ∀a m ∈ A, ∀b i ∈ B, ∀b j ∈ A B j -{s m }, γ j m -β j m -t b ij ≤ R • 1 -X ij m β j m + t b ij -γ j m ≤ R • 1 -X ij m (18) Pour les cantons d'origine, qui n'ont donc pas de canton qui les précède, nous utilisons la durée moyenne de traversée. ∀a m ∈ A, β sm m = α sm m + ε sm m + d b sm (19) ∀a m ∈ A, δ sm m = β sm m + t b sm ( ∀a m ∈ A,∀b i ∈ B, ∀b j ∈ A B i , X i,j m = 1 ⇒ α j m = β i m + d c ij ( 22 ) Pour linéariser cette relation, nous utilisons un double encadrement. ∀a m ∈ A, ∀b i ∈ B, ∀b j ∈ A B i , α j m -β i m -d c ij ≤ R • (1 -X i,j m ) β i m + d c ij -α j m ≤ R • (1 -X i,j m ) (23) Lorsque X i,j m = 1, le double encadrement par 0 permet d'assurer la contrainte de séquencement recherchée. Au contraire, lorsque X i,j m = 0, les inégalités de l'équation (23) sont toujours vérifiées. d) Intervalles horaires des cantons non utilisés: Les AGV ne passant pas par un canton b i voient leurs intervalles de temps être calés au temps zéro, afin de diminuer le nombre de variables non affectées du modèle mathématique à résoudre. Ainsi, il existe une relation entre les variables Y i m et les intervalles horaires d'occupation des cantons. Nous cherchons à imposer les relations : ∀a m ∈ A, ∀b i ∈ B c , Y i m = 0 ⇒ α i m = 0 (24) ∀a m ∈ A, ∀b i ∈ B l , Y i m = 0 ⇒ α i m = 0 ε i m = 0 ∀a m ∈ A, ∀b i ∈ B c , α i m ≤ R • Y i m ( 26 ) ∀a m ∈ A, ∀b i ∈ B l , α i m ≤ R • Y i m ε i m ≤ R • Y i m ( 27 ) Dans les contraintes précédentes, R est une constante suffisamment grande, ce qui assure que les inégalités ( 26 ∀a m ∈ A, ∀b i ∈ B, ∀b j ∈ A B i , X i,j m = 1 ⇒ ψ i c j m = φ i c j m + d c ij (28) X i,j m = 1 ⇒ δ i c j m = ψ i c j m + t c ij (29) Pour linéariser ces relations, nous utilisons un double encadrement. ∀a m ∈ A, ∀b i ∈ B, ∀b j ∈ A B i , ψ i c j m -φ i c j m -d c ij ≤ R • 1 -X ij m φ i c j m + d c ij -ψ i c j m ≤ R • 1 -X ij m (30) δ i c j m -ψ i c j m -t c ij ≤ R • 1 -X ij m ψ i c j m + t c ij -δ i c j m ≤ R • 1 -X ij m (31) g) Traversée des carrefours: Lors du passage d'un canton b i à un canton b j , l'AGV a m traverse le carrefour i c j . L'intervalle horaire correspondant à la traversée de ce carrefour dépend de sa date de sortie du canton b i . Nous souhaitons imposer la relation suivante : ∀a m ∈ A, ∀b i ∈ B, ∀b j ∈ A B i , X i,j m = 1 ⇒ φ i c j m = β i m (32) Pour linéariser cette relation, nous utilisons un double encadrement. ∀a m ∈ A, ∀b i ∈ B, ∀b j ∈ A B i , φ i c j m -β i m ≤ R • (1 -X i,j m ) β i m -φ i c j m ≤ R • (1 -X i,j m ) (33) h) Intervalle horaire des carrefours non utilisés: Les AGV ne passant pas par un carrefour c p voient leurs intervalles de temps être calés au temps zéro, afin de diminuer le nombre de variables non affectées du modèle mathématique à résoudre. Ainsi, il existe une relation entre les variables Z p m et les intervalles horaires d'occupation des carrefours. Nous cherchons à imposer la relation : ∀a m ∈ A, ∀c p ∈ C, (Z p m = 0) ⇒ φ p m = 0 (34) Pour cela, nous devons linéariser l'opérateur d'implication afin d'exprimer la relation dans un format compatible avec le solveur utilisé. La variable φ p m étant positive, il suffit d'imposer une simple inégalité pour la forcer à la valeur nulle. ∀a m ∈ A, ∀b i ∈ B c , φ p m ≤ R • Z p m (35) i) Note: Il devrait être possible de relâcher la contrainte (35) car elle n'est pas nécessaire pour assurer la correction des solutions renvoyées par le solveur. Des jeux d'essais devront être menés pour en mesurer l'impact. 6 ∀a m = a n ∈ A, ∀b i ∈ B, B i m,n + B i n,m = 1 (36) b) Note: Lorsque les AGV considérés n'empruntent pas le canton b i , les variables B i m,n et B i n,m peuvent prendre des valeurs arbitraires sans que cela impacte la correction de la solution générée par le solveur. c) Occupation des cantons: Si les deux AGV considérés passent par le canton b i , on impose qu'ils utilisent cette ressource à des instants différents à l'aide de la contrainte suivante : g) Occupation des carrefours: Les même considérations que précédemment sont utilisées pour interdire la présence de deux AGV différents dans un même carrefour. ∀a m =a n ∈ A, ∀b i ∈ B, ∀c p ∈ C i γ i m +ω j∈ ← - A B p i X j,i n ≤ α i n + R •   4 -Y i m -Y i n -B i m,n - j∈ A B p i X i,j m   ( ∀a m =a n ∈ A, ∀c p ∈ C, δ p m ≤ φ p n + R • 3 -Z p m -Z p n -C p m,n (39) IV. R ÉSULTATS Différents jeux d'essai ont été créés pour tester la modélisation et vérifier que les contraintes du système sont respectées. Pour cela, deux réseaux simples qui illustrent des problématiques particulières à résoudre ont été créés. Le premier réseau est illustré par la figure 1. Il contient une grue de quai et une grue de stockage. Ce réseau est utilisé par trois jeux d'essai : • jeu1 : une mission est gérée par un AGV. La mission est un transfert de la grue de quai (haut de la figure) vers la grue de stock (bas de la figure). Il permet de vérifier la capacité du modèle mathématique à fournir des solutions à la problématique du routage. • jeu2a : deux missions sont gérées par deux AGVs. La première mission est un transfert de la grue de quai vers la grue de stock. La deuxième mission est un transfert de la grue de stock vers la grue de quai. Il permet de vérifier la gestion de carrefour et des circulations verticales. • jeu2b : deux missions sont gérées par deux AGVs. Les deux missions sont un transfert de la grue de quai vers la grue de stock. Ces résultats permettent de voir que les contraintes physiques du système sont bien respectées, notamment au niveau des contraintes de sécurité. En effet, il est important que plusieurs AGVs ne peuvent pas se trouver dans le même canton ou carrefour au même moment. Ils permettent également de mettre en évidence la limite de résolution optimale rapidement atteinte par le solveur. A partir de quatre AGVs, le solveur n'est plus en mesure de trouver la solution optimale. Il faut cependant noter qu'une solution (non optimale) est tout de même trouvée par celui-ci en moins d'une seconde pur le jeu3. Malheureusement, le constat est que ce modèle souffre du passage à l'échelle. Les travaux à venir visent à contourner ce problème. Deux pistes sont poursuivies actuellement. La première consiste à proposer des heuristiques spécifiques ; l'intérêt reside dans l'obtention rapide d'une solution. Une autre piste consiste à affiner le modèle mathématique pour tenter de faire chuter la combinatoire. 4 ) 4 Variables auxiliaires Y et Z: Afin de pouvoir formuler plus facilement les contraintes de sécurité, nous définissions un ensemble de variables auxiliaires permettant d'en simplifier l'expression. a) Origine et destination: Les AGV doivent respecter les cantons de départ et d'arrivée qui leur sont affectés. ∀m ∈ A, Y sm m = 1 (10) ∀m ∈ A, Y tm m = 1 (11) b) Note: Ces contraintes sont redondantes puisqu'elles découlent des équations (3) et (5) et de celles du paragraphe suivant. En effet, l'AGV est toujours sur son parcours dans un canton de départ ou d'arrivée. Ainsi, il est toujours utile de fixer les variables dont on connait a priori la valeur dans un modèle mathématique. c) Parcours des cantons: De manière évidente, lorsqu'un AGV a m sort du canton b i pour entrer dans le canton b j (c'està-dire lorsque X i,j m = 1), il passe par b i et b j . Inversement, si l'AGV a m passe par b i , il doit y entrer et en sortir. i c j m vaut 1 . 1 i c j m lorsque X i,j m vaut 1, mais de ne pas imposer réciproquement le passage du canton b i au canton b j lorsque Z En effet, le carrefour i c j peut également être un point de raccordement entre deux autres cantons b k et b l différents de b i et b j , dans le cas de structures de carrefours en étoile par exemple. Dans ce cas, i c j représente le même carrefour que k c l . 5) Intervalles horaires d'occupation des cantons et carrefours: Nous introduisons dans cette section des contraintes permettant de définir les intervalles horaires de présence des AGV dans chacun de ces emplacements. Une fois ces intervalles de temps connu, il sera possible d'imposer les contraintes de sécurité permettant d'interdire à deux AGV de se trouver dans le même canton ou carrefour au même moment. a) Intervalle horaire d'occupation d'un canton: On rappelle que l'intervalle horaire d'occupation d'un canton b i par un AGV a m est dénoté par [α i m , γ i m ], où les bornes temporelles sont exprimées en secondes, à partir du temps zéro correspondant à la date de réception des missions par la couche macro. Celles-ci sont contraintes par la durée de traversée nominale d b i j et la durée de sortie du canton considéré en fonction du canton traversé précédemment. 20) b) Origine: Les AGVs occupent leur canton d'origine dès le temps zéro. Le début de l'intervalle d'occupation de ces cantons doit donc être initialisé à zéro. ∀a m ∈ A, α sm m = 0 (21) c) Succession de cantons: Lorsqu'un AGV a m passe d'un canton b i à un canton b j , les intervalles horaires d'occupation des cantons b i et b j doivent se succéder, en prenant en compte la durée d c ij de traversée du carrefour i c j . Nous cherchons à imposer la relation : cela, nous devons linéariser l'opérateur d'implication afin d'exprimer la relation dans un format compatible avec le solveur utilisé. Les variables α i m et ε i m étant positives, il suffit d'imposer une simple inégalité pour les forcer à la valeur nulle. ) et (27) sont toujours vérifiées lorsque Y i m = 1. Au contraire, lorsque Y i m = 0, les variables α i m et ε i m sont contraintes à devenir nulles par un double encadrement par 0. e) Note: Il devrait être possible de relâcher les contraintes (26) et (27) car elles ne sont pas nécessaires pour assurer la correction des solutions renvoyées par le solveur. Des jeux d'essais devront être menés pour en mesurer l'impact. f) Intervalle horaire d'occupation d'un carrefour: On rappelle que l'intervalle horaire d'occupation d'un carrefour c p par un AGV a m est dénoté par [φ p m , δ p m ], où les bornes temporelles sont exprimées en secondes, à partir du temps zéro correspondant à la date de réception des missions par la couche macro. Celles-ci sont contraintes par la durée de traversée nominale d c ij et par la durée de sortie t c ij du carrefour i c j considéré en fonction du canton traversé précédemment et du canton de destination. 37) L'équation (36) implique que si deux AGV a m et a n sont affectés au même canton b i , et que l'AGV a m se dirige vers le carrefour c p , le terme 4 -Y i m -Y i n -B i m,n -j∈ A B p i X i,j mest nul. Dans le cas où l'AGV a n se dirige également vers le carrefour c p , l'équation (37) implique ainsi que γ i m ≤ α i n et donc que l'AGV a m quitte le canton b i avant l'arrivée de l'AGV a n dans ce même canton. Dans le cas où l'AGV a n ne se dirige pas vers le carrefour c p , l'équation (37) implique que γ i m < α i n . L'ingélaité stricte dans ce cas empêche deux AGV de se croiser entre un carrefour et un canton (ce qui n'est pas possible). Si le terme est strictement positif (parce que l'un ou l'autre des AGV ne passe pas par le canton b i , ou parce que a m est séquencé après a n quand B i m,n = 0), la contrainte (37) est trivialement vérifiée compte tenu de la grande valeur de la constante R.d) Note: Il est connu que la valeur de R a un impact sur les performances du solveur utilisé. Des jeux d'essais devront être menés pour le mesurer. La valeur de ω a également un impact sur les performances du solveur. Une valeur trop petite peut fournir des solutions non réalisables, tandis qu'une valeur trop grande peut amener le solveur a écarter des bonnes solutions.e) Ordre d'occupation des carrefours: L'ordre d'occupation du carrefour c p par les AGV a m et a n est caractérisé par les variables C p m,n (qui vaut 1 lorsque a m se présente avant a n ) et C p n,m (inversement). Un des AGV doit obligatoirement être ordonnancé avant l'autre, ce qui s'exprime par la contrainte suivante :∀a m = a n ∈ A, ∀c p ∈ C, C p m,n + C p n,m = 1(38)f) Note: Lorsque les AGV considérés n'empruntent pas le carrefour c p , les variables C p m,n et C p n,m peuvent prendre des valeurs arbitraires sans que cela impacte la correction de la solution générée par le solveur. Fig. 1 . 1 Fig. 1. Topologie du premier réseau Le deuxième réseau est illustré par la figure 2. Il contient deux grues de quai et trois grues de stocks. Ce réseau est utilisé par un seul jeu d'essai (jeu3). Dans ce jeu, quatre missions sont gérées par quatre AGVs. Les détails des missions sont décrites Fig. 2 . 2 Fig. 2. Topologie du deuxième réseau la gestion d'une flotte d'AGV dans un envirronement portuaire. Lorsqu'un ordre de transport arrive des autorités portuaires (TOS), il convient de trouver une solution pour transporter le conteneur de son origine vers son lieu de stockage. C'est un problème classique de routage d'une flotte de véhicule. Il constitue un problème complexe et ouvert s'incrivant dans les problèmes d'optimisation combinatoire. Il s'agit de trouver les routes pour chaque mission et de respecter les contraintes horaires proposées. Le modèle proposé résout et optimise le flux de véhicule à partir d'une position initiale quelconque. + exprimées en secondes. De plus, chaque paire de cantons (b i , b j ) ∈ B × A B i est associé à la durée de parcours nominale de j lorsque l'AGV vient de i dénotée par d b ij ∈ N + , et une durée de sortie de j dénotée par t b ij ∈ N + exprimées en secondes. L'ensemble B est partitionné en cantons courts B c et longs B l tels que B = B c ∪ B l . On dénote B c et B l leurs cardinaux respectifs. b) Carrefours (crossroads c p ).: On dénote par C = {c 1 , c 2 , . . . , c C } = {c p } p∈[[1,C]] l'ensemble des cantons du système étudié. On note C son cardinal. D'après nos hypothèses, pour chaque paire de cantons (b i , b j ) ∈ B× A B i , une durée de parcours dénotée d c ij ∈ N + , et une durée de sortie dénotée t c ij ∈ N + , exprimées en secondes, sont associées à la traversée du carrefour lorsque l'AGV va du canton b i au canton b j . c) Relations entre cantons et carrefours.: La topologie considérée correspond à un enchaînement de cantons et carrefours, sous la forme d'un graphe biparti 1 .Pour tout canton b i ∈ B, on dénote par C i l'ensemble des carrefours accessibles 2 depuis b i .Pour tout canton b i ∈ B, on dénote par A B i l'ensemble des cantons accessibles depuis b i . A B i , il existe un unique carrefour c p tel que b j ∈ A B p i . On dénote cet unique carrefour entre les cantons b i et b j par l'expression i c j . d) Note.: Il pourrait être intéressant de définir également l'ensemble ← -A B j caractérisant les cantons depuis lesquels un AGV peut venir lorsqu'il entre dans le canton b j . Pour tout canton b i ∈ B, et tout carrefour c p ∈ C i accessible depuis ce canton, on dénote par A B p i l'ensemble des cantons accessibles depuis b i en passant par le carrefour c p , et ← -A B p j l'ensemble des cantons depuis lesquels un AGV peut venir lorsqu'il entre dans b j en passant par le carrefour p. La topologie est telle qu'il ne peut y avoir qu'un unique carrefour entre deux cantons accessibles entre-eux. Autrement dit, on ne peut passer d'un canton à un autre que par au plus un seul carrefour. Formellement, pour tout canton b i ∈ B, et tout canton b j ∈ Il faut noter que l'existence d'un carrefour i c j n'implique pas obligatoirement l'existence d'un carrefour j c i dans l'autre sens, compte-tenu de notre remarque précédente, puisque certains cantons ne peuvent être parcourus que dans un seul sens. e) Missions (AGV a m , a n ).: Nous faisons l'hypothèse que le moteur d'affectation a affecté à chaque mission un AGV. Nous unifions donc dans notre modélisation les vocab- ulaires de mission et d'AGV. On dénote par A = {a 1 , a 2 , . . . , a A } = {a m } m∈[[1,A]] l'ensemble des AGV du système étudié. On note A son cardinal. 1,B]] l'ensemble des cantons du système étudié. On note B son cardinal. D'après nos hypothèses, chaque canton b i ∈ B est associé à une durée de parcours nominale moyenne dénotée par d b i ∈ N + , et une durée moyenne de sortie dénotée par t b i ∈ N ) Contraintes de sécurité: Les contraintes de sécurité permettent d'interdire à deux AGV d'être présents au même moment dans le même canton ou le même carrefour. Pour les exprimer, nous utilisons les intervalles horaires définis précédemment, ainsi que des variables de séquencement permettant d'indiquer dans quel ordre deux AGV empruntant la même ressource doivent la réserver. du canton b i par les AGV a m et a n est caractérisé par les variables B i m,n (qui vaut 1 lorsque a m se présente avant a n ) et B i n,m (inversement). Un des AGV doit obligatoirement être ordonnancé avant l'autre, ce qui s'exprime par la contrainte suivante : a) Ordre d'occupation des cantons: L'ordre d'occupation Il ne peut y avoir deux cantons successifs qui ne soient pas reliés par un carrefour et inversement. Si l'on considère un canton b i qui ne puisse être parcouru que dans un seul sens, le carrefour en amont de ce canton ne sera pas considéré comme accessible.

01744818

en

2024/03/05 22:32:07

2017

https://hal.science/hal-01744818/file/PhysRevB95_075402_2017.pdf

T Garandel R Arras X Marie P Renucci L Calmels Electronic structure of the Co(0001)/MoS2 interface, and its possible use for electrical spin injection in a single MoS2 layer Keywords: 72.25.Dc, 72.25.Hg, 72.25.Mk, 73.20.At, 73.63.Rt, 75.70.Ak, 75.70.Cn The ability to perform efficient electrical spin injection from ferromagnetic metals into twodimensional semiconductor crystals based on transition metal dichalcogenide monolayers is a prerequisite for spintronic and valleytronic devices using these materials. Here, the hcp Co(0001)/MoS2 interface electronic structure is investigated by first-principles calculations based on the density functional theory. In the lowest energy configuration of the hybrid system after optimization of the atomic coordinates, we show that interface sulfur atoms are covalently bound to one, two or three cobalt atoms. A decrease of the Co atom spin magnetic moment is observed at the interface, together with a small magnetization of S atoms. Mo atoms also hold small magnetic moments which can take positive as well as negative values. The charge transfers due to covalent bonding between S and Co atoms at the interface have been calculated for majority and minority spin electrons and the connections between these interface charge transfers and the induced magnetic properties of the MoS2 layer are discussed. Band structure and density of states of the hybrid system are calculated for minority and majority spin electrons, taking into account spin-orbit coupling. We demonstrate that MoS2 bound to the Co contact becomes metallic due to hybridization between Co d and S p orbitals. For this metallic phase of MoS2, a spin polarization at the Fermi level of 16 % in absolute value is calculated, that could allow spin injection into the semiconducting MoS2 monolayer channel. Finally, the symmetry of the majority and minority spin electron wave functions at the Fermi level in the Co-bound metallic phase of MoS2 and the orientation of the border between the metallic and semiconducting phases of MoS2 are investigated, and their impact on spin injection into the MoS2 channel is discussed. I.INTRODUCTION Triggered by the success of graphene, the field of two-dimensional (2D) semiconductor (SC) crystals based on transition metal dichalcogenide (TMDC) monolayers encounters a spectacular development [1]. This new class of exciting materials presents several original characteristics. Their direct band gap [2] allows investigations by optical techniques, and the strong spin-orbit coupling combined with the lack of inversion symmetry result in nonequivalent valleys in the reciprocal space, that can be selectively addressed by circularly polarized light due to optical selection rules [3]. In a sense, the valley index could constitute a novel degree of freedom to carry and process information ("valleytronics" [4]). The exploration of the spin and valley degrees of freedom by all-optical experiments have been carried in the past years [5,6,7]. However, electrical spin injection yet remains elusive in these systems [8]. The ability to inject spin polarized currents would pave the way for new spintronic and spinoptronic devices [9,10,11], where one can imagine to take benefit from optical and spin properties of these materials. It could also benefit to future valleytronic devices, where electrical generation and control of the valley index is required. Due to the unique correlation between the spin and valley indices of charge carriers, this point is directly linked to electrical injection of spin polarized (and energy selected) carriers in TMDC. The problem of electrical injection into 2D semiconductors like MoS 2, MoSe2, WSe2, or WS2 is thus a new challenge in this field, after the recent exploration of electrical spin injection into tridimensional (3D) semiconductors such as GaAs [9,10,11], Si [12], Ge [13], and in 2D Graphene [14]. The first experimental proof of electrical spin injection in 2D semiconductors has just been reported very recently: Ye et al. have shown that spin polarized holes can be injected in WSe2 from the ferromagnetic semiconductor GaMnAs that acts as a spin aligner [15]. Unfortunately, this ferromagnetic material has a Curie temperature near 200K, far below room temperature. An alternative way consists in using ferromagnetic metals (FM) like Cobalt, Iron or Nickel, with Curie temperatures well above room temperature. In the context of the electrical spin injection into 3D semiconductors in the diffusive regime, it is well established that one has to overcome the impedance mismatch [16] between the FM and the SC layers: this is usually performed by inserting an oxide layer or a tunnel barrier [17]. The electrical spin injection from a ferromagnetic electrode into a single TMDC layer should, however, be totally different from the case of spin injection in a 3D semiconductor and the problem has to be reconsidered. In particular, the semiconductor nature of MoS 2 just below the metallic contact is questionable because covalent bonds could be formed between the metal and the TMDC monolayer, as discussed by Allain et al. [8]. Modification of the magnetic properties of MoS2 below the contact, in particular a spin-polarization of the electron states near the Fermi level and induced magnetic moments, could also occur due to the direct contact between MoS2 and the magnetic layer. First-principles calculations of the electronic structure based on the density functional theory (DFT) constitute a tool of choice to understand the bonding mechanisms between the FM and the TMDC layers and to give a clear description of the spin-polarized electron states at their interface. Most of the first-principles studies on interfaces between metals and a single TMDC layer have focused on non-magnetic metals like Ir, Pd, Ru, In, Ti, Au, Mo or W [18,19]. Only few studies have concerned interfaces between Fe, Co or Ni and a TMDC monolayer: Dolui et al. have studied the Giant Magnetoresistance of Fe/MoS2/Fe magnetic tunnel junctions and demonstrated that MoS2 becomes conductive when the MoS2 spacer only contains one or two layers, due to the strong interaction between Fe and S atoms; this behavior could, however, be due to the fact that Fe is present on both sides of the MoS2 layer, and the case of a single Fe/MoS2 interface has not been investigated there [20]. Considering an interface between a single Co atomic layer and a MoS2 sheet, Chen et al. showed that the electronic structure of these two monolayers is drastically modified by a strong interface binding [21]; their results are however strongly influenced by the extreme thinness of the Co layer (which is so thin that it even becomes half-metallic), while Co electrodes in real Co/MoS2-based devices would certainly be thicker and contain several Co atomic layers. Leong et al. and Co/MoS2 interfaces [23]. Their study of the Fe(111)/MoS2 interface is based on a supercell in which a 3x3 slab of Fe(111) and a 4x4 cell of MoS2 are stacked together. For Co/MoS2, they used a supercell in which a MoS2 monolayer is sandwiched between face-centered cubic (fcc) Co layers and an atomic structure which should not correspond to that of a real Co/MoS2 interface, not only because Co actually crystallizes in the hexagonal compact (hcp) structure, but also because their multilayer is based on 4x4 Co(111) atomic layers superimposed on a 3x3 MoS2 single layer: considering the experimental values of the lattice parameters (0.2507 nm for hcp Co and 0.312 nm for MoS2), this interface corresponds to a relatively high lattice mismatch of 6.9%. Moreover, the MoS2 layer would only be bound to Co on one of its two sides, in a realistic MoS2/Co contact. For all these reasons, the genuine atomic structure of the Co/MoS2 interface may probably be different from that used by these authors. In the present paper, we considered supercells built from 5x5 Co(0001) atomic layers with the hcp stacking and a 4x4 MoS2 single layer. These stacking would correspond to the very small lattice mismatch of 0.4% and would be more realistic for calculating the physical properties of the Co/MoS2 interface between a Co magnetic electrode and a single MoS2 layer. After a brief description of the first-principles methods that we have used, we will first describe the atomic structure of the Co(0001)/MoS2 interface, before giving details on the electronic states, magnetic moments, and charge transfers at the interface. We finally discuss the physical properties of the Co(0001)/MoS2 interface in the perspective of spin injection in the 2D semiconductor MoS2, and give insights on the possible utilization of the low strained Co/MoS2 contact for in plane spin transport in lateral channels [17]. II.FIRST-PRINCIPLES METHODS The ground state energy, the charge and the spin densities of all the supercells have been calculated self-consistently using the full-potential augmented plane waves + local orbitals (APW+lo) method implemented in the code WIEN2k [START_REF] Blaha | WIEN2k, an augmented plane wave+local orbitals program for calculating crystal properties[END_REF]. The Kohn-Sham equation has been solved in the framework of the density functional theory (DFT), using the parametrization proposed by Perdew, Burke and Ernzerhof for the exchange and correlation potential that was treated within the generalized gradient approximation (GGA) [START_REF] Perdew | [END_REF]. In all our supercells, we used atomic sphere radii of 1.8, 1.8, and 2.0 atomic units (a. u.), respectively for Co, S and Mo atoms. The largest wave vector Kmax used for expanding the Kohn-Sham wave functions in the interstitial area between atomic spheres is given by the dimensionless parameter RminKmax=6.0, where Rmin is the smallest atomic sphere radius of the supercell; this corresponds to an energy cut-off of 151 eV. The irreducible wedge of the Co/MoS2 supercell two-dimensional Brillouin zone was sampled with a k-mesh of typically 24 different k-vectors, generated with a special k-grid used to perform Brillouin zone integrations with the modified tetrahedron integration method. To model the Co/MoS2 interface, we used rather big symmetric supercells consisting in a Co slab with hcp stacking and a thickness of five 5x5 (0001) monolayers (MLs), covered on each of its two sides by a 4x4 MoS2 single layer, followed by vacuum. The thickness of the Co layer is sufficient to recover most of the electronic structure of bulk hcp Co at the center of the slab (shape of the density of states curves, value of the spin magnetic moment) and the thickness of the vacuum separation between periodically adjacent MoS2 layers is above 1 nm, large enough to avoid interaction effects between them. We have chosen to use a periodic slab with a single MoS2 layer on both sides of Co to get also identical surfaces on both sides of the Co slab and vacuum separation, which avoids artefacts such as charge transfer between surfaces across the Co layer. This kind of huge supercells contain 125 Co, 64 S and 32 Mo atoms and up the 36 non-equivalent atoms, depending on the relative positions of S and Co atoms at the interface. This high number of atoms is a necessary requirement to treat the problem of the realistic low-strained single MoS2 contact that can be found, for example, at the source or the drain of a spin-field effect transistor (FET) based on a MoS2 channel. The lattice parameter that we chose for the Co(5x5)/MoS2(4x4) unit cell corresponds to four times the lattice parameter calculated for MoS2 (0.319 nm). The atomic structure of all the different MoS2/Co(5MLs)/MoS2 supercells that we have considered has been obtained by minimizing the forces acting on the different atoms. The minimum energy is achieved when all the atoms have reached their equilibrium position in the supercell. London dispersion forces are not included in the calculation. Once the atomic structure has been calculated, we can choose to include spin-orbit coupling effects at each self-consistent loop, to check whether these effects modify the electronic structure of the Co(0001)/MoS2 interface. This is performed within the second-order perturbation theory, for which we chose to include unoccupied electron states up to the maximum energy of 48 eV above the Fermi level, the magnetization being oriented along the Co(0001) axis. III. ATOMIC STRUCTURE OF THE Co(0001)/MoS2 INTERFACE We considered three different supercells, which correspond to three different manners of superimposing the 4x4 unit cell of 1H-MoS2 (shown on Figure 1a) on the 5x5 unit cell of Co(0001) (Fig. 1b). The first supercell (further labelled Supercell1) corresponds to the case where one of the interface S atoms of the 4x4 MoS2 cell (for instance, the S atom at the corners of the 4x4 MoS2 cell shown in Fig. 1a) is located just above one of the fcc hollow atomic sites of the Co(0001) surface. The second supercell (Supercell2) corresponds to the case where one of the interface S atoms is on a top atomic site (i. e. just above an atom of the surface Co ML). We finally considered the last supercell (Supercell3), for which one of the interface S atoms is above one of the hcp hollow sites of the Co(0001) surface (i. e. just above an atom of the subsurface Co ML). These three supercells are represented in Figure 2. The atomic structure of these three supercells has been calculated self-consistently without including spinorbit coupling. After all the atoms have reached their equilibrium position, we observed that the ground state energy is the lowest for Supercell1, and respectively 0.0827 eV and 0.813 eV higher for Supercell2 and Supercell3. These energy differences correspond to the whole supercells, which all contain 2 MoS 2/Co interfaces with 16 MoS2 formula units in the 4x4 MoS2 cell. Consequently, the difference per MoS2 formula unit, between the ground state energies of the different supercells is only of 2.6 meV between Supercell2 and Supercell1, and of 25.4 meV between Supercell3 and Supercell1. From now, we will only consider the physical properties of Supercell1, which corresponds to the lowest energy interface structure. remaining S and Co interface atoms. These interatomic distances are very close to those which have been measured for bulk cobalt sulfides (0.232 nm for CoS2 with the pyrite structure [26]). Each of the two This clearly shows that bonding between the MoS2 single layer and the Co(0001) surface has a covalent nature and is not due to the Van der Walls interaction. Each of the interface Co atoms are bound to a single S atom, except the four ones marked by a star in Fig. 3. The atomic layers present a small warping near the Co/MoS2 interface, due to the different numbers of chemical bounds formed by the non-equivalent interface S atoms. Based on the averaged values of the z-coordinates (the z-axis being perpendicular to the interface) calculated for the atoms in the different atomic layers, we can estimate the average distance between successive atomic layers: we obtain an average distance of 0.205 nm between the interface Co and interface S atomic layers, and average distances of 0.160 and 0.154 nm between the Mo layer and the interface S and external S layers, respectively. These later distances can be compared to the distance of 0.157 nm calculated between the S and Mo atomic layers of an isolated MoS2 sheet. The undulation of the successive atomic layers (difference between the highest and lowest z-coordinates) is respectively of 0.011 nm, 0.029 nm, 0.026 nm and 0.023 nm for the interface Co, interface S, Mo and external S atomic layers. IV. ELECTRONIC STRUCTURE OF THE Co(0001)/MoS2 INTERFACE The majority and minority spin band structures of the Co/MoS2 slab are shown in Figures 4b and4c. They contain a huge number of bands, some of them corresponding to cobalt bands and others to MoS2 bands, folded on themselves, with important band gap opening at the center and the edges of the two-dimensional Brillouin zone. The shaded areas which appear in these band structures correspond to the Co d-band continuum (below -0.5 eV for majority spin and across the Fermi level for minority spin electrons). Additional bands, which do not correspond to the folded bands of Co or MoS2 also appear in this figure; this is for instance the case between -0.5 eV and 0.5 eV for majority spin electrons. These new bands correspond to interface Bloch states involving covalently bound Co and S atoms. They strongly modify the physical properties of MoS2, giving a metallic behavior to this layer, induced by interface covalent bonding. Some of these interface bands have a non-negligible dispersion near the Fermi level (electrons in the metallic phase of MoS2 at the Co/MoS2 interface will not have a high effective mass). Each energy band in Figs. 4b and4c is drawn with small circles, the radius of which is proportional to the contribution of the MoS2 layer to the corresponding Bloch states. The band of the Co/MoS2 slab corresponding to the bottom of the conduction band of the MoS2 single layer appears near 0.32 eV above the Fermi level (it is indicated with a blue arrow on Fiq.4b). This band is more clearly visible for minority spin (Fig. 4b) than for majority spin electrons for which it is located in the continuum of Co d-bands (Fig. 4c). The difference between the energy of this band and the Fermi level corresponds to the height of the Schottky barrier [19], the value of which can be estimated at 32 . 0 = B φ eV for the Co/MoS2 interface. Our calculation of the Schottky barrier height is performed at the DFT level. As discussed by Zhong et al. [27], this seems to give results in a better agreement with experimental ones than those computed with the GW method, probably due to the fact that many-electron effects are greatly depressed by the charge transfer at the MoS2/metal interface, which significantly screens electron-electron interaction [27]. Figure 5 shows the density of states (DOS) of the MoS2 layer at the Co(0001)/MoS2 interface. This DOS curve is continuous between -8 eV and energies well above the Fermi level, confirming the metallic character of the MoS2 layer when it is covalently bound to the cobalt surface. This holds for majority as well as for minority spin electrons. Despite the strong modification of the electronic structure of MoS2, important DOS peaks can be identified in Fig. 5 as belonging to the valence and conduction bands of the isolated MoS2 layer: The DOS curves of the isolated MoS2 sheet has also been represented in Fig. 5, where it has been shifted to coincide to the main DOS peaks of MoS2 at the Co/MoS2 interface. The bottom of the conduction band in the shifted DOS curve of the isolated MoS2 layer is near 0.32 eV above the Fermi level, which confirms the value of the Schottky barrier height estimated from the band structure of the supercell. This estimation of the Schottky barrier height is larger than the values measured experimentally (between 60 meV in [28] and 121 meV [29]). These measurements are however performed on exfoliated MoS2 flakes. In these cases, the flakes are exposed to air before Co is deposited. It means that impureties and molecules can be trapped at the interface, resulting in localized states within the gap that can modify the pinning of the fermi level, and thus the Schottky barrier height. It would be interesting to compare our result with a full epitaxial Co/MoS2 hybrid system elaborated in ultra-high vacuum, when it will be available. Note that our estimation of the Schottky barrier height of the Co/MoS2 interface is comparable to values found in previous studies for similar interfaces involving another metal, like the Ti/MoS2 system [19]. The majority and minority spin densities of states give access to the spin polarization S P near the Fermi level in the MoS2 layer. This important quantity, defined as the difference between the majority and the minority spin densities of states divided by their sum, clearly when Mo and S atoms keep the same positions as in Supercell1 and Co atoms have all been removed. The majority spin space-dependent charge transfer can finally be obtained from ( ) ( ) ( ) ( ) { } r r r r 2 , , MoS Co n n n n ↑ ↑ ↑ ↑ + - = ∆ . The minority spin charge transfer is similarly given by 7 shows a top and a side view (the direction of observation corresponds to the blue arrow in Fig. 3) of the calculated three-dimensional majority and minority spin charge transfers interface is due to an excess of minority spin electrons, concomitant with a lack of majority spin electrons on interface Co atoms. Similarly, the magnetic moment of Mo atoms is due to an excess of majority spin and a lack of minority spin electrons for the Mo atoms which have a positive spin magnetic moment (right hand side of Figs. 7b and7d), and mostly to an excess of minority spin electrons for Mo atoms which have a negative magnetic moment (left hand side of Fig. 7d). ( ) ( ) ( ) ( ) { } r r r r 2 , , MoS Co n n n n ↓ ↓ ↓ ↓ + - = ∆ . Figure VII. PERSPECTIVES: TOWARDS ELECTRICAL SPIN INJECTION IN MoS2 In the context of spin injection into bulk semiconductor materials, it has been established that the so-called conductivity mismatch [16] between the ferromagnetic metal injector and the semiconductor constitutes a fundamental obstacle for efficient spin injection at the ferromagnetic metal/semiconductor interface in the diffusive regime. To circumvent this problem, it has been shown that a thin tunnel barrier must be introduced between FM and SC, inducing an effective spin dependent resistance [17]. A possible tunnel barrier that can be exploited is the natural Schottky barrier that occurs between the two materials. As spin injection in tunnel regime is desirable, a careful engineering of the doping in the semiconductor close to the interface is required in order to make the Schottky barrier thin enough to behave as a tunnel barrier. It has been realized successfully in Fe/GaAs system [10] where GaAs was gradually n-doped up to 10 19 cm -3 at the interface, resulting in a very efficient electrical spin injection from Fe into GaAs in tunnel regime. In the context of TMDCs, the general problem of electrical contacts on TMDCs for transport in a two-dimensionnal channel is a challenging task (tackled for example by phase engineering techniques [31,32], as well as using contacts based on heterostructures involving graphene [22]. Concerning spin injection with Co/MoS2 (or other FM/TMDC [33]) interfaces, one has to consider the Schottky barrier between the metallic phase of MoS2 just below the contact, labeled hereafter (MoS2)*, and the semiconductor MoS2 channel out of the contact. As in GaAs, one could imagine to strongly increase the doping in MoS2 close to the (MoS2)*/MoS2 one-dimensional border, see Figure 8. Even if, up to now, spatially controlled chemical doping of a single layer TMDC is still a challenge, such an in-plane localized doping has been already obtained with the help of additional gate electrodes developed successfully for in-plane P-I-N junctions in TMDC Light Emitting Diodes [34]. Considering the electrical spin injection tunnel process at this (MoS2)*/MoS2 (n-doped) one-dimensional interface, one has to compare, for majority and minority spin, the compatibility of the symmetries of the electron wave functions in (MoS2)* at the Fermi level, with the ones in the conduction band of the MoS2 single layer semiconductor channel, in order to estimate the efficiency of the tunneling process. The conduction electron wavefunctions in the isolated MoS2 channel exhibit a strong Mo dz 2 character (as well as a smaller Mo s character), and a S (px+py) character. The DOS at the Fermi level calculated for (MoS2)* is qualitatively different from the DOS at the bottom of the conduction band of MoS2. Due to the strong hybridization between Co and S atomic orbitals at the Co/MoS2 interface, Bloch electron states at the Fermi level result from a non-trivial combination of atomic orbitals that involve Mo-dz 2 , S-(px+py) and also other orbitals, see Table 1. However, the contribution of Mo-dz 2 and S-(px+py) atomic orbitals is not negligible and still represents 33% and 37% of the total DOS of (MoS2)* at the Fermi level, respectively for majority and minority spin electrons. The symmetry of the majority and minority spin electron states at the Fermi level in (MoS2)* is thus partly compatible with the symmetry of Bloch states in the conduction band in the MoS2 channel. The efficiency of electrical injection in the MoS2 channel in the tunnel regime should also depend on the direction of the one dimensional border between (MoS2)* and MoS2. We know that the Bloch vector of electrons in the conduction band of the MoS2 channel, after tunneling through the Schottky barrier, corresponds to one of the K-valleys. We also know, using the simple model of a free electron with mass m and energy E propagating in a plane towards a straight one-dimensional potential step where the potential jumps from 0 to E U > , that the penetration depth that characterizes the exponential decay of the wave function after the step is given by 2 1 2 // 2 2 2 2 -                   + - m k E U m h h ; // k is the component of the electron two- dimensional Bloch vector parallel to the straight border, and TABLE CAPTIONS: Table 1: Total and partial (for the most important atomic orbitals) Mo and S majority and minority spin density of states at the Fermi level for the Co/MoS2 interface. All the results are given in the same arbitrary unit. FIGURE CAPTIONS have studied the Ni(111)/MoS2 interface, but they mostly focus their study on the consequences of the insertion of a graphene layer between Ni and MoS2, than on a detailed description of the Ni(111)/MoS2 interface [22]. Finally, Yin et al. have recently calculated the electronic structure of the Fe/MoS2 Co/MoS2 interfaces in the supercell involves 25 Co and 16 S atoms. One of these 16 interface S atoms (further labelled S3) is covalently bound to 3 different but equivalent interface Co atoms (labelled Co3); 3 of the 16 interface S atoms (labelled S2) are each bound to 2 different but equivalent Co atoms (Co2), and all the 12 remaining interface S atoms are bound to a single Co atom. The position of all these atoms of the Co/MoS2 interface is shown in Figure 3. The distance between interface S and Co atoms is of 0.236 nm between S3 and Co3, 0.234 nm between S2 and Co2, and of 0.221 or 0.222 nm between the 12 . Red and green areas respectively correspond to a local excess and a local lack of electrons. As expected, we see on this figure that charge transfers occur between interface Co and S atoms along the Co-S covalent bonds. This figure shows that the reduction of the Co magnetic moment at the . It follows that the straight (MoS2)*/MoS2 border should be perpendicular to the (ΓK) direction of the two-dimensional Brillouin zone, in order to get a spin-polarized current with maximum intensity reaching a K-valley in the MoS2 was granted access to the HPC resources of CALMIP supercomputing center under the allocation p1446 (2014-2016). Figure 1 : 1 Figure 1: Atomic structure (top views) of (a): the 4x4 unit cell of MoS2, and (b): the 5x5 unit cell of a 5MLs Co(0001) slab. Figure 2 : 2 Figure 2: Side views of the atomic structure of the Co/MoS2 supercells corresponding to (a): Supercell1 (b): Supercell2, and (c): Supercell3. Figure 3 : 3 Figure 3: Top view of the atomic structure of the Co(0001)/MoS2 interface for Supercell1. The atoms Co2 (dark blue spheres), Co3 (medium blue), S2 (red) and S3 (orange) discussed in section III are indicated. The Co atoms marked by a star are those which are not directly bound to a S atom. Figure 4 : 4 Figure 4: Band structure of (a): the 4x4 MoS2 single layer supercell, (b): the Co/MoS2 slab for majority spin, (c): the Co/MoS2 slab for minority spin, (d): the 5x5 Co(0001) slab for majority spin and (e): the 5x5 Co(0001) slab for minority spin. The radius of the red circles in panels (b) and (c) is proportional to the MoS2 contribution to the electron states. The blue arrow in panel (a) indicates the minimum of the conduction band in MoS2; its corresponding position in the band structure of the Co/MoS2 slab is also shown with a blue arrow in panel (b), where it corresponds to the height of the Schottky barrier. Figure 5 : 5 Figure 5: Contribution of a MoS2 monolayer to the density of states of Supercell1 (dark curves).The density of states of an isolated MoS2 layer is also represented after an energy shift (red curves). The upper and lower parts of the figure respectively correspond to majority and minority spin electrons. Figure 6 : 6 Figure 6: Spin polarization in the MoS2 layer, calculated without (red curve) and with (dark curve) spin-orbit coupling. Figure 7 : 7 Figure 7: (a): Top view and (b): side view of the majority spin charge transfer Figure 8 : 8 Figure 8: Sketch representing (a): the bottom view (below the Co contact) and (b): the side view of the physical area involving the Co(0001) contact, (MoS2)*, the Schottky barrier (MoS2)*/ MoS2 obtained by suitable doping and the MoS2 channel. The one-dimensional border which should be designed to lower the exponential decay of wave functions in the Schottky barrier is indicated by red a dashed line in (a), together with the (ΓK) direction of MoS2 (red arrow). The corresponding Schottky barrier profile is represented in panel (c). Figure 3 Figure 4 Figure 5 345 Figure 3 Figure 6 6 Figure 6 ACKNOWLEDGEMENTS The authors acknowledge the Université Fédérale de Toulouse-Midi-Pyrénées and Région Midi-Pyrénées for the PhD grant SEISMES as well as the grant NEXT n° ANR-10-LABX-0037 in the framework of the « Programme des Investissements d'Avenir". This work indicates if a spin-polarized current can be injected through the Co/MoS2 interface to a MoS2 channel. DOS curves shown in Fig. 5 The estimation of the spin polarization at the Fermi level for Co/MoS2 interface is of importance for spin injection. As pointed out by Mazin [30], an accurate determination of the electric current spin polarization would however require complementary DFT-based calculations (including the transmittance of the whole complex structure and the effects of the bias voltage). V. SPIN MAGNETIC MOMENTS AT THE Co(0001)/MoS2 INTERFACE The spin magnetic moment of Co atoms is on average 8% lower at the Co(0001)/MoS2 interface than in bulk hcp Co (1.69 µB), with values that depend on the kind of interface S atom to which they are covalently bound: The interface Co atoms that show the highest spin magnetic moment (1.66 µB and 1.63 µB) are the 4 atoms which are not bound to S atoms, followed by Co3 (1.62 µB) and Co2 atoms (1.57 µB). All the other interface Co atoms have a spin-magnetic moment between 1.48 and 1.50 µB: the lowering of the interface Co atom magnetic moment is more important when Co atoms are more strongly bound to S atoms, with a shorter Co-S bond length. All the interface S atoms have a small spin magnetic moment with the same sign as the Co atom magnetic moments, with values between 0.012 and 0.016 µB. The spin magnetic moment of S atoms in the external S layer is even smaller (0.003 to 0.004 µB). The spin magnetic moment of Mo atoms (between -0.029 and -0.024 µB) is antiferromagnetically coupled to the Co and S magnetic moments, except when these Mo atoms are bound to one or to two S2 atoms (in this case, Mo spin magnetic moments respectively take the positive values 0.008 and 0.050 µB). VI. CHARGE TRANSFER AT THE Co(0001)/MoS2 INTERFACE To calculate the charge transfer between atoms induced by covalent bonding at the Co(0001)/MoS2 interface, we proceeded as follows: first, we calculated the majority spin electron density ( ) VIII. CONCLUSIONS In this paper we have investigated the electronic structure of a single low-strained hcp Co(0001)/MoS2 interface, using first-principles calculations based on the functional density theory, in order to estimate the potentiality of a cobalt injector for electrical spin injection into a MoS2 monolayer, in view of spintronic devices as spin-FET. We first described the lowest energy atomic structure and show that interface S atoms are covalently bound to one, two or three interface Co atoms. A lower spin magnetic moment is observed for Co atoms at the interface, together with a small magnetization of S atoms. Mo atoms also hold small magnetic moments which can takes positive as well as negative values. The induced magnetic moments have been interpreted in terms of majority and minority spin charge transfers at the interface. Band structures and density of states curves have been calculated for minority and majority spin electrons in the hybrid system, taking into account spin-orbit coupling. We demonstrate that MoS2 just below the cobalt contact becomes metallic due to hybridization with Co d orbitals. TABLES: FIGURES:

01744852

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2024/03/05 22:32:07

2018

https://hal.science/hal-01744852/file/Template_ISBI2018-copy.pdf

Sudhanya Chatterjee Olivier Commowick Onur Afacan Simon K Warfield Christian Barillot MULTI-COMPARTMENT MODEL OF BRAIN TISSUES FROM T2 RELAXOMETRY MRI USING GAMMA DISTRIBUTION Keywords: T 2 relaxometry, microstructure, brain The brain microstructure, especially myelinated axons and free fluids, may provide useful insight into brain neurodegenerative diseases such as multiple sclerosis (MS). These may be distinguished based on their transverse relaxation times which can be measured using T 2 relaxometry MRI. However, due to physical limitations on achievable resolution, each voxel contains a combination of these tissues, rendering the estimation complex. We present a novel multi-compartment T 2 (MCT2) estimation based on variable projection, applicable to any MCT2 microstructure model. We derive this estimation for a three-gamma distribution model. We validate our framework on synthetic data and illustrate its potential on healthy volunteer and MS patient data. INTRODUCTION MRI voxels of the human brain are heterogeneous in terms of tissue types due to the limited imaging resolution and physical constraints. Each voxel in the white matter (WM) contains a large number of myelinated and non-myelinated axons, glial cells and extracellular fluids [START_REF] Tomasch | Size, distribution, and number of fibres in the human corpus callosum[END_REF][START_REF] Lancaster | Three-pool model of white matter[END_REF]. For example, every square millimeter of the corpus callosum in a human brain has more than 100,000 fibers (myelinated and non-myelinated) of varying diameters [START_REF] Tomasch | Size, distribution, and number of fibres in the human corpus callosum[END_REF]. These tissues can be distinguished based on their T 2 relaxation times. Myelin being a tightly wrapped structure has a very short T 2 relaxation time of 10 milliseconds (ms) [START_REF] Lancaster | Three-pool model of white matter[END_REF]. The estimated T 2 relaxation time of the myelinated axons is 40ms [START_REF] Lancaster | Three-pool model of white matter[END_REF]. The ventricles and tissue injury regions contain free fluids which have a high T 2 relaxation time (>1000ms). The T 2 relaxation values between those of myelin and myelinated axons and the free fluids correspond to the glial cells and extracellular tissues [START_REF] Lancaster | Three-pool model of white matter[END_REF]. An ability to obtain the condition of these tissues can help us gain better insights into the onset and progress of neurodegenerative diseases such as multiple sclerosis (MS). Myelin water fraction (MWF) has been computed from T 2 relaxometry images using a variety of approaches [START_REF] Mackay | Magnetic resonance of myelin water: An in vivo marker for myelin[END_REF][START_REF] Akhondi-Asl | Fast myelin water fraction estimation using 2D multislice CPMG[END_REF]. Most of these methods primarily focus on the MWF estimation. However, MWF alone might not be able to convey the en-tire information since it is a relative measurement. For example, in MS patients a decrease in MWF in a WM lesion might be caused by myelin loss or fluid accumulation due to tissue injury or both. Hence for relative measurements like water fraction (WF), all the WF maps should be observed simultaneously for a complete understanding of the tissue condition. Here we propose an estimation framework to obtain brain microstructure information using a multi-compartment tissue model from T 2 relaxometry MRI data. The T 2 space is modeled as a weighted mixture of three continuous probability density functions (PDF), each representing the tissues with short, medium and high T 2 relaxation times. We estimate the PDF parameters using variable projection (VARPRO) approach [START_REF] Golub | Separable nonlinear least squares: the variable projection method and its applications[END_REF]. We derive this generic estimation framework for gamma PDFs. We validate the proposed method using synthetic data against known ground truth. We then illustrated it on a healthy subject and MS patient. METHOD AND MATERIALS Theory Signal model The T 2 space is modeled as a weighted mixture of three PDFs, representing each of the three T 2 relaxometry compartments: short-, medium-and high-T 2 . Thus the voxel signal at the i-th echo time (t i ) is given as: s (t i ) = M 0 3 j=1 w j ∞ 0 f j (T 2 ; p j ) EPG (T 2 , T E, i, B 1 ) dT 2 (1) Each compartment is described by a chosen PDF, f j (T 2 ; p j ), where p j = {p j1 , . . . , p jn } ∈ R n are the PDF parameters. In Eq. (1), w j is the weight of the j-th distribution with j w j = 1. ∆T E, B 1 and M 0 are the echo spacing, field inhomogeneity and magnetization constant respectively. Imperfect rephasing of the nuclear spins after application of refocusing pulses leads to the generation of stimulated echoes [START_REF] Hennig | Calculation of Flip Angles for Echo Trains with Predefined Amplitudes with the Extended Phase Graph (EPG)-Algorithm: Principles and Applications to Hyperecho and TRAPS Sequences[END_REF]. Hence the T 2 decay is not purely exponential. The stimulated echoes are thus obtained using the EPG algorithm [START_REF] Prasloski | Applications of stimulated echo correction to multicomponent T2 analysis[END_REF]. EPG(•) is the stimulated echo computed at t i = i ∆TE where i = {1, . . . , m} and m is the number of echoes. Optimization M 0 and w j can be combined into a single term α j ∈ R + without any loss of generality. In that case, the weight w j corresponding to each compartment is obtained as w j = α j / i α i . In the most general case, the least squares minimization problem is thus formulated as: α, p, B1 = arg min α,p,B1 m i=1   y i - 3 j=1 α j λ j (t i ; p, B 1 )   2 = arg min α,p,B1 Y -Λ (p, B 1 ) α 2 2 (2) where Y ∈ R m is the observed signal and m is the number of echoes; α ∈ R + 3 ; Λ ∈ R m×3 ; p = {p 1 , p 2 , p 3 } ∈ R k , where k = 3n. In Eq. ( 2), each element of Λ, Λ ij =λ (t i ; p j , B 1 ), is computed as: Λ ij = ∞ 0 f j (T 2 ; p j ) EP G (T 2 , T E, i, B 1 ) dT 2 (3) Due to the EPG formulation, there is no closed form derivative solution for the optimization of B 1 [START_REF] Prasloski | Applications of stimulated echo correction to multicomponent T2 analysis[END_REF]. Hence, we opt for an alternate optimization scheme where we iterate between optimization of {p, α} with a fixed value of B 1 and optimization for B 1 using the obtained {p, α} values. The terms Λ (p, B 1 ) and α in Eq. ( 2) are linearly separable. Hence we can use the VARPRO approach to solve for {p, α} [START_REF] Golub | Separable nonlinear least squares: the variable projection method and its applications[END_REF]. The unknown α is substituted by Λ (p) + Y, where Λ (p) + is the Moore-Penrose generalized inverse of Λ (p). The VARPRO cost function is computed as: arg min p I -Λ (p) Λ (p) + Y 2 2 (4) where, I -Λ (p) Λ (p) + is the projector on the orthogonal complement of the column space of Λ (p). Since p ∈ R k , the Jacobian matrix J ∈ R k×m and its columns are computed as shown in [START_REF] Golub | Separable nonlinear least squares: the variable projection method and its applications[END_REF]. To compute the elements of J, we need to obtain ∂Λ/∂p ji , ∀i, j [START_REF] Golub | Separable nonlinear least squares: the variable projection method and its applications[END_REF]. After solving Eq. ( 4) for p, the values of α are obtained as Λ (p) + Y. The optimization for {α, p} and B 1 is performed alternatively until convergence. B 1 is optimized using a gradient free optimizer (BOBYQA), as it does not have any closed form solution [START_REF] Prasloski | Applications of stimulated echo correction to multicomponent T2 analysis[END_REF]. Multi-compartment model using gamma PDF The previous estimation framework is generic as it does not depend on the chosen PDF. We choose here to use gamma PDF for f j (•) for j = {1, 2 ,3} since their non-negativity and skewed nature are well suited to describe the compartments used to model the T 2 space. The mean T 2 values of myelin, myelinated axons, inter-and extra-cellular and free fluids in the brain are well studied in the literature [START_REF] Lancaster | Three-pool model of white matter[END_REF][START_REF] Mackay | Magnetic resonance of myelin water: An in vivo marker for myelin[END_REF]. Hence we parameterized each f j in terms of its mean (µ j ) and variance (v j ) rather than the usual shape and scale parameter representation (refer Eq. ( 5)). Using this parametric form of the gamma PDF makes the choice of optimization bounds convenient. f (T 2 ; µ j , v j ) = T (µ 2 j /vj )-1 2 Γ µ 2 j /v j (v j /µ j ) µ 2 j v j exp -T 2 v j /µ j (5) Hence we have, There are almost no echoes available which correspond to the high-T 2 compartment. The robustness and accuracy of the implementations to simultaneously estimate the weights and all the PDF parameters has been found to be not reliable [START_REF] Kj Layton | Modelling and estimation of multicomponent T 2 distributions[END_REF]. p = {µ s , v s , µ m , v m , µ h , v h } Hence we choose to estimate only the mean of the gamma PDF corresponding to the medium-T 2 compartment. Using the VARPRO approach we thus estimate four parameters of the signal model: mean of the medium-T 2 gamma PDF (µ m ) and the three weights corresponding to each compartment. Hence only ∂Λ/∂µ m is required for computing the Jacobian matrix and is obtained as: ∂Λ ∂µ m = ∞ 0 f (T 2 ; µ m , v m ) µ m v m 2 log T 2 µ m v m - (6) 2Ψ µ 2 m v m + 1 - T 2 v m EP G (T 2 , T E, i, B 1 ) dT 2 where Ψ(•) is the digamma function. The remaining gamma PDF parameter values are pre-selected for the three compartments based on histology findings reported in the literature [START_REF] Mackay | Magnetic resonance of myelin water: An in vivo marker for myelin[END_REF] and are set as {µ s , µ h } = {30, 2000} ms and {v s , v m , v h } = {50, 100, 6400} ms 2 . We assume a reasonable bound on µ m of 100-125ms for its optimization. The minimization problem in Eq. ( 4) is solved for µ m using the analytically obtained derivative in Eq. ( 6) with a gradient based optimizer [START_REF] Svanberg | A class of globally convergent optimization methods based on conservative convex separable approximations[END_REF]. The short-T 2 compartment here indicates the condition of myelin and myelinated axons [START_REF] Lancaster | Three-pool model of white matter[END_REF]. The medium-T 2 compartment's WF conveys information on the condition of axons, glial cells and extracellular fluids [START_REF] Lancaster | Three-pool model of white matter[END_REF]. The condition of free fluids (such as in ventricles and fluid accumulation due to tissue injuries) is indicated by the high-T 2 WF values. Experiments Synthetic data. MS patient data. The method was finally tested on T 2 relaxometry MRI data of a MS patient. The observations from our estimation maps were compared with the pathological findings on MS lesion reported in the literature [START_REF] Lassmann | Heterogeneity of multiple sclerosis pathogenesis: Implications for diagnosis and therapy[END_REF][START_REF] Cr Guttmann | The evolution of multiple sclerosis lesions on serial MR[END_REF]. We observed whether the estimation maps obtained from our method were able to provide insight into MS lesion which corroborate with the pathological findings. The acquisition details are the same as for the healthy volunteer data. The ventricles and other regions with free fluids have a higher µ m compared to the normal appearing white matter (NAWM) tissues. This is relevant as free fluids have a higher T 2 value compared to the relatively tightly bound tissues present in NAWM. MS patient data. Two lesions are present in the MR image of the MS patient shown in Fig. 3 that are marked with red and blue arrows. We observe absence of short-T 2 WF in both lesions. The lesions and their neighboring tissues have higher w m values than NAWM tissues. The w h map shows fluid accumulation in lesion-2 but not in lesion-1 . The estimated medium-T 2 gamma PDF map shows a higher PDF mean for both lesions compared to the NAWM. In lesion-1 the estimated µ m increases (with respect to the neighboring NAWM) as it approaches the core of the lesion, but is less than the µ m estimated at the ventricles where there is free fluid. RESULTS Synthetic DISCUSSION Our method was successfully validated against synthetic data with known ground truth for all SNR values (refer Fig. 1). High w s values in the genu of the CC of healthy volunteer data (refer Fig. 2) is due to the high myelin and myelinated fibers density in this region compared to any other part of the brain [START_REF] Tomasch | Size, distribution, and number of fibres in the human corpus callosum[END_REF]. In the estimation maps of the MS patient (refer Fig. 3), the absence of short-T 2 WF in the lesions and in its neighboring regions can be explained by demyelination of the nerve fibers caused by MS [10][3]. Demyelination at the onset of MS is followed by macrophage intervention leading to an increased cellular activity in the MS lesion regions [START_REF] Lassmann | Heterogeneity of multiple sclerosis pathogenesis: Implications for diagnosis and therapy[END_REF] whose T 2 relaxation time is greater than myelin and myelinated axons but less than those of free fluids [START_REF] Lancaster | Three-pool model of white matter[END_REF]. This phenomenon might explain the high w m values in the lesions and in neighboring regions. Demyelination is followed by progressive axonal damage and fluid accumulation (due to tissue injuries) in MS lesions [START_REF] Lassmann | Heterogeneity of multiple sclerosis pathogenesis: Implications for diagnosis and therapy[END_REF]. The extent of axonal damage and fluid accumulation in the lesions can provide useful information regarding the lesion state and its response to a treatment. Lesion-2 has fluid accumulation unlike lesion-1 , possibly indicating that the two lesions are in different stages. The continuous axonal damage in the MS lesions [10][11] explains the higher µ m in the lesion regions compared to the neighboring NAWM as a higher µ m value indicates tissues with less tightly bound water. The increment in µ m values in lesion-1 as we approach the lesion core from the lesion boundary might also indicate a reduction in axon density. This is in accordance with the pathology of MS lesion evolution [10][11]. CONCLUSION We proposed a generic estimation method to obtain estimates of tissue microstructure in brain by modeling the T 2 spectrum as a weighted mixture of three gamma PDFs. The maps estimated from our method can be effective in understanding the heterogeneity of lesions [START_REF] Lassmann | Heterogeneity of multiple sclerosis pathogenesis: Implications for diagnosis and therapy[END_REF] in MS patients and be used as potential biomarkers to have information on the MS lesion growth stage. As a part of the future study we intend to validate the observations by applying the proposed estimation framework on more healthy controls and MS patient datasets. data. The results of the synthetic data simulation are shown in Fig 1. It shows that with increasing SNR the weights estimation gets more accurate. The bars around the mean value are the 95% confidence intervals (CI) which is obtained as 1.96 times the standard deviation of the estimates. The CIs of the estimation improve with with increasing SNR for all three WFs. The ground truth lies in the CI of the mean estimated weights for all three compartments. Fig. 1 . 1 Fig. 1. Mean of estimated weights with a 95% confidence interval for 100 signal averages for the synthetic data study. Healthy volunteer data. The estimation maps for the healthy volunteer are shown in Fig 2. The genu of the corpus callosum (CC) has higher w s values compared to any other region Fig. 2 . 2 Fig. 2. Estimation maps for a healthy volunteer. Fig. 3 . 3 Fig. 3. Estimation maps for MS patient data. Lesion-1 and lesion-2 are marked with red and blue arrows respectively. where, (•) s , (•) m and (•) h are the PDF parameters describing the short-, medium-and high-T 2 compartments respectively. Due to practical limitations such as feasible acquisition time, coil heating and specific absorption rate (SAR) guidelines, T 2 relaxometry MRI sequences have limitations on the shortest echo time and number of echoes per acquisition. The high-T 2 compartment aims at capturing of free fluids in the brain, and hence has a T 2 relaxation time larger than 1 second[START_REF] Mackay | Magnetic resonance of myelin water: An in vivo marker for myelin[END_REF]. A standard T 2 spin echo multi-contrast sequence has the shortest T 2 acquisition (first echo) at around 8 -10ms and has 20 -40 acquired echoes. Hence for the short-T 2 compartment, we usually have a very limited (around 3-4) number of echoes. The proposed method was first validated against synthetic data generated following a known ground truth, composed of three gamma PDFs with parameters, {µ s , µ m , µ h } = {25, 120, 1900} and {v s , v m , v h } = {40, 90, 6000}. The weights chosen for each compartment were, {w s , w m , w h } = {0.2, 0.7, 0.1}. The B 1 , M 0 and T 1 values considered for this simulation were 1.3, 950 and 1000 respectively. The experiments are carried out for SNR values ranging from 5 to 100 in steps of 5. We simulated T 2 relaxometry data with the following parameters: first echo (TE 0 ) is at 9ms; TE= 9ms; 32 echoes, 100 signal averages. Healthy volunteer data. The method was tested on T 2 relaxometry data acquired on a healthy volunteer (male, age: 26) with the following acquisition details: Siemens 3T MRI scanner; 2D multislice CPMG sequence; 32 echoes; TE 0 = 9ms; echo spacing of 9ms; TR = 3000ms; single slice acquired; slice thickness of 4mm; in plane resolution of 1.04mm × 1.04mm; matrix size of 192 × 192.

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2024/03/05 22:32:07

2018

https://insu.hal.science/insu-01744885/file/Laurent_et_al-2018-Journal_of_Metamorphic_Geology.pdf

Valentin Laurent email: [email protected] Pierre Lanari Inès Naïr Romain Augier Abdeltif Lahfid Laurent Jolivet Exhumation of eclogite and blueschist (Cyclades, Greece):Pressure-temperature evolution determined by thermobarometry and garnet quilibrium modelling Keywords: Subduction zone, Exhumation of eclogite and blueschist, Garnet equilibrium modelling, XMAPTOOLS, Slab rollback High-pressure rocks such as eclogite and blueschist are metamorphic markers of paleosubduction zones, and their formation at high-pressure and low-temperature conditions is relatively well understood since it has been the focus of numerous petrological investigations in the past 40 years. The tectonic mechanisms controlling their exhumation back to the surface are, however, diverse, complex and still actively debated. Although the Cycladic Blueschist Unit (CBU, Greece) is among the best worldwide examples for the preservation of eclogite and blueschist, the proposed P-T evolution followed by this unit within the Hellenic subduction zone is quite different from one study to another, hindering our comprehension of exhumation processes. In this study, we present an extensive petrological dataset that permits refinement of the shape of the P-T trajectory for different subunits of the CBU on Syros. High-resolution quantitative compositional mapping has been applied to support the thermobarometric investigations, which involve semi-empirical thermobarometry, garnet equilibrium modelling and P-T isochemical phase diagrams. The thermodynamic models highlight the powerful use of reactive bulk compositions approximated from local bulk compositions. The results were also | INTRODUCTION Comprehension of the dynamic processes controlling the exhumation of high-pressure and lowtemperature (HP-LT) metamorphic rocks during subduction is partly based on the reconstruction of detailed Pressure-Temperature-time-deformation (P-T-t-d) paths of tectonic units that have undergone a complete burial-exhumation cycle. The Cycladic Blueschist Unit (CBU), cropping out in the Cycladic archipelago (Greece), is one of the best worldwide examples of a fossilized subduction channel. Syros Island, located in the central part of the Cyclades, is mainly composed of the CBU and is famous for its excellent preservation of HP-LT metamorphic rocks such as eclogites and blueschists. Consequently, this island has been the focus of petrological, structural and geochronological studies aimed at constraining the tectonometamorphic evolution of the CBU subduction complex (e.g. [START_REF] Cliff | Geochronological challenges posed by continuously developing tectonometamorphic systems: insights from Rb-Sr mica ages from the Cycladic Blueschist Belt, Syros (Greece)[END_REF][START_REF] Keiter | A new geological map of the Island of Syros (Aegean Sea, Greece): Implications for lithostratigraphy and structural history of the Cycladic Blueschist Unit[END_REF][START_REF] Keiter | Structural development of high-pressure metamorphic rocks on Syros island (Cyclades, Greece)[END_REF][START_REF] Lagos | High precision Lu-Hf geochronology of Eocene eclogite-facies rocks from Syros, Cyclades, Greece[END_REF][START_REF] Laurent | Strain localization in a fossilized subduction channel: Insights from the Cycladic Blueschist Unit (Syros, Greece)[END_REF][START_REF] Laurent | Extraneous argon in high-pressure metamorphic rocks: Distribution, origin and transport in the Cycladic Blueschist Unit (Greece)[END_REF][START_REF] Lister | White mica 40 Ar/ 39 Ar age spectra and the timing of multiple episodes of high-P metamorphic mineral growth in the Cycladic eclogite-blueschist belt, Syros, Aegean Sea, Greece[END_REF][START_REF] Philippon | Tectonics of the Syros blueschists (Cyclades, Greece): From subduction to Aegean extension[END_REF][START_REF] Schumacher | Glaucophane-bearing marbles on Syros, Greece[END_REF]Soukis & Stöckli, 2013;[START_REF] Tomaschek | Zircons from Syros, Cyclades, Greece -recrystallization and mobilization of zircon during high-pressure metamorphism[END_REF]Trotet, Jolivet, & Vidal, 2001a;[START_REF] Trotet | Exhumation of Syros and Sifnos metamorphic rocks (Cyclades, Greece). New constraints on the PT paths[END_REF]. However, after a decade of investigations, the tectonometamorphic evolution of the CBU of Syros is still actively debated, as attested by the different shapes of P-T paths proposed in the literature from burial to exhumation (Figure 1). For example, [START_REF] Trotet | Exhumation of Syros and Sifnos metamorphic rocks (Cyclades, Greece). New constraints on the PT paths[END_REF] used a multi-equilibrium approach with the TWEEQU software [START_REF] Berman | Thermobarometry using multi-equilibrium calculations: a new technique, with petrological applications[END_REF] to characterize the P-T paths followed by eclogite-, blueschist-and greenschist-facies metamorphic rocks from different tectonic subunits on Syros. Their results lead to the conclusion that, irrespective of the intensity of retrogression, the different units have all undergone the same metamorphic peak conditions around 20 kbar and 550 °C but have followed distinct exhumation evolutions after peak metamorphism (Figure 1a). In this scenario, the preservation of eclogitic parageneses at the top of the lithological pile of Syros is explained by cooling during exhumation. A stronger retrogression under greenschist-facies conditions is observed further down and is associated Accepted Article This article is protected by copyright. All rights reserved. | GEOLOGICAL CONTEXT | Aegean domain The Aegean domain corresponds to a collapsed segment of the Hellenic belt above a north-plunging subduction zone (Figure 2; [START_REF] Jolivet | Exhumation of deep crustal metamorphic rocks and crustal extension in arc and back-arc regions[END_REF][START_REF] Le Pichon | The Aegean Sea[END_REF][START_REF] Ring | The Hellenic subduction system: high-pressure metamorphism, exhumation, normal faulting, and large-scale extension. Annual Review[END_REF]. The tectonometamorphic evolution of this domain is traditionally described in two main stages (Figure 2; [START_REF] Jolivet | Cenozoic geodynamic evolution of the Aegean[END_REF][START_REF] Ring | The Hellenic subduction system: high-pressure metamorphism, exhumation, normal faulting, and large-scale extension. Annual Review[END_REF]. First, the late Cretaceous-Eocene convergence between Africa and Eurasia plates led to the formation of the Hellenides-Taurides belt. During this episode, a series of oceanic and continental nappes entered the subduction zone and were thrust on top of each other in an overall HP-LT metamorphic context [START_REF] Bonneau | Subduction, collision et schistes bleus; l'exemple de l'Egee (Grece)[END_REF]. Then, inception and acceleration of African slab retreat from 35-30 Ma led to southward migration of the subduction front, crustal collapse of the belt in the back-arc domain and extensional reworking of the nappe stack [START_REF] Jolivet | Correlation of syn-orogenic tectonic and metamorphic events in the Cyclades, the Lycian nappes and the Menderes massif. Geodynamic implications[END_REF][START_REF] Jolivet | Aegean tectonics: Strain localisation, slab tearing and trench retreat[END_REF][START_REF] Le Pichon | The Aegean Sea[END_REF][START_REF] Lister | Metamorphic core complexes of Cordilleran type in the Cyclades, Aegean Sea, Greece[END_REF]. Extension in the back-arc domain was characterized by a distributed deformation oriented N-S over a wide region covering the entire Aegean Sea, part of western Anatolia and the Rhodope Massif in the north [START_REF] Gautier | Ductile crust exhumation and extensional detachments in the central Aegean (Cyclades and Evvia Islands)[END_REF][START_REF] Jolivet | Cenozoic geodynamic evolution of the Aegean[END_REF][START_REF] Jolivet | Ductile extension and the formation of the Aegean Sea[END_REF][START_REF] Ring | The Hellenic subduction system: high-pressure metamorphism, exhumation, normal faulting, and large-scale extension. Annual Review[END_REF][START_REF] Urai | Alpine deformation on Naxos (Greece)[END_REF]. Extensional tectonics were also characterized by more localized deformation with the development of large-scale detachments and metamorphic core complexes (MCCs) in which the exhumation of HP-LT units was completed in a low-pressure and hightemperature (LP-HT) environment (Figure 2; [START_REF] Gautier | Structure and kinematics of upper Cenozoic extensional detachment on Naxos and Paros (Cyclades Islands, Greece)[END_REF][START_REF] Jolivet | Ductile extension and the formation of the Aegean Sea[END_REF][START_REF] Lister | Metamorphic core complexes of Cordilleran type in the Cyclades, Aegean Sea, Greece[END_REF][START_REF] Urai | Alpine deformation on Naxos (Greece)[END_REF]. The Cycladic Archipelago is located in the centre of the Aegean domain and corresponds to the deepest exhumed parts of the Hellenides-Taurides belt (Figure 2). In this archipelago, the Cycladic Blueschist Unit, belonging to the Pindos oceanic domain [START_REF] Bonneau | Correlation of the Hellenide nappes in the south-east Aegean and their tectonic reconstruction[END_REF][START_REF] Bonneau | Subduction, collision et schistes bleus; l'exemple de l'Egee (Grece)[END_REF], reached peak-pressure conditions during the formation of the Hellenides at 53-48 Ma (Figure 2; [START_REF] Lagos | High precision Lu-Hf geochronology of Eocene eclogite-facies rocks from Syros, Cyclades, Greece[END_REF][START_REF] Laurent | Extraneous argon in high-pressure metamorphic rocks: Distribution, origin and transport in the Cycladic Blueschist Unit (Greece)[END_REF][START_REF] Lister | White mica 40 Ar/ 39 Ar age spectra and the timing of multiple episodes of high-P metamorphic mineral growth in the Cycladic eclogite-blueschist belt, Syros, Aegean Sea, Greece[END_REF][START_REF] Tomaschek | Zircons from Syros, Cyclades, Greece -recrystallization and mobilization of zircon during high-pressure metamorphism[END_REF][START_REF] Uunk | Understanding phengite argon closure[END_REF]. This HP-LT metamorphic unit was exhumed during the Eocene within the subduction channel between a top-to-the south thrust at the base and top-to-the east/northeast synorogenic detachment at the top, the Vari Detachment (Figure 2; [START_REF] Augier | Exhumation kinematics of the Cycladic Blueschists unit and backarc extension, insight from the Southern Cyclades (Sikinos and Folegandros Islands, Greece)[END_REF][START_REF] Brun | Exhumation of high-pressure rocks driven by slab rollback[END_REF][START_REF] Huet | Thrust or detachment? Exhumation processes in the Aegean: insight from a field study on Ios (Cyclades, Greece)[END_REF][START_REF] Jolivet | Subduction tectonics and exhumation of high-pressure metamorphic rocks in the Mediterranean orogens[END_REF][START_REF] Laurent | Strain localization in a fossilized subduction channel: Insights from the Cycladic Blueschist Unit (Syros, Greece)[END_REF][START_REF] Ring | The Hellenic subduction system: high-pressure metamorphism, exhumation, normal faulting, and large-scale extension. Annual Review[END_REF][START_REF] Ring | The Hellenic subduction system: high-pressure metamorphism, exhumation, normal faulting, and large-scale extension. Annual Review[END_REF]. In the Cyclades, regional-scale detachments such as the North Cycladic Detachment System (NCDS), the Naxos-Paros Detachment (NPD) or the West Cycladic Detachment System (WCDS) have accommodated back-arc extension (Figure 2; [START_REF] Grasemann | Miocene bivergent crustal extension in the Aegean: Evidence from the western Cyclades (Greece)[END_REF]Jolivet et al., 2010). Eclogite and blueschist of the CBU are best preserved on Syros Island, where the synorogenic Vari Detachment is exposed [START_REF] Laurent | Strain localization in a fossilized subduction channel: Insights from the Cycladic Blueschist Unit (Syros, Greece)[END_REF]Soukis & Stöckli, 2013;Trotet et al., 2001a). | Geology of Syros Located in the Cycladic Archipelago (Figure 2), Syros is mainly composed of metasedimentary and metabasite from the CBU. To the SE of the island, a large-scale klippe of Pelagonian affinity, locally referred as the Vari Unit, is also exposed, limited by the Vari detachment (Figure 3; e.g. Soukis & Stöckli, 2013). [START_REF] Laurent | Strain localization in a fossilized subduction channel: Insights from the Cycladic Blueschist Unit (Syros, Greece)[END_REF] have subdivided the stack of the CBU on Syros in three subunits, delimited by extensional top-to-the east shear zones and characterized by their lithology and predominant metamorphic facies, which are from bottom to top as follows (Figure 3): (1) The Posidonia Subunit is composed of the structurally lower felsic gneiss of Komito overlain by albitic micaschists with intercalated, quite rare, boudins of metabasite and thin marble layers. This subunit has been pervasively overprinted in the greenschist-facies, except in few places where occurrences of blueschist-and eclogite-facies parageneses are seldom observed, (2) The Chroussa Subunit is composed of a lithostratigraphic sequence of alternating micaschists, thick marble layers and metabasites. This subunit is in part overprinted in the greenschist-facies while other places show well preserved eclogite-and blueschist-facies parageneses, (3) The Kampos Subunit is mainly composed of a tectonic mélange of metabasites wrapped in serpentinites and minor metasedimentary rocks. Within this subunit, eclogite-and blueschist-facies parageneses are spectacularly preserved, apparently escaping significant retrogression in the greenschist facies. Eclogite parageneses are therefore recognized within all three subunits [START_REF] Laurent | Strain localization in a fossilized subduction channel: Insights from the Cycladic Blueschist Unit (Syros, Greece)[END_REF], implying that despite their entirely different degrees of retrogression, these three subunits have undergone similar HP-LT metamorphic and peak P-T conditions [START_REF] Trotet | Exhumation of Syros and Sifnos metamorphic rocks (Cyclades, Greece). New constraints on the PT paths[END_REF]. Finally, structurally positioned above the CBU [START_REF] Keiter | Structural development of high-pressure metamorphic rocks on Syros island (Cyclades, Greece)[END_REF][START_REF] Keiter | A new geological map of the Island of Syros (Aegean Sea, Greece): Implications for lithostratigraphy and structural history of the Cycladic Blueschist Unit[END_REF]Soukis & Stöckli, 2013;Trotet et al., 2001a), the Vari Unit is formed by a greenschist mylonitic unit overlain by the orthogneiss of Vari intruding amphibolite-facies metabasite. High-pressure imprinting is not recognized and apparently lacking as in other outcrops of Pelagonian rocks [START_REF] Laurent | Strain localization in a fossilized subduction channel: Insights from the Cycladic Blueschist Unit (Syros, Greece)[END_REF]Soukis & Stöckli, 2013). | METHODS | EPMA compositional mapping X-ray maps have been acquired to measure the compositional variability of the studied minerals at the local scale (from half a mm to a few µm) and to investigate the relationships between compositional zoning and microstructures. This technique has proved to be efficient in the characterization of compositional zoning related to P-T changes [START_REF] Kohn | Retrograde net transfer reaction insurance for pressure-temperature estimates[END_REF][START_REF] Lanari | Diachronous evolution of the alpine continental subduction wedge: evidence from P-T estimates in the Briançonnais Zone houillère (France-Western Alps)[END_REF][START_REF] Lanari | Deciphering high-pressure metamorphism in collisional context using microprobe mapping methods: Application to the Stak eclogitic massif (northwest Himalaya)[END_REF][START_REF] Loury | Late Paleozoic evolution of the South Tien Shan: Insights from P-T estimates and allanite geochronology on retrogressed eclogites (Chatkal range, Kyrgyzstan)[END_REF]. A JEOL JXA-8230 instrument was used at ISTerre (University of Grenoble-Alpes, France) to acquire X-ray compositional maps and spot analyses required for the analytical standardization of the maps [START_REF] De Andrade | Quantification of electron microprobe compositional maps of rock thin sections: an optimized method and examples[END_REF]. Analytical conditions for mapping were 15 keV accelerating voltage, 100 nA specimen current and 15 keV accelerating voltage and 12 nA specimen current for spot analyses. Compositional mapping was carried out with dwell time of 200 ms and a step size (corresponding to the pixel size in the final images) of 2 μm. The X-ray compositional maps were processed using the program XMAPTOOLS 2.2.1 [START_REF] Lanari | XMapTools: A MATLAB\copyright-based program for electron microprobe X-ray image processing and geothermobarometry[END_REF][START_REF] Lanari | Quantitative compositional mapping of Accepted Article This article is protected by copyright. All rights reserved. mineral phases by electron probe micro-analyser[END_REF]. The classification assigned each pixel to a mineral, which was then standardized using the high-resolution spot analyses as internal standards. Each intensity map was converted during this step to a map of oxide weight-percentage composition. Local bulk compositions were generated from the oxide weight-percentage maps by averaging pixels with a density correction (Lanari & Engi, 2017). Finally, the structural formula was calculated for each pixel of the mapped area on the basis of 12 anhydrous oxygen for garnet, 11 for white mica, 12.5 for epidote, 6 for clinopyroxene, 14 for chlorite and 8 for feldspar. Accepted Article This article is protected by copyright. All rights reserved. | Bulk rock composition and reactive bulk composition Bulk rock composition of each sample was measured at the CRPG (Nancy, France) where major element concentrations were obtained by ICP-OES analyses. Results are reported in Table 1. In the literature, most of the P-T isochemical phase diagrams (or pseudosections) rely on the measured bulk rock composition as an approximation of the reactive (or effective) bulk composition, which is defined as the composition of the equilibration volume at a given stage of the rock history. However, it has been demonstrated that this approximation can lead to erroneous P-T estimates if the sample contains zoned minerals (see Lanari & Engi, 2017 for a recent review). In such cases it is critical to restrict the investigation to a more reduced scale where chemical equilibrium may have been established. As the diffusion rates of the major cations in the intergranular medium are largely unknown, a qualitative approximation of the size of the equilibration volume is required. In this study, all the equilibrium models involving Gibbs free energy minimizations were computed using as reference both (1) the bulk rock composition measured by ICP-OES and (2) a local bulk composition derived from thin-section analysis and compositional maps. In both cases, the reference composition is used as the first reactive bulk composition, i.e. to estimate the P-T condition of the first stage. Then, the reactive bulk composition evolves along the P-T trajectory because of garnet fractionation (Konrad-Schmolke, O'Brien, [START_REF] Konrad-Schmolke | Garnet growth at high-and ultra-high pressure conditions and the effect of element fractionation on mineral modes and composition[END_REF][START_REF] Moynihan | An automated method for the calculation of P-T paths from garnet zoning, with application to metapelitic schist from the Kootenay Arc, British Columbia, Canada[END_REF][START_REF] Spear | Metamorphic fractional crystallization and internal metasomatism by diffusional homogenization of zoned garnets[END_REF]. The local bulk compositions were calculated based on the mode of the petro-textural domains (porphyroblasts with their inclusions, mineral-matrix, domains showing late retrogression) observed within the thin section and their local bulk compositions determined with XMAPTOOLS; this is somewhat equivalent to what has been done in other studies [START_REF] Loury | Late Paleozoic evolution of the South Tien Shan: Insights from P-T estimates and allanite geochronology on retrogressed eclogites (Chatkal range, Kyrgyzstan)[END_REF][START_REF] Marmo | Fractionation of bulk rock composition due to porphyroblast growth: effects on eclogite facies mineral equilibria, Pam Peninsula, New Caledonia[END_REF]Warren & Waters, 2006). Mineral abundances were estimated by thin-section optical image analysis. The composition of each local domain is evaluated from the compositional map using XMAPTOOLS and a density correction function. | Thermometric/thermobarometric methods | Raman Spectrometry of Carbonaceous Material (RSCM) geothermometry The RSCM method is an empirical geothermometric approach based on the quantification of the degree of organization of carbonaceous material (CM) reached during metamorphism (Beyssac, Goffé, Chopin, & Rouzaud, 2002a). Due to the irreversible character of graphitization, CM structure is not affected by retrogression reactions and allows metamorphic peak temperature (RSCM-T) to be calculated, using an area ratio (R2 ratio) of different peaks of the Raman spectra (Beyssac et al., 2002a). The reader interested in the graphitization process during HP-LT metamorphism in terms of physico-chemical transformation of CM is referred to [START_REF] Beyssac | Graphitization in a highpressure, low temperature metamorphic gradient: A Raman micro-spectroscopy and HRTEM study[END_REF]. RSCM-T are determined with a calibration-attached accuracy of  50 °C in the range 330-640 °C (Beyssac et al., 2002a). Relative uncertainties on RSCM-T obtained from samples selected within the same section are however much smaller and bracketed to 10-15 °C (e.g. [START_REF] Augier | Exhumation kinematics of the Cycladic Blueschists unit and backarc extension, insight from the Southern Cyclades (Sikinos and Folegandros Islands, Greece)[END_REF][START_REF] Gabalda | Thermal structure of a fossil subduction wedge in the Western Alps[END_REF][START_REF] Brovarone | Stacking and metamorphism of continuous segments of subducted lithosphere in a highpressure wedge: the example of Alpine Corsica (France)[END_REF]. Raman spectra Accepted Article This article is protected by copyright. All rights reserved. were obtained using the Renishaw inVia Reflex system (BRGM-ISTO, Orléans). RSCM analyses were conducted on thin sections prepared on CM-rich metasedimentary rocks cut in the structural X-Z plane. To avoid defects on the CM related to thin-section preparation, analyses were all performed below the surface of the section by focusing the laser beam beneath dominantly quartz and calcite. Between 11 and 25 spectra were recorded in order to bring out the inner structural heterogeneity of CM within samples. Internal dispersion of RSCM-T presents generally unimodal temperature distributions with quite low dispersion. The investigated samples are 19 CM-bearing marble and metapelite samples collected within the three subunits composing the CBU on Syros, ensuring the determination of the large-scale thermal structure of this island (Figure 3). | Thermodynamic modelling The P-T evolution of the metamorphic rocks was reconstructed based on forward thermodynamic models. As garnet growth strongly fractionates the reactive bulk composition, the program GRTMOD (Lanari et al. 2017) was used to retrieve the P-T information stored in garnet compositional zoning. GRTMOD searches the optimal P-T conditions for the composition of each successive growth zone using a fractional crystallization model and with possible resorption of the previous zones (a complete description is provided in Lanari et al. 2017). As the maximum temperature reached by the sample is ~ 550 °C (see below), intragranular diffusion is not expected to have significantly altered the compositional zoning of the large porphyroblast that is interpreted as growth zoning [START_REF] Caddick | Preservation of garnet growth zoning and the duration of prograde metamorphism[END_REF]. As the compositions of the co-existing phases such as omphacite and phengite are less sensitive to changes in the reactive bulk composition (e.g. [START_REF] Airaghi | Microstructural vs compositional preservation and pseudomorphic replacement of muscovite in deformed metapelites from the Longmen Shan (Sichuan, China)[END_REF] for phengite), the bulk rock composition or local bulk composition was used to generate the P-T diagrams showing mineral isopleths. The Gibbs free energy minimizations and the isochemical phase diagrams were computed using the program Theriak-Domino (de [START_REF] Capitani | The computation of chemical equilibrium in complex systems containing non-ideal solutions[END_REF][START_REF] De Capitani | The computation of equilibrium assemblage diagrams with Theriak/Domino software[END_REF]) in the chemical system (±MnO)-Na 2 O-CaO-K 2 O-FeO-Fe 2 O 3 -MgO-Al 2 O 3 -SiO 2 -TiO 2 -H 2 O. The internally consistent thermodynamic database JUN92.bs [START_REF] Berman | Internally-consistent thermodynamic data for minerals in the system Na 2 O-K 2 O-CaO-MgO-FeO-Fe 2 O 3 -Al 2 O 3 -SiO 2 -TiO 2 -H 2 O-CO 2[END_REF] and subsequent updates) was used for modelling. A comparison of the results obtained with other thermodynamic databases is shown in Figure S1. Four models were systematically derived for each sample, using the bulk rock composition (WR) and the local bulk compositions (LB), in both either considering or not MnO. | PETROGRAPHY AND MINERAL CHEMISTRY A petrographic description of analysed samples based on the results of compositional mapping is presented here. Four samples, characteristic of the entire CBU from top to base, were selected for this study (from a collection of 138). Two were collected in the Kampos Subunit to determine burial and peak-pressure conditions. Additionally, one sample was selected in Chroussa Subunit to characterize the transition from blueschist-to greenschist-facies P-T conditions and one sample preserving HP parageneses was collected from Posidonia Subunit to compare the maximum pressure conditions recorded in this subunit and in the Kampos Subunit. Typical textures and mineral assemblages of the four samples are shown in Figure 4. All mineral abbreviations are after Whitney and Evans (2010). Accepted Article This article is protected by copyright. All rights reserved. | Sample SY1401 (Kampos Subunit) | Outcrop and sample description The studied outcrop is located south of Syros airport and shows a 50 m-long retrogression gradient from well-preserved HP rocks, including very fresh eclogite, to strongly overprinted HP rocks in the greenschist-facies (Figures 3 and5). This apparent metamorphic transition is accompanied by an increasing gradient of deformation toward the more retrogressed rocks (Figure 5a). This outcrop corresponds to the contact zone between the Kampos and Chroussa subunits, which is interpreted as an extensional top-to-the east shear zone, namely the Kastri Shear Zone (see Laurent et al., 2016 for details). Well-preserved eclogite shows a weak deformation and still exhibits the original structure of a pillow-lava breccia (Figure 5b-d). While only a slight foliation is observed, a lineation oriented N90 °E is marked by intense stretching (Figure 5c). Markers of asymmetric ductile deformation, such as shear bands, are rare and visible only at the microscopic scale, showing an incipient top-to-the east sense of shear (Figure 6a). The eclogitic metapillow-lava breccia studied here is typical of rocks from the Kampos Subunit with only weak subsequent overprinting. In this sense, this sample appears as one of the best candidates to determine peak-pressure conditions and possibly prograde P-T conditions of the CBU on Syros. The metamorphic assemblage of this sample includes garnet, white mica, epidote, clinopyroxene and less abundant quartz, albite, apatite, rutile and sphene (Figures 4 and6). Garnet occurs as 0.5-1.5 mm porphyroblasts and contains inclusions of epidote, white mica, clinopyroxene, albite and quartz (Figures 4 and6). | Petrography Chemical composition mapping of 400,000 pixels covers an area of 1600*1000 μm and was acquired around a 1.3 mm diameter garnet showing a large diversity of mineral inclusions (Figure 6). This garnet is comprised within an eclogitic matrix and surrounded by a pressure shadow, dividing the Xray map into three distinct petro-textural domains (Figure 6d). The eclogitic matrix is mainly composed of omphacite, epidote and white mica with less abundant quartz, apatite and sphene (Figure 6b,c). The pressure-shadow on the garnet is mainly composed of white mica associated with albite (Figure 6). Compositional zoning is observed in garnet porphyroblasts with, from core to rim, a decrease of almandine and spessartine (XAlm 0.650.57 and XSps 0.0450.02 ) and an increase of grossular and pyrope (Figure 7; XGrs 0.240.30 and XPrp 0.070.11 ). The rim composition of garnet is also affected by the presence of phengite inclusions with a relatively higher grossular, pyrope and spessartine content that evidences garnet re-equilibration around them (Figure 7). Accepted Article This article is protected by copyright. All rights reserved. The composition of white mica shows a more complex pattern with four distinct compositional groups based on the Si and Na contents (Figure 7). Within the omphacitic matrix and in the pressure-shadow, phengite shows a strong increase in celadonite from core to rim and at the contact with the garnet porphyroblast (Figure 7; phengite 1: XCel =0.35-0.45 and phengite 2: XCel =0.45- 0.65 ). Phengite 1 is also observed as an inclusion in the garnet rim. In contrast, Si-poor phengite 3 and phengite-paragonite mixing compositions are only observed as inclusions in garnet cores (Figure 7). In the eclogitic matrix, clinopyroxene corresponds to omphacite, with some grains showing a compositional zoning marked by an increase of the jadeite-content from core to rim (Al_M in Figure 7; XJd 0.360.46 ). In contrast, inclusions of clinopyroxene observed in the core of the garnet have lower jadeite content (XJd =0.30-0.35 ) similar to the core of some grains of the matrix (Figure 7). Two distinct compositions of epidote are observed in the map with higher zoisite-content for the grains included into the garnet (XZo =0.60-0.75 ) than those of the omphacitic matrix (XZo =0.30-0.45 ). Plagioclase has homogeneous and nearly pure albitic composition (XAb =0.99 ) and is found both included in the core of the garnet and in the pressure-shadow where it has grown into the cleavage of phengite and around a few grains of remnant omphacite and quartz. Finally, less abundant rutile is observed in some garnet rims, while apatite and sphene accessory minerals are only observed in the omphacitic matrix (Figure 6). | Petrographic interpretations The compositional zoning preserved in garnet, phengite and omphacite can be used in a qualitative way to reconstruct the P-T history of this sample. Several lines of evidence suggest the prograde growth of garnet under increasing pressure conditions, (1) plagioclase inclusions are only observed in the core of garnet, suggesting that the transformation of plagioclase into omphacite was not complete at the initiation of garnet growth, (2) the chemical composition of omphacite and phengite inclusions in the core of garnet have respectively lower XJd and XCel content than inclusions in the rim, which again suggests an increase in pressure during garnet growth, (3) the compositional zoning in both omphacite and phengite grains in the matrix shows a similar trend (e.g. increase in XJd and XCel, respectively) and also supports the record of a prograde trajectory. The only evidence of retrogression observed in this sample is the presence of albite in the pressure shadow. Albite occurs only in omphacite-poor domains and is interpreted as a late feature, formed at the expense of omphacite and quartz. This interpretation is supported by observations of remnant quartz and omphacite grains in the pressure shadow that are systematically surrounded by newly grown albite (Figure 6). | Sample SY1460 (Kampos Subunit) | Outcrop and sample description This sample was collected in the Kampos Subunit on the south coast of Syros, and is structurally positioned just below the Vari Detachment (Figure 3; [START_REF] Laurent | Strain localization in a fossilized subduction channel: Insights from the Cycladic Blueschist Unit (Syros, Greece)[END_REF]Soukis & Stöckli, 2013;Trotet et al., 2001a). This outcrop is composed of metabasite preserving eclogite-and blueschistfacies parageneses that are ductilely sheared with syn-blueschist top-to-the east kinematics (Laurent et al., 2016). This sample corresponds to a foliated and stretched boudin of eclogite hosted in blueschist- Accepted Article This article is protected by copyright. All rights reserved. facies rocks. Textural analysis shows that a matrix composed of omphacite and white mica with minor calcite, sphene and epidote formed ~ 95 vol% of the sample volume and comprised a few large garnet (0.5-1 mm) and glaucophane (5-10 mm) crystals (Figures 4 and8). | Petrography The X-ray map of 204,750 pixels covering an area of 910*900 μm was acquired around a 0.9 mm diameter garnet of almandine type showing inclusions of clinopyroxene and rutile and a core to rim zonation (Figure 9). Almandine content decreases toward the rim (XAlm 0.660.59 ) while grossular and pyrope content show a general increase (XGrs 0.240.28 and XPrp 0.060.09 ). The spessartine content displays a more complex, oscillatory zoning pattern (XSps 0.0420.058 ; Figure 9). Two compositional groups of white mica are observed in this sample (phengite 1 and 2, Figure 9). Phengite 1 is characterized by a relatively high-celadonite (XCel =0.60-0.73 ) and occurs as fine grain cores surrounded by phengite 2 rims, which has lower celadonite (XCel =0.45-0.53 ). Most of the matrix grains show phengite 2 composition (Figure 9). Omphacite can be divided into two compositional groups. Omphacite 1 has lower jadeite and higher diopside content (XJd =0.36-0.40 , XDi =0.35-0.38 ) than omphacite 2 (XJd =0.46-0.54 , XDi =0.27-0.32 ), and is observed in the core of the omphacite 2 in both matrix and garnet (Figure 9). The composition of glaucophane is homogeneous (XGln =0.95- 0.99 ) and only a few epidote grains are observed in the matrix with a few allanitic cores (Figure 8). | Petrographic interpretations Similar to sample SY1401, prograde P-T conditions are suggested by the compositional zoning preserved in garnet and omphacite. However, in this case the compositional zoning of phengite shows a different record, with lower XCel contents observed at the rim of garnet grains. This feature suggests later re-equilibration of phengite grains in the mineral matrix under lower pressure conditions and possibly higher temperatures. Oscillatory zoning in garnet may reflect variations in the garnet-forming reaction or transportcontrolled growth [START_REF] Kohn | Geochemical Zoning in Metamorphic Minerals A2[END_REF]. It is an open question if the assumption of chemical equilibrium used in the thermodynamic models is valid or not for this specific case. It is important to note here that only Mn is affected by oscillatory zoning and to a relatively low extent (between XSps 0.045 and XSps 0.06 ) that would not necessarily be detected without using quantitative maps. If the equilibrium volume for Mn was smaller than the equilibration volume of the other major elements when garnet grew, then the Mn component needs to be excluded from the thermodynamic computations. Accepted Article This article is protected by copyright. All rights reserved. | Sample SY1407 (Chroussa Subunit) | Outcrop and sample description The outcrop is located in the southeast of the island within the Chroussa Subunit (Figure 3). It is mainly composed of albite-epidote-blueschist-facies (AEBS metamorphic facies of [START_REF] Evans | Phase relations of epidote-blueschists[END_REF] metasedimentary rock, hosting metabasite boudins in which eclogite and blueschist parageneses are preserved. This sample corresponds to an AEBS-facies metasedimentary rock showing a clear lineation oriented N70 °E marked by the stretching of glaucophane and albite. Textural analysis shows that this sample is composed of a matrix of quartz, white mica, glaucophane, albite and epidote with minor chlorite, rutile, and apatite in which large garnet occurs (0.5-1.5 mm, Figures 4 and10). | Petrography The compositional mapping of 288,750 pixels covers an area of 1100*1050 μm and was acquired around a 1.5 mm diameter garnet comprising mainly quartz inclusions (Figure 10). The composition of the garnet porphyroblast changes from core to rim (Figure 11). The core shows comparatively low almandine, spessartine and pyrope (XAlm =0.61-0.62 , XSps =0.075-0.085 , XPrp =0.038-0.04 ) and high grossular (XGrs =0.24-0.26 ). A decrease of both almandine and spessartine is observed through the mantle to rim (XAlm 0.660.59 , XSps =0.120.04 ) coupled to an increase of pyrope and grossular (XPrp 0.0460.061 , XGrs 0.170.28 ). The composition of white mica corresponds in a large majority to paragonite, with only a few phengite grains observed as inclusions in garnet and single grains in the mineral-matrix (Figure 11). Glaucophane grains, observed in the matrix, show textural equilibrium relationships with albite and epidote (Figure 10). | Petrographic interpretations In this sample, compositional zoning is only observed in garnet but with a more complex pattern than in samples SY1401 and SY1460. The texture and compositional zoning of the garnet core differs from the mantle and rim and is interpreted as a relict preserved from either the protolith or as a first growth stage followed by intense resorption. As rocks of the CBU are all characterized by a single metamorphic history, the garnet core is more likely the result of a first stage of garnet growth followed by intense garnet resorption before a new stage of garnet growth. The lack of index mineral inclusions in the core (most of the inclusions are quartz) does not allow the coexisting assemblage to be determined. The presence of albite inclusions in the garnet rim suggests that the second growth episode took place at moderate pressure conditions, possibly during exhumation. This is consistent with fine-grained phengite inclusions in the garnet rim that show intermediate Si-content and with the presence of glaucophane, epidote and albite in textural equilibrium in the matrix. Similar features have been interpreted as strong evidence of retrogression in the albite-epidote-blueschist facies [START_REF] Evans | Phase relations of epidote-blueschists[END_REF]. Following these observations, compositional zoning of the garnet in SY1407 may be interpreted as a first event of garnet growth, followed by an intense stage of resorption and a second growth episode that took place after partial exhumation of the rock. Accepted Article This article is protected by copyright. All rights reserved. | Sample SY1418 (Posidonia Subunit) | Outcrop and sample description This sample was collected in the SW of Syros, in the Posidonia Subunit, where few metabasite lenses occur within the gneiss of Komito (Figure 3). Metabasite appears pervasively retrograded under greenschist-facies conditions. However, [START_REF] Laurent | Strain localization in a fossilized subduction channel: Insights from the Cycladic Blueschist Unit (Syros, Greece)[END_REF] have shown that within metre-scale metabasite boudins, HP relicts are preserved with the occurrence of eclogite-and blueschist-facies parageneses. Sample SY1418 corresponds to one of these preserved HP metabasite occurrences of the Posidonia Subunit and has been selected to estimate maximum P-T conditions undergone by the Posidonia Subunit. Textural analysis shows that this sample is formed by a matrix of fine-grained glaucophane and omphacite with minor rutile, Fe-oxide, white mica, epidote and albite hosting porphyroblasts of garnet, glaucophane, quartz, calcite and chlorite (Figures 4 and12). Pressure shadows are composed of quartz, phengite, epidote and glaucophane with minor albite and represent 10 vol% of the sample (Figure 12). | Petrography The compositional mapping of 160,801 pixels covers an area of 802*802 μm and was acquired around a 0.6 mm diameter garnet showing numerous inclusions of quartz, epidote and Fe-oxide (Figure 12). This garnet is compositionally zoned with increasing almandine and pyrope (XAlm 0.620.70 , XPrp 0.0380.09 ) and decreasing spessartine and grossular toward the rim (XSps 0.080.004 , XGrs 0.240.17 ; Figure 13). The composition of white mica is homogeneous and corresponds to phengite (XCel= 0.40 ). The minor presence of omphacite is also observed as fine grains in the main glaucophane matrix (Figure 12). | Petrographic interpretations Omphacite compositions suggest that this sample has reached eclogite-facies conditions. Then, it has recorded a strong retrogression as attested by the practically complete destabilisation of omphacite (+quartz) into albite. The chlorite-albite assemblage is characteristic of low-pressure conditions. Compositional zoning is again restricted to garnet and does not provide a clear picture of the recorded P-T history. However, the higher proportion of garnet in this sample (~ 35 vol%; Figure 12) suggests crystallization under HP conditions with absence of significant resorption. | THERMOBAROMETRY RESULTS | RSCM geothermometry RSCM geothermometry has been applied to 19 samples distributed all over the island (Figure 3). Detailed results, including R2 ratio, number of spectra, RSCM-T and standard deviation are presented in Table 2. In addition, measured RSCM temperatures are all reported on the metamorphic map of Syros (Figure 3). Maximum temperatures recorded in the three subunits of Syros yielded very similar RSCM-T. This temperature is then compared with peak temperature estimates obtained using thermodynamic modelling. Accepted Article This article is protected by copyright. All rights reserved. Results show that RSCM-T range from 489 to 564 °C in the Posidonia Subunit, 485 to 581 °C in the Chroussa Subunit and 510 to 561 °C in the Kampos Subunit. On average, RSCM-T are equivalent within error in each subunit with 537 ± 20 °C in the Posidonia Subunit, 540 ± 19 °C in the Chroussa Subunit and 530 ± 17 °C in the Kampos Subunit (Table 2). Moreover, the maximum RSCM-T measured in each subunit is also equivalent within error, showing that these three subunits have experienced quite similar peak temperatures during their metamorphic history. However, the RSCM-T measured within a single subunit varies up to 100 °C, as observed in the Posidonia and Chroussa subunits (Table 2). There is no apparent correlation between the internal RSCM-T variations and the structural positions of the sample. In some cases, varying RSCM-T were estimated in samples collected at the same structural level, within a single and consistent lithology (Figure 3). The significance of partial discrepancies in the temperatures is beyond the scope of this study. Such variations are generally explained by the presence of an inherited component of CM (Beyssac et al., 2002a) or to the concentration of structural defects caused by pervasive deformation (see [START_REF] Aoya | Extending the applicability of the Raman carbonaceous-material geothermometer using data from contact metamorphic rocks[END_REF] for further discussions). | Empirical thermobarometry The garnet-omphacite thermometer of [START_REF] Ravna | The garnet-clinopyroxene Fe 2+ -Mg geothermometer: an updated calibration[END_REF] was used to estimate the temperature of garnet and co-existing omphacite, whereas the pressure information was extracted from the assemblage garnet-omphacite-phengite (Waters & Martin, 1996). The P-T maps have been calculated in XMAPTOOLS for samples SY1401 and SY1460, following the strategy described in [START_REF] Lanari | Deciphering high-pressure metamorphism in collisional context using microprobe mapping methods: Application to the Stak eclogitic massif (northwest Himalaya)[END_REF][START_REF] Lanari | XMapTools: A MATLAB\copyright-based program for electron microprobe X-ray image processing and geothermobarometry[END_REF]. In sample SY1401, petrographic observations suggest that garnet records prograde to peakpressure conditions. In order to derive maximum pressure of garnet growth, the rim composition of garnet and the phengite 1 composition of white mica were used (Figure 7, Tables 3 and4), as our petrographic observations suggest that they grew in equilibrium during the same P-T stage. Results predict peak P-T conditions for garnet growth between 22-24 kbar and 500-560 °C (Figure 14). In sample SY1460, omphacite and phengite are also observed in textural equilibrium with the rim composition of the garnet (Figure 9). The phengite 2 composition has been used for thermobarometry and estimated P-T conditions range from 19 to 21 kbar and 500 to 560 °C for the growth of the garnet rim (Figure 14). As a preliminary conclusion, results of empirical thermobarometry using compositional maps suggest maximum pressure and temperature for the garnet growth of the CBU of Syros at 22  2 kbar and 530  30 °C. Note that the assumption of chemical equilibrium between the different groups of garnet, phengite and omphacite are based on the petrographic observation and tested in the following using Gibbs free energy minimization. Accepted Article This article is protected by copyright. All rights reserved. | Garnet equilibrium modelling GRTMOD computations were performed on samples SY1401, SY1460, SY1407 and SY1418 using several compositions (WR and LB, Table 1) including or excluding MnO and using the JUN92.bs thermodynamic database (see the methods, section 3.3.2). The results are reported in P-T diagrams (Figure 15) with error bars that represent the uncertainty in the optimal P-T conditions resulting from the uncertainties in chemical analyses, taken as the spacing between the garnet isopleths (see Lanari et al., 2017). The residual values (C o in the original publication, Lanari et al., 2017) represent the quality of the isopleths intersection, i.e. the quality between the modelled and observed garnet compositions. A value of C o < 0.04 ensures a good fit of the model and the fit is excellent where C o < 0.01. Isopleths of Si-content in phengite and Al-content in omphacite were calculated using the local bulk composition and are shown in the P-T diagrams of samples SY1401 and SY1460 (Figure 15). In sample SY1401, predictions fall in a narrow P-T range using the LB and considering MnO or not (see the discussion below, section 6.1.2). A first event of garnet growth is constrained at 16-18 kbar and 450-500 °C (core) and a second event is suggested at 22-24 kbar and 550-580 °C using the LB composition (rim, Figure 15). For this model, 20 vol% of garnet is produced (core 2 vol% and rim 18 vol%; Figure 15). In sample SY1460, garnet P-T predictions can only be made using the LB composition and residual values are systematically high for all the models (above >0.04) suggesting either a smaller equilibrium volume, or kinetic phenomena (see discussion). In this model ~9 vol% of garnet is produced (2 vol% core and 7 vol% rim). In sample SY1407, the best defined models were obtained using the LB composition and suggest garnet growth for core, mantle and rim at lower pressure conditions of 10-12 kbar at 500-560 °C. A less well defined but still acceptable model obtained using the WR and including MnO suggest garnet growth for core and mantle at 18-20 kbar and 470-500 °C (dark blue squares in Figure 15). In this last case, 2.5 vol% of garnet is produced (core ~0.13 vol%, mantle 0.57 vol% and rim 1.85 vol%). Finally, in sample SY1418, the only garnet P-T predictions that are characterized by low residuals suggest homogeneous P-T conditions for core and rim around 18  2 kbar and 450  50 °C (Figure 15). In this case, 12 vol% of garnet is produced (core 0.12 vol% and rim 11.8 vol%). The models including MnO and with a garnet rim predicted at higher pressure show much higher garnet mode of 33 vol% (core 12 vol% and rim 21vol%, Figure 15). The main result of these models is that three main growth events of garnet are partially recorded in the four investigated samples (Figure 15). First, a prograde pulse of garnet growth is predicted from the core garnet composition of SY1401 and SY1418 at 17  2 kbar and 450  50 °C (Figure 15). Then, the maximum pressure conditions recorded by garnet are estimated at 22-24 kbar and 550-580 °C from the rim garnet composition of SY1401 (circles in Figure 15). While these estimates are only retrieved in one sample, the residual values show a good fit of the model, and these conditions are consistent with maximum pressure conditions retrieved from the Si-content in phengite for SY1401 and from the Al-content in omphacite for SY1401 and SY1460 samples (Figures 7,9 and 15). Finally, a last event of garnet growth is recorded in SY1407. This retrograde garnet growth event is well constrained between 9-12 kbar and 500-570 °C, from both core, mantle and rim compositions of the garnet (Figure 15). P-T isochemical phase diagrams Results obtained using garnet equilibrium modelling were compared with P-T isochemical phase diagrams to test mutual consistency between predicted and observed mineral assemblages in sample SY1401 (Figure 16). Two phase diagrams using the LB and WR composition were calculated at P-T conditions encompassing the first stage of core garnet crystallization determined by equilibrium modelling at 16-18 kbar and 450-500 °C (Figures 15 and16). Two additional isochemical phase diagrams were then elaborated for P-T conditions surrounding the second stage of rim garnet crystallization constrained at 22-24 kbar and 550-580 °C (Figure 16). To calculate these diagrams, LB and WR compositions were modified to account for fractionation of garnet that crystallized during stage 1. For the first stage of garnet growth (core), the predicted assemblages are consistent with the minerals trapped as inclusions in garnet cores for both LB and WR compositions (Figure 16). Phengite, paragonite, omphacite and epidote are all predicted to be in equilibrium and observed in the garnet core (Figures 6 and7), while rutile and quartz are not observed in the garnet porphyroblast that has been mapped but in other garnet cores of the same sample. Only chlorite is predicted in equilibrium but has not been clearly identified within the garnet core, consistent with a garnetforming reaction involving chlorite. Furthermore, on the compositional map of SY1401, albite is observed within the core of the garnet but not predicted in equilibrium for these P-T conditions. Albite was not completely destabilized when the garnet nucleated, suggesting significant P-T overstepping with incomplete re-equilibration of the mineral matrix. If the matrix was only partially re-equilibrated at the time garnet nucleated, the inclusion phases in the garnet core do not reflect the equilibrium assemblage from which this garnet nucleated. In this scenario, the deviation from an equilibrium model cannot be approximated using an OS model (e.g. [START_REF] Castro | Reaction overstepping and re-evaluation of peak P-T conditions of the blueschist unit on Sifnos, Greece: implications for the Cyclades subduction zone[END_REF][START_REF] Spear | Overstepping the garnet isograd: a comparison of QuiG barometry and thermodynamic modeling[END_REF]. In this study, the P-T conditions of garnet nucleation for three samples showing different matrix assemblages fall within a narrow P-T window of <30 °C and <2 kbar. In the OS model, the garnet composition is fully controlled by the Gibbs free energy of the matrix assemblage (assuming equilibrium). Different matrix assemblagesfor different bulk rock compositionsare likely to produce different degrees of overstepping (as reported by [START_REF] Castro | Reaction overstepping and re-evaluation of peak P-T conditions of the blueschist unit on Sifnos, Greece: implications for the Cyclades subduction zone[END_REF], which are not observed in this study. For the second stage of garnet (rim), the two models produce different results as shown by the GRTMOD results (Figures 15 and16). For the modified LB composition, the results show an excellent correlation between the predicted assemblage in equilibrium and petrographic observations with phengite, omphacite, epidote and quartz that are all observed as inclusions near the garnet rim and/or within the matrix (Figures 6 and7). Rutile is not present in the mapped area but is commonly observed as an inclusion in the rim of other porphyroblasts of the same sample. Additionally, results obtained using the fractionated WR composition confirm that the calculated pressure prediction for the growth episode of the garnet rim is too high, as coesite is predicted above 25 kbar and is not identified within the CBU (Figure 16). Accepted Article This article is protected by copyright. All rights reserved. | DISCUSSION AND CONCLUSIONS While Syros Island is a world-class reference for the preservation of HP-LT metamorphic rocks such as eclogite and blueschist, the P-T path of the CBU in this island is not completely understood (Figure 1), hindering our global understanding of exhumation processes and deep subduction dynamics. This study complements results obtained in previous thermobarometric studies conducted in the CBU [START_REF] Ashley | Geothermobarometric history of subduction recorded by quartz inclusions in garnet[END_REF][START_REF] Dragovic | Pulsed dehydration and garnet growth during subduction revealed by zoned garnet geochronology and thermodynamic modeling, Sifnos, Greece[END_REF][START_REF] Dragovic | Using garnet to constrain the duration and rate of water-releasing metamorphic reactions during subduction: An example from Sifnos, Greece[END_REF][START_REF] Groppo | Glaucophane schists and associated rocks from Sifnos (Cyclades, Greece): New constraints on the P-T evolution from oxidized systems[END_REF][START_REF] Huet | Coupled phengite 40 Ar-39 Ar geochronology and thermobarometry: P-T-t evolution of Andros Island (Cyclades, Greece)[END_REF][START_REF] Parra | Relation between the intensity of deformation and retrogression in blueschist metapelites of Tinos Island (Greece) evidenced by chlorite-mica local equilibria[END_REF][START_REF] Schumacher | Glaucophane-bearing marbles on Syros, Greece[END_REF][START_REF] Trotet | Exhumation of Syros and Sifnos metamorphic rocks (Cyclades, Greece). New constraints on the PT paths[END_REF]. Our new P-T estimates for the CBU on Syros highlight a two-step exhumation history in the subduction channel, resembling ones previously retrieved on Tinos and Andros islands [START_REF] Huet | Coupled phengite 40 Ar-39 Ar geochronology and thermobarometry: P-T-t evolution of Andros Island (Cyclades, Greece)[END_REF][START_REF] Parra | Relation between the intensity of deformation and retrogression in blueschist metapelites of Tinos Island (Greece) evidenced by chlorite-mica local equilibria[END_REF] but which have not been described on Syros thus far. | P-T path and comparison of the methods | P-T path of the CBU on Syros In this study, different independent thermobarometric methods such as RSCM peak-temperatures, empirical thermobarometry and thermodynamic modelling have been used to reconstruct the P-T path of the CBU on Syros (Figure 17). The most important result of our study is the good agreement between the different methods used in different samples, which allows the P-T evolution of the CBU to be accurately determined (Figure 17). All methods used in this study provide results in line with the petrological observations. Peak metamorphic temperatures measured with the RSCM method are in good agreement with maximum temperatures estimated for both empirical and garnet thermobarometry at 530  30 °C (Figure 17). Additionally, minimum peak metamorphic P-T conditions obtained with empirical thermobarometry (22-24 kbar for 500-560 °C and 19-21 kbar for 500-560 °C) fall in the same range of maximum P-T conditions calculated with GRTMOD (22-24 kbar and 550-580 °C, Figure 17). The retrograde path is constrained through an event of garnet growth occurring at 10-12 kbar and 500-570 °C that suggests a period of relatively isobaric heating. This late event is only recorded in retrogressed samples from the basal and mid-part of the CBU on Syros. RSCM-T estimates provide also an upper limit to the maximum amount of heating during this period. The P-T trajectory is consequently well constrained, at least for the prograde to peak part and the retrograde event (Figure 17). | Bulk composition and kinetic sensitivity in equilibrium models As the isochemical equilibrium phase diagrams and isopleth positions critically depend on the bulk composition used for modelling, WR compositions and LB composition extracted from compositional maps were measured. Both compositions were systematically tested, considering or not MnO (Figure 15). While MnO has a direct impact on the modal abundance of garnet produced, the models computed either with WR or LB and assuming a MnO-free system generally have lower residue values (or at least similar, Figure 15). A significant amount of MnO is measured in all garnet cores and in the mantle composition of the SY1407 garnet (XSps 0.06-0.11 ; Table 3). Our results show relative mutual consistency between WR and LB compositions without MnO (light blue colour in Figure 15) that yield close predictions in samples SY1401, SY1407 and SY1418. When considering MnO, Accepted Article This article is protected by copyright. All rights reserved. results show more variations between the predicted garnet P-T equilibrium using WR or LB compositions as shown in samples SY1407 and SY1418, demonstrating that considering MnO in the models is critical in this study. Note that a significant higher fraction of garnet is predicted for models including MnO in sample SY1418 (from 12 to 33 vol%). To compare the observed and predicted modes of garnet, it is necessary to include MnO, as garnet is the main carrier of Mn in these rocks (see also [START_REF] Tinkham | Metapelite phase equilibria modeling in MnNCKFMASH: the effect of variable Al2O3 and MgO/(MgO+ FeO) on mineral stability[END_REF]. Moreover, this work highlights the powerful use of LB composition calculated from the composition of local assemblages identified on compositional maps as an approximation of reactive bulk compositions for thermodynamic modelling. Because of the local heterogeneities of metamorphic rocks and the compositional zoning of the rock-forming minerals, the measured wholerock composition is not always adapted for modelling (e.g. [START_REF] Lanari | Deciphering high-pressure metamorphism in collisional context using microprobe mapping methods: Application to the Stak eclogitic massif (northwest Himalaya)[END_REF][START_REF] Loury | Late Paleozoic evolution of the South Tien Shan: Insights from P-T estimates and allanite geochronology on retrogressed eclogites (Chatkal range, Kyrgyzstan)[END_REF]Warren & Waters, 2006). In this study, several examples suggest that the models based on LB provide better results (lower residuals for similar P-T conditions). This result is in line with small equilibration volumes, which would, for example, be smaller than hand specimen sample used to estimate the bulk rock composition. The quantitative mapping strategy employed in this study helps to ensure that the composition of each textural-domain is accurately determined (Lanari & Engi, 2017;[START_REF] Lanari | Quantitative compositional mapping of Accepted Article This article is protected by copyright. All rights reserved. mineral phases by electron probe micro-analyser[END_REF]. This would not be the case by using only qualitative maps and EPMA spot analyses. Sample SY-14-60 is an interesting case study as the compositional maps show evidence of transport-controlled growth with the oscillatory zoning of MnO (Figure 9). In this case, high residuals are obtained in GRTMOD (Figure 15), suggesting that the part of the rock considered in the models never reached whole-scale chemical equilibrium during the entire P-T cycle. Other potential cause for MnO oscillatory zoning in the garnet, such as externally-derived fluid, is not likely due to the very low solubility of Mn in the fluid at HP conditions. All other samples show compositional zoning more typical of growth zoning suggesting equilibrium-controlled growth (Figures 7,11 and 13). In this case, the successive compositions of garnet can be related to differences in P-T conditions and the models have lower residuals (Figure 15). | P-T evolution of the CBU In light of our results, the shape of the P-T path of the CBU on Syros is constrained by prograde, peak and retrograde garnet growth stages, maximum RSCM temperature and peak pressure crystallization of the garnet-omphacite-phengite assemblage (Figure 17). Preservation of prograde mineral parageneses constraining the burial P-T conditions of HP-LT metamorphic rocks is rare and often preserved only in garnet cores. In our study, late burial P-T conditions of the CBU in the subduction zone is constrained by a garnet growth stage that was recorded in three samples at 17  2 kbar and 450  50 °C (Figures 15 and17). On Sifnos, the other Cycladic island where HP-LT eclogites and blueschists are well preserved, prograde garnet growth event has been also retrieved at ~ 20 kbar and 450-500 °C [START_REF] Dragovic | Using garnet to constrain the duration and rate of water-releasing metamorphic reactions during subduction: An example from Sifnos, Greece[END_REF][START_REF] Groppo | Glaucophane schists and associated rocks from Sifnos (Cyclades, Greece): New constraints on the P-T evolution from oxidized systems[END_REF]. The timing of this early growth of garnet is well constrained at 53.4  2.6 Ma [START_REF] Dragovic | Pulsed dehydration and garnet growth during subduction revealed by zoned garnet geochronology and thermodynamic modeling, Sifnos, Greece[END_REF]. Accepted Article This article is protected by copyright. All rights reserved. We have obtained consistent peak metamorphic P-T conditions from the three subunits of the CBU using independent thermobarometric methods at 22  2 kbar and 530  30 °C (Figure 17). These conditions differ significantly from the results of [START_REF] Schumacher | Glaucophane-bearing marbles on Syros, Greece[END_REF] who proposed lower peak metamorphic conditions at 15-16 kbar and 500 °C (Figure 1). Our results are thus in line with peak P-T conditions proposed on Syros by [START_REF] Trotet | Exhumation of Syros and Sifnos metamorphic rocks (Cyclades, Greece). New constraints on the PT paths[END_REF] and with other recent studies conducted on Sifnos [START_REF] Ashley | Geothermobarometric history of subduction recorded by quartz inclusions in garnet[END_REF][START_REF] Dragovic | Using garnet to constrain the duration and rate of water-releasing metamorphic reactions during subduction: An example from Sifnos, Greece[END_REF][START_REF] Dragovic | Pulsed dehydration and garnet growth during subduction revealed by zoned garnet geochronology and thermodynamic modeling, Sifnos, Greece[END_REF][START_REF] Groppo | Glaucophane schists and associated rocks from Sifnos (Cyclades, Greece): New constraints on the P-T evolution from oxidized systems[END_REF]. The shape of the retrograde P-T path is characterized by a two-step exhumation history delimited by a phase of isobaric heating for the two lowermost subunits, Chroussa and Posidonia (Figure 17). The first step of exhumation was achieved under HP-LT conditions for all units, permitting the preservation of eclogite-and blueschist-facies parageneses. Such shape of the P-T path for early exhumation following peak P-T conditions is also retrieved on Sifnos Island [START_REF] Groppo | Glaucophane schists and associated rocks from Sifnos (Cyclades, Greece): New constraints on the P-T evolution from oxidized systems[END_REF]. Moreover, this retrograde evolution of the CBU on Syros is close to that proposed by [START_REF] Lister | White mica 40 Ar/ 39 Ar age spectra and the timing of multiple episodes of high-P metamorphic mineral growth in the Cycladic eclogite-blueschist belt, Syros, Aegean Sea, Greece[END_REF], except for their unconstrained retrograde P-T loops (Figures 1 and17). After this first step of exhumation, a last garnet growth event was recorded at lower P-T conditions (Figures 15 and17). In our interpretation, this retrograde garnet growth event is related to a phase of relatively isobaric heating at 10-12 kbar and 500-570 °C (Figure 17). Because samples of the Kampos Subunit are not very retrogressed, we suggest that rocks of this subunit have not been affected by this heating and have continued to be exhumed under HP-LT conditions. Recently proposed (but not constrained) for the CBU on Syros by [START_REF] Lister | White mica 40 Ar/ 39 Ar age spectra and the timing of multiple episodes of high-P metamorphic mineral growth in the Cycladic eclogite-blueschist belt, Syros, Aegean Sea, Greece[END_REF], this heating phase was previously retrieved on Tinos and Andros islands [START_REF] Huet | Coupled phengite 40 Ar-39 Ar geochronology and thermobarometry: P-T-t evolution of Andros Island (Cyclades, Greece)[END_REF][START_REF] Parra | Relation between the intensity of deformation and retrogression in blueschist metapelites of Tinos Island (Greece) evidenced by chlorite-mica local equilibria[END_REF]. On Tinos, this phase was estimated at 9-10 kbar and between 400 and 550 °C, while on Andros, it was estimated at lower conditions of 7 kbar and 300-420 °C [START_REF] Huet | Coupled phengite 40 Ar-39 Ar geochronology and thermobarometry: P-T-t evolution of Andros Island (Cyclades, Greece)[END_REF][START_REF] Parra | Relation between the intensity of deformation and retrogression in blueschist metapelites of Tinos Island (Greece) evidenced by chlorite-mica local equilibria[END_REF]. Finally, our results do not permit determining the second phase of exhumation that followed the heating period. However, consistent P-T conditions of this late phase of ductile exhumation have been estimated on Syros, Sifnos, Tinos and Andros, and were utilized to draw a complete P-T path of the CBU (Huet et al., 2016;[START_REF] Parra | Relation between the intensity of deformation and retrogression in blueschist metapelites of Tinos Island (Greece) evidenced by chlorite-mica local equilibria[END_REF][START_REF] Trotet | Exhumation of Syros and Sifnos metamorphic rocks (Cyclades, Greece). New constraints on the PT paths[END_REF]Figure 17). To conclude, the P-T path proposed in this study for the CBU on Syros differs significantly from the preceding ones (Figures 1 and17). Our new estimates partly reinforce the conclusion of [START_REF] Trotet | Exhumation of Syros and Sifnos metamorphic rocks (Cyclades, Greece). New constraints on the PT paths[END_REF], who suggested that the P-T evolution was not the same throughout the nappe pile and that the best-preserved parts at the top had been exhumed along a colder path than the lower units. Here, however, we show in addition that the evolutions of the three subunits were similar during the first part of exhumation, from 22 kbar to 10 kbar and that they diverged from that point, with the lower units being heated at constant pressure before their final exhumation, a precision not attained with the methods used by [START_REF] Trotet | Exhumation of Syros and Sifnos metamorphic rocks (Cyclades, Greece). New constraints on the PT paths[END_REF]. In our interpretation, the increasing gradient of retrogression in the greenschist-facies conditions observed toward the base of the CBU is directly linked to these different P-T evolutions. The geodynamic implications of these results are discussed in the following section. Accepted Article This article is protected by copyright. All rights reserved. | Implications for exhumation processes and deep dynamics in the Hellenic subduction zone The tectonometamorphic history of the CBU has been the focus of many works in the last decades. In this section, we synthetize these studies in an attempt to deepen understanding of exhumation processes and deep dynamics in the subduction zone. The conditions of peak metamorphic P-T conditions in the CBU on Syros have been debated in the literature (Figure 1, e.g. [START_REF] Schumacher | Glaucophane-bearing marbles on Syros, Greece[END_REF][START_REF] Trotet | Exhumation of Syros and Sifnos metamorphic rocks (Cyclades, Greece). New constraints on the PT paths[END_REF]. In light of our results and considering recent studies conducted on Sifnos, it seems now clear that the CBU was buried deeply in the subduction zone at a minimum of 60-65 km, assuming lithostatic pressure [START_REF] Ashley | Geothermobarometric history of subduction recorded by quartz inclusions in garnet[END_REF][START_REF] Dragovic | Using garnet to constrain the duration and rate of water-releasing metamorphic reactions during subduction: An example from Sifnos, Greece[END_REF][START_REF] Dragovic | Pulsed dehydration and garnet growth during subduction revealed by zoned garnet geochronology and thermodynamic modeling, Sifnos, Greece[END_REF][START_REF] Groppo | Glaucophane schists and associated rocks from Sifnos (Cyclades, Greece): New constraints on the P-T evolution from oxidized systems[END_REF][START_REF] Trotet | Exhumation of Syros and Sifnos metamorphic rocks (Cyclades, Greece). New constraints on the PT paths[END_REF]. Burial of the CBU is characterized by top-to-the SW sense of shear [START_REF] Philippon | Tectonics of the Syros blueschists (Cyclades, Greece): From subduction to Aegean extension[END_REF]Roche, Laurent, Cardello, Jolivet, & Scaillet, 2016). For example, [START_REF] Philippon | Tectonics of the Syros blueschists (Cyclades, Greece): From subduction to Aegean extension[END_REF] identified on Syros prograde top-to-the SW sense of shear affecting pseudomorphs of lawsonite, while kilometric SW verging folds have been interpreted as prograde drag folds on Sifnos [START_REF] Roche | Anatomy of the Cycladic Blueschist Unit on Sifnos Island (Cyclades, Greece)[END_REF]; see also [START_REF] Aravadinou | Ductile nappe stacking and refolding in the Cycladic Blueschist Unit: insights from Sifnos Island (south Aegean Sea)[END_REF], for the early top-to-the south component of the burial path). Moreover, the P-T path proposed in this study for the CBU on Syros shows interesting similarities with those previously proposed for Tinos and Andros islands [START_REF] Huet | Coupled phengite 40 Ar-39 Ar geochronology and thermobarometry: P-T-t evolution of Andros Island (Cyclades, Greece)[END_REF][START_REF] Parra | Relation between the intensity of deformation and retrogression in blueschist metapelites of Tinos Island (Greece) evidenced by chlorite-mica local equilibria[END_REF]. The main characteristic of these paths is the two-step exhumation history delimited by an isobaric phase of heating (Figure 17). The first part of the retrograde path constrains the P-T conditions of a syn-orogenic phase of exhumation from eclogite-to blueschist-facies conditions. On Syros, the timing of this first exhumation step, accommodated within the subduction channel while the overall regime was still under compression, is constrained between 50 and 35 Ma by 40 Ar/ 39 Ar and Rb/Sr ages [START_REF] Cliff | Geochronological challenges posed by continuously developing tectonometamorphic systems: insights from Rb-Sr mica ages from the Cycladic Blueschist Belt, Syros (Greece)[END_REF][START_REF] Laurent | Extraneous argon in high-pressure metamorphic rocks: Distribution, origin and transport in the Cycladic Blueschist Unit (Greece)[END_REF][START_REF] Lister | White mica 40 Ar/ 39 Ar age spectra and the timing of multiple episodes of high-P metamorphic mineral growth in the Cycladic eclogite-blueschist belt, Syros, Aegean Sea, Greece[END_REF]. Syn-orogenic exhumation kinematic indicators are well exposed on Syros, Sifnos and Tinos, showing a clear top-to-the E/NE sense of shear observed in preserved HP rocks [START_REF] Gautier | Ductile crust exhumation and extensional detachments in the central Aegean (Cyclades and Evvia Islands)[END_REF][START_REF] Jolivet | Ductile extension and the formation of the Aegean Sea[END_REF][START_REF] Laurent | Strain localization in a fossilized subduction channel: Insights from the Cycladic Blueschist Unit (Syros, Greece)[END_REF][START_REF] Philippon | Tectonics of the Syros blueschists (Cyclades, Greece): From subduction to Aegean extension[END_REF][START_REF] Roche | Anatomy of the Cycladic Blueschist Unit on Sifnos Island (Cyclades, Greece)[END_REF]Trotet et al., 2001a). The main structure accommodating this top-to-the NE deformation is the Vari Detachment described on Syros and Tinos and located at the roof of the CBU. [START_REF] Ring | The Hellenic subduction system: high-pressure metamorphism, exhumation, normal faulting, and large-scale extension. Annual Review[END_REF] interpreted this deformation as a synconvergent extensional shearing at the top of an extruding wedge, contemporaneous with thrusting at the base of the wedge. This is consistent with a syn-orogenic exhumation of the CBU between a basal thrust, observed on Ios, and a detachment at the top, the Vari Detachment [START_REF] Huet | Thrust or detachment? Exhumation processes in the Aegean: insight from a field study on Ios (Cyclades, Greece)[END_REF][START_REF] Jolivet | Cenozoic geodynamic evolution of the Aegean[END_REF][START_REF] Jolivet | Subduction tectonics and exhumation of high-pressure metamorphic rocks in the Mediterranean orogens[END_REF]. A pause in the exhumation accompanied by isobaric heating at the base of the crust has also been previously shown on Tinos and Andros. This study recognizes the record of this re-heating period in Syros, except for the top of the CBU where HP-LT parageneses are the best preserved (i.e. in Kampos Subunit; Figure 17). This phase coincides with a change in subduction dynamics, and the settling of a faster slab retreat, coeval with the migration of the subduction front southward within the more external zones (Phyllite-Quartzite Nappe in Crete and on the Peloponnese; [START_REF] Jolivet | Cenozoic geodynamic evolution of the Aegean[END_REF][START_REF] Jolivet | Subduction tectonics and exhumation of high-pressure metamorphic rocks in the Mediterranean orogens[END_REF]. Slab retreat then induced large-scale back-arc extension leading to thermal re-equilibration of the lithosphere from a cold syn-orogenic regime in the subduction zone to a Accepted Article This article is protected by copyright. All rights reserved. warmer post-orogenic regime in the back-arc domain [START_REF] Jolivet | Cenozoic geodynamic evolution of the Aegean[END_REF][START_REF] Parra | Relation between the intensity of deformation and retrogression in blueschist metapelites of Tinos Island (Greece) evidenced by chlorite-mica local equilibria[END_REF]. The timing of this isobaric heating has been determined on Tinos and Andros between 37 and 30 Ma [START_REF] Bröcker | High-Si phengite records the time of greenschist facies overprinting: implications for models suggesting mega-detachments in the Aegean Sea[END_REF][START_REF] Bröcker | 40Ar/39Ar and oxygen isotope studies of polymetamorphism from Tinos Island, Cycladic blueschist belt, Greece[END_REF][START_REF] Huet | Coupled phengite 40 Ar-39 Ar geochronology and thermobarometry: P-T-t evolution of Andros Island (Cyclades, Greece)[END_REF] and needs further temporal constraints on Syros. An interesting feature is that the degree of heating was practically identical on Tinos and Andros (~ 120-150 °C) but seems to be lower on Syros (~ 60-70 °C), which probably allows better preservation of HP-LT metamorphic rocks on Syros. Finally, the last part of the retrograde path of the CBU on Syros, Tinos and Andros constrains the P-T conditions of the post-orogenic phase of exhumation from blueschist-to greenschist-facies conditions (Figure 17; [START_REF] Huet | Coupled phengite 40 Ar-39 Ar geochronology and thermobarometry: P-T-t evolution of Andros Island (Cyclades, Greece)[END_REF][START_REF] Parra | Relation between the intensity of deformation and retrogression in blueschist metapelites of Tinos Island (Greece) evidenced by chlorite-mica local equilibria[END_REF]. The timing of this last ductile exhumation phase is constrained between 30 and 20 Ma by 40 Ar/ 39 Ar and Rb/Sr ages [START_REF] Bröcker | The geological significance of 40Ar/39Ar and Rb-Sr white mica ages from Syros and Sifnos, Greece: a record of continuous (re) crystallization during exhumation?[END_REF][START_REF] Bröcker | Rb-Sr isotope studies on Tinos Island (Cyclades, Greece): additional time constraints for metamorphism, extent of infiltration-controlled overprinting and deformational activity[END_REF], 2005;[START_REF] Bröcker | 40Ar/39Ar and oxygen isotope studies of polymetamorphism from Tinos Island, Cycladic blueschist belt, Greece[END_REF][START_REF] Forster | Several distinct tectono-metamorphic slices in the Cycladic eclogite-blueschist belt, Greece[END_REF][START_REF] Laurent | Extraneous argon in high-pressure metamorphic rocks: Distribution, origin and transport in the Cycladic Blueschist Unit (Greece)[END_REF]. This post-orogenic phase of exhumation is mainly accommodated by opposing-sense sets of detachment such as the North Cycladic Detachment System (NCDS, Jolivet et al., 2010), associated with syn-greenschist top-to-the NE sense of shear, or the West Cycladic Detachment System (WCDS, [START_REF] Grasemann | Miocene bivergent crustal extension in the Aegean: Evidence from the western Cyclades (Greece)[END_REF], a syn-greenschist top-to-the SW system of detachments outcropping on Serifos, Kythnos and Kea islands. Then, final exhumation stages of the CBU toward the surface were accommodated by post-metamorphic brittle deformation through low-angle and steeply dipping normal faults (e.g. [START_REF] Jolivet | Ductile extension and the formation of the Aegean Sea[END_REF][START_REF] Jolivet | Correlation of syn-orogenic tectonic and metamorphic events in the Cyclades, the Lycian nappes and the Menderes massif. Geodynamic implications[END_REF][START_REF] Keiter | A new geological map of the Island of Syros (Aegean Sea, Greece): Implications for lithostratigraphy and structural history of the Cycladic Blueschist Unit[END_REF][START_REF] Mehl | From ductile to brittle: evolution and localization of deformation below a crustal detachment (Tinos, Cyclades, Greece)[END_REF][START_REF] Philippon | Tectonics of the Syros blueschists (Cyclades, Greece): From subduction to Aegean extension[END_REF]Ring, Thomson, & Bröcker, 2003;[START_REF] Ring | The Hellenic subduction system: high-pressure metamorphism, exhumation, normal faulting, and large-scale extension. Annual Review[END_REF]. The multi-stage exhumation process of the CBU within the Hellenic subduction zone is strongly governed by slab rollback. Expanding to the general aspects of subduction zones, we suggest that such metamorphic evolution of HP-LT units, with two exhumation stages delimited by a phase of isobaric heating, should be regarded as a characteristic feature of exhumation driven by slab rollback. To better understand the transition from crustal thickening (syn-orogenic) to back-arc extension (postorogenic) and the dynamics of the subduction channel, it is of paramount importance to know (1) when and at which depth heating has started, (2) whether or not the HP-LT metamorphic units has first been exhumed along a cooling path similar to that retrieved on Syros, and (3) what is the mechanical significance of the depth at which this isobaric heating took place. Accepted Article This article is protected by copyright. All rights reserved. Figure 15: Garnet equilibrium modelling using the program GRTMOD (Lanari et al., 2017). Results are calculated using the thermodynamic database JUN92.bs and both measured bulk rock composition (WR, square) and local bulk compositions (LB, circle) considering or not MnO (dark vs. light blue colour). Solid lines are the error bars on the optimal P-T conditions (see text). Numbers refer to the residual for each growth stage (core / rim or core / mantle / rim). Raman peak temperature estimates (RSCM-T) are represented in grey (2 σ weighted error) for samples of the same subunit. Figure 16: P-T isochemical phase diagrams of SY1401 showing the good consistency between the predicted mineral assemblage in equilibrium for the two events of garnet growth and the mineral assemblage observed in the corresponding textural domain of the compositional map. Note that no consistency is observed for the second stage of rim garnet growth using the fractionated WR composition. Abbreviations: LB, Local Bulk; WR, Bulk Rock. Accepted Article This article is protected by copyright. All rights reserved. Accepted Article This article is protected by copyright. All rights reserved. Table 2: RSCM peak temperature results. For each sample, the total number of measured Raman spectra (n) is shown together with the mean calculated temperature and the mean R2 ratio. The associated standard deviation (SD), related to the intra-sample heterogeneity, is also indicated. Peak temperature results are also represented on the Figure 2. T (°C) Posidonia Subunit Chroussa Subunit Kampos Subunit Sample n R2 Accepted Article This article is protected by copyright. All rights reserved. Accepted Article This article is protected by copyright. All rights reserved. Figure 3 : 3 Figure 3: Geological and metamorphic maps of Syros and associated cross-sections showing the large-scale structure of Syros (after Laurent et al. 2016). Location of samples and RSCM metamorphic peak temperature are shown on the metamorphic map. Figure 4 : 4 Figure 4: Thin-section pictures of typical textures and mineralogical assemblages of analysed samples for thermobarometry and thermodynamic modelling. Figure 5 : 5 Figure 5: Outcrop and sample description of SY1401 (Kampos Subunit). a) The Kastri Shear Zone, in which a strong gradient of deformation and retrogression is observed from eclogite-to greenschistfacies rocks. b, c, d) Decreasing scale pictures from outcrop to sample SY1401. Pictures b and c show stretched eclogites preserving a pillow-lava breccia structure and picture d corresponds to SY1401 before sampling. Figure 6 : 6 Figure 6: Petrography of sample SY1401. a) Thin section photography showing the location of the mapped area. The white arrow shows incipient top-to-the east shear deformation. b) BSE image of the mapped area. c) Map of the minerals (obtained from the classification of the X-ray maps). d) Schematic representation of the mapped area with the distinct petro-textural domains and their estimated proportion in the sample. Figure 7 : 7 Figure 7: X-Ray compositional map of SY1401. a, b, c, d) Composition of the garnet is zoned from core to rim. e) Composition of white mica defines four distinct groups in the map. f) Composition of clinopyroxene. Figure 8 : 8 Figure 8: Petrography of sample SY1460. a) BSE image of a drilled rock-section of SY1460 locating the mapped area. b) BSE image of the mapped area. c) Map of the minerals (obtained from the classification of the X-ray maps). d) Schematic representation of the mapped area with the distinct petro-textural domains and their estimated proportion in the sample. Figure 9 : 9 Figure 9: X-Ray compositional map of SY1460. a, b, c, d) Composition of the garnet is zoned from core to rim. e) White mica defines two phengite compositions. f) Composition of clinopyroxene. Figure 10 : 10 Figure 10: Petrography of sample SY1407. a) BSE image of a drilled rock-section of SY1407 locating the mapped area. b) BSE image of the mapped area. c) Map of the minerals (obtained from the classification of the X-ray maps). d) Schematic representation of the mapped area with the distinct petro-textural domains and their estimated proportion in the sample. Figure 11 : 11 Figure 11: X-Ray compositional map of SY1407. a, b, c, d) Composition of the garnet is zoned from core to rim. e) White mica corresponds mainly to paragonite with only few phengite. Figure 12 : 12 Figure 12: Petrography of sample SY1418. a) BSE image of a drilled rock-section of SY1418 locating the mapped area. b) BSE image of the mapped area. c) Map of the minerals (obtained from the classification of the X-ray maps). d) Schematic representation of the mapped area with the distinct petro-textural domains and their estimated proportion in the sample. Figure 13 : 13 Figure 13: X-Ray compositional map of SY1418. a, b, c, d) Composition of the garnet is zoned from core to rim. e) White mica composition corresponds to phengite. Figure 14 : 14 Figure14: P-T map calculated using the garnet-omphacite thermometer of[START_REF] Ravna | The garnet-clinopyroxene Fe 2+ -Mg geothermometer: an updated calibration[END_REF] together with the garnet-omphacite-phengite barometer of Waters and Martin (1996) (from XMAPTOOLS). Figure 17 : 17 Figure 17: P-T path of the Cycladic Blueschist Unit on Syros. a) RSCM peak temperatures and results of empirical thermobarometry. b) Thermodynamic modelling of garnet growth. c) Compiled P-T constraints for the CBU on Syros, Sifnos, Tinos and Andros islands. d) Proposed P-T path for the CBU on Syros. Table 1 : 1 Bulk rock (WR) and local bulk (LB) compositions used for thermodynamic modeling. SY1401 SY1460 SY1407 SY1418 WR LB WR LB WR LB WR LB SiO 2 52.64 51.54 50.76 51.34 67.39 65.61 48.52 49.75 TiO 2 0.69 0.44 0.78 0.90 0.37 0.48 2.51 1.65 Al 2 O 3 13.97 16.45 14.11 15.63 14.21 15.94 14.28 13.20 FeO 8.15 9.42 6.29 7.48 4.71 4.61 15.58 17.63 MnO 0.17 0.22 0.12 0.21 0.10 0.25 0.24 0.54 MgO 5.21 5.06 5.76 5.13 2.42 1.62 6.04 4.43 CaO 10.39 9.72 9.25 8.04 2.78 1.84 5.59 4.97 Na 2 O 4.67 4.23 5.51 5.29 4.04 4.52 3.28 3.20 K 2 O 1.70 1.85 2.50 2.47 0.53 0.21 0.16 0.08 Total 97.57 98.93 95.08 96.49 96.55 95.09 96.20 95.46 Grt core Grt mantle Grt rim Grt core Grt rim Grt core Grt mantle Grt rim Grt core Grt rim SY1401 SY1460 SY1407 SY1418 SiO 2 38.57 38.70 39.35 37.26 37.86 35.65 35.42 35.85 36.70 37.81 TiO 2 21.44 21.50 22.04 22.08 22.91 22.01 21.35 21.55 19.73 20.52 Al 2 O 3 28.73 29.26 27.77 30.25 27.34 28.63 30.02 27.71 28.97 31.59 FeO 3.15 2.00 0.75 2.80 1.61 3.49 5.09 2.10 3.08 0.28 MnO 1.58 1.73 2.56 1.45 2.21 1.00 1.17 1.46 1.02 2.18 MgO 7.93 8.37 9.73 7.22 9.58 8.89 6.02 10.13 7.85 7.15 CaO 0.01 0.01 0.01 0.04 0.03 0.03 0.03 0.03 0.02 0.02 Na 2 O 0.01 0.01 0.01 0.00 0.00 0.01 0.01 0.01 0.04 0.03 K 2 O 101.40 101.58 102.21 101.10 101.54 99.71 99.11 98.84 97.41 99.58 Atom site distribution (12 anhydrous-oxygen basis) Si(O/T) 3.03 3.03 3.03 2.96 2.95 2.89 2.91 2.91 3.03 3.03 Al(Y) 1.99 1.98 2.00 2.06 2.10 2.10 2.06 2.06 1.92 1.94 Fe(X) 1.89 1.92 1.79 2.01 1.78 1.94 2.06 1.88 2.00 2.12 Mg(X) 0.18 0.20 0.29 0.17 0.26 0.12 0.14 0.18 0.13 0.26 Mn(X) 0.21 0.13 0.05 0.19 0.11 0.24 0.35 0.14 0.22 0.02 Ca(X) 0.67 0.70 0.80 0.61 0.80 0.77 0.53 0.88 0.70 0.61 Xalm 0.64 0.65 0.61 0.67 0.61 0.63 0.67 0.61 0.66 0.70 Xprp 0.06 0.07 0.10 0.06 0.09 0.04 0.05 0.06 0.04 0.09 Xsps 0.07 0.04 0.02 0.06 0.04 0.08 0.11 0.05 0.07 0.01 Xgrs 0.23 0.24 0.27 0.21 0.27 0.25 0.17 0.29 0.23 0.20 Table 3 : 3 Garnet compositions used for thermodynamic modelling. Table 4 : 4 White mica compositions used for thermodynamic modelling. ACKNOWLEDGEMENTS This work has received funding from the European Research Council (ERC) under the seventh Framework Program of the European Union (ERC Advanced Grant, grant agreement No 290864, RHEOLITH) and from the Institut Universitaire de France. It is a contribution of the Labex VOLTAIRE. The authors are grateful to S. Janiec and J.G. Badin (ISTO) for the preparation of thin sections and V. Magnin and V. Batanova (ISTerre) for the acquisition of X-Ray compositional maps. Finally, we thank B. Dragovic for insightful and helpful reviews and D.Whitney for careful and constructive editorial handling. Accepted Article This article is protected by copyright. All rights reserved. Accepted Article This article is protected by copyright. All rights reserved. Earth and Planetary Sciences, 38, 45-76. Ring, U., Thomson, S., & Bröcker, M. (2003). Fast extension but little exhumation: the Vari detachment in the Cyclades, Greece. Geological Magazine,140,[245][246][247][248][249][250][251][252]. Accepted Article This article is protected by copyright. All rights reserved. SUPPORTING INFORMATION Additional Supporting Information may be found online in the supporting information tab for this article. Figure S1. Garnet equilibrium modelling using the program GRTMOD (Lanari et al., 2017). 2008), c) [START_REF] Lister | White mica 40 Ar/ 39 Ar age spectra and the timing of multiple episodes of high-P metamorphic mineral growth in the Cycladic eclogite-blueschist belt, Syros, Aegean Sea, Greece[END_REF]. Mineral abbreviations are after Whitney and Evans (2010). Facies: AEBS, albite epidote blueschist; AM, amphibolite; EA, epidote-amphibolite; EB, epidote-blueschist; EC, eclogite; GS, greenschist; LB, lawsonite-blueschist; LC, lawsonite-chlorite; PA, pumpellyite-actinolite; PrAc, prehnite-actinolite; PrP, prehnite-pumpellyite; ZE, zeolite [START_REF] Peacock | The importance of blueschist→ eclogite dehydration reactions in subducting oceanic crust[END_REF]. Figure captions Figure 2: Tectonic reconstructions of a N-S section of the Aegean domain highlighting the structure of the Hellenic subduction zone, the southward retreat of the slab with time and the evolution of the Cycladic Blueschist Unit from burial to final exhumation (after [START_REF] Jolivet | Cenozoic geodynamic evolution of the Aegean[END_REF]. Abbreviations: NCDS, North Cycladic Detachment System; PND, Paros-Naxos Detachment, WCDS, West Cycladic Detachment System, NAF, North Anatolian Fault.

01744978

en

2024/03/05 22:32:07

2018

https://inria.hal.science/hal-01744978/file/pods050withoutcopyright.pdf

Serge Abiteboul Pierre Bourhis Victor Vianu Explanations and Transparency in Collaborative Workflows Keywords: data-centric workflows, collaboration, views, explanations We pursue an investigation of data-driven collaborative workflows. In the model, peers can access and update local data, causing side-e↵ects on other peers' data. In this paper, we study means of explaining to a peer her local view of a global run, both at runtime and statically. We consider the notion of "scenario for a given peer" that is a subrun observationally equivalent to the original run for that peer. Because such a scenario can sometimes di↵er significantly from what happens in the actual run, thus providing a misleading explanation, we introduce and study a faithfulness requirement that ensures closer adherence to the global run. We show that there is a unique minimal faithful scenario, that explains what is happening in the global run by extracting only the portion relevant to the peer. With regard to static explanations, we consider the problem of synthesizing, for each peer, a "view program" whose runs generate exactly the peer's observations of the global runs. Assuming some conditions desirable in their own right, namely transparency and boundedness, we show that such a view program exists and can be synthesized. As an added benefit, the view program rules provide provenance information for the updates observed by the peer. INTRODUCTION Consider peers participating in a collaborative workflow. Such peers are typically willing to publicly share some data and actions, but keep others private or disclose them only to selected participants. During a run of the workflow, a peer observes side e↵ects of other peers' actions, but may wish to be provided with a more informative explanation of the workflow. At runtime, one would like to explain to each peer the side e↵ects she observes, in terms of the unfolding run. Statically, one would like to provide the peer with a program specifying all the transitions she may observe. In this paper, we consider the problem of providing peers with such runtime and static explanations. In particular, we identify two natural properties of workflows, transparency and boundedness, that are of interest in their own right and greatly facilitate the explanation task. We use the data-driven collaborative workflow model of [START_REF] Abiteboul | Collaborative data-driven workflows: think global, act local[END_REF]. In contrast to process-centric workflows, datadriven workflows treat data as first-class citizens [START_REF] Nigam | Business artifacts: An approach to operational specification[END_REF]. In our collaborative workflow model, each peer sees a view of a global database, that hides some relations, columns of other relations (projection), and tuples (selection). The workflow is specified by datalog-style rules, with positive and negative conditions in rule bodies, and insertions/deletions in rule heads. An event in the system is an instantiation by some peer of a rule in the workflow's program. Consider a run of a workflow and a particular peer, say p. To be able to understand a run from p's perspective, it is useful to isolate the portion of the run relevant to p from other computations that may be occurring in the system. Towards this goal, we introduce the notion of scenario for p, which is a subrun that is observationally equivalent for p to the original run. Such a scenario includes events visible at p, but also events initiated by other peers, that have no immediate side-e↵ects visible at p, but eventually enable events with visible side-e↵ects. Among possible scenarios, minimal ones are desirable because they exclude redundant or useless (from the peer's viewpoint) events. We show that computing minimal scenarios for a peer is generally hard (coNP-complete). Moreover, despite being observationally equivalent to the original run, scenarios can be misleading by di↵ering considerably from what occurs in the actual run. To overcome these issues, we consider an additional property of scenarios, called faithfulness, that guarantees tighter consistency between the scenario and the actual run. Moreover, faithful scenarios turn out to be particularly well behaved. They form a semiring with respect to natural operators, which enables e cient computation of minimal faithful scenarios, as well as their incremental maintenance. We show that every run has a unique minimal faithful scenario for each peer, that can be computed efficiently. We use minimal faithful scenarios as the natural semantic and computational basis for explaining runs. We then turn to the ambitious goal of providing static specifications of the runs as seen from a peer's perspective, which we call view programs for that peer. While such programs cannot generally exist for informationtheoretic reasons, we consider some natural properties of workflows allowing to construct view programs that define precisely a peer's observations of the workflow runs. The properties we consider, transparency and boundedness, are often desirable in practice, for technical and even ethical reasons. In layman terms, an algorithm is transparent (for a specific purpose) if it discloses its motivation and actions. In workflows involving human participants (often the case for collaborative workflows), one might want to require transparency for particular users; indeed, one may be compelled by law to do so in certain settings. Intuitively, a workflow is transparent for a peer if the data that the peer sees at each point in a run is su cient to determine all possible future transitions visible at that peer. For example, if a CEO vetoes the hiring of Alice, and as a consequence it becomes certain that she cannot be hired in the future, this information must be disclosed to her in the next transition she sees. Boundedness is a more technical condition. For an integer h, h-boundedness of a run for a peer p limits to h the number of consecutive events invisible but relevant to p that the other peers can perform. (This does not prevent them from performing arbitrarily long sequences of events irrelevant to that particular peer.) We show that one can decide whether, for a given h, a workflow program only produces transparent and hbounded runs for a particular peer. Furthermore, when this is the case, one can construct a "view program" for a peer that specifies exactly the transitions that peer may see. The rules of the program also provide the peer with provenance information, consisting of the facts visible to that peer that have led to the transition. Because of boundedness, the provenance always involves a bounded number of tuples, which allows their static specification in the bodies of rules of the view program. The synthesis of view-programs described in Section 5 is related in spirit with partner synthesis in services modeled as Petri Nets [START_REF] Wolf | Does my service have partners? Trans. Petri Nets and Other Models of Concurrency[END_REF][START_REF] Lohmann | Wendy: A tool to synthesize partners for services[END_REF][START_REF] Sürmeli | Synthesizing cost-minimal partners for services[END_REF]. With practical considerations in mind, we lastly present some design guidelines for producing transparent and hbounded programs for a specified peer. We also show that, for a large class of programs, one can force transparency and h-boundedness by rewriting each program so that, modulo minor di↵erences, it has the same transparent and h-bounded runs as the original and filters out runs violating these properties. The article is organized as follows. The model is described in Section 2. Scenarios are considered in Section 3, faithfulness in Section 4, transparency and view programs in Section 5, and design methodology for transparent and bounded programs in Section 6. Related work and conclusions are considered in two last sections. Some of the proofs are relegated to an appendix. COLLABORATIVE WORKFLOW In this section, we recall the collaborative workflow model of [START_REF] Abiteboul | Collaborative data-driven workflows: think global, act local[END_REF], introducing minor extensions. We start with some basic terminology. We assume an infinite data domain dom with a distinguished element ? (intuitively denoting an undefined value), and including an infinite set P 1 of peers. We also assume an infinite domain of variables var disjoint from dom. A relation schema is a relation symbol together with a sequence of distinct attributes. We denote the sequence of attributes of R by att(R). A database schema is a finite set of relation schemas. A tuple over R is a mapping from att(R) to dom. An instance of a database schema D is a mapping I associating to each R 2 D a finite set of tuples over R, i.e., a relation over R. We denote by Inst(D) the set of instances of D. We assume that each relation schema R is equipped with a unique key, consisting, for simplicity, of a single attribute K (the same for all relations). An instance I 2 Inst(D) is valid (for the key constraints) if, for each R 2 D, no tuple in I(R) has value ? for attribute K, and there are no distinct tuples u, v in I(R) with the same key. We denote by Inst K (D) the set of valid instances of D. We can apply to each instance I 2 Inst(D) the following chase step up to a fixpoint J denoted chase K (I): for some R, some A, and distinct u, v in I(R), with u(K) = v(K), u(A) 6 =?, and v(A) =?, replace v by v 0 identical to v except that v 0 (A) = u(A). Note that the chase turns some invalid instances into valid ones, while for others it terminates with invalid instances. More precisely, an instance is turned into a valid one i↵ it contains no two tuples with the same key and distinct non-null values for the same attribute. In this case, the result of the chase is unique. For technical reasons, we associate to each relation R (with attribute K), a unary relational view Key R , that consists of the projection of R on K, i.e., for each R and instance I, I(Key R ) = ⇡ K (I(R)). We recall the notion of full conjunctive query with negation (FCQ ¬ query for short). It is adapted to our context to take into account the view relations for keys. A term is a variable or a constant. A literal is of the form (¬) R(x), (¬) Key R (y), x = y, or x 6 = y, where x is a sequence of terms of appropriate arity, x is a variable, and y a term. A FCQ ¬ query is an expression A 1 ^... ^An (for n 0) where each A i is a literal and such that each variable occurs in some positive literal R(u) (a safety condition). Observe that the use of a literal Key R (k) is syntactic sugar, since it can be replaced by R(k, z 1 , ...) where the z i are new. On the other hand, the use of ¬Key R (k) is not. For attributes A, B, and a in dom (possibly ?) , A = a and A = B are elementary conditions. A condition is a Boolean combination of elementary conditions. Peer views of the database will be defined using projections and selections. Note that, in order to enable powerful static analysis, the views in [START_REF] Abiteboul | Collaborative data-driven workflows: think global, act local[END_REF] did not use selections. The more powerful views used here better capture realistic peer views. Collaborative schema. We now define collaborative workflow schemas, extending the definition in [START_REF] Abiteboul | Collaborative data-driven workflows: think global, act local[END_REF]. Starting from a global database schema D and a finite set P of peers participating in the workflow, the collaborative schema specifies, for each peer p, a view of the database. The view consists of selection-projection views of a subset of the relations in D. A view of R 2 D for a peer p (if provided) is denoted R@p. The view allows p to see only some of the attributes of R (projection on att(R@p)) and only some of the tuples (specified by a selection denoted (R@p)). The view of a global instance I of D at p is denoted I@p. Before providing the formal definition, we introduce the following notation. For a relation R and an instance J over a subset att(J) of att(R), J ? denotes the instance of R obtained by padding all tuples of J with the value ? on all attributes in att(R) att(J). Definition 2.1. For a global database schema D, a collaborative schema S consists of a finite set P of peers, and for each p 2 P, a view schema D@p such that: • each relation in D@p is of the form R@p for R in D, • for each R@p, K 2 att(R@p) ✓ att(R). To each relation R@p 2 D@p is associated a selection condition (R@p) over att(R). For an instance I over D, the view instance of I at peer p, denoted I@p, is the instance over D@p defined by: for each R@p in D@p, • I@p(R@p) = ⇡ att(R@p) ( (R@p) (I(R)) ). Furthermore, we impose the following (losslessness) condition: For each I 2 Inst K (D) and R 2 D, I(R) = chase K ⇣ [ {(I@p(R@p)) ? | p 2 P, R@p 2 D@p} ⌘ The losslessness property guarantees that for each tuple in I(R), and each A in R, the value of the tuple for A is visible at some peer. The global instance can therefore be recovered from its peer views using the chase. One can e↵ectively check whether a collaborative schema has the losslessness property. Let S be a collaborative schema, with global schema D and set P of peers. A peer p can perform two kinds of updates on a valid instance I: • A deletion of the form Key R@p (k), where R@p 2 D@p and k is a key value in I@p(R@p). The deletion results in removing from I(R) the tuple with key k. • An insertion of the form +R@p(u), where R@p 2 D@p and u is a tuple over att(R@p) such that: -(i) J = chase K (I [ {R(u ? )} ) is valid, and -(ii) u is subsumed by some tuple v in J@p(R@p). Then J is the result of the insertion. Note that the semantics of an update requested by some peer is specified on the global instance. This circumvents the view update problem. Observe also that a peer can delete only a tuple the peer sees, and that, if an insertion succeeds, the tuple the peer inserted is part of its view after the update (by (ii)). Some subtleties of updates are illustrated next. Example 2.2. Suppose the database consists of a single relation R over KAB, and we have two peers p, q, att(R@p) = KAB, att(R@q) = KA, (R@p) is A = ?, and (R@q) is true. The losslessness condition is not satisfied by this schema. For example, consider the global instance I obtained using the sequence of inserts: +R@p(k, ?, c); +R@q(k, a). It consists of a single tuple, R(k, a, c). Note that as a result of the second insertion, the tuple with key k disappears from the view of p. Moreover, I cannot be reconstructed from the collective views of the peers (and the value "c" is lost). The losslessness condition prevents such anomalies. Moreover, it allows treating the global instance as a virtual rather than materialized database, represented in a distributed fashion by the views of the peers. Collaborative workflow. A collaborative workflow specification (in short workflow spec) W consists of a collaborative schema S and a workflow program for W, i.e., a finite set of "update rules" for each peer p of W. The rules are defined using the auxiliary notion of "update atom" as follows. An update atom at p is an insertion atom or a deletion atom. An insertion atom at p is of the form +R@p(x) where R@p 2 D@p and x is a tuple of variables and constants, of appropriate arity. A deletion atom p is an expression Key R@p (x), where R@p 2 D@p and x is a variable or constant. A rule at peer p is an expression Update :-Cond where: • Cond is a FCQ ¬ query over D@p, and • Update is a sequence of update atoms at p such that: If the sequence includes two updates of the same relation R of tuples with keys x, x 0 , respectively, then x and x 0 are not both the same constant and the body includes a condition x 6 = x 0 . In the previous definition, the conditions x 6 = x 0 impose that no two updates in the same rule a↵ect the same tuple. As a consequence, the order of the update atoms in a rule is irrelevant. An example of rule is the following, where Assign(x, y) says that employee x is assigned to project y, and HR is the Human Relations peer: Key Assign@HR (x), + Assign@HR(x 0 , y) : Assign@HR(x, y), Replace@HR(x, x 0 ), x 6 = x 0 The rule allows the HR peer to replace employee x by employee x 0 on project y. If P is the program of a workflow spec W for a collaborative schema S, we speak simply of the (workflow) program P when S and W are understood. Let P be a program. To simplify, we assume that a run of P starts from the empty instance (note that an arbitrary "initial" instance can be constructed by the peers due to losslessness). The global instance then evolves under transitions caused by updates in rule instantiations, defined next. Let ↵ = Update(ȳ) :-Cond(x) be a rule of P at some peer p where x are the variables occurring in Cond and ȳ are the variables in Update. A valuation ⌫ of ↵ for a global instance I is a mapping from ȳ [ x to dom such that I@p |= Cond(⌫(x)). For a valuation ⌫ of a rule ↵ at some p for some I, the instantiation ⌫↵ is called an event; p the peer of this event, denoted peer (⌫↵), and ↵ its rule. We define the transition relation `e among valid global instances of D as follows. For I, J, and some event e as above, I `e J if all insertions and deletions in Update(⌫(ȳ)) are applicable, and J is obtained from I by applying them in any order. A run of P is a finite sequence ⇢ = {(e i , I i )} 0in , such that ; `e0 I 0 , where ; is the empty instance; and for each 0 < i  n, I i 1 `ei I i . Additionally, we require: for each event e i = ⌫↵, if x is a variable occurring in the head of ↵ but not in its body, then x must be instantiated to a globally fresh value, i.e. ⌫(x) does not occur in const(P ) or in I 1 . . . I i 1 . We denote by Runs(P ) the set of runs of P . Let ⇢ = {(e i , I i )} 0in be a run of P . Note that the event sequence (e 1 ...e n ), denoted e(⇢), uniquely determines the run ⇢. By slight abuse, we sometimes call run a sequence (e 1 ...e n ) of events that yields a run. It is occasionally useful to consider runs starting from an arbitrary initial instance I rather than ;. A run on initial instance I is defined in the obvious manner, by replacing ; with I in the previous definition. Normal-form programs. We show a normal form for workflow programs that will be particularly useful in the next sections. A workflow program P is in normal form if (i) each rule whose head contains a deletion Key R@q (x) also contains a literal R@q(x, u) in its body, and (ii) rule bodies do not contain negative literals of the form ¬R@q(x, u) or positive literals of the form Key R@q (x). Intuitively, (i) simply makes explicit that deletion updates are e↵ective. (Recall that this is imposed by the definition of deletion.) As for (ii), it allows distinguishing between the cases when a fact R@q(k, u) is false because no tuple in R@q has key k, or because R@q(k, v) is a fact for some v 6 = u. Thus, rules in normal form provide more refined information. We next show that this does not limit the expressivity of the model. Proposition 2.3. For each workflow program P , one can construct a normal-form program P nf , and a function ✓ from the rules of P nf to rules of P , such that ⇢ = {(e i , I i )} 0in is a run of P i↵ ⇢ nf = {(f i , I i )} 0in is a run of P nf for some {f i } 0in such that peer(e i ) = peer(f i ) and rule(e i ) = ✓(rule(f i )). Proof. Informally, P nf is constructed as follows. For each rule containing a deletion Key R@q (x) in its head, a literal R@q(x, u) is added to the body, where u consists of distinct new variables. This guarantees (i). For (ii), a literal Key R@q (x) occurring in the body of some rule of P can be replaced by a literal R@q(x, u), where u consists of distinct new variables. Now suppose a literal ¬R@q(x, u) occurs in the body of some rule r of P . First, an instantiation ⌫ of this rule may hold because ¬Key R@q (⌫(x)) holds. Then this can be captured by a rule obtained from r by replacing ¬R@q(x, u) by ¬Key R@q (x). Next, an instantiation ⌫ of r may hold because R@q(⌫(x), v) holds for some v and ⌫(u(A)) 6 = v(A) for some attribute A 6 = K of R@q. Then this can be captured by a rule obtained from r by replacing ¬R@q(x, u) by R@q(x, z) where z is a tuple of distinct new variables, and adding the condition u(A) 6 = z(A). Note that the previous construction replaces r with a set Rules(r) of rules in P nf , corresponding to the different cases. The mapping ✓ is defined by ✓(r 0 ) = r for each r 0 2 Rules(r). Rules r of P that are not modified are included as such in P nf , and ✓(r) = r. VIEWS AND SCENARIOS In this section, we isolate the portions of a run that are relevant to a particular peer and accurately mirror what is actually occurring in the workflow from that peers viewpoint. Towards this goal of filtering out irrelevant events, we introduce the notions of subruns and scenarios. In the next section, we further consider scenarios called faithful that adhere more closely to the actual run and also turn out to have desirable properties from a semantic and computational viewpoint. We first define the p-view of a run, for a peer p. Intuitively, this is the most basic view of a run, consisting essentially of the observations of peer p. Let ⇢ = {(e i , I i )} 0in be a run of P . An event e i is visible at p if peer(e i ) = p or peer(e i ) 6 = p and I i 1 @p 6 = I i @p. Otherwise, e i is invisible (or silent) at p. The p-view of an event e, denoted e@p, is e if peer(e) = p and ! if peer(e) 6 = p, where ! is a new symbol (standing for "world"). Definition 3.1. Let ⇢ = {(e i , I i )} 0in be a run of P , p a peer, and ⇢ 0 = {(e i @p, I i @p)} 0in . The sequence obtained from ⇢ 0 by deleting all (e i @p, I i @p) such that e i is invisible at p, is called the view of ⇢ at p, and denoted ⇢@p. We denote Runs@p(P ) = {⇢@p | ⇢ 2 Runs(P )}. Thus, ⇢@p consists of all transitions caused by p marked with e i @p, as well as all transitions of events visible at p caused by other peers, marked with !. We next consider subruns of a given run, with the goal of isolating the portion of the run that is relevant to p, and which can be used as a sound basis for providing the peer with additional information. For example, in Section 5 we discuss richer views that include provenance information for the updates observed by a peer, extracted from subruns relevant to the peer. A subrun b ⇢ of ⇢ is a run such that e(b ⇢) is a subsequence of e(⇢). In a subrun, only some of the events of ⇢ are retained, in the same order as they occur in ⇢. Observe that the instances in b ⇢ are typically di↵erent than those in ⇢. Of course, not all subsequences of e(⇢) yield subruns. If a subsequence ↵ of e(⇢) does yield a subrun, it is denoted run(↵) (or ↵ by slight abuse). We are interested in those subruns of a run that are compatible with p's observations of the run. This is captured by the notion of "scenario". We will, in particular, be interested by "minimal" such scenarios that, in some sense, explain what can be observed by p in a non-redundant manner. Definition 3.2. Let P be a workflow program, p a peer of P , and ⇢ = {(e i , I i )} 0in a run of P . A sce- nario of ⇢ at p is a subrun b ⇢ of ⇢ such that ⇢@p = b ⇢@p. A scenario b ⇢ of ⇢ at p is minimal if there is no sce- nario ⇢ of ⇢ at p for which e(⇢) is a strict subsequence of e(b ⇢). A minimal scenario is minimum if there is no shorter scenario of ⇢ at p. Clearly, ⇢ itself is a scenario for ⇢ at p, but is likely to include portions that are irrelevant to p's observations, which motivates the study of minimal and minimum scenarios. We next show that finding a minimum scenario is hard. (We will see that the problem is hard also for minimal scenarios.) In the proof, and throughout the paper whenever convenient, we use propositions in workflow programs as syntactic sugar. A proposition x can be simulated by a unary relation R x (with key K), a literal (¬)x by (¬)R x (0), an insertion or deletion of this literal by +R x (0) or Key R x (0). Theorem 3.3. It is np-complete, given a workflow program P , a run ⇢ of P , a peer p and an integer N , whether there exists a scenario of ⇢ at p, of length at most N . Moreover, this holds even for workflows with ground positive rules and no deletions. Proof. Membership in np is obvious. For hardness, we use a reduction from the hitting set problem. An instance of Hitting Set consists of a finite set V = {v 1 , . . . , v n }, a set {c 1 , . . . , c k } of subsets of V , and an integer M < n. It is np-complete whether there exists W ⇢ V of size at most M such that W \ c j 6 = ; for 1  j  k [START_REF] Garey | Computers and Intractibilitiy: A Guide to the Theory of NP-Completeness[END_REF]. From an instance of Hitting Set, we construct a program and a run ⇢ as follows. The schema consists of propositions V i (1  i  n), C j (1  j  k) , and OK. There are two peers p, q. Peer q sees all propositions and p only OK. The rules are the following: (a) +V i @q :-, for each i 2 [1..n], (b) +C j @q :-V i @q, for each i 2 [1..n], j 2 [1..k], v i 2 c j , (c) +OK@q :-C 1 @q, . . . , C k @q Observe that the workflow program uses only propositions. Note that all rules belong to peer q but p can observe the value of OK. Intuitively, firing a subset of the (a)rules designates a subset W of V . With W chosen, the (b)-rules designate the sets c j hit by some element in W . If all sets are hit, rule (c) is enabled. Consider firing first all (a)-rules, followed by all (b) rules, and ending with rule (c). It is clear that this yields a run; call it ⇢. Intuitively, ⇢ corresponds to picking the trivial hitting set W = V . It is easy to check that there exists a hitting set W of size at most M i↵ there exists a scenario of ⇢ for p, of length at most M + k + 1. For the specific runs ⇢ used in the proof, some minimal scenario can always be found e ciently in a greedy manner. (Start with ⇢. First, remove one (a)-rule at a time together with the (b) rules depending on it, and check if the remaining sequence is still a scenario. Then when no (a) rule can be further removed, keep only one (b) rule for each j. The resulting scenario is minimal.) However, this cannot be done for arbitrary runs. Indeed, the next result shows that testing whether a scenario is minimal is hard (see Appendix for proof). Theorem 3.4. It is coNP-complete, given a workflow program P , a run ⇢ of P , and a peer p, whether ⇢ is a minimal scenario of ⇢ at p. Moreover, this holds even for workflows with ground positive rules and no deletions. The lack of a unique minimal scenario of runs for a given peer is problematic when richer views need to be defined starting from several candidate minimal scenarios. Moreover, as seen in the next section, even minimal scenarios can provide misleading explanations about what occurs in the global run. We will propose a natural restriction called faithfulness, that overcomes the problems of unrestricted scenarios. FAITHFUL SCENARIOS A scenario b ⇢ of a run ⇢ at peer p produces the same observations as ⇢ at p, but is allowed to achieve this by means that di↵er considerably from what occurred in the original run. This can be misleading, as illustrated further in Examples 4.1 and 4.2. We therefore consider an additional property of subruns, called faithfulness, that guarantees tighter consistency between what happens in the subrun and in the actual run. The idea, that we will pursue in the next section, is that the workflow system wants to be more transparent for particular individual peers. Furthermore, as we shall see, faithful scenarios turn out to be particularly well behaved: each run has a unique minimal faithful scenario for p, computable in polynomial time, that explains what happens at p in a concise, non-redundant way. We also demonstrate useful properties of faithful scenarios, in particular that they are closed under intersection and union. Before defining faithful scenarios, we illustrate some of the discrepancies that may arise between arbitrary scenarios and actual runs. C i , i 2 [1, k]. Consider a run that derives OK. Suppose the run starts with V 1 @q :-; C 5 @q :-V 1 @q ; V 2 @q :-; C 5 @q :-V 2 @q. Then a scenario could ignore C 5 @q :-V 1 @q although this is the event that actually derived C 5 @q. Example 4.2. Consider a workflow with peers cto, ceo, assistant, and applicant and propositions ok and approval. The peers cto, ceo, and assistant all see ok and approval, and applicant sees only approval. Consider the run ⇢ consisting of the following events: e: +ok@cto :f : ok@cto :g: +ok@ceo :h: +approval@assistant :-ok@assistant The subrun e h is a scenario of ⇢ at peer applicant. It indicates that the applicant's request was approved because it was ok'd by the cto. This subrun is misleading, since in the actual workflow, the cto retracted its ok and the request was approved by the ceo. This arises because the subrun ignores the deletion of ok@cto. Let p be a fixed peer of some workflow program P and ⇢ a run. We next introduce the notion of "p-faithful subrun of ⇢", that prevents the kinds of anomalies previously illustrated. First, the definition is driven by the intuition that tuples in a given relation with a fixed key k represent evolving objects in the workflow. Objects identified by k can go through several lifecycles, occurring between the creation and deletion of a tuple with key k. Faithfulness requires that boundaries of lifecycles of events in the subrun be the same as those in the run, eliminating anomalies such as Example 4.2. It also requires that the events a↵ecting relevant attributes of object k in the same lifecycle be the same in the subrun as in the actual run, eliminating alternative subruns as in Example 4.1. To define faithful subruns formally, we use the following auxiliary definitions. We assume wlog that all programs are in normal form. Let P be a program and ⇢ = {(e i , I i )} 0in a run of P . We say that k occurs as a key of R in an event e i of peer q, if it occurs in a literal R@q(k, ū) or ¬Key R@q (k) in the body of e i , or as an update +R@q(k, ū) or Key R@q (k) in the head of e i . We say that k occurs as a key of R in a sequence ↵ of events if it occurs as a key of R in some event of ↵. We denote the set of such keys by K(R, ↵). Let k 2 K(R, e(⇢)) where We now define the ingredient of faithfulness concerning boundaries of lifecycles. For a subrun b ⇢ of ⇢, the requirement applies to e(b ⇢) alone. In fact, it will be useful to define this notion for arbitrary subsequences of e(⇢). Observe that boundaries of R-lifecycles of k in ⇢ that do not contain events in ↵ need not be included in ↵. ⇢ = {(e i , I i )} 0in . A closed R-lifecycle of k in ⇢ is an interval [i 1 , i 2 ] ✓ [0, n] such that e i1 inserts in We now consider the last ingredient of faithfulness. Its definition relies on the auxiliary concept of the set of attributes of R that are "relevant" to peer q, that is denoted att(R, q). Specifically, the values on att(R, q) determine whether a given tuple is seen by q, and provides the visible values. Formally, for a peer q with selection condition (R@q), we define att(R, q) = att(R@q) [ att( (R@q)), where att( (R@q)) is the subset of attributes of R used in (R@q). The last ingredient of faithfulness focuses on events that modify existing tuples with a given key. Intuitively, modification faithfulness for a peer p requires that, within a lifecycle of key k, all updates of attributes relevant to p must be included in the subsequence. It also requires that updates of attributes relevant to other peers participating in the subsequence be included as well. Definition 4.4. Let ⇢ = {(e i , I i )} 0in be a run of P and p a fixed peer. A subsequence ↵ = {e ij } 0jm of e(⇢) is modification faithful for p if the following holds for each e ij , for each R and each k 2 K(R, e ij ): if e i belongs to the same R-lifecycle of k in ⇢ as e ij , i < i j , peer(e ij ) = q, and e i contains an insertion that turns in ⇢ some attribute in att(R, q) [ att(R, p) of an existing tuple with key k from ? to some other value, then e i also belongs to ↵. Observe that boundary faithfulness is independent of the fixed peer p but modification faithfulness is dependent on p. We can now define the notion of faithful subsequence and subrun of a run. Observe that faithfulness rules out the counterintuitive scenarios in Examples 4.1 and 4.2. For instance, gh is an applicant-faithful subsequence of the run in Example 4.2, whereas eh is not. Moreover, note that gh is a subrun. This is not coincidental. We next show the following key fact: p-faithful subsequences of e(⇢) always yield scenarios of ⇢ for p. The proof is provided in the appendix. Lemma 4.6. Let ⇢ be a run of W, p a peer, and ↵ a p-faithful subsequence of e(⇢). Then (i) ↵ yields a subrun of ⇢, and (ii) run(↵) is a scenario of ⇢ for p. We next show the existence of a unique minimal pfaithful scenario of ⇢ for p, that can be computed in ptime. To this end, we define an operator T p (⇢, •) on subsequences of e(⇢). For each subsequence ↵ of e(⇢), T p (⇢, ↵) consists of the subsequence of e(⇢) obtained by adding to ↵ all events of ⇢ whose presence is required by boundary and modification p-faithfulness due to events that are already in ↵. Observe that (almost by definition) a subsequence ↵ of ⇢ is boundary and modification p-faithful i↵ it is a fixed-point of T p (⇢, •), i.e., T p (⇢, ↵) = ↵. Let ⌧ be the subsequence relation on sequences of events. Note that T p (⇢, •) is monotone with respect to ⌧, i.e., for ↵ ⌧ , T p (⇢, ↵) ⌧ T p (⇢, ). Consider the increasing sequence ↵ 0 = ↵, ↵ 1 = T p (⇢, ↵ 0 ), ↵ 2 = T p (⇢, ↵ 1 ), ... and let T ! p (⇢, ↵) = ↵ n where n is the minimum integer for which ↵ n = ↵ n+1 . Now we have: Theorem 4.7. Let P be a program schema. For each run ⇢ of P there is a unique minimal p-faithful scenario ⇢ of ⇢. Moreover, ⇢ equals run(T ! p (⇢, ↵)), where ↵ consists of all events of ⇢ visible at p, and can be computed from ⇢ in polynomial time. Proof. Clearly, T ! p (⇢, ↵) is p-faithful, because it is a fixed-point of T p (⇢, •) and contains all events of ⇢ visible at p. By Lemma 4.6, it also yields a scenario for p. Consider any p-faithful scenario ⇢ of ⇢ for p. By definition of p-faithfulness, ⇢ must include the events in ↵. From the fact that ↵ ⌧ e(⇢), T p (⇢, •) is monotone, and e(⇢) is a fixed-point of T p (⇢, •), it follows that T ! p (⇢, ↵) ⌧ e(⇢). Thus, ⇢ = T ! p (⇢, ↵) is the unique minimal p-faithful scenario of ⇢. Clearly, T ! p (⇢, ↵) can be computed in polynomial time from ⇢. We now consider two natural ways of combining subsequences of a run ⇢, that are useful in many practical situations: "multiplying" them by taking the intersection of their events, and "adding" them by taking the union of their events. We will show that p-faithful subsequences of a run ⇢ are closed under both operations, and form a semiring. Formally, let ↵ 1 and ↵ 2 be subsequences of e(⇢) for a run ⇢. Their product, denoted ↵ 1 ⇤↵ 2 is the subsequence consisting of the events in ⇢ occurring in both ↵ 1 and ↵ 2 . Their addition, denoted ↵ 1 + ↵ 2 , is the subsequence consisting of the events of ⇢ in ↵ 1 or ↵ 2 . Clearly, if ⇢ 1 and ⇢ 2 are subruns of ⇢, e(⇢ 1 ) + e(⇢ 2 ) and e(⇢ 1 ) ⇤ e(⇢ 2 ) are not guaranteed to yield subruns of ⇢. Addition and multiplication of sequences are both of interest in practice. Closure under intersection is the core reason for the existence of a unique minimal pfaithful scenario for a peer. As we will see, addition is useful in incremental maintenance of minimal p-faithful scenarios. We next show the following (see Appendix for proof). Theorem 4.8. Let ⇢ be a run of W. The set of pfaithful scenarios of ⇢ is closed under addition and multiplication, and forms a semiring. Incremental evaluation To conclude the section, we discuss how to maintain incrementally the minimum pfaithful scenario of a run, leveraging the closure under addition of p-faithful scenarios. More specifically, for a run ⇢, we wish to maintain incrementally T ! p (⇢, ↵), where ↵ consists of the events of ⇢ visible at p. To do so, we additionally maintain some auxiliary information, consisting of T ! p (⇢, f ) for each event f in ⇢. Intuitively, T ! p (⇢, f ) represents an "explanation" of the individual event f by a minimal boundary and modification pfaithful subrun of ⇢ containing f . Note that f need not be visible at p. Suppose the current run is ⇢ and we have computed T ! p (⇢, ↵) and T ! p (⇢, f ) for each event f in ⇢. Now suppose that a new event e arrives. We need to compute the following (where ⇢.e denotes the concatenation of ⇢ and e): (i) T ! p (⇢.e, f ), for each event f in ⇢.e. (ii) T ! p (⇢.e, ↵ 0 ), where ↵ 0 = ↵.e if e is visible at p and ↵ 0 = ↵ otherwise. Consider (i). For f = e, T ! p (⇢.e, e) consists of e together with all events in T ! p (⇢, g) for some g 2 T p (⇢.e, e). For f 6 = e, there are two cases, depending on whether e is the right-boundary of an open lifecyle of a key occurring in T ! p (⇢, f ). If this is the case, meaning that e is in T p (⇢.e, T ! p (⇢, f )), then T ! p (⇢.e, f ) consists of e together with the events in T ! p (⇢, f ) and those in T ! p (⇢, g) for some g 6 = e in T p (⇢.e, e). Otherwise, T ! p (⇢.e, f ) = T ! p (⇢, f ). Now consider (ii) . Suppose e is visible at p, or is the right-boundary of an open lifecycle of a key in T ! p (⇢, ↵), meaning that e is in T p (⇢.e, T ! p (⇢, ↵)). Then T ! p (⇢.e, ↵ 0 ) consists of e together with the events in T ! p (⇢, ↵) and in T ! p (⇢, g) for some g 6 = e in T p (⇢.e, e). Otherwise, T ! p (⇢.e, ↵ 0 ) = T ! p (⇢, ↵). Observe that the incremental maintenance algorithm outlined above requires only single applications of the operator T p (⇢, •), avoiding computations of its least fixpoint from scratch. This is similar in spirit to results on incremental maintenance of recursive views by nonrecursive queries (e.g., see [START_REF] Dong | Incremental maintenance of recursive views using relational calculus/sql[END_REF]). TRANSPARENCY & VIEW-PROGRAMS The goal. Let P be a workflow program, and p a peer. We have defined, for each run ⇢ of P , the view ⇢@p of the run as seen by peer p. As a next step, we would like to also provide p with a "view-program" P 0 whose runs are precisely the views Runs(P )@p. Intuitively, we wish such a program to provide p with an "explanation" of the global workflow in terms it can understand and access. The view-program will use the same schema D@p as p, and the fictitious peer ! (world) that represents the environment. The rules of P 0 consist of the rules of p in P , together with new rules for peer !, that define the transitions caused by other peers with visible side-e↵ects at p. More precisely, a program P 0 is a view-program for P at p if: • P 0 is over global schema D@p and uses two peers p and !, both with schema D@p, and all selection conditions true. • the rules of peer p are the same in P and P 0 . • (completeness) for each run ⇢ of P , there exists a run ⇢ 0 of P 0 such that ⇢@p is obtained from ⇢ 0 by replacing all !-events with !. • (soundness) for each run ⇢ 0 of P 0 , there exists a run ⇢ of P , such that ⇢@p is obtained from ⇢ 0 by replacing all !-events with !. We illustrate the notion of view-program with an example. Example 5.1. Consider a program P with peers hr, ceo, cfo, and Sue, using the following rules: +Cleared@hr(x) :-+cfoOK@cfo(x) :-+Approved@ceo(x) :-Cleared@ceo(x), cfoOK@ceo(x) +Hire@hr(x) :-Approved@hr(x) Note that there are no rules for Sue. Suppose hr, cfo, and ceo see all relations, but Sue sees only relations Cleared and Hire. A view-program P 0 for Sue operates on the schema Cleared and Hire and has the following rules: +Cleared@!(x) :-( †) +Hire@!(x) :-Cleared@!(x) It is easy to check that P 0 is sound and complete for P and peer Sue, and so it is a view-program for P at Sue. Remark 5.2. Observe that soundness and completeness of view-programs amounts to equivalence of P 0 and P with respect to the views of their linear runs. However, consider the following subtlety in the above example. By soundness of P 0 and rule ( †), if Sue sees Cleared(x), then there exists a run of P in which Sue sees Hire(x) inserted in the next transition visible to her. However, it is not the case that this is possible in every run of P , because the transition is also dependent on relation cfoOK, invisible to Sue. Enforcing this stronger property requires a more stringent notion of equivalence based on the trees of runs rather than just linear runs. We later show how this can be done by forbidding the use of information invisible to the given peer that affects the view of the peer. Intuitively, this makes the collaborative workflow more transparent to the peer. We formally introduce the notion of transparency further. Note that one can trivially define a sound program for P at p (by keeping only the rules of P at peer p) and a complete one (by adding to the rules of p all rules at ! that insert or delete up to M arbitrary tuples in relations of D@p, where M is the maximum number of updates in the head of a rule in P ). Clearly, these are of little interest. Ideally, one would like a program that is both sound and complete. Unfortunately, as shown next, it is not generally possible to construct such a program. Proposition 5.3. There exists a program P and a peer p, such that there exists no view-program for P at p. Proof. (sketch) The program P uses three binary relations R, S, T that a peer q sees, whereas peer p sees only R, T . Suppose p has a rule inserting an arbitrary tuple in R (so p can construct arbitrary instances over R). Peer q has two rules to add to S pairs in the transitive closure of R. Finally, q has a rule to transfer a tuple (0, 1) from S to T . No view-program can exist for P at p because the insertion of the tuple (0, 1) in T @p is conditioned by the existence of a directed path of arbitrary length from 0 to 1 in R@p. This cannot be expressed by any rule with a bounded number of literals in its body. Theorem 5.4. It is undecidable, given a program P and a peer p, whether there exists a view-program for P at p. Proof. (sketch) We use the following observation: (?) It is undecidable, given a program Q whose schema includes a unary relation U , whether there exists a run of Q leading to an instance where U is non-empty. The proof of (?) is by a straightforward reduction from the Post Correspondence Problem, known to be undecidable [START_REF] Post | Recursive unsolvability of a problem of Thue[END_REF]. Intuitively, Q attempts to construct a solution to an instance of the PCP. If it succeeds, a fact is inserted in U (details omitted). Consider the program P in the proof of Proposition 5.3, modified so that the rule to transfer the tuple S(0, 1) to T (0, 1) is controlled by non-emptiness of U . Now add to P the rules of an arbitrary program Q whose schema contains U (and no other relations of P ). Then a viewprogram at p exists for the resulting program i↵ there is no run of Q leading to some non-empty U . This is undecidable by (?). Transparency and Boundedness. Clearly, an obstacle towards obtaining a view-program for P at p is that updates visible at p may depend on information unavailable to p. To overcome this di culty, we consider a property of programs called "transparency". Intuitively, transparency of a program P for peer p guarantees that the possible updates of a view instance I@p caused by other peers are determined by I@p. Put di↵erently, the other peers must disclose to p all information that they use in order to modify p's view of the data. The only action that can depend on hidden information is the creation of new values, which is constrained by the global history. It turns out that transparency of a program P for p does not alone guarantee the existence of a viewprogram of P for p. This is because the other peers can still perform arbitrarily complex computations hidden from p. For instance, the program used in the proof of Proposition 5.3 is transparent for p but does not have a view-program for p. To control the complexity of computations a↵ecting p, we introduce a notion of "boundedness" of P with respect to p, that limits the number of steps invisible but relevant to p that are carried out by other peers. As we will see, transparency together with boundedness guarantee the existence of a view-program and its e↵ective construction. We next define transparency, then turn to boundedness. To formalize the notion of transparency, we first define "fresh" instances, obtained as the results of events visible at p. We use the following notation. If ↵ is a sequence of events of P yielding a run on initial instance I, we say that ↵ is applicable at I and denote by ↵(I) the last instance in the run. Definition 5.5. Let P be a program and p a peer. An instance I is p-fresh if I = ; or there exists an instance I 0 and an event e of P that is applicable to I 0 and visible at p, such that e(I 0 ) = I. We can now define transparency. We will use the notion of minimum p-faithful run, defined as follows. A run ↵ on initial instance I is a minimum p-faithful run if ↵ = T ! p (↵, v), where v consists of the events of ↵ that are visible at p. In other words, ↵ is its own minimum p-faithful scenario for p. To deal with new values, we will need the following. For a sequence ↵, let adom(↵) consist of all values occurring in ↵, and new(↵) consist of all values a for which there is an event e in ↵ such that a occurs in the head but not in the body of e. Let const(P ) denote the set of constants used in program P , together with ?. Definition 5.6. A program P is transparent for p if for all p-fresh instances I, J such that I@p = J@p the following holds. For every sequence ↵ of events such that adom(J)\new(↵) = ;, if ↵ is a minimum p-faithful run on I such that all its events but the last are silent at p, then the same holds on J, and ↵(I)@p = ↵(J)@p. Intuitively, transparency implies that the computation as seen from p depends only on what p sees, except for the specific choice of new values. The definition is illustrated by the following example. Example 5.7. Consider again Example 5.1. It is easy to see that the program in the example is not transparent for Sue. Intuitively, this is because the relation cfoOK carries information that Sue does not see, yet it impacts Sue's view of the workflow. Now consider the following program (obtained by eliminating cfoOK): +Cleared@hr(x) :-+Approved@ceo(x) :-Cleared@ceo(x) +Hire@hr(x) :-Approved@hr(x) At first glance, the program may now appear to be transparent for Sue. However, this is not the case. Indeed, consider the instances I, J containing the following facts: I: Cleared(Sue); Approved(Sue) J: Cleared(Sue) Clearly, I and J are Sue-fresh since both can be obtained by an application of the Sue-visible event +Cleared@hr(Sue). In addition, I@Sue = J@Sue = {Cleared@Sue(Sue)}. The sequence consisting of the single event +Hire@hr(Sue) :-Approved@hr(Sue) is a minimum Sue-faithful run on instance I. Transparency requires it to also be applicable on J, which is not the case. Intuitively, in order for transparency to hold, Sue-freshness must ensure that pre-existing information invisible to Sue, such as the fact Approved(Sue) in instance I, cannot be used in later events leading to transitions visible by Sue. This can be achieved in various ways. We illustrate one approach, that is also adopted in the design methodology for transparent programs in Section 6. We introduce an additional binary relation Stage visible by all peers, that inhibits the use of any information computed prior to the latest Sue-visible update. Relation Stage is either empty or contains a single tuple Stage(0, s) where s is a value refreshed by peer Sue prior to events by other peers. The invisible relation Approved is extended with an extra column holding the current value of s. The program is the following: +Stage@Sue(0,s) :-¬ Key Stage@Sue (0) +Cleared@hr(x), Key Stage@Sue (0) :-Stage@hr(0,s) +Approved@ceo(x,s) :-Cleared@ceo(x), Stage@ceo(0,s) +Hire@hr(x), Key Stage@hr (0) :-Approved@hr(x,s), Stage@hr(0,s) Observe that Stage(0,s) is deleted by each event visible at Sue, which forces its re-initialization with a fresh s before any event using invisible relations can proceed, preventing the use of previous invisible facts. It is easy to check that the program is now transparent for Sue. We next introduce the boundedness property. Definition 5.8. Let P be a program, p a peer, and h an integer. P is h-bounded for p if for each instance I and sequence ↵ of events applicable to I that yields a minimum p-faithful run at p such that all its events but the last are invisible at p, |↵|  h. Intuitively, this bounds the number of consecutive events that are silent but relevant to p. Note that the bound applies only to minimum p-faithful subruns. Thus, the other peers are still allowed to carry out arbitrarily long silent computations that do not a↵ect p. We next consider the decidability of transparency and boundedness. It is easy to show: Theorem 5.9. It is undecidable, given a program P and peer p, (i) whether P is transparent for p, and (ii) whether there exists h such that P is h-bounded for p. Proof. Straightforward, using (?) in the proof of Theorem 5.4. On the other hand, as shown next, it is decidable if a program is h-bounded for some given h. Moreover, for programs that are h-bounded, transparency is decidable. In particular, it is decidable whether a program is simultaneously h-bounded and transparent. Indeed, we will see that the two together guarantee the existence of a view program that can be e↵ectively constructed. We first show that h-boundedness is decidable (see Appendix for proof). Intuitively, this holds because violations of h-boun-dedness are witnessed by minimum p-faithful runs of bounded length, on initial instances of bounded size. Theorem 5.10. It is decidable in pspace, given a program P , peer p, and integer h, whether P is h-bounded for p. Next, we show that transparency is decidable for hbounded programs. Theorem 5.11. The following are decidable in pspace: (i) given a program P that is h-bounded for peer p, whether P is transparent for p, and (ii) given a program P , a peer p and an integer h, whether P is h-bounded and transparent for p. The proof is provided in the appendix. Clearly, (ii) follows from (i) and Theorem 5.10. The proof of (i) relies on the existence of short counterexamples for violations of transparency by h-bounded programs. Remark 5.12. Observe that the p-fresh instances used in the definition of transparency need not be reachable in actual runs of P . Thus, the definition has a "uniform" flavor, reminiscent of uniform containment for Datalog. Limiting transparency to reachable instances would yield a much weaker requirement, leading to undecidability of p-transparency even for h-bounded programs. Previously, we assumed that the boundedness parameter h is given. There are various ways to obtain h. One approach is heuristic: by examining traces of runs, one can "guess" h and then test h-boundedness using Theorem 5.10. Another possibility, briefly considered in Section 6, is to provide syntactic restrictions on the program P that ensure h-boundedness for some h computable from P . Alternatively, we introduce in Section 6 the means of ensuring by design transparency and hboundedness of a program, for a given peer and desired h. View-programs and provenance. We show that for each program P and peer p such that P is h-bounded and transparent for p, one can construct a view-program of P for p. As discussed earlier, the view program uses peers p and ! (for world). It contains the rules for p, and additional rules for ! that define the side-e↵ects observed by p as a result of actions by other peers. We describe the construction of the rules for !. Intuitively, the rules specify, for each instance I@p visible at p, the possible updates to I@p caused by minimal p-faithful runs of length up to h starting from I. The body of each rule specifies the tuples of I@p causing the update, so intuitively provides the provenance of the update in terms of the data visible at p. The view program P @p. We outline informally the construction of the view-program of P for p, that we denote P @p. Let C h+1 = {a 1 , ..., a m }, where C h+1 is the set of constants (polynomial in P ) defined in the proof of Theorem 5.10. For each i 2 [1, m], let x i be a new distinct variable. Let ⌫ be the mapping defined by ⌫(a i ) = x i if a i 6 2 const(P ) and ⌫(a i ) = a i otherwise. Consider a p-fresh instance I and a sequence of events ↵ of P , both over C h+1 , such that the tuples in I(R) use only keys in K(R, ↵) for each relation R, and ↵ is a minimal p-faithful run of P on I in which all events but the last are invisible at p. Let J = ↵(I). Observe that by boundedness of P for p, |↵|  h, and so there are finitely many triples (I, ↵, J) as above. For each such triple (I, ↵, J), the view program P @p contains a rule for peer ! constructed as follows. For each relation R: • (positive body) for each t in I@p(R), add R@!(⌫(t)) to the body. • (negative body) for each a i in K(R, ↵) that is not a key value in I@p(R), add ¬Key R@! (⌫(a i )) to the body. • (inequalities) for all a i , a j where i 6 = j, the body contains the inequality ⌫(a i ) 6 = ⌫(a j ). • (head insertions) For each t in J@p(R) I@p(R), add +R@!(⌫(t)) to the head. • (head deletions) For each a i in I@p(Key R@p ) J@p(Key R@p ), add Key R@! (⌫(a i )) to the head. Observe that P @p is a syntactically valid program with schema D@p and peers p and !. We next show that the construction is correct (see Appendix for proof). Theorem 5.13. Given a program P that is h-bounded and transparent for peer p, the program P @p is a viewprogram of P for p. Moreover, as suggested in Remark 5.2, the view-program P @p constructed above is sound and complete not only with respect to the linear runs of P as viewed by p, required by the definition, but also with respect to its tree of runs as viewed by p. We omit the formal development. TRANSPARENT PROGRAM DESIGN Transparency and h-boundedness for a given peer may be desirable goals for some applications. There are various ways to achieve them. It is of course possible to first design the workflow program and then test it a posteriori for transparency and h-boundedness. However, it may be preferable in practice to specify directly view programs that are transparent and h-bounded by design. We begin the section by showing how this can be done by following some simple design guidelines. We then show that a large class of programs can be transformed so as to make them transparent and h-bounded, by filtering out the runs that violate these properties while preserving the runs that satisfy them (modulo some minor di↵erences). Before proceeding, we introduce some notions used throughout the section. Consider a run ⇢, and a subsequence e.↵.e 0 of consecutive events of ⇢, of which the only events visible at p are e and e 0 . Then ↵.e 0 is a p-stage. In the following, each non-trivial stage (↵ 6 = ✏) will be equipped with a unique id, in order to identify the facts produced during that stage. Intuitively, transparency is obtained by controlling the provenance of facts produced in that stage that lead to the visible event e 0 . Generating the stage ids can be done using a binary relation Stage, visible by all peers. Stage is either empty or contains a single tuple with key 0. A fact Stage(0, s) indicates that the current stage id is s. When a peer q carries out an event visible at p, it deletes the current fact Stage(0, s) (if such exists). A special rule, that can be performed by any peer q, inserts a new tuple Stage(0, s 0 ) with a fresh value s 0 . Specifically, the rule is +Stage@q(0, z) :-¬ Key Stage@q (0). All rules generating events invisible at p are guarded by an atom Stage(0, x) (so all p-invisible events of the stage are preceded by the event creating a new stage id). Note that the above assumes that each peer q can tell whether its updates are visible at p. This is not always the case, but holds under certain conditions, such as (C1) further. Transparency and boundedness by design. We introduce design guidelines to guarantee transparency and h-boundedness for a designated p. It turns out to be rather subtle to guarantee transparency while allowing other peers to perform arbitrary computations that do not impact p. In order to ensure transparency of a program P for a peer p, we impose the following restrictions on program specifications, to be followed in the design process: (C1) Each peer that sees a relation R visible at p (including p) sees it fully. Formally, for each relation R@p 2 D@p, if R@q 2 D@q then att(R@q) = att(R) and (R@q) = true. (C2) The program maintains the Stage relation as previously described. Note that, because of (C1), every non-trivial update of a relation in D@p, caused by any peer, is also visible at p. Thus, every peer can tell when it performs an update visible at p. As noted previously, this enables the maintenance of relation Stage. (C3) The relations in D are separated in two disjoint classes: p-transparent and p-opaque. The relations that p sees are all p-transparent. The ptransparent relations that p does not see include an attribute, StageID, that contains the id of the stage in which the tuple was created. (C4) If an event modifies some p-transparent relation, (i) only positive facts from p-transparent relations with the current stage id can occur in its body, and (ii) all the updates in the head are either updates of p-visible relations, creations of tuples with new keys in a p-transparent relation, or modifications of tuples in such a relation that have been created during the same stage and are visible by the peer performing the event. When p is understood, we simply speak of transparent and opaque relations/facts. It is straightforward to specify syntactic criteria to guarantee (C1-C2-C3-C4). For instance, (C4)(ii) can be ensured as follows: if +R@q(x, u) occurs in the head of a rule for a transparent p-invisible relation R, then either x is a variable that does not occur in the body (so it generates a new key) or R@q(x, v) occurs in the body, where v(StageID) is the current stage id provided by relation Stage. Condition (C1) is natural in many applications where peers are doing some computations about a peer p, for which transparency is desired. For instance, p may be a customer, a job applicant, a participant in a crowdsourcing application, etc. Intuitively, (C1) prevents a peer from unknowingly performing some update that is visible at p. We briefly elaborate on the motivation of (C4). Clearly, the use of opaque relations in rule bodies may lead to non-transparent computations. The restriction disallowing deletions from p-invisible transparent relations in heads of rules simplifies the presentation, but such deletions could be allowed at the cost of a more complex construction. The following example illustrates the motivation for prohibiting simultaneous updates of transparent and opaque relations in rule heads. Example 6.1. Consider a workflow program P with a peer p, a p-visible relation R, and a p-invisible opaque relation T . Note that there is no transparent invisible relation. Suppose that P includes the following rules: +R@q(Sue, hire), +T@q(Sue, hire) :-+R@q(Sue, reject), +T@q(Sue, reject) :-The other peers may have silently computed for an arbitrarily long time, and derived T (Sue, reject). This precludes application of the first rule. Intuitively, they have ruled out a possible future event for Sue without letting her know, thus violating transparency. It is straightforward to see that a program satisfying (C1-C2-C3-C4) is transparent for p. Note that other peers may perform arbitrary computations as long as they do not a↵ect what p sees. Observe that the transparent program shown in Example 5.7 follows the previous design guidelines. We next show how to guarantee h-boundedness within a stage ↵.e 0 immediately following an event e. We wish to ensure that the minimum p-faithful subrun of ↵.e 0 contains no more than h events, leading to the activation of the visible event e 0 . This could be easily done by limiting the length of the entire stage to h + 1 using a propositional counter. However, this brute-force solution would be overly restrictive, because one often wishes to allow within the same stage an unbounded number of events that a↵ect only p-opaque relations, or p-transparent relations in events not leading to e 0 . Achieving this requires a more careful approach, in which the "steps" in each stage are identified by ids consisting of fresh values. More precisely, each event within the stage ↵.e 0 is called a step, and is identified by a step id. We will use the notion of step-provenance of a fact, i.e. the set of step ids in that stage that contribute to deriving the fact. In more detail, we equip each p-transparent relation invisible at p, say Q, with h additional columns B 1 • • • B h that are used to record the step-provenance of each fact in the relation (ids of the steps contributing to its creation). When an update +Q(u, b 1 , ..., b h ) is performed, its set of non-? B i 's is set to the concatenation of all distinct non-? values of the B i 's in the body of the event, augmented with a new id for the current step (shared by all insertions in the head). Recall that by (C4), there is no key deletion from Q. (In some sense, all the facts in Q are logically deleted when a new stage is entered.) Thus, a p-transparent event can be activated only if there is "enough room", i.e., if a sequence of at most h events of the stage are su cient to enable this event. In particular, the last update of the stage has at most h non-? B i 's. The events of the corresponding steps provide a p-equivalent sequence of length at most h, so the minimal p-faithful scenario for that stage has length at most h. Thus, the resulting run is h-bounded for p. In summary, we can show: Theorem 6.2. Each program obtained using the aforementioned guidelines is transparent and h-bounded for p. Boundedness by acyclicity. We next show how to guarantee boundedness for a certain class of programs using an acyclicity condition. We consider programs with single updates in heads of rules (which we call linear-head programs), satisfying (C1). For such a program, we define the p-graph of P as follows. Its nodes are the relations in D, and there is an edge (intuitively "depends on") from R to Q if Q is invisible at p and there is a rule at some peer q whose head is +R@q(u) or Key R@q (h) and its body contains Q@q(v) or ¬Key Q@q (k). The program P is p-acyclic if for each R@p 2 D@p, the subgraph of its p-graph induced by the nodes reachable from R is acyclic. We can show the following (see Appendix for proof). Enforcing transparency and boundedness. We next show, given a program P and peer p, how to rewrite P into a transparent and h-bounded program for p that has essentially the same behaviour as P except that it filters out the runs that are either not transparent or not h-bounded for p. We already defined these properties for programs. We need to define them for runs. Definition 6.4. Let P be a program and p a peer. A run ⇢ of P is transparent for p if for each stage ↵.e 0 of ⇢, the minimum p-faithful subrun ↵ 0 .e 0 of ↵.e 0 has the following property. For any p-fresh instance J such that I@p = J@p, and adom(J) \ new(↵ 0 .e 0 ) = ;, ↵ 0 .e 0 is a minimum p-faithful run on J, all its events but e 0 are silent at p, and ↵ 0 .e 0 (I)@p = ↵ 0 .e 0 (J)@p. We say that ⇢ is h-bounded for p if |↵ 0 .e 0 |  h for every ↵ 0 .e 0 as above. The set of transparent and h-bounded runs of P for p is denoted tRuns p,h (P ). Our rewriting technique applies to the programs satisfying certain conditions, that we call transparencyform. Unlike the conditions used in the design guidelines, transparency-form does not require separating transparent and opaque relations at the schema level, instead allowing to make a more refined distinction at the fact level. In this more permissive setting, runs are no longer guaranteed to be transparent and h-bounded. However, we show how the violating runs can be filtered out. Definition 6.5. A normal-form program is in transparencyform (TF for short) for p if it satisfies (C1-C2) and: (C3') For each rule of a peer q 6 = p, if its head contains an update +R@q(x, ȳ) for some R that p does not see, either x is a variable that does not appear in the body (key creation) or the body contains an atom R@q(x, z). (C4') For each relation R that p does not see, and each peer q such that R@q 2 D@q, the selection (R@q) uses only attributes in att(R@q). As discussed earlier, condition (C1) is meant to guarantee that a peer knows when it is performing an event visible at p, which enables maintaining the relation Stage. (C3') is a natural condition that essentially comes down to preventing the "reuse" of a key after it has been deleted. The motivation for (C4') is more subtle. The presence of a fact in the view of some peer q may depend (because of selections) on some values that q does not see and that have been derived in a non-transparent manner. This may lead to violations of transparency that cannot be filtered out. We will show the main result of this section for programs satisfying (C1-C2-C3'-C4'). We first need to introduce the notion of "run projection". Definition 6.6. (run projection) Let P be a program over schema D. Let ⇧ be a schema consisting of a subset of the relations in D, each having a subset of its attributes in D (always containing the key). ⇧ induces a projection function on runs, defined as follows. The projection ⇧(⇢) of a run ⇢ of P is obtained from ⇢ by removing facts and updates over relations not in ⇧, projecting out the missing attributes in facts and updates over relations in ⇧, and removing events with resulting empty heads. We say that ⇧ is the identity for peer p if for every run of ⇢ of P , ⇧(⇢)@p = ⇢@p. We extend this definition to a set Runs of runs and denote the result ⇧(Runs). We now state the main result of the section: Theorem 6.7. Let P be a TF program, p a peer, and h an integer. One can construct a program P t that is transparent and h-bounded for p, and a projection function ⇧ that is the identity for p, such that the runs of P that are transparent and h-bounded for p are exactly the projections of the runs of P t , i.e., tRuns p,h (P ) = ⇧(Runs(P t )). The construction of the program P t is outlined further. From Theorems 5.13 and 6.7 it also follows that, for an arbitrary TF program P , we can obtain a view program that specifies precisely the views at p of the runs of P that are transparent and h-bounded for p. Corollary 6.8. Let P be a TF program, p a peer, and h an integer. One can construct a view program P t p for P t and p such that Runs(P t p ) = {⇢@p | ⇢ 2 tRuns p,h (P )}. Remark 6.9. Note that, if the peers attempt to perform a non-transparent computation, the transformed program P t will prevent carrying out the run and the computation may block. In practice, one might want to let the computation proceed and simply send an alert. Alternatively, one might wish to perform some "recovery", e.g., roll back to the state at the beginning of the stage. It is possible to modify P t to implement such an alert or a roll-back. Program construction. We next outline the construction of the program P t from the given TF program P . Intuitively, we need to identify, in each stage of a run of P , the "transparent facts" that have been obtained in a transparent manner within that stage. Transparent facts can only be created or updated by "transparent events", in which all the facts used in the body are transparent. More precisely, a positive fact is transparent at a particular time within a stage if it is p-visible, or if all events that participate in defining that fact up to that time, within that stage, are transparent. A negative fact is transparent if it concerns a p-visible relation or if its key was transparently created and transparently deleted in the same stage (recall that by (C3') keys cannot be reused after being deleted). We next enrich the schema of P in order to keep track of the transparent facts. There are some subtleties in the process: (i) a p-invisible tuple may have portions that are transparent and portions that are not, (ii) stepprovenance has to be recorded at the level of attributes rather than of the entire fact, and (iii) the system remembers which are the keys that were created and then deleted transparently during that stage. The schema is modified as follows. Each relation R of P has a corresponding relation R t in P t . Tuples in R t will use the same keys as in R; intuitively, the tuple of key k of R t will hold information about the tuple of key k in R. For each attribute A of R, R t has an attribute tA. For each q, tA has the same visibility as A, i.e. tA 2 att(R t @q) i↵ A 2 att(R@q). For the key k, besides the attribute tK, the relation R t includes an attribute dK with the same visibility as K. Intuitively, the attribute tA indicates if the value of the corresponding attribute was produced transparently (its value is ?) or not (it has the particular value 1). Each p-visible fact is transparent by definition. The attribute dK is turned to 1 when the tuple is deleted transparently. Finally, for each A 2 att(R), R t has attributes A s 1 , ..., A s h in R t , where A s 1 holds the step id of the event that defined this attribute, and the others provide the step-provenance of that event (the list of step ids that lead to it). A rule r in R at q is transformed into a set of transparent rules in P t by adding new atoms and updates as follows. For each atom R@q(k, u) in the body, we add an atom R t @q(k, ...) to the body to record extra information for k, and for each atom ¬Key R@q (k), an atom R t @q(k, ...) to the body to record extra information about k, including the fact that the deletion of k was transparent. For each update +R@q(k, u), there exists an update +R t @q(k, ...), and for each update Key R@q (k) an update +R t @q(k, ...). The information included in +R t @q is explained further. Consider a fact R@q(k, u) in a p-invisible relation. Suppose that the tuple with key k in R t @q satisfies: for each A in att(R@q), the value of tA is ?, the tuple stage (as provided by K s 1 ) is the current stage id, and dK is ?. Then R@q(k, u) holds transparently. Now, consider ¬Key R@q (k) holds. Suppose that the tuple with key k in R t @q satisfies: tK is ?, the tuple stage (as provided by K s 1 ) is the current stage id, and dK is 1. Then ¬Key R@q (k) holds transparently. To detail the use of the A s i attributes, consider the firing of a transparent event and let H be the set of step-IDs occurring in its body augmented with the current step-ID. Then: • | H | h. • If the event modifies a non-key attribute A (there is a single step in the minimum p-faithful subrun of a stage that may do that), the set of non-? values in the A s i attributes of the resulting tuple is H. • If the event creates a tuple with a new key, the set of non-? values of the K s i attributes of the created tuple equals H. • If the event deletes a tuple already recording a set H 0 of step-IDs in its attributes K s i , the values in H H 0 are added in still-available places in the K s i . Note that this imposes that runs can only progress transparently as long as there is enough space available to record h step ids, which guarantees h-boundedness. It is also important to observe that, in transparent events, all updates are e↵ective. This is guaranteed because the program is in TF. The program also allows non-transparent events. These may come from the use of some non-transparent fact in the body. They may also come from the use of only transparent facts in the body, but such that the number of step ids occurring in them plus one is larger than h. When an event is not transparent, it is not allowed to modify a visible relation; it can only update other relations in an opaque manner. The program P can be modified to incorporate the above information, allowing to trace transparency status and step ids. All the necessary information can be maintained as outlined above. Each rule of P yields at most exponentially many new rules resulting from a case analysis on the transparency status of the attributes, and the number of steps constributing to their generation. One can show correctness of the construction, which yields Theorem 6.7 and Corollary 6.8. RELATED WORK A survey on data-centric business process management is provided in [START_REF] Hull | Data management perspectives on business process management: tutorial overview[END_REF], and surveys on formal analysis of data centric workflows are presented in [START_REF] Calvanese | Foundations of data-aware process analysis: a database theory perspective[END_REF][START_REF] Deutsch | Automatic verification of database-centric systems[END_REF]. Although not focused explicitly on workflows, Dedalus [START_REF] Alvaro | Dedalus: Datalog in time and space[END_REF][START_REF] Hellerstein | The declarative imperative: experiences and conjectures in distributed logic[END_REF] and Webdamlog [START_REF] Abiteboul | A rule-based language for web data management[END_REF][START_REF] Abiteboul | Viewing the web as a distributed knowledge base[END_REF] are systems supporting distributed data processing based on condition/action rules. Local-as-view approaches are considered in a number of P2P data management systems, e.g., Piazza [START_REF] Tatarinov | The piazza peer data management project[END_REF] that also consider richer mappings to specify views. Update propagation between views is considered in a number of systems, e.g., based on ECA rules in Hyperion [START_REF] Arenas | The hyperion project: from data integration to data coordination[END_REF]. Finite-state workflows with multiple peers have been formalized and extensively studied using communicating finite-state systems (called CFSMs in [START_REF] Parosh | Verifying programs with unreliable channels[END_REF][START_REF] Brand | On communicating finite-state machines[END_REF], and ecompositions in the context of Web services, as surveyed in [START_REF] Hull | Web services composition: A story of models, automata, and logics[END_REF][START_REF] Hull | Tools for composite web services: a short overview[END_REF]). Formal research on infinite-state, datadriven collaborative workflows is still in an early stage. The business artifact model [START_REF] Nigam | Business artifacts: An approach to operational specification[END_REF] has pioneered datadriven workflows, but formal studies have focused on the single-user scenario. Compositions of data-driven web services are studied in [START_REF] Deutsch | Verification of communicating data-driven web services[END_REF], focusing on automatic verification. Active XML [START_REF] Abiteboul | The Active XML project: an overview[END_REF] provides distributed datadriven workflows manipulating XML data. A collaborative system for distributed data sharing geared towards life sciences applications is provided by the Orchestra project [START_REF] Green | Orchestra: facilitating collaborative data sharing[END_REF][START_REF] Ives | The orchestra collaborative data sharing system[END_REF]. Our model is an extension of the collaborative datadriven workflow of [START_REF] Abiteboul | Collaborative data-driven workflows: think global, act local[END_REF]. The results in [START_REF] Abiteboul | Collaborative data-driven workflows: think global, act local[END_REF] focus on the ability of peers to reason about temporal properties of global runs based on their local observations, and are orthogonal to the present investigation. To enable static analysis, the model of [START_REF] Abiteboul | Collaborative data-driven workflows: think global, act local[END_REF] uses more restricted views that those considered here. Attaching provenance to facts derived in a rule-based language is considered in, e.g., [START_REF] Zaychik Mo Tt | Collaborative access control in Webdamlog[END_REF][START_REF] Abiteboul | A formal study of collaborative access control in distributed datalog[END_REF]. The paper [START_REF] Bourhis | Analyzing data-centric applications: Why, what-if, and how-to[END_REF] studies a notion of explanation in a model of datacentric workflows with a single user, no views, and no abstraction. They consider a notion of explanation that has some similarities to our faithful explanations, but is much simpler. Their results do not apply here. There has been extensive work on causality and explanations in logic and AI. More specific to data-centric workflows, the relationship between provenance, explanations, and causality is considered in [START_REF] Cheney | Causality and the semantics of provenance[END_REF]. The focus is on provenance of data resulting from complex processes, such as scientific workflows. The synthesis of view-programs described in Section 5 is related in spirit to partner synthesis in services modeled as Petri Nets [START_REF] Wolf | Does my service have partners? Trans. Petri Nets and Other Models of Concurrency[END_REF][START_REF] Lohmann | Wendy: A tool to synthesize partners for services[END_REF][START_REF] Sürmeli | Synthesizing cost-minimal partners for services[END_REF]. The issue of transparency of algorithms is gaining increased attention, see e.g., the Data Transparency Lab (datatransparencylab.org/) in the US, and the Transalgo Lab starting in France. Data transparency has been studied in di↵erent contexts. For instance, the causality of machine-learning-based decisions is studied in [START_REF] Datta | Algorithmic transparency via quantitative input influence: Theory and experiments with learning systems[END_REF]. Workflow transparency sometimes refers to the ability of considering a business process independently of the workflow implementing it, an aspect not considered here. Data transparency has also been considered in the context of workflows in [START_REF] Wolter | Collaborative workflow management for egovernment[END_REF], where an architecture for providing transparency in human-centric eGovernment workflows is proposed. CONCLUSION In this paper, we formally studied the problem of providing explanations of data-driven collaborative workflows to peers participating in the workflows, exploring semantic and computational issues. We identified faithful scenarios for a peer p as a particularly appealing basis for explanations from a semantic viewpoint. In a first contribution, we show that faithful scenarios form a semiring with respect to union and intersection, implying the existence of a unique minimal faithful scenario for each peer, computable in polynomial time, and enabling incremental maintenance of scenarios. In a second contribution, we identified desirable properties of workflows, namely transparency and boundedness, that guarantee the existence of a view program for a peer p, and showed how such a program can be constructed. Finally, we studied how programs satisfying transparency and h-boundedness for some peer p can be designed. We also show how, under certain restrictions, runs violating transparency or h-boundedness can be filtered out. A remaining open question is whether it is possible to perform such a filtering for arbitrary workflows. It is possible to implement workflow programs by relying on a master server that has access to all the information, receives the updates, propagates them to appropriate peers, and controls transparency and boundedness for certain peers. Blockchain technology provides an alternative to such a central authority. It is, in spirit, an excellent match with collaborative workflows. A blockchain-based implementation of collaborative workflows is therefore a promising research direction with challenging technical issues, notably with respect to performance and access control. hold for the valuation mapping all variables to true. Let R be a relation of arity n + 2, with key K, attributes A x for each x 2 X, and a last attribute A q . For each variable x 2 X, there is a peer p x that sees the projection of R over K, A x . There is a peer q that sees K, A q . In addition, there is a peer p that sees the projection of R on K with the selection p = (A q = 1) ^( _ ' ) where = ^x2X (A x = 1), and ' checks that the formula ' is true for the valuation ⌫ of X such that ⌫(x) = true i↵ A x = 1. The program consists of all ground rules of the form r xi : +R@p xi (0, 1) :-(for each x i 2 X) e : +R@q(0, 1) :-Consider the run ⇢ consisting of r x1 . . . r xn e. Observe that p sees R@p(0) only after the last event, because of the condition A q = 1 in its selection condition on R. We prove that ' is not satisfiable i↵ ⇢ is a minimal scenario of ⇢ at p. First suppose that ' is satisfiable. Let ⌫ be a valuation satisfying '. Consider the subsequence of r xi 1 . . . r xi k e obtained by keeping only the events r xi j such that ⌫(x ij ) is true. By (*), it is shorter than the original sequence. Let ⇢ ⌫ be the corresponding run. It is easy to see that ⇢ ⌫ is a strict subrun of ⇢ and that ⇢ ⌫ @p = ⇢@p. Thus ⇢ is not minimal at p. Now suppose that ' is not satisfiable. Let ⇢ 0 be a scenario of ⇢ at p. Because ' is not satisfiable, ' can never be satisfied, so in order for p to see R@p(0), it is necessary that hold, so ⇢ 0 must contain all events in ⇢. Thus, ⇢ is minimal. Proof of Lemma 4.6 Let ⇢ = {(e i , I i )} 0in and ↵ = {e ij } 0jm . We prove by induction on h (0  h  m) that ( †) ↵| h = {e ij } 0jh yields a subrun {(e ij , I 0 j )} 0jh of ⇢ and I 0 h @p = I i h @p. Suppose ( †) holds. Then ↵ yields a subrun of ⇢, establishing (i). Additionally, I 0 j @p = I ij @p for 0  j  m. This together with the fact that ↵ includes all events in ⇢ visible at p implies that run(↵)@p = ⇢@p. Thus, run(↵) is a scenario of ⇢ for p, establishing (ii) We now prove ( †). For the basis, let h = 0. We need to show that (a) e i0 is applicable to the empty instance and (b) I 0 0 @p = I i0 @p. For (a), suppose the body of e i0 contains a literal R@q(k, u). Then i 0 belongs to an Rlifecycle of k in ⇢, whose left boundary must be included in ↵, a contradiction. Thus the body of e i0 contains no literal of the form R@q(k, u) and is applicable to the empty instance, proving (a). Now consider (b). Consider t 2 I 0 0 @p(R@p) with key k. Then e i0 must insert in R a tuple t 0 with the same key, such that t 0 @p = t. Thus, i 0 belongs to the R-lifecycle of k in ⇢, and in fact it must be its left boundary (otherwise, by boundary faithfulness, e i0 must be preceded by another event in ↵, a contradiction). It follows that I 0 0 and I i0 both contain t 0 , so t 2 I i0 @p(R@p). Thus, I 0 0 @p ✓ I i0 @p. Conversely, let t 2 I i0 @p(R@p) with key k. Let h 0 (h 0  i 0 ) be the minimum in the same R-lifecycle of k in ⇢, for which a tuple with key k is visible by p in I h 0 @p(R@p). It follows that e h 0 is visible at p so it is included in ↵, so h 0 = i 0 and e i0 must insert a tuple t 0 with key k. Moreover, i 0 must also be the left boundary of the Rlifecycle of k (or else, that left boundary would have to be included in ↵ prior to e i0 ). It follows that t 0 @p = t. Thus, t 2 I 0 0 (R@p), and I i0 ✓ I 0 0 . This completes the basis. For the induction step, suppose {e ij } 0jh yields a subrun {(e ij , I 0 j )} 0jh of ⇢ where I 0 h @p = I i h @p, for h < m, and consider e i h+1 . For (a), we need to show that e i h+1 is applicable in I 0 h . Let R@q(k, u) occur in the body of e i h+1 . Then i h belongs to an R-lifecycle of k in ⇢, and, by modification faithfulness, all prior events of the Rlifecycle that a↵ect attributes in att(R, q) of tuples with key k are included in ↵. It follows that I 0 h (R) and I i h (R) both contain tuples with key k, that agree on att(R, q). Thus, since R@q(k, u) holds in I i h , it also holds in I 0 h . Next, suppose ¬Key R @q(k) occurs in the body of e i h+1 . Suppose k 2 I i h (Key R ). Then, similarly to the previous case, I 0 h (R) and I i h (R) both contain tuples with key k, that agree on att(R, q), so ¬Key R @q(k) holds in I 0 h . Now suppose that k 6 2 I i h (Key R ) but k 2 I 0 h (Key R ). Let v < h be the left boundary of the R-lifecycle in run(↵| h ) to which h belongs. It follows that i v belongs to an R-lifecycle of k in ⇢ but i h does not, so the Rlifecycle has a right boundary in ⇢ occurring before i h , which by boundary faithfulness must also belong to ↵. This contradicts the fact that h is in an R-lifecyle of k in run(↵| h ). So, k 6 2 I 0 h (Key R ) and ¬Key R @q(k) holds in I 0 h . In summary, e i h+1 is applicable in I 0 h . Now consider (b). Let t 2 I 0 h+1 @p(R@p) where t has key k. Thus, I 0 h+1 (R) contains a tuple t 0 with key k such that t 0 @p = t. From boundary and modification faithfulness it easily follows that I i h+1 (R) contains a tuple t 00 with key k that agrees with t 0 on att(R, p), so t 00 @p = t 0 @p = t and t 2 I i h+1 @p(R@p). Thus, I 0 h+1 @p ✓ I i h+1 @p. Conversely, let t 2 I i h+1 @p(R@p) with key k. Similarly to the base case, let h 0 be the minimum in the same R-lifecycle of k in ⇢, for which a tuple with key k is visible by p in I h 0 @p(R@p). It follows that e h 0 is visible at p so is included in ↵. Clearly, e h 0 must contain an insertion of a tuple with key k into R. From boundary and modification faithfulness it follows that I i h+1 (R) and I 0 h+1 (R) contain tuples with key k that agree on att(R, p), so t 2 I 0 h+1 @p(R@p). Thus, I i h+1 @p ✓ I 0 h+1 @p. This completes the induction and the proof of ( †). 2 Example 4 . 1 . 41 Consider again the worlflow used in the proof of Theorem 3.3. Suppose that p also sees the propositions Definition 4 . 3 . 43 Let ⇢ = {(e i , I i )} 0in be a run of some workflow program P . A subsequence ↵ = {e ij } 0jm of e(⇢) is boundary faithful if for every e ij and k 2 K(R, e ij ) such that i j belongs to an R-lifecycle 1 of k in ⇢, the left boundary event of the R-lifecycle belongs to ↵, and the right boundary event of the R-lifecycle also belongs to ↵, if the R-lifecycle is closed. Definition 4 . 5 . 45 A subsequence ↵ of e(⇢) is p-faithful if it contains all events of ⇢ that are visible at p, is boundary faithful, and modification faithful for p. A subrun b ⇢ of ⇢ is p-faithful if e(b ⇢) is p-faithful. Theorem 6 . 3 . 63 Let P be a linear-head program over schema D satisfying (C1). Let b be the maximum number of facts in a rule body of P , d = |D|, and a the maximum arity of a relation in D plus one. If P is pacyclic then it is h-bounded for p, where h = (ab + 1) d . R a new tuple with key k, this tuple is not deleted between i 1 and i 2 , and e i2 deletes it. An open R-lifecycle of k in ⇢ is an interval [i 1 , 1) such that e i1 inserts in R a new tuple with key k and this tuple is not deleted later on in ⇢. In both cases, we say that e i1 is the left boundary event of the lifecycle, and we say that e i2 is the right boundary event of the closed lifecycle [i 1 , i 2 ]. Observe that k K(R, ei j ) need not belong to an Rlifecycle containing ij, because k may occur in a negative literal ¬Key R@q (k). Acknowledgment. Serge Abiteboul and Pierre Bourhis are supported by the ANR Project Headwork, ANR-16-CE23-0015, 2016-2021. Victor Vianu is supported in part by the National Science Foundation under award IIS-1422375. APPENDIX A. APPENDIX We provide proof outlines for several results of the paper. Proof of Theorem 3.4 Membership in coNP is immediate. For hardness, we reduce the problem of testing unsatisfiability of a Boolean formula to testing whether a scenario for some peer p is minimal. Let ' be a formula over some set X = {x 1 , ..., x n } of variables. We assume without loss of generality that (*) ' does not Proof of Theorem 4.8 We first note the following useful fact, that follows immediately from the definition of T p (⇢, •). Lemma A.1. The operator T p (⇢, •) is additive: for all subsequences ↵ 1 , ↵ 2 of e(⇢), T p (⇢, ↵ 1 +↵ 2 ) = T p (⇢, ↵ 1 )+ T p (⇢, ↵ 2 ). We now turn to the proof of the theorem. Let ⇢ 1 an ⇢ 2 be p-faithful scenarios of ⇢. Consider first e(⇢ 1 ) + e(⇢ 2 ). By definition, e(⇢ 1 ) + e(⇢ 2 ) contains all events of ⇢ visible at p. By additivity of T p (⇢, •), T p (⇢, e(⇢ 1 ) + e(⇢ 2 )) = T p (⇢, e(⇢ 1 )) + T p (⇢, e(⇢ 2 )) = e(⇢ 1 ) + e(⇢ 2 ). Thus, e(⇢ 1 ) + e(⇢ 2 ) is also a fixed-point of T p (⇢, •) and so it is a p-faithful scenario of ⇢. Now consider e(⇢ 1 ) ⇤ e(⇢ 2 ). Since ⇢ 1 and ⇢ 2 are pfaithful scenarios, e(⇢ 1 ) and e(⇢ 2 ) are fixed-points of T p (⇢, •) and contain all events of ⇢ visible at p. Since e(⇢ 1 ) ⇤ e(⇢ 2 ) ⌧ e(⇢ 1 ) and e(⇢ 1 ) ⇤ e(⇢ 2 ) ⌧ e(⇢ 2 ), it follows that T ! p (⇢, e(⇢ 1 ) ⇤ e(⇢ 2 )) ⌧ e(⇢ 1 ) and T ! p (⇢, e(⇢ 1 ) ⇤ e(⇢ 2 )) ⌧ e(⇢ 2 ), so . Since e(⇢ 1 ) ⇤ e(⇢ 2 ) also contains all events visible at p, by Lemma 4.6, e(⇢ 1 ) ⇤ e(⇢ 2 ) yields a p-faithful scenario of ⇢. Finally, observe that multiplication distributes over addition, ✏ is the additive identity and ⇢ the multiplicative identity. 2 Proof of Theorem 5.10 We begin with two technical lemmas. The first essentially says that various properties of sequences of events are invariant under isomorphism. Lemma A.2. Let I be an instance and ↵ = e 1 . . . e n a sequence of events applicable at I. Let f be a bijection on dom that is the identity on const(P ). We denote by f (↵) the sequence of events obtained by applying f to every value occurring in ↵. Then the following hold: (minimum) p-faithful run on f (I), and the events visible at p are the same in the two runs. Proof. Straightforward induction on the length of ↵. We also need the following. Recall that, for each relation R, K(R, ↵) denotes the set of values occurring as keys of relation R in some event of ↵. For an instance I, we denote by I|K(↵) the instance retaining, for each relation R, only the tuples in I(R) with keys in K(R, ↵). Lemma A.3. Let I be an instance, ↵ a sequence of events, and I|K(↵) ✓ J ✓ I. The following hold: (i) if ↵ is a (minimum) p-faithful run on I then it is also a (minimum) p-faithful run on J, and the events visible at p are the same in the two runs. (ii) if ↵ is a (minimum) p-faithful run on J such that adom(I) \ new(↵) = ; then ↵ is also a (minimum) p-faithful run on I, and the events visible at p are the same in the two runs. Proof. Also by induction on the length of ↵. The only subtlety concerns new values. For (i), note that, if an event of ↵ creates a new value on I, that value is also new in the run on J, since J ✓ I. For (ii), the converse holds because adom(I) \ new(↵) = ; ensuring that the new values created in ↵ do not occur in I. We can now prove Theorem 5.10. By definition, P is not h-bounded i↵ ( ‡) there is an instance I and sequence ↵ of events, of length h + 1, that yields a minimim p-faithful run on initial instance I, such that all of its events but the last are silent at p. Let c m be the maximum number of values occurring in a sequence ↵ of events of length at most m and an instance I such that the tuples in I(R) use only keys in K(R, ↵) for each relation R. Let cm = |const(P )| + c m and C m consist of const(P ) together with c m additional distinct constants (so |C m | = cm ). By Lemmas A.2 and A.3 (i), it is sucient to check ( ‡) for sequences ↵ of events of length at most h + 1 and instances I, both using only values in C h+1 . This establishes decidability. The pspace upper bound follows from the fact that ch+1 is polynomial in h and P , which yields a non-deterministic pspace test. This completes the proof. 2 Proof of Theorem 5.11 Clearly, (ii) follows from (i) and Theorem 5.10. Consider (i). Let P be a program that is h-bounded for p. By a slight reformulation of the definition of transparency, P is transparent for p i↵ the following holds. ( †) For all instances I 1 , I 2 and events e 1 , e 2 such that e i is applicable at I i and visible at p (i = 1, 2) and e 1 (I 1 )@p = e 2 (I 2 )@p, and for each sequence ↵ of events such that adom(e 2 (I 2 )) \ new(↵) = ;, if ↵ is a minimum p-faithful run on e 1 (I 1 ) such that all its events but the last are silent at p, then the same holds on e 2 (I 2 ), and ↵(e 1 (I 1 ))@p = ↵(e 2 (I 2 ))@p. We show that ( † 0 ): ( †) holds i↵ it holds for all instances I 0 1 , I 0 2 such that, for each relation R, I 0 i (R) contains at most c|↵|+2 tuples. For suppose this holds. Since P is h-bounded, it is su cient to check ( †) for instances I 1 and I 2 with at most ch+2 tuples in each relation. The existence of counterexamples can be easily checked by a nondeterministic pspace algorithm. This completes the proof. We now show ( † 0 ). The "only if" part is trivial. Consider the "if" part. Suppose ( †) holds for all instances I 0 1 , I 0 2 such that, for each relation R, I 0 i (R) contains at most c|↵|+2 tuples. Let I 1 , I 2 be arbitrary instances, e 1 , e 2 events such that e i is applicable at I i and visible at p (i = 1, 2), e 1 (I 1 )@p = e 2 (I 2 )@p, ↵ is a minimum p-faithful run on e 1 (I 1 ) such that all but its last event are silent at p, and adom(e 2 (I 2 )) \ new(↵) = ;. We can assume without loss of generality that adom(I 2 )\ new(↵) = ;; otherwise, we can rename the values in I 2 and e 2 that occur in the intersection by a bijection that is the identity on const(P 1 and I 0 2 contains at most c|↵|+2 tuples. We next show that I 0 1 , I 0 2 satisfy the conditions of ( †). By Lemma A.3 (i), e i is applicable to I 0 i and is visible at p (i = 1, 2). Moreover, ↵ is a minimum pfaithful run on e 1 (I 0 1 ) and all but its last event are silent at p. We show that e 1 (I 0 1 )@p = e 2 (I 0 2 )@p. Let t 2 e 1 (I 0 1 )@p(R@p) with key k for some R. Observe that k 2 K 1,2 . Since e 1 (I 0 1 ) ✓ e 1 (I 1 ) and e 1 (I 1 )@p = e 2 (I 2 )@p, t 2 e 2 (I 2 )@p(R@p) and there is t 0 2 e 2 (I 2 )(R), with the same key k as t, such that t = t 0 @p. Suppose there is no tuple with key k in I 2 , so t 0 is created by e 2 . Then t 0 is also in e 2 (I 0 2 ) and t 2 e 2 (I 0 2 )@p(R@p). Now suppose there is a tuple t 00 with key k in I 2 (R). Since k 2 K 1,2 , t 00 2 I 0 2 (R) and so t 0 2 e 2 (I 0 2 )(R) and t 2 e 2 (I 0 2 )@p(R@p). We have shown that e 1 (I 0 1 )@p ✓ e 2 (I 0 2 )@p. The converse holds by symmetry, so e 1 (I 0 1 )@p = e 2 (I 0 2 )@p. Also, adom(e 2 (I 0 2 )) \ new(↵) = ;. Since I 0 1 and I 0 2 satisfy the condition of ( †), it follows that ↵ is a minimum p-faithful run on e 2 (I 0 2 ) such that all but its last event are silent at p, and ↵(e 1 (I 0 1 ))@p = ↵(e 2 (I 0 2 ))@p. By Lemma A.3 (ii), ↵ is also a minimum p-faithful run on e 2 (I 2 ) such that all but its last event are silent at p. It remains to show that ↵(e 1 (I 1 ))@p = ↵(e 2 (I 2 ))@p. Let t 2 ↵(e 1 (I 1 ))@p(R@p) with key k for some R. Thus, there exists a tuple t 0 2 ↵(e 1 (I 1 ))(R), with key k, such that t = t 0 @p. First suppose k 6 2 K 1,2 , then t 0 also belongs to e 1 (I 1 ). Since e 1 (I 1 )@p = e 2 (I 2 )@p, t 2 e 2 (I 2 )@p(R@p). Thus, there is t 00 2 e 2 (I 2 )(R) with key k, such that t 00 @p = t. Since k 6 2 K 1,2 , ↵ does not a↵ect t 00 , so t 00 2 ↵(e 2 (I 2 ))(R) and t 2 ↵(e 2 (I 2 ))@p(R@p). Now suppose k 2 K 1,2 . If there is no tuple in e 1 (I 1 )(R) with key k, then ↵ creates t 0 on any initial instance on which it is applicable, so t 0 2 ↵(e 2 (I 2 ))(R) and t 2 ↵(e 2 (I 2 ))@p(R@p). Suppose there is a tuple t 00 in e 1 (I 1 )(R) with key k. Since k 2 K 1,2 , t 00 2 e 1 (I 0 1 )(R). It follows that t 0 2 ↵(e 1 (I 0 1 ))(R) and t 2 ↵(e 1 (I 0 1 ))@p(R@p). Since ↵(e 1 (I 0 1 ))@p = ↵(e 2 (I 0 2 ))@p, t 2 ↵(e 2 (I 0 2 ))@p. Since ↵ is applicable to e 2 (I 2 ) and I 0 2 ✓ I 2 , it follows that ↵(e 2 (I 0 2 ))@p ✓ ↵(e 2 (I 2 ))@p, and t 2 ↵(e 2 (I 2 ))@p(R@p). In both cases, t 2 ↵(e 2 (I 2 ))@p(R@p), thus, ↵(e 1 (I 1 ))@p ✓ ↵(e 2 (I 2 ))@p. The converse holds by symmetry. Hence, ↵(e 1 (I 1 ))@p = ↵(e 2 (I 2 ))@p, which concludes the proof. 2 Proof of Theorem 5. [START_REF] Datta | Algorithmic transparency via quantitative input influence: Theory and experiments with learning systems[END_REF] We need to show that P @p is sound and complete for P and p. Consider completeness. For runs ⇢ of P and ⇢ 0 of P @p, we denote by ⇢@p ⇠ ⇢ 0 that fact that ⇢@p is obtained from ⇢ 0 by replacing all !-events with !. Let ⇢ = {(e i , I i )} 0in be a run of P . We can write e(⇢) as ↵ 1 .e i1 .↵ 2 .e i2 . . . ↵ n .e in .↵ n+1 where e ij are the events visible at p (1  j  n) and ↵ j are sequences of events invisible at p (1  j  n + 1). We define a sequence of events E 1 . . . E n yielding a run of P @p, such that Consider a fixed j > 1 for which e ij is not en event of p (the case j = 1 is a straightforward variation). Let e = e ij , e 0 = e i (j 1) , ↵ = ↵ j , I = I i (j 1) , I 0 = I i (j 1) 1 , and J = I ij . Let ↵ be the subsequence of ↵ such that ↵.e is the unique minimum p-faithful subrun of ↵.e on initial instance I. Since P is h-bounded for p, | ↵.e|  h. Let Ī = I|K(e 0 . ↵.e). By Lemma A.3 (i), ↵.e is a minimum p-faithful run of P on Ī, all events of ↵ are invisible at p, and e is visible at p. Also, Ī is a p-fresh instance, since it is easily seen that Ī = e 0 (I 0 |K(e 0 .↵.e)) and e 0 is visible at p. Let J = ↵.e( Ī). Observe that | Ī|  c h+1 . By Lemma A.2 we can assume without loss of generality that Ī and ↵.e use only constants in C h+1 . Consider the rule of P @p corresponding to the triple ( Ī, ↵.e, J). Consider the event E j obtained by applying to the variables of the rule the valuation ⌫ 1 . Clearly, the event is applicable to Ī@p and E j ( Ī@p) = J@p. It remains to show that E j (I@p) = J@p. Let Īc = I Ī. By definition, Īc contains no tuple a↵ected by ↵.e, so J = J [ Īc . Similarly, no tuple of Īc @p is a↵ected by E j . It follows that E j (I@p) = E j ( Ī@p) [ Īc @p = J@p [ Īc @p = J@p. This establishes completeness of P @p. Now consider the soundness of P @p. Let ⇢ p = {(E i , I i )} 0in be a run of P @p. We show that there exists a run ⇢ of We prove the statement by induction on j. Consider j = 0. Thus, E 0 (;) = I 0 . If E 0 is an event of p then the statment holds. Otherwise, by definition, there exists a p-fresh instance I and a minimum p-faithful run ↵.e of P on I, such that the tuples in I(R) use keys in K(R, ↵) for each relation R, and ↵.e(I)@p = E 0 (I@p). By construction, since E 0 has no positive atoms in its body, I@p = ;. By transparency of P , ↵.e is also a minimum p-faithful run of P on ; in which all events but e are invisible at p, and ↵.e(;)@p = ↵.e(I)@p = E 0 (;). This completes the base of the induction. For the induction step, let 0 < j < n suppose there is a run ⇢ j of P such that ⇢ j @p ⇠ ⇢ p | j . Let e(⇢ j ) = ↵ j and ↵ j (;) = J. So J is a p-fresh instance and J@p = I j . From the definiton of E j+1 , it can be shown similarly to the base case that there exists a p-fresh instance I over D, and a minimum p-faithful run ↵.e of P on initial instance I, in which all events but e are invisible at p, such that I@p = I j and ↵.e(I)@p = I j+1 . We would like to obtain a run corresponding to ⇢ p | j+1 by concatenating ↵ j with ↵.e. However, it could be that new(↵.e) \ adom(↵ j ) 6 = ;. Observe that new(↵.e) \ adom(I j ) = ; since adom(I j ) ✓ adom(I). Thus, there are two cases to handle: (i) new(↵.e) contains values in adom(⇢ p | j 1 ), (ii) new(↵.e) contains values in adom(↵ j ) adom(⇢ p | j 1 ) Case (i) can be handled by applying to ↵.e a bijection f on dom that is the identity on const(P ) [ adom(I) [ adom(I j+1 ) and such that adom(f (new(↵.e)))\adom(⇢ p | j 1 ) = ;, and using Lemma A.2. Case (ii) can then be avoided by applying to ↵ j a bijection g on dom that is the identity on const(P )[adom(⇢ p | j ) and such that adom(g(↵ j ))\ new(↵.e) = ;, and using again Lemma A.2. Thus, we can assume that new(↵.e) \ adom(↵ j ) = ;. By transparency of P , since I and J are p-fresh, I@p = J@p, and adom(J) \ new(↵.e) = ;, it follows that ↵.e is a run of P on J and ↵.e(J)@p = ↵.e(I)@p = I j+1 . Thus, ↵ j .↵.e yields a run of P and run(↵ j .↵.e) ⇠ ⇢ p | j+1 . This completes the induction and the proof of soundness of P @p. 2 Proof of Theorem 6.3 Consider an instance I and a sequence ↵.e of events applicable to I that yields a minimum p-faithful run at p, such that all its events but e are invisible at p. Observe that no event of ↵ has a relation visible at p in the head. Since ↵.e is a minimum p-faithful run, ↵.e = T ! p (↵.e, e). Let R be the relation occurring in the head of e. Since e is visible at p, R@p 2 D@p. It can be shown that T ! p (↵.e, e) = T g p (↵.e, e), where g is the maximum length of a path in the p-graph of P , starting from relation R. Intuitively, this is because every productive application of T p (↵.e, •) to an event corresponds to traversing at least one edge in the p-graph (from the relation in the head to some in the body). It follows that T p (↵.e, e) can only be applied g times before converging. Moreover, given a set E of events, T p (↵.e, E) adds to E at most b • |E| lifecycles of keys, each containing at most a events. It follows that |↵.e|  (ab + 1) g  (ab + 1) d . 2 Proof of Theorem 6.7 (sketch) Let P be a TF program, p a peer, and h an integer. Let P t be the program constructed from P as previously described. We show that (*) P t is transparent and h-bounded for p, and (**) each run of P that is transparent and h-bounded for p is the projection of a run of P t . Towards (*), consider a p-fresh instance I and a minimal p-faithful sequence ↵ of events of P t such that only the last one is visible for p. Transparency is satisfied by construction. For h-boundedness, observe that the subrun consisting of the events corresponding to the stepIDs occurring in the B i attributes of the last event of ↵ is observationally equivalent to ↵ for p. Therefore h-boundedness holds as well. Towards (**), first observe that the projection simply removes from the runs of P t , the relations R t for each R. Consider a transparent and h-bounded run ⇢ of P . Let its stages be ↵ i for i 2 [1. .n], and I i be the instance reached after ↵ i for each i. We construct a corresponding run ⇢ 0 of P t . For this, for each i, consider the minimal faithful subrun ↵ 0 i of ↵ i on input I i 1 . It is transparent and, by h-boundedness, its length is less or equal to h. We can therefore extend the events of P to transparent events of P t to capture the events in ↵ 0 i . Because its length is less that h, we dispose of enough space in the B i . For the other events, it is irrelevant whether we extend them using transparent or non transparent events. We thus obtain a run ⇢ 0 such that ⇢ = ⇧(⇢ 0 ), which concludes the proof. 2

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https://unilim.hal.science/hal-00765058/file/Schneider_HLUDOWICUS_Places.pdf

M Gravel S Kaschke Jens Schneider Places of Power in the Realm of Louis the Pious 1 ANR-DFG project "HLUDOWICUS. La productivité d'une crise : Le règne de Louis le Pieux (814-840) et la transformation de l'Empire carolingien -Produktivität einer Krise: Die Regierungszeit Ludwigs des Frommen (814-840) und die Transformation des karolingischen Imperium" (2008-2011) Nobody will deny the importance of space or the nearly constitutional function of several royal palaces in the ninth century, like Aachen during the reign of Charlemagne and Louis the Pious, like Compiègne for Charles the Bald or Regensburg for Louis the German. But today, in 2009, we have now had two thirds of a century of scholarship in which authority in the Early Middle Ages has generally been conceived as based on people and relations between them. Since the 1940s there has been a consensus among historians to focus on the relationships of the important noble families, a consensus which persisted through all epistemological fashions: the key word is network. German historical tradition after the war established the idea of the 'Personenverbandsstaat', a conception of a medieval constitution which is not a state in the modern sense 2 . This conception has been criticised and more or less abolished because, to point out just one reason, of the self-representation of the early medieval aristocracy as feeling a higher responsibility for the res publica3 . As Karl Ferdinand Werner demonstrated, this seems to have been an important marker of distinction for the noblesse4 . I do not pursue this idea here because it leads to a major theoretical argument, as part of which historians as Hans-Werner Goetz, Jörg Jarnut and Walter Pohl have defended the position that there were structures of 1 Paper presented at the International Medieval Congress, Leeds, 2009, July 14 th , as part of the session 'Louis the Pious and the Crisis of the Carolingian Empire' organized by the ANR-DFG project HLUDOWICUS. Among the important work published since 2010 on the problems this paper deals with I should like to mention GRAVEL, Distances, rencontres, communications, and MACLEAN, Palaces, Itineraries. to be published in M. Gravel & S. Kaschke (ed.), Politische Theologie und Geschichte unter Ludwig dem Frommen -Histoire et théologie politiques sous Louis le Pieux, Ostfildern (Thorbecke) 2013 2 administration and government that we can already call a state 5 . I just want to stress the point that it is too simple to say that early medieval society was only people getting along with each other. So, 'Personenverbandsstaat' is history. Yet, historians continue to analyse society, both from an institutional or constitutional point of view -the rules and norms of a society -and from an anthropological perspective, analysing the rituals and taboos organizing life in groups or clans. Much important and admirable work has been done in the study of elites and réseaux de parenté 6 . Braudel's conception of space 7 as a determinant factor of history is far away in today's historical research. I do not intend to deny for my part the importance of studying these networks, nor do I claim to proceed without noticing political actors and their interactions. But I think that different approaches are possible and necessary. Ethnological work has shown the importance of places for the representation of a society, but also for everyday life. This seems particularly true for the memory of the foundation of places. American Maya Indians of the classical period (4 th to 10 th century) lived in proper city-states showing an absolute respect for their foundation places; there were no signs of expansion although they were powerful enough to control dependant smaller cities. Instead, they proceeded to a permanent re-foundation within the limits of their city-states 8 . The Yanomami Indians in South America showed, on the other hand, dynamic spatial rituals 9 . They used the same name for both their habitat and their community. Once they moved to the next place, they would use another name for themselves, the old name becoming a period in the past. Another well-documented example is that of the Australian Aborigines who do not mark the frontiers of their territory but who have their own rituals of memorizing and remembering it. These are the famous 'songlines': secret traditional songs that contain a very well-defined way to 'walk their land', as they put it 10 . At the same time the song is a way of paying respect to the land and the ancestors. These three cases may show how earlier societies performed their given space. Joseph Morsel pointed out the meaning of the Aborigines' songlines as a technique that bound a community Christaller insists on one important point, which is that every central place is dynamic, in other words subject to change. Then there is to mention the work of Eckhard Müller-Mertens who developed the method of cartographical visualization of the 'Reichsstruktur' especially for the tenth century but proceeded to its application on other periods afterwards 16 . Every historian working on places or itineraries knows his maps, which are, though, limited on the material aspects, that's to say the king's movements and property. More recently, 'Topographies of Power' is a collection of some rather important case studies. Janet Nelson's analysis of Aachen as the constructed Carolingian place of power par excellence goes much further than Müller-Mertens, Falkenstein or Binding17 . Aachen has nevertheless not always been the lieu de mémoire that we are used to18 , just as Metz seems to lose political weight in the later ninth and tenth century: the concrete importance of these two Carolingian palaces has to be reviewed. Frans Theuws, taking the example of Maastricht in his contribution to the same volume, describes in anthropological terms the function of a relay, a centre of communicating spaces 19 . His point is to define the role of a centre by analysing both the more or less constructed symbolic past of a place and, at the same time, how it serves as a place of exchange, meaning both trade and social exchange. This essay is focussed on Louis the Pious and the question of which places were the most important for royal administration and authority. Based on the cited studies I put forward five criteria for a place of power in the Early Middle Ages. I suggest the following categories: (1) Episcopal sees (sedes episcoporum) (2) mints (3) royal palaces (palatia) (4) royal assemblies (placita) (5) royal charters (diplomata) These categories will be examined in the following. Of course, this is no more than a rough sketch of a possible approach that needs to be refined; other categories might be added, the main routes for instance, as well as other vectors of communication. Sedes episcoporum Even if nowadays one does not believe anymore in the long term constancy of the territorial shape of the dioceses, the ecclesiastical civitas remains an eminent functional structure in the early middle ages 20 . In the Frankish kingdom of the ninth century Liège on the river Meuse had just replaced Maastricht as Episcopal see 21 . In the North of modern-day France, Arras and Tournai cannot be taken into consideration because they were administrated by the bishops of Cambrai and Noyon until the eleventh century 22 . Mints Rosamond Palatia Far more than the mints, the royal palaces can be considered as symbols of royal or imperial authority 26 . The palatial system is older than the Carolingians: we know that the Merovingian kings used palaces in their cities, but also for their hunting expeditions, especially in the I now want to come back to my initial goal, that is to undertake a systematic approach to the presence of the king's body. In other words, I wish to ask how we might define his preferred and privileged places, the space covered by his immediate authority. to be published in M. Gravel & S. Kaschke (ed.), Politische Theologie und Geschichte unter Ludwig dem Frommen -Histoire et théologie politiques sous Louis le Pieux, Ostfildern (Thorbecke) 2013 The five categories discussed above were the Episcopal sees, the mints, the royal palaces, the assemblies and the enactment of charters. There are only two places which answer to four of the five criteria. It will not come as a surprise that these are Aachen and Worms. Mainz would be a third but we do not have any certain evidence for a royal palace there at the time of Louis the Pious; the emperor might easily have stayed at the monastery of St Albans or rather at his nearby palace at Ingelheim 34 . Worms is one of the important palaces where assemblies took place and charters were given; it is an Episcopal see and it had a prestigious royal palace, rebuilt after it burned in 790/91 35 . Aachen was not a bishopric but it had a famous royal palace and a mint. Given that in Aachen by far the highest number of assemblies were held and highest number of charters given it seems to be justified to declare Aachen as the real sedes regni at the time of Louis the Pious, just as medieval authors did 36 . By a remarkable distance it is followed by Worms, then Mainz which had a mint, too, and finally, given the number of assemblies held there, Nijmegen and Compiègne. So far, so good: but where is the crisis? Did nothing change in Louis' whereabouts? It has been said, especially by German historians working on the charters, that his reign can be seen as the succession of three phases: 'the energetic beginning, the "stagnation and paralysing tension in the 820s", and finally the crisis of the last decade', as Mayke de Jong has put it in a deconstructivist perspective 37 . As a consequence of the crisis in the 820s, the centre of Louis and reinstallation on the throne in 83513 .If we focus on the importance of space, in particular distinct places for the exertion and the representation of power in his realm, we should ask what kind of criteria and what kind of geographical, or -better -spatial, categories may be appropriate in defining places of power 14 . To avoid the network idea of important bishoprics and monasteries to describe important places I tried to establish another, more systematic approach. One of the most important inspirations certainly is the theory of central places, published by Walter Christaller for the first time in 1933 15 . He defines, from a geographical point of view, nine criteria to describe a central place: institutions of (political) administration, cultural and ecclesiastical (!) institutions, institutions of public health and welfare, of social and economic live, of trade and monetary circulation, of production, of labour market, of traffic, and of market regulation. to be published in M. Gravel & S. Kaschke (ed.), Politische Theologie und Geschichte unter Ludwig dem Frommen -Histoire et théologie politiques sous Louis le Pieux, Ostfildern (Thorbecke) 2013 4 McKitterick recently reminded us that mint distribution 'mirrors the concentration of the king's movements'23 . Thanks to the cooperation with a project on early medieval minting we have a revised list of the mint places under Louis the Pious 24 . On the first map we can observe a rather homogeneous net all over the empire but nothing east of the Rhine, except Regensburg on the river Danube. The mint places are essentially bishoprics and only a few trading places like Dorestad on the lower Rhine and Quentovic on the Channel coast. In Aquitaine there is a well-known mint at Melle (Metallum) near Poitiers, but not Limoges, though it had been known as a mint place at the end of the eighth century when Charlemagne struck coinage for his son Louis king of Aquitaine 25 . On the other hand we can see that coins were minted at Dax, which had lost its Episcopal see at the time of Louis the Pious. Ardennes and the Vosges, in Lotharingia 27 . Still, it was significant if a king built a new palace20 Cf MAZEL (ed.), L'espace du diocèse; PATZOLD, Raum der Diözese; ID., Episcopus. Charlemagne did in Paderborn and Nijmegen, or if he chose to enlarge an inherited palace as was done by Louis' sons, Louis the German with Frankfurt and Charles the Bald with Compiègne 28 . The difficulties in compiling a reliable catalogue of royal palaces are well known, as well as the problems of distinction between a completely equipped palatium, a smaller aula or just a villa 29 . For the present purpose, the extremely useful list of charters of Louis the Pious containing some 417 items, established by Jens Peter Clausen and Theo Kölzer, served as a basis, since the new edition prepared at Bonn is not yet published 30 .4. PlacitaAnother issue of Kölzer's editing project for the Monumenta Germaniae historica is the study of the assemblies held by Louis the Pious. From Daniel Eichler's table of 61 assemblies, the one held at Compiègne from October to November 833 has not been counted for the present purpose because Louis the Pious was absent 31 . It was then when Lothar and the Frankish bishops listed all the sins and faults of the dethroned emperor 32 . Eichler counts one each at Worms in the East and Compiègne in the West. Outside this frequently visited space there seem to have been limitrophic assemblies held on the Loire, the Saône and Rhone valley or at Augsburg and Paderborn. The later on in his reign, we deduce from the dark grey points, the more Louis the Pious was obliged to travel to the South-West of his empire. DiplomataWe get confirmation of this phenomenon in analysing the production of the chancery of Louis the Pious. The above-mentioned list by Kölzer contains 417 charters in total. Our third map shows the places where royal charters were given in four chronological phases, from yellow to red. At the same time, the map allows us to compare the principal places of enactment. If we relate the number of issued charters to the frequency and length of the visits Louis paid to these places, we can confirm that he preferred Aachen by far. Kölzer lists 191 charters given in this palace: Aachen is thus 'unbestrittene Hauptresidenz', followed by the middle Rhine region with Ingelheim (23), Frankfurt (23) and Worms (11)33 . At Thionville, Louis the Pious gave 12 charters and held five assemblies. These numbers confirm that the Thionville palace was an important element of the described triangular space where Louis spent most of his time: already more important than Metz that is outside of this central region. 5 possible criteria of a place of power Frankish Empire under Louis the Pious (814-840) Aachen Compiègne Frankfurt Ingelheim Mainz Thionville Worms Episcopal see x x Mint x x Royal palace x x x x ? x x Assemblies 18 5 2 5 2 5 4 Charters 191 9 23 23 12 11 Criteria 4 3 3 3 3 3 4 21 KUPPER, Liège, 78. 22 MERIAUX, Gallia irradiata, 20. 23 MCKITTERICK, Charlemagne, 168. 24 DFG projekt 'Die Merowingischen Monetarmünzen als interdisziplinär-mediaevistische Herausforderung' (2007-2009). Cf JARNUT / STROTHMANN (ed.), Monetarmünzen. -Thanks to Jürgen Strothmann, University of 28 BAUTIER, Le poids de la Neustrie, 557; BINDING, Deutsche Königspfalzen, 118; ZOTZ, Le palais et les elites, Paderborn (now University of Siegen). 233; BARBIER, Le sacré dans le palais franc; MCKITTERICK, Charlemagne, 161. 25 COUPLAND, Trading Places, 221; cf ID., Money and coinage, 24 and Coinages of Pippin, 197. 26 Among the enormous bibliography I only mention BARBIER, Le système palatial; BRÜHL, Palatium und Civitas; EHLERS (ed.), Places of Power; ID. (ed.), Orte der Herrschaft; FENSKE / JARNUT / WEMHOFF (ed.), Splendor palatii; STAAB (ed.), Die Pfalz; ZOTZ, Thomas, Pfalz und Pfalzen, in: RGA² 22 (2003) , 640-645. 27 HENNEBICQUE [R. LE JAN], Espaces sauvages; WENSKUS, Reinhard, Forst, § 2: Historisches, in: RGA² 9 (1995), 348-350. to be published in M. Gravel & S. Kaschke (ed.), Politische Theologie und Geschichte unter Ludwig dem Frommen -Histoire et théologie politiques sous Louis le Pieux, Ostfildern (Thorbecke) 2013 as Paderborn, Augsburg, Vannes, Orléans, Tours, Le Palais-sur-Vienne, Langres, Tramoyes et Chalon-sur-Saône; two at Attigny, Frankfurt and Mainz; three at Quierzy; four at Worms; five at Nijmegen, Ingelheim, Compiègne and Thionville; and nearly four times as many again (18) held at Aachen. The second map thus shows, somewhat surprisingly, a concentration of Louis' 'parliamentary' activity, to use an anachronistic term, more or less along the middle and lower Rhine valley. After the nearly exclusive assembly place of Aachen in the first years we can observe a kind of triangle with the edge points at Nijmegen in the North, Frankfurt-Ingelheim-29 ZOTZ, Vorbemerkungen; BINDING, Deutsche Königspfalzen, 21-26. See for such a catalogue BARBIER, La fortune du prince (583 items, forthcoming). 30 KÖLZER, Kaiser Ludwig der Fromme, unpaginated appendix [40-65]. -Thanks to professor Theo Kölzer, University of Bonn. 31 EICHLER, Fränkische Reichsversammlungen, 111-113. 32 Relatio episcoporum a. 829 (see the present volume), 40-41. Cf DE JONG, penitential state, 228-234. to be published in M. Gravel & S. Kaschke (ed.), Politische Theologie und Geschichte unter Ludwig dem Frommen -Histoire et théologie politiques sous Louis le Pieux, Ostfildern (Thorbecke) 2013 9 5. activities would have been in the Eastern part of the Empire 38 . Is this so? Instead of a conclusion, a last map may bring some clarification. Louis the Pious has been officially and ritually dethroned in 833, a date that can be considered as a kind of institutional caesura. One might ask whether the royal charters given from 835 on indicate a change in Louis' government. Our map showing a correlation of the actum places with the chronological evolution of the charter production clearly confirms the observation Kölzer already made on the basis of his list, that Aachen becomes less important after 834 39 . If we compare the total number of 191 charters to only 17 in the last seven years, being 20 % of his reign from 814 to 840, we can ascertain that Aachen was losing its central status.The same observation is true for Frankfurt where he had built a new palace but did not have the occasion to spend much time: only four charters after 833. Worms, Ingelheim and Nijmegen each show two to three charters, rather a quantité négligeable compared to their former importance. Quierzy and Thionville are two palaces that already figured on the maps shown here; they maintain their status with five and six charters after 834.The fourth and last map we made finally resumes the six most frequent enactment places of the charters given from 834 until his death on the 20 th of June 840. The surprise here is Attigny, and especially Poitiers, both places where Louis did not give many charters before 833. With four and six charters in his last years they seem to have become more important for the Emperor. After the Attigny assembly of 822 when he sent out Lothar to Italy and Pippin to Aquitaine, Louis convoked another assembly there in November 834 40 . The Poitou region was familiar to the former King of Aquitaine who was born in the palace of Chasseneuil, some eight kilometres from Poitiers. Certainly, one should not overstrain this observation. Louis spent his last winter in Poitiers together with his wife Judith because he held a military campaign against Pippin II whose claim to his father's throne Louis would not accept 41 . Still, this short analysis of the charters proves that we cannot state that Louis the Pious moved his presence and political action to the East. The maps shown here, even if they represent a preliminary sketch, indicate a clear shift to the South-West after 833: more and more assemblies were held in what would become the Francia occidentalis of Louis' son Charles the Bald. The political equilibrium between the different geographical parts of his empire that has been seen as one big issue of the reign of Louis the Pious 42 appears so as the result of his last years' activities.The observation of Aquitaine as a region somehow apart during the process of supposed fragmentation shall have to be proved. There is the testimony of the assemblies, the charters and the mints, but the question is if, or why, Aquitaine was left out by Louis as long as Pippin I st was King there. It is remarkable that the only coins bearing a choronymic indication of the mint place during the reign of Louis the Pious are the AQVITANIA coins minted under Pippin I st in Bordeaux and perhaps Bourges, if we believe Simon Coupland 43 . The old Emperor trying to reintegrate the regnum of his beginnings recalls the case of Charles the Simple in Western Francia who, once he had acquired his 'greater heritage' 44 , took refuge in Lotharingia in the last years of his reign.All the maps shown here are part of the HLUDOWICUS project. I would like to thank Rémi Crouzevialle, University of Limoges, for the cartographical collaboration. 34 I am obliged to professor Ludger Körntgen, Mainz, for confirming this opinion. Cf BRÜHL, Palatium und Civitas, 108-111, who suggested a 'Klosterpfalz' in Mainz. 35 BRÜHL, Palatium und Civitas, 128; EICHLER, Fränkische Reichsversammlungen, 60. 36 Regino Prumiensis, Chronicon, 98 (a. 869); cf Nithard, Histoire, 116 (lib. 4, c. 1), a scribe's marginal note: sedes prima Frantię. 37 Paper given at the International Medieval Congress, Leeds, 2009, session 706 organized by HLUDOWICUS on map 2: assemblies map 3: charter production and actum places July, 14 map 1: mint places map 4: frequent actum places after 834 th : 'Capitularies, Charters and the So-called Crisis of the Reign of Louis the Pious'. 38 KÖLZER, Kaiser Ludwig der Fromme, 32. 39 KÖLZER, Kaiser Ludwig der Fromme, 32. to be published in M. Gravel & S. Kaschke (ed.), Politische Theologie und Geschichte unter Ludwig dem Frommen -Histoire et théologie politiques sous Louis le Pieux, Ostfildern (Thorbecke) 2013 13 40 DEPREUX, Lieux de rencontre, 214; EICHLER, Fränkische Reichsversammlungen, 42. 41 Annales Bertiniani, 34-35 (a. 839). 42 BAUTIER, Le poids de la Neustrie, 555. 43 COUPLAND, Coinages of Pippin, 205-207. to be published in M. Gravel & S. Kaschke (ed.), Politische Theologie und Geschichte unter Ludwig dem Frommen -Histoire et théologie politiques sous Louis le Pieux, Ostfildern (Thorbecke) 2013 14 44 Actes de Charles III le Simple: largiore vero hereditate indepta (eschatocol, used from 911 on). I would like to thank Simon MacLean, St Andrews, who kindly accepted to read and correct this paper. MAYER, Ausbildung der Grundlagen; TELLENBACH (ed.), Studien und Vorarbeiten. Nithard, Histoire,[142][143][144] c. 7). WERNER, Naissance de la noblesse, MORSEL, Construire l'espace, 295. For a survey see SCHNEIDER, Suche nach dem verlorenen Reich, 242-251; cf HEIMANN / SCHNEIDER, Kloster -Landschaft. 13 SCHIEFFER, Die Karolinger, 131; DE JONG, penitential state, 214- 259. 14 Cf DE JONG / THEUWS (ed.), Topographies of power; EHLERS (ed.), Places of Power. 15 CHRISTALLER, Die zentralen Orte; cf STEUER, Zentralorte. MÜLLER-MERTENS, Die Reichsstruktur. NELSON, Aachen; MÜLLER-MERTENS, Die Reichsstruktur; FALKENSTEIN, Charlemagne et Aix-la-Chapelle; BINDING, Deutsche Königspfalzen. MCKITTERICK, Charlemagne, 158.

01745104

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2024/03/05 22:32:07

2013

https://hal.science/hal-01745104/file/2013-june-pimrc-workshop_salah.pdf

Vineeth S Varma email: [email protected] Salah Eddine Elayoubi email: [email protected] Merouane Debbah email: [email protected] Samson Lasaulce email: [email protected] On the Energy Efficiency of Virtual MIMO Systems The major motivation behind this work is to optimize the sleep mode and transmit power level strategies in a small cell cluster in order to maximize the proposed energy efficiency metric. We study the virtual multiple input multiple output (MIMO) established with each base station in the cluster equipped with one transmit antenna and every user equipped with one receive antennas each. The downlink energy efficiency is analyzed taking into account the transmit power level as well as the implementation of sleep mode schemes. In our extensive simulations, we analyze and evaluate the performance of the virtual MIMO through zero-forcing schemes and the benefits of sleep mode schemes in small cell clusters. Our results show that for certain configurations of the system, implementing a virtual MIMO with several transmit antennas can be less energy efficient than a system with sleep mode using OFDMA with a single transmitting antenna for serving multiple users. I. INTRODUCTION The energy consumed by the radio access network infrastructure is becoming a central issue for operators [START_REF] Saker | System selection and sleep mode for energy saving in cooperative 2G/3G networks[END_REF]. The goal of this work is to provide insights on how to design green radio access networks, especially in the framework of virtual MIMO systems. Indeed, classical network architectures are focused on integrated, macro base stations, where each cell covers a predetermined area, and inter-cell interference is reduced by the means of fixed frequency reuse patterns [START_REF] Elayoubi | Uplink intercell interference and capacity in 3G LTE systems[END_REF]. Heterogeneous Networks (HetNets) introduced a new notion of small cells where pico or femto base stations are deployed within the coverage area of the macro base stations [START_REF] Saker | Optimal control of wake up mechanisms of femtocells in heterogeneous networks, Selected Areas in Communications[END_REF]. Virtual MIMO is a step forward in this context that allows distributed systems of base stations/antennas that cover a common area and cooperate in order to increase the overall spectral efficiency [START_REF] Wang | Cooperative MIMO channel models: A survey[END_REF]. This paper focuses on these latter solutions and aims at addressing the problem from an energy efficiency point of view. For classical macro networks, early works focused on designing energy-efficient power control mechanisms [START_REF] Goodman | Power Control for Wireless Data[END_REF]. Therein, the authors define the energy-efficiency of a communication as the ratio of the net data rate (called goodput) to the radiated power; the corresponding quantity is a measure of the average number of bits successfully received per joule consumed at the transmitter. This metric has been used in many works. Although fully relevant, the performance metric introduced in [START_REF] Goodman | Power Control for Wireless Data[END_REF] ignored the fact that transmitters consume a constant energy regardless of their output power level [START_REF] Desset | Flexible power modeling of LTE base stations[END_REF]. The impact of this constant energy has been studied for single user point-to-point MIMO systems in [START_REF] Varma | An Energy Efficient Framework for the Analysis of MIMO Slow Fading Channels[END_REF]. Sleep mode mechanisms have thus been regarded as a solution for this issue; they consist in deactivating network resources that have low traffic load, eliminating thus both the variable and constant parts of the energy consumption [START_REF] Saker | System selection and sleep mode for energy saving in cooperative 2G/3G networks[END_REF]. This mechanism has been applied to macro networks [START_REF] Saker | System selection and sleep mode for energy saving in cooperative 2G/3G networks[END_REF], as well as to heterogeneous networks with macro and small cells [START_REF] Saker | Optimal control of wake up mechanisms of femtocells in heterogeneous networks, Selected Areas in Communications[END_REF]. Our aim in this paper is to extend this concept to virtual MIMO networks, where an antenna that is not significantly contributing to the network capacity (for a given configuration of user positions and radio channels) is put into sleep mode. The remainder of this paper is organized as follows. In section II, we present the system model and the resource allocation scheme. Section III presents our energy efficiency metric and optimizes it for a given system and channel configuration, using sleep mode mechanisms. Section IV presents some numerical examples and section V eventually concludes the paper. II. SYSTEM MODEL The wireless system under consideration is the downlink in a virtual MIMO system within a small cell cluster. To be precise, each of the small cell base stations are connected to a central processor and so they act as antennas for the virtual MIMO as shown in Fig 1 . We refer to the set of these base Fig. 1. An example illustration of a 2×2 virtual MIMO with g i,j representing the channel between BS antenna i and user j. stations as the "cluster". Each user is equipped with a single receive antenna. In order to eliminate interference zero-forcing is implemented. We consider a block-fading channel model where the channel fading stays is assumed to stay constant for the duration of the block and changes from block to block. The base stations require the channel state information available at the user end in order to implement the zero-forcing technique. Therefore, in each block channel a training and feedback mechanism happens, after which data is transmitted. We also assume that every base station is capable of entering into a "sleep-mode". In this mode, the base station does not send any pilot signals and therefore does not perform the training or feedback actions consuming a lesser quantity of power compared to the active base stations. Let there be M base stations in the cluster and K users. Define K = {1, 2, . . . , K} and M = {1, 2, . . . , M } the sets of users and base station antennas. A. Power consumption model As the transmit antennas are not co-located, each of them have an individual power budget. When a base station is active, it consumes a constant power of b due to the power amplifier design and training or feedback costs. Additionally, it consumes a power P m x m 2 proportional to the radiated power, where P m ≤ P max and x m ≤ 1 is the signal transmitted and P max is the power constraint [START_REF] Saker | System selection and sleep mode for energy saving in cooperative 2G/3G networks[END_REF] [START_REF] Desset | Flexible power modeling of LTE base stations[END_REF]. When it is placed on sleep mode, it is assumed that it only consumes power c where c < b. Denote by s the sleep mode state vector of the cluster with elements s m ∈ {0, 1}. The base station m is in sleep mode when s m = 1 and active when s m = 0. Thus the power consumption of the m-th base station is cs i + (1 -s i )(b + P m x m 2 ). The total power consumption of the cluster is given by: P tot = M m=1 cs m + (1 -s m )(b + P m x m 2 ) (1) For any given state of the cluster, we define ω(s) as the total number of base stations that are active. This value can be calculated as ω(s) = Mm s m . If M < K, zero-forcing can not be used. However, if M > K, and ω(s) ≥ K, then the zero-forcing technique can be implemented by choosing K base stations to transmit the data signals after all ω(s) active base stations train and obtain feedback on their channels. B. The zero-forcing scheme As there are K users connected to the ω(s) active base stations, there are a total of ω(s) × K number of channels. Let ζ = {1, 2, . . . , ω(s)} be the set of active base stations. We denote by G with elements g l,k ∈ R the path fading between base station l ∈ ζ and user k ∈ K, while H with elements hl,k ∈ C denotes the fast fading component, resulting in a signal at k given by: y k = ω(s) l=1 g l,k P l σ 2 hl,k x l + z (2) where x is the signal transmitted with x l as its elements; z is the normalized noise and σ 2 the noise strength. Note that g l,k can be determined based on the user location while hl,k are i.i.d. zero-mean unit-variance complex Gaussian random variables. We define H(G, H) as the combined channel matrix with elements hl,k = √ g l,k hl,k . In our work, as we focus on the small-cell scenario where the antennas are distributed over the cell in a dense manner, we assume that every user can have a similar level of signal strength. Define N = {1, 2, . . . , N } as the set of transmitting antennas that perform zero forcing beam-forming. We define β ∈ N → ζ as the function that describes which base stations in ζ will be picked to transmit the data signals. Given BS j ∈ N , the corresponding label for the BS in ζ is given by β(j). Given ω(s) active base stations that perform training and receive feedback on H, we define the effective channel matrix as H( H, β), where its elements h j,k = hβ(j),k . For zeroforcing, we require that the number of transmitting antennas is equal to the number of receiving antennas or K = N . With this constraint, if H is an invertible matrix, we define: x = (H( H, β)) -1 u (3) where u is a vector of length K, where u k which is the signal received by the k-th user. In this work, we take u k 2 = 1 so that each user obtains identical signal strengths. This constraint has several benefits: 1) This results in a very "fair" beam forming scheme as all users experience equal signal strength and thus similar data rates. 2) As the base station antennas are spread over the cell, there is no user on the "cell edge" or "cell center". In this situation, equal SINRs for all users can result in less congestion when user traffic patterns are taken into account 3) Finally, the resulting system is far less complex and easier to analyze than one with arbitary values for u k 2 . With these definitions, we can now precode the transmitted signal as: x = x α( H, β) (4) where α( H, β) = max(x m ). This pre-coding works only if all the P j are equal, and so we chose P j = P 0 . From this point onwards, P 0 represents the transmit power level of the system with this pre-coding. The signal received by each of the K users is given by: y k = P 0 σ 2 u k α( H, β) + z (5) Thus the SINR at each user is now given by γ k = P 0 α( H, β) 2 σ 2 (6) III. ENERGY EFFICIENCY OPTIMIZATION This work aims at minimizing the energy consumption by base stations. If each user in the network is connected to download some data, then the total energy consumed by the network is the total power consumed multiplied by the total duration for which the user stays connected. Energy efficiency (EE) is a metric that is often used to measure this, and maximizing the energy efficiency leads to minimizing the total energy consumed. A. Defining the EE metric Before defining the EE, we first calculate the total power consumption of the network. From ( 1) and (4), the total power consumed is given by: P tot (P 0 , H, β) = M m=1 cs m + (1 -s m ) ×   b + P 0 (H -1 ( H, β)u) β -1 (m) α( H, β) 2   (7) Here we define ∀m ∈ M; β -1 (m) = j if j ∈ N exists s.t β(j) = m 0 otherwise. (8) and () j is the j -th element if j = 0 and is 0 if j = 0. In this scenario, we define the instantaneous energy efficiency as: η(P 0 , H, β) = k f (γ k (P 0 , H, β)) P tot (P 0 , H, β) (9) where f () gives the effective throughput as a function of the SINR. f (γ k ) = log(1 + γ k ) for example. However when we study the base station energy efficiency for a longer duration, the effects of fast fading in H gets averaged and in this case a more reasonable definition for the EE is: η(P 0 , G, β) = E H[ k f (γ k (P 0 , H(G, H), β))] E H[P tot (P 0 , H(G, H), β)] (10) B. Optimization w.r.t the transmit power In this section, we show some properties of our energy efficiency metric w.r.t P 0 . If the goal of a system is to be energy efficient using power control, then one important question arises: Is there a unique power for which the energy efficiency is maximized ? We answer this question with the following proposition: Proposition 1: Given a certain path loss matrix G and a selection of transmitting base stations β in the virtual MIMO system, the average EE η(P 0 , G, β) is maximized for a unique P * 0 and is quasi-concave in P 0 . Proof: Consider the SINR for each user γ k . It can be observed that ∂γ k (P0, H,β)) ∂P0 is a constant. So if f () is concave in γ k , it is also concave in P 0 . Now consider the numerator of (10), E H[ k f (γ k (P 0 , H(G, H), β))] . This is a weighted sum of several concave functions and is hence also concave itself. Similarly, ∂Ptot(P0, H,β) ∂P0 can also be verified to be a constant. Hence, ∂ E H[Ptot (P0, H,β)] ∂P0 is a constant. Thus η(P 0 , G, β) is the ratio of a concave function of P 0 to a linear function of P 0 . This is known to be quasi-concave and has a unique maximum P * 0 satisfying: ∂ η(P * 0 , G, β) ∂P 0 = 0 (11) This proposition guarantees that given a certain choice of transmit antennas and a path fading matrix, the energy efficiency can always be optimized w.r.t the transmit power level P 0 . C. Optimizing the selection of transmitting base stations Given a certain sleep mode state s, there are ω(s) base stations active that train and obtain feedback. From this set ζ, K base stations have to be picked for zero-forcing. This choice is mathematically expressed by β. The β that optimizes the energy efficiency depends on the channel state H. The following proposition details the method of choosing the β that optimizes EE. Proposition 2: When P0 b → 0, the β * that maximizes η(P 0 , G, β) is obtained by: β * = arg min[α( H, β); β ∈ {N → K}] (12) Proof: By observing ( 6) it can be seen that γ k (P 0 , H, β) is maximized by picking β * . And so k f (γ k (P 0 , H, β)) is maximized when β = β * . When P0 b → 0, for β * and any β 1 we have: lim P 0 b →0 η(P 0 , G, β * ) -η(P 0 , G, β 1 ) = (13) E H[ k f (γ k (P 0 , H, β * )) -k f (γ k (P 0 , H, β 1 ))] M m=1 cs m + (1 -s m )b ≥ 0 (14) This implies that we pick β such that α( H, β) is minimized for optimizing EE when b >> P 0 . From a practical point of view, the above result is useful as it proposes an algorithm to pick the right base stations based on the CSI obtained from all the base stations that are active. The condition b >> P 0 is most applicable when the users are close to the base stations resulting in a low P 0 being used for maximizing EE. IV. NUMERICAL RESULTS In this section we use simulations and numerical calculations to study the effectiveness of our proposal as well as the advantages offered. For the purpose of a thorough numerical study, we will analyze two kinds of system settings A and B. For both the configurations the common parameters are: 1) c = b 10 W 2) P max = 2 W 3) f (γ) = B log(1 + γ) 4) σ 2 = 1 mW Where B = 10 6 hz is the bandwidth. The fast fading co-efficient we consider is hi,j = o(π m,k )Ω + 0.1ξ. Where ξ ∈ CN (0, 1), a is the direct line of sight factor which plays a dominant role in most small cell networks, o π m,k ∈ 0, 1 is the shadow factor and o(pi m,k ) = 1 with probability π m,k . Here π m,k is the probability that the receiver k has line of sight with the BS antenna m. We take π m,k = 0.5 ∀(k, m) for our simulations. The presented results study the case of two users K = 2 served by a small cell cluster of three base stations, i.e M = 3. In addition to zero-forcing, when there are two users a single base station could also alternately use Orthogonal Frequency-Division Multiple Access (OFDMA) to serve the two users and keep the other two BS in sleep mode (i.e . Our numerical simulations study all these possible scenarios and compare their respective EE performances. In both of the settings presented below, we study two main regimes of interest: 1) b = 1W : This regime represents the futuristic case where power amplifier efficiencies are quite high and the constant power consumed is lower than the maximum RF output power. 2) b = 10W : This regime represents the more current state of the art w.r.t power amplifier efficiency where in small cell antennas, a large portion of the power is lost as a fixed cost. We also consider two possible values of Ω, the line of sight factor. The case Ω = 10 is representative of pico-cells that are deployed externally, whereas the case Ω = 0 represents femtocells deployed internally and no line of sight communication is possible. A. Setting A The deployment of antennas and the user locations are shown in Fig 2 . In this setting we take g m,k = 1 ∀m, k. In Fig 3 we study the EE of a VMIMO system with a very efficient power amplifier. In this figure, we notice that using all available base station antennas is more efficient when line of sight communications are possible. In this case, no BS is in sleep mode and all the antennas train and obtain feedback on their channels. The choice of β is very much relevant in this scenario. However in the regime where there is no direct line of sight (users are inside buildings), it becomes more efficient to just use 2 BS antennas and put one on sleep mode. As the configuration is symmetric, the choice of the base station in sleep mode is not relevant. In Fig 4 we study the EE of a VMIMO system with an inefficient power amplifier. In this figure, we notice that using all available base station antennas is not efficient even when line of sight communications are possible. In this case, having one BS in sleep mode and obtaining a 2×2 virtual MIMO with the other remaining antennas is the most efficient. Suprisingly, in the case of no line of sight, i.e Ω = 0, we observe that using OFDMA with one BS active is the most efficient solution. This is explained by the relative inefficiency of zero-forcing in the low SNR regime, causing less energy to be spent by having two BS antennas in sleep mode and just one antenna transmitting for the two users in orthogonal frequencies. B. Setting B The deployement of antennas and the user locations are shown in Fig 5 . In this setting we take g 1,1 = g 2,1 = 4, g 3,1 = g 1,2 = g 2,2 = 0.1 and g 3,2 = 10. In Fig 6 , similarly to what was done in the previous setting, we study the EE of a VMIMO system with a very efficient power amplifier. In this figure, for both Ω = 0 and Ω = 1 we see that having to use 2 BS antennas and put one on sleep mode is the most efficient. In this setting, the configuration of BS and users are asymmetric and the BS to be put in sleep mode has to be chosen carefully. BS 1 and 2 are symmetric and are close to user 1, but 3 is closer to user 2. In this case In this setting, we see from Fig 7 that unlike in Setting A, using OFDMA to divide resources between the two users is not as efficient as ZF due to the higher SNR when served by nearby BS antennas. Like in Fig 6, choosing s 1 = 1 or s 2 = 1 and zero-forcing is always the most efficient solution. V. CONCLUSION This paper studies the performance of virtual MIMO systems from an energy efficiency perspective. It defines an energy efficiency metric that takes into account the capacity as well as the energy consumption, and considers both fixed and variable parts of this latter. We optimize the power allocations of the different antennas and show that sleep mode can bring a significant energy efficiency gain. This involves deactivating antennas that do not have a significant contribution to the system capacity, for a given number of users and radio channel conditions. This work is applicable only for the specific case of a small cell cluster with a centralized network and CSIT. Thus, many extensions of the proposed work are possible. The most relevant extension is to apply the proposed framework Fig. 2 . 2 Fig. 2. Setting A schematic Fig. 3 .Fig. 4 . 34 Fig. 3. Setting A: EE v.s P 0 for b = 1 W Fig. 5 . 5 Fig. 5. Setting B schematic Fig. 7 . 7 Fig. 7. Setting B: EE v.s P 0 for b = 10 W VI. ACKNOWLEDGMENTS This work is a joint collaboration between Orange Labs, Laboratoire des signaux et systémes (L2S) of Supélec and the Alcatel Lucent Chair of Supélec. This work is part of the European Celtic project "Operanet2".

01745111

en

2024/03/05 22:32:07

2013

https://hal.science/hal-01745111/file/blackseacom2013_mhri.pdf

Mariem Mhiri email: [email protected] Vineeth S Varma email: [email protected] Maël Le Treust email: [email protected] Samson Lasaulce email: [email protected] Abdelaziz Samet email: [email protected] On the benefits of repeated game models for green cross-layer power control in small cells (Invited Paper) Keywords: distributed power control, energy efficiency, repeated game, channel state information In this paper, we consider the problem of distributed power control for multiple access channels when energy-efficiency has to be optimized. In contrast with related works, the presence of a queue at each transmitter is accounted for and globally efficient solutions are sought. To this end, a repeated game model is exploited and shown to lead to solutions which are distributed in the sense of the decision, perform well globally, and may rely on limited channel state information at the transmitter. I. INTRODUCTION Designing energy-efficient communication systems has become a critical issue in modern day wireless networks. The problem treated in this work deals with power control when energy efficiency (EE) has to be optimized. This metric (EE) has been defined in [START_REF] Goodman | Power control for wireless data[END_REF] as a ratio of the net data rate (goodput) to the transmit power level. The problem was formulated as a non-cooperative game where each transmitter aims at selfishly maximizing its individual energy-efficiency. The considered solution is the Nash equilibrium (NE) which is shown to be unique but generally Pareto inefficient. To deal with this inefficiency, an operating point (OP) was proposed in [START_REF] Treust | A repeated game formulation of energyefficient decentralized power control[END_REF] where repeated game was exploited. Authors in [START_REF] Treust | A repeated game formulation of energyefficient decentralized power control[END_REF] showed that when playing with the developed OP according to a cooperation plan, only channel state information (CSI) is needed and transmitters can improve the social welfare (sum of utilities). Recently, a generalized EE metric has been proposed in [START_REF] Varma | A cross-layer approach for energy-efficient distributed power control[END_REF] for two important transport layer protocols (Transmission Control Protocol (TCP) and User Datagram Protocol (UDP)). The new EE metric is based on a cross-layer approach and takes into account the effects of the presence of a queue with a finite size at the transmitter. An interference channel system was studied and it was shown that a unique NE exists for a noncooperative game. In this paper, we consider the problem of distributed power control with the new EE metric according to UDP protocol developed in [START_REF] Varma | A cross-layer approach for energy-efficient distributed power control[END_REF] and for multiple access channels (MAC) system. Our goal is to find another unique solution concept which is efficient and may rely on limited CSI at the transmitter. We refer to a repeated game model (RG) developed in [START_REF] Treust | A repeated game formulation of energyefficient decentralized power control[END_REF] and try to apply the results on the crosslayer power control game. One of the major mathematical distinction between the two metrics used is the presence of a constant power term in the denominator of the EE metric. Although it appears to be a small change, the structure of the equilibrium solution is quite different. The optimal SINR when using the [START_REF] Goodman | Power control for wireless data[END_REF] metric is independent of the channel state. This property is lost when accounting for the constant power consumption, and motivates us to propose a new OP for the cross-layer metric. The main contributions of this work are: 1) Study the RG when using the cross layer EE as the utility of the game; 2) Establish the threshold on the game length beyond which the equilibrium policy can be pareto-optimal; 3) Propose a new OP that is efficient and can be reached in a distributed manner. This paper is structured as follows. In section II-A, we introduce the system model under study. Then, we define (in section II-B) the static power control game. This is followed (section II-C) by a review of the non-cooperative one-shot game. In section III, we give the formulation of the RG model. In section IV, we introduce the new OP and an equilibrium for the finite RG is proposed. Numerical results are presented and discussed in section V. Finally, concluding remarks are proposed in section VI. II. PROBLEM STATEMENT A. System model The communication network under study is that of a MAC system, where N small transmitters are communicating with a receiver and are operating in the same frequency band. Transmitter i ∈ {1, . . . , N } sends a signal √ p i x i with power p i ∈ [0, P max i ] where P max i > 0 is the maximum transmit power. The channel gain of the link between transmitter i and the destination is denoted as g i . Thus, the baseband signal received is written: y i = g i √ p i x i + N j=1 j =i g j √ p j x j + n i , (1) with n i is additive white Gaussian noise (AWGN) with mean 0 and variance σ 2 i . We assume that σ 2 i is identical for all the transmitters such that: σ 2 i = σ 2 . Therefore, the resulting SINR γ i at the receiver is given by: γ i (p) = p i |g i | 2 σ 2 + 1 L N j=1 j =i p j |g j | 2 , (2) where p = (p 1 , p 2 , . . . , p N ) is the power vector which will describe later the power actions of the N transmitters and L refers to the spreading factor [START_REF] Varma | A cross-layer approach for energy-efficient distributed power control[END_REF]. We assume that the described system is based on the IP (Internet Protocol) stack where packets arrive from an upper layer into a finite memory buffer of size K (in packets). Here, the considered protocol is UDP for which the packet arrival process follows a Bernoulli process with a constant probability q, independent from the SINR. This results in an effective packet loss denoted by Φ(γ i ) and an energy efficiency η i given by: η i (p i , p -i ) = Rq(1 -Φ(γ i (p))) b + qp i (1 -Φ(γ i (p))) f (γ i (p)) , (3) where p -i = (p 1 , .., p i-1 , p i+1 , .., p N ), R is the used throughput (in bit/s) and b represents the fixed consumed power when the radiated power is zero [START_REF] Varma | A cross-layer approach for energy-efficient distributed power control[END_REF]. B. Static power control game The major motivation behind this work is in order to establish an efficient equilibrium point to which a completely distributed system can converge to. A non-cooperative game has been introduced in [START_REF] Varma | A cross-layer approach for energy-efficient distributed power control[END_REF] where the existence of a unique Nash equilibrium was proved. Here, we are looking for more efficient solutions which are distributed in the sense of the decision making, but may rely on limited channel state information at the transmitter. As motivated in [START_REF] Varma | A cross-layer approach for energy-efficient distributed power control[END_REF], the power control can be modeled by a strategic form game (see e.g., [START_REF] Lasaulce | Game Theory and Learning for Wireless Networks: Fundamentals and Applications[END_REF]). Definition 2.1: The game is defined by the ordered triplet G = N , (A i ) i∈N , (u i ) i∈N where • N is the set of players. Here, the players of the game are the sources/transmitters, N = {1, . . . , N }; • A i is the set of actions. Here, the action of source/transmitter i consists in choosing p i in its action set A i = [0, P max i ]; • u i is the utility function of each user according to UDP given by: u i (p i , p -i ) = η i (p i , p -i ) (4) The function f : [0, +∞) → [0, 1] is a sigmoidal efficiency function which corresponds to the packet success rate verifying f (0) = 0 and lim x→+∞ f (x) = 1. The function Φ identifies the packet loss due to both bad channel conditions and the finiteness of the packet buffer. This can be calculated as: Φ(γ i ) = (1 -f (γ i ))Π K (γ i ) (5) where Π K (γ i ) is the stationary probability that the buffer is full and is given by: Π K (γ i ) = ω K (γ i ) 1 + ω(γ i ) + . . . + ω K (γ i ) (6) with: ω(γ i ) = q(1 -f (γ i )) (1 -q)f (γ i ) (7) In [START_REF] Varma | A cross-layer approach for energy-efficient distributed power control[END_REF], the authors prove that the non-cooperative game with rational players, G, allows for a unique pure Nash equilibrium (NE). This NE is the set of powers from which no player has anything to gain by changing only his own strategy unilaterally. This is explained in the following section. C. Review of the non-cooperative game The non-cooperative power control game has been investigated in [START_REF] Varma | A cross-layer approach for energy-efficient distributed power control[END_REF] where the quasi-concavity of the utility function given in (4) was proved. Accordingly, as the NE represents the fundamental solution for a non-cooperative game, existence and uniqueness of such a solution have been studied and demonstrated as well. Thus, the optimal power denoted as p * i is obtained by setting ∂u i /∂p i to zero, which leads to solve the following equation: bγ i Φ (γ i ) + q 1 -Φ(γ i ) f (γ i ) 2 [f (γ i ) -p i γ i f (γ i )] = 0, ( 8 ) where γ i = dγ i dp i = γ i p i , f = df dγ i and Φ = dΦ dγ i . However, the NE solution is not always Pareto efficient for many scenarios. An example is presented in Fig. 1 where we stress that the NE is far from the Pareto frontier. Motivated by the need to design an efficient solution relying on limited CSI at the transmitter, we move to the repeated game framework. III. REPEATED POWER CONTROL GAME In repeated games (RG), as the name suggests, the same game is played several times. The long-term interactions between the players in such a situation is studied under the RG framework. The players react to past experience by taking into account what happened in all previous stages and make decisions about their future choices [START_REF] Hart | Robert aumann's game and economic theory[END_REF], [START_REF] Sorin | Repeated games with complete information[END_REF]. The resulting utility of each player is an average of the utility of each stage. A game stage t corresponds to the instant in which all players choose their actions simultaneously and independently and thus a profile of actions can be defined by p(t) = (p 1 (t), p 2 (t), . . . , p N (t)). When assuming full monitoring, this profile choice is observed by all the players and the game proceeds to the next stage [START_REF] Sorin | Repeated games with complete information[END_REF]. The sequence of actions p i (t) of a transmitter i at time t defines his history denoted as h(t) = p i (t) = (p i (1), p i (2), . . . , p i (t -1)) and which lies in the set H t = P t-1 i . Before playing stage t, all histories are known by all the players [START_REF] Treust | A repeated game formulation of energyefficient decentralized power control[END_REF]. According to the above descriptions, a pure strategy δ i,t of player i ∈ N is a mapping from H t to the action set A i = [0, P max i ] specifying the action to choose after each history [START_REF] Treust | A repeated game formulation of energyefficient decentralized power control[END_REF], [START_REF] Sorin | Repeated games with complete information[END_REF]: δ i,t : H t → [0, P max i ] h(t) → p i (t) (9) We define the joint strategy δ = (δ 1 , δ 2 , . . . , δ N ) as the vector of all the players strategies. In this paper, we are interested in the finite repeated game, i.e the game is played for a finite number of steps (T steps). The utility function of each player results from averaging over the instantaneous utilities over all the game stages. At each stage t, the instantaneous utility of player i is a function of the profile of actions of all the players p(t). Definition 3.1: The utility function of the i th player for the finite RG is the arithmetic average of the sum of the utilities for the initial T first stages [START_REF] Sorin | Repeated games with complete information[END_REF], [START_REF] Aumann | Long-term competition-a gametheoretic[END_REF]. We have [START_REF] Treust | A repeated game formulation of energyefficient decentralized power control[END_REF]: v T i (δ) = 1 T T t=1 u i (p(t)) for the finite RG (10) where T ≥ 1 defines the number of game stages in the finite RG. An equilibrium solution of the RG is defined in the following manner: Definition 3.2: A joint strategy δ satisfies the equilibrium condition for the finite repeated game if for all players i ∈ N , for all other strategies δ i , we have v T i (δ) ≥ v T i (δ i , δ -i ). It means that no deviating strategy δ i can increase the utility v T i (δ) of any one player. This equilibrium solution is exactly what we are interested in, as a strategy δ satisfying the above condition would be precisely what rational players in a RG would play. In a RG with complete information and full monitoring, the Folk theorem characterizes the set of possible equilibrium utilities [START_REF] Treust | A repeated game formulation of energyefficient decentralized power control[END_REF], [START_REF] Sorin | Repeated games with complete information[END_REF]. It states that the set of Nash equilibrium in a RG is precisely the set of feasible and individually rational outcomes of the one-shot game (non-cooperative game) [START_REF] Hart | Robert aumann's game and economic theory[END_REF], [START_REF] Sorin | Repeated games with complete information[END_REF]. In a RG, interesting patterns of behavior between players can be seen and studied. This includes: rewarding and punishing, cooperation and threats, transmitting information and concealing [START_REF] Hart | Robert aumann's game and economic theory[END_REF]. IV. AN OPERATING POINT AND REPEATED GAME CHARACTERIZATION A. New OP for the game G Consider the operating point (OP) described in [START_REF] Treust | A repeated game formulation of energyefficient decentralized power control[END_REF]. It is identified by a subset of points of the achievable utility region such that p i |g i | 2 = p j |g j | 2 for all (i, j) ∈ N . The optimal subset of powers consists of the solutions of the following system of equations: ∀(i, j) ∈ N , ∂u i ∂p i (p) = 0 with p i |g i | 2 = p j |g j | 2 (11) with u i is the utility function defined in (4). Due to the presence of the parameter b which we consider different from 0, it can be observed that there will be N different solutions corresponding to equation (11) in terms of p i and thus the operating point from [START_REF] Treust | A repeated game formulation of energyefficient decentralized power control[END_REF] is not well defined when using the utility defined in [START_REF] Varma | A cross-layer approach for energy-efficient distributed power control[END_REF]. To deal with this problem, a new OP is proposed. The new OP consists in setting p i |g i | 2 to a constant denoted as α that can be optimized. We propose to determine a unique optimal α by maximizing the expected sum utility over all the channel states as follows: α = arg max E g N i=1 u i (α, g) (12) When playing at the OP, the power played by the i th player, denoted as pi , is given by: pi = α |g i | 2 (13) In the following section, we focus on the characterization of the finite RG. B. Repeated power control game characterization As a first step, we determine the minimum number of stages (T min ) corresponding to the finite RG. When the number of stages in the game is less than T min , the equilibrium of the RG is to simply play at the NE. However, if T > T min , a more efficient equilibrium point can be reached. Assuming that channel gains |g i | 2 lie in a compact set [ν min i , ν max i ] [2], we have the following proposition: Proposition 4.1 (Equilibrium solution for the finite RG): For a finite RG, if T > T min , then the corresponding equilibrium solution is given by [START_REF] Treust | A repeated game formulation of energyefficient decentralized power control[END_REF]: δ i,t : pi for t ∈ {1, 2, . . . , T -T min } p * i for t ∈ {T -T min + 1, . . . , T } P max i for any deviation detection (14) where T min is: Tmin=        Aν max i bν min i +γ i σ 2 B - Gν max i bν min i + αH Eν min i bν max i +γ * i ( σ 2 + 1 L j =i p * j ν max i ) F - Cν min i bν max i + γ i( σ 2 + 1 L j =i p max j ν max i ) D        (15) The proof for this proposition is given in Appendix A, as well as the quantities A, B, C, D, E, F , G and H. γ * i is the SINR at the NE while γi and γ i are the SINRs related to the utility max and the utility minmax respectively (see Appendix A). V. NUMERICAL RESULTS We consider a scenario with a MAC where N transmitters are communicating with their corresponding receiver (e.g. base station). The efficiency function is assumed to be f (x) = e -c/x where c = 2 R R 0 -1 with R and R 0 are the throughput and the used bandwidth and supposed to be 1 Mbps and 1 MHz respectively. The other parameters are set as follows: • σ 2 = 10 -3 W • b = 10 -2 W • K = 10 • q = 0.5 • P max i = P max = 10 -1 W The channel gains are assumed to be |g i | 2 = 1 and |g j | 2 = 0.5. Our simulations consist in showing firstly the advantage of the OP regarding the NE of the one shot game. Thus, we plot the achievable utility region, the NE and the proposed OP when considering a system of two transmitters and a spreading factor L = 2. In Fig. 2, the region delimited by the Pareto frontier and the minmax level defines, according to the Folk theorem, the possible set of equilibrium utilities of the RG. In addition, we highlight that the new OP dominates in terms of Pareto the NE and it is Pareto efficient. Fig. 3 represents the ratio w F RG w N E for the finite RG as a function of the number of stages. We have: w F RG w N E = N i=1 ( T -Tmin t=1 ũi (p(t)) + T t=T -Tmin+1 u * i (p(t))) N i=1 T t=1 u * i (p(t)) (16) We consider the same system as in Fig. 2 (2 transmitters and a spreading factor L = 2). We proceed to an averaging over channel gains lying in a compact set such that 10 log 10 νmax νmin = 20. According to equation (15), the minimum number of stages T min is equal to 4800. According to this figure, we deduce that the social welfare can be improved when playing a RG. 3. Improvement of the social welfare in finite repeated game vs the Nash equilibrium. While the efficiency of the RG while using the traditional metric defined in [START_REF] Goodman | Power control for wireless data[END_REF] seems to be higher, it requires a longer game than in the cross layer model. Fig. 3 plots the improvement of the social welfare as defined in ( 16). This improvement obtained is compared for case when using the metric defined in [START_REF] Goodman | Power control for wireless data[END_REF] to the cross-layer metric used. The required time for profiting from the RG scenario is much lower in the cross-layer case, but the improvement seems to be relatively smaller. However, note that the NE in the crosslayer game itself is more efficient than the NE in [START_REF] Goodman | Power control for wireless data[END_REF] and so in absolute terms, the proposed OP is still quite efficient and can be utilized for shorter games. This validates our approach and shows that the RG formulation is a useful technique for efficient distributed power control. VI. CONCLUSION In this paper, we study an efficient solution for a relevant game with a new EE metric considering a cross-layer approach and taking into account the effects of the presence of a queue with a finite size at the transmitter. As the NE is generally inefficient in terms of Pareto, we design a new OP and exploit a repeated game model to improve the performance of a MAC system. We contribute to express the analytic form for the minimum number of stages in a finite RG. Moreover, our approach provides an efficient solution relying on limited CSI at the transmitter when comparing to the NE and contributes to considerable gains in terms of social welfare for the finite RG. As a first step, we determine the power p i maximising u i and which we denote as ṗi . This amounts to reduce ∂u i /∂p i to 0. We recall that we consider the following notations: γ i = dγi dpi = γi pi , f = df dγi and Φ = dΦ dγi . The power ṗi maximising u i is then the solution of the following equation: b γ i p i Φ (γi)+q 1-Φ(γ i ) f (γ i ) 2 [f(γi)-γif (γi)]=0 (17) Therefore, the expression of the maximum utility function writes as: ui( ṗi,p-i)= Rq(1-φ( γi )) b+ ṗi q(1-φ( γi )) f ( γi ) , with: γi= ṗi |g i | 2 σ 2 + 1 L j =i p j |g j | 2 In a second step, we are interested in studying the behavior of ui ( ṗi , p -i ) as a function of p j for j = i; which amounts to calculating the sign of ∂ ui( ṗi,p-i) ∂pj ,which is shown to be negative in [START_REF] Varma | A cross-layer approach for energy-efficient distributed power control[END_REF]. Therefore ∂ ui( ṗi,p-i) ∂pj < 0. As ui is a decreasing function of p j , it reaches its maximum when p j = 0 and it is minimum when p j = p max j (for all j = i). A. Expression of ūi The utility ui reaches its maximum when p j = 0. When substituting p j = 0 in the SINR expression γi , this allows the determination of the optimal power ṗi : b |g i | 2 σ 2 Φ (γi)+q 1-Φ(γ i (p i )) f (γ i (p i )) 2 [f(γi(pi))-γif (γi(pi))]=0 (18) As the latter equation is a function of the SINR, the solution will be in terms of SINR and will be denoted as γi . The corresponding optimal power is pi = γiσ 2 |gi| 2 . Then, we have: ūi= Rq(1-φ(γ i )) b+ γi σ 2 |g i | 2 q(1-φ(γ i )) f (γ i ) B. Expression of u i We proceed as described previously and determine the optimal SINR denoted as γ i which is the solution of the following equation: b|g i | 2 σ 2 + 1 L j =i p max j |g j | 2 Φ (γi)+q 1-Φ(γ i ) f (γ i ) 2 [f(γi)-γif (γi)]=0 (19) Then, we have: ui= Rq(1-φ( γ i )) b+ γ i |g i | 2 ( σ 2 + 1 L j =i p max j |g j | 2 ) q(1-φ( γ i )) f ( γ i ) C. Existence proof of γ i and γi Both equations ( 18) and ( 19) are resulting from the same equation (17) for two different forms of the SINR (γ i for p j = 0 and γ i for p j = p max j ). Showing the existence of these two solutions amounts to prove the existence of the solution of equation (17). However, according to the study established in [START_REF] Varma | A cross-layer approach for energy-efficient distributed power control[END_REF], it was proved that u i is quasi-concave in (p i , p -i ) and then it exists γ + such that the first derivative of u i regarding to p i is strictly positive on [0, γ + ] and strictly negative on [γ + , +∞] for all p j ∈ [0, p max j ] : the first derivative is continous and is equal to zero in γ + . According to the utility which we are studying (max or minmax), γ + is either γi (eq. ( 18)) or γ i (eq. ( 19)). D. Proof The SINR γi refers to the SINR when playing the new OP. In order to simplify expressions, we use the following notations: A = Rq(1 -φ(γ i )) B = q(1-φ(γi)) f (γi) C = Rq(1 -φ( γ i )) D = q(1-φ( γi)) f ( γi) E = Rq(1 -φ(γ * i )) F = q(1-φ(γ * i )) f (γ * i ) G = Rq(1 -φ(γ i )) H = q(1-φ(γi)) f (γi) The inequality (20) becomes: A|g i | 2 b|g i | 2 +γ i σ 2 B + T s=T -T min +1 Eg C|g i | 2 b|g i | 2 + γ i( σ 2 + 1 L j =i p max j |g j | 2 ) D ≤ G|g i | Fig. 1 . 1 Fig. 1. Pareto inefficiency of the NE. Fig. 2 . 2 Fig. 2. Pareto dominance and Pareto efficiency of the proposed OP regarding the NE. Fig. Fig.3. Improvement of the social welfare in finite repeated game vs the Nash equilibrium. While the efficiency of the RG while using the traditional metric defined in[START_REF] Goodman | Power control for wireless data[END_REF] seems to be higher, it requires a longer game than in the cross layer model. utilities max and minmax are expressed respectively as follows: ūi = maxp -i maxp i ui(pi,p-i) ui = minp -i maxp i ui(pi,p-i) From [ 2 ] 2 , we have:ūi (p(t)) + T s=T -Tmin+1 E g { u i (p(s))} ≤ ũi (p(t)) + T s=T -Tmin+1 E g {u * i (p(s))}

01745187

en

2024/03/05 22:32:07

2018

https://inria.hal.science/hal-01745187/file/pmc-gs-maxflow.pdf

Zdravko I Botev Pierre L'ecuyer Reliability Estimation for Networks with Minimal Flow Demand and Random Link Capacities Keywords: network reliability, stochastic flow network, Conditional Monte Carlo, permutation Monte Carlo, generalized splitting We consider a network whose links have random capacities and in which a certain target amount of flow must be carried from some source nodes to some destination nodes. Each destination node has a fixed demand that must be satisfied and each source node has a given supply. We want to estimate the unreliability of the network, defined as the probability that the network cannot carry the required amount of flow to meet the demand at all destination nodes. When this unreliability is very small, which is our main interest in this paper, standard Monte Carlo estimators become useless because failure to meet the demand is a rare event. We propose and compare two different methods to handle this situation, one based on a conditional Monte Carlo approach and the other based on generalized splitting. We find that the first is more effective when the network is highly reliable and not too large, whereas for a larger network and/or moderate reliability, the second is more effective. Introduction Network reliability estimation problems are commonplace in various application areas such as transportation, communication, and power distribution systems; see for example [START_REF] Gertsbakh | Models of network reliability[END_REF]. In many of those problems, the states of certain network components are subject to uncertainty and there is a set of conditions under which the network is operational, and one wishes to estimate the network unreliability, defined as the probability u that the network is in a failed state (i.e., is not operational). When u is very small, a standard (crude) Monte Carlo (MC) approach that merely generates the component states, computes the indicator function that the network is operational or not, and averages over n independent runs to estimate u, is unsatisfactory because the relative error (defined as the standard deviation of the estimator divided by the expected value u) of the MC estimator goes to infinity when u → 0. One reliability problem that has received a lot of attention is the static network reliability estimation problem, in which each link of the network is failed with a given probability and the network is operational when a given (specific) subset of the nodes are all connected. Effective estimation methods have been developed for this problem when u is small; see [START_REF] Botev | Static network reliability estimation via generalized splitting[END_REF][START_REF] Botev | Static network reliability estimation under the Marshall-Olkin copula[END_REF][START_REF] Elperin | Estimation of network reliability using graph evolution models[END_REF][START_REF] Fishman | A Monte Carlo sampling plan for estimating network reliability[END_REF][START_REF] Gertsbakh | Models of network reliability[END_REF][START_REF] Lomonosov | Combinatorics and reliability Monte Carlo[END_REF] and the references therein. The model considered in this paper is more general. Instead of having only a binary state (up or down), each link has a random capacity that can take many possible values, there is a fixed demand that must be satisfied at certain nodes (called the destination nodes), a fixed supply is available at some other nodes (the source nodes), and the network is operational when it can carry the flow to satisfy all the demands. As a special case, there can be a single source node and a single destination node, with a fixed demand, and the network is operational when the maximum flow that can be sent from the source to the destination reaches the demand. We will describe our methods in this particular setting to simplify the notation, but the methods apply to the general setting as well. The case of links with binary states is a special case. The several methods developed for this special case do not readily apply to the network flow setting considered here, but we show how two of the best available methods for the binary case, permutation Monte Carlo (PMC) and generalized splitting (GS), can be adapted to this problem. The adaptation is not straightforward. The PMC method [START_REF] Elperin | Estimation of network reliability using graph evolution models[END_REF][START_REF] Gertsbakh | Models of network reliability[END_REF][START_REF] Lomonosov | Combinatorics and reliability Monte Carlo[END_REF]] constructs an artificial continuous-time Markov chain (CTMC) defined as follows. Each capacity is assumed to have a discrete distribution over a finite set of possible values. This can approximate a continuous distribution if needed. We assume that all links start at their minimal capacity, and the capacity of one link may increase each time the CTMC has a jump. The CTMC is constructed so that the probability that the network is failed at time 1 is equal to u. PMC generates the discrete-time Markov chain (DTMC) underlying the CTMC, i.e., only the sequence of states that are visited until the network is operational, and conditional on that sequence it computes the probability that the network is failed at time 1, as an estimator of u. This conditional probability can be computed by exploiting the property that the failure time has a phase-type conditional distribution, whose cumulative distribution function (cdf) and density can be expressed in terms of matrix exponentials. We show how to adapt and apply the PMC principle to our problem. The CTMC construction is quite different than for the binary case. We also prove, under certain conditions, that the resulting PMC estimator has bounded relative error (BRE) when u → 0 for a given network. GS [START_REF] Botev | Efficient Monte Carlo simulation via the generalized splitting method[END_REF] is a rare-event estimation method where the rare event is the intersection of a nested sequence of events and its probability is the product of conditional probabilities. Each conditional probability is estimated thanks to resampling strategies, making the overall estimation more accurate than a direct estimation of the rare-event probability itself. The application of GS to this problem was discussed in [START_REF] Botev | Reliability of stochastic flow networks with continuous link capacities[END_REF] for the situation in which the capacities have a continuous distribution, and experimental results were reported for a small example. But the GS algorithm proposed there does not work in general when the capacities have a discrete distribution. We show however that GS can be applied in the discrete case if we combine it with the same CTMC construction as for PMC. The GS algorithm does not have BRE in the asymptotic regime when u → 0, but it becomes more efficient that PMC when the size of the network increases. The relative error typically increases (empirically) as O(log u). The remainder of the paper is organized as follows. In Section 2, we formulate the network flow model considered in this paper. In Section 3, we construct a CTMC which permits one to apply PMC to this model, for the case where each capacity is distributed over a finite set. In Section 4, we explain how to apply GS to this model. We report numerical experiments in Section 5. Our experimental results agree with the fact that PMC has BRE when u → 0, under appropriate conditions. It can accurately estimate extremely small values of u when the network is not too large. When the network gets larger and u is not too small, on the other hand, GS becomes more effective than PMC. The model Let G = (V , E ) be a graph with a set of nodes V and a set of links E with cardinality m = |E |. For i = 1, . . . , m, link i has a random integer-valued flow capacity X i with discrete marginal distribution p i (x) = P[X i = x] over the set X i = {c i,0 , . . . , c i,b i }, 0 ≤ c i,0 < c i,1 < • • • < c i,b i < ∞. This is a standard assumption; see [START_REF] Botev | Reliability of stochastic flow networks with continuous link capacities[END_REF] and the references therein. Thus, the random network state X = (X 1 , . . . , X m ) belongs to the space X = ∏ m i=1 X i and has joint pdf p(x) = P(X = x), for x ∈ X . We also make the standard independence assumption (see [START_REF] Alexopoulos | Capacity expansion in stochastic flow networks[END_REF][START_REF] Bulteau | A new importance sampling Monte Carlo method for a flow network reliability problem[END_REF][START_REF] Chou | An efficient and robust design optimisation of multi-state flow network for multiple commodities using generalised reliability evaluation algorithm and edge reduction method[END_REF]) that p(x) = ∏ m i=1 P[X i = x i ] and that the nodes do not fail. To keep the notation and the exposition simple, in the remainder of the paper we describe the model and the methods under the assumption that there is a single source and a single destination. The generalization to multiple sources and destinations is straightforward, as explained below. The fixed demand level at the destination is d net > 0 and the maximum flow that can be carried from the source to the destination is a random variable Ψ(X), which is a function of the link capacities. The well-known max-flow min-cut theorem says that the maximum value of a flow from a source to a destination is equal to the minimum capacity of a cut in the network. Efficient algorithms are available to compute Ψ(X); for example the Ford-Fulkerson algorithm. We are interested in estimating the unreliability of the flow network, defined here as u = P[Ψ(X) < d net ] = ∑ {x∈X :Ψ(x)<d net } p(x); that is, the probability that the maximum flow Ψ(X) fails to meet the demand. This problem was considered in [START_REF] Fishman | Monte Carlo: Concepts, algorithms, and applications[END_REF], for example. In the particular case where X i = {0, 1} for each i and d net = 1, we have an instance of the static network reliability problem mentioned in the introduction, with the source and destination as the selected set of nodes to be connected. To generalize to multiple sources and destinations, we would assume a fixed demand d i at each destination node i, a fixed supply s i at each source node i, and the event {Ψ(X) < d net } would be replaced by the event that the network does not have sufficient capacity to send flow to satisfy all the demands from the available supplies. For small networks, it is possible to compute and store most of the minimal cutsets or pathsets and use them to obtain exact or approximate values for u; see [START_REF] Jane | Computing multi-state two-terminal reliability through critical arc states that interrupt demand[END_REF][START_REF] Zuo | An efficient method for reliability evaluation of multistate networks given all minimal path vectors[END_REF] for example. But for large networks, no polynomial-time algorithm is known for computing u exactly [START_REF] Colbourn | The combinatorics of network reliability[END_REF], and one must rely on approximations or on estimation via Monte Carlo. Of particular interest is the situation in which the network is highly reliable, i.e., u is a very small rare-event probability, because crude Monte Carlo then becomes ineffective. Several Monte Carlo variance-reduction methods have been proposed for network reliability estimation in rare-event situations; see, e.g., [START_REF] Botev | Static network reliability estimation via generalized splitting[END_REF][START_REF] Botev | Static network reliability estimation under the Marshall-Olkin copula[END_REF][START_REF] Cancela | On the RVR simulation algorithm for network reliability evaluation[END_REF][START_REF] Elperin | Estimation of network reliability using graph evolution models[END_REF][START_REF] Gertsbakh | Models of network reliability[END_REF][START_REF] Ramirez-Marquez | A Monte-Carlo simulation approach for approximating multi-state two-terminal reliability[END_REF][START_REF] L'ecuyer | Approximate zero-variance importance sampling for static network reliability estimation[END_REF][START_REF] Tuffin | An adaptive zero-variance importance sampling approximation for static network dependability evaluation[END_REF] and the references given there. Most of these methods are for the special case of independent links with binary states and nodes that never fail. Some have been extended to links with three possible states [START_REF] Gertsbakh | Permutational methods for performance analysis of stochastic flow networks[END_REF][START_REF] Gertsbakh | Network reliability Monte Carlo with nodes subject to failure[END_REF][START_REF] Gertsbakh | Ternary networks: Reliability and Monte Carlo[END_REF], but this remains restrictive. We now describe how two of the most efficient methods, PMC and GS, can be adapted to our model. Reformulating the model as a CTMC and applying PMC We now show how to construct an artificial CTMC for this static model, which will permit us to apply PMC as described in the introduction. This CTMC construction differs from that used in [START_REF] Botev | Static network reliability estimation under the Marshall-Olkin copula[END_REF][START_REF] Gertsbakh | Models of network reliability[END_REF]. Constructing the CTMC For each i, let P(X i = c i,k ) = p i (c i,k ) = r i,k > 0 for k = 0, . . . , b i . Define independent exponential random variables Y i,1 , . . . ,Y i,b i with rates λ i,1 , . . . , λ i,b i , respectively, where the λ i,k still have to be chosen. Suppose that the capacity of link i is c i,0 from time T i,0 = 0 to time T i,1 = min(Y i,1 , . . . ,Y i,b i ) (exclusive), after that it is c i,1 from time T i,1 to time T i,2 = min(Y i,2 , . . . ,Y i,b i ), it is c i,2 from time T i,2 to time T i,3 = min(Y i,3 , . . . ,Y i,b i ), and so on, and finally it is c i,b i from time T i,b i to T i,b i +1 = ∞. Under this process, the capacity of link i at time γ ≥ 0 is given by X i (γ) = c i,k for T i,k ≤ γ < T i,k+1 and 0 ≤ k ≤ b i (1) = max k {c i,k : T i,k ≤ γ}. (2) The times T i,1 , . . . , T i,b i are not necessarily all distinct; often, many of them are equal, so that the number of jumps at which the capacity changes can be much smaller than b i . For example, if T i,1 = Y i,b i , then we have T i,1 = T i,2 = • • • = T i,b i . As another example, if b i = 3 and Y i,2 < Y i,1 < Y i,3 , then 0 < T i,1 = T i,2 < T i,3 and the capacity of link i jumps from c i,0 to c i,2 at time T i,2 = Y i,2 and jumps again from c i,2 to c i,3 at time T i,3 = Y i,3 . In general, the process {X i (γ), γ ≥ 0} has an upward jump at each of the distinct jump times T i,k . To show that this process is a CTMC, suppose that we are at time γ ≥ 0 and X i (γ) = c i,k . Then we know that Y i,k ≤ γ and that Y i, > γ for all > k. The Y i, for < k can be anything, but they have no influence on the process trajectory after time γ. This means that the current state X i (γ) contains all the relevant information that needs to be known at time γ to generate the future of the process. The capacity X i (γ) of link i at time γ ≥ 0 satisfies P[X i (γ) ≤ c i,k ] = P[min(Y i,k+1 , . . . ,Y i,b i ) > γ] = exp[-γ(λ i,k+1 + • • • + λ i,b i )]. If we select the λ i,k 's so that the last expression equals r i,0 + • • • + r i,k for each k when γ = 1, then X i (1) has the exact same distribution as X i , the capacity of link i in the original static model. This is equivalent to having λ i,k+1 + • • • + λ i,b i = -ln(r i,0 + • • • + r i,k ). To achieve this, it suffices to put λ i,b i = -ln(r i,0 + • • • + r i,b i -1 ) = -ln(1 -r i,b i ) (3) and then for k = b i -1, . . . , 1 (in descending succession): λ i,k = -ln(r i,0 + • • • + r i,k-1 ) -λ i,k+1 -• • • -λ i,b i . (4) Note that (4) can be rewritten as λ i,k = -ln(r i,0 + • • • + r i,k-1 ) + ln(r i,0 + • • • + r i,k ), which can never be negative. We have proved the following. Proposition 1. If we select λ i,b i , λ i,b i -1 , . . . , λ i,1 according to ( 3) and ( 4) and the process X i (•) as in ( 2), then λ i,k ≥ 0 for each k, {X i (γ), γ ≥ 0} is a CTMC process, and X i (1) has exactly the same discrete distribution as the capacity of link i in the original model: P[X i (1) = c i,k ] = r i,k for k = 0, . . . , b i . As a result, X(1) = (X 1 (1), . . . , X m (1) ) has the same distribution as X and one has u = P[Ψ(X(1)) < d net ]. Applying PMC Under the assumption that all links are independent, a simple way of applying PMC to this model is as follows. Generate all the Y i,k 's independently with their rates λ i,k , put them in a large vector Y = (Y 1,1 , . . . ,Y 1,b 1 , . . . ,Y m,1 , . . . ,Y m,b m ) of size κ = b 1 + • • • + b m , and sort this vector in increasing order to obtain Y π(1) ≤ Y π(2) ≤ • • • ≤ Y π(κ) , where π( j) = (i, k) if Y i,k is in position j in the sorted vector, so that π = (π(1), . . . , π(κ)) can be seen as the permutation of the pairs (i, k) that corresponds to the sort. This permutation gives an ordering of the κ pairs (i, k). When scanning those pairs in the given order, each pair (i, k) corresponds to a potential capacity increase for link i. The capacity increases if and only if no pair (i, k ) for k > k has occurred before. Conditional on π, one can add those pairs in the given order and update the capacities accordingly, until the maximum flow in the network reaches d net . Suppose this occurs when adding the pair (i, k) = π(C) for some integer C > 0. Let T C = Y π(C) . The (unbiased) conditional (PMC) estimator of u is then P[Ψ(X(1)) < d net | π] = P[T C > 1 | π] = P[T C > 1 | π(1), . . . , π(C)] = P[A 1 + • • • + A C > 1], where A 1 = Y π(1) is an exponential random variable with rate Λ 1 = ∑ m i=1 ∑ b i k=1 λ i,k , each A j = Y π( j) -Y π( j-1 ) is an exponential random variable with rate Λ j = Λ j-1λ π( j-1) for j = 2, . . . ,C, and these A j 's are independent. Given π and C, T C = A 1 + • • • + A C is the sum of C independent exponential random variables with rates Λ 1 , . . . , Λ C , which has a phase-type distribution, whose complementary cdf is given by 1 -F(γ | π) = P[T C > γ | π] = e t 1 exp(Q γ)1 (5) where e t 1 = (1, 0, . . . , 0), 1 = (1, . . . , 1) t (the t means "transposed"), and Q =         -Λ 1 Λ 1 0 • • • 0 0 -Λ 2 Λ 2 0 . . . 0 0 . . . . . . 0 . . . 0 0 -Λ C-1 Λ C-1 0 • • • 0 0 -Λ C         . Reliable and fast computation of ( 5) is discussed in [START_REF] Botev | Static network reliability estimation via generalized splitting[END_REF][START_REF] Botev | Static network reliability estimation under the Marshall-Olkin copula[END_REF]. To compute the critical number C at which the flow reaches the demand, we must be able to update efficiently the maximum flow in the network each time we increase the capacity of one link. We do this as explained in Section 4 of [START_REF] Botev | Reliability of stochastic flow networks with continuous link capacities[END_REF]. We refer to this algorithm as the incremental maximum flow algorithm. To estimate u by PMC, for a fixed threshold d net , we simulate n independent realizations W 1 , . . . ,W n of W = W (π) = P[T C > 1 | π] (6) and take the average Wn = (1/n) ∑ n i=1 W i . Compared with the crude Monte Carlo estimator that would take the indicator I = I[Ψ(X(1)) < d net ] in place of W , it is always true that Var[W ] < Var[I], because W = E[I | π]. The estimators discussed so far are for a single (fixed) demand d net . With PMC, it is also possible to estimate u = u(d net ) as a function of the demand d net , over some interval, using the same simulations for all demands. To do this, for any given permutation π, we can compute C = C(d net ) as a function of the demand over the interval of interest. This would be a step function, often with just a few jumps. Then we compute W for all values of C that are visited over this interval. This provides an estimator W (d net ) of u(d net ) as a function of d net . By averaging the n realizations of this estimator, we obtain a functional estimator of u(d net ) over the interval of interest. Improved PMC The PMC strategy described earlier can be improved by removing some useless jumps. First, whenever c i,k ≥ d net for k < b i , we can immediately remove all the jumps (i, k + 1), . . . , (i, b i ), because when the capacity of a link has reached d net , it is useless to increase it further. Capacity levels larger than d net can in fact be all reset to d net right away in the model, the probability of d net in the new model being taken as the the probability of values larger than d net in the initial model. For simplicity, when the demand is fixed, we assume in our algorithm that this has been done already, so that c i,b i ≤ d net for all i, and then there is no need to remove those useless capacity levels. Second, the jump times Y i,k that do not change the capacity of link i can also be removed. That is, whenever π( j) = (i, k) and the capacity of link i has already reached a value c i,k > c i,k , i.e., (i, k ) = π( j ) for some j < j, then there is no need to consider the pair (i, k) when it is encountered in the permutation, so we can remove the corresponding jump. Let π be the permutation obtained after removing all those pairs (i, k) from the sorted vector, and C the corresponding value of C in this reduced permutation. As soon as the max flow reaches d net , we have found C. When we encounter π( j) = (i, k) and the previous capacity of link i was c i,k < c i,k , the capacity of link i jumps to c i,k and we must decrease Λ j by λi,k = λ i,k +1 + • • • + λ i,k , because the jumps that correspond to (i, k + 1), . . . , (i, k) can now be removed from consideration. draw Y i,1 , . . . ,Y i,b i with the appropriate rates λ i,k 3: let L i = {(i, 1), . . . , (i, b i )} 4: min ← (i, b i ) and λi,b i ← λ i,b i 5: for k = b i -1 to 1 do 6: if Y i,k > Y min then 7: remove (i, k) from the list L i 8: λmin ← λmin + λ i,k 9: S i,k ← 0 // this jump is deactivated S i,k ← 1 // this jump is activated 14: merge the sorted lists L 1 , . . . , L m into a single list sorted by increasing order, Y π(1) , . . . ,Y π( κ) 15: Λ 1 ← λ 1,1 + • • • + λ 1,b 1 + • • • + λ m,b m 16: j ← 0 17: X ← (c 1,0 , . . . , c m,0 ) 18: while maximum flow Ψ(X) < d net do 19: j ← j + 1 20: if S π( j) = 1 then 21: (i, k) ← π( j) // this jump has not been removed or executed 22: Λ j+1 ← Λ j -λi,k 23: S i,k ← 0 24: X i ← c i,k // increase capacity of i-th link 25: Filter() // do nothing (default), or FilterSingle, FilterAll, etc. 26: C ← j // the critical jump number 27: return W ← P A 1 + • • • + A k-1 > 1 | π, C . Algorithm 1 describes this reduced version of PMC in a more formal way. It returns one realization of W . Indentation delimits the scope of the loops. In the first for loop, for each link i, the algorithm generates the exponential random variables Y i,k and then immediately eliminates those that correspond to (useless) jump times at which the capacity of the link does not change. This preliminary filtering is very easy and efficient to apply and may eliminate a significant fraction of the jumps, especially for links that have many capacity levels. The remaining jumps are sorted in a single list (for all links) and each one receives a Boolean tag S π( j) , initialized to 1, which means that this jump is currently scheduled to occur. Then these jumps are "executed" in chronological order, by increasing the corresponding capacities, until the critical jump number C is found. After that, W can be computed. The Boolean variables S π( j) are used in the optional Filter() subroutine, which can be used to try to eliminate further useless jumps after a jump is executed and the corresponding capacity is increased (this is discussed in Section 3.4). Algorithm 1 would be invoked n times, independently, and u would be estimated by the average of the n realizations of W . Other variants of the algorithm can be considered and some might be more efficient, but this is not completely clear. For example, instead of generating all the variables Y i,k at the beginning, one may think of generating the permutation π directly without generating those Y i,k , as was done in [START_REF] Gertsbakh | Models of network reliability[END_REF] for the binary case. This appears complicated and we did not implement it. Removing jumps having no impact on maximum flow In Algorithm 1, in the case where Filter() does nothing, all pairs (i, k) for which the capacity of link i increases are retained in π and the corresponding jumps are executed. But it sometimes occurs that increasing the capacity of link i to c i,k (or more) is useless because it can no longer have an impact on the event that the maximum flow exceeds the demand or not. In this case, one can cancel (deactivate) all the future jumps related to the capacity of link i. In our implementation, these future jumps are canceled by setting their Boolean variables S i,k to 0. Increasing the capacity of link i is useless in particular if it is already possible to send d net units of flow between the two nodes connected by link i. This obviously happens if the capacity of link i is already of d net , which is trivial to verify, but under our assumption (made at the beginning of Section 3.3) that a link has no capacity level above d net , this cannot happen, and our algorithm ignores this possibility. Increasing the capacity of link i is also useless when d net units of flow can be sent in total, either directly on link i or indirectly via other links. This is generally harder (more costly) to verify. To detect it, one can run a max-flow algorithm to compute how much flow can be sent between these two nodes. This can be done each time the capacity of a link is increased. Algorithm 2 does this only for the link i whose capacity has just been increased, at each step j. Since the link i generally changes at every step j, we have a different max-flow problem (for a different pair of nodes) at each step. For this reason, in our implementation we recompute the max-flow from scratch at each step j. Of course, this brings significant overhead. Algorithm 3 is even more ambitious: it computes the max-flow between nodes for all pairs of nodes. Then for each link i for which the current max-flow between the corresponding two nodes meets the demand, it cancels all future jumps associated with that link. Doing this at each step j might be too costly, so in our implementation the user selects a positive integer ν and does it only at every ν steps, i.e., when j is a multiple of ν. We compute the max-flow for all pairs of nodes using the algorithm of [START_REF] Gusfield | Very simple methods for all pairs network flow analysis[END_REF], which is a simplified variant of the Gomory-Hu method [START_REF] Gomory | Multi-terminal network flows[END_REF]. This algorithm computes the all-pairs max-flow by applying |V | -1 times a max-flow algorithm for one pair of nodes, which is generally more efficient than applying a max-flow algorithm for each link. Algorithm 3 also recomputes the maximum flows from scratch each time it is called, rather than reusing computations from the previous time and just updating the max flows. (In fact, the |V | -1 pairs of nodes for which the max-flow is computed in the algorithm change from one call to the next, and we are not aware of an effective incremental algorithm that would reuse and just update the previous computations.) Algorithm 2 : FilterSingle f ← compute maximum flow between terminal nodes of link i if f ≥ d net then for k = 1 to b i do if S i,k = 1 then S i,k ← 0 Λ j+1 ← Λ j+1 -λi,k Algorithm 3 : FilterAll if j mod ν = 0 then {F v,w } ← max flow between all pairs of nodes (v, w), computed via Gusfield's algorithm for all i = (v, w) ∈ E do if F v,w ≥ d net then for k = 1 to b i do if S i,k = 1 then S i,k ← 0 Λ j+1 ← Λ j+1 -λi,k Bounded relative error for PMC For a single run, the crude MC estimator I = I[Ψ(X(1)) < d net ] of u, which is a Bernoulli random variable with mean u, has variance u(1 -u), so its relative error (RE) is RE[I] = u(1 -u)/u ≈ u -1/2 → ∞ when u → 0. With n runs, the variance is divided by n and the RE by √ n. When u is very small, we may need an excessively large n to obtain a sufficiently small RE. With PMC, the RE is sometimes much better behaved than with MC. In this section, we obtain conditions under which the PMC estimator W has bounded relative error (BRE), i.e., RE[W ] remains bounded when u → 0. The proofs have some similarity with those in [START_REF] Botev | Static network reliability estimation under the Marshall-Olkin copula[END_REF]. Suppose the probabilities r i,k = r i,k (ε) in our model depend on some parameter ε in a way that u = u(ε) → 0 when ε → 0. In what follows, the quantities in the model are assumed implicitly to depend on ε. A non-negative quantity that may depend on ε is O(1) if it remains bounded when ε → 0. It is Θ(1) if it is bounded and also bounded away from 0, when ε → 0. In our setting, the vector Y and the permutation π have finite length κ and C is bounded by κ. The number of possible permutations is therefore finite. Let p(π) be the probability of permutation π. Proposition 2. (i) If p(π) = Θ(1) for all π, then the PMC estimator has BRE. (ii) This holds in particular if λ i,k i /λ j,k j = Θ(1) for all i, j, k i , k j (we then say that the rates are balanced). Proof. 1), which implies BRE. (i) Note that u ≥ P[T C > 1 | π]p(π) = W (π)p(π) for any π. If p(π) = Θ(1), then W (π)/u = O(1) and max π W (π)/u = O(1). Therefore E[W 2 /u 2 ] = O( (ii) Note that p(π) = ∏ κ j=1 λ π( j) /Λ π( j) . Under the given assumption, λ π( j) /Λ π( j) = Θ(1) for all j, which implies that p(π) = Θ(1) for all π. As a concrete illustration of an asymptotic regime in which the r i,k depend on ε, we define a regime similar to one that has been widely used for highly reliable Markovian systems [START_REF] Nakayama | General conditions for bounded relative error in simulations of highly reliable Markovian systems[END_REF][START_REF] Rubino | Markovian models for dependability analysis[END_REF][START_REF] Shahabuddin | Importance sampling for the simulation of highly reliable Markovian systems[END_REF]. Suppose that link i is operating at capacity c i,k . with probability r i,k = a i,k ε d i,k for some constants a i,k > 0 and d i,k > 0 independent of ε, for all k ∈ {0, . . . , b i -1} (that is, not at full capacity). This implies that r i,b i = 1 -∑ b i -1 j=0 r i,k = Θ(1) for all i. That is, the event that any link is not at full capacity is a rare event. This implies that failure to meet the demand is a rare event, and therefore RE[I] → ∞ when ε → 0. More specifically, any state vector x = (c 1,k 1 , . . . , c m,k m ) for which Ψ(x) < d net has probability P(X = x) = ∏ m i=1 r i,k i = ε d(x) (1 + o(1)) for some d(x) > 0. Let d min = min{d(x) : Ψ(x) < d net }. Then u = ∑ {x:Ψ(x)<d net } P(X = x) = Θ(ε d min ). On the other hand, we have Proposition 3. In the setting just defined, the PMC estimator has BRE. Proof. Recall that for 1 ≤ i ≤ m, λ i,b i = -ln(∑ b i -1 k=0 r i,k ) = Θ(ln(ε)). Moreover, for all k < b i , λ i,k = ln(∑ k =0 r i, )-ln(∑ k-1 =0 r i, ) = Θ(ln(ε)). The conditions of Proposition 2 (ii) are then verified, hence the result. The results of this section apply to the improved PMC variants as well; the proofs are easily adapted. A generalized splitting algorithm Botev et al. [START_REF] Botev | Reliability of stochastic flow networks with continuous link capacities[END_REF] have explained how to adapt the GS algorithm proposed and studied in [START_REF] Botev | Efficient Monte Carlo simulation via the generalized splitting method[END_REF][START_REF] Botev | Static network reliability estimation via generalized splitting[END_REF] to the stochastic flow problem considered here, but for the situation where the capacities have a continuous distribution. The aim of the algorithm is to obtain a sample of realizations of X which is approximately a sample from the distribution of X conditional on Ψ(X) < d net . The estimator is then given by the realized sample size (which is random) divided by its largest possible value. The algorithm uses intermediate demand levels d net = d τ < • • • < d 1 < d 0 , where d 0 is the maximal possible flow, achieved when each link i is at its maximal capacity c i,b i . These levels and their number τ are fixed a priori and chosen so that P[Ψ(X) < d t | Ψ(X) < d t-1 ] ≈ 1/s for t = 1, . . . , τ -1, and at most 1/s for t = τ, where s is a small integer also fixed (usually and in all our experiments in this paper, s = 2). The levels are estimated by pilot runs, as explained in [START_REF] Botev | Efficient Monte Carlo simulation via the generalized splitting method[END_REF][START_REF] Botev | Static network reliability estimation via generalized splitting[END_REF]. The algorithm starts by sampling X from its original distribution. If Ψ(X) < d 1 , it resamples each coordinate of X conditional on Ψ(X) < d 1 , via Gibbs sampling, repeats this s times, and keeps the states X for which Ψ(X) < d 2 (their number is in {0, . . . , s}). At each level t = 3, . . . , τ, this type of resampling is applied to each state that has been retained at the previous step (for which Ψ(X) < d t-1 ), by resampling that state twice from its distribution conditional on Ψ(X) < d t-1 , and retaining the states for which Ψ(X) < d t . At the last level, we count the number N of chains for which Ψ(X) < d τ , and return W = N/s τ-1 as an estimator of u. This is repeated n times independently, to produce n independent realizations of W , say W 1 , . . . ,W n , whose average Wn is an unbiased estimator of u. This estimator does not have BRE, because the RE increases with the number of levels; the RE is typically (roughly) proportional tolog u (see [START_REF] Botev | Dependent failures in highly-reliable static networks[END_REF] for a proof of this result in an idealized setting). It can also handle large networks. In general, this GS algorithm is not directly applicable when the capacities have discrete distributions, because then Ψ(X) also has a discrete distribution and it may happen that this distribution is too coarse (e.g., all the probability mass is on just a few possible values). Then it may be impossible to select levels d t for which P[Ψ(X) < d t | Ψ(X) < d t-1 ] ≈ 1/s for t = 1, . . . , τ -1. It is nevertheless possible to apply GS in that case by constructing the vector Y as for PMC in the previous section, and resampling this vector instead of X. Recall that Y is a vector of κ independent exponential random variables. The GS algorithm will operate similarly as the one described above, except that now the levels 0 = γ 0 < γ 1 < • • • < γ τ = 1 are on T C , and the resampling at each step t is for Y and is conditional on T C > γ t-1 . This is valid because {Ψ(X(γ t )) < d net } = {T C > γ t }. The corresponding GS procedure operates in the same way as the GS algorithm with anti-shocks in [START_REF] Botev | Static network reliability estimation under the Marshall-Olkin copula[END_REF]. We first generate Y from its original distribution. Then at each level t, we take each state (realization or modification of Y) that has been retained at the previous step (for which T C > γ t-1 ), we resample all its coordinates s times (i.e., for s Gibbs sampling steps, where each step starts from the result of the previous step) from its distribution conditional on T C > γ t-1 , to obtain two new states, and we retain the states for which T C > γ t . At the last level, we count the number N of chains for which T C > γ τ = 1, and return W = N/s τ-1 as an estimator of u. The resampling of Y conditional on T C > γ t-1 via Gibbs sampling can be done in a similar way as in [START_REF] Botev | Static network reliability estimation via generalized splitting[END_REF][START_REF] Botev | Static network reliability estimation under the Marshall-Olkin copula[END_REF]. We first select a permutation π of the κ coordinates of the vector Y. Then for j = 1, . . . , κ, we resample Y π( j) as follows: If π( j) = (i, k), the current capacity X i (γ t-1 ) of link i is less than c i,k (or equivalently min(Y i,k , . . . ,Y i,b i ) > γ t-1 ), and by changing the current capacity of link i to c i,k (or equivalently changing Y i,k to 0) we would have T C < γ t-1 (the maximum flow would meet the demand), then we resample Y i,k from its exponential density truncated to (γ t-1 , ∞). Otherwise we resample Y i,k from its original exponential density. To sample from the truncated density, it suffices to generate Y i,k from the original density and add γ t-1 . Numerical Examples In this section, we provide some numerical examples that compare the PMC and GS algorithms, and show how they behave when u → 0. In these examples, we parameterize the models by ε in a way that u = u(ε) → 0 when ε → 0, exactly as in Section 3.5, in the asymptotic regime when the probability that links are not operating at full capacity is getting close to zero. For all variants of PMC, we used formula (2) in [START_REF] Botev | Static network reliability estimation under the Marshall-Olkin copula[END_REF] with high precision arithmetic to compute (5). Experimental setting We used the same experimental protocol as in [START_REF] Botev | Static network reliability estimation under the Marshall-Olkin copula[END_REF], comparing four methods. Method PMC refers to Algorithm 1 without any filtering step. PMC-Single and PMC-All refer to PMC combined with the filtering as in Algorithms 2 and 3 respectively. Method GS refers to generalized splitting, implemented as described in Section 4. The splitting levels were determined via the adaptive Algorithm 3 of [START_REF] Botev | Static network reliability estimation via generalized splitting[END_REF], with n 0 = 500 and s = 2. The levels were estimated using a single run of the adapative algorithm, and these same levels were used for every independent replication of the GS algorithm. For each example and method, we report the unreliability estimate Wn , its empirical relative error RE[ Wn ] = S n /( √ n Wn ) where S 2 n is the empirical variance, and the work-normalized relative variance (WNRV) of Wn , defined as WNRV[ Wn ] = T × RE 2 [ Wn ], where T is the total CPU time (in seconds) for the n runs of the algorithm. One must keep in mind that T and the WNRV depend on the software and hardware used for the computations. The experiments were run on Intel Xeon E5-2680 CPUs, on a linux cluster. The sample size for every algorithm was n = 5 × 10 4 . For each example we use the following model. Each link i has the capacity levels {0, 1, . . . , b i }, i.e., c i,k = k for k = 0, . . . , b i . We take r i,k = P(X i = c i,k ) = ρ b i -k-1 ε for k < b i and r i,b i = 1 - ∑ b i -1 k=0 ρ b i -k-1 ε , where ρ, ε and {b i } are model parameters. A 4 × 4 lattice graph Our first example uses the 4 × 4 lattice graph, which has 16 nodes and 24 links. The flow has to be sent from one corner to the opposite corner. We take b i = 8, ρ = 0.6 and d net = 10, and let ε range from 10 -4 to 10 -13 . Table 1 reports the values of Wn , RE[ Wn ] and WNRV[ Wn ], for methods GS and PMC, for some values of ε. Figure 1 shows plots of RE[ Wn ] and WNRV[ Wn ] for all four methods. We see that for this small example, PMC-All always has the smallest RE, followed by PMC-Single. These REs increase very slowly when ε decreases, for the values we have tried. They should eventually stabilize when ε → 0. In terms of work-normalized relative variance, i.e., when taking the computing time into account, GS is the most effective method when ε is not too small, but when ε gets smaller, GS requires a larger simulation effort while the effort required by PMC variants remains approximately stable, so these PMC methods eventually catch up, in agreement with our asymptotic results. Among them, PMC-Single has the smallest WNRV. ε considered, PMC-All has the smallest RE while GS wins in terms of WNRV. ε = 10 -4 ε = 10 -5 ε = 10 -6 ε = 10 -7 ε = 10 - A dodecahedron network In this example we use the well-known dodecahedron network (Figure 3), with 20 nodes and 30 links, often used as a standard benchmark in network reliability estimation [START_REF] Botev | Static network reliability estimation via generalized splitting[END_REF][START_REF] Cancela | A recursive variance-reduction algorithm for estimating communication-network reliability[END_REF][START_REF] Cancela | Rare event analysis by Monte Carlo techniques in static models[END_REF][START_REF] Cancela | Analysis and improvements of path-based methods for Monte Carlo reliability evaluation of static models[END_REF][START_REF] Tuffin | An adaptive zero-variance importance sampling approximation for static network dependability evaluation[END_REF]. Here we took ρ = 0.7, b i = 4 and d net = 5. Note that when ε is very small, most of the failures will occur because there is not enough capacity in the three links connected to node 1 (links 1, 2, 3), or not enough capacity in the three links connected to node 20 (links [START_REF] Rubino | Markovian models for dependability analysis[END_REF][START_REF] Shahabuddin | Importance sampling for the simulation of highly reliable Markovian systems[END_REF][START_REF] Tuffin | An adaptive zero-variance importance sampling approximation for static network dependability evaluation[END_REF]. These are the two bottleneck cuts. Table 2 reports the values of Wn , RE[ Wn ], and WNRV[ Wn ], for the GS and PMC methods, for different values of ε. We see that the estimates Wn agree very well across the two methods. Figure 4 shows RE[ Wn ] and WNRV[ Wn ] as functions of ε, for all four methods. We see that PMC-All has by far the smallest RE for all ε, and it also wins in terms of WNRV, except for ε > 10 -5 where GS wins. The latter case is approximately when u ≥ 7 × 10 -11 , which is already pretty small. When ε decreases, the WNRV increases for GS in part because the RE increases, but also because the computing time increases. The figure shows what happens when ε gets very small. Algorithm 1 : 1 PMC algorithm for multi-state flow network 1: for i = 1 to m do 2: Figure 1 : 1 Figure 1: RE (left) and WNRV (right) for four methods, for the 4 × 4 lattice graph. Figure 2 : 2 Figure 2: RE (left) and WNRV (right) for four methods, for the 6 × 6 lattice graph. Figure 3 : 3 Figure 3: A dodecahedron graph with 20 nodes and 30 links (figure taken from [3]). 10 - 3 3 10 -5 10 -7 10 -9 10 -11 10 -13 10 -15 10 -17 10 -5 10 -7 10 -9 10 -11 10 -13 10 -15 10 - Figure 4 : 4 Figure 4: RE (left) and WNRV (right) for four methods, for the dodecahedron example 8 Wn for PMC 2.99 × 10 -5 2.98 × 10 -6 2.99 × 10 -7 2.99 × 10 -8 2.99 × 10 -9 RE[ Wn ] for PMC 3.16 × 10 -2 3.34 × 10 -2 3.41 × 10 -2 3.69 × 10 -2 3.74 × 10 -2 WNRV[ Wn ] for PMC 3.17 × 10 -2 3.20 × 10 -2 3.17 × 10 -2 3.62 × 10 -2 3.54 × 10 -2 Wn for GS 2.98 × 10 -5 2.99 × 10 -6 2.99 × 10 -7 2.98 × 10 -8 2.99 × 10 -9 RE[ Wn ] for GS 3.43 × 10 -2 3.32 × 10 -2 3.15 × 10 -2 3.30 × 10 -2 4.33 × 10 -2 WNRV[ Wn ] for GS 1.07 × 10 -2 1.43 × 10 -2 1.68 × 10 -2 2.35 × 10 -2 2.86 × 10 -2 Table 1 : 1 Estimation of u, RE, and WNRV for some values of ε, for the 4 × 4 lattice example 5.3 6 × 6 lattice graph Figure 2 shows plots of RE[ Wn ] and WNRV[ Wn ] for a 6 × 6 lattice graph, with 36 nodes and 60 links. Again, the flow has to be sent from one corner to the opposite corner. Here, for all values of 10 -1.2 Method GS PMC PMC-All 10 -0.4 Relative Error 10 -1.6 10 -1.4 PMC-Single Work Normalized Relative Variance 10 -1 10 -0.8 10 -0.6 Method GS PMC 10 -1.2 PMC-All PMC-Single 10 -6 10 -8 10 -10 10 -12 10 -6 10 -8 10 -10 10 -12 7.06 × 10 -13 7.06 × 10 -15 7.05 × 10 -17 RE[ Wn ] for PMC 8.63 × 10 -2 7.21 × 10 -2 6.68 × 10 -2 5.97 × 10 -2 5.86 × 10 -2 WNRV[ Wn ] for PMC 1.85 × 10 -1 1.37 × 10 -1 1.14 × 10 -1 8.95 × 10 -2 8.57 × 10 -2 Wn for GS 7.07 × 10 -9 7.06 × 10 -11 7.07 × 10 -13 7.07 × 10 -15 7.05 × 10 -17 RE[ Wn ] for GS 3.95 × 10 -2 4.30 × 10 -2 4.58 × 10 -2 5.17 × 10 -2 4.97 × 10 -2 WNRV[ Wn ] for GS 3.13 × 10 -2 4.52 × 10 -2 6.00 × 10 -2 8.81 × 10 -2 1.06 × 10 -1 Table 2 : 2 Estimation of u, RE, and WNRV for some values of ε, for the dodecahedron example Acknowledgments This work has been supported by the Australian Research Council under DE140100993 Grant to Z. I. Botev, an NSERC-Canada Discovery Grant, a Canada Research Chair, and an Inria International Chair to P. L'Ecuyer. P. L'Ecuyer acknowledges the support of the Faculty of Science Visiting Researcher Award at UNSW. We are grateful to Rohan Shah, who performed the numerical experiments.

01745195

en

2024/03/05 22:32:07

2017

https://hal.science/hal-01745195/file/laera_19774.pdf

Davide Laera email: [email protected] Thierry Schuller Kevin Prieur Daniel Durox Sergio M Camporeale Sébastien Candel Flame Describing Function analysis of spinning and standing modes in an annular combustor and comparison with experiments Keywords: This article reports a numerical analysis of combustion instabilities coupled by a spinning mode or a standing mode in an annular combustor. The method combines an iterative algorithm involving a Helmholtz solver with the Flame Describing Function (FDF) framework. This is applied to azimuthal acoustic coupling with combustion dynamics and is used to perform a weakly nonlinear stability analysis yielding the system response trajectory in the frequency-growth rate plane until a limit cycle condition is reached. Two scenarios for mode type selection are tentatively proposed. The first is based on an analysis of the frequency growth rate trajectories of the system for different initial solutions. The second considers the stability of the solutions at limit cycle. It is concluded that a criterion combining the stability analysis at the limit cycle with the trajectory analysis might best define the mode type at the limit cycle. Simulations are compared with experiments carried out on the MICCA test facility equipped with 16 matrix burners. Each burner response is represented by means of a global FDF and it is considered that the spacing between burners is such that coupling with the mode takes place without mutual interactions between adjacent burning regions. Depending on the nature of the mode being considered, two hypotheses are made for the FDFs of the burners. When instabilities are coupled by a spinning mode, each burner features the same velocity fluctuation level implying that the complex FDF values are the same for all burners. In case of a standing mode, the sixteen burners feature different velocity fluctuation amplitudes depending on their relative position with respect to the pressure nodal line. Simulations retrieve the spinning or standing nature of the self-sustained mode that were identified in the experiments both in the plenum and in the combustion chamber. The frequency and amplitude of velocity fluctuations predicted at limit cycle are used to reconstruct time resolved pressure fluctuations in the plenum and chamber and heat release rate fluctuations at two locations. For the pressure fluctuations, the analysis provides a suitable estimate of the limit cycle oscillation and suitably retrieves experimental data recorded in the MICCA setup and in particular reflects the difference in amplitude levels observed in these two cavities. Differences in measured and predicted amplitudes appear for the heat release rate fluctuations. Their amplitude is found to be directly linked to the rapid change in the FDF gain as the velocity fluctuation level reaches large amplitudes corresponding to the limit cycle, underlying the need of FDF information at high modulation amplitudes. Introduction Many recent studies focus on combustion instabilities coupled by azimuthal modes in annular systems. There are yet few comparisons between predictions and well controlled experiments. The present investigation aims at filling this gap by developing a nu-• Two different operating conditions are considered, one leading to a spinning limit cycle and another one leading to a standing limit cycle. • This framework is then used to calculate the limit cycles of standing and spinning solutions and to compare the calculated oscillation with measurements on a laboratory scale test facility "MICCA" developed at EM2C laboratory. The amplitudes and phase relationships of pressure in the plenum and chamber and the heat release rate signals are compared for two different operating conditions. • Finally, two scenarios are tentatively proposed to explain the mode type selection. The first considers that the frequency and growth rate trajectories of initially spinning and standing modes determine the solution at the limit cycle. The second suggested by a reviewer considers that it is the stability of the limit cycle which determines the observed oscillation. At this point, it is worth briefly reviewing some recent investigations of instabilities in annular devices. Combustion instabilities coupled by azimuthal modes are often studied by theoretical or numerical means. On the numerical level one finds a growing number of massively parallel large eddy simulations of annular chambers with multiple burners [START_REF] Staffelbach | Large eddy simulation of self excited azimuthal modes in annular combustors[END_REF][START_REF] Fureby | Les of a multi-burner annular gas turbine combustor[END_REF][START_REF] Wolf | Using les to study reacting flows and instabilities in annular combustion chambers[END_REF] . These calculations retrieve features of azimuthally coupled combustion instabilities observed in experiments in engine-like conditions [START_REF] Worth | Self-excited circumferential instabilities in a model annular gas turbine combustor: global flame dynamics[END_REF] . However, the complexity of the situation precludes direct comparison between calculations and observations. One difficulty in performing such comparisons lies in the definition of the flame model. The first investigations of combustion instabilities coupled to azimuthal modes were performed by combining a low-order acoustic model of an annular combustor with a time delayed n -τ flame response (see for example [START_REF] Stow | Thermoacoustic oscillations in an annular combustor[END_REF][START_REF] Evesque | Low-order acoustic modelling for annular combustors: validation and inclusion of modal coupling[END_REF] ). This kind of model is also assumed in the recent analytical studies developed in [START_REF] Parmentier | A simple analytical model to study and control azimuthal instabilities in annular combustion chambers[END_REF][START_REF] Bauerheim | An analytical model for azimuthal thermoacoustic modes in an annular chamber fed by an annular plenum[END_REF] to analyze the linear stability of annular chambers fed by an annular plenum with multiple discrete burners. Both spinning and standing modes are predicted depending on the circumferential symmetry of the system. Circumferential instabilities of industrial combustors were analyzed in [START_REF] Krebs | Thermoacoustic stability chart for high-intensity gas turbine combustion systems[END_REF][START_REF] Campa | Prediction of the thermoacoustic combustion instabilities in practical annular combustors[END_REF] by means of a Helmholtz solver approach. In these studies, the flame response was modeled by a n -τ description with parameters retrieved from CFD calculations of the steady combustion process. These previous studies carried out with linear tools could not account for finite amplitude effects that determine the oscillation frequency and level at the limit cycle. These features were considered for example in [START_REF] Pankiewitz | Time domain simulation of combustion instabilities in annular combustors[END_REF][START_REF] Stow | A time-domain network model for nonlinear thermoacoustic oscillations[END_REF] by combining different numerical strategies for acoustic propagation with a nonlinear flame model in the time domain. Numerical control strategies for annular configurations featuring spinning limit cycles were developed in [START_REF] Morgans | Model-based control of combustion instabilities in annular combustors[END_REF][START_REF] Illingworth | Adaptive feedback control of combustion instability in annular combustors[END_REF] . Spinning and standing modes were observed in [START_REF] Pankiewitz | Time domain simulation of combustion instabilities in annular combustors[END_REF] depending on whether the mean flow velocity was neglected or considered in the simulations. For a circumferential instability in an axisymmetric geometry, the spinning waveform was always preferred to the standing mode in the simulations carried out in [START_REF] Stow | Thermoacoustic oscillations in an annular combustor[END_REF] . However, no comparison with experiments were reported in these works. On a theoretical level, criteria for appearance of spinning or standing modes have been derived by considering the dynamics of azimuthal modes coupled by a nonlinear flame model expressed in terms of pressure perturbations alone. One of the first analysis was carried out in [START_REF] Schuermans | Non-linear combustion instabilities in annular gas-turbine combustors[END_REF] with a saturation function linking heat release and pressure fluctuations. In this framework, the dynamics of azimuthal modes is described by two harmonic oscillators which are nonlinearly coupled. The stability of standing and traveling waves at limit cycle can then be assessed. A further simplification was later introduced by assuming that the system behaves like a Van der Pol oscillator [START_REF] Noiray | Investigation of azimuthal staging concepts in annular gas turbines[END_REF][START_REF] Ghirardo | Azimuthal instabilities in annular combustors: standing and spinning modes[END_REF][START_REF] Noiray | On the dynamic nature of azimuthal thermoacoustic modes in annular gas turbine combustion chambers[END_REF] . Results indicate that the spinning or standing nature of the unstable mode originates from the nonlinearity and non-uniformity of the flame response and can be influenced by different factors, like transverse velocity fluctuations [START_REF] Dawson | Flame dynamics and unsteady heat release rate of self-excited azimuthal modes in an annular combustor[END_REF][START_REF] Oconnor | Recirculation zone dynamics of a transversely excited swirl flow and flame[END_REF] or turbulence, which can stochastically disturb the limit cycle amplitudes. This nonlinear flame model was used for example in [START_REF] Bothien | Analysis of azimuthal thermo-acoustic modes in annular gas turbine combustion chambers[END_REF] to reproduce the dynamical behavior observed in a real engine. Another comparison is presented in [START_REF] Noiray | Deterministic quantities characterizing noise driven hopf bifurcations in gas turbine combustors[END_REF] between numerical and experimental growth rates calculated by means of a system identification technique, but the oscillation levels of the different pressure signals are not shown. Recently, Ghirardo et al. [START_REF] Ghirardo | State-space realization of a describing function[END_REF][START_REF] Ghirardo | Weakly nonlinear analysis of thermoacoustic instabilities in annular combustors[END_REF] managed to introduce in their time domain model a more reliable FDF, linking heat release and pressure disturbances by a time-invariant nonlinear operator. Criteria for self-sustained thermo-acoustic instabilities coupled by spinning and standing modes were then derived by examining the stability of the analytical solutions at limit cycles. This framework was then tested with experimental data from Bourgouin et al. [START_REF] Bourgouin | Characterization and modeling of a spinning thermoacoustic instability in an annular combustor equipped with multiple matrix injectors[END_REF] where a stable spinning mode is observed at limit cycle. Their analysis confirmed that for the operating condition explored, there was a stable spinning solution and that standing solutions, if they existed, were unstable. In all of these previous studies there are no direct comparisons between predictions and measurements for pressure and heat release rate oscillations and only limited validations of model predictions for different operating conditions. The difficulties encountered in these various investigations are compounded by the presence of multiple flames which respond collectively over a wide frequency range, and by the modal density in the annular geometry when the size of the system is large like in industrial gas turbine combustors. One possible simplification consists in considering that the heat release from the different burners is uniformly distributed over the circumference of the annular chamber. Following this approach, Bourgouin et al. [START_REF] Bourgouin | Characterization and modeling of a spinning thermoacoustic instability in an annular combustor equipped with multiple matrix injectors[END_REF] developed an analytical one-dimensional framework to represent the dynamics of the laboratory scale MICCA annular combustor. Assuming a simplified flame response, the spinning instability recorded during experiments was reproduced in terms of frequency and amplitude of velocity fluctuations at the limit cycle. A theoretical interpretation was also proposed for the angular shift observed in the experiments between the nodal lines in the plenum and in the chamber. However, this analysis was carried out for a fixed frequency and fixed oscillation level corresponding to the values observed in the experiment at the limit cycle. On the experimental level there are relatively few data sets corresponding to instrumented conditions that can be used to benchmark models and simulations. Most of the measurements performed on real systems consist of unsteady pressure signals with no access to the flame dynamics [START_REF] Krebs | Thermoacoustic stability chart for high-intensity gas turbine combustion systems[END_REF][START_REF] Bothien | Analysis of azimuthal thermo-acoustic modes in annular gas turbine combustion chambers[END_REF][START_REF] Mastrovito | Analysis of pressure oscillations data in gas turbine annular combustion chamber equipped with passive damper[END_REF] . Both spinning and standing mode patterns were observed in the laboratory scale annular device equipped with low swirl injectors that was developed in the engineering department of Cambridge University [START_REF] Worth | Self-excited circumferential instabilities in a model annular gas turbine combustor: global flame dynamics[END_REF][START_REF] Worth | Modal dynamics of self-excited azimuthal instabilities in an annular combustion chamber[END_REF] . This setup allows heat release rate measurements and flame dynamics analysis through optical windows, but the flame transfer function was not determined and pressure signals were not recorded in the combustion chamber. This article is organized as follows. A novel procedure is derived in Section 2 that combines a Helmholtz solver with sixteen independent FDFs. This is used to determine the limit cycle conditions by means of a weakly nonlinear stability analysis. Depending on the nature of the mode being considered, the FDFs are assumed to operate at equal or at different velocity fluctuation levels for the examination of the dynamics of spinning and standing modes, respectively. This numerical procedure is validated in Appendix A in an idealized geometrical configuration with a simplified flame model by retrieving the amplitude and stability properties of the theoretical limit cycles [START_REF] Noiray | Investigation of azimuthal staging concepts in annular gas turbines[END_REF] . Section 3 describes the MICCA combustor experiment with 16 matrix laminar injectors and the numerical framework used for the stability analysis. The FDF determined for one of the matrix injectors is presented in Section 4 for the first operating condition leading to a spinning mode limit cycle in the MICCA combustor. The same type of analysis is repeated in Section 5 for the second operating point leading to a standing mode limit cycle. The frequency and amplitude of velocity fluctuations predicted at limit cycle are used to reconstruct the pressure oscillations in the plenum and in the chamber and the heat release rate fluctuation signal. These signals are then compared with microphone records and photomultiplier measurements at two different flame locations. A mode type selection analysis is presented in Section 6 . For the two operating points, frequencygrowth rate trajectories are calculated and the stability properties of the simulated limit cycles are discussed. These analyses are tentatively used to determine the limit cycle structure. Weakly nonlinear stability numerical analysis The methodology used to assess the stability of the annular combustor acoustic modes and determine the limit cycle state of the system follows that developed for the FDF based weakly nonlinear stability analysis of single burner setups [START_REF] Noiray | A unified framework for nonlinear combustion instability analysis based on the flame describing function[END_REF][START_REF] Palies | Nonlinear combustion instability analysis based on the flame describing function applied to turbulent premixed swirling flames[END_REF][START_REF] Silva | Combining a Helmholtz solver with the flame describing function to assess combustion instability in a premixed swirled combustor[END_REF][START_REF] Han | Prediction of combustion instability limit cycle oscillations by combining flame describing function simulations with a thermoacoustic network model[END_REF] . In the time domain one has to consider a wave equation including a damping term defined by a first order time derivative of the pressure fluctuations p multiplied by a damping rate δ ( s -1 ) [START_REF] Kinsler | Fundamentals of acoustics[END_REF] and a source term representing effects of the unsteady heat release rate disturbances ˙ q : ∂ 2 p ∂t 2 + 4 πδ ∂ p ∂t -∇ • c 2 ∇ p = (γ -1) ∂ ˙ q ∂t , (1) where it has been assumed that the mean pressure p is essentially constant so that ρc 2 is also constant. Assuming that all fluctuations are harmonic x = ˆ x exp (-iωt) , one obtains the following equation in the frequency domain: ω 2 c 2 ˆ p + i ω4 πδ c 2 ˆ p + ρ∇ • 1 ρ ∇ ˆ p = i γ -1 c 2 ω ˆ ˙ q, ( 2 ) where ω denotes the complex angular frequency. The mean density ρ, speed of sound c and specific heat ratio γ distributions are specified. In this frequency domain, the analysis is carried out by using the finite element method (FEM) based on the commercial software COMSOL Multiphysics . This code solves the classical Helmholtz equation in which heat release rate fluctuations ˆ ˙ q are treated as pressure sources. A nonlinear description of the interaction between combustion and acoustics is required to capture the limit cycles of a thermoacoustic system. If the flame is compact with respect to the wavelength of the unstable mode, the dynamics of the flame may be represented in terms of a global FDF F [START_REF] Silva | Combining a Helmholtz solver with the flame describing function to assess combustion instability in a premixed swirled combustor[END_REF] , where the FDF gain and phase lag are function of the amplitude of the incoming perturbation [START_REF] Noiray | A unified framework for nonlinear combustion instability analysis based on the flame describing function[END_REF][START_REF] Han | Simulation of the flame describing function of a turbulent premixed flame using an open-source les solver[END_REF] . In the frequency domain F links relative heat release rate fluctuations ˆ ˙ Q/ ˙ Q to relative velocity fluctuations | ˆ u / u | measured at a reference point of the system. The FDF is a complex function expressed in terms of a gain G and phase ϕ as follows: F (ω r , | u / u | ) = ˆ ˙ Q (ω r , | u / u | ) / Q | ˆ u / u | = G (ω r , | u / u | ) exp iϕ(ω r , | u / u | ) , (3) where | u / u | stands for the relative velocity fluctuation level, with u the root-mean-square of the velocity signals taken at the reference position in the injector unit j and ω r corresponds to the real part of the complex frequency ω. A weakly nonlinear approach is used to couple Eq. ( 3) with Eq. ( 2) retrieving the solution of the nonlinear problem as a perturbation of a linear problem. This is achieved by linearizing the FDF by fixing a velocity fluctuation level | u / u | . The finite element discretization of this set of linearized equations along with the boundary conditions results in the following eigenvalue problem [START_REF] Nicoud | Acoustic modes in combustors with complex impedances and multidimensional active flames[END_REF][START_REF] Laera | A finite element method for a weakly nonlinear dynamic analysis and bifurcation tracking of thermo-acoustic instability in longitudinal and annular combustors[END_REF] : [ A ] P + ω [ B (ω) ] P + ω 2 [ C ] P = [ D (ω) ] P , ( 4 ) where P is the pressure eigenmodes vector. The matrices [ A ] and [ C ] contain coefficients originating from the discretization of the Helmholtz equation, [ B ( ω)] is the matrix of the boundary conditions and of the damping and [ D ( ω)] represents the source term due to the unsteady heat release. With the introduction of the heat release rate the eigenvalue problem defined by Eq. ( 4) becomes nonlinear and is solved with an iterative algorithm. At the k th iteration Eq. ( 4) is first reduced to a linear eigenvalue problem around a specific frequency k : ( [ A ] + k [ B ( k ) ] -[ D ( k ) ] ) P + ω 2 k [ C ] P = 0 , ( 5 ) where k = ω k -1 is the previous iteration result. The software uses the ARPACK numerical routine for large-scale eigenvalue problems. This is based on a variant of the Arnoldi algorithm, called the implicit restarted Arnoldi method [START_REF] Lehoucq | ARPACK users' guide: solution of large-scale eigenvalue problems with implicitly restarted Arnoldi methods[END_REF] . This procedure is iterated until the error defined by = | ω kk | becomes lower than a specific value, typically 10 -6 . Once convergence is achieved, the real part of ω yields the oscillation frequency, f = -( ω)/2 π Hz, while the imaginary part of ω corresponds to the growth rate α = (ω) / 2 π s -1 that allows the identification of unstable modes: p ∝ exp ( 2 παt -i 2 π f t ) . If α is positive, the acoustic mode is unstable and the amplitude of fluctuations grows with time. If α is negative, the acoustic mode is stable and perturbations decay with time. The eigenvalue procedure is repeated by incrementing the amplitude level until a limit cycle condition is reached when α = 0 . Differently from Silva et al. [START_REF] Silva | Combining a Helmholtz solver with the flame describing function to assess combustion instability in a premixed swirled combustor[END_REF] approach for single burner setups, in a multi-burner annular combustor the linearization is performed for each of the FDFs assumed in the model. In the present numerical framework, the distribution of velocity fluctuation levels between the FDFs is fixed a priori depending on the spinning or standing nature of the azimuthal mode under investigation: • In a spinning mode, the nodal line rotates at the speed of sound, however, the oscillation amplitude is uniform all around the chamber and the velocity fluctuation level | u / u | is the same for each burner. In this case, the FDFs of the individual burners have the same complex value [START_REF] Bourgouin | Characterization and modeling of a spinning thermoacoustic instability in an annular combustor equipped with multiple matrix injectors[END_REF] . Mathematically this is formulated as follows: | u / u | ( ) = C. ( 6 ) In this expression, is a vector containing the angular coordinates of the reference points where the velocity fluctuation is specified allowing the calculation of the FDF. Each reference point lies on the injector axis 20 mm below the injector outlet. • In a standing mode, each injector operates with a different amplitude of oscillation depending on its relative position with respect to the nodal line [START_REF] Durox | Nonlinear interactions in combustion instabilities coupled by azimuthal acoustic modes[END_REF] . As a consequence, different am- | u / u | ( ) = ψ ( ) /ψ max | u / u | j , (7) where ψ ( )/ ψ max is the normalized azimuthal eigenmode structure. The numerical implementation of this model in the Helmholtz solver framework is not trivial because the mode ψ is the solution of the eigenvalue analysis. The frequency of this mode and consequently also the pressure distribution is highly influenced by the amount of heat release rate fluctuations considered in the model as shown in [START_REF] Laera | Impact of heat release distribution on the spinning modes of an annular combustor with multiple matrix burners[END_REF] (see the trajectory maps in Fig. 9 ). This results in an iterative procedure where the solution at the k th iteration is obtained using the ψ distribution computed at the k th -1 iteration. This procedure is iterated until the maximum error defined by = | ψ ( ) k -ψ ( ) k -1 | is lower than a threshold value, typically 10 -3 . It should be noticed that at each k th iteration, the nonlinear eigenvalue problem defined by Eq. ( 4) is solved in order to compute the new mode structure ψ [START_REF] Laera | Impact of heat release distribution on the spinning modes of an annular combustor with multiple matrix burners[END_REF] . A validation of the proposed numerical procedure is discussed in Appendix A. The case investigated considers an annular cavity with a uniform distribution of heat release and a simplified model for the nonlinear flame response to pressure disturbances. Analytical solutions were derived for both spinning and standing limit cycles together with conditions for their stability [START_REF] Noiray | Investigation of azimuthal staging concepts in annular gas turbines[END_REF] . Confrontations between numerical simulations carried out in this work and analytical expressions perfectly match for both the rotating and standing modes validating the numerical procedure. Experimental setup and numerical representation We now briefly describe the MICCA combustor shown in Fig. 1 a. This system comprises an annular plenum connected by 16 injectors to an annular chamber formed by two cylindrical concentric quartz tubes of 200 mm length. Each injector consists of a cylinder of d br = 33 mm diameter and l br = 14 mm length exhausting gases through a l inj = 6 mm thick perforated plate featuring 89 holes of d p = 2 mm diameter located on a 3 mm square mesh. The system fed by a propane/air mixture allows the stabilization of a set of laminar conical flames above the 16 injectors as shown in Fig. 2 for one injector. Two operating conditions are investigated. Figure 2 a shows flames stabilized above a single matrix injector for a stoichiometric mixture φ= 1, while flames obtained for a slightly richer mixture at φ= 1.11 shown in Fig. 2 b have a higher longitudinal extension. These images were taken for stable operation in a single matrix injector setup. When the MICCA combustor is operated at condition A, a thermo-acoustic instability coupled to a spinning mode with stable limit cycle is observed [START_REF] Bourgouin | Characterization and modeling of a spinning thermoacoustic instability in an annular combustor equipped with multiple matrix injectors[END_REF] . When the MICCA is operated at condition B, a stable limit cycle coupled to a standing mode is found [START_REF] Durox | Nonlinear interactions in combustion instabilities coupled by azimuthal acoustic modes[END_REF] . Slanted [START_REF] Bourgouin | A new pattern of instability observed in an annular combustor: the slanted mode[END_REF] self-sustained combustion oscillations coupled to azimuthal modes were also identified in this setup when the flow operating conditions were modified. The heat release rate is distributed in the simulations over a small volume located at the exit of each burner. It consists of a cylindrical volume of height l f and diameter d f . A previous study on the same combustor indicates that the dynamics of perturbations is influenced by the extension of the flame domain [START_REF] Laera | Impact of heat release distribution on the spinning modes of an annular combustor with multiple matrix burners[END_REF] . For conical flames, the flame volume dimensions are deduced by processing the images of the flame region under steady conditions shown in Fig. 2 as described in [START_REF] Laera | Impact of heat release distribution on the spinning modes of an annular combustor with multiple matrix burners[END_REF] . The numerical representation of the system is shown schematically in Fig. 3 a. The plenum consists of an annular cavity linked to the combustion chamber volume by sixteen injection units as in the real configuration. A model is used to represent the matrix injectors shown in Fig. 3 b. The body of each burner has the same dimensions as the real system. The perforated plate is replaced by a cylindrical volume having the same l inj = 6 mm thickness as the perforated plate and a base area with a diameter of d inj = 18.9 mm corresponding to the total flow passage area of the injector. The total height of the burner is l br + l in j = 20 mm. The combustion chamber is modeled as an annular duct open to the atmosphere with an augmented length of 41 mm to account for an end correction resulting in a total length of l cc = 241 mm. The value of this correction was determined experimentally by submitting the MICCA chamber to harmonic acoustic excitations near the unstable resonant mode and by scanning a microphone along a longitudinal axis above the combustion chamber annulus [START_REF] Laera | Impact of heat release distribution on the spinning modes of an annular combustor with multiple matrix burners[END_REF] . All other boundaries are treated as rigid adiabatic walls. The combustor operates at atmospheric conditions. The temperature of the plenum is set equal to 300 K. Following previous studies [START_REF] Palies | Nonlinear combustion instability analysis based on the flame describing function applied to turbulent premixed swirling flames[END_REF][START_REF] Kinsler | Fundamentals of acoustics[END_REF] , the damping rate is deduced from a resonance response of the system by imposing an external perturbation with a loudspeaker and measuring the resonance sharpness. These measurements were carried out under cold flow conditions avoiding any corrections to account for absorption or generation of acoustic energy by the flame [START_REF] Palies | Nonlinear combustion instability analysis based on the flame describing function applied to turbulent premixed swirling flames[END_REF] . This, however, introduces some uncertainty since the value of the damping rate under hot conditions may differ from that estimated at room temperature. Figure 4 shows two acoustic response curves measured in the plenum ( Fig. 4 a) and in the combustion chamber ( Fig. 4 b) providing the resonance frequency bandwidth f at half-power. The damping rate δ appearing in Eq. ( 2) is deduced from the frequency bandwidth 2 δ = f ( s -1 ). The azimuthal thermo-acoustic instabilities coupled by spinning and standing modes are discussed in what follows. Numerical simulations are compared with experiments and the stability of the numerical predictions is evaluated. Analysis of operating point A For operating condition A corresponding to a stoichiometric propane/air mixture φ= 1 with a bulk flow velocity measured in the cylindrical body of each injector equal to u b = 0.49 m s -1 , the system sustains a well-established spinning limit cycle coupled to the first azimuthal mode at a frequency of 487 Hz [START_REF] Bourgouin | Characterization and modeling of a spinning thermoacoustic instability in an annular combustor equipped with multiple matrix injectors[END_REF] . In the numerical model, effects of steady combustion are taken into account through a temperature distribution in the gas stream. The temperature was measured, along a longitudinal axis passing in the center of one burner location, with a movable thermocouple and varies from 1470 K near the flame zone to 1130 K at the end of the combustion chamber. The flame domain consists of a cylindrical volume of height l f = 4 mm and diameter d f = 36 mm ( Fig. 3 b). These dimensions are deduced by processing the image shown in Fig. 2 a as in [START_REF] Laera | Impact of heat release distribution on the spinning modes of an annular combustor with multiple matrix burners[END_REF] . This gives a flame volume V f = 4.18 cm 3 which, for a global thermal power per burner of ˙ Q = 1.44 kW, yields a heat release rate per unit volume equal to ˙ q = 3 . 2 × 10 8 W m -3 . For each burner, the interaction between combustion and acoustics is expressed by making use of a global FDF determined experimentally in a single burner setup comprising the same injector and equipped with a driver unit, a hot wire and a photomultiplier to measure velocity and heat release rate fluctuations respectively [START_REF] Noiray | A unified framework for nonlinear combustion instability analysis based on the flame describing function[END_REF][START_REF] Boudy | Describing function analysis of limit cycles in a multiple flame combustor[END_REF] . The reference point for the velocity fluctuation measurements is located in the injection unit 20 mm below combustor backplane. Figure 5 shows the FDF used in this analysis. Measurements of the gain and phase lag were carried out for five velocity fluctuation levels ranging from | u / u | = 0.1 to | u / u | = 0.5 (white disks in Fig. 5 ). One difficulty is to gather FDF data at large forcing amplitudes. Due to limitations of the equipment used to modulate the flame, it was not possible to cover the full frequency and amplitude ranges. A well-resolved FDF is used in the simulations by interpolation between the experimental points and extrapolation where experimental samples are missing. At very high amplitude levels that are reached in the MICCA experiment, the flames are disrupted and it is reasonable to represent this behavior by a fast drop in their response to external perturbation. Data are ex- trapolated at these levels with the best fit curve of a smooth fourth order polynomial function using data gathered at lower levels. This is carried out for each forcing frequency. There is a certain amount of uncertainty introduced by this process, but it is based on measured data points while most of the theoretical investigations are based on simplified representations. The nonlinear n -τ models [START_REF] Dowling | Nonlinear self-excited oscillations of a ducted flame[END_REF][START_REF] Li | Time domain simulations of nonlinear thermoacoustic behaviour in a simple combustor using a wave-based approach[END_REF][START_REF] Laera | A weakly nonlinear approach based on a distributed flame describing function to study the combustion dynamics of a fullscale lean-premixed swirled burner[END_REF] , simplified third order polynomials of heat release as a function of pressure [START_REF] Noiray | Investigation of azimuthal staging concepts in annular gas turbines[END_REF] or time invariant nonlinear representations of heat release rate as a function of pressure [START_REF] Ghirardo | Weakly nonlinear analysis of thermoacoustic instabilities in annular combustors[END_REF] do not feature all the complexity of the nonlinear flame response considered in the present study. Heat release rate fluctuations are assumed to be driven in the MICCA annular combustor by the fluctuating mass flow rates due to axial velocity perturbations through the corresponding injector [START_REF] Wolf | Using les to study reacting flows and instabilities in annular combustion chambers[END_REF][START_REF] Krebs | Thermoacoustic stability chart for high-intensity gas turbine combustion systems[END_REF] . This assumption is reasonable as the injectors are well separated in the configuration explored and there is no visible mutual interaction [START_REF] Worth | Modal dynamics of self-excited azimuthal instabilities in an annular combustion chamber[END_REF] . The reference points for the velocity fluctuations in the numerical domain are taken on the axis of the burner at the same distance from the flame domain as in the experiments. Dynamics of an initially spinning mode In a first stage, following the experimental observations the stability analysis is carried for the first azimuthal mode (1A) of the MICCA chamber starting the simulations with a spinning mode structure and the velocity distribution described by Eq. ( 6) . Figure 6 compares the system trajectories plotted in a frequencygrowth rate plane for two different values of the damping rate δ= 0 s -1 and δ= 12.5 s -1 in Eq. ( 2) . Each curve is colored with respect to the velocity fluctuation level that varies from | u / u | = 0.10 to | u / u | = 0.63 by steps of 0.01. Symbols are only plotted every 10 increments and at the trajectory endpoint to ease reading. The dynamical trajectories of the system are controlled by three free parameters, the frequency f , the growth rate α and the relative velocity amplitude level | u / u | . The 1A mode is found to be linearly unstable. For small velocity perturbations, the system features the highest growth rate of about 280 s -1 and a frequency around 400 Hz. A reduction of the growth rate and a substantial increase of the instability frequency is observed when | u / u | is augmented. This is due to the reduction of the FDF gain when the velocity fluctuation level increases, as can be observed in Fig. 5 a. If no damping is considered, the limit cycle condition is reached at a velocity fluctuation level that nullifies the gain of the FDF. When damping is considered the limit cycle is reached for an amplitude level | u / u | = 0.61, which corresponds to a FDF gain G = 0.08 at a frequency of f = 473 Hz. This frequency is close to that observed at limit cycle in the experiments f = 487 Hz. It is worth noting that at the velocity fluctuation level that nullifies the FDF gain, the sys- tem shows a negative growth rate of α= -12.5 s -1 which equals the damping rate that was determined experimentally. This checks that the dissipation rate is well represented in the numerical procedure. The structure of the unstable mode is now investigated at the limit cycle. Since the sixteen burners have the same FDF and are assumed to operate at the same amplitude level, the circumferential symmetry of the system defined by the annular geometry is conserved. The nonlinear stability analysis leads to degenerate solutions featuring two azimuthal modes sharing the same frequency and spatial structures shifted by π /2 as in the linear case [START_REF] Bauerheim | Symmetry breaking of azimuthal thermo-acoustic modes in annular cavities: a theoretical study[END_REF] . It is thus possible to add these two solutions at each amplitude level and obtain a spinning mode. The result is shown in Fig. 7 b (bottom) in the form of a pressure phase evolution plotted along the azimuthal direction at a mid-height position in the plenum and in the combustion chamber backplane. The phase evolutions feature a shift of 0.14 rad between the plenum and the combustion chamber. This angular shift is also observed in experiments and confirmed theoretically [START_REF] Bourgouin | Characterization and modeling of a spinning thermoacoustic instability in an annular combustor equipped with multiple matrix injectors[END_REF][START_REF] Laera | Impact of heat release distribution on the spinning modes of an annular combustor with multiple matrix burners[END_REF] . The pressure magnitude | ˆ p | shown in Fig. 7 a is obtained by plotting pressure contour lines computed by the Helmholtz solver on a cylindrical surface passing through the middle of the combustion chamber, the plenum, the burners and the microphone waveguides. The pressure iso-lines are deformed in the vicinity of the burners due to the near field acoustic interactions with the injectors, heat release zone and waveguides. However, a spectral analysis of the pressure distribution p = p (θ ) , not shown here, features small harmonic levels. At the burners locations, the harmonic content remains within 6% of the signal amplitude and falls to 1% one centimeter away from the chamber backplane as highlighted by the pressure iso-lines plotted in Fig. 7 a. This is confirmed observing the pressure distribu- tion p (θ ) = | ˆ p | cos arg( ˆ p ) along the azimuthal direction plotted in Fig. 7 b (top). Deformations appear in the distribution taken at the backplane of the combustion chamber, whereas the influence vanishes in the plenum. Stability of the limit cycle The stability of the spinning equilibrium point is now investigated following ideas developed in [START_REF] Ghirardo | Weakly nonlinear analysis of thermoacoustic instabilities in annular combustors[END_REF] : limit cycles coupled to spinning modes are stable if the derivative of the FDF with respect to the amplitude of the spinning mode oscillation ( | u / u | sp ) is negative. For a constant damping rate, the derivative of the FDF around the equilibrium point can be approximated by the derivative of the growth rate, indicating that the stability criterion can be reformu- ∂α ∂| ˆ p | | u / u | sp < 0 (8) The growth rate α plotted in Fig. 21 c as a function of the perturbation level u / | ū | features a negative derivative around the spinning oscillation equilibrium point reached for u / | ū | = 0.61. Eq. ( 8) is thus satisfied and one can conclude that the predicted spinning mode is stable as observed in the experiments. Comparisons with experiments Spinning pressure oscillations at the limit cycle corresponding One may however note that this harmonic content remains weak. The pressure peak at 974 Hz reaches 10 Pa. In the numerical calculation, signals are reconstructed by considering only the first harmonic found at 473 Hz in the simulations. Nevertheless, a good match is found in terms of amplitude and phase for the four sensors indicating that the fundamental dominates and proving that the numerical procedure is able to predict the difference in amplitude levels between the two cavities of the system. Again, a small phase mismatch can be observed when the comparison is carried out over a longer duration. Time resolved heat release rate signals are deduced in the experiments from two photomultipliers (H1 and H2) equipped with an OH * filter and placed at locations shown in Fig. 1 c. The heat release rate fluctuation ˙ Q can also be deduced from the simulation: ˙ Q ˙ Q = ˆ u u F (ω r , | u / u | ) exp (iω r t ) , (9) where ˆ u / ū is the calculated velocity oscillation level at limit cycle, F is the FDF that needs to be evaluated at the same forcing level, ˙ Q is the mean heat release rate, ω r is the angular frequency at limit cycle and t denotes the time. Figure 9 shows the recorded heat release rate signals (dashed lines) and the numerical reconstructions (solid lines). The two photomultipliers records feature nearly the same amplitudes and a phase shift of 1.63 rad being close to the theoretical value of π /2. However, the amplitude is Table 1 Sensitivity of the limit cycle oscillation frequency f , the pressure amplitude recorded by microphone MC1 in the chamber, the FDF gain and phase lag and the relative level of heat release rate measured by photomultiplier H1 with respect to the velocity fluctuation level | u / ū | reached at limit cycle. continuous lines with rectangular marks in Fig. 9 . They feature the same phase shift as that recorded by the photomultipliers. In terms of amplitude, the two reconstructions share the same amplitude, but the value is about four times lower than the one recorded in the experiments. The reason for this sizable difference is now investigated with the help of Eq. ( 9) . Given the relatively low damping of the system ( δ = 12 . 5 s -1 ), the limit cycle is reached for a large velocity oscillation level | u / u | = 0.61 higher than 0.5, in a range where the FDF gain is slightly extrapolated and features a steep slope as can be seen in Fig. 5 1 . This is not the case for the corresponding heat release rate oscillation ˙ Q / ˙ Q due to the rapid drop of the FDF at large perturbation amplitudes. Data in Table 1 indicate that a reduction of 5% of the level of velocity fluctuation at the limit cycle results in a variation of 170% of the FDF gain G and, correspondingly, in the resulting heat release rate oscillation amplitude. The phase of the FDF and the frequency of the resonant mode remain as a first approximation unaffected by these changes. The amplitude differences between measurements and numerical predictions reduce when the heat release rate signals are reconstructed for an oscillation level | u / u | = 0.58 as shown by the continuous lines with circular marks in Fig. 9 . A perfect match in amplitude between the experimental and numerical signals is still not achieved, but differences are notably reduced. These tests confirm the strong sensitivity of the predicted level of heat release rate reached at limit cycle due to small uncertainties on the velocity oscillation level in the injector. This feature reflects that small uncertainties on the data gathered for FDF at high forcing levels may lead to large deviations of the predicted heat release rate fluctuations due to the rapid drop of the FDF gain with the forcing amplitude when the flame is disrupted. | u / u | f (Hz) MC1 (Pa) Gain ϕ ( × π) | ˙ Q / ˙ Q | 0. Analysis of operating point B For an equivalence ratio φ= 1.11 and a bulk flow velocity equal to u b = 0.66 m s -1 , the system features a well-established selfsustained combustion oscillation associated to standing mode at a frequency of 498 Hz that was fully characterized in [START_REF] Durox | Nonlinear interactions in combustion instabilities coupled by azimuthal acoustic modes[END_REF] . Two standing mode patterns have been observed in the system for the same operating conditions depending on the run recorded. Figure 10 shows long time exposure photographs of these two modes with the position of the nodal line indicated by the red dashed line. The mode structure shown in Fig. 10 a features a nodal line between burners I -II and between burners IX -X . In this work, this mode structure will be designated as "V type". Figure 10 b shows the mode structure with the nodal line between burners V -VI and burners XIV -XV that will be referred as "H type". The FDF corresponding to this new operating condition is shown in Fig. 11 . The same difficulties to get data for the FDF at high amplitudes, typically | u / ū | > 0 . 5 , persist for this new operating condition. As for operating condition A, the heat release rate is uniformly distributed over each burner by post-processing the flame image shown in Fig. 2 b [START_REF] Laera | Impact of heat release distribution on the spinning modes of an annular combustor with multiple matrix burners[END_REF] recorded for operating regime B in the single burner setup in a thermo-acoustically stable state. This procedure results in a flame volume V f = 6.11 cm 3 that is distributed over a cylinder of height l f = 6 mm, 2 mm longer than that used for operating regime A, and a diameter of d f = 36 mm which, for a global thermal power per burner of ˙ Q = 2.08 kW, yields a heat release rate per unit volume equal to ˙ q = 3.31 W m -3 . The mean temperature in the combustion chamber varies from 1521 K near the flame zone to 1200 K at the end of the combustion chamber. The temperature of the plenum is set equal to 300 K. Dynamics of an initially standing mode The numerical procedure described in Section 2 is used for the analysis of the 1A standing mode dynamics when the system operates at regime B. Without unsteady heat release, the circumferential symmetry of the system defined by the annular geometry is conserved and the eigenvalue analysis yields degenerate solutions. Figure 12 2 , computed under passive flame conditions ( ˙ q = 0) and plotted over a plane located at the burner inlet section. The two modes share the same frequency 472 Hz, but their structures are shifted by π /2. The corresponding distribution used to initialize the simulation is shown in Fig. 12 b where symbols indicate the burners' angular locations. When a distribution of FDFs is introduced ( ˙ q = 0), the rotational symmetry of the system is broken [START_REF] Noiray | Investigation of azimuthal staging concepts in annular gas turbines[END_REF] and the eigenvalue analysis results no more in degenerate azimuthal modes but yield two distinct waves characterized by different modal structures. The frequency and the growth rate of these modes depend on the level of asymmetry considered in the system. At the onset of instability, Fig. 11 (b) indicates that the velocity fluctuation level weakly influences the flame response. The eigenvalue analysis of this weakly asymmetric system yields two waves with different yet close frequencies and growth rates (see Appendix B). Increasing the velocity fluctuation level, the differences increase between the gain and phase values taken by the FDFs from the different burners. This leads to stronger asymmetric configurations in which the two modes resulting from the eigenvalue analysis are characterized by an important shift in growth rate and also frequency. The frequency shift depends on the amount of heat release rate fluctuation considered in the system [START_REF] Silva | Combining a Helmholtz solver with the flame describing function to assess combustion instability in a premixed swirled combustor[END_REF] . In the validation calculations discussed in the Appendix B with a simplified heat release response, the two waves manifest only a growth rate shift, while they share the same frequency. Here the two solutions feature both a frequency and growth rate shifts. As discussed in Section 2 , for each velocity level | u / u | j , the mode structure ψ k computed at the k th iteration is used in Eq. ( 7) to modulate the FDFs. However, in order to be consistent, only the mode structure closer to the one chosen at the first iteration (the ψ 0 1 or ψ 0 2 shown in Fig. 12 ) is followed until the convergence criterion described in Section 2 is satisfied. The dynamical trajectories of the system are first tracked without considering any damping by setting δ= 0 s -1 in Eq. ( 2) to analyze the influence of the chosen initial distribution. These trajectories are plotted in Fig. 13 the different levels, symbols are only plotted every 10 increments and at the trajectory endpoint. Rectangular marks in Fig. 13 indicate results obtained by considering the initial mode structure ψ (0) 1 shown in Fig. 12 a and designated as H type. Circular marks in Fig. 13 indicate results obtained by considering the initial mode structure ψ (0) 2 in Fig. 12 a and designated as V type. Regardless of the eigenmode used to initialize the simulations the frequencies and growth rates are the same. It is worth noting that, in contrast to spinning mode calculations, a limit cycle condition cannot be reached without accounting for a finite damping level regardless of the velocity oscillation level considered in the system. At each point of the trajectory, flames close to the nodal line always experience a small velocity fluctuation | u / u | . This small oscillation leads to a finite heat release rate fluctuation and this causes the mode to become unstable. With the introduction of a damping term, a limit cycle is obtained as shown in Fig. 14 where the dynamical trajectories without damping, δ= 0 s -1 (circular marks), and with a damping rate of δ= 12.5 s -1 (rectangular marks) are compared. Each curve is colored in this figure by the velocity fluctuation level that varies from | u / u | = 0.1 to | u / u | = 0.9 with steps of 0.01. The limit cycle condition α= 0 is reached in the simulation for a fluctuation level | u / u | = 0.86 at a frequency f = 478 Hz. This value is close to that recorded at the limit cycle in the experiments f = 498 Hz. As discussed in [START_REF] Noiray | Investigation of azimuthal staging concepts in annular gas turbines[END_REF] , the amplitude of a standing limit cycle is greater than the amplitude of a spinning limit cycle that would settle for the same operating conditions and the same flame response model. Even though the operating conditions differ, it is found here that the oscillation level | u / u | = 0 . 86 of the limit cycle of operat- ing condition B associated to standing mode is larger than the one | u / u | = 0 . 61 found for the operating condition A featuring a limit cycle with a spinning pattern. In agreement with experiments shown in Fig. 10 , two different modal structures with a nodal line shifted by π /2 are predicted at the limit cycle depending on the chosen initial condition. Figure 15 a shows the H type limit cycle pressure distribution calculated by the Helmholtz solver. This solution is obtained from the modal pressure distribution ψ (0) 1 shown in Fig. 12 as initial condition. Figure 15 b shows the V type limit cycle pressure distribution obtained assuming the initial distribution ψ (0) 2 . The distribution of the velocity fluctuation level | u / u | reached at limit cycle for both modal structures is shown in Fig. 16 . It differs from a pure sinusoid due to the local deformations of the pressure field near the injector outlets as already discussed for operating condition A. These deformations are taken into account in the ψ function used to fix the velocity oscillation level in the different burners. Symbols in Fig. 16 indicate the levels reached at the sixteen burner positions. One may note that for the largest velocity fluctuations, flow reversal conditions can be reached during part of the oscillation cycle in the injectors located close to the pressure anti-nodal lines. It is next interesting to compare the location of the pressure nodal line. There is no precise experimental determination of its angular position. Examining the long time records shown in Fig. 10 , one finds that in both configurations the nodal line is always located between two burners in a region indicated in grey in Fig. 17 . The same figure also shows the predicted nodal line plotted as a red dashed line for the two structures. For the standing limit cycle featuring a V type structure, the nodal line is predicted between two burners in Fig. 17 Stability of the limit cycle The stability of the standing limit cycle is now investigated. According to Ghirardo et al. [START_REF] Ghirardo | Weakly nonlinear analysis of thermoacoustic instabilities in annular combustors[END_REF] , a standing limit cycle is stable if three necessary and sufficient conditions are satisfied. In essence, the first condition requires that at the limit cycle, the growth rate decreases when the velocity level is increased. The growth rate trajectory obtained by varying the maximum amplitude of the velocity oscillation level | u / u | plotted in Fig. 23 c (continuous curve with rectangular marks) indicates that the derivative of the growth rate around the standing equilibrium point is negative. This test confirms that the first stability condition is satisfied. The second criterion defines a condition on the orientation of the nodal line. Ghirardo et al. [START_REF] Ghirardo | Weakly nonlinear analysis of thermoacoustic instabilities in annular combustors[END_REF] have demonstrated that this condition is always fulfilled in configurations with a large number of burners. Although the MICCA combustor features a finite number of burners, this second criterion may be regarded as satisfied considering that heat release rate fluctuations take place in a flame domain, which nearly covers the entire surface area of the combustion chamber. The third condition discusses the stability of the standing wave pattern. Stable standing modes need to comply with the following inequality: N 2 n = 2 π 0 F (ω r , | u / u | i ψ (θ )) /Z(θ ) sin (2 nθ ) dθ > 0 . ( 10 ) In this expression, n is the azimuthal mode order, which is here equal to n = 1 and the mode ψ is chosen such that there is a pressure anti-node at θ = π / 4 . The H type mode has the closest structure fulfilling this condition and is used for the calculation. The FDF values F in Eq. ( 10) are calculated at the limit cycle oscillation frequency f = 480 Hz with the corresponding velocity distribution. The quantity Z ( θ ) designates the impedance, i.e., the ratio of pressure in the flame zone to the velocity at the reference point. Results for F (| u / u | i ψ (θ )) /Z(θ ) are plotted in red in Fig. 18 . It consists of a piecewise function taking constant values over the angular extensions of the flame zones and zero values in the angular extensions between two flames. In the same graph the function sin (2 n θ ) is represented in black. The consequence is that the component in red in Fig. 18 changes sign with amplitude resulting in a positive overall integral for N 2 n . The predicted standing limit cycle is thus found to be stable as observed in experiments. This is due to the fact that the FDF of the matrix burners investigated in the present study features a phase lag which is sensitive to | u / u | as shown in Fig. 11 b. The situation slightly differs from that investigated in Appendix A with a simplified flame response characterized by a monotonically decreasing gain but a constant phase lag independent of the input amplitude level. In this case, limit cycles coupled to standing waves are found to be unstable. Comparisons with experiments Calculations for the standing limit cycle are now compared to measurements. The frequency and amplitude of velocity fluctuations predicted at limit cycle at one reference point of the system are used to reconstruct time resolved pressure fluctuations in the plenum and in the chamber and heat release rate fluctuations. The analysis is only carried out for results corresponding to the V structure type. Figure 19 (a The amplitude and the phase shift between microphones located near the anti-nodal line, i.e., MP1 and MP5, is well captured by the simulations. Calculations also retrieve the amplitudes of the microphones located near the nodal line, i.e., MP3 and MP7, however, with a small phase mismatch. This is due to the shift of θ ∼ 3 °between the angular position of the sensors and of the nodal line shown in Fig. 17 a. In the numerical reconstruction, microphones MP3 and MP7 are located on the two sides of the nodal line and their signals are consequently in phase opposition. In the experiments, these records are nearly in phase as shown by the green and blue dashed lines in Fig. 19 (a) (top). Increasing the interrogation period one finds a small phase mismatch between experiments and simulations due to the 20 Hz difference between the measured and predicted limit cycle oscillation frequencies. Figure 19 (a-bottom) shows the same type of comparison for pressure signals in the combustion chamber. Again, numerical reconstructions and measurements are, respectively, identified by continuous and dashed lines. In the experiments, the signals feature a some harmonic content and significantly differ from one to another depending on the azimuthal position of the sensors. The analysis of the spectrum content of the signals shown in Fig. 19 (bbottom) reveals the second harmonic peak at 992 Hz. A previous study on the same combustor featuring standing mode oscillations [START_REF] Durox | Nonlinear interactions in combustion instabilities coupled by azimuthal acoustic modes[END_REF] has shown that this frequency is associated with the first longitudinal mode of the combustion chamber (1L). For the pressure signals located close to the nodal line, the amplitude of the second harmonic peak is comparable to the amplitude of the 1A mode putting in evidence a competition between the 0L-1A plenum oscillation at 495 Hz and a longitudinal mode associated to the 1L-0A mode of the chamber. This yields the distorted signals recorded by microphones MC3 and MC7 shown in Fig. 19 (a-bottom). The other microphones located closer to the pressure anti-nodes feature a second harmonic at 992 Hz with an amplitude of about one order of magnitude lower than that of the first harmonic [START_REF] Durox | Nonlinear interactions in combustion instabilities coupled by azimuthal acoustic modes[END_REF] . This yields the roughly sinusoidal signals with a peak amplitude at 495 Hz of 60 Pa recorded by microphones MC1 and MC5 shown in Fig. 19 (a-bottom). In the FDF framework, the numerical signals can only be reconstructed for the first harmonic oscillation. Overall, a good match is found in terms of both amplitude and phase for the four sensors indicating that the numerical procedure is able to pre- dict the difference in amplitude levels between the two cavities of the system. Again, a small phase mismatch is observed when the comparison is carried out over a longer duration period. The numerical heat release rate signals (continuous lines) reconstructed with Eq. ( 9) are compared in Fig. 20 with the two OH * light intensity signals (dashed lines) recorded by the photomultipliers (see Fig. 1 (c)). The two photomultipliers signals feature different amplitudes. The injector close to the nodal line features the largest heat release rate oscillations reaching ˙ Q / ˙ Q = 0 . 5 measured by H1. A drop of about 50% of the peak amplitude is observed for the injector close to the anti-nodal line (H2 signal in Fig. 20 ). It is also worth noting that the signal is in this case not a pure sinusoidal wave indicating the presence of harmonics. The two signals measured by the photomultipliers are nearly in phase due to the position of the sensors with respect to the nodal line. The small phase shift is due to the presence of harmonics. As for the pressure signals, the simulated heat release rate signals only consider the first harmonic. The reconstructed numerical signals have the same phase shift as those recorded by the photomultipliers. This indicates that the phase of the FDF is well captured. The numerical procedure also retrieves the amplitude drop between the two signals, however, the predicted amplitude peak and the one recorded in the experiments differ. For the H1 signal, differences between simulations and experiments are mainly due to the mismatch on the position of the nodal line shown in Fig. 17 . As already discussed for operating condition A, the heat release rate oscillation level strongly depends on the predicted value of | u / u | and on the gain of the FDF in Eq. ( 9) . For high velocity fluctuation levels, the FDF gain rapidly drops as shown in Fig. 11 a. As a consequence, the numerical prediction of the H2 signal is quite sensitive to small uncertainties on the velocity oscillation level at the limit cycle. These uncertainties lead to important variations of the FDF gain G and consequently in the predicted heat release rate oscillation at limit cycle. The origin of the differences observed at limit cycles between measurements and simulations is thus the same as for the spinning mode calculations studied in the previous section. Mode type selection Following the suggestion of a reviewer, an investigation of the possible scenarios leading to the spinning and standing limit cycles analyzed in the previous sections is now proposed. Analysis for the operating point A The stability analysis is repeated for operating condition A, but instead of assuming an initially spinning mode as done in Section 4.1 , simulations are initiated with a standing mode structure and the velocity distribution described by Eq. ( 7) . The corresponding trajectory (black continuous line with rectangular marks) plotted in a frequency-growth rate-velocity fluctuation level space is compared in Fig. 21 a with the spinning mode trajectory calculated in Section 4.1 (red continuous line with circular marks). At the starting point, i.e. for | u / u | = 0.1, the two trajectories perfectly match. This is due to the fact that in the linear regime, i.e. for values of | u / u | ≤0.1, the FDF is simply a transfer function and its gain and phase do not depend on the input amplitude level. This also means that in simulations started with a standing mode structure, each burner of the chamber operates with the same velocity fluctuation. Projecting these trajectories on a frequency-growth rate ( f -α) plane, as shown in At the equilibrium point one finds that the two solutions share the same frequency, but differ in the level of velocity fluctuations: the standing mode limit cycle is obtained at higher values of | u / u | with respect to the spinning mode. One may then first discuss the stability of the two equilibrium points. The spinning limit cycle has been already proved to be stable in Section 4.2 . To prove the stability of the standing mode solution, the three criteria summarized in Section 5.2 are applied to this new standing equilibrium point. This analysis with the same experimental FDF shown in Fig. 5 has been recently carried out by Ghirardo et al. [START_REF] Ghirardo | Weakly nonlinear analysis of thermoacoustic instabilities in annular combustors[END_REF] . According to this study, a standing limit cycle assumed to occur for a value of | u / u | st = 0.5 and for a frequency of f = 487 Hz has been shown to be unstable and this was accepted as the main reason why the spinning mode prevails in this case. One may note, however, that the amplitude level adopted in [START_REF] Ghirardo | Weakly nonlinear analysis of thermoacoustic instabilities in annular combustors[END_REF] differs from the value obtained from the present numerical results where | u / u | st 0 . 86 and the frequency is slightly at variance with that deduced from the present calculations. The stability analysis of the limit cycle at | u / u | st 0 . 86 and f = 473 Hz is thus repeated here. The standing mode remains unstable as observed from Fig. 22 where inequality of Eq. ( 10) is graphically verified as described in Section 5.2 . In addition to the stability analysis of the equilibrium solutions, a second possible mode type selection scenario may be considered by analyzing the two trajectories shown in Fig. 21 c. One may observe that the growth rate of the standing mode trajectory is always greater than the one of the spinning mode, with the exception of the range with | u / u | → 0.1 where the spinning mode trajectory has a greater growth rate. This region is highlighted in the zoomed box in Fig. 21 c. Noting that the two dynamical trajectories do not cross ( Fig. 21 a), one may conclude that the nature of the limit cycle is determined in the region where the FDF begins to depend on the input amplitude. According to this scenario, the spinning mode of oscillation is selected in this case because its growth rate slightly exceeds that of the standing mode. Once on that trajectory the velocity fluctuation level remains uniformly the same for all injectors and this leads to a spinning mode limit cycle. Increasing the value of | u / u | on the spinning mode path all the flames respond to the same velocity fluctuation level. A jump to the standing mode solution might be possible in principle but this would require a strong perturbation that would change the velocity distribution and provide the nonuniformity in amplitude pertaining to the standing mode oscillation. The difference in growth rates of the standing and spinning solutions is relatively small and one cannot be totally sure that this explains why the spinning mode is experimentally observed in this case. One can say at least that the range where the bifurcation takes place corresponds to the region where the FDF map shown in Fig. 5 is interpolated between experimental data. The errors in that region are lower than those made when the amplitude of oscillation becomes large near the limit cycle of the standing mode. One may conclude from this analysis that the selection between spinning and standing modes may be explained by examining the stability of the final limit cycles in agreement with criteria derived by Ghirardo et al. [START_REF] Ghirardo | Weakly nonlinear analysis of thermoacoustic instabilities in annular combustors[END_REF] . However, the mode type selection is perhaps not only a matter of limit cycle stability. Simulations also reveal that the spinning mode features a growth rate exceeding that of the standing mode in a narrow region where the FDF starts to change with the input amplitude level. While the difference is minute, an alternative scenario has been developed to explain the spinning mode selection. A jump between the spinning and standing limit cycles seems to be possible but would require a sufficiently strong perturbation modifying the azimuthal distribution in velocity fluctuation. This mechanism would then be somewhat similar to the mode switching process documented in Noiray et al. [START_REF] Noiray | A unified framework for nonlinear combustion instability analysis based on the flame describing function[END_REF] . Analysis for operating point B As for the previous configuration, the stability analysis is now repeated for operating point B with an initial spinning mode structure and the velocity distribution in Eq. ( 6) . The predicted trajectory (red continuous line with circular marks) is compared with the trajectory calculated in Section 5.1 in the three-dimensional 23 c, one finds that the growth rate deduced from the standing mode distribution is always greater than the growth rate of the spinning mode distribution. According to this criterion, the standing mode prevails in this situation leading to a limit cycle with a standing pattern in agreement with experiments. f -α -| u / u | space in The stability of the two possible equilibrium points is now assessed. The standing equilibrium point was already proved to be stable in Section 5.2 . However according to Eq. ( 8) , the spinning equilibrium point is also found to be stable. This state is however not observed in the experiment. As a conclusion, the observed stable limit cycle state is in agreement with the stability criteria developed in Ghirardo et al. [START_REF] Ghirardo | Weakly nonlinear analysis of thermoacoustic instabilities in annular combustors[END_REF] , but these criteria are insufficient for this operating condition to fully determine which state is observed in the experiment. This case indicates without ambiguity that the dynamical trajectories need to be considered to fully determine the state of oscillation observed in the experiments. The main findings for the stability and trajectory analyses of the two operating points are summarized in Table 2 . It is concluded that a criterion combining the stability analysis at the limit cycle with the trajectory analysis (line 7 in Table 2 ) might best define the mode type at the limit cycle. Conclusion A procedure combining a Helmholtz solver and the FDF framework is proposed to analyze the dynamics of an annular combustor. The FDF is determined experimentally, together with the damping rate. This method is first validated by making use of an idealized annular combustor model characterized by a simplified nonlinear flame response. Subsequently, the dynamics of spinning and standing combustion instabilities observed in a laboratory scaled annular combustor are then investigated numerically. Spinning and standing initial solutions are considered first without effects of unsteady heat release. For the spinning mode dynamics, effects of unsteady heat release is treated by assuming that the velocity oscillation level | u / u | is the same in each burner and that the flame response described by the FDF operates at this level in the 16 injectors. For the standing mode dynamics, all injectors operate at different velocity oscillation levels | u / u | (θ ) depending on their relative position θ with respect to the nodal line of the analyzed mode. The distribution of | u / u | over the injectors is deduced from the pressure distribution computed by the code using an iterative procedure. Calculations of the dynamical trajectories of spinning and standing modes are carried out for increasing velocity oscillation levels until a limit cycle is reached when the calculated growth rate equals the damping rate. This numerical procedure is used to tentatively determine the modal structure at the limit cycle. Results indicate that the system trajectories and the stability of the limit cycle need both to be considered to fully determine the final state observed in the experiments for the two operating conditions explored. The problem of mode type selection however deserves further investigation, since the present results correspond to special cases with no proof of generality. Results at limit cycle for pressure, velocity and heat release rate signals are for the first time compared to detailed experimental data. The main findings of this study are: • The predicted instability frequencies match those measured in the experiments for the two different operating conditions investigated. • The proposed numerical strategy allows to capture the correct pressure signals recorded in the plenum and in the chamber with a good match in terms of phase shift and amplitude for both operating conditions featuring a stable spinning limit cycle and a stable standing limit cycle. • For the standing limit cycle, the pressure signals close to the nodal line are less well reproduced than those at the anti-nodal line, but the two possible positions of the nodal lines observed in the experiments between two burners are well captured. • The stability of the predicted spinning and standing limit cycles has been proven numerically by making use of recent theoretical elements and match the states observed in the experiments. • Heat release rate signals at spinning and standing limit cycles are well reproduced in terms of phase shift and of heat release distribution between the different burners, but the predicted amplitudes differ from those measured. This is mainly attributed to the fact that the limit cycle is reached for oscillation states where the gain of the FDF rapidly drops and nonlinearly with the velocity fluctuation level | u / u | . This means that even small uncertainties on | u / u | lead to large variations of the FDF gain and, consequently, to large uncertainties on the amplitude of the predicted heat release rate fluctuations. • Two scenarios are investigated to deduce the preferred modal type that will settle at limit cycle. The first scenario based on a stability analysis of the predicted limit cycle oscillation distinguishes the final outcome in one case but fails to do so in another case. A new scenario relying on a calculations of frequency-growth rate trajectories of initially spinning and standing modes appears to identify the solution type. In one case the difference in growth rate is relatively small and may not be completely meaningful. The combination of these two scenarios might perhaps give the answer but this admittedly needs to be confirmed by further investigations. This study shows that the numerical procedure developed herein adequately reproduces the dynamics of combustion instabilities coupled to azimuthal modes provided that the FDF can be determined accurately at sufficiently large perturbation levels. The lack of such data is a source of uncertainty. Acknowledgment The development of the annular combustor MICCA was funded by the Agence Nationale de la Recherche , Contract No. ANR-08-BLAN-0027-01 , and by Safran Tech. Davide Laera was supported by a research fellow grant provided by the Politecnico di Bari (Bari, Italy) for a six months period at EM2C. The authors wish to thank the reviewers for their critical and constructive examination of this article. Appendix A. Validation of the numerical procedure Theoretical results from Noiray et al. [START_REF] Noiray | Investigation of azimuthal staging concepts in annular gas turbines[END_REF] are used validate the proposed numerical procedure. The main elements of this analytical model are reproduced here. The geometry considered is an idealized annular combustion chamber of radius R where pressure fluctuations are assumed to be only function of time and azimuthal coordinate θ . After proper normalization of the flow variables, the normalized pressure disturbances p obey to: ∂ 2 p ∂t 2 + ζ ∂ p ∂t - ∂ 2 p ∂θ 2 = ∂Q ∂t , ( 11 ) where Q represents the normalized volumetric heat release rate disturbance and ζ accounts for damping. It is further assumed that the pressure of any azimuthal mode of order n can be expressed as: p(θ , t ) = A (t ) sin (nt ) cos (nθ ) + B(t ) cos (nt ) sin (nθ ) , (12) where A and B are slowly varying functions with time. For a nonlinear flame response linked to the pressure by: Q = β p -κ p 3 , (13) Noiray et al. [START_REF] Noiray | Investigation of azimuthal staging concepts in annular gas turbines[END_REF] found that the dynamics of this system was determined by a set of two-nonlinear coupled differential equations: dA dt = (β -ζ ) A - κ 32 9 A 2 + 3 B 2 A , ( 14 ) dB dt = (β -ζ ) B - κ 32 9 B 2 + 3 A 2 B. ( 15 ) The spinning and standing limit cycles correspond to the fixed points of this system. Spinning modes are found for A = B = 2 (β -ζ ) / 3 κ. Standing modes correspond to solutions with one Figure 24 shows the computational domain used for the Helmholtz solver simulations. It consists of an annular duct with a length of 0.2 m. The radial extension of the annulus is set to 1 mm to avoid any low frequency radial components. Furthermore, a velocity node is assumed at each boundary to exclude the rising of longitudinal components. Under passive flame conditions ( Q = 0 ), the eigenmode analysis leads to degenerate solutions due to the circumferential symmetry of the system. In this case, the two first azimuthal modes ψ 1 and ψ 2 share the same frequency and are shifted by π /2. These two modes are orthogonal and form a basis used to describe the pressure field in the system. The heat release rate is assumed to be distributed in a flame sheet volume extending 4 mm in the longitudinal direction and covering the entire inlet section indicated in blue in Fig. 24 . The Helmholtz solver requires the transposition of the nonlinear flame model in the frequency domain. Taking the Fourier transform of Eq. ( 13) yields: ˆ Q ω r , | ˆ p | = ˆ p ( ω ) 3 4 κ| ˆ p | 2 -β , ( 16 ) where ˆ Q and ˆ p are the dimensionless Fourier transforms of heat release rate and pressure disturbances taken at the inlet. The weakly nonlinear stability analysis around the first azimuthal mode of the system (1A) is repeated for different pres- For the spinning mode calculation, the same forcing level is imposed at each injector. The circumferential symmetry of the system defined by the annular geometry is thus conserved. Consequently, regardless of the pressure fluctuation level | ˆ p | considered in the model, the weakly nonlinear stability analysis always yield degenerate solutions. This indicates that the two degenerate modes ψ 1 and ψ 2 have the same amplitude, a result in line with [START_REF] Noiray | Investigation of azimuthal staging concepts in annular gas turbines[END_REF] . also equal to that found analytically [START_REF] Noiray | Investigation of azimuthal staging concepts in annular gas turbines[END_REF] . If the pressure fluctuation level is increased further, damping of acoustic energy becomes greater than the energy gained and the oscillation amplitude ceases to grow. For | ˆ p | = 1 the FDF saturates at 3 / 4 κβ, the growth rate is negative (damping rate) and is equal to α = -ζ / 2 . For the standing mode calculation, the value of the FDF gain depends on the angular position and is fixed by the distribution analytical predictions [START_REF] Noiray | Investigation of azimuthal staging concepts in annular gas turbines[END_REF] . The stability of the predicted limit cycles is now investigated. By inspection, one observes that the stability condition of Eq. ( 8) is satisfied for the growth rate behavior shown in Fig. 25 a, indicating that the simulated spinning limit cycle is found to be stable in agreement with analytical results [START_REF] Noiray | Investigation of azimuthal staging concepts in annular gas turbines[END_REF] . The stability of the standing mode is analyzed by determining the Jacobian matrix. Since B= 0 when A = 0 and vice versa, the only non-zero components of the Jacobian matrix are the coefficients: For each point of the red trajectory B= 0, indicating that this mode has no influence on the standing equilibrium point of this system [START_REF] Noiray | Investigation of azimuthal staging concepts in annular gas turbines[END_REF] . For a given oscillation level | ˆ p | max , the growth rate α ψ 2 value on the red curve corresponds to the Jacobian coefficient J 22 . At the equilibrium point J 22 is found to be equal to λ 2 = ( βζ ) / 3 in agreement with the analytical value [START_REF] Noiray | Investigation of azimuthal staging concepts in annular gas turbines[END_REF] . The eigenvalues λ 1 and λ 2 of the computed Jacobian matrix have opposite signs indicating that the standing mode is a saddle point and corresponds thus to an unstable limit cycle in agreement with theoretical predictions. These simulations show that the proposed numerical procedure suitably retrieves the amplitude and the stability properties of both standing and spinning limit cycle in an idealized configuration where analytical results are available. plitudes of velocity fluctuations | u / u | have to be considered to represent the response of the flame above each injector. This leads to consider different FDF gains and phase lags values for each injector. In the proposed methodology, for each level of velocity fluctuations | u / u | j , the distribution of | u / u | over the injectors is determined from the pressure distribution computed by the code by setting: Fig. 1 . 1 Fig. 1. (a) Photograph of the MICCA combustor with a close up view of a matrix injector and a waveguide outlet. (b) Schematic representation of the experimental setup. (c) Top view of the MICCA chamber with microphone and photomultiplier measurement locations indicated. Fig. 2 . 2 Fig. 2. Images of the flame region recorded above one matrix injector under stable operation for the two conditions investigated. Fig. 3 . 3 Fig. 3. (a) Top view schematic representation of the model. (b) Longitudinal A-A cut showing geometrical details of the matrix injector model and of the flame domain. (c) Three dimensional model of the MICCA chamber with the details of the unstructured mesh comprising approx. 130,0 0 0 tetrahedral elements. Fig. 4 . 4 Fig. 4. Acoustic response of the MICCA combustor from 300 Hz to 500 Hz measured by microphones located in the (a) plenum and (b) combustion chamber. The frequency bandwidth f determined at half maximum provides the damping rate in both volumes. Fig. 5 . 5 Fig. 5. Interpolated Flame Describing Function (FDF) obtained for operating point A: φ= 1.00 and u b = 0.49 m s -1 . Experimental data points are displayed as white dots. (a) Gain. (b) Phase ϕ. Fig. 6 . 6 Fig. 6. Dynamical trajectories in the frequency ( f ) -growth rate ( α) plane colored by the velocity fluctuation level | u / u | for two damping rates δ = 0 (rectangular marks) and δ= 12.5 s -1 (circular marks). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article). Fig. 7 . 7 Fig. 7. (a) Pressure mode magnitude | ˆ p | with pressure contour lines plotted on a cylindrical surface equidistantly located from the lateral walls. (b) Top: pressure structure ˆ p = | ˆ p | cos arg( ˆ p ) distribution along the azimuthal direction in the plenum (circular marks) and in the combustion chamber (rectangular marks). Bottom: pressure phase evolution in the azimuthal direction in the plenum (circular marks) and in the combustion chamber (rectangular marks). Fig. 8 . 8 Fig. 8. (a) Four pressure signals recorded by microphones in the plenum (top) and in the combustion chamber (bottom) under spinning limit cycle conditions (dashed lines) compared with numerical reconstructions (solid lines). (b) Spectral content of the pressure signals in the plenum (top) and in the combustion chamber (bottom). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article). Fig. 9 . 9 Fig. 9. Time resolved heat release rate signals recorded by two photomultipliers in the combustion chamber (dashed lines) compared with numerical reconstructions (solid lines) for | u / u | = 0.61 (rectangular marks) and for | u / u | = 0.58 (circular marks). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article). to | u / u | = 0.61 are now compared to measurements. In the experiments, these fluctuations are recorded by microphones placed at four positions equidistantly separated on the external perimeter of the plenum and at four positions in the backplane of the combustion chamber as shown in Fig.1 c. Figure8(a) displays microphone measurements in the plenum (dashed lines) compared with the numerical reconstructions (continuous lines) at the same locations. The phase shift between two microphones signals corresponds to their relative position confirming the spinning nature of the mode. Well-established sinusoidal signals with a peak of about 260 Pa at a frequency of 487 Hz, as shown in Fig.8(b) (top), are found in the experiments. The amplitude and the phase shift between microphones is well captured in the simulation. Increasing the examination period, a small phase mismatch between experiments and simulations appears due to the 15 Hz difference between the numerical and experimental frequencies. Figure 8 ( 8 bottom) shows experimental (dashed lines) and numerical signals (solid lines) in the combustion chamber. The microphones mounted on waveguides at a distance of 170 mm away from the backplane of the combustion chamber measure a delayed signal with a time lag τ m -b = 0.5 ms. Since this delay is not negligible compared to the oscillation period of the instability ( 2 ms), it is taken into account in the data processing. It is first worth noting that the pressure fluctuation level only reaches 60 Pa near the chamber backplane and experimental signals are not purely symmetric relative to the ambient mean pressure, indicating the presence of harmonics as revealed in the spectral content of the pressure signals shown in Fig. 8 (b) (bottom). not constant with time indicating the presence of harmonics. As for the pressure signals, the simulated heat release rate signals only consider the oscillation at the fundamental frequency. The reconstructed numerical signals with | u / u | = 0.61 correspond to the Fig. 10 . 10 Fig. 10. Long time exposure photographs of the flames at a limit cycle coupled to a stable standing azimuthal mode. The velocity nodal line is shown as a dashed red line. (a) V type mode structure with a nodal line observed between burners I -II and IX -X . (b) H type mode structure with with a nodal line between burners VIII -IX and XIV -XV [37] . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article). a. A small change of the velocity fluctuation level | u / u | weakly alters the corresponding pressure oscillation level in the plenum and the chamber because these quantities scale linearly with | u / u | as shown in Table Fig. 11 . 11 Fig. 11. Interpolated Flame Describing Function (FDF) obtained for operating point B: φ= 1.11 and u b = 0.66 m s -1 . Experimental data points are displayed as white dots. (a) Gain. (b) Phase ϕ. Fig. 12 . 12 Fig. 12. (a) Pressure modulus | p | of the f = 472 Hz degenerate modes computed under passive flame conditions and plotted in a plane located at the burner inlet section. (b) Pressure distribution ψ used to initialize the computation of the standing mode. Symbols indicate the angular position of the sixteen burners. a shows the pressure modulus | p | of the degenerates modes, ψ (0) 1 and ψ (0) in a frequency-growth rate plane and colored with respect to the maximum amplitude level | u / u | j , which is varied from | u / u | = 0.1 to | u / u | = 0.9. The subscript j will be omitted in what follows. Unless indicated otherwise, | u / u | always refers to the maximum velocity fluctuation level considered in the simulation for the standing mode analysis. In order to emphasize Fig. 13 . 13 Fig. 13. Dynamical trajectories colored by the maximum velocity oscillation level | u / ū | in absence of damping δ= 0 s -1 . Square marks indicate results obtained for the H type standing mode structure. Circular marks indicate results obtained for the V type mode structure. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article). Fig. 14 . 14 Fig. 14. Dynamical trajectories in the frequency ( f ) -growth rate ( α) plane colored by the velocity fluctuation level | u / u | without damping δ= 0 (circular marks) and with damping δ= 12.5 s -1 (square marks) distributed uniformly in the numer- ical domain. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article). Fig. 15 . 15 Fig. 15. (a) Pressure modal distribution | ˆ p | / | ˆ p | max featuring a H type (a) or V type (b) mode structure calculated by the Helmholtz solver with pressure contour lines plotted on a cylindrical surface passing through the middle of the combustion chamber, the plenum, the burners and the microphone waveguides. Fig. 16 . 16 Fig. 16. Velocity distribution at the burners' positions for the standing mode at limit cycle with the V type (black dotted line) and H type (red dotted line) mode structure. Circular symbols indicate the burners' positions oriented as shown in Fig. 17 . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article). Fig. 17 . 17 Fig. 17. Indication of the predicted angular position of the numerical pressure nodal line (red dashed line) together with the angular region (highlighted in grey) in which the experimental nodal line is recorded for both observed standing mode structures (a) V type (b) H type. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article). a. A shift of π /2 is observed for the H type structure in Fig. 17b. Fig. 18 . 18 Fig. 18. The piecewise function F (ω r , | u / u | i ψ (θ )) /Z(θ ) for the combustor under analysis (red line) together with a sin (2 n θ) (black line). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article). -top) displays pressure measurements in the plenum (dashed lines) compared with the numerical reconstructions (continuous lines) at the same locations in the numerical domain. Microphones MP3 and MP7, located at θ MP 3 = 78.8 °and θ MP 7 = 258.7 °, respectively, detect pressure oscillations close to the nodal line and they consequently feature low fluctuation levels in the experiments. Microphones MP1 and MP5 located near the anti-nodal line record a well-established sinusoidal signal with a peak amplitude of 350 Pa at a frequency of 498 Hz as shown in Fig. 19 (b-top). Given the standing nature of the recorded mode, the phase shift between two microphone signals corresponds to their relative position with respect to the nodal line. Records of microphones located at opposite sides of the nodal line are shifted by π . Fig. 19 . 19 Fig. 19. (a) Four pressure signals recorded by microphones in the plenum (a) and in the combustion chamber (b) at limit cycle (dashed lines) compared with numerical reconstructions (solid lines). (b) Spectral content of the pressure signals in the plenum (top) and in the combustion chamber (bottom). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article). Fig. 20 . 20 Fig. 20. Time resolved heat release rate signals for the standing mode recorded by H1 and H2 photomultipliers in the combustion chamber (dashed lines) compared with numerical reconstructions (solid lines). Fig. 21 . 21 Fig. 21. (a) Trajectories f -α -| u / u | of solutions of Eq. (2) for operating regime A initiated with a standing mode structure (black continuous line with rectangular marks) and a spinning mode structure (red continuous line with circular marks). Projection of the trajectories on f -α plane (b) and α -| u / u | plane (c). Circle markers indicate results for an initially spinning mode structure, square markers indicate results for a standing mode structure. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article). Fig. 22 . 22 Fig. 22. The piecewise function F (ω r , | u / u | i ψ (θ )) /Z(θ ) for the standing equilibrium point at | u / u | st 0 . 86 and f = 473 Hz (red line) together with a sin (2 n θ) (black line). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article). Fig. 23 a. As already observed during the analysis of the trajectories of the system dynamics operating in regime A, the two trajectories obtained in regime B perfectly match for low velocity fluctuation levels when | u / u | = 0.1. The paths then diverge leading to different limit cycles. In distinction with regime A studied in Fig. 21 b, the two trajectories plotted in the frequencygrowth rate plane in Fig. 23 b overlap only in the linear range. When | u / u | > 0 . 1 , the two trajectories diverge leading to limit cycles with different frequencies and amplitudes. Analyzing the trajectories in the α -| u / u | plane in Fig. Fig. 23 . 23 Fig. 23. Trajectories f -α -| u / u | of solutions of Eq. (2) for operating regime B initiated with a standing mode structure (black continuous line with rectangular marks) and a spinning mode structure (red continuous line with circular marks). Projection of the trajectories on f -α plane (b) and α -| u / u | plane (c). Circle markers indicate results for an initially spinning mode structure, square markers indicate results for a standing mode structure. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article). Prediction of trajectory analysis Spinning mode at f = 473 Hz with peak pressure of 260 Pa in the plenum and 57 Pa in the chamber Standing mode at f = 478 Hz with a peak pressure of 345 Pa in the plenum and 48 Pa in the chamber Criterion combining stability at limit cycle and trajectory analysis Spinning mode Standing mode Experimental limit cycle oscillation Spinning mode at f = 487 Hz with a peak pressure of 260 Pa in the plenum and 60 Pa in the chamber Standing mode at f = 498 Hz with a peak pressure of 350 Pa in the plenum and 60 Pa in the chamber Fig. 24 . 24 Fig. 24. Computational domain used for the validation of the numerical procedure. The flame domain is highlighted in blue. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article). • sure amplitude levels starting from | ˆ p | = 0 and incrementing this value until a limit cycle condition is reached. Considering that the heat release rate is continuously distributed in the flame domain, and designating the angular coordinate by θ , one gets the following pressure distributions: Spinning mode: | ˆ p | ( θ ) =C, where C is a real positive constant, • Standing mode: | ˆ p | ( θ ) = | ˆ p | j ψ ( θ ) . Figure 25 a 25 shows results for the growth rate α as a function of the pressure fluctuation level | ˆ p | , which is varied from 0 to 1. For | ˆ p | → 0 the FDF corresponds to a linear flame transfer function (FTF). The value of the growth rate α = 1 / 2(β -ζ ) derived from the linear stability analysis from [16] is retrieved by the simulation when | ˆ p | → 0 . Increasing the pressure fluctuation amplitude, the growth rate α drops until a limit cycle condition is reached α = 0 for a pressure amplitude | ˆ p | = 2 (βζ ) / (3 κ ) , which is Fig. 25 . 25 Fig. 25. (a) Spinning mode calculations. Growth rate α of the two degenerate modes ψ 1 and ψ 2 as a function of the pressure oscillation level | ˆ p | . The limit cycle (black rectangular symbol) is reached for | ˆ p | = Ā = B = 2 (βζ ) / 3 κ. (b) Standing mode calculations. Growth rate α of mode ψ 1 plotted as a function of the maximum pressure oscillation level | ˆ p | . The limit cycle (black rectangular symbol) is reached for | ˆ p | = A = 4 / 3 (βζ ) /κ. Due to symmetry, results corresponding to the distribution ψ 2 are the same. β= 0.15, ζ = 0.05 and κ= 0.2 for both spinning and standing calculations. | ˆ p | ( θ ) . As a consequence, modes ψ 1 and ψ 2 are not degenerate. In the calculation, for each pressure level | ˆ p | j , the eigenmode ψ 1 is used to distribute | ˆ p | over θ ( B= 0 and A = 0). Due to symmetry, results with the distribution ψ 2 are the same. Figure 25 b shows results of the simulation for the growth rate α as a function of the maximum pressure oscillation level | ˆ p | max , which is varied from 0 to 1. Analytical results for the growth rate α = 1 / 2(β -ζ ) at vanishing perturbation amplitudes are again captured by the simulation when | ˆ p | max → 0. The growth rate then decays until a limit cycle is reached α= 0 when the maximum pressure oscillation level reaches | ˆ p | max = 4 / 3 (βζ ) /κ, a value again in agreement with J 11 = 11 λ 1 = ∂ ˙ A ∂A A= A , B=0 and J 22 = λ 2 = ∂ ˙ B ∂B A= A , B=0 . (17)In these expressions, the dot operator indicates the time derivative. The evaluation of the Jacobian coefficient J 11 starts from the calculation of the time derivative of the mode amplitude A . This can easily be done by recalling the definition of the growth rate ˙ A = αA . This product is calculated for each amplitude using numerical values of the growth rate α and amplitude A determined by the code. Computations of ˙ A (black dashed line) are plotted in Fig. 26 . 26 Fig. 26. Numerical evaluation of the Jacobian matrix coefficients for the standing mode. Calculations make use of results for the growth rate α of the mode ψ 1 (black continuous line), for the growth rate α ψ 2 of the ψ 2 mode when the pressure fluc- tuation level | ˆ p | is distributed following the ψ 1 distribution (red continuous line) and ˙ A (black dashed line) as a function of | ˆ p | max . β= 0.15, ζ = 0.05 and κ= 0.2. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article). Fig. 26 26 Fig. 26 together with the growth rate α (black continuous line) as a function of the maximum pressure fluctuation level | ˆ p | max . Following its definition, J 11 corresponds to the slope of the black dashed curve taken at the equilibrium point | ˆ p | max = A . The computed value λ 1 = ζβ coincides with the one predicted analytically [16] . For A → 0, it is also worth noting that the slope of the black dashed curve is equal to the growth rate α = (βζ ) / 2 predicted by a linear stability analysis. The red continuous curve in Fig. 26 26 Fig. 26 corresponds to the growth rate α ψ 2 trajectory of the ψ 2 mode when the pressure fluctuation level | ˆ p | is distributed following the ψ 1 distribution. When | ˆ p | max → 0, the black and red trajectories converge to the same value α ψ 2 = α = (βζ ) / 2 and the two modes are again found to be degenerate. Increasing the oscillation level | ˆ p | max , the black and red continuous trajectories diverge. For each point of the red trajectory B= 0, indicating that this mode has no influence on the standing equilibrium point of Table 2 2 Synthesis of the stability and trajectory analyses. Operating point A Operating point B Stability of spinning mode based Stable when criterion is applied at | u / u | = 0.61 and Stable when criterion is applied at | u / u | = 0.59 and on Eq. (8) f = 473 Hz f = 466 Hz Stability of standing mode based Unstable when criterion is applied at | u / u | = 0.86 and Stable when criterion is applied at | u / u | = 0.86 and on Eq. (10) f = 473 Hz f = 478 Hz Prediction of stability analysis Spinning mode Spinning or standing mode Trajectory analysis Spinning mode features a growth rate that is slightly Standing mode growth rate is larger than that of the greater than that of the standing mode over a limited spinning mode for all oscillation amplitudes range of amplitudes

01745246

en

2024/03/05 22:32:07

2017

https://hal.science/hal-01745246/file/unet-sara.pdf

Sara Berri email: [email protected] Vineeth Varma Samson Lasaulce email: [email protected] Mohammed Said Radjef email: [email protected] Jamal Daafouz email: [email protected] Studying Node Cooperation in Reputation Based Packet Forwarding within Mobile Ad hoc Networks Keywords: Mobile ad hoc networks, Packet forwarding, Cooperation, Evolutionary game theory, ESS, Replicator dynamics In the paradigm of mobile Ad hoc networks (MANET), forwarding packets originating from other nodes requires cooperation among nodes. However, as each node may not want to waste its energy, cooperative behavior can not be guaranteed. Therefore, it is necessary to implement some mechanism to avoid selfish behavior and to promote cooperation. In this paper, we propose a simple quid pro quo based reputation system, i.e., nodes that forward gain reputation, but lose more reputation if they do not forward packets from cooperative users (determined based on reputation), and lose less reputation when they chose to not forward packets from non-cooperative users. Under this framework, we model the behavior of users as an evolutionary game and provide conditions that result in cooperative behavior by studying the evolutionary stable states of the proposed game. Numerical analysis is provided to study the resulting equilibria and to illustrate how the proposed model performs compared to traditional models. Introduction A mobile ad hoc network (MANET) is a wireless multi-hop network formed by a set of mobile independent nodes. A key feature about MANETs is that they are self organizing and are without any established infrastructure. The absence of infrastructure implies that all networking functions, such as packet forwarding, must be performed by the nodes themselves [START_REF] Basagni | Mobile ad hoc networking[END_REF]. Thus, multi-hop communications rely on mutual cooperation among network's nodes. As the nodes of an ad hoc network have limited energy, the nodes may not want to waste their energy by forwarding packets from other nodes. If all the nodes are controlled by a central entity, this will not be a major issue as cooperation can be a part of the design, but in applications where each node corresponds to an individual user, it is crucial to develop mechanisms that promote cooperation among the nodes. Several works in the literature provide solutions based on incentive mechanisms, such as those based on a credit concept [START_REF] Buttyán | Enforcing Service Availability in Mobile Ad-Hoc WANs[END_REF], [START_REF] Buttyán | Stimulating Cooperation in Self-Organizing Mobile Ad Hoc Networks[END_REF], [START_REF] Krzesinski | Promoting Cooperation in Mobile Ad Hoc Networks[END_REF] etc., whose idea being that nodes pay for using some service and they are remunerated when they provide some service (like packet forwarding). Others like [START_REF] Jianl | HEAD: A Hybrid Mechanism to Enforce Node Cooperation in Mobile Ad Hoc Networks[END_REF], [START_REF] Li | Game-Theoretic Analysis of Cooperation Incentive Strategies in Mobile Ad Hoc Networks[END_REF] use reputationbased mechanisms to promote cooperation. Game theory has been a vital tool in literature to study the behavior of self-serving individuals in serval domains including MANETs. In [START_REF] Félegyházi | Game Theory in Wireless Networks: A Tutorial[END_REF], [START_REF] Félegyházi | Nash Equilibria of Packet Forwarding Strategies in Wireless Ad Hoc Networks[END_REF], [START_REF] Jaramillo | A Game Theory Based Incentivize Cooperation in Wireless Ad hoc Networks[END_REF], etc. the interaction among nodes in packet forwarding is modeled as a one shot game based on prison's dilemma model, extended then to repeated game. Furthermore, evolutionary game theory is introduced in [START_REF] Seredynski | Evolutionary Game Theoretical Analysis of Reputation-based Packet Forwarding in Civilian Mobile Ad Hoc Networks[END_REF], [START_REF] Seredynski | Analysing the Development of Cooperation in MANETs using Evolutionary Game Theory[END_REF], [START_REF] Tang | When Reputation Enforces Evolutionary Cooperation in unreliable MANETs[END_REF], to study the dynamic evolution of system composed of nodes and to analyze how cooperation can be ensured in a natural manner. In [START_REF] Seredynski | Evolutionary Game Theoretical Analysis of Reputation-based Packet Forwarding in Civilian Mobile Ad Hoc Networks[END_REF] the evolutionary game theory is applied to study cooperation in packet forwarding in mobile ad hoc networks. Here, the authors used the prison's dilemmabased model [START_REF] Félegyházi | Game Theory in Wireless Networks: A Tutorial[END_REF] and the aim was to implement several strategies in the game and to evaluate performance, by observing their evolution over time. The aforementioned works rely on incentive mechanisms, which has been proved to improve nodes cooperation. However, implementing such solutions often result in a large computational complexity during the game. We would like to find an answer to the following question, "Is it possible to achieve global cooperation in packet forwarding by a simple and natural way?". In this paper, we model the nodes interaction in a MANET as an evolutionary game by proposing a new formulation of the packet forwarding and reputation model. We introduce a simple reputation system with a quid pro quo basis, wherein, reputation is gained by forwarding packets and is lost when refusing to forward. However, a key feature is that the reputation loss depends on the packet source. If the packet is from a node with low reputation, less reputation is lost by not forwarding that packet. This simply means that selfish users will naturally have low reputation, while users are encouraged to help other cooperative users, resulting in a significantly different model from the likes of [START_REF] Seredynski | Evolutionary Game Theoretical Analysis of Reputation-based Packet Forwarding in Civilian Mobile Ad Hoc Networks[END_REF], [START_REF] Tang | When Reputation Enforces Evolutionary Cooperation in unreliable MANETs[END_REF] etc. With this model, we study two node classes, one which try to maintain a certain high reputation, and another class which disregard their reputation. We show that nodes are likely to cooperating by means of evolutionary game theory concepts and provide numerical results showing how the proposed model improves network performance. The novel reputation model we propose will naturally result in the cooperative users cooperating among each other and refusing to forward packets from selfish users, thereby eliminating the need for a third party to punish selfish behavior. The remainder of the paper is structured as follows. In Sec. 2 we formulate the reputation and game models. We propose to analyze the evolutionary game in Sec. 3, by providing the associated equilibrium, and studying strategies evolution. This allows us to determine condition ensuring global cooperative behavior. The numerical results are presented in Sec. [START_REF] Krzesinski | Promoting Cooperation in Mobile Ad Hoc Networks[END_REF]; it has to be noted that the results are the same for any game settings satisfying the provided conditions and not only for the given examples. Finally, Sec. 5 presents the conclusion. Problem formulation and proposed game model In this section, we provide a game model to study the packet forwarding interaction. We consider a packet forwarding game, where the players are the nodes, each of them can be cooperative, by forwarding other nodes' packets, or noncooperative, by dropping other nodes' packets. Thus, the players have to choose a strategy s i from the strategy set S={C, NC}. The actions C and NC mean cooperative and non-cooperative, respectively. The two player packet forwarding game can be defined in its strategic-form as following. G (2) =< {1, 2}, {S i } i∈I , {u i } i∈I >, (1) where: • I = {1, 2} is the set of players (two players), that are the network nodes; • S i is the set of pure strategies of player i ∈ I, which is the same for all the players, corresponding to S={C, NC}; • u i is the utility of player i ∈ I, that depends on its behavior and that of its opponent. To demonstrate the utility formulation, we consider the case of a pair of nodes from the network, within which a node may act as a sender and a relay (and vice versa). Thus, the players' utility can be represented by a payoff matrix as given by [START_REF] Buttyán | Enforcing Service Availability in Mobile Ad-Hoc WANs[END_REF]. A = C NC C λ -1 -1 NC λ 0 , (2) where: λ > 0 is a coefficient representing the benefit associated to successfully sending a packet while spending a unit of energy. The first player actions are along the rows and the second players along the columns. Naturally, when λ < 1 no nodes are motivated to cooperate as the energy cost relative to the gain from having packets relayed is too high. In the interesting case (the case where a MANET framework is feasible) of λ > 1, the outcome of the proposed game can be characterized by the well-known Nash Equilibrium (NE), which is the strategy profile from which no player has interest in changing unilaterally its strategy. The resulting strategy profile is beneficial for players when they act individually. However, the NE of the packet forwarding game is inefficient, corresponding to drop all the time, and provides for players 0 as utility. Thus, to overcome this problem we propose to add to the game (1) a reputation model, that defines the reward and the cost in terms of reputation according to the node decision, cooperative or non-cooperative. On the other hand, it would be better to model the interactions among all the N nodes and not just the two-player case. To deal with this, we propose to introduce evolutionary game theory, where the dynamical evolution of game strategies is studied through pairwise interactions. In the following section, we provide the reputation model we propose, and construct the new packet forwarding game including the reputation mechanism as an integrated system. That means the game is played taking into account the reputation, which we show can be interpreted as a constraint on the strategy space, while the nodes aim to maximize their utility function. Reputation model We assume that there is a reputation system introduced in order to discourage selfish behavior and reward cooperative behavior by separating these two classes of nodes. The reputation system is represented as a function depending on the own action and the opponent's action. The reputation increases by a certain margin δ r whenever a node relays the packet from another node, chooses C as action. Reputation is lost whenever a node refuses to relay a packet, by choosing NC as action. However, the loss of reputation from refusing to relay the packet from a node with low reputation δ b is smaller than the loss incurred by refusing to relay the packet from a well reputed node δ g . For ease of notation the reputation of a user i ∈ I is given by R i (t), and if R i (t) > 0 the node has a good reputation and otherwise bad reputation. The change in reputation is given by: R i (t + 1) = R i (t) + d i (t)δ r -(1 -d i (t))(δ g 1(R j > 0) + δ b 1(R j ≤ 0)), (3) where: d i (t) ∈ {0, 1} is the decision to relay or not at time t. 0 corresponds to the action NC, and 1 to C. j is a random variable indicating the sender requesting i to relay. 1 is the indicator function, it is one when the condition inside the brackets is satisfied and 0 otherwise. We consider two primary classes of nodes based on their reputation value. The set H of "Hawks" who are selfish (non-cooperative) and don't care about reputation. As a result these nodes never relay packets, i.e., s i =NC ∀i ∈ H, but use the network and try to make the other nodes relay their packets, and so we have R i (t) < 0 ∀i ∈ H. These nodes will always have d i (t) = 0 for all t and therefore will also have a low reputation. The other class of nodes are the set D of "Doves", who try to maintain a positive reputation. These nodes will have a strategy s such that on average their reputation gain is positive. Let us denote the dove population share (fraction of users who are in the dove class) by p. The population share of hawks will simply be given by 1 -p. Utility maximization In this subsection, we present how reputation system is integrated in the packet forwarding game in order to improve game outcomes and avoiding the noncooperative situation. As even the doves do not want to waste energy, they will not attempt to transmit ever single packet, but only such that their average reputation gain is at least 0 (reputation must be an increasing function). We assume that even a cooperative node, i.e., the Dove class, does not relay packets all the time. The doves have a mixed strategy to relay messages, and it relays messages from other doves with probability s d , and from hawks with a probability s h , i.e., the action C is chosen with different probabilities depending on the opponent's class. As a result, the net utility is given by the number of times their packets get forward subtracted by the energy cost paid is given by them. The expected payoff of doves is given by the formula (4). U (D, p) = (λ -1)ps d -s h (1 -p). (4) This must be maximized over s d , s h while maintaining a positive reputation, i.e., E[R i (t + 1) -R i (t)] ≥ 0 or p(s d δ r -(1 -s d )δ g ) + (1 -p)(s h δ r -(1 -s h )δ b ) ≥ 0. ( 5 ) Therefore for a given population share of doves p, we can find the strategy of doves by solving the following optimization problem. max s d ,s h U (D, p) p(s d δ r -(1 -s d )δ g ) + (1 -p)(s h δ r -(1 -s h )δ b ) ≥ 0 0 ≤ s d ≤ 1, 0 ≤ s h ≤ 1. (6) Hawks have the same utility function, but don't have a reputation constraint, therefore, the corresponding expected payoff is given by the formula [START_REF] Félegyházi | Game Theory in Wireless Networks: A Tutorial[END_REF]. U (H, p) = λps h . (7) Thus, the expected payoff of any individual is given by (8): U (p, p) = pU (D, p) + (1 -p)U (H, p), (8) where p is the population profile. If λ > 1, U (D, p) is maximized trivially by choosing s d = 1. Therefore, s h will be the smallest such that constraint (5) holds. p(s d δ r -(1 -s d )δ g ) + (1 -p)(s h δ r -(1 -s h )δ b ) ≥ 0 ⇒ pδ r + (1 -p)(s h δ r -(1 -s h )δ b ) ≥ 0 ⇒ pδ r + (1 -p)(s h (δ r + δ b ) -δ b ) ≥ 0 ⇒ s h ≥ (1-p)δ b -pδr (δ b +δr)(1-p) . Thus, we have: s h = max (1 -p)δ b -pδ r (δ b + δ r )(1 -p) , 0 . (9) Note that the introduction of s h < 1 is one of the main novelties of this paper, which can be different from s h = 1 as defined in the traditional forwarding game payoff [START_REF] Félegyházi | Game Theory in Wireless Networks: A Tutorial[END_REF], [START_REF] Félegyházi | Nash Equilibria of Packet Forwarding Strategies in Wireless Ad Hoc Networks[END_REF], [START_REF] Jaramillo | A Game Theory Based Incentivize Cooperation in Wireless Ad hoc Networks[END_REF], [START_REF] Seredynski | Evolutionary Game Theoretical Analysis of Reputation-based Packet Forwarding in Civilian Mobile Ad Hoc Networks[END_REF], [START_REF] Tang | When Reputation Enforces Evolutionary Cooperation in unreliable MANETs[END_REF], etc. where the cooperative nodes forward packet all the time without making distinction among the opponent nodes that can be cooperative, belonging to D, or non-cooperative, belonging to H. Furthermore, the proposed reputation system is simpler and can be defined as a constraint when nodes take decision purely based on the reputation class of the packet source node. Evolutionary game formulation We can formally define the resulting evolutionary game with the strategic form G =< {D, H}, {(s d , s h )} × {0}, p ∈ [0, 1], {u c } c∈{D,H} >, (10) where: • {D, H} are the reputation classes (or population types); • {(s d , s h )} is the set of strategies playable by D, with always playing 0 or NC strategy; • p is the population share of class D; • u c is the utility of class D or H as defined in ( 4) and [START_REF] Félegyházi | Game Theory in Wireless Networks: A Tutorial[END_REF]. Our objective in the following section is to study the evolution of strategies in this game, and analyze possible equilibrium points. Evolutionary game analysis Evolutionary game theory study the dynamic evolution of a given population based on two main concepts: evolutionary stable strategy (ESS) and replicator dynamics. Let p the initial population profile. We assume that a proportion ε of this population plays according to another profile q (population of mutants), while the other individuals keep their initial behavior p. Thus, the new population profile is (1 -ε)p + εq. The expected payoff of a player that plays according to p is U (p, (1 -ε)p + εq), and it is equal to U (q, (1 -ε)p + εq) for the one playing according to q. Definition 1 [START_REF] Smith | The Logic of Animal Conflict[END_REF] A strategy p ∈ ∆ is an evolutionary stable strategy (ESS), if : ∀q ∈ ∆, ∃ ε = ε(q) ∈ (0, 1), ∀ε ∈ (0, ε) U (p, (1 -ε)p + εq) > U (q, (1 -ε)p + εq), (11) ε is called invasion barrier of the strategy p, which may depend on q. The replicator dynamics is the process that specifies how a population is distributed over the pure strategies set in a game evolving in time. Definition 2 (Replicator dynamics). The replicator dynamics is given by ( 12) [START_REF] Jonker | Evolutionary Stable Strategies and Game Dynamics[END_REF]: ṗi = p i [U (i, p) - |S| j=1 p j U (j, p)], i ∈ {1, . . . , |S|}. (12) The system [START_REF] Tang | When Reputation Enforces Evolutionary Cooperation in unreliable MANETs[END_REF] describes the replication process in continuous time. It gives the percentage of individuals newly playing strategy s i in the next period, it depends on the initial value p i (t 0 ). Using the relation [START_REF] Tang | When Reputation Enforces Evolutionary Cooperation in unreliable MANETs[END_REF], the replicator dynamics of the proposed game is: ṗ = p(1 -p)(p(λ -1)(1 -s h ) -s h ). ( 13 ) For the evolutionary game G, we have the following results. Theorem 1. When λ > 1, the evolutionary game G admits exactly two ESS at p = 0 with an invasion barrier ε = min δ b δrq(λ-1) , 1 , and p = 1 with an invasion barrier ε = 1. When the initial configuration is such that p < p T , the replicator dynamics takes the system to p = 0 and when p > p T , the replicator dynamics takes the system to p = 1 with p T = δ b δ b + λδ r corresponding to the mixed NE. Proof. First, we can easily verify that p T corresponds to a mixed NE by noticing that the utilities of H and D classes are identical at this point. Next, we use the definition 1, to prove the results stated in Theorem 1 corresponding to the invasion barrier. Let x = (1 -ε)p + εq, and Ū = U (p, x) -U (q, x). U (p, x) and U (q, x) are defined using the relation [START_REF] Félegyházi | Nash Equilibria of Packet Forwarding Strategies in Wireless Ad Hoc Networks[END_REF]. Ū = p(x(λ -1) -s h (1 -x)) + (1 -p)(λxs h ) -q(x(λ -1) -s h (1 -x)) -(1 -q)(λxs h ) = (p -q)(x(λ -1) -s h (1 -x)) -(p -q)(λxs h ) = (p -q)(x(λ -1)(1 -s h ) -s h ). ( 14 ) 1. In the first case, p = 0, this gives s h = δ b δ b +δr . We can solve for the condition when Ū > 0 ⇒ -q(εq(λ -1)(1 -s h ) -s h ) > 0 ⇒ (-εq(λ -1)(1 -s h ) + s h ) > 0 ⇒ (-εq(λ -1)δ r + δ b ) > 0 ⇒ ε < δ b δ r q(λ -1) . (15) Thus, from the definition 1 we conclude that p = 0 is an ESS with ε = min δ b δrq(λ-1) , 1 as an invasion barrier. Note that p = 0 is an ESS only if the population share of D decreases, and that of H increases, i.e., the replicator dynamics is negative or ṗi < 0. If s h = s h , ṗ < 0 gives the following result: ṗ < 0 ⇒ p(1 -p)(p(λ -1) - (1 -p)δ b -pδ r (δ b + δ r )(1 -p) (p(λ -1) + 1)) < 0 ⇒ p((δ b + δ r λ)p -δ b ) δ b + δ r < 0 ⇒ p < p T (16) 2. Now we prove that p = 1 is an ESS. In this case s h = 0. Thus, the following result: Ū = (1 -ε(1 -q))(λ -1), we have λ > 1 ⇒ Ū > 0. This implies, according to definition 1, that p = 1 is an ESS ε = 1 as an invasion barrier. This occurs if ṗ > 0, i.e., when: p((δ b + δ r λ)p -δ b ) δ b + δ r > 0 ⇒ > δ b δ b + δ r = p T (17) 4 Numerical analysis In this section, we present numerical application of the proposed evolutionary game including a reputation system. All the results are based on the replicator dynamics which describes how the population evolves, and allows one to determine others performance metrics such as expected utility of players and the number of forwarded packets. We study the effect of the proposed reputation model on the evolutionary stable strategy of the game. Fig. 1 presents the results, we consider two scenarios: 1) The curve in solid line corresponds to results provided by the proposed game model including a reputation system, and assuming that the cooperative nodes forward packets of non-cooperative nodes with some probability s h . 2) The curve in dashed line corresponds to results provided by putting s h = 1, meaning that cooperative nodes forward all the time, which corresponds to the previous packet forwarding game introduced in [START_REF] Félegyházi | Game Theory in Wireless Networks: A Tutorial[END_REF], [START_REF] Félegyházi | Nash Equilibria of Packet Forwarding Strategies in Wireless Ad Hoc Networks[END_REF], [START_REF] Jaramillo | A Game Theory Based Incentivize Cooperation in Wireless Ad hoc Networks[END_REF], [START_REF] Seredynski | Evolutionary Game Theoretical Analysis of Reputation-based Packet Forwarding in Civilian Mobile Ad Hoc Networks[END_REF], etc. It is seen that using our new formulation, which integrates a reputation mechanism as a constraint, the system could converge towards a cooperative state, by carefully choosing the game settings. Thus, global cooperation could be guaranteed after a given time. Whereas, when the game does not include the reputation constraint, the population converges to the strategy non-cooperation, which is the unique evolutionary stable strategy of the game, regardless of the initial condition and game settings. The results given by the Figure 1 can be used to characterize the expected utility of players. Fig. 2 presents the results of both cases s h = s h and s h = 1. From these figures, we observe that the utilities evolve over time in the same way that the proportion of the considered population, corresponding to D in that case. Thus, the proposed model provides better results, it promotes cooperation among nodes. Indeed, in order to show the influence level of these results on network performance, we consider a network composed of 50 nodes, randomly placed in surface of 1000m × 1000m, with a transmission range equals to 150m, and plot normalized number of forwarded packets within a network, using the proposed game model with constraint, and that introduced in previous works [START_REF] Félegyházi | Game Theory in Wireless Networks: A Tutorial[END_REF], [START_REF] Félegyházi | Nash Equilibria of Packet Forwarding Strategies in Wireless Ad Hoc Networks[END_REF], [START_REF] Jaramillo | A Game Theory Based Incentivize Cooperation in Wireless Ad hoc Networks[END_REF], [START_REF] Seredynski | Evolutionary Game Theoretical Analysis of Reputation-based Packet Forwarding in Civilian Mobile Ad Hoc Networks[END_REF], etc. defined without any constraint and assuming that the cooperative nodes forward all the time, i.e., s h = 1. We assume that all the nodes need to send 10 packets to a given destination. Fig. 3 represents the results for the following game settings: λ = 3, δ r = 3 and δ b = 1. It clearly shows a direct influence, because the number of forwarded packets is strongly linked to the cooperative nodes proportion in packet forwarding. Remark: While setting δ b = 0 can indeed make p = 1 the only ESS, this may not be a suitable reputation model for the MANET framework due to several reasons. Firstly, setting δ b = 0 will completely discourage D from forwarding packets from H class, which may also include new users to the MANET, thereby discouraging new users as they might be unable to send their packets without increasing their reputation. Secondly, note that D may not always forward packets from D as in practice the channel conditions between the nodes also play a big role in determining the resource cost and therefore the utility gained by forwarding (which we have not accounted for in this work). Accounting for channel fading due to path loss or small-scale fading will therefore, be a relevant extension of this work. These considerations show that reputation model parameters must be carefully designed in practice. The contribution of this paper is to propose a the packet forwarding game [START_REF] Félegyházi | Game Theory in Wireless Networks: A Tutorial[END_REF], introducing a reputation system, which modifies reputation based on the reputation class of the packet source, i.e., cooperative or noncooperative. The aim is to motivate node cooperation using a simple and efficient mechanism. As a smaller reputation is lost by not forwarding packets from selfish users (classified by the reputation system), cooperative users will effectively forward the packets from other cooperative users and may avoid forwarding packets from selfish users. Effectively, we have demonstrated using evolutionary game theory concepts that, global cooperation in the network can be achieved under some conditions we stated related to the game settings with a low computational complexity. Finally, through simulations, we have shown that in terms of the number of forwarded packets in the MANET, the proposed game model provides significant gains over the game model where the cooperative nodes forward packet regardless of the opponent's behavior. As an extension of the present work, we propose to study the multi-hop case, where the interaction involves more than two players. Another relevant extension would be to account for channel fluctuations and the resulting utility function which might result in the cooperative class users forwarding packets from selfish users and not another cooperative user despite the reputation losses, due the channel conditions being favorable. 1 Fig. 1 . 11 Fig. 1. Evolutionary dynamics of the Doves, nodes that play the strategy 'Cooperation' with a proability s * h in the proposed game model, and previous packet forwarding game model where s h = 1. We plot for several initial frequency values. 1 Fig. 2 . 12 Fig. 2. The expected utility of D in the proposed game previous packet game model, where s h = 1. We plot for several initial frequency values 0.7 and 0.3. 1 Fig. 3 . 13 Fig. 3. Normalized number of forwarded packets for an Ad hoc network of 50 nodes, in the proposed game model and previous packet forwarding game model, where s h = 1. We plot for two initial frequency values 0.7 and 0.3. The present work is supported by the LIA project between CRAN, Lorea and the International University of Rabat.

01745270

en

2024/03/05 22:32:07

2017

https://hal.science/hal-01745270/file/unet6-1.pdf

Daniel Bonilla Licea email: [email protected] Vineeth S Varma Samson Lasaulce email: [email protected] Jamal Daafouz email: [email protected] Mounir Ghogho email: [email protected] Des Mclernon email: [email protected] Robust trajectory planning for robotic communications under fading channels come Introduction Traditionally in wireless literature, the trajectory of the mobile node is assumed to be an exogenous variable and the communication resources are optimized based only on the wireless parameters. However, we have seen an emergence of new technology like unmanned aerial or ground vehicles, drones and mobile robots which have communication objectives in addition to their destination or motion based objectives [START_REF] Licea | Trajectory planning for energy-efficient vehicles with communications constraints[END_REF]. Several works [START_REF] Ooi | Minimal energy path planning for wireless robots[END_REF] have studied trajectory optimization problems when the communication constraint is that of having a target SNR. However, we are interested in the case where the communication requirement is downloading a certain number of bits within a given time. Previously, we have studied the problem where a mobile robot (MR) must download (or upload) a given amount of data from an access point and also reach a certain destination within a given time period in [START_REF] Licea | Trajectory planning for energy-efficient vehicles with communications constraints[END_REF]. However, in [START_REF] Licea | Trajectory planning for energy-efficient vehicles with communications constraints[END_REF], we did not account for wireless channel fading and in fact assumed that the wireless signal strength is determined purely based on the path loss. In this work, we want to relax this strong assumption and account for small-scale fading and shadowing effects. In this article we will show how to design offline a robust reference trajectory under limited amount of information and high uncertainty about the wireless channel. This trajectory will allow the MR to reach the goal point and completely transmit the content of its buffer to the access point (AP) with a sufficiently high probability. In practice, this reference trajectory will be preloaded on the MR prior to the execution on the task and it will serve the MR as guide which may need to be slightly modified according to the wireless channel measurements collected by the MR while executing its task. This adaptation mechanism is outside the scope of this article and we will only focus on the design of the reference trajectory. Future works will address the online adaptation mechanism. The main contributions of this paper are as follows. -Trajectory planning of a MR starting from an arbitrary point, which must reach a certain target point and download a certain number of bits from a nearby access point. -Optimization of the trajectory to minimize a cost function which depends on the amount of data left in the buffer to be downloaded and the energy consumed. -Considering a robust cost function which accounts for the random fluctuations of the wireless channel due to small-scale fading and shadowing effects. Note that the first two contributions were also provided in [START_REF] Licea | Trajectory planning for energy-efficient vehicles with communications constraints[END_REF] for the much more simpler case in which only path-loss is assumed to determine the wireless signal. The rest of the paper is structured in the following manner. We provide the model for the wireless communication system and the robot motion in Section 2. We then provide the problem statement in Section 3 and provide a solution concept in Section 4. Finally, we provide numerical simulations in Section 5. System Model The position of the MR is given by p(t) ∈ R 2 , at any time t ∈ R ≥0 . We assume that the robot starts at position s, i.e. p(0) = s. The MR and the AP communicate with a frame duration T during which the channel fading is assumed to be a constant, i.e we assume a block fading model. The robot has a buffer with state b(k) ∈ Z ≥0 denoting the number of bits it must transmit at the discretized time k = t T . The initial buffer size is the total file size and is assume to be given by N , i.e., b(0) = N . The robot is equipped with a wireless system to communicate with an access point at p AP satisfying the following properties. Communications system The MR will move among dynamic scatterers and the bandwidth used for the communication will be lower than the coherence bandwidth. As a consequence the wireless channel between the MR and the access point (AP) will experience time-varying and flat multipath (small scale) fading as well as shadowing (large-scale fading). With loss of generality, we assume, from now on, that the communication problem consists in uploading data from the MR to the AP. The signal received by the AP at time t can be written as y AP (t, p(t)) = h(p(t), t)s(p(t)) p(t) -p AP α/2 2 x(t) + n AP (t), (1) where p AP is the location of the AP, h(p(t), t) represents the time-varying smallscale fading which we assume to be Nakagami distributed and s(p(t)) represents the shadowing term which we assume to be lognormal distributed [START_REF] Cai | A Two-Dimensional Channel Simulation Model for Shadowing Processes[END_REF]. Nakagami fading is well suited to model the behavior of the multipath fading in many practical scenarios [START_REF] Simon | Digital Communications over Fading Channels[END_REF]. Without loss of generality we assume E[|h(p(t))| 2 ] = 1 and so the p.d.f. of |h(p(t))| becomes f h (z, m) = 2m m Γ (m) z 2m-1 exp -mz 2 , ( 2 ) where m is the shape factor of the Nakagami distribution. As mentioned before, the shadowing term s(p(t)) is lognormal distributed and so we have log (s(p(t))) ∼ N 0, σ 2 s with σ 2 s being the its variance. Also, the normalized spatial correlation of the shadowing is r(p, q) = exp - p -q 2 β , (3) where β is the decorrelation distance which will be unknown to the MR prior to the execution of the trajectory. Now, the coefficient α in (1) is the power path loss coefficient which usually takes values between 2 and 6 depending on the environment; x(t) is the signal transmitted by the robot with average power E[|x(t)| 2 ] = P and n AP (t) ∼ CN (0, σ 2 n ) is the zero mean additive white Gaussian (AWGN) noise at the AP's receiver. From [START_REF] Licea | Trajectory planning for energy-efficient vehicles with communications constraints[END_REF] we have that the signal-to-noise ratio (SNR) at the AP (in dB) is: Γ dB (p(t)) = 10 log 10 P σ 2 n + 20 log 10 (s(p(t))) + 20 log 10 (|h(p(t), t)|) -10α log 10 ( p(t) -p AP 2 ) . (4) As a result, the number of bits in the MR's buffer is given by: b(k) =     N - k j=0 R Γ (p(jT ))     + (5) where and a + = a for a > 0 and a + = 0 for a ≤ 0; Γ (p(jT )) is the estimate of Γ (p(jT )) which is Γ dB (p(jT )) in linear scale, N is the initial number of bits in the buffer and R Γ (p(kT )) is the number of bits in the payload of the packet transmitted during the duplexing period k. As mentioned above, the number R Γ (p(kT )) of bits transmitted in the payload is computed by the MR according to its most recent SNR estimate. So we have (for b(k) = 0): R Γ = R j , ∀ Γ ∈ [η j , η j+1 ), j = 0, 1, • • • , J (6) with R j < R j+1 , η j < η j+1 , R 0 = 0, η 0 = 0 and η 1 must be above the sensitivity of the AP's receiver. Mobile robot We assume the MR to be omnidirectional and its velocity is assumed to be controlled directly. This results in its motion described by ṗ(t) = u(t), (7) where p(t) is the MR position at time t and u(t) is the control input which is bounded by: u(t) 2 ≤ u max , (8) Finally, the mechanical energy spent by the MR between t 0 and t 1 while using the control signal u(t) is: E mechanical (t 0 , t 1 , u) = m t1 t0 u(t) 2 dt. ( 9 ) where m is the mass of the MR. Problem statement The objective of the robot is to depart from a starting point s to a goal point g within a time t f and transmit the all the content from its buffer to the AP. The desired trajectory is such that it consumes little mechanical energy from the robot and also allows the robot the transmit all the content of the buffer quickly. In addition we want that when the MR follows this trajectory it succeeds in emptying its buffer with a high probability. We assume that the only knowledge available to the MR (and the designer) about the environment (prior to the execution of the trajectory) is the position of the starting and goal points (i.e., s and g); an estimate of the path loss coefficient α, but we assume no knowledge about the severity of the small-scale fading (i.e., about the shaping factor m in (2)). Solving the general problem with no approximation is very hard due to the large amount of stochastic perturbations, the shadowing correlation and the large number of terms in the sum of [START_REF] Malmirchegini | On the Spatial Predictability of Communication Channels[END_REF]. This results in a very complicated expression for the probability of the buffer to be empty at t f . Therefore, we look at the most likely buffer state given by b (k) =     N - k j=0 R (Γ (p(jT )))     + (10) where R (Γ (p(jT ))) is the statistical mode of R (Γ (p(jT ))), i.e., R (Γ (p(kT ))) = max argmax R∈{Rj } J j=0 Pr (R (Γ (p(kT ))) = R) . ( 11 ) This results in the following optimization problem minimize u θ 1 tf 0 u(t) 2 2 u 2 max dt + θ 2 t f T k=0 T b(k) N s.t. ṗ(t) = u(t) u(t) 2 ≤ u max , p(0) = s, p(t f ) = g, t f T k=0 R (Γ (p(kT ))) ≥ r R N. (12) The optimization target is a convex combination of the energy spent in motion by the robot (9) and of a second term which estimates how quickly the buffer is emptied. This second term is a sum over the most likely number of bits left in the buffer at time instant t = kT (i.e., E [b(k)]). The coefficients {θ k } 2 k=1 of the convex combination determine the relative importance of each optimization criterion. Note that due to the stochastic nature of the channel we can not ensure that when the MR follows the reference trajectory it will always be able to empty its buffer but we can ensure that this happens with a certain probability. As calculating the actual probability of failing to meet the communication requirement constitutes a very hard task as explained above, we introduce r R ≥ 1 which is an overestimation parameter selected by the designer. The final constraint in (12) ensures that the sum of the statistical mode of the bits transmitted in the payload over all the trajectory is equal to an overestimation of the initial number of bits in the buffer, i.e., r R N . So when the trajectory is actually executed, the probability that the buffer will be emptied will be high and by increasing the overestimation parameter r R we can reduce the probability of the MR failing to empty its buffer when it reaches the goal point g. The term b(t) is a discreet and deterministic function of the MR's position. This difference makes the problem much more feasible to solve. Now, to solve the optimization problem (12) we first define the region A j as: A j = {p | R (Γ (p)) = R j }. ( 13 ) Due to the wireless channel model the region A J is circular while the shape of region A j , for j = 1, 2, • • • , J -1, is a ring with inner and outer radii of r j+1 and r j respectively. And r j is given by: r j = min r | R (Γ (r[cos(θ) sin(θ)] -p AP )) = R j (14) The radii r j are computed from the channel statistics which can be estimated using the techniques presented in [START_REF] Malmirchegini | On the Spatial Predictability of Communication Channels[END_REF]. Nevertheless, for lack of space we do not provide here the details on how to compute it. We also define u j (t) as any control law that takes the vehicle through the regions {A k } j k=0 . The set of all control laws u j (t) will be denoted as U j and U = ∪ J j=0 U j is the set of all control laws. One simple way to solve (12) is to first solve it with the additional constraint u ∈ U j , once for each different value of j = 1, 2, • • • , J. We will denote as u * j (t) the optimum control law that solves (12) under the additional constraint u ∈ U j and u * (t) as control law that solves (12) under the constraint u ∈ {u * j (t)} J j=1 . Therefore to solve (12) we will calculate all the optimum control signals u * j . In order to minimize the mechanical energy term in the optimization target of (12) the optimum control law u * j (t) must make the robot enter and exit the convex hull of each region {A n } j n=0 at most once. These input and output points to the convex hull of the area A j are denoted by i j and o j respectively. We regroup these points in the following set C j = {s, i 1 , i 2 , • • • , i j , o j , o j-1 , • • • , o 1 , g} and index them as follows: c j 0 = s, c j n = i n , for n = 1, 2, • • • , j, c j n = o 2j+1-n , for n = j + 1, j + 2, • • • , 2j, c j 2j+1 = g. (15) where s and g are the starting and goal points for the robot. In addition, t n is the time instant in which the robot is at p j n and: τ n t f = (t n+1 -t n ), n = 0, 1, • • • , 2j (16) where: 2j n=0 τ n = 1, τ n > 0, (17) Note that the coefficients {τ n } 2j n=0 determine the portion of time t f that the robot takes to go from c j n-1 to c j n . Let us also write the points belonging to C j in polar coordinates as: c j n = r j n [cos(φ j n ) sin(φ j n )] T . ( 18 ) From the definition of i n and o n we know that they lie in a circle of radius r n which can be computed from the p.m.f. of R (Γ (p(kT ))). Therefore we know {r j n } 2j n=1 and as a consequence the only unknowns to uniquely determine C j are the angles 5 {φ j n } 2j n=1 , where the φ j n is the angle of c j n respect to the AP. Now, the optimum control law u * j (t) takes the robot from c j 0 up to c j 2j+1 in ascending order through each point in C j . We can also see that the second term in the optimization target of (12) depends only the time spent in each region A j (i.e., on the durations τ k t f ) and not on the shape of the particular path followed by the robot nor by its velocity profile. So, the velocity profile and the path must be selected to minimize the mechanical energy (i.e., the first term in the optimization target ( 12)). To do so the vehicle must go from c j n-1 to c j n in a time τ n t f (to be determined) using minimum energy. Using calculus of variations [START_REF] Kirk | Optimal control theory: An introduction[END_REF] we can show that this is achieved by: u j (t) = c j n -c j n-1 τ n-1 t f ∀ t ∈ [t n-1 , t n ). ( 19 ) Therefore if we add the constraint u ∈ U j and then we optimize {τ n } 2j n=0 and the angles {φ j n } 2j n=1 we obtain u * j (t). Now, if we use the constraint u ∈ U j and select u j (t) to take the form (19) then the optimization target of problem (12) becomes: J {τ n } 2j n=0 , {φ j n } 2j n=1 = θ 1 2j+1 n=1 c j n -c j n-1 2 u 2 max τ n-1 t f + θ 2 t f T k=0 T b(k) N . (20) And using (13), ( 19) and the constraint in (12) we have the following approximation: τ j t f T R j + j-1 n=0 (τ 2j-n + τ n )t f T R n ≥ r R N (21) So, taking into account ( 19)-( 21) the optimization problem (12) becomes: minimize {τn} 2j n=0 ,{φ j n } 2j n=1 J {τ n } 2j n=0 , {φ j n } 2j n=1 s.t. 2j+1 n=1 τ n = 1, τ n > 0, r 2 n +r 2 n-1 -2rnrn-1 cos(φ j n -φ j n-1 ) τ 2 n t 2 f ≤ u 2 max , n = 0, 1, • • • , 2j c j n = r j n [cos(φ j n ) sin(φ j n )] T , c j 0 = s c j 2j+1 = g, τj t f T R j + j-1 n=0 (τ 2j-n + τ n )t f T R n ≥ r R N (22) where the first line of constraints ensures that the coefficients {τ k } 2j k=0 determine the portion of the total time t f taken to go from one point in C j to the next one. The next line of constraints establishes the maximum velocity of the robot. The final constraint is the robust constraint which will allow the designer to obtain a high probability of the MR emptying completely its buffer. To solve the optimization problem (22) we first express the angles {φ j n } 2j n=1 as function of the durations {τ n } 2j n=0 . This is achieved by deriving the optimization target of ( 22), see more details in [START_REF] Licea | Trajectory planning for energy-efficient vehicles with communications constraints[END_REF]. Then we use simulated annealing algorithm (SAA) [START_REF] Russell | Artificial Intellingence: A Modern Approach[END_REF] to optimize the durations {τ n } 2j n=0 . This concludes the discussion about the optimization of the trajectory and in the next section we present some simulations to better understand its behaviour and observe its performance. Simulations In this section we present some simulations to gain some insight about the trajectories obtained by the method presented in this paper. We select 10 log 10 P σ 2 n = 33dB. Now, the initial number of bits in the buffer b(0) = 600N s while the possible amount of bits transmitted in one packet can be R 0 = 0, R 1 = 4N s , R 2 = 16N s , R 3 = 64N s where N s is the number of symbols transmitted in one packet. Note that such values for the number of bits transmitted in the payload can be obtained using a rectangular M-QAM modulation. Now, regarding the thresholds {η j } J j=0 we fix them so that the bit error rate is at least 10 -3 . Now regarding the channel we select the path loss coefficient as α = 2, shadowing variance σ 2 s = 2.5 and then for the decorrelation distance we select β = 10λ, where λ is the wavelength of the RF carrier used for communications. We select the starting and the goal points to be s = [8λ 0] and g = [9 -6]λ while we locate the access point at the origin. Then the time to reach the goal point is t f = 20s, the period between packets T = 100ms and the maximum velocity of the MR is 10λ per second. First of all we consider for references a trajectory that goes from s to g using minimum energy. This is achieved by a linear path between both points and a constant velocity profile. We will denote such trajectory as T 0 . Then we consider a trajectory T 1 optimized according to (22) with θ 1 = 1, θ 2 = 0 and r R = 1. This trajectory is optimized to use minimum energy while satisfying constraint (21). Then we also consider another trajectory T 1 optimized according to (22) with θ 1 = 0, θ 2 = 1 and r R = 1. This trajectory is optimized to empty the buffer as quick as possible. In Fig. 1 we can observe the paths corresponding to the trajectories T 0 , T 1 and T 2 . We first note that the path corresponding to T 1 is shorter than the path corresponding to T 2 which agrees with the fact that the trajectory T 1 is optimized to minimize the energy consumed (while satisfying constraint (21)). Then regarding the shape of the paths we see that the path of T 2 reaches A 2 through the shortest path, this is done in order improve as quick as possible the transmission rate in order to empty the buffer as soon as possible. Now, regarding the path for T 1 the robot reaches A 2 by moving in an orthogonal direction with respect to the vector gs, by doing so the robot minimizes the amount of deviation from g which reduces then the distance total distance travelled and consequently the energy spent. When we observe the velocity profiles of both trajectories in Fig. 2 we first note that the period with highest velocity takes place from t = 0 until t = τ 1 this is because the robot is rushing to get out from A 0 to start transmitting as many bits as possible. Then we also observe that the minimum velocity occurs when the robot reaches the inner most area of the trajectory (in this case A 2 ). This is in order to spend as much time as possible in that area with the best channel conditions in the trajectory. Then, in table 1 we observe the average time in which the buffer is emptied E[t empt ], the probability of success P S (i.e., the probability of emptying the buffer when reaching g) and the amount of mechanical energy used normalized by mλ 2 . As it is expected the trajectory T 0 uses minimum energy but its probability of success is very low (0.0868). On the other hand the probability of success for the optimized trajectories T 1 and T 2 is much higher, 0.7206 and 0.9347 respectively, but due to the larger paths and velocities their energy consumption is higher. Now, we observe the effect of the robustness parameter r R , see (21). To do so we consider two more trajectories. The first one, denoted T 3 , is optimized according to (22) with θ 1 = 0.3, θ 2 = 0.7 and r R = 1. While the second trajectory, denoted T 4 , is optimized according to (22) with θ 1 = 0.3, θ 2 = 0.7 and r R = 1.5. We observe in Fig. 3 that their path is really similar (the path corresponding to T 4 is slightly larger) but their velocity profiles are clearly different as we can observe in Fig. 4. The trajectory T 4 spends a larger time in the area A 2 in order to increase the average data rate and therefore increase the probability of success. But by doing so the robot has to move quicker when it gets out from A 2 in order to reach g in time. By comparing the probabilities of success of T 4 with T 3 in table 1 we observe that increasing the robustness parameter r R indeed increases the probability of success although it also increases the energy consumption. Note that all the optimized predefined trajectories were able to produce a relatively large probability of success in a fading channel without the use of any kind of diversity. This large probability of success was achieved by optimizing the trajectories using only first order statistics of the wireless channel. In the future we will take into account channel measurements to develop an online mechanism which further improves the success probability while reducing the amount of mechanical energy. Conclusions We have formulated the problem of robust trajectory optimization for an MR with a target point to reach and a certain number of bits to transmit within a given time. Due to small scale fading and shadowing effects, obtaining a suitable reference trajectory offline is non-trivial. Therefore, we consider the most likely buffer state at each time determined based on the statistical mode and optimize the desired metric by introducing an overestimation parameter for robustness. This approach results in an optimization problem with a feasible solution. Fig. 1 . 2 Fig. 2 . 122 Fig.1. Paths corresponding to trajectories T0 (green), T1 (blue) and T2 (magenta). Starting point s represented by a circle, goal point s represented by a triangle and AP location at the origin. We observe as well the delimitation of the areas {Aj} 3 j=0 . Fig. 3 . 3 Fig. 3. Paths corresponding to trajectories T3 (green) and T4 (dashed red). 4 Fig. 4 . 44 Fig. 4. Velocity profiles of trajectory T3 (top) and T4 (bottom). Table 1 . 1 Performance of different trajectories trajectory E[tempt] (s) PS Energy/(mλ 2 ) (J) T0 14.46 0.0868 0.1859 T1 10.41 0.7206 0.8139 T2 7.72 0.9347 4.8924 T3 7.68 0.8652 3.8928 T4 7.59 0.9262 4.7909 The present work is supported by the LIA project between CRAN, Lorea and the International University of Rabat.

01745316

en

2024/03/05 22:32:07

2017

https://hal.science/hal-01745316/file/gr2017-pub00054936.pdf

Spiegelberger Lucie Bezombes email: [email protected] Stéphanie Gaucherand Christian Kerbiriou Marie-Eve Reinert Thomas Spiegelberger Joseph William Bull Cara Clark Christian Küpfer Frank Lupi Charles K Minns Serge Muller Ecological equivalence assessment methods: what trade-offs between operationality, scientific basis and comprehensiveness? Keywords: Biodiversity offset, ecological equivalence, ecological equivalence assessment methods, no net loss, mitigation hierarchy, compensation. 1976 US Fish and Wildlife Service, USA Species population terrestrial habitats Biodiversity Offsets Programme, international Terrestrial habitats Habitat Terrestrial habitats Biotope Ecological equivalence assessment methods: what trade-offs between operationality, scientific basis and comprehensiveness? Introduction Biodiversity erosion has accelerated in recent decades [START_REF] Sala | Global Biodiversity Scenarios for the Year 2100[END_REF] and has become a major environmental concern as biodiversity loss is identified as a major driver of ecosystem change [START_REF] Hooper | A global synthesis reveals biodiversity loss as a major driver of ecosystem change[END_REF]. Alongside "classic" answers such as species and ecosystems protection and conservation, biodiversity compensation is increasingly used to counteract impacts from development. It is applied worldwide and has legal status in some countries (e.g., the United States, Canada, Australia, Germany, France and the United Kingdom). Compensation mechanisms remain country-dependent (McKenney & Kiesecker 2010; Commissariat Général au Développement Durable (CGDD) 2012) but are usually integrated in the mitigation hierarchy, after avoidance and reduction of impacts. Efforts have been put into enhancing biodiversity compensation, and biodiversity offset in particular. Biodiversity offset is a way of compensating for biodiversity losses (Business and Biodiversity Offsets Programme, BBOP 2012a) with the aim of achieving "no net loss" (NNL) of biodiversity (ten [START_REF] Ten Kate | Biodiversity offsets: Views, experience, and the business case[END_REF]). Concerns about offset practices have been expressed in the literature for many years [START_REF] Race | Fixing Compensatory Mitigation: What Will it Take?[END_REF] as offset is the last lever on which it is possible to act in order to achieve NNL [START_REF] Gibbons | Offsets for land clearing: No net loss or the tail wagging the dog?[END_REF]. Notably, frameworks have been established to guide offset measures design in order to achieve NNL of biodiversity (Business and Biodiversity Offsets Programme, BBOP). One of the main conditions is that biodiversity gains should be comparable, or equivalent to biodiversity losses [START_REF] Gardner | Biodiversity Offsets and the Challenge of Achieving No Net Loss[END_REF]. When this happens, "ecological equivalence" is reached. Ecological equivalence is one of the most widely discussed conceptual challenges in the related scientific literature [START_REF] Gonçalves | Biodiversity offsets: from current challenges to harmonized metrics[END_REF]. A particularly controversial aspect is how ecological equivalence should be assessed. A number of essential considerations that should be taken into account in order to evaluate equivalence have been identified [START_REF] Quétier | Assessing ecological equivalence in biodiversity offset schemes: Key issues and solutions[END_REF][START_REF] Bull | Biodiversity offsets in theory and practice[END_REF][START_REF] Quetier | No net loss of biodiversity or paper offsets? A critical review of the French no net loss policy[END_REF]), which we summarize in four key groups: ecological, spatial, temporal and uncertainty considerations. Ecological considerations gather (i) issues related to the choice of biodiversity components for which losses and gains are quantified, also called target biodiversity [START_REF] Quétier | Assessing ecological equivalence in biodiversity offset schemes: Key issues and solutions[END_REF] and (ii) the set of indicators that is used to quantify those biodiversity components, also known as currency [START_REF] Bull | Biodiversity offsets in theory and practice[END_REF] or metrics (Business and Biodiversity Offsets Programme, BBOP 2012a). Spatial considerations relate to the integration of impacted and compensatory sites landscape context in equivalence assessment. Landscape context gives information about landscape components influencing biodiversity (e.g., connectivity and metapopulation functioning; [START_REF] Beier | Do Habitat Corridors Provide Connectivity[END_REF] which are notably important to locate offset sites [START_REF] Kiesecker | A Framework for Implementing Biodiversity Offsets: Selecting Sites and Determining Scale[END_REF][START_REF] Saenz | A Framework for Implementing and Valuing Biodiversity Offsets in Colombia: A Landscape Scale Perspective[END_REF]. According to the BBOP (2012b) "a biodiversity offset should be designed and implemented in a landscape context to achieve the expected measurable conservation outcomes". Temporal considerations are related to the time lag (also called delay) between the moment when impact on biodiversity occurs and the moment when offset measures become fully effective [START_REF] Maron | Can offsets really compensate for habitat removal? The case of the endangered red-tailed black-cockatoo[END_REF], ensuing interim losses of biodiversity [START_REF] Dunford | The use of habitat equivalency analysis in natural resource damage assessments[END_REF]. One current solution to avoid or reduce interim losses is to implement compensation ahead of impacts (e.g., by using mitigation banks; [START_REF] Wende | Mitigation banking and compensation pools: improving the effectiveness of impact mitigation regulation in project planning procedures[END_REF]. But when no bank system is available, assessment of equivalence should take into account temporal considerations [START_REF] Laitila | A method for calculating minimum biodiversity offset multipliers accounting for time discounting, additionality and permanence[END_REF]. Finally, considerations on uncertainty refer to the lack of confirmed knowledge and hindsight when assessing equivalence, and particularly in this article we focus on the risk of failure when implementing offset measures [START_REF] Moilanen | How Much Compensation is Enough? A Framework for Incorporating Uncertainty and Time Discounting When Calculating Offset Ratios for Impacted Habitat[END_REF][START_REF] Curran | Is there any empirical support for biodiversity offset policy?[END_REF]. This risk mostly depends on the species or ecosystems concerned by offset [START_REF] Tischew | Evaluating Restoration Success of Frequently Implemented Compensation Measures: Results and Demands for Control Procedures[END_REF], the type of offset implemented [START_REF] Anderson | Ecological restoration and creation: a review[END_REF]) such as habitat restoration, protection, creation or enhancement [START_REF] Levrel | Compensatory mitigation in marine ecosystems: which indicators for assessing the "no net loss" goal of ecosystem services and ecological functions?[END_REF]) and the ecological engineering techniques used [START_REF] Jaunatre | Can ecological engineering restore Mediterranean rangeland after intensive cultivation? A large-scale experiment in southern France[END_REF]. Equivalence Assessment Methods (EAMs) exist worldwide and are used by developers or authorities to evaluate biodiversity losses and gains (e.g., State of Florida 2004;Gibbons et al. 2009;Darbi & Tausch 2010). They are specifically conceived to ensure that offset measures are sufficient to reach ecological equivalence. Although every EAM seeks to ensure NNL of targeted biodiversity, none is fully satisfactory and principles underlying some EAMs have been discussed [START_REF] Mccarthy | The habitat hectares approach to vegetation assessment: An evaluation and suggestions for improvement[END_REF][START_REF] Gordon | Perverse incentives risk undermining biodiversity offset policies[END_REF]. Notably, depending on the method used, calculations result in different offset surfaces for the same impact [START_REF] Bull | Comparing biodiversity offset calculation methods with a case study in Uzbekistan[END_REF]. It seems rather difficult or even impossible to move toward an unanimous worldwide method, mainly because of (i) diversity in offset policies between countries (McKenney & Kiesecker 2010), (ii) disparity between development projects and the resources committed to biodiversity conservation (Regnery et al. 2013b), and (iii) disparities in biodiversity status context and conservation issues. Nonetheless, exploring interactions between the characteristics underlying EAMs could highlight ways of improving equivalence assessment. Thus, we characterized existing EAMs regarding three "challenges" that we identified to be determinant in EAMs effectiveness to meet NNL. In this article, we call these three "challenges" operationality, scientific basis and comprehensiveness. On one hand, operationality is needed by developers and public authorities to carry out standardized assessments in a small amount of time, at reasonable costs [START_REF] Laycock | Biological and operational determinants of the effectiveness and efficiency of biodiversity conservation programs[END_REF]) and in consistence with the skills level of structures involved in mitigation studies. On the other hand, growing awareness comes from the scientific sphere that equivalence assessment should be grounded on scientific basis, including evidence based biodiversity evaluation, objective and transparent metrics and calculation [START_REF] Gonçalves | Biodiversity offsets: from current challenges to harmonized metrics[END_REF] and feedbacks from previous offset related experiences [START_REF] Maron | Can offsets really compensate for habitat removal? The case of the endangered red-tailed black-cockatoo[END_REF][START_REF] Pöll | Challenging the practice of biodiversity offsets: ecological restoration success evaluation of a large-scale railway project[END_REF]. Despite the importance of both operationnality and scientific basis challenges, they are often seen as not fully compatible. Finally, comprehensiveness is a transversal challenge addressing the fact that EAMs development should take into account all four key equivalence considerations, as highlighted by [START_REF] Quétier | Assessing ecological equivalence in biodiversity offset schemes: Key issues and solutions[END_REF]. We can hypothesize that it is an obstacle for operationality and that it is more compatible with scientific basis. The objective of this paper is to provide elements of reflection for the development of future EAMs contributing to design offset measures that lead to NNL, by exploring two main questions: (i) Is there a common structure underlying all EAMs and what elements of such structure could be used as basis when developing an EAM? (ii) What are the synergies and trade-offs in achieving operationality, scientific basis and comprehensiveness. Particularly, is operationality necessarily in contradiction with both other challenges? Is it possible to combine all three challenges in one EAM accepted by both operational and scientific spheres? These EAMs were chosen because they were either published in a scientific journal or had accessible guidelines that could be used to understand how they were constructed and for what purpose. Only main EAMs were analyzed, but we are aware that there are variants adapted to specific cases and that different versions of guidelines are used simultaneously [START_REF] Duel | The habitat evaluation procedure as a tool for ecological rehabilitation of wetlands in The Netherlands[END_REF][START_REF] Tanaka | How to Assess "No Net Loss" of Habitats-A Case Study of Habitat Evaluation Procedure in Japan's Environmental Impact Assessment[END_REF]. The EAM selection intended to give an overview of the current EAMs diversity and also of EAMs commonly used. Thus this is not an exhaustive sample but rather a representative one as it covers North America, Australia and Western Europe which are three main zones where offset policies are well-established [START_REF] Madsen | State of biodiversity markets: offset and compensation programs worldwide[END_REF]. The sample also covers all kind of ecosystems (terrestrial, aquatic, marine or wetlands). Material and method Analysis of EAMs structure In Uncertainty: how do EAMs take into account the risk of offset failure? Finally, we identified the "compensation unit" used in each EAM, which is the currency calculated for a site and then compared between impacted sites (loss of biodiversity units) and offset sites (gains of biodiversity units). Synergies and trade-off between the three EAMs challenges Twelve criteria were defined, covering a large range of characteristics related to how operationality, scientific basis and comprehensiveness are taken into account in EAMs. A description of those criteria and the working hypothesis underlying their choice are specified in Table 2. In our work EAMs are considered operational when they have pre-defined indicators ("Indicators set up"), are rapid to implement ("Implementation rapidity"), when data needed are easily available ("Data availability") and when "like for unlike" offset designs (exchangeability between biodiversity impacted and compensated) are possible ("Exchangeability"). EAMs are considered to have scientific basis when all the indicators used to assess biodiversity are based on scientific documentation ("Biodiversity indicators"), when the metrics used are quantitative and appropriate to the biodiversity component being assessed ("Biodiversity indicator metrics"), when spatial considerations are taken into account with dedicated indicators ("Spatial considerations") and when uncertainty is taken into account based on previous feedbacks ("Uncertainty considerations"). Finally, EAMs are considered comprehensive when they include all key equivalence considerations ("Key equivalence consideration"), when they target species, habitats and ecosystem functions ("Biodiversity components"), when they require various types of data (from the literature, GIS, field data, etc., "Data type") and when they evaluate biodiversity with a relevant set of indicators ("Indicators number"). Each criterion was defined by 3 or 4 modalities (see Appendix B for modalities details). For most modalities, data could be derived from the published version of EAMs. However, to complete certain modalities (e.g., those relating to "Implementation Rapidity") we interviewed experts who either use the EAM in the field or have contributed to its construction (see Appendix C, experts' names and functions are given when they agreed to be cited). When divergent answers were obtained for a given EAM, priority was given to the answer obtained from EAMs developers which was the case for the UMAM, CRAM, UK pilot and German Ökokonto (see Appendix C). We found some mismatches between experts' answers and theoretic guidelines, but this could be explained by differences in EAMs variants or case-by-case practices. In these cases, we decided to stick to the theoretical guidelines (see Appendix D). A score from 1 to 3 or 4 (depending on the number of modalities) was then given to each criterion, where 1 is the lowest level of challenge achievement, and 4 the highest (see Appendix B). For example, an EAM that require only very easy to access data will receive a 4 for the "Data availability" criterion. This scoring system was deliberately simple and linear to give all modalities a similar weight. The aim of this scoring was to highlight synergies and trade-offs between these criteria, and beyond, between the three challenges. We suppose that some correlations between particular criteria will occur, as for example, if large data collection (Data Type) is required, data availability may be low. Moreover, when users have to choose indicators (Indicators set up), they can a priori choose a combination of qualitative and quantitative discrete or continuous metrics (Biodiversity indicator metrics) which would imply a correlation between these criteria. However, it remains theoretical as in practice users could very well choose only indicators with qualitative metrics. Data analysis A principal component analysis (PCA) was performed on all criteria scores (see Appendix C), in order to analyze how EAMs addressed operationality, scientific basis and comprehensiveness. Mean scores were calculated for each challenge (ScoreOp, ScoreScBs and ScoreComp) as the relative mean of the scores attributed to the four criteria describing the challenges, expressed as percentage challenge achievement. These mean scores were added as supplementary variables in the PCA (so that they do not contribute to PCA axis construction). Correlations between criteria were assessed by a nonparametric measure of rank correlation, Spearman rank coefficient (rho), as a complement to PCA, in order to identify oppositions and synergies between criteria underlying the challenges. Criteria were considered correlated for rho ≥ ±0.5 [START_REF] Freckleton | On the misuse of residuals in ecology: regression of residuals vs. multiple regression[END_REF]. The PCA also allows identification of EAMs groups according to the challenge they best achieve. All analyses used R software version 3.1.2 with the corresponding FactoMineR package [START_REF] Husson | Package 'FactoMineR[END_REF]. Results EAMs general structure The analysis of the 13 EAMs indicates that they all share a common structure to calculate losses and gains of biodiversity (Figure 1). They all consider two sites (impacted site and offset site) at two time points (before and after impact or offset measures). One or several indicators are chosen as surrogates to qualify or quantify the targeted biodiversity components, which differ from one EAM to another depending on the context. Two main EAM types can be identified according to the range of biodiversity they target: "specialized", using indicators for a specific ecosystems (for example Australian endemic vegetation for the Habitat Hectare method or Florida's wetlands for UMAM ) and "generalist" using general indicators adapted to a wide range of ecosystems (e.g., terrestrial ecosystems for PilotUK) (Table 3). A benchmark can be used if there is an identified reference state for the targeted biodiversity (e.g., for Habitat Hectare the benchmark is "the same vegetation type in a mature and longundisturbed state", and for UMAM it is a "reference standard wetland" considered as in good ecological quality). A quantitative value based on these indicators is attributed to the site before and after impacts (to calculate biodiversity losses) or offset (to calculate biodiversity gains) and is multiplied by the related site areas. This combination of biodiversity "quality" and "quantity" constitutes the "compensation unit". A tiny majority of EAMs (8 out of 13) evaluate ecological equivalence by attributing "compensation units" to impacted and offset sites (Table 3), allowing biodiversity losses and gains to be assessed and compared on the same basis. There are no specific rules for offsetting one compensation unit by another, only that the number of units exchanged in the offsetting process must be at least equal. The other five EAMs go one step further by using specific rules to size offset measures. This can be done by integrating temporal or uncertainty related ratios to increase the compensatory site area (e.g., Habitat Evaluation Procedure, UMAM; Table 3), or by assessing losses and gains every year during impacts and offset (Figure 1) from the moment impacts occur and the moment when offset measures are considered as effective with a discounted rate (Resource, Habitat, Landscape Evaluation Analysis and Habitat Evaluation Procedure). In all cases, the only values that were calculated based on real measures of the current state of the sites are the one related to the impacted site before impact and to the offset site before offset measures. All other values (after impact or offset measures) are calculated based on predictions. Some EAMs provide a basis for such predictions (i.e. Resource, Habitat, and Landscape Evaluation Analysis), but most of the time, the user has to find a way to make predictions as accurate as possible. 3.2. Trade-off and synergies between the three EAM challenges Correlations among criteria between and within challenges The relationship between criteria and EAMs can be correctly summarized by the two first PCA axes according to the amount of variation explained by these two first axes (64%). There is no clear opposition between scores of operationality, scientific basis and comprehensiveness Positive correlations between criteria related to the same challenges also occur. It is the case for three out of four criteria related to operationality (Data Availability ~ Indicators Setup, rho = 0.86; Indicators Setup ~ Implementation Rapidity, rho = 0.78; and Data Availability ~ Implementation Rapidity, rho = 0.54). This means that it is easy to combine these criteria in order to obtain a good level of operationality. However there is no positive correlation between criteria related to comprehensiveness and negative correlation for criteria related to scientific basis Biodiversity Indicator Metrics ~ Biodiversity Indicators, rho = -0.64) implying difficulties to develop scientific basis in every aspects. Groups of EAMs defined by the challenge they best achieve The PCA highlights the existence of a few groups of EAMs characterized by similar scores for a small number of criteria. Because three criteria (out of the four) related to operationality contributed the most to axis 1, EAMs on the right side of the PCA graph on Figure 2b can be considered as operational ones (HabHect, PilotUK, SomersetHEP, UMAM, CRAM, Ökokonto, LdClEval, and FishHab). They have pre-defined indicators, are rapid to implement (less than 1 week or between 1 week and 6 months) and data used are free and quick to collect, or specific data-bases exist for these methods. On the left side of axis 1, a group of five EAMs (HEP, PilotBBOP, HEA, REA, LEA, Figure 2b) was defined mainly by two other criteria that contribute to axis 1: BiodivIndMc (90%) and DataTp (73%) (Figure 2a). These EAMs need complex data to be implemented (data can come from the literature, GIS, simple field visits, field inventories or field monitoring and modeling) and indicators metrics can be a combination of qualitative and quantitative data (both discrete and continuous). Criteria contributing the most to axis 2 (Figure 2a) are Uncertainty Consideration (86%) and Exchangeability (76%) on the upper side and Spatial Consideration (68%) on the lower side. Quite surprisingly, no EAM combines very well both spatial and uncertainty considerations. Furthermore, EAMs trouble making the integration of uncertainty science based: only the Canadian Fish Habitat method (isolated on axis 2 upper extremity) uses a ratio based on existing data-bases providing scientific feedbacks on previous offset measures (highest score for Uncertainty Consideration) in order to adjust the offset surface areas. A group of three EAMs (HabHect, CRAM and LdClEval) appears clearly on Figure 2b being characterized by high scores for Spatial Consideration, meaning that spatial indicators (e.g., connectivity) are taken into account in the calculation of the compensation unit. Indeed, it make less sense to evaluate impacted and compensatory sites values within a particular landscape context when equivalence is assessed in a "like for unlike" perspective. Finally, no group of EAMs can be characterized by high scientific basis as every criterion related to scientific basis contributes to the PCA graph in a different direction (Figure 2a) involving high scores for this challenge apportioned among EAMs. Discussion We analyzed the structure of existing EAMs and assessed the possible synergies and trade-offs between criteria underlying the way EAMs address operationality, scientific basis and comprehensiveness. The studied EAMs share a common structure to evaluate sites biodiversity and to size offset although they handle ecologic, spatial, temporal considerations and uncertainty in various ways. There is no clear trade-off in challenge achievement but some criteria within or between challenges are negatively correlated. No EAM perfectly addressed all three challenges and groups of EAMs were identified according to criteria or challenge they best achieved. EAMs general structure We identified three main aspects of EAMs common structure that should be considered when developing an EAM and discuss the way they could be improved. Target biodiversity All EAMs evaluated biodiversity losses and gains by combining biodiversity "quality" and area. Biodiversity "quality" is expressed in terms of three main components: species (e.g., threatened, endemic, patrimonial), habitat (e.g., protected ecosystems, wetlands, species habitat) and functionalities (e.g., connectivity, wetland functions). Only 5 EAMs out of 13 focus on ecosystem functionalities in addition to species and habitats, while scientists currently strongly encourage assessing biodiversity functionality, notably in order to better integrate "ordinary" biodiversity in offset processes (Regnery et al. 2013b). Offsetting ecosystem functionalities and "ordinary" biodiversity is also beginning to appear in offset policies: for example, the French consultative process "Grenelle de l'Environnement" (2007) specifies that "ordinary" biodiversity should be evaluated by Environmental Impact Assessment (EIA), notably for the role played as ecological corridors, and be compensated for if impacted [START_REF] Quetier | No net loss of biodiversity or paper offsets? A critical review of the French no net loss policy[END_REF]. That is why at least part of the "compensation units" should be based on ecosystems functionalities. This should be done in consistency with offset policies which influence considerably the biodiversity components targeted (e.g., the US Wetland Mitigation policy requires offset for wetlands, in Europe the Birds and Habitats Directives requires offset for specific birds species or habitats [START_REF]Council Directive 92/43/EEC of 21 May 1992 on the conservation of natural habitats and of wild fauna and flora[END_REF][START_REF]Directive 2009/147/EC of the European Parliament and of the Council of 30 November 2009 on the conservation of wild birds on the conservation of wild birds (codified version)[END_REF] and the offset measures outcomes (e.g., wetland functionalities restoration, species population conservation). According to the targeted biodiversity (either imposed by offset policies or chosen as best surrogate for all biodiversity) the use of "specialized" or "generalist" EAMs is more or less appropriate. Specialized EAMs seem best indicated to maximize the accuracy of equivalence assessment when impacts concern a limited geographic zone composed of a single type of ecosystem. Generalist EAMs are probably more appropriate for projects impacting biodiversity over a large area including various habitat types such as wetlands, forests, rivers, meadows, etc., in order to embrace a global view of the site's biodiversity. Indicators Indicators chosen as surrogates of biodiversity are at the very heart of EAMs in a sense that they enable calculation of the "compensation units" [START_REF] Bekessy | The biodiversity bank cannot be a lending bank[END_REF]. Even when the same type of ecosystem is targeted, the set of indicators is different from one EAM to the other, involving various approaches of ecosystem evaluation. This is for example the case for UMAM Predictions To assess biodiversity losses and gains, predictions have to be made, since offset measures have to be sized mostly before the project can be conducted in order to obtain permits. Predictions concern biodiversity state after impact (effect of habitat destruction or fragmentation on onsite and surrounding biodiversity) and after offset (biodiversity trajectory and likelihood of offset success). The fact that half of the assessment of equivalence is based on prediction means that this assessment is far from precise, especially since accuracy of forecasting is often low. Modeling techniques (e.g., [START_REF] Meineri | Combining correlative and mechanistic habitat suitability models to improve ecological compensation[END_REF] adapted to EAMs could greatly increase efficiency in assessing losses and gains (Resource/Habitat Evaluation Analysis already requires use of modeling, although quite simple). Another way to make more accurate predictions and reduce uncertainty would be for EAM users to take advantage of feedback from previous impacts or offset measures in similar habitats or for the same species or taxa [START_REF] Walker | The restoration and re-creation of species-rich lowland grassland on land formerly managed for intensive agriculture in the UK[END_REF][START_REF] Tischew | Implementation of Basic Studies in the Ecological Restoration of Surface-Mined Land[END_REF][START_REF] Tischew | Evaluating Restoration Success of Frequently Implemented Compensation Measures: Results and Demands for Control Procedures[END_REF]). This could be achieved by drawing tendencies from data [START_REF] Specht | Data management challenges in analysis and synthesis in the ecosystem sciences[END_REF] generated by all EIA individually for a large set of projects. 4.2. Trade-offs and synergies between the three EAM challenges: why do they exist and how could they be overcome (or not)? Based on their average scores, the EAM challenges we identified as operationality, scientific basis and comprehensiveness are not incompatible but still no EAM combines all these challenges perfectly. This is due to some trade-offs occurring between few criteria within and between challenges. Compromises tend to favor operationality The majority of analyzed EAMs showed high operational scores (8 out of 13 EAMs have mean scores of operationality from 64% to 85%, see Appendix E). These more operational EAMs (HabHect, PilotUK, SomersetHEP, UMAM, CRAM, Ökokonto, LdClEval, and FishHab) use a system of predefined indicators, are mostly specialized and are quick to implement. They are reproducible and easy to use but are very context dependent. For project developers, one priority is to propose offset measures that will be accepted by decision-makers, and that can be rapidly implemented at a reasonable cost [START_REF] Cuperus | Ecological compensation in Dutch highways planning[END_REF]. To this end, operational tools are needed and EAMs with predefined indicators seem therefore more suitable, with a higher likelihood of acceptance if assessment is science-based. Most EAMs having predefined indicators with a scoring system rely on rapidly collected and inexpensive (or free) data, and therefore are rapid to implement (UMAM, CRAM, Habitat Hectare, UK Pilot method). However, this can imply compromising on some criteria related to other challenges as it precludes largescale data collection and modeling, which are elements contributing to comprehensiveness. In addition, the use of rapidly collected data implies that indicator metrics are qualitative which leads to a lower level of scientific basis. Therefore, less operational EAMs (HEP, HEA, REA, LEA, PilotBBOP) which better combine both other challenges are often used for large-scale "voluntary" offset (BBOP 2014a(BBOP , 2014b) ) or accidental impacts [START_REF] Roach | Policy evaluation of natural resource injuries using habitat equivalency analysis[END_REF] which should be subject to less temporal, financial and legislative constraints than "classic" development project. Heterogeneity in the integration of scientific basis Trade-offs between criteria within a challenge concern especially scientific basis (EAMs have high scores for one or some criteria related to this challenge but never all of them). Depending Improving synergies between scientific basis and comprehensiveness There are neither trade-offs nor strong synergies between criteria related to scientific basis and comprehensiveness. Existing knowledge could largely benefit to a better combination of these challenges achievement in order to better assess equivalence in the design phase of offset measures. Notably, key equivalence considerations are well identified in literature [START_REF] Norton | Biodiversity Offsets: Two New Zealand Case Studies and an Assessment Framework[END_REF][START_REF] Bull | Biodiversity offsets in theory and practice[END_REF][START_REF] Gardner | Biodiversity Offsets and the Challenge of Achieving No Net Loss[END_REF]) and science-based solutions have already been suggested to integrate delay and uncertainties in offset design [START_REF] Moilanen | How Much Compensation is Enough? A Framework for Incorporating Uncertainty and Time Discounting When Calculating Offset Ratios for Impacted Habitat[END_REF][START_REF] Laitila | A method for calculating minimum biodiversity offset multipliers accounting for time discounting, additionality and permanence[END_REF][START_REF] Cochrane | Modeling with uncertain science: estimating mitigation credits from abating lead poisoning in Golden Eagles[END_REF]. Both ecological and spatial considerations should be addressed using the multiplicity of existing indicators covering a wide range of species and habitats (e.g., [START_REF] Andreasen | Considerations for the development of a terrestrial index of ecological integrity[END_REF][START_REF] Biggs | A biodiversity intactness score for South Africa[END_REF][START_REF] Regnery | Tree microhabitats as indicators of bird and bat communities in Mediterranean forests[END_REF]). Combining operationality, scientific basis and comprehensiveness Finally, our study aimed to identify if all challenges could be combinable in one EAM accepted by both operational and scientific spheres. One issue that affects all 3 challenges is data: operationality relies on data availability, comprehensiveness on data diversity which influences the accuracy of biodiversity assessment (e.g., species conservation status, Bensettiti et al. 2012 ), and scientific basis on data provenance (data updating is notably crucial and even more important with global changes modifying ecosystems dynamics, [START_REF] Vitousek | Human Domination of Earth's Ecosystems[END_REF]. We therefore suggest one main avenue to develop EAMs combining the three challenges: the creation and use of biodiversity offset dedicated data-bases gathering relevant information concerning key equivalence considerations (e.g., risks associated to offset failure based on previous feedback) for at least species and ecosystems frequently targeted in offset procedures. In this way, EAMs implementation could be based on a large amount of data which would be available for users and which could be regularly updated with recent knowledge. This would require a certain investment both in time and money, but would also make information coming from scientific documentation available (for example ecological corridor identification based on the species dispersal ability). An important aspect remains the data interpretation, and tendencies should be established (some data could, for instance, be contradictory) so that the data is used in the most efficient way. Such data-bases could be developed by public authorities at regional or national level (French government intend to create such data base gathering data from all EIA). Moreover, some companies (Virah-Sawmy et al. 2014) own a large amount of land and therefore have the possibility to offset their impacts on biodiversity on their own land. In this purpose, biodiversity issues (e.g., ecosystems maps or species lists) can better be identified in advance for their offset needs (e.g., French biodiversity observatories in alpine ski resorts). In this way, offset measures could be anticipated and launched before impacts occur to reduce time lags, and the offset site location could be made consistent with biodiversity issues improving sites integration in landscape context. Conclusion All studied EAMs share a general framework to assess ecological equivalence where equivalence key considerations (ecological, spatial, temporal and uncertainties) are taken into account in different ways, which influence EAMs operationality, scientific basis comprehensiveness. The analysis of these three "challenges" revealed that operationality tends to be favored in EAMs development, while there is heterogeneity in the integration of scientific basis in EAMs. No EAM is fully satisfying as none combines all challenges perfectly. One way of better combining operationality, scientific basis and comprehensiveness is to develop and use offset dedicated data-bases providing hindsight on local context and previous offset measures. The common structure underlying EAMs suggests that, even though some aspects could be improved, no better solution has yet been found. In developing EAMs, it might be useful to think "out of the box" and invent new structures. Finally, demonstrating ecological equivalence does not guaranty alone offset measures design that reaches the "no net loss" objective. Some issues related to what is really done in practice like offset long-term duration, maintenance and governance, remain of great importance. The way indicators are defined in the method. Predefined indicators make EAMs more standardized and lead to repeatable and comparable equivalence evaluation [START_REF] Quétier | Assessing ecological equivalence in biodiversity offset schemes: Key issues and solutions[END_REF]. Data availability (DataAv) Level of data cost and time to collect data that are needed to fill in indicators. Inexpensive and rapid to collect data will provide more guaranties that EAMs will be widely used than expensive and long to collect data (a parallel can be drawn with river health assessment [START_REF] Boulton | An overview of river health assessment: philosophies, practice, problems and prognosis[END_REF] Implementation rapidity (ImpRp) Cumulative time needed to both collect data and implement EAMs. Rapid method implementation notably reduces the risk of biodiversity losses related to delay in offset measures design [START_REF] Bas | Improving marine biodiversity offsetting: A proposed methodology for better assessing losses and gains[END_REF]. . Exchangeability (Exchg) EAMs adaptation to allow a certain degree of exchangeability between biodiversity impacted and compensated (like for like or like for unlike offset). Developers have more flexibility in designing offsets with like for unlike (or similar) offsets [START_REF] Quétier | Assessing ecological equivalence in biodiversity offset schemes: Key issues and solutions[END_REF]Quétier et al. 2014;Bull et al. 2015). Scientific basis (ScBs) Biodiversity indicators (BiodivInd) On which basis biodiversity indicators were set up in EAMs. The use of indicators based on defensible scientific documentation provides more guaranties that biodiversity evaluation is rigorous (indicator has been demonstrated to be a good surrogate of targeted biodiversity component) and consensual (there is a global agreement among scientific community) [START_REF] Mccarthy | The habitat hectares approach to vegetation assessment: An evaluation and suggestions for improvement[END_REF][START_REF] Gonçalves | Biodiversity offsets: from current challenges to harmonized metrics[END_REF]. Biodiversity indicator metrics (BiodivIndMc) Type of metrics (qualitative, quantitative discrete or continuous) used to inform biodiversity indicators. Quantitative metrics (e.g. number of bat species, height of vegetation) give losses and gain calculation more accuracy and transparency [START_REF] Noss | Indicators for monitoring biodiversity: a hierarchical approach[END_REF]) whereas qualitative metrics are more subject to interpretation bias and subjective judgment. Spatial consideration (SpCd) The way spatial consideration (impacted or compensatory sites insertion in landscape) is taken into account in the method. Measuring landscape components (connectivity, fragmentation…) with appropriate indicators is essential for integrating the effect of surrounding landscape on sites biodiversity (e.g. significance of species richness) to losses and gain comparison [START_REF] Quétier | Assessing ecological equivalence in biodiversity offset schemes: Key issues and solutions[END_REF][START_REF] Gardner | Biodiversity Offsets and the Challenge of Achieving No Net Loss[END_REF]. Uncertainty consideration (UnCd) The way uncertainty (probability of offset failure) is taken into account in the method. As all offsets have a chance of failing to meet expectations, uncertainty can be considered by weighting gains calculation according to the probability of offset success [START_REF] Moilanen | How Much Compensation is Enough? A Framework for Incorporating Uncertainty and Time Discounting When Calculating Offset Ratios for Impacted Habitat[END_REF]. In this purpose, using of area-based offset multipliers is frequent but they are relevant only when based on feedbacks about previous offset measures [START_REF] Tischew | Evaluating Restoration Success of Frequently Implemented Compensation Measures: Results and Demands for Control Procedures[END_REF]. Comprehensiveness (Comp) Key equivalence considerations (EqCd) Number of key equivalence considerations (ecological, spatial, temporal, uncertainty) taken into account in the method. These four considerations have been identified in the literature to be essential when calculating equivalence in order to design offset achieving "no net loss" [START_REF] Moilanen | How Much Compensation is Enough? A Framework for Incorporating Uncertainty and Time Discounting When Calculating Offset Ratios for Impacted Habitat[END_REF][START_REF] Quétier | Assessing ecological equivalence in biodiversity offset schemes: Key issues and solutions[END_REF][START_REF] Bull | Biodiversity offsets in theory and practice[END_REF][START_REF] Gardner | Biodiversity Offsets and the Challenge of Achieving No Net Loss[END_REF] Target Biodiversity (TgBiodiv) Target biodiversity components evaluated in EAMs. In order to capture biodiversity complexity, losses and gains should be evaluated for a maximum of biodiversity components: species populations, ecosystems (or habitats) and functionalities [START_REF] Noss | Indicators for monitoring biodiversity: a hierarchical approach[END_REF][START_REF] Pereira | Essential Biodiversity Variables[END_REF]. Data type (DataTp) Type of data needed to fill in indicators (data from literature, GIS, simple field visit, inventories…). Using all kind of data provides various types of information at different scales and accuracy leading to a more comprehensive losses and gains assessment. Number of indicators (NbInd) Number of indicators used to evaluate biodiversity at impacted and compensatory sites. The multidimensional nature of biodiversity makes it complicated to evaluate and using one single indicator (or proxy) has been demonstrated to be insufficient [START_REF] Bull | Biodiversity offsets in theory and practice[END_REF]. Multiple indicators are preferable to capture a maximum of biodiversity components (diversity, functionality…) [START_REF] Andreasen | Considerations for the development of a terrestrial index of ecological integrity[END_REF]. of the analysis No general rule, consideration treated on a case by case basis. Habitat Unit (HU)=HSI* habitat areal extent HSI is the observed indicator compared to the optimal condition. Resource and Habitat Equivalency Analysis (REA / HEA) (NOAA 1995[START_REF] Washington | Scaling compensatory restoration action: Guidance document for natural resource damage assessment under the Oil Pollution Act of 1990[END_REF] Habitat resource (e.g., species population) or service (e.g., primary production) No general rule, consideration treated on a case by case basis. Resource or service is calculated for each year of the analysis, and at least during all impact duration, and until offset effectiveness. No general rule, consideration treated on a case by case basis. Discounted Resource/ Service Acre Year = proxy value * discounted rate * site area Canadian method Fish Habitat (FishHab) (Minns et al. 2001) Lacustrine habitats condition for fish productivity No general rule, consideration treated on a case by case basis. Not taken into account in offset sizing No general rule, consideration treated on a case by case basis. Habitat Suitability Index (HIS) as surrogate of fish habitat productivity. Calculated from an Habitat Suitability Matrix (HSM) model Habitat Hectare (HabHect) (Parkes et al. 2003) Native vegetation condition Indicators of landscape context Not taken into account in offset sizing No general rule, consideration treated on a case by case basis. Habitat Hectare=Habitat Wetland integrity and functionality Indicators of landscape context and location Multiplier to size offset related to offset effectiveness delay. Multiplier to size offset related to probability of offset success Delta= mean of the three indicators final category score Indicators are scored from 0 to 10 (10 is the benchmark) Landscape Equivalency Analysis (LEA) (Bruggeman et al. 2005) Species population Species population is modeled for different landscape evolution scenarios. Species population is calculated for each year of the analysis. No general rule, consideration treated on a case by case basis. The Habitat Evaluation Procedure (HEP) was developed in the late seventies in USA by the US Fish and Wildlife Service, in order to calculate comparable Habitat Units (HUs) and use them as a basis for sizing optimal offsets. This EAM focuses on habitats. It is stipulated in HEP that an area can have various habitats (with measurable areal extents) and that they can have different suitability for species that may occur in that area. The habitat suitability is quantified in HEP via Habitat Suitability Index Models (HSIs). To calculate HSIs, user has first to select species of interest (they can be patrimonial and endangered species, umbrella species etc. depending on the issues on the site). Then, for each species, a HIS has to be chosen to best reflect species condition in its habitat (or the habitat suitability for this species). It must be in an index form: Discounted Landscape Service 𝐼𝑛𝑑𝑒𝑥 𝑣𝑎𝑙𝑢𝑒 = 𝑉𝑎𝑙𝑢𝑒 𝑜𝑓 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑜𝑓 𝑐𝑜𝑚𝑝𝑎𝑟𝑎𝑖𝑠𝑜𝑛 so for an HSI it will be: 𝐻𝑆𝐼 = 𝑆𝑡𝑢𝑑𝑦 𝑎𝑟𝑒𝑎 ℎ𝑎𝑏𝑖𝑡𝑎𝑡 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛 𝑂𝑝𝑡𝑖𝑚𝑢𝑚 ℎ𝑎𝑏𝑖𝑡𝑎𝑡 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛 The "Optimum habitat condition" is a benchmark found in literature or measured in the field. Metrics for "Study area habitat condition" can be for example species abundance or biomass/unit area, but must reflect the habitat suitability for this species. The next step consists in calculating cumulative Habitat Units (HU's) for the species, for each year of the evaluation (e.g. each year of project): 𝐶𝑢𝑚𝑢𝑙𝑎𝑡𝑖𝑣𝑒 𝐻𝑈 ′ 𝑠 = ∑ 𝐻𝑆𝐼 𝑎 * ℎ𝑎𝑏𝑖𝑡𝑎𝑡 𝑎𝑟𝑒𝑎𝑙 𝑒𝑥𝑡𝑒𝑛𝑡 𝑎 𝑝 𝑖=1 Where: 𝐻𝑆𝐼 𝑎 is the species' HSI at year i 𝑎𝑟𝑒𝑎𝑙 𝑒𝑥𝑡𝑒𝑛𝑡 𝑎 is the area of available habitat for species at the year i. It is calculated in different ways depending whether the species habitat include only one vegetation cover type, or more than one. There three possibilities: (i) species habitat includes one cover type (e.g. forest), (ii) species habitat includes several cover types, but each one provides all of species requirements (i.e. shelter, food), (iii) species habitat includes several cover types, but each one provides only one species requirement (e.g. forest/shelter and meadow/food). If HSI value is not available for every year, user can decompose the period of analysis in smaller period and calculate Cumulative HU with a specific formula. Finally, Average Annual Habitat Units (AAHU's) are calculated as follow for each year of the evaluation (e.g. each year of project): 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝐴𝑛𝑛𝑢𝑎𝑙 𝐻𝑎𝑏𝑖𝑡𝑎𝑡 𝑈𝑛𝑖𝑡𝑠 (𝐴𝐴𝐻𝑈 ′ 𝑠) = 𝐶𝑢𝑚𝑢𝑙𝑎𝑡𝑖𝑣𝑒 𝐻𝑈′𝑠 𝑃𝑒𝑟𝑖𝑜𝑑 𝑎𝑛𝑎𝑙𝑦𝑠𝑖𝑠 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑦𝑒𝑎𝑟𝑠 This value is the one used in the losses and gains calculation, as follow: 𝐿𝑜𝑠𝑠𝑒𝑠 𝑜𝑟 𝐺𝑎𝑖𝑛𝑠 = 𝐴𝐴𝐻𝑈 ′ 𝑠 𝑤𝑖𝑡ℎ -𝐴𝐴𝐻𝑈 ′ 𝑠 𝑤𝑖𝑡ℎ𝑜𝑢𝑡 Where: 𝐴𝐴𝐻𝑈 ′ 𝑤𝑖𝑡ℎ is the AAHU's for the impacted site or the compensatory site with impact or offsets. 𝐴𝐴𝐻𝑈 ′ 𝑤𝑖𝑡ℎ𝑜𝑢𝑡 is the AAHU's for the impacted site or the compensatory site without impact or offsets (initial state of area before impact and before offsets). This evaluation takes into consideration the natural evolution of both impacted and compensatory site (without any impact or offset). There are two main equations to size offsets, depending on the compensation goal: -In-kind: the HU lost are offset for each evaluation species (the list of target species is identical to the list of negatively impacted species). -Equal replacement: the HU lost are offset through a gain of an equal number of HU's (the list of target species may or may not be identical to the list of negatively impacted species). 𝑂𝑝𝑡𝑖𝑚𝑢𝑚 𝑐𝑜𝑚𝑝𝑒𝑛𝑠𝑎𝑡𝑖𝑜𝑛 𝑎𝑟𝑒𝑎 = -𝐴 * ∑ 𝐿𝑜𝑠𝑠𝑒𝑠(𝑖) 𝑛 𝑖=1 ∑ 𝐺𝑎𝑖𝑛𝑠(𝑖) 𝑛 𝑖=1 Where: A is the size of candidate compensation study area i is the species number, and n is the total number of identified species.  Resource and Habitat Equivalency Analysis (NOAA 1995[START_REF] Washington | Scaling compensatory restoration action: Guidance document for natural resource damage assessment under the Oil Pollution Act of 1990[END_REF] Resource Equivalency Analysis (REA) and Habitat Equivalency Analysis (HEA) are EAMs developed initially to size offset for accidental impacts on resource (REA) or service (HEA) (i.e. oil spill on salt marsh) by the National Oceanic and Atmospheric Administration, in the United States. These EAMs are based on two restoration actions: one primary restoration on the impacted site, and one compensatory restoration on the compensatory site. The latest aims to offset the impacted site's interim losses. Only one proxy is needed to represent the level of resource or service lost. In REA, it is understood by "resource" a particular species population. So the proxy chosen by the user can be abundance for example. In HEA, it is understood by "service" a particular function of a habitat. The proxy is also chosen by the user and can be the primary productivity of salt marsh for example. First, user has to determine the benchmark, which is the level of resource or service on the impacted site before the accident occurred. Then, losses are calculated according to a "recovery function" representing the evolution of resource or service level from the accident to the benchmark on the impacted site with primary restoration. The same way, gains are calculated according to a "maturity function" representing the evolution of resource or service level from the beginning of offsets to the benchmark for the compensatory site. Losses and gains are calculated each year for a certain amount of time, which at minimum must last until the level of resource or service has reached the benchmark. A discounted rate is used in these EAMs in order to take in consideration the relation the public has with the resource or service losses and gains. In Environmental policies, this discounted rate is often 3% (Commissariat Général du Développement Durable (CGDD) 2011). With the application of a discounted rate, losses have an increasing value over time, and in the contrary, gains have a decreasing value over time. Losses and gains are calculated as follow (the unit is discounted resource or service acre year). Where: i is the year when primary restoration starts on the impacted site (area 1) j is the year when offsets start on the compensatory site (area 2) b is the year when the calculation is stopped (benchmark level have to be reached). R is the % of resource or service lost or gained compared to the benchmark in average at the nth year. D is the discounted rate Ecological equivalence is achieved when losses = gains. The equations to size offsets that achieve equivalence is as follow: Losses 𝐴𝑟𝑒𝑎 2 = ∑ ( 𝑅 𝑡 * 𝐷 𝑏 𝑛=𝑖 ) * 𝑎𝑟𝑒𝑎 1 ∑ ( 𝑅 𝑡 * 𝐷 𝑏 𝑛=𝑗 )  Canadian method Fish Habitat (Minns et al. 2001) In Canada, the Department of Fisheries and Oceans provide tools for managing and protecting Canada's fishery resources. Section 35 of the Fisheries Act is a general prohibition forbidding the Harmful Alteration, Disruption or Destruction (HADD) of fish habitats. If all mitigation measures cannot prevent a HADD, an authorization is required and proponents are then obligated to develop a set of compensatory actions that will result at least in no net loss of fish productivity. The Fish Habitat method is an EAM allowing to size offset so they can achieve no net loss. The method is mainly intended to be used to assess development projects occurring in large inland lakes. It involves the use of a "Habitat Suitability Matrix" (HSM) model implemented as a software package with many features (but regulatory users only use basic elements). "The essence of the approach is the idea that the habitat preferences of individual fish species and life stages can be quantified and aggregated into habitat suitability indices [HSI] that in turn can be used as surrogate measures of fish habitat productivity" (Minns et al. 2001). To calculate HSI's, the HSM model uses pooled matrices representing the aggregate habitat preferences of many species. Species lists are identified and are grouped by life stage, trophic level regime and thermal preference. HIS values are generated for specific combinations of water depth, substrate and vegetation cover that can be assigned to individual habitat patches. HIS values range between 0 and 1, which represent a percentage of the benchmark (1 is the benchmark value). HIS's as surrogates of fish productivity are calculated for three areas: -the area of habitat lost due to development activity (𝐴 𝐿𝑜𝑠𝑠 ) -the area modified, directly and indirectly, as a result of the development activity (𝐴 𝑀𝑜𝑑 ) -the area created or modified elsewhere to compensate for the development activity (𝐴 𝐶𝑜𝑚𝑝 ) To achieve ecological equivalence, the result of the following equation has to be neutral (no net loss of biodiversity) or positive (net gain): 𝛥𝑃 𝑛𝑜𝑤 = [(𝑃 𝑀𝑜𝑑 -𝑃 𝑁𝑜𝑤 ) * 𝐴 𝑀𝑜𝑑 ] -(𝑃 𝑀𝑎𝑥 * 𝐴 𝐿𝑜𝑠𝑠 ) + [(𝑃 𝐶𝑜𝑚 -𝑃 𝑁𝑜𝑤 ) * 𝐴 𝐶𝑜𝑚 ] Where: 𝛥𝑃 𝑛𝑜𝑤 is the net change of natural productivity of fish habitat 𝑃 𝑀𝑎𝑥 is the maximum potential unit area productivity rate (or productive capacity) 𝑃 𝑁𝑜𝑤 is the present unit area productivity rate 𝑃 𝑀𝑜𝑑 is the modified unit area productivity rate in affected areas 𝑃 𝐶𝑜𝑚 is the compensation unit area productivity rate in affected areas.  Habitat Hectare (Parkes et al. 2003) The "Habitat Hectare" approach has been first developed by (Parkes et al. 2003) for the Victorian Department of Natural Resources and Environment in Australia. Here we will name "Habitat Hectare" this particular EAM, even though other EAMs are based on the same principle (e.g. UK pilot method, BBOP pilot method). This principle consists in multiplying a value reflecting the quality of the site with the site area. Habitat Hectare focuses on terrestrial biodiversity related to native vegetation. A site is evaluated according to several indicators (listed below), some related to the site condition, and others related to landscape context. Each indicator is scored as a percentage of a benchmark (Pre-European vegetation condition). A pre-defined weight is attributed to each indicator. The site final score is called the "Habitat Score" (HS) and it is calculating summing all indicators scores. It must be multiplied by the site area (in hectare). Four Habitat Scores are calculated: 𝐻𝑆 𝐴 for the current score of the habitat that will be impacted (area 1) 𝐻𝑆 𝐵 is the predicted score for the habitat after impacts (area 1) 𝐻𝑆 𝐶 is the current score of the habitat proposed for offsets (area 2) 𝐻𝑆 𝐷 is the predicted score of the habitat after offsets (area 2) Ecological equivalence is achieved when: (𝐻𝑆 𝐴 * 𝑎𝑟𝑒𝑎 1 ) -(𝐻𝑆 𝐵 * 𝑎𝑟𝑒𝑎 1 ) (losses) = (𝐻𝑆 𝐷 * 𝑎𝑟𝑒𝑎 2 ) -(𝐻𝑆 𝐶 * 𝑎𝑟𝑒𝑎 2 ) (gains). The equations to size offsets that achieve equivalence is: 𝐴𝑟𝑒𝑎 2 = 𝐻𝑆 𝐴 * 𝑎𝑟𝑒𝑎 1 -𝐻𝑆 𝐵 * 𝑎𝑟𝑒𝑎 1 𝐻𝑆 𝐷 -𝐻𝑆 𝐶 This calculation has to be done for each impacted habitat.  Uniform Mitigation Assessment Method (State of Florida 2004) The Uniform Mitigation Assessment Method (UMAM) is a "rapid assessment method" developed specifically for Florida's wetlands in order to assess their functionality. User have to score each indicator between 0 and 10 depending on the indicator condition (the guidance help the user defined it). An average score is calculated by category and the average of these score is the site final score called Delta. Four Deltas are calculated. 𝛥 𝐴 for the current score of the site that will be impacted (area 1) 𝛥 𝐵 is the predicted score for the site after impacts (area 1) 𝛥 𝐶 is the current score of the site proposed for offsets (area 2) 𝛥 𝐷 is the predicted score of the site after offsets (area 2) The method includes two multipliers for gains calculation: -the T-factor reflects the time lag associated with mitigation (the period of time between when the functions are lost at an impact site and when those functions are replaced by the mitigation), and the additional mitigation needed to account for the deferred replacement of wetland or surface water functions. It determined with a correspondence grid between years and scores. -The mitigation risk, evaluated to account for the degree of uncertainty that the proposed conditions will be achieved, resulting in a reduction in the ecological value of the mitigation assessment area. The risk is scored on a scale from 1 (for no or de minimus risk) to 3 (high risk), on quarter-point (0.25) increments. Losses and gains are calculated as follow: 𝐿𝑜𝑠𝑠𝑒𝑠 = (𝛥 𝐴 -𝛥 𝐵 ) * 𝑎𝑟𝑒𝑎 1 𝐺𝑎𝑖𝑛𝑠 = [ ( 𝛥 𝐷 -𝛥 𝐶 ) 𝑇-𝑓𝑎𝑐𝑡𝑜𝑟 * 𝑟𝑖𝑠𝑘 ] * 𝑎𝑟𝑒𝑎 2 Ecological equivalence is achieved when losses = gains The equations to size offsets that achieve equivalence is as follow: 𝐴𝑟𝑒𝑎 2 = (𝛥 𝐴 -𝛥 𝐵 ) * 𝑎𝑟𝑒𝑎 1 ( 𝛥 𝐷 -𝛥 𝐶 ) * (𝑇 -𝑓𝑎𝑐𝑡𝑜𝑟 * 𝑟𝑖𝑠𝑘)  Landscape Equivalency Analysis (Bruggeman et al. 2005) This EAM has been developed to calculate ecological credit for species mitigation bank in the United States. It is elaborated on the same principles than REA and HEA. The method aim to assess a landscape conservation value (service) for metapopulation, evaluated through two main indicators: abundance and genetic variance. Those indicators are calculated for three landscape evolution scenario. A landscape is modeled as "habitat patches [which] are distinguished by greater habitat quality than surrounding areas. Area outside of the habitat patch that allow low occupancy rates (lower habitat quality) are classify as the matrix" (Bruggeman et al. 2005). As in REA and HEA, a discounted rate is used. Abundance and genetic variance are modeled for: -the B scenario (benchmark) where there is no habitat loss -the M scenario (mitigation) where a conservation bank is added -the W scenario (withdrawal) where impacts sites in the landscape require the withdrawal of credit from the mitigation bank (several choices are possible). Where: i is the year when impacts occurred and when credits are bought to a conservation bank. x is the year when calculation is stopped Nb, Nw and Nm are the abundance calculated in scenario B, W and M respectively, in average for the tth year Gb, Gw and Gm are the abundance calculated in scenario B, W and M respectively, in average for the tth year D is the discounted rate The credits are calculated so that the "landscape configurations that provide equivalent levels of services despite changes in landscape structure that result from losing a patch or changing matrix quality" (Bruggeman et al. 2005).  BBOP pilot method (Business and Biodiversity Offsets Programme (BBOP) 2009) The BBOP has proposed a methodology detailed in the Biodiversity Design Offset Handbook Apendix C. It has been designed for voluntary biodiversity offsets, and it based on "Habitat Hectare" principles. There are two versions of the method, one focusing on habitats, and the other focusing on species. It is recommended to use in priority the habitat version and the species version as a complement. All terrestrial habitats can be assessed. For the habitat version, no indicators are imposed, they are to the choice of user, but the methodology provides guidance in this choice. The 10 to 20 indicators used to assess both impacted and compensatory sites have to be chosen according to ecological, spatial (e.g. connectivity), political (e.g. protected species) and social (e.g. emblematic species) issues. First, they have to be informed for a "benchmark area", chosen as well by the user for its ideal habitat condition. Each indicator value found in the "benchmark area" will be its maximum value. Each indicator has also to be weighted depending on the importance it has in the habitat assessment (the sum of weights equals 100). Four Habitat Scores are calculated. 𝐻𝑆 𝐴 for the current score of the site that will be impacted (area 1) 𝐻𝑆 𝐵 is the predicted score for the site after impacts (area 1) 𝐻𝑆 𝐶 is the current score of the site proposed for offsets (area 2) 𝐻𝑆 𝐷 is the predicted score of the site after offsets (area 2) This calculation has to be done for each impacted habitat. 𝐻𝑆 For the species version, only one indicator has to be chosen, representing the species population (e.g. abundance). A benchmark value is also fixed. The calculation is the same as the one in the habitat version.  Land Clearing Evaluation (Gibbons et al. 2009) The Land Clearing Evaluation (LCE) is the EAM behind the calculation of credits in the context of Biobanking in the New South Wales state in Australia (Department of Environment Climate Change and Water 2009). LCE focuses on terrestrial biodiversity related to native vegetation which will be cleared. A site (whether a proposal site for clearance or a biobank site) is evaluated according to three values: the Regional Value (RV), the Landscape Value (LV) and the Site Value (SV). RV represents the site conservation significance of vegetation at the regional scale. The two latest are calculated using native vegetation biodiversity variables, scored as a percentage of a benchmark (Pre-European vegetation condition). The score goes from 0 to 3 according in which category the variable is. A pre-defined weight is attributed to each variable. Each value calculation is detailed as follow:  Regional Value: It is the same equation for both clearing and offset sites. Where: z is a zone with the same vegetation type and the same condition R is the per cent of the vegetation type in the zth zone that is remaining relative to its predicted pre-European distribution  Landscape Value: Land Clearing Evaluation (Gibbons et al., 2009) 19 Canadian method Fish Habitat (Minns et al., 2001) 20 Other: (1) 22 1-On which biodiversity component(s) does the method focus? 23 Please choose one answer. If your answer is not already mentioned, please precise it in "Other". 25 Species (e.g. protected species) ( calculated as the relative mean of the related criteria scores, see Appendix E), as shown with their projection on Figure2a. However, when considering each criterion separately, negative and positive correlations between criteria related to different challenges or within a single challenge occur.As we expected, criteria related to operationality are negatively correlated to criteria related to scientific basis and also comprehensiveness. Some of these correlations are quite intuitive and confirm what we assumed (Implementation Rapidity ~ Data Type, rho = -0.74; Data Availability ~ Data Type, rho = -0.58; and Indicator Setup ~ Biodiversity Indicator Metrics, rho = -0.87). Using large data collection leads to low implementation rapidity and low data availability. The other correlations constitute less expected results: data needed for filling in indicators with qualitative metrics is more available than for filling in indicators with quantitative metrics (Data Availability ~ Biodiversity Indicator Metrics, rho = -0.83); furthermore spatial considerations are more taken into account when assessing equivalence in a "like for like" perspective (Exchangeability ~ Spatial Consideration, rho = -0.65). As any individual criteria within scientific basis and comprehensiveness are not correlated, those challenges could be combined. Surprisingly, positive correlations also occur between criteria related to operationality and scientific basis (Implementation Rapidity ~ Biodiversity Indicators, rho = 0.66) and between operationality and comprehensiveness (Indicators Setup ~ Number of Indicators, rho= 0.64). In other terms, using scientifically based indicators do not slow down the implementation rapidity and using a set of several well adapted indicators is easier if they have been previously pre-defined. on the context and resources, scientific basis are integrated in EAMs either through development of scientifically documented biodiversity indicators (Land Clearing Evaluation) and landscape context integration (Habitat Hectare, CRAM), or through the use of ratios reflecting uncertainty based on feedbacks (e.g., Fish Habitat). The heterogeneity in the integration of scientific basis can be explained by differences in knowledge and resources available depending on the EAMs developer organism. Developing EAMs with solid scientific basis for every criterion requires researchers to be involved in EAMs design, alongside offset stakeholders and experts.Besides, both EAMs integrating best scientific basis (BBOP pilot method and Land Clearing Evaluation, see Appendix E) included researchers in their design phase. The number of research projects focusing on improving offset design is increasing[START_REF] Gonçalves | Biodiversity offsets: from current challenges to harmonized metrics[END_REF] but there is still a gap between complex and technically advanced tools developed by researchers, such as software implemented for identifying important areas for connectivity (e.g., "Graphab",[START_REF] Foltête | A software tool dedicated to the modelling of landscape networks[END_REF] or "Circuitscape",[START_REF] Koen | Landscape connectivity for wildlife: development and validation of multispecies linkage maps[END_REF] and what is actually used in practice by consultancies and developers. Therefore we strongly encourage researchers to publish or propose research tools and methods available for developers and authorities in the context of biodiversity offset. Fig 1 : 1 Fig 1: Equivalence Assessment Method (EAM) general structure. Two sites are considered: the impacted site (dark grey boxes) and the potential offset site (light grey boxes). Site values are calculated for each site (center boxes) thanks to various indicators, and "compensation units" are obtained by multiplying these values by the site areas. Solid arrows and regular font correspond to features shared by most EAMs. Dotted arrows and italics correspond to main options for EAMs. Fig 2a : 2a Fig 2a: Principal Component Analysis (PCA) variable graph. Criteria relative to operationality are in italic, criteria relative to scientific bases are in regular and criteria relative to comprehensiveness are in bold. Average scores for each challenge (Operationality, Scientific Basis and Comprehensiveness) are represented with dotted arrows. Fig 2b : 2b Fig 2b: Principal Component Analysis (PCA) individuals graph. order to evaluate how EAMs are structured we first conducted a qualitative bibliographic study. We started from Quétier's & Lavorel's publication (2011) to described EAMs characteristics according to the four key equivalence considerations: (i) Ecological: what components of biodiversity do EAMs evaluate? (ii) Spatial: how do EAMs take into account the landscape context? (iii) Temporal: how do EAMs take into account time lags? And (iv) Table 1 : 1 Context of selected EAMs implementation EAM name, code and reference Structure and Country where EAM was implemented initially Offset policy in which EAM can be implemented Type of impacts for which EAM can be implemented Habitat Evaluation Procedure US Fish and Wildlife Service, United US Conservation Banking Development project impacting terrestrial (HEP) (US Fish and Wildlife Service States or aquatic biodiversity (USFWS) 1980) Resource and Habitat Equivalency National Oceanic and Atmospheric Damage Assessment and Accidental impacts on biodiversity Analysis Administration, United States Restoration Program (REA / HEA) (NOAA 1995, 1997) Canadian method Fish Habitat Department of Fisheries and Oceans, Canada's National Fish Habitat Development project impacting lacustrine (FishHab) (Minns et al. 2001) Canada Compensation Program habitats Habitat Hectare Victorian Department of Natural BushBroker Program Projects impacting native vegetation. (HabHect) (Parkes et al. 2003) Resources and Environment, Australia Uniform Mitigation Assessment Florida Department of Environmental US Wetland and Stream Development project impacting wetlands Method Protection, United States Mitigation Banking and wetlands mitigation banks (UMAM) (State of Florida 2004) Landscape Equivalency Department of Fisheries and Wildlife, US Conservation Banking Credits for endangered species mitigation Analysis (LEA) (Bruggeman et al. Michigan State University, United banks 2005) States BBOP pilot method (PilotBBOP) Business and Biodiversity Offsets Every non-constraining offset Development project impacting (Business and Biodiversity Offsets Programme, international policy biodiversity Programme (BBOP) 2009) Land Clearing Evaluation New South Wales Government, BioBanking Proposals to clear native vegetation (LdClEval) (Gibbons et al. 2009) Australia German Ökokonto Baden-Württemberg Region, Nature Conservation Law Development project impacting (Ökokonto) (Darbi & Tausch 2010) Germany biodiversity Californian Rapid Assessment California Wetlands Monitoring US Wetland and Stream Development project impacting wetlands Method Workgroup, United States Mitigation Banking and wetlands mitigation banks (CRAM) (California Wetlands Monitoring Workgroup (CWMW) 2013) Pilot method in United Kingdom Department for Environment, Food & UK Environmental Impact Development project impacting terrestrial (PilotUK) (Department for Rural Affairs, England Assessment biodiversity Environment, Food & Rural Affairs 2012) Somerset Habitat Evaluation Somerset County Council. England National Planning Policy Development project impacting terrestrial Procedure Framework (NPPF) biodiversity (SomersetHEP) (Burrows 2014) Table 2 : 2 Description of criteria related to operationality, scientific basis and comprehensiveness and working hypothesis underlying criteria choices. EAM Criteria Description and working hypothesis challenge Indicators set up (IndSetup) (Op) Operationality Ecological credits (E) for conservation bank are calculated as follow (the unit is discounted landscape service year): With the abundance indicator: 𝐸 = ∑ 𝑥 𝑡=𝑖 1 (1 + 𝐷 2 ) ( 𝑁𝑚 𝑡 -𝑁𝑤 𝑡 𝑁𝑏 𝑡 ) With the genetic variance indicator: 𝐸 = ∑ 𝑥 𝑡=𝑖 1 (1 + 𝐷) ( 𝐺𝑏 𝑡 -𝐺𝑤 𝑡 𝐺𝑏 𝑡 ) -∑ 𝑥 𝑡=𝑐 1 (1 + 𝐷) ( 𝐺𝑏 𝑡 -𝐺𝑚 𝑡 𝐺𝑏 𝑡 ) Where:x is the number of chosen indicators 𝑉 𝐴 , 𝑉 𝐵 , 𝑉 𝐶 , 𝑉 𝐷 are the values of the nth indicator 𝐶 𝑛 is the weight of the nth indicator L is the % of the nth indicator increase thanks to offset * offset success probability Ecological equivalence is achieved when:(𝐻𝑆 𝐴 * 𝑎𝑟𝑒𝑎 1 ) -(𝐻𝑆 𝐵 * 𝑎𝑟𝑒𝑎 1 ) (losses) = (𝐻𝑆 𝐷 * 𝑎𝑟𝑒𝑎 2 ) -(𝐻𝑆 𝐶 * 𝑎𝑟𝑒𝑎 2 ) (gains).The equations to size offsets that achieve equivalence is as follow:𝐴𝑟𝑒𝑎 2 = 𝐻𝑆 𝐴 * 𝑎𝑟𝑒𝑎 1 -𝐻𝑆 𝐵 * 𝑎𝑟𝑒𝑎 1 𝐻𝑆 𝐷 -𝐻𝑆 𝐶 𝐻𝑆 𝐷 = ∑ 𝑥 𝑛=1 [( (𝑉 𝐷 * 𝐿)+ 𝑉 𝐷 𝑉 𝑏 ) 𝑛 * 𝐶 𝑛 ] 𝐴 = ∑ 𝑥 𝑛=1 [( 𝑉 𝐴 𝑉 𝑏 ) 𝑛 * 𝐶 𝑛 ] 𝐻𝑆 𝐵 = ∑ 𝑥 𝑛=1 [( 𝑉 𝐵 𝑉 𝑏 ) 𝑛 * 𝐶 𝑛 ] 𝐻𝑆 𝐶 = ∑ 𝑥 𝑛=1 [( 𝑉 𝐶 𝑉 𝑏 ) 𝑛 * 𝐶 𝑛 ] -How was the method developed? 32 It was developed by researchers or by a governmental organization 36It was developed by a collective group (e.g. researchers, consultants, administration...) at the regional (or 26 27 Natural habitat(s) (e.g. wetlands or old-growth forest) 28 Natural Habitat(s) + species (e.g. wetland with patrimonial species) 29 Natural Habitat(s) + species + ecosystem functions 30 Other: 31 33 2Please choose one answer 34 35 It was developed by consultants 37 38 state) level 39 It was developed by a collective group at the national (or federal) level 40 3-What is the kind of data used in the method? Field monitoring Uniform Mitigation Assessment Method (State ofFlorida, 2004) German Ökokonto(Darbi & Tausch 2010) Simple field data Field inventories This research was financed by the French government "CIFRE" grant for PhD students and Electricité de France (EDF). 1994 National Oceanic and Atmospheric Administration, USA Natural Resources and Environment, Australia Habitat Evaluation Procedure The variables used are listed below. Each variable is scored from 0 to 3 according to which category its value stands. Variables: (1) % Cover of native vegetation within a 1.75 km radius of the site (1000 ha) (2) % Cover of native vegetation within a 0.55 km radius of the site (100 ha) (3) % Cover of native vegetation within a 0.2 km radius of the site (10 ha) (4) Connectivity value (5) Total adjacent remnant area (6) % Within riparian area 𝐿𝑉 𝑐𝑙𝑒𝑎𝑟𝑖𝑛𝑔 𝑠𝑖𝑡𝑒 = ∑ (𝑆 𝑣 * 𝑊 𝑣 ) Where: 𝑆 𝑣 is the score for vth variable (1-6) 𝑊 𝑣 is the weighting for the vth variable (1-6)  Site Value: The variables used are listed below. Each variable is scored from 0 to 3 according to which category its value stands. * zone area) z Where: z is a zone with the same vegetation type and the same condition 𝑆 𝑣 is the score for vth variable (a-j) 𝑊 𝑣 is the weighting for the vth variable (a-j) A is a constant weighting given to the interaction terms (authors used 5) k = (sd + se + sf)/3 c is the maximum score that can be obtained given the variables that occur in the benchmark for the vegetation type zone area is the total area of the nth vegetation zone in hectares. Clearance is accepted only if the gain in each value on offset site is higher or equal than the losses in each value on clearing site (meaning if ecological equivalence is achieved): 𝑅𝑉 𝐷 ≥ 𝑅𝑉 𝐴 𝐿𝑉 𝑜𝑓𝑓𝑠𝑒𝑡 𝑠𝑖𝑡𝑒 ≥ 𝐿𝑉 𝑐𝑙𝑒𝑎𝑟𝑖𝑛𝑔 𝑠𝑖𝑡𝑒 𝑆𝑉 𝐷 -𝑆𝑉 𝐶 ≥ 𝑆𝑉 𝐴 -𝑆𝑉 𝐵 Where: A is the current value of site proposed for clearing B is the predicted value of site after proposed clearing C is the current value of site proposed for offsets D is the predicted value of site proposed for offsets. The equations, metrics and variables detailed here, as well as the data that underpinned them, were codified into a computer software tool to facilitate LCE application for users.  German method Ôkokonto (Darbi & Tausch 2010) In Germany, mitigation modalities are settled in each Land for five environmental components: biotopes and species, water, soil, landscape, and air and climate. The general mitigation method is called Ôkokonto (it is not the only mitigation method used in Germany). Here the modalities for the Bade-Wurtemberg Land are detailed for the biotopes and species component. The method focuses on biotopes, with the assumption that species can be protected through their habitat protection. In Germany, a biotope is a uniform geographic unit from a vegetation typology and/or landscape point of view. The Ökokonto-Verordnung decree indexes and classifies the Land's biotopes (with a total of 223). Each biotope is classified according to: -a "normal" value expressed in EcoPoints/m² corresponding to the average biotope's condition -a value range allowing to take into account the changing biotope's condition In reality, there are two sets of value range with a "normal" value: one called "realistic" and the other elaborated to take into account certain environmental measures uncertainty. For example, a biotope could be characterized as follow: -with a "realistic" value range from 20 to 30 EcoPoints/m² and a 25 EcoPoints/m² "normal" value. -with an "uncertainty" value range from 15 to 25 EcoPoints/m² and a 20 EcoPoints/m² "normal" value. The number of EcoPoints/m² a biotope will get is determined depending on three criteria: its degree of "naturalness", the role it has for endangered or patrimonial species and its distinctiveness in the local scale. A software allows these three criteria combination to calculate the different values above for each biotope. The values go from 1 to 64 EcoPoints/m². To calculate biodiversity losses and gains, four biotope values (V) are needed: 𝑉 𝐴 for the current value of the biotope that will be impacted (area 1) 𝑉 𝐵 is the predicted value for the biotope after impacts (area 1) 𝑉 𝐶 is the current value of the biotope proposed for offsets (area 2) 𝑉 𝐷 is the predicted value of the biotope after offsets (area 2) Ecological equivalence is achieved when: (𝑉 𝐴 * 𝑎𝑟𝑒𝑎 1 ) -(𝑉 𝐵 * 𝑎𝑟𝑒𝑎 1 ) (losses) = (𝑉 𝐷 * 𝑎𝑟𝑒𝑎 2 ) -(𝑉 𝐶 * 𝑎𝑟𝑒𝑎 2 ) (gains). The equations to size offsets that achieve equivalence is as follow: . Distinctiveness includes parameters such as species richness, diversity, rarity (at local, regional, national and international scales) and the degree to which a habitat supports species rarely found in other habitats. The Habitat Score is calculated as (Condition * Distinctiveness). It must be multiplied by the site area (in hectare). Four Habitat Scores are calculated: 𝐻𝑆 𝐴 for the current score of the habitat that will be impacted (area 1) 𝐻𝑆 𝐵 is the predicted score for the habitat after impacts (area 1) 𝐻𝑆 𝐶 is the current score of the habitat proposed for offsets (area 2) 𝐻𝑆 𝐷 is the predicted score of the habitat after offsets (area 2) Ecological equivalence is achieved when: In addition, the UK pilot EAM includes four multipliers which aim to take into account spatial, temporal and uncertainty dimensions in offset sizing: -R1: offset probability of success -R2: duration for the offset to be effective -R3: offset location (ecological network) -R4: condition of hedgerows on impacted site The equations to size offsets that achieve equivalence is as follow: This calculation has to be done for each impacted habitat.  Californian Rapid Assessment Method (California Wetlands Monitoring Workgroup (CWMW) 2013) The Californian Rapid Assessment Method (CRAM) is a "rapid assessment method" like UMAM, but developed specifically for California's wetlands. There are two primary purposes for using CRAM. It is used to assess the ambient condition of a population of wetlands or to assess the condition of an individual wetland or wetland project. Wetland type must be identified following the guideline classification. Where: 𝑠𝑐𝑜𝑟𝑒 𝐵𝐿𝐶 = [𝑏𝑢𝑓𝑓𝑒𝑟 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛 * (%𝐴𝐴 𝑤𝑖𝑡ℎ 𝑏𝑢𝑓𝑓𝑒𝑟 * 𝑎𝑣𝑒𝑟𝑎𝑔𝑒 𝑏𝑢𝑓𝑓𝑒𝑟 𝑤𝑖𝑑𝑡ℎ) 0.5 ) 0.5 ] + 𝑎𝑞𝑢𝑎𝑡𝑖𝑐 𝑎𝑟𝑒𝑎 𝑎𝑏𝑢𝑛𝑑𝑎𝑛𝑐𝑒 𝑠𝑐𝑜𝑟𝑒 𝐵𝑆 = 𝑚𝑒𝑎𝑛(𝑝𝑙𝑎𝑛𝑡 𝑐𝑜𝑚𝑚𝑢𝑛𝑖𝑡𝑦) + ℎ𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝑖𝑛𝑡𝑒𝑟𝑠𝑝𝑒𝑟𝑠𝑖𝑜𝑛 + 𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝑏𝑖𝑜𝑡𝑖𝑐 𝑠𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑒 Four CS are calculated for each Assessment Areas (AA). The (AA) is the portion of the wetland that is assessed using CRAM. An AA might include a small wetland in its entirety. But, in most cases the wetland will be larger than the AA. Rules are therefore explained in the guideline to delineate the AA, which must only represent one type of wetland. 𝐶𝑆 𝐴 for the current score of the AA that will be impacted (AA 1) 𝐶𝑆 𝐵 is the predicted score for the AA after impacts (AA 1) 𝐶𝑆 𝐶 is the current score of the site proposed for offsets (AA 2) 𝐶𝑆 𝐷 is the predicted score of the site after offsets (AA 2) Ecological equivalence is achieved when: The equations to size offsets that achieve equivalence is as follow: CRAM Scores are comparable only between the same types of wetland.  Somerset Habitat Evaluation Procedure (Burrows 2014) This EAM has been adapted from the US Fish and Wildlife Service's method to be usable in the English context. It is based on the same principle: the calculation of Habitat Units (HU) which are the product of a Habitat Suitability Index (suitableness of habitat for species) and the total area of habitat affected or required for the species. Habitats are classified into over 400 categories with an Integrative Habitat System (IHS) using hierarchical Habitat Codes. The IHS provides as well Matrix, Formation and Land Use/Management added to the Habitat Code. Each habitat category is scored on a scale from 0 (poor) to 6 (excellent), according to its condition to support species, no matter the distinctiveness (i.e. broadest, priority level). Then the Matrix score (from 0 to 6) is added or subtracted depending on the contribution the "matrix" has on habitat suitability. Matrix here represents certain elements like scrubs or single trees which can influence habitat suitability for species. Formation and Management are scored between 0 and 1, depending on their effect on habitat and are multipliers (e.g. a species could require grazed grassland). All these information are gathered in a database for each habitat (ongoing). So IHS is calculated as follow: The IHS obtained is finally multiplied by the Density Band (scored 1, 2 or 3, according to the occurrence of the species in the habitat). HUs on site are calculated as follow (area is in hectare): 𝐻𝑈𝑠 = (𝐼𝐻𝑆 * 𝐷𝑒𝑛𝑠𝑖𝑡𝑦 𝐵𝑎𝑛𝑑) * 𝐴𝑟𝑒𝑎 The HUs calculation has not to be done necessarily for all species impacted, but some umbrella species should be chosen to represent a habitat. Two HUs are calculated on each impacted and compensatory sites: 𝐻𝑈𝑠 𝑟𝑒𝑞𝑢𝑖𝑟𝑒𝑑 for HUs lost due to impacts and that have to be offset 𝐻𝑈𝑠 𝑟𝑒𝑡𝑎𝑖𝑛𝑒𝑑/𝑒𝑛ℎ𝑎𝑛𝑐𝑒𝑑 for HUs retained or enhanced due to onsite or offsite offsets 𝐻𝑈 𝑟𝑒𝑞𝑢𝑖𝑟𝑒𝑑 = 𝐻𝑈𝑠 𝑏𝑒𝑓𝑜𝑟𝑒 𝑖𝑚𝑝𝑎𝑐𝑡 * 𝑅𝑖𝑠𝑘𝑠 𝐻𝑈 𝑟𝑒𝑡𝑎𝑖𝑛𝑒𝑑/𝑒𝑛ℎ𝑎𝑛𝑐𝑒𝑑 = 𝐻𝑈𝑠 𝑎𝑓𝑡𝑒𝑟 𝑜𝑓𝑓𝑠𝑒𝑡 -𝐻𝑈𝑠 𝑏𝑒𝑓𝑜𝑟𝑒 𝑜𝑓𝑓𝑠𝑒𝑡 Where: "𝑅𝑖𝑠𝑘𝑠" include delivery and temporal risks about offset measures. They are scored with specific grids provide by DREFA, and depend on the type of habitat. Ecological equivalence is achieved when: 𝐻𝑈 𝑟𝑒𝑞𝑢𝑖𝑟𝑒𝑑 -𝐻𝑈 𝑟𝑒𝑡𝑎𝑖𝑛𝑒𝑑 𝑜𝑟 𝑒𝑛ℎ𝑎𝑛𝑐𝑒𝑑 (𝑙𝑜𝑠𝑠𝑒𝑠) = 𝐻𝑈 𝑟𝑒𝑡𝑎𝑖𝑛𝑒𝑑 𝑜𝑟 𝑒𝑛ℎ𝑎𝑛𝑐𝑒𝑑 (𝑔𝑎𝑖𝑛𝑠) It is considered that any impact on a species population affected by the development must be replaced by habitat enhancement or creation that is accessible to that particular population. Appendix B: Description of challenges, criteria and modalities used to characterize EAMs. Challenge Criteria Modalities and Scoring Operationality (Op) Indicator set up (IndSetup) -IndSetup1: user has to choose one or several indicators -IndSetup2: indicators are predefined without a scoring system -IndSetup3: indicators are predefined with a scoring system Data availability (DataAv) DataAv1: data are costly in terms of both time and money DataAv2: data are costly in terms of time (e.g., repeated data collection in the field) but not money DataAv3: data are cost-free (e.g., open-access data-bases) and rapid to collect (e.g., simple indicators measured in the field) DataAv4: Specific data-bases exist for the method (e.g., giving biodiversity units for specific habitat) so data is free and rapid to collect Implementation rapidity (ImpRp) The time to implement method is: -ImpRp1: greater than 1 year -ImpRp2: between 6 months and 1 year -ImpRp3: between 1 week and 6 months -ImpRp4: less than 1 week Exchangeability (Exchg) Exchg1: EAM only allows calculation for "like for like" offset Exchg2: EAM allows calculation for "like for unlike" offset when "like for like" offset is not relevant Exchg3: The method can be adapted to compute "like for like" offset, or "like for unlike" offset when "like for like" offset is not relevant Scientific basis (ScBs) Biodiversity indicators (BiodivInd) -BiodivInd1: Indicators have to be chosen by users, based on examples and advice given in the EAM guideline -BiodivInd2: Indicators are fixed in the method and based on expert opinion -BiodivInd3: Indicators are fixed in the method and based on scientific documentation Biodiversity indicator metrics (BiodivIndMc) BiodivIndMc1: metric is qualitative BiodivIndMc2: metric is quantitative discrete only or combined with qualitative BiodivIndMc3: metric is quantitative continuous only or combined with quantitative discrete BiodivIndMc4: metric is a combination of the three Spatial consideration (SpCd) -SpCd1: spatial consideration is not taken into account in the theoretical guidelines, but is used on a case-by-case basis in practice -SpCd2: a ratio is used to adjust the surface area that will need to be offset -SpCd3: some indicators include these issues directly (e.g., connectivity indicators) Uncertainty consideration (UnCd) UnCd1: uncertainty is not taken into account in the theoretical guidelines, but is considered on a case-by-case basis in practice UnCd2: a ratio is used to adjust the offset surface area (this ratio is the result of expert opinion) UnCd3: some indicators include this consideration directly (e.g., contribution to a site value) UnCd4: a ratio is used to adjust the offset surface area. This ratio is based on scientific literature or an existing data-base which provides scientific feedback on previous restoration actions 94 Please choose one answer Some indicators include this consideration directly (e.g. connectivity indicators) A ratio to adjust the surface area that will need to be offset is used 98 99 9.b-If the method takes into account uncertainty consideration, how is it incorporated in the 100 method? 101 Please choose one answer 103 A ratio is used to adjust the surface of offset area. This ratio is the result of a negotiation on a case by No, It only allows "like for like" offsets (e.g. offsets targets the species impacted) 113 Yes, it allows "like for unlike" offsets (e.g. one species impacted can be offset with another species with 114 the same value) 115 When using the method, it can be adapted to compute "like for like" offsets, or "like for unlike" offsets 116 when "like for like" offsets are not relevant To fill in the form for another method, please click on "Send" and then "Send another answer". 123  Comments about experts' answers to the questionnaire 124 (1) Only one "other" EAM was filled in (Biobanking in Australia), but it has the same principles as Land 125 Clearing Evaluation since this EAM constituted the base for the development of Biobanking in the New 126 South Wales, Australia. Experts were solicited specifically for their experience in the EAMs suggested in 127 the list, which explains why we did not obtain a lot of "other" answers. 129 Experts' contribution was the most important for questions 3, 4, 5, 6, 7 and 9 because the answers 130 required more than a bibliographic study of EAM to be filled in. Notably, to answer questions 3 to 5, expert 131 needs to have tested EAM in practice. Questions 6, 7 and 9 required precision about how EAMs were 132 designed, which is not always explicitly said in theoretical guidelines. 133 Divergences between published documents analysis and experts answers, and between different experts 134 answers when several answers where obtained for the same EAM, concerned particularly questions 8 and 135 9. Indeed, we found out after dialogue with expert that there is certain flexibility for spatial considerations * Answers which were subject to dialogue with expert. The main aspects for which we needed to ask precisions to experts concern the "Equivalence Considerations", and as consequence the "Spatial and uncertainty Considerations". Indeed, in practice, it is usual that ratios taking into account delay or risks are used to adjust the offset area, but in is done as an adaptation in case by case (in addition to the EAM baseline). (1) We did not have an answer for this EAM, because it is almost the same as HEA except for all that concern biodiversity targeted and indicator used. We used only the theoretical guideline to attribute modalities to this EAM. Appendix E: EAMs challenges average scores and EAMs final scores, expressed as a percentage of challenge achievement.

01745351

en

2024/03/05 22:32:07

2012

https://hal.science/hal-01745351/file/%5BZhao12-Nanoarch%5D%20Crossbar%20Architecture%20Based%20on%202R%20Complementary%20Resistive%20Switching%20Memory%20Cell.pdf

W S Zhao Y Zhang J O Klein D Querlioz D Ravelosona C Chappert J M Portal M Bocquet H Aziza D Deleruyelle C Muller Crossbar Architecture Based on 2R Complementary Resistive Switching Memory Cell Keywords: Crossbar, Resistive Switching, complementary cell, I Emerging non-volatile memoires (e.g. STT-MRAM, OxRRAM and CBRAM) based on resistive switching are under intense R&D investigation by both academics and industries. They provide high performance such as fast write/read speed, low power and good endurance (e.g. >10 12 ) beyond Flash memories. However the conventional access architecture based on 1 transistor + 1 memory cell limits its storage density as the selection transistor should be large enough to ensure enough current for the switching operation. This paper describes a design of crossbar architecture based on 2R complementary resistive switching memory cell. This architecture allows fewer selection transistors, and minimum contacts between memory cells and CMOS control circuits. The complementary cell and parallel data sensing mitigate the impact of sneak currents in the crossbar architecture. We performed transient simulations based on two memory technologies: STT-MRAM and OxRRAM to validate the functionality of this design by using CMOS 65 nm design kit and memory compact models. INTRODUCTION Modern computing systems suffer from rising static power due to the high leakage currents, which increase exponentially following the fabrication node miniaturization of CMOS technology (e.g. <90 nm) [1]. According to ITRS 2011, the static power will start to play the major role of whole power consumption in the next years [2]. In order to relieve this power issue, emerging non-volatile memoires (NVM) based on resistive switching are under intense R&D investigation by both academics and industries. Spin transfer torque magnetic random access memory (STT-MRAM) [3]; Conductive-Bridge RAM (CBRAM) [A-4] and Oxide Resistive RAM (OxR-RAM) [A-5] are among the most promising technologies. They promise to provide much higher performances than Flash memory such as fast write/read speed, low power and good endurance (e.g. >10 12 ). Since 2009, a number of NVM preindustrial prototypes [6][7][8][START_REF] Tappertzhofen | Capacity based nondestructive readout for complementary resistive switches[END_REF][10] were presented and the commercial products should be available soon. Even though these emerging NVM are based on different physics, they hold many common features. For instance, they are two terminal nanoscale devices; their resistances vary to present '0' and '1'; their memory cell are implemented at backend of line (BEOL) process [A-6-10] etc. Thereby they use the same access structure 1T (transistor)+ 1R (resistive memory) shown in Fig. 1a, and then benefit the existing peripheral control circuits of Dynamic RAM (DRAM). However this structure presents some drawbacks: the transistor is normally much larger than the minimum size in order to obtain the sufficient current for fast memory cell switching; there are lots of large interconnects between CMOS circuits and memory cell due to the thick metals (e.g. M3). They make the density of these NVM technologies lower than Flash memory. For example, Fig. 2b shows the 65 nm layout implementation of the conventional STT-MRAM access design. In order to get fast switching speed (e.g. 10 ns), the selection transistor should be large enough to ensure a high current (e.g. 100 A). The CMOS circuits impose definitively the density of STT-MRAM density instead of magnetic tunnel junction (MTJ) [A-11]. Crossbar architectures were proposed to relax the density limitation of two-terminal resistive switching devices imposed by the CMOS [A-8-11]. There are only interconnects between CMOS circuits and memory cell on the edge of the crossbar array. A number of memory cells share the same selection transistor (see Fig. 2). The cell area efficiency can be greatly improved, and the back-end process of NVM defines then the density instead of CMOS circuits. However, the conventional crossbar architectures suffer from sneak currents and low data access speed. The former is the most critical as the data sensing can be disturbed completely by the sneak currents. Moreover, there are parasite resistances throughout a large-65nm 580nm 410nm scale crossbar array, which leads to lots of sensing errors [A-6-12]. These issues are difficult to surmount and few efficient design solutions addressing this issue have been reported previously in the literature. In this paper, we present a new crossbar architecture based on complementary resistive switching memory. Combining with parallel data sensing, the impact of sneak currents and parasite resistance can be mitigated to ensure the correct data sensing. In order to validate the functionality of this design, we performed transient simulations based on two memory technologies: STT-MRAM and OxRRAM by using CMOS 65 nm design kit and memory compact models. The rest of the paper is organized as follows. In the next section, we introduce the principles of STT-MRAM and OxR-RAM. In Sections III, we describe the design details of complementary crossbar architecture. In Sections IV, we show the transient simulation of the crossbar architecture by using CMOS 65 nm design kit and compact models of the STT-MRAM and OxR-RAM. Finally, a discussion and concluding remarks are provided in section V. II. EMERGING RESISTIVE SWITCHING MEMORIES In this section, we introduce briefly the principles of STT-MRAM and OxR-RAM, which were studied as resistive switching memory cell in the proposed crossbar architecture. A. STT-MRAM technology principle Magnetic RAM (MRAM) promises stable non-volatility, fast write/read access speed and infinite endurance etc [A-2-3]. The MRAM storage element, MTJ nanopillar is mainly composed of three thin films: a thin oxide barrier and two ferromagnetic (FM) layers (see Fig. 3a). As a result of the tunnel magnetoresistance (TMR) effect [A-12], the nanopillar resistance, RP or RAP, depends on the relative orientation, Parallel (P) or Anti-Parallel (AP), of the magnetization of the two FM layers. By using crystalline MgO barriers, the TMR ratio=RAP-RP/RP of MTJ nanopillars can reach more than 600% at room temperature [A-13-14]. This allows the state of MTJs to be detected easily by CMOS sense amplifiers [A-15]. Spin transfer torque (STT) is one of the most promising switching approaches thanks to its high power efficiency and fast writing speed [A-4-5, 16]. This switching mechanism greatly simplifies the CMOS switching circuit, as only a bidirectional current is required. One time the current through the MTJ exceeds the critical current; the MTJ will switch its state (see Fig. 3b). It opens the door to build up the first true universal memory with MRAM, which should provide both large capacity (> Gigabit) and high speed (<ns). Recent progress demonstrate perpendicular magnetic anisotropy (PMA) in CoFeB/MgO structures provides a high energy barrier E to deal with the issue of thermal stability of in-plane anisotropy, which also presents the advantages of low threshold current, high speed operation and high TMR ratios comparing with in-plane anisotropy. B. OxRRAM technology principle A large number of oxide-based materials showing a resistive switching are reported in the literature [Waser07] [Seo04] [Kim10]. Among them, metal oxides like HfO2, Ta2O5, NiO, TiO2 or Cu2O are promising candidates due their compatibility with CMOS processes and high on/off resistance ratio. In its simplest form, resistive memory element relies on a Metal/Insulator/Metal (MIM) stack (Fig. 4) that can be easily integrated into Back-End Of Line (BEOL). The MIM structure is generally composed of an active layer, usually a non-stoechiometric dielectric. The bipolar behavior is mainly due to an asymmetrical geometry of the electrode. In recent years, many studies have highlighted the good performances of non-stoichiometric HfOx [Lee08-IEDM] films used as switching layer (also used in memristors demonstrated by ). Besides, an additional buffer layer, also called interface layer, such as Al2O3 or Ti may play an important role in the reliability and the reduction of the programming voltage [Lee08-IEDM] [Lee10]. OxRRAM technology is still in its "infancy" since the physics of resistance switching is not yet fully understood. So far, it is broadly accepted that the electro migration of oxygen vacancies plays a critical role in the resistance switching [Ahnl07]. After an initial electroforming process, the memory element may be reversibly switched between reset (high resistance) and set states (low resistance). Electroforming stage corresponds to a voltage-induced resistance switching from an initial very high resistance state (virgin state) to a conductive state. In the literature, unipolar, bipolar and non-polar electrical behaviors are reported. In the case of bipolar switching, addressed in this paper, bipolar voltage sweeps are required to switch the memory element (Fig. 5). Resistive switching in an OxRRAM element corresponds to an abrupt change between a High Resistance State (RHRS or OFF state) and a Low Resistance State (RLRS or ON state). This resistance change is achieved by applying specific threshold voltage to the structure (i.e. VSET and VRESET). III. COMPLEMENTARY CROSS-BAR ARCHITECTURE The cross point architecture is shown in Fig. 6, which is composed of three parts: a memory array for data storage, bit line and word line drivers, read decoder and sense amplifiers for read operation. The operating mechanisms will be detailed in the following subsections: A. Cell structure and operation Every cell is composed of two resistive switching elements (2R) as shown in inset of Fig. 6. For every cross-point, a common word line (WL) is connected to the bottom electrode (BE) of both resistive switching elements while their top electrode (TE) are respectively connected to bit-line (BL) and complementary bit-line ( BL ). Cell programming operation are performed in two phases, as follow:  1 st Phase: For the selected word, the common WL is grounded and all the bit-line and complementary bit-line are biased to VDD and a current flow from top to bottom electrode, to set the resistive elements to a low resistance state (RLRS) as shown Fig. 7.a.  2 nd Phase: For the selected word, the common WL is set to VDD while BL and BL are biased to complementary value in order to selectively reset (high resistance state -RHRS) either the top resistive element or the bottom resistive element. In this case the current flows from the bottom to the top electrode. At the end of the programming operation both elements have opposite resistance state. WL 3 BL 0 BL 1 BL 2 BL 3 BL 0 BL 1 BL 2 BL 3 ! "# ! "# $ "# ! "# ! "# $"# $"# The read operation is performed through a differential sensing of the bit-line and complementary bit-line and will be described in the sense-amplifier sub-section. B. Array architecture As shown in Fig. 6, the array is divided in M word per word line (e.g. 2). Each word is composed of N bits (e.g. 2). There is one driver circuit associated with each word-line and bit-line to ensure the proper biasing conditions for all modes of operations (write, read, unselected). A read decoder connects the N sense-amplifier to the selected word during the read phase. C. Driver circuit When dealing with bipolar resistive switching element, one may be able to apply bipolar voltage between the top and bottom electrode as well as bidirectional current to properly achieved programming phase. Moreover, as described in previous sub-section, voltage and current must remain below threshold to do not changed resistance value of unselected cell. To achieve this set of conditions, drivers have to be connected to the bit-line and the word-line. Row & column decoders together with the control logic activate the drivers. Decoder and control logic are similar to other well-known memory circuits and are not described here. A driver is composed (Fig. 8) of one PMOS to connect a line to VDD and two NMOS to connect a line to gnd or VDD/2. It is important to note that the driver sizing is crucial to control the current and voltage levels applied to the cell, which is a mandatory step to properly determine the RHRS and RLRS values. D. Sense amplifier definition The read operation of data stored in cross-point resistive switching memory is currently one of the major challenges to develop this approach. Indeed, sneak path or destructive read with CRS element [S Tappertzhofen11] are a strong limit to develop this type of architecture. Moreover, the resistance ratio (RHRS/RLRS) and the process variations have to be considered when designing a sense solution. A sense amplifier performing with high reliability is then required. Fig. 9 shows a pre-charge based sense amplifier (PCSA), which has demonstrated the best tolerance to different sources of variation [A-13], while keeping high speed and low power. In this SA, the read operation is performed in two phases: 1 st Phase: The SA is first connected to the bit-line of the selected word with SEN set to '1' and the circuit is precharged with PCH equals '0'. 2 nd Phase: The data stored in the 2R cell can be evaluated to logic level at the output Q as PCH is changed to '1' and WL is pulled down to '0'. ! " " # $%&# ! " " # ! "! # $ ! "! # $ $%&# ! " " # ! " " # ! %! # $ ! "! E. Sneak current mitigation As mentioned previously, this structure is designed to mitigate the impact of sneak currents and parasite resistance. Thanks to the complementary configuration in the memory cell, there are always the same numbers of unselected transistors and the same distance of wire connection (see Fig. 7), this balanced structure allows the impact of parasite resistance to be neglected during data sensing. Parallel data sensing allows the impact of sneak currents to be mitigated as all the bit lines are set to the same voltage potential. We performed a simple simulation of 4×4 crossbar array and evaluated the impact of sneak currents on data sensing for different architectures. The worst case is implemented, where the cell to address is ROFF and all the other cells are RON. The sensing of this cell suffers from the most important impact of sneak current. RON is fixed to 10K ohm; the parasite resistance in each wire (BLs and WLs) connected is set to 10 ohm. Fig. 10a shows that the current value for sensing in series (black square curve) is much higher than the expected current value (red point curve). This confirms that the sneak currents can perturb completely the sensing operation if there are no any mitigation solutions. However, the current value for sensing in parallel (blue triangle curve) equals nearly to the expected current value as the ROFF/RON is lower than 10. This can be explained by the reduction of sneak currents in each memory cell (blue square curve) and the decreasing of sensing current difference between ROFF and RON, as shown in Fig. 10b. IV. VALIDATION WITH BIPOLAR OXRRAM AND STT-MRAM MEMORY CELL The aim of this section is to present the architecture validation with array simulation on both NVM technologies by using compact models and CMOS 65 nm design kit. In the following, we describe briefly the compact models. Transient simulation of the proposed architecture with STT-MRAM and OxRRAM are then presented. The static behaviors of STT switching in PMA MTJ is mainly based on the calculation of threshold or critical current Ic0, which can be expressed by the Eq.A-2 [A-17], A. Resistive Memory Cell modeling 1) STT-MRAM compact model E g e V H M g e I B K S B c        2 ) ( 0 0   (A-2) where E is the barrier energy (see also Eq.1), α is the magnetic damping constant, γ is the gyromagnetic ratio, e is the elementary charge, μB the Bohr magneton, V the volume of the free layer and kB the Boltzmann constant. The switching dynamics of STT in PMA MTJ is presented in [A-18] and Eq.A-3 shows the dependence of switching current Iwrite value with the duration. ) ( ) 1 ( ] ) 4 ln( 2 [ 1 0 2 I I P P em P C c write free ref ref B           (A-3) where C≈0.577 is the Euler's constant, ξ=E/kBT the activation energy in units of kBT, Pref, Pfree the tunneling spin polarizations of the reference and free layers, we assume that Pref=Pfree=P for this compact model, m is the magnetic moment of free layer. Fig. 11 shows the transient simulation of this compact model, which could also verify the agreement of the dynamic behavior between physical models and experimental measurements. We found that the switching delay is inversely proportional to the writing current as described in Eq. A-3. 2) OxRRAM Compact Model The proposed OxRRAM modeling approach relies on a physical model accounting for both set and reset operations in bipolar resistive switching devices. In considering electric field-induced migration of oxygen vacancies within the switching layer, the model enables continuously accounting for both set and reset operations into a single master equation in which the resistance is controlled by the diameter of the conduction pathways (): T k V q E a T k V q E a b c e l l b c e l l e e d t d                       m a x (1) In equ. 1,  represents the creation/destruction rate, Ea the activation energy and VCell the cell voltage. Based on this expression, the cell current can be expressed as a function of the size of this conductor path:     HRS HRS LRS Cell Cell L V I              2 max 2 4 (2) Where LRS andHRS represents the conductivity in the Low Resistance State and the High Resistance State. This physical model demonstrates its flexibility to match the switching voltages, the levels current and the dynamics dependence on various technologies [NVMTS11] [START_REF] Aziza | Bipolar OxRRAM memory array reliability evaluation based on fault injection[END_REF]. This last point is a key point to perform a realistic circuit analysis. In this way, the model carte has been adjusted to fit at the behavior of the most aggressive component from literature: Set/Reset voltage below 1V [Cagli11-IEDM] and programming time around 10ns [Lee08-IEDM]. Table I summarizes the cell operation parameters for very short programming pulse. In order to keep the same data read access speed with that of data programming, the pulse duration of "En_Read" (see also Fig. 6) is set to ~1.1ns. The word address changes between two "En_Read" pulses during ~100 ps and the data stored in this 4×4 cross-point STT-MRAM can be detected word by word in ~5ns. It is noteworthy that the sensing speed can be accelerated up to ~200 ps/word [20], which would lead to an asymmetric delay between the programming and reading operations. Nevertheless, this asymmetric delay is nearly ubiquitous in non-volatile memories and it may present some advantages in terms of power and access speed as the nonvolatile memories are read more frequently than programmed. 2) OxRRAM array simulation To validate the efficiency of the proposed architecture with the OxRRAM technology, an array composed of 32 word-lines of 2 words of 16 bits including line drivers and PCSAs, as shown Fig. 11, is fully simulated. Moreover, as depicted in the inset Fig. 11, WL and BL resistance and capacitance are modeled through RC element distributed all over the array. The simulation is divided in two: a write phase, where all the words in the array are successively written with the pattern given Fig. 12, followed by the selective read of the top-left word in the array. The write time is set to 20ns with 10ns by phase. The read time is also set to 20ns with 10ns pre-charge and 10ns to generate a logic value on the output. It is important to note that the write and read timing value gives a strong margin versus the capability of the PCSA, as depicted on simulation results given Fig. 13. The Fig. 13 gives an overview of the simulation results, with the writing phase of: a cell in the selected word, a cell in an unselected word, a cell in a unselected bit-line and of a read phase. The Fig. 13(a) shows the behavior of the cell00 while in a first phase the cell is programmed (cell both OxRRAM elements are set and only top OxRRAM element is reset). Moreover, this simulation results validates the fact that the cell00 remains unchanged, during the write phase of the second word (word01) on the same word-line (WL0) or during the write phase of the fisrt word (word10) of the second word-line (WL1) 1 0 1 0 1 0 1 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 sharing the same bit-lines. The Fig. 13(b) validates the ability of the architecture to successfully read in parallel a full word (word00) with typically cell00='0' and cell01='1'. V. CONCLUSION This paper describes a generic design of crossbar architecture based on 2R complementary resistive switching memory cell. This architecture allows fewer selection transistors, and minimum contacts between memory cells and CMOS control circuits. The complementary cell and parallel data sensing mitigate the impact of sneak currents in the crossbar architecture. This generic architecture is implemented on two emerging technologies STT-MRAM and OxRRAM. Compact model of STT-MRAM and OxRRAM resistive elements are developed to simulate full array on both technologies. Simulation results of a 4×4 STT-MRAM array and a 32×32 OxRRAM array (16 bits word length) validate successfully the functionality of the proposed architecture using CMOS 65nm design kit. Figure 1 . 1 Conventional access approach based on 1Transistor and 1 resistive memory cell. (a) Circuit diagram (b) Layout implementation at 65 nm; the size of selection transistor is about 56 F 2. Figure 2 . 2 Figure 2. The layout implementation promises the best area efficiency, where the die area per storage bit is 2F 2 and the selection transistor is shared by a number of MTJs associated in the same word (e.g. 4). Figure 3. (a) Vertical structure of an MTJ nanopillar composed of CoFeB/MgO/CoFeB thin films. (b) Spin transfer torque switching mechanism: the MTJ state changes from parallel (P) to anti-parallel (AP) as the positive electron flow direction IP->AP>IC0, on the contrast, its state will return as the negative electron flow direction IAP->P>IC0. Figure 4 . 4 Figure 4. OxRRAM memory element stack overview. Figure 5 . 5 Figure 5. Typical I-V characteristic of a bipolar OxRRAM memory device. Figure. 6 : 6 Figure. 6: Proposed Cross Point architecture (4 × 2 words array with 2 bits word length). It includes three parts, a cross-point array of 2R cell for data storage, word line (left side) and bit line (bottom) driver and read circuits (top). Figure. 7 . 7 Figure. 7. Operation process in the cross point array: (a) For the selected word, the WL=gnd and both bit-line are biased to VDD (write 1 st phase). (b) For the selected word, WL=VDD while the bit-line are set to opposite values (write 2 nd phase). (c) For the unselected words, both the word-lines and the bit-lines are biased to VDD/2 to ensure that unselected resistive elements do not set or reset. Figure. 8 . 8 Figure. 8. Bit-line and word-line driver description. Two NMOS and one PMOS are connected to each bit-line and word-line to ensure correct biasing during all phases of operation (Write 1 st and 2 nd phase, Read 1 st and 2 nd phase). Figure. 9 : 9 Figure. 9: Pre-Charged Sense Amplifier (PCSA) for data sensing: MPC0 and MPC1 serve to pre-charge the bit-line to VDD; MPA0, MPA1, MNA0 and MNA1 constitute the amplifier; MNE0 and MNE1 play the role of "Enable". . (a) The comparison of expected current value, sensing current in series and sensing current in parallel VS the ROFF/RON ratio (b) Parallel sensing currents to address ROFF and RON in the crossbar array, the sneak currents in unaddressed RON memory cell VS the ROFF/RON ratio. A CoFeB/MgO/CoFeB STT-MTJ compact model has been recently developed based on the physical theories and experimental measurements of perpendicular magnetic anisotropy (PMA) MTJ [A-16]. It integrates the physical models of static, dynamic and stochastic behaviors. The major parameters are shown in the Table.I. Figure. 10 . 10 Figure.10. Transient simulation of the PMA MTJ demonstrates the integration of dynamic model and helps us to evaluate the tradeoff between CMOS die area and switching speed. . (a) Configuration of the small crossbar array (b) Parallel sensing of a 4×4 cross-point array within ~5ns (~1.2ns/word). Fig. 11 11 Fig. 11 (a) and (b) demonstrate the mixed simulation of parallel writing /reading for this 4×4 cross-point resistive Figure. 13 : 13 Figure. 13: Simulation results with (a) write phase of a selected cell, behavior of an unselected cells and read phase of Word00 (cell00 and cell01) TABLE I . I CELL OPERATION PARAMETERS. VTE (V) VBE (V) STATE SET (Write) 0.8@10ns 0 ONOFF RESET (Erase) 0 -0.8V@10ns OFFON Read <0.4@10ns Sensing ON or OFF B. Architecture simulation The aim of this section is to validate the functionality of the architecture for both technologies 1) STT-MRAM array simulation TABLE I PARAMETERS I For instance, it takes only one cycle of switching duration, ~1.1 ns driven by the signal "EN_Write" to program a word to "0000" or "1111". Thanks to the fast computing speed of the PMA MTJ compact model, the simulation of this 4×4 cross-point memory can be performed in ~30 minutes in a medium performance CAD server (two Xeon: 4-Core, 12MB cache, 2.4GHz and 8GB 1.3GHz RAM). switching memory and confirm the expected operations shown in the section III. AND VARIABLES PRESENT IN THE FITTING FUNCTIONS Parameter Description Default Value Area MTJ surface 65 nm x 65 nm TMR(0) TMR ratio with 0 Vbias 120% V Volume of free layer surface x1.3 nm R•A Resistance-area product 10 Ωµm 2 Vwrite Writing voltage 1.5 V Vread Reading voltage 1.2 V Jc0 Critical current density 5.7 x 10 5 A/cm 2 ACKNOWLEDGMENT The authors wish to acknowledge support the French national projects NANOINNOV-SPIN, PEPS-NVCPU, ANR-MARS.

01745369

en

2024/03/05 22:32:07

2017

https://hal.science/hal-01745369/file/TWV_ARXIV_final%285%29.pdf

Chao Zhang Vineeth S Varma Samson Lasaulce Rapha : El Visoz Interference Coordination via Power Domain Channel Estimation A novel technique is proposed which enables each transmitter to acquire global channel state information (CSI) from the sole knowledge of individual received signal power measurements, which makes dedicated feedback or inter-transmitter signaling channels unnecessary. To make this possible, we resort to a completely new technique whose key idea is to exploit the transmit power levels as symbols to embed information and the observed interference as a communication channel the transmitters can use to exchange coordination information. Although the used technique allows any kind of low-rate information to be exchanged among the transmitters, the focus here is to exchange local CSI. The proposed procedure also comprises a phase which allows local CSI to be estimated. Once an estimate of global CSI is acquired by the transmitters, it can be used to optimize any utility function which depends on it. While algorithms which use the same type of measurements such as the iterative waterfilling algorithm (IWFA) implement the sequential best-response dynamics (BRD) applied to individual utilities, here, thanks to the availability of global CSI, the BRD can be applied to the sum-utility. Extensive numerical results show that significant gains can be obtained and, this, by requiring no additional online signaling. I. INTRODUCTION Interference networks are wireless networks which are largely distributed decision-wise or information-wise. In the case of distributed power allocation over interference networks with multiple bands, the iterative water-filling algorithm (IWFA) is considered to be one of the wellknown state-of-the art distributed techniques [START_REF] Yu | Distributed multiuser power control for digital subscriber lines[END_REF] [3] [START_REF] Mertikopoulos | Distributed learning policies for power allocation in multiple access channels[END_REF]. IWFA-like distributed algorithms have at least two attractive features: they only rely on local knowledge e.g., the individual signalto-interference plus noise ratio (SINR), making them distributed information-wise; the involved : L2S (CNRS-CentraleSupelec-Univ. Paris Sud), Gif-sur-Yvette, France. ˚CRAN (CNRS-Univ. of Lorraine), Nancy, France. ; Orange Labs, Issy-les-Moulineaux, France. The material in this paper was presented in part at the 2015 EUSIPCO Conference [START_REF] Varma | Power modulation: Application to inter-cell interference coordination[END_REF]. computational complexity is typically low. On the other hand, one drawback of IWFA and many other distributed iterative and learning algorithms (see e.g., [START_REF] Rose | Learning equilibria with partial information in decentralized wireless networks[END_REF] [START_REF] Lasaulce | Game Theory and Learning for Wireless Networks: Fundamentals and Applications[END_REF]) is that convergence is not always ensured [START_REF] Mertikopoulos | Distributed learning policies for power allocation in multiple access channels[END_REF] and, when converging, it leads to a Nash point which is globally inefficient. One of the key messages of the present paper is to show that it is possible to exploit the available feedback signal more efficiently than IWFA-like distributed algorithms do. In the exploration phase 1 , instead of using local observations (namely, the individual feedback) to allow the transmitters to converge to a Nash point, one can use them to acquire global channel state information (CSI). This allows coordination to be implemented, and more precisely global performance criteria or network utility to be optimized during the exploitation phase. As for complexity, it has to be managed by a proper choice of the network utility function which has to be maximized. To obtain global CSI, one of the key ideas of this paper is to exploit the transmit power levels as information symbols and to exploit the interference observed to decode these information symbols. In the literature of power control and resource allocation, there exist papers where the observation of interference is exploited to optimize a given performance criterion. In this respect, an excellent monograph on power control is [START_REF] Chiang | Power control in wireless cellular networks[END_REF]. Very relevant references include [START_REF] Stańczak | Distributed utility-based power control: Objectives and algorithms[END_REF] and [START_REF] Schreck | Compressive rate estimation with applications to device-to-device communications[END_REF]. In [START_REF] Stańczak | Distributed utility-based power control: Objectives and algorithms[END_REF], optimal power control for a reversed network (receivers can transmit) is designed, in which the receiver uses the interference to estimate the cross channel, assuming perfect exchange of information between the transmitters. In [START_REF] Schreck | Compressive rate estimation with applications to device-to-device communications[END_REF], the authors estimate local CSI from the received signal but in the signal domain and in a centralized setting. To the best to the authors' knowledge, there is no paper where the interference measurement is exploited as a communication channel the transmitters can utilize to exchange information or local CSI (namely, the channel gains of the links which arrive to a given receiver), as is the case under investigation. In fact, we provide a complete estimation procedure which relies on the sole knowledge of the individual received signal strength indicator (RSSI). The proposed approach is somewhat related to the Shannontheoretic work on coordination available in [START_REF] Larrousse | Coded power control: Performance analysis[END_REF] [START_REF] Larrousse | Coordination in distributed networks via coded actions with application to power control[END_REF], which concerns two-user interference 1 IWFA operates over a period which is less than the channel coherence time and it does so in two steps: an exploration phase during which the transmitters update in a round robin manner their power allocation vector; an exploitation phase during which the transmitters keep their power vector constant at the values obtained at the end of the exploration phase. As for IWFA, unless mentioned otherwise, we will assume the number of time-slots of the exploitation phase to be much larger than that of the exploration phase, making the impact of the exploration phase on the average performance negligible. channels when one master transmitter knows the future realizations of the global channel state. It is essential to insist on the fact that the purpose of the proposed estimation scheme is not to compete with conventional estimation schemes such as [START_REF] Caire | Multiuser MIMO achievable rates with downlink training and channel state feedback[END_REF] (which are performed in the signal-domain), but rather, to evaluate the performance of an estimation scheme that solely relies on information available in the power-domain. Indeed, one of the key results of the paper is to prove that global CSI (without phase information) can be acquired from the sole knowledge of a given feedback which is the SINR or RSSI feedback. The purpose of such a feedback is generally to adjust the power control vector or matrix but, to our knowledge, it has not been shown that it also allows global CSI to be recovered, and additionally, at every transmitter. This sharply contrasts with conventional channel estimation techniques which operate in the signal domain and use a dedicated channel for estimation. The main contributions and novelty of this work are as follows: § We introduce the important and novel idea of communication in the power domain, i.e., by encoding the message on the transmit power and decoding by observing the received signal strength. This can be used in fact to exchange any kind of low-rate information and not only CSI. § This allows interfering transmitters to exchange information without requiring the presence of dedicated signaling channels (like direct inter-transmitter communication), which may be unavailable in real systems (e.g., in conventional Wifi systems or heterogeneous networks). § Normal (say high-rate) communication can be done even during the proposed learning phase with a sub-optimal power control, i.e., communication during the learning time in the proposed scheme is similar to communication in the convergence time for algorithms like IWFA. § We propose a way to both learn and exchanged the local CSI. Global CSI is acquired at every transmitter by observing the RSSI feedback. § The proposed technique accounts for the presence of various noise sources which are nonstandard and affect the RSSI measurements (the corresponding modeling is provided in Sec. II). By contrast, apart from a very small fraction of works (such as [START_REF] Mertikopoulos | Distributed learning policies for power allocation in multiple access channels[END_REF] [13] [START_REF] Coucheney | Distributed optimization in multi-user MIMO systems with imperfect and delayed information[END_REF]), IWFA-like algorithms assume noiseless measurements. § We conduct a detailed performance analysis to assess the benefits of the proposed approach for the exploitation phase, which aims at optimizing the sum-rate or sum-energy-efficiency. As (imperfect) global CSI is available, globally efficient solutions become attainable. The proposed work can be extended in many respects; the main extensions are marked as (‹). II. PROBLEM STATEMENT AND PROPOSED TECHNIQUE GENERAL DESCRIPTION Channel and communication model: The system under consideration comprises K ě 2 pairs of interfering transmitters and receivers; each transmitter-receiver pair will be referred to as a user. Our technique directly applies to the multi-band case, and this has been done in the numerical section. In particular, we assess the performance gain which can be obtained with respect to the IWFA. However, for the sake of clarity and ease of exposition, we focus on the single-band case, and explain in the end of Sec. IV, the modifications required to treat the multiband case. From this point on, we will therefore assume the single-band case unless otherwise stated. In the setup under study, the quantities of interest for a transmitter to control its power are given by the channel gains. The channel gain of the link between Transmitter i P t1, ..., Ku and Receiver j P t1, ..., Ku is denoted by g ij " |h ij | 2 , where h ij may typically be the realization of a complex Gaussian random variable, if Rayleigh fading is considered. In several places in this paper we will use the K ˆK channel matrix G whose entries are given by the channel gains g ij , i and j respectively representing the row and column indices of G. Each channel gain is assumed to obey a classical block-fading variation law. More precisely, channel gains are assumed to be constant over each transmitted data frame. A frame comprises T I `TII `TIII consecutive time-slots where T m P N, m P tI, II, IIIu, corresponds to the number of time-slots of Phase m of the proposed procedure; these phases are described further. Transmitter i, i P t1, ..., Ku, can update its power from time-slot to time-slot. The corresponding power level is denoted by p i and is assumed to be subject to power limitation as: 0 ď p i ď P max . The K´dimensional column vector formed by the transmit power levels will be denoted by p " pp 1 , ..., p K q T , T standing for the transpose operator. Feedback signal model: We assume the existence of a feedback mechanism which provides each transmitter, an image or noisy version of the power received at its intended receiver for each time-slot. The power at Receiver i on time-slot t is expressed as ω i ptq " g ii p i ptq `σ2 `ÿ j‰i g ji p j ptq. ( 1 ) where σ 2 is the receive noise variance and p i ptq the power of Transmitter i on time-slot t. We assume that the following procedure is followed by the transmitter-receiver pair. Receiver i: measures the received signal (RS) power ω i ptq at each time slot and quantizes it with N bits (the RS power quantizer is denoted by Q RS ); sends the quantized RS power p ω i ptq as feedback to Transmitter i through a noisy feedback channel. After quantization, we assume that for all i P t1, ..., Ku, p ω i ptq P W , where W " tw 1 , w 2 , . . . , w M u such that 0 ď w 1 ă w 2 ă ¨¨¨ă w M and M " 2 N . Transmission over the feedback channel and the dequantization operation are represented by a discrete memoryless channel (DMC) whose conditional probability is denoted by Γ. The distorted and noisy version2 of ω i ptq, which is available at Transmitter i, is denoted by r ω i ptq P W ; the quantity r ω i ptq will be referred to as the received signal strength indicator (RSSI). With these notations, the probability that Transmitter i decodes the symbol w given that Receiver i sent the quantized RS power w k equals Γpw |w k q. In contrast with the vast majority of works on power control and especially those related to the IWFA, we assume the feedback channel to be noisy. Note also that these papers typically assume SINR feedback whereas the RSSI is considered here. The reasons for this is fourfold: 1) if Transmitter i knows p i ptq, g ii ptq, and has SINR feedback, this amounts to knowing its RS power since ω i ptq " g ii p i ptq ˆ1 `1 SINR i ptq ˙where SINR i ptq " g ii p i ptq σ 2 `ÿ j‰i g ji p j ptq ; 2) Assuming an RS power feedback is very relevant in practice since some existing wireless systems exploit the RSSI feedback signal (see e.g., [START_REF] Sesia | The UMTS Long Term Evolution: From Theory to Practice[END_REF]); 3) The SINR is subject to higher fluctuations than the RS power, which makes SINR feedback less robust to distortion and noise effects and overall less reliable; 4) As a crucial technical point, it can be checked that using the SINR as the transmitter observation leads to complex estimators [START_REF] Lasaulce | Technique de coordination d'émetteurs radio fondé sur le codage des niveaux de puissance d'émission[END_REF], while the case of RS power observations leads to a simple and very efficient estimation procedure, as shown further in this paper. Note that, here, it is assumed that the RS power is quantized and then transmitted through a DMC, which is a reasonable and common model for wireless communications. Another possible model for the feedback might consist in assuming that the receiver sends directly received signal power over an AWGN channel; depending on how the feedback channel gain fluctuations may be accounted for, the latter model might be more relevant and would deserve to be explored as well (‹). Proposed technique general description: The general power control problem of interest consists in finding, for each realization of the channel gain matrix G, a power vector which maximizes a network utility of the form upp; Gq. For this purpose, each transmitter is assumed to have access to the realizations of its RSSI over a frame. One of the key ideas of this paper is to exploit the transmit power levels as information symbols and exploit the observed interference (which is observed through the RSSI or SINR feedback) for inter-transmitter communication. The corresponding implicit communication channel is exploited to acquire global CSI knowledge namely, the matrix G and therefore to perform operations such as the maximization of upp; Gq. The process of achieving the desired power control vector is divided into three phases (see Fig. 1). In Phase I, a sequence of power For every time-slot of Phase I, each transmitter transmits at a prescribed power level which is assumed to be known to all the transmitters. One of the key observations we make in this paper is that, when the channel gains are constant over several time-slots, it is possible to recover local CSI from the RSSI or SINR; this means that, as far as power control is concerned, there is no need for additional signaling from the receiver for local CSI acquisition by the transmitter. Thus, the sequences of power levels in Phase I can be seen as training sequences. Technically, a difference between classical training-based estimation and Phase I is that estimation is performed in the power domain and over several time-slots and not in the symbol domain (symbol duration is typically much smaller than the duration of a time-slot) within a single time-slot. Also note that working in the symbol domain would allow one to have access to h ij but the phase information on the channel coefficients is irrelevant for the purpose of maximizing a utility function of the form upp; Gq. Another technical difference stems from the fact that the feedback noise is not standard, which is commented more a little further. By denoting pp i p1q, ..., p i pT I qq, i P t1, ..., Ku, the sequence of training power levels used by Transmitter i, the following training matrix can be defined: P I " ¨p1 p1q . . . p K p1q . . . . . . . . . p 1 pT I q . . . p K pT I q ‹ ‹ ‹ ' . (2) With the above notations, the noiseless RS power vector ω i " pω i p1q, ..., ω i pT I qq T can be expressed as: ω i " P I g i `σ2 1. (3) where g i " pg 1i , .., g Ki q T and 1 " p1, 1, ..., 1q T . To estimate the local CSI g i from the sole knowledge of the noisy RS power vector or RSSI r ω i we propose to use the least-squares (LS) estimator in the power domain (PD), abbreviated as LSPD, to estimate the local CSI as: r g LSPD i " `PT I P I ˘´1 P T I `r ω i ´σ2 1 ˘. ( 4 ) where σ 2 is assumed to be known from the transmitters since it can always be estimated through conventional estimation procedures (see e.g., [START_REF] Lasaulce | Training-based channel estimation and de-noising for the UMTS TDD mode[END_REF]). Using the LSPD estimate for local CSI therefore assumes that the training matrix P I is chosen to be pseudo-invertible. A necessary condition for this is that the number of time-slots used for Phase I verifies: T I ě K. Using a diagonal training matrix allows this condition to be met and to simplify the estimation procedure. It is known that the LSPD estimate may coincide with the maximum likelihood (ML) estimate. This holds for instance when the observation model of the form r ω i " ω i `z where z is an independent and additive white Gaussian noise. In the setup under investigation, z represents both the effects of quantization and transmission errors over the feedback channels and does not meet neither the independence nor the Gaussian assumption. However, we have identified a simple and sufficient condition under which the LSPD estimate maximizes the likelihood P pr ω i |g i q. This is the purpose of the next proposition. Proposition III.1. Denote by G ML i the set of ML estimates of g i , then we have piq G ML i " arg max g i T I ź t"1 Γ ´r ω i ptq ˇˇQ RS ´eT t P I g i `σ2 ¯¯; piiq r g LSPD i P G ML i when for all , arg max k Γpw |w k q " ; where e t is a column vector whose entries are zeros except for the t th entry which equals 1. Proof. See Appendix A. The sufficient condition corresponding to piiq is clearly met in classical practical scenarios. Indeed, as soon as the probability of correctly decoding the sent quantized RS power symbol (which is sent by the receiver) at the transmitter exceeds 50%, the above condition is verified. It has to be noted that G ML i is not a singleton set in general, which indicates that even if the LSPD estimate maximizes the likelihood, the set G ML i will typically comprise a solution which can perform better e.g., in terms of mean square error. If some statistical knowledge on the channel gains is available, it is possible to further improve the performance of the channel estimate. Indeed, when the probability of g i is known it becomes possible (up to possible complexity limitations) to minimize the mean square error E}p g i ´gi } 2 . The following proposition provides the expression of the minimum mean square error (MMSE) estimate in the power domain (PD) . Proposition III.2. Assume that @i P t1, ..., Ku, p ω i and r ω i belong to the set Ω " tw 1 , ..., w M T I u, where w 1 " pw 1 , w 1 , ..., w 1 q T , w 2 " pw 1 , w 1 , ..., w 2 q T ,..., w M T I " pw M , w M , ..., w M q T (namely, vectors are ordered according to the lexicographic order and have T I elements each). Define G m as G m :" x P R K `: Q RS `PI x `σ2 1 ˘" w m ( . (5) Then the MMSE estimator in the power domain expresses as: r g MMSEPD i " M T I ÿ m"1 T I ś t"1 Γ pr ω i ptq|w m ptqq ż Gm φ i ´gi ¯gi dg 1i ...dg Ki M T I ÿ m"1 T I ś t"1 Γ pr ω i ptq|w m ptqq ż Gm φ i ´gi ¯dg 1i ...dg Ki , (6) where φ i represents the probability density function (p.d.f.) of g i and w m ptq is the t-th element of w m . Proof. See Appendix B. In the simulation section (Sec. V), we will compare the LSPD and MMSEPD performance in terms of estimation SNR, sum-rate, and sum-energy-efficiency. While the MMSEPD estimate may provide a quite significant gain in terms of MSE over the LSPD estimate, it also has a much higher computational cost. Simulations reported in Sec. V will exhibit conditions under which choosing the LSPD solution may involve a marginal loss w.r.t. the MMSEPD solution e.g., when the performance is measured in terms of sum-rate. Therefore the choice of the estimator can be made based on the computation capability, the choice of utility for the system under consideration, or the required number of time-slots (MMSEPD allows for a number of timeslots which is less than K, whereas this is not possible for LSPD). Note that some refinements might be brought to the proposed estimator e.g., by using a low-rank approximation of the channel vector (see e.g., [START_REF] Lasaulce | Training-based channel estimation and de-noising for the UMTS-TDD mode[END_REF]), which is particularly relevant if the channel appears to possess some sparseness. IV. PHASE II: LOCAL CSI EXCHANGE IN THE POWER DOMAIN Phase II comprises T II time-slots. The aim of Phase II is to allow Transmitter i, i P t1, ..., Ku, to exchange its knowledge about local CSI with the other transmitters; the corresponding estimate will be merely denoted by r g i " pr g 1i , ..., r g Ki q T , knowing that it can refer either to the LSPD or MMSEPD estimate. The proposed procedure is as follows and is also summarized in Fig. 3. Transmitter i quantizes the information r g i through a channel gain quantizer called Q II i and maps the obtained bits (through a modulator) into the sequence of power levels p II i " pp i pT I 1q, ..., p i pT I `TII qq T . From the RSSI observations r ω II j " pr ω j pT I `1q, ..., r ω j pT I `TII qq T , Transmitter j (j ‰ i) can estimate (through a decoder) the power levels used by Transmitter i. To facilitate g i Phase I Ý ÝÝÝÝ Ñ r g i Quantizer Ý ÝÝÝÝÝÝ Ñ Q II i pr g i q Modulator Ý ÝÝÝÝÝÝ Ñ p II i Ó Eq.p1q r g j i Dequantizer& Demodulator Ð ÝÝÝÝÝÝÝ Ý r p II i Decoder 7j, j‰i Ð ÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝ Ý r ω II j Fig. 2: The figure summarizes the overall processing chain for the CSI the corresponding operations, we assume that the used power levels in Phase II have to lie in the reduced set P " tP 1 , ..., P L u with @ P t1, ..., Lu, P P r0, P max s. The estimate Transmitter j has about the channel vector g i will be denoted by r g j i " `r g j 1i , ..., r g j Ki ˘T. The corresponding channel matrix estimate is denoted by r G j . In what follows, we describe the proposed schemes for the three operations required to exchange local CSI namely, quantization, power modulation, and decoding. The situation where transmitters have different estimates of the same channel is referred to as a distributed CSI scenario in [START_REF] Kerret | Degrees of freedom of the network MIMO channel with distributed CSI[END_REF]. Assessing analytically the impact of distributed CSI on the sum-rate or sum-energy-efficiency is beyond the scope of this paper but constitutes a very relevant extension of it (‹); only simulations accounting for the distributed CSI effect will be provided here. It might be noticed that the communication scenario in Phase II is similar to the X-channel scenario in the sense that each transmitter wants to inform the other transmitters (which play the role of receivers) about its local CSI, and this is done simultaneously. All the available results on the X-channel exploit the channel structure (e.g., the phase information) to improve performance (e.g., by interference alignment [START_REF] Maddah-Ali | Communication Over MIMO X Channels: Interference Alignment, Decomposition, and Performance Analysis[END_REF] or filter design). Therefore, knowing how to exploit the X-channel scenario in the setup under consideration (which is in part characterized by the power domain operation) in this paper, appears to a relevant extension (‹). Channel gain quantization operation Q II i : The first step in Phase II is for each of the transmitters to quantize the K´dimensional vector r g i . For simplicity, we assume that each element of the real K´dimensional vector r g i is quantized by a scalar quantizer into a label of N II bits. This assumption is motivated by low complexity but also by the fact that the components of r g i are independent in the most relevant scenarios of interest. For instance, if local CSI is very well estimated, the estimated channel gains are close to the actual channel gains, which are typically independent in practice. Now, in the general case of arbitrary estimation noise level, the components of r g i will be independent when the training matrix P I is chosen to be diagonal, which is a case of high interest and is motivated further in Sec. V. Under the channel gain (quasi-) independency, vector quantization would bring (almost) no performance improvement. The scalar quantizer used by Transmitter i to quantize r g ji is denoted by Q II ji . Finding the best quantizer in terms of ultimate network utility (e.g., in terms of sum-rate or sum-energy-efficiency) does not appear to be straightforward (‹). We present two possible quantization schemes in this section. A possible, but generally sub-optimal approach, is to determine a quantizer which minimizes distortion. The advantage of such approach is that it is possible to express the quantizer and it leads to a scheme which is independent of the network utility; this may be an advantage when the utility is unknown or changing. A possible choice for the quantizer Q II i is to use the conventional version of the Lloyd-Max algorithm (LMA) [START_REF] Lloyd | Least squares quantization in PCM[END_REF]. However, this algorithm assumes perfect knowledge of the information source to be quantized (here this would amount to assuming the channel estimate to be noiseless) and no noise between the quantizer and the dequantizer (here this would amount to assuming perfect knowledge of the RS power). The authors of [START_REF] Djeumou | Practical quantize-and-forward schemes for the frequency division relay channel[END_REF] proposed a generalized version of the Lloyd-Max algorithm for which noise can be present both at the source and the transmission but the various noise sources are assumed to verify standard assumptions (such as independence of the noise and the source), which are not verified in the setting under investigation; in particular, the noise in Phase I is the estimation noise, which is correlated with the transmitted signal. Deriving the corresponding generalized Lloyd-Max algorithm can be checked to be a challenging task, which is left as an extension of the technical solutions proposed here (‹). Rather, we will provide here a special case of the generalized Lloyd-Max algorithm, which is very practical in terms of computational complexity and required knowledge. The version of the Lloyd-Max algorithm we propose will be referred to as ALMA (advanced Lloyd-Max algorithm). ALMA corresponds to the special case (of the most generalized version mentioned previously) in which the algorithm assumes noise on the transmission but not at the source (although the source can be effectively noisy). This setting is very well suited to scenarios where the estimation noise due to Phase I is negligible or when local CSI can be acquired reliably by some other mechanism. In the numerical part, we can observe the improvements of the proposed ALMA with respect to the conventional LMA. Just like the conventional LMA, ALMA aims at minimizing distortion by iteratively determining the best set of representatives and the best set of cells (which are intervals here) when one of the two is fixed. The calculations for obtaining the optimal representatives and partitions are given in Appendix C for both the special case of no source noise as well as for the general case. Solving the general case can be seen from Appendix C to be computationally challenging. To comment on the proposed algorithm which is given by the pseudo-code of Algorithm 1, a few notations are in order. We denote by q P t1, ..., Qu the iteration index (where Q is the upper bound on the number of iterations) and define R " 2 This minimization operation requires some statistical knowledge. Indeed, the probability that the dequantizer decodes the representative v pqq ji,r given that v pqq ji,n has been transmitted needs to be known; this probability is denoted by π ji pr|nq and constitutes one of the inputs of Algorithm 1. The second input of Algorithm 1 is the p.d.f. of g ji which is denoted by φ ji . The third input is given by the initial choice for the quantization intervals that is, the set ! u p0q ji,1 , ..., u p0q ji,R`1 ) . Convergence of ALMA to a global minimum point is not guaranteed and finding sufficient condition for global convergence is known to be non-trivial. However, local convergence is guaranteed; an elegant and general argument for this can be found in [START_REF] Beaude | Crawford-Sobel meet Lloyd-Max on the grid[END_REF]. Conducting a theoretical analysis in which global convergence is tackled would constitute a significant development of the present analysis (‹), which is here based on typical and realistic simulation scenarios. At this point two comments are in order. First, through ( 7)-( 8), it is seen that ALMA relies on some statistical knowledge which might not always be available in practice. This is especially the case for π ji and γ ji since the knowledge of channel distribution information (CDI, i.e., φ ji ) is typically easier to be obtained. The CDI may be obtained by storing the estimates obtained during past transmissions and forming empirical means (possibly with a sliding window). If the CDI is time-varying, a procedure indicating to the terminals when to update the statistics might be required. Second, if we regard Phase II as a classical communication process, then the amount of information sent by the source is maximized when the source signal is uniformly Inputs: π ji , φ ji pg ji q, ! u p0q ji,1 , ..., u p0q ji,R`1 ) Outputs: u ‹ ji,1 , ..., u ‹ ji,R`1 ( , v ‹ ji,1 , ..., v ‹ ji,R`1 ( Initialization: Set q " 0. Initialize the quantization intervals according to ! u p0q ji,1 , ..., u p0q ji,R`1 ) . Set u p´1q ji,r " 0 for all r P t1, ..., Ru. while max r ||u pqq ji,r ´upq´1q ji,r || ą δ and q ă Q do Update the iteration index: q Ð q `1. For all r P t1, 2, .., Ru set v pqq ji,r Ð R ÿ n"1 π ji pr|nq ż u pq´1q ji,n`1 u pq´1q ji,n g ji φ ji pg ji qdg ji R ÿ n"1 π ji pr|nq ż u pq´1q ji,n`1 u pq´1q ji,n φ ji pg ji qdg ji . ( 7 ) For all r P t2, 3, .., Ru set u pqq ji,r Ð R ÿ n"1 rπ ji pn|rq ´πji pn|r ´1qs ´vpqq ji,n ¯2 2 R ÿ n"1 rπ ji pn|rq ´πji pn|r ´1qs v pqq ji,n . (8) end @r P t2, ..., Ru, u ‹ ji,r " u pqq ji,r , u ‹ ji,1 " 0 and u ‹ ji,R`1 " 8 @r P t1, ..., Ru, v ‹ ji,r " v pqq ji,r Algorithm 1: Advanced Lloyd-Max algorithm (ALMA) distributed. It turns out minimizing the (end-to-end) distortion over Phase II does not involve this. Motivated by these two observations we provide here a second quantization scheme, which is simple but will be seen to perform quite well in the numerical part. We will refer to this quantization scheme as maximum entropy quantizer (MEQ). For MEQ, the quantization interval bounds are fixed once and for all according to: @r P t1, ..., Ru, @pj, iq P t1, ..., Ku 2 , ż u ji,r`1 u ji,r φ ji pg ji qdg ji " 1 R . ( 9 ) The representative of the interval ru ji,r , u ji,r`1 s is denoted by v ji,r and is chosen to be its centroid: v ji,r " ż u ji,r`1 u ji,r g ji φ ji pg ji qdg ji ż u ji,r`1 u ji,r φ ji pg ji qdg ji . ( 10 ) We see that each representative has the same probability to occur, which maximizes the entropy of the quantizer output, hence the proposed name. To implement MEQ, only the knowledge of φ ji is required. Additionally, the complexity involved is very low. Power modulation: To inform the other transmitters about its knowledge of local CSI, Transmitter i maps the K labels of N II bits produced by the quantizer Q II i to a sequence of power levels pp i pT I `1q, p i pT I `2q, . . . , p i pT I `TII qq. Any one-to-one mapping might be used a priori. Although the new problem of finding the best mapping for a given network utility arises here and constitutes a relevant direction to explore (‹), we will not only develop this here. Rather, our main objective here is to introduce this problem and illustrate it clearly for a special case which is treated in the numerical part. To this end, assume Phase II comprises T II " 2 time-slots, K " 2 users, and that the users only exploit L " 2 power levels during Phase II say P " tP min , P max u. Further assume 1´bit quantizers, which means that the quantizers Q II ji produce binary labels. For simplicity, we assume the same quantizer Q is used for all the four channel gains g 11 , g 12 , g 21 , and g 22 : if g ij P r0, µs then the quantizer output is denoted by g min ; if g ij P pµ, `8q then the quantizer output is denoted by g max . Therefore a simple mapping scheme for Transmitter 1 (whose objective is to inform Transmitter 2 about pg 11 , g 21 q) is to choose p 1 pT I `1q " P min if Qpg 11 q " g min and p 1 pT I `1q " P max otherwise; and p 1 pT I `2q " P min if Qpg 21 q " g min and p 1 pT I `2q " P max otherwise. Therefore, depending on the p.d.f. of g ij , the value of µ, the performance criterion under consideration, a proper mapping can chosen. For example, to minimize the energy consumed at the transmitter, using the minimum transmit power level P min as much as possible is preferable; thus if PrpQpg 11 q " g min q ě PrpQpg 11 q " g max q, the power level P min will be associated with the minimum quantized channel gain that is Qpg 11 q " g min . Power level decoding: For every time-slot t P tT I `1, ..., T I `TII u the power levels are estimated by Transmitter i as follows r p ´iptq P arg min p ´iPP K´1 ˇˇˇˇÿ j‰i p j r g ji ´pr ω i ptq ´pi ptqr g ii ´σ2 q ˇˇˇˇ, (11) where p ´i " pp 1 , .., p i´1 , p i`1 , .., p K q. As for every j, r g ji is known at Transmitter i, the above minimization operation can be performed. It is seen that exhaustive search can be performed as long as the number of tests, which is L K´1 , is reasonable. For this purpose, one possible approach is to impose the number of power levels which are exploited over Phase II to be small. In this respect, using binary power over Phase II is not only relevant regarding complexity issues but also in terms of robustness against the various possible sources of noise. As for the number of interfering users using the same channel (meaning operating on the same frequency band, at the same period of time, in the same geographical area), it will typically be small and does not exceed 3 or 4 in real wireless systems. More generally, this shows that the proposed technique can accommodate more than 4 users in total; For example, if we have 12 bands, having 48 " 12 ˆ4 users would be manageable by applying the proposed technique for each band. As our numerical results indicate, using [START_REF] Larrousse | Coordination in distributed networks via coded actions with application to power control[END_REF] as a decoding rule to find the power levels of the other transmitters generally works very well for K " 2. When the number of users is higher, each transmitter needs to estimate K ´1 power levels with only one observation equation, which typically induces a non-negligible degradation in terms of symbol error rate. In this situation, Phase II can be performed by scheduling the activity of all the users, such that only 2 users are active at any given time-slot in Phase II. Once all pairs of users have exchanged information on their channel states, Phase II is concluded. Remark 1. Note that the case where only one user is active at a time is a special case of the decoding scheme assumed here. The advantage of our more general decoding scheme is that it can be used when strict SINR feedback is used [START_REF] Varma | Power modulation: Application to inter-cell interference coordination[END_REF] instead of RSSI; indeed when only one user is active at a time, the SINR becomes an instantaneous SNR and cannot convey any coordination information. Concerning the setting with RSSI feedback, the drawback of our assumption is that in the presence of noise on the RS power feedback, the performance of Phase II may be limited when the cross channel gains are very small. If this turned out to be a crucial problem, allowing only one user to be active at a time is preferable. Remark 2 (required number of time-slots). The proposed technique typically requires K `K " 2K time-slots for the whole exploration phase (Phases I and II). It therefore roughly require the same amount of resources as IWFA, which indeed needs about 2K or 3K SINR samples to converge to Nash equilibrium. While channel acquisition may seem to take some time, please note that regular communication is uninterrupted and occurs in parallel. As already mentioned, the context in which the proposed technique and IWFA are the most suited is a context where the channel is constant over a large number of time-slots, which means that the influence of the exploration phase on the average performance is typically negligible. Nonetheless, some simulations will be provided to assess the optimality loss induced by using power levels to convey information. Remark 3 (extension to the multi-band scenario). As explained in the beginning of this paper, Phases I and II are described for the single-band case, mainly for clarity reasons. Here, we briefly explain how to adapt the algorithm when there are multiple bands. In Phase I, the only difference exists in choosing the training matrix. With say S bands to transmit, for each band s P t1, ..., Su, the training matrix P s I has to fulfill the constraint S ÿ s"1 p s i ptq ď P max where p s i is the power Transmitter i allocates to band s. In Phase II, each band performs in parallel like the single-band case. Since there are power constraints for each transmitter, the modulated power should satisfy S ÿ s"1 p s i ptq ď P max . Remark 4 (extension to the multi-antenna case). To perform operation such as beam-forming, the phase information is generally required. The proposed local CSI estimation techniques (namely, for Phase I) do not allow the phase information or the direction information to be recovered; Therefore, another type of feedback should be considered for this. However, if another estimation scheme is available or used for local CSI acquisition and that scheme provides the information phase, then the techniques proposed for local CSI exchange (namely, for Phase II) can be extended. An extension which is more in line with the spirit of the manuscript is given by a MIMO interference channel for which each transmitter knows the interference-plus-noise covariance matrix and its own channel. This is the setup assumed by Scutari et al in their work on MIMO iterative water-filling [START_REF] Scutari | The MIMO iterative waterfilling algorithm[END_REF]. Remark 5 (type of information exchanged). One of the strengths of the proposed exchange procedure is that any kind of information can be exchanged. However, since SINR or RSSI is used as the communication channel, this has to be at a low-rate which is given by the frequency at which the power control levels are updated and the feedback samples sent. V. NUMERICAL ANALYSIS In this section, as a first step (Sec. V-A), we start with providing simulations which result from the combined effects of Phases I and II. To make a coherent comparison with IWFA, the network utility will be evaluated without taking into account a cost possibly associated with the exploration or training phases (i.e., Phases I and II for the proposed scheme or the convergence time for IWFA). The results are provided for a reasonable scenario of small cell networks which is similar to those already studied in other works (see e.g., [START_REF] Samarakoon | Ultra dense small cell networks: Turning density into energy efficiency[END_REF] for a recent work). As a second step (Sec. V-B and V-C), we study special cases to better understand the influence of each estimation phase and the different parameters which impact the system performance. As shown in Fig. 3, the considered scenario assumes K " 9 small cell base stations with maximal transmit power P max " 30 dBm. One or two bands are assumed, depending on the scenario considered. One user per cell is assumed, which corresponds to a possible scenario in practice (see e.g., [START_REF] Samarakoon | Ultra dense small cell networks: Turning density into energy efficiency[END_REF] [25] [START_REF] Moustakas | Power optimization in random wireless networks[END_REF]). We also use this setup to be able to compare the proposed scheme with IWFA whose performance is generally assessed for the most conventional form of the interference channel, namely, K transmitter-receiver pairs. The normalized receive noise power is σ 2 " 0 dBm. This corresponds to SNRpdBq " 30 where the signal-to-noise ratio is defined by A. Global performance analysis: a simple small cell network scenario SNRpdBq " 10 log 10 ˆPmax σ 2 ˙. ( 12 ) Here and in all the simulation section, we set the SNR to 30 dB by default. RS power measurements are quantized uniformly in a dB scale with N " 8 bits and the quantizer input dynamics or range in dB is rSNRpdBq ´20, SNRpdBq `10s. The DMC Γ is constructed with error probability to the two nearest neighbors, i.e., for the symbols w 1 ă w 2 ă ¨¨¨ă w M (with M " 2 N ), Γ pw i |w j q " if |i ´j| " 1 and Γ pw i |w j q " 0 if |i ´j| ą 1. In this section " 1%; the quantity will be referred to as the feedback channel symbol error rate (FCSER). For all pi, jq and s (s always being the band index) the channel gain g s ij on band s is assumed to be exponentially distributed namely, its p.d.f. writes as φ s ij pg s ij q " , ISD being the inter site distance. this section, the system performance is assessed in terms of sum-rate, the sum-rate being given by: u sum-rate pp 1 , ..., p K ; Gq " K ÿ i"1 S ÿ s"1 logp1 `SINR s i pp 1 , ..., p K ; Gqq. ( 13 ) where p i " pp 1 i , ..., p S i q represents the power allocation vector of Transmitter i, SINR s i is the SINR at Receiver i in band s and expresses as SINR s i " g s ii p s i σ 2 `ÿ j‰i g s ji p s j . Fig. 4a, represents the average sum-rate against the ISD. The sum-rate is averaged over 10 4 realizations of the channel gain matrix G and the inter site distance is the distance between two neighboring small base stations. Three curves are represented. The top curve corresponds to the performance of the sequential best-response dynamics applied to the sum-rate (referred to as Team BRD) in the presence of perfect global CSI. The curve in the middle corresponds to Team BRD which uses the estimate obtained by using the most simple association proposed in the paper namely, LSPD for Phase I and the 2´bit MEQ for Phase II. The LSPD estimator uses K time-slots and the K´dimensional identity matrix P I " P max I K for the training matrix. The 2´bit MEQ uses binary power control (L " 2) and 2K time-slots to send the information, i.e., g i ); this corresponds to the typical number of time-slots IWFA needs to converge. At last, the bottom curve corresponds to IWFA using local CSI estimates provided by Phase I. It is seen that about 50% of the gap between IWFA and Team BRD with perfect CSI can be bridged by using the proposed estimation procedure. When the interference level is higher, the gap becomes larger. Fig. 4b depicts exactly the same scenario as Fig. 4a except that only one band is available to the small cells i.e., S " 1. Here the gap can be bridged at about 65% when using Team BRD with the proposed estimation procedure. In this section, some choices have been made: a diagonal training matrix and the LSPD estimator has been chosen for Phase I and the MEQ has been chosen for Phase II. The purpose of the next sections is to explain these choices, and to better identify the strengths and weaknesses of the proposed estimation procedures. Fig. 4: The above curves are obtained in the scenario of Fig. 4 in which K " 9 transmitter-receiver pairs, SNRpdBq " 30, the FCSER is given by " 0.01, N " 8 quantization bits for the RSSI, and L " 2 power levels. Using the most simple estimation schemes proposed in this paper namely LSPD and MEQ can bridge the gap between the IWFA and the team BRD with perfect CSI, about 50% when S " 2 and about 65% when S " 1. B. Comparison of estimation techniques for Phase I In Phase I, there are two main issues to be addressed: the choice of the estimator and the choice of the training matrix P I . To compare the LSPD and MMSEPD estimators, we first consider the estimation SNR (ESNR) as the performance criterion to compare them. The estimation SNR of Transmitter i is defined here for the case S " 1 and is given by: ESNR i " Er}G} 2 s Er}G ´r G i } 2 s . ( 14 ) where }.} 2 stands for the Frobenius norm and r G i is the global channel estimate which is available to Transmitter i after Phases I and II. In this section, we always assume a perfect exchange in Phase II to conduct the different comparisons. This choice is made to isolate the impact of Phase I estimation techniques on the estimation SNR and the utility functions which are considered for the exploitation phase. After extensive simulations, we have observed that the gain in terms of ESIR by using the best training matrix (computed by an exhaustive search over all the matrix elements) is found to be either negligible or quite small when compared to the best diagonal training matrix (computed by an exhaustive search over the diagonal elements); see e.g., Fig. 6a for such a simulation. Therefore, for the rest of this paper, we will restrict our attention to diagonal training matrices for reducing the computational complexity without any significant performance loss. To conclude about the choice of the training matrix, we assess the impact of using power levels to learn local CSI instead of using them to optimize the performance of Phase I. For this, we compare in Fig. 6b the scenario in which a diagonal training matrix is used to learn local CSI, with the scenario in which the best training matrix in the sense of the expected sum-rate (over Phase I). Global channel distribution information is assumed to be available in the latter scenario. The corresponding choice is feasible computationally speaking for small systems. Fig. 5a represents for K " 2, S " 1, and SNRpdBq " 30, the estimation SNR (in dB) against the signal-to-interference ratio (SIR) in dB SIRpdBq which is defined here as SIRpdBq " 10 log 10 ˆEpg 11 q Epg 21 q ˙" 10 log 10 ˆEpg 22 q Epg 12 q ˙. (15) The three curves in red solid lines represent the MMSEPD estimator performance while the three curves in blue dashed line represent the LSPD estimator performance. The performance gap between MMSEPD and LSPD depends on the quality of the RSSI at the transmitters. When RS power measurements are quantized with N " 8 bits and the feedback channel symbol error rate is " 1%, the gap in dB is very close to 0. Using MMSEPD instead of LSPD becomes much more relevant in terms of ESNR when the quality of feedback is degraded. Indeed, for N " 2 bits and " 10%, the gap is about 5 dB. Note that having a very small number of RSSI quantization bits and therefore significant feedback quality degradation may also occur in classical wireless systems where the feedback would be binary such as an ACK/NACK feedback. Indeed, an ACK/NACK feedback can be seen as the result of a 1´bit quantization of the RSSI or SINR. The proposed technique might be used to coordinate the transmitters just based on this particular and rough feedback. Even though the noise on the RSSI is correlated with the signal and is not Gaussian, we observe that MMSEPD and LSPD (which can be seen as a zero-forcing solution) perform similarly when the noise becomes negligible. At last note that the ESNR is seen to be independent of the SIR; this can be explained by the used training matrix, which is diagonal. The above comparison is conducted in terms of ESNR but not in terms of final utility. To assess the impact of Phase I on the exploration phase, two common utility functions are considered namely, the sum-rate and the sum-energy-efficiency (sum-EE) which is defined as: u sum-EE pp 1 , ..., p K ; Gq " K ÿ i"1 S ÿ s"1 f pSINR s i pp 1 , ..., p K ; Gqq S ÿ s"1 p s i . ( 16 ) where the same notations as in [START_REF] Anandkumar | Robust rate maximization game under bounded channel uncertainty[END_REF] are used; f is an efficiency function which represents the packet success rate or the probability of having no outage. Indeed, the utility function u sum-EE corresponds to the ratio of the packet success rate to the consumed transmit power and has been used in many papers (see e.g., [27] [28] [29] [30] [31]). Here we choose the efficiency function of [START_REF] Belmega | Energy-efficient precoding for multiple-antenna terminals[END_REF]: f pxq " exp ´´c x ¯with c " 2 r ´1 " 1, r being the spectral efficiency. Fig. 5b depicts for K " 2, S " 1, N " 2, " 10% the average relative utility loss ∆u in % against the SIR in dB. The average relative utility loss in % is defined by ∆up%q " 100E « upp ‹ 1 , ..., p ‹ K ; Gq ´upr p ‹ 1 , ..., r p ‹ K ; Gq upp ‹ 1 , ..., p ‹ K ; Gq ff . (17) where upp ‹ 1 , ..., p ‹ K ; Gq is the best sum-utility which can be attained when every realization of G is known perfectly. The latter is obtained by performing exhaustive search over 100 values equally spaced in r0, P max s and this for each draw of G; the average is obtained from 10 4 independent draws of G. The utility upr p ‹ 1 , ..., r p ‹ K ; Gq is also obtained with exhaustive search but by using either the LSPD or MMSEPD estimator and assuming Phase II to be perfect. Fig. 5b shows that even under severe conditions in terms observing the RS power at the transmitter, the MMSEPD and LSPD estimators have the same performance in terms of sum-rate. This holds even though the gap in terms of ESNR is 5 dB (see Fig. 5a). Note that the relative utility loss is about 3% showing that the sum-rate performance criterion is very robust against channel estimation errors. When one considers the sum-EE, the relative utility loss becomes higher and is the range 15% ´20% and the gap between MMSEPD and LSPD becomes more apparent this time and equals about 5%. The observations made for the special setting considered here have been checked to be quite general and apply for more users, more bands, and other propagation scenarios: unless the RSSI is very noisy or when only an ACK/NACK-type feedback is available, the MMSEPD and LSPD estimators perform quite similarly. Since the MMSEPD estimator requires more knowledge and more computational complexity to be implemented, the LSPD estimator seems to be the best choice when the quality of RSSI is good as it is in current cellular and Wifi systems. To conclude this section, we provide the counterpart of Fig. 6b for phase 2 in Fig. 7. The scenario in which a diagonal training matrix is used to exchange local CSI, with the scenario in which power control is to maximize the expected sum-rate (over Phase II). But here, the expectation is not taken over local CSI since it is assumed to be known. The corresponding choice is feasible computationally speaking for small systems. In this section, we assume Phase I to be perfect. Again, this choice is made to isolate the impact of Phase II estimation techniques on the estimation SNR and the utility functions which are considered for the exploitation phase. When L " 2 and we quantize with 1-bit, we map the smallest representative of the quantizer to the lowest power and the largest to the highest power level in P and the other element. If L ą 2, the power levels belong to the set " 0, 1 C. Comparison of quantization techniques for Phase II L ´1 P max , 2 L ´1P max , ..., P max * are picked and the representatives are mapped in the order corresponding to their value. In Phase II, the most relevant techniques to be determined is the quantization of the channel gains estimated through Phase I. For K " 2 users, S " 1 band, L " 2 power levels, and SNRpdBq " 30, Fig. 8a provides ESNR(dB) versus SIR(dB) for the three channel gain quantizers mentioned in this paper: ALMA, LMA, and MEQ. The three quantizers are assumed to quantize the channel gains with only 1 bit. Since only two power levels are exploited over Phase II, this means that the local CSI exchange phase (Phase II) comprises K time-slots. The three top curves of Fig. 8a correspond to N " 8 RS power quantization bits and " 1% while the three bottom curves correspond to N " 2 bits and " 10%. First of all, it is seen that the obtained values for ESNR are much lower than for Phase I. Even in the case where N " 8 and " 1%, the ESNR is around 10 dB whereas it was about 40 dB for Phase I. This shows that the limiting factor for the global estimation accuracy will come from Phase II; additional comments on this point are provided at the end of this section. Secondly, Fig. 8a shows the advantages offered by the proposed ALMA over the conventional LMA. Fig. 8b depicts for K " 2, S " 1, N " 8, " 1% the average relative utility loss ∆u in % against the SIR in dB for ALMA and MEQ. The two bottom (resp. top) curves correspond to the sum-rate (resp. sum-EE). The relative utility loss is seen to be comparable to the one obtained for Phase I. Interestingly, MEQ is seen to induce less performance losses than ALMA, showing that the ENSR or distortion does not perfectly reflect the need in terms optimality for the exploration phase. This observation partly explains why we have chosen MEQ in Sec. V-A for the global performance evaluation; many other simulations (which involve various values for K, N , S, , etc) not provided here confirm this observation. An important comment made previously is that Phase II constitutes the bottleneck in terms of estimation accuracy for the final global CSI estimate available for the exploitation phase. Here, we provide more details about this limitation. Indeed, even when the quality of the RSSI is good, the ESNR only reaches 10 dB and even increasing the quantization bits by increasing the Fig. 9: The power level decoding scheme proposed in this paper is simple and has the advantage of being usable for the SINR feedback instead of RSSI feedback. However, the proposed scheme exhibits a limitation in terms of coordination ability when the inference is very low. The consequence of this is the existence of a maximum ESNR for Phase II. Here we observe that despite increasing the number of quantization bits or time slots used, the ESNR is bounded. power modulation levels or time slots used does not improve the ESNR as demonstrated by the following figures. For N " 8 RS power quantization bits and " 1%, SNRpdBq " 30, Fig. 9a shows the ESNR versus the number of channel quantization bits used by MEQ. It is seen that the ESNR reaches a maximum whether a high interference scenario (SIRpdBq " 0) or a low interference scenario (SIRpdBq " 10) is considered. In Fig. 8a, the ESNR was about 9 dB when the 1´bit MEQ is used and the SIR equals 0 dB. Here we retrieve this value and see that the ESNR can reach 13 dB when the 4´bit MEQ is implemented, meaning that 16 power levels are used in Phase II. Now, when the SIR is higher, using the 2´bit MEQ is almost optimal. If the RSSI quality degrades, then using only 1 or 2 bits for MEQ is always the best configuration. Another approach would be to increase the number of channel gain quantization bits and still only use two power levels over Phase II by increasing the number of time-slots used in Phase II. Fig. 9b assumes exactly the same setup as Fig. 9a but here it represents the ESNR as a function of the number of time-slots used in Phase II. Here again, an optimal number of time-slots appears for the same reason as for Fig. 9a. Both for Fig. 9a and Fig. 9b, one might wonder why the ESNR is better when the interference is high. This is due to the fact that when the interference is very low, the decoding operation of the power levels of the others becomes less reliable. The existence of maximum points in Fig. 9a and Fig. 9b precisely translates the tradeoff between the channel gain quantization noise and power level decoding errors. VI. CONCLUSION First, we would like to remind a few comments about the scope and originality of this paper. One of the purposes of this paper is to show that the sole knowledge of the received power or SINR feedback is sufficient to recover global CSI. The proposed technique comprises two phases. Phase I allows each transmitter to estimate local CSI. Obviously, if there already exists a dedicated feedback or signalling channel which allows the transmitter to estimate local CSI, Phase I may be skipped. But even in the latter situation, the problem remains to know how to exchange local CSI among the transmitters. Phase II proposes a completely new solution for exchanging local CSI, namely using power modulation. Phase II is based in particular on a robust quantization scheme of the local channel gains. Phase II is therefore robust against perturbations on the received power measurements; it might even be used for 1´bit RSSI which would correspond to an ACK/NACK-type feedback, showing that even a rough feedback channel may help the transmitters to coordinate. Note that the proposed technique is general and can be used to exchange and kind of information and not only local CSI. Second, we summarize here a few observations of practical interest. For Phase I, two estimators have been proposed for Phase I: the LSPD and the MMSEPD estimators. Simulations show that using the MMSEPD requires some statistical knowledge and is more complex, but is well motivated when the RS power is quantized roughly or the feedback channel is very noisy. Otherwise, the use of the LSPD estimator is shown to be sufficient. During Phase II, transmitters exchange local CSI by encoding it onto their power level and using interference as a communication channel; Phase II typically requires K time-slots at least (assuming all transmitters simultaneously communicate in Phase II), which makes 2K time-slots for the whole estimation procedure. This is typically the number of time-slots needed by IWFA to converge, when it converges. For Phase II, three estimation schemes are provided which are in part based on one of the two quantizers ALMA and MEQ; the quantizers are computed offline but are exploited online. MEQ seems to offer a good trade-off between complexity and performance in terms of sum-rate or sum-energy-efficiency. In contrast with Phase I in which the estimation SNR typically reaches 40 dB for good RS power measurements, the estimation SNR in Phase II is typically around 10 dB, showing that Phase II will constitute the bottleneck in terms of estimation quality of global CSI. This is due to fact that the cross channel gains may be small when they fluctuate (this would not occur in the presence of Rician fading), which generates power level decoding errors. As explained, one way of improving the estimation SNR over Phase II is to activate only one user at a time, but then the proposed power level decoding scheme would only apply to RSSI feedback and not to SINR feedback anymore. In Phase III, having global CSI, each transmitter can apply the BRD to the sum-utility instead of applying it to an individual utility as IWFA does, resulting in a significant performance improvement as seen from our numerical results. ¯¯ [START_REF] Lasaulce | Training-based channel estimation and de-noising for the UMTS TDD mode[END_REF] where e t is a column vector whose entries are zeros except for the t th . In [START_REF] Lasaulce | Training-based channel estimation and de-noising for the UMTS TDD mode[END_REF], (a) holds as the estimation and feedback process g i to p ω i to r ω i (represented in Fig. 1) is Markovian, (b) holds because the DMC is separable and (c) holds because Pr ´p ω i |g i ¯is a discrete delta function that is zero everywhere except when Q RS ´PI g i ¯" p ω i . From [START_REF] Lasaulce | Training-based channel estimation and de-noising for the UMTS TDD mode[END_REF], the set of the ML estimators can now be written as G ML i " # arg max g i T 1 ź t"1 Γ ´r ω i ptq |Q RS ´eT t P I g i `σ2 ¯¯+ ( 19 ) which is the first claim of our proposition. Now, we look at the LS estimator, which is know from (4) to be P I g LSPD i `σ2 1 " r ω i (20) or equivalently: e T t P I g LSPD i `σ2 " r ω i ptq (21) If for all , arg max k Γpw |w k q " , then the ML set can be evaluated based on [START_REF] Maddah-Ali | Communication Over MIMO X Channels: Interference Alignment, Decomposition, and Performance Analysis[END_REF] as G ML i " ! g i |@t, Q RS ´eT t P I g i `σ2 ¯" r ω i ptq ) (22) Therefore, we observe that if G ML i is given as in [START_REF] Djeumou | Practical quantize-and-forward schemes for the frequency division relay channel[END_REF], then from (21), we have g LSPD i P G ML i , our second claim. Pr `r g k ji " v ji,r |r g ji " r x ˘" R ÿ n"1 Pr `r g k ji " v ji,r |Q II i pr g ji q " v ji,n ˘Pr `QII i pr g ji q " v ji,n |r g ji " r x " R ÿ n"1 πpr|nq Pr `QII i pr g ji q " v ji,n |r g ji " r x ˘ [START_REF] Haddad | Spectral efficiency of energy efficient multicarrier systems[END_REF] where we know Pr `QII i pr g ji q " v ji,n |r g ji " r x ˘" $ & % 1 if r x P ru ji,n , u ji,n`1 q 0 if r x R ru ji,n , u ji,n`1 q (32) Substituting ( 32) and ( 31) in [START_REF] Bacci | A game-theoretic approach for energy-efficient contention-based synchronization in OFDMA systems[END_REF], we get Er `gji ´r g k ji ˘2s " γ ji pr x|xq φ ji pxq px ´vji,r q 2 dxdr x. For fixed transition levels u ji,n , the optimum representatives v ji,r 1 are obtained by setting the partial derivatives of the distortion Er `gji ´r g k ji ˘2s, with respect to v ji,r 1 , to zero. That is BEr `gji ´r g k ji ˘2s Bv ji,r 1 " R ÿ n"1 π ji pr 1 |nq ż 8 x"0 ż u ji,n`1 r x"u ji,n 2γ ji pr x|xqφ ji pxq px ´vji,r 1 q dxdr x " 0 which results in with u ji,1 " 0 and u ji,R`1 " 8 as the boundary conditions. Solving the above conditions is very difficult as the variable to solve is inside the integral as an argument of γ. Therefore we consider the special case where γ ji pp x|xq " δpx ´p xq where δ is the Dirac delta function which is 0 at all points except at 0 and whose integral around a neighborhood of 0 is 1. This corresponds Fig. 1 : 1 Fig. 1: The flowchart of the proposed scheme rate (bps/Hz) Team BRD with perfect global CSI Team BRD with estimated global CSI (LSPD+MEQ) IWFA with estimated local CSI (LSPD) rate (bps/Hz) Team BRD with perfect global CSI Team BRD with estimated global CSI (LSPD+MEQ) IWFA with estimated local CSI (LSPD) (b) S " 1 Using MMSEPD instead of LSPD in Phase I becomes useful in terms of ESNR when the RSSI quality becomes too rough (bottom curves). LSPD perf. in terms of sum-EE MMSEPD perf. in terms of sum-EE LSPD perf. in terms of sum-rate MMSEPD perf. in terms of sum-rate (b) The figure provides the relative utility loss under quite severe conditions in terms of RSSI quality (N " 2, " 10%). Fig. 5 : 5 Fig. 5: Comparing MMSEPD and LSPD assuming perfect Phase II. MMSEPD with best training matrix MMSEPD with best diagonal matrix LSPD with best training matrix LSPD with best diagonal matrix (a) Scenario: K " 2, S " 1, and SNRpdBq " 30, " 0, N " 2 quantization bits. Using a diagonal training matrix typically induces a small performance loss in terms of ESNR even in worst-case scenarios. Optimality loss induced in Phase I when using power levels to learn local CSI instead of maximizing the expected sum-rate. This loss may be influential on the average performance when the number of time-slots of the exploitation phase is not large enough. Fig. 6 : 6 Fig. 6: Influence of the training matrix. Fig. 7 : 7 Fig.7: Optimality loss induced in Phase II when using power levels to exchange local CSI instead of maximizing the expected sum-rate. This loss may be influential on the average performance when the number of time-slots of the exploitation phase is not large enough. ALMA with N=8 and ε=1% LMA with N=8 and ε=1% MEQ with N=8 and ε=1% ALMA with N=2 and ε=10% MEQ with N=2 and ε=10% LMA with N=2 and ε=10% (a) Performance measured by ESNR considering good (three top curves) and bad (three bottom curves) RSSI quality conditions. ALMA perf. in terms of sum-rate MEQ perf. in terms of sum-rate ALMA perf. in terms of sum-EE MEQ perf. in terms of sum-EE (b) Performance measured by relative utility loss, with utility being the sum-EE or sum-rate. Fig. 8 : 8 Fig.8: Performance analysis of conventional LMA, ALMA and MEQ assuming Phase I to be perfect. ESNR against quantization bits used in MEQ. ) = 0 and L = 2 power levels SIR (dB) = 10 and L = 2 power levels (b) ESNR against T II Pr ´p ω i " w m |g i ¯TI ź t" 1 Γ 1 Γ 11 APPENDIX A PROOF OF PROPOSITION III.1 Proof: From Section II, we have p ω i P Ω and r ω i P Ω, where Ω is a discrete set. Therefore, we can rewrite the likelihood probability Pr ´r ω i |g i ¯as follows Pr ´r ω i |g i ¯paq " Pr pr ω i |p ω i " w m q Pr ´p ω i " w m |g i pbq " pr ω i ptq |p ω i ptqq pcq " ´r ω i ptq |Q RS ´eT t P I g i `σ2 v ji,r 1 " R ÿ n" 1 π ji pr 1 |nq ż 8 x" 0 ż u ji,n` 1 r x"u ji,n xγ ji pr x|xqφ ji pxqdr xdx R ÿ n" 1 π ji pr 1 |nq ż 8 x" 0 żpπ ji pr|n 1 ´1q ´πji pr|n 1 qq ż 8 0γ 18011808 u ji,n`1 r x"u ji,n γ ji pr x|xqφ ji pxqdr xdx .(34)For fixed representatives v ji,r , the optimum transition levels u ji,n 1 are obtained by setting the partial derivatives of the distortion Er `gji ´r g k ji ˘2s with respect to u ji,n 1 , to zero. We use the second fundamental theorem of calculus, i.e., d dx ż x a f ptqdt " f pxq to obtain u ji,n 1 for all n 1 P t2, .., Ru as BEr `gji ´r g k ji ˘2s Bu ji,n1 " ji pu ji,n 1 |xqφ ji pxq pv ji,r ´xq 2 dx " 0 N II . For each channel gain estimate r g ji to be quantized, we denote by v ji " ! v ji,1 , ..., v pqq pqq ji,R ) the set of representatives and by ! u pqq ji,1 , ..., u pqq ji,R`1 ) (with u pqq ji,1 " 0 and u pqq ji,R`1 " 8) the set of interval bounds which defines how the set r g ji lies in (namely r0, `8q) is partitioned. At each iteration, the choice of the set of representatives or intervals aims at minimizing the end-to-end distortion E|r g ji ´gji | 2 . TABLE I : I Acronyms used in Sec. V Interference MS 8 Cell size: d ˆd SBS 7 SBS 8 SBS 9 MS 7 Inter-site distance: d MS 9 d MS 6 SBS 4 SBS 5 SBS 6 MS 4 MS 5 MS 1 SBS 1 SBS 2 SBS 3 MS 2 MS 3 Fig. 3: Small cell network configuration assumed in Sec. V-A Transmitter i and Receiver j and d 0 " 5 m is a normalization factor. The normalized coordinates of the mobile stations MS 1 , ..., MS 9 are respectively given by: p3.8, 3.2q, 1 Erg s ij s corresponds to the well-known Rayleigh fading assumption. Here, Epg s ij q models the path loss exp ˆ´g s ij Erg s ˙; this ij s effects for the link ij and depends of the distance as follows: Epg s ij q " ˆd0 d ij ˙2 where d ij is the distance between p7.9, 1.4q, p10.2, 0.7q, p2.3, 5.9q, p6.6, 5.9q, p14.1, 9.3q, p1.8, 10.6q, p7.1, 14.6q, p12.5, 10.7q; the real coordinates are obtained by multiplying the former by the ratio ISD d 0 Note that, for the sake of clarity, it is assumed here that the RS power quantizer and DMC are independent of the user index, but the proposed approach holds in the general case. PROOF OF PROPOSITION III.2 Proof: After the RSSI quantization, the M T I different levels of p ω i or r ω i are w 1 , w 2 , .., w M T I forming the set Ω. Define by h : Ω Ñ G which maps the observed RSSI feedback to a channel estimate, where G :" tg 1 , g 2 , ..., g M T I u, such that hpw m q " g m . That is, when transmitter i observes the RSSI feedback r ω i to be w m , local channel estimate r g i is g m . Based on the above definitions, we have that The term Pr ´r g i " g n |g i " x ¯can be further expanded as Pr ´r g i " g n , r ω i " w , p ω i " w m |g i " x " Pr ´r g i " g n |r ω i " w ¯Pr pr ω i " w |p ω i " w m q Pr ´p ω i " w m |g i " x ¯(24) Now we know that the mapping hpq is deterministic and results in hpw m q " g m . Therefore, Pr ´r g i " g n |r ω i " w ¯" δ n, , where δ n, is the Kronecker delta function such that δ n, " 0 when n ‰ and δ n, " 1 when n " . Additionally, we also know that Pr pr ω i " w |p ω i " w m q " Γ pw ptq|w m ptqq by definition (where w m ptq is the t-th component of w m ) . This results in [START_REF] Samarakoon | Ultra dense small cell networks: Turning density into energy efficiency[END_REF] being simplified to Recall that p ω i " Q RS `PI g i ˘by definition of the quantizer. Define by resulting in Now, we can simplify ( 23) using ( 27) and ( 25) into For a fixed DMC, we can find the g MMSE i which will minimize the distortion by taking the derivative of the distortion over g n : To minimize distortion, this derivative should be equal to zero. The g n minimizing the distortion is by definition, the MMSE of the channel given r ω i " w n . Therefore by rearranging (29), we can find the expression for the MMSE given in the proposition III.2. APPENDIX C CALCULATIONS FOR THE ALMA As defined in the main text, r g k ji P tv ji,1 , ..., v ji,R u and the p.d.f. of r g ji is denoted by γ ji in general. Note that when r g ji belongs to a discrete set, we can replace the integrals and γ ji with a sum and discrete probability function without any significant alteration to our results and calculations. Denoting the p.d.f of g ji by φ ji , the distortion between g ji and r g k ji can be written as Pr `r g k ji " v ji,r |r g ji " r x ˘γji pr x|xq φ ji pxq px ´vji,r q 2 dxdr x which is the distortion observed by transmitter k when transmitter i communicates g ji in Phase II. As the transmitter i estimates g ji as r g ji , the quantization operation Q II i is performed resulting in r g ji being quantized into a certain representative v ji,n , if r g ji P ru ji,n , u ji,n`1 q. Given that the transmitter i operates at a power level corresponding to v ji,n , the transmitter k will decode v ji,r with a probability πpr|nq as defined in Section IV. Now we can expand the term Pr `r g k ji " v ji,r |r g ji " r x ˘in the following manner. to the case where the channel is perfectly estimated after phase I. This directly transforms (34) to [START_REF] Chiang | Power control in wireless cellular networks[END_REF] of the ALMA, and we can simplify (35) into rπ ji pr|n 1 ´1q ´πji pr|n 1 qs φ ji pu ij,n 1 q pv ji,r ´uij,n 1 q 2 (36) We have R ÿ r"1 rπ ji pr|n 1 ´1q ´πji pr|n 1 qs pu ij,n 1 q 2 " 0 since R ÿ r"1 π ji pr|n 1 q " 1, resulting in u ij,n 1 " ř R r"1 rπ ji pr|n 1 ´1q ´πji pr|n 1 qs v 2 ji,r 2 ř R r"1 rπ ji pr|n 1 ´1q ´πji pr|n which is (8) used in the ALMA.

01741741

en

2024/03/05 22:32:07

2017

https://hal.science/hal-01741741/file/cerino2017.pdf

PharmD Mathieu Cerino email: [email protected] MD Svetlana Gorokhova MD Pascal Laforet Ben Rabah Emmanuelle Yaou Jean Salort-Campana Shahram Pouget Bruno Attarian Jean-François Eymard Anne Deleuze Boland PhD Rabah Ben Yaou MD Emmanuelle Salort-Campana MD Jean Pouget MD Shahram Attarian MD Bruno Eymard PhD Jean-François Deleuze PhD Anne Boland MD Anthony Behin MD Tanya Stojkovic PhD Gisele Bonne MD Nicolas Levy Marc Bartoli Martin Krahn email: [email protected] Genetic Characterization of a French Cohort of GNE-mutation negative inclusion body myopathy patients with exome sequencing Keywords: exome, hIBM, GNE, NGS, diagnosis, myopathy ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Genetic characterization of a French cohort of GNE-mutation negative inclusion body myopathy patients using exome sequencing (Abstract) INTRODUCTION: Hereditary inclusion body myopathy (hIBM) refers to a group of clinically and genetically heterogeneous diseases. The overlapping histochemical features of hIBM with other genetic disorders lead to low diagnostic rates with targeted single-gene sequencing. This is true for the most prevalent form of hIBM, GNEpathy. Thus, we used whole exome sequencing (WES) to evaluate whether a cohort of clinically suspected GNEpathy patients undiagnosed by targeted GNE analysis could be genetically characterized. METHODS: 20 patients with hIBM but undiagnosed by targeted GNE sequencing were analyzed using WES before data filtering on 306 genes associated with neuromuscular disorders. RESULTS: 7 patients out of 20 were found to have disease-causing mutations in genes associated with hIBM, or genes allowing for hIBM in the differential diagnosis, or associated with unexpected diagnosis. DISCUSSION: NGS is an efficient strategy in the context of hIBM, resulting in a molecular diagnosis for 35% of the patients initially undiagnosed by targeted GNE analysis. INTRODUCTION Hereditary inclusion body myopathies (hIBM) represent a heterogeneous group of muscular disorders defined by the relatively nonspecific criterion of rimmed vacuoles on muscle biopsy 1 . GNEpathy 2 , caused by mutations in GNE (UDP-N-acetylglucosamine-2-epimerase/Nacetylmannosamine kinase, MIM*603824) [START_REF] Eisenberg | The UDP-Nacetylglucosamine 2-epimerase/N-acetylmannosamine kinase gene is mutated in recessive hereditary inclusion body myopathy[END_REF] is the most common form of hIBM, with many clinical features overlapping with other forms of hIBM, implicating other genes or forms with yet unknown underlying genetic defects. Targeted analysis of GNE in a large recentlydescribed French cohort with suspected GNEpathy provides only a 20% diagnostic yield (32 of 164 patients) [START_REF] Cerino | Novel pathogenic variants in a French cohort widen the mutational spectrum of GNE myopathy[END_REF] . In the present study, we evaluated the extent to which a cohort of clinically suspected GNEpathy patients undiagnosed by GNE targeted analysis may be genetically characterized, by implicating other genes previously known to cause neuromuscular disorders using whole exome sequencing (WES) associated with data filtering for 306 genes of interest. METHODS We selected 20 unrelated index cases (IC) with clinically suspected GNEpathy associated with rimmed vacuoles on muscle biopsy samples, but for which no GNE disease-causing mutation had been identified by direct targeted sequencing. Samples had been prepared and stored by the Center of Biological Resources, Department of Medical Genetics, La Timone Hospital, Marseille, and were used following the ethical recommendations of our institution and according to the Declaration of Helsinki. All included patients gave their written consent prior to the genetic study, in accordance with French law. WES was performed using the SureSelect Human All Exon Kit version 5 (Agilent Technologies, Santa Clara, California) and the HiSeq 2000 (Illumina, San Diego, California). Sequencing data were processed on the Illumina pipeline (CASAVA1.8.2) before using GATK 5 variant calling and ANNOVAR 6 annotation using the GRCh37/hg19 Human genome version, coverage statistics were computed using VarAFT (Variant Analysis and Filtration Tool ; http://varaft.eu, 2016), which uses BedTools 7 .VarAFT was also used to sort and filter the obtained variants. Our initial analysis strategy focused on 306 genes previously reported to cause neuromuscular disorders, and selected from the Gene Table of Neuromuscular Disorders 8 (including groups 1 to 5 and the main differential diagnosis genes) as previously described 9,10 . A mean overall sequencing depth of 106X and a mean coverage of the coding exons of 95% (at 20X depth) and 91% (at 30X depth) was obtained for these 306 genes. Predicted pathogenicity of identified variants was determined using UMD-predictor [START_REF] Salgado | UMD-Predictor: a High Throughput Sequencing Compliant System for Pathogenicity Prediction of any Human cDNA Substitution[END_REF] , SIFT (Sort Intolerant From Tolerant human Protein) [START_REF] Vaser | SIFT missense predictions for genomes[END_REF] , PolyPhen-2 (Polymorphism Phenotyping v2) [START_REF] Adzhubei | A method and server for predicting damaging missense mutations[END_REF] and HSF (Human Splicing Finder) [START_REF] Desmet | Human Splicing Finder: an online bioinformatics tool to predict splicing signals[END_REF] softwares. Regarding HSF [START_REF] Desmet | Human Splicing Finder: an online bioinformatics tool to predict splicing signals[END_REF] in silico results, we defined four types of predicted splicing effects: 1) Probably damaging: associated with predicted strong splicing effect due to broken donor site (DS) or acceptor site (AS) and/or new DS/AS creation and/or strong possibility of broken Exonic Splicing Enhancer (ESE) site; 2) Possibly damaging: associated with predicted medium splicing effect relating to newly created DS/AS and/or medium possibility of broken ESE site; 3) Uncertain: associated with predicted mild splicing effect due to newly created DS/AS and/or low possibility of broken ESE site; and 4) Not affected: predicted weak or no splicing effect. The overall pathogenicity score for each variant was determined according to the American College of Medical Genetics (ACMG) guidelines [START_REF] Richards | Standards and guidelines for the interpretation of sequence variants: a joint consensus recommendation of the American College of Medical Genetics and Genomics and the Association for Molecular Pathology[END_REF] . We established four groups of patients based on the degree of certainty of molecular diagnosis using the ACMG guidelines. The group with "definite diagnosis" consisted of the following patients: 1) Those carrying a homozygous variant classified as "pathogenic" using ACMG guidelines in a gene known to cause an autosomal recessive form of disease; 2) Compound heterozygotes carrying two variants classified as pathogenic; 3) Patients carrying one variant classified as pathogenic in a gene known to cause an autosomal dominant form of disease. The group with "probable diagnosis" was composed of patients carrying variants that were classified as "likely pathogenic" by ACMG guidelines. Patients carrying variants found to be pathogenic by certain prediction tools, but classified as "variants of uncertain significance" by ACMG guidelines were placed in the group with "possible diagnosis". For those patients in the "no established diagnosis" group, no variant compatible with the patient's phenotype was found. All disease-causing variants identified by WES were confirmed using direct targeted sequencing (Genetic analyzer 3500XL; Thermo Fisher Scientific, Waltham, Massachusetts) and the following gene sequence references: ACTA1 (NM_001100), CAPN3 (NM_000070), DES (NM_001927), FLNC (NM_001458), GYG1 (NM_004130), MYH2 (NM_017534), TARDBP (NM_007375), TTN (NM_001267550) and VCP (NM_007126). RESULTS All phenotypic and mutational data are detailed in Table 1. A definite diagnosis was obtained for seven index cases (ICs). Patient P1 harbored a previously reported mutation in TTN (Titin, MIM*188840) associated with hereditary myopathy with early respiratory failure (HMERF) [START_REF] Palmio | Hereditary myopathy with early respiratory failure: occurrence in various populations[END_REF] . The homozygous status of this mutation is consistent with the parental consanguinity. For Patients P2 and P5, the same heterozygous mutation in VCP (Valosin-Containing Protein, MIM *601023), previously described in the literature [START_REF] Stojkovic | Clinical outcome in 19 French and Spanish patients with valosin-containing protein myopathy associated with Paget's disease of bone and frontotemporal dementia[END_REF] , was discovered and associated with similar onset and clinical features (distal myopathy of upper and lower limbs). Compound heterozygous known mutations in TTN 18, [START_REF] Hackman | Truncating mutations in C-terminal titin may cause more severe tibial muscular dystrophy (TMD)[END_REF] were found in patient P3, whereas patient P4 harbored a previously described heterozygous variant in DES (Desmin, MIM*125660) [START_REF] Bär | Conspicuous involvement of desmin tail mutations in diverse cardiac and skeletal myopathies[END_REF] leading to cardiomyopathy and myofibrillar abnormalities on the muscle biopsy, features that were retrieved in patient P4. For patient P6, a known FLNC (Filamin C, MIM *102565) mutation was found [START_REF] Vorgerd | A mutation in the dimerization domain of filamin c causes a novel type of autosomal dominant myofibrillar myopathy[END_REF] . Surprisingly, we identified compound heterozygous mutations for the GYG1 (Glycogenin 1, MIM*603942) gene in patient P7, associated with polyglucosan body myopathy type 2. In this patient, we found a previously described GYG1 variant with a proven deleterious effect on splicing [START_REF] Malfatti | A new muscle glycogen storage disease associated with glycogenin-1 deficiency[END_REF] , associated with a novel GYG1 mutation leading to a frameshift of the reading frame and the introduction of a premature translation termination codon. Further investigations allowed additional clinical and histo-immunological features thus suggesting a polyglucosan body myopathy (data not shown). A probable diagnosis was obtained for patients P8 and P9, with novel compound heterozygous TTN Our study illustrates that a NGS approach is more efficient than the gene-by-gene strategy for several reasons. First, it allowed us to explore genes responsible for disorders within the differential diagnosis of hIBM, including VCP and DES. Second, our strategy permitted sequencing of large-sized genes, such as TTN and FLNC, which is not routinely performed, leading to the identification of variants in five index cases. Third, this approach allowed us to modify the incorrect diagnosis of hIBM in one patient, initially based on the presence of rimmed vacuoles on muscle biopsy, to a different muscle disorder caused by variants in the gene GYG1. Thus, NGS has the potential to alter a misdiagnosis due to a misleading muscle biopsy. Although using WES to explore a subset of genes might not provide as much target sequence coverage as a sequencing strategy specifically designed for these genes 10 , there are several advantages of using this approach. The sequencing results for samples where no pathogenic variants were identified in the initially explored genes can be reanalyzed to explore additional genes or all genes in the whole exome. In this way, further analyses are ongoing for the cases among our cohort that remain without genetic characterization. Another advantage of WES over targeted exome sequencing is its versatility and ability to be applied to many different diseases, as different sets of genes can be assessed without the need to develop and test a specific sequencing strategy 25, [START_REF] Dias | An analysis of exome sequencing for diagnostic testing of the genes associated with muscle disease and spastic paraplegia[END_REF] . In conclusion, the exome-based sequencing strategy described here is an efficient way to diagnose such genetically heterogeneous disorders as hIBMs. Finally, 9 9 mutations and a heterozygous TARDBP (Tar DNA-Binding Protein, MIM *605078) variant respectively while two novel heterozygous variants in the FLNC and the ACTA1 (Actin, Alpha, skeletal muscle 1, MIM *102610) genes fulfilled the possible diagnosis overall pathogenicity score in patients P10 and P11 respectively. ICs remained without a molecular diagnosis following mutational analysis of the 306 genes of interest. Considering only the first (definite) group of patients, the yield of diagnosed patients was 35% in this cohort (7/20). DISCUSSION Next-Generation Sequencing (NGS) is already used by many genetics laboratories and is being used with increasing frequency as the standard initial analysis for myopathies and other heterogeneous genetic disorders. The molecular diagnosis yield of 35% obtained in this study is consistent with other reports showing a range of 25 to 50 percent for rare genetic disorders diagnosis by WES 23,24 . Frequency in 1000G, ESP and ExAC databases of all the variants described in Table1is lower than 0.2%. * Additional variant with uncertain significance found for patient P9: MYH2: c.2090A>G (p.His697Arg). † Variant affecting the same nucleic and amino acid positions as another variant, c.1127G>A (p.Gly376Asp), previously described in the literature[START_REF] Conforti | TARDBP gene mutations in south Italian patients with amyotrophic lateral sclerosis[END_REF][START_REF] Solski | A novel TARDBP insertion/deletion mutation in the flail arm variant of amyotrophic lateral sclerosis[END_REF][START_REF] Lattante | TARDBP and FUS Mutations Associated with Amyotrophic Lateral Sclerosis: Summary and Update[END_REF] in two different familial amyotrophic lateral sclerosis cases with a similar phenotype presentation as patient P9 (upper and lower limb weakness with no cognitive impairment). Table 1 : Pathogenicity assessment for the identified variants in patients with definite, probable and possible diagnoses Patient Gender Phenotype / Genetic inheritance Muscle biopsy Genes/Variants Status UMD- predictor 11 SIFT 12 PolyPhen-2 13 ACMG Guidelines 15 Splicing prediction (HSF 14 ) Pathogenic variants described in literature Patients with definite diagnosis 1 In bold: pathogenicity prediction strong and moderate; NP: not performed (UMD-predictor, SIFT and PolyPhen-2 algorithms do not provide a pathogenicity score for variants creating a stop or a frameshift); UMD-predictor 11 : Universal Mutation Database predictor; SIFT 12 : Sort Intolerant From Tolerant; PolyPhen-2 13 : Polymorphism Phenotyping v2; HSF 14 : Human Splicing Finder; ACMG 15 : American College of Medical Genetics; AD: Autosomal Dominant; AR: Autosomal Recessive; HMREF: Hereditary Myopathy with early REspiratory Failure; Rimmed vacuoles P1 F PM of lower limbs at onset (40yo) evolving towards HMREF / AR Cytoplasmic inclusions Disruption of the intermyofibrillar TTN: c. 95195C>T (p.Pro31732Leu) HOZ Pathogenic Damaging Probably damaging Pathogenic NP YES 16 network P2 F DM of upper and lower limbs (tibialis anterior muscle) / AD No axial muscle weakness Rimmed vacuoles network Disruption of the intermyofibrillar VCP: c.410C>T (p.Pro137Leu) HTZ Pathogenic Damaging Probably damaging Pathogenic Not affected YES 17 Early onset (childhood) DM of lower limbs (tibial Rimmed vacuoles TTN: c.102271C>T (p.Arg34091Trp) HTZ Pathogenic Damaging Probably damaging Pathogenic NP YES 18 P3 F muscular dystrophy) slowly evolving / AR Dystrophic muscle biopsy TTN: c.107647delT (p.Ser35883Glnfs*10) HTZ NP NP NP Pathogenic NP YES 19 Late onset (45yo) DM of Rimmed vacuoles P4 M upper and lower limbs with cardiac involvement / Atrophic fibers Disorganized DES: c.1360C>T (p.Arg454Trp) HTZ Pathogenic Damaging Probably damaging Pathogenic Not affected YES 20 AD myofibrillar network P5 M DM of upper and lower limbs. Onset at 30yo / AD Rare rimmed vacuoles (<5) VCP: c.410C>T (p.Pro137Leu) HTZ Pathogenic Damaging Probably damaging Pathogenic Not affected YES 17 Late onset (45yo) DM of P6 F lower limbs and pelvic girdle myopathy Rimmed vacuoles FLNC: c.8130G>A (p.Trp2710*) HTZ NP NP NP Pathogenic NP YES 21 / AD P7 F Late onset (45yo) DM of upper and lower limbs with slow evolution / AR Rimmed vacuoles (on the initial biopsy) recharacterized as polyglucosan bodies (on the second biopsy) GYG1: c.143+3G>C (p.Asp3Glufs*4) GYG1: c.996_1005del10 (p.Tyr332fs*1) Comp. HTZ NP NP NP NP NP NP Pathogenic Pathogenic Probably damaging NP 22 NO YES Patients with probable diagnosis Early onset (14yo) DM of lower limbs (tibial TTN: c.15346C>T (p.Arg5116*) HTZ NP NP NP Pathogenic NP NO muscular dystrophy) P8 M evolving towards hamstring muscle with quadriceps sparing Rimmed vacuoles TTN: c. 107680G>A (p.Gly35894Arg) HTZ Pathogenic Damaging Probably damaging Likely pathogenic NP NO / AR P9 * M Late onset (50yo) DM of upper and lower limbs / AD Rimmed vacuoles No inflammation Atrophic fibers TARDBP: c.1127G>T (p.Gly376Val) HTZ Pathogenic Tolerated Benign pathogenic Likely damaging Possibly NO † Patients with possible diagnosis P10 M Limb girdle muscular dystrophy / AD Rare rimmed vacuoles (<5) FLNC: c.6526C>T (p.Arg2176Cys) HTZ Pathogenic Tolerated Probably damaging Uncertain significance Not affected NO P11 M DM of lower limbs with very slow evolution / AD Rimmed vacuoles ACTA1: c.437C>T (p.Ala146Val) HTZ Pathogenic Tolerated Probably damaging Uncertain significance Uncertain NO DM: Distal Myopathy; PM: Proximal Myopathy; yo: years old. HOZ: homozygous; HTZ: heterozygous; Comp. HTZ: compound heterozygous (with confirmed segregation analysis). This study was supported by the France Génomique infrastructure (grant no. ANR-10-INBS-09) managed by the National Research Agency (ANR) part of the Investment for the Future program, Fondation Maladies Rares, FHU-MaRCHE, the APHM, Inserm, the "Bureau des PU-PH de l'Assistance Publique -Hôpitaux de Marseille (AP-HM)", and by a Grant FP7/2007-2013 from the European Community Seventh Framework Programme (Grant Agreement No. 2012-305121) "Integrated European omics research project for diagnosis and therapy in rare neuromuscular and neurodegenerative diseases" (NEUROMICS). The authors sincerely thank Christel Castro, Jean-Pierre Desvignes, David Salgado, Christophe Béroud, Eric Salvo, Rafaelle Bernard, Jocelyn Laporte, Johann Bohm and Mark Lathrop for their contributions to this work. We also thank the patients and their referring physicians for their participation.

01745366

en

2024/03/05 22:32:07

2019

https://hal.science/hal-01745366/file/LSE-FDM.pdf

Weizhu Bao Émi Carles email: [email protected] AND Su § Chunmei Qinglin Tang email: [email protected] ERROR ESTIMATES OF A REGULARIZED FINITE DIFFERENCE METHOD FOR THE LOGARITHMIC SCHR ÖDINGER EQUATION * Keywords: Logarithmic Schrödinger equation, logarithmic nonlinearity, regularized logarithmic Schrödinger equation, semi-implicit finite difference method, error estimates, convergence rate AMS subject classifications. 35Q40, 35Q55, 65M15, 81Q05 We present a regularized finite difference method for the logarithmic Schrödinger equation (LogSE) and establish its error bound. Due to the blow-up of the logarithmic nonlinearity, i.e. ln ρ → -∞ when ρ → 0 + with ρ = |u| 2 being the density and u being the complex-valued wave function or order parameter, there are significant difficulties in designing numerical methods and establishing their error bounds for the LogSE. In order to suppress the round-off error and to avoid blow-up, a regularized logarithmic Schrödinger equation (RLogSE) is proposed with a small regularization parameter 0 < ε ≪ 1 and linear convergence is established between the solutions of RLogSE and LogSE in term of ε. Then a semi-implicit finite difference method is presented for discretizing the RLogSE and error estimates are established in terms of the mesh size h and time step τ as well as the small regularization parameter ε. Finally numerical results are reported to confirm our error bounds. 1. Introduction. We consider the logarithmic Schrödinger equation (LogSE) which arises in a model of nonlinear wave mechanics (cf. [START_REF] Bia Lynicki-Birula | Nonlinear wave mechanics[END_REF]), (1.1) i∂ t u(x, t) + ∆u(x, t) = λu(x, t) ln(|u(x, t)| 2 ), x ∈ Ω, t > 0, u(x, 0) = u 0 (x), x ∈ Ω, where t is time, x ∈ R d (d = 1, 2, 3) is the spatial coordinate, λ ∈ R\{0} measures the force of the nonlinear interaction, u := u(x, t) ∈ C is the dimensionless wave function or order parameter and Ω = R d or Ω ⊂ R d is a bounded domain with homogeneous Dirichlet or periodic boundary condition 1 fixed on the boundary. It admits applications to quantum mechanics [START_REF] Bia Lynicki-Birula | Nonlinear wave mechanics[END_REF][START_REF] Gaussons | Solitons of the logarithmic Schrödinger equation[END_REF], quantum optics [START_REF] Buljan | Incoherent white light solitons in logarithmically saturable noninstantaneous nonlinear media[END_REF][START_REF] Krolikowski | Unified model for partially coherent solitons in logaritmically nonlinear media[END_REF], nuclear physics [START_REF] Hefter | Application of the nonlinear Schrödinger equation with a logarithmic inhomogeneous term to nuclear physics[END_REF], transport and diffusion phenomena [START_REF] Hansson | Propagation of partially coherent solitons in saturable logarithmic media: A comparative analysis[END_REF][START_REF] Martino | Logarithmic Schrödinger-like equation as a model for magma transport[END_REF], open quantum systems [START_REF] Hernandez | General properties of Gausson-conserving descriptions of quantal damped motion[END_REF][START_REF] Yasue | Quantum mechanics of nonconservative systems[END_REF], effective quantum gravity [START_REF] Zloshchastiev | Logarithmic nonlinearity in theories of quantum gravity: Origin of time and observational consequences[END_REF], theory of superfluidity and Bose-Einstein condensation [START_REF] Avdeenkov | Quantum bose liquids with logarithmic nonlinearity: Self-sustainability and emergence of spatial extent[END_REF]. The logarithmic Schrödinger equation enjoys three conservation laws, mass, momentum and energy [START_REF]Semilinear Schrödinger equations[END_REF][START_REF] Cazenave | Équations d'évolution avec non linéarité logarithmique[END_REF], like in the case of the nonlinear Schrödinger equation with a power-like nonlinearity (e.g. cubic): M (t) : = u(•, t) 2 L 2 (Ω) = Ω |u(x, t)| 2 dx ≡ Ω |u 0 (x)| 2 dx = M (0), P (t) : = Im Ω u(x, t)∇u(x, t)dx ≡ Im Ω u 0 (x)∇u 0 (x)dx = P (0), t ≥ 0, E(t) : = Ω |∇u(x, t)| 2 dx + λF (|u(x, t)| 2 ) dx ≡ Ω |∇u 0 (x)| 2 + λF (|u 0 (x)| 2 ) dx = E(0), (1.2) where Im f and f denote the imaginary part and complex conjugate of f , respectively, and (1.3) F (ρ) = ρ 0 ln(s)ds = ρ ln ρ -ρ, ρ ≥ 0. On a mathematical level, the logarithmic nonlinearity possesses several features that make it quite different from more standard nonlinear Schrödinger equations. First, the nonlinearity is not locally Lipschitz continuous because of the behavior of the logarithm function at the origin. Note that in view of numerical simulation, this singularity of the "nonlinear potential" λ ln(|u(x, t)| 2 ) makes the choice of a discretization quite delicate. The second aspect is that whichever the sign of λ, the nonlinear potential energy in E has no definite sign. In fact, whether the nonlinearity is repulsive/attractive (or defocusing/focusing) depends on both λ and the value of the density ρ := ρ(x, t) = |u(x, t)| 2 . When λ > 0, then the nonlinearity λρ ln ρ is repulsive when ρ > 1; and respectively, it is attractive when 0 < ρ < 1. On the other hand, when λ < 0, then the nonlinearity λρ ln ρ is attractive when ρ > 1; and respectively, it is repulsive when 0 < ρ < 1. Therefore, solving the Cauchy problem for (1.1) is not a trivial issue, and constructing solutions which are defined for all time requires some work; see [START_REF] Carles | Universal dynamics for the defocusing logarithmic Schrödinger equation[END_REF][START_REF] Cazenave | Équations d'évolution avec non linéarité logarithmique[END_REF][START_REF] Guerrero | Global H 1 solvability of the 3D logarithmic Schrödinger equation[END_REF]. Essentially, the outcome is that if u 0 belongs to (a subset of) H 1 (Ω), (1.1) has a unique, global solution, regardless of the space dimension d (see also Theorem 2.2 below). Next, the large time behavior reveals new phenomena. A first remark suggests that nonlinear effects are weak. Indeed, unlike what happens in the case of a homogeneous nonlinearity (classically of the form λ|u| p u), replacing u with ku (k ∈ C \ {0}) in (1.1) has only little effect, since we have i∂ t (ku) + ∆(ku) = λku ln |ku| 2 -λ(ln |k| 2 )ku . The scaling factor thus corresponds to a purely time-dependent gauge transform: ku(x, t)e -itλ ln |k| 2 solves (1.1) (with initial datum ku 0 ). In particular, the size of the initial datum does not influence the dynamics of the solution. In spite of this property which is reminiscent of linear equations, nonlinear effects are stronger in (1.1) than in, say, cubic Schrödinger equations in several respects. For Ω = R d , it was established in [START_REF] Cazenave | Stable solutions of the logarithmic Schrödinger equation[END_REF] that in the case λ < 0, no solution is dispersive (not even for small data, in view of the above remark), while if λ > 0, the results from [START_REF] Carles | Universal dynamics for the defocusing logarithmic Schrödinger equation[END_REF] show that every solution disperses, at a faster rate than for the linear equation. In view of the gauge invariance of the nonlinearity, for Ω = R d , (1.1) enjoys the standard Galilean invariance: if u(x, t) solves (1.1), then, for any v ∈ R d , so does u(x -2vt, t)e iv•x-i|v| 2 t . A remarkable feature of (1.1) is that it possesses a large set of explicit solutions. In the case Ω = R d : if u 0 is Gaussian, u(•, t) is Gaussian for all time, and solving (1.1) amounts to solving ordinary differential equations [START_REF] Bia Lynicki-Birula | Nonlinear wave mechanics[END_REF]. For simplicity of notation, we take the one-dimensional case as an example. If the initial data in (1.1) with Ω = R is taken as u 0 (x) = b 0 e -a 0 2 x 2 +ivx , x ∈ R, where a 0 , b 0 ∈ C and v ∈ R are given constants satisfying α 0 := Re a 0 > 0 with Re f denoting the real part of f , then the solution of (1.1) is given by [START_REF] Ardila | Orbital stability of Gausson solutions to logarithmic Schrödinger equations[END_REF][START_REF] Carles | Universal dynamics for the defocusing logarithmic Schrödinger equation[END_REF] ( 1.4) u(x, t) = b 0 r(t) e i(vx-v 2 t)+Y (x-2vt,t) , x ∈ R, t ≥ 0, with (1.5) Y (x, t) = -iφ(t) -α 0 x 2 2r(t) 2 + i ṙ(t) r(t) x 2 4 , x ∈ R, t ≥ 0, where φ := φ(t) ∈ R and r := r(t) > 0 solve the ODEs [START_REF] Ardila | Orbital stability of Gausson solutions to logarithmic Schrödinger equations[END_REF][START_REF] Carles | Universal dynamics for the defocusing logarithmic Schrödinger equation[END_REF] φ = α 0 r 2 + λ ln |b 0 | 2 -λ ln r, φ(0) = 0, r = 4α 2 0 r 3 + 4λα 0 r , r(0) = 1, ṙ(0) = -2 Im a 0 . (1.6) In the case λ < 0, the function r is (time) periodic (in agreement with the absence of dispersive effects). In particular, if a 0 = -λ > 0, it follows from (1.6) that r(t) ≡ 1 and φ(t) = φ 0 t with φ 0 = λ ln(|b 0 | 2 ) -1 , which generates the uniformly moving Gausson as [START_REF] Ardila | Orbital stability of Gausson solutions to logarithmic Schrödinger equations[END_REF][START_REF] Carles | Universal dynamics for the defocusing logarithmic Schrödinger equation[END_REF] ( 1.7) u(x, t) = b 0 e λ 2 (x-2vt) 2 +i(vx-(φ0+v 2 )t) , x ∈ R, t ≥ 0. As a very special case with b 0 = e 1/2 and v = 0 such that φ 0 = 0, one can get the static Gausson as (1.8) u(x, t) = e 1/2 e λ|x| 2 /2 , x ∈ R, t ≥ 0. This special solution is orbitally stable [START_REF] Cazenave | Stable solutions of the logarithmic Schrödinger equation[END_REF][START_REF] Cazenave | Orbital stability of standing waves for some nonlinear Schrödinger equations[END_REF]. On the other hand, in the case λ > 0, it is proven in [START_REF] Carles | Universal dynamics for the defocusing logarithmic Schrödinger equation[END_REF] that for general initial data (not necessarily Gaussian), there exists a universal dynamics. For extensions to higher dimensions, we refer to [START_REF] Ardila | Orbital stability of Gausson solutions to logarithmic Schrödinger equations[END_REF][START_REF] Carles | Universal dynamics for the defocusing logarithmic Schrödinger equation[END_REF] and references therein. Therefore, (1.1) possesses several specific features, which make it quite different from the nonlinear Schrödinger equation. Different numerical methods have been proposed and analyzed for the nonlinear Schrödinger equation with smooth nonlinearity (e.g. cubic nonlinearity) in the literature, such as the finite difference methods [START_REF] Bao | Uniform error estimates of finite difference methods for the nonlinear Schrödinger equation with wave operator[END_REF][START_REF]Optimal error estimates of finite difference methods for the Gross-Pitaevskii equation with angular momentum rotation[END_REF], finite element methods [START_REF] Akrivis | On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schrödinger equation[END_REF][START_REF] Karakashian | A space-time finite element method for the nonlinear Schrödinger equation: the continuous Galerkin method[END_REF] and the time-splitting pseudospectral methods [START_REF] Bao | Numerical solution of the Gross-Pitaevskii equation for Bose-Einstein condensation[END_REF][START_REF] Taha | Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, nonlinear Schrödinger equation[END_REF]. However, they cannot be applied to the LogSE (1.1) directly due to the blow-up of the logarithmic nonlinearity, i.e. ln ρ → -∞ when ρ → 0 + . The main aim of this paper is to present a regularized finite difference method for the LogSE (1.1) by introducing a proper regularized logarithmic Schrödinger equation (RLogSE) and then discretizing the RLogSE via a semi-implicit finite difference method. Error estimates will be established between the solutions of LogSE and RLogSE as well as their numerical approximations. The rest of the paper is organized as follows. In Section 2, we propose a regularized version of (1.1) with a small regularization parameter 0 < ε ≪ 1, and analyze its properties, as well as the convergence of its solution to the solution of (1.1). In Section 3, we introduce a semi-implicit finite difference method for discretizing the regularized logarithmic Schrödinger equation, and prove an error estimate, in which the dependence of the constants with respect to the regularization parameter ε is tracked very explicitly. Finally, numerical results are provided in Section 4 to confirm our error bounds and to demonstrate the efficiency and accuracy of the proposed numerical method. Throughout the paper, we use H m (Ω) and • H m (Ω) to denote the standard Sobolev spaces and their norms, respectively. In particular, the norm and inner product of L 2 (Ω) = H 0 (Ω) are denoted by • L 2 (Ω) and (•, •), respectively. Moreover, we adopt A B to mean that there exists a generic constant C > 0 independent of the regularization parameter ε, the time step τ and the mesh size h such that A ≤ C B, and c means the constant C depends on c. 2. A regularized logarithmic Schrödinger equation. It turns out that a direct simulation of the solution of (1.1) is very delicate, due to the singularity of the logarithm at the origin, as discussed in the introduction. Instead of working directly with (1.1), we shall consider the following regularized logarithmic Schrödinger equation (RLogSE) with a samll regularized parameter 0 < ε ≪ 1 as (2.1) i∂ t u ε (x, t) + ∆u ε (x, t) = λu ε (x, t) ln (ε + |u ε (x, t)|) 2 , x ∈ Ω, t > 0, u ε (x, 0) = u 0 (x), x ∈ Ω. 2.1. Conserved quantities. For the RLogSE (2.1), it can be similarly deduced that the mass, momentum, and energy are conserved. Proposition 2.1. The mass, momentum, and 'regularized' energy are formally conserved for the RLogSE (2.1): M ε (t) := Ω |u ε (x, t)| 2 dx ≡ Ω |u 0 (x)| 2 dx = M (0), P ε (t) := Im Ω u ε (x, t)∇u ε (x, t)dx ≡ Im Ω u 0 (x)∇u 0 (x)dx = P (0), t ≥ 0, E ε (t) := Ω |∇u ε (x, t)| 2 + λF ε (|u ε (x, t)| 2 ) (x, t)dx ≡ Ω |∇u 0 (x)| 2 + λF ε (|u 0 (x)| 2 ) dx = E ε (0), (2.2) where (2.3) F ε (ρ) = ρ 0 ln(ε + √ s) 2 ds = ρ ln (ε + √ ρ) 2 -ρ + 2ε √ ρ -ε 2 ln (1 + √ ρ/ε) 2 , ρ ≥ 0. Proof. The conservation for mass and momentum is standard, and relies on the fact that the right hand side of (2.1) involves u ε multiplied by a real number. For the energy E ε (t), we compute d dt E ε (t) = 2 Re Ω ∇u ε • ∇∂ t u ε + λu ε ∂ t u ε ln(ε + |u ε |) 2 -λu ε ∂ t u ε (x, t)dx + 2λ Ω ∂ t |u ε | ε + |u ε | 2 -ε 2 ε + |u ε | (x, t)dx = 2 Re Ω ∂ t u ε -∆u ε + λu ε ln(ε + |u ε |) 2 (x, t)dx = 2 Re Ω i|∂ t u ε | 2 (x, t)dx = 0, t ≥ 0, which completes the proof. Note however that since the above 'regularized' energy involves L 1 -norm of u ε for any ε > 0, E ε is obviously well-defined for u 0 ∈ H 1 (Ω) when Ω has finite measure, but not when Ω = R d . This aspect is discussed more into details in Subsections 2.3.3 and 2.4. The Cauchy problem. For α > 0 and Ω = R d , denote by L 2 α the weighted L 2 space L 2 α := {v ∈ L 2 (R d ), x -→ x α v(x) ∈ L 2 (R d )}, where x := 1 + |x| 2 , with norm v L 2 α := x α v(x) L 2 (R d ) . In the case where Ω is bounded, we simply set L 2 α = L 2 (Ω). Regarding the Cauchy problems (1.1) and (2.1), we have the following result. 0 ∈ H 1 0 (Ω) ∩ L 2 α , for some 0 < α ≤ 1. • There exists a unique, global solution u ∈ L ∞ loc (R; H 1 0 (Ω) ∩ L 2 α ) to (1.1), and a unique, global solution u ε ∈ L ∞ loc (R; H 1 0 (Ω) ∩ L 2 α ) to (2.1). • If in addition u 0 ∈ H 2 (Ω), then u, u ε ∈ L ∞ loc (R; H 2 (Ω)). • In the case Ω = R d , if in addition u 0 ∈ H 2 ∩L 2 2 , then u, u ε ∈ L ∞ loc (R; H 2 ∩L 2 2 ) . Proof. This result can be proved by using more or less directly the arguments invoked in [START_REF] Carles | Universal dynamics for the defocusing logarithmic Schrödinger equation[END_REF]. First, for fixed ε > 0, the nonlinearity in (2.1) is locally Lipschitz, and grows more slowly than any power for large |u ε |. Therefore, the standard Cauchy theory for nonlinear Schrödinger equations applies (see in particular [START_REF]Semilinear Schrödinger equations[END_REF]Corollary 3.3.11 and Theorem 3.4.1]), and so if u 0 ∈ H 1 0 (Ω), then (2.1) has a unique solution u ε ∈ L ∞ loc (R; H 1 0 (Ω)). Higher Sobolev regularity is propagated, with controls depending on ε in general. A solution u of (1.1) can be obtained by compactness arguments, by letting ε → 0 in (2.1), provided that we have suitable bounds independent of ε > 0. We have i∂ t ∇u ε + ∆∇u ε = 2λ ln (ε + |u ε |) ∇u ε + 2λ u ε ε + |u ε | ∇|u ε |. The standard energy estimate (multiply the above equation by ∇u ε , integrate over Ω and take the imaginary part) yields, when Ω = R d or when periodic boundary conditions are considered, 1 2 d dt ∇u ε 2 L 2 (Ω) ≤ 2|λ| Ω |u ε | ε + |u ε | |∇|u ε || |∇u ε |dx ≤ 2|λ ∇u ε 2 L 2 (Ω) . Gronwall lemma yields a bound for u ε in L ∞ (0, T ; H 1 (Ω)), uniformly in ε > 0, for any given T > 0. Indeed, the above estimate uses the property Im Ω ∇u ε • ∆∇u ε dx = 0, which needs not be true when Ω is bounded and u ε satisfies homogeneous Dirichlet boundary conditions. In that case, we use the conservation of the energy E ε (Proposition 2.1), and write ∇u ε (t) 2 L 2 (Ω) ≤ E ε (u 0 ) + 2|λ| Ω |u ε (x, t)| 2 |ln (ε + |u ε (x, t)|)| dx + 2ε|λ| u ε (t) L 1 (Ω) + 2|λ|ε 2 Ω |ln (1 + |u ε (x, t)|/ε)| dx 1 + ε|Ω| 1/2 u ε (t) L 2 (Ω) + Ω |u ε (x, t)| 2 |ln (ε + |u ε (x, t)|)| dx 1 + Ω |u ε (x, t)| 2 |ln (ε + |u ε (x, t)|)| dx, t ≥ 0, where we have used Cauchy-Schwarz inequality and the conservation of the mass M ε (t). Writing, for 0 < η ≪ 1, Ω |u ε | 2 |ln (ε + |u ε |)| dx ε+|u ε |>1 |u ε | 2 (ε + |u ε |) η dx + ε+|u ε |<1 |u ε | 2 (ε + |u ε |) -η dx u ε L 2 (Ω) + u ε 2+η L 2+η (Ω) + u ε 2-η L 2-η (Ω) 1 + ∇u ε dη/2 L 2 (Ω) , where we have used the interpolation inequality (see e.g. [START_REF] Nirenberg | On elliptic partial differential equations[END_REF]) u L p (Ω) u 1-α L 2 (Ω) ∇u α L 2 (Ω) + u L 2 (Ω) , p = 2d d -2α , 0 ≤ α < 1, we obtain again that u ε is bounded in L ∞ (0, T ; H 1 (Ω)), uniformly in ε > 0, for any given T > 0. In the case where Ω is bounded, compactness arguments show that u ε converges to a solution u to (1.1); see [START_REF]Semilinear Schrödinger equations[END_REF][START_REF] Cazenave | Équations d'évolution avec non linéarité logarithmique[END_REF]. When Ω = R d , compactness in space is provided by multiplying (2.1) with x 2α u ε and integrating in space: d dt u ε 2 L 2 α = 4α Im x • ∇u ε x 2-2α u ε (t) dx x 2α-1 u ε L 2 (Ω) ∇u ε L 2 (Ω) , where we have used Cauchy-Schwarz inequality. Recalling that 0 < α ≤ 1, x 2α-1 u ε L 2 (Ω) ≤ x α u ε L 2 (Ω) = u ε L 2 α , and we obtain a bound for u ε in L ∞ (0, T ; H 1 (Ω) ∩ L 2 α ) which is uniform in ε. Uniqueness of such a solution for (1.1) follows from the arguments of [START_REF] Cazenave | Équations d'évolution avec non linéarité logarithmique[END_REF], involving a specific algebraic inequality, generalized in Lemma 2.4 below. Note that at this stage, we know that u ε converges to u by compactness arguments, so we have no convergence estimate. Such estimates are established in Subsection 2.3. To prove the propagation of the H 2 regularity, we note that differentiating twice the nonlinearity in (2.1) makes it unrealistic to expect direct bounds which are uniform in ε. To overcome this difficulty, the argument proposed in [START_REF] Carles | Universal dynamics for the defocusing logarithmic Schrödinger equation[END_REF] relies on Kato's idea: instead of differentiating the equation twice in space, differentiate it once in time, and use the equation to infer H 2 regularity. This yields the second part of the theorem. To establish the last part of the theorem, we prove that u ∈ L ∞ loc (R; L 2 2 ) and the same approach applies to u ε . It follows from (1.1) that d dt u(t) 2 L 2 2 = -2 Im R d x 4 u(x, t)∆u(x, t)dx = 8 Im R d x 2 u(x, t) x • ∇u(x, t)dx ≤ 8 u(t) L 2 2 x • ∇u(t) L 2 (R d ) . (2.4) By Cauchy-Schwarz inequality and integration by parts, we have x • ∇u(t) 2 L 2 (R d ) ≤ d j=1 d k=1 R d x 2 j ∂u(x, t) ∂x k ∂u(x, t) ∂x k dx = -2 R d u(x, t) x • ∇u(x, t)dx - R d |x| 2 u(x, t)∆u(x, t)dx ≤ 1 2 x • ∇u(t) 2 L 2 (R d ) + 2 u(t) 2 L 2 (R d ) + 1 2 u(t) 2 L 2 2 + 1 2 ∆u(t) 2 L 2 (R d ) , which yields directly that x • ∇u(t) L 2 (R d ) ≤ 2 u(t) L 2 (R d ) + u(t) L 2 2 + ∆u(t) L 2 (R d ) . This together with (2.4) gives that d dt u(t) L 2 2 ≤ 4 x • ∇u(t) L 2 (R d ) ≤ 4 u(t) L 2 2 + 8 u(t) L 2 (R d ) + 4 ∆u(t) L 2 (R d ) . Since we already know that u ∈ L ∞ loc (R; H 2 (R d )), Gronwall lemma completes the proof. Remark 2.1. We emphasize that if u 0 ∈ H k (R d ), k ≥ 3, we cannot guarantee in general that this higher regularity is propagated in (1.1), due to the singularities stemming from the logarithm. Still, this property is fulfilled in the case where u 0 is Gaussian, since then u remains Gaussian for all time. However, our numerical tests, in the case where the initial datum is chosen as the dark soliton of the cubic Schrödinger equation multiplied by a Gaussian, suggest that even the H 3 regularity is not propagated in general. 2.3. Convergence of the regularized model. In this subsection, we show the approximation property of the regularized model (2.1) to (1.1). 2.3.1. A general estimate. We prove: Lemma 2.3. Suppose the equation is set on Ω, where Ω = R d , or Ω ⊂ R d is a bounded domain with homogeneous Dirichlet or periodic boundary condition, then we have the general estimate: (2.5) d dt u ε (t) -u(t) 2 L 2 (Ω) ≤ 4|λ| u ε (t) -u(t) 2 L 2 (Ω) + ε u ε (t) -u(t) L 1 (Ω) . Before giving the proof of Lemma 2.3, we introduce the following lemma, which is a variant of [12, Lemma 9. |Im ((f ε (z 1 ) -f ε (z 2 )) (z 1 -z 2 ))| ≤ |z 1 -z 2 | 2 , z 1 , z 2 ∈ C. Proof. Notice that Im [(f ε (z 1 ) -f ε (z 2 )) (z 1 -z 2 )] = 1 2 [ln(ε + |z 1 |) -ln(ε + |z 2 |)] Im(z 1 z 2 -z 1 z 2 ). Supposing, for example, 0 < |z 2 | ≤ |z 1 |, we can obtain that |ln(ε + |z 1 |) -ln(ε + |z 2 |)| = ln 1 + |z 1 | -|z 2 | ε + |z 2 | ≤ |z 1 | -|z 2 | ε + |z 2 | ≤ |z 1 -z 2 | |z 2 | , and |Im(z 1 z 2 -z 1 z 2 )| = |z 2 (z 1 -z 2 ) + z 2 (z 2 -z 1 ))| ≤ 2|z 2 | |z 1 -z 2 |. Otherwise the result follows by exchanging z 1 and z 2 . Proof. (Proof of Lemma 2.3) Subtracting (1.1) from (2.1), we see that the error function e ε := u ε -u satisfies i∂ t e ε + ∆e ε = λ u ε ln(ε + |u ε |) 2 -u ln(|u| 2 ) . Multiplying the error equation by e ε (t), integrating in space and taking the imaginary parts, we can get by using Lemma 2.4 that 1 2 d dt e ε (t) 2 L 2 (Ω) = 2λ Im Ω [u ε ln(ε + |u ε |) -u ln(|u|)] (u ε -u)(x, t)dx ≤ 2|λ| e ε (t) 2 L 2 (Ω) + 2|λ| Ω e ε u [ln(ε + |u|) -ln(|u|)] (x, t)dx ≤ 2|λ| e ε (t) 2 L 2 (Ω) + 2ε|λ| e ε (t) L 1 (Ω) , where we have used the general estimate 0 ≤ ln(1 + |x|) ≤ |x|. Convergence for bounded domain. If Ω has finite measure, then we can have the following convergence behavior. Proposition 2.5. Assume that Ω has finite measure, and let u 0 ∈ H 2 (Ω). For any T > 0, we have (2.6) u ε -u L ∞ (0,T ;L 2 (Ω)) ≤ C 1 ε, u ε -u L ∞ (0,T ;H 1 (Ω)) ≤ C 2 ε 1/2 , where C 1 depends on |λ|, T , |Ω| and C 2 depends on |λ|, T , |Ω| and u 0 H 2 (Ω) . Proof. Note that e ε (t ) L 1 (Ω) ≤ |Ω| 1/2 e ε (t) L 2 (Ω) , then it follows from (2.5) that d dt e ε (t) L 2 (Ω) ≤ 2|λ| e ε (t) L 2 (Ω) + 2ε|λ||Ω| 1/2 . Applying Gronwall's inequality, we immediately get that e ε (t) L 2 (Ω) ≤ e ε (0) L 2 (Ω) + ε|Ω| 1/2 e 2|λ|t = ε|Ω| 1/2 e 2|λ|t . The convergence rate in H 1 follows from the property u ε , u ∈ L ∞ loc (R; H 2 (Ω)) and the Gagliardo-Nirenberg inequality [START_REF] Leoni | A first course in Sobolev spaces[END_REF], ∇v L 2 (Ω) v 1/2 L 2 (Ω) ∆v 1/2 L 2 (Ω) , which completes the proof. Remark 2.2. The weaker rate in the H 1 estimate is due to the fact that Lemma 2.3 is not easily adapted to H 1 estimates, because of the presence of the logarithm. Differentiating (1.1) and (2.1) makes it hard to obtain the analogue in Lemma 2.3. This is why we bypass this difficulty by invoking boundedness in H 2 and interpolating with the error bound at the L 2 level. If we have u ε , u ∈ L ∞ loc (R; H k (Ω)) for k > 2, then the convergence rate in H 1 (Ω) can be improved as e ε L ∞ (0,T ;H 1 (Ω)) ε k-1 k , by using the inequality (see e.g. [START_REF] Nirenberg | On elliptic partial differential equations[END_REF]): v H 1 (Ω) v 1-1/k L 2 (Ω) v 1/k H k (Ω) . 2.3.3. Convergence for the whole space. In order to prove the convergence rate of the regularized model (2.1) to (1.1) for the whole space, we need the following lemma. Lemma 2.6. For d = 1, 2, 3, if v ∈ L 2 (R d ) ∩ L 2 2 , then we have (2.7) v L 1 (R d ) ≤ C v 1-d/4 L 2 (R d ) v d/4 L 2 2 , where C > 0 depends on d. Proof. Applying the Cauchy-Schwarz inequality, we can get for fixed r > 0, v L 1 (R d ) = |x|≤r |v(x)|dx + |x|≥r |x| 2 |v(x)| |x| 2 dx r d/2 |x|≤r |v(x)| 2 dx 1 2 + |x|≥r |x| 4 |v(x)| 2 dx 1 2 |x|≥r 1 |x| 4 dx 1 2 r d/2 v L 2 (R d ) + r d/2-2 v L 2 2 . Then (2.7) can be obtained by setting r = v L 2 2 / v L 2 (R d ) 1/2 . Proposition 2.7. Assume that Ω = R d , 1 ≤ d ≤ 3, and let u 0 ∈ H 2 (R d ) ∩ L 2 2 . For any T > 0, we have u ε -u L ∞ (0,T ;L 2 (R d )) ≤ C 1 ε 4 4+d , u ε -u L ∞ (0,T ;H 1 (R d ))) ≤ C 2 ε 2 4+d , where C 1 depends on d, |λ|, T , u 0 L 2 2 and C 2 depends on additional u 0 H 2 (R d ) . Proof. Applying (2.7) and the Young's inequality, we deduce that ε e ε (t) L 1 (R d ) ≤ εC d e ε (t) 1-d/4 L 2 (R d ) e ε (t) d/4 L 2 2 ≤ C d e ε (t) 2 L 2 (R d ) + ε 8 4+d e ε (t) 2d 4+d L 2 2 , which together with (2.5) gives that d dt e ε (t) 2 L 2 (R d ) ≤ 4|λ|(1 + C d ) e ε (t) 2 L 2 (R d ) + 4C d |λ|ε 8 4+d e ε (t) 2d 4+d L 2 2 . Gronwall lemma yields e ε (t) L 2 (R d ) ≤ ε 4 4+d e ε (t) d 4+d L 2 2 e tC d,|λ| . The proposition follows by recalling that u ε , u ∈ L ∞ loc (R; H 2 (R d ) ∩ L 2 2 ). Remark 2.3. If we have u ε , u ∈ L ∞ loc (R; L 2 m ) for m > 2, then by applying the inequality ε v L 1 (R d ) ε v 1-d 2m L 2 (R d ) v d 2m L 2 m v 2 L 2 (R d ) + ε 4m 2m+d v 2d 2m+d L 2 m , which can be proved like above, the convergence rate can be improved as u ε -u L ∞ (0,T ;L 2 (R d )) ε 2m 2m+d . Remark 2.4. If in addition u ε , u ∈ L ∞ loc (R; H s (R d )) for s > 2, then the conver- gence rate in H 1 (R d ) can be improved as e ε L ∞ (0,T ;H 1 (R d ))) ≤ Cε 2m 2m+d s-1 s , by using the Gagliardo-Nirenberg inequality: ∇v L 2 (R d ) ≤ C v 1-1/s L 2 (R d ) ∇ s v 1/s L 2 (R d ) . The previous two remarks apply typically in the case of Gaussian initial data. Convergence of the energy. In this subsection we will show the convergence of the energy E ε (u 0 ) → E(u 0 ). Proposition 2.8. For u 0 ∈ H 1 (Ω) ∩ L 1 (Ω), the energy E ε (u 0 ) converges to E(u 0 ) with |E ε (u 0 ) -E(u 0 )| ≤ 4 ε|λ| u 0 L 1 (Ω) . Proof. It can be deduced from the definition that |E ε (u 0 ) -E(u 0 )| = 2|λ| ε u 0 L 1 (Ω) + Ω |u 0 (x)| 2 [ln(ε + |u 0 (x)| -ln(|u 0 (x)|)] dx -ε 2 Ω ln (1 + |u 0 (x)|/ε) dx ≤ 4 ε|λ| u 0 L 1 (Ω) , which completes the proof. Remark 2.5. If Ω is bounded, then H 1 (Ω) ⊆ L 1 (Ω). If Ω = R d , then Lemma 2.6 (and its natural generalizations) shows that H 1 (R) ∩ L 2 1 ⊆ L 1 (R), and if d = 2, 3, H 1 (R d ) ∩ L 2 2 ⊆ L 1 (R d ). Remark 2.6. This regularization is reminiscent of the one considered in [START_REF] Carles | Universal dynamics for the defocusing logarithmic Schrödinger equation[END_REF] in order to prove (by compactness arguments) that (1.1) has a solution, (2.8) i∂ t u ε (x, t) + ∆u ε (x, t) = λu ε (x, t) ln ε + |u ε (x, t)| 2 , x ∈ Ω, t > 0. With that regularization, it is easy to adapt the error estimates established above for (2.1). Essentially, ε must be replaced by √ ε (in Lemma 2.3, and hence in its corollaries). 3. A regularized semi-implicit finite difference method. In this section, we study the approximation properties of a finite difference method for solving the regularized model (2.1). For simplicity of notation, we set λ = 1 and only present the numerical method for the RLogSE (2.1) in 1D, as extensions to higher dimensions are straightforward. When d = 1, we truncate the RLogSE on a bounded computational interval Ω = (a, b) with homogeneous Dirichlet boundary condition (here |a| and b are chosen large enough such that the truncation error is negligible): (3.1) i∂ t u ε (x, t) + ∂ xx u ε (x, t) = u ε (x, t) ln(ε + |u ε (x, t)|) 2 , x ∈ Ω, t > 0, u ε (x, 0) = u 0 (x), x ∈ Ω; u ε (a, t) = u ε (b, t) = 0, t ≥ 0, 3.1. A finite difference scheme and main results on error bounds. Choose a mesh size h := ∆x = (b -a)/M with M being a positive integer and a time step τ := ∆t > 0 and denote the grid points and time steps as x j := a + jh, j = 0, 1, • • • , M ; t k := kτ, k = 0, 1, 2, . . . Define the index sets T M = {j | j = 1, 2, • • • , M -1}, T 0 M = {j | j = 0, 1, • • • , M }. Let u ε,k j be the approximation of u ε (x j , t k ), and denote u ε,k = (u ε,k 0 , u ε,k 1 , . . . , u ε,k M ) T ∈ C M+1 as the numerical solution vector at t = t k . Define the standard finite difference operators δ c t u k j = u k+1 j -u k-1 j 2τ , δ + x u k j = u k j+1 -u k j h , δ 2 x u k j = u k j+1 -2u k j + u k j-1 h 2 . Denote X M = v = (v 0 , v 1 , . . . , v M ) T | v 0 = v M = 0 ⊆ C M+1 , equipped with inner products and norms defined as (recall that u 0 = v 0 = u M = v M = 0 by Dirichlet boundary condition) (u, v) = h M-1 j=1 u j v j , u, v = h M-1 j=0 u j v j , u ∞ = sup j∈T 0 M |u j |; u 2 = (u, u), |u| 2 H 1 = δ + x u, δ + x u , u 2 H 1 = u 2 + |u| 2 H 1 . (3.2) Then we have for u, v ∈ X M , (3.3) (-δ 2 x u, v) = δ + x u, δ + x v = (u, -δ 2 x v). Consider a semi-implicit finite difference (SIFD) discretization of (3.1) as following (3.4) iδ c t u ε,k j = - 1 2 δ 2 x (u ε,k+1 j + u ε,k-1 j ) + u ε,k j ln(ε + |u ε,k j |) 2 , j ∈ T M , k ≥ 1. The boundary and initial conditions are discretized as (3.5) u ε,k 0 = u ε,k M = 0, k ≥ 0; u ε,0 j = u 0 (x j ), j ∈ T 0 M . In addition, the first step u ε,1 j can be obtained via the Taylor expansion as (3.6) u ε,1 j = u ε,0 j + τ u 1 (x j ), j ∈ T 0 M , where u 1 (x) := ∂ t u ε (x, 0) = i u ′′ 0 (x) -u 0 (x) ln(ε + |u 0 (x)|) 2 , a ≤ x ≤ b. Let 0 < T < T max with T max the maximum existence time of the solution u ε to the problem (3.1) for a fixed 0 ≤ ε ≪ 1. By using the standard von Neumann analysis, we can show that the discretization (3.4) is conditionally stable under the stability condition (3.7) 0 < τ ≤ 1 2 max{| ln ε|, ln(ε + max j∈TM |u ε,k j |)} , 0 ≤ k ≤ T τ . Define the error functions e ε,k ∈ X M as (3.8) e ε,k j = u ε (x j , t k ) -u ε,k j , j ∈ T 0 M , 0 ≤ k ≤ T τ , where u ε is the solution of (3.1). Then we have the following error estimates for (3.4) with (3.5) and (3.6). Theorem 3.1 (Main result). Assume that the solution u ε is smooth enough over Ω T := Ω × [0, T ], i.e. (A) u ε ∈ C [0, T ]; H 5 (Ω) ∩ C 2 [0, T ]; H 4 (Ω) ∩ C 3 [0, T ]; H 2 (Ω) , and there exist ε 0 > 0 and C 0 > 0 independent of ε such that u ε L ∞ (0,T ;H 5 (Ω)) + ∂ 2 t u ε L ∞ (0,T ;H 4 (Ω)) + ∂ 3 t u ε L ∞ (0,T ;H 2 (Ω)) ≤ C 0 , uniformly in 0 ≤ ε ≤ ε 0 . Then there exist h 0 > 0 and τ 0 > 0 sufficiently small with h 0 ∼ √ εe -CT | ln(ε)| 2 and τ 0 ∼ √ εe -CT | ln(ε)| 2 such that, when 0 < h ≤ h 0 and 0 < τ ≤ τ 0 satisfying the stability condition (3.7), we have the following error estimates e ε,k ≤ C 1 (ε, T )(h 2 + τ 2 ), 0 ≤ k ≤ T τ , e ε,k H 1 ≤ C 2 (ε, T )(h 2 + τ 2 ), u ε,k ∞ ≤ Λ + 1, (3.9) where Λ = u ε L ∞ (ΩT ) , C 1 (ε, T ) ∼ e CT | ln(ε)| 2 , C 2 (ε, T ) ∼ 1 ε e CT | ln(ε)| 2 and C depends on C 0 . The error bounds in this Theorem show not only the quadratical convergence in terms of the mesh size h and time step τ but also how the explicit dependence on the regularization parameter ε. Here we remark that the Assumption (A) is valid at least in the case of taking Gaussian as the initial datum. Define the error functions e ε,k ∈ X M as (3.10) e ε,k j = u(x j , t k ) -u ε,k j , j ∈ T 0 M , 0 ≤ k ≤ T τ , where u ε is the solution of the LogSE (1.1) with Ω = (a, b). Combining Proposition 2.5 and Theorem 3.1, we immediately obtain (see an illustration in the following diagram): u ε,k O(h 2 +τ 2 ) / / O(ε)+O(h 2 +τ 2 ) * * U U U U U U U U U U U u ε (•, t k ) O(ε) u(•, t k ) Corollary 3.2. Under the assumptions of Proposition 2.5 and Theorem 3.1, we have the following error estimates e ε,k ≤ C 1 ε + C 1 (ε, T )(h 2 + τ 2 ), e ε,k H 1 ≤ C 2 ε 1/2 + C 2 (ε, T )(h 2 + τ 2 ), 0 ≤ k ≤ T τ , (3.11) where C 1 and C 2 are presented as in Proposition 2.5, and C 1 (ε, T ) and C 2 (ε, T ) are given in Theorem 3.1. Error estimates. Define the local truncation error ξ ε,k j ∈ X M for k ≥ 1 as ξ ε,k j = iδ c t u ε (x j , t k ) + 1 2 δ 2 x u ε (x j , t k+1 ) + δ 2 x u ε (x j , t k-1 ) -u ε (x j , t k ) ln(ε + |u ε (x j , t k )|) 2 , j ∈ T M , 1 ≤ k < T τ , (3.12) then we have the following bounds for the local truncation error. Lemma 3.3 (Local truncation error). Under Assumption (A), we have ξ ε,k H 1 h 2 + τ 2 , 1 ≤ k < T τ . Proof. By Taylor expansion, we have (3.13) ξ ε,k j = iτ 2 4 α ε,k j + τ 2 2 β ε,k j + h 2 12 γ ε,k j , where α ε,k j = 1 -1 (1 -|s|) 2 ∂ 3 t u ε (x j , t k + sτ )ds, β ε,k j = 1 -1 (1 -|s|)∂ 2 t u ε xx (x j , t k + sτ )ds, γ ε,k j = 1 -1 (1 -|s|) 3 ∂ 4 x u ε (x j + sh, t k+1 ) + ∂ 4 x u ε (x j + sh, t k-1 ) ds. By the Cauchy-Schwarz inequality, we can get that α ε,k 2 = h M-1 j=1 |α ε,k j | 2 ≤ h 1 -1 (1 -|s|) 4 ds M-1 j=1 1 -1 ∂ 3 t u ε (x j , t k + sτ ) 2 ds = 2 5 1 -1 ∂ 3 t u ε (•, t k + sτ ) 2 L 2 (Ω) ds - 1 -1 M-1 j=0 xj+1 xj (|∂ 3 t u ε (x, t k + sτ )| 2 -|∂ 3 t u ε (x j , t k + sτ )| 2 )dxds = 2 5 1 -1 ∂ 3 t u ε (•, t k + sτ ) 2 L 2 (Ω) ds - 1 -1 M-1 j=0 xj+1 xj ω xj ∂ x |∂ 3 t u ε (x ′ , t k + sτ )| 2 dx ′ dωds ≤ 2 5 1 -1 ∂ 3 t u ε (•, t k + sτ ) 2 L 2 (Ω) + 2h ∂ 3 t u ε x (•, t k + sτ ) L 2 (Ω) ∂ 3 t u ε (•, t k + sτ ) L 2 (Ω) ds ≤ max 0≤t≤T ∂ 3 t u ε L 2 (Ω) + h ∂ 3 t u ε x L 2 (Ω) 2 , which yields that when h ≤ 1, α ε,k ≤ ∂ 3 t u ε L ∞ (0,T ;H 1 (Ω)) . Applying the similar approach, it can be established that β ε,k ≤ 2 ∂ 2 t u ε L ∞ (0,T ;H 3 (Ω)) . On the other hand, we can obtain that γ ε,k 2 ≤ h 1 -1 (1 -|s|) 6 ds M-1 j=1 1 -1 ∂ 4 x u ε (x j + sh, t k+1 ) + ∂ 4 x u ε (x j + sh, t k-1 ) 2 ds ≤ 4h 7 M-1 j=1 1 -1 ∂ 4 x u ε (x j + sh, t k+1 ) 2 + ∂ 4 x u ε (x j + sh, t k-1 ) 2 ds ≤ 8 7 ∂ 4 x u ε (•, t k-1 ) 2 L 2 (Ω) + ∂ 4 x u ε (•, t k+1 ) 2 L 2 (Ω) ≤ 4 u ε 2 L ∞ (0,T ;H 4 (Ω)) , which implies that γ ε,k ≤ 2 u ε L ∞ (0,T ;H 4 (Ω)) . Hence by Assumption (A), we get ξ ε,k τ 2 ∂ 3 t u ε L ∞ (0,T ;H 1 (Ω)) + ∂ 2 t u ε L ∞ (0,T ;H 3 (Ω)) + h 2 u ε L ∞ (0,T ;H 4 (Ω)) C0 τ 2 + h 2 . Applying δ + x to ξ ε,k and using the same approach, we can get that |ξ ε,k | H 1 τ 2 ∂ 3 t u ε L ∞ (0,T ;H 2 (Ω)) + ∂ 2 t u ε L ∞ (0,T ;H 4 (Ω)) + h 2 u ε L ∞ (0,T ;H 5 (Ω)) C0 τ 2 + h 2 , which completes the proof. For the first step, we have the following estimates. Lemma 3.4 (Error bounds for k = 1). Under Assumption (A), the first step errors of the discretization (3.6) satisfy e ε,0 = 0, e ε,1 H 1 τ 2 . Proof. By the definition of u ε,1 j in (3.6), we have e ε,1 j = τ 2 1 0 (1 -s)u ε tt (x j , sτ )ds, which implies that e ε,1 τ 2 ∂ 2 t u ε L ∞ (0,T ;H 1 (Ω)) τ 2 , |e ε,1 | H 1 τ 2 ∂ 2 t u ε L ∞ (0,T ;H 2 (Ω)) τ 2 , and the proof is completed. Proof. [Proof of Theorem 3.1] We prove (3.9) by induction. It follows from Lemma 3.4 that (3.9) is true for k = 0, 1. Assume (3.9) is valid for k ≤ n ≤ T τ -1. Next we need to show that (3.9) still holds for k = n + 1. Subtracting (3.4) from (3.12), we get the error equations (3.14) iδ c t e ε,m j = - 1 2 (δ 2 x e ε,m+1 j +δ 2 x e ε,m-1 j )+r ε,m j +ξ ε,m j , j ∈ T M , 1 ≤ m ≤ T τ -1, where r ε,m ∈ X M represents the difference between the logarithmic nonlinearity (3.15) r ε,m j = u ε (x j , t m ) ln(ε + |u ε (x j , t m )|) 2 -u ε,m j ln(ε + |u ε,m j |) 2 , 1 ≤ m ≤ T τ -1. Multiplying both sides of (3.14) by 2τ (e ε,m+1 j + e ε,m-1 j ), summing together for j ∈ T M and taking the imaginary parts, we obtain for 1 ≤ m < T /τ , (3.16) e ε,m+1 2 -e ε,m-1 2 = 2τ Im(r ε,m + ξ ε,m , e ε,m+1 + e ε,m-1 ) ≤ 2τ r ε,m 2 + ξ ε,m 2 + e ε,m+1 2 + e ε,m-1 2 . Summing (3.16) for m = 1, 2, • • • , n (n ≤ T τ -1), we obtain e ε,n+1 2 + e ε,n 2 ≤ e ε,0 2 + e ε,1 2 + 2τ e ε,n+1 2 + 2τ n-1 m=0 ( e ε,m 2 + e ε,m+1 2 ) + 2τ n m=1 r ε,m 2 + ξ ε,m 2 . (3.17) For m ≤ n, when |u ε,m j | ≤ |u ε (x j , t m )|, we write r ε,m j as |r ε,m j | = e ε,m j ln(ε + |u ε (x j , t m )|) 2 + 2u ε,m j ln ε + |u ε (x j , t m )| ε + |u ε,m j | ≤ 2 max{ln(ε -1 ), | ln(ε + Λ)|}|e ε,m j | + 2|u ε,m j | ln 1 + |u ε (x j , t m )| -|u ε,m j | ε + |u ε,m j | ≤ 2|e ε,m j |(1 + max{ln(ε -1 ), | ln(ε + Λ)|}). On the other hand, when |u ε (x j , t m )| ≤ |u ε,m j |, we write r ε,m j as |r ε,m j | = e ε,m j ln(ε + |u ε,m j |) 2 + 2u ε (x j , t m ) ln ε + |u ε (x j , t m )| ε + |u ε,m j | ≤ 2 max{ln(ε -1 ), | ln(ε + 1 + Λ)|}|e ε,m j | + 2|u ε (x j , t m )| ln 1 + |u ε,m j | -|u ε (x j , t m )| ε + |u ε (x j , t m )| ≤ 2|e ε,m j |(1 + max{ln(ε -1 ), | ln(ε + 1 + Λ)|}) , where we use the assumption that u ε,m ∞ ≤ Λ + 1 for m ≤ n. Thus it follows that r ε,m 2 | ln(ε)| 2 e ε,m 2 , when ε is sufficiently small. Thus when τ ≤ 1 2 , by using Lemmas 3.3, 3.4 and (3.17), we have e ε,n+1 2 + e ε,n 2 e ε,0 2 + e ε,1 2 + τ n-1 m=0 ( e ε,m 2 + e ε,m+1 2 ) + τ n m=1 r ε,m 2 + ξ ε,m 2 (h 2 + τ 2 ) 2 + τ | ln(ε)| 2 n-1 m=0 ( e ε,m 2 + e ε,m+1 2 ). We emphasize here that the implicit multiplicative constant in this inequality depends only on C 0 , but not on n. Applying the discrete Gronwall inequality, we can conclude that e ε,n+1 2 e CT | ln(ε)| 2 (h 2 + τ 2 ) 2 , for some C depending on C 0 , which gives the error bound for e ε,k with k = n + 1 in (3.9) immediately. To estimate |e ε,n+1 | H 1 , multiplying both sides of (3.14) by 2(e ε,m+1 j -e ε,m-1 j ) for m ≤ n, summing together for j ∈ T M and taking the real parts, we obtain |e ε,m+1 | 2 H 1 -|e ε,m-1 | 2 H 1 = -2 Re r ε,m + ξ ε,m , e ε,m+1 -e ε,m-1 = 2τ Im r ε,m + ξ ε,m , -δ 2 x (e ε,m+1 + e ε,m-1 ) = 2τ Im δ + x (r ε,m + ξ ε,m ), δ + x (e ε,m+1 + e ε,m-1 ) ≤ 2τ |r ε,m | 2 H 1 + |ξ ε,m | 2 H 1 + |e ε,m+1 | 2 H 1 + |e ε,m-1 | 2 H 1 . (3.18) To give the bound for δ + x r ε,m , for simplicity of notation, denote u ε,m j,θ = θu ε (x j+1 , t m ) + (1 -θ)u ε (x j , t m ), v ε,m j,θ = θv ε,m j+1 + (1 -θ)v ε,m j , for j ∈ T M and θ ∈ [0, 1]. Then we have δ + x r ε,m j = 2δ + x (u ε (x j , t m ) ln(ε + |u ε (x j , t m )|)) -2δ + x (u ε,m j ln(ε + |u ε,m j |)) = 2 h 1 0 [u ε,m j,θ ln(ε + |u ε,m j,θ |)] ′ (θ)dθ - 2 h 1 0 [v ε,m j,θ ln(ε + |v ε,m j,θ |)] ′ (θ)dθ = I 1 + I 2 + I 3 , where I 1 := 2δ + x u ε (x j , t m ) 1 0 ln(ε + |u ε,m j,θ |)dθ -2δ + x u ε,m j 1 0 ln(ε + |v ε,m j,θ |)dθ, I 2 := δ + x u ε (x j , t m ) 1 0 |u ε,m j,θ | ε + |u ε,m j,θ | dθ -δ + x u ε,m j 1 0 |v ε,m j,θ | ε + |v ε,m j,θ | dθ, I 3 := δ + x u ε (x j , t m ) 1 0 (u ε,m j,θ ) 2 |u ε,m j,θ |(ε + |u ε,m j,θ |) dθ -δ + x u ε,m j 1 0 (v ε,m j,θ ) 2 |v ε,m j,θ |(ε + |v ε,m j,θ |) dθ. Then we estimate I 1 , I 2 and I 3 , separately. Similar as before, we have |I 1 | ≤ 2|δ + x u ε (x j , t m )| 1 0 ln ε + |u ε,m j,θ | ε + |v ε,m j,θ | dθ + 2 δ + x e ε,m j 1 0 ln(ε + |v ε,m j,θ |) dθ = 2|δ + x u ε (x j , t m )| 1 0 ln 1 + |u ε,m j,θ | -|v ε,m j,θ | ε + min{|u ε,m j,θ |, |v ε,m j,θ |} dθ + 2 δ + x e ε,m j 1 0 ln(ε + |v ε,m j,θ |) dθ ≤ 2 ε |δ + x u ε (x j , t m )| |e ε,m j | + |e ε,m j+1 | + 2 δ + x e ε,m j max{ln(ε -1 ), | ln(ε + 1 + Λ)|} 1 ε |e ε,m j | + |e ε,m j+1 | + ln(ε -1 ) δ + x e ε,m j , and |I 2 | = δ + x u ε (x j , t m ) 1 0 |u ε,m j,θ | ε + |u ε,m j,θ | - |v ε,m j,θ | ε + |v ε,m j,θ | dθ + δ + x e ε,m j 1 0 |v ε,m j,θ | ε + |v ε,m j | dθ ≤ |δ + x e ε,m j | + |δ + x u ε (x j , t m )| 1 0 ε|u ε,m j,θ -v ε,m j,θ | (ε + |u ε,m j,θ |)(ε + |v ε,m j,θ |) dθ ≤ |δ + x e ε,m j | + |δ + x u ε (x j , t m )| ε 1 0 |u ε,m j,θ -v ε,m j,θ |dθ |δ + x e ε,m j | + 1 ε |e ε,m j | + |e ε,m j+1 | . In view of the inequality that (u ε,m j,θ ) 2 |u ε,m j,θ |(ε + |u ε,m j,θ |) - (v ε,m j,θ ) 2 |v ε,m j,θ |(ε + |v ε,m j,θ |) = (u ε,m j,θ ) 2 -u ε,m j,θ v ε,m j,θ |u ε,m j,θ |(ε + |u ε,m j,θ |) + u ε,m j,θ v ε,m j,θ |u ε,m j,θ |(ε + |u ε,m j,θ |) - (v ε,m j,θ ) 2 |v ε,m j,θ |(ε + |v ε,m j,θ |) ≤ |u ε,m j,θ -v ε,m j,θ | ε + u ε,m j,θ (v ε,m j,θ ) 2 (u ε,m j,θ -v ε,m j,θ ) + εv ε,m j,θ (u ε,m j,θ |v ε,m j,θ | -|u ε,m j,θ |v ε,m j,θ ) |u ε,m j,θ ||v ε,m j,θ |(ε + |u ε,m j,θ |)(ε + |v ε,m j,θ |) ≤ 4|u ε,m j,θ -v ε,m j,θ | ε , we can obtain that I 3 |δ + x e ε,m j | + 1 ε |e ε,m j | + |e ε,m j+1 | . Thus we can conclude that |δ + x r ε,m j | 1 ε |e ε,m j | + |e ε,m j+1 | + ln(ε -1 ) δ + x e ε,m j . Summing (3.18) for m = 1, 2, • • • , n (n ≤ T τ -1), we obtain |e ε,n+1 | 2 H 1 + |e ε,n | 2 H 1 ≤ |e ε,0 | 2 H 1 + |e ε,1 | 2 H 1 + τ n m=1 |r ε,m | 2 H 1 + |ξ ε,m | 2 H 1 + τ |e ε,n+1 | 2 H 1 + τ n-1 m=0 (|e ε,m | 2 H 1 + |e ε,m+1 | 2 H 1 ). Thus when τ ≤ 1/2, by using Lemmas 3.3 and 3.4, we have |e ε,n+1 | 2 H 1 + |e ε,n | 2 H 1 |e ε,0 | 2 H 1 + |e ε,1 | 2 H 1 + τ n m=1 1 ε 2 |e ε,m | 2 H 1 + |ξ ε,m | 2 H 1 + τ | ln(ε)| 2 n-1 m=0 |e ε,m | 2 H 1 + |e ε,m+1 | 2 H 1 e CT | ln(ε)| 2 ε 2 (h 2 + τ 2 ) 2 + τ | ln(ε)| 2 n-1 m=0 (|e ε,m | 2 H 1 + |e ε,m+1 | 2 H 1 ). Applying the discrete Gronwall's inequality, we can get that |e ε,n+1 | 2 H 1 e CT | ln(ε)| 2 (h 2 + τ 2 ) 2 /ε 2 , which establishes the error estimate for e ε,k H 1 for k = n + 1. Finally the boundedness for the solution u ε,k can be obtained by the triangle inequality u ε,k ∞ ≤ u ε (•, t k ) L ∞ (Ω) + e ε,k ∞ , and the inverse Sobolev inequality [START_REF] Thomée | Galerkin finite element methods for parabolic problems[END_REF] e ε,k ∞ e ε,k H 1 , which completes the proof of Theorem 3.1. Case II: A general initial data, i.e. u 0 in (1.1) is chosen as (4.2) u 0 (x) = tanh(x)e -x 2 , x ∈ R, which is the multiplication of a dark soliton of the cubic nonlinear Schrödinger equation and a Gaussian. Notice that in this case, the logarithmic term ln |u 0 | 2 is singular at x = 0. The RLogSE (2.1) is solved numerically by the SIFD (3.4) on domains Ω = [-12 , 12] and Ω = [-16, 16] for Case I and II, respectively. To quantify the numerical errors, we introduce the following error functions: e ε (t k ) := u(•, t k ) -u ε (•, t k ), e ε (t k ) := u ε (•, t k ) -u ε,k , e ε (t k ) := u(•, t k ) -u ε,k , e ε E := |E(u) -E ε (u ε )|. (4.3) Here u and u ε are the exact solutions of the LogSE (1.1) and RLogSE (2.1), respectively, while u ε,k is the numerical solution of the RLogSE (2.1) obtained by the SIFD (3.4). The 'exact' solution u ε is obtained numerically by the SIFD (3.4) with a very small time step, e.g. τ = 0.01/2 9 and a very fine mesh size, e.g. h = 1/2 15 . Similarly, the 'exact' solution u in Case II is obtained numerically by the SIFD (3.4) with a very small time step and a very fine mesh size as well as a very small regularization parameter ε, e.g. ε = 10 -14 . The energy is obtained by the trapezoidal rule for approximating the integrals in the energy (1.2) and (2.2). 4.1), which confirms the error bounds in Corollary 3.2. Conclusion. In order to overcome the singularity of the log-nonlinearity in the logarithmic Schrödinger equation (LogSE), we proposed a regularized logarithmic Schrödinger equation (RLogSE) with a regularization parameter 0 < ε ≪ 1 and established linear convergence between RLogSE and LogSE in terms of the small regularization parameter. Then we presented a semi-implicit finite difference method Theorem 2 . 2 . 22 Let λ ∈ R and ε > 0. Consider (1.1) and (2.1) on Ω = R d , or bounded Ω with homogeneous Dirichlet or periodic boundary condition. Consider an initial datum u 3.5], established initially in [13, Lemme 1.1.1]. Lemma 2.4. Let ε ≥ 0 and denote f ε (z) = z ln(ε + |z|), then we have 4 . 4 Numerical results. In this section, we test the convergence rate of the regularized model (2.1) and the SIFD(3.4). To this end, we take d = 1, Ω = R and λ = -1 in the LogSE (1.1) and consider two different initial data: Case I: A Gaussian initial data, i.e. u 0 in (1.1) is chosen as(4.1) u 0 (x) = 4 -λ/πe ivx+ λ 2 x 2 , x ∈ R, with v = 1.In this case, the LogSE (1.1) admits the moving Gausson solution (1.7) with v = 1 and b 0 = 4 -λ/π as the exact solution. 4. 1 . 1 Convergence rate of the regularized model. Here we consider the error between the solutions of the RLogSE (2.1) and the LogSE (1.1). Fig. 4.1 shows e ε , e ε H 1 , e ε ∞ (the definition of the norms is given in (3.2)) at time t = 0.5 for Cases I & II, while Fig. 4.2 depicts e ε E (0.5) for Cases I & II and time evolution of e ε (t) with different ε for Case I. For comparison, similar to Fig. 4.1, Fig. 4.3 displays the convergent results from (2.8) to (1.1). From Figs. 4.1, 4.2 & 4.3 and additional numerical results not shown here for brevity, we can draw the following conclusions: (i) The solution of the RLogSE (2.1) converges linearly to that of the LogSE (1.1) in terms of the regularization parameter ε in both L 2 -norm and L ∞ -norm, and respectively, the convergence rate becomes O( √ ε) in H 1 -norm for Case II. (ii) The regularized energy E ε (u ε ) converges linearly to the energy E(u) in terms of ε. (iii) The constant C in (2.6) may grow linearly with time T and it is independent of ε. (iv) The solution of (2.8) converges at O( √ ε) to that of (1.1) in both L 2 -norm and L ∞ -norm, and respectively, the convergence rate becomes O(ε 1/4 ) in H 1 -norm for Case II. Thus (2.1) is much more accurate than (2.8) for the regularization of the LogSE (1.1). (v) The numerical results agree and confirm our analytical results in Section 2. 2 || e ε || H 1 Fig. 4 . 1 :Fig. 4 . 2 : 214142 Fig. 4.1: Convergence of the RLogSE (2.1) to the LogSE (1.1), i.e. the error e ε (0.5) in different norms vs the regularization parameter ε for Case I (left) and Case II (right). 4. 2 . 1 || e ε || H 1 Fig. 4 . 3 : 21143 Fig. 4.3: Convergence of the RLogSE (2.8) to the LogSE (1.1), i.e. the error e ε (0.5) in different norms vs the regularization parameter ε for Case I (left) and Case II (right). 2 Fig. 4 . 4 : 244 Fig. 4.4: Convergence of the SIFD (3.4) to the RLogSE (2.1), i.e. errors e ε (0.5) vs τ (with h = 75τ /64) under different ε for Case I initial data. * This work was partially supported by the Ministry of Education of Singapore grant R-146-000-223-112 (MOE2015-T2-2-146) (W. Bao). for discretizing RLogSE and proved second-order convergence rates in terms of mesh size h and time step τ . Finally, we established error bounds of the semi-implicit finite difference method to LogSE, which depend explicitly on the mesh size h and time step τ as well as the small regularization parameter ε. Our numerical results confirmed our error bounds and demonstrated that they are sharp.

01745418

en

2024/03/05 22:32:07

2017

https://hal.science/hal-01745418/file/Portal.2017.TNANO.Design%20and%20Simulation%20of%20a%20128%20kb%20Embedded%20Nonvolatile%20Memory%20Based%20on%20a%20Hybrid%20RRAM%20%28HfO2%20%2928%20nm%20FDSOI%20CMOS%20Technology..pdf

Jean-Michel Portal email: [email protected] Marc Bocquet Santhosh Onkaraiah Mathieu Moreau Hassen Aziza Damien Deleruyelle Kholdoun Torki Elisa Vianello Alexandre Levisse Bastien Giraud Olivier Thomas Design and Simulation of a 128 kb Embedded Nonvolatile Memory Based on a Hybrid RRAM (HfO_2 )/28 nm FDSOI CMOS Technology Keywords: Embedded non-volatile memory, memory architecture, resistive switching memory, RRAM, I des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. two strong markets: the high performance, high density, high capacity memories for computation in servers, and the low energy, low cost embedded-memories devoted to autonomous connected nodes. The growth of these two markets is strongly related to each other and is at the basis of Internet of Things market. High performance computing applications for servers require high quantities of NVM [START_REF] Lynch | Big data: How do your data grow?[END_REF] in order to store and process tremendous amount of data generated by the autonomous connected nodes that are spread in the environment [START_REF] Gubbia | Internet of Things (IoT): A vision, architectural elements, and future directions[END_REF]. To this aim, high performance Volatile and Non-Volatile memories are used. In classic computing, the first elements of memory hierarchy are the volatile memories: SRAM and DRAM. The DRAM memory is used as a buffer between NVM and SRAM cache memories. At the end of the hierarchy are NVM memories HDD [START_REF] Shiroishi | Future Options for HDD Storage[END_REF] and NAND Flash technologies [START_REF] Helm | A 128Gb MLC NAND-Flash device using 16nm planar cell[END_REF]. The main constraints of these memories are (i) overall power consumption (from the DRAM refreshment to the NVM programming operations), (ii) communication cost between processing units and memories, and (iii) production costs per bit. Embedded memories for autonomous connected objects are more focused in reducing energy consumption and the overall chip cost. These device operations are based on classical memory hierarchy. Von Neumann or Harvard architectures are considered. SRAM memories are used to store the variables while NVM is used to store the instruction code. Embedded NVM are based on NOR flash technologies [START_REF] Ogura | A 90nm Floating Gate "B4-Flash" Memory Technology-Breakthrough of the Gate Length Limitation on NOR Flash Memory[END_REF] [6] [START_REF] Fastow | A 45nm NOR Flash Technology with Self-Aligned Contacts and 0.024µm2 Cell Size for Multi-level Applications[END_REF] down to 40nm commercial CMOS technologies. For both server applications (standalone NVM) and autonomous connected nodes (embedded NVM), scaling down the technology nodes is a real struggle. NAND flash technology complexity (air gap [START_REF] Seo | Highly reliable M1X MLC NAND flash memory cell with novel active air-gap and p+ poly process integration technologies[END_REF], vertical stacking [START_REF] Park | Three-Dimensional 128 Gb MLC Vertical nand Flash Memory With 24-WL Stacked Layers and 50 MB/s High-Speed Programming[END_REF], multiple patterning [START_REF] Pikus | Advanced multi-patterning and hybrid lithography techniques[END_REF]) and production costs are becoming unaffordable. On the embedded side, NOR flash is facing extremely complex structure [START_REF] Do | Scaling of split-gate flash memory and its adoption in modern embedded non-volatile applications[END_REF], and the co-integration with advanced CMOS nodes leads to reliability issues due to the high voltages needed for NVM programming operations. Moreover, scaling down floating gate technologies leads to Design and Simulation of a 128kb Embedded Non-Volatile Memory based on a Hybrid RRAM (HfO 2 ) / 28nm FDSOI CMOS Technology Jean-Michel Portal, Marc Bocquet, Santhosh Onkaraiah, Mathieu Moreau, Hassen Aziza, Damien Deleruyelle, Kholdoun Torki, Elisa Vianello, Alexandre Levisse, Bastien Giraud, Olivier Thomas, Fabien Clermidy reduced endurance and retention [START_REF] Grupp | The bleak future of NAND flash memory[END_REF]. In order to continue the scaling of NVM technologies, for both Embedded and Standalone applications, emerging resistive switching technologies (i.e. RRAM) are extensively investigated [START_REF] Clermidy | Resistive memories: Which applications?[END_REF] because of their Back End of Line (BEoL) compatible structure and their high scalability [START_REF] Lee | Scaling trends and challenges of advanced memory technology[END_REF]. Among all the RRAM technologies, the Oxide-Based RRAM (OxRAM), because of their simple stack structure, fast switching, and common material appear as a promising solution for Flash memories replacement [START_REF] Benoist | Advanced CMOS Resistive RAM Solution as Embedded Non-Volatile Memory[END_REF]. Several metal oxides can be used to obtain the OxRAM behavior such as AlOx, NiOx, TaOx, TiOx or HfOx [16] [17] [18] [START_REF] Kwon | Atomic structure of conducting nanofilaments in TiO2 resistive switching memory[END_REF]. Integrating RRAMs as Flash memories replacement can be a solution either for Standalone Memories or for Embedded Memories. In the context of Standalone Memories, the supporting CMOS technology is totally dedicated to the memory integration and, as a consequence, the associated CMOS technology does not suffer from the memory requirements. In the context of Embedded Memories, CMOS technology is a limitation due to integration severe constraints. For example, in NOR Flash integration at advanced CMOS nodes, High Voltage Thick Oxide Transistors are integrated at the cost of additional masks, higher thermal budget and higher occupied area [START_REF] Tanzawa | High-voltage transistor scaling circuit techniques for high-density negative-gate channel-erasing NOR flash memories[END_REF]. The co-integration of RRAM at sub 30nm CMOS technologies leads to reliability issues on the Thin Oxide Transistors and is usually solved by using the IO transistors (thicker gate oxide process with no impact on the thin oxide transistor performances) [START_REF] Fackenthal | 19.7 A 16Gb ReRAM with 200MB/s write and 1GB/s read in 27nm technology[END_REF]. The direct impact is a higher maximum voltage, but also a larger footprint (up to 15 times for an equivalent drive). Several Embedded RRAM Memories are presented considering thick gate oxide transistors, in [START_REF] Sheu | A 4Mb embedded SLC resistive-RAM macro with 7.2ns read-write random-access time and 160ns MLC-access capability[END_REF] a 4Mb HfO2-based RRAM memory in 180nm CMOS technology with write verify technic is demonstrated, in [START_REF] Chang | A High-Speed 7.2-ns Read-Write Random Access 4-Mb Embedded Resistive RAM (ReRAM) Macro Using Process-Variation-Tolerant Current-Mode Read Schemes[END_REF] a 4Mb embedded RRAM memory is demonstrated with process variation control circuits in 130nm CMOS technology. In [START_REF] Chang | A Low-Voltage Bulk-Drain-Driven Read Scheme for Sub-0.5 V 4 Mb 65 nm Logic-Process Compatible Embedded Resistive RAM (ReRAM) Macro[END_REF] a 4Mb macro using a unipolar RRAM is embedded in 65nm CMOS technology. A 1Mb RRAM memory embedded 28nm bulk CMOS is demonstrated in [START_REF] Chang | Embedded 1Mb ReRAM in 28nm CMOS with 0.27-to-1V Read Using Swing-Sample-and-Couple Sense Amplifier and Self-Boost-Write-Termination Scheme[END_REF] with write assist circuits. However, in these papers the reliability issue on MOS transistors during programming operations and during the forming step is not considered. Others studies using thick oxide transistors and Conductive Bridge Random Access Memories (CBRAM) [START_REF] Fackenthal | 19.7 A 16Gb ReRAM with 200MB/s write and 1GB/s read in 27nm technology[END_REF] or Phase Change Memories (PCM) [START_REF] Borghi | r. a. t. a. 1. w. throughput[END_REF] are reported. In [START_REF] Shen | High-K metal gate contact RRAM (CRRAM) in pure 28nm CMOS logic process[END_REF], a testchip with 28nm thin-gate oxide bitcells was presented and the MOS transistors reliability is shown for a 3V programming voltage. This paper presents the architecture of an embedded 128kb memory cut based on a hybrid RRAM (HfO 2 ) and thin-gate oxide 28nm Fully Depleted Silicon On Insulator (FDSOI) CMOS technology suitable for NOR Flash replacement. Validation of the proposed architecture is performed through post-layout simulation by using RRAM compact model calibrated on CEA-Leti RRAM samples [START_REF] Vianello | Resistive Memories for Ultra-Low-Power embedded computing design[END_REF] and implemented under transistor-level simulator Eldo [START_REF]Mentor Graphics Website[END_REF] and CMOS 28nm FDSOI design kit from STMicroelectronics. The rest of the paper is organized as follows: In the next section, the RRAM technology is briefly introduced together with the compact model calibrated on state of the art devices. In section III, the bit-cell and the memory array organization are described with the different modes of operation (FORMING/SET, RESET and READ). Section IV is dedicated to the full memory macro-cell description, including peripheral circuits, addressing hierarchy and scheduler. The validation of the macro-cell through simulation is presented in Section V. Finally, Section VI gives some concluding remarks and highlights on the proposed embedded memory. II. RRAM OVERVIEW: TECHNOLOGY AND COMPACT MODEL RRAM based on HfO 2 are studied as part of the bit-cell of the proposed memory architecture. The RRAM stack is composed of a 5 nm thick HfO 2 resistive switching layer embedded in-between a TiN/Ti Top Electrode (TE) and a TiN Bottom Electrode (BE). The resistive switching layers are deposited by Atomic Layer Deposition (ALD), whereas the metallic electrodes are deposited by Physical Vapor Deposition (PVD) [START_REF] Vianello | Resistive Memories for Ultra-Low-Power embedded computing design[END_REF]. RRAM modeling used in this study is based on the work presented in [START_REF] Bocquet | Robust compact model for bipolar oxidebased resistive switching memories[END_REF]. This approach relies on electric fieldinduced creation/destruction of oxygen vacancies within the switching layer. The model enables continuous accounting for FORMING, SET and RESET operations into a single master equation in which the resistance is controlled by the radius of a conductive filament (namely r CF ). After calibration, the model satisfactorily matches quasistatic and dynamic experimental data measured on actual HfO 2 -based memory elements. Moreover, to account for the variability of RRAM technology, two corners cases were simulated. They include the two extreme behaviors observed experimentally: one favoring the SET mechanism and slowing the RESET, and the other being the exact opposite. The Fig. 1 show the quasi-static behavior of RRAM devices and the good modeling correlation. The corners encompass the full range of features that ensure to take into account the worst case of FORMING, SET and RESET. The Fig. 2.a describes the dependence of the switching time according to the amplitude of the programming pulse voltage. It is important to note that rapid operations -low consumptions -need higher CMOS standard voltage. Furthermore, the Fig. 2.b. underlines the interest of high voltage applied during the RESET to ensure a high resistance value. These two central behaviors are perfectly captured by the model implemented and extreme behaviors are included within in the corner simulations. III. BIT-CELL AND MEMORY ARRAY ORGANIZATION In this section, the bit-cell structure is detailed together with memory array organization. Based on the array arrangement, biasing conditions for selected and unselected cells in the array are discussed for the different modes of operation. A. Bit-cell structure The bit-cell is based on a 2T1R structure [START_REF] Jovanovic | Design considerations for reliable OxRAM-based non-volatile flipflops in 28nm FD-SOI technology[END_REF], as described in Fig. 3.a and exhibits one NMOS and one PMOS to access the Top Electrode (TE) of the RRAM, whereas the Bottom Electrode (BE) is connected to the Reset Line (RL). In addition to the classical Bit-Line (BL) and Word-Line (WL), two others lines are present namely Reset-Line (RL) and Set-Line (SL). As depicted Fig. 3.b, to avoid any CMOS core process modification, the RRAM elements are introduced in the Metal Insulator Metal (MIM) stack between the first and second 8x metallization level in place of the decoupling capacitance. On one hand, this solution is fully compatible with the standard CMOS 28nm FDSOI process flow but on the other hand this integration scheme is achieved at the cost of a larger RRAM foot-print on the upper metallization level. The additional lines RL and SL are used to perform FORMING/SET and RESET/READ operations on two different paths, as illustrated Fig. 4. Doing so FORMING/SET voltages are applied on the top electrode of the RRAM through the PMOS (Fig. 4.a), whereas ground (gnd) is applied on the top-electrode through a NMOS and RESET voltage is applied on the RL during a RESET operation (Fig. 4.b). With this scheme, degradation of the voltage level for the different operations is reduced, since positive voltages are applied through PMOS transistor whereas ground voltages are applied through NMOS. Moreover, the compliance current, during FORMING / SET operations, is defined by the sizing of the PMOS transistor and by the V DDM biasing. Given that the current flowing through the cell during the RESET operation must be above the FORMING / SET current, the NMOS transistor has to be larger than the PMOS for an equivalent biasing. B. Memory array organization The arrangement of the memory array is given in Fig. 5.a with the schematic view and Fig. 5.b with the corresponding layout view. The RL and WL lines are shared per row whereas BL and SL are shared per column. To access a cell for a given operation, biasing of all the four access lines is mandatory, whereas inhibition voltages must be applied on unselected cells. Depending, on the operation unselected cells are inhibited by turning OFF the access transistor (NMOS and PMOS) with V GS below the threshold voltage or by having a voltage difference on the RRAM close to zero volt. Fig. 6 summarizes the biasing of the different access lines for four cases: selected bit-cell, unselected bit-cell on the same row, unselected bit-cell on the same column, unselected bit-cell on the rest of the array without any common lines with the selected cell. Three potentials are used, V DD equal to V DDM in the array, ground (gnd) and a high voltage (HV). HV takes a value close to 2.5×V DDM for the FORMING operations and a value close to 2×V DDM for the SET and RESET operations. Fig. 6 depicts a single cell access, but this memory array organization enables to perform all the programming operations by selecting an entire or partial row or a full or partial array. This feature offers the capability to have the best compromise between speed and consumption. In the proposed memory macro-cell, the array is defined by the bank size. A bank is composed of 1024 cells organized on 32 rows per 32 columns, in other words, 32 words of 32 bits. To program a word, a RESET operation is first performed on the bits to reset and followed by a SET operation on the remaining bits of the word. Similarly to NOR Flash memory, two programming phases are used to write a word. But, in our case, both operations are selective contrary to Flash memory, where erase operation is applied to all bits in the word followed by a selective write operation. In this section the full macro-cell architecture is detailed, starting with the peripheral blocks to address the bank array, up to the full macro-cell hierarchy including scheduler finite state machine to generate timing and internal signals. It is worth to note that the macro-cell communication bus is fully compatible with AMBA 3 AHB lite protocol [START_REF]AMBA 3 AHB-Lite Protocol Specification Documentation v1.0[END_REF]. A. Peripheral block description The macro-cell architecture is massively multi-bank, thus all peripheral circuits to program and read the content of the bit-cells are introduced at bank level. In this subsection, levelshifters used for programming operations as well as sense amplifier used for reading operation are detailed. Banks are powered with two power supply V DD =V DDana and a higher voltage for FORMING/SET/RESET operations named HV. During programming operation, level shifters are used to drive HV on RL and SL access lines, while classic buffers are used to drive gnd/V DDana on BL and WL access lines. Table I gives a summary of possible biasing of the different access lines. The architecture of level shifters acting on SL and RL are represented in Fig. 7 and Fig. 8 respectively with their schematic and layout views. Level shifter structures are designed with cascade MOS since the voltage difference between any MOS transistor nodes has to be below or equal to VDD to ensure reliability and avoid gate-oxide breakdown. Only bit-cells on a common row are activated at a time during programming operation, thus since SL level shifters are shared per column, they only have to drive single cell FORMING/SET compliance current, limiting the sizing of their output stage. On the contrary, RL level shifters are shared per row, thus during a RESET operation, they may drive up to the 32 bitcells of the addressed word. Thus, the sizing of the RL level shifter output stage is able to drive 32 times the current of a single cell. It is to be noted on the layout view of the RL level shifter (Fig. 8.b), that a single output buffer is multiplied 32 times to form the complete output stage. Similar to the SL level shifter, the input of the RL level shifter is driven by logic gate biased in the V DD domain. In order to sense the value of each bit-cells, sense amplifiers are added on top of the corresponding columns. Applying a current I READ through the resistive element and comparing the resulting voltage V READ with a voltage reference V REF gives the logic value of the bit-cell. I READ is generated thanks to a Wilson current mirror structure. An operational full swing amplifier is used to discriminate between V READ and V REF . Since, RRAM technology is still in development, the reference voltage V REF is provided externally for characterization purpose. Indeed, by trimming V REF value, knowing I READ , it is possible to extract the resistance value of all the bit-cells in the array and thus extract Low Resistance State (LRS) and High Resistance State (HRS) distributions on chip. Doing so at the end of the characterization procedure, V REF value is set between the voltage distribution corresponding to LRS and HRS distributions. Moreover, an external pad gives a direct access to the bit-cell content to extract RRAM resistance value in order to verify the value extracted from V REF trimming. This pad can be connected to the sense amplifier of the first column (BL 0 ) of each bank. A bank is composed of 32 by 32 bit-cells to avoid voltage drop on the access line. Indeed, since the whole circuit is designed with GO1 devices the voltage budget is limited, especially during the FORMING steps. Thus, all peripheral circuits, i.e. level shifters, sensing circuitry are implemented at bank level, as depicted Fig. 10 with the layout view. Moreover, addressing signal gates all control signals at bank level. Thus, unselected bank are completely inactive, with no internal signal change, preventing any disturb or extra powerconsumption. B. Bank organization and addressing hierarchy For the selected bank an acknowledgement signal is generated to properly apply all control signal and to avoid any temporal drift between signals due to hierarchy routing. The top-analog circuit is divided into four sectors of eight pages and each page contains four banks. Control signals, generated by the digital scheduler, are enabled at each level of the hierarchy. Moreover, buffers are inserted on all input signals (data, address and control) and tri-state buffer are inserted on the data-out signals, to prevent delay issue. C. Scheduler description The scheduler is a Finite State Machine (FSM) compatible with AMBA 3 AHB lite protocol, which generates all necessary timing and internal signal to drive the top analog circuit. The timing can be trimmed since a programmable timer gives the time reference. Indeed, the targeted bus clock is in the range of a few ten's of MHz, in other words a few hundreds of 'ns' period, while FORMING/RESET/SET operations are in a range of 'ms' to 'µs' depending on the targeted HV voltage and RRAM technology variability. Thus, to enhance yield, timing has to be programmable in order to fit the voltage/duration dependency of the RRAM technology for the outlier cells of the memory array. The scheduler states are Ready, Read, Write. The Write state is decomposed in a second FSM, since a write operation can be of two kinds FORMING or RESET/SET. Moreover, a write/verify procedure is embedded in the scheduler. After a write, which is a RESET operation followed by a SET operation, a read is performed and data are compared to data in, in case of mismatch, up to 10 cycles of RESET/SET operations can be applied. After 10 cycles, if Write operation still fails the HRESP signal indicates an error. This write/verify procedure has been implemented to tackle cycleto-cycle variability of the RRAM technology. The full layout of the 128kb Non-Volatile Memory based on a Hybrid RRAM (HfO2) / 28nm FDSOI CMOS Technology is given Fig. 11 and its main simulated features are summarized Table II • READ sequentially each addressed word of the bank to retrieve the inverse checkerboard pattern. Fig. 12 shows the FORMING operation of a full bank, all the words are sequentially formed, it is worth to note that the FORMING process duration for a word is 100µs, thus for a full bank, it represents 3.2ms. The Fig. 12 highlights the voltage, current and Conductive Filament (CF) radius variation for the first bit-cell of the first word (Cell0,0) and for the last bit-cell of the last word (Cell31,31). For all the bitcell, the compliance current during FORMING is set to 62µA, representing an overall current of nearly 2mA for a full word. The evolution of the CF radius clearly shows the FORMING operation. The biasing conditions extracted from the simulation are: • All the BL and WL remains to VDDana=1.2V during the FORMING process, • SL is set to HVFORMING=2.75V, instead of 2.8V due to voltage drop in the SL level-shifter, • RL is set to 5mV, instead of ground due to voltage drop in the RL level-shifter for the selected word, whereas it is set to HV FORMING for unselected words. Thus the selected RRAM are biased to 2.75V during FORMING. One can notice a very low leakage on the already formed cell of 0.76µA (see I CELL (cell 0,0 ) during the FORMING of the word 31 in Fig. 12). Fig. 13 shows simulation results for the overall READ/PROGRAM process on a bank (READ to verify FORMING, PROGRAM checkerboard pattern, READ to verify checkerboard pattern, PROGRAM inverse checkerboard pattern, READ to verify inverse checkerboard pattern). Functionality is validated, since the programmed patterns are successfully read. Fig. 14 and Fig. 15 highlight respectively a PROGRAM operation and a READ operation. The PROGRAM operation is divided into two steps, in a first step bit-cells corresponding to DATA_IN='1' are RESET, whereas in a second step bitcells corresponding to DATA_IN='0' are SET. On Fig. 14 DATA_IN 0 ='1', thus the cell 0,0 is RESET, with a current of 87µA for a voltage on the RRAM of 2.32-0.11 = 2.21V. In the SET step, the cell remains unchanged. Each steps, RESET and SET, takes 80ns for a global PROGRAM time of 170ns per word. Global current consumption is in a range of 1.5mA (SET all the cell) to 2.9mA (RESET all the cell). It is important to notice that the RESET current, as shown in Fig. 14, is not constant during the RESET phase. Fig. 15 exhibits the two steps of a READ operation, in the first step the current source is enabled to pre-charge the bit-line for 40ns, during the second step the sense amplifier is activated for 2ns to differentiate between HRS and LRS state. The current through the cell during the READ operation is below 2µA. The READ operation duration together with the voltage on the RRAM (worst case 0.8V for a HRS RRAM) allow to avoid any disturb on the cell. This assumption is validated since there is no CF radius change during the READ process presented Fig. 15. The same simulations are performed for the corner cases obtained from the worst-case measurements on the RRAM cell as depicted in Section II together with the SS corner of the CMOS core process. The To guarantee the functionality of the macro-cell in the worstcase scenarios, timing has to be adapted, due to the voltage/timing characteristic of the RRAM. Doing so for nominal voltage configuration, timings and energy per cell are strongly degraded versus typical simulation as represented Fig. 16 (Typical (V 1 ) versus Corners (V 1 )). To recover typical results at nominal voltages for worst-case scenarios, voltages have to be increased as represented Fig. 16 by V 2 and V 3 voltage sets. However, the maximal stress voltage defined as gate/source or gate/drain voltage ramps up to 1.8V for Moreover, it is interesting to note that the best efficiency is achieved with higher voltages but this track of optimization must be carefully used considering reliability issue of the standard logic MOS devices. Since January 2015 he is the project leader of Silicon Impulse an IC competence center helping companies to design innovative products based on the latest low-power semiconductor technologies. He is author or co-author of 75 articles in international refereed journals and conferences and 25 patents. Figure 1 : 1 Figure1: Experimental I(V) characteristics for Electroforming, Set, and Reset measured on a large number of memory elements reflecting the device-todevice variability presented in[START_REF] Vianello | Resistive Memories for Ultra-Low-Power embedded computing design[END_REF] and simulation results including corners definition. Figure 2 : 2 Figure 2: Experimental (a) switching time for Forming, Set and Reset operations as a function of voltage and (b) RHRS as a function of stop voltage during Reset operation presented in [28] and corresponding simulation results. Figure 3 : 3 Figure 3: (a) Schematic view of the bit-cell with 2T1R and four access lines (BL, WL, SL and RL) (b) Layout of the bit-cell, with a RRAM element introduces in the Metal Insulator Metal (MIM) stack to avoid CMOS process modification at the cost of large cell area. Figure 4 : 4 Figure 4: (a) FORMING/SET are performed through the PMOS, SL is set to the positive operation voltage, RL is grounded whereas NMOS transistor is off (b) RESET is performed through NMOS, RL is set to the positive operation voltage, BL is grounded whereas PMOS transistor is off. For READ operation, BL is set to the read voltage and RL is grounded, here also the PMOS transistor is off. To reduce voltage degradation, two different paths are used through PMOS and NMOS, in order to fit the bipolar behavior of the RRAM. Figure 5 : 5 Figure 5: (a) Schematic view of a two by two bit-cell array (b) Layout view of a two by two bit-cell array. SL and BL lines are shared in column; RL and WL are shared in row. Figure 6 : 6 Figure 6: (a) Array biasing condition for a FORMING/SET operation performed on a single cell (b) Array biasing condition for a RESET operation performed on a single cell (c) Array biasing condition for a READ operation performed on a single cell. Biasing condition of the selected cell can be extended to any cells on a row to perform parallel operations. Biasing conditions can also be applied on the full array to perform global operation. MACRO-CELL ARCHITECTURE OVERVIEW Figure 7 : 7 Figure 7: (a) Schematic view of the level shifter used to drive SL access line (b) Layout view of the level shifter used to drive SL access line. The SL level shifter output swing is defined between VDDM and HV. The level shifter input is controlled with standard voltage logic (VDD domain). Figure 8 : 8 Figure 8: (a) Schematic view of the level shifter used to drive RL access line (b) Layout view of the level shifter used to drive RL access line. The RL level shifter output swing is defined between gnd and HV. The level shifter input is controlled with standard voltage logic (VDD domain). Figure 9 : 9 Figure 9: (a) Schematic view of the sensing circuitry connected to a bit-line including read current generation, output direct access and operational amplifier (b) Layout view of the 32 sensing circuitry associated to the 32 bitline of a single bank. Fig. 9 . 9 Fig.9.a illustrates the sense amplifier architecture, including CDMA standing for Current Direct Memory Access to provide I OUT on the first column of each bank from an external PAD for characterization purpose, doing so internal current mirror is disconnected. Finally, it is worth noting that internal current mirror as well as operational amplifier can be disconnected from V DD when no sensing operation is required. The layout of the sense amplifier together with current mirror is given Fig.9.b. Figure 10 : 10 Figure 10: Layout view of a full bank including a 32 by 32 bit-cells with surrounding peripherals circuits, i.e. 32 SL level shifters below the array, 32 BL drivers and 2 sensing circuitry on top of the array and 32 RL level shifter and 32 WL drivers on the right of the array. Figure 11 : 11 Figure 11: Layout view of the full macro-cell with top analog (right) and scheduler (left). . Figure 12 : 12 Figure 12: Simulation results (corner Typical) of the FORMING process for a full bank. Current through RRAM of the cell0,0 (Word 0, Bit 0) and of the cell31,31 (Word 31, Bit 31), CF radius evolution during forming, as well as access-line voltage to the cell0,0 and cell31,31 are plotted. Figure 13 : 13 Figure 13: Simulation results (corner Typical) of the full PROGRAM/READ process for a full bank. CF radius evolution of the cell0,0 RRAM (Word 0, Bit 0) and of the cell31,31 RRAM (Word 31, Bit 31), during PROGRAM&READ, as well as access-line voltage to the cell0,0 and cell31,31 are plotted. Finally, the similarity of the DATA_IN and DATA_OUT values shows the success of PROGRAM and READ operations. Figure 14 : 14 Figure 14: Simulation results (corner Typical) of a PROGRAM operation on cell0,0. Current and CF radius of the cell0,0 RRAM (Word 0, Bit 0) and of the cell31,31 RRAM (Word 31, Bit 31), during PROGRAM, as well as access-line voltage to the cell0,0 and cell31,31 are plotted. The program operation is composed of two steps, RESET/SET. Fig. 16 . 16 summarizes the time and energy per cell for different operations, comparing typical (CMOS & RRAM TT) and worst-case results using the corners case (CMOS SS & RRAM corner Slow FORMING/SET and Slow-corner RESET) for the nominal voltages, defined as V1 (HV FORMING =2.8V, HV SET/RESET =2.4V) and for two others voltages V2 (HV FORMING =3.0V, HV SET/RESET =2.6V) and V3 (HV FORMING =3.2V, HV SET/RESET =2.8V). Figure 15 : 15 Figure 15: Simulation results (corner Typical) of a READ operation for a full word. CF radius evolution of the cell0,0 RRAM (Word 0, Bit 0) and of the cell31,31 RRAM (Word 31, Bit 31), during PROGRAM&READ, as well as access-line voltage to the cell0,0 and cell31,31 are plotted. Pre-charge and sense phase are detailed. Figure 16 : 16 Figure 16: Operation duration and energy per cell for typical and corners cases, for 3 different voltage sets. FORMING and 1 . 1 4V for SET/RESET with set V 2 and respectively 2V and 1.6V with set V 3 . These voltages remain acceptable for the FDSOI 28nm standard logic CMOS technology. VI. CONCLUSION In this paper, a full 128kb Embedded Non-Volatile Memory based on a Hybrid RRAM (HfO 2 ) & 28nm FDSOI CMOS Technology is presented. The key points of the architecture are the use of standard logic MOS exclusively, avoiding any high voltage MOS usage, program/verify procedure to mitigate cycle to cycle (C2C) variability and direct bit-cell read access for characterization purpose. This architecture has been fully validated through an extensive set of postlayout simulations at different voltage levels and using typical and the most pessimistic corners (MOS SS and RRAM worst FORMING/SET and RESET corners). The memory is functional at all corners for the different set of voltages owing to the time/voltage dependencies of RRAM. 25 papers in international conferences and journals. He is the main author of a book chapter and the main inventor or coinventor of 15 patents. Fabien Clermidy obtained his Ph.D in microelectronic from INPG, Grenoble in 1999 and his supervisor degree in 2011. He is a pioneer in designing Network-on-Chip based multicore. He was the leader of the second generation of Networkon-Chip based multicore dedicated to 3GPP-LTE. At this period, his team elaborated one of the first 3D multi-core prototypes embedding a WIDE-IO DRAM memory called WIOMING. He is currently managing the digital circuit laboratory implied in the development of new architectures using emerging technologies such as 3D TSV, 3D monolithic integration and emerging memories. He has published 2 books, more than 75 journal and conferences papers and is author or co-author of[START_REF] Benoist | Advanced CMOS Resistive RAM Solution as Embedded Non-Volatile Memory[END_REF] patents. Alexandre Levisse received his B.S. (Electrical Engineering) degree in 2012 and his M.S. (Electrical Engineering) degree in 2014, both from Aix-Marseille University, France. He is currently PhD student in CEA-Leti (Grenoble, France) and IM2NP (Aix-Marseille University, France). His research interests include emerging resistive memories with emphasis on circuit design, architecture and crossbar architecture. Elisa Vianello received the Ph.D. degree in microelectronics from the University of Udine, Udine, Italy, and the Polytechnic Institute of Grenoble, Grenoble, France, in 2009. She has been a Scientist with CEA-LETI, Grenoble, since 2011. Olivier Thomas received the M.S. Electrical Engineering degree in 2001 and the Ph.D. degree in microelectronics in 2004. He joined the CEA-LETI Laboratory in the Center for Innovation in Micro & Nanaotechnology (MINATEC), Grenoble, France in 2005. He was first involved in the development of low-power and low-leakage design solutions for digital wireless applications in 65nm Partially-Depleted SOI technology in collaboration with STMicroelectronics. From 2006 to 2010, he was in charge of low power SRAM and Digital design projects in Thin Film SOI technologies. His work was focused on efficient and simple multiple-VT design solutions. From 2010 to 2012, he was a visiting researcher at Berkeley Wireless Research Center (BWRC) of University of California at Berkeley. He worked on methodologies to characterize on large-scale static/dynamic SRAM performances. Back to CEA-LETI, from 2012 to 2014, he launched and led a advanced memory design group at LETI. TABLE I VOLTAGE I SWING ON THE BIT-CELL ACCESS LINES Access lines Voltage swing WL gnd to VDDana BL gnd to VDDana SL VDDana to HV RL gnd to HV scaled Flash memories. In 2005 he joined the Memory group of Im2np (Institut Matériaux Microélectronique Nanosciences de Provence) and became associate professor at Aix-Marseille Université. His research topics include nanoscale electrical characterization by scanning probe microscopy and physical modeling of emerging memory devices such as RRAM. In 2016 he became Professor at the Institut des Nanotechnologies de Lyon (INSA de Lyon) where he currently works on plastic electronics. Hassen Aziza received his B.S. and M.S. degrees in Electrical Engineering, both from University of Marseille, France. He received his Ph.D. degree in 2002 from the University of Marseille, France. Hassen Aziza is currently associate professor at Aix-Marseille University-IM2NP laboratory (Institute of materials, microelectronics and nanosciences of Provence). His research fields cover design, test and reliability of conventional non-volatile memories (Flash & EEPROM) as well as emerging memories (Resistive RAM). He is (co)author of more than 90 papers in international conferences and journals and is (co)inventor of 4 patents. Marc Bocquet received the Ph.D. degree in micro and nanoelectronics from University Grenoble, Grenoble, France, in 2009. He became an Associate Professor with the University of Marseille -Polytech'Marseille, in 2010, and he is a member of the Memories Team, IM2NP. He has conducted several studies on understanding the physical mechanisms in dielectrics to link the physical and chemical properties to the electricalperformance/reliability of memory devices: flash and resistive memory. Kholdoun Torki received the Ph.D. degree in Microelectronics from the Institut National Polytechnique de Grenoble, France, in 1990. He joined CMP as senior engineer in 1990, later on joining CNRS/CMP in 1994. He is currently Technical Director at CMP since 2002. His research interest includes Deep Submicron design methodologies, Non Volatile Memory CMOS co-integration, and 3D-IC integration. He authored and co-authored more than 100 scientific papers, coauthored 2 patents, designed more than 30 ASIC circuits, and participated or coordinated 15 European and National projects. He is member of the Board of Directors at iRoC Technologies. Bastien Giraud received the Ph.D. degree in 2008 from Telecom ParisTech France. The PhD thesis focused on SRAM design in Double Gate FDSOI. In 2009, he was postdoctoral researcher at UC Berkeley working on low power circuits and SRAM variability. From 2010, he works at CEA/Leti as a circuit designer specialized in memory and low power circuit in advanced technologies. His research interests include resilient memory with assist technics, energy efficiency, specific design technics and non-volatile memories. His current research are focused on SRAM ULV and robust, smart CAM, logic in memory, crossbar, using advanced CMOS technologies and non-volatile RRAM technologies such as CBRAM, OxRAM and PCRAM. He has published more than Jean-Michel Damien Deleruyelle received the Ph.D. degree in micro and nanoelectronics from Aix-Marseille University in 2004 after a thesis carried out at CEA-Leti (Grenoble, France) on ultra-

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https://hal.science/hal-01745507/file/Potal.2012.JOLPE.AuthorVer.Non-Volatile%20Flip-Flop%20Based%20on%20Unipolar%20ReRAM%20for%20Power-Down%20Applications.pdf

Jean-Michel Portal email: [email protected] Marc Bocquet Damien Deleruyelle Christophe Muller Non-Volatile Flip-Flop Based on Unipolar ReRAM for Power-Down Applications Keywords: Low-power, Power-down, Flip-Flop, Non volatile memory, Resistive switching Memory, ReRAM In this paper, we propose a new architecture of non-volatile Flip-Flop based on ReRAM unipolar resistive memory element (RNVFF). This architecture is proposed in the context of powerdown applications. Flip-Flop content is saved into ReRAM memory cell before power-down and restored after power-up. To simulate such a structure a compact model of unipolar ReRAM was developed and calibrated on best in class literature data. The architecture of the RNVFF, based on the insertion of a non-volatile memory block before a master-slave Flip-Flop, is detailed. The save and restore processes are described from the succession of four operating modes (normal, save, read, reset) needed by the save and restore processes. Finally, the structure is fully validated through electrical simulations, when the data to save is either '0' or '1'. INTRODUCTION A major challenge in nomad applications is the reduction of power consumption. The mainstream of power reduction is driven since many years by transistor downscaling and concomitant voltage reduction. A side effect of this reduction is the increase of leakage current in sub-threshold regime with more than 40% of active mode energy dissipation due to power leakage [START_REF] Kursun | Multi-Voltage CMOS Circuit Design[END_REF][START_REF] Sery | Life is CMOS: Why chase life after?[END_REF] of idle transistors. To overcome this issue, solutions based on process changes have been proposed such as high- oxide associated with a metal gate [START_REF] Robertson | High dielectric constant gate oxides for metal oxide Si transistors[END_REF]. Another well-known solution to save power is to power-down subcircuits of System on Chip (SoC) during idle state. However, when sub-circuits are powered-down, the data saved in the Flip-Flops are lost with a subsequent high power budget required for saving/restoring their contents together with sub-threshold leakage current. Numerous design solutions have been proposed to maintain Flip-Flop contents such as multithreshold voltages MOS transistors used with power gating techniques [START_REF] Jiao | Low-Leakage and Compact Registers with Easy-Sleep Mode[END_REF]. The basic principle to save the Flip-Flop's content during power-down relies on a retention circuit also known as balloon circuit [START_REF] Matsuya | A 1-V high-speed MTCMOS circuit scheme for power-down application circuits[END_REF]. The scheme of a retention Flip-Flop with balloon latch is reproduced in Fig. 1 [START_REF] Matsuya | A 1-V high-speed MTCMOS circuit scheme for power-down application circuits[END_REF]. Using this technique, the master-slave Flip-Flop is connected either to virtual ground or VDD while a balloon latch is connected to real ground and VDD. During power-down, the data of the slave latch in the Flip-Flop is memorized in the balloon latch while the Flip-Flop is disconnected from the ground or VDD thanks to a switch inserted between the real and the virtual ground line. The integration of Non-Volatile Flip-Flop (NVFF) in SoC may also be a solution to lower power consumption. The recent emergence of innovative low voltage memory concepts paves the way for novel NVFF solutions as already demonstrated with either ferroelectric FeRAM [START_REF] Yan | A design of ferro-DFF for non-volatile systems[END_REF][START_REF] Wang | A Compare-and-write Ferroelectric Nonvolatile Flip-Flop for Energy-Harvesting Applications[END_REF] or magnetoresistive MRAM [START_REF] Zhao | Spin-MTJ based Non-Volatile Flip-Flop[END_REF][START_REF] Guillemenet | On the use of magnetic RAMs in field-programmable gate arrays[END_REF][START_REF] Yamamoto | Nonvolatile delay Flip-Flop based on spin-transistor architecture and its power-gating applications[END_REF] memories and recently with bipolar memristive devices [START_REF] Robinett | A memristor-based nonvolatile latch circuit[END_REF]. Fig. 2 depicts FeRAM-based NVFF [START_REF] Wang | A Compare-and-write Ferroelectric Nonvolatile Flip-Flop for Energy-Harvesting Applications[END_REF] solution, in which a non-volatile back-up module based on the insertion of two FeRAM memory cells is used to save and restore the Flip-Flop content during power-down. This back-up module is connected to the output of the slave stage of the Flip-Flop. The MRAM-based solution [START_REF] Zhao | Spin-MTJ based Non-Volatile Flip-Flop[END_REF] illustrated in Fig. 3 is based on the insertion of two MRAM cells in the master stage of the Flip-Flop. Here again, the modified non-volatile master stage enables storing data during the power-down phase. Even if both technologies are compatible with CMOS standard processes, they rely either on complex stack of magnetic layers for MRAM or on a high temperature crystallization ferroelectric oxide for FeRAM. The main purpose of this paper is to show how emerging memory concept relying on unipolar resistive switching (namely ReRAM standing for Resistive Random Access Memory) may also represent an interesting solution for implementation in NVFF (RNVFF for ReRAM NVFF). This solution could benefit from the good compatibility between ReRAM and CMOS technologies. The paper is organized as follows. Section 2 is dedicated to unipolar ReRAM physical model description and its calibration on best in class literature data. In section 3, the save/restore processes of RNVFF are detailed together with the architecture of the Flip-Flop. Section 4 presents simulation results that validate the efficiency of the solution. Finally, section 5 gives concluding remarks. UNIPOLAR RERAM PHYSICAL MODEL OVERVIEW Introduction NiO-based unipolar resistive switching device (ReRAM) is a good candidate for distributed memory applications due to its simple MIM (metal/Insulator/metal) structure, good compatibility with standard CMOS processes, low operating voltage (i.e. below 1 V in [START_REF] Kim | Electrical observations of filamentary conductions for the resistive memory switching in NiO films[END_REF]) and fast programming time (i.e. in the sub-10 ns range in [START_REF] Tsunoda | Low Power and High Speed Switching of Ti-doped NiO ReRAM under the Unipolar Voltage Source of less than 3 V[END_REF]). For the particular class of ReRAM devices relying on thermochemical mechanisms, the memory effect is due to creation/destruction of conductive filaments (CF) within the oxide providing two resistive states named low resistance state (LRS) and high resistance state (HRS). In unipolar ReRAM, the same voltage polarity enables switching either from HRS to LRS (set) and from LRS to HRS (reset). The main drawback is the "electroforming" stage required to create initial CFs within a virgin dielectric oxide. In fact, this process requiring a higher voltage compared to set and reset voltages could be a strong issue when embedding ReRAM in CMOS logic. However, recent works have proposed technological solutions that enable reducing forming voltages to the level of set voltage, paving the way toward "forming-free" devices [START_REF] Bruchhaus | Memristive Switches with Two Switching Polarities in a Forming Free Device Structure[END_REF]. Unipolar ReRAM physical model description The proposed RNVFF circuit relies on a compact model accounting for both set and reset operations in NiO-based unipolar resistive switching devices [START_REF] Bocquet | Self-consistent physical modeling of set/reset operations in unipolar resistive-switching memories[END_REF]. The initial physical model takes into account two mechanisms: redox reactions (i.e. electrochemical oxidation/reduction processes) and thermal diffusion. Set operation is governed by a local reduction process leading to the creation of CFs, whereas reset operation involves both oxidation reaction and thermal diffusion. Nevertheless, considering involved activation energies, the oxidation mechanism may be neglected, the diffusion process mainly governing the reset operation. The description of set and reset operations relies on a self-consistent kinetics equation (eq. 1) linking diffusion and reduction velocities diff and red respectively to the dimensionless concentration of metallic species CNi. The local diffusion velocity diff (eq. 2) of the metallic species explains the thermal rupture of CF during reset operation [START_REF] Russo | Self-Accelerated Thermal Dissolution Model for Reset Programming in Unipolar Resistive-Switching Memory (ReRAM) Devices[END_REF]. In equation 2, Ea is the activation energy governing the thermally-assisted exodiffusion of metallic species, kdiff is the thermal diffusion rate and TCF represents the temperature of CF: Ni T k Ea diff diff C e k CF b       (2) Besides, equation 3 gives a simplified expression of the reduction velocity red (expressed by diff red Ni dt dC     (1) classical Butler-Volmer equation [START_REF] Bard | Electrochemical Methods: Fundamentals and Applications[END_REF]), in which  is the asymmetry factor, k0 is the reaction rate, E0 is the free energy of the reaction at equilibrium, VCell is the applied voltage and TOx is the oxide temperature:   Ni T k V q E 0 red C 1 e k v Ox b Cell 0          (3) In equation 2, the local CF temperature TCF(x) in x direction increases along with the applied voltage (VCell) due to Joule effect as described in equation 4. In this latter equation, CF is the CF conductivity, Kth is the CF thermal conductivity, tOx is the oxide thickness and Tamb is ambient temperature. Solving the 1D heat equation, the CF temperature is given by:                         2 2 Ox 2 Ox Cell th CF amb CF x 4 t t V K T x T (4) Finally, it must be underlined that the present physical model enables continuously accounting for both creation (set) and destruction (reset) of conductive filaments. This numerical feature is a key point for a model dedicated to be implemented in computer-aided design tools. ReRAM model calibration and model card extraction Before simulating circuits integrating ReRAM devices, the physical model was confronted to quasi-static and dynamic I(V) characteristics measured on actual devices [START_REF] Kim | Electrical observations of filamentary conductions for the resistive memory switching in NiO films[END_REF][START_REF] Cagli | Evidence for threshold switching in the set process of NiO-based ReRAM and physical modeling for set, reset, retention and disturb prediction[END_REF][START_REF] Lee | 2-stack 1D-1R Cross-point Structure with Oxide Diodes as Switch Elements for High Density Resistance RAM Applications[END_REF]. Fig. 4 shows quasi-static set and reset I(V) characteristics measured on NiO-based memory elements by several authors [START_REF] Kim | Electrical observations of filamentary conductions for the resistive memory switching in NiO films[END_REF][START_REF] Cagli | Evidence for threshold switching in the set process of NiO-based ReRAM and physical modeling for set, reset, retention and disturb prediction[END_REF][START_REF] Lee | 2-stack 1D-1R Cross-point Structure with Oxide Diodes as Switch Elements for High Density Resistance RAM Applications[END_REF]. The physical model shows an excellent agreement with experimental data for both set and reset operations, which demonstrates its flexibility to match electrical data reported on various technologies. Moreover, Fig. 5 reports experimental and simulated evolutions of reset current Ireset as a function of the maximum current ISetMax used in preceding set operation [START_REF] Nardi | Control of filament size and reduction of reset current below 10 μA in NiO resistance switching memories[END_REF]. The proposed model well catches the universal Ireset = f(ISetMax) trend observed on various NiO-based technologies and confirms the scalability of the reset current [START_REF] Nardi | Control of filament size and reduction of reset current below 10 μA in NiO resistance switching memories[END_REF]. Besides, as reported in ref. [START_REF] Bocquet | Self-consistent physical modeling of set/reset operations in unipolar resistive-switching memories[END_REF], the model is also able to fit the evolution of programming voltages along with ramp speed to describe the dynamic behavior of memory elements. Among I(V) characteristics shown in Fig. 4, data published by Kim et al. [START_REF] Kim | Electrical observations of filamentary conductions for the resistive memory switching in NiO films[END_REF] exhibiting switching voltages compatible with 65 nm CMOS technology VDD (Fig. 4c) are selected to extract the model card for design purpose. This latter model card also fulfilled the condition of achieving set and reset operations in 10 ns under 1.2 V bias are reported in ref. [START_REF] Tsunoda | Low Power and High Speed Switching of Ti-doped NiO ReRAM under the Unipolar Voltage Source of less than 3 V[END_REF]. RNVFF ARCHITECTURE WITH SAVE/RESTORE PROCESS DESCRIPTION Introduction In power-down applications, Flip-Flop with non-volatile capability might be an alternative solution to power gating technique. The architecture of the non-volatile Flip-Flop is presented in section 3.2, while the different operating phases of the save and restore processes are described in section 3.3. RNVFF architecture Proposed RNVFF solution relies on the implementation of a non-volatile memory (NVM) block connected to the input of a conventional master-slave Flip-Flop. As illustrated Fig. 6 the NVM block is composed of routing components (input tri-states inverters and output multiplexer). In between lies a branch that connects a central point (MEM) to VDD through two serial PMOS (MP1 and MP2) on one hand and to ground through a ReRAM memory cell on the other hand. It has to be noticed that the area of this latter branch is reduced since ReRAM element may be processed in the back-end of line on the top of CMOS level. The input tri-states inverter enables connecting and isolating the test feature with a multiplexer on their input. Therefore the multiplexer of the NVM block could be mixed with the scanmultiplexer to introduce a minimal delay overhead. To summarize, the structure of the NVM block is very simple with two routing elements (input tristates inverter and output multiplexer) and a branch with two transistors and one ReRAM element (i.e. 2T/1R structure). Considering that the output multiplexer may be mixed with a scan multiplexer, the area overhead introduced by the structure is one tri-states inverter and a 2T/1R branch. Save and restore processes description To save and restore Flip-Flop content, four successive operating modes are required:  The normal mode in which the Flip-Flop works in a conventional manner;  The save mode in which Flip-Flop content is saved in NVM block;  The read mode in which NVM block content is restored in the Flip-Flop;  The reset mode in which ReRAM memory cell turns back to a high resistance state while the Flip-Flop works in a conventional way. Active path in normal operating mode is illustrated by solid red line in Fig. 7. In this mode the ReRAM cell is isolated from the input (IN) of the structure thanks to the input tri-states inverter, which is open (SAVE_EN = '0'). It has to be mentioned, that in this operating mode, the ReRAM cell remains in its initial high resistance state (i.e. HRS). The Flip-Flop works in a conventional manner by connecting the input (IN) of the structure to the input (D) of the Flip-Flop thanks to the output multiplexer (READ_EN = '1'). The delay overhead introduces is due to the extra-load of the tri-states input inverters and the routing through the output multiplexer. Again, the actives paths of the save operating mode are illustrated by the solid red line in Fig. 8. In this mode the ReRAM cell is connected to the input (IN) of the structure thanks to the input tri-states inverter, which is turned on (SAVE_EN = '1'). The MEM central point is now the complement of the input (IN) value. When the tri-states inverter is activated, the ReRAM cell is switched to its low resistance state LRS (IN = '0' and MEM = '1') or remains in a high resistance state HRS (IN='1' and MEM='0'). The Flip-Flop continues to work in a conventional manner and store the value of the input (IN) thanks to the output multiplexer (READ_EN = '1'). So during the save mode, the ReRAM resistance state reflects the value of the Flip-Flop. Logic '1' corresponds to a high resistance state whereas logic '0' corresponds to a low resistance state. Right after the save mode, the Flip-Flop can be completely powered-down since ReRAM cell stores the information. After power-up, a read mode is mandatory to restore Flip-Flop content. Active path in read operating mode is illustrated in Fig. 9. In this mode, the output multiplexer (READ_EN = '0') enables connecting the MEM node to the input (D) of the Flip-Flop, while a clock edge restores the content of the Flip-Flop. The resistive bridge (voltage divider) composed by MP1 (turned-on with RESET_EN = '0') and MP2 on one side and the ReRAM cell on the other side determines the voltage on the node MEM. If the ReRAM cell is in its LRS state then the MEM voltage is grounded and a '0' is restored in the Flip-Flop. In contrast, if the MEM voltage is closed to VDD, a '1' is restored in the Flip-Flop. MP2 is introduced in the branch to limit the voltage on the node MEM when close to VDD in order to initiate the switching of the ReRAM cell to a HRS state during the read mode, if needed (ReRAM cell in a LRS state). Active paths of the reset operating mode are illustrated in Fig. 10. In this mode, the Flip-Flop works again in a conventional way thanks to the output multiplexer that connects the input (IN) of the structure to the input of the Flip-Flop (READ_EN = '1'). Doing so, the ReRAM cell is isolated from the rest of the circuit and can turn back to a HRS state, if needed, thanks to the application of a voltage close to VDD through MP1 (turned-on with RESET_EN = '0') and MP2. When RESET_EN = '1', the reset process is stopped and the whole structure turns back to a normal operating mode. Conclusion In conclusion, it is important to note that the standard functionality of the Flip-Flop is guaranteed in all modes, except during read mode. Moreover, the ReRAM state is HRS in all modes when the content to save is '1' with a minimal leakage current consumption. When the content to save is '0' then the current consumption is restricted to the save, read and reset modes with the switching of the ReRAM cell from HRS to LRS and respectively from LRS to HRS. It is also important to underline that no biasing is necessary during power-off to preserve ReRAM state. RNVFF SIMULATION RESULTS To validate the RNVFF functionality, the full structure is simulated under electrical simulator using a low power CMOS 65 nm design kit and the unipolar ReRAM compact model fitted on best in class literature data. VDD is nominal for the technology and set to 1.2 V during all operating modes except during power-off where it is set to 0 V. All operating modes are simulated for input values of '1' and '0'. MP2 has a minimal length (L=0.06 µm) and a double width (W=0.24 µm) and MP1 has minimal dimensions (L=0.06 µm and W=0.12 µm). The tri-states inverter is composed of NMOS and PMOS transistors with a double length (L=0.12 µm) to limit current during set process and standard width (WPMOS=0.15 µm, WNMOS=0.12 µm). The output multiplexer and the Flip-Flop are standard-cells from the library. i.e. normal mode, save mode, power-off, read mode, reset mode and again normal mode. Fig. 12 shows the chronograms of the input (IN) , the Flip-Flop input (D) and the current through the ReRAM cell (IReRAM) to save and restore a logic '0'. As described in the previous section, the ReRAM is set during the save mode with a current of 12 µA during 2 ns. During the read and reset modes, the ReRAM is reset during 5 ns with a current decreasing from 9 µA to 0 µA. During the read mode, the input IN is set to '1' while the read process forces a '0' value at the input D of the Flip-Flop, validating successfully the save and restore processes for a data equal to '0'. Fig. 13 shows the chronograms of the input (IN), the Flip-Flop input (D) and the current through the ReRAM cell (IReRAM) to save and restore a logic '1'. As previously described, the ReRAM cell remains in HRS during the save mode with a current below 0.1 µA during 2 ns. During the read and reset modes, the ReRAM cell remains in HRS during 5 ns with a current decreasing from 0.4 µA to 0 µA. During the read mode, the input IN is set to '0' while the read process enforces a '1' value at the input D of the Flip-Flop, validating successfully the save and restore processes of a data equal to '1'. In conclusion, the simulation results validate successfully the functionality of the RNVFF in all operating modes. The simulation also demonstrates that the current consumption of this structure is restricted to the save and restore processes of a logic '0'. Indeed, the ReRAM cell remains always in HRS, when the data to save and restore is equal to '1'. CONCLUSION In this paper, a new architecture of non-volatile Flip-Flop based on unipolar Resistive RAM is proposed. This latter architecture is dedicated to power-down applications, in which the content of the Flip-Flop is saved as resistance states in a ReRAM device before power-down and restored after power-up. The overall save and restore processes are detailed together with the architecture of the proposed structure. One may notice that this architecture relies on a non-volatile memory block inserted at the front of a Flip-Flop. The first advantage of such a structure is a better compatibility between the ReRAM memory element and the CMOS level as compared to MRAM or FeRAMbased solutions. Moreover, the use of such a structure does not require any biasing during power-off in comparison to retention Flip-Flop employing a balloon latch. Another point to underline is the low power consumption during all operating modes, except when the cell is set or reset (corresponding to save and read/reset of a '0' content). Finally, the full structure is successfully validated with electrical simulation using a 65 nm CMOS design kit and the unipolar compact model calibrated on best in class data from the literature. FIGURES AND TABLES Figure 1. Architecture of a classical balloon latch used with power-gating technique (redrawn from [START_REF] Matsuya | A 1-V high-speed MTCMOS circuit scheme for power-down application circuits[END_REF]). ISet IReset [START_REF] Cagli | Evidence for threshold switching in the set process of NiO-based ReRAM and physical modeling for set, reset, retention and disturb prediction[END_REF] Simu. this work V Cell (V) ICell (A) VCell (V) Symbols: [START_REF] Kim | Electrical observations of filamentary conductions for the resistive memory switching in NiO films[END_REF] Lines: Simulations central point (MEM) to the input of the structure (IN). PMOS transistors of this inverter are adequately sized to provide a current limitation through the ReRAM element when connecting the input IN to the central point MEM. The two inputs multiplexer enables bypassing NVM block, when the signal READ_EN = '1', otherwise (READ_EN = '0') the output of the NVM block is connected to the input (D) of the Flip-Flop. It is worth noting that in most of SoC, Flip-Flops integrate a scan Fig. 11 11 Fig. 11 presents the chronograms of VDD and control signals, i.e. SAVE_EN, READ_EN and Figure 2 . 2 Figure 2. Flip-Flop architecture with a back-up module based on FeRAM memory for energy harvesting application (redrawn from[START_REF] Wang | A Compare-and-write Ferroelectric Nonvolatile Flip-Flop for Energy-Harvesting Applications[END_REF]). Figure 3 . 3 Figure 3. Architecture of a latch with a non-volatility capability based on two magnetic tunnel junctions MTJ (redrawn from[START_REF] Zhao | Spin-MTJ based Non-Volatile Flip-Flop[END_REF]). Figure 4 . 4 Figure 4. Experimental I(V) characteristics measured on a NiO-based memory structure reported in (a) [18], (b) [19], and (c) [12] and corresponding simulations using the presented ReRAM physical model. Figure 5 . 5 Figure 5. Maximum current during the reset operation (IReset) as a function of the maximum current during the preceding set operation (ISetMax). Experimental data were extracted from [20]. Figure 6 .Figure 7 . 67 Figure 6. RNVFF architecture with Non-volatile block based on ReRAM cell connected to the Flip-Flop input D. Figure 8 .Figure 9 . 89 Figure 8. RNVFF active path during save operating mode. Figure 10 .Figure 11 . 1011 Figure 10. RNVFF active path during reset operating mode. Figure 12 .Figure 13 . 1213 Figure 12. Chronograms of data signals and ReRAM current during the save & restore processes using a RNVFF, with IN='0'.

01745537

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2024/03/05 22:32:07

2017

https://hal.science/hal-01745537/file/doc00028799.pdf

Ali Castaings Alain Bouscayrol email: [email protected] Walter Lhomme R Trigui Power Hardware-In-the-Loop simulation Keywords: hardware-in-the-loop, energy management, supercapacitor, battery, fuel cell, vehicle come The urban travel demand is significantly growing. According to the International Energy Agency the 2012 concentration of CO2 was about 40% higher than in the mid-1800s (IEA, 2013). It is then important to find out alternatives to conventional thermal vehicles. Several solutions have been depicted such as battery electric vehicles or Fuel Cell (FC) vehicles [START_REF] Chan | Electric, Hybrid, and Fuel-Cell Vehicles: Architectures and Modeling[END_REF]. However, each solution has some limitations. FCs have some power transfer issues [START_REF] Bernard | Fuel-Cell Hybrid Powertrain: Toward Minimization of Hydrogen Consumption[END_REF] while batteries have some lifetime issues [START_REF] Omar | Lithium iron phosphate based battery -Assessment of the aging parameters and development of cycle life model[END_REF]. Multi-sources vehicles represent an interesting alternative as they enable to take advantage of the properties of the different sources [START_REF] Ehsani | Modern Electric, Hybrid Electric, and Fuel Cell Vehicles: Fundamentals, Theory, and Design[END_REF]. However, they represent very complex systems. It is then difficult to manage such systems. Several works has been done on Energy Management Strategies (EMSs) of multi-sources vehicles. Two approaches have been depicted [START_REF] Salmasi | Control Strategies for Hybrid Electric Vehicles: Evolution, Classification, Comparison, and Future Trends[END_REF][START_REF] Wirasingha | Classification and Review of Control Strategies for Plug-In Hybrid Electric Vehicles[END_REF], rule-based approach [START_REF] García | Operation mode control of a hybrid power system based on fuel cell/battery/ultracapacitor for an electric tramway[END_REF][START_REF] Thounthong | Energy management of fuel cell/battery/supercapacitor hybrid power source for vehicle applications[END_REF] and optimization-based approach [START_REF] Yu | An innovative optimal power allocation strategy for fuel cell, battery and supercapacitor hybrid electric vehicle[END_REF][START_REF] Odeim | Power Management Optimization of a Fuel Cell/Battery/Supercapacitor Hybrid System for Transit Bus Applications[END_REF]. The main issues are related to real-time applications. Moreover, the EMSs have to ensure the physical limitations (i.e. overcharge or depleting) of the sources for any driving condition. As a consequence, it is important to find testing procedures to assess the EMSs before their implementation in a real vehicle. Power Hardware-in-The-Loop (P-HIL) simulation [START_REF] Bouscayrol | Hardware-in-the-loop simulation[END_REF] has been used in several applications for testing components before their implementation in a real system. P-HIL has thus been used for testing EMSs of hybrid and electric vehicles in real-time conditions [START_REF] Allègre | Flexible real-time control of a hybrid energy storage system for electric vehicles[END_REF][START_REF] Castaings | Practical control schemes of a battery/supercapacitor system for electric vehicle[END_REF][START_REF] Odeim | Power Management Optimization of an Experimental Fuel Cell/Battery/Supercapacitor Hybrid System[END_REF]. The objective of this paper is to present a P-HIL simulation of a FC-battery-Supercapacitors (SCs) vehicle. The developed platform enables to assess an EMS in real-time conditions (e.g. various driving conditions). The control organization of the P-HIL simulation is achieved by using Energetic Macroscopic Representation (EMR) [START_REF] Bouscayrol | Graphic Formalisms for the Control of Multi-Physical Energetic Systems[END_REF]. The second section is devoted to the description of the P-HIL simulation. The control organization is presented in the third section. The results are given in the last section before the conclusion. II. P-HIL SIMULATION OF THE STUDIED SYSTEM A. P-HIL principle Hardware-In-the Loop simulation consists in adding some actual elements (hardware) in the simulation loop [START_REF] Bouscayrol | Hardware-in-the-loop simulation[END_REF]. In Power HIL, some power elements can be tested before their implementation on the real system. It is useful for testing the subsystem and its control in real-time conditions. (Figure 1.a). In P-HIL simulation, the power part is split into two parts, the part under test (with its control) and the emulated part. An interface (interf. in ). The emulation system must have the same behavior than the simulated system. The control references of the emulation system come from the simulation of the emulated system. Also, the interface must be faster than the emulated system to emulate without delay. The next part is devoted to the control organization of the P-HIL system. Different parts have to be interconnected. Indeed, there are the models simulations, the emulation subsystems with their control, the tested subsystems and their control. A graphical formalism, Energetic Macroscopic Representation (EMR) is used as a tool for achieving the subsystems interconnection. First, EMR is based on action-reaction principle. It enables to ensure a physical connection between the elements. Second, EMR approach is based on causality principle. It enables to deduce the control structure of the system and to use real-time models for the emulation subsystems. III. CONTROL ORGANIZATION A. Real part The control of the system is achieved by using Energetic Macroscopic Representation (EMR). EMR highlights energetic properties of the components of a system to develop control schemes [START_REF] Bouscayrol | Graphic Formalisms for the Control of Multi-Physical Energetic Systems[END_REF]. There are several pictograms to represent the system model (see Appendix). By using EMR approach, the control part is organized in two levels, the "local control" part and the "global control" part (i.e. EMS). The main interest of using EMR is that the "local control" part of the system can be systematically deduced by "mirror" effect from its EMR. The EMR and the control part of the "real part" of the system are depicted Figure 4. Local control part The local control is represented by the light blue blocks in Global control part The global control part corresponds to the Energy Management Strategy (EMS). That aims to use the degrees of freedom of the control in the best way. There are two kinds of EMSs for multi-sources vehicles; rule-based EMSs and optimization-based EMSs [START_REF] Salmasi | Control Strategies for Hybrid Electric Vehicles: Evolution, Classification, Comparison, and Future Trends[END_REF] Figure 4: EMR and control organization of the "real part" B. Emulated parts The EMR and its control organization of the emulated parts are depicted Figure 5. The purple blocks correspond to the simulated part of the P-HIL simulation. As it can be noticed, the control references come from the simulation of the real components models (purple pictograms). Also this is a reduced-scale P-HIL simulation. As a result, some adaption coefficients are taken into account [START_REF] Allègre | Flexible real-time control of a hybrid energy storage system for electric vehicles[END_REF]. These coefficients enable to pass from the full-scale simulated models to the reference signals of the reduced-scale system (1). The reduced-scale coefficients values are given in Table 1. VALIDATION OF AN ENERGY MANAGEMENT STRATEGY A. Principle The experimental setup is presented Figure 6. A dSPACE 1005 card is used as an interface between the power part and the computer board. The EMS is an optimization-based strategy. It consists in minimizing the hydrogen consumption while improving the battery lifetime [START_REF] Castaings | Comparison of energy management strategies of a battery/supercapacitors system for electric vehicle under real-time constraints[END_REF]. The first test is achieved on a standard driving cycle (WLTC class 2, low velocity phase Figure 7) where the EMS parameters have been identified. This corresponds to "ideal" driving conditions. The second test is carried out using a real driving cycle (Figure 8) coming from results on the instrumented car (Tazzari Zero) [START_REF] Depature | Efficiency Map of the Traction System of an Electric Vehicle from an On-Road Test Drive[END_REF]. This test enables to assess the robustness of the EMS when varying the driving conditions. The parameters of the full scale and reduced scale systems are given in Table 1. This is interesting for its lifetime (Figure 9). As depicted in Figure 10 the EMS enables to respect the SCs voltage limitations. This is important for ensuring the system safety. This aspect has been assessed thanks to the P-HIL platform. The same trends can be noticed for the real driving cycle (Figure 11 and Figure 12). The key point is that the EMS still enables to reach interesting performances while ensuring the system safety. However, as the parameters were not computed on this driving cycle, the SCs tend to be discharged at the end of the driving cycle. This can cause some repeatability issues. Indeed, if the same driving cycle is repeated, the results won't be the same as the previous one. Csc=19 F usc-M=44 V | usc-m=0.65usc-M uSC-0=0.9usc-M Battery 24 cells (3.3 V / 20 Ah / 820 W) SoCb-M=100 % | SoCb-m=90 % SoCb-0=95 % ki-sbat=1/17 ku-sbat=1 Figure 1.a) is required for connecting the simulation signals and the power signals. The interface has then power and signals elements. (Figure 1.b Figure 1 1 Figure 1: P-HIL, (a) principle, (b) practical scheme Figure 2 : 2 Figure 2: Architecture of the studied vehicle P-HIL organization The objective of the P-HIL simulation is to assess the system controllability in real-time and to validate the FC and SCs behavior. In the presented work, the emulated parts are the battery branch and the traction part. The corresponding emulation systems are depicted Figure 3. For the battery branch, the battery is replaced by a SCs bank. The SCs bank has to reflect two battery characteristics  the battery SoC limitations : this depends on the SCs bank size  the battery voltage dynamics. The SCs voltage has higher dynamics than the battery voltage ones. If the battery model is accurate enough, the battery voltage dynamics can be reflected by the SCs. The traction part is emulated by a current source composed of a DC-DC converter, a smoothing inductor and a SCs bank. The main dynamics of the inverter current are taken into account in the traction model (cf. section III). Figure 3 : 3 Figure 3: P-HIIL system architecture Figure 4 . 4 It manages the system components to track the reference of the DC bus voltage. The right duty cycles of the converters (αb, αfc and αsc) are then defined. In addition, the local control points out the control requirements. In the studied case 4 sensors and 4 controllers (closed-loop control) are required as well. The inversion of an accumulation element is performed via a closed-loop control (crossed blue parallelogram). A conversion element is directly inverted with an open-loop control (blue parallelogram). The inversion of a coupling element depicts degrees of freedom that correspond to the output of the EMS (global control). Figure 5 : 5 Figure 5: EMR and control organization of the emulated parts Figure 6 : 6 Figure 6: Experimental setup Figure 9 : 9 Figure 9: FC branch and traction currents Figure 11: FC branch and traction currents

01745567

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2024/03/05 22:32:07

2018

https://hal.science/hal-01745567/file/lasaulce-tarbouriech-v3%283%29.pdf

Samson Lasaulce email: [email protected] Sophie Tarbouriech email: [email protected] Information constraints in multiple agent problems with i.i.d. states . In wireless communications, it may represent the state of the global communication channel. The approach used is to exploit Shannon theory to characterize the achievable long-term utility region. Two scenarios are described. In the first scenario, the number of agents is arbitrary and the agents have causal knowledge about the state. In the second scenario, there are only two agents and the agents have some knowledge about the future of the state, making its knowledge non-causal. Chapter overview This chapter concerns the problem of coordination among agents. Technically, the problem is as follows. We consider a set of K ≥ 2 agents. Agent k has a utility, payoff, or reward function u k (x 0 , x 1 , ..., x K ) where x k , k ≥ 1, is the action of Agent k while x 0 is the action of an agent called Nature. The Nature's actions correspond to the system state and is assumed to be non-controlled; more precisely, Nature corresponds to an independent and identically distributed (i.i.d.) random process. The problem studied in this chapter is to characterize the long-term utility region under various assumptions in terms of observation at the agents. By long-term utility for Agent k we mean the following quantities: U k (σ 1 , ..., σ K ) = lim T →+∞ 1 T E T ∑ t=1 u i (X 0 (t), ..., X K (t)) (1) where σ k = (σ k,t ) t≥1 is a sequence of functions which represent the strategy of Agent k, x k (t) is the action chosen by Agent k at time or stage t ≥ 1, t being the time or stage index; concerning notations, as far as random variables are concerned, capital letters will stand for random variables whereas, small letters will stand for realizations. Note that, implicitly, we assume sufficient conditions (such as utility boundedness) under which the above limit exists. The functions σ k,t , k ∈ {1, ..., K}, map the available knowledge to the action of the considered agent. The available knowledge depends on the information assumptions made (e.g., the knowledge of the state can be causal or non-causal). We will distinguish between two scenarios. In the first scenario, agents are assumed to have some causal knowledge (in the wide sense) about the state whereas, in the second scenario non-causal knowledge (i.e., some knowledge about the future) about the state is assumed. The second scenario is definitely the most difficult one technically, which is why only two agents will be assumed. Remarkably, the long-term utility region, whenever available, can be characterized in terms of elegant information constraints. For instance, in the scenario of non-causal state information, determining the long-term utility region amounts to solving a convex optimization problem whose non-trivial constraints are the derived information-theoretic constraints. Introduction An important example, which illustrates well how the results reported in this chapter can be used, is given by the problem of power control in wireless networks (see Fig. 1). Each transmitter has to adapt its transmit power not only to the fluctuations of the quality of the link (or channel gain) between itself and its respective receiver but also to the transmit power levels of the other transmitters that uses the same radio resources (and therefore create interference). This problem is a multi-agent problem where the agents are the transmitters, the actions of the agents are their transmit power level, and the system state is given by the set of channel gains of the various links in presence; channel gains are typically non-controlled variables (they do not depend on the transmit power levels) and evolve in a random manner; in practice, each transmitter has a partial and imperfect knowledge of the system state. Now, if the agents (namely, the transmitters in the considered example) have a certain performance criterion, which will be referred to as a utility function for the general setup considered in the chapter, the important problem of knowing the best achievable utilities appears. For instance, a transmitter might be designed to maximize its communication rate. The best data rate of a given transmitter would be obtained if all the other transmitters would be silent (i.e., when they don't transmit) and when the transmitter perfectly adapts its power to the channel gain fluctuations of the link between itself and its intended receiver. Obviously, in the real life, several transmitters will transmit at the same time, hence the need to coordinate as well as possible, which leads to the problem of characterizing the best performance possible in terms of coordination. This precisely corresponds to the problem of characterizing the long-term utility region i.e., the set of possible achievable points (U 1 ,U 2 , ...,U K ) for a given definition for the strategies. In Sections. 3 and 4, we will consider two different definitions for the strategies, each of them corresponding to a given observation structure that is, to some given information assumptions. Fig. 1 The problem of power control in wireless networks is a typical application for the results provided in this chapter. The agents are the transmitters, the agents' actions are given by the transmit power level, and the agent utility function may be its communication rate with its intended receiver. General problem formulation This chapter aims at describing a few special instances of a general problem which has been addressed in several recent works [START_REF] Larrousse | Coded Power Control: Performance Analysis[END_REF], [START_REF] Larrousse | Implicit coordination in two-agent team problems. Application to distributed power allocation[END_REF], [START_REF] Larrousse | Coordination in state-dependent distributed networks: The two-agent case[END_REF], [START_REF] Larrousse | Coordination in State-Dependent Distributed Networks[END_REF], [START_REF] Agrawal | A framework for decentralized power control with partial channel state information[END_REF], [START_REF] Larrousse | Coordination in distributed networks via coded actions with application to power control[END_REF], [START_REF] Treust | Joint empirical coordination of source and channel[END_REF]. We consider K ≥ 2 agents, where Agent k ∈ {1, . . . , K} produces time-t action x k (t) ∈ X k 1 for t ∈ {1, . . . , T }, T ≥ 1, the set X k representing the set of actions for Agent k. Each agent has access to some observations associated with the chosen actions and the realization of a random process {X 0,t } T t=1 = {X 0,1 , ..., X 0,T } ∈ X T 0 . In the motivating example described in the introduction, the random process was given by the global wireless channel state i.e., the set of qualities of all the links in presence. In a control problem, the random process may represent a non-controlled perturbation or some uncertainty. All agents' actions and the random process also affect the agents' individual stage or instantaneous utility functions u 1 , ..., u K where for all k ∈ {1, ..., K} the function u k writes: u k : X 0 × X 1 × ... × X K → R (x 0 , x 1 , ..., x K ) → u k (x 0 , x 1 , ..., x K ) . (2) One of the main goals of the chapter is to explain how to determine the set of feasible expected long-term utilities: U (T ) k = E 1 T T ∑ t=1 u k (X 0,t , X 1,t , ..., X K,t ) , (3) that are reachable by some strategies for the agents. The set of feasible utilities is fully characterized by the set of feasible averaged joint probability distributions on the (K +1)-uple {(X 0,t , X 1,t , ..., X K,t )} T t=1 . Indeed, denoting by P X 0,t X 1,t ...X K,t the joint probability distribution of the time (K + 1)-uple (X 0,t , X 1,t , ..., X K,t ), we have U (T ) k = 1 T T ∑ t=1 E [u k (X 0,t , X 1,t , ..., X K,t )] = 1 T T ∑ t=1 ∑ x 0 ,...,x K P X 0,t X 1,t ...X K,t (x 0 , x 1 , ..., x K )u k (x 0 , x 1 , ..., x K ) = ∑ x 0 ,...,x K u k (x 0 , x 1 , ..., x K ) 1 T T ∑ t=1 P X 0,t X 1,t ...X K,t (x 0 , x 1 , ..., x K ). Therefore the problem of characterizing the long-term utility region amounts to determining the set of averaged distributions P (T ) (x 0 , x 1 , ..., x K ) = 1 T T ∑ t=1 P X 0,t X 1,t ...X K,t (x 0 , x 1 , ..., x K ) (4) that can be induced by the agents' strategies. For simplicity, and in order to obtain closed-form expressions, we shall focus on the case where T → ∞ [START_REF] Gossner | Optimal use of communication resources[END_REF], [START_REF] Larrousse | Coded Power Control: Performance Analysis[END_REF]. We consider two types of scenarios with two different observation structures. In the first scenario, referred to as the non-causal state information scenario, the agents 1 Throughout the chapter, we assume that all the alphabets such as X k are finite. observe the system states non-causally. That means, at each stage t ∈ {1, . . . , T } they have some knowledge about the entire state sequence X T 0 = (X 0,1 , . . . , X 0,T ). In the second scenario, referred to as the causal state information scenario, the agents learn the states only causally and therefore, at any stage t, the agents have some knowledge about the sequence X t 0 = (X 0,1 , . . . , X 0,t ), where throughout the chapter we use the shorthand notations A m and a m for the tuples (A 1 , . . . , A m ) and (a 1 , . . . , a m ), when m is a positive integer. 3 Coordination among agents having causal state information Limiting performance characterization Firstly, we define the information structure under consideration. At every instant or stage t, Agent k is assumed to have an image or a partial observation S k,t ∈ S k of the nature state X 0,t with respect to which all agents are coordinating. In the case of the wireless power control example described in the introduction, this might be the knowledge of local channel state information, e.g., a noisy estimate of the direct channel between the transmitter and the associated receiver. The observations S k,t are assumed to be generated by a memoryless channel. By memoryless it is meant that the joint conditional probability on sequences of realizations factorizes the product of individual conditional probabilities. Denoting by k the transition probability for the observation structure of Agent k, the memoryless condition can be written as: P(s T K |x T 0 ) = T ∏ t=1 k (s k (t)|x 0 (t)). (5) The strategy or the sequence of decision functions for Agent k, σ k,t , is defined by: σ k,t : S t k -→ X k (6) (s k (1), s k (2), ..., s k (t)) -→ x k (t) (7) where S k is observation alphabet for Agent k. As mentioned in Section 2, the problem of characterizing the long-term utility region amounts to determining the achievable correlations measured in terms of joint distribution, hence the notion of implementability for a distribution. Definition 1. (Implementability) The probability distribution Q(x 0 , x 1 , ..., x N ) is im- plementable if there exist strategies (σ 1,t ) t≥1 , ..., σ K,t ) t≥1 such that as T → +∞, we have for all x ∈ X , 1 T T ∑ t=1 P X 0,t ...X K,t (x 0 , ..., x K ) -→ Q(x 0 , ..., x K ) (8) where P X 0,t ...X K,t is the joint distribution induced by the strategies at stage t. The following theorem is precisely based on the notion of implementability and characterizes the achievable long-term utilities that are implementable under the information structure [START_REF] Agrawal | A framework for decentralized power control with partial channel state information[END_REF]; for this, we first define the weighted utility function w as a convex combination of the individual utilities u k : w = K ∑ k=1 λ k u k . ( 9 ) Theorem 1. [5] Assume the random process X 0,t to be i.i.d. following a probability distribution ρ and the available information to the transmitters S k,t to be the output of a discrete memoryless channel obtained by marginalizing the joint conditional probability . An expected payoff w is achievable in the limit T → ∞ if and only if it can be written as: w = ∑ x 0 ,x 1 ,...x N , u,s 1 ,...s N ρ(x 0 )P U (u) (s 1 , ..., s K |x 0 )× ∏ K k=1 P X k |S k ,U (x k |s k , u) w(x 0 , x 1 , ..., x K ). ( 10 ) where U is an auxiliary variable, which can be optimized, and P X k |S k ,U (x k |s k , u) is the probability that Agent k, chooses action x k after observing s k , u. The auxiliary variable U is an external lottery known to the agents beforehand, which can be used to achieve better coordination e.g., in presence of individual constraints or at equilibrium. Theorem 1 allows us find all the achievable utility vectors (U 1 , ...,U K ). Indeed, the long-term utility region being convex (this readily follows from a time-sharing argument), its Pareto boundary can be found by maximizing the weighted utility w. Of course, remains the problem of determining the strategies allowing to operate at a given arbitrary point of the utility region. Since, this problem is non-trivial and there does not exist any methodology for this, we provide an algorithm which allows one to find a suboptimal strategies. Indeed, the associated multilinear optimization problem is too complex to be solved and to overcome this we resort to an iterative technique which is much less complex but is suboptimal. An algorithm to determine suboptimal strategies One of the merits of Theorem 1 is to provide the best performance achievable in terms of long-term utilities when agents have an arbitrary observation structure. However, Theorem 1 does not provide practical strategies which would allow a given utility vector to be reached. Finding "optimal" strategies consists in finding good sequences of functions as defined per [START_REF] Agrawal | A framework for decentralized power control with partial channel state information[END_REF], which is an open and promising direction to be explored. More pragmatically, the authors of [START_REF] Agrawal | A framework for decentralized power control with partial channel state information[END_REF] proposed to restrict to stationary strategies which are merely functions of the form f k : S k → X k . This choice is motivated by practical considerations such as computational complexity and it is also coherent with the current state of the literature. The water-filling solu-tion is a special instance of this class of strategies. To find good decision functions, the idea, which is proposed in [START_REF] Agrawal | A framework for decentralized power control with partial channel state information[END_REF], is to exploit Theorem 1. This is precisely the purpose of this section. The first observation we make is that the best performance only depends on the vector of conditional probabilities P X 1 |S 1 ,U , ..., P X K |S K ,U and the auxiliary variable probability distribution P U , the other quantities being fixed. It is therefore relevant to try to find an optimum vector of lotteries for every action possible and use it to take decisions. Since this task is typically computationally demanding, a possible and generally suboptimal approach consists in applying a distributed algorithm to maximize the expected weighted utility. The procedure proposed in [START_REF] Agrawal | A framework for decentralized power control with partial channel state information[END_REF] is to use the sequential best response dynamics (see e.g., [START_REF] Lasaulce | Game Theory and Learning for Wireless Networks: Fundamentals and Applications[END_REF]). The idea is to fix all the variables (that are probability distributions here) expect one and maximize the expected weighted utility with respect to the only possible degree of freedom. This operation is then repeated by considering another variable. The key observation to be made is then to see that when the distributions of the other agents are fixed, the best distribution for Agent k boils down to a function of s k , giving us a candidate for a decision function which can be used in practice. To describe the algorithm of [START_REF] Agrawal | A framework for decentralized power control with partial channel state information[END_REF] (see also Fig. 2), we first rewrite the expected weighted utility in the following manner: W = ∑ x 0 ,x 1 ,...x K ,u,s 1 ,...s K ρ(x 0 )P U (u)× (11) Γ (s 1 , ..., s K |x 0 , x 1 , ..., x K )× (12) K ∏ k=1 P X k |S k ,U (x k |s k , u) w(x 0 , x 1 , ..., x K ) = ∑ i k , j k ,u δ i k , j k ,u P X k |S k ,U (x k |s k , u) (13) where i k , j k , u are the respective indices of x k , s k , u and δ i k , j k ,u = ∑ i 0 ρ(x i 0 )Γ k (s k |x i 0 ) ∑ i -k u k (x i 0 , x i 1 , ...x i K )× ∑ j -k ∏ k =k Γ k (s j k |x 0 ) ∏ k =k P X k |S k ,U (x k |s k , u) P U (u) (14) where i -k , j -k are the indices which represent i k , j k being constant, while all the other indices are summed over. To make the description of the algorithm clearer, we have also assumed the independence of the observation channels as well as independence of the signal with the strategies chosen by the agents, i.e., Γ (s 1 , ..., s K |x 0 , x 1 , ..., x K ) = Γ 1 (s 1 |x 0 ) × . . . ×Γ K (s K |x 0 ) . Written under this form, for every agent, optimizing the expected weighted utility in a distributed manner im-plies giving a probability 1 for the optimal coefficient δ i k , j k ,u , and every player does that turn by turn. ! Fig. 2 Pseudo-code of the Algorithm proposed in [START_REF] Agrawal | A framework for decentralized power control with partial channel state information[END_REF] to find suboptimal strategies. To conclude, note that the above algorithm always converges. This can be proved e.g., by induction or by calling for an exact potential game property (see e.g., [START_REF] Mondrere | Potential games[END_REF], [START_REF] Lasaulce | Game Theory and Learning for Wireless Networks: Fundamentals and Applications[END_REF]). Coordination between two agents having noncausal state information Limiting performance characterization As explained previously, the problem of characterizing the utility region in the case where the state is known non-causally to the agents is much more involved technically. Even in the case of two agents, one may have to face with an open problem, depending on the observation structure assumed for the agents. Here, we consider an important case for which the problem can be solved, as shown in [START_REF] Larrousse | Coordination in State-Dependent Distributed Networks[END_REF]. Therein, the authors consider an asymmetric observation structure. In the case of non-causal state information, agents' strategies are sequences of functions that are defined as follows. For Agent 1 the strategy is defined by: σ 1,t : S T 1 × Y t-1 1 -→ X 1 (15) (s 1 (1), ..., s K (T ), y 1 (1), ..., y 1 (t -1)) -→ x 1 (t) (16) and for Agent 2 the strategy is defined by: σ 2,t : S T 2 × Y t-1 2 -→ X 2 (17) (s 2 (1), ..., s 2 (t), y 2 (1), ..., y 2 (t -1)) -→ x 2 (t) (18) where y k (t) ∈ Y k is the observation Agent k has about the triplet (x 0 (t), x 1 (t), x 2 (t)) whereas s k (t) ∈ S k is the observation Agent k has about the state x 0 (t). Note that distinguishing between the two observations s k and y k is instrumental. Indeed, it does not make any sense physically speaking to assume that an agent might have some future knowledge about the actions of the other agents, which is why the feedback signal is strictly causal. On the other hand, assuming some knowledge about the future of the non-controlled state x 0 perfectly makes sense, as motivated the chapter abstract and the works quoted in the list of references. More precisely the observation is assumed to be the output of a memoryless channel whose transition law is denoted by Γ : Pr Y 1 (t) = y 1 (t),Y 2 (t) = y 2 (t) X t 0 = s t 0 , X t 1 = x t 1 , X t 2 = x t 2 ,Y t-1 1 = y t-1 1 ,Y t-1 2 = y t-1 2 = Γ (y 1 (t), y 2 (t)|sx 0 (t), x 1 (t), x 2 (t)). (19) We now provide the characterization of the set of implementable probability distributions both for the considered non-causal strategies. Theorem 2. [START_REF] Larrousse | Coordination in State-Dependent Distributed Networks[END_REF] The distribution Q is implementable if and only if it satisfies the following condition2 I Q (S 1 ;U) ≤ I Q (V ;Y 2 |U) -I Q (V ; S 1 |U). (20) where U and V are auxiliary random variables and Q is any joint distribution that factorizes as Q(x 0 , s 1 , s 2 , u, v, x 1 , x 2 , y 1 , y 2 ) = ρ(x 0 ) (s 1 , s 2 |x 0 )P UV X 1 |S 1 (u, v, x 1 |s 1 )P X 2 |US 2 (x 2 |u, s 2 )Γ (y 1 , y 2 |x 0 , x 1 , x 2 ) (21) In practice, to plot the utility region, one typically has to solve a convex optimization problem. To be illustrative, we consider the special case of [START_REF] Larrousse | Coded Power Control: Performance Analysis[END_REF] namely, Y = X 1 . Denoting by H the entropy function, the problem of finding the Pareto-frontier of the utility region exactly corresponds to solving the following optimization problem: minimize -∑ x 0 ,x 1 ,x 2 Q(x 0 , x 1 , x 2 )w(x 0 , x 1 , x 2 ) subject to H Q (X 0 ) + H Q (X 2 ) -H Q (X 0 , X 1 , X 2 ) ≤ 0 -Q(x 0 , x 1 , x 2 ) ≤ 0 -1 + ∑ x 0 ,x 1 ,x 2 Q(x 0 , x 1 , x 2 ) = 0 -ρ(x 0 ) + ∑ x 1 ,x 2 Q(x 0 , x 1 , x 2 ) = 0 . The above problem can be shown to be convex (see [START_REF] Larrousse | Coded Power Control: Performance Analysis[END_REF]). In the next section we exploit this result to assess the performance gain brought by implementing coordination for distributed power control in wireless networks. Application to distributed power control Here we apply the results of the previous section to the wireless power control problem. A flat-fading interference channel (IC) with two transmitter-receiver pairs is considered. Transmissions are assumed to be time-slotted and synchronized; the timeslot or stage index is denoted by t ∈ N * . For k ∈ {1, 2} and " = -k" (-k stands for the terminal other than k), the signal-to-noise plus interference ratio (SINR) at Receiver k on a given stage writes as SINR k = g kk x k σ 2 +g k x -k where x k ∈ X IC i = {0, P max } is the power level chosen by Transmitter k, g k represents the channel gain of link k , and σ 2 the noise variance. If Transmitter 1 is fully informed of x 0 = (g 11 , g 12 , g 21 , g 22 ) for the next stage and Transmitter 2 has no transmit CSI while both transmitters want to maximize the average of a common stage payoff which is w IC (x 0 , x 1 , x 2 ) = ∑ 2 k=1 f (SINR k (x 0 , x 1 , x 2 ) ), there may be an incentive for Transmitter 1 to inform Transmitter 2 what to do for the next stage; a typical choice for f is f (a) = log(1 + a). Since Transmitter 1 knows the optimal pair of power levels to be chosen on the next stage, say (x * 1 , x * 2 ) ∈ arg max (x 1 ,x 2 ) w(x 0 , x 1 , x 2 ), a simple coded power control (CPC) policy for Transmitter 1 consists in transmitting on stage t at the level Transmitter 2 should transmit on stage t + 1. Therefore, if Transmitter 2 is able to observe the actions of Transmitter 1, power levels will be optimally tuned half of the time. Such a simple policy, which will be referred to as semicoordinated PC (SPC), may outperform (in terms of average payoff) pragmatical PC policies such as the one for which the maximum power level is always chosen by both Transmitters ((x 1 , x 2 ) = (P max , P max ) is the Nash equilibrium of the static game whose individual utilities are u k = f (SINR k )). The channel gain of the link between Transmitter k and Receiver is assumed to be Bernouilli distributed: g k ∈ {g min , g max } is i.i.d. and Bernouilli distributed g k ∼ B(p k ) with P(g k = g min ) = p k . The utility function is either f (a) = log(1 + a) or f (a) = a. We define SNR[dB] = 10 log 10 P max σ 2 and set g min = 0.1, g max = 1.9, σ 2 = 1. The low and high interference regimes (LIR for low interference regime, HIR and for high interference regime) are respectively defined by (p 11 , p 12 , p 21 , p 22 ) = (0.5, 0.9, 0.9, 0.5) and (p 11 , p 12 , p 21 , p 22 ) = (0.5, 0.1, 0.1, 0.5). At last, Y ≡ X 1 and we define two reference PC policies : full power control (FPC) policy x k = P max for every stage ; the semi-coordinated PC (SPC) policy x 2 = P max , x † 1 ∈ arg max x 1 w IC (x 0 , x 1 , P max ). Fig. 3 and 4 depict the relative gain in % in terms of average payoff versus SNR[dB] which is obtained by costless optimal coordination and information-constrained coordination. Compared to FPC, gains are very significant whatever the interference regime and provided the SNR has realistic values. Compared to SPC, the gain is of course less impressive since SPC is precisely a coordinated PC scheme but, in the HIR and when the communication cost is negligible, gains as high as 25% can be obtained with f (a) = log(1 + a) and 45% with f (a) = a. Conclusion In this chapter, we have described an information-theoretic framework to characterize the limiting performance of a multiple agent problem. More precisely, the theoretical performance analysis has been conducted in terms of long-term utility region. We have seen that the problem amounts to finding the set of implementable joint distribution over the system state and actions. Both in the scenarios of causal and non-causal state information, auxiliary random variables appear in the characterization of implementable joint distribution. To be able to assess numerically the limiting performance for given utility functions, an optimization problem has to be solved. In the causal state information scenario, the problem is multilinear and the challenge is due to the dimension of the vectors involved. In the non-causal state information scenario, the problem to be solved is a convex problem; more precisely, the information constraint function which translates the agent capabilities in terms of coordination is a convex function of the joint distribution. Note that although the state is not controlled and evolves randomly, the general problem of characterizing the utility region for any number of agents is not trivial. Of course, the problem Fig. 4 The difference with Fig. 3 is that the reference power control policy is the semi-coordinated power control policy (SPC), which is already a CPC policy. Additionally, the top curve is obtained with f (a) = a. is even more difficult in the case of controlled states, which therefore constitutes one possible non-trivial extension of the results reported in this chapter. Another interesting research direction would be to consider the case where the state and actions are continuous. A first attempt to this has be made in [START_REF] Agrawal | Implicit coordination in two-agent team problems[END_REF]. Interestingly, the corresponding problem can be shown to be strongly connected to the famous Witsenhausen problem [START_REF] Witsenhausen | A counterexample in stochastic optimum control[END_REF], [START_REF] Sahai | Demystifying the Witsenhausen counterexample[END_REF], which is a typical decentralized control problem where control and communication intervene in an intricate manner. Fig. 3 3 Fig.3Relative gain in terms of expected payoff ("CPC/FPC -1" in [%]) vs SNR[dB] obtained with CPC (with and without communication cost) when the reference power control policy is to transmit at full power (FPC). expected payoff & HIR Costless communication case & LIR Information-constrained expected payoff &LIR Costless communication & HIR & f(a)=a The notation I Q (A; B) indicates that the mutual information should be computed with respect to the probability distribution Q.

01745578

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2024/03/05 22:32:07

2007

https://hal.science/hal-01745578/file/Bocquet%20-%20ICMTD07%20-%20fixed%20charge%20%26%20trapping%20-%20HfAlO%20IPD.pdf

J Grampeix F Buckley J.-P Martin M Colonna G Gély G Pananakakis B Ghibaudo De Salvo M Bocquet email: [email protected] G Molas H Grampeix J Buckley F Martin J P Colonna M Gély G Ghibaudo B De Salvo S Deleonibus Intrinsic fixed charge and trapping properties of HfAlO interpoly dielectric layers L'archive ouverte pluridisciplinaire Introduction In order to meet the performance requirements of future generations of Flash memory [1], one of the nearest major improvements will concern the scaling of the InterPoly Dielectric (IPD) stack. For the 45nm and 35nm nodes, in order to compensate the loss of the vertical sidewalls of the poly-Si floating gate and keep high the coupling ratio [START_REF] Alessandri | Proc. Of the 208th ECS Meeting[END_REF][START_REF] Houdt | [END_REF], the IPD thickness should be reduced. In a previous work [START_REF] Molas | Proc. of ESSDERC[END_REF], we proposed HfAlO high-k materials to replace the nitride layer in ONO interpoly dielectric stacks for future Flash memories, arguing the advantages both in terms of coupling and insulating properties. We showed that the leakage current was strongly governed by the trapping in the high-k layer, with a strong temperature activation. Indeed, a Poole-Frenkel conduction, probably assisted by the traps in the HfAlO layer, was identified. In this paper, we further developed the analysis of HfAlO high-k materials, embedded in OHO stacks, by focusing on the trapping properties and fixed charges. In particular, we investigate: (1) the intrinsic negative fixed charge in the oxides, (2) the trapping phenomena which take place in the high-k dielectrics with various compositions and thicknesses during a gate stress, (3) the retention properties of these layers, and (4) finally, we present simulations based on a Shockley Read Hall (SRH) approach which allow us to model the electron trapping in the OHO stacks. Experimental results Sample description The schematic of the triple layer capacitors studied in this work is shown in Figure 1. The high-k films were sandwiched between two HTO (High Thermal Oxide) deposited at 730°C, with a thickness of 4nm. Three different high-k materials, deposited by ALCVD, were studied: Hafnium Oxide (HfO 2 ), Aluminum Oxide (Al 2 O 3 ) and Hafnium Aluminate (HfAlO). In HfAlO films, the Hf concentrations were controlled by the HfCl 4 :Al(CH 3 ) 3 deposition cycle ratio, which are respectively: 9:1 (94% of Hf), 1:4 (31% of Hf), and 1:9 (27% of Hf). Different high-k physical thicknesses ranging between 3nm and 9nm were fabricated by controlling the number of ALD deposition cycles. Intrinsic negative fixed charge In this section we investigate the intrinsic fixed charge of the HfAlO layers. We can observe that the flatband voltages of OHO samples are shifted compared to the 10nm-thick HTO reference sample, due to the presence of intrinsic fixed charge in the high-k layers. The shift increases monotonically when the Al concentration of the HfAlO alloy increases. The origin of the intrinsic negative fixed charge in HfAlO materials is nowadays still not clear [START_REF] Lee | [END_REF]. In amorphous Al 2 O 3 layer, it was suggested that the Al 2 O 3 could be dissociated into (AlO 4/2 ) -and Al 3+ [6] and that, at the SiO 2 /Al 2 O 3 interface, the charge compensation does not take place. The inset of Figure 2 shows that V FB is a linear function of the HfAlO thickness, which suggests a surface rather than a volume distribution of the fixed charge according to Equation 1, being in agreement with previous works reported in the literature [7][8][9]. ( ) H H H H H SiO T fb t t t t V ε ρ ε σ ρ σ ε ⋅ ⋅ + ⋅ + ⋅ + ⋅ = ∆ 2 2 2 (1) with : : charge density at high-k/Bottom HTO interface. : volume charge density in high-k. t H : thickness of high-k layer. t T : thickness of HTO top layer. 2) localised at the bottom HTO /High-k interface. ε H : high-k dielectric constant. ε SiO2 : SiO 2 dielectric constant. Trapping properties In this section, we investigate in more details the charge trapping phenomena of OHO samples during a gate stress. The gate current density as a function of the electric field is reported in Figure 3. On the same graph is also reported the time evolution of the gate current at constant gate bias (J G -Time) on virgin devices. The hysteretic behaviour, and the continuous decreasing of the leakage current with the elapsing time demonstrate that trapping phenomena take place in the high-k materials. Note that the charge trapping could be an issue for IPD applications, as it may degrade the reliability of the memory, generating threshold voltage instabilities. To evaluate more precisely the trapping capabilities of the interpoly stacks, we monitored the evolution of the flatband voltages as a function of time when the devices were submitted to different gate stresses (Figure 4). A continuous V FB shift is observed, showing the progressive electron trapping in the stack as the stress time increases. It clearly appears that for a given stress condition, the trapping capability increases with the Hf concentration. As already reported in the literature [START_REF] Molas | Proc. of ESSDERC[END_REF]10], this result could be correlated with the crystalline structure of the high-k materials: the larger the Hf concentration, the more crystalline the layer, and hence, the higher the trapping capability. Based on these measurements, we extracted the charges trapped in the gate stack after programming. We assume that the charges are localized at the interface between the Bottom HTO and the HfAlO layer. Indeed, the further the traps are from the cathode, the slower the charging kinetics. Making this assumption, it appears that the extracted trapped charge value does not depend on the HfAlO thickness (Figure 5). This confirms (see Equation 1) that charges are mainly trapped at the high-k interface and that the bulk contribution is negligible at the first order. The stressing conditions are performed at constant V G /EOT. We assume that the charges are localized at the interface between the Bottom HTO and the HfAlO layer. Retention characteristics In this section, we investigate the dynamic discharging of the charges previously trapped in the OHO layers by a writing stress. Figure 6 plots the room temperature retention characteristics of OHO samples with different compositions of the HfAlO layer. We observe that for the Hf-rich sample, the electron discharging rate is quite fast, whereas the HfAlO 1:4 and 1:9 keep most of the charge after 10 6 s. Figure 6 also shows the strong temperature activation. Indeed, the charge loss is strongly accelerated at 85°C (within a factor 2 for the 9:1 sample). Nevertheless, the trend observed at room temperature is still conserved, i.e.: the 1:4 and 1:9 HfAlO layers exhibit the same charge decay, while for the 9:1 HfAlO sample, only 50% of trapped charge remains after 10 6 s. These observations are consistent with experiments performed on SONOS-like structures with high-k trapping layers [11]. Modelling In this part, we introduce an analytical model to qualitatively explain the trapping characteristics of our IPD stacks. To this aim, we use the SRH model presented in [10], and we focus on the Hf-rich (9:1) OHO samples. The simulations are performed assuming that: The trapped charge is localized at the bottom HTO / HfAlO interface, which is consistent with our experimental data. Note that in a more realistic approach, we should take into account the charge trapped in the HfAlO bulk, characterised by slower trapping time constants. The thermalization of electrons in the SiO 2 layer close to the cathode is neglected. In fact, in our range of programming voltages, the electron paths in the conduction band of the HTO, after tunneling, is inferior to 1nm (Figure 7), which is indeed shorter than the mean free path of electrons in SiO 2 [12]. In other words, we assume that the electron energy remains constant till the trapping in HTO/HfAlO interface states happens. The gap (E G =5.65eV) and the permittivity (εr=17) of 9:1 HfAlO were extracted by ellipsometry and by C-V G measurements, respectively. We also consider that the bottom and top HTO are 4.5nm and 5nm thick respectively, which is in agreement with TEM observations. The shift between the conduction band of Si and HfAlO, E C , and the trapping cross section, , were fixed based on literature data: E C =2eV [13,14], =10 -18 cm 2 [START_REF] Fernandes | Proc. of ESSDERC[END_REF]10]. N st , the trap density, is adjusted to fit the experimental saturation level. Fig. 8: Modeling of the trapping characteristics of OHO stack (HfAlO thickness is 6nm, concentration is 9:1), based on structure and parameters reported in Figure 7. Based on these assumptions, we use the following equations to simulate the programming characteristics: st N ft Vth Ct × ∆ = ( ) ( ) ( ) 1 dft ft cn ep ft en cp dt = -× + -× + cn/en and cp/ep are the electron and hole capture/emission rates which govern carrier exchanges between the traps and substrate. ft: trap occupation probability according to Shockley-Read-Hall statistics model [START_REF] Ielmini | [END_REF]. Ct: trap to gate coupling capacitance. Figure 8 shows the experimental and the modelling trapping characteristics. We observe a very good correlation between the simulation and the experimental data for the three programming voltages, which validates our theoretical approach. Conclusions In this paper we investigated the intrinsic fixed charge and trapping phenomena happening under stress of HfAlO based interpoly dielectric stacks. We demonstrated that the fixed charge content increases with the Al concentration of the HfAlO layer. We showed that the trapping capability when the device is submitted to a constant voltage stress increases as the Hf ratio of the compound increases. Based on programming measurements, we proved that in our devices the electron trapping mainly occurs at the first interface, between the bottom HTO and the HfAlO layer, rather than in the volume of the high-k dielectrics. We also observed that the discharging rate of the previously trapped charges is more important for Hf-rich alloys. Finally, an analytical model based on a SRH approach allowed us to fit our experimental data and to extract the main trapping parameters of HfAlO high-k materials. Fig. 1 : 1 Fig. 1: Schematic showing the capacitor device stack studied in this work. Various high-k were investigated: HfO 2 , Al 2 O 3 and HfAlO with different Hf:Al ratios (9:1, 1:4 and 1:9). Figure 2 Fig. 2 : 22 Figure 2 plots the capacitance-voltage characteristics of the studied OHO triple layers in the virgin state. Fig. 3 : 3 Fig.3: Current density J G versus equivalent electric field E G of HTO /HfAlO 9:1 -9nm/HTO stack. J G -time measurements on virgin devices are also represented for different electric fields. Inset: J G -time measurements performed at different applied electric field (1MV/cm, 6MV/cm and 10MV/cm) as a function of time. Fig. 4 :Fig. 5 : 45 Fig. 4: Programming characteristics of OHO samples with various HfAlO compositions. The HfAlO thickness is 6nm. Fig. 6 : 6 Fig. 6: Room temperature and 85°C retention characteristics of OHO samples with various compositions of HfAlO. The HfAlO thickness is 6nm. The programming conditions are fixed to have an initial flatband voltage shift of 1.5V for each sample. 2 Fig. 7 : 27 Fig. 7: Energy band diagram of OHO stack at V G =10V simulated in this work. Fitting parameters are indicated. The HfAlO thickness is 6nm, the concentration is 9:1. The charges trapped in the HTO-HfAlO interface are responsible for the different electric field values in the bottom and top oxide layers. Table 1 1 summarizes the number of equivalent fixed charge localised at the bottom HTO/High-k interface, calculated from the V FB shifts. HfAlO Number of fixed charge composition nb/cm 2 9:1 3•10 12 1:4 3.5•10 12 1:9 4•10 12 Table 1 : 1 Number of equivalent fixed charge (extracted from the characteristics reported in Figure Acknowledgments Part of this work was supported by the MEDEA+ NEMESYS project.

01745625

en

2024/03/05 22:32:07

2018

https://hal.science/hal-01745625/file/Finite%20element%20method-based%20kinematics%20and%20closed-loop%20control%20of%20soft%2Ccontinuum%20manipulators.pdf

Frederick Largilliere Alexandre Kruszewski Zhongkai Zhang Rochdi Merzouki Christian Duriez email: [email protected] Finite element method-based kinematics and closed-loop control of soft, continuum manipulators Thor Morales Bieze, FEM-based kinematics and closed-loop control of soft, continuum manipulators Thor Morales Bieze, Frederick Largilliere, Alexandre Kruszewski, Zhongkai Zhang, Rochdi Merzouki and Christian Duriez Abstract-This paper presents a modeling methodology and experimental validation for soft 1 manipulators to obtain forward and inverse kinematic models under quasistatic conditions. It offers a way to obtain the kinematic characteristics of this type of soft robots that is suitable for offline path planning and position control. The modeling methodology presented relies on continuum mechanics which does not provide analytic solutions in the general case. Our approach proposes a real-time numerical integration strategy based on Finite Element Method (FEM) with a numerical optimization based on Lagrangian Multipliers to obtain forward and inverse models. To reduce the dimension of the problem, at each step, a projection of the model to the constraint space (gathering actuators, sensors and endeffector) is performed to obtain the smallest number possible of mathematical equations to be solved. This methodology is applied to obtain the kinematics of two different manipulators with complex structural geometry. An experimental comparison is also performed in one of the robots, between two other geometric approaches and the approach that is showcased in this paper. A closed-loop controller based on a state estimator is proposed. The controller is experimentally validated and its robustness is evaluated using Lypunov stability method. Index Terms-Soft manipulators, Continuum robots, Soft robots, Finite Element Method and Robotic control I. INTRODUCTION For the past four decades, rigid-link manipulators have been successfully deployed in the industrial environment. Their rigid bodies and high-torque joints are perfectly fitted to perform tasks that involve accurate positioning while carrying considerable payloads. However, as the applications for these systems move away from this structured environment, traditional rigid manipulators have been less successful. Indeed, their rigid and bulky bodies is a problem for adaptation to dynamic environments. Roboticists, trying to cope with the new applications for manipulators, have turned their attention to nature, seeking for inspiration to design new robot manipulators. Soft manipulators are robots often inspired by the morphology and functionality of biological agents like octopus tentacles and elephant trunks [START_REF] Neppalli | Octarm-a soft robotic manipulator[END_REF] [2] [3] [4] or tendrils [5] [6]. This type of manipulator deforms continuously to achieve a certain pose and can exhibit key advantages over their rigid counterpart with suitable design: they are lighter and therefore have less energy consumption, present a bigger power-to-weight ratio as well as a natural compliance because of their material properties. This compliance also gives the manipulators the ability to better adapt themselves to dynamic work environments and to work side by side with humans, without the concern of hazardous collisions. Because of these characteristics, soft manipulators have found a niche of applications in the medical field [START_REF] Shiva | Tendon-based stiffening for a pneumatically actuated soft manipulator[END_REF] [8] [START_REF] Simaan | A dexterous system for laryngeal surgery[END_REF] [10] [START_REF] Haraguchi | A pneumatically driven surgical manipulator with a flexible distal joint capable of force sensing[END_REF]. The compliant nature of soft continuum manipulators comes with the issues of modeling and control of their behavior, which is highly non-linear. A popular approach to model continuum robots has been the modification of methods already established to model rigid manipulators. In [START_REF] Hannan | Analysis and initial experiments for a novel elephant's trunk robot[END_REF], Hannan and Walker presented the development of the kinematic model for a trunk robot. The model considers that each section bends with constant curvature. This approach has been used to express the kinematics of continuum trunks [START_REF] Jones | Kinematics for multisection continuum robots[END_REF] and tendrillike continuum robots [START_REF] Bardou | Control of a multiple sections flexible endoscopic system[END_REF]. The constant curvature models can be used even when the shape of the continuum robot does not conform to a circular arc. In [START_REF] Mahl | A variable curvature continuum kinematics for kinematic control of the bionic handling assistant[END_REF], the kinematics of the bionic handling assistant are obtained by modeling each section of the robot as a finite number of serially connected circular arcs with different parameters each. The models mentioned, while producing close-form analytic models, are only based on the geometry of the robot, without consideration for the mechanics of the structure, necessary to properly describe the deformation of this type of robot. A. Model requirements In contrast with rigid robots, soft manipulator kinematics not only depend on the geometry of the robot, but also on its mechanical properties, in particular the stiffness of the structure. While rigid manipulator kinematics can be used to solve positioning problems with the assumption of resistance/counter-actuation to gravity or load effects, soft manipulators easily comply to these forces and deform. To answer the same problems of positioning, it is then necessary to take into account the current deformation (ie change of geometry) induced by these forces to obtain a kinematic relation between position of end-effector and position of actuators. (Fig. 1). The degrees of freedom in a rigid manipulator are determined by the joints of the manipulator and are often all actuated. In soft manipulators, the degrees of freedom are generated by the deformation of the continuum and their number is infinite. (It can be noted that it is disconnected from Fig. 1: In this example a tendon is pulled to create the motion of an elastic soft robot. Starting with the same geometry, the material stiffness has an influence on the kinematics (output vs input displacements). the number of actuators). Usually this problem is addressed by a discretization of the degrees of freedom of the continuum, through methods provided by computational mechanics. Another difference in soft manipulators, compared to rigid ones, is that if a load is carried by a soft manipulator, this load will deform the robot and modify its kinematics. B. Related Work This recapitulation of previous work is centralized in the use of continuum mechanics in the modeling of soft manipulators. A discussion on the use of mechanics-based methods to describe soft continuum robots can start by mentioning the work of Chirikjian [START_REF] Chirikjian | A modal approach to hyperredundant manipulator kinematics[END_REF] [START_REF]Kinematically optimal hyper-redundant manipulator configurations[END_REF], who used continuum mechanics to define a curve that describes the pose of hyper-redundant robots without considering actuation. This work laid the basis for subsequent work on continuum manipulator models. Given the uprising trend in the design of manipulators composed by a single elongated backbone actuated by tendons, researchers have explored beam theory-based approaches to describe the pose of this particular type of robots. Rucker et al. present in [START_REF] Rucker | Statics and dynamics of continuum robots with general tendon routing and external loading[END_REF] the potential of this theory applied to the modeling of manipulators with tendon routing. In [START_REF] Smoljkic | Compliance computation for continuum types of robots[END_REF] the compliance model of continuum robots is obtained by considering the robot as a single section of Cosserat rod. In [START_REF] Jones | Three dimensional statics for continuum robotics[END_REF], Jones presents a static model in three-dimensional space and in [START_REF] Giorelli | A two dimensional inverse kinetics model of a cable driven manipulator inspired by the octopus arm[END_REF], this theory was implemented to compute the inverse kinematic model of a tendon-driven tentacle manipulator in two dimensions under Euler-Bernoulli beam hypothesis. While providing models suitable for fast computing, beam theory is limited in its application by the shape of the backbone, i.e. when the backbone cannot be assimilated as a uniform beam, this modeling approach loses relevance. In particular, when the body of the robot is actuated intrinsically by pneumatic or hydraulic actuators it tends to have a more complex shape, and its behavior cannot be described by beam theory methods. Finite Element Analysis is increasingly used in the field of soft robots. In [START_REF] Connolly | Mechanical programming of soft actuators by varying fiber angle[END_REF], a direct finite element simulation is used to observe the behavior of soft pneumatic actuators. In this article, the development of a method to compute the kinematic model of soft manipulators that relies on the finite element model of the structure of the robot is presented. By using different types of elements (tetrahedral, beam, or shell elements), the methodology can be used on robots of very complex shapes. Contrary to the majority of models currently available in the literature, this approach also models two types of actuators, which enables this technique to be used as part of the control of the robot as well as in off-line analysis. Moreover, gravity and payload carried by the end-effector can be accounted for by this approach. This article presents the following contribution towards the kinematic modeling of soft manipulators: • A FEM-based modeling approach that accounts for complex structural shapes and the mechanics of the employed material. • The model of two actuation systems (i.e. pulling on cables and pneumatics) currently implemented in the majority of designs of soft manipulators. • The integration of sensors in the simulation that allows for an observer of the manipulator in the configuration space. • The validation of the modeling approach using two completely different deformable manipulators. • The experimental comparison of this approach to two other geometric-based models. • An experimental study on the robustness of the model under external loading. • A closed-loop controller based on a state estimator and the robustness analysis of the closed-loop system. In section II, the formulation of the static equilibrium and the constraints for end-effector, actuators and sensors is explained. Section III shows the projection of the model in the constraint space and the convex optimization process used to solve the reduced model. Section IV presents the experimental validation of the forward and inverse kinematic models and finally, in Section V, a discussion about the results and limitations of the model, as well as the perspectives for future work are presented. II. METHODS In this section, the development of the Finite Element Method (FEM) of soft manipulators is presented. The method relies on the constitutive law of the material from which the robot is made. This constitutive law can be directly measured by conducting stress/strain mechanical tests to a material sample in the ideal case. When the strain/stress tests cannot be done, an approximation of the constitutive law can be obtained in the simulation. The main idea is to tune these material parameters qualitatively by approximating the deformation seen in reality by that observed on simulation in which the deformation of the real robot is matched given a known static load. A similar approach is presented in [START_REF] Coevoet | Registration by interactive inverse simulation: application for adaptive radiotherapy[END_REF] in the context of radiotherapy. After measuring the constitutive law, a volumebased approach is used with tetrahedral elements. Then, the mathematical formulation of the constraints is introduced using Lagrange multipliers. In this method, the end-effector, actuators, and sensors use constraint models. A. FEM model of soft and continuum robots Depending on the shape of the robot, one could use volume, surface or linear elements to compute the non-linear deformation of the structure. In this paper, volumetric elements 2 are used. A non-linear formulation accounts for the large displacements and rotations of the structure. In continuum mechanics, this is considered as the case of large strain but small stress. More sophisticated FEM models can be proposed in the future, according to the constitutive law and the solicitation of the employed material (i.e. large stress). The computation in real-time with such models will be even more challenging, but the principles of the method described in this paper would still apply. The corotational implementation of volume FEM, presented in [START_REF] Felippa | A systematic approach to the element-independent corotational dynamics of finite elements[END_REF], is particularly suitable for linear elasticity under the hypothesis of large displacements. The shape of the robot is meshed using linear tetrahedral elements, but the same method could be used with other elements, shape functions and more advanced material laws. In the FEM, the nodes at the vertices of the elements represent the degrees of freedom of the manipulator. Even for a considerable amount of nodes, the approach is fast to compute, numerically stable and a free implementation in C++ is available in the open-source framework SOFA [START_REF] Faure | Sofa: A multi-model framework for interactive physical simulation[END_REF]. During each step i of the simulation, the following linearization of the internal forces is updated: f(x i ) ≈ f(x i-1 ) + K(x i-1 )dx (1) where f provides the volumetric internal stiffness forces at a given position x of the nodes, K(x) is the tangent stiffness matrix that depends on the actual positions of the nodes and dx is the difference between positions dx = x i -x i-1 . This linearization is valid as long as the displacement of the nodes dx is small. The lines and columns that correspond to fixed nodes are removed from the system to get a full rank for matrix K. In f and K, the rows (and columns for K) contain the component of the internal forces (x, y, z) for the nodes, in the order corresponding to their numbering in the mesh. In this paper, the study is limited to quasi-static behavior on purpose, since the simulation step required to capture high frequency vibrations is not feasible. Thus, in a first approach, the assumption is that the control of the robot is performed at low velocities, so that the inertia effects in the motion of the robot can be neglected. One seeks to establish static equilibrium at each step from the first law of Newton: f ext + f(x i ) + H T λ = 0 (2) 2 The method has also been tested with beam elements. where f ext represents the external forces (e.g. gravity) and H T λ gathers the contributions of the end-effector, actuators and the contact forces as Lagrange multipliers (see the following sections). The way H is obtained is explained in sections II-B and II-C but its computation is performed with the values obtained from the previous step. We then use the expression H(x i-1 ) and through the linearization explained in (1), we obtain the following formulation : -K(x i-1 )dx = f ext + f(x i-1 ) + H(x i-1 ) T λ (3) The variables dx and λ are both unknown and are found during the optimization process. It is noted that the matrix K is highly sparse. In the implementation, a conjugate gradient solver is used and preconditioned by a sparse LDL T decomposition. For a mesh composed of about 1000 nodes and about 3000 tetrahedral elements, a refresh rate of 60Hz is obtained with the implementation available in SOFA. B. Constraint for the end-effector To set the Lagrange multiplier on the end-effector, a point or a set of points of the robot needs to be considered as the endeffector. It could be any point(s) mapped on the finite element mesh. For each point, the constraint objective is to reduce the difference between the end-effector position and its desired position p des . Thus, a function δ e (x) : R 3n → R 3 with n being the number of nodes, evaluates this difference along x, y and z. If the end-effector corresponds to a node i of the mesh, the function is: δ e (x) = x i -p des , where x i is the position of node i. If the effector is set inside an element, we use: δ e (x) = n ∑ i=0 φ i (p e f f )x i (4) where p e f f is the position of the end-effector in the rest configuration of the FEM model and φ i is the shape (interpolation) function associated to node i. If several points are used for end-effector position, the vector δ e (x) gathers the evaluation of the difference for all the points. The function is then R 3n → R 3m , where m is the number of end-effector points. The matrix H used for the end-effectors corresponds to H e (x) = ∂ δ e (x) ∂ x . The matrix H e is highly sparse: A row, that corresponds to a component of a point of the end-effector, will contain non null values on a very small number of columns: As the point is mapped on a single tetrahedral element, there is a maximum of 4 non-null value per row. Of course, the column should match with the components of the nodes, given the fact that the nonnull values are gathered in 3x3 diagonal block matrices. Finally, an important point is the effort value that is put on the Lagrange multiplier that corresponds to the terminal effector. The value of λ e will depend on the load applied on the end-effector. Two cases can be considered: (I) if the points defined as end-effector move freely in the space, there is no physical interaction, so the contribution of the constraint vanishes λ e = 0. (II) if one or several points of the end-effector carries one object l which mass creates a load that could deform the structure. In such cases, the corresponding load should be set on λ e = m l g. with m l the mass of the object and g the gravity field. C. Actuator constraint model In this work, the actuator model takes into account its physical characteristics. Two types of actuators have been implemented in the framework: Tendon (cable) and pneumatic actuators. The contributions of these actuator constraints are unknown before the optimization process. However, given the type of actuation, the constraint is not set the same way. a) Cable actuator: In a first case (Fig 2 ), a cable is used to actuate the structure. The cable can simply be attached at one point of the structure, but it can also go through several other points (frictionless guides are considered) In all cases, the unknown λ a is the stretching force inside the cable. This force is unilateral (λ a ≥ 0). Let's suppose now that the points are numbered starting from the extremity where the cable is being pulled. The matrix H is computed this way: At each point p, we take the direction of the cable before d b = x p -x p-1 x p -x p-1 and after d a = x p+1 -x p x p+1 -x p . To obtain the constraint direction that is applied to the point, we use d p = d a -d b . Note that the direction of the final point is equal to the direction "before" as d a does not exist. These constraint directions are mapped on the nodes using the interpolation:     . . . f n . . .     =     . . . φ n (α, β , γ) d p . . .     λ a = H T a λ a (5) A function δ a (x) is defined to provide the length of the cable, given the position of the constrained node(s). The actuator stroke can also be included by imposing δ a (x) ∈ [δ min δ max ]). Through the use of this function, we get H a = ∂ δ a (x) ∂ x . b) Pneumatic actuator: The formulation is compatible with pressure-based actuation of cavities that are placed on the structure, as seen in Fig. 3. In that case, the effort λ a is the pressure exerted on the wall of the cavity. As the pressure is uniform inside the cavity, a single constraint can be set for each pneumatic actuator. All triangles of the cavity wall will be involved: For each triangle t, the area and the normal direction are computed. If this result is multiplied by the pressure, one obtains the force applied by the pneumatic actuator on the nodes t of this triangle. Consequently, the contribution of each triangle is added in the corresponding column of H T a . Fig. 3: Pressure actuation In the particular case of a pneumatic actuation, λ a provides the difference of pressure inside the cavity compared to the atmospheric pressure. Usually, pneumatic actuators only provide positive pressure so λ a ≥ 0. However, in some cases, it is also possible to create both negative and positive pressure using vacuum/pressure actuation. In that case, there is no particular constraint on the unknown value of λ a , despite an eventual limit (max / min) of pressure that can be achieved by the actuator. The approach to the modeling of fluidic actuators can also account for hydraulic actuators, by accounting for the weight distribution of the liquid at any given configuration, as presented in [START_REF] Rodríguez | Real-time simulation of hydraulic components for interactive control of soft robots[END_REF]. D. Sensor constraint model In order to relate the end-effector position and the geometry of the manipulator, one needs sensors that can measure the geometric state of the robot. In this study, the sensors used can measure the lengths of the sections that compose the manipulator, but also that can be easily integrated in the design. String potentiometers offer a good solution to acquire information on the real geometric state of the robot. As in the case of the cable actuator, the string of the sensor is routed through several frictionless guides, at n points x n . In the model, the measure of the lengths read by the sensor will be n-1 ∑ i=1 x i+1 -x i (6) which evaluates the distance between each sensor guide after the position of the nodes have been updated. A function δ s is defined to represent the difference between the current lengths of the sensors and the desired lengths. The matrix H s that gathers the directions of the sensor constraint is obtained in the same way as for the cable actuator, shown in section II-C. III. REDUCED MODEL IN THE CONSTRAINT SPACE The classical resolution of a FEM problem (like solving the static equilibrium of the structure described at equation 3) provides a forward model: it allows to compute the displacements of the structure, given the values of the efforts put on the actuator λ a . However, in the case of position control, the actuation λ a is the unknown. Yet, for controlling the motion of the soft robot, an inverse model is needed, which is challenging to compute in real-time as the size of the system is in the range of several thousands degrees of freedom. In this work, another approach is used, based on the projection of the mechanics in the constraint space that drastically reduces the size of the optimization problem. This approach, initially developed in [START_REF] Duriez | Control of elastic soft robots based on real-time finite element method[END_REF], is generalized. A new formulation of the inverse problem in the form of a quadratic programming (QP) optimization (developed in [START_REF] Largilliere | Real-time control of soft-robots using asynchronous finite element modeling[END_REF]) is used. A. Reduced compliance in constraint space As stated above, the optimization process relies on a projection of the mechanics in the constraint space. Each constraint has a direction that is set by a line of the matrices H e and H a This matrix is sparse, as the direction of the constraints is mapped on few nodes of the FE mesh. The values of the effort applied by the actuators λ a are not known at the beginning of the optimization process, whereas the value of λ e is supposed to be known. The first step consists of obtaining a free configuration x free of the robot which is found by solving the equation 3 while considering that there is no actuation applied to the deformable structure. In practice, the known value of λ e is used and λ a = 0 is imposed. The linear equation 3 is solved using a LDL T factorization of the matrix K. Given this new free position x free for all the nodes of the mesh, one can evaluate the values of δ free e = δ e (x free ), the shift between the effector point(s) position and the desired position introduced in section II-B. One can also evaluate δ free a = δ a (x free ) the position of the actuated points without actuation effort. From the FEM formulation of the problem that uses a large matrix K, a formulation that accounts for the directions of the constraints placed for actuators and end-effectors is derived. Using the Schur complement of matrix K in the Lagrange multiplier-augmented system [29], a small formulation of δ e is obtained. This variable expresses the difference between the desired position for the end-effector and its current position in terms of the actuators contributions λ a : δ e = H e K -1 H T a W ea λ a + δ free e (7) The Schur complement also provides similar formulations for the difference between a desired sensor or actuator position and its current position: δ a = H a K -1 H T a W aa λ a + δ free a (8) δ s = H s K -1 H T a W sa λ a + δ free s (9) This step is central in the method. It consists in projecting the mechanics into the constraint space. As the constraints are the inputs (effector position shift and sensor length shift) and outputs (effort to apply on the actuators) of the inverse problem, the smallest possible projection space for the inverse problem is obtained. It allows for a projection that drastically reduces the size of the search space without loss of information. Indeed, section III-B shows how the matrices W ea and W aa provides the mechanical coupling equations between actuators and effector point(s). After this projection, the optimization is processed in the reduced constraint space to get the values of λ a . This part is described in the section III-C. The final configuration of the soft robot, at the end of the time step, is obtained as : x t = x free + K -1 H T a λ a . (10) It should be emphasized that one of the main difficulties is to compute W ea and W aa in a fast manner. No pre-computation is possible as their value changes at each iteration. However, this type of projection problem is frequent when solving friction contact on deformable objects. Thus, several strategies are already implemented in SOFA [START_REF] Faure | Sofa: A multi-model framework for interactive physical simulation[END_REF]. B. Coupled Kinematic Equations Using the compliance operator W ea , one can get a measure of the mechanical coupling between effector and actuator, and with W aa , the coupling between actuators. For instance, the displacement δ i e created on the end-effector (along a direction stored on the line i of matrix H e ) by a unitary force λ j a applied by the actuator (which is stored at the line j of matrix H a ) is directly obtained by ∆δ i e = w i j ea λ j a + δ i,free e . As the motion is created by deformation, the motion of actuator j is influenced by actuator k. Through the same principle, actuator k also influences the displacement of the effector. To get a kinematic link between actuators and effector, the method needs to account for the mechanical coupling that can exist between actuators. It is captured by W aa that can be inverted if actuators are defined on independent degrees of freedom. Thus one can get a kinematic link by rewriting equation [START_REF] Kato | Tendondriven continuum robot for endoscopic surgery: Preclinical development and validation of a tension propagation model[END_REF] as: δ e = W ea W -1 aa (δ a -δ free a ) + δ free e (11) Equation ( 11) is composed of a reduced number of linear equations that relate the displacement of the actuators to the displacement of the effector. Consequently, matrix W ea W -1 aa is equivalent to the Jacobian matrix of a rigid manipulator. This matrix is a local linearization provided by the FEM model on a given position. It needs to be updated for deformations with large displacements. C. Inverse model by convex optimization The goal of the optimization is to find how to actuate the structure so that the end-effector of the robot reaches a desired position. This was initially proposed in [START_REF] Largilliere | Real-time control of soft-robots using asynchronous finite element modeling[END_REF]. It consists in reducing the norm of δ e which actually measures the shift between the end-effector and its desired position. Thus, computing min( 12 δ T e δ e ) leads to a Quadratic Programming (QP) problem: min 1 2 λ a T W T ea W ea λ a + λ a T W T ea δ free e ( 12 ) sub ject to (course of actuators) : δ min ≤ δ a = W aa λ a + δ free a ≤ δ max and (case of unilateral effort actuation) : λ a ≥ 0 (13) The use of a minimization allows to find a solution even when the desired position is out of the workspace of the robot. In such a case, the algorithm will find the point that minimizes the distance to the desired position while staying in the limits introduced by the course of the actuators. In practice, the QP solver available in the Computational Geometry Algorithms Library (CGAL) [START_REF] Fabri | Cgal: The computational geometry algorithms library[END_REF] is used. The matrix of the QP W T ea W ea is symmetric. If the number of actuators is equal or inferior to the size of the end-effector space, the matrix is also definite. In such a case, the solution of the minimization is unique. In the case when the number of actuators is greater than the degrees of freedom of the effector points, the matrix of the QP is only semi-definite. Consequently, the solution could be non-unique. A new criterion for the minimization can be introduced, based on the deformation energy. Indeed, this energy E de f is linked to the mechanical work of the forces exerted by the actuators. E de f can be computed by evaluating the dot product between λ a and the displacements of the actuators ∆δ a = δ aδ free a due to the actuator forces E de f = λ a T ∆δ a = λ a T W aa λ a . Yet, matrix W aa is positive-definite if the actuators are placed on different nodes of the FEM or with different directions (i.e. if there is no linear dependencies between lines of H a . Thus, one can add this energy in the minimization process by replacing [START_REF] Hannan | Analysis and initial experiments for a novel elephant's trunk robot[END_REF] with: min 1 2 λ a T (W T ea W ea + εW aa )λ a + λ a T W T ea δ free e ( 14 ) with ε chosen sufficiently small so that the deformation energy does not disrupt the quality of the effector positioning. In practice, ε = tr(W T ea W ea ) tr(W aa ) * 10 -3 so that the term εW aa does not alter the value of the QP matrix. Thanks to this modification, the QP matrix becomes positive-definite and a unique solution of the problem can be found. The CBHA is the bionic continuum manipulator component of the RobotinoXT, a didactic mobile platform designed by Festo Robotics. The system is shown in Fig. 4 (a). The bionic continuum manipulator is formed by 2 serially connected sections of pneumatic actuators, an axially rotating wrist and a compliant gripper. Without actuation, the manipulator has a length of 206mm, with each section having a length of 103mm. The width at the base of the manipulator is 100mm long and the top has 80mm of width. In our study, the end of the second section is considered as the end-effector. IV. KINEMATIC MODELS OF Each section of the manipulator is composed of a parallel array of pneumatic actuators, as shown in Fig. 4 (b). By applying different pressures to the bellows, each section can bend or extend independently. The pose of the manipulator is obtained as the contribution of the poses of the 2 sections. In order to sense the state of the robot, string potentiometers measure the lengths of the actuators. B. Forward kinematic models The forward kinematic model of a soft manipulator deals with the problem of finding the end-effector position, given a defined configuration of the manipulator. For a rigid manipulator, this configuration is simply the set of variables associated with the joints of the robot. In contrast with the rigid robots, the variables that express the configuration of a soft manipulator change with respect to the structure of the robot and its type of actuation, and therefore, cannot be obtained in a straightforward manner. The FEM-based methods explained before provide an easy way to obtain the kinematic relation between the end-effector and the configuration of the manipulator. • FEM-based model Given the intrinsic nature of the CBHA, the configuration of the robot is represented by the lengths of the pneumatic actuators that correspond to an end-effector position. Of course, the description of the robot could be given in the actuator space directly, using in this case Equation 7, to attain a pressureto-position model, but that requires a precise control over the actuation (in this case the pressure inside the cavities) in order to obtain a good estimation of the position of the end-effector. Instead, Equation 9, which is reproduced here for clarification, is used to relate the end-effector position to the configuration of the manipulator represented by the lengths of each pneumatic actuator, given by the sensors. This representation is clearer in the context of kinematic modeling, and allows for a position-to-position model which is less sensitive to unknown hardware parameters. δ s = H s K -1 H T a W sa λ a + δ free s (15) In this approach, no geometrical assumptions are needed. Each part of the robot is modeled in detail using shell and tetrahedral elements, as presented in Fig. 5. The mesh used in the model of the pneumatic cavities is composed by 3528 elements. Once the constraints have been incorporated in the model, the convex optimization finds each actuator contribution required to have the desired sensor lengths. The final position of the end-effector is obtained after the position of the nodes of the mesh is updated. • Constant curvature model This model of the CBHA, which was developed in [START_REF] Escande | Modelling of multisection bionic manipulator: Application to robotinoxt[END_REF] and [START_REF] Escande | Geometric modelling of multisection bionic manipulator: Experimental validation on robotinoxt[END_REF] and validated in [START_REF] Escande | Kinematic calibration of a multisection bionic manipulator[END_REF], works under the assumption that, after actuation, the resulting pose of each section in the robot can be represented by an arc section with constant curvature (Fig. 6). The evolution from end-to-end of a section i is described, in terms of backbone parameters, by 2 coupled rotations and one translation in the homogeneous transformations: i j T =     c 2 φ i cθ i + s 2 φ i cφ i sφ i (cθ i -1) cφ i sθ i x i cφ i sφ i (cθ i -1) s 2 φ i cθ i + c 2 φ i sφ i sθ i y i -cφ i sθ i -sφ i sθ i cθ i z i 0 0 0 1     (16) where the notations s and c mean sine and cosine respectively. The cartesian coordinates of the end of the bending section i are given by (x i , y i , z i ), where x i = r i cφ i (1cθ i ), y i = r i sφ i (1cθ i ) and z i = r i sθ i . The backbone variables φ i , θ i and r i can be expressed in terms of the actuator lengths in order to have the correct kinematic relations : φ i = tan -1 ( √ 3(l 3 -l 1 ) 2l 1 -l 2 -l 3 ) θ i = D i 3d i r i = (l 1 +l 2 +l 3 )d i D i (17) with D i = 2 l 2 1 + l 2 2 + l 2 3 -l 1 l 2 -l 1 l 3 -l 2 l 3 (18) The parameter d i represents the diameter of section i. In this model, each section is considered to be a cylinder with constant radius. The lengths of each actuator in the section i are represented by l 1 , l 2 and l 3 . • Hybrid model In this approach, developed in detail in [START_REF] Lakhal | Hybrid approach for modeling and solving of kinematics of compact bionic handling assistant manipulator[END_REF], the CBHA is considered as 17 vertebrae serially connected. Between each pair of vertebrae, an inter-vertebra section is modeled as a 3UPS-1UP joint (3 universal-prismatic-spherical joints and one universal-prismatic joint). The behavior of a sub-structure composed by 2 vertebrae and an inter-vertebra is represented by a parallel robot with 3 DoF, as shown in Fig. 7. Fig. 7: Sub-structure of the CBHA modeled as a parallel robot. The parallel robot consists of an upper and a lower platform connected by 3 limbs and a central leg. The limbs are modeled by a UPS joint in which only the prismatic part is active allowing the control of the position and orientation of the upper vertebra, with respect to the lower vertebra. The central leg is modeled as a passive UP joint and is used to constraint the rotation about the longitudinal axis of the parallel robot, as well as any shearing motion between the vertebrae. The position and orientation of the upper vertebra k +1, with respect to the lower vertebra k is given by the transformation matrix k k+1 T =     cθ k sθ k sψ k sθ k cψ k 0 0 cψ k -sψ k 0 -sθ k sψ k cθ k cθ k cψ k z k 0 0 0 1     (19) where the angles θ k and ψ k represent pitch and roll angles, respectively, and the notations s and c mean sine and cosine respectively. In this model, the prismatic variable q n,k shown in Fig. 7 represents the length of each inter-vertebra, which is a percentage of the total length of the actuator. This percentage can be obtained by considering the minimum and maximum elongation of each inter-vertebra. This development is presented in detail in [START_REF] Lakhal | Hybrid approach for modeling and solving of kinematics of compact bionic handling assistant manipulator[END_REF]. C. Experimental validation and model comparison In order to validate the model, a set of 50 end-effector positions distributed inside the task space of the manipulator were selected. For each position, the correspondent configuration of the robot was recorded using the string potentiometers that are placed along the structure of the robot. The set of lengths recorded were used as an input for the forward kinematic model. The experiment assumes zero-end-effector payload. The results are compared to those of the Constant Curvature and also the Hybrid approach. This comparison is presented in Fig. 8 and9. The results show that the constraint approach is more accurate in estimating the position of the end-effector, with a Root-Mean-Square (RMS) error of 4.66mm, compared to 12.87mm and 17.09mm of error for the Constant curvature and the Hybrid approaches, respectively. We hypothesize that the imprecise measurement of displacement for each vertebra may be the cause of the hybrid approach being less precise than ours, as there were only 6 string potentiometers available to chart the displacements. Moreover, this model was initially developed to be able to inverse it, more than for the pure precision of the forward kinematic model. Nevertheless, the FEM model still has a few limitations in its development. These limitations represent the main source of error in the results: for the moment, the constitutive law used to model the material of the trunk is only an approximation. Another source of error comes from the geometry of the trunk itself. When the trunk is bent at a maximum angle, the outer walls of the pneumatic cavities collide with each other, as shown in Fig. 10. The consideration of these collisions is not yet implemented in the simulation. The generic nature of the approach showcased in this article is illustrated by obtaining the inverse kinematics of two different soft manipulators. The simulation of the inverse model provides the position control of the robots in open loop that can be used to pilot directly the robot, as in the case of the parallel soft robot. D. Inverse kinematic model In this section, an experimental validation of the modeling methodology is conducted two different soft robots: • A parallel soft robot made of silicone, actuated with tendons (cables) controlled in position, • The Compact Bionic Handling Assistant (CBHA). The inverse model provided by convex optimization in realtime allows to teleoperate the robots in open-loop: Given a desired input position of the effector, the desired output for the actuators is computed. For the soft parallel robot, a desired position of the tendons is provided. 1) Modeling and feed-forward control of a parallel soft robot: This experiment is based on a 3D soft robot, made of silicone, which design is inspired by parallel robots with closed kinematic chains (Fig. 11). In its rest position, the dimensions of the robot are 180 × 180 × 130mm The robot naturally deforms and sinks under the action of gravity, but 4 unilateral actuators (servo-motors that are connected to the structure of the robot with cables) are placed on each leg to prevent and pilot the deformation. The effector position is placed on the upper part of the robot. Its trajectory is defined in 3D (and can interactively be changed by a user) and the algorithm provides the position to apply to the servomotors. The Young modulus of the silicone, measured experimentally, is used to parametrize the robot. The FEM model of the robot is composed of 4147 Tetrahedrons and 1628 Nodes. When projected in the constraint space, the size of the system is highly reduced: 3 equations for the effector, and 4 equations for the actuators. The convex optimization that leads to the inverse model can be performed in real-time. The most timeexpensive part of the computation is the projection expressed in equations ( 7) to ( 8) (50 ms on a Core i7, 2.8GHz), but when using the Graphics Processing Unit (GPU) method described in [START_REF] Courtecuisse | Preconditionerbased contact response and application to cataract surgery[END_REF], it significantly reduces the computation time of the projection (15ms in this case). To validate the method, a study of the discrepancy between the desired positions and the obtained positions is conducted on static positions distributed across a workspace of 25mm × 25mm × 50mm around the rest-position of the robot (see Fig. 12). The measurements are performed using a motion capture system based on infrared cameras 3 . On a sample of 28 positions, a mean error of 1.4mm is obtained with a standard deviation of 0.6 mm and a maximum error of 2.9 mm. This illustrates the precision that can be achieved using such FEM approaches. It can be noted that these results are obtained using an open-loop and with a position to position control: for a given position of the effector, the algorithm finds a position for the actuator cables. This is a favorable case for FEM precision because the partial differential equations are enforced with Dirichlet boundary conditions. 2) Inverse kinematics of the CBHA: Considering the kinematic relationship for the CBHA given in section IV-D, that is the link between the actuator lengths and the position of the end-effector, the inverse kinematic model, solved by the convex optimization will give the actuator lengths that result from a predefined end-effector position. The FEM analysis applied to model this soft robot is detailed in A domain decomposition strategy is applied in order to perform the computation of the model (Equation ( 3)) and the projection in the constraint space (Equation ( 7)). After the actuator contribution required to achieve the desired position of the end-effector is applied to the model, and the position of the nodes is updated, the readings from the sensors in the simulation, given by Equation 6, will give the lengths of the pneumatic actuators that represent the output of the inverse kinematic model. To validate the method, a set of 50 end-effector positions are selected inside the task space of the robot and the corresponding set of lengths for each position is recorded by the sensors of the robot. The same set of positions is used as inputs for the inverse model, and the resultant length of each actuator is estimated. This study is summarized in Table 1, where l 1 , . . . , l 6 represent the lengths of the actuators and their values are in mm, µ represents the mean error and σ is the standard deviation. The results are presented in Fig. 13. The results show a mean error between 2.43mm and 4.08mm across all lengths, which represents between 1.21% and 2.04% of the total length of the manipulator. As in the case of the parallel robot, the set of actuator contributions (in this case the pressures applied to the cavities) obtained from the optimization process can be used as input for the real robot to obtain a feed-forward control. However, as explained before, there are some considerable discrepancies between the pressure calculated by the simulation and the pressure applied to the cavities, mainly caused by the way the pressure is regulated in the robot. This leads to less accurate results. In order to improve the results in terms of efforts (pressure,force) in the inverse model, one can use more advanced constitutive models for the materials. One of these models is the St Venant-Kirchhoff hyper-elastic model. The stress/strain relationship in the St Venant-Kirchhoff model is represented by the Second Piola-Kirchhoff stress tensor S that has the form: S = λ tr(E)I + 2µE ( 20 ) where E is the Lagrange-Green strain tensor and λ and µ are the Lamé constants that can be approximated from the Young's modulus and Poisson's ratio of the material in question. We have conducted tests on the parallel soft robot using the St. Venant-Kirchhoff model to compare the results to those obtained using the corotational formulation. In the tests we observed very small errors in the displacement output (3.24% of a total cable stroke of 50mm). In the case of the force output we observed bigger errors (16.02% of the tension in the cable) related to the errors made by the corotational formulation in the stress computation. E. Deflection of end-effector under external loading As explained in section II, one of the advantages of this modeling approach is the ability to predict the deflection of the robot under external loading, given a good representation of the material mechanics. If the load is known a priori, the value of the force acting on the end-effector λ e can be used in equation [START_REF] Zhao | Design and analysis of a kind of biomimetic continuum robot[END_REF] to compute the position that accounts for said force. In order to validate this modeling feature, a set of experiments were conducted on both manipulators using known loads. First, an initial configuration for the manipulator without loading is selected and the position of the end-effector is measured, then the load is applied and the new position of the end-effector is recovered after the robot achieves static equilibrium. The same load is then applied to the model of the manipulator using the same initial pose and the resulting endeffector position is also recovered. In the case of the CBHA, the model of the sensors presented in section II-D is used to apply the configuration of the real robot measured by the string potentiometers to the simulation model. A vector that connects initial and final end-effector positions represents the deflection caused by the loading. In order to assess the repeatability of the measurements, the loading sequence described before is performed 40 times for each loading value and the average value is then used for the model validation. A standard deviation of 0.4838mm is obtained across all the measurements. The comparison between measured and model deflections for both manipulators is presented in Fig. 14 and15. In the figures, the blue line represents the compliance to loading of the manipulators and the red line is the prediction of the model. In the case of the CBHA, the maximum error is 4.107mm with an average error of 2.1047mm. Nevertheless, Fig. 15 shows the CBHA presents a strain hardening/necking stages of plastic behavior at the beginning of the loading profile which corresponds to the compliance of the plastic material from which the manipulator is made of (polyamide nylon), and therefore the model prediction is accurate only for a small region of the profile. In order to improve the model predictive capabilities for the CBHA in particular, two constitutive laws could be implemented to account for the different behaviors, but this would modify the way the inverse FEM is formulated. In contrast, the maximum error in the case of the parallel manipulator is 2.06mm with an average error of 2.01mm. The reason we obtain better results is because the material of the parallel robot conforms better to the assumption of high deformation and low stress, while also being an elastic material with no plastic behavior. V. FEM-BASED CLOSED-LOOP CONTROL OF CONTINUUM ROBOTS In section IV-D2, the relationship between the sensor lengths and the end-effector position of the CBHA was obtained based on the FEM simulation of the robot, however, in order to control the motion of the robot, the set of pressures applied to the actuators is to be computed. Indeed, the relationship given by equation ( 7) can be used to control directly the robot in open-loop, but as explained in IV-B this requires an accurate control over the pressures applied to the robot. Moreover, non-linear behaviors like the hysteresis and strainrate dependency of the material (which is not considered in the model) render the feedforward control of the manipulators unusable in real applications. Controllers for soft manipulators have been investigated in the past with the intention of rejecting non-linear behaviours and model uncertainties that result from the complex dynamics of the manipulators. Control based on energy formulations [START_REF] Ivanescu | Dynamic control for a tentacle manipulator with sma actuators[END_REF], model-less approaches [START_REF] Yip | Model-less feedback control of continuum manipulators in constrained environments[END_REF] and feedback controllers [START_REF] Penning | Towards closed loop control of a continuum robotic manipulator for medical applications[END_REF] [40] have been proposed before with the intention of achieving accurate positioning of the manipulators in presence of nonmodeled dynamics. In this section, a closed-loop control strategy based on a state estimator is proposed. A. Closed-loop control design The closed-loop controller is designed to ensure the correct configuration of the robot, given a desired end-effector position. A reference computation is performed to transform the desired position to the correspondent configuration. Assuming that the external forces are constant, the discrete model of the system, derived form of equation ( 9), takes the form: δ s,k+1 = δ s,k + J sa (x k )∆λ a,k+1 (21) where J sa = W sa is the Jacobian matrix between sensors and actuators. When the desired sensor lengths are provided by the reference computation, we can propose the closed-loop control approach shown in Fig. 16 In Fig. 16 the blue blocks represents the computations performed by simulation. Two simulations executing simultaneously are implemented in the closed-loop system: One main simulation that computes the Inverse kinematic model and a second simulation that acts as a state estimator for the system. The state estimator is the Forward kinematic model simulation of the robot that computes an estimated configuration for the robot based on the lengths of the sensors. This configuration is used to update the state of the Inverse kinematic model at each simulation step. In this way, we make sure that the configurations of both simulation model and the manipulator are similar before the estimation of the Jacobian is computed. The tracking error e k in the closed-loop system is computed as: e k = δ s,k -δ d s,k (22) with δ d s,k represents the desired lengths of the sensors and δ s,k represents the current lengths in the robot. We define the control vector v k as: where Ĵsa (x k ) is the estimated Jacobian matrix between the sensors and actuators and r k = ∆λ a,k+1 . Using Eq. 23, the kinematic model can be rewritten as: v k = Ĵsa (x k )r k (23) δ s,k+1 = δ s,k + v k (24) The control law is based on proportional integrative strategy, therefore, the control vector v k is designed in the sensor space as: v k = -k p e k -k i h k (25) with k p and k i being the proportional and integrative gains of the controller, respectively. The integrative term h at time k + 1 is computed as: h k+1 = h k + e k (26) Then, the control allocation based on a Quadratic Programming (QP) formulation [START_REF] Johansen | Control allocation-a survey[END_REF] is employed to find a unique solution to: r k = Ĵ+ sa (x k )v k (27) where Ĵ+ sa is the pseudo-inverse of the estimated Jacobian. In practice, as Ĵsa (x k ) may not be fully invertible, we introduce a variable O defined as O = Ĵ+ sa (x k )r k -v k (28) Using O, the QP problem formulation (III-C) becomes: min u k (O T O) (29) the resulting r k will be the best possible inversion of Eq. 23 in the least square sense. In addition, the QP formulation allows to define constraints like actuator saturation or positive direction of actuation. Using Eq. 25 in Eq. 27, r k is rewritten as r k = -Ĵ+ sa (x k )(k p e k + k i h k ) (30) Using Eq. 30 in Eq. 21, the closed-loop system is defined as: e k+1 = e k + J sa (x k )r k (31) which in the ideal case in which Ĵsa is invertible, can be written as: e k+1 = e k + v k (32) The system of Eq. 32 is a simple first order discrete model that can be controlled with any standard controller. We choose the control strategy to be based on proportional-integrative control law because we want to improve the convergence rate and remove any steady state error (in the sensor space at least). After testing, the selected gain values are k p = 0.14 and k i = 0.0003 as a compromise between the rise time of the signal and its overshot. Fig 17 shows the response of one actuator length of the simulated robot and the real robot given a pre-computed set points corresponding to an end-effector position inside the task space of the robot. The position is chosen so that the actuators are far from their saturation points. The model simulation and the real robot have different initial condition. The experiment was performed for 2500 simulation steps with a simulation step of 0.1 seconds. After 1000 simulation steps, the set points are changed in both the simulation and the real robot. The results show that both, the simulation of the robot and the robot itself have the same settling time t s ≈ 400 simulation steps. We can also see that the curve that represents the measured value of the lengths in the robot jumps between two values. This behavior is a consequence of the poor resolution of the string potentiometers. Fig 17 also shows a different behavior in the transitory stage of the curve of the measured lengths. This behavior can be attributed to different factors; first, there is the time required to compute the configuration of the manipulator from the measured sensor lengths; second, there is a time delay for the desired pressure to be applied to the robot, and finally, the plasticity of the material from Fig. 17: Comparison of real and estimated actuator length of the CBHA. A second set point is applied to the system after 1000 simulation steps in order to observe the performance of the controller. The step is 0.1s for the experiment. which the manipulator is built (polyamide-nylon) which is not accounted for in our FEM model. On the other hand, the pneumatic valves that control the pressure inside the actuators have a small dead zone, so, when the manipulator starts its motion from a zero-pressure condition, very small increments in the pressure do not produce any motion until this dead zone is surpassed, which is not considered in the FEM. A second experiment was performed using the real robot in the loop. In this experiment an external unknown force was applied to the manipulator in order to see the uncertainty rejection capabilities of the controller. Fig. 18 shows the results of this experiment. B. Robustness analysis Because of modeling uncertainties, the estimated Jacobian matrix Ĵsa (x k ) is, in general, different from the Jacobian of the robot J sa (x k ). We introduce the vector ω k that represents the disparities between the real Jacobian and the estimated Jacobian. We call this vector the inversion error and is defined as: ω k = [I -J sa (x k ) Ĵ+ sa (x k )]v k (33) Then, the closed-loop system is re-written as: e k+1 = e k + v k + ω k = e k -k p e k -k i h k + ω k (34) The disturbed closed-loop system is: e k+1 h k+1 = I -k p I -k i I I I e k h k + I 0 ω k (35) It can be disturbing that we end up with such a simple linear system. We emphasize to the reader that the non-linearities are taken into account by the two simulation blocks (FKM and IKM in Fig. 16) in the closed-loop control. In Eq. 35, we are writing the system in terms of e k and h k and if the model was perfect, the system would be trivial. However, we can have modeling errors, that is why, in the following, we will prove that the control is robust to these modeling uncertainties ω k . To simplify the notation of the problem, we define the following vectors: X k+1 = e k+1 h k+1 , X k = e k h k , D = I 0 , F = k p k i (36) Also I -k p I -k i I I I = A -BF (37) where A = I 0 I I , B = I 0 (38) Using this notation, matrix ω k is written as: ω k = [I -J sa (x k ) Ĵ+ sa (x k )]FX k (39) We assume that the error in the Jacobian estimation is bounded by a bounding parameter γ such that: ω T k ω k = X T k F T [I -J sa (x k ) Ĵ+ sa (x k )] T [I -J sa (x k ) Ĵ+ sa (x k )]FX k ≤ γ 2 X T k F T FX k (40) with [I -J sa (x k ) Ĵ+ sa (x k )] T [I -J sa (x k ) Ĵ+ sa (x k )] ≤ γ 2 I (41) For the proof of stability, we use Lyapunov's second method of stability [START_REF] Lyapunov | The general problem of the stability of motion[END_REF]. We define the Lyapunov candidate function as: V = X T k PX k ( 42 ) where P is an unknown Lyapunov matrix with the properties P T = P > 0 (43) From Eq. 42 and the notation given in Eq. 36, the variation of the Lyapunov function is defined as: ∆V = X T k+1 PX k+1 -X T k PX k (44) Using Eq. 38, Eq. 44 is re-defined as: ∆V = ((A -BF)X k + Dω k ) T P((A -BF)X k + Dω k ) -X T k PX k (45) By A -BF = C (46) Eq. 45 is written as: ∆V = (CX k + Dω k ) T P(CX k + Dω k ) -X T k PX k = X T k C T PCX k + X T k C T PDω k + ω T k D T PCX k + ω T k D T PDω k -X T k PX k (47) Reverting the notation in Eq. 38, Eq. 47 can be written in matrix form as: ∆V = X k ω k T (A -BF) T P(A -BF) -P (A -BF) T PD D T P(A -BF) D T PD X k ω k (48) For the proof, we introduce an accessory parameter α ≥ 0 in Eq. 40, such that: ϒ = αω T ω -αγ 2 X T k F T FX k < 0 (49) From Eq. 49, the left hand side of the inequality is written in matrix form as: ϒ = X k ω k T -αγ 2 F T F 0 0 αI X k ω k < 0 (50) Adding and subtracting this term to Eq. 48 allow us to find a bounding for ∆V as: ∆V -ϒ + ϒ = X k ω k T Q X k ω k + ϒ ( We know from Eq. 49 that ϒ < 0. Therefore, if Q is definite negative, then ∆V < 0. To prove the closed-loop system to be stable, the values for matrix P > 0 and α ≥ 0 need to be found such as matrix Q is definite negative, given the predefined values of the boundary parameter γ and the tuned controller parameter k p and k i . To this end, a Linear Matrix Inequality [START_REF] Boyd | Linear matrix inequalities in system and control theory[END_REF] Solver called SeDuMi [START_REF] Sturm | Using sedumi 1.02, a matlab toolbox for optimization over symmetric cones[END_REF] is used in the software Matlab. In order to describe the LMI given by Eq. 46, Yalmip [START_REF] Lofberg | Yalmip: A toolbox for modeling and optimization in matlab[END_REF], a toolbox for optimization that is compatible with Matlab is employed. Given a value of γ = 0.98 and the gain values k p = 0.14 and k i = 0.0003, the LMI was solved successfully. The resulting matrix P and the parameter α that make matrix Q negative definite are: Using the LMI solver, we can also compute the maximum value of γ, which provides an insight on the robustness of the closed-loop system. After some iterations we have: max γ = 0.98685 < 1 (54) Recalling Eq. 39, if ω T ω > 1, then matrices J sa (x k ) and Ĵ+ sa (x k ) do not have the same sign, which means that the robot Jacobian and the estimated Jacobian indicate opposite directions. In our case, γ = 0.98685 is close to the limit case. The proposed closed-loop system is robustly stable and can handle high Jacobian inversion errors in the change of control variables. VI. CONCLUSIONS AND FUTURE WORK This paper presents a modeling methodology to obtain the kinematic relationships of soft manipulators. The kinematic equations are derived from a FEM model (or any equivalent physics based model) that can be obtained from the geometry and the material properties of a soft manipulator. After a projection in a small constraint space, a set of coupled equations relate the position of the end-effector to the contribution of actuators and displacement of sensors. The validity of the method is demonstrated in two different manipulators with complex geometry. In the case of the CBHA, the results were compared to those obtained with two geometric models developed for the same robot. While the model of the material used does not take into account the properties of viscosity, this consideration is only due to the absence of knowledge of these specific properties for the material used. Indeed, the framework used allows for modeling viscoelasticity with Prony series [START_REF] Marchesseau | Multiplicative jacobian energy decomposition method for fast porous visco-hyperelastic soft tissue model[END_REF]. In general, a viscoelastic model is characterized by a rate-independent term, which in this case is the shear modulus representing the elastic behavior, and a rate-dependent modulus. The rate-dependent modulus of the material is defined by the Prony series based on time; faster strain rates will induce higher modulus than static loads. The limitations on the use of Prony series come with the determination of the required coefficients, since it involves stress relaxation tests performed under controlled temperature and loading speed. Another way to model viscoelasticity behaviour is to introduce a ratedependent damping effect using Rayleigh equation. Rayleigh damping is a viscous damping that is proportional to a linear combination of mass and stiffness. Using Rayleigh damping, The internal forces in the robot (equation 1) takes the form: f(x i ) ≈ f(x i-1 ) + K(x i-1 )dx + B(x i-1 )dx (55) where the Rayleigh damping matrix is computed as: B = αM + β K (56) where M and K are the mass and stiffness matrices, respectively, and α and β are the coefficients of proportionality. The problem of position control for soft manipulators was solved by obtaining the inverse kinematic relationships of two different types of robots. The implementation of the simulation of the model was then used to directly pilot one of the manipulators given a desired position of the end-effector in feed-forward control. The feed-forward control of the robots relies entirely on its model. Because the lack of material parameters, the openloop system does not account for non-linear behaviors such as viscosity. The closed-loop controller proposed in this papers was proven to be able to reject these model uncertainties and improve the overall behavior of the manipulator. Moreover, the proposed controller can be used even when high Jacobian inversion errors are present. It is important to remark that the method is no longer viable when we leave the quasi-static motion case, and for the moment, the sampling rate required to capture vibrations in the robot is not feasible. Nevertheless, this first approach to the kinematics and control for soft manipulator opens up some interesting perspectives for future work: • The model of the tendons does not account for the friction between the cable and the guides it passes through. Including a term in the formulation of the direct model to account for the friction can be done, but the way it will change the inverse model should be investigated. • Given the information provided by the FEM model, a study on the impedance control of the robot is feasible. The information regarding the compliance of the robot can be directly extracted from the FEM. Fig. 2 : 2 Fig. 2: Tendon actuation. d b and d a on the figure, represent the direction of the tendon before and after the cable guide, respectively, which are used to compute the normal forces at the guides. Fig. 4: The RobotinoXT by Festo Robotics. (left) The anatomy of the Compact Bionic Handling Assistant. (right) A section of the manipulator, composed by 3 pneumatic actuators and their correspondent length sensor. Fig. 5 : 5 Fig. 5: Visual model of the trunk and the underlying finite element model. Fig. 6 : 6 Fig. 6: Constant curvature model Fig. 8 : 8 Fig. 8: X/Y view of the results from the model comparison. Fig. 9 : 9 Fig. 9: X/Z view of the results from the model comparison. Fig. 10 : 10 Fig. 10: Collision of the outer wall of the cavities. The collisions occur on the orange edges depicted in the bent actuator (right). Fig. 11 : 11 Fig. 11: Deformable parallel manipulator. Fig. 12 : 12 Fig. 12: Comparison of desired trajectory and measured trajectory of the parallel manipulator Fig. 13 : 13 Fig. 13: Comparison of measured and estimated lengths of one of the sensors given a predefined set of end-effector positions for the CBHA Fig. 14 :Fig. 15 : 1415 Fig. 14: Comparison between measured and predicted deflections caused by external loading on the parallel manipulator Fig. 16 : 16 Fig. 16: Closed-loop control of the CBHA based on IKM and FKM simultaneous and the controller Fig. 18 : 18 Fig. 18: Measured lengths of the CBHA in closed-loop. An external force is applied to the manipulator after 1050 time steps. The time step is 0.1s for the experiment. BF) T P(A -BF) -P + αγ 2 F T F (A -BF) T PD D T P(A -BF) DPD T -αI P = 646.4512 1.2983 1.2983 0.0087 and α = 4655 (53) TABLE 1 : 1 Statistical analysis of the error between measured and estimated lengths for the CBHA l(mm) l 1 l 2 l 3 l 4 l 5 l 6 µ(mm) 3.2 2.43 3.86 4.08 3.6 3.69 σ (mm) 1.55 1.76 2.05 2.12 2.56 2.06 In the literature, these manipulators are usually classified as continuum robots. However, their main characteristic of interest in this paper is that they create motion by deformation, as opposed to the classical use of articulations. The positioning precision provided by the motion capture system is less than 0.1mm VII. AUTHOR DISCLOSURE STATEMENT No competing financial interests exist. VIII. ACKNOWLEDGMENTS This research was part of the project COMOROS supported by ANR (Tremplin-ERC) and the Conseil Régional Hautde-France and the European Union through the European Regional Development Fund (ERDF).

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2024/03/05 22:32:07

2011

https://hal.science/hal-01745633/file/Tirano.2011.SISSC.Abstract.pdf

On the electrical variability of resistive-switching memory devices based on NiO oxide Resistive-switching memories (so-called RRAM) are increasingly investigated since they gather low cost, high integration capabilities together with good performances [1]. RRAM memories based on transition metal oxide are promising candidate because of a simple metal/oxide/metal stack allowing the integration of memory elements into the back end of line [2]. Before substituting conventional Flash memories, RRAM devices must fulfill reliability requirements and the variability of their electrical characteristics has to be properly apprehended. In this paper the variability on forming and reset characteristics are reproduced by 1D modeling. By using this approach, this paper aims at identifying the key physical parameters that may explain the intrinsic spread of electrical characteristics. As depicted in Fig. 1, RRAM devices consist of a NiO active layer (25 nm thick) sandwiched between two platinum electrodes (25 nm thick). Such devices exhibit unipolar resistive switching effect between two conductivity states, i.e. a low resistance state (LRS) and a high resistance state (HRS) [3]. To identify the conduction mechanism in the pristine state, current-voltage characteristics were accurately measured at various temperatures. Below the forming voltage (zone III in Fig. 2a), the current does not show evident thermal activation that may be in agreement with a trap-assisted tunneling (TAT) mechanism described here in terms of barrier height (B) and average distance between traps (w) [6]. Figure 3 shows the max/min and median dc IV characteristics measured on several RRAM either during the forming stage (Fig. 3a) or during the reset operation (Fig. 3b). Finally in Fig. 4 forming and reset voltages are monitored with respect to environing temperature (these are first data ever shown in the literature, to our knowledge, on this aspect). To apprehend the impact of physical parameters on the intrinsic variability of electrical characteristics, a self-consistent physical model accounting for both forming and reset operations was used [4]. This model takes into account two distinct mechanisms: a redox reaction (i.e. electrochemical oxidation/reduction processes) during forming, and thermal diffusion and dissolution of conductive filaments (CFs) during reset [5]. In Table 1, the impact on electrical characteristics is evidenced for each physical parameters. Microscopical parameters related to TAT conduction (B and w) and the free energy of the reduction reaction at equilibrium (Ered) mainly affect the forming voltage. In contrast, the activation energy of the LRS (Ea) involved in the dissolution of CFs and retention characteristic, together with electrical parameters related to the CFs (resistivityCF0, thermal conduction KthCF, thermal coefficientT) largely affect the reset voltage and the reset current. Another important parameter is the current compliance during forming Icomp that strongly influences reset current and resistance in LRS [7]. Indeed Icomp must be controlled to avoid larger spread of electrical characteristics [8]. Provided these elements, the mean values (ie µ) of different parameters were calibrated on the median IV, for forming and RESET (Fig 3 ); then slight variation (ie ) on each of them allowed a good fit on min/max characteristics, as evidenced in Fig. 3 and Fig. 5 on large set of data: in Table II the set of µ and  are highlighted. Note that in Table II a quantitative view of the impact of each microscopical parameters is provided in the last column. Eventually in Fig. 4 we compare modeling result on random set of physical parameters (with µ and  from Table 2) with data. We obtain a fine agreement with data, that could be improved enlarging the statistical ensemble of data. To summarize, the present work enables apprehending the impact of physical parameters on the variability of NiO-based RRAM. The important variability of current in pristine state may be linked on variation of NiO layer during fabrication process which impacts the barrier height (B) and average distance between traps (w). And the modification of filament structure could impact the electrical parameters (CF0 and KthCF) and can explain the variability in LRS for current and reset voltage. Also the impact of each microscopical parameter has been quantitatively put in relation with the macrocopical electrical counter-part. S .Tirano a,b , M.Bocquet a , Ch.Muller a , D.Deleruyelle a , L.Perniola b , V.Jousseaume b , B. De Salvo b , G.Reimbold b a IM2NP, UMR CNRS 6242, Aix-Marseille Université, F-13451 Marseille Cedex 20, France b CEA-LETI, MINATEC Campus, 17 rue des Martyrs, F-38054 Grenoble, France Figure 1 : 1 Figure 1: Schematic description of the NiO-based RRAM memory elements. Figure 2 : 2 Figure 2: a) Typical current-voltage characteristic measured from the pristine state. Zone I: no resistance switching; Zone II onset resistance switching; Zone III switching event to LRS. b) Temperature-dependent evolution of current at low voltage. Figure 3 : 3 Figure 3: Extreme-value current-voltage characteristics measured either in the forming stage a) or during the reset operation b). Each experimental curve is satisfactorily fitted by the unipolar switching model. Figure 4 : 4 Figure 4: Forming and reset voltage behavior with respect to temperature for experimental and simulation. Table 1 : 1 Electrical impact of parameters for forming and reset operation Figure 5 : 5 Figure 5: Experimental and simulated dispersions of forming (left) and reset (right) 20 events on each. Table 2 : 2 Mean value (µ) and standard deviation () of physical parameters used to fit experimental data (Fig 3-4). Impact on electrical characteristics due to an independent variation of each physical parameters in forming and reset operations

01745644

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2024/03/05 22:32:07

2011

https://hal.science/hal-01745644/file/Muller.2011.CICC.Design%20Challenges%20for%20Prototypical%20and%20Emerging%20Memory%20Concepts%20Relying%20on%20Resistance%20Switching_Invited.pdf

Ch Muller D Deleruyelle O Ginez J-M Portal M Bocquet Design Challenges for Prototypical and Emerging Memory Concepts Relying on Resistance Switching Keywords: Prototypical and emerging memory concepts, resistance switching, operation mechanisms, memory design, embedded or distributed memory circuits Integration of functional materials in memory architectures led to emerging concepts with disruptive performances as compared to conventional charge storage technologies. Beside floating gate solutions such as EEPROM and Flash, these alternative devices involve voltage or current-controlled switching mechanisms between two distinct resistance states. The origin of the resistance change straightforwardly depends upon the nature and fundamental physical properties of functional materials integrated in the memory cell. After a general overview of non volatile memories, this paper is focused on prototypical and emerging memory cells and on their ability to withstand a downscaling of their critical dimensions. In addition, despite different maturity levels, a peculiar attention is turned toward common guidelines helpful for designing embedded or distributed resistive switching memory circuits. I. OVERVIEW Currently, the microelectronic industry is facing new technological challenges for continuously improving performances of memory devices in terms of access time, storage density, endurance or power consumption. The major bottleneck to overcome is the downsizing of memory cell dimensions necessary to integrate a larger number of elementary devices and subsequently more functionalities on a constant silicon surface. This drastic size reduction is constrained, in particular, by the intrinsic physical limits of materials integrated in the memory architecture. Beside volatile random access memories (RAM) such as Dynamic (DRAM) or Static (SRAM), non volatile memory technologies may be subdivided in two categories depending upon the mechanism used to store binary data. A first group of solid state devices is based on charge storage in a polysilicon floating gate. In this family, gathering usual EEPROM and Flash technologies [START_REF] Van Houdt | Physics of Flash memories[END_REF], new concepts or architectures are required to satisfy CMOS "More Moore" scaling trends. For instance, Si1-xGex "strained silicon" technology enables boosting the charge carrier mobility and discrete charge trapping improves data retention and enables an extension towards "multi-bits" storage. Presently, Flash technology still remains the undisputed reference, regardless of its NAND (dense and cheap) or NOR (fast) architecture. However, as the scaling of the conventional floating gate cell becomes ever more complicated below 32 nm node, opportunities for alternative concepts are rising. In the last two decades, several major semiconductor Companies have explored new solutions integrating a functional material, whose fundamental physical property enables data storage. Older technology, called FRAM for Ferroelectric RAM [START_REF] Scott | Ferroelectric memories[END_REF], is in small volume production for several years by Fujitsu and Texas Instruments which license Ramtron patents. With their DRAM-like architecture FRAM memories integrate ferroelectric capacitors that permanently store electrical charges. Thanks to their low voltage operations, fast access time and low power consumption, FRAM devices mainly address "niche" applications such as contactless smart cards, RFID tags and nomad devices. Excluding FRAM technology, recent R&D efforts also led to a new class of disruptive technologies based on resistive switching mechanisms. These memories, presenting two stable resistance states controlled by an external current or voltage, attract much attention for future solid state devices. The physical origin of the resistance change depends upon the nature and fundamental physical property of materials integrated in the memory cell. As a result, a broad panel of new concepts is currently rising: magnetoresistive memories (MRAM) and derived concepts; phase change memories (PCM); resistive memories (RRAM) including oxide resistive (OxRRAM) and "nanoionic" memories (CBRAM and PMC) [START_REF] Ch | Emerging memory concepts: materials, modeling and design[END_REF]. As compared to other technologies, PCM and RRAM memories are more favourably positioned as alternatives to Flash for sub-22 nm nodes. Nevertheless, these two technologies have different maturity levels: while first PCM products are available (e.g. Omneo TM by Micron, former Numonyx), RRAM memories are still at early R&D stage. Following 2010 ITRS's report, MRAM, PCM and FRAM are categorized as "prototypical" memories whereas RRAM is an "emerging" memory [START_REF]Potential and Maturity of Selected Emerging Research Memory Technologies[END_REF]. Hence, alternative non volatile memory concepts, either evolutionary or revolutionary, are being explored to satisfy the need for higher storage capacity and better performances. Within a broad panel of innovative solutions, this review specifically addresses resistive switching technologies. After a brief description of memory cells and their ability to withstand downscaling of their critical dimensions, peculiar attention is turned toward common guidelines helpful for designing embedded or distributed memory circuits. II. MEMORY MARKETS FOR RESISTIVE TECHNOLOGIES To penetrate the markets currently covered by SRAM, DRAM and Flash memories, prototypical and emerging concepts are facing stringent requirements of process compatibility and scalability at material, device and circuit levels. Status and outlook may be proposed as follows:  MRAM arising from spintronics may be viewed as a credible candidate to replace existing technologies in many applications requiring standalone or embedded solutions combining Flash-like non volatility, SRAM-like fast nanosecond switching and DRAM-like infinite endurance. For future, "Spin Torque Transfer" (STT-MRAM) concept appears as a promising technology able to merge aforementioned advantages and scalable for aggressive technological nodes (cf. §III).  PCM is probably the most advanced alternative memory concept in terms of process maturity, storage capacity and access time. Furthermore, the phase change memory element exhibits an excellent ability to withstand a downsizing of its critical dimensions. Considering limitations in terms of high reset current and low performances in retention, PCM is rather positioned as a NOR Flash substitute. Nevertheless, it may be also envisaged to rethink PCM subsystem architecture to bring the technology within competitive range of DRAM. To exploit PCM's scalability as a DRAM alternative, new design solutions are expected to balance long latencies, high energy writing and finite endurance [START_REF] Lee | Architecting phase change memory as a scalable DRAM alternative[END_REF].  RRAM concept is in an earlier stage of development compared with MRAM and PCM. Since RRAM memory elements can be integrated into Back-End Of Line (BEOL), this technology is of particular interest for high density storage with possibly multi-levels threedimensional architectures. Metal-organic complex-based RRAM [START_REF] Ch | Resistance change in memory structures integrating CuTCNQ nanowires grown on dedicated HfO2 switching layer[END_REF], transition metal oxide (TMO) based Ox-RRAM [START_REF] Sawa | Resistive Switching in Transition Metal Oxides[END_REF][START_REF] Dumas | Resistive switching characteristics of NiO films deposited on top of W and Cu pillar bottom electrodes[END_REF] and chalcogenide-based CBRAM appear as promising candidates [START_REF] Kozicki | Nanoscale memory elements based on solid-sate electrolytes[END_REF]. However, retention and endurance still remain to be demonstrated for CBRAM devices, while the high reset current is an issue in TMO-based memory devices. As a consequence, RRAM concepts, still in their infancy, require further academic and industrial investigations (i) to validate integration and scalability capabilities; (ii) to uncover the physical origin of resistance switching which remains sometimes controversial; (iii) to model the reliability. Beside conventional standalone (Fig. 1a) and embedded (Fig. 1b) memories, emerging resistive concepts also enable designing innovative electronic functions such as field programmable gate array (FPGA) or logic devices (e.g. Flip Flop) in which non volatile memory circuits are distributed over a single chip (Fig. 1c). This latter distributed implementation requires CMOS compatible, low cost, low voltage and low power non volatile memory elements. Before investigating novel architectures, it is of primary importance to develop design kits compatible with microelectronics design suites like Cadence or Mentor. Design kits dedicated to memories require first compact models describing cell's properties and verification tools that enable designing and checking the circuit layout [START_REF] Ch | Emerging memory concepts: materials, modeling and design[END_REF]. III. MAGNETORESISTIVE MEMORIES, MRAM In magnetoresistive devices, data are no longer stored by electrical charges, as in semiconductor-based memories, but by a resistance change of a complex magnetic nanostructure [START_REF] Tehrani | Status and outlook of MRAM memory technology[END_REF]. MRAM cells integrate a magnetic tunnel junction (MTJ) consisting of a thin insulating barrier (i.e. tunnel oxide) separating two ferromagnetic (FM) layers. Using lithographic processes, junctions are etched in the form of sub-micron sized pillars connected to electrodes. In conventional FIMS (Field Induced Magnetic Switching), each MTJ is located at the intersection of two perpendicular metal lines (Fig. 2a). MTJ resistance depends upon the relative orientation of magnetizations in the two FM layers: the magnetization in FM reference layer is fixed whereas the one in FM storage layer is switchable (Fig. 2b). Magnetization reversal in FM storage layer is controlled by two external magnetic fields produced by currents injected in the surrounding metal lines. Tunnel magneto-resistance (TMR) is due to tunnelling of electrons through the thin oxide layer sandwiched between two FM films having either antiparallel (high resistance state "0") or parallel (low resistance state "1") magnetizations. In MRAM technology three major issues are identified: (i) low resistance discrimination between "0" and "1" states (i.e. small sensing margin); (ii) high sensitivity to disturb during writing ("bit fails"); (iii) high currents (few mA) required for magnetization reversal. To overcome these issues, different solutions are proposed:  To enlarge the sensing margin, the tunnel barrier in amorphous aluminium oxide AlOx of first device generations was progressively replaced by crystallized magnesium oxide MgO.  Everspin Company (former Freescale), world's first volume MRAM supplier, already sells a 4, 8 and 16 Mb MRAM chips based on 180 nm CMOS technology and relying on the "Toggle" concept to limit disturbs during writing [START_REF] Andre | A 4 Mb 0.18 µm 1T1MTJ Toggle MRAM with balanced three input sensing scheme and locally mirrored unidirectional write drivers[END_REF]. Thanks to a new free layer magnetic nanostructure called "synthetic antiferromagnet" (SAF), an appropriate bit orientation and a current pulse sequence, MRAM cell bit state is programmed via Savtchenko switching. In that configuration, a single metal line alone cannot switch the bit, providing greatly enhanced selectivity over the conventional FIMS MRAM.  In October 2010, French startup Crocus Technology and TowerJazz completed the first stage integration of "Thermally-Assisted" MRAM into 130 nm CMOS platform. In TA-MRAM cell, a current injected in MTJ during writing induces a Joule heating in FM layers and the subsequent temperature increase facilitates the magnetization reversal [START_REF] Prejbeanu | Thermally assisted switching in exchange-biased storage layer magnetic tunnel junctions[END_REF]. TA-MRAM concept enables (i) shrinking the memory cell size with only one metal line to produce magnetic field; (ii) reducing the power consumption thanks to limited writing current; (iii) improving bit fails immunity.  More recently, STT ("Spin Torque Transfer") switching was proposed to solve aforementioned issues and make MRAM memory compatible with more aggressive nodes. Large Companies (e.g. Renesas, Hynix, IBM, Samsung…) and few startups (e.g. Grandis, Avalanche, Everspin, Crocus…) are actively working on this new concept and state they can push the integration limits beyond 45 and 65 nm nodes for standalone and embedded memories respectively [START_REF] Nagai Hide | Spin-Transfer Torque writing technology (STT-RAM) for future MRAM[END_REF]. In STT writing mode a spin-polarized current, which flows through MTJ, exerts a spin torque on the magnetization of FM storage layer [START_REF] Diao | Spintransfer torque switching in magnetic tunnel junctions and spintransfer torque random access memory[END_REF]. Technologically, STT concept requires the integration of a polarizer to select the spin of electrons injected into the magnetic nanostructure and ensuring the magnetization reversal. As compared to conventional MRAM cells, STT-RAM solution is said to consume less power, to improve bit selectivity and to reduce the memory cell size to approximately 6-9 F 2 . Similarly to DRAM, EEPROM or NOR Flash memories and whatever the writing mode (Toggle, TA or STT), MRAM core-cell is based on the association of 1 select MOS transistor with 1 magnetic junction (i.e. 1T/1MTJ). To individually access each memory cell, magnetic tunnel junctions are located at each intersection of a bottom "Word Line" (WL) connected to the transistor gate and an upper "Bit Line" (BL) (Fig. 2c). In contrast to multi-layer structure proposed for RRAM technology, such 1T/1MTJ cells disable stacking memory layers due to disturbs between neighbouring cells during magnetic field-assisted writing. Polarization of "Word", "Bit" and "Source" (SL) lines used to select one core-cell in the memory matrix straightforwardly depends on the writing mode. In TA-MRAM, writing requires a heating current injection in MTJ obtained in applying a positive voltage on the gate of the select transistor (WL) of the addressed core-cell whereas the source line is grounded. Concomitantly, a current is injected in single metal line (BL) to create a magnetic field that enables switching of FM storage layer. In STT writing mode, a positive voltage is applied on the select transistor gate (WL) of the addressed cell [START_REF] Hosomi | A novel nonvolatile memory with spin torque transfer magnetization switching: spin-RAM[END_REF]. Since writing does not require magnetic field, only bipolar voltage is applied between BL and SL to inject a top-down or bottom-up spin polarized current enabling magnetization reversal. Beside memory architectures, TA [START_REF] Guillemenet | Non-volatile run-time field-programmable gate arrays structures using thermally assisted switching magnetic random access memories[END_REF] or STT [START_REF] Zhao | Spin-MTJ based non volatile flip-flop[END_REF] MTJs may be also integrated in reconfigurable FPGA. Using a heterogeneous design there is no need to load the configuration data from an external non volatile memory as required in SRAM-based FPGA. Non volatility enables reducing both power consumption and configuration time required at each start-up of the circuit in comparison with classical static random access memory-based FPGAs. IV. PHASE CHANGE MEMORIES, PCM Since the 1970s, chalcogenide phase change materials attract much attention for data storage applications [START_REF] Ovshinsky | Reversible electrical switching phenomenon in disordered structures[END_REF], particularly in rewriteable optical media such as compact, digital versatile or more recently Blu-ray discs. In addition, they offer a great potential for non volatile phase change memories abbreviated as PCM [START_REF] Lankhorst | Low-cost and nanoscale non-volatile memory concept for future silicon chips[END_REF][START_REF] Cho | A 0.18 µm 3.0 V 64 Mb nonvolatile Phase transition Random Access Memory (PRAM)[END_REF]. This concept exploits the resistance change (few orders of magnitude) due to reversible amorphous (state "0") to crystalline (state "1") phase transitions. As depicted in Fig. 3, to reach amorphous state (i.e. reset operation), GST material is heated above the melting temperature TM (typically 600-700°C) and then rapidly cooled. In contrast, GST material is placed in the crystalline state (i.e. set operation) when "slowly" (few tens of ns) cooled from a temperature in between melting point TM and crystallization temperature TX (Fig. 3). Technologically, a layer of chalcogenide alloy (e.g. Ge2Sb2Te5, GST) is sandwiched between top and bottom electrodes: the phase change is induced in GST programmable volume through an intense local Joule effect caused by a current injected from bottom electrode contact acting as a "heater" (Fig. 4a). As illustrated in Fig. 3, improvement of PCM performances needs to overcome three main issues:  Design innovative cell architectures that enable reducing reset current which remains quite high in existing prototypes (typically 0.5 mA).  Propose new chalcogenide materials exhibiting a higher crystallization temperature to improve data retention (spontaneous amorphous-to-crystalline transition) and crystallizing rapidly to decrease switching time.  Fabricate confined phase change volume to limit heat spread toward neighbouring cells during reset operation. As compared to MRAM technology, PCM memories have a better ability to withstand size reduction. Indeed, although the endurance is still limited (less than 10 10 cycles), the memory cell is much more compact (typically 10 F 2 ) and downsizing should be extended beyond 25 nm node by shrinking programmable volume. PCM is rather positioned as a NOR Flash substitute: on this market segment, PCM technology displays significant improvements in terms of endurance, access time and bit alterability (i.e. state switching without intermediate erase operation required in floating gate technologies). PCM technology basically uses 1X/1R core-cells (Fig. 4b), X being one access device associated with one phase change resistor R. As shown in Fig. 4c, each core-cell is addressed thanks to WL connected to access device, BL and SL. Considering the high reset current required to switch chalcogenide material from crystalline to amorphous phase, access device must be able to drive high current during this operation. Access may be controlled by a bipolar junction transistor (BJT) [START_REF] Gill | Ovonic unified memorya highperformance nonvolatile memory technology for stand-alone memory and embedded applications[END_REF], a diode [START_REF] Oh | Full integration of highly manufacturable 512Mb PRAM based on 90nm technology[END_REF] or a MOS transistor [START_REF] Hyung-Rok On | Enhanced write performance of a 64 Mb phase-change random access memory[END_REF]. Dedicated peripheral circuits have to be designed to write [START_REF] Bedeschi | An 8 Mb demonstrator for high-density 1.8 V phasechange memories[END_REF]. Current passing through the cell is measured thanks to a mirror biased structure and the current is compared to a reference 1BJT/1R dummy cell. V. REDOX MEMORIES, RRAM Last ten years have seen the emergence of new memories labelled as RRAM for Resistive RAM and based on various mechanisms of resistance switching excluding phase change as used in PCM (cf. §IV). In its simplest form, RRAM element relies on MIM structures (Metal/Insulator/Metal) whose conductivity can be electrically switched between high (state "0") and low (state "1") resistance states (Fig. 5a). RRAM memory elements are gaining interest for (i) their intrinsic scaling characteristics compared to the floating gate Flash devices; (ii) their potential small size; (iii) their ability to be organized in dense crossbar arrays (Fig. 5b). Hence, RRAM concept is seen as a promising candidate to replace Flash memories at or below 22 nm technological nodes. Following the latest ITRS classification published in 2010 on "Potential and Maturity of Selected Emerging Research Memory Technologies" [START_REF]Potential and Maturity of Selected Emerging Research Memory Technologies[END_REF], different "flavours" of RRAM memory elements are disclosed depending on the mechanism involved in the resistance change [START_REF] Waser | Nanoionics-based resistive switching memories[END_REF][START_REF] Waser | Redox-Based Resistive Switching Memories -Nanoionic Mechanisms, Prospects, and Challenges[END_REF]. In the ITRS's report, all RRAM concepts are categorized as "Redox RAM" that encompasses a wide variety of MIM structures and materials sharing reduction/oxidation (i.e. redox) electrochemical processes [START_REF] Waser | Redox-Based Resistive Switching Memories -Nanoionic Mechanisms, Prospects, and Challenges[END_REF]. Redox mechanisms can operate in the bulk I-layer, along conductive filaments formed within the I-layer, and/or at the I-layer/metal contact interfaces in MIM structure. The following sub-categories are proposed:  In "Fuse/anti-fuse" memories, the resistance switching is driven by local reduction/oxidation mechanisms leading to formation/dissolution of conductive filaments within a transition metal oxide [START_REF] Sawa | Resistive Switching in Transition Metal Oxides[END_REF][START_REF] Dumas | Resistive switching characteristics of NiO films deposited on top of W and Cu pillar bottom electrodes[END_REF][START_REF] Waser | Nanoionics-based resistive switching memories[END_REF][START_REF] Waser | Redox-Based Resistive Switching Memories -Nanoionic Mechanisms, Prospects, and Challenges[END_REF]. Operations do not depend upon applied voltage polarity (unipolar switching).  In "valence change mechanism", the resistance switching relies on redox electrochemistry that changes the conductivity of the I-layer. Field-induced oxygen vacancies drift plays a predominant role in few specific oxidebased memory elements: TiO2-based "Memristors" [START_REF] Strukov | The missing memristor found[END_REF] recently demonstrated by Hewlett-Packard Company; CMOx TM elements developed by Unity Semiconductor [START_REF] Meyer | Oxide dual-layer memory element for scalable non-volatile cross-point memory technology[END_REF]; NiO layers obtained from nickel oxidation in small via structures [START_REF] Courtade | Integration of resistive switching NiO in small via structures from localized oxidation of nickel metallic layer[END_REF][START_REF] Courtade | Method for manufacturing a memory element comprising a resistivity-switching NiO layer and devices obtained thereof[END_REF][START_REF] Goux | Coexistence of the bipolar and unipolar resistive switching modes in NiO cells made by thermal oxidation of Ni layers[END_REF]. Memory operations require polarity reversal (i.e. bipolar operations).  CBRAM ("Conductive Bridge RAM") [START_REF] Symanczyk | Conductive bridging memory development from single cells to 2Mbit memory arrays[END_REF] and PMC ("Programmable Metallization Cells") [START_REF] Kozicki | Nanoscale memory elements based on solid-sate electrolytes[END_REF] belong to "nanoionic" memories. MIM-like memory elements consist in an inert electrode (W, Pt ...), an ionic conductor used as solid electrolyte (WO3, MoO3, GeSe, Ag-GeSe...) and an active electrode (Ag, Cu...) producing, through an electrochemical reaction, ions (Ag + , Cu + …) drifting within solid electrolyte. For this type of mechanism, a polarity change of applied voltage is mandatory (i.e. bipolar operations). As previously underlined, RRAM memories are still in a R&D stage and only few memory circuits have been published in the literature. As memory elements may be integrated into BEOL, RRAM technology is of particular interest for high density storage with possibly multi-levels threedimensional architectures (Fig. 6a) [START_REF] Kügeler | High density 3D memory architecture based on the resistive switching effect[END_REF][START_REF] Baek | Multilayer cross-point binary oxide resistive memory (OxRRAM) for post-NAND storage application[END_REF]. Nevertheless, it has to be stressed out that from a memory cell and array design point of view, unipolar operations are preferred over bipolar operations. For this promising class of memories, two cell architectures are generally proposed depending on the nature of RRAM element: 1T/1R (i.e. one transistor associated with one resistive element) for CMOS-based active matrix; 1R enabling crossbar-type memory passive matrix. The first demonstrators integrated memory cells with size of 4-8 F 2 for active matrix and (4/n) F 2 for a passive matrix (i.e. crossbar) with n storage layers [START_REF] Waser | Nanoionics-based resistive switching memories[END_REF][START_REF] Symanczyk | Conductive bridging memory development from single cells to 2Mbit memory arrays[END_REF]. Nevertheless, in passive crossbars, the memory elements cannot be electrically isolated while neighboring cells are addressed. This problem can be solved by adding serial elements with a specific (high) non-linearity (e.g. diode as shown in Fig. 6b) at each resistively switching cell, depending on the resisting properties and the array size. Samsung Company already demonstrated a "non-CMOS" solution with a two-layer architecture based on 1D/1R memory cells associating a diode with a nickel oxide-based resistive element exhibiting unipolar switching [START_REF] Lee | 2-stack 1D-1R cross-point structure with oxide diodes as switch elements for high density resistance RAM applications[END_REF]. Such a crossbar memory matrix was easily designed with the help of BL and WL to access each cell individually. CBRAM derives from parent technology PMC, which was developed by Axon TC in collaboration with Arizona State University. In 2007, Qimonda/Altis/Infineon consortium demonstrated a 2 Mb CBRAM test chip with read-write control circuitry implemented in a 90 nm technological node with read/write cycle time less than 50 ns [START_REF] Dietrich | A nonvolatile 2-Mbit CBRAM memory core featuring advanced read and program control[END_REF]. The corresponding CBRAM circuit was developed using 8 F 2 corecells associating 1 MOS transistor with 1 Conductive Bridging Junction (i.e. 1T/1CBJ). The chip design was based on a fast feedback regulated CBJ read voltage and on a novel program charge control using dummy cell bleeder devices. It has to be mentioned that Adesto startup has acquired intellectual property and patents related to CBRAM technology from Qimonda in October 2010. In addition, Adesto announced in November 2010 a manufacturing partnership with Altis Semiconductor. The low power resistive switching at voltages below 1 V, the ability to scale to minimum geometries below 20 nm and the multi-level capability make CBRAM a very promising non volatile emerging memory technology. Nevertheless, as compared to prototypical MRAM, FRAM and PCM memories, CBRAM concept requires refresh operations due to the poor retention capability in low resistance state (contrary to the high resistive state which is usually much more stable in time). Thus, a refresh voltage is applied to the CBRAM memory element at a predetermined time to strengthen and stabilize low resistance state. This smart refresh enables preserving the resistance margin ΔR necessary to unambiguously discriminate high and low resistance states during read. This refresh is performed without destroying data stored in CBRAM element whereas in DRAM a rewriting of respective state is mandatory. To conclude, RRAM technology is very promising and once again many design solutions exist for implementation. However, depending on RRAM concept, specific peripheral circuits are required to guaranty reliable memory operations. VI. TWO GUIDING PRINCIPLES For both prototypical (MRAM, PCM) and emerging (RRAM) technologies, two guiding principles are shared for designing generic resistive switching memory circuits:  The first guideline is related to the implementation of bistable resistive elements in a memory array. In most of cases, memory cells rely on the association of a select/access device with a resistive element. Depending on the resistive concept (magnetic, phase change or redox), the select/access device can be a BJT, a MOS transistor or a diode. Even if MOS transistor is frequently preferred, access device is adapted to electrical characteristics of resistive element (set and reset currents and voltages, resistance levels, memory window…). To access each 1T/1R memory cell individually, suited bias conditions are applied to Bit (BL), Source (SL) and Word (WL) Lines of the addressed cell, other lines being grounded and/or floating. For illustration, Hush and Baker [START_REF] Hush | Complementary bit PCRAM sense amplifier and method of operation[END_REF] have proposed an architecture for sensing the resistance state of a programmable conductor random access memory element (Fig. 7a). The design is based on complementary memory elements, one holding the resistance state being sensed and the other holding a complementary resistance state. A sense amplifier detects voltages discharging through high and low resistance elements to determine the resistance state.  Resistance in "0" and "1" states represents the second common guideline (Fig. 7b). Indeed, reliable read operations require an unambiguous discrimination of low and high resistances, below and above the median resistance RRef defined as R/2 = [RHigh -RLow]/2. Moreover, the bit cell resistance distributions must be narrow to avoid any overlap between "0" and "1" states (i.e. large R/ ratio, with  the distribution standard deviation). Finally, as the maximum current consumed during read operation is linked to RLow, resistance in "1" state must be as large as possible to decrease overall power consumption. Figure 7. (a) Architecture that enables sensing the resistance state of a programmable conductor random access memory element using complementary resistive elements (dashed square). (b) Bit cell resistance distributions of "0" and "1" resistance states. VII. CONCLUSION In summary, reliable memory operations require a close matching between electrical characteristics of resistive elements and design rules. In other words, to reach suitable sensing margin (i.e. large R) shifted toward high resistances altogether with narrow bit cell resistance distributions (i.e. low ), it is of primary importance to:  Control materials microstructure, critical dimensions of memory element, process variability…;  Understand physical mechanisms explaining resistance switching;  Develop relevant compact models to be implemented in electrical simulators. Hence, the development of prototypical and emerging memory concepts emphasizes the necessity to make stronger links between memory cell materials and processes, modeling and circuit design. Figure 1 . 1 Figure 1. Circuit/architecture of non volatile systems: (a) Conventional implementation with logic block and standalone non volatile memory (NVM) on separated chips. This configuration enables integrating separately optimized technologies but requires tradeoffs in terms of cost and communication speed between chips. (b) Embedded implementation with NVM and logic block on a single monolithic substrate. (c) Implementation with distributed NVM circuits on a single chip. Figure 2 . 2 Figure 2. (a) To write MRAM bit, currents are passing through perpendicular metal lines surrounding MTJ. (b) The resultant magnetic fields enable programming a bit in reversing magnetization of upper FM storage layer (select MOS transistor is OFF). To read a bit, a current is passing through the MTJ and its resistance is sensed (select MOS transistor is ON). (c) Schematic diagram of series-parallel architecture for 1T/1MTJ cells. Figure 3 . 3 Figure 3. Temperature profiles used for set and reset operations in PCM.Innovations are mandatory to improve data retention, reduce power con- Figure 4 . 4 Figure 4. (a) Conventional PCM architecture in which two adjacent memory cells are coupled to a common digit line. Access MOS transistor driven by a word line WL is connected to phase change element through a conductive plug acting as a "heater". (b) 1 access device/1 resistor memory cell. (c) Electrical schematic arrangement of a PCM memory array. Figure 5 . 5 Figure 5. (a) Scheme of typical RRAM memory element: resistive switching layer is sandwiched between top and bottom electrodes to form simple MIM (Metal/Insulator/Metal) structures. (b) Crossbar-type memory architecture in which memory elements are located at the intersections of per- Figure 6 . 6 Figure 6. Three-dimensional view (a) and schematic (b) of a two-layer crossbar memory array integrating 1 diode/1 resistor (1D/1R) core-cells. 1D/1R cell enables isolating a memory element while neighbouring cells are addressed. and read data in PCM memory cell: during writing, access device injects current into the storage material and thermally induces phase change, which is detected during reading. Contrary to MRAM concept, a special emphasis is required on write drivers. The time-dependent temperature profiles used to switch chalcogenide material have to be carefully controlled through the monitoring of set and reset currents injected in the programmable volume. For instance, Woo Yeong Cho et al. have proposed a solution based on a current mirror source belonging to a local column constituted of 1NTMOS/1R cells[START_REF] Cho | A 0.18 µm 3.0 V 64 Mb nonvolatile Phase transition Random Access Memory (PRAM)[END_REF]. The main advantage of such a structure is the reuse of the current source for all columns through the entire memory chip. For read operations, PCM memory circuits require sense amplifier able to discriminate the two resistance states. As compared to MRAM devices, the resistance margin R is larger (at least 1 decade) and constraints on sense amplifier are slightly relaxed. Bedeschi et al. have proposed a solution based on current measurement mode implemented in 8 Mb PCM memory matrix using a BJT as selector

01745647

en

2024/03/05 22:32:07

2018

https://hal.science/hal-01745647/file/Truchetet-Pradeu_Robustness%20and%20repair_Online%20version.pdf

Eric Vivier Marie-Elise Truchetet Thomas Pradeu email: [email protected] Re-thinking our understanding of immunity: Robustness in the tissue reconstruction system Seminars in Immunology (2018) Issue on "Redundancy and Robustness", guest edited by Keywords: Tissue repair, tissue regeneration, robustness, redundancy, plasticity, surveillance Robustness, understood as the maintenance of specific functionalities of a given system against internal and external perturbations, is pervasive in today's biology. Yet precise applications of this notion to the immune system have been scarce. Here we show that the concept of robustness sheds light on tissue repair, and particularly on the crucial role the immune system plays in this process. We describe the specific mechanisms, including plasticity and redundancy, by which robustness is achieved in the tissue reconstruction system (TRS). In turn, tissue repair offers a very important test case for assessing the usefulness of the concept of robustness, and identifying different varieties of robustness. Introduction Robustness can be defined as the maintenance of specific functionalities of a given system against internal and external perturbations [START_REF] Csete | Reverse engineering of biological complexity[END_REF][START_REF] Kitano | Biological robustness[END_REF]. The term, routinely used in engineering (e.g. [START_REF] Maynard | Architectural elements of language engineering robustness[END_REF]), is now pervasive in the life sciences [START_REF] Whitacre | Biological Robustness: Paradigms, Mechanisms, and Systems Principles[END_REF]. Systems and processes as diverse as bacterial chemotaxis, biochemical networks, cells, organisms, and ecosystems, among many others, have been described as robust [START_REF] Stelling | Robustness of cellular functions[END_REF][START_REF] Barkai | Robustness in simple biochemical networks[END_REF][START_REF] Alon | Robustness in bacterial chemotaxis[END_REF][START_REF] Wilmers | Understanding ecosystem robustness[END_REF][START_REF] Wagner | Robustness and evolvability in living systems[END_REF]. For example, a plane is robust when it continues to fly despite severe turbulence (for example thanks to the flexibility of its wings), and a bacterial cell is robust to modifications in genetic regulation when it tolerates a high number of these modifications [START_REF] Isalan | Evolvability and hierarchy in rewired bacterial gene networks[END_REF]. The notion of robustness, however, is very broad, and often elusive. To make it more precise, it has long been emphasized (e.g., [START_REF] Lesne | Robustness: confronting lessons from physics and biology[END_REF]) that two crucial questions must systematically be addressed when talking about robustness: first, what is robust, and second to what is it robust? In other words, a system is not robust in general; rather, it is robust to a certain kind of perturbations that can occur at a given level (or at a limited number of levels). The most stirring applications of the concept of robustness are those where talking about robustness seems directly operative, that is, sheds a new and important light on a given phenomenon, as illustrated by several cases including bacterial chemotaxis [START_REF] Alon | Robustness in bacterial chemotaxis[END_REF]. The aim of the present paper is to ask whether the concept of robustness can illuminate the processes of tissue repair and tissue regeneration, and whether, in turn, tissue repair and tissue regeneration offer a promising basis to better define the notion of robustness applied to biological phenomena. We are therefore interested in robustness at a particular level, namely that of tissues, and against a particular set of perturbations, namely damages made on tissues (physical or chemical aggressions, infectious agents, or "internal" stresses). Our focus on repair and regeneration at the tissue level is justified by the recent wealth of data on this issue [START_REF] Eming | Inflammation and metabolism in tissue repair and regeneration[END_REF], and by the obvious clinical interest of this topic, especially in the age of regenerative medicine [START_REF] Pang | An overview of the therapeutic potential of regenerative medicine in cutaneous wound healing[END_REF], but it is important to keep in mind that repair occurs also at other levels (including genetic [START_REF] David | Base-excision repair of oxidative DNA damage[END_REF] and cellular [START_REF] Tang | Self-repairing cells: How single cells heal membrane ruptures and restore lost structures[END_REF] level) in the organism. The idea that repairing oneself is fundamental to the organism's unity and individuality has been suggested at least since the 19 th century, particularly by physiologist Claude Bernard [START_REF] Bernard | Lectures on the phenomena of life common to animals and plants[END_REF]. More recently, the concept of robustness has been commonly associated with repair and regeneration [START_REF] Ninov | Current advances in tissue repair and regeneration: the future is bright[END_REF][START_REF] Galliot | Trends in tissue repair and regeneration[END_REF][START_REF] Bateson | Plasticity, Robustness, Development and Evolution[END_REF][START_REF] Laurent | Immune-Mediated Repair: A Matter of Plasticity[END_REF]. Much remains to be said, however, about how robustness and tissue repair can shed light one on the other. Tissue repair and regeneration involve a horde of components and pathways, including structural (e.g., fibroblasts, ECM, etc.) and immunological (e.g., neutrophils, macrophages, etc.) ones [START_REF] Eming | Inflammation and metabolism in tissue repair and regeneration[END_REF][START_REF] Laurent | Immune-Mediated Repair: A Matter of Plasticity[END_REF][START_REF] Gurtner | Wound repair and regeneration[END_REF][START_REF] Eming | Evolution of immune pathways in regeneration and repair: Recent concepts and translational perspectives[END_REF]. For this reason, we propose the concept of the "tissue reconstruction system" (TRS) to embrace all the different aspects of this phenomenon (see Figure 1). Repair is essential for the survival and maintenance of the body [START_REF] Bernard | Lectures on the phenomena of life common to animals and plants[END_REF][START_REF] Gurtner | Wound repair and regeneration[END_REF]. Failures in the repair process can lead to various pathological conditions, including fibrotic diseases, ulcers, hypertrophic and keloid scars, as well as cancers [START_REF] White | Inflammation, wound repair, and fibrosis: reassessing the spectrum of tissue injury and resolution[END_REF][START_REF] Menke | Impaired wound healing[END_REF][START_REF] Serra | From Inflammation to Current and Alternative Therapies Involved in Wound Healing[END_REF]. Repair is continuously occurring, to some degree, in organisms (e.g., skin renewal), in response to their constant exposure to damages of different types (physical, chemical, radiological, etc.). Even though there exists to a large extent a continuum between repair and regeneration [START_REF] Eming | Wound repair and regeneration: mechanisms, signaling, and translation[END_REF], the two phenomena can be considered distinct in several respects. Regeneration describes the capacity to regrow complex organs entirely, generally with the implication of several cell types [START_REF] Galliot | Trends in tissue repair and regeneration[END_REF][START_REF] Brockes | Comparative aspects of animal regeneration[END_REF][START_REF] Alvarado | Bridging the regeneration gap: genetic insights from diverse animal models[END_REF][START_REF] Birnbaum | Slicing across Kingdoms: Regeneration in Plants and Animals[END_REF]. In mammals, for example, the renewal of the epidermis is a form of repair, because it involves a single cell type (keratinocytes), whereas for the liver one can talk about regeneration as it involves several cell types (hepatocytes, sinusoidal endothelial cells, stellate cells, Kupffer cells, etc.) [START_REF] Michalopoulos | Liver Regeneration[END_REF]. Many repair mechanisms have been conserved across different taxa, including Drosophila, zebrafish, chick, and mammals [START_REF] Eming | Evolution of immune pathways in regeneration and repair: Recent concepts and translational perspectives[END_REF][START_REF] Eming | Wound repair and regeneration: mechanisms, signaling, and translation[END_REF]. The capacity to regenerate many complex organs such as limbs, however, is found only in a subset of living things [START_REF] Eming | Wound repair and regeneration: mechanisms, signaling, and translation[END_REF][START_REF] Brockes | Comparative aspects of animal regeneration[END_REF]. One important aim of this paper is to better clarify the similarities and differences between repair and regeneration, thanks to the concept of robustness. We explain here how robustness can help better characterize the process of tissue reconstruction, through a description of the specific mechanisms, including plasticity and redundancy, by which robustness is achieved in the TRS. We also demonstrate that different repair-associated disorders (such as fibrosis, ulcers, and cancers) can be understood as the result of deregulated robustness. In turn, we show that the TRS offers a remarkable test case to defining the notion of robustness in a more precise and operational way, and more specifically to distinguishing different forms of robustness (structural vs. functional; preventive vs. corrective; partial vs. complete; dysfunctional vs. as a dysfunction). What is robustness? With the increasing attention paid recently to systems biology and complex systems, many living processes or systems have been described as "robust" [START_REF] Csete | Reverse engineering of biological complexity[END_REF][START_REF] Kitano | Biological robustness[END_REF][START_REF] Kitano | Towards a theory of biological robustness[END_REF]. The exact meaning of the word "robustness" often remains, however, elusive. The term originated in physics [START_REF] Lesne | Robustness: confronting lessons from physics and biology[END_REF], and engineering [START_REF] Alon | Biological networks: the tinkerer as an engineer[END_REF] (though the engineering-related meaning is itself rooted in the physiology of the 19 th and 20 th century, including the work of Claude Bernard [START_REF] Kourilsky | The natural defense system and the normative self model[END_REF]). (On the relationship between biology and engineering, see [START_REF] Calcott | Engineering and Biology: Counsel for a Continued Relationship[END_REF]). In general, robustness is defined as the maintenance of specific functionalities of the system against internal and external perturbations. Two major requirements for any claim about biological robustness are to determine what exactly the robust system is, and against which type(s) of perturbations it is said to be robust. Importantly, robustness does not amount to conservation or absence of change. Robustness allows changes in the structure and components of the system owing to perturbations, but the key idea is that robustness leads to the maintenance of specific functions. It is likely that robustness is an evolved trait [START_REF] Wagner | Robustness and evolvability in living systems[END_REF][START_REF] Wagner | Robustness, evolvability, and neutrality[END_REF][START_REF] Wagner | Robustness and evolvability: a paradox resolved[END_REF]. Moreover, there are often trade-offs between robustness and other traits. In particular, systems that are evolved to be robust against certain perturbations can be extremely fragile to unexpected perturbations (see, e.g., [START_REF] Kitano | Biological robustness[END_REF][START_REF] Whitacre | Biological Robustness: Paradigms, Mechanisms, and Systems Principles[END_REF]). Despite the fact that, historically, the concept of robustness took root to some extent in the concept of homeostasis, the two notions are different. Homeostasis is about maintaining constant (or almost constant, within a certain range) a value (e.g., body temperature in homeothermic animals) [START_REF] Cannon | Organization for Physiological Homeostasis[END_REF][START_REF] Kotas | Homeostasis, Inflammation, and Disease Susceptibility[END_REF]. Robustness, in contrast, is about maintaining a given function F against given types of perturbations (P1, P2, etc.). Examples of robust processes or systems in biology abound [START_REF] Whitacre | Biological Robustness: Paradigms, Mechanisms, and Systems Principles[END_REF]. These include chemotaxis in bacteria [START_REF] Barkai | Robustness in simple biochemical networks[END_REF][START_REF] Alon | Robustness in bacterial chemotaxis[END_REF], cell cycle in budding yeast [START_REF] Li | The yeast cell-cycle network is robustly designed[END_REF], reliable development despite noise and environmental variations [START_REF] Félix | Robustness and evolution: concepts, insights and challenges from a developmental model system[END_REF], ecosystem reconstruction after a catastrophic event [START_REF] Wilmers | Understanding ecosystem robustness[END_REF], among many others. As shown by Kitano [START_REF] Kitano | Biological robustness[END_REF], the four main mechanisms that ensure robustness are: system control, alternative mechanisms, modularity, and decoupling. System control consists in negative and positive feedbacks that enable the system to reach robustness against some perturbations. An example is bacterial chemotaxis, in which negative feedback plays a major role [START_REF] Yi | Robust perfect adaptation in bacterial chemotaxis through integral feedback control[END_REF]. Robustness can also be realized by alternative (or "fail-safe") mechanisms, that is, multiple routes to achieve a given function, which is to say that the failure of one of these routes can be compensated by another. This includes redundancy (where identical or nearly identical components can realize a given function) and diversity (where heterogeneous components can realize a given function). There are now many examples of these phenomena in the immune system (e.g., [START_REF] Vély | Evidence of innate lymphoid cell redundancy in humans[END_REF]). Modularity is another important dimension of robustness: robustness is often achieved by modules, that is, flexible sets of components that collectively realize a given function, rather than by individual components [START_REF] Hartwell | From molecular to modular cell biology[END_REF]. Finally, decoupling is the prevention of undesired connection between low-level variations and highlevel functionalities. An example is the buffer mechanisms that decouple genetic variations from phenotypic expression, e.g., HSP chaperones [START_REF] Rutherford | Between genotype and phenotype: protein chaperones and evolvability[END_REF]. Here we focus on how the concept of robustness can be applied to the immune system and the TRS across the living world. Robustness has not been widely mentioned in immunology, though some exceptions exist (e.g., [START_REF] Kourilsky | The natural defense system and the normative self model[END_REF][START_REF] Feinerman | Variability and Robustness in T Cell Activation from Regulated Heterogeneity in Protein Levels[END_REF][START_REF] Jonjic | Functional plasticity and robustness are essential characteristics of biological systems: lessons learned from KLRG1-deficient mice[END_REF][START_REF] Mantovani | The chemokine system: redundancy for robust outputs[END_REF]). In particular, Mantovani [START_REF] Mantovani | The chemokine system: redundancy for robust outputs[END_REF] proposed that robustness provides a conceptual framework to understand intriguing aspects of the chemokine system, most prominently its redundancy (see also Mantovani, this special issue). Germain, Altan-Bonnet, and colleagues have explored theoretically and experimentally the mechanisms through which T cells can be both robust and adaptable to variations in protein expression [START_REF] Feinerman | Variability and Robustness in T Cell Activation from Regulated Heterogeneity in Protein Levels[END_REF]. Kourilsky has proposed to understand the immune system as conferring robustness to the whole organism via its capacity to systematically detect and respond to internal as well as external perturbations [START_REF] Kourilsky | The natural defense system and the normative self model[END_REF]. The question raised here is different and complementary, in so far as robustness is examined at the tissue level, and we ask which exact roles the immune system plays in this tissue-level robustness. In what follows, we detail how the TRS works, mainly via five processes, namely plasticity, functional redundancy, constant surveillance, restraint, and dynamic adjustment. We then show how pathologies associated with dysfunctions in tissue repair (e.g, fibrosis, ulcers, and cancer) can be understood as resulting from a deregulation of one or several of these five processes. We propose that the TRS offers a remarkable test case to define the notion robustness in a more precise and operational way, and more specifically to distinguishing different forms of robustness (structural vs. functional; preventive vs. corrective; partial vs. complete; dysfunctional vs. as a dysfunction). Importantly, we will consider both "repair" (defined as the partial reconstruction of an organ or tissue) and "regeneration" (defined as the complete reconstruction of a complex organ or tissue) examples, and explain how the concept of robustness helps clarify the differences between repair and regeneration. The mechanisms that mediate tissue reconstruction Tissue reconstruction is a complex and dynamic process, comprising overlapping, highly orchestrated stages -namely inflammation, tissue formation, and tissue remodeling [START_REF] Gurtner | Wound repair and regeneration[END_REF]. Tissue reconstruction involves many molecular and cellular components, which tightly interact. Understanding the interactions between these components and how they are regulated both spatially and temporally is a major aim for anyone interested in tissue repair, regeneration, and repair-associated pathologies. We show here that the TRS exhibits five key features that participate in robustness, and which are shared by many actors involved in the TRS: the TRS is plastic, redundant, under constant surveillance, restrained, and continuously dynamic. Plasticity in the TRS First, a major feature of the TRS is the plasticity of the cells involved in tissue reconstruction. The word "plasticity" is used with different and sometimes confusing meanings in the scientific literature. Here we understand cell plasticity in two different and important senses [START_REF] Laurent | Immune-Mediated Repair: A Matter of Plasticity[END_REF]. The first sense is intra-lineage cell plasticity, that is, changes in cell function and phenotype within a given cell lineage -for example, M1 macrophages turning into M2 macrophages. This is sometimes called "functional plasticity" [START_REF] Galli | Phenotypic and functional plasticity of cells of innate immunity: macrophages, mast cells and neutrophils[END_REF]. The second sense is translineage cell plasticity, that is, the switch from one lineage to another -e.g., from macrophages to fibroblasts [START_REF] Chang-Panesso | Cellular plasticity in kidney injury and repair[END_REF]. This can also be called plasticity by "transdifferentiation" [START_REF] Das | Monocyte and macrophage plasticity in tissue repair and regeneration[END_REF] or by "reprogramming" -a phenomenon now known to occur in some non-immune cells [START_REF] Plikus | Regeneration of fat cells from myofibroblasts during wound healing[END_REF]. Actors of plasticity in tissue reconstruction are diverse, from immune to nonimmune cells. In what follows, we describe the main cellular actors in the repair process, with a particular emphasis on how they illustrate the phenomenon of plasticity. We show that this plasticity is central to the functioning of the TRS. Far from being "one-shot" weapons, long-living neutrophils -which are central players in tissue reconstruction -are remarkably plastic. Indeed, neutrophils can differentially switch phenotypes, and display distinct subpopulations under different microenvironments [START_REF] Yang | The Diverse Biological Functions of Neutrophils, Beyond the Defense Against Infections[END_REF]. At the inflammatory stage of the repair process, neutrophils can play either a pro-resolving or an anti-resolving role. In addition to this intra-lineage plasticity, repair-associated neutrophils are capable of trans-lineage plasticity (plasticity by transdifferentiation) [START_REF] Balta | Qualitative and quantitative analysis of PMN/T-cell interactions by InFlow and super-resolution microscopy[END_REF][START_REF] Takashima | Neutrophil plasticity: acquisition of phenotype and functionality of antigen-presenting cell[END_REF][START_REF] Matsushima | Neutrophil differentiation into a unique hybrid population exhibiting dual phenotype and functionality of neutrophils and dendritic cells[END_REF][START_REF] Hampton | The lymph node neutrophil[END_REF]. Type 1 macrophages (M1) drive the early inflammatory responses that lead to tissue destruction, whereas type 2 macrophages ("M2" or "alternatively activated reparative macrophages") exert a central role in wound healing [START_REF] Biswas | Macrophage plasticity and interaction with lymphocyte subsets: cancer as a paradigm[END_REF][START_REF] Mantovani | Macrophage plasticity and polarization in tissue repair and remodelling[END_REF][START_REF] Nahrendorf | The healing myocardium sequentially mobilizes two monocyte subsets with divergent and complementary functions[END_REF][START_REF] Wynn | Macrophages in Tissue Repair, Regeneration, and Fibrosis[END_REF][START_REF] Wynn | Quantitative assessment of macrophage functions in repair and fibrosis[END_REF][START_REF] Mills | Anatomy of a Discovery: M1 and M2 Macrophages[END_REF]. Generation of a pro-type 2 microenvironment gradually leads to the switch from inflammatory to pro-repair macrophages. These cells promote tissue repair by producing pro-reparative cytokines and participate in a pro-type 2 microenvironment. A wide range of macrophage subtypes exists [START_REF] Das | Monocyte and macrophage plasticity in tissue repair and regeneration[END_REF][START_REF] Mantovani | Macrophage plasticity and polarization in tissue repair and remodelling[END_REF][START_REF] Jenkins | Local macrophage proliferation, rather than recruitment from the blood, is a signature of TH2 inflammation[END_REF][START_REF] Chávez-Galán | Much More than M1 and M2 Macrophages, There are also CD169+ and TCR+ Macrophages[END_REF]. Efficient tissue repair requires inflammatory macrophages, tissue repair macrophages, and resolving macrophages (producers of resolvins, IL-10 and TGF-b) [START_REF] Das | Monocyte and macrophage plasticity in tissue repair and regeneration[END_REF][START_REF] Wynn | Macrophages in Tissue Repair, Regeneration, and Fibrosis[END_REF][START_REF] Wang | Molecular mechanisms that influence the macrophage m1m2 polarization balance[END_REF]. Beyond intra-lineage plasticity, macrophages might participate actively in the tissue-remodeling phase of repair process by transdifferentiation into other cell types, notably endothelial cells [START_REF] Yan | Vascular endothelial growth factor modified macrophages transdifferentiate into endothelial-like cells and decrease foam cell formation., Vascular endothelial growth factor modified macrophages transdifferentiate into endothelial-like cells and decrease foam cell formation[END_REF]. Innate Lymphoid Cells (ILCs) are a recently discovered family of immune cells that includes three subsets: ILC1, ILC2 and ILC3 [START_REF] Eberl | The brave new world of innate lymphoid cells[END_REF][START_REF] Spits | Innate lymphoid cells--a proposal for uniform nomenclature[END_REF][START_REF] Vivier | The evolution of innate lymphoid cells[END_REF]. ILC2-secreted amphiregulin, a protein shown to orchestrate tissue repair [START_REF] Zaiss | Emerging functions of amphiregulin in orchestrating immunity, inflammation, and tissue repair[END_REF], promotes wound healing by acting directly on fibroblasts, leading to ECM deposit. ILC responses to different stimuli allow intra-lineage plasticity between the different subsets [START_REF] Ohne | IL-1 is a critical regulator of group 2 innate lymphoid cell function and plasticity[END_REF][START_REF] Silver | Inflammatory triggers associated with exacerbations of COPD orchestrate plasticity of group 2 innate lymphoid cells in the lungs[END_REF]. This plasticity between different ILC subtypes might allow for rapid innate immune responsiveness in repair [START_REF] Almeida | Innate lymphoid cells: models of plasticity for immune homeostasis and rapid responsiveness in protection[END_REF][START_REF] Zhang | Cutting Edge: Notch Signaling Promotes the Plasticity of Group-2 Innate Lymphoid Cells[END_REF]. Overall, cell plasticity is a pivotal process by which tissue reconstruction is achieved. This is confirmed by the fact that, as detailed below, inappropriate realizations of cellular plasticity (excess or insufficiency) may lead to various disorders. Functional redundancy in the TRS Functional redundancy is another important feature of the TRS. Functional redundancy describes a situation in which different elements have similar functions or similar effects on a trait [START_REF] Whitacre | Biological Robustness: Paradigms, Mechanisms, and Systems Principles[END_REF]. Though some forms of functional redundancy occur in every organism as part of normal functioning, this phenomenon has often been observed in pathological contexts, where it appears that an organism deficient in one cell type can "compensate" this deficiency thanks to other cell types or other molecules or pathways [START_REF] Edelman | Degeneracy and complexity in biological systems[END_REF]. The TRS often displays "degeneracy", which refers to the existence of structurally diverse but functionally similar components [START_REF] Edelman | Degeneracy and complexity in biological systems[END_REF]. Overall, the TRS is characterized by a high level of redundancy, even though some components and pathways seem to be pivotal in the reconstruction process. ILCs have potent immunological functions in experimental conditions, but their contributions to immunity in natural conditions are unclear. It has been shown that SCID patients with IL2RG and JAK3 mutations and ILC-deficient had no particular susceptibility to disease [START_REF] Vély | Evidence of innate lymphoid cell redundancy in humans[END_REF]. Thus, ILCs appear to be dispensable in humans who have a functional adaptive immune system, at least in the context of modern medicine and hygiene conditions [START_REF] Vély | Evidence of innate lymphoid cell redundancy in humans[END_REF][START_REF] Rankin | Complementarity and redundancy of IL-22-producing innate lymphoid cells[END_REF]. Functional redundancy allows the evocation of an overall type 1 or 2 immune response rather than talking more restrictively about type 1 or 2 neutrophils/macrophages/T cells. Those cells often produce the same types of molecules (albeit sometimes with different temporal patterns). This redundancy is not only important to maintain robustness against perturbations; it also creates feedback loops (and thereby a virtuous or vicious circle, depending on the situation), participating in the establishment of a local microenvironment that displays particular features. Besides immunological redundancy, immune cells participate in the secretion of structural molecules such as matrix metalloproteinases (MMP) altogether with fibroblasts, pericytes, and endothelial cells. The relative importance of macrophage and other immune cell contribution to tissue reconstruction compared to the aforementioned structural cells might depend on the nature of the tissue and the injury. Constant surveillance Tissue reconstruction is an active process where some actors are on constant standby. It is of major importance at the level of DNA repair, as DNA lesions occurring during reprogramming are monitored by a surveillance mechanism called the zygotic checkpoint [START_REF] Ladstätter | A Surveillance Mechanism Ensures Repair of DNA Lesions during Zygotic Reprogramming[END_REF]. At the tissue level, some cells, including various types of immune cells [START_REF] Fan | Hallmarks of Tissue-Resident Lymphocytes[END_REF], are highly specialized in the surveillance of damages. Of crucial importance are tissue-resident sentinel cells, as they are present and on standby before any damages. ILCs are found preferentially on epithelial barrier surfaces such as the skin, lungs, and gut, where they protect against infection and maintain the integrity of the barriers. ILCs are tissue-resident sentinels enriched at mucosal surfaces. They exert a constant surveillance on epithelia, and have a complex crosstalk with their microenvironment. They are highly involved in tissue repair through their sentinel position and the cytokines they produce [START_REF] Klose | Innate lymphoid cells as regulators of immunity, inflammation and tissue homeostasis[END_REF][START_REF] Rak | IL-33-Dependent Group 2 Innate Lymphoid Cells Promote Cutaneous Wound Healing[END_REF]. Different tissues often have their preferential sentinels, such as NK cells in the liver, or Langerhans cells in the skin. Cells of the innate but also adaptive immune system are involved in this surveillance. In particular, tissue resident memory T cells (TRM) -which reside in tissues without recirculating through the blood or lymph, and constitute a transcriptionally and phenotypically unique T cell lineage -have been shown to be key guardians against viral infections [START_REF] Mueller | Tissue-resident memory T cells: local specialists in immune defence[END_REF]. Cells traditionally seen as non-immune such as epithelial cells (ECs) play an important role in this collaborative surveillance process. They line body surface tissues and provide a physicochemical barrier to the external environment. This barrier is not a mere passive mechanical protection. Frequent microbial and non-microbial challenges cause activation of ECs, with release of cytokines and chemokines as well as alterations in the expression of cell-surface ligands. Epithelial stress is rapidly sensed by tissue-resident immune cells, which can directly interact with self-moieties on ECs and initiate both local and systemic immune responses. ECs are thus key drivers of immune surveillance at body surface tissues [START_REF] Dalessandri | Beneficial Autoimmunity at Body Surfaces -Immune Surveillance and Rapid Type 2 Immunity Regulate Tissue Homeostasis and Cancer[END_REF]. Restraint of the TRS Detecting and responding to damages is so central for the organism's survival that the TRS is always on alert, ready to be triggered. But at the same time this system also constitutes a potential threat for the organism (inflammation, tissue formation, and tissue remodeling can all go awry, with potentially dramatic consequences), and must therefore be constantly kept under control. Numerous cells restrain the TRS through negative feedback, active production of pro-resolving molecules, and other dynamic mechanisms. These cells are important at all stages but they are particularly crucial for the pro-resolving phase after inflammation. Pro-resolving neutrophils demonstrate the ability to: (i) produce several pro-resolving mediators (as lipoxins), (ii) form NETs and aggregated NETs, according to a cell-density dependent sensing mechanism, which dismantles the pro-inflammatory gradient by degrading the inflammatory cytokines and chemokines, (iii) store and release the proresolving protein annexin A1 [START_REF] Jones | The role of neutrophils in inflammation resolution[END_REF]. Inflammation resolution is partly mediated by the clearance of apoptotic neutrophils by macrophages through efferocytosis [START_REF] Stark | Phagocytosis of apoptotic neutrophils regulates granulopoiesis via IL-23 and IL-17[END_REF]. Non-apoptotic neutrophils can leave the injury site by reverse transmigration. Recently described resolving macrophages (producers of resolvins, IL-10 and TGF-b) are important actors of repair regulation. In mice, some regulatory T cells (Tregs) are able to produce amphiregulin, favoring the resolving phase of the inflammation process [START_REF] Burzyn | A special population of regulatory T cells potentiates muscle repair[END_REF]. Depletion of muscle Tregs has profound impact on muscle regeneration with loss of regenerative fibers, collagen deposition and fibrosis, leading to a disorganized tissue structure. In the absence of Tregs, effector T cell infiltrate increases in the injured muscle and the switch from inflammatory to anti-inflammatory macrophage diminishes. Dynamic adjustment of the TRS TRS is a highly dynamic process implying a large recruitment of various cells, with movements in a tri-dimensional matrix, and with many back and forth between different steps that are not fixed and can often overlap. The dynamic character of the TRS is visible at the level of the recruited cells, but also of the resident cells. Standby periods are not to be considered totally at rest. Resident cells are never completely motionless. Moreover, cells are constantly replaced in a dynamic process. Tissues are continuously exposed to potentially hazardous environmental challenges in the form of inert material and microbes. In the epidermis, for example, Langerhans cells (LC) form a dense network of cells capable of capturing antigens and migrating to the lymph node after crossing the basement membrane into the dermis, and they are able to promote tolerance or immune responses [START_REF] Seneschal | Human epidermal Langerhans cells maintain immune homeostasis in skin by activating skin resident regulatory T cells[END_REF]. Velocity of migration is partly regulated by the microenvironment, and skin Tregs display a much slower migration compared to effector CD4+ T cells, although acute inflammation results in a rapid increase in their motility [START_REF] Mueller | Tissue-resident T cells: dynamic players in skin immunity[END_REF]. Gradients of chemokines largely participate in cell recruitment when damages occur. CD14+ monocytes and neutrophils are very mobile cells, highly and promptly recruited in case of injury [START_REF] Achachi | UV Radiation Induces the Epidermal Recruitment of Dendritic Cells that Compensate for the Depletion of Langerhans Cells in Human Skin[END_REF]. The recruitment of neutrophils during the inflammatory phase is linked to a sharply regulated communication system based on the CXC chemokine/CXC receptors balance [START_REF] Nair | Study Investigators, Safety and efficacy of a CXCR2 antagonist in patients with severe asthma and sputum neutrophils: a randomized, placebo-controlled clinical trial[END_REF]. The injury triggers the production of G-CSF that converts the CXCR4 dominant signaling to that of CXCR2 in the bone marrow microenvironment, leading to the release of more mature neutrophils into the peripheral blood stream [START_REF] Silvestre-Roig | Neutrophil heterogeneity: implications for homeostasis and pathogenesis[END_REF]. Functional aberrancy in these systems leads to impaired wound healing [START_REF] Su | Chemokine Regulation of Neutrophil Infiltration of Skin Wounds[END_REF]. In a collaborative pathway, the release of chemoattractant factors by neutrophils, such as lactoferrin, attracts monocytes and activates macrophages [START_REF] Frangogiannis | Regulation of the inflammatory response in cardiac repair[END_REF]. A counterpart to recruitment is obviously needed to ensure robustness of a tissue, or else an overabundance of cells could lead to tissue destruction. Efferocytocis, transmigration, and specific apoptosis allow recruited cells to be cleaned up after damages. As the rest of the paper will show, two kinds of consequences follow from this analysis of the five key features of the TRS. First, it offers an important basis to re-think some tissue reconstruction-associated pathologies as dysfunctions of robustness. Second, it offers a test case to assess the usefulness of the notion of robustness in physiological and pathological conditions, and leads to distinguishing different forms of robustness. Dysfunctions of the tissue reconstruction system It has been suggested by Kitano and others that the concept of robustness can shed light on certain pathological processes [START_REF] Kitano | Biological robustness[END_REF]. Pathologies could result from robustness as a dysfunction (the process under consideration is robust, but this robustness is detrimental to the organism, as happens for example in AIDS or some cancers, where the robustness of a system is "hijacked" [START_REF] Kitano | Biological robustness[END_REF][START_REF] Kitano | Biological robustness in complex host-pathogen systems[END_REF][START_REF] Kitano | Cancer as a robust system: implications for anticancer therapy[END_REF]) or a dysfunctional robustness, which is to say a rupture of robustness (i.e., the process should be robust, but is not). This approach applies very well to the dysfunctions of the TRS. Mechanisms that mediate tissue reconstruction to ensure robustness are constantly challenged. These mechanisms are sometimes overwhelmed, leading to various consequences depending on the situation, from the rupture of robustness to the promotion of the disease thanks to robustness-associated mechanisms, and to an excess of robustness. The final consequence of each situation is a pathological process. Through concrete examples (ulcers, fibrosis, and cancers), we will illustrate these different threats to the TRS to ensure robustness, emphasizing in each case exactly which mechanisms are challenged (see Table 1, which present several additional examples). Ulcers, or rupture of robustness Pathological situations of insufficient repair such as ulcers underlie that the TRS mechanisms ensuring robustness are overwhelmed. Since the robustness of the tissue can be jeopardized, it is important to analyze the various components listed earlier. The value of a more detailed analysis of component robustness is dual. This makes it possible to precisely identify vulnerabilities, which vary depending on the clinical situation, but also to work out innovative therapeutic strategies. As we saw, cell plasticity is a crucial dimension of the TRS and it is especially true for neutrophils. This is confirmed by the fact that incapacity of neutrophils to switch plastically from one state to the other can contribute to ulcers, e.g., skin or gastric ulcers. Impossibility of tuning the response toward a pro-resolving phase by experimentally blocking neutrophils in a pro-inflammatory state directly leads to chronic inflammation and deregulation of the TRS [START_REF] Whitmore | Cutting Edge: Helicobacter pylori Induces Nuclear Hypersegmentation and Subtype Differentiation of Human Neutrophils In Vitro[END_REF], while the reintroduction of very plastic cells in damaged tissues can overcome this defect. Understanding that in this case the ulcer is a rupture of the robustness due to insufficient cellular plasticity allows to consider completely new therapeutic options. Several studies in animal models showed that adipose tissuederived stem cell sheet application to mucosal or skin wounds accelerates wound healing and decreases the degree of fibrosis [START_REF] Perrod | Cell Sheet Transplantation for Esophageal Stricture Prevention after Endoscopic Submucosal Dissection in a Porcine Model[END_REF][START_REF] Kato | Allogeneic Transplantation of an Adipose-Derived Stem Cell Sheet Combined With Artificial Skin Accelerates Wound Healing in a Rat Wound Model of Type 2 Diabetes and Obesity[END_REF]. While inflammation has to be regulated to ensure the completion of the TRS, a failure in that process can lead to a rupture of robustness. A deficiency of efferocytosis has been identified as a causative agent of sterile chronic granulomatous disease in mice [START_REF] Zeng | An efferocytosis-induced, IL-4-dependent macrophage-iNKT cell circuit suppresses sterile inflammation and is defective in murine CGD[END_REF]. In chronic ulcers, favoring the resolving phase (e.g., through efferocytosis, pro-Treg therapeutics, or resolving compounds) could be an innovative strategy [START_REF] Chan | Interleukin 2 Topical Cream for Treatment of Diabetic Foot Ulcer: Experiment Protocol[END_REF][START_REF] Reis | Lipoxin A4 encapsulated in PLGA microparticles accelerates wound healing of skin ulcers[END_REF][START_REF] Lohmann | Glycosaminoglycan-based hydrogels capture inflammatory chemokines and rescue defective wound healing in mice[END_REF]. Even though defects of inflammatory regulation are clearly involved in the deregulation of the TRS, therapeutic avenues to counteract these defects are still in their infancy, and mostly limited to animal models. A better understanding of the crucial place of that mechanism in the global robustness of the TRS will tend to raise the interest for therapeutics targeting regulation. In ulcers, the loss of epithelial cells disturbs the ongoing surveillance of the TRS. In the eye, the inflammation of the cornea leads to damages of this protective barrier. Given that the cornea is an avascular tissue and contains few immune cells, corneal resident cells function as sentinel cells as well as immune modulators during corneal inflammation. They are able to sense bacterial infection through toll like receptor (TLR)-mediated detection. As a consequence, a loss of substance (i.e., a very significant injury) could lead to the disappearance of key first-line sentinel cells, normally responsible for the recruitment of other crucial cells in the repair process [START_REF] Allen | The Silent Undertakers: Macrophages Programmed for Efferocytosis[END_REF][START_REF] Fukuda | Corneal Fibroblasts as Sentinel Cells and Local Immune Modulators in Infectious Keratitis., Corneal Fibroblasts as Sentinel Cells and Local Immune Modulators in Infectious Keratitis[END_REF]. Other resident cells are involved in this mechanism (see Table 1). Targeting sentinels could constitute a new therapeutic avenue in the treatment of chronic ulcers. Finally, due to the reduction of the dynamic flow to the damage site, new cells cannot come from the upstream and revitalize the system in an ulcer. Promoting the migration and proliferation of cells could accelerate wound healing [START_REF] Liu | CXCR4 antagonist delivery on decellularized skin scaffold facilitates impaired wound healing in diabetic mice by increasing expression of SDF-1 and enhancing migration of CXCR4-positive cells, Wound Repair Regen[END_REF][START_REF] Kai | Accelerated Wound Healing on Skin by Electrical Stimulation with a Bioelectric Plaster[END_REF]. Each of the five components of robustness can be compromised depending on the type of ulcer. For clinicians, thinking according to our classification and identifying which mechanism is deficient can therefore change very concretely their therapeutic management. Fibrosis or excess of robustness Keloid and hypertrophic scars can also be seen as the result of a dysfunction in the fundamental mechanisms of the TRS. One could consider fibrosis as a kind of hypertophic scar, and as such fibrosis could follow from a deregulated TRS as well. In a normal repair cycle, the resolution of damage-induced inflammation allows the system to rebuild itself efficiently. In contrast, the absence of resolution means the persistence of inflammation and also, especially in fibrosis, a disconnection between the levels of resolution and remodeling. An excess of plasticity can also be pathological. For example, epithelial-mesenchymal transition (EMT) reflects a high level of cell plasticity essential during embryogenesis and wound healing, but EMT can be aberrantly regulated in fibrosis [START_REF] Skrypek | Epithelial-to-Mesenchymal Transition: Epigenetic Reprogramming Driving Cellular Plasticity[END_REF][START_REF] Gasparics | Alterations in SCAI Expression during Cell Plasticity, Fibrosis and Cancer[END_REF]. Cell plasticity could also be a hurdle for achieving some cell therapy. A pro-fibrotic microenvironment results in systematic M2 polarization even if macrophages of another type are injected. In contrast, the infusion of stabilized pro-resolving macrophages is associated with reduced kidney interstitial fibrosis and inflammation, as well as preservation of the phenotype and functions of macrophages [START_REF] Guiteras | Macrophage Overexpressing NGAL Ameliorated Kidney Fibrosis in the UUO Mice Model[END_REF]. Thus, a precise knowledge of the proper physiopathology of the studied condition is crucial to understanding whether a higher or a lower plasticity is needed. The cause of fibrosis is sometimes attributed to the persistence of damage triggers such as chronic infection. Nevertheless, in hepatitis C, it is the inadequacy of the TRS response rather than the persistence of the infection that is at stake [START_REF] Rios | Chronic hepatitis C liver microenvironment: role of the Th17/Treg interplay related to fibrogenesis[END_REF]. Tregs or proresolving cells have also been suspected to be involved in more general fibrotic processes, such as systemic sclerosis (SSc) [START_REF] Ugor | Increased proportions of functionally impaired regulatory T cell subsets in systemic sclerosis[END_REF][START_REF] Bhattacharyya | Endogenous ligands of TLR4 promote unresolving tissue fibrosis: Implications for systemic sclerosis and its targeted therapy[END_REF]. From this point of view, promoting the resolution of inflammation could be considered as a key aim to reverse fibrosis [START_REF] Brennan | Specialized pro-resolving mediators in renal fibrosis[END_REF][START_REF] Grabiec | The role of airway macrophages in apoptotic cell clearance following acute and chronic lung inflammation[END_REF]. A constant monitoring is an essential element of the TRS responsiveness. The fact that it is provided by resident cells guarantees this prompt response when damages occur. However, in some cases, including fibrosis, this surveillance can be over-stimulated and associated with an overly sustained response. As described before, innate immune signaling via TLRs is a key driver of persistent fibrotic response. Chronic signaling on resident mesenchymal cells underlies the switch from a self-limited repair response to non-resolving pathological fibrosis characteristic of systemic sclerosis. Limiting the responsiveness of resident cells to innate stimulation could be of interest to prevent fibrotic processes [START_REF] Bhattacharyya | Endogenous ligands of TLR4 promote unresolving tissue fibrosis: Implications for systemic sclerosis and its targeted therapy[END_REF]. Resident cells themselves can also be responsible for the excessive stimulation without any clear external trigger [START_REF] Cheng | Guards at the gate: physiological and pathological roles of tissue-resident innate lymphoid cells in the lung[END_REF][START_REF] Hams | IL-25 and type 2 innate lymphoid cells induce pulmonary fibrosis[END_REF][START_REF] Li | Skin-Resident Effector Memory CD8+CD28-T Cells Exhibit a Profibrotic Phenotype in Patients with Systemic Sclerosis[END_REF]. A static TRS cannot result in normal repair. Indeed, different but more or less intricate phases must follow one another. Nonetheless, an excess of migration of pro-fibrotic cells into the tissue can be detrimental in a normal repair process. This happens, for example, in the lungs with fibrocytes. These cells enter the lungs in response to their chemoattractant CXCL12, and differentiate into fibroblasts or myofibroblasts, leading to excessive deposition of collagen-rich extracellular matrix. It has been shown that inhibiting the flow of fibrocytes to the lungs by a peptide called R1R2 attenuates pulmonary fibrosis by reducing the invasion of fibrocytes through basement membrane-like proteins [START_REF] Chiang | R1R2 peptide ameliorates pulmonary fibrosis in mice through fibrocyte migration and differentiation[END_REF]. Cancer or hijacking of robustness Cancerous tumors have been related to deregulated repair by Dvorak, who describes them as "wounds that do not heal" [START_REF] Dvorak | Tumors: wounds that do not heal. Similarities between tumor stroma generation and wound healing[END_REF]. It is now well established that an inflammatory microenvironment promotes cancer [START_REF] Mantovani | Cancer-related inflammation[END_REF]. It has also been suggested that the formation and maintenance of a cancerous tumor could be seen as a robust process [START_REF] Kitano | The theory of biological robustness and its implication in cancer[END_REF]. Here we consider cancer as evolving from a damaged tissue, where the TRS could act to prevent the expansion of the injury and promote repair. The cancerous tumor, to drive its own development, hijacks some properties of the TRS that normally ensure robustness in physiological conditions. Epithelial-mesenchymal transition (EMT) is an important process in embryonic development, fibrosis, but also in cancer metastasis. The activation of EMT in cancer allows cells to acquire migratory, invasive, and stem-like properties. SCAI is characterized as a tumor suppressor inhibiting metastasis in different human cancer cells, and which is thought to be reduced in some tumors. SCAI expression decreases in a model of endothelialmesenchymal transition, which suggests that it could be important for cell plasticity. Nevertheless, its role in cancer remains to be further investigated, as its expression could be associated with high or low progression of the tumor depending on the type of cancer [START_REF] Gasparics | Alterations in SCAI Expression during Cell Plasticity, Fibrosis and Cancer[END_REF]. Macrophage polarization could influence immune checkpoint therapy resistance. The plasticity of macrophages is used by cancerous tumors, and targeting this plasticity could be of interest to increase the response to immunotherapy such as ipilimumab [START_REF] Soto-Pantoja | Unfolded protein response signaling impacts macrophage polarity to modulate breast cancer cell clearance and melanoma immune checkpoint therapy responsiveness[END_REF]. Redundancy may explain cancer resistance to certain treatments. Recently developed immunotherapies do not target the tumor cells as such; instead, they promote the local immune responses in the tumor microenvironment, which has some important consequences on key immunological actors of the TRS. Redundancy of the TRS becomes central in these conditions [START_REF] Lavi | Redundancy: a critical obstacle to improving cancer therapy[END_REF]. For example, IL-6 belongs to a family of cytokines with highly redundant functions, which use the glycoprotein 130 chain for signal transduction. It has an important role in the pathophysiology of multiple myeloma, where it supports the growth and survival of the malignant plasma cells in the bone marrow. Because of this redundancy, targeting IL-6 is highly difficult. Antibodies against glycoprotein 130 constitute a better option, as they can overcome this redundancy [START_REF] Burger | Due to interleukin-6 type cytokine redundancy only glycoprotein 130 receptor blockade efficiently inhibits myeloma growth[END_REF]. Insofar as cancer may be seen as both a cause and consequence of tissue damages, cancer cells could activate the TRS. In particular, cancerous tumors can use the above described restrain mechanisms of the TRS to induce a type of immune tolerance that will be at their own advantage. Tumor associated macrophages (TAMs) are one of the main actors of this phenomenon. Different therapeutic strategies have been proposed to address this problem, such as the suppression of TAM recruitment, their depletion, the switch of M2 TAMs into antitumor M1 macrophages, and the inhibition of TAM-associated molecules [START_REF] Santoni | Triple negative breast cancer: Key role of Tumor-Associated Macrophages in regulating the activity of anti-PD-1/PD-L1 agents[END_REF]. Immune surveillance could be considered as insufficient in cancer. The TRS lets cancer cells grow and develop as if it were incapable of "seeing" them. Resident memory CD8+T cells (TRMs) represent a recently described subset of long-lived memory T cells that remain in the tissues, do not recirculate, and are therefore very important actors in immunosurveillance. It has been shown that TRMs were present in human non-small cell lung tumor tissues, and their frequency was correlated with better overall survival than other infiltrating immune cells. In that case, the cancer misleads the immunosurveillance system, which suggests that strategies increasing the number of TRMs or activating them such as vaccines could be developed following this concept [START_REF] Granier | Tissue-resident memory T cells play a key role in the efficacy of cancer vaccines[END_REF]. Prior to metastatic cell arrival, a premetastatic niche in distant organs could be an important step in the metastatic cascade. This phenomenon suggesting highly dynamic process from cancer cells could be preceded by neutrophil migration and recruitment. As such the dynamic property of the TRS is used to prepare the basis for tumor cell engraftment in parenchyma [START_REF] Donati | Neutrophil-Derived Interleukin 16 in Premetastatic Lungs Promotes Breast Tumor Cell Seeding[END_REF]. Overall, the concept of robustness helps better understand TRS-associated pathologies, either as a deficiency in the fundamental processes by which robustness is normally realized (plasticity, etc.), or as an emerging, local form of robustness that is detrimental to the organism. Conclusion: the virtues of thinking about tissue reconstruction in terms of robustness In light of the various physiological and pathological examples examined in this paper, we propose that it is extremely fruitful to conceive of the tissue reconstruction system in terms of robustness, for three main reasons. First, the recognition by Kitano and others [START_REF] Kitano | Biological robustness[END_REF] of different robustness-promoting mechanisms (system control, alternative mechanisms, modularity, decoupling) constitutes a useful conceptual framework to better describe the TRS and its dysfunctions in pathological situations. For example, plasticity and redundancy of immune components within the TRS have been described by scientists who were supportive of the concept of robustness [START_REF] Mantovani | The chemokine system: redundancy for robust outputs[END_REF][START_REF] Mantovani | Macrophage plasticity and polarization in tissue repair and remodelling[END_REF], and it seems likely that continuing to apply this concept will reveal even more plasticity and redundancy. Moreover, thinking in terms of robustness helps understanding that even a situation that could seem static, such as skin renewal for example, is in fact the outcome of a highly dynamic, continuously ongoing, process, and that it is pivotal to study in detail the mechanisms ensuring this process. It also suggests that the "default state" of the TRS is to be on alert, which means that tissue reconstruction is always active, though under constant restraint. As soon as the brake is lifted, the whole process of tissue repair (i.e., inflammation, tissue formation, and tissue remodeling) is triggered, which guarantees a higher capacity to react to various and often inevitable damages. Of course, this constant activation is energetically costly, but one should keep in mind that it is a low-level activation, and that it is probably essential for survival. From a pathological point of view, the emphasis on the redundancy of the TRS, for example, is of the utmost importance. It shows that it is often entirely inadequate to hope for important benefits by intervening on just one actor or pathway. Indeed, in many cases, although some cells or pathways seemed crucial in a pathological process in vitro, their inhibition in vivo does not lead systematically to the pathological phenotype, because of the redundancy of some components and pathways. In other words, some pathological conditions reveal the role of certain cells or pathways, which are not normally indispensable, but become central when other components are missing. For example, alarmins IL-25, IL-33, and TSLP have a high level of redundancy, which makes anti-fibrotic treatments very difficult to develop. Blocking a single pathway is most often ineffective, so it is more promising to consider modulation of the response at a very early stage, or to identify common pathways that could be targeted. Moreover, acting on one of the mechanisms of robustness while others play a more significant role is not effective in achieving repair. This is probably what happens when one treats fibrosis with immunosuppressive therapy in cases where restoring cellular plasticity would in fact be more adequate. As a general rule, then, one should never draw hasty conclusions about whether or not some actors have an important role in the TRS before having tested them in real-life pathological conditions. Second, the example of the TRS can, in turn, help us make some crucial conceptual distinctions about robustness. On the basis of the examples explored here, one can indeed distinguish functional vs. structural robustness, partial vs. complete robustness, and corrective vs. preventive robustness (Figure 2). Functional robustness in the case of the TRS means that tissue function (or, at least, one tissue function) is restored, but not tissue structure. For example, after significant skin injury, a scar will form, which will restore the protective function of the skin, but the initial structure of the skin will not be restored. In contrast, regeneration often leads to the restoration of both the structure and the function of the tissue. For example, adult zebrafish fins, including their complex skeleton, regenerate exactly to their original form within two weeks after an amputation. Importantly, some forms of complete tissue regeneration can also be observed in mammalian embryos, but this capacity is subsequently lost for most tissues (the most significant exception in humans is the liver, which can indeed regenerate, though it does not always recover its initial structure). Along similar lines, it is important to emphasize that robustness to a given challenge and at a certain level can be more or less effective. Robustness is partial when tissue function and/or structure is not completely restored, as illustrated by most cases of tissue repair in mammals, for example. Robustness is said to be complete when tissue function and/or structure is entirely restored, as illustrated by cases such as fin regeneration in zebrafish already mentioned, limb regeneration in many amphibians [START_REF] Brockes | Amphibian Limb Regeneration: Rebuilding a Complex Structure[END_REF], or tissue regeneration in many echinoderms [START_REF] Carnevali | Regeneration in Echinoderms: repair, regrowth, cloning[END_REF]. Furthermore, robustness can be corrective or preventive. It is corrective when it consists in the active restoration of a strongly disturbed state. For example, tissue repair or regeneration after significant damages is a form of corrective robustness, because it follows a major perturbation (damages), and it involves, as we saw, a complex, dynamic, and regulated interplay of many different components. In contrast, robustness is preventive when it occurs in the absence of a major perturbation while minimizing the risk of a major perturbation and its detrimental consequence. For example, epithelial repair occurs continuously in the body, which requires an extremely rich orchestration of events [START_REF] Peterson | Intestinal epithelial cells: regulators of barrier function and immune homeostasis[END_REF]. This preventive robustness helps insure that the skin is always sufficiently "sealed off" and at the same time sufficiently smooth to achieve its functions. When this process is interrupted, for example in ulcers, the organism is at a high risk of being damaged and invaded by pathogens or toxic substances. Of course, there will be a grey zone here, because it is not always clear whether a perturbation is major or not, and different tissues are likely to perceive perturbations differently. For example, the liver is constantly exposed to toxic chemicals that could endanger the rest of the body, and its regenerative capacities are certainly evolutionarily related to this particular exposition [START_REF] Michalopoulos | Liver Regeneration[END_REF]. An additional distinction seems important to better grasp the role of robustness in pathological contexts. Dysfunctions in robustness of the TRS can indeed be understood along two different lines. In some cases, the tissue fails to be robust, presumably because one or several important components or pathways of the TRS are not working properly. This is what we call a dysfunction of robustness. For example, we saw that the incapacity to realize cell plasticity can sometimes lead to the failure of tissue reconstruction. Yet, in other cases, the tissue is robust, but this robustness is, in this specific context, detrimental to the organism. This is what we call robustness as a dysfunction. For example, a tumor can constitute a robust tissue, which is well vascularized, nourished, and constantly repaired, often via the co-optation of classical physiological mechanisms to the benefit of the tumor itself. Here again, this distinction between a dysfunction of robustness and dysfunctional robustness might prove useful in other contexts, beyond the case of the TRS. A third and final consequence concerns the very understanding of immunity. Since the beginnings of immunology, immunity has been conceived primarily as a form of defense -most often against pathogens. Yet, if the perspective offered in this paper is correct, then immunity needs to be re-defined within a much wider context. Immune processes, we submit, concern not only defense, but also the construction (development) and reconstruction (constant repair; occasional repair after a significant damage; regeneration) of the organism,. Indeed, a typical immune system in nature is constantly busy surveying, renewing, and repairing the body,. This is not to say, obviously, that immune defense is not important, and has not been a major selective pressure in the evolution of immune systems. Our suggestion is that immune systems have evolved under a multidimensional complex selective pressure, which includes a capacity to develop and repair as well as a capacity to defend against different sorts of threats. The way scientists traditionally delineate the immune system reflects an intellectual decision. This does not mean, of course, that the immune system is not "real", but rather that there exist many different ways to divide up living entities into different "systems". In the present paper, we have argued in favor of another intellectual decision by suggesting that it is more appropriate to focus on a functionally defined "system" of interest (namely the tissue reconstruction system) than on traditionally defined systems (such as the immune system). Repair and defense are probably just two sides of the same coin -a lesson that thinking immunity in terms of robustness might help us keep in mind. EMT and SCAI in renal fibrosis [START_REF] Gasparics | Alterations in SCAI Expression during Cell Plasticity, Fibrosis and Cancer[END_REF] -MET/EMT with the tumorinitiating ability required for metastatic colonization [START_REF] Yao | Mechanism of the mesenchymal-epithelial transition and its relationship with metastatic tumor formation[END_REF] -Plasticity between the epithelial and the mesenchymal states rather than a fixed phenotype [START_REF] Liao | Revisiting epithelial-mesenchymal transition in cancer metastasis: the connection between epithelial plasticity and stemness[END_REF] -UPR in macrophage polarization and plasticity with shift to M1-like profile [START_REF] Soto-Pantoja | Unfolded protein response signaling impacts macrophage polarity to modulate breast cancer cell clearance and melanoma immune checkpoint therapy responsiveness[END_REF] Functional redundancy ILC redundancy [START_REF] Vély | Evidence of innate lymphoid cell redundancy in humans[END_REF] -IL-25, IL-33, and TSLP redundancy and fibrosis [START_REF] Vannella | Combinatorial targeting of TSLP, IL-25, and IL-33 in type 2 cytokine-driven inflammation and fibrosis[END_REF][START_REF] Gieseck | Type 2 immunity in tissue repair and fibrosis[END_REF] -Targeting porcupine in kidney fibrosis and Wnt O-acylation [START_REF] Madan | Experimental inhibition of porcupine-mediated Wnt O-acylation attenuates kidney fibrosis[END_REF] -IL-6 and glycoprotein 130 in the pathophysiology of multiple myeloma [START_REF] Burger | Due to interleukin-6 type cytokine redundancy only glycoprotein 130 receptor blockade efficiently inhibits myeloma growth[END_REF] Constant surveillance -Loss of substance (i.e., a very significant injury) and disappearance of sentinel cells in ulcers [103] -Langerhans cells and hypoxia [START_REF] Pierobon | Regulation of Langerhans cell functions in a hypoxic environment[END_REF] -Fibronectin-EDA and tenascin-C sensed by TLR4 on resident cells and fibrotic processes [START_REF] Bhattacharyya | Endogenous ligands of TLR4 promote unresolving tissue fibrosis: Implications for systemic sclerosis and its targeted therapy[END_REF] -ILC2s in pulmonary fibrosis [START_REF] Cheng | Guards at the gate: physiological and pathological roles of tissue-resident innate lymphoid cells in the lung[END_REF][START_REF] Hams | IL-25 and type 2 innate lymphoid cells induce pulmonary fibrosis[END_REF]] -CD8+CD28-T cells and profibrotic cytokine IL-13 in the skin of systemic sclerosis (SSc) patients [START_REF] Li | Skin-Resident Effector Memory CD8+CD28-T Cells Exhibit a Profibrotic Phenotype in Patients with Systemic Sclerosis[END_REF] -TRMs in human non-small cell lung tumor tissue [START_REF] Granier | Tissue-resident memory T cells play a key role in the efficacy of cancer vaccines[END_REF] -Role of amphiregulin in orchestrating responses to tumors [START_REF] Zaiss | Emerging functions of amphiregulin in orchestrating immunity, inflammation, and tissue repair[END_REF] Restraint -Resolution deficiency and sterile chronic granulomatous disease [98-101] -Imbalance Treg/Th17 in pyoderma gangrenosum [START_REF] Caproni | The Treg/Th17 cell ratio is reduced in the skin lesions of patients with pyoderma gangrenosum[END_REF] -Chronic hepatitis C and hepatic fibrosis with the Th17/Treg balance Table 1. Main mechanisms involved in the robustness of the tissue reconstruction system (TRS), and its dysfunctions in major pathological situations (ulcer, fibrosis, and cancer). preventive robustness. The four cases presented here are merely illustrations, among others, showing how these distinctions can be applied to real-life cases. In severe skin injury, robustness is corrective, partial, and functional. In liver regeneration in mammals, robustness is corrective, almost complete, and functional. In the continuous renewal of epithelia, robustness is preventive, complete, and functional. Finally, in limb regeneration in salamander, robustness is corrective, complete, and functional. Figure 2 2 Figure 2 sumps up the three distinctions proposed here (structural vs. functional; partial vs. Figures and Tables - [START_REF] Rios | Chronic hepatitis C liver microenvironment: role of the Th17/Treg interplay related to fibrogenesis[END_REF] -Role of TRegs in SSc[START_REF] Ugor | Increased proportions of functionally impaired regulatory T cell subsets in systemic sclerosis[END_REF] -SSc and TLRs with persistence of the response[START_REF] Bhattacharyya | Endogenous ligands of TLR4 promote unresolving tissue fibrosis: Implications for systemic sclerosis and its targeted therapy[END_REF]] -Resolving inflammation against fibrosis and specialized pro-resolving lipid mediators [112] -Macrophages and efferocytosis [113] -TAMs recruitment in triple negative breast cancer [124] -Tregs in tumor progression [138] -Tregs and cancer cell clearance [139] -Tregs and cancer immunotherapies with IL-2 [140] -To target immune checkpoints such as CTLA4, PD1 or TIGIT to both interfere with Treg function and enhance effector responses at the same time [Cancer cells and use of the dynamic potential of neutrophils [126] -CCL26 in colorectal cancer cells invasion by inducing TAM infiltration [142] -Inhibitors of the receptor tyrosine kinase c-MET and impairment of the mobilization and recruitment of neutrophils into tumors [143] Figure 1 . 1 Figure1. Overview of the "tissue reconstruction system" (TRS). Many various components and pathways are involved in both tissue repair and tissue regeneration. Crucial components of the TRS include structural (e.g., fibroblasts, extracellular matrix, etc.) and immunological (e.g., neutrophils, macrophages, etc.) components. The concept of TRS is intended to embrace all the main entities and mechanisms responsible for tissue repair and tissue regeneration. Figure 2 . 2 Figure 2. The exploration of the tissue reconstruction system (TRS) leads to distinguishing different types of robustness, namely structural vs. functional robustness, partial vs. complete robustness, and corrective vs. preventive robustness. The four cases presented here are merely illustrations, among others, showing how these distinctions can be applied to real-life cases. In severe skin injury, robustness is corrective, partial, and functional. In liver regeneration in mammals, robustness is corrective, almost complete, and functional. In the continuous renewal of epithelia, robustness is preventive, complete, and functional. Finally, in limb regeneration in salamander, robustness is corrective, complete, and functional. Acknowledgements We would like to thank Cécile Contin-Bordes, Paôline Laurent, Alberto Mantovani, Jean-François Moreau, and Derek Skillings for discussions about tissue repair and robustness. Thomas Pradeu has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme -grant agreement n°637647 -IDEM.

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https://hal.science/hal-01745684/file/214.pdf

Nicolas Gauvrit email: [email protected] Jean-Charles Houillon Jean-Paul Delahaye Generalized Benford's Law as a Lie Detector may come IntroductIon During the late 19th century, an intriguing phenomenon was discovered by [START_REF] Newcomb | note on the frequency of use of the different digits in natural numbers[END_REF]: The first significant digit (leftmost nonzero digit) of seemingly random numbers often fails to follow a flat distribution with an equal proportion of 1s, 2s, … , 9s, as one would expect, but instead follows a decreasing distribution, with more 1s than 2s, more 2s than 3s, and so forth. The same phenomenon was later rediscovered and detailed by [START_REF] Benford | the law of anomalous numbers[END_REF]. According to what is now referred to as Benford's law or Newcomb-Benford law (NBL), the distribution of the first significant digit X of a "random" number follows a logarithmic law given by P(X = d) = Log(1+1/d), where Log stands for the base 10 logarithm and d stands for a digit (in the range of 1-9; see Table 1). Many real-world datasets approximately conform to NBL (Hill, 1998;[START_REF] Nigrini | Benford's Law: Applications for forensic accounting, auditing, and fraud detection[END_REF]. For instance, the distance between earth and known stars (Alexopoulos & Leontsinis, 2014) or exoplanets [START_REF] Aron | crime-fighting maths law confirms planetary riches[END_REF], crime statistics (Hickman & Rice, 2010), the number of daily-recorded religious activities [START_REF] Mir | the Benford law behavior of the religious activity data[END_REF], earthquake depths [START_REF] Arroucau | Benford's law of first digits: From mathematical curiosity to change detector[END_REF], interventional radiology Dose-Area Product data [START_REF]the novel application of Benford's second order analysis for monitoring radiation output in interventional radiology[END_REF], financial variables [START_REF] Ausloos | Benford's law and theil transform of financial data[END_REF], and internet traffic data (Arshadi & Jahangir, 2014), were found to conform to NBL. In psychology, NBL was found relevant in the study of gambling behaviors [START_REF] Chou | Benford's law and number selection in fixed-odds numbers game[END_REF], brain activity recordings (Kreuzer et al., 2014), language [START_REF] Dehaene | cross-linguistic regularities in the frequency of number words[END_REF][START_REF] Delahaye | Culturomics: Le numérique et la culture [culturomics: the numerical and the culture[END_REF], or perception [START_REF] Beeli | Frequency correlates in grapheme-color synaesthesia[END_REF]. this is an open access article under the cc By-nc-nd license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Prop (%) 30.1 17.6 12.5 9.69 7.92 6.69 5.80 5.12 4.58 the SenSItIvIty and SpecIfIcIty of Benford analySIS Human pseudorandom productions are in many ways different from true randomness [START_REF] Nickerson | the production and perception of randomness[END_REF]. For instance, participants' productions show an excess of alternations [START_REF]Bias and processing capacity in the generation of random time intervals[END_REF] or are overly uniform (Falk & Konold, 1997). As a consequence, fabricated data might fit NBL to a lesser extent than genuine data [START_REF] Banks | the apparent magnitude of number scaled by random production[END_REF]. Haferkorn ( 2013 These results support the so-called Benford analysis, which uses a measure of discrepancy from NBL to detect fraudulent or erroneous data [START_REF] Bolton | statistical fraud detection: A review[END_REF][START_REF] Kumar | data mining for detecting carelessness or mala fide intention[END_REF][START_REF] Nigrini | Benford's Law: Applications for forensic accounting, auditing, and fraud detection[END_REF]. It has been used to audit industrial and financial data [START_REF] Rauch | deficit versus social statistics: empirical evidence for the effectiveness of Benford's law[END_REF][START_REF] Rauch | liBor manipulation -empirical analysis of financial market benchmarks using Benford's law[END_REF] Hsü (1948), [START_REF] Kubovy | response availability and the apparent spontaneity of numerical choices[END_REF]Hill (1988) provided direct experimental evidence that human-produced data conform poorly to NBL. Another study went even further in formulating meaningful tasks by using the type of data known to exhibit a Benford distribution. [START_REF] Burns | sensitivity to statistical regularities: People (largely) follow Benford's law[END_REF] asked participants to guess real-world values, such as the US gross national debt or the peak summer electricity consumption in Melbourne. He found that although participants' first digit responses did not perfectly follow a logarithmic law, they conformed to the logarithmic distribution better than to the uniform distribution. Burns concluded that participants are not too bad at producing a distribution that conforms to NBL as soon as the task involves the type of realworld data that do follow NBL. One limitation of Burns' ( 2009) study is that it only works at a population level. We cannot know from his data if a particular individual would succeed in producing a pseudorandom series conforming to NBL, since each participant produced a single value. Nevertheless, his and Diekmann's studies certainly suggest that using Benford's law to detect fraud is questionable in general since humans may be able to produce data confirming to NBL, in which case a Benford test will yield many undetected frauds, lacking sensitivity. As mentioned above, not all random variables or real-world datasets conform to NBL (and when they do, it is generally only in an approximate manner). Because many real-world datasets do not conform to NBL, a Benford test used to detect fraud not only may have low sensitivity but may also have low specificity. GeneralIzed Benford'S law Several researchers (e.g., [START_REF] Schmeiser | survival distributions satisfying Benford's law[END_REF] have studied conditions under which a distribution seems more likely to satisfy NBL. Fewster ( 2009) provided an intuitive explanation of why and when the law applies and concluded that any dataset that smoothly spans several orders of magnitude tends to conform to NBL. Data limited to one or two orders of magnitude would generally not conform to the law. To pursue the question of why many data conform to NBL further, the conservative version of the NBL may be a better starting point than the mere first-digit analysis. Recall that in the conservative version, a random variable X conforms to NBL if Frac(Log(X)) ~ U([0,1[). In an attempt to show that the roots of NBL ubiquity should not be looked for in the specific properties of the logarithm, Gauvrit and Delahaye (2008,2009) defined a generalized Benford's law (GBL) associated with any function f as follows: A random variable X conforms to a GBL as- sociated with function f if Frac(f(X)) ~ U([0,1[). The classical NBL thus appears as a special case of GBL, associated with function Log. Testing several mathematical and real-world datasets, Gauvrit and Delahaye (2011) found that several of them fit GBL better than NBL. Of 12 datasets they studied, six conformed to the classical NBL, while Study 1 In Study 1, we examined human pseudorandom productions in four different realistic settings, such as those where a classical Benford's law has been initially observed, and compared these responses with true sample values. Participants A sample of 169 adults (63 women) took part in this experiment. Participants were recruited via social networks and e-mails. Ages ranged from 13 to 73 years (M Age = 40.9, SD = 11.6). Method Participants were randomly assigned to one of four groups: cities (n = 41), numbers (n = 44), stars (n = 36), or tuberculosis (n = 48). In each group, participants were informed that a series of 30 numbers had been selected at random from a dataset, and were instructed to produce what they thought would be a credible outcome-that is, to supply 30 plausible numbers. In the cities group, the list was the set of popula- Measures For each set, X of 30 values' (either fabricated or real samples) observed distributions of the fractional parts of f(X), with f(X) = Log(X), f(X) = π × X 2 and f(X) = √(X) Results As expected, fabricated data were usually less consistent with GBL than real data (see Figure 1, Table 2). There is only one exception to this feature: GBL associated with square function in the case of numbers. Depending on the context, different computational variants of GBL seemed more appropriate for segregating true values from fabricated ones. In the case of the largest US cities, for instance, human and real data did not significantly differ in terms of conformity to NBL or GBL with the square function, but they differed when calculated with a square-root function. To analyze the specificity and sensitivity of a fraud detection tool based on GBL, we drew receiver operating characteristic (ROC) curves (see Figure 2) and computed the areas under the curves (AUCs). As shown in Table 3, different sets of data resulted in different patterns. For the cities condition, classical NBL was barely efficient, whereas square root yielded better results. With the Plouffe database, all GBLs were relevant, although the one associated with function π × X 2 appeared to be the best one. Discussion Overall, NBL appeared to be a better means than the other tested variants of GBL for distinguishing between fabricated and real data. However, NBL is not always the best measure, as the cities condition showed. Even when NBL was an efficient measure, such as in the number condition, some other GBL may have been even better or at least as good as NBL, for example, GBL associated with square, for which the AUC was greater than the NBL AUC. Depending on the type of data one tests, different types of GBL could thus be adviced, either replacing or complementing classical Benford analysis. A further argument in favor of the GBL analysis is that, with the growing popularity of the Benford analysis, potential swindlers might become aware of the necessity of conforming to the NBL. Alternative methods complementing the classical analysis (for another such method, see [START_REF] Miller | data diagnostics using secondorder tests of Benford's law[END_REF] could thus prove useful, especially in view of the fact that it would be particularly difficult to fabricate data conforming to a whole set of variants of GBL. Study 2 One possible reason why [START_REF] Diekmann | not the first digit! Using Benford's law to detect fraudulent scientific data[END_REF] found that humans were able to produce accurate values (r) is that the notion was familiar to the participants (students in the social sciences), a feature that may have had an impact on the outcome. If this is true, our positive results in Study 1 might have been the result of too low a familiarity with the material at hand. In Study 2, we investigated the possible effects of familiarity, as well as that of cognitive effort, on the accuracy of the Benford analysis. Participants A sample of 124 first-year psychology students (103 women) from a distant learning program volunteered in the experiment. Ages ranged from 22 to 55 years (M Age = 38.27; SD = 9.08). Participants were recruited by e-mail and voluntarily accepted to participate. We chose distant learning students as participants because, contrary to ordinary students, they have various backgrounds and previous working experiences in a diversity of fields, warranting greater variation in the familiarity with the material. Method The experiment was performed online using a Google Form (https:// www.google.com/forms/about/). We used country population data, as this was believed to grant a somewhat larger variation in familiarity. Participants were asked to produce series of only 20 data points to lower the risk of tiredness. Participants were randomly assigned to one of two groups (no time pressure or time pressure condition). Each group included 62 participants. They were informed that they would have to supply a list of 20 values that could be the numbers of inhabitants of 20 randomly selected countries in the world. They were asked to try to guess what these populations could be. They were told that a true sampling would be performed for comparison with their answers. In the no time pres-sure condition, the instruction was to be "as accurate as possible, taking as much time as needed. " In the time pressure condition, the instruction was to be "as fast and accurate as possible. " The former condition is known to conduce to superior cognitive effort than the latter (e.g., [START_REF] Maule | the effects of time pressure on human judgment and decision making[END_REF]. Self-reported data on level of expertise in the field of country populations were also collected using a 6-point Likert scale, from 0 = absolutely naïve about country populations to 5 = expert in country populations. This measure serves as an indication of the participants' familiarity with the material. Sixty-two true samples of 20 country populations were selected from real-world data (found at http://data.okfn.org/data/core/ population#data) to be compared with participants' productions. Measures For each set X of 20 values (either fabricated or real samples), observed distributions of Frac(Log(X)) were computed. The discrepancy from uniformity was assessed in each case using the Kolmogorov-Smirnov statistic D. Results Reported expertise was rather low (M = 1.3; SD = .97; range of 0-4). To assess the effect of expertise, we performed an analysis of variance (ANOVA), with the dependent variable D and the independent variable Level of Expertise (6). No significant effect was found, F(4, 119) = 0.64, p = .63. The same procedure yielded nonsignificant statistics for GBL associated with the square-root function, F(4, 119) = 1.29, p = .28, and with GBL associated with function π × X 2 , F(4, 119) = 0.20, p = .94. Correlation analysis showed nonsignificant links between level of expertise and D, for classical NBL, r(122) = -.11, p = .20 , GBL associated with a square-root function, r(122) = -.14, p = .10, and also with function πX 2 , r(122) = -.06, p = .53. To assess the effect of cognitive effort, we performed an ANOVA with a dependent variable D and an independent variable Group (2). There was a significant influence, F(2, 183) = 7.33, p < .001, but a Tukey HSD test showed that the two groups did not differ significantly from one another (p = .46), although they both differed from real data values.The same procedure was used to assess the effect of cognitive effort related to other variants of GBL. No effect was found for GBL associated with the square-root function, F(2, 183) = 1.53, p = .22, or with function πX 2 , F(2, 183) = 0.01, p = .99. Discussion Familiarity does not appear to influence the quality of participants' responses in terms of GBL. The experimental procedure used to assess the possible effect of time pressure or cognitive effort yielded no significant results either. These results thus suggested that producing fraudulent data that would remain undetected under the Benford analysis is not necessarily a matter of familiarity or cognitive effort. It is, however, fair to note two limitations. First, time pressure, although widely used to increase cognitive effort, probably does not result in large differences, especially if a task requires little effort in general. Concerning familiarity with the material, no participants declared themselves experts (score of 5 on the scale) about country populations, most of the sample lying in the range of 0-3, so that we would not have detected a specific effect only appearing with true experts. concludInG dIScuSSIon We performed the first investigation of the generalized Benford analysis, an equivalent of the classical Benford analysis, but based on the broader GBL. Results from Study 1 rendered mild support for the generalized Benford analysis, including the classical Benford analysis. They also draw attention to the fact that different types of data yielded different outcomes, suggesting that the best way of detecting fraud using GBL associated with some function f would be obtained either by finding the function f that best matches the particular data at hand or by combining different analyses. Although the classical Benford analysis was validated in our studies, it occasionally failed at detecting humanproduced data as efficiently as other generalized Benford analysis. The present positive results could have been the result of our sample characteristic, in which participants, contrary to real swindlers, might have put little effort into the task since the stakes were low. Plus, the participants were not highly familiar with the material at hand. To rule out the possibility that our results resulted from such features and GBL would be inapplicable in real situations, Study 2 aimed at demonstrating that cognitive effort and familiarity with the material have little effect on the participants' responses. The data supported this view, although further studies (including higher levels of cognitive pressure and true experts) would be recommended. With Benford analysis having become more common in fraud detection, new complementary analyses are needed [START_REF] Miller | data diagnostics using secondorder tests of Benford's law[END_REF]. The GBL analysis potentially provides a whole set of such fraud detection methods, which means making it more difficult, even for informed swindlers intentionally conforming to NBL, to remain undetected. http://www.ac-psych.org 2017 • volume 13(2) • 121-127 127 Falk, r., & Konold, c. (1997). Making sense of randomness: implicit encoding as a basis for judgment. Psychological Review, 104, 301-318. doi: 10.1037/0033-295X.104.2.301 Fewster, r. M. (2009). A simple explanation of Benford's law. The American Statistician, 63, 27-32. doi: 10.1198Statistician, 63, 27-32. doi: 10. /tast.2009Statistician, 63, 27-32. doi: 10. .0005 gauvrit, n., & delahaye, J.-P. (2008)). Pourquoi la loi de Benford n'est pas mystérieuse [Why Benford's law is not mysterious]. Humaines, 182, 7-16. gauvrit, n., & delahaye, J.-P. (2009). loi de Benford général- Criminology, 26, 333-349. doi: 10.1007Criminology, 26, 333-349. doi: 10. /s10940-010-9094-6 hill, t. P. (1988)). random-number guessing and the first digit phenomenon. Psychological Reports, 6, 967-971. doi: 10.2466Reports, 6, 967-971. doi: 10. / pr0.1988Reports, 6, 967-971. doi: 10. .62.3.967 hill, t. P. (1998)). the first digit phenomenon: A century-old observation about an unexpected pattern in many numerical tables applies to the stock market, census statistics and accounting data. American Scientist, 86, 358-363. hsü, e. h. (1948). An experimental study on "mental numbers" and a new application. Journal of General Psychology, 38, 57-67. doi: 10.1080/00221309.1948.9711768 Kreuzer, M., Jordan, d., Antkowiak, B., drexler, B., Kochs, e. F., & schneider, g. (2014). Brain electrical activity obeys Benford's law. Anesthesia & Analgesia, 118, 183-191. doi: 10.1213/ Mathématiques et Sciences ) compared algorithm-based and human-based trade orders and concluded that algorithm-based orders approximated NBL better than human-based orders. Hales, Chakravorty, and Sridharan (2009) showed that NBL is efficient in detecting fraudulent data in an industrial supply-chain context. , to gauge the scientific publication process (de Vries & Murk, 2013), to separate natural from computer-generated images (Tong, Yang, & Xie, 2013), or to detect hidden messages in images' .jpeg files (Andriotis, Oikonomou, & Tryfonas, 2013).As a rule, the Benford analysis focuses on the distribution of the first digit and compares it to the normative logarithmic distribution.However, a more conservative version of Benford's law states that numerical values or a variable X should conform to the following property: Frac(Log(X)) should follow a uniform distribution in the range of[0,1[. Here, Frac(x) stands for x-Floor(x), Floor(x) being the largest integer inferior or equal to x. The logarithmic distribution of the first digit is a mathematical consequence of this version[START_REF]the first digit problem[END_REF]. 10 conformed to a GBL with function f(x) = π × x 2 , and nine with square-root function. On the other hand, none conformed to GBL with function Log o Log. These findings suggest that a GBL associated with the relevant function-depending on the context-might yield more specific or sensitive tests for detecting fraudulent or erroneous data.We addressed this question in two studies. In both studies, each participant produced a whole series of values, allowing analyzing the resulting distribution at an individual level. In Study 1, we examined three versions of GBL in four different situations in order to compare the sensitivity and specificity of different types of GBL analyses. Study 2 explored the potential effects of variations of familiarity with the material and of cognitive effort on the productions. tions of the 5,000 most populated US cities. Numbers corresponded to the dataset of numerical constants published by Simon Plouffe and described in the experiment as an extensive encyclopedia of mathematical constants. Participants assigned to the stars group were told that the dataset was the list of distances from earth to all known visible stars expressed in light-years. Last, the tuberculosis group dealt with the set of known country-wise incidences of tuberculosis as measured in 2012.Real samplings of 30 numbers from the corresponding databases were also performed. The set of populations of the biggest US cities came from an online dataset (http://factfinder2.census.gov/). Numbers were randomly selected from the Simon Plouffe database of numerical values (http://www.plouffe.fr). The distances to the stars were read from the HYG2.0 dataset (http://www.astronexus.com/hyg) and multiplied by 3.262 to render them in light-years instead of parsecs.Lastly, the tuberculosis dataset was downloaded from the World Health Organization's (WHO) website (http://www.who.int/tb/country/data/ download/en/). , were computed. The two last functions were selected on the basis of previous studies indicating that they led to satisfying fits with several numerical datasets(Gauvrit & Delahaye, 2009).The deviation from GBL was measured by the Kolmogorov-Smirnov statistic D-that is, the maximum difference between the cumulative distribution function of Frac(f(X)) and the cumulative distribution function of U([0,1[). D thus serves as a proximal measure of conformity to GBL. This statistic is a classical measure of distance between distributions that grounds the classical Kolmogorov-Smirnov test. isée [generalized Benford's law]. Mathématiques et Sciences Humaines, 186, 5-15. doi: 10.4000/msh.11034 gauvrit, n., & delahaye, J.-P. (2011). scatter and regularity implies Benford's law… and more. in h. Zenil (ed.), Randomness through computation (pp. 53-69). london, england: World scientific. gauvrit, n., & Morsanyi, K. (2014). the equiprobability bias from a mathematical and psychological perspective. Advances in Cognitive Psychology, 10, 119-130. doi: 10.5709/acp-0163-9 haferkorn, M. (2013). humans vs. algorithms -who follows newcomb-Benford's law better with their order volume? in F. A. rabhi & P. gomber (eds.), Enterprise Applications and Services in the Finance Industry (pp. 61-70). Berlin, germany: springer. hales, d. n., chakravorty, s. s., & sridharan, v. (2009). testing Benford's law for improving supply chain decision-making: A field experiment. International Journal of Production Economics, 122, 606-618. doi: 10.1016/j.ijpe.2009.06.017 hickman, M. J., & rice, s. K. (2010). digital analysis of crime statistics: does crime conform to Benford's law? Journal of Quantitative tAble 1 . 1 Proportion of 1s, 2s,…, 9s as First significant digit in a series conforming to nBl Note. NBL = Newcomb-Benford law. Digit 1 2 3 4 5 6 7 8 9 Author Note P., oikonomou, g., & tryfonas, t. (2013). JPeg steganography detection with Benford's law. Digital Investigation, 9, 246-257. doi: 10.1016/j.diin.2013.01.005

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2018

https://hal.sorbonne-universite.fr/hal-01745694/file/EA_2018_0nline%20Ms.pdf

Catherine Debiemme-Chouvy email: [email protected] Ahmed Fakhry Françoise Pillier Electrosynthesis of polypyrrole nano/micro structures using an electrogenerated oriented polypyrrole nanowire array as framework Keywords: Polypyrrole, nanostructures, nanowire, microstructure, electrochemical synthesis The purpose of this paper is to show that it is possible to increase the diameter and the length of the nanostructures of a framework formed of oriented polypyrrole nanowires that has been prepared by a templateless electrochemical method based on the use of a pyrrole solution containing a high concentration of weak-acid anion and a low concentration of non-acidic anion. The dimensions of the initial nanowires are increased by performing an additional electrosynthesis in a 'classical' monomer solution. Depending on the polarization time of this last synthesis (a few tens of seconds), wires with various diameters, from one hundred up to several hundred nanometers, are obtained. In addition to the variation of the nanowire size, these findings confirm, as outlined in the reaction mechanism we have proposed, that the base of the nanowires is surrounded by a thin non-conductive polymer i.e. by an overoxidized polypyrrole film. Actually this paper shows a proof-of-concept. Indeed one can imagine that the second polymeric electrodeposit could be performed using an organic monomer solution, using functionalized pyrrole monomer to fabricate a biosensor having large specific area, and/or using anions which could be drugs. Introduction Polypyrrole (PPy) is one of the most widely used conducting polymers, because of its numerous advantages such as its biocompatibility, environmental stability or ease of preparation even under nanostructured form. Polypyrrole nanostructures are generally synthesized either by a chemical or by an electrochemical route using soft or hard templates [START_REF] Malinauskas | Conducting polymer-based nanostructurized materials: electrochemical aspects[END_REF][START_REF] Yin | Controlled Synthesis and Energy Applications of One-Dimensional Conducting Polymer Nanostructures: An Overview[END_REF][START_REF] Tran | One-Dimensional Conducting Polymer Nanostructures: Bulk Synthesis and Applications[END_REF][START_REF] Heinze | Electrochemistry of Conducting Polymers-Persistent Models and New Concepts[END_REF][START_REF] Long | Recent advances in synthesis, physical properties and applications of conducting polymer nanotubes and nanofibers[END_REF][START_REF] Lee | Highly aligned ultrahigh density arrays of conducting polymer nanorods using block copolymer templates[END_REF][START_REF] Piraux | Self-supported three-dimensionally interconnected polypyrrole nanotubes and nanowires for highly sensitive chemiresistive gas sensing[END_REF][START_REF] Li | Conducting polymer nanomaterials: electrosynthesis and applications[END_REF][START_REF] Lu | Polypyrrole micro-and nanowires synthesized by electrochemical polymerization of pyrrole in the aqueous solutions of pyrenesulfonic acid[END_REF][START_REF] Shen | Facile synthesis of polypyrrole nanospheres and their carbonized products for potential application in high-performance supercapacitors[END_REF][START_REF] Duvail | Physical properties of magnetic metallic nanowires and conjugated polymer nanowires and nanotubes[END_REF] or sacrificial oxidative templates such as MnO 2 [START_REF] Zhang | Reactive Template Synthesis of Polypyrrole Nanotubes for Fabricating Metal/Conducting Polymer Nanocomposites[END_REF][START_REF] Benhaddad | Chemical synthesis of hollow sea urchin like nanostructured polypyrrole particles through a core-shell redox mechanism using a MnO2 powder as oxidizing agent and sacrificial nanostructured template[END_REF][START_REF] Wang | Facile synthesis of hierarchical conducting polypyrrole nanostructures via a reactive template of MnO2 and their application in supercapacitors[END_REF][START_REF] Dubal | Growth of polypyrrole nanostructures through reactive templates for energy storage applications[END_REF]. For instance, PPy nanotubes can be synthesized by using a self-degraded methyl orange (MO) template method [START_REF] Yang | Facile fabrication of functional polypyrrole nanotubes via a reactive self-degraded template[END_REF][START_REF] Yang | Electrochemical synthesis of functional polypyrrole nanotubes via a self-assembly process[END_REF][START_REF] Sapurina | Polypyrrole nanotubes: The tuning of morphology and conductivity[END_REF][START_REF] Kopecka | Polypyrrole nanotubes: mechanism of formation[END_REF]. However It has also been established that polypyrrole nanostructures can also be prepared without the use of any template [START_REF] Debiemme-Chouvy | Nanostructured polypyrrole materials: Focus on templateless synthetic methods and on some applications[END_REF][START_REF] Chronakis | Conductive polypyrrole nanofibers via electrospinning: Electrical and morphological properties[END_REF][START_REF] Tran | A template-free route to polypyrrole nanofibers[END_REF][START_REF] Turco | Templateless synthesis of polypyrrole nanowires by non-static solution-surface electropolymerization[END_REF][START_REF] Tran | Toward an understanding of the formation of conducting polymer nanofibers[END_REF]. Notably In that respect, in our previous works [START_REF] Debiemme-Chouvy | Template-free one-step electrochemical formation of polypyrrole nanowire array[END_REF][START_REF] Fakhry | Templateless electrogeneration of polypyrrole nanostructures: impact of the anionic composition and pH of the monomer solution[END_REF][START_REF] Fakhry | Mechanism of formation of templateless electrogenerated polypyrrole nanostructures[END_REF], we brought to light that different polypyrrole nanostructures can be synthesized by using a one-step electrochemical synthesis, without using any template. Indeed, nanostructured polypyrrole films, which are superhydrophilic, are electrogenerated in the presence of (i) a high concentration of weak-acid anions conferring to upon the monomer solution a pH between 6 and 10, and (ii) a low concentration of non-acidic anions such as perchlorate ions. Notably, Depending on the perchlorate concentration of a 0.15 M pyrrole solution containing 0.2 M monohydrogenophosphate, different PPy nanostructures can be synthesized. In the absence or in the presence of a very low concentration of perchlorate ions (< 10 -4 M), an ultrathin non-conductive overoxidized PPy (OPPy) film is obtained [START_REF] Debiemme-Chouvy | One-step electrochemical synthesis of a very thin overoxidized polypyrrole film[END_REF][START_REF] Fakhry | Electrochemical Characterisations of Ultra Thin Overoxidized Polypyrrole Films Obtained by One-Step Electrosynthesis[END_REF]. This finding was notably determined in particular by XPS analyses [START_REF] Debiemme-Chouvy | One-step electrochemical synthesis of a very thin overoxidized polypyrrole film[END_REF] and EIS studies [START_REF] Fakhry | Electrochemical Characterisations of Ultra Thin Overoxidized Polypyrrole Films Obtained by One-Step Electrosynthesis[END_REF]. Notice that due to its compactness, this OPPy film prevents the diffusion to the electrode surface of large redox species such as ascorbic acid and dopamine [START_REF] Debiemme-Chouvy | A very thin overoxidized polypyrrole membrane as coating for fast time response and selective H 2 O 2 amperometric sensor[END_REF] or Fe(CN) 6 4-and Ru(NH 3 ) 6 2+ [START_REF] Debiemme-Chouvy | Characterization of a very thin overoxidized polypyrrole membrane: application to H 2 O 2 determination[END_REF] avoiding their electro-oxidation. It is not the case for On the contrary, small molecules such as H 2 O 2 and H 2 O can diffuse across this OPPy film, and therefore be oxidized at the electrode surface [START_REF] Debiemme-Chouvy | A very thin overoxidized polypyrrole membrane as coating for fast time response and selective H 2 O 2 amperometric sensor[END_REF][START_REF] Debiemme-Chouvy | Characterization of a very thin overoxidized polypyrrole membrane: application to H 2 O 2 determination[END_REF]. In the presence of a higher concentration of perchlorate ions (> 10 -4 M), an oriented nanowire (NW) array or a network of more or less interconnected nanofibers is formed [START_REF] Fakhry | Templateless electrogeneration of polypyrrole nanostructures: impact of the anionic composition and pH of the monomer solution[END_REF]. M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT 3 The oxidation of pyrrole (Py) leads to the formation of polypyrrole under its oxidized form, doped with anions (A -), according to the following reaction [START_REF] Genies | Spectroelectrochemical study of polypyrrole films[END_REF][START_REF] Qiu | Electrochemically initiated chain polymerization of pyrrole in aqueous-media[END_REF]: ( where γ stands for the doping level of the polymer, it is generally in the range of 0.25 to 0.33 [START_REF] Diaz | Electrochemical preparation and characterization of conducting polymers[END_REF][START_REF] Salmon | Chemical modification of conducting polypyrrole films[END_REF]. In the presence of weak-acid anions, the protons released during Py oxidation (reaction (1)) are captured by these anions: ( The mechanism that we have previously proposed in order to explain the formation of a nanostructured polypyrrole film is summarized in Figure 1. It is based on the variation of the interfacial concentration of anions. Indeed, during pyrrole oxidation protons are released during pyrrole oxidation, and collected by the weak-acid anions present in the solution (reaction (2)), which results in a drastic decrease, or elimination, of anions at the electrode/solution interface. As pyrrole oxidation requires the presence of anions (reaction (1)), which are no longer available, this reaction cannot occur anymore. Instead, water oxidation takes place, leading to the formation of hydroxyl radicals. These radicals can either react with the already formed polypyrrole film, resulting in its overoxidation [START_REF] Debiemme-Chouvy | An insight into the overoxidation of polypyrrole materials[END_REF], or with themselves, leading to the formation of H 2 O 2 molecules. These molecules are subsequently oxidized into O 2 molecules that form nanobubbles inside the polymer, protecting it against the hydroxyl radical action and therefore preventing locally its overoxidation allowing the conservation of some conductive zones (step 2 in Fig. 1). After bubbles evolution, the electrooxidation of the monomers takes place at these zones, leading to the formation of nanorods/nanowires (step in Figure 1). Obviously, the diameter of these PPy nanostructures depends on the size of the O 2 bubbles. Notice that this process differs from the - - ) ( n - e 2) - n ) ((2 H 1) - 2(n ] A n , [(Py) A n Py n γ + + + γ → γ + + + γn AH H A - → + + M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT 4 one involving gas bubbles as template for PPy microstructure electrosynthesis [START_REF] Qu | Preparation of polypyrrole microstructures by direct electrochemical oxidation of pyrrole in an aqueous solution of camphorsulfonic acid[END_REF][START_REF] Mazur | Polymerization at the Gas/Solution interface: Preparation of polymer microstructures with gas bubbles as templates[END_REF][START_REF] Parakhonskiy | Polypyrrole Microcontainers: Electrochemical Synthesis and Characterization[END_REF]. Therefore, the nature of the anions present in the monomer solution and the release of protons during Py electropolymerization are the two main parameters involved in polypyrrole nanostructure formation. Under potentiostatic conditions, using a Pt anode, the diameter of the PPy nanowires is about 80 nm and their length depends on the electrode polarization time [START_REF] Debiemme-Chouvy | Template-free one-step electrochemical formation of polypyrrole nanowire array[END_REF]. It has been shown that PPy nanostructures are also electrogenerated in the presence of a low monomer concentration and a high non-acidic anion concentration [START_REF] Fakhry | Templateless electrogeneration of polypyrrole nanostructures: impact of the anionic composition and pH of the monomer solution[END_REF]. The oriented PPy nanowire array could be used as framework for the growth of larger and longer nano/micro structures. Therefore, a second electrosynthesis could be done, using for example functionalized monomer in order to prepare a biosensor [START_REF] Baur | Immobilization of biotinylated biomolecules onto electropolymerized poly(pyrrole-nitrilotriacetic acid)-Cu2+ film[END_REF][START_REF] Baur | Label-Free Femtomolar Detection of Target DNA by Impedimetric DNA Sensor Based on Poly(pyrrolenitrilotriacetic acid) Film[END_REF][START_REF] Giroud | Impedimetric Immunosensor Based on a Polypyrrole-Antibiotic Model Film for the Label-Free Picomolar Detection of Ciprofloxacin[END_REF][START_REF] Gondran | Electrogenerated poly(pyrrole-Iactosyl) and poly(pyrrole-3 '-sialyllactosyl) interfaces: toward the impedimetric detection of lectins[END_REF][START_REF] Haddour | Electrogeneration of a poly(pyrrole)-NTA chelator film for a reversible oriented immobilization of histidine-tagged proteins[END_REF][START_REF] Kazane | Highly Sensitive Bisphenol-A Electrochemical Aptasensor Based on Poly(Pyrrole-Nitrilotriacetic Acid)-Aptamer[END_REF][START_REF] Xu | Label-free impedimetric thrombin sensor based on poly(pyrrole-nitrilotriacetic acid)aptamer film[END_REF], or using organic solvent containing anions which could be drugs [START_REF] Guiseppi-Elie | Electroconductive hydrogels: Synthesis, characterization and biomedical applications[END_REF][START_REF] Novak | Overoxidation of polypyrrole in propylene carbonate -An in situ FTIR study[END_REF][START_REF] Samanta | Electroresponsive nanoparticles for drug delivery on demand[END_REF][START_REF] Sirivisoot | Electrically controlled drug release from nanostructured polypyrrole coated on titanium[END_REF]. The aim of the present work is to show that it is possible to vary the diameter and the length of oriented polypyrrole nanowires by performing an additional electrosynthesis in a classical pyrrole solution. Moreover, it allows to confirm provides confirmation that the nanowire base is surrounded by an overoxidized polypyrrole (non-conductive polymer) layer. These experiments also allow us to determine whether or not the polypyrrole nanowires are conductive over their entire surface (top and sides). Therefore, the strategy developed in the present work involves two steps. The first one requires is the electrogeneration of oriented nanowires, which requires the use of a monomer solution containing a high concentration of weak-acid anions and a low concentration of perchlorates. The second step is based on implies an anodic polarization in a classical monomer solution, without any weak-acid anions. Experimental The polypyrrole films were synthesized using preliminarily distilled pyrrole. All the solutions were prepared with bi-distilled water. Py, K 2 HPO 4 , KCH 3 COO, K 2 CO 3 and LiClO 4 were purchased from Aldrich. M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT 5 The electrochemical experiments were performed in a classical three-electrode electrochemical cell. A platinum foil was used as counter electrode and a saturated calomel electrode (SCE) was used as the reference electrode. A double junction was used in order to avoid chloride diffusion into the Py solution. The working electrode was a Pt electrode (S = 0.07 cm 2 ) for electrochemical studies and Au/mica substrate for SEM observations. The preparation of the substrates was the following: a thin gold film (∼ 80 nm) was deposited under low pressure (10 -4 Pa) by thermal evaporation on a mica substrate. All the electrosyntheses were performed under potentiostatic conditions, in 0.15 M Py + 0.2 M K 2 HPO 4 + 10 -3 M LiClO 4 for the first synthesis and in 0.15 M Py + 0.2 M LiClO 4 for the second one, during given times named t 1 and t 2, respectively. The pH of the first monomer solution is 8.9. The superhydrophilic character of the PPy films was checked by a drop test experiment: a small pure water droplet is deposited onto the film and the spreading of the drop is analyzed. When the film is nanostructured the drop spreads out whereas it does not for classical cauliflower-like films. As far as the electropolymerization of Py is concerned, an Autolab PGSTAT30 potentiostat (Ecochemie) controlled with the GPES software was employed. The film morphology was examined under a field emission gun scanning electron microscope (FEG-SEM), Ultra55 Zeiss, operating at 5 kV. Results and discussion Firstly, the framework composed of oriented PPy nanowires was synthesized by performing an electropolymerization of pyrrole monomers under potentiostatic conditions at 0.75 V/SCE in a solution composed of 0.15 M pyrrole, 0.2 M K 2 HPO 4 and 10 -3 M LiClO 4 (see the SEM micrograph in Fig. 1). The average diameter of the nanowires is about 80 nm. Notice that the framework can also be obtained using acetate or carbonate instead of Some of these polypyrrole films have been observed by SEM. In Figure 4 are reported micrographs obtained for a Au/mica substrate first polarized during 600 s and then only a part of its surface has undergone a second polarization of 50 seconds in the 'classical' Py solution. M A N U S C R I P T A C C E P T E D Micrograph A shows the film after the first polarization whereas micrographs B-D show the film after the second polarization. Notice that micrographs A and B were done at the same magnitude. From these micrographs, it is clear that the growth of the polymer occurs on the whole surface of the nanowires, since their diameter increases from 80 nm to 200 nm and their length increases from 500 nm [START_REF] Debiemme-Chouvy | Template-free one-step electrochemical formation of polypyrrole nanowire array[END_REF] up to 1500 nm. Besides, the SEM micrographs presented in Figure 5 show different kinds of (nano)structures depending on the second synthesis duration, the first synthesis duration being 200 s. When the second polarization duration increases, the polypyrrole nanowire diameter increases too. Finally, for 300 seconds of polymerization, the PPy film has a cauliflower-like structure (see micrographs E and F in Fig. 5) because during the electrode polarization the diameter of the nanowires has so increased that their coalescence has occurred. The diameter and the length of the polypyrrole nanostructures obtained after the second synthesis versus the polymerization duration are plotted in Figure 6. The nanowire diameter increases with the second polarization time, leading to a decrease of the space between the nanowires until they coalesce and form a 2D film that is no more superhydrophilic, confirming that the PPy film is no longer nanostructured. Before coalescence, the growth rate in terms of fiber length is about 17 nm s -1 , which is almost 20 times faster than the rate determined with the Py solution used for the framework synthesis (first synthesis) [START_REF] Debiemme-Chouvy | Template-free one-step electrochemical formation of polypyrrole nanowire array[END_REF]. This finding could be explained by the fact that the Py electropolymerization is limited by the concentration of the anions present at the electrode/solution interface [START_REF] Li | Effect of anion concentration on the kinetics of electrochemical polymerization of pyrrole[END_REF]. are plotted in Fig. 7A. As it can be noticed from this figure, for a given t 2 , the relationship between Q 1 and Q 2 is linear and does not pass through the origin: Q 2 = α + β Q 1 (3) Whatever t 1 is, the first synthesis leads to PPy nanowire arrays having the same NW density and the same NW diameter, as these parameters only depend on the nature of the electrode, the applied potential, and the Py solution composition. Therefore, the variation of t 1 i.e. Q 1 only leads to the variation of the PPy nanowire length [START_REF] Debiemme-Chouvy | Template-free one-step electrochemical formation of polypyrrole nanowire array[END_REF]. As the relationship between the charge passed during the second synthesis and the one passed during the first synthesis (which determines the length of the wires) is linear, one can conclude that the Py polymerization takes place on the entire surface of the nanowires and not only on their top. In this latter case, Q 2 should be independent of the wire length i.e. of Q 1 . This conclusion is in good agreement with the SEM observations (Figs. [START_REF] Heinze | Electrochemistry of Conducting Polymers-Persistent Models and New Concepts[END_REF][START_REF] Long | Recent advances in synthesis, physical properties and applications of conducting polymer nanotubes and nanofibers[END_REF][START_REF] Lee | Highly aligned ultrahigh density arrays of conducting polymer nanorods using block copolymer templates[END_REF]. α and β parameters of equation ( 3) versus the polarization time of the second synthesis are depicted in Figures 7B and7C, respectively. These relationships are linear. Q 2 corresponds to Py oxidation that leads to the generation of PPy at a place that can be divided into three zones, named a, b, and c in Figure 8. Q 2 = Q a + Q b + Q c ( 4 ) As it can be noticed from Figure 8, zone a corresponds to the enlargement of the nanowires. Therefore, Q a is function of Q 1 . Zones b and c correspond to the elongation of the nanowires and the enlargement of the nanowires formed during the second synthesis, respectively. Therefore both Q b and Q c are independent from Q 1 , which sets the length of the nanowire. α is proportional to t 2 as evidenced by Fig. 7B, in good agreement with Fig. 6. Consequently, taking into account that α = Q b + Q c and β = a t 2 , the combination of equations ( 3) and (4) gives M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT Q 2 = Q b + Q c + a t 2 Q 1 (5) All these findings allow us to confirm that the PPy nanowires of the framework (first synthesis) are conductive on their entire surface i.e. on their top and their sides, as it is possible to deposit another polypyrrole film all around the nanostructures whose diameter increases with the polarization time of the second synthesis (Fig. 6). Moreover, these findings confirm that the layer which surrounds the nanowire base is non-conductive i.e. it is an overoxidized PPy film, as shown in the mechanism proposed in our previous work regarding the formation of the different polypyrrole nanostructures (see Fig. 1). Conclusions It is possible to use oriented polypyrrole nanowires obtained by a one-step templateless electrochemical method as framework to perform a second synthesis in a pyrrole solution that does not contain a high concentration of weak-acid anions. Depending on the polarization duration of this synthesis, the diameter of the nanowires varies from a few tens of nanometers up to several hundred nanometers, and after a threshold of polarization time cauliflower-like structures are obtained. These findings are important because they allow us to confirm the global mechanism we proposed to achieve the electrogeneration of PPy nanostructure without the use of any template, just by using a monomer solution containing a high concentration of weak-acidic anions having pKa > 6. Finally, this paper shows a proof-of-concept, indeed one can imagine that the second electrodeposit could be performed using functionalized pyrrole monomer to fabricate electrochemical biosensors with large specific area, for example or using an organic solvent, and/or using anions which could be drugs. Figures Figures 2A and 2B (curves (a)) show the anodic current versus time responses for the first and Finally, a series of PPy framework was prepared by varying the polarization durations (t 1 ), leading to different anodic charges (Q 1 ). A second polarization, lasting a given time (t 2 = 10, 15, 20, 30, 40 or 50 seconds), was then M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT 8 performed in the 'classical' Py solution (solution containing only perchlorates as anions). The anodic charges after the second synthesis (Q 2 ) vs. anodic charges after the first synthesis (Q 1 ) Figure 1 .Figure 2 .Figure 3 . 123 Figure 1. General mechanism of PPy nanowire electrogeneration. Left: Drawing showing the evolution of the polymer deposit with the anodic polarization time, in Py aqueous solution containing a high concentration of K 2 HPO 4 and a low concentration of LiClO 4 . Right: SEM micrograph (side view, 60°) of PPy nanowires electrosynthesized at 0.75 V/SCE for 200 s in 0.15 M Py + 0.2 M K 2 HPO 4 + 10 -3 M LiClO 4 aqueous solution. Steps 1, 3, 4: Py oxidation ; step 2 : water oxidation leading to OH and O 2 formation. O 2 locally protect the PPy film against the action of OH which overoxidizes PPy. Steps 1, 2, 3 last around a few seconds. Figure 4 .Figure 5 .Figure 6 . 1 Figure 7 . 21 Figure 8 . 45617218 Figure 4. SEM micrographs of PPy. A) after a first synthesis at 0.75 V/SCE for 600 s in 0.15 M Py + 0.2 M K 2 HPO 4 + 10 -3 M LiClO 4 aqueous solution. B-D) after a second synthesis at 0.75 V/SCE for 50 s in 0.15 M Py + 0.2 M LiClO 4 aqueous solution. A,B,C top view, D side view (60°). Scale bar: 200 nm (A), (B), (D) ; 100 nm (C). COO -, we have obtained the same results i.e. nanostructured PPy films (see Table1). Then, another polypyrrole deposit was performed onto these nanostructures. This second electrosynthesis was conducted in a monomer solution containing only perchlorates as anions, solution which generally leads to the formation of a PPy film with cauliflower-like structure. ACCEPTED MANUSCRIPT monohydrogenophosphate. Indeed by replacing HPO 4 2-by CO 3 2-or CH 3 M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT Nanostructured film ; (2) Overoxidized PPy film ; (3) pH adjusted with KOH

01745729

en

2024/03/05 22:32:07

2013

https://hal.science/hal-01745729/file/001_WP802.pdf

Built-In Self-Test Structure (BIST) for Resistive RAMs Characterization: Application to Bipolar OxRRAMs Hassen AZIZA, Marc BOQUET, Mathieu MOREAU and J-Michel PORTAL IM2NP-UMR CNRS 6242, Aix-Marseille University, France, [email protected] Problem Formulation: Resistive Random Access Memory (ReRAM) is a form of nonvolatile storage that operates by changing the resistance of a specially formulated solid dielectric material [START_REF] Gibbons | Switching properties of thin Nio films[END_REF]. Among ReRAMs, Oxide-based Resistive RAMs (so-called OxRRAM) are promising candidates due their compatibility with CMOS processes and high ON/OFF resistance ratio. Common problems with OxRRAM are related to high variability in operating conditions and low yield. OxRRAM variability mainly impact ON/OFF resistance ratio. This ratio is a key parameter to determine the overall performance of an OxRRAM memory. In this context, the presented built-in structure allows collecting statistical data related to the OxRRAM memory array (ON/OFF resistance distributions) for reliability assessment of the technology. Built-In Self-Test (BIST) structure: Fig. 1a presents the elementary array used for simulation which is constituted by a 3×3 1T/1R cell matrix, a row decoder, a column decoder and a sense amplifier for the read operation. The OxRRAM compact model used to build the memory array satisfactorily matches quasi-static and dynamic experimental data measured on actual HfO 2 -based devices (Fig. 1b). Notice that the OxRRAM model allows a variability analysis based on OxRRAM card model parameters variation. The card model parameter variations are chosen to feet experimental data [START_REF] Aziza | Evaluation of OxRAM cell variability impact on memory performances through electrical simulations[END_REF]. Fig. 2 presents the single-ended sense amplifier (solid line). The circuit working principle is quite simple. During a READ operation, the OxRRAM cell is biased throw the row decoder (with V read > 0 and WL high). The current cell value I cell is generated according to the memory state (RON≈7kΩ after a SET operation and ROFF≈25 kΩ after a RESET operation). Simulation results: To validate the BIST structure, a variability analysis is conducted through 500 Monte Carlo simulations. The elementary matrix presented in Fig. 1, embedding the BIST structure, is considered for simulations. As a result of cell variability, circuit performance exhibits much wider variability. In Fig. 3a, RON and ROFF distributions are plotted (RON/ROFF variability being correlated with actual silicon results). In Fig. 3b, V IN SET and V IN RESET distributions are also plotted. These last distributions are similar to RON and ROFF distributions. This trend is confirmed by the correlation curve presented in Fig. 4. Therefore, the modified sense amplifier structure can be used as a powerful tool to track any resistance variations but also to characterize the memory array variability. To summarize, a built in self-test structure is presented for Resistive RAM characterization and variability evaluation. The area overhead introduced by structure is relatively low as the structure is integrated in the sense amplifiers. The normal mode of operation of the memory is preserved. Besides, extracted values are given in a digital data format, so the extraction process does not required any analog pin on the tester, making it fully digital tester compliant or easily observable via the random logic. Fig. 1 (Fig. 2 12 Fig. 1 (a) OxRRAM memory array (a) I-V characteristic measured on HfO2-based devices and corresponding simulation using the OxRRAM model & SET voltage as a function of the programming ramp The comparator input V IN is directly proportional to I cell and therefore to the OxRRAM resistance (two distinct values are available for V IN : V IN SET for RON and V IN RESET for ROFF). So that the sense amplifier operates properly, on the one hand the difference between V IN SET and V IN RESET must be the highest and on the other hand V REF has to be set exactly between V IN SET and V IN RESET values. At a circuit level, V IN is the parameter to consider in terms of memory functionality. To extract V IN value, a variable voltage reference source and a multiplexer are incorporated in the sensing circuit (dotted part of the circuit) and the READ operation is modified as follow:• V REF increases step by step from 0 to V dd (V REF increase is controlled by a shift register),• V IN value is detected when the sense amplifier output switches (the shifting process stops when V REF > V IN , i.e Rd_REG signal becomes active), • V IN is available at the circuit output in a numerical value when BIST_EN signal is high.

01745768

en

2024/03/05 22:32:07

2018

https://inria.hal.science/hal-01745768/file/paper-hal.pdf

Tien-Duc Cao email: [email protected] Ioana Manolescu email: [email protected] Xavier Tannier email: [email protected] Searching for Truth in a Database of Statistics The proliferation of falsehood and misinformation, in particular through the Web, has lead to increasing energy being invested into journalistic fact-checking. Fact-checking journalists typically check the accuracy of a claim against some trusted data source. Statistic databases such as those compiled by state agencies are often used as trusted data sources, as they contain valuable, high-quality information. However, their usability is limited when they are shared in a format such as HTML or spreadsheets: this makes it hard to find the most relevant dataset for checking a specific claim, or to quickly extract from a dataset the best answer to a given query. We present a novel algorithm enabling the exploitation of such statistic tables, by (i) identifying the statistic datasets most relevant for a given fact-checking query, and (ii) extracting from each dataset the best specific (precise) query answer it may contain. We have implemented our approach and experimented on the complete corpus of statistics obtained from INSEE, the French national statistic institute. Our experiments and comparisons demonstrate the effectiveness of our proposed method. INTRODUCTION The media industry has taken good notice of the value promises of big data. As more and more important and significant data is available in digital form, journalism is undergoing a transformation whereas technical skills for working with such data are increasingly needed and also present in newsrooms. In particular, the areas of data journalism and journalistic fact checking stand to benefit most from efficient tools for exploiting digital data. Statistics databases produced by governments or other large organizations, whether commercial or administrative ones, are of particular interest to us. Such databases are built by dedicated personnel at a high cost, and consolidated carefully out of multiple inputs; the resulting data is typically of high value. Another important quality of such data is that it is often shared as open data. Example of such government open data portals http://data.gov (US), http://data.gov.uk (UK), http://data.gouv.fr and http://insee.fr (France), http://wsl.iiitb.ac.in/sandesh-web/sandesh/index (India) etc. While the data is open, it is often not linked, that is, it is not organized in RDF graphs, as recommended by the Linked Open Data best practice for sharing and publishing data on the Web [START_REF] W3c | Best Practices for Publishing Linked Data[END_REF]. Instead, such data often consists of HTML or Excel tables containing numeric data, links to which appear in or next to text descriptions of their contents. This prevents the efficient exploitation of the data through automated tools, capable for instance to answer queries such as "find the unemployment rate in my region in 2015". In a prior work [START_REF] Cao | Extracting Linked Data from Statistic Spreadsheets[END_REF], we have devised an approach to extract from high-quality, statistic Open Data in the "tables + text description" frequently used nowadays, Linked Open Data in RDF format; this is a first step toward addressing the above issues. Subsequently, we applied our approach to the complete set of statistics published by INSEE, a leading French national statistics institute, and republish the resulting RDF Open Data (together with the crawling code and the extraction code 1 ). While our approach is tailored to some extent toward French, its core elements are easily applicable to another setting, in particular for statistic databases using English, as the latter is very well-supported by text processing tools. In this work, we introduce novel algorithms for searching for answers to keyword queries in a database of statistics, organized in RDF graphs such as those we produced. First, we describe a dataset search algorithm, which given a set of user keywords, identifies the datasets (statistic table and surrounding presentations) most likely to be useful for answering the query. Second, we devised an answer search algorithm which, building on the above algorithm, attempts to answer queries, such as "unemployment rate in Paris in 2016", with values extracted from the statistics dataset, together with a contextualization quickly enabling the user to visually check the correctness of the extraction and the result relevance. In some cases, there is no single number known in the database, but several semantically close ones. In such cases, our algorithm returns the set of numbers, again together with context enabling its interpretation. We have experimentally evaluated the efficiency and the effectiveness of our algorithms, and demonstrate their practical interest to facilitate fact-checking work. In particular, while political debates are broadcast live on radio or TV, fact-checkers can use such tools to quickly locate reference numbers which may help them publish live fact-checking material. In the sequel, Section 2 outlines the statistic data sources we consider, their organization after our extraction [START_REF] Cao | Extracting Linked Data from Statistic Spreadsheets[END_REF], and our system architecture. Section 3 defines the search problem we address, and describes our search algorithms. Section 4 presents our experimental evaluation. We discuss related work in Section 5, then conclude. 1. The outline, when present, is a short paragraph on the Web page from where D i can be downloaded. The outline extends the title with more details about the nature of the statistic numbers, the interpretation of each dimension, methodological details of how numbers were aggregated etc. INPUT DATA AND ARCHITECTURE A dataset consists of header cells and data cells. Data cells and header cells each have attributes indicating their position (column and row) and their value. For instance, in Figure 1, "Belgium", "Bulgaria" etc. up to "France", "Oct-2016" to "Oct-2017", and the two fused cells above them, are header cells, while the other are data cells. Each data cell has a closest header cell on its column, and another one on its row; these capture the dimension values corresponding to a data cell. To enable linking the results with existing (RDF) datasets and ontologies, we extract the table contents into an RDF graph. We create an RDF class for each entity type, and assign an URI to each entity instance, e.g., dataset, table, cell, outline etc. Each relationship is similarly represented by a resource described by the entity instances participating to the relationship, together with their respective roles. System architecture A scraper collects the Web pages publicly accessible from a statistic institute Web site; Excel and HTML tables are identified, traversed, and converted into RDF graphs as explained above. The resulting RDF data is stored in the Apache Jena TDB server 2 , and used to answer keyword queries as we explain next. SEARCH PROBLEM AND ALGORITHM Given a keyword-based query, we focus on returning a ranked list of candidate answers, ordered in the decreasing likelihood that they contain (or can be used to compute) the query result. A relevant candidate answer can be a data cell, a data row, column or even an entire dataset. For example, consider the query "youth unemployment in France in August 2017". An Eurostat dataset3 is a good candidate answer to this query, since, as shown in Figure 1, it contains one data cell, at the intersection of the France row with the Aug-2017 column. Now, if one changes the query to ask for "youth unemployment in France in 2017", no single data cell can be returned; instead, all the cells on the France row qualify. Finally, a dataset containing 2017 French unemployment statistics over the general population (not just youth) meets some of the search criteria (2017, France, unemployment) and thus may deserve to appear in the ranked list of results, depending on the availability of better results. This task requires the development of specific novel methods, borrowing ideas from traditional IR, but following a new methodology. This is because our task is very specific: we are searching for information not within text, but within tables, which moreover are not flat, first normal form database relations (for which many keyword search algorithms have been proposed since [START_REF] Hristidis | DISCOVER: Keyword Search in Relational Databases[END_REF]), but partially nested tables, in particular due to the hierarchical nature of the headers, as we explained previously. While most of the reasoning performed by our algorithm follows the two-dimensional layout of data in columns and tables, bringing the data in RDF: (i) puts a set of interesting, high-value data sources within reach of developers and (ii) allows us to query across nested headers using regular path queries expressed in SPARQL (as we explain in Section 3.5). We describe our algorithms for finding such answers below. Dataset search The first problem we consider is: given a keyword query Q consisting of a set of keywords u 1 , u 2 , . . ., u m and an integer k, find the k datasets most relevant for the query (we explain how we evaluate this relevance below). We view each dataset as a table containing a title, possibly a comment, a set of header cells (row header cells and column header cells) and a set of data cells, the latter containing numeric data 4 . At query time, we transform the query Q into a set of of keywords W = w 1 , w 2 , . . . , w n using the method described in Section 3.2. Offline, this method is also used to transform each dataset's text to words and we compute the score of each word with respect to a dataset, as described in Section 3.3. Then, based on the word-dataset score and W , we estimate datasets' relevance to the query as we explain in Section 3.4. Text processing Given a text t (appearing in a title, comment, or header cell of the dataset, or the text consisting of the set of words in the query Q), we convert it into a set of words using the following process: • First, t is tokenized (separated into distinct words) using the KEA5 tokenizers. Subsequently, each multi-word French location found in t that is listed in Geonames 6 , is put together in a single token. • Each token (word) is converted to lowercase, stop words are removed, as well as French accents which complicate matching. • Each word is mapped into a word2vec vector [START_REF] Mikolov | Distributed Representations of Words and Phrases and their Compositionality[END_REF], using the gensim [START_REF] Radim | Software Framework for Topic Modelling with Large Corpora[END_REF] tool. Bigrams and trigrams are considered following [START_REF] Mikolov | Distributed Representations of Words and Phrases and their Compositionality[END_REF]. We had trained the word2vec model on a generaldomain French news web page corpus. Word-dataset score For each dataset extracted from the statistic database, we compute a score characterizing its semantic content. A first observation is that datasets should be returned not only when they contain the exact words present in the query, but also if they contain very closely related ones. For example, a dataset titled "Duration of marriage" could be a good match for the query "Average time between marriage and divorce" because of the similarity between "duration" and "time". To this effect, we rely on word2vec [START_REF] Mikolov | Distributed Representations of Words and Phrases and their Compositionality[END_REF] which provides similar words for any word in its vocabulary: if a word w appears in a dataset D, and w is similar to w ′ , we consider w ′ also appears in D. The score score (w ) of a dataset D w.r.t the query word w is 1 if w appears in D. . If w does not appear in D: • If there exists a word w ′ , from the list of top-50 similar words of w according to word2vec, which appears in D, then score (w ) is the similarity between w and w ′ . If there are several such w ′ , we consider the one most similar to w. • If w is the name of a Geonames place we can't apply the above scoring approach because "comparable" places (e.g., cities such as Paris and London) will have high similarity in the word2vec space. As the result, when user asks for "unemployment rate Paris", the data of London might be returned instead of Paris's. Let p be the number of places that Geonames' hierarchy API 7 returns for w (p is determined by Geonames and depends on w). For instance, when querying the API with Paris, we obtain the list Île-de-France, France, Europe. Let w ′ i be the place at position i, 1 ≤ i ≤ p in this list of returned places, such that w ′ i appears in D. Then, we assign to D a score for w equal to (p + 1 -i)/(p + 1), that is, the most similar place according to Geonames has rank p/(p + 1), and the least similar has the rank 1/(p + 1). If D contains several of the places from the w's hierarchy, we assign to D a score for w corresponding to the highest-ranked such place. • 0 otherwise. Based on the notion of word similarity defined above, we will write w ≺ W to denote that the word w from dataset D either belongs to the query set W , or is close to a word in W . Observe that, by definition, for any w ≺ , we have score (w ) > 0. We also keep track of the location(s) (title, header and/or comment) in which a word appears in a dataset; this information will be used when answering queries, as described in Section 3.4. In summary, for each dataset D and word w ∈ D such that w ≺ W , we compute and store tuples of the form: These tuples are encoded in JSON and stored in a set of files; each file contains the scores for one word (or bigram) w, and all the datasets. Relevance score function Content-based relevance score function. This function, denoted д 1 (D,W ), quantifies the interest of dataset D for the word set W ; it is computed based on the tuples (w, score (w ), location(w, D), D) where w ≺ W . We experimented with many score functions that give high ranking to datasets that have many matching keywords (see details in section 4.2.2). These functions are monotonous in the score of D with respect to each individual word w. This enables us to apply Fagin's threshold algorithm [START_REF] Fagin | Optimal Aggregation Algorithms for Middleware[END_REF] to efficiently determine the k datasets having the highest д 1 score for the query W . Location-aware score components. The location -title (T), row or column headers (HR or HC), or comments (C) -where a keyword occurs in a dataset can also be used to assess the dataset relevance. 7 http://www.geonames.org/export/place-hierarchy.html For example, a keyword appearing in the title often signals a dataset more relevant for the search than if the keyword appears in a comment. We exploit this observation to pre-filter the datasets for which we compute exact scores, as follows. We run the TA algorithm using the score function д 1 to select r × k datasets, where r is an integer larger than 1. For each dataset thus retrieved, we compute a second, refined score function д 2 (see below), which also accounts for the locations in which keywords appear in the datasets; the answer to the query will consists of the top-k datasets according to д 2 . The second score function д 2 (D,W ) is computed as follows. Let w ′ be a word appearing at a location loc ∈ {T, HR, HC, C} such that w ′ ≺ W . We denote by w ′ loc, D (or just w ′ loc when D is clear from the context) the existence of one or several located occurrence of w ′ in D in loc. Thus, for instance, if "youth" appears twice in the title of D and once in a row header, this leads to two located occurrences, namely youth T, D and youth HR, D . Then, for loc ∈ {T, HR, HC, C} we introduce a coefficient α loc allowing us to calibrate the weight (importance) of keyword occurrences in location loc. To quantify D's relevance for W due to its loc occurrences, we define a location score component f loc (D,W ). In particular, we have experimented with two f loc functions: • f sum loc (D,W ) = α w ≺W scor e (w l oc, D ) loc • f count loc (D,W ) = α count {w ≺W } loc where for score (w loc, D ) we use the value score (w ), the score of D with respect to w (Section 3.3). Thus, each f loc "boosts" the relevance scores of all loc occurrences by a certain exponential formula, whose relative importance is controlled by α loc . Further, the relevance of a dataset increases if different query keywords appear in different header locations, that is, some in HR (header rows) and some in HC (header columns). In such cases, the data cells at the intersection of the respective rows and columns may provide very precise answers to the query, as illustrated in Figure 1: here, "France" is present in HC while "youth" and "17" appear in HR. To reflect this, we introduce another function f H (D,W ) computed on the scores of all unique located occurrences from row or column headers; we also experimented with the two variants, f sum H and f count H introduced above. Putting it all together, we compute the content-and location-aware relevance score of a dataset for W as: д 2 (D,W ) = д 1 (D,W ) + Σ loc ∈ {T,HR,HC,C} f loc (D,W ) + f H (D,W ) Finally, we also experimented with another function д * (D,W ) defined as: д * 2 (D,W ) =      д 2 (D,W ), if f T (D,W ) > 0 0, otherwise д * 2 discards datasets having no relevant keyword in the title. This is due to the observation that statistic dataset titles are typically carefully chosen to describe the dataset; if no query keyword can be connected to it, the dataset is very likely not relevant. Data cell search We now consider the problem of identifying the data cell(s) (or the data rows/columns) that can give the most precise answer to the user query. Such an answer may consist of exactly one data cell. For example, for the query "unemployment rate in Paris", a very good answer would be a data cell D r,c whose closest row header cell contains "unemployment rate" and whose closest column header cell contains "Paris". Alternatively, query keywords may occur not in the closest column header cell of D r,c but in another header cell that is its ancestor in D. For instance, in Figure 1, let D r,c be the data cell at the intersection of the Aug-17 column with the France row: the word "youth" occurs in an ancestor of the Aug-17 header cell, and "youth" clearly characterizes D r,c 's content. We say the closest (row and column) header cells of D r,c and all their ancestor header cells characterize D r,c . Another valid answer to the "unemployment rate in Paris" query would be a whole data row (or a whole column) whose header contains "unemployment" and "Paris". We consider this to be less relevant than a single data cell answer, as it is less precise. We term data cell answer an answer consisting of either a data cell, or a row or column; below, we describe how we identify such answers. We identify data cells answers from a given dataset D as follows. Recall that all located occurrences in D, and in particular those of the form w HR and w HC for w ≺ W , have been pre-computed; each such occurrence corresponds either to a header row r or to a header column c. For each data cell D r,c , we define #(r , c) as the number of unique words w ≺ W occurring in the header cells characterizing D r,c . Data cells in D may be characterized by: (1) Some header cells containing HR occurrences (for some w ≺ W ), and some others containing HC occurrences; (2) Only header cells with HR occurrences (or only header cells with HC ones). Observe that if D holds both cell answers (case 1) and row-or column answers (case 2), by definition, they occur in different rows and columns. Our returned data cell answers from D are: • If there are cells in case 1, then each of them is a data cell answer from D, and we return cell(s) with highest #(r, c) values. • Only if there are no such cells but there are some relevant rows or columns (case 2), we return the one(s) with highest #(r , c) values. This is motivated by the intuition that if D has a specific, one-cell answer, it is unlikely that D also holds a less specific, yet relevant one. Concretely, we compute the #(r , c) values based on the (word, score, location, dataset) tuples we extract (Section 3.3). We rely on SPARQL 1.1 [START_REF] W3c | SPARQL Protocol and RDF Query Language[END_REF] queries on the RDF representation of our datasets (Section 2) to identify the cell or row/column answer(s) from each D. SPARQL 1.1 features property paths, akin to regular expressions; we use them to identify all the header cells characterizing a given D r,c . Note that this method yields only one element (cell, row or column) from each dataset D, or several elements if they have the exact same score. An alternative would have been to allow returning several elements from each dataset; then, one needs to decide how to collate (inter-rank) scores of different elements identified in different datasets. We consider that this alternative would increase the complexity of our approach, for unclear benefits: the user experience is often better when results from the same dataset are aggregated together, rather than considered as independent. Suggesting several data cells per dataset is then more a question of result visualization than one pertaining to the search method. EVALUATION This section describes our experimental evaluation. Section 4.1 describes the dataset and query workload we used, which was split into a development set (on which we tuned our score functions) and a test set (on which we measured the performance of our algorithms). It also specifies how we built a "gold-standard" set of answers against which our algorithms were evaluated. Section 4.2 details the choice of parameters for the score functions. Datasets and Queries We have developed our system in Python (61 classes and 4071 lines). Crawling the INSEE Web site, we extracted information out of 19,984 HTML tables and 91,161 spreadsheets, out of which we built a Linked Open Data corpus of 945 millions RDF triples. We collected all the articles published online by the fact-checking team "Les Décodeurs",8 a fact-checking and data journalism team of Le Monde, France's leading national newspaper, between March 10th and August 2nd, 2014; there were 3,041 articles. From these, we selected 75 articles whose fact-checks were backed by INSEE data; these all contain links to https://www.insee.fr. By reading these articles and visiting their referenced INSEE dataset, we identified a set of 55 natural language queries which the journalists could have asked a system like ours. 9We experimented with a total of 288 variants of the д 2 function: • д 1 was either д 1a , д 1b or д 1c ; • д 2 relied either on f sum loc or on f count loc ; for each of these, we tried different value combinations for the coefficients α T , α HC , α HR and α C ; • we used either the д 2 formula, or its д * 2 variant. We built a gold-standard reference to this query set as follows. We ran each query q through our dataset search algorithm for each of the 288 д 2 functions, asking for k = 20 most relevant datasets. We built the union of all the answers thus obtained for q and assessed the relevance of each dataset as either 0 (not relevant), 1 ("partially relevant" which means user could find some related information to answer their query) or 2 ("highly relevant" which means user could find the exact answer for their query); a Web front-end was built to facilitate this process. Experiments We specify our evaluation metric (Section 4.2.1), then describe how we tuned the parameters of our score function, and the results of our experiments focused on the quality of the returned results (Section 4.2.2). Last but not least, we put them into the perspective of a comparison with the baselines which existed prior to our work: IN-SEE's own search system, and plain Google search (Section 4.2.3). Figure 2: Screen shot of our search tool. In this example, the second result is a full column; clicking on "Tous les résultats" (all results) renders all the cells from that column. Evaluation Metric. We evaluated the quality of the answers of our runs and of the baseline systems by their mean average precision (MAP 10 ), widely used for evaluating ranked lists of results. MAP is traditionally defined based on a binary relevance judgment (relevant or irrelevant in our case). We experimented with the two possibilities: • MAP h is the mean average precision where only highly relevant datasets are considered as relevant • MAP p is the mean average precision where both partially and highly relevant datasets are considered relevant. We also experimented with some modified variants that take into account the sum of matching keywords: 1). • We computed the MAP scores obtained on the full development set for all 288 combinations of parameters, and plotted them from the best to the worst (Figure 3; due to the way we plot the data, two MAP h and MAP p values shown on the same vertical line may not correspond to the same score function). The figure shows that the best-performing 15 combinations leads to scores higher than 0.80, indicating that any of these could be used with pretty good results. These two results tend to show that we can consider our results as stable, despite the relatively small size of the query set. Running time. Processing and indexing the words (close to those) appearing in the datasets took approximately three hours. We ran our experiments on a machine with 126GB RAM and 40 CPUs Intel(R) Xeon(R) E5-2640 v4 @ 2.40GHz. The average query evaluation time over the 55 queries we identified is 0.218 seconds. Comparison against Baselines. To put our results into perspective, we also computed the MAP scores of our test query set on the two baselines available prior to our work: INSEE's own dataset search system 11 , and Google search instructed to look only within the INSEE web site. Similarly to the evaluation process of our system, for each query we selected the first 20 elements returned by these systems and manually evaluated each dataset's relevance to the given query. Table 2 depicts the MAP results thus obtained, compared against those of our system. Google's MAP is very close to ours; while our work is obviously not placed as a rival of Google in general, we view this as validating the quality of our results with (much!) smaller computational efforts. Further, our work follows a white-box approach, whereas it is well known that the top results returned by Google are increasingly due to other factors beyond the original PageRank [START_REF] Brin | The Anatomy of a Large-Scale Hypertextual Web Search Engine[END_REF] information, and may vary in time and/or with result personalization, Google's own A/B testing etc. We end this comparison with two remarks. (i) Our evaluation was made on INSEE data alone due to the institute's extensive database on which fact-checking articles were written, from which we derived our test queries. However, as stated before, our approach could be easily adapted to other statistic Web sites, as we only need the ability to crawl the tables from the Web site. As is the case for INSEE, this method may be more robust than using the category-driven navigation or the search feature built in the Web site publishing the statistic information. (ii) Our system, based on a fine-granularity modeling of the data from statistic tables, is the only one capable of returning cell-level answers (Section 3.5). We show such answers to the users together with the header cells characterizing them, so that users can immediately appreciate their accuracy (as in Figure 2). RELATED WORK AND PERSPECTIVES In this work, we focused on how to improving the usability statistic tables (HTML tables or spreadsheets) as reference data sources against which claims can be fact-checked. Other works focused on building textual reference data source from general claims [START_REF] Bar-Haim | Stance Classification of Context-Dependent Claims[END_REF][START_REF] Levy | Context Dependent Claim Detection[END_REF], congressional debates [START_REF] Thomas | Get out the vote: Determining support or opposition from Congressional floor-debate transcripts[END_REF] or tweets [START_REF] Rajadesingan | Identifying Users with Opposing Opinions in Twitter Debates[END_REF]. Some works focused on exploiting the data in HTML and spreadsheet tables found on the Web. Tschirschnitz et al. [START_REF] Tschirschnitz | Detecting Inclusion Dependencies on Very Many Tables[END_REF] focused on detecting the semantic relations that hold between millions of Web tables, for instance detecting so-called inclusion dependencies (when the values of a column in one table are included in the values of a column in another table). Closest to our work, M. Kohlhase et 11 Available at https://insee.fr al. [START_REF] Kohlhase | XLSearch: A Search Engine for Spreadsheets[END_REF] built a search engine for finding and accessing spreadsheets by their formulae. This is less of an issue for the tables we focus on, as they contain plain numbers and not formulas. Google's Fusion Tables work [START_REF] Gonzalez | Google Fusion Tables: Data Management, Integration, and Collaboration in the Cloud[END_REF] focuses on storing, querying and visualizing tabular data, however, it does not tackle keyword search with a tabular semantics as we do. Google has also issued Google Tables as a working product 12 . In March 2018, we tried to use it for some sample queries we addressed in this paper, but the results we obtained were of lower quality (some were irrelevant). We believe this may be due to Google's focus on data available on the Web, whereas we focus on very high-quality data curated by INSEE experts, but which needed our work to be easily searchable. Currently, our software is not capable of aggregating information, e.g., if one asks for unemployed people from all departements within a region, we are not capable of summing these numbers up into the number corresponding to the region. This could be addressed in future work which could focus on applying OLAP-style operations of drill-down or roll-up to further process the information we extract from the INSEE datasets. Figure 1 : 1 Figure 1: Sample dataset on French youth unemployment. RDF data extracted from the INSEE statistics INSEE publishes a set D of datasets D 1 , D 2 , . . . etc. where each dataset D i consists of (w, score (w ), location(w, D), D) where location(w, D) ∈ {T, HR, HC, C} indicates where w appears in D: T denotes the title, HR denotes a row header cell, HC denotes a column header cell, and C denotes an occurrence in a comment. 4. 2 . 2 22 Parameter Estimation and Results. We experimented with the following flavors of the д 1 function :• д 1b (D,W ) = 10 w ≺W scor e (w ) • д 1d (D,W ) = 10 count {w ≺W } • д 1f (D,W ) = w ≺W10 scor e (w ) • д 1c (D,W ) = w ≺W score (w ) + д 1b (D,W ) • д 1e (D,W ) = w ≺W score (w ) + д 1d (D,W ) • д 1д (D,W ) = w ≺W score (w ) + д 1f (D,W )A randomly selected development set of 29 queries has been used to select the best values for the 7 parameters of our system : α T , α C , α HR , α HC , α H , as well as the different versions of д 1 and д 2 . For this purpose, we ran a grid search with different values of these parameters, selected among {3, 5, 7, 8, 10}, on the development query set, and applied the combination obtaining the best MAP results on the test set (composed of the remaining 26 queries).We found that a same score function has lead to the best MAP h and the best MAP p on the development query set. In terms of the notations introduced in Section 3.4, this best-performing score function is obtained by:• Using д 1,c ; • Using f sumand the coefficient values α T = 10, α C = 3, α HR = 5, α HC = 5 and α H = 7; • Using the д * 2 variant, which discards datasets lacking matches in the title. On the test set, this function has lead to MAP p = 0.76 and MAP h = 0.70 .10 https://en.wikipedia.org/wiki/Information_retrieval#Mean_average_precision Figure 3 : 3 Figure 3: MAP results on the development set for 288 variants of the score function. . . etc. where each dataset D i consists of a table (HTML table or a spreadsheet file), a title D i .t and optionally an outline D i .o. The title is a short nominal phrase stating what D i contains, e.g., "Seasonally adjusted youth (under 25s) unemployment" in Figure Table 1 : 1 Results on the first and second development set.Given that our test query set was relatively small, we performed two more experiments aiming at testing the robustness of the parameter selection on the development set:• We used a randomly selected subset of 17 queries among the 29 development queries, and used it as a new development set. The best score function for this new development set was the same; further, the MAP results on the two development sets are very similar (see Table Dev. set 29 queries Dev. set 17 queries MAP p 0.82 0.83 MAP h 0.78 0.80 Table 2 : 2 Comparing our system against baselines. Our system INSEE search Google search MAP p 0.76 0.57 0.76 MAP h 0.70 0.46 0.69 https://gitlab.inria.fr/cedar/insee-crawler, https://gitlab.inria.fr/cedar/excel-extractor https://jena.apache.org/documentation/tdb/index.html http://ec.europa.eu/eurostat/statistics-explained/images/8/82/Extra_tables_Statis-tics_explained-30-11-2017.xlsx This assumption is backed by an overwhelming majority of cases given the nature of statistic data. We did encounter some counterexamples, e.g. http://ec.europa.eu/eurostat/cache/metadata/Annexes/mips_sa_esms_an1.xls. However, these are very few and thus we do not take them into account in our approach. https://github.com/boudinfl/kea http://www.geonames.org/ http://www.lemonde.fr/les-decodeurs/ This was not actually the case; our system was developed after these articles were written. https://research.google.com/tables

01745785

en

2024/03/05 22:32:07

2018

https://theses.hal.science/tel-01745785/file/76250_CASTAN_2018_archivage.pdf

Alain ) Clémence Marie Linh Antoine Feng Adrien Khaled Arnaud, Guillaume, Yiting, Philippe Fatima Lucile Jean-No Merci Léonard, Gader Merci À Hocine Cora Jubba Lorenzo Alexandre Henry Matthieu Carolina Juan Ouafi Viviane Riccardo Gilles Au Picm Merci À Loïc Leandro Fatima Chiara Anna Sanghyuk D'avoir, Entre Autre Yvan, Jérôme, Léo, Laurence, Cyril À Nada Frédérics Jean-Luc Bérengère Anne Léa Sacha Caro Arthur Je Remercie Chaudement Lise Matchilde, Leslie... Mais je remercie Adrien Jean Mathieu, Camille Si- Mon Moly Alex Colas Ce travail de thèse a été effectué au sein d'une collaboration entre trois laboratoires: le Laboratoire de Chimie Inorganique (LCI) à l'Université Paris-Sud, le Laboratoire d'Etude des Microsctructures (LEM) à l'ONERA, et le Laboratoire de Physique des Interfaces et des Couches Minces (LPICM) à l'Ecole Polytechnique. Ainsi, je souhaite tout d'abord remercier les directeurs de ces trois laboratoires: Talal Mallah, Alphonse Finel, et Pere Roca i Cabarrocas, de m'avoir permis d'y travailler pendant ces trois années. Je tiens à remercier les membres de mon jury de thèse, pour avoir accepté de juger ce travail avec bienveillance. Mes deux rapporteurs, Sofie Cambré et Shigeo Maruyama, ainsi que Vincent Jourdain, examinateur de ce travail, et Pierre Mialane, qui a présidé mon jury. Ce travail a été encadré par Vincent Huc et Annick Loiseau. Je vous remercie tous les deux pour m'avoir permis d'explorer ce sujet avec beaucoup de liberté, de présenter mes travaux, et de participer à des formations, sans aucune restriction. Annick, merci pour ta présence pendant la rédaction de ce manuscrit, nos longues conversations auront largement contribué à le rendre plus intéressant et m'ont beaucoup appris! Merci à vous deux pour votre soutien pendant les dernières répétitions de ma soutenance! Le travail présenté dans ce manuscrit étant le fruit d'une collaboration entre plusieurs équipes de recherche, j'aimerais aussi remercier les personnes qui ont contribué, de près ou de loin, à l'encadrement de cette thèse. Merci à Costel, d'avoir construit des manips qui peuvent tout faire, et pour ta bienveillance. Un grand merci à Laure, dont je regrette de n'avoir sollicité l'aide que très tardivement. Ton implication m'a permis d'explorer d'autres pistes et de ne pas perdre le moral quand les ABP me rendaient la vie dure! Je souhaite aussi exprimer toute ma gratitude envers les capitaines de la team "solubilité", Hakim et Christophe. Merci pour toutes les conversations scientifiques enrichissantes qui ont pu donner un peu de sens à certains de mes résultats! Je souhaite aussi exprimer toute ma reconnaissance envers Maoshuai He, qui m'a beaucoup appris sur la littérature, et la croissance en général. Je n'aurais pas pu présenter tous ces résultats sans l'aide expérimentale précieuse de nombreuses personnes. Je souhaite d'abord remercier Frédéric Fossard, qui m'a appris à me servir d'un TEM, à tenter de dompter (un peu) le Zeiss, et pour les nombreuses manips qu'il a effectuées sur le Titan de Centrale, ou l'ARM à Paris 7. Merci aussi pour ton aide dans la préparation de ma soutenance! Merci à Amandine d'avoir développé une super méthode de transfert qui aura débloqué beaucoup de choses! Merci à Ahmed pour les tubes triés, ainsi qu'à Jean Sébastien-Lauret et ses étudiants Lucile et Géraud pour l'aide sur ces manips. Pour les diverses caractérisations présentées dans ce manuscrit: un grand merci à François Brisset pour l'EDX, à Ileana pour de magnifiques images TEM au PICM. Merci à Diana Dragoe pour l'XPS, pour sa disponibilité pour les manips comme pour les discussions sur l'interprétation des résultats. Je souhaite enfin remercier chaleureusement Sandra Mazerat, qui a effectué toutes les images AFM qui sont présentées ici. Merci pour ta patience, et d'avoir contribué à calmer ma détresse en passant des échantillons quand tu n'en avais pas vraiment le temps. Je tiens à remercier tout particulièrement ma camarade de thèse, Salomé. Notre entraide pendant ces trois années m'a été indispensable, que ce soit pour les manips, les articles, les questions existentielles sur notre sujet partagé, et le soutien moral pendant les temps sombres des échantillons contaminés. Merci aussi pour tout le reste, les conférences-voyage, tout ce fromage, la bonne ambiance généralisée tout simplement! Travailler dans trois laboratoires permet de rencontrer beaucoup de gens! Ainsi j'aimerais remercier les membres du LCI, du LEM et du PICM. Au LCI, merci à tous les étudiants croisés, et notamment Virgile, Irene, Charlotte, Benjamin (ou Introduction Materials of low dimensions have seen a surge of research interest during the past decades, due to the curiosity raised by their size-dependent properties. The allotropic family of carbon which was only considered to be composed of diamond, graphite, and amorphous carbon for a very long time soon welcomed new members. From the 1980's to the early 2000's, three additional forms of carbon were discovered: fullerenes, carbon nanotubes, and graphene, sparking scientific enthusiasm around the world. It was now possible to study low dimension carbon allotropes (2D with graphene, 1D with CNTs, and 0D with fullerenes). Single-walled carbon nanotubes (SWCNTs) were widely studied right after their first observation in 1993. Their outstanding optical and electronic properties were shown to strongly depend on their atomic structure. The fact that SWCNTs can either be semiconducting or metallic, depending solely on their diameter and chiral angle, which dictates the angle at which a graphene sheet is hypothetically rolled to form the nanotube, is a telling example. The ambition to exploit the structurally-dependent properties of SWCNTs made it indispensable to have access to single-structure samples, or at least samples with narrow diameter and chiral angle distributions. The usual synthesis routes of SWCNTs (arc discharge, laser ablation, chemical vapor deposition(CVD)) usually lead to random chirality samples. Therefore, efforts for either structurally selective growth techniques, or effective sorting methods, were of crucial importance. The work presented in this thesis focuses on the first option. Attaining the goal of structurally selective SWCNT growth requires a fine understanding of the mechanisms at play during growth. The complexity of these mechanisms and the difficulty to observe them at the atomic scale make this understanding a true scientific challenge. Over the past twenty years, theoretical works together with experimental trial and error have provided pieces of answers to a few of the many questions surrounding the SWCNT growth mechanisms. INTRODUCTION Experimentally, CVD became the most used synthesis technique, since it allows the fine tuning of a wide range of parameters (temperature, pressure, carbon source, ambient composition, growth time. . . ). Though a large number of factors have an influence of SWNCT growth, and its potential selectivity, the crucial importance of the catalyst nanoparticle was underlined by experimental studies, as well as theoretical calculations. For instance, most advanced atomistic simulations of the nucleation and growth have shown that the carbon content in the catalyst nanoparticle can drive the adhesion of the SWCNT wall to its surface, as well as the growth mode. Simultaneously, experimentally, the use of bimetallic catalyst systems has been a widespread approach for selective growth. While some studies put forward the argument of a synergistic effect between the two used metals, some other results are claimed to rely on the formation of highly refractory alloys, their solid and crystalline state being assumed to be able to drive a selective growth. In a majority of cases, the phenomena responsible for the observed selectivity with regard to the bimetallic catalyst system are not very well understood. The idea that the carbon content in the nanoparticle -driven in certain conditions by its carbon solubility limit -can have an influence on the growth may be linked to the various experimental results showing either selective growth using bimetallic nanoparticles, or variations in the grown SWCNT populations depending on the nature of the catalyst. INTRODUCTION 3 preparation of alloyed bimetallic catalyst nanoparticles with various bimetallic combinations in similar experimental conditions. The bimetallic particles we were primarily interested in, are nanoalloys with tunable carbon solubility. To reach this objective, we have chosen to work with chemically synthesized catalyst precursors. These precursors are synthesized as stable nanoparticle dispersions, and can be deposited on a substrate and subsequently reduced into metallic nanoparticles. The idea behind this strategy is twofold: first, the precursor nanoparticles are directly synthesized with the metal atoms mixed within the structure, potentially facilitating the formation of alloyed nanoparticles; second, the precursor type is chosen because of its versatility: using the same methodology, precursors with a wide variety of bimetallic combinations could be obtained and used for SWCNT growth in the same conditions. This could enable the study of SWCNT growth with nanoparticles with tuned carbon solubilities in identical conditions, allowing a study of the phenomena at play. Another crucial point in SWCNT growth study is their characterization. SWC-NTs are often observed indirectly, relying on resonance phenomena (Raman spectroscopy), or optical transitions (optical absorption and photoluminescence excitation spectroscopy). Although many research groups have focused on the various characterization techniques to observe SWCNTs and evaluate the selectivity of a growth, there are no reliable standardized characterization methodologies. Therefore, selectivity assessment is usually performed using the most practical, available, or less time consuming technique, with little concern for accuracy. Another aspect of the present work is therefore to try to evaluate the methods used for determining the diameter distribution of a SWCNT sample, using two different characterization techniques. This manuscript will be divided in five chapters. The first chapter is an overview of the literature on SWCNTs, their synthesis, and characterization. The focus will be set on selective growth of SWCNTs using CVD, and selectivity evaluation. The second chapter will present the different chemically synthesized catalyst precursors that were used in the present work, and how they will be integrated in the SWCNT growth process. The two characterization techniques used for selectivity evaluation (transmission electron microscopy (TEM), and Raman spectroscopy) will then be presented in detail. The results of the present work will be articulated in three chapters. In the first chapter (Chapter 3), a study employing Prussian blue (PB), and Prussian blue analog (PBA) nanoparticles as catalyst precursors for SWCNT growth will be discussed. The INTRODUCTION second (Chapter 4) will consist of a comparative study of TEM and Raman spectroscopy for the determination of the diameter distribution of a SWCNT growth sample. Finally, preliminary results on two other chemically synthesized catalyst precursors will be given in Chapter 5. Chapter 1 Introduction and state of the art Since their discovery, the growth of SWCNTs has been a tremendously active research field. The ultimate goal being the structural control of SWCNTs during growth, so as to exploit their outstanding physical properties to their full potential. Over the years, this task has proven to be a huge challenge, and has underlined that the understanding of the complex mechanisms behind SWCNT growth were of crucial importance. The goal of selective SWCNT growth has also put forward the importance of accurate, reliable, and standardized characterization techniques for selectivity assessment. This chapter is divided into three main sections. The first is a general introduction to SWCNTs, where their atomic structure and physical properties will be discussed, as well as the various ways in which they can be synthesized. The second section will focus on the vast subject of selective SWCNT growth by CVD. The current knowledge on the effects of growth conditions, and the influence of the catalyst nanoparticles on the resulting SWCNT sample will be first presented. Then, we will present the various characterization techniques that can be used for the assessment of growth selectivity, and their use in the literature will be discussed. In the third and final section, a few theories regarding SWCNT growth mechanisms, stemming either from theoretical or experimental works (and both) will be introduced, along with the resulting potential strategies for controlled growth. Finally, we will present the general objectives of this thesis' work. An introduction to SWCNTs Historically, the first known forms of carbon were graphite, diamond, and amorphous carbon. The existence of these carbon based materials was mentioned by Le Chatelier in 1908 [START_REF] Colas | Leçons sur le carbone[END_REF]. Other forms of carbon were discovered much later in the XX th century, starting with the discovery of fullerenes in 1985 [START_REF] Kroto | C60: Buckminsterfullerene[END_REF], and of carbon nanotubes in the early 1990's. Though tubular carbon structures had been observed as early as 1953 [START_REF] Davis | An unusual form of carbon[END_REF], the discovery of CNTs is attributed assembled in bundles, in order to minimize their energy through Van der Waals interactions [START_REF] Thess | Crystalline ropes of metallic carbon nanotubes[END_REF], or individualized. MWCNTs are concentric SWCNTs with an inter-wall distance of 3.4 Å, which corresponds to the distance between two graphene sheets, interacting via Van der Waals forces. Electronic and optical properties Because of their structural similarities, the electronic structure of SWCNTs can easily be derived from that of graphene. Before going into the electronic structure of SWCNTs, we should therefore give some details on the structure of graphene. As explained above, the unit cell of the honeycomb lattice of graphene is defined in the direct space by the ( -→ a 1 , -→ a 2 ) basis. In the reciprocal space, the lattice is defined by ( - → b 1 , - → b 2 ) , expressed in an orthogonal base as: - → b 1 = ( 2π √ 3a , 2π a ), - → b 2 = ( 2π √ 3a , - 2π a ) (1.6) The first 2D Brillouin zone is a hexagon, comprising three notable high symmetry points: Γ, K, and M, which are respectively located at the center of the hexagon, the corners, and the centers of the edges (see Figure 1.4.b). The band structure of graphene was first calculated using the tight-binding (TB) approach in 1947 [START_REF] Wallace | The band theory of graphite[END_REF]. In the graphene band structure, the π and π * bands join at the six K points at a zero energy. Close to the K points, the dispersion relation is linear, forming Dirac cones, as shown in Figure 1.4.a. Calculations of the density of sates (DOS) for graphene, which corresponds to the number of allowed states in a given dE energy interval, show that it reaches zero at the K points. Moreover, the Fermi energy is exactly at this energy since the π subband contains two electrons (taking the spin into account). This makes graphene a so-called zero-gap semiconductor. When forming a SWCNT from a graphene sheet, we essentially add periodic boundary conditions in the circumferential direction of the nanotube, i.e. the -→ C h direction: - → k • -→ C h = p2π, p ∈ Z (1.7) This leads to the quantization of the wave vectors in this direction. Along the tube axis, a continuum of -→ k are allowed (for an infinite tube), but only discrete values are allowed in the direction perpendicular to it. This defines lines of allowed wave vectors "cutting" the 2D graphene Brillouin zone. The resulting nanotube band structure is the intersection between the planes defined by the Chapter 1. Introduction and state of the art Further calculations and experimental studies were therefore conducted in order to obtain a more accurate Kataura plot, which is crucial for SWCNT sample characterization. This will be discussed in Chapter 2. Synthesis methods After the publication of Sumio Iijima's work in 1991, various methods for the synthesis of carbon nanotubes were developed. Those methods are divided into two categories, depending on the temperature range at which they operate: high temperature, and medium temperature. There are two CNT growth methods at high temperature (laser ablation, and arc discharge or electric arc), and the medium temperature methods consist of variations of the catalytic chemical vapor deposition (CCVD, or CVD for short) method. High temperature methods There are two high temperature routes, which both rely on the sublimation of graphite under an inert atmosphere at temperatures higher than 3000-4000 • C (close to the sublimation temperature of graphite), and the condensation of the obtained evaporated carbon under a high temperature gradient. The difference between those synthesis methods, known as arc discharge and laser ablation, lies in the way the graphite is sublimated. The arc discharge method is directly derived from the method developed by Krätschmer and Huffman in 1990 [START_REF] Krätschmer | Solid c60: A new form of carbon[END_REF] for the mass production of fullerenes. In this process, a voltage difference is applied between two graphite electrodes placed in a chamber under argon or helium partial pressure (around 600 mbar). The electrodes, usually cooled with water, are then progressively brought closer to each other, until the creation of an electric arc, increasing the temperature to about 6000 • C. The graphite from the anode then sublimates, forming a plasma, and the carbon atoms move towards colder regions in the chamber. A nanotube deposit is formed on the cathode, as a result of this rapid condensation within a high temperature gradient. When using pure graphite electrodes, the product of this synthesis contains various carbon based materials, including amorphous carbon, fullerenes, and MWC-NTs. SWCNTs were obtained with this technique by co-evaporating graphite and metals [START_REF] Iijima | Single-shell carbon nanotubes of 1-nm diameter[END_REF], [START_REF] Bethune | Cobalt-catalysed growth of carbon nanotubes with single-atomic-layer walls[END_REF](often Fe, Ni, or Co). The yield in SWCNTs was still very low, until Journet et al. achieved gram-scale production of SWCNTs by adding small amounts of yttrium in the Ni catalyst [START_REF] Journet | Large-scale production of single-walled carbon nanotubes by the electric-arc technique[END_REF]. In the case of the laser ablation technique, the graphite target is sublimated using a focused laser beam (continuous [START_REF] Maser | Production of high-density singlewalled nanotube material by a simple laser-ablation method[END_REF] or pulsed [START_REF] Guo | Self-assembly of tubular fullerenes[END_REF]). The graphite target is placed in a quartz tube, that is heated to 1200 • C under an argon or helium flow. The vaporized carbon moves through the furnace with the gas flow, the nanotubes form in the gas phase, and condense on a water-cooled copper collector. As for the electric arc technique, SWCNTs are obtained when including metals in the graphite target. For both of these techniques, in the case of SWCNT growth, the product only contains a portion, as high as it may be, of SWCNTs usually assembled in bundles at the surface of the catalyst particles [START_REF] Gavillet | Root-growth mechanism for single-wall carbon nanotubes[END_REF]. Consequently, the SWCNT diameter is not correlated to the catalyst nanoparticle size. Other forms of carbon are found, as well as metal catalyst nanoparticles. In order to rid the SWCNTs of these impurities, purification steps are needed. Moreover, the lack of control over the many parameters that have an influence on SWCNT growth has pushed researchers to focus on other techniques for selective growth of SWCNTs. Medium temperature methods: CVD In CVD methods, nanotubes are formed by the catalytical decomposition of a carbon-containing gas at the surface of a metal catalyst nanoparticle, in a furnace at temperatures below 1200 • C. CVD had been used for a long time to grow carbon nanofibers [START_REF] Rodriguez | A review of catalytically grown carbon nanofibers[END_REF], and was then adapted to grow MWCNTs [START_REF] José-Yacamán | Catalytic growth of carbon microtubules with fullerene structure[END_REF] and SWCNTs [START_REF] Dai | Single-wall nanotubes produced by metal-catalyzed disproportionation of carbon monoxide[END_REF]. CVD can either be operated with catalyst nanoparticles formed in situ in the gaseous phase, or with supported catalysts. The general principle of CVD is to heat a catalytic material inside a furnace, and flowing a carboncontaining gas through the furnace over a certain time period. The carbon precursor molecules decompose catalytically at the surface of the catalysts, and the carbon is dissolved, and diffuses through the catalyst nanoparticle. A solid sp 2 tubular carbon structure is formed when the carbon content in the particle reaches a saturation point and precipitates at the surface of the nanoparticle. Carbon precursors can be hydrocarbons C x H y (methane (CH 4 ) being the most widely used), or oxygen-containing molecules, such as carbon monoxide (CO), or ethanol. In the first case, the growth is conducted in reductive conditions (dihydrogren (H 2 ) is a product of C x H y decomposition), and in the second, the growth takes place in oxidative conditions. Here, contrary to the case of the which are finally activated (formation of effective catalyst), usually by a reduction step, or spontaneously during growth by introduction of the carbon precursor. Various methods can be used for the first step of this process, the most widely used are impregnation [START_REF] Hafner | Catalytic growth of single-wall carbon nanotubes from metal particles[END_REF]- [START_REF] Bachilo | Narrow (n, m)-distribution of single-walled carbon nanotubes grown using a solid supported catalyst[END_REF], precipitation [START_REF] Loebick | Effect of chromium addition to the co-mcm-41 catalyst in the synthesis of single wall carbon nanotubes[END_REF], [START_REF] Loebick | Effect of manganese addition to the co-mcm-41 catalyst in the selective synthesis of single wall carbon nanotubes[END_REF], and atomic layer deposition (ALD) [START_REF] He | Growth mechanism of single-walled carbon nanotubes on iron-copper catalyst and chirality studies by electron diffraction[END_REF], [START_REF] He | Selective growth of swnts on partially reduced monometallic cobalt catalyst[END_REF]. Impregnation and precipitation are similar methods: a metal complex is put in contact with a porous, or mesoporous support; the resulting solid can be calcinated, then reduced. Co-impregnation, and co-precipitation are used for the formation of bimetallic catalysts. ALD is a thin film deposition technique that relies on the adsorption and subsequent thermolysis of gaseous metallic compounds. In the case of flat substrates, many different methods of catalyst formation can be used. They can be directly deposited on the substrate before being introduced in the furnace [START_REF] Cheung | Diameter-controlled synthesis of carbon nanotubes[END_REF], or formed in situ prior to or at the same time as the introduction of the carbon precursor. The in situ formation on CVD catalyst nanoparticles can be done in many different ways: dewetting of a continuous thin film [START_REF] Takagi | Singlewalled carbon nanotube growth from highly activated metal nanoparticles[END_REF], calcination and/or reduction of pre-deposited precursors [START_REF] Yang | Chirality-specific growth of single-walled carbon nanotubes on solid alloy catalysts[END_REF], usually deposited by dip-coating, spin-coating, or just drying drops of solutions containing the precursors at the surface of the substrate. Thus, in most cases, the catalyst precursor comes in the form of a thin film. There are three main variations of the classical CVD process, where the decomposition of the carbon precursor is enhanced in order to increase the yield, or reduce the growth temperature. The plasma-enhanced CVD (PECVD) process uses a plasma to decompose the carbon precursor into reactive intermediates like radicals [START_REF] Li | Preferential growth of semiconducting single-walled carbon nanotubes by a plasma enhanced cvd method[END_REF]. Hot filaments can also be used, at the gas inlets, to enhance the decomposition of the carbon-containing gas molecules (HFCVD) [START_REF] Makris | Carbon nanotubes growth by hfcvd: Effect of the process parameters and catalyst preparation[END_REF]. The photothermic effect of a focused laser spot can also be used to locally heat the substrate during growth (LCVD) [START_REF] Bondi | Laser assisted chemical vapor deposition synthesis of carbon nanotubes and their characterization[END_REF]. In the past three decades, many research groups have worked on optimizing CVD for SWCNT growth with various goals (yield, defect-free SWCNTs, length, absence of byproducts such as amorphous carbon and carbon onions...) and some success. There have been some groundbreaking steps in SWCNT synthesis research. The "supergrowth" (SG) process, for example, that uses water stimulation of catalytic activity to obtain dense vertically aligned SWCNT forests in short growth times (2.5 mm high forest in 10 minutes) [START_REF] Hata | Water-assisted highly efficient synthesis of impurity-free single-walled carbon nanotubes[END_REF], and that is now industrially implemented for mass production of SWCNTs, was a major Chapter 1. Introduction and state of the art advancement. The presence of water decreases the formation of amorphous carbon at the surface of the catalyst nanoparticles, preventing their deactivation. This process produces SWNCTs with random chiralities over a wide range of diameters going from 1 to 6 nm. The development of the HiPco process [START_REF] Nikolaev | Gas-phase catalytic growth of singlewalled carbon nanotubes from carbon monoxide[END_REF], which uses a flowing catalyst CVD setup at high pressure, was also a major step for commercialization of SWCNTs. The use of CO as a carbon source, instead of a hydrocarbon, allowed to obtain sub-nanometer diameter SWCNTs. Lastly, the sale of chirality-enriched SWCNT samples has been made possible after the development of the CoMoCAT process in the early 2000's [START_REF] Resasco | A scalable process for production of single-walled carbon nanotubes (swnts) by catalytic disproportionation of co on a solid catalyst[END_REF], where a smart engineering of the catalyst coupled with growth at low temperatures is very probably the key to this claimed selectivity. The catalyst consists of a mix of molybdenum and cobalt in small proportions, during growth, the formation of a molybdenum carbide leads to the segregation of very small Co nanoparticles with a well-controlled size distribution. Now that yield and the presence of defects is not a concern for CVD growth, the ultimate goal for researchers has naturally shifted towards structural control during growth. When it comes to controlling the structure of the SWC-NTs, interesting results have been published, and though certain groups claim to be very close to the goal of perfect single-chirality growth, a lot of effort and progress remain to be made. Selective growth of SWCNTs by CVD SWCNT applications and related challenges This section's aim is not to go into details about all possible applications of SWCNTs, but rather to show the wideness of potentially affected fields, and the challenges they bring to the field of SWCNT growth. We can refer the reader to book chapters, or reviews that go into great detail on this subject [START_REF] Endo | Potential applications of carbon nanotubes[END_REF]- [START_REF] Baughman | Carbon nanotubesthe route toward applications[END_REF]. With their outstanding electronic properties competing with currently used materials, both metallic and semiconductor SWCNTs are great candidates for integration in electronic devices. Metallic SWCNTs have been used to produce highly conductive flexible and transparent films [START_REF] Mustonen | Uncovering the ultimate performance of single-walled carbon nanotube films as transparent conductors[END_REF], [START_REF] Kaskela | Highly individual swcnts for high performance thin film electronics[END_REF], and semiconducting SWCNTs for the fabrication of high performance field effect transistors (FETs) [START_REF] Avouris | Carbon-based electronics[END_REF]- [START_REF] Qiu | Scaling carbon nanotube complementary transistors to 5-nm gate lengths[END_REF]. Coupled with their optical properties, SWCNTs have been successfully integrated in optoelectronic devices such as solar cells [START_REF] Cui | Air-stable highefficiency solar cells with dry-transferred single-walled carbon nanotube films[END_REF], and light-emitting diodes as the emitting material [START_REF] Wang | High-performance carbon nanotube lightemitting diodes with asymmetric contacts[END_REF]. The fluorescence [START_REF] Cherukuri | Mammalian pharmacokinetics of carbon nanotubes using intrinsic near-infrared fluorescence[END_REF]- [START_REF] Yomogida | Industrial-scale separation of high-purity single-chirality singlewall carbon nanotubes for biological imaging[END_REF], or the strong Raman scattering [START_REF] Cottenye | Raman tags derived from dyes encapsulated inside carbon nanotubes for raman imaging of biological samples[END_REF] of SWCNTs has also been exploited in bio-imaging applications. Moreover, SWCNTs have also been widely studied for gas sensing applications [START_REF] Qi | Toward large arrays of multiplex functionalized carbon nanotube sensors for highly sensitive and selective molecular detection[END_REF]- [START_REF] Bondavalli | Gas fingerprinting using carbon nanotubes transistor arrays[END_REF]. In order to exploit the properties of SWCNTs within these ever-evolving potential applications, the availability of SWCNT samples with a controlled and specific chirality is an absolute necessity. In the case of nanoelectronic devices, pure semiconducting or metallic SWCNT samples can also be interesting. In any case, obtaining those SWCNT samples implies one of two technical challenges: they either require effective post-synthesis sorting methods, or direct selective growth. The following sections will review the progress and achievements in selective CVD growth of SWCNTs. Defining growth selectivity Since the ultimate goal to synthesize SWCNTs with pre-defined (n, m) is extremely challenging, selective growth can be broken down into different categories depending on the structural parameter, or property that the synthesized SWCNTs have in common. As explained in the previous section, CVD allows the synthesis of nanotubes with a very wide range of structures. Two ways to narrow down the pool of chiralities are attempting to control the diameter, and attempting to control the chiral angle of synthesized SWCNTs. Controlling just one of these two structural parameters, meaning succeeding to narrow down the diameter or chiral angle distribution of the sample, leads to either diameter-selective (or diameterspecific when the selectivity is high) growth, or chiral angle-selective growth. When both are controlled, the growth is, of course, considered chirality-or (n, m)-selective, or specific. Potential applications of SWCNTs often exploit their electronic properties. For these applications, it is very interesting to have samples of either semiconducting or metallic SWCNTs. A lot of recent research focuses on trying to obtain semiconducting-, or metallic-enriched SWCNT samples, by optimizing the semiconducting to metallic (SC/M) ratio. Influence of CVD parameters on selectivity Since the first SWCNT growth by CVD, research groups have conducted countless studies on the influence of all possibly controllable parameters on the resulting SWCNTs, and growth selectivity. In CVD, the original widely spread hypothesis concerning the control of SWCNT diameter was that is was dominated by catalyst nanoparticle size. The control of chirality distribution, or SC/M ratio of a sample is more of an empirical search, since no hypotheses were initially set on the matter. This section gives a non-exhaustive list of the CVD parameters that have been shown to affect selectivity. Growth temperature It is generally observed that increasing the temperature tends to lead to the growth of large-diameter SWCNTs [START_REF] He | Low temperature growth of swnts on a nickel catalyst by thermal chemical vapor deposition[END_REF]- [START_REF] Li | Diameter tuning of single-walled carbon nanotubes with reaction temperature using a co monometallic catalyst[END_REF]. The most widely used argument to explain this, is a thermally-facilitated coarsening of the catalyst nanoparticles, as the nanoparticle diameter is assumed to determine the SWCNT diameter. However, a contradicting experimental study by Yao et al. [START_REF] Yao | Temperaturemediated growth of single-walled carbon-nanotube intramolecular junctions[END_REF] showed that the diameter of an individual SWCNT could be tuned through growth temperature changes, with a decrease in diameter when the temperature was increased. Choice of carbon feedstock and feeding rate There are very contradicting results regarding the effect of carbon feeding rate on the diameter distribution of grown SWCNTs. This might also be highly dependent on the carbon source. While certain groups have claimed that reducing the carbon feeding rate led to smaller diameter nanotubes, others have shown the opposite. Saito et al. [START_REF] Saito | Selective diameter control of single-walled carbon nanotubes in the gasphase synthesis[END_REF] showed, using a floating catalyst reactor, that increasing the flow rate of ethylene (the carbon source), yields in shifting the diameter distribution towards smaller diameters. On the contrary, Geohegan et al. [START_REF] Geohegan | Flux-dependent growth kinetics and diameter selectivity in single-wall carbon nanotube arrays[END_REF] performed experiments in which increasing acetylene (C 2 H 2 ) flow rate in a pulsed-CVD apparatus led to a shift of the diameter distribution towards higher diameters, over an overall very broad range of diameters (sub-nanometer, to 6 nm). When it comes to the choice of carbon feedstock, the carbon source used for CVD growth can have a very significant impact on the resulting SWCNT population. He et al. have conducted studies on the comparison between CO and CH 4 as carbon sources in the same experimental conditions and using the same Fe catalyst where they found that CO had a tendency to favor the growth of small diameter (sub-nanometer to around 1.6 nm) SWCNTs with large chiral angles, while CH 4 led to growth of SWCNTs with diameters ranging from 1.0 to 4.5 nm [START_REF] He | Chiral-selective growth of single-walled carbon nanotubes on stainless steel wires[END_REF], [START_REF] He | Diameter and chiral angle distribution dependencies on the carbon precursors in surfacegrown single-walled carbon nanotubes[END_REF]. Wang et al. have compared the use of four different gas sources (CO, ethanol, methanol, and C 2 H 2 ) for SWCNT growth in the same experimental conditions and noticed strong variations in the chirality distributions of the obtained samples [START_REF] Wang | n, m) selectivity of single-walled carbon nanotubes by different carbon precursors on co-mo catalysts[END_REF]. The choice of carbon feedstock has also proven to have an effect on the SC/M ratio of the SWCNT sample [START_REF] Parker | Increasing the semiconducting fraction in ensembles of single-walled carbon nanotubes[END_REF], [START_REF] Wang | Direct enrichment of metallic single-walled carbon nanotubes induced by the different molecular composition of monohydroxy alcohol homologues[END_REF]. Parker et al. showed SC percentages going from less than 20 % to between 50 -70 % when switching from CH 4 to isopropanol (from estimations based on FET devices) [START_REF] Parker | Increasing the semiconducting fraction in ensembles of single-walled carbon nanotubes[END_REF]. The general observation is that the percentage of semiconducting tubes is increased by an increase in oxygen content in the carbon precursor molecules. The explanation proposed for this phenomenon is an in situ selective etching of metallic SWCNT by the oxygen present in the reactor, which is a known method for selective growth [START_REF] Liu | Controlled growth of semiconducting and metallic single-wall carbon nanotubes[END_REF]. CVD ambient and selective etching The fact that metallic SWCNTs are more chemically reactive to oxidation [START_REF] Liu | Chirality-dependent reactivity of individual single-walled carbon nanotubes[END_REF], has led some research groups to try to optimize the CVD ambient for selective etching of metallic SWCNTs [START_REF] Yu | Selective removal of metallic single-walled carbon nanotubes by combined in situ and post-synthesis oxidation[END_REF]. Introducing the suitable amount of oxygen (through carbon-containing molecules such as ethanol [START_REF] Ding | Selective growth of well-aligned semiconducting singlewalled carbon nanotubes[END_REF], water [START_REF] Che | Selective synthesis and device applications of semiconducting single-walled carbon nanotubes using isopropyl alcohol as feedstock[END_REF], [START_REF] Zhou | General rules for selective growth of enriched semiconducting single walled carbon nanotubes with water vapor as in situ etchant[END_REF]), or hydrogen (etching through formation of hydrocarbons instead of carbon oxides) [START_REF] Li | High-quality, highly concentrated semiconducting single-wall carbon nanotubes for use in field effect transistors and biosensors[END_REF], can substantially increase the semiconducting SWCNT content of a given sample (up to an estimated 93 % of SC SWCNTs for [START_REF] Li | High-quality, highly concentrated semiconducting single-wall carbon nanotubes for use in field effect transistors and biosensors[END_REF], for instance). UV irradiation has also been used to successfully etch metallic SWCNTs during growth [START_REF] Hong | Direct growth of semiconducting single-walled carbon nanotube array[END_REF]. This in situ etching has also been used to grow metallic-enriched SWCNT samples. Hou et al. optimized CVD conditions in order to simultaneously grow small-diameter semiconducting SWCNTs and large-diameter metallic SWCNTs, after which they introduced hydrogen in the reactor to selectively etch the semiconducting tubes [START_REF] Hou | Preparation of metallic single-wall carbon nanotubes by selective etching[END_REF] (estimated 88 % metallic SWCNT content), leading to the direct preparation of conducting SWCNT films. This study also shows the limitations of the selective etching methods: since the etching is also diameterdependent, it can only be fully efficient if the growth is highly diameter-selective. Chapter 1. Introduction and state of the art Choice of substrate In the case of supported CVD growth, the catalyst nanoparticles are in contact, and therefore interact, with a substrate. The effect of the substrate on catalytic activity of nanoparticles has been demonstrated in the literature. For instance, the catalytic activity of gold nanoparticles for CO oxidation was demonstrated to be much lower when supported on Al 2 O 3 than on other chemically inert metal oxides [START_REF] Haruta | Gold catalysts prepared by coprecipitation for low-temperature oxidation of hydrogen and of carbon monoxide[END_REF]. The observed lattice-guided alignment of SWCNTs grown on single-crystal substrates such as quartz [START_REF] Yuan | Horizontally aligned single-walled carbon nanotube on quartz from a large variety of metal catalysts[END_REF] and sapphire [START_REF] Han | Template-free directional growth of singlewalled carbon nanotubes on a-and r-plane sapphire[END_REF] is also indicative of an influence of the substrate on SWCNT growth. The strength and extent of the anchoring of the catalyst nanoparticles onto the substrate varies, leading to different effects. Some influence of the substrate on growth selectivity has also been demonstrated. Ishigami et al. showed preferential growth of near zigzag nanotubes on A-plane sapphire, and near armchair on R-plane sapphire [START_REF] Ishigami | Crystal plane dependent growth of aligned single-walled carbon nanotubes on sapphire[END_REF], hinting to the importance of substrate choice. Influence of the catalyst on selectivity While a significant amount of studies emphasize on the control of CVD parameters and ambient for selective SWCNT growth, the choice of catalyst has also been put forward as one of the key parameters for achieving selectivity. Obviously so, the catalyst nanoparticle plays a key role in the growth mechanism, and has therefore to be strongly taken into account. Size and morphology The size and morphology of catalyst nanoparticles seem to play important roles in the selectivity of SWCNT growth. Harutyunyan et al. showed that by modifying CVD ambient, they could alter the morphology of Fe catalyst nanoparticles, with an impact on the electronic properties of the grown SWCNTs [START_REF] Harutyunyan | Preferential growth of single-walled carbon nanotubes with metallic conductivity[END_REF]. Many papers in the literature focus on the effect of the size of the catalyst particles and SWCNT growth selectivity. There is experimental evidence of the control of SWCNT diameter through catalyst size control [START_REF] Kim | Heat-driven size manipulation of fe catalytic nanoparticles for precise control of single-walled carbon nanotube diameter[END_REF]- [START_REF] Song | Synthesis of bandgapcontrolled semiconducting single-walled carbon nanotubes[END_REF]. In these studies, the proposed explanation for this was that by controlling the diameter of the catalyst nanoparticles, and assuming a matching between nanoparticle diameter and SWCNT diameter, the SWCNT diameter distribution would inevitably be controlled. However, Fiawoo et al. [START_REF] Fiawoo | Evidence of correlation between catalyst particles and the single-wall carbon nanotube diameter: A first step towards chirality control[END_REF] have shown experimental proof of the existence of two growth modes. By defining the d SW CN T /d N P aspect ratio, they were able to discriminate between the tangential mode (d SW CN T /d N P 0.75), and the perpendicular mode (d SW CN T /d N P < 0.75). This implies that controlling SWCNT diameter through nanoparticle size control can only be done in the tangential growth mode. Moreover, in situ HRTEM experiments have shown that catalyst nanoparticles tend to continuously rearrange during CVD growth, modifying the d SW CN T /d N P ratio. This indicates that controlling SWCNT diameter may be done by controlling other growth parameters. Composition The interaction between carbon and the catalyst nanoparticle is at the heart of the SWCNT growth process. Since this interaction strongly depends on the nature of the catalyst, and its composition, an influence of these factors on growth selectivity is expected. Regarding the nature of the catalyst, meaning the element used as a catalyst, many results in the literature attest to the differences between metals when it comes to selectivity. For example, Chen et al. conducted a comparative study between monometallic Co-MCM-41 (MCM-41 is a commercially available mesoporous silica substrate) and Ni-MCM-41 [START_REF] Chen | Single-wall carbon nanotube synthesis by co disproportionation on nickel-incorporated mcm-41[END_REF]. It was found that the Co catalyst led to the growth of small diameter SWC-NTs with a narrow diameter distribution, whereas the Ni catalyst, in the same conditions, did not exhibit any selectivity. This was explained by the higher nucleation rate of Ni nanoparticles under CO. This experimental proof of the effect of the nature of the catalyst metal shows that tuning catalyst nanoparticle properties may be an interesting approach to selective SWCNT growth. Using bimetallic catalysts is a good way to tune the properties of the catalyst nanoparticles, and it gives access to more possibilities. The use of bimetallic nanoparticles as catalysts for the growth of SWCNTs has been a popular option since the development of the CoMoCAT process [START_REF] Resasco | A scalable process for production of single-walled carbon nanotubes (swnts) by catalytic disproportionation of co on a solid catalyst[END_REF]. It was shown to produce SWCNT samples enriched in (6, 5) and (7, 5) chiralities (representing 38% of the SWCNT sample considering that metallic SWCNTs represented 1/3 of all nanotubes) [START_REF] Bachilo | Narrow (n, m)-distribution of single-walled carbon nanotubes grown using a solid supported catalyst[END_REF], and its selectivity was explained by a synergistic effect of Co and Mo [START_REF] Alvarez | Synergism of co and mo in the catalytic production of single-wall carbon nanotubes by decomposition of co[END_REF], sparking interest towards bimetallic catalysts. Since then, a significant amount of studies reporting selective growth from bimetallic catalysts has been published, using a wide variety of bimetallic combinations. A number of papers have reported near-armchair chiral selectivity, using, to name Chapter 1. Introduction and state of the art a few, FeCo [START_REF] Miyauchi | Fluorescence spectroscopy of single-walled carbon nanotubes synthesized from alcohol[END_REF], FeCu [START_REF] He | Predominant (6, 5) single-walled carbon nanotube growth on a copper-promoted iron catalyst[END_REF], FeRu [START_REF] Li | Selective synthesis combined with chemical separation of single-walled carbon nanotubes for chirality selection[END_REF], CoMo [START_REF] Wang | n, m) selectivity of single-walled carbon nanotubes by different carbon precursors on co-mo catalysts[END_REF], CoPt [START_REF] Liu | High temperature selective growth of single-walled carbon nanotubes with a narrow chirality distribution from a copt bimetallic catalyst[END_REF], FePt [START_REF] He | Environmental transmission electron microscopy investigations of pt-fe 2 o 3 nanoparticles for nucleating carbon nanotubes[END_REF], FeTi [START_REF] He | Fe ti o based catalyst for large-chiral-angle single-walled carbon nanotube growth[END_REF]. Some diameter control was also shown using CoRh [START_REF] Thurakitseree | Diameter controlled chemical vapor deposition synthesis of single-walled carbon nanotubes[END_REF], CoCr [START_REF] Loebick | Effect of chromium addition to the co-mcm-41 catalyst in the synthesis of single wall carbon nanotubes[END_REF], CoMn [START_REF] Loebick | Effect of manganese addition to the co-mcm-41 catalyst in the selective synthesis of single wall carbon nanotubes[END_REF], CoCu [START_REF] Cui | Synthesis of subnanometer-diameter vertically aligned single-walled carbon nanotubes with copper-anchored cobalt catalysts[END_REF] bimetallic systems. Pioneering work by Chiang et al. demonstrated the influence of the composition of a Ni x Fe 1-x alloy on the selectivity of SWCNT growth. Using a microplasma reactor, the authors were able to synthesize monometallic Fe and Ni nanoparticles, as well as bimetallic alloy nanoparticles with various compositions, with similar sizes. The changes in composition of the Ni x Fe 1-x alloys led to changes in the chirality distribution of the resulting SWCNT sample [START_REF] Chiang | Linking catalyst composition to chirality distributions of as-grown single-walled carbon nanotubes by tuning ni x fe 1-x nanoparticles[END_REF]. Using a Ni 0.27 Fe 0.73 catalyst led to a narrower (n, m) distribution, with a single dominating [START_REF] Novoselov | Electric field effect in atomically thin carbon films[END_REF][START_REF] Iijima | Helical microtubules of graphitic carbon[END_REF] population, while the Ni 0.5 Fe 0.5 catalyst led to less selective growth. Semiconductor enriched samples were obtained using a specific alloy composition [START_REF] Chiang | Nanoengineering ni x fe1-x catalysts for gas-phase, selective synthesis of semiconducting single-walled carbon nanotubes[END_REF], highlighting the crucial role of catalyst composition in growth selectivity. All of these studies attest to the significant interest directed towards catalyst composition optimization for selective SWCNT growth. However, the term "bimetallic" usually indicates that two metals were used for the preparation of the catalyst, or that the catalyst precursor contains two metals. For a long time, and aside from a few exceptions, the so-called bimetallic catalyst nanoparticles were rarely characterized, whether it is before or after SWCNT synthesis. When they are characterized, we can distinguish three types of bimetallic catalysts (see examples extracted from the literature in Figure 1.9). On rare occasions, the catalyst nanoparticles are proven to be nanoalloys (Figure 1.9.c), where the two metals involved in the catalyst synthesis form homogeneously mixed bimetallic nanoparticles [START_REF] Liu | High temperature selective growth of single-walled carbon nanotubes with a narrow chirality distribution from a copt bimetallic catalyst[END_REF], [START_REF] Chiang | Linking catalyst composition to chirality distributions of as-grown single-walled carbon nanotubes by tuning ni x fe 1-x nanoparticles[END_REF]. Here, the composition of the catalyst is thought to play an important role in the selectivity. In most cases, however, the bimetallic catalyst displays phase-segregation due to its fabrication process, and can take one of two general forms. The first one, analogous to the CoMoCAT catalyst, consists of small monometallic clusters anchored onto a metallic, or metal oxide matrix [START_REF] Loebick | Effect of chromium addition to the co-mcm-41 catalyst in the synthesis of single wall carbon nanotubes[END_REF], [START_REF] Loebick | Effect of manganese addition to the co-mcm-41 catalyst in the selective synthesis of single wall carbon nanotubes[END_REF], [START_REF] He | Predominant (6, 5) single-walled carbon nanotube growth on a copper-promoted iron catalyst[END_REF], [START_REF] He | Fe ti o based catalyst for large-chiral-angle single-walled carbon nanotube growth[END_REF] (Figure 1.9.b). In this particular case, the selectivity of the growth is owed to the size control of monometallic catalyst nanoparticles by the matrix. The second one consists of phase-segregated nanoparticles, with a "support" side made of an alloy or solid solution of the two metals, or simply one of the metals, and an effective catalyst side made of the other metal [START_REF] Cui | Synthesis of subnanometer-diameter vertically aligned single-walled carbon nanotubes with copper-anchored cobalt catalysts[END_REF], or a metal carbide [START_REF] He | Environmental transmission electron microscopy investigations of pt-fe 2 o 3 nanoparticles for nucleating carbon nanotubes[END_REF] (Figure 1.9.a). Table 1.1 -Non-exhaustive list of bimetallic catalysts used for selective SWCNT growth, and experimental details. In most cases, the given catalyst preparation refers to the first step in the preparation process (impregnation, precipitation...), which is usually followed by a calcination step, and reduction. [START_REF] He | Precise determination of the threshold diameter for a single-walled carbon nanotube to collapse[END_REF][START_REF] Bethune | Cobalt-catalysed growth of carbon nanotubes with single-atomic-layer walls[END_REF] [START_REF] Yang | Chirality-specific growth of single-walled carbon nanotubes on solid alloy catalysts[END_REF] or (16,0) [START_REF] Yang | Growing zigzag (16, 0) carbon nanotubes with structuredefined catalysts[END_REF] chirality-specific growths, depending on growth temperature (1030 • C or 1050 • C) using a cobalttungsten catalyst (Co-W). The highly refractory property of tungsten was supposed to be responsible for the high stability of the catalyst nanoparticles under growth conditions. The reason for the selectivity of the growth according to experimental observations and density functional theory (DFT) calculations (at 0 K without considering the presence of carbon during the growth process) by Y. Li et al. is based on a matching between the nanotube wall and a specific plane of the solid catalyst nanoparticle. However, there is little proof that the catalyst nanoparticles remain solid under CVD conditions. The actual structure of the catalyst nanoparticles during SWCNT growth is also still debated [START_REF] An | Chirality specific and spatially uniform synthesis of single-walled carbon nanotubes from a sputtered co-w bimetallic catalyst[END_REF]. Interestingly, the two studies cited here [START_REF] Yang | Chirality-specific growth of single-walled carbon nanotubes on solid alloy catalysts[END_REF], [START_REF] An | Chirality specific and spatially uniform synthesis of single-walled carbon nanotubes from a sputtered co-w bimetallic catalyst[END_REF] showed selectivity toward the same chirality population in two very different experimental setups, indicating that the Co-W bimetallic combination, whatever its structure during CVD growth, can be of great interest for understanding the role of the catalyst in growth selectivity. We should note here that the use of the word "epitaxy" may not be adapted to the growth of SWCNTs. Epitaxy is a term used for the growth of an atomic layer on top of a flat crystalline surface, the atomic layer forms with the same orientation and lattice as the crystalline surface. To transpose this physical concept to SWCNT growth is probably not exactly accurate. Growth selectivity characterization Since their discovery, a number of techniques have been used for SWCNT characterization. The aim of this section is not to make an exhaustive presentation of these techniques, but rather to present the ones that are the most commonly used for SWCNT growth sample characterization, and selectivity assessment. The discussion will focus more on their potential biases in selectivity assessment than technical details of the techniques themselves. Resonant Raman spectroscopy Raman spectroscopy is one the most used characterization techniques in the field of SWCNT growth. It is based on the inelastic scattering of light by matter. The technique will be presented in more detail in Chapter 2. SWCNTs show several Raman-active modes, that provide information on their structure. The most widely used SWCNT Raman mode for selectivity assessment is the radial breathing mode (RBM), which corresponds to a vibration of the carbon atoms in the direction perpendicular to the tube axis [START_REF] Dresselhaus | Raman spectroscopy of carbon nanotubes[END_REF]. The RBM frequency (ω RBM ) is inversely proportional to the SWCNT diameter, and it can therefore be extracted from the Raman spectrum, using an empirical law of general formula ω RBM = A/d t + B, where A and B are constants determined experimentally or by calculations. Since the RBM is a resonant mode, it only appears on the Raman spectrum if the energy of the laser used to Chapter 1. Introduction and state of the art probe the sample is equal to, or within a resonance window around an optical transition energy of the SWCNT. It is therefore possible, using the Kataura plot, to determine whether the probed SWCNT is metallic or semiconducting. With careful calibration of the Kataura plot, an adapted ω RBM = f (d t ) law, and fine tuning of resonant conditions by adjusting the excitation energy, it is also possible to unambiguously determine the (n, m) indices of the SWCNT. The resonant aspect of Raman spectroscopy implies that using one excitation wavelength is not sufficient for probing all SWCNT populations, leading to potential blind spots when characterizing a growth sample. The Raman crosssection is also known to be chirality dependent [START_REF] Doorn | Resonant raman excitation profiles of individually dispersed single walled carbon nanotubes in solution[END_REF], which can have a strong impact on quantitative selectivity data. Transmission electron microscopy Historically, TEM is the first technique used to identify and characterize nanotubes, as it was through TEM that they were first discovered. As Raman spectroscopy, this technique will be presented in more detail in the next chapter. The basic principle of TEM is to shine an electron beam on a thin sample, and collect the beam transmitted through the sample. TEM imaging is one of the most robust SWCNT characterization methods. The many techniques related to TEM allow the unambiguous structural characterization of SWCNT samples. A routine TEM with limited resolution can give basic information on nanotubes: diameter, number of walls, length. With aberration-corrected microscopes, it is also possible to determine the chirality of SWCNTs in atomically resolved imaging mode. With the adapted experimental techniques, it is possible to acquire an electron diffraction (ED) pattern from an isolated SWCNT, from which the chirality of the SWCNT can be extracted [START_REF] Lambin | Structural analysis by elastic scattering techniques[END_REF]. The advantage of TEM is that it allows the direct observation of all SWCNTs, regardless of their electronic or optical properties. TEM is also a very powerful tool to make joint observations of the SWCNT and its catalyst nanoparticle, whether it be post-synthesis, or in situ observation. Optical absorption spectroscopy Optical absorption spectroscopy (OAS) is a very common technique for the characterization of a wide variety of materials. The sample is exposed to a monochromatic light beam, and the intensity of the radiation at the exit of the sample is compared to the incident intensity. This is done over a range of wavelengths, to form an absorption spectrum. For SWCNTs, an absorption peak Chapter 1. Introduction and state of the art This broadening is due to the heterogeneity of the sample's chirality distribution, and to environmental effects. When the sample contains a range of chiralities, absorption occurs for all allowed transitions over a range of energies, corresponding to several unresolved (n, m) specific peaks. Bundling effects, or the interaction with other compounds also contribute to peak broadening. The absorption cross-section is also chirality-dependent [START_REF] Vialla | Chirality dependence of the absorption cross section of carbon nanotubes[END_REF], making it difficult to quantitatively determine the chirality or diameter distribution, or the SC/M ratio. Moreover, enven though absorption experiments have been conducted on isolated SWCNTs, a large amount of SWCNTs are necessary to obtain measurable absorption peaks, meaning that this characterization technique is not well adapted to all sample types. Nonetheless, a non-negligible advantage of this technique is that in principle, it enables the observation of all SWCNTs, regardless of chirality, diameter, or SC/M character. Photoluminescence excitation spectroscopy The process through which light is absorbed by matter, leading to the formation of an excited state, which is relaxed to a ground state by emission of light with a lower energy, is called photoluminescence (PL). PL in SWCNTs was first reported in 2002 [START_REF] O'connell | Band gap fluorescence from individual single-walled carbon nanotubes[END_REF]. In the specific case of SWCNTs, the PL process can be described as follows: a photon is absorbed, creating an electron-hole pair, that is annihilated by emission of a lower energy photon. The absorption occurs at an allowed optical transition energy (E ii ), corresponding to the distance between a pair of vHSs, and the emission at a lower transition energy, mostly E SC 11 . The time between absorption and emission is a few nanoseconds, implying that "fluorescence" is a more precise term define the process, but the more general term PL is mostly used [START_REF] Lefebvre | Photoluminescence: Science and applications[END_REF]. In PL excitation (PLE) spectroscopy, a map of PL emission intensity (emission energy along the x-axis) as a function of excitation energy is plotted. An intense spot on a PLE map corresponds to an intense emission, as a result of an excitation at a higher transition energy. For each (n, m) population present in the sample, PLE provides a measurement of the corresponding (E SC 11 , E SC 22 ) pair. Those (E SC 11 , E SC 22 ) spots are present in a certain energy domain on the PLE map called the "fingerprint region". The chirality distribution of the sample can be extracted from this region (see Figure 1.10). The main drawback of PLE for the assessment of growth selectivity is that since the PL phenomenon is specific to semiconducting SWCNTs, only those can be detected. Bundled nanotubes do not luminesce, making this technique suitable for isolated SWCNTs only. More extensive work on chirality-dependent PL intensity is also needed for accurate quantitative analysis of the chirality distribution [START_REF] Tsyboulski | Structure-dependent fluorescence efficiencies of individual singlewalled carbon nanotubes[END_REF]. Moreover, it has been shown that the luminescence depends strongly on the environment. Atomic force microscopy Atomic force microscopy (AFM) is often used for the measurement of SWCNT diameters supported on flat substrates. It is a common characterization technique for evaluating the topography of a sample. The principle of AFM resides in the detection of forces involved when a surface is approached and scanned by a tip [START_REF] Binnig | Atomic force microscope[END_REF]. The force field between the extremity of this tip and the atoms beneath it is measured. Since the force field depends on the distance between the tip and the surface, their measurement gives insights into the topography of the sample. When used for SWCNT characterization, AFM is used for length and diameter measurement. The length of a supported straight SWCNT is easily extracted from a topographic image of the substrate, and the diameter is determined by measuring the height of the nanotube laying flat on the substrate surface. AFM cannot give access to the chirality, or semiconducting or metallic character of the SWCNT. Contrary to TEM, the number of walls cannot be evaluated. SC/M ratio evaluation through device fabrication In a significant amount of studies, the performance of electronic devices fabricated using as-grown SWCNTs is used as a way to evaluate the SC/M ratio of a given sample. As mentioned previously, an application of interest of SWCNTs is their use in FETs, the fabrication of which relies on the obtention of high quality semiconducting SWCNTs. A FET is an electronic device composed of a semiconducting channel connected on either side to a source, and a drain electrode. A voltage is applied between the source and a third electrode, called the gate, which is separated from the channel by a dielectric material. This voltage controls the current through the channel. A way to measure the performance of a FET is to calculate the on-current to off-current ratio (I on /I of f ): a high I on /I of f (above 10 4 at minimum) means a high on-current, combined with a low leakage current, which is associated with high performance. Usual (I on /I of f ) ratios for SWCNT-based FETs are in the 10 5 -10 7 range [START_REF] Avouris | Carbon-based electronics[END_REF]. Chapter 1. Introduction and state of the art In a SWCNT-based FET, the channel material is either an array of semiconducting SWCNTs, or an individual semiconducting SWCNT. The direct integration of synthesized SWCNTs in these electronic devices can therefore be a way to indirectly measure the semiconductor content of a sample. The general idea behind this measurement is that the presence of metallic SWCNTs will have a tendency to degrade the performance of the devices, leading to low I on /I of f ratios. Use of characterization techniques for selectivity evaluation As we have established, a few characterization techniques can be used to determine whether a SWCNT growth is selective. Those techniques give access to different types of information, with different sources of bias, and there is no perfectly accurate way to evaluate the diameter distribution, chirality distribution, or SC/M ratio of a SWCNT growth sample. In order to have a better understanding of the use of these techniques for selectivity assessment by research groups, we looked closely at more than 50 papers on the CVD growth of SWCNTs, published by a dozen of research groups from the early 2000's to 2016, with a focus on selective growth. We will quickly review the methodologies for selectivity assessment affiliated with the previously presented techniques, the use of cross-characterization in general, and finally the occurrence of TEM and Raman cross-characterization in those papers. Selectivity evaluation methods related to characterization techniques As explained previously, Raman is used in selectivity assessment to determine the diameter distribution, SC/M ratio, and sometimes (n, m) distribution of samples. In most cases, Raman is simply employed as a qualitative evaluation technique to assess the evolution of diameter range [START_REF] Li | Diameter tuning of single-walled carbon nanotubes with reaction temperature using a co monometallic catalyst[END_REF] or SC/M ratio [START_REF] Ding | Selective growth of well-aligned semiconducting singlewalled carbon nanotubes[END_REF] with growth condition changes. One can easily assess a shift in diameter distribution towards higher diameters through an increase in growth temperature by observing a global shift of RBM signals towards lower frequencies, for instance [START_REF] Miyauchi | Fluorescence spectroscopy of single-walled carbon nanotubes synthesized from alcohol[END_REF], [START_REF] Wang | Low temperature growth of single-walled carbon nanotubes: Small diameters with narrow distribution[END_REF]. Raman is also used as a quantitative selectivity assessment method. Depending on the sample type, different methodologies are applicable. Here, we will distinguish two types of samples: diluted SWCNT mats on flat substrates, or "bulk" SWCNT samples, obtained from powder supported catalysts, or by floating catalyst CVD, or dispersed in solution. If the sample is supported on a flat substrate, it is necessary to acquire multiple spectra (mappings) to obtain a global view of the sample. In the case of supported catalysts in the form of powders, or unsupported catalysts (so-called "bulk" samples), the amount of SWCNTs probed under one laser spot is much higher, and one spectrum is sufficient to get an overview of the sample. In the latter case, quantitative selectivity evaluation is usually performed by looking at RBM peak intensities, relying on the hypothesis that peak intensity is directly related to the abundance of a certain species. This can only be accurate if the chirality dependence of Raman cross-sections is taken into account (see Chapter 2.3.2.3). When it comes to using Raman for SC/M ratio evaluation, various methods have been implemented over the years. Some groups relied on a comparative study of RBM intensities attributed to metallic or semiconducting SWCNTs [START_REF] Harutyunyan | Preferential growth of single-walled carbon nanotubes with metallic conductivity[END_REF], [START_REF] Chiang | Nanoengineering ni x fe1-x catalysts for gas-phase, selective synthesis of semiconducting single-walled carbon nanotubes[END_REF] (see Figure 1.11). In other cases, the SC/M ratio is calculated on global spectra of a "bulk" sample, by counting the number (n, m) indexed RBM peaks corresponding to semiconducting tubes, and those corresponding to metallic tubes, without taking into account multiple SWCNTs were actually contributing to the peak intensity [START_REF] Zoican Loebick | Selective synthesis of subnanometer diameter semiconducting single-walled carbon nanotubes[END_REF]. In the case of flat substrates, when acquiring Raman mappings on a less abundant sample, the usual method is to acquire multiple Raman spectra, and count each RBM peak as representative of one resonating SWCNT, not taking into account their relative intensities. The hypothesis that one RBM peak counts as one SWCNT cannot be systematically verified, and is used for simplification. This methodology has been widely used in the past few years for the determination of chirality distribution using four [START_REF] Yang | Growing zigzag (16, 0) carbon nanotubes with structuredefined catalysts[END_REF], [START_REF] An | Chirality specific and spatially uniform synthesis of single-walled carbon nanotubes from a sputtered co-w bimetallic catalyst[END_REF], or three excitation wavelengths [START_REF] Zhao | Chemical vapor deposition synthesis of near-zigzag single-walled carbon nanotubes with stable tube-catalyst interface[END_REF], diameter distribution [START_REF] He | Diameter and chiral angle distribution dependencies on the carbon precursors in surfacegrown single-walled carbon nanotubes[END_REF], or SC/M ratio using four [START_REF] Liu | Controlled growth of semiconducting and metallic single-wall carbon nanotubes[END_REF], or two [START_REF] Zhang | Growth of semiconducting single-wall carbon nanotubes with a narrow band-gap distribution[END_REF], [START_REF] Qin | Growth of semiconducting single-walled carbon nanotubes by using ceria as catalyst supports[END_REF] excitation wavelengths. The value(s) of the excitation wavelength(s), as well as their number, varies from one study to another. Figure 1.11 shows an example of SC/M ratio determination from a "bulk" sample (left), using relative intensities of RBM peaks, and from SWCNTs on a flat substrate (right), where each RBM peak counts as one SWCNT in the statistical SC/M ratio evaluation. TEM is mostly used as a way to attest that SWCNTs are grown, and to quickly verify the diameter range of a sample without statistical study [START_REF] Zoican Loebick | Selective synthesis of subnanometer diameter semiconducting single-walled carbon nanotubes[END_REF], [START_REF] Ciuparu | Uniformdiameter single-walled carbon nanotubes catalytically grown in cobaltincorporated mcm-41[END_REF]- [START_REF] Li | Selective synthesis of large diameter, highly conductive and high density single-walled carbon nanotubes by a thiophene-assisted chemical vapor deposition method on transparent substrates[END_REF]. Some groups determine the diameter distribution of a sample from TEM images [START_REF] Zhang | Growth of semiconducting single-wall carbon nanotubes with a narrow band-gap distribution[END_REF], [START_REF] Liu | Diameter-selective growth of single-walled carbon nanotubes with high quality by floating catalyst method[END_REF]. Even though measuring a SWCNT diameter from a TEM image is not trivial [START_REF] Fleurier | Transmission electron microscopy and uv-vis-ir spectroscopy analysis of the diameter sorting of carbon nanotubes by gradient density ultracentrifugation[END_REF], no information is given on the method used. In very of the amount of a certain SWCNT chirality [START_REF] Wang | Selective synthesis of (9, 8) single walled carbon nanotubes on cobalt incorporated tud-1 catalysts[END_REF], or the intensities are "corrected" by taking into account (n, m)-dependent quantum efficiency [START_REF] Wang | Pressureinduced single-walled carbon nanotube (n, m) selectivity on co-mo catalysts[END_REF]. On the rare occasions where AFM is featured in SWCNT growth papers, the technique is systematically used to determine the diameter distribution of a sample, through height measurement [START_REF] Ding | Selective growth of well-aligned semiconducting singlewalled carbon nanotubes[END_REF], [START_REF] Kim | Size engineering of metal nanoparticles to diameter-specified growth of single-walled carbon nanotubes with horizontal alignment on quartz[END_REF]. Finally, there is no pre-defined procedure to determine the SC/M ratio from electronic devices. In a few papers, the fabrication of FETs is only used to prove the capacities of the grown SWCNTs as a channel material. Quantitative determination of the SC/M ratio is measured in several different ways. For multiple-SWCNT devices with over 500 tubes between the source and the drain, Ding et al. suggested that the I on /I of f directly represented the average SC/M ratio [START_REF] Ding | Selective growth of well-aligned semiconducting singlewalled carbon nanotubes[END_REF]. For individual-SWCNT devices, it is possible to count the SC devices, using a somewhat arbitrary threshold I on /I of f value. Certain groups considered a device to be semiconducting if I on /I of f was above 10 [START_REF] Zhang | Diameter-specific growth of semiconducting swnt arrays using uniform mo2c solid catalyst[END_REF], [START_REF] Li | Importance of diameter control on selective synthesis of semiconducting single-walled carbon nanotubes[END_REF]. On the other hand, Li et al. considered a device to be metallic when I on /I of f was below 100 [START_REF] Li | Selective synthesis of large diameter, highly conductive and high density single-walled carbon nanotubes by a thiophene-assisted chemical vapor deposition method on transparent substrates[END_REF]. For each presented technique, aside from AFM and ED, the methodology for statistical, or quantitative selectivity assessment is never quite clear. Different groups tend to use different selectivity evaluation methodologies when using the same characterization technique, inevitably leading to different results. Whether or not the shift from one result to another is significant is difficult to establish. In summary, absolute SC/M ratio values, chirality enrichment percentages, or mean diameters are to be compared from one study to another with a bit of caution, even if the characterization technique is the same. SWCNT sample cross-characterization in the literature Considering that there is no absolute characterization technique, cross-characterization is indispensable for selectivity assessment. In almost all of the selected studies, more than one technique is used for the characterization of SWCNT samples. For a wide majority, more than two techniques are used, and including device fabrication, up to six characterization techniques can be used. When it comes to selectivity assessment, more than one technique are used in a very large majority of cases. However, studies comparing quantitative, or statistical data coming from more than one characterization technique only represent a quarter of the studies with selectivity claims. In most cases, quantitative evaluation is done using a technique of choice, and another (or a few others) are relied upon for consistency. For instance, TEM images showing one or two SWCNTs can be used to ensure that the diameter range determined by a statistical diameter analysis conducted through AFM is reliable [START_REF] Zhang | Diameter-specific growth of semiconducting swnt arrays using uniform mo2c solid catalyst[END_REF], or absorption can be used to qualitatively confirm a quantified chirality enrichment measured by PLE [START_REF] Liu | High temperature selective growth of single-walled carbon nanotubes with a narrow chirality distribution from a copt bimetallic catalyst[END_REF]. This makes it difficult to estimate and compare the accuracy of quantitative methods for selectivity assessment. Focus on TEM and Raman In an effort to try to evaluate the relative accuracy of available characterization techniques for the quantitative measurement of growth selectivity, a part of the work presented in this thesis will focus on the comparison between conventional TEM, and multi-wavelength resonant Raman spectroscopy for the determination of the diameter distribution of a SWCNT growth sample. When looking at the number of publications with growth selectivity claims in the literature, Raman and TEM (conventional TEM, and electron diffraction combined) seem to be, by far, the two most used characterization techniques, with more than two thirds of the selected papers featuring TEM, and more than three quarters featuring Raman (the next most used characterization technique is OAS, used in less than half of the selected papers). It is striking, however, to see that quantitative data are extracted from TEM and Raman in only a fraction of these publications. Diameter distributions are extracted from conventional TEM data in only 30% of the time, and quantitative selectivity evaluation is performed in only 45% of papers featuring Raman spectroscopy (SC/M ratio calculation, diameter distribution determination, and (n, m) indexing combined). This is even more surprising considering that electronic devices, AFM, and PLE are used for quantitative evaluation in a wide majority of cases. Though cross-characterization between TEM and Raman is often performed, comparison of statistical data from the two methods is rarely seen. We found two papers featuring statistical data for both TEM and Raman regarding SWCNT sample diameter distributions. Song et al. determined the diameter distributions of several SWCNT samples by measuring the diameter of 50 tubes by TEM, and counting RBM peaks from several Raman spectra using one excitation wavelength [START_REF] Song | Synthesis of bandgapcontrolled semiconducting single-walled carbon nanotubes[END_REF]. The two distributions for each sample (mean diameter ranging from 0.9 nm to 1.5 nm) appear to be close to identical, with mean [START_REF] He | Diameter and chiral angle distribution dependencies on the carbon precursors in surfacegrown single-walled carbon nanotubes[END_REF]. The two distributions are similar, but their shape differ and there is a slight shift in mean diameter (0.2 nm). This is explained by the fact that the Raman characterization is done using only one excitation wavelength, potentially setting aside multiple SWCNT populations. From the conclusions of these two publications, it is unclear whether TEM and Raman can coincide for diameter distribution assessment. In both cases, close results are obtained, but not satisfyingly enough in one of the papers. Moreover, the Raman characterization in both cases is conducted with only one excitation wavelength, inevitably making the probing of the SWCNT samples incomplete. It would therefore be interesting to conduct a comparative study between TEM and Raman for diameter distribution assessment, using multiple wavelengths, and types of samples. Theoretical and experimental insights into SWCNT nucleation and growth Since nanotubes were first synthesized, researchers have proposed various theories for their growth mechanism. Throughout the years, models have been both confirmed and refuted by experimental results, highlighting the complexity of the matter. But the aim of chirality selective growth has kept this motivation to understand the CVD growth process very much alive. Unfortunately, though the understanding of the mechanisms driving SWCNT nucleation and growth is the theoretical key to chirality selective growth, many questions on the subject are yet to be answered [START_REF] Amara | Modeling the growth of single-wall carbon nanotubes[END_REF]. In this section, we will first give a chronological overview of the proposed growth mechanisms for nanotubes. The latest theoretical insights on growth mechanisms will then be discussed, and the resulting suggestions for achieving selective growth will be presented. Chapter 1. Introduction and state of the art The VLS growth model In the early 1970's, when Baker et al. [START_REF] Baker | Formation of filamentous carbon from iron, cobalt and chromium catalyzed decomposition of acetylene[END_REF] grew carbon nanofibers from the catalytic decomposition of C 2 H 2 on metallic particles, they developed a model to understand their growth mechanism. They made the hypothesis that the vaporliquid-solid (VLS) model proposed to explain the growth of Si whiskers [START_REF] Wagner | Vapor-liquid-solid mechanism of single crystal growth[END_REF] was adaptable to the case of carbon nanofibers (CNF). This growth mechanism can be decomposed into three steps: 1. Adsorption and dissociation of the carbon-containing molecules at the surface of the catalyst nanoparticle, to form elementary carbon 2. Dissolution of the carbon in the bulk of the particle, leading to the formation of a liquid metal-carbide, and diffusion of the carbon in the bulk of the particle 3. When the catalyst particle becomes supersaturated with carbon, precipitation of solid carbon at the surface of the particle The continuous supply of carbon keeps the fiber growth going until deactivation of the catalyst nanoparticle. According to Baker et al., the driving force of the diffusion of carbon through the bulk of the particle was a temperature gradient caused by the exothermicity of carbon precursor dissociation at the surface of the nanoparticle, and the endothermicity of the precipitation of solid carbon. This interpretation, however, could not explain CNF growth using carbon precursors whose decomposition is endothermic, which is the case for CH 4 [START_REF] Evans | Growth of filamentary carbon on metallic surfaces during the pyrolysis of methane and acetone[END_REF] for instance. It was later demonstrated that the driving force of the mechanism was in fact a chemical potential gradient [START_REF] Tibbetts | Why are carbon filaments tubular?[END_REF] between the carbon atoms brought to the surface of the nanoparticle by the decomposition of the carbon feedstock molecules, and the carbon sp 2 wall [START_REF] Snoeck | Filamentous carbon formation and gasification: Thermodynamics, driving force, nucleation, and steady-state growth[END_REF]. In some ways, SWCNT growth is similar to CNF growth, and the VLS mechanism still serves as a foundation to explain SWCNT growth today. The considered objects here are different, essentially much smaller, so we can expect different behaviors. Over the years, a few variations of this model were proposed to explain SWCNT growth. HRTEM evidence [START_REF] Gavillet | Root-growth mechanism for single-wall carbon nanotubes[END_REF]. In this case, the mechanism (root-growth) was applied to both CVD and high temperature synthesis of SWCNTs, and an emphasis was put on the competition between growth and deactivation of the catalyst by encapsulation by a graphitic shell. Today, the experimental and theoretical study of the growth mechanism is far from over, and many questions remain debated, or simply unanswered. Here is a non-exhaustive list of these questions: • Does the carbon diffuse through the bulk or the surface of the catalyst? • What is the physical state of the particle? • What is the chemical state of the particle? • What allows the nucleation, and growth of a SWCNT? • How is the growth stopped? From the VLS to the VSS model As mentionned above, one of the big debates concerns the physical state of the catalsyt nanoparticle during the growth. According to the VLS model, the nanoparticle should be a liquid droplet of metal supersaturated with carbon. However, starting with the first in-situ HRTEM experiments during growth, various groups have claimed that catalyst nanoparticles could be solid during the growth process for MWCNTs [START_REF] Helveg | Atomicscale imaging of carbon nanofibre growth[END_REF], and later SWCNTs [START_REF] Hofmann | In situ observations of catalyst dynamics during surface-bound carbon nanotube nucleation[END_REF], [START_REF] Yoshida | Atomicscale in-situ observation of carbon nanotube growth from solid state iron carbide nanoparticles[END_REF]. There have also been experiments and theoretical arguments supporting surface diffusion of carbon. This goes against the VLS model, which implies bulk diffusion of carbon through catalyst nanoparticles. This gave rise to a new model to explain SWCNT growth: the vapor-solid-solid (VSS) model (first proposed for the growth of GaAs nanofibers [START_REF] Persson | Solid-phase diffusion mechanism for gaas nanowire growth[END_REF]), where the catalyst nanoparticle is crystalline, and the carbon diffuses through the surface of the catalyst only. This model encouraged the development of a strategy for chirality specific growth based on so-called epitaxy [START_REF] Reich | Control the chirality of carbon nanotubes by epitaxial growth[END_REF]. DFT simulations have been used to support this strategy, by calculating the interaction energy between nanotube caps with various chiralities and a flat metal surface on which the cap is lying (implying a perpendicular growth mode), and looking for lattice matching [START_REF] Reich | Epitaxial growth of carbon caps on ni for chiral selectivity[END_REF]. This kind of calculation is performed at T = 0 K, and is therefore not appropriate to model the state of a crystalline metal surface at typical growth temperatures. Chapter 1. Introduction and state of the art Furthermore, this strategy relies on the idea that the cap formed during nucleation will necessarily determine the chirality of the resulting SWCNT. This idea is realistic because the breaking and reforming of a C-C bond to heal or make a defect that would modify the tube structure is quite energy consuming. However, in situ studies have shown that short nanotubes at the very beginning of growth tend to be unstable [START_REF] Hofmann | In situ observations of catalyst dynamics during surface-bound carbon nanotube nucleation[END_REF]. Statistical TEM studies on SWCNT lengths conducted after growth evidenced the existence of a length threshold of about 5 mn after which durable growth could occur [START_REF] Fiawoo | Evidence of correlation between catalyst particles and the single-wall carbon nanotube diameter: A first step towards chirality control[END_REF]. Page et al. [START_REF] Page | Mechanisms of singlewalled carbon nanotube nucleation, growth, and healing determined using qm/md methods[END_REF] showed that the number of pentagons, hexagons, or heptagons formed strongly depended on carbon feedstock decomposition rate. A fast decomposition leads to a high growth rate, which occurs with the formation of a high density of defects (pentagons and heptagons), eventually leading to structure modification of the nanotube during growth. A slower decomposition rate will give more time for healing defects during growth, leading to a better chance of conserving the cap chirality through the growth process. More recently, with a similar objective of so-called expitaxial-like growth, studies aiming at finding adequate lattice matching between certain SWCNT chiralities and crystalline planes on tungsten-based catalysts have been conducted [START_REF] Yang | Chirality-specific growth of single-walled carbon nanotubes on solid alloy catalysts[END_REF], [START_REF] Zhang | Arrays of horizontal carbon nanotubes of controlled chirality grown using designed catalysts[END_REF]. However, there is no direct way to prove that the specific lattice planes matching with certain tube chiralities actually exist in the CVD growth conditions. During CVD, high temperature, the presence of carbon-containing molecules (and possibly other compounds) in the system, and the presence of carbon dissolved in the particle strongly affect its properties. We should also note that the fact that in situ TEM experiments have allowed the observation of crystalline catalysts during growth does not mean that it is the case in "real" CVD conditions (higher pressure and feedstock flow rate, possibly different substrate...). To summarize, if the catalyst is crystalline during growth (which is possible and clearly depends on the catalyst size and composition, as well as CVD parameters), and if the cap formation determines SWCNT chirality (by somehow promoting a slow growth rate), then so-called epitaxial (which does not constitute an actual epitaxy, but rather an orientation relation between tube and particle at most) growth could be a viable strategy. Aside from these conclusions solely concerning cap structure engineering, computer simulations on the nucleation of SWCNTs can give very insightful information on the role of the [START_REF] Amara | Understanding the nucleation mechanisms of carbon nanotubes in catalytic chemical vapor deposition[END_REF]. The method allows to take into account the carbon chemical potential (µ C ) gradient, which is the thermodynamic driving force of the nucleation, which is not the case for in molecular dynamics studies. The carbon chemical potential is experimentally fixed by the thermochemistry driving the carbon precursor decomposition at the surface of the catalyst, which can essentially be controlled by CVD parameter tuning. The calculation of adsorption isotherms on Ni nanoparticles revealed the existence of an optimal µ C window for possible SWCNT growth. It was also shown that sp 2 carbon segregation was initiated when C content in the particle attained a solubility limit. The findings were in good agreement with the criterium experimentally determined according to which a metal can be a good catalyst for SWCNT growth if it has a non-zero but low carbon solubility [START_REF] Deck | Prediction of carbon nanotube growth success by the analysis of carbon-catalyst binary phase diagrams[END_REF]. In 2012, Diarra et al. used the same model to simulate the very early stages of SWCNT growth [START_REF] Diarra | Importance of carbon solubility and wetting properties of nickel nanoparticles for single wall nanotube growth[END_REF]. They proceeded by fixing short nanotube sections to a small Ni nanoparticle (around 80 C atoms and 80 Ni atoms total), with varying chiralities within a diameter range of 0.7-0.8 nm, heating them, and letting the systems relax to their equilibrium state. Those systems were then used as starting points for GCMC simulations, where a wide range of parameter (temperature, µ C ) combinations were tested, only to keep the 10 where growth was successful. These calculations allowed the atomic-level observation of a tangential growth mechanism. Since it had been previously demonstrated that carbon solubility played an important role in the nucleation process, they tried to understand how this solubility was affected by size or temperature. This paper also discussed the wetting properties of sp 2 carbon on a Ni nanoparticle as a function of its carbon content. As demonstrated much earlier for the wetting properties of bulk metal droplets on graphite [START_REF] Naidich | Wetting of graphite by nickel as affected by the liquid-phase dissolution process of carbon[END_REF], dewetting was favored with increasing C content. In light of the simulations of SWCNT growth, the growth phenomenon was interpreted as a balance between two antagonist phenomena: dewetting of the team shows a strong dependence of d SW CN T /d N P on carbon content inside the particle. It is interesting to note that these two growth modes result in two different tube-catalyst interaction schemes, tangential growth implying surface interaction, and perpendicular growth only a line interaction. These two types of interactions mean two ways of trying to influence growth by controlling the tube-nanoparticle interaction. We can extract strategies for controlling the growth of SWCNTs from these findings. A possibly interesting way to move towards chirality-specific growth would be twofold: first a good control of the nanoparticle diameter distribution, and second a control of the growth mode [START_REF] Amara | Modeling the growth of single-wall carbon nanotubes[END_REF]. The control of catalyst nanoparticle size has to be done through control of the nanoparticle synthesis and optimization of their deposition on the substrate, or of CVD parameters, in order to prevent coalescence. The control of the growth mode, is achievable in two ways: first through CVD parameters, and second through fine tuning of carbon solubility in the catalyst nanoparticles. For the latter, of course, modifying the catalyst nanoparticle's properties through its composition could be a promising route towards growth selectivity through growth mode control. Aside from selective growth considerations, tuning the catalyst's interaction with carbon through its composition could be an interesting experimental way to better understand the role of carbon solubility in SWCNT growth in general. Objectives of this work In this chapter, we have seen that the research on selective growth in the last decades has been extremely abundant and fruitful. However, many questions remain regarding what exactly is responsible for growth selectivity, and the different effects growth parameters and catalyst nature have on growth selectivity. We have established that it is sometimes difficult to extract trends from the literature, for a few reasons. First, there are clearly contradicting results regarding the effect of some growth parameters. This can be explained by the second reason for which the study of literature on SWCNT growth is a quite complex manner: growth conditions obviously vary drastically from one study to another. Each laboratory has its own CVD reactor, employs different carbon sources, at varying flowrates, in varying temperature and pressure conditions. It is clear, when analyzing the literature, that the use of bimetallic catalysts for Chapter 1. Introduction and state of the art selective growth may be an interesting route. This, judging both from promising experimental results, and theoretical calculations. Modulating the catalyst properties and its interactions with carbon atoms through nanoalloying has indeed been shown to be a potentially interesting strategy. We have shown in this chapter that the comparison of different bimetallic combinations and their performances for selective growth is rendered very impractical by the fact that growths are conducted in different conditions, the catalyst nature is not always clear, the catalyst nanoparticles are not always alloys, and their preparation methods vary drastically from one study to another. In order to have experimental insights into the effect of catalyst composition on SWCNT growth selectivity, it is necessary to elaborate a SWCNT growth method enabling the use of a wide variety of bimetallic combinations as catalysts with similar sizes and morphologies in the same conditions, without changing the catalyst preparation method depending on the chosen combination. We have evidenced that in a majority of cases, when catalyst nanoparticles are fabricated through physical routes (Table 1.1), the bimetallic catalyst nanoparticles exhibit phase segregation, meaning that the effective catalyst, in so-called bimetallic catalysts, is in fact monometallic. It is therefore important to make sure that the preparation method leads to nanoalloys. To achieve this, we have chosen to focus on chemically-synthesized catalyst precursor nanoparticles with the two metals already mixed within the structure. This route is original, because it allows for the deposition of preformed catalyst precursors with controlled sizes and composition. The first objective of this work is therefore the elaboration of this SWCNT synthesis route, using various chemically-synthesized catalyst precursors. We have chosen, in a first attempt, to work using Prussian blue (PB), and its analogs as catalyst precursors. Three catalyst systems have been selected: a pure iron catalyst, obtained by the reduction of PB, and two bimetallic systems obtained by the reduction of PB analogs (PBAs). To test the ability to produce nanoalloys with the developed method, we selected a first system whose bulk binary phase diagram predicts the formation of a solid solution in the case of a 1:1 stoichiometry, and a system for which a phase-segregation is expected at a 1:1 stoichiometry. The first system is the Fe-Ni bimetallic system [START_REF] Swartzendruber | The fe-ni (iron-nickel) system[END_REF], and the second is the Cr-Ni bimetallic system [START_REF] Turchi | Modeling of ni-cr-mo based alloys: Part i-phase stability[END_REF] (see Appendix B for binary phase diagrams). In the case of the Ni-Fe system, we expect the formation of a nanoalloy with a lower carbon solubility limit than pure iron, which will enable a comparison based on carbon solubility. Two other chemically synthesized precursor types ("cyanosols", and polyoxometalates (POMs)) were also selected for their potential to form other nanoalloys of interest for SWCNT growth. An Fe-Pd system, with a even lower carbon solubility than Fe-Ni will be studied through the "cyanosol" (which will be defined in Chapter 2) route, and a Co-W system will be studied using POM precursors. The second objective will be to evaluate the catalysts' abilities to grow SWCNT with any kind of selectivity, and to try to optimize the synthesis conditions with selectivity aims. The comparison of the different prepared catalyst systems with regard to SWCNT growth will be attempted. Our review of the literature has also led to the realization that there is a strong need for a standardized characterization methodology for selectivity assessment of CVD-grown SWCNT samples. The third objective of this work will be to try to evaluate the accuracy of the two available characterization techniques (TEM and Raman spectroscopy). This will be done through a comparative study on various SWCNT samples, focusing on the determination of the diameter distributions of those samples. Chapter 2 Experimental methods The previous chapter ended by a brief presentation of the objectives of this work. With the goal to obtain alloyed bimetallic catalyst nanoparticles, we chose to work with chemically synthesized catalyst precursors with defined sizes, structures, and compositions. The first section of this chapter will give a brief description of the catalyst precursor "families" that have been chosen for this study, along with an explanation on how they will be synthesized and/or used as SWCNT growth catalyst precursors. The first presented family (Prussian blue and its analogs) is the one that was the most extensively studied. The other two families were studied as preliminary work. The second section of the chapter is a presentation of the two characterization techniques that have been used throughout this work for SWCNT sample characterization. The characterization methodology for selectivity assessment, and more precisely diameter distribution measurement of SWCNT samples, will also be discussed. Chemically synthesized nanoalloy catalyst precursors This section will focus on the three different types of catalyst precursors that have been chosen for this thesis. The first type, Prussian blue (PB) and its analogs (PBA), is the one that has been the main focus of this work. The second is, in a way, a derivative of the PBA family, known as the "cyanosol" family. The third type of catalyst precursors, polyoxometalates (POM), are studied here because a specific compound from this family has been proven to lead to some selectivity in SWCNT growth. Chapter 2. Experimental methods ) and a ferric (Fe(III)) salt (chloride or nitrate, for instance) in aqueous solution. The origin of its color and its crystallographic structure were uncovered much later. Though structures were proposed with the emergence of X-ray diffraction experiments in the 30's [START_REF] Keggin | Structures and formulae of the prussian blues and related compounds[END_REF], the structure was only accurately resolved in 1977 by Buser et al. [START_REF] Buser | The crystal structure of prussian blue: Fe4 [fe (cn) 6] 3. xh2o[END_REF]. They also showed that PB had the same structure as the Turnbull blue, obtained from the coprecipitation of K 3 [Fe III (CN) 6 ] and a Fe(II) salt in water. Nanoparticles of Prussian blue and its analogs PB crystallizes in a face centered cubic (FCC) lattice (generally considered as Fm3m, but Pm3m space group), with a 10.17 Å lattice parameter, where the motif consists of alternating Fe III and Fe II linked by cyanide bridges, K atoms are present in variable amounts in tetrahedral interstitial sites. The fact that Fe III and Fe II are linked by cyanide bridges implies a substantial spacing between the metallic atoms. For this reason, zeolithic water molecules can be present in the interstitial tetrahedral sites of the structure. The Fe II (CN) 6 content in the structure varies from 3/4 ("insoluble" form) to 1 ("soluble" form), depending on the amount of potassium in the interstitial sites (ranging from 0 to 1). In the occurrence of Fe II (CN) 6 vacancies, Fe(III) ions complete their coordination sphere with water molecules. Figure 2.1 shows a hypothetical PB structure without any vacancies that may be close to the "soluble" form, and the real structure containing vacancies and zeolithic water molecules. The color of PB stems from a metal-to-metal intervalence charge transfer from low spin Fe(II) to high spin Fe(III) atoms, leading to a strong absorption band at 700 nm. Aside from its use as a pigment, PB has been widely studied for its interesting electrochemical properties that may be exploited in biosensors and electrochromic devices [START_REF] Itaya | Electrochemistry of polynuclear transition metal cyanides: Prussian blue and its analogues[END_REF]. PB serves as a generic model for a much bigger family of compounds called PBAs. precursors for SWCNT growth. The two former require the use of organic solvents, and stabilizing surfactants or polymers cover the surface of the particle. The latter leads to the formation of particles embedded in a solid template, and additional steps are needed to retrieve them. In all cases, processing the obtained objects as bare nanoparticles involves additional purification steps that may lead to their alteration, and coalescence. Since 2005, a method has been developed in our laboratory by Catala, Mallah et al. to synthesize self-standing charged PBA particles in aqueous solution. This method is obviously advantageous because it does not utilize stabilizing agents such as surfactants. It is also extremely simple to operate and allows the synthesis of nanoparticles of many different PBA systems under similar experimental conditions. The stability of the nanoparticle dispersions ranges from several weeks to a few months, depending on the PBA system. The obtained particles are individualized in solution because of repulsive Coulomb interactions due to their negative charge. This stability relies on the fine tuning of transition metal salt concentrations. Published studies during the PhD of Daniela Brinzei present the principle of the synthesis for Ni 2+ -based PBA particles (CsNiFe [START_REF] Brinzei | Magnetic behaviour of negatively charged nickel (ii) hexacyanoferrate (iii) coordination nanoparticles[END_REF], and CsNiCr [START_REF] Brinzei | Spontaneous stabilization and isolation of dispersible bimetallic coordination nanoparticles of cs x ni [cr (cn) 6][END_REF]). The particles exhibit a hydrodynamic diameter of 5-7 nm depending on system and experimental conditions. A chemical analysis of the particles post-retrieving using cetyltrimethylammonium cations (CTA + ) shows that their general formula is Cs x Ni II [M' III (CN) 6 ] y , with x = 0.4 -0.5, and y = 0.95 -1. Their overall negative charge is therefore due to both the inherent negative charge of the core of the particle according to the alkali content in the structure and the very low amount of hexacyanometallate vacancies, as well as to the negative surface charges of the coordinated hexacyanometallates. Since then, the method has been applied and optimized to a number of other PBA systems (KFeFe, CsNiCo, KFeRu, etc.) with equal success for obtaining individualized, stable particles with a controlled size and overall 1:1 -1:0.95 metallic ratio. The synthesis method is extremely simple: an aqueous solution of the hexa-aquo metal salt (M) (concentration ranging from 0.5 mM to 4 mM) is quickly added to an aqueous solution of equal concentration of the hexacyanometallate (M') containing the alkali cation in excess (when its presence is necessary for obtaining negatively charged particles) under vigorous stirring. The PBA particles are obtained dispersed and stable in water. Depending on the system, the synthesis is done either at ambient temperature or at 2 • C. Methods have also been developed in the lab to tune the size of the particles by adding shell layers to a pre-existing PBA particle seed, with the same or different PBA system, leading to either simply bigger particles with unchanged composition [START_REF] Prado | Tailored coordination nanoparticles: Assessing the magnetic single-domain critical size[END_REF], or core-shell PBA heterostructures [START_REF] Prado | Magnetic anisotropy of cyanidebridged core and core-shell coordination nanoparticles probed by x-ray magnetic circular dichroism[END_REF]. In summary, a simple method is an efficient synthesis solution to a wide range of PBA nanoparticles with controlled sizes. It is applicable to monometallic, such as PB, or bimetallic systems, like NiFe and NiCr, with a 1:1 stoichiometry. Further control over the size and composition can also be considered by adding shells to a pre-existing PBA structure, leading to the synthesis of potential trimetallic PBAs, the only liability here would be the increasing size of the nanoparticles. One could also consider the potential tuning of the nanoparticle compositions by playing with the metal salts concentrations, or specifically by trying to tune the content of the alkaline cation in the PBA nanoparticle structure, which is partly responsible for the vacancy content in the structure. However, this tuning is quite unfortunately limited with this synthesis method. This is why the work will focus on 1:1 stoichiometry compounds in the first instance, while future works could focus on metal stoichiometry tuning. Use as catalyst precursors It has been established in the first chapter of this manuscript that a good catalyst precursor for exploring potentially selective SWCNT growth should be small in size, with a narrow size distribution and controlled composition, and give access to many bimetallic combinations in the same conditions. So as to conserve their small size, the particles need to be protected from coalescence, which starts by synthesizing non-aggregated particles. All of these conditions seem to be fulfilled by PBA nanoparticles synthesized by the co-precipitation method developed by Catala et al [START_REF] Brinzei | Magnetic behaviour of negatively charged nickel (ii) hexacyanoferrate (iii) coordination nanoparticles[END_REF], [START_REF] Brinzei | Spontaneous stabilization and isolation of dispersible bimetallic coordination nanoparticles of cs x ni [cr (cn) 6][END_REF]. They are also free of stabilizing agents, preventing potential contamination and deactivation of the catalysts without involving purification steps. The open question that remains is the possibility of reducing these particles into bimetallic alloys while keeping a good control of the size distribution. As explained previously, manufacturing nanoalloy nanoparticles for the growth of SWCNTs is challenging. One of the main challenges is to avoid phase segregation during the catalyst fabrication process. Using precursor nanoparticles where the two metals are already mixed within the structure is obviously advantageous. Provided that the desired solid solution or alloy is thermodynamically allowed to form at the nanoscale for the considered metallic combination. PBAs have already been successfuly used for that purpose. Yamada et al. have grown Pt-Co nanoparticles 10-15 nm in diameter with various metallic Pt:Co ratios by thermal reduction under H 2 of Pt IV /Co tetracyanoplatinate PBA nanoparticles synthesized themselves by the reverse micelle method [START_REF] Yamada | Novel synthetic approach to creating ptco alloy nanoparticles by reduction of metal coordination nano-polymers[END_REF]. Another example in the literature reports the formation of supported FeNi bimetallic nanoparticles from PBAs embedded in mesoporous silica [START_REF] Folch | A coordination polymer precursor approach to the synthesis of nife bimetallic nanoparticles within hybrid mesoporous silica[END_REF]. The formed bimetallic nanoparticles are, in those cases, too large to serve as seeds for the growth of SWCNTs, and it is not entirely clear whether the formed particles are truly alloyed. However, their synthesis demonstrates the feasibility of reducing PBAs into metallic compounds. The capabilities of a reductive treatment of small bimetallic PBA nanoparticles synthesized by the co-precipitation method in water after deposition on a substrate to lead to the formation of nanoalloy catalyst is one of the questions the work presented here attempts to answer. Method In the present work, the PB and PBA nanoparticles were synthesized using the spontaneous stabilization method in water. Three systems were tested: one monometallic (PB, ie Fe), and two bimetallic (NiFe and NiCr PBAs). A process for the deposition of the as-obtained nanoparticles was developed in order to obtain a dense monolayer of the PB/PBA nanoparticles on a SiO 2 /Si wafer. This process, given in more detail in Chapter 3 of this manuscript, relies on the anchoring of the PB/PBA nanoparticles onto a silane molecule comprising a terminal pyridine group previously attached to the wafer through a self-assembling process. Then, the aim is to form nanoalloy nanoparticles in situ using a reductive pretreatment (H 2 atmosphere), directly followed by the CVD growth of SWCNTs. The fact that the effective catalyst nanoparticles are formed directly inside the CVD chamber ensures that they do not undergo any possible chemical transformation between their formation and their use as catalysts nanoparticles. "Cyanosol" nanoparticles General presentation In the 1990's, a research group in Princeton University used sol-gel chemistry to synthesize amorphous cyanide-bridged transition metal polymers they named "cyanogels" [START_REF] Heibel | Use of sol-gel chemistry for the preparation of cyanogels as ceramic and alloy precursors[END_REF], which are essentially an amorphous form of PBAs. They are prepared by the reaction of a chlorometalate [M II Cl 4 ] 2-and a hexacyanometalate [M' II/III CN 6 ] 4-/3-in a sol-gel process. The chloride ligands of M are partially substituted by cyanide ligands, bridging M and M' in a PBA-like structure. First, the formation of a sol takes place: small clusters of coordination networks nucleate (cyanosol), then the clusters collide and bond with each other, so as to gradually form a bulk 3D amorphous and porous negatively-charged network called a gel (cyanogel). Therefore, for forming small size catalyst nanoparticles, the first step is the synthesis of the so-called cyanosols which consists in stopping the sol-gel process in its very early stage, so as to freeze the reactional medium as a colloidal nanoparticle suspension. This was achieved by Bocarsly et al. in 2009 [START_REF] Burgess | Stabilizing cyanosols: Amorphous cyanide bridged transition metal polymer nanoparticles[END_REF]. Being a transition state before the formation of the cyanogel, cyanosol nanoparticles are considered to be very unstable and tend to aggregate. The method employed by Bocarsly et al. was to prevent the strong interactions between the nanoparticles by replacing their counter ions by CTA + . Figure 2.3.a shows a schematic representation of the synthesis process. The obtained nanoparticles are soluble in organic solvents. This cyanosol synthesis method has been studied for a few bimetallic systems: M = Pd, Pt, and M' = Fe, Co, Ru, leading to the preparation of nanoparticles with sizes ranging from 4 to 5 nm (see Figure 2.3). These studies showed that the time at which precipitation was performed was crucial to obtain isolated particles. Very recent work has been conducted on cyanosol nanoparticles in our laboratory with the aim of synthesizing cyanosol nanoparticles with a method similar to the PBA particle co-precipitation synthesis. Small (2-4 nm) free-standing PdFe and PtFe cyanosol nanoparticles were obtained in water, with stabilities ranging from a few days to several months, that had not been reported. These nanoparticles constitute an interesting candidate as catalyst precursors for SWCNT growth. characterized with HRTEM-EDX showing that the obtained 5±1 nm nanoparticles were crystalline, and that both metals were present in the particles. Though the size of the obtained nanoparticles is bigger than the sizes needed for selective tangential growth, and their composition homogeneity is not entirely unambiguously demonstrated, these results are encouraging, and could be optimized for other metal combinations in order to obtain monodisperse nanoalloy catalyst nanoparticles for SWCNT growth. Another potential advantage of cyanosol compounds as catalyst precursors for SWCNT growth is their enhanced composition flexibility. In comparison with PBAs, whose composition ranges are limited by the imposed FCC crystalline structure (depending on the vacancies and presence of alkali cations, the two extreme stoichiometries (M:M') are 1.0 and 1.5) the amorphous cyanosols open a wider range of metallic ratios. The synthesis of 2:1 Pd:Co cyanosol nanoparticles is an example illustrating this flexibility [START_REF] Burgess | Stabilizing cyanosols: Amorphous cyanide bridged transition metal polymer nanoparticles[END_REF]. This offers the possibility to tune the catalyst nanoparticles' composition over a broader stoichiometry range. Method Since the cyanosol systems are chemically similar to the PBA systems, our intent here is to mimic the methodology applied for the PBA systems. The feasibility of using the same deposition process on a SiO 2 /Si wafer, as well as for the formation of nanoalloy catalyst nanoparticles and consequent SWCNT growth will be tested. The cyanosol bimetallic system used in the present work is FePd. The POM used in this approach and described in [START_REF] Yao | Two hexa-tm-containing (tm= co2+ and ni2+) (p 2w12)-based trimeric tungstophosphates[END_REF], is made of three hexavacant Dawson-type phosphotungstate P 2 W 12 sub-units linked by WO(H 2 O) entities, forming a P 6 W 39 shell that contains six vacant sites to incorporate Co 2+ atoms. A solution containing the clusters is simply dropped onto a SiO 2 wafer, and the Co-W catalyst nanoparticles are formed by first calcinating the POM layer in air, and second reducing it at about 1000 • C under hydrogen, leading to the formation of Co-W bimetallic nanoparticles. Figure 2.5 shows the structure of the POM used in this approach, and the schematic of the SWCNT implemented growth process. In order to test the feasibility of nanoalloy catalyst formation, and subsequent SWCNT growth from this type of POM, we will first attempt to deposit the POMs in a homogeneous layer on top of a SiO 2 /Si wafer. Since the POMs used here are clusters with smaller sizes than the nanoparticles of PBAs or cyanosols, the feasibility of the SWCNT growth will have to be tested. Polyoxometalates (POMs) Method Synthesis of SWCNTs from precursors The SWCNT CVD growths presented in this manuscript have been performed in a HFCDV reactor, at the LPICM in Ecole Polytechnique. Figure 2.7 presents a schematic of the reactor. The reactor is equipped with two gas inlets, each equipped with their own filament. While the first inlet is used for the introduction of the carbon precursor -methane (CH 4 ) in the case of this work-the second is used for the introduction of hydrogen (H 2 ). The filaments are at a fixed distance from the sample, and favor the local decomposition of the carbon precursor and the formation of atomic hydrogen before their contact with the sample. The power of the filaments can be independently controlled, leaving more room for optimization, and allowing the control of the amount of activated hydrogen and carbon radicals. The role of activated H 2 is to perform a reductive pretreatment of the catalyst precursor nanoparticles before the growth, as well as to continuously etch the amorphous carbon formed on the reactor sidewall during growth [START_REF] Kim | The role of catalytic nanoparticle pretreatment on the growth of vertically aligned carbon nanotubes by hot-filament chemical vapor deposition[END_REF]. The reactor is composed of a cylindrical quartz tube enclosed in an 80 mm wide tubular heater with an approximately 250 mm length uniformly heated. The sample can be placed in two positions: in the cold zone, or in the hot zone (blue part on the left, and orange part of the tube on Figure 2.7, respectively). The filaments are made of 0.4 mm thick tungsten wire. The residual base pressure of the reactor is 10 -6 mbar, preventing the presence of adsorbed water molecules on the walls. First, the sample is placed in the cold zone, and the reactor is pumped to its residual pressure. The reactor is heated at the desired growth temperature, while the sample is kept in the cold zone. H 2 is then introduced at a 100 sccm (standard cubic centimeters per minute) flow rate, and the corresponding filament power of 160 W is turned on. Once the pressure of the chamber is stabilized at 90 mbar, the sample is moved to the hot zone. At this point, the chamber only contains H 2 , and the reductive pretreatment takes place. At the end of the desired pretreatment time, we introduce the carbon precursor at a 20 sccm flow rate, with a filament power of 120 W, and increase the pressure to 100 mbar. The growh of SWCNTs takes place, for usually 30 minutes. At the end of the growth, the sample is placed back in the cold zone, the gas inlets are closed, the chamber is pumped, the filaments and the heater turned off. When the chamber has cooled down to a temperature below 150 • C, the sample can be taken out of the chamber for characterization. Characterization of SWCNT samples The various characterization methods that are used for SWCNTs have been presented in brief in the previous chapter. Here, we will discuss in more detail the two methods that were available and used for this work which are TEM and Raman spectroscopy, along with the developed methodology used for these two methods regarding growth selectivity assessment. Transmission electron microscopy Historically, the invention of electron microscopes stemmed from the limitations in the resolution of optical microscopes. The wave-like characteristic of electrons was first theorized by de Broglie in 1925. Electron diffraction experiments conducted in 1927 confirmed their wave nature. Considering electrons like optical photons with a smaller wavelength was then possible, and their use as the "seeing" radiation for microscopes with the aim of decreasing the resolution was conceivable. Ruska and Knoll were the first to put forth this idea in 1932, and the first TEMs were commercially available a few years later [START_REF] Williams | Transmission electron microscopy: a textbook for materials science[END_REF]. Why use electrons? In optical microscopy, the resolution is defined by the shortest distance between two points that can be distinguished as separate entities. The Rayleigh criterion defining resolution in classical optical microscopy gives δ, this smallest resolved distance by the following equation: δ = 0.61λ µ sin β (2.1) Where λ is the wavelength of the radiation, µ the refractive index of the medium, and β the semi-angle of the magnifying lens collection (µ sin β is the numerical aperture(NA)). The wavelength of the radiation is clearly a limitation for resolution: in the case of visible light, with roughly 400nm < λ < 700nm, and taking NA equal to 1, the resolution cannot go below a few hundreds of nanometers. De Broglie's work on the wave characteristic of electrons showed that they could be used to overcome this intrinsic limitation of optical microscopy. Considering an electron (mass m 0 , charge e, and particle momentum p), it can be associated to a wave with a wavelength λ = h/p. The relativistic wavelength of an electron accelerated through a potential difference V is given by the following: λ = h 2m 0 eV (1 + eV 2m 0 c 2 ) (2.2) A first information is given by this relation, which is that increasing accelerating voltage leads to a decrease in electron wavelength, which means higher resolution. Secondly, considering an accelerating voltage of 100 kV, a theoretical resolution of about 4 pm could be reached, going below the atomic radius. This theoretical resolution has not yet been reached, because of our inability to fabricate ideal electron lenses. Decades of development have allowed for the TEM resolution to decrease significantly, and the recent emergence of aberrationcorrected microscopes has brought it down to 50 pm. In a TEM, a high energy electron beam is formed, then the electrons interact with the specimen, some of the electrons will be transmitted through the specimen, and an image of the specimen is projected onto a screen. The interaction between electrons and matter is many-fold, and is at the heart of the versatility of TEM as a characterization technique. These interactions are detailed in the next section. Electron-matter interactions Figure 2.8 depicts the various signals that can be generated from the interaction between an electron beam and a thin specimen. When an electron interacts with a specimen, it can be either transmitted, or scattered. The transmitted beam is parallel to the incident beam. The scattering processes can be subdivided into two categories: elastic scattering, when the electron's energy is unchanged through the process, and inelastic scattering, when the electron looses a measurable amount of energy. Secondary Elastic scattering The most important part of elastic scattering (and of scattering as a whole) is electron diffraction: when interacting with a crystalline specimen, the electron Chapter 2. Experimental methods beam is scattered and the scattered waves interfere to form diffraction patterns with high intensities in well-determined directions. Those diffraction patterns can be acquired and studied in a TEM, and carry structural information on the specimen. The interference between diffracted beams and the transmitted beam give rise to high resolution imaging, which will be explained later. The second, less significant, type of elastic scattering is the large-angle deviation elastic scattering. The interactions between the incident beam and the specimen are Coulombic, and it interacts with the electron cloud as well as the nucleus. When the incident electrons interact with atom nuclei, large-angle deviation elastic scattering (backscattering when the deviation angle reaches 180 • ) occurs. Certain microscopes are equipped to detect these electrons for imaging purposes. Inelastic scattering When colliding with electrons from the specimen, the incident electrons give away a part of their kinetic energy, which constitutes an inelastic interaction and results in the generation of various signals. This electron energy loss can be measured by collecting the inelastically scattered electrons and dispersing them depending on their kinetic energy, this is known as electron energy loss spectroscopy (EELS). The lower energy loss interactions correspond to valence to conduction band transitions, then surface and volume plasmons. Energy losses at higher energies correspond to the ejection of core electrons, which are characteristic of the atom and make an elemental analysis of the sample possible. In this latter case, the vacancy created is filled by an electron from an outer-shell of the electron cloud. This relaxation is either accompanied by the emission of an X photon, or the ejection of a weakly bound electron called an Auger electron. The energies of the X-ray photons generated by these inelastic interactions are characteristic of the atom, and their measurement through X-ray dispersive spectroscopy (EDX) can therefore give quantitative chemical information on the sample. The low mean free path of Auger electrons implies that they only leave the specimen if they are generated at its surface, and are not used for characterization in TEM. The electrons ejected by the collisions with incident electrons are called secondary electrons, and are generally, as the Auger electrons, not used for characterization. In the case of a semiconducting specimen, the energy obtained from the incident electron can generate an electron-hole pair: an electron from the valence band is promoted to the conduction band. The hole-electron recombination is accompanied by the emission of a photon. This process is known as cathodoluminescence (CL), and gives insights into the optical and electronic structure of the material. It is mostly studied in SEMs, which are operated at much lower voltages than TEMs. Basic operation principles of TEM Drawing an analogy with optical microscopy is very useful for understanding how a TEM works. In optical microscopy, an incident light beam passes through (and interacts with) a thin object, and an image of this object is formed through a simple optical lens system. In TEM, the basic concept is exactly the same, aside from the fact that electrons are used instead of photons, and electromagnetic lenses drive the electrons through a column under high vacuum. Electromagnetic lenses consist of copper coils that create a magnetic field between two polepieces. The intensity of the current in the coils allows the control of the electron's trajectory, and the modulation of the focalization of the beam. They can be considered as convex optical lenses with controllable focal length. A TEM can therefore be understood in a first approach, by analogy with the optical microscope, by using a geometric optics formalism. Figure 2.9 shows the schematic representations of a TEM, using geometric optics to depict image formation. The process of obtaining the image of an object with a TEM can be divided into four steps: 1. Electron beam formation 2. Electron-sample interaction 3. Image formation Image acquisition The interaction between the electron beam and the specimen has already been discussed, we therefore only focus on the other three steps in the following. Forming the electron beam Depending on the microscope, different electron sources can be used. The two main kinds of electron sources are thermoionic and field-emission sources. The first generates electrons when heated, and the latter when a strong electric potential is applied between the source and an anode. When heating any material and if the energy thus provided to its electrons is higher than their workfunction, the material is turned into an electron source. However, if the material is heated too much, it can melt or evaporate. A good thermoionic source is therefore either a high-melting point material, or a material with a very low workfunction. In the first microscopes, a tungsten filament was used as a thermoionic source because of its refractory characteristic; today the microscopes that still use thermoionic sources use lanthanum hexaboride (LaB 6 ) crystals because of its low workfunction. The main disadvantage of thermoionic sources is their "high" energy dispersion (∆E = 1.5 eV), which limits the resolution of the microscope because of chromatic aberrations. The lifetime of a LaB 6 source is also quite low and it has to be changed often. Field-emission sources are commonly referred to as FEGs (for field-emission guns). They are based on the fact that an electric field is increased at sharp points. By applying a several kV potential difference between a 100 nm tip (tungsten single crystal) and an anode, an electrostatic field as high as 10 10 V/m is obtained. This induces the generation of an electron current through a tunneling effect. Impurities and oxidation at the surface of the tip may hinder the field-emission effect. Operating in ultra-high vacuum conditions (P< 10 -9 Pa) while keeping the tip at room temperature can be a solution, this is the principle of the cold-FEG. Another solution is to heat the tip, which actually leads to a hybrid thermoionic and field emission process. Such FEGs are known as Schottky, and their tip is a tungsten single crystal coated with zirconium oxide (ZrO 2 ) which enhances electron emission (low worfunction) and helps preserving the tip. FEGs provide the TEM with a much more temporally coherent electron source (∆E = 0.7 eV for a Schottky FEG, and ∆E = 0.3 eV for a cold FEG) with a much higher current density, and they last longer. The electrons are then progressively accelerated by a series of anodes, with the chosen accelerating voltage which is typically between 80 and 300 kV. Whatever the type of electron source, we can consider it to be punctual, meaning that the electron beam is a spherical wave before entering the condenser. The condenser is a series of electromagnetic lenses that control the size and convergence angle of the beam illuminating the sample. In conventional TEM, the condenser allows for parallel illumination of the specimen (planar wave). Image formation When the planar wave of incident electrons interacts with the specimen through the electrostatic atomic potential V( -→ r ), the wave of electrons at the exit of the specimen Ψ t ( -→ r ) is the solution of the Schrödinger equation. When considering a perfect crystal, the solution is written as a sum of planar waves in the reciprocal space. After the electron beam has traversed the specimen, the objective lens will first form a magnified image of the object. The objective lens forms the Fourier transform (FT) of Ψ t ( -→ r ) in its back focal plane, and the magnified image is formed by inverse Fourier transformation. The image is then projected onto the screen (or detector) by the projection system. An area-selection diaphragm can be placed in the diffraction plane (back focal plane of objective lens) for different imaging modes. Selecting the transmitted beam leads to bright field (BF) imaging, the diffracting parts of the specimen Chapter 2. Experimental methods are dark. Selecting the diffraction spot from a specific crystallographic plane family is known as the dark field (DF) mode. Acquisition system In order to obtain an image from the observed object, electron intensity needs to be converted into light intensity. This is done by two different technologies on traditional TEMs. The image can be directly projected and viewed onto a fluorescent screen, or digitalized and observed on a computer screen via a CCD (for charge-coupled device) camera. The fluorescent screen is located at below the projection system, and is coated with a material emits photons (in the 450 to 550 nm wavelength range) when excited with electrons. As shown in the previous section, the signals generated by the interaction between the electron beam and the sample are not limited to images or diffraction patterns. In order to collect those various signals, TEMs are equipped with other adapted detectors such as X-ray detectors, and high-angle annular dark field (HAADF) detectors, which will be mentioned later in this manuscript. HRTEM In conventional TEM, a certain part of the electron beam is selected after it has passed through the specimen. In so-called high resolution TEM (HRTEM) mode, the image is constructed through the interference of diffracted and transmitted electrons. A high resolution image is an interferential image, whose contrast is determined by the phase shift between the interfering waves. In the case of a very thin specimen, of ∆z thickness, the exit electron wavefunction only undergoes a phase shift Φ( -→ r ) defined as σV p ( -→ r )∆z, where σ is the relativistic interaction constant, and V p ( -→ r ) is the projected atomic potential responsible for the scattering of the electron beam. The transmitted wavefuncion can therefore be written as: Ψ t ( - → r ) = Ψ 0 ( - → r ) exp iσV p ( - → r )∆z (2.3) Within the thin and weakly diffracting object consideration, the exponential can be approximated using a Taylor expansion, which leads to: Ψ t ( - → r ) = Ψ 0 ( - → r ).[1 + iσV p ( - → r )∆z] = Ψ 0 ( - → r ) + iΨ 0 ( - → r )σV p ( - → r )∆z) (2.4) The transmitted wavefunction is therefore the superposition of two planar waves, one being the incident wave propagated through ∆z, and the other a wave having undergone a diffusion of small amplitude, and with a π/2 phase shift. This defines the so-called weak phase approximation which applies in the case of SWCNT observation, because they are very thin objects (two atomic layers), with very light atoms, therefore diffracting the electron beam weakly. The observed image in TEM does not directly correspond to the exit wavefunction, because it is affected by the imperfections of the electron source and of the lenses of the microscope, the most important for the HRTEM mode being the objective lens, called aberrations. The phase of the electron waves is affected by the spherical aberration, defocus (∆f ), and astigmatism of this lens, while the chromatic aberrations only affect their amplitude. We only consider the phase shift induced by aberrations, and the most important components are the spherical aberration (C s ), and the defocus. The function that represents this effect of the microscope on the phase of the electron wave is called the coherent transfer function and is defined as follows: T (ν) = exp[-iπλ(C s λ 2 ν 4 + ∆f ν 2 )] (2.5) Where ν is the spatial frequency, λ the wavelength, and C s is the spherical aberrations coefficient. In the reciprocal space, the transfer of the microscope is represented by the multiplication of the FT of the exit wavefunction by the transfer function. Within the weak phase approximation, and considering a normalized incident wavefunction, this leads, in the real space, to the following image wavefunction: Ψ i ( - → r ) = F T -1 [Ψ( - → r )] = 1 + iF T -1 [exp(iχ(ν)).F T (Φ( - → r ))] (2.6) Where Ψ( -→ r ) is the wavefunction in the object focal plane of the objective lens. Without taking into account magnification, we then define the image intensity by the square of the amplitude of Ψ i ( -→ r ), which leads to: I( - → r ) = 1 -2F T -1 [sin(χ(ν)).F T -1 (Φ( - → r ))] (2.7) The contrast is defined by Γ( -→ r ) = I( -→ r ) -1. The contrast is optimized in a HRTEM image when the phase shift induced by the microscope is the same for all spatial frequencies ν used for image formation. This is achieved at the so-called Scherzer defocus: ∆f = -1.2(C s λ) 1/2 (2.8) TEM applied to SWCNT characterization Depending on the resolution of the microscope, the information that can be extracted from the TEM image of a SWCNT varies. Using conventional TEM, it is possible to measure the diameter of the nanotubes with significant precision. Using HRTEM on a microscope with a high enough resolution (better than the C-C distance), the chirality of the nanotube can be unambiguously determined. Image of a SWCNT and diameter determination A SWCNT, being made of a rolled-up atomic sheet of graphene, the weak phase object approximation applies, if the nanotube is observed in a longitudinal projection. The mean projected potential is higher at the edges of the nanotube. This defines two parallel lines of highest atomic density. When the TEM has a "low" resolution, only these lines contribute to the contrast image, and the diameter of the SWCNT can be extracted from this image. A study of the contrast images of SWCNTs by Loiseau et al. using image simulations [START_REF] Loiseau | Understanding carbon nanotubes[END_REF], [START_REF] Stadelmann | Ems-a software package for electron diffraction analysis and hrem image simulation in materials science[END_REF] has shown that this fringe image varies substantially with defocus. Around the Scherzer defocus, only one dark fringe (called Fresnel fringe) is observed on each side of the SWCNT. When going away from the Scherzer defocus, the fringes tend to widen until other sets of fringes start to appear. The highest contrast is obtained for the Scherzer defocus, and this is where the diameter should be measured. Fleurier et al. [START_REF] Fleurier | Transmission electron microscopy and uv-vis-ir spectroscopy analysis of the diameter sorting of carbon nanotubes by gradient density ultracentrifugation[END_REF] have established a method for the measurement of nanotube diameter from a TEM image. An image is taken of the nanotube at (or very close to) the Scherzer defocus, and an intensity profile is extracted from the image, perpendicularly to the nanotube axis. The diameter is then defined as the distance between the two inflection points of the derivative of the contrast intensity, meaning at the midpoint between the minimum of the dark fringe, and the maximum of the subsequent light fringe. Figure 2.10 shows at TEM image of SWCNTs, and an extracted intensity contrast with a diameter measurement. The method for diameter measurement was validated by TEM image simulations, and it is estimated to give an accurate value of the diameter with an error of 0.5Å. Chapter 2. Experimental methods Selectivity assessment methodology In this work, the main use of TEM for characterization of SWCNT growth samples was the measurement of diameters. For a given sample, the diameter distribution was systematically determined using the same method, so as to accurately compare samples with each other. For each considered sample, we took images of SWCNTs on three different squares of the TEM grid when possible. In some occasions, the grid is badly damaged from the transfer process, and only a few squares remain to be observed, and in other cases, when the growth yield is too low, SWCNTs are present on only a very limited number of squares. On each square, various locations containing nanotubes were randomly selected, and we recorded images containing all the present SWCNTs. All the images were recorded at the same magnification (160 k). The diameters of all the suspended SWCNTs on theses images with a large enough clean section were measured. At least one hundred diameters were measured for each sample, and the diameter distribution histogram was plotted with a 0.1 nm binning. TEM was also a way to simply ensure that the growth product consisted only of SWCNTs, or that SWCNTs were the great majority of products. It is difficult or impossible to determine the percentage of MWCNTs with the other available characterization techniques (SEM and Raman). STEM-EDX for catalyst characterization Though the subject of this section is to present TEM for SWCNT characterization, it seemed important to mention a TEM-derived technique that was used for catalyst characterization in this work, which is scanning transmission electron microscopy (STEM) coupled with EDX. The electron beam can be focused on the specimen, and used as a probe that scans the specimen (similar to SEM). As mentioned previously, EDX consists in the measurement of the X-ray photons generated by the inelastic interactions between the electron beam and the specimen, which have an energy that is characteristic of the energy difference between the two electron shells involved in the transition (as explained previously, the ionized atom returns to its ground state by filling the hole in its inner shell). This energy difference is characteristic of the atom. Phonons are collective vibrational modes of atoms in a solid. They have a strong impact on thermal, and mechanical properties, as well as electron transport. Their study provides valuable information on the probed material. One of the two main techniques available for phonon study is Raman spectroscopy. It consists in shining a monochromatic light using a laser onto a sample. The scattered light is then collected, the Rayleigh component is filtered out, and a grating is used to disperse the scattered light, which is then detected with a CCD detector. The intensity of the scattered light is plotted versus the Raman shift ν 1 . When the excitation wavelength is equal to, or close, to an optical transition energy, the Raman signal is enhanced through electron-phonon coupling. This is known as the resonant Raman effect, which is indispensable to SWCNT characterization and can enhance the signal by a factor of about 10 3 [START_REF] Dresselhaus | Raman spectroscopy of carbon nanotubes[END_REF]. In the case of SWCNTs, both the resonant process and the existence of vHSs in the DOS contribute to the strong enhancement of the Raman intensity. This explains how a single resonant SWCNT can give rise to a measurable Raman signal, which contains information on its phonon structure and electronic properties. The resonant aspect of Raman spectroscopy for SWCNTs has very important implications in the viability of their characterization. Raman spectroscopy for SWCNT characterization The phonon structure of SWCNTs can be theoretically derived from the graphene phonon structure, as for the electronic structure, by the zone-folding approximation. Graphene has a two atom unit cell, and therefore has six phonon branches (three acoustic branches and three optical branches). In the case of SWCNTs, the number of phonon modes depends on the chirality of the nanotube: since the unit cell defined by -→ C h and -→ T contains much more atoms, the number of phonon branches can go up to 150 for an achiral SWCNT, and even more for a chiral tube. The number of Raman-active modes, however, is somewhat independent of diameter and chiral angle. There are 15 Raman-active modes in SWCNTs [START_REF] Saito | Physical properties of carbon nanotubes[END_REF]. Characteristic Raman modes of CNTs The most characteristic Raman signals of SWCNTs, and easiest to measure are the RBM mode, the tangential mode (TM, or the G band), and the D band. Figure 2.16 shows a typical SWCNT Raman spectrum displaying these dispersion, or interaction with a substrate and/or catalyst nanoparticles. The second relation, proposed later by Araujo et al. [START_REF] Araujo | Nature of the constant factor in the relation between radial breathing mode frequency and tube diameter for single-wall carbon nanotubes[END_REF] is: ω RBM = 227 d t 1 + C e d 2 t (2.11) Where ω RBM = 227/d t corresponds to a pristine isolated nanotube grown from water-assisted CVD with negligeable environmental effect, and all other ω RBM values are an upshift from this pristine value, with C e an adjustable constant representing the environmental effect. This relation is considered to be more accurate than the previous law, but the latter is still widely used for SWCNT sample selectivity assessment. Since the RBM is a resonant feature, having determined the diameter using this equation, and knowing the excitation wavelength can hypothetically give even more information on the probed SWCNT. For an RBM signal to appear, we need one of its E ii to be equal to the excitation energy E L , or within the resonant window around E L . Reporting to the Kataura plot, the chirality of the resonant nanotube can hypothetically be determined. This (n,m) indexation process however requires precise calibration based on Raman characterization of isolated SWCNTs of known chirality, for instance. The semiconducting or metallic character of the probed nanotube can nonetheless be quite accurately determined with this method. The G band corresponds to vibrations in the direction parallel to the nanotube surface (see Figure 2.16.c). This vibration mode is also present in 2D graphite, but its multiband structure for SWCNTs is unique. The G band is located on the Raman spectrum between 1585 cm -1 and 1595 cm -1 , the peak observed at the highest Raman shift is called the G + band, and the others are called G -. In the case of metallic nanotubes, the G -band has a broad and asymmetrical profile called the Breit-Wigner-Fano lineshape [START_REF] Brown | Origin of the breit-wigner-fano lineshape of the tangential g-band feature of metallic carbon nanotubes[END_REF]. The G band is useful in SWCNT characterization to control sample heating by laser irradiation. It is known to shift towards lower frequencies when the sample is heated [START_REF] Chiashi | Cold wall cvd generation of single-walled carbon nanotubes and in situ raman scattering measurements of the growth stage[END_REF]. Monitoring the frequency of the G band allows for proper setting of the laser power during acquisition. At lower Raman shifts, at about 1350 cm -1 is the D band. Its frequency depends on the excitation wavelength. It is theoretically not Raman-active, and its presence on the Raman spectrum stems from a double-resonance process where electrons are scattered by defects in the sp 2 carbon structure. For this reason, its intensity is indicative of the quantity of defects in the probed SWC-NTs. It can also be indicative of the presence of other carbonated structures, like amorphous carbon. The D band is quite useful to have insights into the crystalline quality of nanotubes. A commonly used factor for this is the ratio of the intensity of the D band, over that of the G band (I D /I G ), it can be used to compare samples, but does not have quantitative value. Selectivity assessment by Raman spectroscopy Diameter distribution assessment As explained in the previous section, the RBM is a specific spectroscopic signature of a SWCNT. Being able to determine the diameter of the probed nanotube directly means that by accumulation of RBM frequencies from SWCNTs from a given sample gives access to the diameter distribution of the sample. The diameter selectivity can therefore be assessed through RBM acquisition. The RBM frequency is very sensitive to the environment surrounding the SWCNT. For a SWCNT within a bundle, the RBM frequency can be increased by up to 20 cm -1 [START_REF] Venkateswaran | Probing the single-wall carbon nanotube bundle: Raman scattering under high pressure[END_REF], a study comparing two different surfactants as a dispersion medium for SWCNTs in solution shows substantial shifts in RBM frequencies [START_REF] Maultzsch | Radial breathing mode of single-walled carbon nanotubes: Optical transition energies and chiralindex assignment[END_REF], and shifts of up to 12 cm -1 have been observed between suspended SWCNTs and SWCNTs supported on SiO 2 [START_REF] Zhang | Substrate-induced raman frequency variation for single-walled carbon nanotubes[END_REF]. Since ω RBM is sensitive to the environment, the law used to determine the diameter of the SWCNT has to reflect the environment representative of the sample. Table 2.2 gives a non-exhaustive overview of the different w RBM = A/d t + B found in the literature, testifying to the wide range of possible choices. The resonant aspect of Raman spectroscopy implies that only the SWCNTs with an E ii within the resonance window around the excitation energy E L may be detected. Using only one laserline would drastically restrict the pool of detectable SWCNTs within the actual sample total population. To prevent this restriction and have an assessment that is as accurate as possible, using several laserlines is crucial. [START_REF] Araujo | Resonance raman spectroscopy of the radial breathing modes in carbon nanotubes[END_REF]. SWCNT sample type A (cm -1 nm) B (cm -1 ) Supergrowth CVD SWCNTs [START_REF] Araujo | Nature of the constant factor in the relation between radial breathing mode frequency and tube diameter for single-wall carbon nanotubes[END_REF] 227 0 Alcohol CVD SWCNTs on quartz [START_REF] Araujo | Third and fourth optical transitions in semiconducting carbon nanotubes[END_REF] 217 15 CVD SWCNTs on SiO 2 (Fe catalyst) [START_REF] Jorio | Structural (n, m) determination of isolated single-wall carbon nanotubes by resonant raman scattering[END_REF] 248 0 Free-hanging isolated SWCNT [START_REF] Paillet | Raman active phonons of identified semiconducting singlewalled carbon nanotubes[END_REF] 204 27 HiPco SDS-dispersed SWCNTs [START_REF] Bachilo | Structure-assigned optical spectra of single-walled carbon nanotubes[END_REF] 223.5 12.5 HiPco SDS-dispersed SWCNTs [START_REF] Jorio | Resonance raman spectroscopy (n, m)-dependent effects in small-diameter single-wall carbon nanotubes[END_REF] 227 17.3/11.8 CVD SWCNTs on SiO 2 235.9 5.5 (Fe, Ni, Co-W catalyst) [START_REF] Zhang | n, m) assignments and quantification for single-walled carbon nanotubes on sio 2/si substrates by resonant raman spectroscopy[END_REF] Semiconducting to metallic ratio assessment Having determined the diameter of a SWCNT from its measured RBM frequency, and knowing the E L , makes it possible, by reporting back to the Kataura plot, to discriminate a semiconducting nanotube from a metallic one. Using the same statistical approach as for diameter assessment, the selectivity on electronic properties can also be assessed. And as for the RBM frequencies, the E ii of SWCNTs are affected by their environment. The E ii are for instance observed to increase for an isolated SWCNT, as opposed to a bunddled SWCNT [START_REF] Fantini | Optical transition energies for carbon nanotubes from resonant raman spectroscopy: Environment and temperature effects[END_REF], and were seen to vary when using different surfactants for solution dispersion [START_REF] Maultzsch | Radial breathing mode of single-walled carbon nanotubes: Optical transition energies and chiralindex assignment[END_REF]. Using the Kataura plot adapted to the SWCNT sample is therefore important. As mentioned in Chapter 1, the original version of the Kataura plot [START_REF] Kataura | Optical properties of single-wall carbon nanotubes[END_REF] is based on a simple nearest-neighbor tight-binding model, within the zone folding approximation. As revealed by experimental results very early on, the importance of self-energy and excitonic effects result in an overall blue-shift of the E ii . As a result, calculations taking the many-body effects into account were performed. Some groups used a so-called extended tight-binding model (ETB) [START_REF] Jiang | Chirality dependence of exciton effects in singlewall carbon nanotubes: Tight-binding model[END_REF], [START_REF] Popov | Comparative study of the optical properties of single-walled carbon nanotubes within orthogonal and nonorthogonal tight-binding models[END_REF], including a large number of neighbors, and σπ hybridization, while others used ab initio calculations, directly describing excitonic effects [START_REF] Spataru | Excitonic effects and optical spectra of single-walled carbon nanotubes[END_REF]. Kataura plots were also elaborated by fitting experimental data. Araujo et al. used resonant Raman spectroscopy with a tunable laser to measure the E ii of as-grown alcohol-assisted and SG SWCNTs, and fitted the obtained data with an analytic equation, leading to the construction of an experiment-based Kataura plot [START_REF] Araujo | Resonance raman spectroscopy of the radial breathing modes in carbon nanotubes[END_REF], [START_REF] Araujo | Third and fourth optical transitions in semiconducting carbon nanotubes[END_REF], [START_REF] Araujo | The role of environmental effects on the optical transition energies and radial breathing mode frequency of single wall carbon nanotubes[END_REF]. Liu energies [START_REF] Liu | An atlas of carbon nanotube optical transitions[END_REF]. Michel et al. plotted a "normalized" Kataura plot by comparing E ii calculated by a non-orthohonal TB model, and coupled ED and Raman spectroscopy of individual free-standing SWCNTs [START_REF] Michel | Indexing of individual single-walled carbon nanotubes from raman spectroscopy[END_REF]. Aside from relative shifts of optical transition energies due to environmental effects, these plots were in overall agreement. In this study, we chose to determine the SC/M ratio of our samples using an easily accessible calculated ETB Kataura plot from [START_REF] Sato | Discontinuity in the family pattern of single-wall carbon nanotubes[END_REF], shown in Figure 2.17. One should keep in mind that this SC/M ratio evaluation is only meant to compare samples, and not to establish any absolute values, which would require to construct a Kataura plot adapted to our samples. Assessing the resonance window for RBMs is also crucial for proper delimitation of SC and M diameter ranges around a given excitation energy. The resonant window is affected by the environment, as well as nanotube chirality. Fantini et al. have shown that the resonance window was 112 meV wide in the case of bundled SWCNTs, and only 65 meV in the case of SWCNTs dispersed in solution using SDS [START_REF] Fantini | Optical transition energies for carbon nanotubes from resonant raman spectroscopy: Environment and temperature effects[END_REF]. Calculations by Park et al. [START_REF] Park | Raman resonance window of single-wall carbon nanotubes[END_REF] have shown that the resonance window is always several tens of meV, and increases with diameter. Souza Filho et al. gave a much higher values [START_REF] Souza Filho | Anomalous twopeak g-band raman effect in one isolated single-wall carbon nanotube[END_REF] of about 200 meV for CVD Chapter 2. Experimental methods grown SWCNTs on SiO 2 . The resonance window is defined as the full width at half maximum energy range over which a given SWCNT is in resonance, meaning that there can be a Raman signal outside of this resonance window. The experiments conducted to determine this resonance window for a given SWCNT in a given environment are performed using a tunable laser over a wide energy range. In our experiments, an RBM peak could be counted outside of the resonance window. For this reason, and because the SiO 2 substrate may have a strong impact on this value, we chose a wider "resonance" width around the excitation energy of ± 100 meV. Blind spots and uncertainties Since resonant Raman spectroscopy is not a characterization technique that provides a direct view of the SWCNT's diameter, or chirality, as opposed to HRTEM or ED, the results obtained from its analysis have to be considered with great caution. Several factors can have a non-negligible impact on the end result when using Raman spectroscopy for growth selectivity assessment for instance. When it comes to determining the diameter of a SWCNTs by measuring its RBM frequency, some uncertainty is expected. Aside from the uncertainty in the measurement of the frequency, which is related to the spectrometer and the grating that is used during the experiment, an additional uncertainty is due to the ω RBM → d t relation that is used to calculate the diameter based on the RBM frequency. As explained previously, this relation depends on the specific type of SWCNT sample that is being probed. The error linked to the relation that is used will also obviously have an impact when trying to determined the metallic or semiconducting properties of the probed SWCNT, and even more so when trying to determine its (n,m) indices. When characterizing not an individual SWCNT, but an ensemble of SWCNTs with a distribution of diameters or (n,m) indices, even more care should be taken for the analysis of the Raman characterization. First, the pool of SWC-NTs that can be probed is determined by the excitation wavelength(s) used, as only SWCNTs in resonance or quasi-resonance with the excitation energy will be detected. Moreover, determining the diameter distribution of the sample may be rendered difficult due to the differences in Raman cross-sections from one SWCNT to another. Further, attempting to evaluate the SC/M ratio, or the (n,m) distribution of a SWCNT sample is an even more challenging task. Indeed, the E ii of a SWCNT, which determine its resonance conditions, depend on the environment. In order to discriminate between a semiconducting or a metallic SWCNT when knowing the RBM frequency and the excitation wavelength for instance, requires very good knowledge of the sample. The means of bias of resonant Raman spectroscopy for the evaluation of the diameter distribution of an as-grown SWCNT sample will be further discussed in Chapter 4. Methodology All Raman spectroscopy characterizations were performed using a HORIBA LabRam ARAMIS spectrometer, using a x100 objective, with four excitation wavelengths (473 nm, 532 nm, 633 nm, and 785 nm). The spectra were acquired for 1s using a 1800 groove per mm grating, giving a spectral resolution of 0.7 cm -1 . In order to obtain reliable statistical data for diameter distribution assessment, we proceeded systematically with the same method. Raman mappings were recorded on the surface of the characterized wafer. The dimensions of the mappings were 50 µm by 50 µm, with a 5 µm step in both directions, which amounted to a total of 121 recorded spectra for each mapping. We used a large enough step to avoid detecting the same SWCNT twice in a row, but small enough not to avoid going over a too large area in one mapping, and possibly losing focus conditions (see Figure 2.18). The mappings were done on random locations on the substrate with the four excitation wavelengths. Depending on the yield of a given CVD growth, the number of mappings varied from one sample to the other, aiming at counting at least one hundred, to several hundreds of RBM peaks (when possible) total. The same number of mappings was of course recorded with each laser. The laser power and acquisition time were carefully adjusted in order to have significant RBM signal in a short enough time without heating the sample. The spectrometer was calibrated for a Si signal at 520.7 cm -1 . The obtained data were analyzed as follows: for every spectrum of every mapping, each RBM peak (with an intensity higher than three times the background intensity) was counted as one SWCNT. They were fitted with a Lorentzian profile, and their frequency was measured. We consider the uncertainty on the RBM frequency to be of 1 cm -1 . The diameters of all the counted SWCNTs Chapter 3 Growth of SWCNTs using Prussian Blue and its analogs as catalyst precursors It has been established in the first chapter of this manuscript that catalyst engineering is a key point for selective growth of SWCNTs using CVD. Here, we present a new general synthesis method with the aim of producing homogeneous bimetallic alloy nanoparticles, with a controlled size and morphology, able to catalyze the growth of SWCNTs. The basics of this catalyst synthesis is to use preformed stoichiometric PBA nanoparticles. Catalyst nanoparticles are then prepared in situ in a hot filament CVD reactor with subsequent high temperature treatment in reducing atmosphere prior to SWCNT growth. With this method, we hope to enable the comparison between catalyst nanoparticles with different behaviors in their catalytic activity, and potential role in growth selectivity. For this reason, multiple catalyst systems are tested. A monometallic iron catalyst system, derived from the original PB structure, as well as two bimetallic systems derived from easily synthesized PBA systems (NiFe, and NiCr) were tested in the following study. The reduction of the NiFe PBA nanoparticles is expected to yield alloyed nanoparticles, having a lower carbon solubility than pure Fe, and the reduction of the NiCr PBA nanoparticles is expected to lead to a phase segregation. These assumptions are based on the bulk binary phase diagrams of both systems (see Appendix B.1). The first aim of this study is to successfully synthesize and characterize several PB and PBA pre-catalyst systems, which will be presented in the first section of this chapter. Then, a crucial objective was to ensure the possibility of forming well-defined catalyst nanoparticles, with a controlled size, and in the case of the bimetallic NiFe system, as nanoalloys, with a controlled composition. This is discussed in the second section of the chapter. The abilities of 3.2. Synthesis and characterization of catalyst precursors: PB and PBA nanoparticles 91 the wafer, forming a homogeneous monolayer. The catalyst precursor-coated wafer is then placed in a CVD reactor, where the catalyst nanoparticles will be formed by reduction under a H 2 atmosphere (at growth temperature) of the PB/PBA nanoparticles into metallic nanoparticles. The carbon precursor in directly introduced after 5 minutes of reductive pretreatment, and SWCNT growth takes place. After growth, the samples can be taken out of the reactor for characterization. In the next sections, we will discuss each step of this process, along with the characterizations that help to understand them, starting from the synthesis of the catalyst precursors, up to the growth of SWCNTs. As explained in Chapter 2, the synthesis method employed here has been developed in our laboratory for the past decade. The three precursors are synthesized in very similar conditions, according to the method presented in [START_REF] Catala | Nanoparticles of prussian blue analogs and related coordination polymers: From information storage to biomedical applications[END_REF]: • K 4 F e II (CN ) 6 + F e(N O 3 ) 3 → K x F e III [F e II (CN ) 6 ] y • K 3 F e III (CN ) 6 + N iCl 2 + CsCl → Cs x N i II [F e III (CN ) 6 ] y • K 3 Cr III (CN ) 6 + N iCl 2 + CsCl → Cs x N i II [Cr III (CN ) 6 ] y The first system is the monometallic PB system, and the other two are bimetallic PBA systems. In all three cases, an alcaline cation is present in the structures' tetrahedral interstitial sites with variable stoichiometry: K + in the case of PB, and Cs + in the cases of the two PBA systems (A cation mentioned in Chater 2.1.1). In order to simplify the notations in this chapter, the three systems will be referred to using the two metal ions on each side of the CN bridges in the PB/PBA structure: FeFe for PB, NiFe, and NiCr for the two PBA systems. Further in the chapter, this notation will also be used for referring to the catalyst systems for SWCNT growth, keeping a clear distinction between the effective catalyst, meaning the nanoparticles obtained from the reduction of the PB/PBA nanoparticles, and the catalyst precursor, meaning PB/PBA system. Chapter 3. Growth of SWCNTs using Prussian Blue and its analogs as catalyst precursors For the synthesis of PB nanoparticles, an aqueous solution of F e(N O 3 ) 3 at 0.5 mM is poured into a F e II (CN ) 6 solution at 0.5 mM under high stirring, after having been cooled down to 2 For all studied systems, the as-obtained PB/PBA nanoparticles are stable in water for at least several weeks without any stabilizing agent, which leaves their surface available to get coordinated to an organic ligand. Characterization of the obtained nanoparticles The as-obtained PB/PBA nanoparticles were fully characterized to measure their sizes, and confirm their structure, and composition. For TEM investigation, a few drops of the solution containing the as-synthesized nanoparticles are deposited on a Cu TEM grid coated with a full carbon membrane. For size measurements using dynamic light scattering (DLS), the measurements were performed directly using the as-obtained solution. For the X-ray powder diffraction (XRPD) and infrared (IR) spectroscopy, the nanoparticles were precipitated by adding an excess of calcium chloride (CaCl 2 ) to the aqueous dispersion. The solution is centrifuged, the resulting powder is rinsed using DI water, and dried under vacuum. A quick presentation, along with experimental details, on all characterization techniques used in this chapter are given in Appendix A. Size DLS allows the measurement of the mean hydrodynamic size of the nanoparticles dispersed in the solution. This measurement is performed as a routine check of our PB/PBA nanoparticle syntheses, it gives a general idea of the sizes and dispersion state of the synthesized objects. 3.2. Synthesis and characterization of catalyst precursors: PB and PBA nanoparticles 95 PB nanoparticles are slightly bigger than the NiFe and NiCr PBA nanoparticles. According to TEM analysis, however, NiFe and NiCr PBA nanoparticles are synthesized with the same mean sizes, whereas DLS gives a slightly lower hydrodynamic size for the NiCr system. Since TEM is a more accurate way to determine the size distributions of the particles, we assume that the difference between the DLS measurements between the NiFe and NiCr systems is due to the enhanced error margin when using DLS for the characterization of low concentration dispersions. The size analysis by DLS and TEM imaging demonstrates our ability to synthesize PB/PBA nanoparticles for the three mono-and bimetallic systems presented in this study. The particles are obtained individualized as stable dispersions in water, with similar, and well-controlled sizes. The obtained sizes are ideal for reduction into metallic nanoparticles, when considering the substantial decrease in lattice parameter between PB/PBA and the zero oxidation number corresponding compounds. In order to confirm that the nanoparticles detected by DLS, and observed by TEM are PB/PBA nanoparticles, the crystallographic structure was studied. Crystallographic structure The crystallographic structures of the PB and PBA nanoparticles were verified using XRPD. The XRPD patterns obtained for the PB and PBA precipitated and dried nanoparticles are presented in Figure 3.4. The PB/PBA FCC structure was confirmed for all three systems, and the lattice planes are indexed on the patterns. The lattice constants were extracted from the patterns, and are given in Table 3.1, along with the calculated correlation domains using the Scherrer relation [212]: τ = Kλ βcos(θ) (3.1) Where τ is the mean crystalline domain size, K is the shape factor (taken as 0.89), λ is the wavelength of the X-ray beam, β is the full width at half maximum (FWHM) of the diffraction peak, and θ is the Bragg angle. The calculated lattice constants are 10.19 Å, 10.31 Å, and 10.51 Å for FeFe, NiFe and NiCr, respectively. They are all in good agreement with values obtained in the literature [START_REF] Buser | The crystal structure of prussian blue: Fe4 [fe (cn) 6] 3. xh2o[END_REF], [START_REF] Brinzei | Spontaneous stabilization and isolation of dispersible bimetallic coordination nanoparticles of cs x ni [cr (cn) 6][END_REF]. The Scherrer domain sizes are calculated as an average on the three most intense peaks for the NiFe and NiCr PBA systems, and directly evaluated from the most intense peak for the FeFe system. The domain sizes are overestimated compared to the sizes determined by TEM, but this is to be expected as the bigger nanoparticles constitute a higher contribution to the intensity of the peaks in the XRD pattern [START_REF] Brinzei | Spontaneous stabilization and isolation of dispersible bimetallic coordination nanoparticles of cs x ni [cr (cn) 6][END_REF]. Moreover, the uncertainty associated with this relation increases when measuring sub-10 nm objects. This error is probably even more marked in the case of the FeFe system, were the calculation was done using only one peak, since the too small amount of powder (directly related to the low concentration of the aqueous dispersion of nanoparticles) derived from the nanoparticle synthesis led to low intensity and a high signal-to-noise ratio on the XRPD pattern. Chemical analysis IR spectroscopy was performed on all systems in order to further confirm the PB/PBA network formation. The spectra are shown in Figure 3.5, in the vincinity of the CN stretching spectral range (2300 -1800 cm -1 ). The spectra obtained for the PB/PBA powders are superimposed with the corresponding hexacyanometallate complex spectrum. In each case, the CN band in the PB/PBA system is blue-shifted in comparison to the CN band of the complex, attesting to the formation of the PB/PBA coordination network by cyanide bridging. In the case of the FeFe system, a broad band is observed around 2072 cm -1 , in good agreement with values seen in the literature [START_REF] Ghosh | Infrared spectra of the prussian blue analogs[END_REF]. For the NiCr PBA, the bridging CN ligands (Ni II -CN-Cr III ) stretching mode can be observed at 2171 cm -1 , and a shoulder corresponding to terminal CN ligands, present at the surface of the nanoparticles at 2130 cm -1 . In the case of the NiFe system, two intense peaks are observed for the bridging CN ligands. This has been previously observed [START_REF] Heurtaux | Nanoparticules de réseaux de coordination à ponts cyanures: Systèmes superparamagnétiques, photomagnétiques et multi-composants[END_REF]. The band at 2164 cm -1 , blue shifted compared to the free CN stretching mode observed at 2118 cm -1 for the hexacyanometalate K 3 Fe III (CN) 6 , corresponds to the CN in Fe III -CN-Ni II . The second band, which is red shifted compared to the free CN in the hexacyanometalate, corresponds to the CN ligands in Fe II -CN-Ni II . We can also observe a shoulder on this last stretching band at 2060 cm -1 , corresponding to the presence of terminal CN ligands (Fe II -CN). In summary, the NiFe system displays a partial reduction of Fe III to Fe II . All the values for the CN vibration bands are given in Table 3.2. For the two PBA systems, SEM-EDX analysis was performed on the powders after precipitation and drying of the nanoparticles. The results of the quantitative analysis of this EDX data are presented in Table 3.3. The obtained atomic fractions roughly correspond to a 1:1 metallic ratio for both systems (Ni:Fe, and Ni:Cr), and a 0.6:1 Cs:M ratio. This characterization indicates that the bimetallic catalyst precursor nanoparticles obtained with this synthesis method are, globally, stoichiometric. as evidenced by DLS. TEM analysis showed narrow size distributions, and individual nanoparticles, with sizes well-suited for use as catalyst precursors for tangential growth of SWCNTs after reduction. The PB/PBA structure was confirmed for all systems through IR spectroscopy and XRPD. The M:M' ratios were determined by a SEM-EDX study, confirming the 1:1 stoichiometry for both bimetallic systems. In the next sections, these FeFe, NiFe, and NiCr PB/PBA nanoparticles are integrated in our process, and all steps are subsequently studied. The next section therefore focuses on wafer preparation, and the PB/PBA nanoparticle deposition step (steps 1 and 2 in Figure 3.1). Sample preparation and PB/PBA precursor deposition Wafer preparation All wafers used in this study are 300 nm thermal SiO 2 coated silicon (SiO 2 /Si) wafers purchased from SiMat. Each sample was cleaned by ultrasonication in dichloromethane (CH 2 Cl 2 ) for 15 minutes, followed by 10 minutes of Ar-O plasma treatment in order to activate the SiO 2 surface. The cleaned SiO 2 /Si wafers were functionalized, according to the synthesis protocol described in [START_REF] Bouanis | Direct synthesis and integration of individual, diameter-controlled single-walled nanotubes (swnts)[END_REF], with a self-assembled monolayer (SAM) of a silane molecule having a terminal pyridine group able to form a coordination bond with the PBA nanoparticles (see Appendix B.2 for details on synthesis). The wafers were immersed in a 10 -3 M solution of the silane molecule in distilled toluene (C 7 H 8 ) for 12 hours. The samples were subsequently rinsed with toluene and dichloromethane followed by annealing at 100 • C for two hours in ambient air. PB/PBA nanoparticle deposition In order to graft the PB/PBA nanoparticles, the silanized wafers were immersed in the as-obtained colloidal PB/PBA solutions for one hour. The excess of nanoparticles was rinsed off with DI water, and the wafers were left to dry under air. The surfaces of the obtained PB/PBA-coated wafers were examined through AFM. Typical AFM images for the three systems are shown in Figure 3.6, for each system, a height line profile is extracted to give an indication of the roughness of the surface. The large 2 µm by 2 µm images clearly show that the PB/PBA nanoparticles are deposited as a homogeneous layer. The observed surfaces appear to be fully covered by nanoparticles, without unwanted deposition of oversized objects, and no zones showing excess thickness are observed. High density of the PB/PBA mat is evidenced by the zoomed-in images also presented in Figure 3.6. The line height profiles show peaks going from, roughly, 1 to 6 nm, in good agreement with the TEM measurements presented in the previous section. We should note that the particle layer seems to be denser for the bimetallic NiFe and NiCr PBA systems, than for the monometallic FeFe PB system. This can be attributed to the difference in concentrations between the mono-and bimetallic PB/PBA nanoparticle dispersions. As it has been said in the previous section, the concentration of the hexa-aquo and the hexacyanometalate complex solutions are four times lower (0.5 mM) in the case of the PB system, than for the PBA systems (2.0 mM), resulting in an overall lower concentration of the particle suspension. Since the deposition rate depends on the concentration, we can assume that a longer deposition time is needed for full coverage for the FeFe system. This could be prevented by a longer deposition time for the PB system, or extracting the nanoparticles, and re-dispersing them with a higher concentration, prior to deposition, but both solutions are time consuming. The density of the nanoparticle layer obtained with a 1 hour deposition time is nonetheless satisfying, as it can help prevent coalescence in the CVD reactor. A control experiment was performed to demonstrate the importance of the anchoring SAM in this process, in the case of the FeFe PB system. A SiO 2 /Si wafer was sonicated for five minutes in a series of solvents (acetone, IPA, CH 2 Cl 2 ), dried, and immersed in the aqueous solution containing the FeFe nanoparticles. After 24 hours, the wafer was taken out of the solution, successively rinsed using DI water and methanol, and dried under vacuum. AFM imaging was performed on the dried wafer in order to evaluate the deposition efficiency. Virtually no nanoparticles were detected (see Appendix B.4), whereas sufficient coverage of the wafer surface is obtained after one hour of immersion time when the wafer is coated with the silane molecule. This confirms the necessity of the SAM, especially in the case of this system, which is obtained as a suspension at a much lower concentration. The presence of the deposited metals was also confirmed by X-ray photoelectron spectroscopy (XPS) for the two bimetallic PBA systems. The global Ni:Fe and Ni:Cr ratios were quantified, so as to confirm that the metallic ratios on Chapter 3. Growth of SWCNTs using Prussian Blue and its analogs as catalyst precursors the surface of the wafers were in agreement with the stoichiometric ratios observed on the PBA powders by EDX. This is an important point. Indeed, since the deposition is directly made from the as-obtained nanoparticle-containing solution, the process presents a risk of anchoring metal cations that have not reacted, which could lead to an effective metallic ratio on the surface that does not correspond to the wanted 1:1 ratio. The XPS-extracted atomic percentages of the present elements, as well as the atomic fractions of the elements of interest in the Cs, M, M' mixture are reported in Table 3.4. In the case of the NiCr system, the 1:1 M:M' ratio was preserved after deposition on the surface. Moreover, the Cs content is left unchanged. However, in the case of the NiFe system, a slight increase in the Ni content (Ni:Fe ratio of 1:0.7), and decrease in the Cs content are observed. First of all, a slight difference can be expected between the EDX and XPS quantifications, owing to the increased error margin of the XPS quantification which can go up to 15 %. However, these changes are significant and though they may be toneddown by the error margin increase, they cannot be fully ignored. The significant decrease in Cs content could be explained by the fact that the EDX characterization was performed on the powder retrieved from the as-synthesized particle suspension, whereas the XPS characterization was performed on the wafer after PBA deposition and thorough rinsing with DI water. It is not excluded that part of the Cs atoms are removed during the rinsing step. When it comes to the increase in Ni content, this may be due to residual adsorption of Ni 2+ cations on the surface. We should also keep in mind that XPS analysis gives a global idea of the chemical composition at the surface of the wafer, and not the composition of individual nanoparticles. Conclusion In summary, the analysis of the surfaces of the wafers after SAM and PB/PBA nanoparticle depositions shows that our process allows for the formation of a homogeneous catalyst precursor nanoparticle layer, with controlled size and composition in the case of the bimetallic systems. In the specific case of the NiFe PBA system, a slight change in the surface composition is observed, as compared to EDX results, which can only be partly attributed to the lower accuracy of XPS when it comes to quantification. The next section focuses on the reductive pre-treatment step that leads to the formation of the effective catalyst nanoparticles, with a focus on the two bimetallic systems. Catalyst nanoparticle formation and characterization After the catalyst precursors have been deposited on the samples, they have to undergo a reductive pretreatment under activated H 2 atmosphere at growth temperature (step 3 in Figure 3.1). The wafers coated by the PB/PBA nanoparticles were placed in the cold zone of the CVD chamber, the chamber was pumped to a 10 -6 mbar pressure, and the temperature was set to 800 • C in the hot zone. After stabilization of the temperature, H 2 was introduced at 100 sccm (standard cubic centimeters per minute) flow rate, and the corresponding filament power of 160 W turned on. Once the pressure in the chamber was stabilized at 90 mbar, the wafers were moved into the hot zone and maintained in the reductive atmosphere for 5 minutes. This treatment is immediately followed by the SWCNT synthesis, by directly introducing CH 4 (without modifying H 2 flow rate) at 20 sccm flow rate and for a corresponding hot filament power of 120 W, while the overall pressure was increased to 100 mbar. In order to characterize the effective catalyst nanoparticles however, the samples were taken out of the CVD chamber right after the 5-minute pretreatment step. Wafer surface analysis after reductive pretreatment The surfaces of the wafers taken out of the CVD reactor before performing SWCNT growth were studied using AFM. In such a way, one can observe the are cooled to ambient temperature and exposed to air, the studied nanoparticles may not be in the exact chemical state as the effective catalyst nanoparticles that remain in the CVD chamber until the end of the SWCNT growth step, and are therefore, not exposed to air and oxidation. Typical AFM images of the catalyst nanoparticle-coated wafers are presented in Figure 3.7. The results are presented in the same manner as the AFM analyses previously shown on the PB/PBA nanoparticle-covered wafers. We can observe that the homogeneity of the nanoparticle layer is preserved after the reductive pretreatment. For all systems, the observed surfaces are fully covered by nanoparticles, evidenced again by the zoomed-in images. Compared to the line height profiles for the PB/PBA deposition step, we can see that, globally, the sizes of the nanoparticles on the surface of the wafers have decreased significantly. This was to be expected since the PB/PBA structures have a much larger lattice constant than the corresponding metals or alloys. After reduction, the catalyst nanoparticles have sizes compatible with tangential growth of SWCNTs, which was the desired result. We should note here that for the NiCr system, the AFM image as well as the featured height profile, reveal that two populations of nanoparticles can be distinguished. Nanoparticles with sizes around roughly 1 -3 nm on one hand, and larger nanoparticles with sizes around 5 -10 nm on the other hand. This is not the case for the FeFe and NiFe systems, which exhibit much more homogeneous sizes. TEM characterization of the catalyst nanoparticles For all catalyst systems, the wafers that were taken out of the CVD chamber before undergoing SWCNT growth were used for TEM characterization. In order to study the effective catalyst nanoparticles, they were transferred onto Cu TEM grids coated with a carbon membrane, using the method previously described in Chapter 2.3.1.5. In this specific case, preventing contamination during TEM observation by residual PMMA and organic solvent impurities was attempted by a more thorough rinsing than for observing SWCNTs (where a PMMA residue can be helpful for maintaining SWCNTs in place over holes in the membrane), and a systemic rinsing in IPA before the acetone evaporated. Chapter 3. Growth of SWCNTs using Prussian Blue and its analogs as catalyst precursors Monometallic FeFe catalyst A rapid TEM characterization of the transferred FeFe catalyst nanoparticles was performed, so as to confirm the narrow size distribution of the catalyst nanoparticles after the reductive pretreatment, as well as the size reduction in comparison to the precursor nanoparticles. The two size distributions (FeFe PB, and FeFe effective catalyst) are displayed in Figure 3.8. The mean nanoparticle size for the effective catalyst nanoparticles is 1.8 ± 0.6 nm, which corresponds to about half of the precursor nanoparticle mean size (3.7 ± 0.9 nm). Since the density of nanoparticles on the TEM grid is extremely low, the sizes of only about 50 nanoparticles were counted (compared to about 200 for the PB nanoparticles). The results are nonetheless consistent with the values extracted from AFM height profile measurements. Bimetallic catalysts A typical TEM image of catalyst nanoparticles after transfer is shown in Figure 3.9, along with the size distribution histograms obtained by statistical analysis of particle sizes for both bimetallic systems (roughly 150 particles for each distribution). Since a thorough rinsing was required for satisfactory observation of the nanoparticles, the surface density found on the TEM grids is extremely low, compared to the aspect of the surface of the wafers observed by AFM. This implies that a lot of material is lost during the transfer process. Chapter 3. Growth of SWCNTs using Prussian Blue and its analogs as catalyst precursors dark field (HAADF) STEM image, elemental maps for the two metallic components, and the relative map obtained by superimposing the two elemental maps, attest that the alloying is effective. Elemental maps were extracted at energies of 6.30-6.50 keV (Fe Kα), and 7.37-7.58 keV (Ni Kα). According to the bulk Ni-Fe binary phase diagram, a solid solution can form at 800 • C for a 1:1 ratio [START_REF] Swartzendruber | The fe-ni (iron-nickel) system[END_REF]. The STEM-EDX data displayed in Figure 3.11.a is in agreement with this, even though arguments based on bulk phase diagrams should be taken with a grain of salt, as they are known to change at the nanoscale [START_REF] Vallee | Size and segregation effects on the phase diagrams of nanoparticles of binary systems[END_REF]. This indicates that our method enables the formation of FeNi alloyed nanoparticles. However, a majority of the observed nanoparticles display a phase segregation, with the appearance of a Ni-rich phase, and a Fe-rich phase. An example of this situation is shown in Figure 3.11.b., where a very clear boundary can be seen between the two phases, like in a so-called Janus configuration. This observed demixion, and the fact that it is not systematic, as some observed nanoparticles are alloyed, raises a few questions. The first question is whether the phase segregation appears upon cooling, and/or exposure to ambient air, or during the reductive pretreatment. In other words, can we suppose that the observed nanoparticles correspond to the actual effective catalyst nanoparticles from which SWCNTs grow in the next step of the process? Since some of the nanoparticles appear to be alloyed and others are not, it is difficult to conclude on whether or not SWCNTs would be grown from alloyed or segregated nanoparticles. Second, we can question the potential role of the size of the nanoparticle in the phase segregation. The study was conducted on nanoparticles whose sizes are not representative of the actual diameter distribution obtained from the analysis of TEM images discussed previously. Figure 3.12 shows a comparison of the nanoparticle size distribution obtained from measurements of around 150 NiFe nanoparticles (showed in Figure 3.9), and the size distribution of the nanoparticles observed in our STEM-EDX study. It clearly shows that the nanoparticles studied by STEM-EDX have sizes that are, for the most part, larger than the mean diameter of the TEM-extracted diameter distribution. It is interesting to note that the three observed "alloyed" nanoparticles have diameters of 2.8 nm, 2.4 nm, and 2.2 nm. They are some of the smallest nanoparticles observed by STEM-EDX. For these small objects, a size-dependent behavior could be expected. We could assume that the behavior of the NiFe nanoparticles that were not observed by STEM-EDX (sizes ranging from 1 to 2 nm) is closer to that of the alloyed nanoparticles (smallest observed nanoparticles) than the Janus nanoparticles. It could also be argued that the nanoparticle displayed in Figure 3.11.a could be a Janus nanoparticle rotated by a 90 • angle (as compared to the particles observed in Figure 3.11.b), whose projection would give the impression of an alloyed system. Regardless, we still have good reason to assume that the nanoparticles are alloyed in CVD conditions. As explained above, the phase segregation could be size-related, as indicated by the measured sizes of the seemingly alloyed nanoparticles. The bulk binary phase diagram predicts a solid solution in growth conditions, and even in the phase-segregated nanoparticles, the Fe (Ni) content in the Ni-rich (Fe-rich) phase is non-zero. This cannot be affirmed in all certainty, but the more probable scenario is that the effective catalyst is a Fe-Ni solid solution with a composition that cannot be predicted. We can therefore expect the behavior of this catalyst with regard to SWCNT growth to differ from that of the FeFe catalyst. We should note that the STEM-EDX analysis allowed to confirm the absence of Cs in the nanoparticles (see spectra in Appendix B.6). The Cs atoms initially present in the PBA structure are expected to sublimate during the pretreatment step, as Cs has a boiling point of 670.8 • C at atmospheric pressure. Moreover, sulfur is found in the nanoparticles. Since the silane molecule used to functionalize the wafer surface contains one atom of sulfur, it is possible that it is the source of the sulfur observed in the nanoparticles. However, the sulfur content the formation of alloyed nanoparticles when the alloy is not thermodynamically favored in the bulk phase. We should also note that again, we were able to confirm the absence of Cs in the nanoparticles, meaning that it will not play any role in the SWCNT growth process further on. Judging from this STEM-EDX study, we can safely assume that SWCNT nucleation and growth from this NiCr catalyst probably arise from the Ni-rich (or pure Ni) phase of the segregated nanoparticle. Indeed, nickel is well known for its ability to catalyze the growth of SWCNTs [START_REF] He | Low temperature growth of swnts on a nickel catalyst by thermal chemical vapor deposition[END_REF], [START_REF] Chen | Single-wall carbon nanotube synthesis by co disproportionation on nickel-incorporated mcm-41[END_REF], [START_REF] Chiang | Linking catalyst composition to chirality distributions of as-grown single-walled carbon nanotubes by tuning ni x fe 1-x nanoparticles[END_REF]. Moreover, the growth of SWCNTs using pure chromium nanoparticles as catalysts has not, to our knowledge, been demonstrated in the literature. A study by Deck et al. even reported a failed attempted CNT growth using a Cr-based catalyst [START_REF] Deck | Prediction of carbon nanotube growth success by the analysis of carbon-catalyst binary phase diagrams[END_REF]. We may therefore consider the effective catalysts to be Ni nanoparticles. This catalyst system could be compared to studies in the literature showing SWCNT growth from segregated nanoparticles, where only one of the metals catalyzes the growth. In a study by He et al. previously mentioned in this manuscript, in situ HRTEM experiments showed that the SWCNT growth stemmed from the Fe-pure shell of a complex Fe-Pt bimetallic catalyst system [START_REF] He | Environmental transmission electron microscopy investigations of pt-fe 2 o 3 nanoparticles for nucleating carbon nanotubes[END_REF]. A study by Cui et al. reported the formation of a phase-segregated Co-Cu bimetallic catalyst system by STEM-EDX investigation, where small Co nanoparticles were anchored onto Cu nanoparticles, and assumed a growth stemming from the Co nanoparticles [START_REF] Cui | Synthesis of subnanometer-diameter vertically aligned single-walled carbon nanotubes with copper-anchored cobalt catalysts[END_REF]. Conclusion In this section, we have showed that our process allows the efficient fabrication of three different catalyst systems. In all cases, the reductive pretreatment led to the formation of a dense layer of catalyst nanoparticles with sizes well-adapted to tangential SWCNT growth. We observed a systematic size reduction in agreement with the difference in lattice parameters between a PB/PBA structure and the corresponding metallic structures. We also confirmed that coalescence was under control both after the reductive pretreatment, and after SWCNT growth. Three catalyst systems with different behaviors have been successfully prepared: a monometallic Fe catalyst with controlled size, a bimetallic NiFe catalyst that can be assumed to be alloyed under growth conditions, and a bimetallic NiCr system that displays phase segregation, by which we can assume a growth stemming from a small monometallic Ni effective catalyst. All of these systems are prepared in identical conditions, and have different behaviors with regard to their interaction with carbon, or evolution with temperature. Chapter 3. Growth of SWCNTs using Prussian Blue and its analogs as catalyst precursors SWCNT growth In this section, we present the results obtained for the growth of SWCNTs (fourth and final step of our process in Figure 3.1) from the three presented catalytic systems elaborated from PB/PBA precursors. We will first focus on SWCNT growth from the FeFe system, which serves as a monometallic reference, and then we will discuss SWCNT growth from the two bimetallic systems. SWCNT growth from the monometallic PB catalyst precursor PB nanoparticles were successfully synthesized, with a controlled size, and their crystallographic structure has been unambiguously confirmed through XRPD and IR characterizations. We have shown that, using a SAM of a silane molecule bearing a pyridine group, we were able to coat SiO 2 /Si wafers with a homogeneous layer of those nanoparticles, as confirmed by AFM measurements. Our AFM and TEM study of the FeFe catalyst nanoparticles after reductive pretreatment of the BP nanoparticles has shown that we obtain a monolayer of catalyst nanoparticles with a controlled size, logically smaller than that of the catalyst precursors. The AFM also shows no sign of coalescence. We have therefore validated our process for the formation of FeFe catalyst nanoparticles. We will now discuss their performance as SWCNT growth catalysts. SWCNT growth feasibility SWCNT growths were performed using the HFCVD reactor presented in Chapter 2.2. We should underline that after the formation of the catalyst nanoparticles by a reductive pretreatment under activated H 2 at growth temperature, the carbon source is directly introduced in the CVD chamber to pursue SWCNT growth. The samples are not in contact with air at any point between their introduction in the chamber (wafer coated with SAM and catalyst precursor) and their removal after the end of the growth process and cooling. The generic conditions of the growth are as follows: H 2 flow rate is maintained at 100 sccm (and the filament power maintained at 160 W), CH 4 is introduced with a flow rate of 20 sccm, the pressure is increased to 100 mbar, the filament at the CH 4 insert is set to 120 W, and the growth lasts 30 minutes. The first growth trial was performed at 800 • C (meaning the pretreatment was also performed at 800 • C). SWCNTs were successfully grown using our Chapter 3. Growth of SWCNTs using Prussian Blue and its analogs as catalyst precursors of SWCNTs present on this surface unit, but we found it was a practical way to compare our samples with regard to yield. The mean number of SWCNTs detected under one laser spot for this sample is 4.2. The good quality of the synthesized SWCNTs is demonstrated by the Raman mapping in Figure 3.14.d recorded in the D and G bands spectral range using a 532 nm laser. The intensity of the D band (around 1350 cm -1) is visibly low, and it rarely exceeds 10 % of the intensity of the G band (around 1590 cm -1). The HRTEM image displayed in Figure 3.14.b, showing the lattice structure of the SWCNTs, also attests to the good crystallinity of the grown SWCNTs. We can also note that the HRTEM image clearly shows a SWCNT linked to the catalyst nanoparticle it grew from (upper right), with a tube diameter to particle size close to 1, indicating a tangential growth mode in this particular case. Right below this, a nanoparticle and a carbon cap, that could have continued as a tangential growth can be observed as well. According study comparing the use of CO and CH 4 for SWCNT growth and their respective effect on growth modes, we could expect tangential growth in the present growth conditions [START_REF] Amara | Modeling the growth of single-wall carbon nanotubes[END_REF]. Unfortunately, observing these occurrences of a tube-particle relation on our transferred samples is rare, and we should note that the lack of statistical data on this specific matter prevents us from drawing any clear conclusions. However, no perpendicular growth tube-particle relations were observed, justifying the assumption that the experimental conditions used in the present study favor the tangential growth mode. We have therefore shown that our growth process does enable the growth of SWCNTs from PB nanoparticles as catalyst precursors. We have obtained high quality SWCNTs with a high yield, and shown the occurrence of a tangential growth mode. Raman study of growth temperature effect SWCNT growths from the FeFe catalyst were then attempted at various temperatures ranging from 700 • C to 1000 • C, with the aim of studying the effect of increasing growth temperature on the resulting SWCNTs. The obtained samples were characterized using SEM and Raman spectroscopy, for the evaluation of relative yield, diameter distribution, SC/M ratio, and SWCNT approximate length. This is a classical phenomenon that has been widely reported in the literature, as discussed in Chapter 1. In most cases, it is explained by an Oswald ripening effect, favored at high temperature because of an increased metal mobility. Iron being a metal with a rather low melting point, this explanation could very well be valid here. Assuming that the SWCNTs grow in tangential mode, a catalyst coarsening phenomenon could explain why the diameter distribution is significantly shifted towards larger diameters at 900 • C, by a global increase in catalyst nanoparticle size. In the present case, however, metal diffusion should be prevented by the presence of hydrogen in the reactor during growth, keeping Oswald ripening and coalescence under control [START_REF] Jeong | Atomic hydrogendriven size control of catalytic nanoparticles for single-walled carbon nanotube growth[END_REF]. Moreover, the size of the catalyst nanoparticles after SWCNT growth at 800 • C were measured by TEM, and the mean diameter was found to be 2.0 ± 0.9 nm (see section 3.4.2.3). The mean diameter is very close to the value obtained from nanoparticles before growth (2 Å shift). As explained previously, some coalescence is observed, but it involves nanoparticles with sizes above 3.0 nm, which are not involved in the tangential growth mode. The predominant population of catalyst nanoparticles does not evolve after 30 minutes at 800 • C under a CH 4 /H 2 ambient, indicating an almost non-existent coalescence, and Oswald ripening. Regarding nanoparticle coarsening phenomena with increasing temperatures (900, and 1000 • C), TEM analyses should be performed to confirm their absence. Since transfers from wafer to grid after SWCNT growth were optimized for high yield SWCNT growths, transferred grids are not available at 900 and 1000 • C. A comparison between catalyst nanoparticle sizes after SWCNT growth at 800 • C and at 1000 • C was performed for a FeRu PBA system in the exact same conditions, and showed no evolution of the catalyst nanoparticle size distribution. This can lead to another possible explanation for the observed evolution in the Raman-extracted diameter distributions. Chapter 3. Growth of SWCNTs using Prussian Blue and its analogs as catalyst precursors As discussed in Chapter 1, Monte Carlo simulations published in the last few years have put carbon content, and solubility in the catalyst nanoparticle at the heart of nucleation and growth mechanisms. A successful nucleation and growth was demonstrated to be the result of a fine balance between sufficient dewetting of the sp 2 carbon wall from the catalyst nanoparticle to ensure cap lift-off, while maintaining a certain adhesion in order for the growth to continue by preventing a detachment of the SWCNT from the catalyst nanoparticle from which it grew [START_REF] Aguiar-Hualde | Probing the role of carbon solubility in transition metal catalyzing single-walled carbon nanotubes growth[END_REF]. It was shown that the carbon content in the catalyst nanoparticle controlled this phenomenon: a low carbon content in the nanoparticle favors wetting, whereas a high carbon content leads to dewetting. In our growth conditions a rather high carbon content is used, and a long growth time, we can consider that the carbon content in the catalyst nanoparticles easily reaches the solubility limit. This implies that we can control the carbon content in the catalyst nanoparticles by tuning the carbon solubility in the nanoparticle. Here, at a fixed nanoparticle composition, two key parameters control the carbon solubility in the catalyst nanoparticles: temperature and size. An increased temperature is known to increase the carbon solubility in metals, and a decrease in the size of nanoparticles also leads to an increase in carbon solubility [START_REF] Magnin | Size dependent phase diagrams of nickel-carbon nanoparticles[END_REF]. To explain the evolution of the Raman-extracted diameter distribution with growth temperature, we can assume the following scenario (see Figure 3.17). Considering that the pool of potentially active catalyst nanoparticles for tangential growth lies within the 0.5 -3.0 nm range: at low temperatures (here 700 • C), small nanoparticles are activated for SWCNT growth, and large nanoparticles are poisoned by carbon encapsulation due to their low carbon solubility and therefore too high adhesion to the carbon wall. At 800 • C, the carbon solubility is increased, and larger catalyst nanoparticles are activated, leading to a slight shift in the diameter distribution towards larger diameters (here: mean Raman-extracted diameter going from 1.11 nm to 1.30 nm). At 900 • C, the carbon solubility in small nanoparticles starts to be high enough to favor tube detachment instead of growth, and even larger nanoparticles are now activated for SWCNT growth. This theory may also explain the absence of SWCNTs at 1000 • C: at this temperature, all nanoparticles have reached the dewetting-favoring regime, precluding any sustainable SWCNT growth. Similar experiments performed by another PhD student using FeRu PBA catalyst precursors show successful growths at 1000 • C of larger diameter SWCNTs with Chapter 3. Growth of SWCNTs using Prussian Blue and its analogs as catalyst precursors information on our samples. Here, Raman characterization is used as a mean to compare samples that were characterized with the exact same method. Effect on SC/M ratio The percentage of semiconducting SWCNTs was extracted from the Raman characterization using the method described in Chapter 2.3.2.3. The results are presented in Table 3.6. First, the results do not differ substantially from one temperature to another. Moreover, all values are close to a 2/3 semiconducting content, corresponding to a random chirality distribution. One can note, however, that the value obtained at 900 • C is above 70 %, we could assume that in a growth were possible in the same conditions at higher temperatures, higher semiconducting contents could be obtained in our samples. These considerations are to be taken with great caution, as Raman spectroscopy may not be suitable to measure the SC/M ratio of a given growth sample, which will be discussed in the next chapter. In summary, our process allowed for the growth of SWCNTs from iron catalyst nanoparticles formed by reduction of PB nanoparticles, demonstrating the effectiveness of this new and original process. We have shown a clear effect of growth temperature on the SWCNTs, by Raman and SEM characterization. An optimum yield is found at 800 • C, and the yield is lower both at lower and higher temperatures, with a zero Raman-extracted "yield" at 1000 • C. Assuming a dominating tangential growth mode, judging both from our growth conditions, and HRTEM experiments, the shift in the diameter distribution can be explained by activation of catalyst nanoparticles of different sizes, linked to the changes in carbon solubility driven by the changes in temperature. The following section aims at testing the feasibility of SWCNT growth with our process from bimetallic PBA precursors, and the study of potential composition effects. SWCNT growth from the bimetallic PB catalyst precursors SWCNT growths were performed in the same generic conditions in the HFCVD reactor, using wafers coated in NiFe and NiCr PBA precursors. According to the STEM-EDX study presented in the previous section, we can assume that the growth from the NiCr catalyst is led by monometallic Ni nanoparticles anchored onto Cr nanoparticles. In the case of the NiFe system, though the exact chemical nature of the effective catalyst is unclear, we can expect a growth from alloyed Chapter 3. Growth of SWCNTs using Prussian Blue and its analogs as catalyst precursors from the NiFe catalyst, after being transferred onto a TEM grid. The two bimetallic catalyst systems led to successful SWCNT growth in usual conditions. As for the monometallic catalyst, growths at various temperatures were attempted, in order to study the influence of the temperature on SWCNT growth from catalysts with different compositions. Raman and SEM study of growth temperature effect As for the FeFe catalyst system, growths were conducted for both the NiFe and NiCr catalyst systems at various temperatures ranging from 700 • C to 1000 • C. Figure 3.19 displays SEM images of the growth samples for the two systems at various temperatures. Despite several trials, the growths at 1000 • C were not successful. As for the FeFe catalyst, we can see a very clear effect of temperature on growth yield, with an optimal growth temperature of 800 • C for both systems. In the case of the NiCr system, the SEM image at 900 • C features only one rather short nanotube, indicating an extremely low density and therefore growth yield. Several Raman spectroscopy mappings on random locations of the sample revealed no detected RBM peaks, i.e. a zero Raman "yield". The calculated Raman-extracted "yields" for all samples will be discussed in the next section. The Raman characterization was performed for all of the successful growth samples, and the resulting diameter distributions are presented in Figure 3.20, along with the diameter distributions obtained for the FeFe catalyst. Looking at the NiFe case, there seems to be little-to-no evolution of the Raman-extracted diameter distribution with increasing temperature. In the NiCr case, the distribution is similar to the ones obtained using the FeFe and NiFe catalysts at 700 • C. At 800 • C, the population of SWCNTs with diameters around 0.8 -1.2 nm decreases drastically, while a population of SWCNTs with larger diameters appears, leading to an overall increase in the mean diameter (which goes from 1.28 nm at 700 • C to 1.44 nm at • C). An overview of the Raman-extracted data for all the presented SWCNT growths is given in Table 3.7. Discussion on the effect of catalyst composition In this section, the three different catalyst systems derived from PB/PBA precursors will be compared, with regard to yield, reproducibility, and growth temperature limit. At 800 • C, we can suppose that the system is in an intermediate temperature condition, where small and large nanopaticles are activated for growth (zone 2 in Figure 3.17): within the pool of potentially active catalyst nanoparticles (size range 0.5 -3.0 nm), a wide range of sizes are active, leading to a higher yield. In that sense, the yield difference between the FeFe and NiFe systems at this temperature also implies a difference in catalyst behavior that could stem from an alloying effect. Another interesting feature here is that the growth temperature limit seems to depend on the catalyst nature. Reproducibility of the Raman spectroscopy results Figure 3.22 shows the superimposition of the Raman-extracted diameter distributions for several different batches of SWCNT growths conducted in the same conditions (at 800 • C), in the cases of the FeFe and NiFe catalyst systems. In the case of the FeFe system, we clearly see that the results are very similar from one growth to another. The same SWCNT growth was performed at least five times with close to identical Raman spectroscopy results. When looking at the NiFe case, we can see that in two cases, the Raman diameter distributions also overlap, but the third is completely different. Similar results have been obtained using the NiCr system. In order to comment on the evolution of the Raman-extracted diameter distribution for the NiFe and NiCr systems, additional experiments should be performed. Indeed, the evolution of the Raman spectroscopy results with increasing temperature would have to be confirmed through growth and characterization ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ System Temperature 700 • C 800 • C 900 • C 1000 • C FeFe XX NiFe XX NiCr XX XX the catalyst nanoparticles barely evolve during the 30-minute SWCNT growth process, as evidenced by TEM measurements. This allowed us to favor the carbon solubility reasoning in comparison to the usually employed Oswald ripening argument to explain the evolution of SWCNT diameter distributions with increasing temperature. The size of the catalyst nanoparticles were also measured after growth at 800 • C for the NiFe and NiCr systems. The mean diameters after growth are 2.1 ± 0.7 nm, and 2.4 ± 1.1 nm, for the NiFe, and NiCr systems, respectively. Here also, the size distributions after growth indicate the absence of catalyst coarsening, and the limited occurrence of coalescence. Moreover, when looking at the size distributions of the effective NiFe (mean size 2.4 nm) and FeFe (mean size 1.8 nm) catalysts at 800 • C, a strict reasoning based on a SWCNT diameter distribution driven by catalyst size would predict a significant difference between the two obtained SWCNT diameter distributions. This is not the case, since the two Raman extracted diameter distributions (see Figure 3.20.a and b) are very similar. For these reasons, we will use the same carbon solubility argument put forward previously in the case of the FeFe catalyst to try to explain these growth temperature limits. In the case of the NiFe system, assuming that the SWCNTs grow from a Ni-Fe alloy of unknown composition, the carbon solubility in the catalyst nanoparticles should be lower, as the carbon solubility limit is lower for Ni than for Fe, than in the FeFe case. A lower carbon solubility within the catalyst nanoparticle would mean a lower carbon content (assuming the solubility limit is always reached) than for the FeFe system at equal temperatures. We could therefore assume that a growth at 1000 • C could be observed, because the higher adherence of the sp 2 carbon to the catalyst nanoparticle could enable to sustain growth at a higher temperature. However, this is not observed, which can imply that the difference in carbon solubility between the FeFe and NiFe catalyst is not sufficient. Chapter 3. Growth of SWCNTs using Prussian Blue and its analogs as catalyst precursors Within this carbon solubility reasoning, the lower growth temperature limit observed for the NiCr system is also surprising. Considering that the SWCNTs grow from a pure, or close to pure, Ni effective catalyst nanoparticle in this specific case, the decreased carbon solubility as compared to the FeFe or NiFe system should delay the detachment of the sp 2 carbon wall from the catalyst nanoparticle (in terms of temperature), which is the opposite of what is observed. Here, the growth temperature limit is decreased. However, this would be considering that the effective catalyst nanoparticles are the same size in all cases. Here, it could be argued that the phase-segregation results in a smaller than expected effective catalyst size. Since a decrease in nanoparticle size has been shown to increase carbon solubility, it could explain the earlier detachment observed in the case of the NiCr system. General conclusion In this chapter, we have shown the synthesis and extensive characterization of nanoparticles of three PB/PBA systems. Using a surface functionalization method, these nanoparticles were successfully deposited on SiO 2 /Si substrates, as evidenced by AFM and XPS characterization. The reduction of these PB/PBA precursor nanoparticles into metallic nanoparticles was then studied. We showed that our new and original process allows for the formation of a homogeneous catalyst nanoparticle layer, with overall controlled sizes for each considered system. The original strategy of using coordination and surface chemistry to anchor pre-formed catalyst precursors with a controlled size, coupled with a reductive thermal pretreatment, is an efficient way to finely control the size of the effective catalyst nanoparticles. The STEM-EDX study of the to bimetallic catalyst systems (NiFe and NiCr) demonstrated the possibility of forming seemingly alloyed nanoparticles when the formation of a solid solution was expected according to the corresponding bulk binary phase diagram. On the contrary, the nanoparticles observed for the NiCr system showed clear phase segregation. Our process allowed for the preparation of three catalyst systems, each showing a different behavior. The formation of catalyst nanoparticle fabrication with three different compositions in the same conditions has rarely been studied with this characterization technique. SWCNT growth was successfully conducted for all catalyst systems at various 3.6. General conclusion 131 temperatures. We observed a clear evolution of the Raman-extracted diameter distribution with growth temperature in the case of the FeFe monometallic catalyst, which could be explained simply by an Oswald ripening effect, but also by considering the temperature as a way to control carbon solubility inside the catalyst nanoparticles, leading to activation of different nanoparticle size populations inducing changes in the SWCNT diameter distribution. Additional experiments regarding the sizes of the catalyst nanoparticles at various temperatures are needed to conclude on the matter. This is further supported by the absence of SWCNT growth at 1000 • C, which could mean that at this temperature, the carbon solubility is such that there is detachment of caps from all nanoparticles. The SC/M ratio did not appear to be strongly modified by growth temperature. The difficulty to determine the actual chemical state of the catalyst nanoparticles during growth, and a lack of strong reproducibility, rendered the interpretation of the Raman spectroscopy results for the two bimetallic catalyst systems quite challenging. However, the STEM-EDX study led to the assumption that SWCNTs grow from a Ni-Fe alloy of unknown composition for the NiFe system, and small pure Ni nanoparticles for the NiCr systems. This led to potential interpretations of the observed growth temperature limits, also based on carbon solubility considerations. The different behaviors observed through Raman characterization for the three catalyst systems show the clear influence of the catalyst composition. Though additional experiments could still enlighten our findings, as well as lead to optimization of growth selectivity, this chapter demonstrated the effectiveness of our process for growing SWCNTs from a variety of catalyst systems, prepared in close to identical conditions. This particular technique for preparing catalyst nanoparticles is simple to implement, efficient, and offers significant improvements compared to usually used physical preparation routes such as co-impregnation coupled with calcination: the composition of each precursor nanoparticle is known and controlled by the inherent structure of the PBA coordination network, the sizes of the precursor nanoparticles are finely controlled, and their deposition and reduction significantly prevent coalescence phenomena. This method can be applied to other PBA systems (bimetallic, or even trimetallic) with more promising compositions. The application of this process, or an adaptation of it, to other families of chemically synthesized catalyst precursors will be presented in Chapter 5. Chapter 3. Growth of SWCNTs using Prussian Blue and its analogs as catalyst precursors In the present chapter, the SWCNTs were only characterized using Raman characterization. The Raman-extracted diameter distributions that were shown in Figure 3.16.b raise questions concerning the potential distortions of results due to the resonant character of the technique. By optimizing a transfer process, we were able to characterize the SWCNTs of the samples with highest yields by TEM, and compared the results to Raman characterization. The results are shown, and extensively discussed in the next chapter. Chapter 4 Comparing TEM and resonant Raman spectroscopy for diameter distribution assessment of a SWCNT sample It was shown in the first chapter of this manuscript that statistical data extracted from more than one characterization technique are rarely compared in publications showing SWCNT growth selectivity. In this chapter, we will explore the comparison between TEM imaging and Raman spectroscopy for determining the diameter distribution of a CVD growth sample, in an effort to validate, or invalidate, our currently used methodology. In the first section, comparisons between TEM-and Raman-extracted diameter distributions on typical CVD growth samples will be shown, and a study on diameter-sorted SWCNTs will then be presented in the next sections, in an attempt to understand the differences observed between TEM-and Raman-extracted diameter distributions on our typical growth samples. Comparing TEM and Raman spectroscopy on typical CVD growth samples The first characterization technique used for our growth sample was Raman spectroscopy because it was a fast way to have an idea of the selectivity depending on growth conditions. It is well adapted to the type of samples that are produced with our technique. TEM characterization was performed later: first by looking at SWCNTs grown through the same process as the usual samples, but on holey SiO 2 membranes deposited on molybdenum TEM grids, and The first sample for which this comparison was performed was a SWCNT growth at 800 • C, in the usual ambient conditions, using PB nanoparticles as a catalyst precursor, but submitting them to a pretreatment under argon. The diameter distributions extracted from the Raman characterization on the wafer on one hand, and the TEM characterization on the other, are presented in Figure 4.1. The distribution extracted from Raman spectra is centered at 1.34 nm, with a standard deviation of 0.26 nm, with more than a quarter of the detected SWC-NTs having a diameter between 1.2 nm and 1.3 nm. Unfortunately, it cannot relevantly be compared with the TEM-extracted diameter distribution, which does not contain enough data to be statistically representative of the sample (only 23 counted SWCNTs). This is due to two main factors. First, it seems that the yield of the growth on the SiO 2 membrane is much lower than the growth on a wafer, in our growth conditions. The difficulty to detect the grown SWCNTs may also be related to the thickness of the SiO 2 membrane, and the fact that it is composed of heavier atoms than a regular carbon membrane, which prevents the detection of the SWCNTs on top of the membrane. Secondly, in order to accurately measure the diameter of a SWCNT, it has to be suspended, and only a few of the observed SWCNTs were crossing the holes in the membrane. This strategy was thus not used further for TEM and Raman comparison. However, looking at this distribution, we can still see that it is shifted towards larger diameters, as compared to the distribution determined by Raman spectroscopy. The mean diameter is 1.81 nm, with a high standard deviation of 0.95 nm. Though it is not statistically representative, this shift is quite surprising, and needs to be examined further. For this, the sample characterized by Raman was transferred onto a TEM grid (using the method described in Chapter 2.3.1.5), with the hope that the higher yield of the growth, and the weaker contrast of the carbon membrane would lead to a much easier detection of the SWCNTs through TEM, and therefore to measurements from a larger number of tubes. TEM images of SWCNTs from this transferred sample, and the corresponding diameter distribution histogram are displayed in Figure 4.2. The transfer method clearly enabled the observation of a significantly larger amount of SWCNTs (45 SWCNTs). The number of measured diameters is of course a lot lower than in the case of the 4.1. Comparing TEM and Raman spectroscopy on typical CVD growth samples 137 locations on the TEM grid, and the fact that the transfer method was not, at the time, fully optimized, leading to significant losses in SWCNTs. After optimization refining, we were able to characterize more samples by TEM, with more statistically representative data. Other samples After more practice with the transfer process, and in particular more careful handling of the PMMA rinsing step, we were able to obtain transferred SWCNT growth samples with less damaged membranes, and thus more available SWC-NTs for diameter measurement. We could therefore compare Raman and TEM characterizations with a better accuracy. This comparison was conducted on two different samples, that were chosen for their significant growth yield, so as to obtain usable TEM grids after transfer. The first sample is a typical growth using the FeFe PB catalyst precursor, at 800 • C in the usual ambient conditions and after undergoing the usual reductive pretreatment. The second is the same CVD growth conducted using the NiFe PBA catalyst precursor. The Raman-extracted diameter distribution is systematically centered at a lower diameter, and is systematically narrower than the TEM diameter distributions. Since Raman spectroscopy does not allow the detection of SWCNTs with large diameters, we also calculated the mean diameters and standard deviations of the two TEM-extracted distributions without taking into account the SWCNTs that could not have been probed with Raman spectroscopy. We considered the lower limit of the RBM spectral range to be 100 cm -1 , which corresponds, using the relation between RBM frequency and SWCNT diameter ω RBM = 248/d t , to an upper diameter limit of 2.5 nm. Aside from these SWCNTs, we distinguished three diameter domains in the distributions: • Domain 1: below 1.3 nm, SWCNT populations seem to be overestimated by Raman spectroscopy and/or underestimated by TEM • Domain 2: between 1.3 nm and 1.7 nm, similar results are given by both characterization techniques 4.1. Comparing TEM and Raman spectroscopy on typical CVD growth samples 139 with a mean difference (TEM-extracted count -Raman-extracted count) of 4.1 % and 4.9 % for the FeFe, and NiFe samples, respectively. 4.1 shows the mean diameters and standard deviations for both samples, as determined by TEM (with and without taking into account the SWCNTs undetected by Raman), and by Raman. In both cases, excluding the SWCNTs with diameters above 2.5 nm in the TEM distribution leads to an evident decrease in the mean diameter, making them a bit closer to the Raman-extracted mean diameters. This modification is however not sufficient to make the mean diameters of the two TEM distributions overlap with their Raman counterparts, as they are still shifted towards higher diameters by about 0.4 nm. In the case of the FeFe sample, the broadness of the distribution is decreased, whereas it is unchanged for the NiFe sample. In both cases, the standard deviations obtained from TEM for SWCNTs with diameters below 2.5 nm are close to the standard deviations obtained from Raman spectroscopy. This indicates that the shift in the diameter distribution is not just attributed to the intrinsic limitation of Raman spectroscopy when it comes to detecting SWCNTs using their RBM mode. In the wide range of diameters where both Raman spectroscopy and TEM imaging enable the detection of SWCNTs, different populations are represented in larger or lower numbers depending on the characterization technique used. In Table 4.2, we show the calculated percentages of the total detected SWCNTs for each characterization technique present in each of the three diameter domains defined previously. In the case of the TEM characterization for the two samples, less than 20 % of the total counted SWCNTs are in domain 1, while around 50 % of the SWCNTs have diameters in domain 3. On the contrary, when considering the SWCNTs detected by Raman spectroscopy, more than 50 % of the SWCNTs are in domain 1, while a little more than 10 % of the detected tubes have diameters higher than 1.7 nm Chapter 4. Comparing TEM and resonant Raman spectroscopy for diameter distribution assessment of a SWCNT sample (domain 3). There are a few possible explanations for this shift, which is, if Potential reasons for observed shift There are potential explanations, involving both TEM and Raman characterizations, to this shift in diameter distributions. We will first list the reasons related with a possible bias in our TEM characterization method (both inherent to TEM, or to the sample preparation process), and will then address the reasons related with Raman spectroscopy. In this section, an approximate dichotomy is made between "small-diameter" and "large-diameter" SWCNTs. These expressions mainly refer to SWCNTs with diameters in the Raman-dominant range (domain 1), and TEM-dominant range (domain 3), respectively. TEM-related Selective SWCNT degradation during transfer process A few studies have given accounts of structural dependence of the reactivity of SWCNTs. Liu et al. showed the dependence of oxidation reactivity of SWCNTs on diameter, chirality, and metallicity [START_REF] Liu | Chirality-dependent reactivity of individual single-walled carbon nanotubes[END_REF]. In a study on covalent functionalization, Strano et al. showed its dependence on metallicity, and the reactivity of SWCNTs was shown to be inversely proportional to their diameter [START_REF] Strano | Electronic structure control of single-walled carbon nanotube functionalization[END_REF]. The latter is explained by curvature-induced strain that is present in small-diameter SWCNTs [START_REF] Niyogi | Chemistry of single-walled carbon nanotubes[END_REF]. The role of metallicity on the reactivity was explained by their electronic band structure [START_REF] Joselevich | Electronic structure and chemical reactivity of carbon nanotubes a chemists view[END_REF]. It therefore seems legitimate to question whether the transfer process can chemically preferably degrade certain SWCNT populations in the sample. The strain in small-diameter SWCNTs make them 4.1. Comparing TEM and Raman spectroscopy on typical CVD growth samples 143 while the SWCNT with a 1.7 nm diameter is intact. This indicates that the latter was indeed more resistant to electron beam degradation. Because of the high sensitivity of SWCNTs to electron beam degradation, all the TEM observations were performed at 80 kV, and using a condenser aperture in order to protect the rest of the sample from beam degradation when focusing on a particular zone. However, it seems that this precaution is not sufficient to prevent all degradation, especially in the case of small-diameter SWCNTs. For this reason, it is possible that some of those SWCNTs are not counted in the diameter distribution determined by TEM, leading to a potential shift of the distribution towards larger diameters. Operator bias: attraction to bigger objects It is well known among experimentators working with TEM that there is a general tendency to be attracted to bigger objects. When operating the TEM at a lower magnitude in order to spot zones on the grid with SWCNTs, a largediameter SWCNT, or a very long SWCNT, can obviously be more easily seen, whereas sub-nanometer SWCNTs require a higher magnification in order to be detected. This can lead to a potential rise in the number of measured largediameter SWCNTs, and a decrease in the number of measured small-diameter SWCNTs, even though the resolution of the microscope allows for the observation of such nanotubes. As a global conclusion, we can say that all of the potential biases related to TEM characterization for diameter distribution assessment, whether they are intrinsic to TEM or due to our characterization process, may render the characterization of small-diameter SWCNTs difficult, possibly leading to a shift of the diameter distribution towards larger diameters. Raman-related As mentioned in Chapter 2, the results of a selectivity assessments based solely on Raman spectroscopy are to be interpreted with caution. This is true for conclusions drawn from every Raman mode (RBM, G, and D bands), but let us focus on the conclusions drawn from the RBM mode with respect to diameter distribution and SC/M ratio. We can distinguish four means of bias and/or uncertainty: Chapter 4. Comparing TEM and resonant Raman spectroscopy for diameter distribution assessment of a SWCNT sample 1. The detectable SWCNT populations are determined by the chosen/available excitation wavelength(s) 2. The Raman cross-section of a given SWCNT is multi-factor-dependent 3. Upper-diameter detection limit (usually no RBM peaks detected for SWC-NTs with diameters above 2.5 nm) 4. The choice of ω RBM → d t law impacts the result Resonance window As explained in Chapter 2, only the SWCNTs in resonance with the excitation wavelength will give rise to measurable Raman signals. In order to attempt to have statistical data that are truly representative of the sample, we need to increase the number of probed SWCNTs as much as possible by using more than one laser line. Without a continuous excitation energy range, however, one has to be aware of the inevitable bias this casts on Raman characterization. The four lasers used in this study cover a very wide range of diameters in both metallic and semiconducting populations. Nevertheless, not all SWCNT populations are detected, which could lead to underestimation and overestimation of certain SWCNT populations through Raman characterization, possibly leading to distortions in the diameter distribution. Intrinsic Raman intensity, or Raman cross-section Even when the pool of detectable SWCNTs is substantial enough to have trustworthy characterization results, the dependence of cross-section on various parameters has a strong impact. In an experimental study on RBM intensities, Jungen et al. [START_REF] Jungen | Raman intensity mapping of single-walled carbon nanotubes[END_REF] model the measured Raman intensity I = φ(αI ii ), by the product of the intrinsic Raman intensity I ii of a given SWCNT, attenuated by α(∆E ii ) that takes into account the distance of the optical transition energy of the SWCNT to the excitation energy ((∆E ii ) = E ii -E L ), and a factor φ = φ(∆Θ, ∆r, j) which is experiment dependent. φ is a function of ∆Θ, the angle between the tube axis and the incident light polarization direction, ∆r is the distance between the scatterrer and the center of the laser spot, and j is the number of identical scatterrers under the laser spot. This gives a general idea of the parameters having an influence on the intensity of a specific RBM peak. We can first note that, using our methodology, j is always considered to be 4.1. Comparing TEM and Raman spectroscopy on typical CVD growth samples 145 equal to 1, which is an approximation which may have a non-negligible impact on the results. Regarding the orientation of the SWCNT relative to the polarization direction, we can consider that it is randomized, due to the disposition of the SWCNTs on the sample. The distance of the optical transition energy of the resonating SWCNT to the excitation energy will come into play later in this chapter. The important factor in this model for the Raman intensity is definitely the intrinsic Raman intensity of the given SWCNT, I ii . Several experimental and theoretical studies on the (n,m)-dependency of I ii (associated with the Raman cross-section), as well as its dependency, for a given SWCNT, on the optical transition, have been performed in the past. They show tremendous influence of the diameter, chiral angle, and optical transition, on the Raman intensity. The study cited above conducted Raman mappings on an as-grown SWCNT sample on SiO 2 with a 532 nm laser, and confronted the plotted Raman intensity as a function of diameter with calculated E ii using a non-orthogonal tight-binding model (excitons and self-energy not taken into account). The results, shown in Figure 4.6.a give striking evidence of this: many SWCNTs in quasi-resonance conditions do not contribute significantly to the Raman signal, while a few high-intensity SWCNTs determine the shape of the Raman intensity profile. Other experimental studies have been conducted in order to determine the diameter distribution [START_REF] Pesce | Calibrating the single-wall carbon nanotube resonance raman intensity by high resolution transmission electron microscopy for a spectroscopy-based diameter distribution determination[END_REF], or the chirality distribution [START_REF] Jorio | Quantifying carbon-nanotube species with resonance raman scattering[END_REF] of a SWCNT sample by correcting RBM intensities with respect to calculated (n,m)-specific Raman RBM intensities, also evidencing the importance of (n,m)-dependence of Raman cross-section. More insight into the specifics of the (n,m)-dependence of the Raman intensity has been gained with experimental studies. Doorn et al. reported, by performing Raman spectroscopy on HiPCo SWCNTs dispersed with SDS, that type II semiconducting SWCNTs ((2n+m)mod3 = 2) had much lower Raman cross-sections than type I semiconducting SWCNTs with similar diameters, by at least an order of magnitude [START_REF] Doorn | Resonant raman excitation profiles of individually dispersed single walled carbon nanotubes in solution[END_REF]. Theoretical calculations using the extended tight-binding model also showed diameter and chiral angle-dependence of the Raman intensity, based on electron-phonon matrix element calculations. They observed a general trend of decreasing Raman cross-section with increasing diameter, while in the same family (2n+m = constant) branch, SWCNTs with lower chiral angles (end of branch) tend to have higher Raman intensities than SWCNTs with high chiral angles [START_REF] Sato | Excitonic effects on radial breathing mode intensity of single wall carbon nanotubes[END_REF] (see all SWCNTs within a 0.6 -2.4 nm diameter range, using the non-orthogonal tight-binding model. All of these factors influencing the intrinsic Raman intensity, or Raman crosssection of a given nanotube inevitably lead to distortions in the diameter distribution. The shape of the distribution, the mean diameter and standard deviation can potentially be greatly affected. Upper-diameter detection limit Within our experimental setup, the Rayleigh scattering signal (or the filtering of this signal), makes it difficult to detect RBM peaks below 100 cm -1 . This is also something to be kept in mind. It means that above a diameter of about 2.5 nm, the SWCNTs in the sample are not accounted for in the distribution, which inevitably contributes to a shift of the distribution towards lower diameters, when compared to TEM, in the case of a sample containing large-diameter SWCNTs. ω RBM -to-diameter relation The laws that are used to determine the diameter from the RBM frequency are not necessarily applicable in the full diameter range of the SWCNT sample. As shown by Araujo et al. [START_REF] Araujo | Resonance raman spectroscopy of the radial breathing modes in carbon nanotubes[END_REF] most of the data used to determine those laws relates to SWCNTs within the diameter range of usually 1 to 2 nm. Most of them are in relative agreement within that range, but their extrapolations diverge for lower and higher diameters without real experimental validation. This might lead to uncertainties regarding small and large diameter SWCNTs. Hypotheses to be tested It is clear that there are potential biases and/or blind spots when using both TEM and Raman spectroscopy for the characterization of a SWCNT growth sample. When it comes to the assessment of the diameter distribution of a growth sample, both characterization techniques have their own sources of bias. Using the above discussion, we have established a list of hypotheses that can all partially explain the shift, and shape differences observed in our diameter distributions: 1. TEM is not adapted to the characterization of small-diameter SWCNTs (typically below 1. The first hypothesis regarding TEM combines the easy degradation of smalldiameter SWCNTs by the electron beam, and the fact that it is easier to detect bigger objects when using TEM. Hypothesis 2 takes into account the possible degradation and loss of SWCNTs during the transfer process. Hypothesis 3 combines prevention of the detection of large-diameter SWCNTs by Raman spectroscopy by Rayleigh scattering, and the diameter-dependent scattering cross sections. The fourth hypothesis hints both at the fact that Raman cross-section is (n,m)-and optical transition-dependent, and that the detected populations are dictated by the choice and availability of excitation lasers. The last possible explanation could be that the methodology applied to our samples for Raman spectroscopy assessment of the diameter distribution is not truly adapted. In order to test these various hypotheses, which mainly rely on a small-diameter SWCNT versus large-diameter SWCNT dichotomy, it seemed necessary to conduct the same study on diameter-sorted SWCNT samples. TEM characterization of diameter-sorted SWC-NTs A general observation when looking at the diameter distributions of the various samples (determined both by TEM and Raman spectroscopy), is that they are a bit broad. The presence of SWCNTs in the sub-nanometer range, as well as over 1.5 nm makes the differences in the two characterization techniques stand out. It is difficult, from these samples, to truly understand where the shift is coming from. In order to better understand the source of the shift, as well as the shape differences between the diameter distributions assessed by TEM and Raman spectroscopy, we wanted to look at SWCNT samples with a narrower diameter distribution in the diameter range of main interest: between 0.8 nm and 1.3 Chapter 4. Comparing TEM and resonant Raman spectroscopy for diameter distribution assessment of a SWCNT sample SWCNT sorting [START_REF] Arnold | Sorting carbon nanotubes by electronic structure using density differentiation[END_REF]), was adapted by Fleurier et al. to DWSNT samples [START_REF] Fleurier | Sorting and transmission electron microscopy analysis of single or double wall carbon nanotubes[END_REF], and further developed by A. Ghedjatti during his PhD thesis [START_REF] Ghedjatti | Etude structurale des nanotubes de carbone double parois[END_REF] (see Appendix C.1 for method). Since the DGU method enables first the sorting of CNTs according to their number of walls, there will be layers containing a very large majority of SWCNTs. Depending on the chosen layer, the center of the diameter distribution of those SWCNTs (as well as the content of DWCNTs) will be shifted. Figure 4.7 shows the results of the sorting on this DWCNT-containing sample performed by Fleurier et al. [START_REF] Fleurier | Sorting and transmission electron microscopy analysis of single or double wall carbon nanotubes[END_REF], displaying the evolution of the absorption spectra, and diameter distributions of both SWCNTs and DWCNTs in different layers of the sorted sample. Some of the first layers contain no DWCNTs, and the SWCNT diameter distribution is quite narrow, and centered around small diameters, making them ideal samples for our study. Ahmed Ghedjatti performed this DGU sorting method on nanotubes from CIRIMAT, with the aim of obtaining, through several sorting rounds, pure DWCNT samples with an enriched semiconducting population. We used the first layer of his first round of sorting, containing small-diameter SWCNTs. Sample characterization scheme In order to test the hypotheses presented in the previous section, we prepared three samples with the diameter-sorted SWCNT solution. Figure 4.8 presents a schematic representation of the preparation process of these samples. First, to determine the diameter distribution of the SWCNTs in the solution using TEM, the solution was directly dropped on a copper TEM grid covered with a holey carbon membrane (sample 1). Second, in order to conduct the Raman characterization in the same conditions as our growth samples, a drop of the diameter-sorted SWCNT solution was deposited onto a SiO 2 /Si wafer, and the Raman spectroscopy characterization was conducted directly on the wafer (sample 3). Before looking into the Raman characterization results, we wanted to make sure that the transfer process was not responsible for the differences observed in the diameter distributions of our growth samples. A piece of the wafer coated with diameter-sorted SWCNTs was used for a transfer onto a TEM grid (sample 2). The TEM-extracted diameter distribution of the transferred diameter-sorted SWCNTs was compared to the diameter distribution extracted from the direct TEM characterization. We then compared the results of the Raman characterization conducted on the SiO 2 /Si wafer to the direct TEM measurements. The numbers of the samples are given by the order in which their respective characterizations will be presented in the next sections. Direct TEM characterization In order to analyze the differences observed between the statistical analysis of TEM and Raman characterizations of our typical SWCNT growth samples, a study was conducted on diameter-sorted SWCNTs. TEM grid preparation In order to determine the diameter distribution of our sample by TEM imaging, a grid was prepared using the solution constituting the first layer of the sorted nanotube sample. The grid was prepared as follows: a copper TEM grid covered with a holey carbon membrane was placed on an absorbing paper, with the carbon membrane (shiny side) facing up. A drop of solution was then dropped on the grid, and it was left to dry in air, covered by a watch glass for at 152 Chapter 4. Comparing TEM and resonant Raman spectroscopy for diameter distribution assessment of a SWCNT sample least two hours. Since the nanotubes are covered by surfactant molecules, the nanotubes are not directly observable. Therefore, the grid has to be rinsed-free of surfactant which may cause contamination in the TEM. The dried grid was placed, with the carbon film facing up, onto a filter paper with 0.45 µL pores within a Büchner funnel mounted with a glass column, plugged to a pump. The glass column was then filled with DI water, and the extraction was regulated so that the water passed very slowly through the Büchner funnel. This is important because a high water flow rate would inevitably lead to a degradation of the thin carbon membrane, preventing the nanotubes to remain on the TEM grid. A final rinsing was then performed using a small volume of ethanol, so as to speed-up the drying process. The rinsed grid was left to dry for a few hours in air. Direct determination of diameter distribution The TEM characterization of the as-obtained nanotube sample was conducted as explained in Chapter 2. Figure 4.9 shows a typical TEM image obtained from the observation of the prepared grid. Out of over 70 images taken from multiple locations on the grid showing hundreds of nanotubes, only four DWCNT were seen, the rest were SWCNTs. Since they were not suspended, they were not counted in the statistical data. It is clear that the percentage of DWCNTs is extremely low in this sample, therefore, their presence cannot induce a significant distortion in the statistical data, either in TEM or Raman characterization. We can clearly observe the presence of remaining organic materials on the SWC-NTs, and that a large quantity of the SWCNTs are still covered in surfactant molecules or other contaminants. One could argue that the fact that some of the SWCNTs are not bare and cannot be measured could also lead to distortions in the statistical data. However, by looking closely at the partly covered SWCNTs in the TEM images, there does not seem to be any sort of diameter-selectivity for a more effective rinsing, meaning that the suspended and bare SWCNTs are representative of the sample. We measured the diameter of 121 SWCNTs from the TEM images of this sample, the diameter distribution histogram is presented in Figure 4.10. The mean diameter value is 1.35 nm, with a standard deviation of 0.63 nm. When setting aside the very few large-diameter SWCNTs (above 2.5 nm, i.e. not detected by Raman), the distribution is quite narrow, with the mean diameter going to 1.15 Chapter 4. Comparing TEM and resonant Raman spectroscopy for diameter distribution assessment of a SWCNT sample our growth sample, the Raman-determined diameter distribution systematically shows the predominance of SWCNTs with diameters between 1.0 nm and 1.3 nm. If this population of SWCNTs were truly predominant in the sample, the present experiments prove that they would have been predominant in our TEM distribution as well. Validation of transfer method An important factor to take into consideration in our comparisons between TEM and Raman characterizations for diameter distribution assessment is that the TEM characterization was always conducted on transferred SWCNTs. This means that our TEM characterization is not a direct observation of the assynthesized sample per se, because the SWCNTs have undergone a long process involving heating, rinsing steps, exposure to various organic solvents and NaOH, etc. The various steps of this process may very well alter the composition of our synthesized SWCNT sample, as explained previously. In order to make sure that the transfer process was a robust and non-altering solution for the TEM observation of our SWCNT growth samples, the following experiment was conducted. Sample preparation We wanted to compare the TEM characterization before and after the transfer process using the diameter sorted SWCNTs, meaning comparing the diameter distribution obtained in the previous section (direct measurement), and the diameter distribution we would obtain from the same sample if it had been transferred from a SiO 2 /Si wafer following the same process as our growth sample. To this end, we had to deposit the SWCNTs from the solution onto a SiO 2 /Si wafer, in a sufficiently homogeneous and diluted mat, so that the Raman characterization methodology for the diameter distribution assessment of our growth samples could be used. This was done using the following procedure: a SiO 2 /Si wafer was cleaned by sonication in dichloromethane for 10 minutes, and was placed for 10 minutes in a plasma cleaner. This is necessary for the surface of the wafer to be hydrophilic for optimal wetting of the solution, but also to ensure the adherence of the SWCNTs to the SiO 2 surface. A drop of solution of the diameter-sorted SWCNTs was dropped onto the wafer, and the drop was spread across the surface by nitrogen blowing. The wafer was dried on a hot plate at about 40 • C for a few minutes. The wafer was then very thoroughly rinsed using DI water, and dried again on a hot plate for a few minutes. A SEM characterization of the sample was conducted so as to verify the state of the SWCNT layer. We obtain a rather dense mat of short SWCNTs (about 0.5 - Comparison with direct measurements The comparison of the diameter distribution obtained from TEM characterization of the diameter-sorted SWCNTs after the transfer process, with the directly-measured diameter distribution is shown in Figure 4.13. For the two samples, the mean diameters and standard deviations are really close, with and without taking into account the SWCNTs with diameters above 2.5 nm, with similar distribution shapes. We should note that the two diameter distributions were obtained from a number of SWCNTs in the same order of magnitude. We can conclude that the transfer process does not appear to modify the diameter distribution of the SWCNT sample. There is no evident decrease in the sub-nanometer population as a result of exposure to various solutions and heating treatments, proving that our transfer process is reliable when it comes to preserving the content of a SWCNT sample. Hypothesis 2 can therefore be overthrown. Raman characterization of sorted SWCNTs For Raman characterization, it is crucial to rinse the sample properly, to prevent the presence of surfactant molecules on the surface of the SWCNTs, which have a significant impact on their RBM frequencies. The same methodology applied to our growth samples was applied here for the Raman spectroscopy characterization. with a shift towards higher diameters of only 0.15 nm for the TEM measurements as compared to the Raman-extracted data. This is much lower than the surprisingly high difference of 0.40 nm seen in our growth samples. When considering the diameter range of the SWCNT sample, we can say that TEM and Raman are in fairly good agreement. However, there seems to be a global shift of the diameter distribution histogram of TEM measurements of 0.1 nm, compared to the Raman-extracted distribution. This is acceptable considering the 0.5 Å error margin associated with the TEM diameter measurement [START_REF] Fleurier | Transmission electron microscopy and uv-vis-ir spectroscopy analysis of the diameter sorting of carbon nanotubes by gradient density ultracentrifugation[END_REF], which could lead to a slight overestimation of diameters. Results by laserline: confrontation with the Kataura plot In order to better understand the diameter distribution obtained by Raman spectroscopy, and which SWCNT populations were responsible for the Raman signal in this sample, we looked at the data collected for each separate laser line. We plotted separate Raman-extracted diameter distributions for each excitation wavelength. The first important observation is that, for this particular sample, a much larger number of RBM peaks were counted for the mappings performed at 532 nm and 633 nm excitation wavelengths than at 473 nm and 785 nm excitation wavelengths. For all excitation wavelengths, the acquisitions were recorded at similar laser powers (between 4 and 10 mW). The RBM peaks counted at 633 nm and 532 nm account for 42.6 % and 47.6 % of the total 437 RBM peaks counted, respectively. On the other hand, the RBM peaks counted at 473 nm and 785 nm account for only 8.2 % and 1.6 % of the total counted RBM peaks, respectively. It therefore appears that much more SWCNTs in our sample are in resonance with the red and green laser energies, or that the SWCNTs that should be in resonance with the other two laser lines within the sample's diameter range are not present, or less present in the sample. Since most of the RBM peaks are detected by the red and green lasers, the distorted shape of the global diameter distribution appears here: in the case of the red laser, two main populations are detected, one below 1.0 nm corresponding to SC-SWCNTs, and another at 1.2 nm corresponding to M-SWCNTs; and in the case of the green laser, one dominating population is detected below 1.0 nm corresponding to M-SWCNTs. This explains the bimodal shape of the global distribution. There is a "void" between 1.0 and 1.1 nm that has to be explained, by something other than the simple absence of this SWCNT population. We seemed to be more adapted to our sample, it served as our main reference, the Kataura plot (1) served as a backup. Assuming a random (n,m) distribution in the sample, the drastic differences in the numbers of counted RBM peaks for each laser line cannot be explained by an absence of resonance conditions for the blue laser, as compared to the red laser, for instance. Also, if we compare the results obtained for the 633 nm and 785 nm lasers, it is surprising to see that a very large number of RBM peaks were detected using the first laser between 1.1 and 1.3 nm whereas only a few were detected for the latter between 0.9 and 1.1 nm. In both cases, SWCNTs with E ii within the same distance of laser line are present. This could be explained by large differences in the RBM cross-sections for the SWCNTs in those two cases. Let us look separately at each laser line, starting with the red laser at 633 nm. In this case, the observed diameter distribution is easily understood: two major features are present within the sample's diameter range, roughly corresponding to the two optical transition energy branches crossing the laser line (E SC 22 , and E M 11 ). The intensity of the feature corresponding to the metallic branch can be explained by the presence of the end of lower branches of two (2n+m) = constant families in the resonance window in the same narrow diameter range. As explained previously, it has been shown that the Raman intensity for SWCNTs in the lower E M 11 L branch is larger than for the upper E M 11 H branch. In the case of the green laser at 532 nm, the most intense bar in the histogram also corresponds to the end of a lower branch of a (2n+m) = constant family within the resonance window between 0.8 nm and 0.9 nm. The count decreases after 0.9 nm even though a large portion of the sample's SWCNTs have diameters of around 1.0 nm. This could be explained by the fact that the only SWCNTs in resonance with the laser line are on the upper branch of E M 11 . The peak between 1.2 -1.3 nm corresponds to the appearance of the E SC 33 branch. However, it is surprising that a large majority of the detected SWCNTs seem to stem from the E M 11 branch around 0.8 -1.0 nm, whereas only a few stem from the E SC 33 branch at higher diameters (1.2 -1.4 nm). Judging from the Kataura plot used here, the lack of Raman signal detected using the blue laser (473 nm) may stem from the fact that, within the diameter range of the sample, only a few SWCNTs are in resonant conditions. In the case of E SC 33 , the SWCNTs may have a too large energy distance to the excitation energy. It is interesting to see that there are barely any SWCNTs detected between 0.9-1.1 nm, which corresponds to the presence of upper branch E M 11 optical transitions, which have much lower Raman cross-sections. The case of the 785 nm laser is quite surprising. A semiconducting SWCNT population around 0.9 -1.2 nm should be in good resonance conditions (similar to the E M 11 branch in the case of the 633 nm laser, with respect to position in the resonance window), but only a few RBM peaks were counted. This is surprising when considering a random chirality distribution in the sample. This hypothesis should be tested using another characterization technique, like ED or HRTEM. In summary, some of the observed Raman data can be partly explained by differences in Raman cross-sections for various SWCNT populations within the sample's diameter range, and supposing a random (n,m) distribution. In general, it appears that most of the populations of SWCNTs that are not detected Chapter 4. Comparing TEM and resonant Raman spectroscopy for diameter distribution assessment of a SWCNT sample by our Raman methodology are semiconducting nanotubes: the E SC 33 of SWC-NTs with diameters between 1.0 and 1.3 nm should be detected in larger numbers using the 473 nm laser, since they fall in the diameter range the most predominant in the sample according to TEM characterization. The same remark can be made for the E SC 22 branch between roughly 0.8 and 1.1 nm using the 785 nm laser: only 7 RBM peaks were counted in total. Some of our data is still difficult to explain, and would require additional characterizations. However, there is little doubt as to whether the choice of excitation wavelengths, and intrinsic Raman intensity have an impact on our resulting distribution. Conclusion and perspectives In this chapter, we have first shown that there is a systematic shift between the diameter distributions assessed by Raman spectroscopy and TEM imaging in our typical CVD grown SWCNT samples. This shift can be as high as 4 Å. Diameter ranges corresponding to Raman dominant, and TEM dominant populations have been identified, on each side of a small range of TEM and Raman partial agreement. To explain the observed shift, we have put forward six hypotheses linked to potential intrinsic biases of both TEM and Raman spectroscopy as characterization techniques, or to potential deficiencies in our methodology. In order to test these hypotheses and gain more insight into the differences between TEM and Raman spectroscopy for diameter distribution assessment, especially in the Raman dominant diameter range, we have conducted a study on diameter-sorted SWCNTs, with a distribution centered around 1.0 -1.2 nm. This study has enabled us to show that TEM is a technique that is perfectly adapted to the characterization of small-diameter SWCNTs, and that the transfer process from SiO 2 /Si wafers used for SWCNT growth is not responsible for the difference in abundance of SWCNTs within the diameter range of 0.5 -1.3 nm between our growth results as measured by TEM and Raman spectroscopy. The differences in the shapes of the distributions hint at potential distortions due to the (n,m) and optical transition dependency of Raman scattering crosssections, but this needs to be verified with complementary characterization techniques. A comparative study using various relations between the RBM frequency and tube diameter has shown that an error of choice of the relation cannot be responsible for the observed shifts. The relative accordance on the diameter range between TEM and Raman spectroscopy for the diameter distribution assessment of the diameter-sorted SWCNT sample, and the large discrepancy observed in the case of our wide diameter distribution growth samples, shows the importance of a cross characterization methodology. It appears that the diameter range can be successfully assessed by Raman spectroscopy when the diameter distribution is narrow, but that in the case of a broad distribution, especially including large-diameter SWCNTs, Raman spectroscopy can give an illusion of selectivity. The distortions observed in the shapes of the diameter distribution histograms also calls for extreme caution when determining the (n,m) distribution of a given sample using only Raman spectroscopy with the present methodology. This accordance obtained on the diameter-sorted sample in this chapter serves as a partial validation for our characterization methodology for the diameter range which is the most delicate to assess with TEM. However, the observed differences in shapes also sheds doubt on its validity. Unpublished work conducted in Esko Kauppinen's group at Aalto University gives similar warnings by comparing Raman spectroscopy and electron diffraction for the evaluation of the SC/M ratio of various SWCNT samples. More experiments would be very insightful additions to this work. First, complementary studies on diameter-sorted SWCNTs with larger diameters could help to verify our third hypothesis regarding the efficiency of Raman spectroscopy for the detection of large-diameter SWCNTs. HRTEM, ED, or PLE studies on the presented diameter-sorted sample would give insights into the potential chirality enrichments of the sample that could explain the seeming over and underestimation of certain populations with Raman spectroscopy that lead to the distortion in the shape of the diameter distribution. Chapter 5 Exploration of two other families of SWCNT catalyst precursors: Cyanosols, and POMs In Chapter 3, the growth of SWCNTs using PB/PBA nanoparticles as catalyst precursors was studied. Here, we present preliminary results regarding, first, the attempt to apply the same process using cyanosol nanoparticles instead of PBAs, and second, the use of POM compounds as inspired by recent publications about Co-W-based catalysts for SWCNT growth. In the first section, we will describe the synthesis and characterization of Fe-Pd-based cyanosol nanoparticles, their deposition on a substrate and subsequent reduction, as well as trials of SWCNT growths from this catalyst precursor system. In the second section, SWCNT growth trials using Co-W-based POM clusters will be shown and discussed. FePd cyanosol nanoparticles as SWCNT growth catalyst precursors Introduction As explained in Chapter 3, the three PB/PBA systems that were studied allowed to conduct SWCNT growths using catalysts with variable carbon solubility. We observed that reducing the carbon solubility of the pure iron catalyst by adding nickel led to different results, but no notable improvements of the catalyst performance. In Chapter 2, we presented the cyanosol family as a practical alternative to PBA synthesis for systems where the simple co-precipitation method did not succeed. This is the case, for instance of the FePd system. As Chapter 5. Exploration of two other families of SWCNT catalyst precursors: Cyanosols, and POMs for the FeNi system, a solid solution is expected to form at typical growth temperatures (800 -1000 • C) according to the Fe-Pd bulk binary phase diagram [START_REF] Franke | Binary systems. part 3: Binary systems from cs-k to mg-zr[END_REF]. We therefore decided to attempt to synthesize and use as catalyst precursors FePd cyanosols nanoparticles. Ideally, a bimetallic nanoalloy FePd catalyst could be formed, with a substantially lower carbon solubility than pure iron, or a NiFe alloy. Synthesis of Fe Characterization of the obtained nanoparticles Size As for the PB/PBA systems, the as-obtained nanoparticle dispersion was characterized by DLS, in order to measure the hydrodynamic size of the cyanosol nanoparticles, the results are shown in Figure 5.1. Only one peak is observed in the hydrodynamic size distribution by number, and it is centered around 2.8 nm. The synthesis method allows to obtain small nanoparticles, that are individualized in solution. We should note here, that in comparison with the PB/PBA systems presented in Chapter 3, the sizes of the obtained cyanosol nanoparticles are smaller (less than half of the hydrodynamic size of 6.3 nm obtained for the PB nanoparticles for instance). A few drops of the as-obtained cyanosol nanoparticle dispersion were dropped onto a TEM grid for analysis. A SEM-EDX analysis of this same powder was performed, so as to gain insight into the composition of the cyanosol nanoparticles. The atomic fraction of Fe, Pd, and Cl were measured. Measuring the atomic fraction of Cl enables 5.1. FePd cyanosol nanoparticles as SWCNT growth catalyst precursors 173 the quantification of the substitution of the Cl ligands by CN ligands in the Pd coordination sphere. The obtained Fe:Pd ratio is 1:0.9, meaning a close to stoichiometric composition, with a slight excess of Fe. This is in agreement with the presence of terminal ligands in the structure, as evidenced above by IR spectroscopy. The Pd:Cl ratio is 1:0.1, which implies that most of the Cl ligands have beeen substituted in the structure. Considering the reactant has four Cl ligands per Pd atom, this ratio indicates that 97.5 % of Cl ligands are substituted, which is compatible with the close to 1:1 metallic ratio (see Appendix B.3 for corresponding EDX spectrum). Conclusion The method developed in our lab leads to the synthesis of FePd cyanosol nanoparticles, with very small controlled sizes, and close to 1:1 metallic ratio. The nanoparticles are stable in water. In the next section, we present the deposition of these particles on a SiO 2 /Si wafer. Sample preparation and cyanosol precursor deposition Wafer preparation The silanization process used in Chapter 3 in the case of PB/PBA precursors is useful but time consuming. Moreover, it adds steps to the global growth process, which may increase the chances of contamination. Here, the synthesis of the cyanosol precursors is performed at a much higher concentration (twenty times higher than for PB, and five times higher than for both PBA systems). Hence, we first tried to deposit the precursor nanoparticles without using the silane SAM step. Before the deposition, the bare SiO 2 /Si wafer were simply sonicated subsequently for five minutes in acetone, IPA, and dichloromethane, and were left to dry in air. The particle deposition was performed right after this cleaning process. Particle deposition The deposition process was conducted as follows: after the bare wafers were cleaned and dried, they were immersed in the as-obtained cyanosol dispersion in DI water, where they remained for three days. After which they were taken 5.1. FePd cyanosol nanoparticles as SWCNT growth catalyst precursors 175 The sample was analyzed by XPS. The presence of Fe and Pd on the surface of the wafer is confirmed, as well as the preservation of the roughly 1:1 metallic ratio determined using EDX in the previous section. Measurements were performed on three different random spots of the wafer for quantitative analysis, and the obtained mean atomic percentages of Fe and Pd were 49.5 %, and 50.5 %, respectively. Since the uncertainty of the XPS is a few percents, we can affirm that the results are in good agreement with the SEM-EDX analysis of the dried cyanosol nanoparticles. The 1:1 metallic ratio is preserved globally, on the surface of the wafer. The XPS analysis also confirms the absence of contamination that could occur during the various steps of the process. The global survey acquired on several points is shown in Appendix B.5. Study of the reduction step As for the PB/PBA systems, we studied the reduction pretreatment of the cyanosol nanoparticles by taking the samples out of the CVD chamber after a 5-minute pretreatment step under H 2 at 800 • C in usual conditions. The wafers were studied by AFM and XPS, and the nanoparticles were transferred onto TEM grids following the process described in Chapter 2 in order to conduct conventional TEM analysis of the nanoparticles, as well as a STEM-EDX study. Wafer surface analysis .5 shows typical AFM images of a SiO 2 /Si wafer coated with the FePd cyanosol nanoparticles after undergoing the reductive pretreatment step. The nanoparticle layer is dense, with small sizes (between 1 and 2 nm roughly). We should note that the image shows very little signs of coalescence. Coalescence seems to be kept under control even without the use of an anchoring SAM, which is very encouraging. However, the drastic size reduction observed in the case of the PB/PBA nanoparticles does not appear here. A more extensive statistical study using TEM imaging should be performed to have more insights into this difference. Here also, XPS analysis was performed to confirm the presence of the two metals, and the Fe:Pd ratio. The quantitative analysis was performed on two random points of the wafer, and the mean Fe and Pd atomic percentage values are 46.8 %, and 53.2 %, respectively. Here, there seems to be a slight excess of Pd on the surface of the wafer, but taking into consideration the uncertainty Discussion, and perspective In this section, we have shown the successful synthesis of FePd cyanosol nanoparticles, with a controlled size and composition. These nanoparticles were then deposited in a homogeneous monolayer on a SiO 2 /Si wafer, without needing to functionnalize the wafer prior to deposition, making the process simpler and less time consuming. The global metallic ratio on the surface of the wafer was analyzed using XPS, and the 1:1 metallic ratio was confirmed. The FePd nanoparticle layer was then reduced in H 2 at 800 • C, and subsequently studied by AFM, XPS, and STEM-EDX after transfer of the nanoparticles onto TEM grids. The aspect of the nanoparticle surface shown by AFM demonstrated the apparent lack of coalescence between the particles, and that the layer remained homogeneous after the pretreatment. The XPS analysis confirmed the 1:1 global metallic ratio on the surface. Unfortunately, STEM-EDX study revealed that a portion of the nanoparticles were pure Pd, while others appeared to be a Pd@Fe core-shells. Since the binary phase diagram predicts a 1:1 solid solution at 800 • C, and the STEM-EDX study is performed after cooling and exposition to ambient air, it is difficult to truly understand the chemical state of the nanoparticles in the CVD chamber. However, core-shell catalysts have proven to be interesting systems for SWCNT growth study [START_REF] He | Environmental transmission electron microscopy investigations of pt-fe 2 o 3 nanoparticles for nucleating carbon nanotubes[END_REF]. An optimization of the process, coupled with more extensive characterization proving the systematic formation of these core-shell nanoparticles could lead to insightful studies. Finally, SWCNT growth was attempted using the deposited FePd cyanosol nanoparticles. Initial successful SWCNT growth results conducted at 800 • C in usual conditions could not be reproduced. Growth attempts at various temperatures ranging from 800 to 1000 • C, with varying pretreatment times were also performed, with no success. More investigation is necessary to understand these results, since they are surprising. First, our pure FeFe catalyst presented in Chapter 3 was a very effective catalyst, and it is a surprise that a growth in the same conditions using the FePd cyanosol system yields no SWCNTs. Additional tests using FeFe cyanosol nanoparticles could be performed to elucidate whether the precursor type, and its deposition process could be responsible for these results. Moreover, additional optimization of the CVD parameters adapted to this specific catalyst system could also be envisioned as a potential solution. The small size of the cyanosol nanoparticle may also be to blame for the lack of SWCNT growth. Since the potential bimetallic catalyst systems that can be obtained through this method are promising, additional work of the optimization of cyanosol nanoparticle synthesis, with the aim of obtaining particle sizes closer to those of the PB/PBA nanoparticles studied in Chapter 3 are undergoing in our laboratory. Co-W POM clusters as SWCNT growth catalyst precursors The work presented in this section consists of preliminary results on the feasibility of SWCNT growth from POM precursors, using our experimental setup. The studies by Yang et al. already discussed in Chapters 1 and 2 of this manuscript feature SWCNT growth by ethanol CVD at atmospheric pressure [START_REF] Yang | Chirality-specific growth of single-walled carbon nanotubes on solid alloy catalysts[END_REF], [START_REF] Yang | Growing zigzag (16, 0) carbon nanotubes with structuredefined catalysts[END_REF]. The chirality specific claims, based on Raman spectroscopy, and the use of a chemically synthesized precursor seemed interesting. We therefore attempted to use similar catalyst precursors to study them with a CH 4 HFCVD. As explained in Chapter 2.1.3.3, the POM precursor used here has an atomically defined W:Co ratio of 34:7 [START_REF] Lisnard | Effect of cyanato, azido, carboxylato, and carbonato ligands on the formation of cobalt (ii) polyoxometalates: Characterization, magnetic, and electrochemical studies of multinuclear cobalt clusters[END_REF], which is different from, but close to the value of the POM precursors used by Yang et al. (36:6). The main difference in composition between the two clusters is that the one used by Yang et al. contains phosphorous (6 atoms per cluster), and the one used here contains silicium (4 atoms per cluster). The structures of the two POMs are quite different, but are similar in sizes. Deposition Since the clusters are negatively charged in aqueous solution, we can assume that they can attach by an attractive Coulomb interaction with pyridines at the surface of a silanized wafer that have been protonated. The deposition of the POM clusters was therefore attempted on silanized wafers, though the conditions are not ideal, since only a small portion of the pyridine groups on the surface are protonated. A 1 mM solution of the POM in DI water was prepared. The silanized wafers were immersed in the solution for 2 hours, before being rinsed with DI water and dried under air at room temperature. AFM and XPS measurements were performed on the surface of the wafer to confirm the deposition of the POMs. the roughness of the silanized SiO 2 . The height measured here is in good agreements with estimates of the POM cluster sizes (2.5 x 1.5 nm). XPS analysis confirmed the presence of both cobalt and tungsten on several random spots of the wafer, further demonstrating the successful deposition of the clusters on the wafer. Some heterogeneity was noticed on the wafer surface, further optimization of the deposition process could improve this. A quaternization of the pyridine groups could ensure a full coverage of positively charged groups on the surface, leading to a better anchoring of the POMs, for instance. AFM analysis was also performed on the POM-coated wafer after undergoing a reductive pretreatment under H 2 at 800 • C (see Figure 5.7.b). The images show that a few clusters are present on the surface, and the height profile shows peaks with heights roughly between 0.8 and 1.4 nm. Since the density of the layer is really low, and the observed objects have sizes approaching the substrate roughness, an AFM image of the silanized wafer surface, along with a height profile, are displayed in Figure 5.7.c. It is clear, when comparing the two images, that there is no ambiguity regarding the deposition. Globally, the size of the resulting nanoparticles seem to be slightly smaller than the sizes of the clusters initially deposited on the surface. More investigations by TEM should be done in order to have more information on the nature of those nanoparticles. SWCNT growth SWCNT growth was attempted in usual conditions at 800 • C and 1000 • C. Raman spectroscopy mappings were performed, confirming the feasibility of the SWCNT growth from this catalyst precursor using our usual process and CVD setup. For both growths, however, the yield was very low. The calculated Raman "yield" (defined in Chapter 3) values are 0.12 for both growth conditions. In the case of the PB/PBA precursors, we observed an optimal yield temperature at 800 • C, and a decrease in yield with increasing temperature, going to zero at 1000 • C. Here, the Raman-extracted yield does not evolve, and interestingly, a growth is possible at 1000 • C, which was not the case for the PB/PBA precursors. The Raman characterization was performed, as for all samples in the present work, using four excitation wavelengths. The diameter distribution histograms extracted from the Raman characterization are presented in Figure 5.8. The mean diameters are 1.30 ± 0.19 nm, and 1.80 ± 0.31 nm for growths at 800 Conclusion In summary, CoW POM clusters were successfully used as catalyst precursors for SWCNT growth at various temperatures using our HFCVD setup. A quick Raman characterization shows an evolution of the diameter distribution with temperature (mean diameter increasing from 1.3 nm to 1.8 nm when increasing growth temperature from 800 • C to 1000 • C), and drastic modification in SC/M ratio. These results are promising in simply demonstrating the feasibility of this specific growth process using a methane CVD reactor, and further studies will certainly lead to more exploration of bimetallic combinations and compositions as catalyst systems, and a better understanding of the phenomena at play in the process. Discussion and perspectives In this brief chapter, we have reported studies on two very different chemically synthesized catalyst precursor systems. First, the cyanosol family was explored as a SWCNT growth catalyst precursor with the FePd system. The synthesis of stiochiometric FePd cyanosol nanoparticles with a controlled composition, and small size, was successfully conducted. The precursor nanoparticles were characterized by DLS, IR, and TEM, and EDX. Their deposition on bare SiO 2 /Si wafers was then performed and analyzed with AFM and XPS, evidencing the successful deposition of a dense layer of nanoparticles with sizes ranging roughly from 1 to 2 nm, and an overall Fe:Pd ratio close to 1:1 on the surface. AFM and XPS were also performed on the wafers after the reductive pretreatment. The 1:1 ratio was kept, and the deposit still consisted of a dense monolayer of small nanoparticles. After reduction in the CVD chamber under H 2 , the "effective" catalyst nanoparticles were studied by STEM-EDX, revealing two nanoparticle types: pure Pd nanoparticles, and Pd@Fe core-shell nanoparticles. Unfortunately, the growth of SWCNTs was not reproducible using this precursor in our usual growth conditions at various temperatures, and with varying reduction times. The fact that this catalyst system did not lead to successful SWCNT growth is still surprising, and further optimization of the growth process and CVD conditions could lead to interesting results. The second catalyst system, derived from Co-W-containing POM clusters, successfully led to the growth of SWCNTs. Here also, further optimization of both the catalyst precursor deposition, and CVD growth are needed to enhance Chapter 5. Exploration of two other families of SWCNT catalyst precursors: Cyanosols, and POMs growth yield. TEM characterization of the catalyst nanoparticles are, of course, a priority in future works, as it is crucial to understand the growth mechanism from this catalyst. Nonetheless, the few results shown here, demonstrating the feasibility of SWCNT growth from these precursors, show a promise in exploiting the richness of the extremely wide and versatile POM family. The experiments presented in this chapter constitute preliminary results. This work will be continued by a PhD student in the coming years. We have already shown the relevance of using chemically synthesized well-defined coordination networks for SWCNT growth and growth mechanism understanding in Chapter 3. With this chapter, we have shown that this relevance goes beyond the PB/PBA family. Our approach, if optimized for each specific compound family, could lead to very promising research works. Conclusion and perspectives The first objective of this work was to explore SWCNT growth selectivity through catalyst design, by developing a new process to prepare bimetallic catalysts with tuned carbon solubility. In this manuscript, we have proposed a new synthesis route to grow SWCNTs by HFCVD using PB/PBA nanoparticles as catalyst precursors. The precursor nanoparticles were synthesized as stable water dispersions, and extensively characterized, confirming their crystallographic structure, composition, and adapted and well controlled sizes. They were successfully used as catalyst precursors for SWCNT growth. Their controlled deposition by coupling surface and coordination chemistry, and controlled reduction under H 2 atmosphere led to the formation of dense layers of catalyst nanoparticles, without signs of coalescence. The STEM-EDX study of the bimetallic systems led to very interesting results. In the case of the NiCr system, for which the bulk binary phase diagram does not forecast the occurrence of a 1:1 solid solution in our growth conditions, we observed a very clear phase segregation in the nanoparticles, and assumed the SWCNT growth stems from a pure Ni effective catalyst. In the case of the NiFe system, for which the bulk binary phase diagram predicts the formation of a 1:1 solid solution in our growth conditions, the catalyst nanoparticles -after cooling and exposure to air -were observed to be either alloy-like, or Janus type nanoparticles. Unlike the phase segregation observed in the NiCr catalyst, the Janus nanoparticles observed for the NiFe system displayed two seeming Ni-Fe solid solutions: a Ni-rich phase on one side, and a Fe-rich on the other. We were therefore able to make the simple assumption that SWCNT growth in this specific case stemmed from a NiFe nanoalloy of unknown composition. Thus, our process allowed us to study three distinct catalyst systems: a pure Fe catalyst, a bimetallic phase-segregated catalyst, and an "alloyed" catalyst. Using the FeFe monometallic reference system, we observed an evolution of the diameter distribution towards larger diameters with increasing growth temperature. Based on a tangential growth mode, we considered two possible explanations for this evolution: a typical Oswald ripening (and coalescence) argument, or a reasoning based on carbon solubility evolution with temperature. In the first scenario, Oswald ripening shifts the global nanoparticle size distribution towards larger diameters, leading to the growth of similarly larger-diameter SWCNTs. In the second scenario, the nanoparticle size distribution is fixed, and temperature has an influence on carbon solubility in the nanoparticle, and therefore the carbon content within the nanoparticle. This leads to the activation, and deactivation (either by poisoning or detachment) of different nanoparticle populations based on their sizes, at different temperatures. The very little evolution of the size distribution of the catalyst nanoparticles before and after CVD growth suggests that, aside from a minor coalescence effect, Oswald ripening does not seem to have a significant impact on the catalyst size distribution, indicating that carbon solubility may be the most probable parameter driving SWCNT growth in this case. Further AFM and TEM characterization in order to elucidate the evolution of catalyst nanoparticle size distributions with temperature are needed to confirm this. The effect of catalyst type and composition was also studied. The reproducibility issues encountered with both bimetallic catalyst systems regarding Raman spectroscopy characterization made a discussion on the evolution of the Ramanextracted diameter distributions, or SC/M ratios difficult. But we noticed differences in growth temperature limits based on the catalyst. Using the carbon solubility argument, we were able to propose an explanation for this. Again, further characterization of catalyst nanoparticle sizes at different temperatures are needed to conclude. The comparison between direct SWCNT diameter measurement by TEM imaging, and our Raman spectroscopy methodology for diameter distribution assessment on our growth samples sparked a few questions on the reliability of both of these techniques for selectivity assessment. The methodology that we applied for Raman spectroscopy characterization is a quite widespread methodology among research groups dealing with selective SWCNT growth, but it is not the methodology proposed by Raman specialists for selectivity assessment. The comparison between TEM-and Raman-extracted diameter distributions systematically showed shifts of about 4 Å towards larger diameter when using TEM on our growth samples. This TEM characterization revealed a quite broad diameter distribution, with a significant portion of SWCNTs with large diameters. We distinguished three diameter ranges based on the relative agreement between TEM and Raman characterizations. By doing so, we established that the broadness of the actual diameter distribution of the sample, and the apparent diameter-dependent adaptability of each technique (Raman spectroscopy seeming more adapted to small diameter SWCNT detection, and the opposite was observed for TEM) were responsible for the observed shift. For a better understanding, a similar comparison was performed using diametersorted SWCNTs with diameters ranging from 0.8 nm to 1.5 nm. The crosscharacterization by TEM and Raman spectroscopy showed a good agreement between the two techniques regarding the diameter range of the distribution. However, distortions in the shape of the distribution were observed in the case of Raman spectroscopy (bimodal), the TEM-extracted distribution having a classical Gaussian shape. By confronting the Raman results to an adapted Kataura plot, we evidenced the absence of observed resonance from certain SWCNT populations within the resonance windows for each excitation wavelength. We assume these discrepancies to be the result of chirality-dependent Raman crosssections, but not all results can be explained by this. More insights could be gained by performing a HRTEM characterization of the diameter-sorted sample, leading to a detailed chirality distribution of the sample. Moreover, additional experiments on diameter-sorted samples with larger diameters could be very insightful, and make this study more complete. In the final chapter, we explored two other chemically prepared catalyst precursor families for SWCNT growth. We first showed the synthesis and characterization of FePd cyanosol nanoparticles. Their deposition and subsequent reduction under H 2 led to the formation of a dense layer of nanoparticles with a controlled diameter, showing no signs of coalescence. Through a STEM-EDX characterization of the nanoparticles after cooling and exposure to air, we evidenced two populations of nanoparticles: pure Pd, and Pd@Fe core-shell nanoparticles. Unfortunately, the first results of SWCNT growth using this precursor were not reproduced. Though the initial goal to form alloy nanoparticles did not seem to be reached, preparing core-shell nanoparticles could be of great interest. Further optimization is needed in this case. Moreover, additional CVD tests should be conducted, and this method could also be applied to a variety of other bimetallic systems (FePt, NiPt, CoMo..). The last precursor system that was studied are W 34 Co 7 O x POM clusters. They were successfully deposited on silanized wafers, and resulted in SWCNT growth. Though the yield was low, interesting temperature effects were observed using Raman spectroscopy. The versatility of the POM family is such that these preliminary results are very promising for application to a wide range of other cluster structures. More optimization of the deposition and CVD parameters could also lead to very promising results. As a general conclusion, though not all goals were attained, the process developed applied to three PB/PBA systems, and its adaptation applied to a cyanosol system, and a POM system, allowed the preparation of five different catalysts of interest. Their comparison in identical experimental conditions led to potential understanding of phenomena at play during SWCNT growth. This method can be applied to many other systems, as attested by studies conducted on three additional Ru-based PBA systems. Our process opens the way to future investigations of the effect of catalyst composition on growth selectivity. In a more general sense, the promise offered by coordination chemistry, and inorganic synthesis for fabrication of SWCNT growth catalyst systems has been demonstrated through this work. Finally, the methodology that is becoming a standard for Raman spectroscopy evaluation of growth selectivity has to be used with great caution, as attested by our comparative study. Cross-characterization seems indispensable when looking at the presented results, especially considering that this methodology is used for chirality distribution assessment. This study shows the necessity for standardized selectivity evaluation methodologies that are cost-effective, practical and not too time-consuming, but above all, reliable, accurate, and reproducible. Appendix D. Résumé de la thèse des particules de catalyseurs, dépendant de leur taille, leur composition, et de la température, est proposée pour expliquer ces deux phénomènes. Dans un second temps, une étude comparée de caractérisation par microscopie électronique en transmission (MET) et spectroscopie Raman multi-longueur d'onde, a été menée dans le but d'évaluer les techniques de caractérisation disponibles dans leurs capacités respectives à permettre une mesure fiable de la distribution de diamètres d'un échantillon de croissance de SWCNTs. Suite à l'observation de différences significatives systématiques entre les résultats obtenus par les deux techniques de caractérisation, plusieurs hypothèses ont été formulées pour tenter d'expliquer ces différences. Une étude complémentaire a été menée sur des SWCNTs triés en diamètres, permettant de tester ces hypothèses. Nous avons pu montrer l'importance du caractère résonant de la caractérisation par spectroscopie Raman dans les différences observés. Cette étude souligne l'importance et la nécessité du développement de méthodes de caractérisation standardisées pour l'évaluation de la sélectivité des croissances de SWCNTs. Enfin, des études préliminaires sur l'utilisation deux autres familles de composés synthétisés par voie chimique comme précurseurs de catalyseurs de croissance de SWCNTs sont présentées dans ce manuscrit. Un système bimétallique FePd de « cyanosol », famille de composés structurellement proche des ABP, a d'abord été étudié. L'adaptabilité de notre procédé à une nouvelle famille de composés a été confirmée. Le second précurseur étudié est un polyoxometallate (POM) avec un ratio atomique Co-W bien défini. L'étude préliminaire par spectroscopie Raman de croissances de SWCNTs à différentes température utilisant ce précurseur a pu confirmer l'intérêt du système bimétallique Co-W d'un part, et de l'utilisation de POMs d'autre part. Cette étude préliminaire confirme l'intérêt de l'utilisation de précurseurs de catalyseurs synthétisés par voie chimique pour la croissance de SWCNTs, et pour l'étude des mécanismes complexes intervenant dans cette dernière. Figure 2 . 6 - 26 Figure 2.6 -Schematic polyhedral structure of the POM used in this study, from [180]. The orange centers correspond to Co atoms, which are surrounded by six oxygen atoms (red) to form CoO 6 octahedra, the blue octahedra are WO 6 , the green tetrahedra are SiO 4 . Figure 2 . 7 - 27 Figure 2.7 -Schematic representation of the HFCVD reactor used the growth of SWCNTs. Figure 2 . 8 - 28 Figure 2.8 -Schematic depiction of electron-matter interaction and the resulting signals. All straight arrows refer to electrons, the elastically scattered electrons are in dark blue, and inelastically scattered electrons and engendered signals (photons, secondary electrons, Auger electrons) are in purple. Figure 2 . 9 - 29 Figure 2.9 -Schematic representation of the basic principle of a TEM. Figure 2 . 14 - 214 Figure 2.14 -Schematic representation of the Zeiss Libra 200 MC microscope. et al. established an empirical Kataura plot based on electron diffraction and Rayleigh scattering measurements of optical transition Figure 2 . 17 - 217 Figure2.17 -Kataura plot from[START_REF] Sato | Discontinuity in the family pattern of single-wall carbon nanotubes[END_REF], based on ETB model calculations. The black circles correspond to metallic SWCNTs, and the blue circles are semiconducting SWCNTs. 3. 2 2 Synthesis and characterization of catalyst precursors: PB and PBA nanoparticles 3.2.1 Synthesis of three different PB/PBA nanoparticle systems: FeFe, NiFe, and NiCr Figure 3 . 4 - 34 Figure 3.4 -XRPD patterns obtained for the three PB/PBA systems, after precipitation of the nanoparticles by CaCl 2 and centrifugation. The peaks marked with a * are attributed to the aluminum sample holder. Figure 3 . 8 - 38 Figure 3.8 -Size distributions of the FeFe PB catalyst precursor nanoparticles (light blue), and of the FeFe effective catalyst nanoparticles (dark blue), evidencing the size decrease of the nanoparticles upon reduction. Figure 3 . 12 - 312 Figure 3.12 -Comparison between the NiFe effective catalyst nanoparticle size distribution extracted from TEM imaging (orange), and the size distribution of the NiFe nanoparticles studied by STEM-EDX (yellow). Figure 3 . 21 - 321 Figure 3.21 -Raman-extracted yields for all catalyst systems at various temperatures. When Raman data was available for several samples, the error bars correspond to the standard deviation of the "yield". 4. 1 . 1 Comparing TEM and Raman spectroscopy on typical CVD growth samples 135 Figure 4 . 4 Figure 4.3 shows the superimposed Raman and TEM diameter distribution histograms for the two samples. For the FeFe (NiFe) sample, the TEM distribution was determined from the measurement of 112 (110) SWCNT diameters, and the Raman distribution was determined by counting 512 (194) RBM peaks. Figure 4 . 4 6.b). In summary, 4.1. Comparing TEM and Raman spectroscopy on typical CVD growth samples 147 Figure 4 . 8 - 48 Figure 4.8 -Schematic representation of the three samples prepared using the diameter-sorted SWCNT solution. The samples are numbered and give the overall progression of this section: we will first have a direct look at the composition of the SWCNT sample by direct TEM characterization (1), then the potential effect of the transfer process will be evaluated by characterization of sample (2), and the TEM characterization will be compared to Raman by looking at sample (3). Figure 4 . 10 - 410 Figure 4.10 -Diameter distribution histogram obtained from measurements from 121 suspended and bare SWCNTs. Figure 4 . 11 - 411 Figure 4.11 -Typical SEM image of diameter-sorted SWCNTs deposited on a SiO 2 /Si wafer. Figure 4 . 13 - 413 Figure 4.13 -Overlap of the diameter distribution histograms for the diametersorted SWCNT sample by direct TEM characterization (black), and after the transfer process (green). Figure 4 . 15 - 415 Figure 4.15 -Superimposition of the diameter distribution histograms obtained by direct TEM characterization (black), and Raman spectroscopy characterization (blue) of the diameter-sorted SWCNT sample. Figure 5 . 5 2 shows a typical TEM image of the Chapter 5. Exploration of two other families of SWCNT catalyst precursors: Cyanosols, and POMs5.1.3.2 Structural and chemical analysisIn order to confirm the formation of the CN-linked polymer, IR spectroscopy was performed on the FePd cyanosol nanoparticles (see Figure5.3 for spectrum in the vincinity of the CN stretching mode). The particles were retrieved using dextran. We can distinguish three different signals: a first band at 2175 cm -1 , and a second at 2118 cm -1 , with a shoulder at 2079 cm -1 . The first band corresponds to the bridging CN ligands (Fe III -CN -Pd II )[START_REF] Pfennig | Synthesis of a novel hydrogel based on a coordinate covalent polymer network[END_REF]. The second signal can be attributed to Fe II -CN -Pd II bridging CN ligands, and the shoulder to terminal Fe III . This indicates a partial reduction of the Fe III to Fe II within the structure. The two bands attributed to bridging CN stretching confirm the formation of the coordination polymer. The frequency of the stretching mode of the terminal CN ligands is in good agreement with the value obtained for the K 4 Fe II (CN) 6 , pure compound, and the band corresponding to the bridging ligands is blue-shifted in comparison to the terminal ligand band. We should note that the relative intensities of the bridging and terminal CN ligands show the abundance of terminal ligands in the structure, in agreement with a high surface to volume aspect ratio. Figure 5 . 3 - 53 Figure 5.3 -IR spectrum of the precipitated and dried FePd cyanosol nanoparticles retrieved with 25 equivalents of dextran. Figure B. 10 - 10 Figure B.10 -XPS survey of the FePd cyanosol nanoparticle layer after deposition on a bare wafer. The presence of Fe, Pd, and K is confirmed, as well as nitrogen, stemming from the CN bridges. Figure B. 11 - 11 Figure B.11 -XPS survey of the Co-W POM layer after deposition on a silanized wafer. The presence of Co, and W is confirmed. Figure B. 13 - 13 Figure B.13 -EDX spectra extracted from three regions of the map (image on the left, 5 nm scale bar) for the segregated NiCr nanoparticles. The topmost spectrum is the mean for the entire circled nanoparticle, the presence of Cr (5.3-3.5 keV) and Ni (7.4-7.6 keV) is confirmed. The second spectrum corresponds to a spot within the Cr-rich region of the nanoparticle: no Ni is detected. The third spectrum corresponds to a spot within the Ni-rich phase of the nanoparticle: no Cr is detected. Further, no signal stemming from Cs is observed, evidencing its absence from the nanoparticle. The signal stemming from copper comes from the Cu-TEM grid. Figure B. 14 - 14 Figure B.14 -EDX spectra extracted from the two Fe-Pd nanoparticles shown in Chapter 5 (image on the left, 3 nm scale bar). Top: core-shell system. The first spectrum is the global spectrum of the entire map, and the second corresponds to the restricted region around the nanoparticles. Fe and Pd are both present within the nanoparticle. Bottom: "pure" Pd nanoparticle. The top spectrum corresponds to the global map, and the bottom one corresponds to the region around the nanoparticle circled in green. A small signal is observed in the Fe-Kα region, but it is very low, and could be attributed to residual noise. In both nanoparticles, the presence of Ti stems from the TEM grid. iii 2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Growth of SWCNTs using Prussian Blue and its analogs as catalyst precursors 3.1 From PB/PBA to SWCNT: general process . . . . . . . . . . . 3.2 Synthesis and characterization of catalyst precursors: PB and PBA nanoparticles . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Synthesis of three different PB/PBA nanoparticle systems: FeFe, NiFe, and NiCr . . . . . . . . . . . . . . . . 3.2.2 Characterization of the obtained nanoparticles . . . . . . 3.2.2.1 Size . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2.2 Crystallographic structure . . . . . . . . . . . . 3.2.2.3 Chemical analysis . . . . . . . . . . . . . . . . . 3.2.2.4 Conclusion . . . . . . . . . . . . . . . . . . . . 3.3 Sample preparation and PB/PBA precursor deposition . . . . . .4.3 STEM-EDX study of bimetallic catalyst nanoparticles . . 3.4.3.1 STEM-EDX study of the NiFe catalyst . . . . . 3.4.3.2 STEM-EDX study of the NiCr catalyst . . . . . 3.4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 SWCNT growth . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 SWCNT growth from the monometallic PB catalyst precursor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1.1 SWCNT growth feasibility . . . . . . . . . . . . 3.5.1.2 Raman study of growth temperature effect . . . 3.5.2 SWCNT growth from the bimetallic PB catalyst precursors122 3.3.1 Wafer preparation . . . . . . . . . . . . . . . . . . . . . . 3.3.2 PB/PBA nanoparticle deposition . . . . . . . . . . . . . 3.3.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Catalyst nanoparticle formation and characterization . . . . . . 3.4.1 Wafer surface analysis after reductive pretreatment . . . 3.4.2 TEM characterization of the catalyst nanoparticles . . . 3.4.2.1 Monometallic FeFe catalyst . . . . . . . . . . . 3.4.2.2 Bimetallic catalysts . . . . . . . . . . . . . . . . 3.4.2.3 Size evolution of catalyst nanoparicles after SWCNT growth . . . . . . . . . . . . . . . . . . . . . . . 33.5.2.1 SWCNT growth feasibility . . . . . . . . . . . . 3.5.2.2 Raman and SEM study of growth temperature effect . . . . . . . . . . . . . . . . . . . . . . . 1.3. Theoretical and experimental insights into SWCNT nucleation and growth 41 catalyst, and the parameters impacting the phenomenon. More realistic calculations, taking into account carbon presence and temperature for instance, have lead to interesting discoveries on nucleation mechanisms, and the role of the catalyst nanoparticle, leading to other, more general strategies for successful and selective SWCNT growth. Amara et al. developed a TB model for Ni-C interactions coupled with grand canonical Monte Carlo (GCMC) simulations to study the nucleation of carbon caps onto Ni nanoparticles 1.3.3 Towards a more general in-between? 2.1.1.1 Prussian blue: structure and properties PB has been known since 1704 and used as a pigment because of its strong characteristic blue color. It is a 3D coordination polymer of general formula K x Fe III [Fe II (CN) 6 ] (3+x)/4 .15H 2 O, and is considered to be the first synthetic coordination compound. It is the resulting compound of the coprecipitation reaction between potassium hexacyanoferrate(II) (K 4 [Fe II (CN) 6 ] POMs are polynuclear molecular clusters made from the assembly of [MO y ] n- polyhedra where M is a metal, with an almost unlimited range of structures. In most cases, M is a group 5 or 6 transition metal in a high oxydation state (Mo VI/V , W VI/V , or V VI/V ). They are synthesized by the condensation of [MO 4 ] n-monomers in an acidic medium. They can be divided into two main compound families: the isopolyoxometalates ([M 2.1.3.1 Definition, synthesis and architecture x O y ] n-), that are formed by the condensation of same-metallic element polyhedra, and heteropolyoxometalates (HPA) ([X z M x O y ] n-), where the condensation occurs around a heteroatom Table 2 . 2 1 -Main characteristics of the two TEMs used for SWCNT characterization in the presented work, adapted from[START_REF] Ghedjatti | Etude structurale des nanotubes de carbone double parois[END_REF]. Microscope Zeiss Libra 200 MC JEOL ARM 200F FEI TITAN 3 G2 Electron source Schottky FEG Cold FEG X-FEG energy-filtered Accelerating 60, 80-200 80-200 60-300 voltage (keV) Corrector Monochromator C s -corrector objective C s -corrector image Energy spread 0.2 0.4 (eV) Defocus spread 7.7 5.5 (nm) Scherzer defocus -64 -3 (nm) Table 2 . 2 2 -A non-exhaustive list of the A and B constant values in the w RBM = A/d t + B law available in the literature, adapted from • C in ice baths. The mixture immediately turns blue, which is a sign of the formation of the PB network, and it is left under high stirring for 30 minutes. The conditions are slightly different for preparing the NiFe and NiCr PBA nanoparticles: the concentration of each solution is multiplied by 4 (2 mM) in comparison to the concentration for the synthesis of PB, and the reaction is performed at room temperature. Moreover, cesium chloride (CsCl) salt is added in stoichiometric proportions to the solution containing the potassium hexacyanometallate [M (CN ) 6 ] x+ . Table 3 . 3 1 -Lattice constants and Scherrer domain sizes extracted from the XRPD patterns for all three PB/PBA systems. System FeFe NiFe NiCr Lattice constant 10.19 10.31 10.51 ( Å) Scherrer domain 7.8 5.9 5.7 size (nm) Table 3 . 3 System FeFe NiFe NiCr Bridging CN (cm -1 ) 2072 2164; 2098 2171 Free CN (cm -1 ) 2060 2130 2 -Frequencies of the CN vibration characteristic band (bridging and non-bridging) for all PB/PBA systems. Table 3 . 3 4 -Atomic percentages of bimetallic PBA nanoparticle layer deposited on SiO 2 /Si, extracted from XPS analysis, and atomic fractions elements of interest in the M, M', Cs mixture. NiFe PBA NiCr PBA Element Global % At. fraction Element Global % At. fraction Si 2p 22.9 - Si 2p 24.6 - O 1s 33.9 - O 1s 37.1 - N 1s 5.6 - N 1s 4.2 - C 1s 35.5 - C 1s 30.9 - Ni 2p 1.2 0.52 Ni 2p 1.2 0.38 Fe 2p 0.8 0.35 Fe 2p - - Cr 2p - - Cr 2p 1.3 0.41 Cs 3d 0.3 0.13 Cs 3d 0.7 0.23 Table 3 . 3 6 -Diameter distributions and SC percentages extracted from Raman spectroscopy characterization of SWCNTs grown from the FeFe catalyst at various temperatures. Temperature Mean diameter Standard deviation %SC ( • C) (nm) (nm) 700 1.11 0.28 65 800 1.30 0.30 61 900 1.82 0.40 73 Table 3 . 3 8 -Report of obtained SWCNT growths for the three catalytic systems at various temperatures. Checkmarks indicate that SWCNT growth is successful, XX indicates a zero Raman "yield". Table 4 . 4 1 -Compared mean diameters and standard deviations of the diameter distributions obtained by TEM and Raman spectroscopy on two typical growth samples. Sample Mean TEM d t (nm) Mean TEM d t < 2.5nm (nm) Mean Raman d t (nm) FeFe 1.73 ± 0.48 FeNi 1.74 ± 0.35 1.67 ± 0.36 1.73 ± 0.35 1.31 ± 0.30 1.35 ± 0.29 Table Table 4 . 4 2 -Comparison of the percentages of the total detected SWCNTs present in each of the delimited 1, 2, and 3 diameter domains defined as d(1) < 1.3 nm, 1.3 ≤ d(2)< 1.7 nm, and 1.7 ≤ d(3) < 2.5 nm, for the two presented samples for TEM and Raman characterizations. FeFe NiFe Domain TEM % Raman % TEM % Raman % 1 19.6 58.0 10.9 51.5 2 32.1 29.7 37.3 36.6 3 48.2 12.3 51.8 11.9 not expected, easily understandable. The next section goes into more detail on what explanations can be brought to understand this observed shift. 3 nm) Chapter 4. Comparing TEM and resonant Raman spectroscopy for diameter distribution assessment of a SWCNT sample 2. The transfer process modifies the sample by degrading small-diameter SWCNTs (below 1.3 nm) 3. Raman spectroscopy is not adapted to the characterization of large-diameter SWCNTs (typically above 2.5 nm) 4. Raman spectroscopy over/under-estimates the presence of certain SWCNT populations 5. ω RBM → d t relationship is not adapted to the sample 6. Our methodology is not adapted to our growth samples Table 4 . 4 3 -Mean diameters and standard deviations of the distributions obtained for the diameter-sorted SWCNT sample by Raman spectroscopy, direct TEM measurement, and TEM measurement after transfer. In the case of TEM measurements, the values are given without consideration of the SWCNTs with diameters above 2.5 nm.Further, the general shape of the histograms differ quite significantly from one Chapter 4. Comparing TEM and resonant Raman spectroscopy for diameter distribution assessment of a SWCNT sample the diameters of the SWCNTs detected by Raman spectroscopy, the bimodal shape of the distribution is preserved. The shape of the distribution being the most important difference between the TEM-and Raman-extracted distributions for the diameter-sorted SWCNT sample, this is an important point. It leads us to consider with more certainty that the distortion in the shape of the distribution may be due to the resonant character of the Raman method, and to multifactor-dependent Raman intensities. Characterization Mean d t Standard deviation technique (nm) (nm) Raman spectroscopy 1.00 0.23 Direct TEM 1.15 0.25 Transferred TEM 1.17 0.33 plotted the separate excitation wavelength-specific diameter distribution Chapter 4. Comparing TEM and resonant Raman spectroscopy for diameter distribution assessment of a SWCNT sample (Kataura plot (2)) is adapted from the Raman spectroscopy studies on "supergrowth" SWCNTs by Araujo et al.[START_REF] Araujo | Resonance raman spectroscopy of the radial breathing modes in carbon nanotubes[END_REF]. Using the methodology developed by Zhang et al. for chirality assignment of SWCNTs based on multi-laser Raman spectroscopy characterization of as-grown SWCNTs on SiO 2 substrates, all E ii were redshifted by 80 meV for semiconducting SWCNTs, and 60 meV for metallic SWCNTs to take into account the environmental effects on the E ii that are considered to be virtually non-existent for as-grown supergrowth SWCNTs[START_REF] Zhang | n, m) assignments and quantification for single-walled carbon nanotubes on sio 2/si substrates by resonant raman spectroscopy[END_REF]. These values are average values derived from comparing the E ii values obtained in a study by Hsieh et al. on specific (n,m) SWCNTs on a SiO 2 substrate[START_REF] Hsieh | Chiral angle dependence of resonance window widths in (2 n+ m) families of single-walled carbon nanotubes[END_REF] to the corresponding values for supergrowth SWCNTs obtained using the equation from[START_REF] Araujo | Resonance raman spectroscopy of the radial breathing modes in carbon nanotubes[END_REF].If we consider the global diameter range of the sample to be roughly delimited by the TEM-extracted diameter distribution, we should detect SWCNTs between 0.8 nm and 1.5 nm. Looking at the four portions of the Kataura plot displayed in Figure 4.18, there are SWCNTs with E ii within a ± 100 meV window around the excitation energy (see Chapter 2.3.2.3) for each excitation energy within the global diameter range of the sample. The equivalent of Figure 4.18 for Kataura plot (1) is shown in Appendix C. Since the Kataura plot (2) III Pd II cyanosol nanoparticles A synthesis route to cyanosol nanoparticles was developed in our lab during the internship of Adrien Barroux in 2016. The objective of this work was to obtain nanoparticles with controlled size and composition, with a method close to the co-precipitation method developed by Catala et al. for the PB/PBA systems. In the case of the Fe-Pd bimetallic system, the Fe III Pd II cyanosol nanoparticles can successfully be synthesized; the nanoparticles are stable in water for a few days up to a few weeks without the introduction of any stabilizing agents.For this synthesis, a 10 mM solution of K 2 Pd II Cl 4 in DI water is poured into a 10 mM solution of K 3 Fe III (CN) 6 under high stirring. The solution turns a light orange, and it is left under stirring for 30 minutes at room temperature. The as-obtained nanoparticle dispersion is directly used for characterization. Acknowledgements The two characterization techniques used for SWCNT sample selectivity assessment were then presented in detail. We established a clear methodology for the evaluation of the diameter distribution of SWCNT growth samples using both Raman spectroscopy, and TEM. Appendix A Characterization techniques, and experimental details The various characterization techniques used in the presented work will be shortly introduced here (in order of appearance in the text), along with experimental details concerning each of them. A.1 Dynamic light scattering (DLS) DLS is a characterization technique that allows the measurement of the hydrodynamic size of objects dispersed in solution. The hydrodynamic size includes intrinsic size of the object and its solvation shell. The measured sizes are therefore larger than the actual size of the objects in solution. This technique, which also gives an idea of the state of aggregation of the objects in solution, is simple and fast, and can be used routinely directly using the as-obtained nanoparticle dispersion. DLS measurements were performed on a Malvern Zetasizer. A.2 TEM: additional experimental details TEM imaging for the PB/PBA nanoparticles was performed using an image corrected FEI TITAN TEM operating at 300 kV. A.3 X-ray powder diffraction (XRPD) When X-rays interact with a crystal, they are scattered. The scattered rays interfere constructively in specific directions corresponding to hkl planes, given by Bragg's law: Where d is the spacing between two diffracting planes from the same family, θ is the incident angle, λ is the wavelength of the monochromatic X-ray beam, and n is an integer. In XRPD, a diffraction pattern giving the diffracted intensity as a function of 2θ. Each diffraction peak corresponds to a hkl plane family, and can be indexed, to determine the crystallographic structure of the material. XRPD analyses were performed on powder deposited on an aluminum plate and recorded on a Phillips Panalytical X'Pert Pro MPD powder diffractor at CuKα (1.5404 Å) radiation equipped with a fast detector. A.4 Infrared spectroscopy IR spectroscopy relies on the fact that chemical compounds absorb IR waves at resonance frequencies characteristic of their structure. It was used in this work to confirm the formation of PB/PBA networks, as well as cyanosol polymers, by looking at the shifts in the frequencies of CN ligand stretching bands. To perform IR spectroscopy, 1 mg of PBA powder is mixed with 99 mg of potassium bromide (KBr), and formed into a pellet. KBr is used because it is transparent in the IR range. The PBA nanoparticles were characterized by transmission IR spectroscopy, performed on a Perkin Elmer Spectrum 100 spectrometer and by reflexion IR performed on a Brüker VERTEX 70. A.5 Scanning electron microscopy (SEM), and SEM-EDX A SEM is a microscope that allows the observation of samples using a focused electron beam that scans their surface. The accelerating voltage is much lower than in the case of a TEM (a few kV, up to about 20 kV), since the electrons do not have to be transmitted through the sample, leading to lower potential resolutions. Various signals are produced through the interaction between the electron beam and the sample. In the most common configuration, SEM is performed by collecting secondary electrons, which will give an image of the topography of the sample. This technique was used for the observation of SWC-NTs grown on SiO 2 /Si substrates. SEM observations of SWCNTs were carried out on HITACHI S 4800 microscope at 1 kV (or 0.5 kV) and 10 µA. The general principle of EDX was presented in Chapter 2 of this manuscript, in the case of its use in a TEM. X-ray photons resulting of the interaction of A.6. Atomic force microscopy (AFM) the electron beam with the sample can also be collected in a SEM, allowing chemical analysis of the sample. This technique was used for determining the Cs:M:M' ratio in PBA compounds. A.6 Atomic force microscopy (AFM) AFM allows the obtention of high resolution images of the topography of samples. AFM was used in the present work to observe the evolution of the topography of SiO 2 /Si substrates after deposition of catalyst precursor nanoparticles or clusters. The basic principle of an AFM is to scan the sample with a tip that is fixed at the end of a cantilever, attached to a piezoelectric scanner that can precisely adjust its height. The cantilever is imposed a frequency close to its resonance frequency (control frequency), and a line by line scan over the sample surface is performed. The attractive and repulsive interactions between the tip and the surface modify the vibration amplitude of the tip. This modification is measured using a laser beam reflected on the cantilever, whose direction is tracked by four photodiodes. The piezeoelectric scanner adjusts the height of the tip with respect to the detected interactions with the surface, so that it returns to its original amplitude. The height is recorded, enabling the formation of topographic images. AFM measurements (tapping mode topography) were performed using di Innova AFM Bruker with NanoDrive v8.02 software. Images were acquired using silicon tips from nanosensors (PPP-NCSTR) with a resonance frequency ranging between 76 and 263 kHz. Images were processed using WsXM software, freely available on internet. A.7 X-ray photoelectron spectroscopy (XPS) XPS is a highly sensitive surface characterization technique, it was used to confirm the presence of certain elements on the surface of SiO 2 /Si substrates after deposition of catalyst precursors and their reduction. Quantitative analysis was also performed after deposition to confirm the metallic ratios obtained by SEM-EDX on powders. XPS is based on the photoelectric effect: when incident X photons hit the surface of the sample, inner shell electrons from the atoms at the surface can be ejected. The sample is irradiated with a monochromatic Xray beam of known energy, the ejected electrons are collected, and their number and kinetic energy are measured. The binding energy of the electrons can be extracted from the measurement of their kinetic energy, by a simple energy conservation equation. An XPS spectrum gives the number of detected electrons as a function of binding energy. The binding energies being characteristic of a specific element, chemical analysis can be performed. This analysis is made quantitative by taking into account the photoionization cross-sections of the different electronic levels. B.5 XPS surveys of catalyst precursors on SiO

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https://inria.hal.science/hal-01745792/file/main.pdf

Arthur Charguéraud email: [email protected] Alan Schmitt email: [email protected] Thomas Wood email: [email protected] JSExplain: A Double Debugger for JavaScript We present JSExplain, a reference interpreter for JavaScript that closely follows the specification and that produces execution traces. These traces may be interactively investigated in a browser, with an interface that displays not only the code and the state of the interpreter, but also the code and the state of the interpreted program. Conditional breakpoints may be expressed with respect to both the interpreter and the interpreted program. In that respect, JSExplain is a double-debugger for the specification of JavaScript. Indeed, in JS, the evaluation of any sub-expression, of any type conversion, and of most internal operations from the specification may result in the execution of user code, hence the raising of an exception, interrupting the normal control flow. Throught its successive editions, the ECMA standard progressively introduced a notation akin to an exception monad ( §1.2). This notation is naturally translated into real code by a proper monadic bind operator of the exception monad. Regarding the state, the standard assumes a global state. A reference interpreter could either assume a global state, modified with side-effects, or thread the state explicitly in purely-functional style. We chose the latter approach for three reasons. First, we already need a monad for exceptions, so we may easily extend this monad to also account for the state. Second, starting from code with an explicit state would make it easier to generate a corresponding inductive definition in a formal logic (e.g., Coq), which we would like to investigate in the future. Third, to ease the reading, one may easily hide a state that is explicitly threaded; the converse, materializing a state that is implicit, would be much more challenging. We thus write our reference interpreter in a purely-functional language extended with syntactic sugar for the monadic notation to account for the state and the propagation of abrupt termination ( §2). For historical reasons, we chose a subset of OCaml as source syntax, but other languages could be used. In fact, we implemented a translator from our subset of OCaml to a subset of JS (a subset involving no side effects and no type conversions). We thereby obtain a JS interpreter that is able to execute JS programs inside a JS virtual machine-JS fans should be delighted. To further improve accessibility to JS programmers, we also translate the source code of our interpreter into a human-readable JS-style syntax, which we call pseudo-JS, and that essentially consists of JS syntax augmented with a monadic notation and with basic pattern matching. Our reference semantics for JS is inherently executable. We may thus execute our interpreter on test suites, either by compiling and executing the OCaml code, or by executing the JS translation of that code. It is indeed useful to be able to check that the evaluation of examples from the JS test suites against our reference semantics produces the desired output. Even more interesting is the possibility to investigate, step by step, the evaluation of the interpreter on a given test case. Such investigation allows to understand why the evaluation of a particular test case produces a particular output-given the complexity of JS, even an expert may easily get puzzled by the output value of a particular piece of code. Furthermore, interactive execution makes it easier for the contributor of a new JS feature to add new test cases and to check that these tests trigger the new features and correctly interact with existing features. In this paper, we present a tool, called JSExplain, for investigating JS executions. This tool can be thought as a double debugger, which displays both the state of the interpreted program and that of the interpreter program. In particular, our tool supports conditional breakpoints expressed simultaneously on the interpreter program and the interpreted program. To implement this tool, we generate a version of our interpreter that is instrumented for producing execution traces ( §3), and we provide a web-based tool to navigate through such traces ( §4). As far as we know, our tool is the first such double debugger, i.e., debugger with specific additions for dealing with programs that interpret other programs ( §5). English Specification of JS To illustrate the style in which the JavaScript standard (ECMA) [START_REF]Ecmascript 2017 language specification[END_REF] is written, consider the description of addition, which will be our running example throughout the paper. In JS, the addition operator casts its arguments to integers and computes their sum, except if one of the two arguments is a string, in which case the other argument is cast to a string and the two strings are concatenated. The ECMA5 presentation (prior to June 2016) appears in Figure 1. First, observe that the presentation describes both the parsing rule for addition and its evaluation rule. Presumably for improved accessibility, the JS standard does not make explicit the notion of an abstract syntax tree (AST). The semantics of addition goes as follows: first evaluate the left branch to a value, then evaluate the right branch to a value, then converts both values (which might be objects) into primitive values (e.g., string, number, ...), then test whether one of the two arguments is a string. If so, cast both arguments to strings and return their concatenation; otherwise cast both arguments to numbers and return their sum. This presentation style used in ECMA5 gives no details about the propagation of exceptions. While the treatment of exceptions is explicit for statements, it is left implicit for expressions. For example, if the evaluation of the left branch raises an exception, the right branch should not be evaluated. It appeared that leaving the treatment of exceptions implicit could lead to ambiguities at what exactly should or should not be evaluated when an exception gets triggered. The ECMA committee hates such ambiguities, because it could (and typically does) result in different browsers exhibiting different behaviors-the nightmare of web-developers. In ECMA6, such ambiguities were resolved by making the propagation of exceptions explicit. Figure 2 shows the updated specification for the addition operator. There are two main changes. First, each piece of evaluation is described on its own line, thereby making the evaluation order crystal clear. Second, the meta-operation ReturnIfAbrupt is invoked on every intermediate result. This metaoperation essentially corresponds to an exception monad. The ECMA6 standards, which aims to be accessible to a large audience, avoids the introduction of the word "monad". Instead, it specifies ReturnIfAbrupt as a "macro", as shown in Figure 3. Essentially, every result consists of a "completion record", which corresponds to a sum type distinguishing normal results from exceptions. In all constructs except try-catch blocks, exceptions interrupt the normal flow of the evaluation. As a result, ECMA6 specification is scattered with about 1100 occurences of ReturnIfAbrupt operations. Realizing the impracticability of that style of specification, the standardization committee decided to introduce in ECMA7 an additional layer of syntactic sugar in subsequent versions of the Evaluation of: AdditiveExpression : AdditiveExpression + MultiplicativeExpression 1. Let lref be the result of evaluating AdditiveExpression. 2. Let lval be GetValue(lref). 3. Let rref be the result of evaluating MultiplicativeExpression. 4. Let rval be GetValue(rref). 5. Let lprim be ToPrimitive(lval). 6. Let rprim be ToPrimitive(rval). 7. If Type(lprim) is String or Type(rprim) is String, then -Return the String that is the result of concatenating ToString(lprim) followed by ToString(rprim). 8. Return the result of applying the addition operation to ToNumber(lprim) and ToNumber(rprim). specification. As detailed in Figure 4, they define the question mark symbol to be a lightweight shorthand for ReturnIfAbrupt steps. The specification of addition in that new style is shown in Figure 5. The presentation of ECMA7 and ECMA8 (Figure 5) is both more concise than that of ECMA6 (Figure 2) and more formal than that of ECMA5 (Figure 1). The use of question marks is to be compared in §2 with the monadic notation that we use for our formal semantics. Requests from the JS Committee The JavaScript standardization body, part of ECMA and known as TC39, includes representatives from browser vendors, major actors of the web, and academics. It aims at defining a common semantics that all browsers should implement. TC39 faces major challenges. On the one hand, it must ensure full backward compatibility, to avoid "breaking the web". In particular, no feature used in the wild ever gets removed from the specification. On the other hand, the committee imposes the rule that no feature may be added to the standard before it has been implemented, shipped, and tested at scale in at least two distinct major browsers. Any member of the committee may propose new features, hence there are many proposals being actively developed, at different stages of formalization [START_REF]TC39 proposals[END_REF]. The rapid evolution of the standard stresses the need for appropriate tools to assist in the rewriting, testing, and debugging of the semantics. In particular, several members with whom we have had interactions expressed their need for several basic tools, such as: • a tool for knowing whether all variables that occur in the specification are properly defined (bound) somewhere; • a tool to perform basic type-checking of the meta-functions and of the variables involved in the specification; • a tool for checking that effectful operations go on a line of their own, to avoid ambiguity in the order of evaluation; • a tool for checking that the behavior is specified in all cases; Evaluation of: AdditiveExpression : AdditiveExpression + MultiplicativeExpression 1. Let lref be the result of evaluating AdditiveExpression. 2. Let lval be ? GetValue(lref). 3. Let rref be the result of evaluating MultiplicativeExpression. 4. Let rval be ? GetValue(rref). 5. Let lprim be ? ToPrimitive(lval). 6. Let rprim be ? ToPrimitive(rval). 7. If Type(lprim) is String or Type(rprim) is String, then a. let lstr be ? ToString(lprim). b. let rstr be ? ToString(rprim). c. Return the String that is the result of concatenating lstr and rstr. 8. let lnum be ? ToNumber(lprim). 9. let rnum be ? ToNumber(rprim). 10. Return the result of applying the addition operation to lnum and rnum. • a tool able to tell which lines from the specification are not covered by any test from the main test suite (test262 [15]); • a tool able to execute step-by-step the specification on concrete JS programs, and to inspect the value of the variables. In particular, step-by-step execution is critically needed to evaluate new features. When the committee decides that a feature proposal is worth integrating, it typically does not accept the proposal as is, but instead modifies the proposal in a way that is amenable to a simple, clear specification without corner cases, carefully trying to avoid harmful interactions with other existing features (or planned features). During this process, at some point the committee members have in their hand a draft of the extended semantics as well as a collection of test cases illustrating the new behaviors that should be introduced. Naturally, they would like to check that their formalization of the extended semantics assigns the expected behavior to each of the test cases. One might argue that such a task could be performed by modifying one of the mainstream browsers. Yet, existing JS runtimes are built with efficiency in mind, with huge code bases involving numerous optimizations. As such, modifying the code in any way is too costly for committee members to investigate variations on a feature request. Even if they could invest the effort, the distance between the English prose specification and the implementation would be too large to have any confidence that the two match, i.e., that the behavior implemented in the code matches the behavior described by the prose. An alternative approach to testing a new feature is to develop an elaboration (local translation) of that feature into plain JS. This can take the form of syntactic sugar adding a missing API, namely a polyfill, or the form of a source to source translation, namely a transpiler. For instance, one might translate so-called "template literals" into simple string concatenation. While polyfills and transpilers are a simple approach to testing new features, they have two major limitations. First, the encoding might be very invasive. For instance, the 2015 version of ECMAScript added proxies, and as a consequence significantly changed the internal methods of the language; the Babel [1] transpiler for proxies [START_REF]Babel proxy plugin[END_REF] is able to simulate this feature in prior version of JS, but at the cost of replacing all field access operations with calls to wrapper functions. Second, the interaction of several new features implemented using these approaches is very difficult to anticipate. Formal Specifications of JS In recent years, two projects, JSCert [START_REF] Bodin | A trusted mechanised javascript specification[END_REF] and KJS [START_REF] Park | KJS: a complete formal semantics of javascript[END_REF] have proposed a formal specification for a significant subset of JS. JSCert provides a big-step inductive definition for ECMA5, (technically, a pretty-bigstep specification [START_REF] Charguéraud | Pretty-big-step semantics[END_REF]), formalized in the Coq proof assistant [START_REF]The Coq proof assistant reference manual[END_REF]. JSCert comes along with a reference interpreter, called JSRef, that is proved correct with respect to the inductive definition. JSRef may be extracted into executable OCaml code for executing tests. KJS describes a small-step semantics for JS as a set of rewriting rules, using the K framework [START_REF] Rosu | An overview of the K semantic framework[END_REF]. This framework has been used to formalize the semantics of several other real-world languages. It provides in particular tool support for executing (syntax-directed) transition rules on a concrete input program. At first sight, it might seem that a formal specification addresses all the requests from the committee. Definitions are thoroughly type-checked; in particular, all variables must be properly bound. Definitions, being defined in a formal language, are ambiguityfree; in particular, the order of evaluation and the propagation of exceptions is precisely specified. The semantics can be executed on concrete input programs; moreover, with some extra tooling, one may execute a set of programs and report on the coverage of the specification by the tests. Given all the nice features of formal semantics, why wouldn't the standardization committee TC39 consider one of these formal semantics as the reference for the language? After discussing with senior members from TC39, we understand that there are (at least) three main reasons why there is no chance for a formal semantics such as JSCert or KJS to be adopted as reference semantics. (1) Formal specifications in Coq or in K use syntax and concepts that are not easily accessible to JS programmers. Yet, the specification is meant to be read by a wide audience. (2) These formal languages have a cost of entry that is too high for committee members to reach the level of proficiency required for contributing new definitions all by themselves. (3) JSCert and KJS come with specifications that can be executed, yet provide no debugger-style support for interactively navigating through an execution and for visualizing the state and the values of the variables, and thus do not help so much in tuning the description of new features. In the present work, we temporarily leave aside the motivation of giving a formal semantics to JS that one could use to formalize properties of the language (e.g., security properties), and rather focus on trying to provide a formal semantics that meets better the day-to-day needs of the TC39 committee. Contribution In this paper, we present a tool, called JSExplain, which aims at providing a formal semantics for JS that addresses the aforementioned limitations of prior work. Our contribution is two-fold. First, we present a specification for JS expressed in a language that, we argue, JS programmers can easily read and write ( §2). Second, we present an interactive tool that supports step-by-step execution of the specification on an input JS program ( §3 and §4). Our tool mimics the features of a debugger, such as navigation controls, state and variable visualization, and conditional breakpoints, but does so for both the interpreter program and the interpreted program. The language in which we display the specification consists of a subset of JS extended with syntactic sugar for monads and basic pattern matching. This language, which we call pseudo-JS, could be the source syntax for our specification. However, for historical and technical reasons, we use as input syntax a subset of OCaml, which is processed using the OCaml type-checker. Our current tool automatically converts the OCaml AST into pseudo-JS code. In the future, we might as well have our reference interpreter be directly in pseudo-JS syntax, and we could typecheck that code either by converting it to OCaml or by reimplementing a basic ML typechecker. A third alternative would be to use the Reason syntax [START_REF][END_REF], a JS-like syntax for OCaml programs. The only difference between the approaches is whether TC39 committee members would prefer to write OCaml style or JS style code. SPECIFICATION LANGUAGE 2.1 Constructs of the Language The input syntax in which we write and display the specification is a purely-functional language that includes the following constructs: variables, constants, sequence, conditional, let-binding, function definition, function application (with support for prefix and infix functions), data constructors, records (including record projections, and the "record-with" construct to build a copy of a record with a number of fields updated), tuples (i.e., anonymous records), and simple pattern matching (only with non-nested patterns, restricted to data constructors, constants, variables, and wildcards). For convenience, let-bindings and functions may bind patterns (as opposed to only variables). We purposely aim for a specification language with a limited number of constructs and a very standard semantics, to minimize the cost of entry into that language. Note that the input code is typechecked in ML. (Polymorphism is used mainly for type-checking options and lists, and operations on them.) As explained earlier ( §1.2), the semantics involves the propagation of exceptions and other abrupt behaviors (break, continue, and return). Their propagation can be described within our small language, using functions and pattern matching. Nevertheless, introducing a little bit of syntactic sugar greatly improves readability. For example, we write "let%run x = e1 in e2" to mean "if_run e1 (fun x -> e2)", where if_run is a function that implements our monad. The monadic operator if_run admits a polymorphic type, hence functions from the specification may return objects of various types. Nevertheless, in practice most functions from the ECMA standard Figure 6: Current input syntax of our specification language: a subset of pure OCaml, extended with monadic notation. are described as returning a "completion triple", which either describe abrupt termination or describe a value. In a number of cases, the value is in fact constrained to be of a particular type. For example, if to_number produces a value, then this value is necessarily a number. The standard exploits this invariant implicitly in formulation such as "let n be the number produced by calling to_number". In constrast, our code needs to explicitly project the number from the value returned. To that end, we introduce specialized monads such as if_number, written in practice "let%number n = e1 in e2". (An alternative approach would be to assign polymorphic types to completion triples, however following this route would require diverging slightly from ECMA's specification in a number of places.) Figure 6 shows the specification of addition in our reference interpreter, in OCaml syntax extended with the monadic notation. This code implements its informal equivalent from Figure 2. In that code, s denotes the state, c denotes the environment (variable and lexical environment, in JS terminology), op corresponds to the operator (here, the constructor C_binary_op_add corresponds to the AST token describing the operator +), v1 and v2 corresponds to the arguments, and w1 and w2 to their primitive values. The function strappend denote string concatenation, whereas "+." denotes addition on floating pointer numbers (i.e., JS's numbers). First observe that, as explained earlier ( §1.1), the state is threaded throughout the code. We show in the next section how to hide the state variables ( §2.2). Observe also that the code also relies on a few auxiliary functions. The function type_compare implements comparison over JS types-to keep our language small and explicit, we do not want to assume a generic comparison function with nontrivial specification. The functions to_primitive_def, to_string, and to_number are internal functions from the specification that implement conversions. These operations might end up evaluating arbitrary user code, and thus could perform side-effects or raise exceptions, hence the need to wrap them in monadic let-bindings. One important feature of this source language is that it does not involve any "implicit" mechanism. All type conversions are explicit in the code, so it is always perfectly clear what is meant. In particular, there is no need to type-check the code to figure out its semantics. In summary, the OCaml code of the interpreter (e.g., Figure 6) is well-suited as a non-ambiguous input language. Note that this code may be compiled using OCaml's compiler in order to run test cases; the current version of our interpreter passes more than 5000 test cases from the official test suite (test262). Translation into Pseudo-JS Syntax Although we believe that it is a desirable feature to have a source langage fully explicit, there is also virtue in pretty-printing the source code of our interpreter in a more concise syntax. The "noise" that appears in the formal specification (e.g. Figure 6) comes from three main sources: (1) every function takes as argument the environment; (2) every function takes as argument and returns a description of the mutable state (a.k.a. heap); (3) values are typically built using numerous constructors, e.g. C_value_prim, which lifts a number (an OCaml value of type float) into a JS value (an OCaml value of type value). Fortunately, we can easily eliminate these three sources of noises. First, the environment is almost always passed unchanged. It may be modified only during the scope of a function call, a with construct, or a block. When it is modified, new bindings are simply pushed into the environment (which behaves like a stack), and subsequently popped. Thus, we may assume, like the ECMA specification does, that the environment is stored in a global state. This saves the need to pass an argument called "c" around. Second, the description of the mutable state is threaded through the code. The "current state" is passed as argument to every function that might perform side-effects, and, symmetrically, the "updated state" is returned to the caller, which binds a fresh name for it. Considering that there is only one version of the state at any given point of an execution, we may assume, like the ECMA specification does, that state to be stored in a global variable. This saves the need to pass values called "s1", "s2"... around. Third, the presence of many constructors is due to the need for casts. Many of these casts could, however, be viewed as "implicit casts" (or "coercions", in Coq's terminology). For a carefully chosen set of casts, defined once and for all, and for a well-typed program with implicit casts, there exists a unique (non-ambiguous) way to insert casts in order to make the program type-check. Although we have previously argued that explicit casts are useful, as they allow giving a semantics that does not depend on type-checking, we now argue that it may also be useful to pretty-print the code assuming implicit casts, in order to improve readability. In summary, we propose to the reader of the specification a version that features implicit state, implicit context, and implicit casts. Given that we are playing the game of pretty-printing syntax, we take the opportunity to switch along the way to a JS-friendly syntax, using brackets and semicolons. This target language, called pseudo-JS syntax, consists of a subset of the JS syntax, extended with monadic notation, and an extended switch construct that is able to bind variables (like OCaml's pattern matching, but restricted to non-nested patterns for simplicity). The pretty-printing of the addition operator in pseudo-JS syntax appears in Figure 7. To illustrate our extended pattern matching syntax feature of pseudo-JS, we show below an excerpt from the main switch that interprets an expression. switch (t) { case Coq_expr_identifier(x): var%run r = identifier_resolution(x); return (r); case Coq_expr_binary_op(e1, op, e2): ... TRACE-PRODUCING EXECUTIONS JSExplain is a tool for interactively investigating execution traces of our JS interpreter executing example JS programs. The interface consists of a web page [START_REF][END_REF] that embeds a JS parser and a traceproducing version of our interpreter implemented in standard JS. So far, we have shown how to translate the OCaml source into pseudo-JS syntax ( §2.2). In this section, we explain how to translate the OCaml source into proper JS syntax, and then how to instrument the JS code in a systematic way for producing execution traces. Figure 8 illustrates the output of translating from our OCaml subset towards JS. Note that this code is not meant for human consumption. We implement monadic operators as function calls, introduce the return keyword where necessary, encode sum types as object literals with a tag field, encode tuples as arrays (encoding tuples as object literals would work too), turn constructor applications into functions calls, implement pattern matching by first switching on the tag field then binding fresh variables to denote the arguments of constructors. We thus obtain an executable JS interpreter in JS which, like our JS interpreter in OCaml, may be used for executing test cases. One interest of the JS version is that it may be easily executed inside a browser, a set up that might be more convenient for a number of users. One limitation, however, is that the number of steps that can be simulated may be limited on JS virtual machines that do not optimize tail-recursive function calls. Indeed, the execution of monadic code involves repeated calls to continuations, whose (tail-call) invocation unnecessarily grows the call stack. To set up our interactive debugger, we produce, from our OCaml source code, an instrumented version of the JS translation. This instrumented code produces execution traces as a result of interpreting an input JS program. These traces store information about all the states that the interpreter goes through. In particular, each event in the trace provides information about the code pointer and the instantiations of local variables from the interpreter code. More precisely, we log events at every entry point of a function, every exit point, and on every variable binding. Each event captures the state, the stack, and the values of all local variables in scope of the interpreter code at the point where the event gets triggered. To reduce noise in the trace, we only log events in the core code of the interpreter, and not the code from the auxiliary libraries. Overall, an execution of the instrumented interpreter on some input JS program produces an array of events. This array can then be investigated using our double debugger ( §4). Figure 9 shows an example snippet of code, giving an idea of the mechanisms at play. Note, again, that this code is not meant for human consumption. The function log_event augments the trace. Consider for instance log_event("Main.js", 4033, ctx_747, "enter"). The first two arguments identify the position in the source file, as a file name and a unique token used to recover the line numbers. The third argument is a context describing values of the local variables, and the fourth argument describes the type of event. When investigating the trace, we need to be able to highlight the corresponding line of the interpreter code. We wish to be able to do so for the three versions of the interpreter code: the OCaml version, the pseudo-JS version, and the plain JS version. To implement this feature, our generator, when processing the OCaml source code, also produces a table that maps, for each version and for each file of the interpreter, event tokens to line numbers. The contexts stored in events are extended each time a function is entered, a new variable is declared, or the function returns (so as to capture the returned value). Contexts are represented as a purelyfunctional linked list of mappings between variable names and values. This representation maximizes sharing and thus minimize the memory footprint of the generated trace. The length of the trace grows linearly with the number of execution steps performed. For example, the simple program "var i = 0; while (i < N) { i++ }" var run_binary_op_add = function (s0, c, v1, v2) { var ctx_747 = ctx_push(ctx_empty, [{key: "s0", val: s0}, {key: "c", val: c}, {key: "v1", val: v1}, {key: "v2", val: v2}]); log_event("JsInterpreter.js", 4033, ctx_747, "enter"); var _return_1719 = if_prim((function () { log_event("JsInterpreter.js", 3985, ctx_747, "call"); var _return_1700 = to_primitive_def(s0, c, v1); log_event("JsInterpreter.js", 3984, ctx_push(ctx_747, [{key: "#RETURN_VALUE#", val: _return_1700}]), "return"); return (_return_1700); }()), function(s1, w1) { ... }); log_event("JsInterpreter.js", 4028, ctx_push(ctx_748, [{key: "#RETURN_VALUE#", val: _return_1718}]), "return"); return (_return_1718); }); log_event("JsInterpreter.js", 4032, ctx_push(ctx_747, [{key: "#RETURN_VALUE#", val: _return_1719}]), "return"); return (_return_1719); }; The fact that these numbers are large reflects the fact that the reference interpreter is inherently vastly inefficient, as it follows the specification faithfully, without any optimization. Due to our use of functional data structures, the memory footprint of the trace should be linear in the length of the trace. We have not observed the memory footprint to be a limit, but if it were we could more carefully select which events should be stored. JSEXPLAIN: A DOUBLE DEBUGGER FOR JS The global architecture of JSExplain is depicted in Figure 10. Starting from our JS interpreter in OCaml, we generate a JS interpreter in JS. We instrument the JS code to produce a trace of events. This compilation is done ahead of time and depicted by solid arrows. When hitting the run button, the flow depicted by the dotted arrows occurs. The web page parses the code from the text area, using the Esprima library [START_REF]ECMAScript parsing infrastructure for multipurpose analysis[END_REF]. This parser produces an AST, with nodes annotated with locations. This AST is then provided as input to the instrumented interpreter, which generates a trace of events. This trace may then be inspected and navigated interactively. For a given event from the execution trace, our interface highlights the corresponding piece of code from the interpreter, and shows the values of the local variables, as illustrated in Figure 11. It also highlights the corresponding piece of code in the interpreted program, as illustrated at the top of Figure 13, and displays the state and the environment of the program at that point of the execution, as illustrated in Figure 12. Recovering the information about the interpreted code is not completely straightforward. For example, to recover the fragment of code to highlight, we find in the trace the closest previous event that contains a call to function with an argument named _term_. This argument corresponds to the AST of a subexpression, and this AST is decorated (by the parser) with locations. Note that, for efficiency reasons, we associate to each event from the trace its corresponding _term_ argument during a single pass, performed immediately after the trace is produced. Similarly, we are able to recover the state and environment associated with the event. The state of the interpreted program consists of four fields: the strictness flag, the value of the this keyword, the lexical environment, and the variable environment. We implemented a custom display for these elements, and also for values of the languages, in particular for objects: one may click on an object to reveal its contents and recursively explore it. We provide several ways to navigate the trace. First, we provide buttons for reaching the beginning or end of the execution, and buttons for stepping one event at a time. Second, we provide, similarly to debuggers, next and previous buttons for skipping function calls, as well as a finish button to reach the end of the current function. These features are implemented by navigating the trace, keeping track of the number of enter and return events. Third, we provide buttons for navigation based not on steps related to the interpreter program but instead based on steps of the interpreted program: source previous and source next find the closest event which induces a change in the location on the subexpression evaluated in the interpreted code, and source cursor finds the last event in the trace for which the associated subexpression contains the active cursor in the "source program" text area. The aforementioned tools are sufficient for simple explorations of the trace, yet we have found that it is sometimes useful to reach events at which specific conditions occur, such as being at a specific line in the interpreter, in the interpreted code, with variables from the interpreter or interpreted code having specific values. We thus provide a text box to enter arbitrary breakpoint conditions to be evaluated on events from the trace. For example, the condition in Figure 13 reaches the next occurrence of a call to run_binary_op_add in a context where the source variable j has value 1. The break point condition may be any JS expression using the following API: I_line() returns the current line of the interpreter, S_line() returns the current line of the source, I('x') returns the value of x in the interpreter, S_raw('x') returns the value of x in the source (e.g. the JS object {tag: "value_number", arg: 5}), and S('x') returns the JS interpretation of the value of x in the source (e.g. the JS value 5). RELATED WORK There are many formal semantics of JavaScript, from pen-and-paper ones [START_REF] Maffeis | An operational semantics for javascript[END_REF], to the aforementioned JSCert [START_REF] Bodin | A trusted mechanised javascript specification[END_REF] and KJS [START_REF] Park | KJS: a complete formal semantics of javascript[END_REF]. As described in §1.4, these semantics are admirable but lack crucial features to be actively used in the standardization effort. To our knowledge, the closest work to the double-debugger approach is the multi-level debugging approach of Kruck et al. [START_REF] Kruck | Multi-level debugging for interpreter developers[END_REF]. They present a debugger for an interpreter for domain-specific languages that lets developers choose the level of abstraction at which they debug their program. An abstraction is a way to display some values (encoded in the host language, or as present in the DSL) as well as showing only stack frames that represent computation at the DSL level. Our technique is more general as it does not focus on domain-specific languages. In fact, our double-debugger approach could be easily adapted to interpreter for other languages than JS. To that end, it suffices to implemented an interpreter for the desired language in the subset of OCaml that we support, and to provide code for extracting and displaying the term and state associated with a given event. We have recently followed that approach and adapted our framework to derive a double-debugger for (a significant subset of) the OCaml programming language. Regarding the translation from OCaml to JS that we implement, one might consider using an existing, general-purpose tool. Js_of_ocaml [START_REF] Vouillon | From bytecode to javascript: the js_of_ocaml compiler[END_REF] converts OCaml bytecode into efficient JS code. Presumably, we could implement the logging instrumentation as an OCaml source-to-source translation and then invoke Js_of_ocaml. Yet, with that approach, we would need to convert the representation of trace events from the encoding of these values performed by Js_of_ocaml into proper JS objects that we can display in the interactive interface. This conversion is nontrivial, as some information, such as the name of constructors, is lost in the process. As we already implemented a translator from OCaml to pseudo-JS, it was simpler to implement a translator from OCaml to plain JS. Another translator from OCaml to JS is Bucklescript [START_REF]Bucklescript[END_REF], which was released after we started our work. Similarly to our translator, Bucklescript converts OCaml code into JS code advertised as readable. Bucklescript also has the limitation that the names of constructors are lost, although presumably this could be easily fixed. Besides, if we wanted to revisit our implementation to base it on Bucklescript, for trace generation we would need to either modify Bucklescript, which is quite complex as it covers the full OCaml language, or to reimplement trace instrumentation at the OCaml level, which should be doable yet would involve a bit more work than at the level of untyped JS code. CONCLUSION AND FUTURE WORK We presented JSExplain to TC39 1 and the committee expressed strong interest. They would like us to extend our specification to cover all of the specification. We have almost finished the formalization of proxies, which are a challenging addition to the language as they change many internal methods. Although all members seem to agree that the current toolset for developing the specification is inappropriate, it requires a strong leadership and a consensus to commit to a new toolchain. Our goal is to cover the current version of ECMAScript, we currently cover ECMAScript 5, and to help committee members use it to formalize new additions to JavaScript. There are numerous directions for future work. (1) We plan to set up a modular mechanism for describing unspecified behaviors (e.g. "for-in" enumeration order) as well as browser-specific behaviors (sometimes browsers deviate from the specification, for historical reasons). (2) We could investigate the possibility of extending the formalization of the standard by also covering the parsing rules of JS; currently, our semantics is expressed with respect to the AST of the input program. (3) To re-establish a link with the original JSCert inductive definition, which is useful for conducting formal proofs about the metatheory of the langage, we would like to investigate 1 https://tc39.github.io/tc39-notes/2016-05_may-25.html#jsexplain-as--tw the possibility of automatically generating pretty-big-step [START_REF] Charguéraud | Pretty-big-step semantics[END_REF] definitions from the reference semantics expressed in our small language, possibly using some amount of annotation to guide the process. (4) To close even further the gap between a formal language and the English prose, we could also investigate the possibility of automatically generating English sentences from the code. Indeed, the prose from the ECMAScript standard is written in such a systematic manner that this should be doable, at least to some extent. Figure 1 : 1 Figure 1: ECMA5 specification of addition. Figure 2 : 2 Figure 2: ECMA6 specification of addition. Figure 4 : 4 Figure 4: ECMA7 and ECMA8 addition for ReturnIfAbrupt. Figure 5 : 5 Figure 5: ECMA7 and ECMA8 specification of addition. and run_binary_op s c op v1 v2 = match op with | C_binary_op_add -> run_binary_op_add s c v1 v2 ... and run_binary_op_add s0 c v1 v2 = let%prim (s1, w1) = to_primitive_def s0 c v1 in let%prim (s2, w2) = to_primitive_def s1 c v2 in if (type_compare (type_of (Coq_value_prim w1)) Coq_type_string) || (type_compare (type_of (Coq_value_prim w2)) Coq_type_string) then let%string (s3, str1) = to_string s2 c (Coq_value_prim w1) in let%string (s4, str2) = to_string s3 c (Coq_value_prim w2) in res_out (Coq_out_ter (s4, (res_val (Coq_value_prim (Coq_prim_string (strappend str1 str2)))))) else let%number (s3, n1) = to_number s2 c (Coq_value_prim w1) in let%number (s4, n2) = to_number s3 c (Coq_value_prim w2) in res_out (Coq_out_ter (s4, (res_val (Coq_value_prim (Coq_prim_number (n1 +. n2)))))) Figure 7 : 7 Figure 7: Generated code for the interpreter in pseudo-JS syntax, with implicit environments, state, and casts. Figure 8 : 8 Figure 8: Snippet of generated code for the interpreter in standard JS syntax, without trace instrumentation. Figure 9 :Figure 10 : 910 Figure 9: Snippet of generated code for the interpreter in standard JS syntax, with trace instrumentation. Figure 11 :Figure 12 : 1112 Figure 11: Display of the variables from the interpreter code Figure 13 : 13 Figure 13: Example of a conditional breakpoint, constraining the state of both the interpreter and the interpreted code. /* new feature */ /* plain JavaScript */ var name = "me"; var name = "me"; `hello ${name}`;"hello " + name; ACKNOWLEDGMENTS We acknowledge funding from the ANR project AJACS ANR-14-CE28-0008 and the CominLabs project SecCloud.

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2024/03/05 22:32:07

2016

https://univ-rennes.hal.science/hal-01439363/file/Mutational%20Spectrum%20in%20Holoprosencephaly.pdf

Dr Christèle Dubourg email: [email protected] Wilfrid Carré Houda Hamdi-Rozé Charlotte Mouden Joëlle Roume Benmansour Abdelmajid Daniel Amram Clarisse Baumann Nicolas Chassaing Christine Coubes Laurence Faivre-Olivier Emmanuelle Ginglinger Marie Gonzales Annie Levy-Mozziconacci Sally-Ann Lynch Sophie Naudion Laurent Pasquier Amélie Poidvin Fabienne Prieur Pierre Sarda Annick Toutain Valérie Dupé Linda Akloul Sylvie Odent Marie De Tayrac Véronique David Mutational Spectrum in Holoprosencephaly Shows That FGF is a New Major Signaling Pathway Keywords: Holoprosencephaly, FGF signaling pathway, multigenic inheritance, targeted NGS, brain malformation ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. INTRODUCTION Holoprosencephaly (HPE; MIM# 236100) is the most frequent congenital brain malformation (1 in 10,000 live births, 1 in 250 conceptuses). It results from incomplete midline division of the prosencephalon between 18 th and 28 th day of gestation, affecting both the forebrain and the face [START_REF] Dubourg | Holoprosencephaly[END_REF][START_REF] Marcorelles | Neuropathology of holoprosencephaly[END_REF]. The clinical spectrum is very wide, ranging from severe HPE with a single cerebral ventricle and cyclopia to clinically unaffected carriers in familial HPE. Three classic anatomical classes have been described, in decreasing order of severity: alobar, semi-lobar, and lobar HPE. The full spectrum of HPE includes also middle interhemispheric variants (MIH) or syntelencephaly, septopreoptic HPE and microforms characterized by midline defects (eg, single maxillary median incisor (SMMI) or hypotelorism) without the brain malformations typical of HPE [START_REF] Barkovich | Analysis of the cerebral cortex in holoprosencephaly with attention to the sylvian fissures[END_REF][START_REF] Hahn | Septopreoptic holoprosencephaly: a mild subtype associated with midline craniofacial anomalies[END_REF][START_REF] Lazaro | Phenotypic and molecular variability of the holoprosencephalic spectrum[END_REF][START_REF] Simon | The middle interhemispheric variant of holoprosencephaly[END_REF]. Not only is HPE highly variable phenotypically, but also very heterogeneous etiologically [START_REF] Bendavid | Holoprosencephaly: An update on cytogenetic abnormalities[END_REF][START_REF] Pineda-Alvarez | Current recommendations for the molecular evaluation of newly diagnosed holoprosencephaly patients[END_REF][START_REF] Roessler | The molecular genetics of holoprosencephaly[END_REF]. HPE may be due to chromosome abnormalities, such as trisomy 13, 18, and triploidy, or may be one of the components of a multiple malformation syndrome, such as Smith-Lemli-Opitz or CHARGE syndrome. The Hartsfield syndrome associates HPE with ectrodactyly, with and without cleft lip and palate. HPE may also result from exposure to maternal diabetes during gestation [START_REF] Johnson | Non-genetic risk factors for holoprosencephaly[END_REF][START_REF] Miller | Risk factors for nonsyndromic holoprosencephaly in the National Birth Defects Prevention Study[END_REF]. Isolated HPE presents a high genetic heterogeneity. To date heterozygous mutations in 15 genes have been identified in HPE patients with four major genes (Sonic hedgehog or SHH MIM# 600725, ZIC2 MIM#603073, SIX3 MIM# 603714, TGIF1 MIM# 602630), and eleven genes that are [START_REF] Arauz | A hypomorphic allele in the FGF8 gene contributes to holoprosencephaly and is allelic to gonadotropin-releasing hormone deficiency in humans[END_REF][START_REF] Bae | Mutations in CDON, encoding a hedgehog receptor, result in holoprosencephaly and defective interactions with other hedgehog receptors[END_REF][START_REF] Bendavid | Holoprosencephaly: An update on cytogenetic abnormalities[END_REF][START_REF] Dupe | NOTCH, a new signaling pathway implicated in holoprosencephaly[END_REF][START_REF] Mouden | Homozygous STIL Mutation Causes Holoprosencephaly and Microcephaly in Two Siblings[END_REF][START_REF] Pineda-Alvarez | Missense substitutions in the GAS1 protein present in holoprosencephaly patients reduce the affinity for its ligand, SHH[END_REF]. These genes encode proteins playing a role in early brain development, which mostly belong to the signaling pathway Shh, and to a lesser extent Nodal and Fgf pathways [START_REF] Arauz | A hypomorphic allele in the FGF8 gene contributes to holoprosencephaly and is allelic to gonadotropin-releasing hormone deficiency in humans[END_REF][START_REF] Mercier | NODAL and SHH dose-dependent double inhibition promotes an HPE-like phenotype in chick embryos[END_REF]. Mutations in SHH, SIX3 and TGIF1 are inherited from an unaffected parent or parent harboring only a microform of HPE in 70% of the cases [START_REF] Mercier | New findings for phenotypegenotype correlations in a large European series of holoprosencephaly cases[END_REF]. It suggests that other events are necessary to develop the disease. Consequently, the mode of inheritance initially described as autosomal dominant with an incomplete penetrance and a variable expression has been redefined [START_REF] Odent | Segregation analysis in nonsyndromic holoprosencephaly[END_REF][START_REF] Mouden | Complex mode of inheritance in holoprosencephaly revealed by whole exome sequencing[END_REF]. HPE is now listed as a polygenic disease having multiple inheritance modes. Among them, polygenic inheritance would require two or more events involving genes from the same or different signaling pathways with functional relationship. This polygenic inheritance plays a role in the variability of the phenotype especially when there is a functional relationship between mutated genes, as this is the case for HPE genes [START_REF] Mercier | NODAL and SHH dose-dependent double inhibition promotes an HPE-like phenotype in chick embryos[END_REF]. This has significant implications for genetic counseling and for risk assessment of patient relatives. Until recently, HPE molecular diagnosis had relied on the detection of point mutations in the four main HPE genes (SHH, ZIC2, SIX3 and TGIF1) by Sanger sequencing and on the search for deletions in either known HPE genes or in the entire genome (using CGH array). Targeted next-generation sequencing (NGS) has been proven in the recent years to be very beneficial clinically, especially for the molecular diagnosis of genetically heterogeneous diseases, such as intellectual disability, hearing loss [START_REF] Shearer | Comprehensive genetic testing for hereditary hearing loss using massively parallel sequencing[END_REF], and ciliopathies like Bardet-Biedl syndrome (M 'Hamdi et al., 2014). Targeted NGS appears to be more suitable for routine clinical practice than whole-exome sequencing as it provides better coverage of particular genes for a lower cost and easier and quicker data interpretation [START_REF] Rehm | Disease-targeted sequencing: a cornerstone in the clinic[END_REF]. Therefore we have developed a targeted NGS panel for the molecular diagnosis of HPE by screening twenty genes positively involved in HPE or defined as candidates for this disorder using the Ion Torrent AmpliSeq and Ion Personal Genome Machine (PGM) strategy. In a cohort of 271 HPE probands tested since the beginning of 2014, we were able to provide a diagnosis in approximately 24% of patients. We also show that components of the FGF signaling pathway are clearly involved in HPE. MATERIALS AND METHODS Sample Collection A total of 257 patients (131 fetuses and 126 living children) with normal conventional karyotype were referred by the French geneticists from the eight different CLAD (Centres Labellisés pour les Anomalies du Développement) of the country, French centers of prenatal diagnosis (CPDPN), fetopathologists from the French Fetopathology Society (SOFFOET), as well as several European centers. The 257 patients are described in Table 1. This cohort includes 130 males and 127 females, who have been diagnosed with alobar (n=62), semilobar (n=54), lobar (n=43), syntelencephaly (n=12), HPE microform (n=80), Hartsfield syndrome (n=3) or Kallmann syndrome (n=3). All samples were obtained with informed consent according to the protocols approved by the local ethics committee (Rennes hospital). Gene selection and panel design Gene Library preparation and DNA sequencing An adapter-ligated library was constructed with the Ion AmpliSeq Library Kit 2.0 (Life Technologies) following the manufacturer's protocol. Briefly, 10 ng of DNA was amplified in two pooled reactions then gathered together. Amplicons were partially digested at primer sequences before ligation with Ion Torrent adapters P1 and A, and the adapter-ligated products were then purified with AMPure beads (Beckman Coulter Genomics, Brea, CA, USA), and PCR-amplified for 7 cycles. The resulting libraries of 11 patients were equalized using the Ion Library Equalizer Kit (Life Technologies) and then pooled. Sample emulsion PCR, emulsion breaking, and enrichment were performed with the Ion PGM Template OT2 200 Kit (Life Technologies), according to the manufacturer's instructions. Briefly, an input concentration of one DNA template copy per Ion Sphere Particles (ISPs) was added to emulsion PCR master mix, and the emulsion was generated with an Ion OneTouch system (Life Technologies). Next, ISPs were recovered, and templatepositive ISPs were enriched with Dynabeads MyOne Streptavidin C1 beads (Life Technologies). The Qubit 2.0 fluorometer (Life Technologies) was used to confirm ISP enrichment. An Ion PGM 200 Sequencing Kit was used for sequencing reactions, as recommended in the protocol, and chips 316 were used to sequence barcoded samples on the Ion Torrent PGM for 500 dNTP-flows. In order to achieve a complete coverage of at least the four main genes for each patient, six fragments, respectively one in SHH, four in ZIC2 and one in SIX3, were systematically studied by Sanger method. Depending on the coverage, analysis of other genes was completed according to the patient phenotype by Sanger sequencing. Bioinformatical analysis The sequencing data produced by the PGM were first processed with the Torrent Suite 4.2.1, Ion Torrent platform-specific pipeline including signal processing, adapter trimming, filtering of poor signal-profile reads and alignment to the hg19 human reference genome with TMAP (Torrent Mapping Alignment Program). Four independent variant calling algorithms from the Torrent suite were used. The four VCF (variant calling format) files were combined and annotated with ANNOVAR (February 2014 build) [START_REF] Wang | ANNOVAR: functional annotation of genetic variants from high-throughput sequencing data[END_REF]. A gene-based annotation identified whether SNPs cause protein-coding changes and the amino acids that were affected based on RefSeq. A filter based annotation identified variants and their associated frequency that were reported in the following databases: dbSNP138, 1000-Genome (1000G), NHLBI-ESP, ExAC (Exome Aggregation Consortium) and ClinVar [START_REF] Landrum | ClinVar: public archive of relationships among sequence variation and human phenotype[END_REF]. ANNOVAR was also used to annotate the predicted functional consequences of missense variants using dbNSFP (database for synonymous SNP's functional predictions) v2.6 (http://sites.google.com/site/jpopgen/dbNSFP) [START_REF] Liu | dbNSFP: a lightweight database of human nonsynonymous SNPs and their functional predictions[END_REF][START_REF] Liu | dbSNP v2.0: a database of human non-synonymous SNVs and their functional predictions and annotations[END_REF]. This database compiles prediction scores and interpretation from ten different algorithms: SIFT, Polyphen2_HDIV, Polyphen2_HVAR, LRT, MutationTaster, MutationAssessor, FATHMM, CADD, MetaSVM and MetaLR (Suppl. Tables S1 andS2). Three conservation scores (GERP++, PhyloP and SiPhy) are also included in dbNSFP v2.6 (Suppl. Tables S1 andS2). The variant annotation was completed with "in-house" data regarding variants frequency within each run, across runs and during previous annotation helping to identify recurring false positives and polymorphisms. Furthermore only variants with a frequency less than 1/1,000 in 1000G, EVS (Exome Variant Server), ExAC held our interest. After variants validation by visualization with IGV (Integrative Genomics Viewer), complementary annotations were performed using Condel v2.0 (Gonzalez-Perez and Lopez-Bigas, 2011) and Alamut Visual v2.4.5 (Interactive Biosoftware) to estimate variant pathogenicity. The information given by different tools were re-examined with caution to provide accurate results: PolyPhen [START_REF] Adzhubei | Predicting Dunctional Effect of Human Missense Mutations Using PolyPhen-2[END_REF], SIFT [START_REF] Kumar | Predicting the effects of coding non-synonymous variants on protein function using the SIFT algorithm[END_REF], Mutation Taster [START_REF] Schwarz | MutationTaster2: mutation prediction for the deep-sequencing age[END_REF], and Align-GVGD [START_REF] Tavtigian | Comprehensive statistical study of 452 BRCA1 missense substitutions with classification of eight recurrent substitutions as neutral[END_REF] were tested for exonic variants. In order to study the effect of potential splice variations, Alamut Visual integrates various splice site prediction methods: SpliceSiteFinder-like [START_REF] Zhang | Statistical features of human exons and their flanking regions[END_REF], MaxEntScan [START_REF] Yeo | Maximum entropy modeling of short sequence motifs with applications to RNA splicing signals[END_REF], NNSPLICE [START_REF] Reese | Improved splice site detection in Genie[END_REF], GeneSplicer [START_REF] Pertea | GeneSplicer: a new computational method for splice site prediction[END_REF], Human Splicing Finder [START_REF] Desmet | Human Splicing Finder: an online bioinformatics tool to predict splicing signals[END_REF], ESEFinder [START_REF] Cartegni | ESEfinder: A web resource to identify exonic splicing enhancers[END_REF], RESCUE-ESE [START_REF] Fairbrother | Predictive identification of exonic splicing enhancers in human genes[END_REF] and EX-SKIP [START_REF] Raponi | Prediction of single-nucleotide substitutions that result in exon skipping: identification of a splicing silencer in BRCA1 exon 6[END_REF] were interrogated. The first five gave scores increasing with the importance of the predicted impact on the splice. Finally, a variant was retained for diagnosis when a majority of tools predicted it as potentially deleterious and/or when family pedigree segregation was consistent. Nucleotide numbering uses +1 as the A of the ATG translation initiation codon in the reference sequence, with the initiation codon as codon 1. We use the tool ProteinPaint (http://pecan.stjude.org) for visualizing amino acid changes corresponding to the retained variants [START_REF] Zhou | Exploring genomic alteration in pediatric cancer using ProteinPaint[END_REF]. Mutation validation All variants with a potential deleterious effect were confirmed by Sanger sequencing. They were submitted to ClinVar (ClinVar accessions SCV000268717 -SCV000268738 on http://www.ncbi.nlm.nih.gov/clinvar/). Segregation analyses were performed whenever DNA was available for additional family members. RESULTS Targeted NGS analysis of the 257 patients identified candidate and diagnosis variants in 23.7% of the cases: mutations with high confidence in their deleterious effect in three of the main genes SHH, ZIC2 and SIX3 were identified in 13.2% of the cases (34/257), and in other tested genes in 10.5% (27/257). For these cases, we were able to give a diagnosis. We also found variants classified as variants of uncertain significance (VUS) in 10% (26/257) of the cases. From these data, the ten first-ranked genes involved in HPE are SHH (5.8%), ZIC2 (4.7%), GLI2 (3.1%), SIX3 (2.7%), FGF8 (2.3%), FGFR1 (2.3%), DISP1 (1.2%), DLL1 (1.2%) and SUFU (0.4%) (Table 1, Fig. 1). All variants were found in a heterozygous state and were held for diagnosis. SHH, ZIC2 and SIX3 retain their position of major genes. Description of the SHH, ZIC2 and SIX3 mutations is provided in Figures 1 and2. As previously described by Mercier et al. [START_REF] Mercier | New findings for phenotypegenotype correlations in a large European series of holoprosencephaly cases[END_REF], our results confirmed that SHH is the major gene implicated in HPE. SHH mutations are mostly missense (Fig. 1) and are inherited in 80% of cases of this study. The spectrum of clinical manifestations associated with SHH mutations is very large and includes severe forms as well as microforms. ZIC2 is the second major gene, which is affected by all types of mutations: missense (42%), frameshift and nonsense (42%) and also splice mutations (16%). ZIC2 alterations are generally associated with severe HPE forms and few facial features and are de novo in 92% of cases in our study. Probands with SIX3 mutation mostly had severe HPE correlated with severe facial features. Like SHH mutations, SIX3 variants are mostly inherited. Altogether, these results support that mutations in SHH and SIX3 are highly inherited, whereas most of the ZIC2 mutations are de novo. GLI2 is mostly involved in midline abnormalities. Six GLI2 heterozygous variants were hold for diagnosis (Fig. 1 and 3, Table 2, Suppl. Table S2). The c.596dupG/p.Ala200Argfs*151 (A200Rfs*151) mutation was identified in a boy with nasal pyriform aperture atresia and was inherited from his asymptomatic mother. The c.790C>T/p.Arg264* (R264*) mutation was identified in a 2-years-old girl with isolated solitary median maxillary central incisor and was inherited from her asymptomatic mother. The c.2064delC/p.Ser690Alafs*5 (S690Afs*5) mutation was identified in a 20-years-old girl with hexadactyly, choanal atresia, hypopituitarism and cerebellar atrophia. This mutation occurred de novo. The c.2237G>A/p.Trp746* (W746*) mutation was identified in a male fetus aborted because of lobar holoprosencephaly, premaxillary agenesis, hexadactyly, pituitary hamartoma, and short femur. Moreover his karyotype revealed a mosaic fragility on chromosome 3 (3p24.1, so very far from TDGF1). This mutation was not inherited from his mother, and DNA from the father was unavailable. The c.4761G>C/p.*1587Tyrext*46 (*1587Y) mutation was found in a 16-year-old boy with hypopituitarism, solitary median maxillary central incisor and choanal atresia. It was inherited from his asymptomatic mother. The c.349G>A/p.Ala117Thr (A117T) variant was found in two brothers, one with hypopituitarism and optic atrophia, the other with bilateral cleft lip and palate. This variant was inherited from the father presenting only subtle hypotelorism. The effect of this variant is uncertain as it involves a moderately conserved amino acid and the physicochemical gap between alanine and threonine is low (Grantham distance = 58). Except the A117T, which is of uncertain clinical significance, all the other variations modify the stop codon. They are inherited in the majority of cases, implicating that these variants in GLI2 clearly show incomplete penetrance. Altogether, the mutations in GLI2 are mostly associated with spectrum linked to midline and characterized by solitary median maxillary central incisor and pituitary insufficiency. Only one is associated with classic HPE. FGF8 reaches the top genes. Six patients of our cohort presented heterozygous variations in FGF8 gene (Fig. 1 and3, Table 2, Suppl. Table S2). A fetus with semilobar HPE presented the c. 356C>T/p.Thr119Met (T119M) variant in FGF8 in association with a splice mutation in FGFR1. The couple had already had a termination of pregnancy due to semilobar HPE and the paternal grandmother presents a right cleft lip. DNA samples were not available, preventing further Sanger validation. The c.317C>A/p.Ala106Glu (A106E) was identified in 4-years old boy with semi-lobar HPE. This variant implicates a highly conserved aminoacid (through 13 species until Fugu) located in the interleukin-1/heparin-binding growth factor domain. It is predicted as possibly damaging by SIFT, PolyPhen and Mutation taster. This mutation occurred de novo. This is the first time that a FGF8 mutation is described in association with syntelencephaly. The c.385C>T/p.Arg129* (R129*) was identified twice in two unrelated families. The first patient is a boy with alobar HPE and the second one is a boy with syntelencephaly. In both cases, the mutation was inherited from the asymptomatic father. The c.617G>A/p.Arg206Gln (R206Q) was also identified twice in two unrelated families. The first case is a 3 year-old girl with microform (pyriform aperture stenosis, solitary median maxillary central incisor, hypotelorism) presenting an additional variant in DLL1 (p.Asp601_Ile602delinsVal). These two variants are also present in her older sister who was operated on for bilateral cleft lip and palate and are inherited from the mother presenting hypotelorism and microretrognathism. So there is an apparent co-segregation of these mutations with minor signs of HPE spectrum in this family. The second case is a female fetus with lobar HPE. Overall, the mutation frequency (2.2%) in FGF8 demonstrates that this gene can be classified as a major gene. FGFR1 is a new major gene in HPE. Six heterozygous variants in FGFR1 (NM_023110.2) were identified in our cohort: five in the intracellular tyrosine kinase domain (TKD, aminoacids 478-767): p.Gly485Val, p.Gly490Arg, p.Gly643Asp, c.1977+1G>A, p.Glu692Lys, and one in the extracellular ligand binding domain (p.Arg250Pro) (Fig. 1 and 3, Table 2, Suppl. Table S2). The c.1454G>T/p.Gly485Val (G485V) and the c.1468G>C/p.Gly490Arg (G490R) were identified in patients with Harstfield syndrome and occurred de novo. The latter has already been reported by Simonis et al. [START_REF] Simonis | FGFR1 mutations cause Hartsfield syndrome, the unique association of holoprosencephaly and ectrodactyly[END_REF]. The c.1928G>A/p.Gly643Asp (G643D) mutation occurred de novo in a patient with nasal pyriform aperture hypoplasia, single central incisor and intellectual deficiency. It involves a highly conserved residue (through 16 species from Caenorhabiditis elegans to Homo sapiens) located in the serine-threonine/tyrosine-protein kinase catalytic domain and the physicochemical gap between glycine and aspartate is important (Grantham distance = 94). AlignGVGD, SIFT and MutationTaster predict a deleterious effect. The c.1977+1G>A variant was identified in a patient with semilobar HPE in association with a variant in FGF8, p.Thr119Met, as described above. The c.1977+1G>A variant is predicted to induce a skipping of exon 17 by all five splice prediction tools. The c.2074G>A/p.Glu692Lys (E692K) mutation was identified in a fetus with HPE and cleft lip and palate, and was inherited from his mother with hypogonadotropic hypogonadism. The c.749G>C/p.Arg250Pro (R250P) mutation was identified in a boy with lobar HPE and bilateral cleft lip and palate. Sanger sequencing suggested a very low proportion of the mutated base (cytosine) to the normal base (guanine) in the father leucocyte DNA (Fig. 4). This was confirmed by NGS sequencing showing mosaicism for the presence of the mutation (GRCh37 genome build: g.38282214C>G) with a frequency of 6% in the peripheral blood, and was perfectly correlated with the phenotype of the father presenting a microform with a right unilateral hypoplasia of the orbicularis of the upper lip and bilateral nasal slot, and MRI showed agenesis of the corpus callosum. The 15-month-old boy now presents diabetes insipidus and septo-optic dysplasia. Mutations in FGFR1 were recently described in Hartsfield syndrome (OMIM 300571), that is a rare and unique association of HPE and ectrodactyly, with or without cleft lip and palate, and variable additional features [START_REF] Hong | Dominant-negative kinase domain mutations in FGFR1 can explain the clinical severity of Hartsfield syndrome[END_REF][START_REF] Simonis | FGFR1 mutations cause Hartsfield syndrome, the unique association of holoprosencephaly and ectrodactyly[END_REF]. Here we identified four FGFR1 mutations in patients presenting HPE without extremities abnormalities. Minor HPE genes present mutations that are associated with a second one in most of the cases. The three HPE minor genes identified by our study are DLL1, DISP1 and SUFU (Fig. 1, Table 2, Suppl. Table S2). In the DLL1 gene, we identified twice the same mutation c.1802_1804del/p.Asp601_Ile602delinsVal (or 601_602del) in two unrelated patients. First, this mutation was found in a patient with semilobar HPE and has already been reported by our group [START_REF] Dupé | NOTCH, a new signaling pathway implicated in holoprosencephaly[END_REF]. Secondly, it was identified in a 3 year-old girl with microform (pyriform aperture stenosis, solitary median maxillary central incisor, hypotelorism). It was found in association with a VUS in FGF8 (R206Q); the two variants perfectly co-segregate with the phenotype in the family and may be implicated in the phenotype as we have shown that Fgf pathway might regulate expression of DLL1 in the chick developing brain [START_REF] Dupé | NOTCH, a new signaling pathway implicated in holoprosencephaly[END_REF]. We also found the c.2117C>T/p.Ser706Leu (S706L) mutation in the DLL1 gene in a fetus with alobar HPE in association with an in-frame deletion in SHH (c.1157_1180del/p.Leu386_Ala393del). The two mutations were however inherited from her asymptomatic father. Regarding the DISP1 gene, we identified two compound heterozygous mutations in a 9-yearold girl with a mild form of lobar HPE, facial dysmorphism and hypotelorism: the c.1087A>G transition leading to a missense mutation p.Asn363Asp (N363D) and the c.1657G>A transition leading to a missense mutation p.Glu553Lys (E553K). The p.Asn363Asp mutation was inherited from the father and the p.Glu553Lys mutation was inherited from the mother [START_REF] Mouden | Complex mode of inheritance in holoprosencephaly revealed by whole exome sequencing[END_REF]. In one polymalformative fetus with bilateral cleft lip and facial dysmorphism suggesting HPE microform, we found a nonsense heterozygous mutation c.2898G>A or p.Trp966* (W966*) in DISP1, associated with a mutation in SUFU (c.1022C>T/p.Pro341Leu) that substitutes the last base of exon 8 and that is predicted deleterious by most bioinformatics prediction tools mutation. Family study unfortunately could not be performed because DNA samples were not available. These results suggest that mutations in minor genes would be found more often in HPE patients with polygenic inheritance. DISCUSSION HPE is a very complex disorder both in clinical and genetic terms involving two or more genetic events. We present here the first large HPE series studied by targeted NGS and we provide a new classification of genes involved in HPE. SHH, ZIC2 and SIX3 remain the top genes in terms of importance with GLI2, and are followed by FGF8 and FGFR1. The fraction of mutations in the major genes (SHH, ZIC2, SIX3) is reduced in the present study compared to previous studies [START_REF] Mercier | New findings for phenotypegenotype correlations in a large European series of holoprosencephaly cases[END_REF]; it is probably due to the present cohort which included more patients with microforms and syntelencephaly. TGIF1 was previously classified as a major HPE gene [START_REF] Mercier | New findings for phenotypegenotype correlations in a large European series of holoprosencephaly cases[END_REF] but did not present any mutation in our study. Similarly PTCH1, GAS1, TDGF1, CDON, FOXH1, NODAL and SHH regulating sequences LMBR1 and RBM33 showed no mutations held for diagnosis in the 257 cases sequenced. New case-control studies need to be performed in larger cohorts to better evaluate their role and diagnosis potential in HPE. Such studies may be much more capable to evaluate the implication of rare variants. The candidate HHAT and SOX2 genes did not present any mutation either. Significantly, the identification of numerous mutations in FGF8 and FGFR1 in our cohort strengthens FGF signaling involvement in HPE. FGF8 is a ligand of the large fibroblast growth factor (FGF) family and is important for gonadotropin releasing hormone (GnRH) neuronal development with human mutations resulting in hypogonadotropic hypogonadism and Kallmann syndrome [START_REF] Falardeau | Decreased FGF8 signaling causes deficiency of gonadotropin-releasing hormone in humans and mice[END_REF][START_REF] Hardelin | The complex genetics of Kallmann syndrome: KAL1, FGFR1, FGF8, PROKR2, PROK2, et al[END_REF]. Our targeted NGS approach demonstrates that mutation in FGF8 occurs more commonly than previously thought [START_REF] Arauz | A hypomorphic allele in the FGF8 gene contributes to holoprosencephaly and is allelic to gonadotropin-releasing hormone deficiency in humans[END_REF][START_REF] Mccabe | Novel FGF8 mutations associated with recessive holoprosencephaly, craniofacial defects, and hypothalamo-pituitary dysfunction[END_REF]. The phenotype associated with FGF8 alterations is variable and mutation can be de novo or inherited. Interestingly, the same inherited nonsense mutation (p.Arg129*) was identified in two unrelated patients, one with a severe HPE and the other with a mild form. It supports that another event could be necessary to lead to severe HPE. We also describe here convincing examples of FGFR1 mutations in patients with isolated HPE. FGFR1 belongs to the tyrosine kinase receptor superfamily and contains an extracellular ligand binding domain with three immunoglobulin (Ig)-like domains (D1-D3) and a cytoplasmic domain responsible for tyrosine kinase activity (Fig. 3). The clinical manifestations of FGFR1 alterations are very heterogeneous since loss-of-function mutations in FGFR1 have been linked to Kallman syndrome (Dode et al., 2003;[START_REF] Albuisson | Kallmann syndrome: 14 novel mutations in KAL1 and FGFR1 (KAL2)[END_REF] Villanueva and de Roux, 2010), hypogonadotropic hypogonadism with or without anosmia [START_REF] Balasubramanian | Prioritizing genetic testing in patients with Kallmann syndrome using clinical phenotypes[END_REF][START_REF] Villanueva | Congenital hypogonadotropic hypogonadism with split hand/foot malformation: a clinical entity with a high frequency of FGFR1 mutations[END_REF][START_REF] Vizeneux | Congenital hypogonadotropic hypogonadism during childhood: presentation and genetic analyses in 46 boys[END_REF], and Hartsfield syndrome [START_REF] Hong | Dominant-negative kinase domain mutations in FGFR1 can explain the clinical severity of Hartsfield syndrome[END_REF][START_REF] Simonis | FGFR1 mutations cause Hartsfield syndrome, the unique association of holoprosencephaly and ectrodactyly[END_REF]. Gain-of-function mutations in FGFR1 have also been identified in about 5% of Pfeiffer syndrome with or without craniosynostosis [START_REF] Chokdeemboon | FGFR1 and FGFR2 mutations in Pfeiffer syndrome[END_REF]. We describe here one case of FGFR1 mutation (p.Glu692Lys) associated both with Kallmann syndrome and HPE. The location of this mutation is consistent with Kallmann syndrome as mutations of neighboring residues (p.Leu590Pro, p.Ile693Phe) were already described in patients with this syndrome [START_REF] Bailleul-Forestier | Dental agenesis in Kallmann syndrome individuals with FGFR1 mutations[END_REF]Dodé et al., 2007). Out of the six FGFR1 mutations described in our study, two were found in patients with Hartsfield syndrome. Previous reports of Hartsfield syndrome implicate FGFR1 mutations in the ATP binding site and the protein tyrosine kinase domain [START_REF] Dhamija | Novel de novo heterozygous FGFR1 mutation in two siblings with Hartsfield syndrome: a case of gonadal mosaicism[END_REF][START_REF] Hong | Dominant-negative kinase domain mutations in FGFR1 can explain the clinical severity of Hartsfield syndrome[END_REF][START_REF] Simonis | FGFR1 mutations cause Hartsfield syndrome, the unique association of holoprosencephaly and ectrodactyly[END_REF]. These mutations would have a dominant-negative activity that would account for the most severe phenotype of Hartsfield syndrome [START_REF] Hong | Dominant-negative kinase domain mutations in FGFR1 can explain the clinical severity of Hartsfield syndrome[END_REF]. Concordantly, the two FGFR1 mutations (p.Gly485Val, p.Gly490Arg) that are associated with Hartsfield syndrome in our cohort are localized in the region coding for ATP binding site (Fig. 3). However, two of the mutations identified in HPE patients without abnormalities of the extremities are also found in the region coding for activation loop of the protein tyrosine kinase domain (p.Gly643Asp; c.1977+1G>A). We hypothesized that these FGFR1 mutations rather lead to a classic loss of function [START_REF] Hong | Dominant-negative kinase domain mutations in FGFR1 can explain the clinical severity of Hartsfield syndrome[END_REF]. FGF8 and FGFR1 are not the only members of the FGF family to be expressed in the early forebrain. Other members should be considered as strong potential candidates for HPE. FGF signaling pathway plays a dominant role in embryonic development and is essential for ventral telencephalon development and digits formation [START_REF] Ellis | ProNodal acts via FGFR3 to govern duration of Shh expression in the prechordal mesoderm[END_REF][START_REF] Gutin | FGF signalling generates ventral telencephalic cells independently of SHH[END_REF][START_REF] Li | FGFR1 function at the earliest stages of mouse limb development plays an indispensable role in subsequent autopod morphogenesis[END_REF]. FGF signaling is involved in maintaining Shh expression in the prechordal tissue, where it plays a crucial role in induction of the ventral forebrain [START_REF] Ellis | ProNodal acts via FGFR3 to govern duration of Shh expression in the prechordal mesoderm[END_REF]. FGFR1 also maintains expression of Shh in the developing limb [START_REF] Li | FGFR1 function at the earliest stages of mouse limb development plays an indispensable role in subsequent autopod morphogenesis[END_REF]. According to our hypothesis, dominant-negative FGFR1 mutations would lead to a more severe downregulation of Shh activity compared to a classic loss of function. It would explain the presence of limb defect in Hartsfield syndrome similar to those observed in the Shh-/knockout mice [START_REF] Chiang | Cyclopia and defective axial patterning in mice lacking Sonic hedgehog gene function[END_REF]. The knowledge of the mode of inheritance in HPE has evolved since the description of an autosomal dominant model with an incomplete penetrance and a variable expression [START_REF] Odent | Segregation analysis in nonsyndromic holoprosencephaly[END_REF] through an autosomal dominant model with modifier genes [START_REF] Roessler | Utilizing prospective sequence analysis of SHH, ZIC2, SIX3 and TGIF in holoprosencephaly probands to describe the parameters limiting the observed frequency of mutant genexgene interactions[END_REF]. Thanks to our NGS strategy targeting twenty genes, we have shown that sixteen per cent of mutations kept for diagnosis was found in association with a second one (FGF8/FGFR1, FGF8/DLL1, DLL1/SHH, DISP1/DISP1, DISP1/SUFU). These cases of double-mutations in two different genes -and even in the same one -strengthen the polygenic inheritance previously illustrated by [START_REF] Mouden | Complex mode of inheritance in holoprosencephaly revealed by whole exome sequencing[END_REF]. Here, a second event in FGF8 was identified in one patient with FGFR1 mutation. In the same way, a gene synergistic interaction between a deleterious FGFR1 allele transmitted from one parent and a loss-of-function allele in FGF8 from the other parent was recently described in two sisters with semilobar and lobar HPE respectively [START_REF] Hong | Dominant-negative kinase domain mutations in FGFR1 can explain the clinical severity of Hartsfield syndrome[END_REF]. Altogether these observations strongly suggest that a cumulative effect on the FGF signaling pathway leads to HPE. We showed that most of mutations were inherited mainly from an asymptomatic parent, which suggests that another event could be necessary to cause HPE. The important and wide variability of expression from an asymptomatic to severe form for a same mutation, the incomplete penetrance and the identification of several mutations in the same patient argue for this oligogenic inheritance. Furthermore, the description of numerous mouse models carrying mutations in two genes of the same or different signaling pathways involved in forebrain development strongly support this mode of inheritance by showing that a cumulative partial inhibition of signaling pathways is necessary to develop HPE [START_REF] Allen | The Hedgehog-binding proteins Gas1 and Cdo cooperate to positively regulate Shh signaling during mouse development[END_REF][START_REF] Krauss | Holoprosencephaly: new models, new insights[END_REF][START_REF] Mercier | NODAL and SHH dose-dependent double inhibition promotes an HPE-like phenotype in chick embryos[END_REF]. However, only a few examples of digenic inheritance in human were reported in the literature until now [START_REF] Hong | Dominant-negative kinase domain mutations in FGFR1 can explain the clinical severity of Hartsfield syndrome[END_REF][START_REF] Lacbawan | Clinical spectrum of SIX3-associated mutations in holoprosencephaly: correlation between genotype, phenotype and function[END_REF][START_REF] Ming | Multiple hits during early embryonic development: digenic diseases and holoprosencephaly[END_REF][START_REF] Mouden | Complex mode of inheritance in holoprosencephaly revealed by whole exome sequencing[END_REF][START_REF] Nanni | The mutational spectrum of the Sonic Hedgehog gene in holoprosencephaly: SHH mutations cause a significant proportion of autosomal dominant holoprosencephaly[END_REF]. The present study demonstrates that digenism would not be so rare in human HPE. Systematic implementation of next generation sequencing in HPE diagnosis will be necessary to account for this multigenic inheritance and to improve genetic counselling. considered as minor genes (PTCH1 MIM#601309, TDGF1 MIM#187395, FOXH1 MIM# 603621, GLI2 MIM# 165230, DISP1 MIM# 607502, FGF8 MIM# 600483, GAS1 MIM# 139185, CDON MIM#608707, NODAL MIM# 601265, DLL1 MIM# 606582 and very recently STIL MIM# 181590) selection was based on their proved or suspected involvement in HPE, or in syndromes including HPE, membership in signaling pathways implicated in HPE, and expression in the developing forebrain compatible with HPE. Known regulatory regions of SHH (LMBR1 MIM# 605522, RBM33) have also been included. The panel was designed with Ion AmpliSeq™ Designer (Life Technologies, ThermoFisher Scientific). It includes coding and flanking intronic sequences (50 base pairs) of the following 20 genes: SHH (NM_000193.2), ZIC2 (NM_007129.3), SIX3 (NM_005413.3), TGIF1 (NM_170695.2), GLI2 (NM_005270.4), PTCH1 (NM_000264.3), GAS1 (NM_002048.2), TDGF1 (NM_003212.3), CDON (NM_016952.4), DISP1 (NM_032890.3), FOXH1 (NM_003923.2), NODAL (NM_018055.4), FGF8 (NM_033163.3), HHAT (NM_018194.4) MIM# 605743, DLL1 (NM_005618.3), SUFU (NM_016169.3) MIM# 607035, SOX2 (NM_003106.3) MIM# 184429, RBM33 (NM_053043.2), LMBR1 (NM_022458.3) and FGFR1 (NM_023110.2) MIM# 136350. It covers 111 kb. Figure 1 . 1 Figure 1. Distribution of mutations held for diagnosis in the top ten holoprosencephaly genes Figure 2 . 2 Figure 2. Mutational landscape of SHH, ZIC2 and SIX3 genes. This comprehensive Figure 3 . 3 Figure 3. Mutational landscape of FGF8, FGFR1 and GLI2 genes, performed with Figure 4 4 Figure 4. p.Arg250Pro (R250P) mutation in FGFR1. (I) Proband. (II) Father. (a) Partial Table 1 . Distribution of holoprosencephaly types and mutations in the cohort of 257 patients. Type All (Male,Female) SHH ZIC2 GLI2 SIX3 FGF8 FGFR1 DISP1 DLL1 SUFU 1 Alobar 62 (24,38) 9,7% 8,1% - 6,5% 1,6% - - 1,6% - Semilobar 54 (26,28) 3,7% 5,6% - 1,9% 3,7% 3,7% - 1,9% - Lobar 43 (27,16) 2,3% 9,3% 2,3% 2,3% 2,3% 2,3% 4,7% - - Syntelencephaly 12 (7,5) 8,3% - - 8,3% 8,3% - - - - Microform 80 (42,38) 6,3% - 8,8% - 1,3% 1,3% 1,3% 1,3% 1,3% Hartsfield 3 (3,0) - - - - 66,7% - - - Kallmann 3 (1,2) - - - - - - - - - TOTAL 257 5,8% 4,7% 3,1% 2,7% 2,3% 2,3% 1,2% 1,2% 0,4% Table 2 . Characteristics of variants identified in GLI2, FGF8, FGFR1, DLL1, DISP1 and SUFU, and associated phenotypes. 2 The GenBank references used for nucleotide numbering were NM_005270.4 for GLI2, NM_033163.3 for FGF8, NM_023110.2 for FGFR1, NM_005618.3 for DLL1, NM_032890.3 for DISP1, NM_016169.3 for SUFU and NM_000193.2 for SHH. Nucleotide numbering uses +1 as the A of the ATG translation initiation codon in the reference sequence, with the initiation codon as codon 1. The deleterious score was given by 10 predictions tools (SIFT, Polyphen2_HDIV, Polyphen2_HVAR, LRT, MutationTaster, MutationAssessor, FATHMM, CADD, MetaSVM and MetaLR).*: For detailed prediction data, see Suppl. TableS2. D, deleterious; P, possibly deleterious; T, tolerated; NPAS, nasal pyriform aperture stenosis; SMMCI, solitary median maxillary central incisor; ND: not determined. Deleteri Gen e Variant (gDNA) Variant (cDNA) (protein) ous score* Patient's phenotype Inheritance Paired mutation or effect hypopituitar GLI2 g.121708913G> A c.349G> A p.Ala117Thr D:2 P:1 T:7 ism, optic atrophia / bilateral father (hypoteloris m) cleft g.121712959dup G c.596du pG p.Ala200Argfs *151 Frameshi ft NPAS mother g.121726436C>T c.790C> T p.Arg264* Stop gain SMMCI mother hexadactyly , choanal g.121743961del C c.2064d elC p.Ser690Alafs *5 Frameshi ft atresia, ism, hypopituitar de novo cerebellar atrophia lobar HPE, premaxillary g.121744134G> A c.2237G >A p.Trp746* Stop gain agenesis, pituitary hamarthom not inherited from the mother a, hexadactyly hypopituitar g.121748251G> C c.4761G >C p.*1587Tyrex t*46 Stop loss ism, choanal SMMCI, mother atresia FGF 8 g.103534509G> T c.317C> A p.Ala106Glu D:9 P:0 T:1 semilobar HPE de novo ACKNOWLEDGMENTS This work was supported by the CHU of Rennes (Innovation Project). We would like to thank the families for their participation in the study, all clinicians who referred HPE cases, the eight CLAD (Centres Labellisés pour les Anomalies du Développement) within France that belong to FECLAD, French centers of prenatal diagnosis (CPDPN) and the SOFFOET for fetus cases, and the "filière AnDDI-Rares". We particularly thank all members of the Molecular Genetics Laboratory (CHU, Rennes) and of the Department of Genetics and Development (UMR6290 CNRS, Université Rennes 1) for their help and advice. We are grateful to Артем Ким for carefully reading this manuscript. The authors acknowledge the Centre de Ressources Biologiques (CRB) Santé BB-0033-00056 (http://www.crbsante-rennes.com) of Rennes for managing patient samples.

01745814

en

2024/03/05 22:32:07

2015

https://inria.hal.science/hal-01745814/file/340025_1_En_20_Chapter.pdf

Igor Mishsky email: [email protected] Nurit Gal-Oz email: [email protected] Ehud Gudes A Topology based Flow Model for Computing Domain Reputation The Domain Name System (DNS) is an essential component of the internet infrastructure that translates domain names into IP addresses. Recent incidents verify the enormous damage of malicious activities utilizing DNS such as bots that use DNS to locate their command & control servers. Detecting malicious domains using the DNS network is therefore a key challenge. We project the famous expression Tell me who your friends are and I will tell you who you are, motivating many social trust models, on the internet domains world. A domain that is related to malicious domains is more likely to be malicious as well. In this paper, our goal is to assign reputation values to domains and IPs indicating the extent to which we consider them malicious. We start with a list of domains known to be malicious or benign and assign them reputation scores accordingly. We then construct a DNS based graph in which nodes represent domains and IPs. Our new approach for computing domain reputation applies a flow algorithm on the DNS graph to obtain the reputation of domains and identify potentially malicious ones. The experimental evaluation of the flow algorithm demonstrates its success in predicting malicious domains. Introduction Malicious botnets and Advanced Persistent Threats (APT) have plagued the Internet in recent years. Advanced Persistent Threat, often implemented as a botnet, is advanced since it uses sophisticated techniques to exploit vulnerabilities in systems, and is persistent since it uses an external command and control (C&C) site which is continuously monitoring and extracting data of a specific target. APTs are generated by hackers but are operated from specific domains or IPs. The detection of these misbehaving domains (including zero day attacks) is difficult since there is no time to collect and analyze traffic data in real-time, thus their identification ahead of time is very important. We use the term domain reputation to express a measure of our belief that a domain is benign or malicious. The term reputation is adopted from the field of social networks and virtual communities, in which the reputation of a peer is derived from evidences regarding its past behavior but also from its relations to other peers [START_REF] Page | Pagerank citation ranking: Bringing order to the web[END_REF]. A domain reputation system can support the decision to block traffic or warn organizations about suspicious domains. Currently lists of domains which are considered legitimate are published by web information companies (e.g., Alexa [START_REF] Gudes | Websites Ranking[END_REF]) while black-lists of malware domains are published by web threat analysis services (e.g., VirusTotal [START_REF]VirusTotal: A free virus, malware and URL online scanning service[END_REF].) Unfortunately the number of domains appearing in both types of lists is relatively small, and a huge number of domains is left unlabeled. Therefore, the problem of assigning reputation to unlabeled domains is highly important. The Domain Name Service (DNS) maps domain names to IP addresses and provides an essential service to applications on the internet. Many botnets use a DNS service to locate their next C&C site. For example, botnets tend to use short-lived domains to evasively move their C&C sites. Therefore, DNS logs have been used by several researchers to detect suspicious domains and filter their traffic if necessary. Choi and Lee [START_REF] Choi | Identifying botnets by capturing group activities in dns traffic[END_REF], analyzed DNS traffic to detect APTs. Such analysis requires large quantities of illegitimate DNS traffic data. An alternative approach was proposed in the Notos system [START_REF] Antonakakis | Building a dynamic reputation model for dns[END_REF] which uses historical DNS information collected passively from multiple recursive DNS resolvers to build a model of how network resources are allocated and operated for legitimate Internet services. This model is mainly based on statistical features of domains and IPs that are used for building a classifier which assigns a reputation score to unlabeled domains. The main difference between the DNS data used for computing reputation and the data used for malware detection is that the first consists of mainly static properties of the domain and DNS topology data, while the latter requires behavioral and time-dependent data. While DNS behavior data may involve private information (e.g., the domains that an IP is trying to access), which ISPs may be reluctant to analyze or share, DNS topology data is much easier to collect. Our research also focuses on computing domain reputation using DNS topology data. Various definitions of the terms trust and reputation have been proposed in the literature as the motivation for a computational metric. Trust is commonly defined following [START_REF] Mui | A computational model of trust and reputation for e-businesses[END_REF] as a subjective expectation an agent has about another's future behavior based on the history of their encounters. The history is usually learned from ratings that peers provide for each other. If such direct history is not available, one derives trust based on reputation. Reputation is defined [START_REF] Mui | A computational model of trust and reputation for e-businesses[END_REF] as the aggregated perception that an agent creates through past actions about its intentions and norms. , where this perception is based on information gathered from trusted peers. These definitions are widely used in state of the art research on trust and reputation as the logic behind trust based reputation computational models in web communities and social networks. However, computing reputation for domains raises several new difficulties: -Rating information if exists, is sparse and usually binary, a domain is labeled either "white" or "black" -Static sources like blacklists and whitelists are often not up-to-date -There is no explicit concept of trust between domains which make it difficult to apply a flow or a transitive trust algorithm. -Reputation of domains is dynamic and changes very fast These difficulties make the selection of an adequate computational model for computing domain reputation a challenging task. The focus of our paper and its main contribution is therefore a flow model and a flow algorithm for computing domain reputation which uses a topology-based network that maps connections of domains to IPs and other domains. Our model uses DNS IP-Domain mappings and statistical information but does not use DNS traffic data. Our approach is based on a flow algorithm, commonly used for computing trust in social networks and virtual communities. We are mainly inspired by two models: the Eigentrust model [START_REF] Kamvar | The eigentrust algorithm for reputation management in p2p networks[END_REF] that computes trust and reputation by transitive iteration through chains of trusting users; and the model by Guha et al. [START_REF] Guha | Propagation of trust and distrus[END_REF] which combines the flow of trust and distrust. The motivation for using a flow algorithm is the hypothesis that IPs and domains which are neighbors of malware generating IPs and domains, are more likely to become malware generating as well. We construct a graph which reflects the topology of domains and IPs and their relationships and use a flow model to propagate the knowledge received in the form of black list, to label domains in the graph as suspected domains. Our preliminary experimental results support our proposed hypothesis that domains (or IPs) connected to malicious domains have a higher probability to become malicious as well. The main contribution of this paper lies in the novelty of the algorithm and the strength of the supporting experimental study. The experimental study supporting our results, uses a sizable DNS database (more than a one million IPs and domains) which proves the feasibility of our approach. The rest of this paper is organized as follows. Section 2 provides a more detailed background on DNS and IP characteristics and on the classical flow models, and then surveys the related work. Section 3 discusses the graph construction, the weight assignment problem and the flow algorithm. Section 4 presents the results of our experimental evaluation and Section 5 concludes the paper and outlines future research. Background and Related Work The domain name system (DNS) translates Internet domains and host names into IP addresses. It is implemented as an hierarchical and distributed database containing various types of data, including host names and domain names, and provides application level protocol between clients and servers. An often-used analogy to explain the Domain Name System is that it serves as the phone book for the Internet by translating human-friendly computer host names into IP addresses. Unlike a phone book, the DNS can be quickly updated, allowing a service's location on the network to change without affecting the end users, who continue to use the same host name. Users take advantage of this when they use meaningful Uniform Resource Locators (URLs). In order to get the IP of a domain, the host usually consults a local recursive DNS server (RDNS).The RDNS iteratively discovers which authoritative name server is responsible for each zone. The result of this process is the mapping from the requested domain to the IP requested. Two categories of models are related to our work. The first category deals with ways to compute domain reputation. The second deals with flow algorithms for the computation of trust and reputation in general. Domain reputation is a relatively new research area. The Notos model for assigning reputation to domains [START_REF] Antonakakis | Building a dynamic reputation model for dns[END_REF] was the first to use statistical features in the DNS topology data and to apply machine learning methods to construct a reputation prediction classifier. Notos uses historical DNS information collected passively from multiple DNS resolvers to build a model of how network resources are allocated and operated for professionally run Internet services. Specifically it constructs a set of clusters representing various types of popular domains statistics and computes features which represent the distance of a specific domain from these clusters. Notos also uses information about malicious domains obtained from sources such as spam-traps, honeynets, and malware analysis services to build a set of features representing how network resources are typically allocated by Internet miscreants. With the combination of these features, Notos constructs a classifier and assigns reputation scores to new, previously unseen domain names.(Note that Notos uses heavily, information about highly popular sites such as Acamai which is not publicly available and therefore make it difficult to compare to.) The Exposure system [START_REF] Leyla | Exposure finding malicious domains using passive dns analysis[END_REF] collects data from the DNS answers returned from authoritative DNS servers and uses a set of 15 features that are divided into four feature types: time-based features, DNS answer-based features, TTL valuebased features, and domain name-based features. The above features are used to construct a classifier based on the J48 decision tree algorithm [START_REF] Witten | Data Mining: Practical Machine Learning Tools and Techniquel[END_REF] in order to determine whether a domain name is malicious or not. Kopis [START_REF] Antonakakis | Detecting malware domains at the upper dns hierarchy[END_REF] is a system for monitoring the high levels of the DNS hierarchy in order to discover the anomaly in malicious DNS activities. Unlike other detection systems such as Notos [START_REF] Antonakakis | Building a dynamic reputation model for dns[END_REF] or Exposure [START_REF] Leyla | Exposure finding malicious domains using passive dns analysis[END_REF],Kopis takes advantage of the global visibility of DNS traffic at the upper levels of the DNS hierarchy to detect malicious domains. After the features are collected it uses the random forest technique as the machine learning algorithm to build the reputation prediction classifier. In the category of flow algorithms for computation of trust in general, two models are of specific interest to our work. The first is Eigentrust [START_REF] Kamvar | The eigentrust algorithm for reputation management in p2p networks[END_REF], a reputation management algorithm for peer-to-peer network. The algorithm provides each peer in the network a unique global trust value based on the peer's history of uploads and thus aims to reduce the number of inauthentic files in a P2P network. The algorithm computes trust and reputation by transitive iteration through chains of trusting users. The page-rank algorithm [START_REF] Page | Pagerank citation ranking: Bringing order to the web[END_REF] uses a similar approach, however it contains special features related to URL referencing. Guha et al. [START_REF] Guha | Propagation of trust and distrus[END_REF] introduce algorithms for implementing a web-of-trust that allows people to express either trust or distrust in other people. Two matrices representing the trust and distrust between people are built using four types of trust relationships. They present several schemes for explicitly modeling and propagating trust and distrust and propose methods for combining the two, using weighted linear combination. The propagation of trust was also used by Coskun et al. [START_REF] Coskun | Friends of an enemy: identifying local members of peer-to-peer botnets using mutual contacts[END_REF] for detecting potential members of botnets in P2P networks. Their proposed technique is based on the observation that peers of a P2P botnet with an unstructured topology, communicate with other peers in order to receive commands and updates. Since there is a significant probability that a pair of bots within a network have a mutual contact, they construct a mutual contact graph. This graph is different than the DNS topology graph we rely on, the attributes and semantics underlying our approach are different and accordingly the algorithm we propose. Wu et al. [START_REF] Wu | Propagating trust and distrust to demote web spam[END_REF] use the distrust algorithm presented by Guha et al. [START_REF] Guha | Propagation of trust and distrus[END_REF] for detecting spam domains but use URL references rather than DNS data to derive the edges between domain nodes. They also discuss trust attenuation and the division of trust between a parent and its "children". Yadav et al. [START_REF] Yadav | Detecting algorithmically generated malicious domain names[END_REF] describe an approach to detect malicious domains based mainly on their names distribution and similarity. They claim that many botnets use the technique of DGA (domain generating algorithm) and they show that domains generated in this form have certain characteristics which help in their detection. The domain names usually have a part in common, e.g. the top level domain (TLD), or a similar distribution of alpha-numeric characters in their names. The success of using the above characteristics in [START_REF] Yadav | Detecting algorithmically generated malicious domain names[END_REF] motivates the construction of domain-domain edges in our graph as well. There are quite a few papers which use DNS data logs to detect Botnets and malicious domains. However these papers use the DNS traffic behavior and not the mapping information used by Notos and in our work. Villarmin et al. [START_REF] Villamarin-Salomon | Bayesian bot detection based on dns traffic similarity[END_REF] provide C&C detection technique motivated by the fact that bots typically initiate contact with C&C servers to poll for instructions. As an example, for each domain, they aggregate the number of non-existent domains (NXDOMAIN) responses per hour and use it as one of the classification features. Another work of this category, presented by Choi and Lee [START_REF] Choi | Identifying botnets by capturing group activities in dns traffic[END_REF] monitor DNS traffic to detect botnets, which form a group activity in similar DNS queries simultaneously. They assume that infected hosts perform DNS queries at several occasions and using this data they construct a feature vector and apply a clustering technique to identify malicious domains. As discussed above, although DNS traffic data has significant features, it is difficult to obtain comparing to DNS topological data. To summarize, although there are some previous works on domain reputation using DNS statistical features (e.g., Notos), and there exist flow algorithms in other trust and reputation domains, the combination of the two as used in this paper is new. The Flow Model The goal of the flow algorithm is to assign domains with reputation scores given an initial list of domains with known reputation (good or bad). The first step is the construction of the Domain-IP graph based on information obtained from a large set of successful DNS transactions represented as A-records. The A-records are used to construct the DNS topology graph, where vertices represent IPs and domains, and the weighted edges represent the strength of their connections. This is described next. In subsection 3.2 we present in detail the flow algorithm, and describe the method for combining good and bad reputation scores. Finally we discuss an optimization of the algorithm needed for large graphs. Constructing the graph and assigning edge weights The DNS topology graph consists of two types of vertices: domains and IPs, deriving four types of edges between them. To construct the graph we use Arecords, and also data available from public sources to estimate the strength of connections between any two vertices, IP or domain, by the amount of common data between them. The IP data for each IP consists of the following five characteristics available from sources such as e.g., WHOIS databae [15]: -Autonomous System number (ASN): a collection of connected Internet Protocol (IP) routing prefixes under the control of one or more network operators that present a common, clearly defined routing policy to the Internet. The ASN number is the indexation of the collection. -Border Gateway Protocol (BGP) prefix: a standardized exterior gateway protocol designed to exchange routing and reachability information between autonomous systems (AS) on the Internet. The protocol defines the routing of of each ASN.BGP prefix as a range of IPs to which it routes. -Registrar: the name of the organization that registered the IP. -Country: the country to which the IP belongs. -Registration date: the date and time at which the IP was registered. For Domain data the key concept is the parent of a domain. A k-Top Level Domain (kTLD) is the k suffix of the domain name [START_REF] Antonakakis | Building a dynamic reputation model for dns[END_REF]. For example: for domain finance.msn.com, the 3TLD is the same as the domain name: finance.msn.com, the 2TLD is .msn.com and the 1TLD is .com. We use the following notation: Set IP is the set containing all the IPs. Set domain is the set containing all the domains. Set parent ⊆ Set domain is the set containing all the parents. Set commonAtt is the set of attributes vectors derived from IP data. Attribute appear in the following order: country, ASN, BGP prefix, registrar, registration date. Missing information is replaced with 'none'. For example (DE, none, none, ST RAT O.DE, none) ∈ Set CommonAtt is the vector element in which the only information available is the country and the registrar. We define a weight function that assigns a weight to each edge in the graph. Let w be a weight function w : (u, v) → [0, 1] used to assign weight to the edge (u, v) where u, v ∈ Set IP ∪ Set domain , for each edge type we consider three alternative weight functions as follows: 1. IP to domain: For ip ∈ Set IP and the list of A-records, let D ip be all the domains mapped to ip. For each d ∈ D ip we define: w(ip, d) = w(d 1 , d 2 ) = 1 |P d | ; 1 log |P d | ; 1. The intuition behind the above definition of weights is that, the effect of a domain reputation on the IPs it is mapped to, increases, as the amount of mapped IPs decreases. We use three approaches for computing weight, which produce 81 different combinations from which a subset was tested. We represent our graph as an adjacency matrix M , in which the value of an entry (i, j) is the weight between vertex i and j computed according to the selected combination. The Flow algorithm The flow algorithm models the idea that IPs and domains affect the reputation of IPs and domains connected to them. This is done by propagating a node's reputation iteratively, so that the reputation in each iteration is added to the total reputation accumulated within a domain or IP node, using some attenuation factor. The attenuation factor is a means to reduce the amount of reputation accumulated by transitivity. The flow algorithm is executed separately to propagate good reputation and bad reputation. The algorithm is presented in two parts, the first is the Basic Flow, which describes the flow algorithm in general, and the second is the Combined Algorithm, which focuses on the way bad reputation and good reputation are combined. Figure 1 outlines the preparation steps prior to the execution of the basic algorithm. The Basic Flow algorithm starts with an initial set of domains which are labeled either bad or good. The parameters of the algorithm are: 1. A matrix M , where each entry represents a weighted edge between the vertices (domains or IPs); M T denotes the transpose of the matrix M . 2. V initial -a vector representing the initial reputation value of each vertex, based on the initial set of labeled vertices. 3. n ∈ N -the number of iterations. 4. atten ∈ [0, 1] -the attenuation factor. 5. θ ∈ [0, 1] -a reputation threshold above which a vertex is considered bad. Algorithm 1 outlines the basic flow algorithm that propagates the reputation from a node to its neighbors and is carried out in three steps: 1. Calculate the matrix for the i th iteration: 2. Calculate the final Matrix after n iterations as the sum of all the matrices with attenuation: M i = (M T • atten) i M f inal = n i=0 (M T • atten) i 3. Calculate the final reputation vector: V f inal = M n • V initial , the reputation scores of all vertices after the propagation. Applying the algorithm separately to propagate bad and good reputation may result in a node that is labeled both as good and bad. The final labeling of such node depends on the relative importance given to each label as done in the combined algorithm. Algorithm 1 Basic procedure Basic(M, n, atten, V initial ) for i = 0 to n do Mi ← (atten • M T ) i M f inal ← n i=0 Mi V f inal ← M f inal • V initial return V f inal The combined algorithm runs the basic flow algorithm twice with V initial = V good and V initial = V bad . Each flow is configured independently with the following parameters: factor, threshold, and number of iterations. We denote n good , n bad ∈ N as the number of iterations for the good and bad flow respectively; atten good , atten bad ∈ R, as the attenuation factor for the good and bad flow respectively; and w ∈ R the weight of the "good" reputation used when combining the results. Algorithm 2 uses of the basic flow algorithm to compute the good and bad reputation of each vertex and merge the results. Set M al is the result set of domains identified as bad. Algorithm 2 Combined 1: V good ← basic(M, n good , atten good , V good ) 2: V bad ← basic(M, n bad , atten bad , V bad ) 3: Set M al ← ∅ 4: for d ∈ Domains do 5: if V bad [d] + w • V good [d] > θ then Set M al ← Set M al ∪ {d} 6: return Set M al The combined algorithms ignores the following observations which lead us to examine another approach: -A reputation score is a value in the range of [0,1], therefore a vertex should not distribute a reputation score higher then 1 to it's neighbors. -Initial labels are facts and therefore domains that were initially labeled as good or bad should maintain this label throughout the propagation process. -A domain gaining a reputation value above a predefined threshold, bad or good, is labeled accordingly and maintain this label throughout the propagation process. The extended algorithm shown as Algorithm 3 is proposed to address these observations. The main new procedure is the N ormalize procedure which normalizes the scores after each iteration. The threshold used is the average of the scores so far (good or bad according to the actual procedure). Optimization for large graphs As we deal with millions of domains and IPs we have to calculate the scores in an efficient way. The most computationally intensive step is the matrix multiplication (see Figure 1). To speed it up we use a special property of our graph which is the existence of Cliques. There are two kinds of cliques in the graph: cliques of IPs which share the same set of common attributes and cliques of domains which share the same parent or the same name server. Since a clique can contain thousands of nodes we calculate the flow within it separately and not as part of the matrix multiplication. A clique in a graph is a subset of its vertices such that every two vertices in the subset are connected by an edge. We define a Balanced Clique as a clique such that all vertices have the same attribute values. This necessarily leads to a clique with a single weight value on all of its edges. Theorem 1. Let M be a matrix representing a weighted directed graph and V a vector representing vertices values, and let BC be the set of vertices that form a balanced clique, such that the weight on every edge in BC is const BC ∈ R; For Algorithm 3 Extended 1: for x ∈ Domains do 2: V bad [x] ← 0; V good [x] ← 0 3: if x ∈ Set bad then V bad [x] ← 1 4: if x ∈ Set good then V good [x] ← 1 5: for i = 1 to n do 6: V good ← V good + (M T ) • V good 7: V bad ← V bad + (M T ) • V bad 8: N ormalize(Set good , V bad , V good ) 9: N ormalize(Set bad , V good , V bad ) 10: Set M al ← ∅ 11: for d ∈ Domains do 12: if V bad [d] + w • V good [d] > θ then Set M al ← Set M al ∪ {d} 13: return Set M al 14: procedure N ormalize(Set1, V1, V2) 15: avg2 ← 0 16: if d∈Set 1 V good [d] > 0 then avg2 ← d∈Set 1 V 2 [d] |{d∈Set 1 :V 2 [d]>0}| 17: for x ∈ Domains do 18: if V1[x] > 1 then V1[x] ← 1 19: if x ∈ Set1 then V2[x] ← 0 20: if V1[x] > 1 -avg2 then V1[x] ← 1 a vertex v ∈ BC, connected only to vertices in BC ((M T ) * V )[v] = const BC • v =i∈BC V [i] (1) Due to space limitation, the proof of this theorem is omitted. Using the property of a balanced clique we devised the following algorithm for computing the scores. The reputation of every clique vertex is the sum of reputation scores of all other vertices of the clique, multiplied by the constant edge weight of the clique. This is shown in algorithm 4. Algorithm 4 AssignBCScores procedure AssignBCScores(BC, constBC , V ) Sum ← i∈BC V [i] for i ∈ BC do V result [i] = (Sum -V [i]) • constBC return V result [i] The complexity of this algorithm is O(|BC|) which is a significant improvement to O(|BC| 2 ), the complexity of the matrix multiplication approach. In our graph, all edges between the same type of vertices (IPs or domains) belong to balanced cliques and therefore this optimization plays a major factor. Experiment results The evaluation of the algorithm uses real data collected from several sources. To understand the experiments and the results, we first describe the data obtained for constructing the graph, and the criteria used for evaluating the results. Data sources We used five sources of data to construct the graph. -A-records: a database of successful mappings between IPs and domains, collected by Cyren [START_REF] Cyren | A provider of cloud-based security solutions[END_REF] from a large ISP over several months. This data consists of over one milion domains and IPs which are used to construct the nodes of the graph. -Feed-framework: a list of malicious domains collected and analyzed by Cyren over the same period of time as the collected A-records. This list is intersected with the domains that appeared in the A-records and serves as the initial known "bad" domains vector. -Whois [15]: a query and response protocol that is widely used for querying databases that store the registered users or assigners of an Internet resource, such as a domain name, an IP address block, or an autonomous system. We use WHOIS to get the IP data, which consists of the five characteristics of IP (ASN, BGP prefix, registrar, country, registration date). -VirusTotal [START_REF]VirusTotal: A free virus, malware and URL online scanning service[END_REF] -a website that provides scanning of domains for viruses and other malware. It uses information from 52 different antivirus products and provides the time a malware domain was detected by one of them. -Alexa: Alexa database ranks websites based on a combined measure of page views and distinct site users. Alexa lists the "top websites" based on this data averaged over a three-months period. We use the set of top domains as our initial benign domains, intersecting it with the domains in the Arecords. This set is filtered to remove domains which appeared as malicious in VirusTotal. We conducted two sets of experiments Tuning-test and Time-test . In the Tuningtest experiment, the DNS graph is built from the entire set of A-records, but the domains obtained by the Feed-Framework are divided into two subsets: an Initial set and a Test set. The Initial set is used as the initial vector of "bad" domains for the flow algorithm. The test set is left out to be identified as bad by the algorithm. Obviously, not all bad domains of the test set can be identified, mainly because the division of domains was done randomly and some of the domains in the test set are not connected to the initial set. Yet, this experiment was very useful to determine the best parameters of the algorithm which were used later in the Time-Test experiment. The time-test experiment, is carried out in two steps corresponding to two consecutive time periods. In the first step, we construct the graph from the data sources described above. We use the feedframework data to set the initial vector of bad domains in the first time period and the Alexa data to set the initial vector of good domains. We execute the flow algorithm (combined or extended) and assign the final score for each node of the graph. To validate the results of the Time-test, we check the domains detected as malicious by the algorithm against data from VirusTotal for the period following the time period used in the first step. We sort the domains by descending bad reputation score and use the score of the domain on the k-th position as the threshold. As the evaluation criteria for this test, we define a Prediction Factor (PF) which evaluate the ability of our algorithm to detect bad domains identified later by VirusTotal. We compute this factor as the ratio of domains labeled as bad that were tagged by VirusTotal and a random set of domains of the same size, that were tagged by VirusTotal. A domain tested with VirusTotal, is considered as tagged only if it was tagged by at least one anti-virus program. A P F i factor considers a domain as tagged by VirusTotal if at least i anti viruses tagged the domain. To compute the prediction factor we extract the set of domains with the khighest bad reputation scores HSet k found by the algorithm and select randomly a set of k domains RSet k : P F i = |HSet k ∩ Set i tagged | |RSet k ∩ Set i tagged | (2) where set i tagged is the set of all domains tagged by at least i anti viruses. A value of P F i indicates the extent to which our algorithm identifies correctly bad domains, compared to the performance of randomly selected domains. A similar approach was used by Cohen et al. [START_REF] Cohen | Early detection of outgoing spammers in large-scale service provider networks[END_REF] that compared the precision of a list of suspicious accounts returned by a spam detector against a randomly generated list. Next we describe the results of the two experiments. The Tuning test There are two main objectives to the Tuning test. The first is to verify that domains with higher bad reputation values are more likely to be detected as bad domains. The second objective is to understand the impact of the different parameters on the final results and configure the evaluation experiment accordingly. We tested the different weight functions discussed in section 3.1. The following combination was found best and was selected to compute the weights on the graph edges: -IP to domain -for ip ∈ Set The tuning test also examines the attenuation of the flow, the number of iterations, and the relative importance of the good domains used to calculate the final score. The tuning test repeats the following steps: 1. Construct the graph from the entire A-records database. 2. Divide the set of 2580 tagged malicious domains from the feed-framework into two sets: an initial set of 2080 domains and a test set consisting of 500 domains. 3. Apply the flow algorithm (combined or extended) using the Initial set to initiate the vector of bad domains and the data from Alexa to initiate the vector of good (as described above). We repeat these steps with different combinations of parameters (see Figure 2). We then compute the reputation scores of all domains, sort the domains by these scores and divide them into 10 bins of equal size. For each bin S i , i = 1..10 we measure the presence of domains that belong to the test set S test (known malicious domains), as |Si∩Stest| |Stest| . In Figure 2 we can see that the 10% of domains with the highest 'bad' reputation score contain the highest amount of malicious domains. Moreover, in all five type of experiments, the bin of domains with the highest scores consists of 20-25 percents of all known malicious domains of S test . The tests we conducted demonstrate the following three claims: 1. Domains with high 'bad' scores have a higher probability to be malicious. 2. The combinations of the "good" and "bad" reputation scores improve the results. 3. After a relatively small number of iterations the results converge. Time-test The time test experiment uses the best parameters as determined by the Tuning test. The first step of the experiment uses data collected during three months: September-November 2014 while the second step uses data collected during the following two months (i.e., December 2014-January 2015). After the first step in which we apply the flow algorithm (either the combined or the extended), and assign bad reputation scores to domains, we validate the domains with the highest 'bad' reputation scores that did not appear in the initial set of malicious domains against the information provided by VirusTotal. The data we used for the time test consists of 2, 385, 432 Domains and 933, 001 IPs constructing the nodes of the graph, from which 956 are tagged as maliciouse and 1, 322 are tagged as benign (Alexa). The resulting edges are 2, 814, 963 Ip to Domain edges, 13, 052, 917, 356 Domain to Domain edges and 289, 097, 046 IP to IP edges. The very large numbers of edges between IP to IP and Domain to Domain, emphasize the importance of the optimized algorithm 4 in Section 3.3. For the validation test we selected the 1000-highest bad reputation domains and calculated the prediction factor P F i for i ∈ {1, 2} (see equation 2). Out of 1000 random domains checked against VirusTotal, only 2.775% were tagged by at least one anti virus product and 0.875% of the random domains were tagged by at least two anti virus product. Table 1 presents the results of the experiment using the Combined and Extended algorithms. We used only 5 iterations since the Tuning test results indicates that this number is sufficient. For the combined algorithm the weight of good reputation score was set to w = -1 and the bad attenuation was set to 1 and 0.8 in tests 1 and 2 respectively. We can see from the table that with the parameters of P2, that from the 1000 domains with the highest score, 323 were tagged by at least one of the anti viruses in VirusTotal (tagged by one), and 203 were tagged by at least two anti viruses (tagged by two) which derive the prediction factor of P F 1 = 11.61 and P F 2 = 23.2, respectively. Due to space limitations we do not present all the results, however the best results were achieved in the second test. For example, out of 200 domains with the highest scores in test 2, 107 domains were tagged which reflect 53.5% of all known malicious domains. Figure 3 demonstrates the ability of our algorithms to predict malicious domains, using the prediction factor (eq. 2). The figure shows that if we consider smaller size of domains with the highest score the prediction rate is better but the overall predicted malicious domains are less. If we consider a larger size we end up with a larger number of domains that we suspect as malicious by mistake( i.e., false positives). Conclusions This paper discusses the problem of detecting unknown malicious domains by estimating their reputation score. It applies a classical flow algorithms for propagating trust, on a DNS topology graph database, for computing reputation of domains and thus discovering new suspicious domains. Our work faces two major challenges: the first is building a DNS graph to represent connections between IPs and domains based on the network topology only and not on the dynamic behavior. The second challenge was the design and implementation of a flow algorithm similar to those used for computing trust and distrust between people. We presented an algorithm that can be deployed easily by an ISP using non private DNS records. The algorithm was evaluated to demonstrate its effectiveness in real-life situations. The results presented in this paper are quite encouraging. In future work we intend to further improve the results by taking more data features into consideration and by conducting further tests with more parameters. Fig. 1 : 1 Fig. 1: The process for computing reputation scores: (1) Create the topology graph, assign weights and represent as a matrix; (2) Create the initial vector used for propagation; (3) Use the vector and the matrix as input to the flow algorithm; (4) output final reputation scores. IP and d ∈ D ip , w(ip, d) = 1 |Dip| . -Domain to IP -for d ∈ Set domain and ip∈ I d , w(d, ip) = 1 |I d | . -IP to IP -for ip 1 , ip 2 ∈ Set commonAtt , s.t. ip 2 = ip 1 , w(ip 1 , ip 2 ) = 1 |Set commonAtt | .-Domain to Domain -for d 1 , d 2 ∈ Set domain , parents P d ⊂ Set parent , andd 1 , d2 ∈ P d , s.t. d 1 = d 2 , w(d 1 , d 2 ) = 1 |P d | .The most effective combination of features for IP data turned to be (none, ASN, BGP-Prefix, Registrar, none), which derives cliques of IPs with the same ASN, BGP-Prefix and Registrar. Fig. 2 : 2 Fig. 2: The percentage of malicious domains by reputation score Fig. 3 : 3 Fig. 3: Prediction factor with respect to the number of highest bad domains Domain to IP: For d ∈ Set domain and a list of A-records, let I d be all the IPs that were mapped to d. For each ip ∈ I d we define: w(d, ip) = 1 |I d | ; 1 log |I d | ; 1. 3. IP to IP: Let commonAtt be a combination of the five attributes of IP data. Let Set commonAtt be the set of all IPs with the attribute combination Common A tt. For each ip 1 , ip 2 ∈ Set commonAtt s.t. ip 1 = ip 2 we define: w(ip 1 , ip 2 ) = Domain to Domain: Let P d be the set of all domains with the same parent domain d. For each d1, d2 ∈ P d s.t. d1 = d2 we define: 2. 1 |Set commonAtt | ; 1 log |Set commonAtt | ; 1. 4. 1 |Dip| ; 1 log |Dip| ; 1. Table 1 : 1 Results for time test experiment Atten bad Atten good Algorithm Tagged by one Tagged by two 1 0.8 1 combined 334 212 2 1 1 combined 323 203 3 none none extended 247 136 Acknowledgement This research was supported by a grant from the Israeli Ministry of Economics, The Israeli Ministry of Science , Technology and Space and the Computer Science Frankel center.

01745856

en

2024/03/05 22:32:07

2018

https://hal.science/hal-01745856/file/2016%201220%20E-health%20platform%20adherence%20factors%20sans%20fig.pdf

Sophie Trijau Vincent Pradel Hervé Servy Pierre Lafforgue Thao Pham email: thao.pham@ap-hm. Patient e-health platform for Rheumatoid Arthritis: accuracy and adherence factors Keywords: personal health records, rheumatoid arthritis, accuracy factors, adherence Background: Personal health records (PHRs) are patient-controlled repositories, capturing health data entered by individuals and providing information related their care. These tools improve treatment adherence but data are scarce concerning tool adherence. The accuracy of the self-recorded data remains controversial. We assessed how support measures improve PHR adoption determined the factors that influence the accuracy of self-recorded data and tool adherence of RA patients. Methods: A controlled randomized study with a PHR tool with integrated electronic health records developed by SANOIA. RA patients with ACR/EULAR 2010 criteria with web access randomized into 3 groups: Group 1 patients were given written information to create and manage a PHR; Group 2 patients received written information and a web technician hotline 48 hours after inclusion; Group 3 patients began their PHR with their rheumatologist during the consultation. Results: 56 RA patients were included (female: 73%, mean age: 57.1, mean DAS28: 3.04, mean RAPID-3: 2.93). Self-reported data accuracy was significantly higher in Groups 2 (73.7%) and 3 (82.4%) than in Group 1 (45.0%), (P = 0.04). Patient adherence was higher in Group 2 (78.9%) compared with Groups 1 (55.0%) and 3 (58.8%) (P = 0.45). Accuracy was correlated to adhesion (P <0.0001). Gender, age, disease duration and activity, treatments, and patient level of interest were not correlated to data accuracy or patient adherence. Conclusion: Information accuracy collected with PHR was relevant and better when patients were initially assisted either by their physician or by non-medical phone support. We also observed better adherence when patients were initially assisted. Background Personal health records (PHRs) are tools enabling patients to collect patient-reported outcomes (PROs) and to report their medical conditions. The aim of such patientcontrolled services is to help individuals play a more active role and to contribute to shared decisions in chronic diseases such as rheumatoid arthritis (RA). These tools have shown their ability to improve treatment adherence [START_REF] Chrischilles | Personal health records: a randomized trial of effects on elder medication safety[END_REF][START_REF] Vrijens | Electronically monitored dosing histories can be used to develop a medication-taking habit and manage patient adherence[END_REF][START_REF] Hogan | Accuracy of data in computer-based patient records[END_REF] but data are scarce concerning adherence to the tool itself. For instance, 153 patients with rheumatic disease consecutively interviewed reported that although they appreciated having access to their online electronic health records, they expressed low confidence rates in the Internet [START_REF] Richter | Changing attitudes towards online electronic health records and online patient documentation in rheumatology outpatients[END_REF]. This lack of confidence may impact patient adhesion. The accuracy of self-recorded data related to medical condition in e-health platforms also remains controversial. Our goal was to assess how support measures, technical or medical, could improve electronic personal health record system (PHR) adoption and to determine the accuracy of self-recorded data and RA patient tool adherence modifying factors. Materials and methods Electronic personal health records tool Sanoia has developed a web tool integrated with electronic personal health records which offers full privacy protection using an innovative anonymity technique [START_REF] Chiche | Evaluation of a prototype electronic personal health record for patients with idiopathic thrombocytopenic purpura[END_REF]. The tool already included numerous factors related to chronic disease such as vaccination records and history. It can also be designed for a specific disease, in our case RA, and proposes an adapted PROs evaluation, such as the Routine Assessment of Patient Index Data (RAPID-3) [START_REF] Pincus | An index of the three core data set patient questionnaire measures distinguishes efficacy of active treatment from that of placebo as effectively as the American College of Rheumatology 20% response criteria (ACR20) or the Disease Activity Score (DAS) in a rheumatoid arthritis clinical trial[END_REF]. Patients Inclusion criteria : outpatients fulfilling the ACR/EULAR 2010 criteria for rheumatoid arthritis [START_REF] Aletaha | Rheumatoid arthritis classification criteria: an American College of Rheumatology/European League Against Rheumatism collaborative initiative[END_REF]. As our aim was to assess factors associated with PHR adhesion and data accuracy, we focused only on patients able to use the e-PHR, so patients without web access were not invited to participate in the study. Study Design We conducted a prospective controlled randomized study. From February to March 2011, the five participating rheumatologists proposed to their consecutive RA patients to use an e-health platform and aimed at collecting patient-related outcomes (PROs) and medical conditions. Each rheumatologist had a randomization list Patients were randomized into 3 groups: Group 1 patients were given simple written information about how to create and manage their file on the Sanoia platform; Group 2 patients received written information and support to manage their files on the platform via a web-technician hotline 48 hours after inclusion; and Group 3 patients started their Sanoia platform files with their rheumatologist during the consultation. Patients were randomized individually by each rheumatologist. Each patient fulfilling inclusion criteria and accepting to participate in the study was assigned sequentially to a group according to his order of inclusion by his/her rheumatologist. This method was preferred to bloc randomization in order to maximize the balance of group effectives even if some rheumatologists include few patients in a context of competitive recruitment. Collected data Adherence Assessments Patients were considered as tool adherent if they connected at least twice and as nonadherent if they connected once or never between baseline (M0) and the 3-month evaluation (M3). Adherence was also assessed at 6 months (M6). Accuracy Assessment We collected the following data: demographics, disease activity data including the disease activity score (DAS-28) and the RAPID-3 [START_REF] Pincus | An index of the three core data set patient questionnaire measures distinguishes efficacy of active treatment from that of placebo as effectively as the American College of Rheumatology 20% response criteria (ACR20) or the Disease Activity Score (DAS) in a rheumatoid arthritis clinical trial[END_REF][START_REF] Prevoo | Modified disease activity scores that include twenty-eight-joint counts. Development and validation in a prospective longitudinal study of patients with rheumatoid arthritis[END_REF], the amount and accuracy of selfrecorded data, ongoing treatment at baseline and 3 months after. For the latter, we focused on medical history, current treatment, consultations, underlying events reporting and other points of interest such as the vaccine situation, smoking status, contraception, imaging. Information accuracy was compared with rheumatologist medical records considered as the gold standard. All the records were assessed by the same reviewer (ST) and scored as following: medical history (0-4 points), current treatment (0-4 points), vaccine status (0-1 point) and known allergies (0-1 point), for a maximum score of 10. A good accuracy score was defined as>9/10. Main outcomes The 2 primary outcomes were the adherence at M3, defined as the proportion of patients who connected at least 2 times between baseline and 3 months, and the accuracy of the patient self-declared data, defined as the proportion of patients with a good accuracy score. Statistical Analysis We compared reporting accuracy and adherence among the groups. Accuracy was assessed as a dichotomous variable. Patients with a good accuracy score, i.e.>9/10, were scored 1. Patients with an accuracy score ≤9/10 or with missing data were scored 0. We also assessed the impact of the following variables on accuracy and adherence: age (age as a continuous variable and age> 60 years yes/no), gender, disease duration, co-morbidities, baseline disease activity (DAS28, RAPID-3), treatments including biologics and corticosteroids and the patient's level of interest in the e-PHR tool. The patient's interest was assessed on a 0-10 numerical scale. Continuous data were described by means (SD) and categorical variables were expressed as frequencies and percentages. To compare groups, we used Chi-square (Fisher when Chi-square application conditions were not met), Mann-Whitney and Krusakl-Wallis tests (depending on categorical/continuous variables and the number of modalities of the categorical variables). SPSS 17.0 version was used for management and statistical analysis. Results We included 56 RA patients, with 20, 19 and 17 patients in Groups 1, 2 and 3 respectively. Their main baseline characteristics were the following: female: 73%, mean age: 57.1 years, mean DAS28: 3.04 and mean RAPID-3: 2.93. Detailed characteristics are reported in Table 1. The proportion of patients who did not use the PHR tool was 35.0%, 21.1% and 19.6% in Groups 1, 2 and 3 respectively. Self-reported data accuracy was significantly higher in Groups 2 (73.7%) and 3 (82.4%) than in Group 1 (45.0%), (p < 0.04) (Figure 1). Moreover, two patients reported medical events that were not in their physician medical records: a history of tuberculosis in a Group 3 patient and a costal chondroma in a Group 2 patient. Patient adherence was higher in Group 2 (78.95%) compared with those of Groups 1 (55.0%) and 3 (58.8%) (P = 0.45) at 3 months. Mean ITT frequency connections are presented in Table 2. Among the patients who connected at least once (N = 13 for Group 1, 16 for Group 2 and 17 for Group 3), the mean number of connections between baseline and M3 was 10.3, 19.3 and 12.2 in Groups 1, 2 and 3 respectively. Adherence remained stable in Group 2 at 6 months (78.9%), whereas it decreased in Groups 1 and 3 (respectively 15.0% and 47.1%). Connection frequencies are presented in Table 2. Accuracy was correlated to adherence (P <0.0001). Gender, age, disease duration, activity of disease (DAS28, RAPID-3), treatments including biologics and corticosteroids, and patient level of interest were not correlated to data accuracy (Table 3). These variables were also not correlated to patient adherence. Discussion This is the first study showing how support measures can influence adoption, adherence to PHR and the accuracy of recorded information. The quality of the information collected with PHR was meaningful and better when patients were initially assisted either by their physician or by non-medical phone support. Agreement between self-report information on PHR and medical records have been assessed with various results with regards to the disease [START_REF] Simpson | Agreement between self-report of disease diagnoses and medical record validation in disabled older women: factors that modify agreement[END_REF][START_REF] Bergmann | Validity of self-reported diagnoses leading to hospitalization: a comparison of self-reports with hospital records in a prospective study of American adults[END_REF][START_REF] Van Gelder | Using Web-Based Questionnaires and Obstetric Records to Assess General Health Characteristics Among Pregnant Women: A Validation Study[END_REF][START_REF] Haapanen | Agreement between questionnaire data and medical records of chronic diseases in middle-aged and elderly Finnish men and women[END_REF][START_REF] De-Loyde | Which information source is best? Concordance between patient report, clinician report and medical records of patient co-morbidity and adjuvant therapy health information[END_REF][START_REF] Van Den Akker | Disease or no disease? Disagreement on diagnoses between self-reports and medical records of adult patients[END_REF]. Comparisons have shown good agreement for diabetes, hypertension, pulmonary disease, cerebrovascular disease and myocardial infarction while other comparisons found low agreement for heart failure, chronic bronchitis, chronic obstructive pulmonary disease, hypertension, osteoporosis, osteoarthritis and RA [START_REF] Simpson | Agreement between self-report of disease diagnoses and medical record validation in disabled older women: factors that modify agreement[END_REF][START_REF] Van Den Akker | Disease or no disease? Disagreement on diagnoses between self-reports and medical records of adult patients[END_REF][START_REF] Okura | Agreement between self-report questionnaires and medical record data was substantial for diabetes, hypertension, myocardial infarction and stroke but not for heart failure[END_REF][START_REF] Engstad | Validity of self-reported stroke : The Tromso Study[END_REF][START_REF] Klungel | Cardiovascular diseases and risk factors in a population-based study in The Netherlands: agreement between questionnaire information and medical records[END_REF][START_REF] Goldman | Evaluating the quality of self-reports of hypertension and diabetes[END_REF][START_REF] Malik | Patient perception versus medical record entry of health-related conditions among patients with heart failure[END_REF][START_REF] Merkin | Agreement of self-reported comorbid conditions with medical and physician reports varied by disease among end-stage renal disease patients[END_REF][START_REF] Skinner | Concordance between respondent self-reports and medical records for chronic conditions: experience from the Veterans Health Study[END_REF][START_REF] Hansen | Agreement between self-reported and general practitioner-reported chronic conditions among multimorbid patients in primary care -results of the MultiCare Cohort Study[END_REF]. In a large community-based cohort, low agreement was found between the 2,893 participating patients and their GP, especially for RA (kappa: 0.17 [0.23-0.11] [START_REF] Van Den Akker | Disease or no disease? Disagreement on diagnoses between self-reports and medical records of adult patients[END_REF]. In RA patients, over-reporting was associated with the male gender, a higher number of diseases and a lower physical and mental quality of life. A higher number of diseases was also associated with underreporting. We did not collect a quality of life variable, but in our RA sample, gender and comorbidities were not correlated with record accuracy. The study sample size was small and possibly not powerful enough to demonstrate a correlation with the potential factors, such as biologic and corticosteroid treatment. The mode of questionnaire administration can affect data quality. Survey responses are different with regards to the type of mode (e.g. self-administered versus interview modes, mail versus telephone interviews, telephone versus face-to-face interviews) [START_REF] Bowling | Mode of questionnaire administration can have serious effects on data quality[END_REF][START_REF] Feveile | A randomized trial of mailed questionnaires versus telephone interviews: response patterns in a survey[END_REF][START_REF] Christensen | Effect of survey mode on response patterns: comparison of face-to-face and self-administered modes in health surveys[END_REF]. As expected, our results confirm that patients that have been supported in the process, either with a technician without medical knowledge or with an MD, have better data collection accuracy. It may appear to be the support itself and not its quality because we found no difference between the methods of support, but the sample size of our study was insufficient to determine differences between the 2 Figure 1 : 1 Figure 1: Proportion of patients with reported data of good accuracy on the e-PHR tool Table and Figure Legends Table 1 : andLegends1 Baseline characteristics Table 2: PHR tool adherence in the 3 groups at M3 and M6 (ITT) Table 3 : 3 Accuracy factors Acknowledgements We gratefully thank C. Foutrier-Morello, J. Fulpin, E. Senbel, S. Steib, S. Trijau for their help in recruiting patients for the study, and the study patients for their contribution. This study was supported by an unrestricted educational grant from UCB Pharma The authors are grateful to Peter Tucker for his careful reading of the manuscript. support groups. Technical or medical support also improved adoption of and adherence to the PHR tool. In conclusion, our results suggest good or very good agreement between selfreported data with a PHR tool and medical records for RA patients. This agreement is improved with a technical or medical process support that also improves adoption and adherence to the tool. Our results confirm that a patient with the right level of support could be a source of reliable data collection, opening new paths that should be confirmed over a longer time period and from an economic point of view.

01435097

en

2024/03/05 22:32:07

2016

https://hal.science/hal-01435097/file/Krakovinsky.2016.ICMTS.pdf

A Krakovinsky email: [email protected] M Bocquet R Wacquez J Coignus D Deleruyelle C Djaou G Reimbold J-M Portal Impact of a Laser Pulse On HfO 2 -based RRAM Cells Reliability and Integrity Keywords: OxRRAM, Laser, Security, Integrity, HfO 2, Simulation, Thermal Attacks, Optical Attacks . Moreover these solutions propose lower switching energy and faster operations compared to the state of the art for Flash, and thus, are seen as an opportunity for the rise of the IoT market. But one of the main concerns regarding IoT is the protection of the data. Contrary to Flash, security of the data in emerging NVM is yet to be evaluated. In order to verify capability of the technology in terms of data integrity, we propose to investigate reliability and integrity of HfO 2 -based Resistive RAM (OxRRAM). This paper details the experimental protocol defined for laser-based attacks, shows that a laser pulse can affect the information stored in a single OxRRAM bit. The occurring phenomenon is then explained by mean of thermal and electrical simulations. I. Introduction Innovative technologies are emerging as solutions for the future of Non Volatile Memories (NVM). One can cite Magnetoresistive Random Access Memories (MRAM), Phase Change RAM (PCRAM) or Resistive RAM (RRAM) as the main technologies of interest. A HfO 2based RRAM solution is considered in this work. Due to its low cost of fabrication and its inherent lower switching energy, RRAM is of high interest for the IoT market. In the following years, billions of smart objects will be interacting between each other. Power consumption is of course an essential issue. But if we consider the amount and nature of the processed data, the security aspect must not be neglected. In other words, emerging NVM must fulfill three essential criteria that are data integrity, data confidentiality and data accessibility. However, most studies still focus on the reliability of these technologies. Therefore none of them have been confronted to attacks such as UV lamp with masking [START_REF] Fournier | Memory Address Scrambling Revealed Using Fault Attacks, Fault Diagnosis and Tolerance in Cryptography (FDTC)[END_REF] or focused laser [START_REF] Skorobogatov | Local Heating Attacks on Flash Memory Devices Hardware-Oriented Security and Trust[END_REF] attacks that have been already shown as sucessful on Flash technology. The first security criterion to be studied for NVM (especifically RRAM) is material integrity. This work proposes to evaluate the impact of external physical constraints on the material. Laser has been aleready proven able to disturb the behaviour of a circuit as well as old Flash technology [START_REF] Skorobogatov | Optical Fault Masking Attacks Fault Diagnosis and Tolerance in Cryptography (FDTC)[END_REF] through fault attacks. As a follow up of these works, it is interesting to attack RRAM in the same way. Section 2 describes the experimental setup whose results are presented in section 3 and 4. Section 5 is focused on the simulation of a laser pulse on a RRAM structure and of RRAM set operation. II. Experimental Setup A. RRAM Principle and Characteristics Parameters A RRAM cell is made of two metal electrodes with a transition metal oxide (TMO) in-between. In our case (see Fig. 1), these are made of a 5nm-thick HfO 2 layer, located between a 10nm-thick Ti top electrode and a 10nm-thick TiN bottom electrode, as described in [START_REF] Vianello | Resistive Memories for Ultra-Low-Power embedded computing design[END_REF]. This technology relies on resistance switching, that is to say, in this case, migrating oxygen ions (as pictured in Fig. 1.) from the oxide to the electrode where a voltage is applied [START_REF] Nardi | Resistive Switching by Voltage-Driven Ion Migration in Bipolar RRAMPart I: Experimental Study[END_REF]. This process creates or dissolves a conductive filament (CF) of oxygen vacancies in the oxide, which is tuning its resistance. The way set and reset (i.e programming and erasing) operations are performed is presented on Fig. 2. V stopreset and the set compliance current I c are parameters which are set experimentally. For each experiment shown in this paper, V stopreset = -1V and I c = 1 mA. B. Instrumentation and Experimental Protocol We use a laser bench with a Nd:YAG source. Three wavelenghths are available : 355 nm (Ultraviolet/UV), 533 nm (Green) and 1064 nm (Infrared/IR). It has a circular 50µm-diameter spot size (as seen in Fig. 3.) that allows shooting the whole cell -whose size is 3 µm up to a 400 µJ energy during a 10 ns pulse. In the first place, to see the influence of laser pulses on the memory cells, their electrical characteristics were evaluated. These were conducted in quasi-static conditions (i.e without considerating temporal aspects) and consisted in 10 cycles of reset/set operations. The resistance value has been measured thanks to a read operation performed after each operation. These preliminary measurements allow verifying that the cells have a regular behavor, in conformity with what has been published. These data will be then used as reference data for the following experiments (See Fig. 4.). The average set and reset voltages are respectively V Set = 0.6 V and V Reset = -0.54 V. Moreover, the average LRS and HRS resistance values are about R LRSmean = 550Ω and R HRSmean =35 kΩ. The experimental protocol of the next step is summed up in Fig. 5. For the need of the experiment, 50% of all devices are left in HRS, and 50% in LRS. A single pulse in either UV or IR has been performed in order to check the potential impact of each wavelength. In the end, a III. Influence on Resistance Values A. Cells state variation and wavelength influence Regarding LRS cells after laser pulse, the first remarkable result is that the LRS remains unaffected. Fig. 6 shows that the cumulative distributions of the resistance values of the cells left initially in the LRS before and right after the pulse are similar. Moreover, both UV and IR data are overlaying. However, contrary to LRS cells (as noticed on Fig. 7), a gap from about a decade in terms of resistance distribution can be seen between HRS cells before (R mean b ef ore = 34 kΩ) and after (R meanaf ter = 3400Ω) they were shot . Besides, half of these cells have switched to the LRS after the pulse. Finally, as far as wavelength is concerned, UV and IR pulses have the same effect on HRS cells. Which means the laser impact on OxRRAM cells is independant from the wavelength used. Therefore, all the following data will not refer to the wavelength used during the experiment. The energy used will rather be taken into account as the most essential parameter. B. Laser Low Resistive State Analysis By confronting the resistance values of LRS cells to HRS cells that switched to the LRS cells after a laser pulse, it can be seen on Fig. 8 that the electrical LRS seems slightly different from the LRS obtained with laser. Indeed, the values of the switching cells (R LRSmean1 = 880Ω) are quite higher than those of the LRS cells (R LRSmean2 = 550Ω). In a nutshell, this analysis showed that a laser pulse can disturb the behaviour of only HRS cells. Nevertheless, the nature of the resistive state obtained after exposition is yet to be confirmed. IV. Impact on Electrical Characteristics A. Cells left in the LRS Even though the resistance values of LRS cells are showing the non-effectiveness of a laser beam on their state, V Set and V Reset measured after the attack may vary from those obtained before. But as expected, laser has not disturbed the cells since V Set and V Reset are matching the reference data (see Fig. 9). B. Cells left in the HRS Concerning the cells left in the HRS state, it has to be figured out whether or not cycling can be performed normally and if the laser set operation can be considered as equivalent to an electrical set operation. The cells that were submitted to a reset operation in a first place were hardly set to the HRS, since more than 80% of our devices first V Reset are above the average voltage reset according to the reference data (as pictured in Fig. 10). This means laser set and electrical set are not equivalent, which is confirmed by the data coming from cells whose first post-laser operation was a set operation. Indeed, their first reset voltage is corresponding to the reference data. Explaining the phenomena responsible for laser switching will require a model of the laser impact on the structure. As well as a model of an electrical set operation so we can compare them. V. Simulations There are two possibilities to explain a change in the oxide structure by the use of a laser beam. The first one is an optical effect provided by the photon injection and the second one is a temperature effect. Indeed, experiments have shown [START_REF] Cabout | Temperature impact (up to 200 C) on performance and reliability of HfO2-based RRAMs, Memory Workshop[END_REF] that high temperature operations tend to Where k is the imaginary part of the refractive index of the material, l the material thickness and λ the wavelength of the laser beam. For the metallization layer only (shown on Fig. 12), c ≈ 1,7 . 10 -15 for λ = 1064 nm and c ≈ 4,3 . 10 -24 for λ = 355 nm. In other words, photons are not able to go through the metallization layer and therefore reach the oxide. Which also means that instead of being caused by an optical effect, the laser switching shall be a consequence of temperature, the main focus of our modeling. A. Laser Modeling The geometry used for this model is presented on Fig. 11. For more accuracy, 1 nm of titanium dioxide has been considered on the top of our structure. The reason is since Regarding the laser impact, the model was designed by using [START_REF] Sands | Pulsed Laser Heating and Melting, Heat Transfer -Engineering applications[END_REF]. Therefore, the transmitted part of the laser energy to the surface of the structure and the heat emitted by each layer are respectively defined as follows : I T = I 0 (1 -R)g(x) (2) S(z) = α × I T L × e -αz (3) Where I 0 is the laser surface power, R the reflectivity of the surface material, g(x) the gaussian profile (whose standard deviation is a third of the laser spot radius) value for the x coordinate, α the optical penetration depth which equals 4πk λ (with k and λ defined as in (1)), z the depth in the layer and I T L the intensity transmitted to the layer top interface. The simulation results show that the maximum temperature in the oxide is 610 K and that it is reached 0.8 ns after the end of the pulse (see also Fig. 12 and13). This temperature is the result of the thermal diffusion coming from the titanium dioxide surface (whose maximum temperature is reached 0.8 ns before the end of the pulse). We might expect higher temperatures provided that the estimated silicon dioxide temperature is about 3000 K after the end of the pulse (beyond SiO 2 melting point which is not taken into consideration in this model). However the duration needed for thermal diffusion from the SiO 2 surface to the HfO 2 layer is about 1 µs since silicon dioxide has a low thermal conductivity ( k = 1.4 W.m -1 .K -1 [START_REF]CRC Handbook of Chemistry and Physics[END_REF]) and the corresponding temperature much lower (about 500 K). The objective of the following electrical set modeling is to verify if the temperature reached during laser exposition is relevant for an electrical set operation. B. Electrical Set Simulation For the electrical set simulation, the model described in [START_REF] Russo | Self-Accelerated Thermal Dissolution Model for Reset Programming in Unipolar Resistive-Switching Memory (RRAM) Devices[END_REF] has been applied to the cell structure. The main objective is to compare the temperature reached during an electrical set operation to the value obtained from thermal modeling. V Set , I c as well as both thermal (k) and electrical (σ) conductivities of the CF are the parameters of interest. By choosing the CF conductivities, we aim to get temperature according to its state after the set operation. In our case, the laser pulses were not able to really set the cells, which means the CF has a structure not far from HfO 2 , whose conductivities are lower than Hf. We then simulated different conductivities values for each couple (V Set ) , I c ) which gave us maps of the temperature reached in the CF in function of k and σ at constant set voltage and compliance current. The values chosen for V Set and I c are respectively 0.5 V/ 0.6 V/ 1.2 V and 1 µA/ 10 µA/ 0.1 mA / 1 mA . Regarding the values of the conductivities, we decided to choose 10 values decreasing linearly (for k) and logarithmically (for σ) from the values given in [START_REF]Properties of Solids; Thermal and Physical Properties of Pure Metals / Thermal Conductivity of Crystalline Dielectrics / Thermal Conductivity of Metals and Semiconductors as a Function of Temperature[END_REF] and [START_REF]CRC Handbook of Chemistry and Physics[END_REF] for Hf. The results presented on Fig. 14 were obtained for V Set = 0.5 V and I c = 10 µA. The temperature calculated with laser modeling (610 K) is reached for k ≈ 4 W.m -1 .K -1 and σ ≈ 5 10 5 S.m -1 . These low conductivities mean that the set hasn't been totally completed since the CF structure is closer to hafnium dioxide than hafnium, explaining the experimental results obtained by laser pulse. VI. Conclusion For the first time, RRAM cells have been disturbed by laser exposition, independently from the wavelength used. It is possible to perform a bitflip only from the HRS to the LRS. Simulations allowed us to explain that this phenomenon is due to the temperature brought by laser heating. Fig. 1 . 1 Fig. 1. Cell layer structure and ion migration description Fig. 2 .Fig. 3 . 23 Fig. 2. Standard I-V characteristic of the 1R cells to be studied. For the set operation, a progressive voltage sweep is applied from '1'. Then the cell switch to the LRS in '2' before its current reaches Ic in '3'. For the reset operation, a voltage sweep is applied from 'A'. The cell switches from the LRS to the HRS in 'B' . The sweep is stopped at Vstopreset in 'C' Fig. 4 . 4 Fig. 4. Boxplot of 1500 LRS/HRS reference values (a) and Set/Reset voltages (b) obtained during preliminary characterisations of the studied cells. The box extends from the lower to upper quartile values of the data, with a line at the median. Fig. 5 . 5 Fig. 5. Experimental Protocol Diagram Fig. 6 .Fig. 9 . 69 Fig. 6. Cumulative distributions of the resistance values of 84 LRS cells before and after a single laser pulse, sorted by wavelength used Fig. 10 . 10 Fig. 10. Cumulative distribution of set and reset voltages of HRS cells after 1 Reset/Set cycle sorted by first operation performed . These results were compared to reference data Fig. 12 . 12 Fig. 12. Temperature repartition 0.8 ns after the 10 ns laser pulse on the whole structure Fig. 13 . 13 Fig. 13. Temperature repartition 0.8 ns after the laser pulse in the area around the hafnium dioxide layer Acknowledgements : This work was performed with the support of the CATRENE CA208 Mobitrust project.

01415935

en

2024/03/05 22:32:07

2016

https://univ-rennes.hal.science/hal-01415935/file/The%20Impact%20of%20Donor%20Type%20on%20Long-Term.pdf

Sandrine Visentin email: [email protected] MD, PhD Pascal Auquier MD, PhD Yves Bertrand MD, PhD André Baruchel MD Marie-Dominique Tabone MD Cécile Pochon MD Charlotte Jubert Maryline Poiree MD, PhD Virginie Gandemer MD Anne Sirvent An L E A Study Sandrine Visentin MD Maryline Poirée MD Jacinthe Bonneau MD, PhD Catherine Paillard MD Claire Freycon MD, PhD Justyna Kanold Virginie Villes MD, PhD Julie Berbis MD Claire Oudin MD Claire Galambrun MD, PhD Isabelle Pellier MD Geneviève Plat MD, PhD Hervé Chambost MD, PhD Guy Leverger MD, PhD Jean-Hugues Dalle MD, PhD Gérard Michel The Impact of Donor Type on Long-Term Health Status and Quality of Life after Allogeneic Hematopoietic Stem Cell Transplantation for Childhood Acute Leukemia: A Leucemie de l'Enfant et de L'Adolescent Study Keywords: hematopoietic stem cell transplantation, late effects, quality of life, childhood leukemia, cord blood transplantation published or not. The documents may come INTRODUCTION Hematopoietic stem cell transplantation (HSCT) has been successfully used to treat children with high-risk or relapsed acute leukemia. Many children and adolescents who undergo HSCT become long-term survivors and may develop long-term complications, such as endocrinopathies, musculoskeletal disorders, cardiopulmonary compromise and subsequent malignancies [START_REF] Faraci | Non-endocrine late complications in children after allogeneic haematopoietic SCT[END_REF][START_REF] Nieder | NHLBI First International Consensus Conference on Late Effects after Pediatric Hematopoietic Cell Transplantation: Long Term Organ Damage and Dysfunction Following Pediatric Hematopoietic Cell Transplantation[END_REF][START_REF] Cohen | Endocrinological late complications after hematopoietic SCT in children[END_REF][START_REF] Chow | Late Effects Surveillance Recommendations among Survivors of Childhood Hematopoietic Cell Transplantation: A Children's Oncology Group Report[END_REF]. When available, an HLA-matched sibling donor (SD) remains the donor of choice for children who require HSCT. However, only approximately 25% of candidates eligible for allogeneic HSCT have an HLA-matched SD. In the absence of a SD, an HLA-matched unrelated volunteer donor (MUD) or unrelated umbilical cord blood (UCB) are alternative transplant sources. In fact, despite the establishment of bone marrow donor registries with more than 25 million volunteers worldwide, finding a MUD remains a problem for many patients. Thus, the use of UCB as an alternative source for HSCT has increased substantially in the last decade, especially for children [START_REF] Ballen | Umbilical cord blood transplantation: the first 25 years and beyond[END_REF]. Currently, it is estimated that several thousand UCB transplantations have been performed. The short-term outcome of children transplanted with UCB (e.g., hematopoietic recovery, acute and chronic graft versus host disease (GvHD), treatment-related mortality, survival and causes of death) have been well described [START_REF] Eapen | Outcomes of transplantation of unrelated donor umbilical cord blood and bone marrow in children with acute leukaemia: a comparison study[END_REF][START_REF] Benito | Hematopoietic stem cell transplantation using umbilical cord blood progenitors: review of current clinical results[END_REF][START_REF] Grewal | Unrelated donor hematopoietic cell transplantation: marrow or umbilical cord blood?[END_REF][START_REF] Zheng | Comparative analysis of unrelated cord blood transplantation and HLA-matched sibling hematopoietic stem cell transplantation in children with high-risk or advanced acute leukemia[END_REF][START_REF] Tang | Similar outcomes of allogeneic hematopoietic cell transplantation from unrelated donor and umbilical cord blood vs. sibling donor for pediatric acute myeloid leukemia: Multicenter experience in China[END_REF]. Although overall survival is comparable, it has been clearly established that the course of the early post-transplant period and principal complications differ with respect to the transplant cell source. The risk of GvHD and related complications is intrinsically higher after MUD transplantation compared with sibling transplantation, even if a recent extensive pediatric study has shown that this risk can be overcome by using intensive prophylaxis with cyclosporine, methotrexate and anti-thymocyte globulin [START_REF] Peters | Stem-Cell Transplantation in Children With Acute Lymphoblastic Leukemia: A Prospective International Multicenter Trial Comparing Sibling Donors With Matched Unrelated Donors-The ALL-SCT-BFM-2003 Trial[END_REF]. UCB transplant induces GvHD to a lesser degree than MUD transplantation, although UCB hematopoietic recovery is slower, thereby resulting in an extended duration of the aplastic phase and subsequent increased risk of severe infection [START_REF] Gluckman | Resultsof Unrelated Umbilical Cord Blood Hematopoietic Stem Cell Transplantation[END_REF][START_REF] Zecca | Chronic graft-versus-host disease in children: incidence, risk factors, and impact on outcome[END_REF]. In contrast, very few studies have assessed long-term post-transplant health status with regard to donor type in a multivariate analysis [START_REF] Bresters | High burden of late effects after haematopoietic stem cell transplantation in childhood: a single-centre study[END_REF][START_REF] Armenian | Long-term health-related outcomes in survivors of childhood cancer treated with HSCT versus conventional therapy: a report from the Bone Marrow Transplant Survivor Study (BMTSS) and Childhood Cancer Survivor Study (CCSS)[END_REF][START_REF] Hows | Comparison of long-term outcomes after allogeneic hematopoietic stem cell transplantation from matched sibling and unrelated donors[END_REF][START_REF] Baker | Late effects in survivors of chronic myeloid leukemia treated with hematopoietic cell transplantation: results from the Bone Marrow Transplant Survivor Study[END_REF][START_REF] Khera | Nonmalignant Late Effects and Compromised Functional Status in Survivors of Hematopoietic Cell Transplantation[END_REF], and to our knowledge, no studies have compared childhood leukemia survivors who received UCB with those who underwent SD or MUD HSCT. Using the data extracted from the French cohort of childhood leukemia survivors (L.E.A., "Leucémie de l'Enfant et de L'Adolescent"), our primary objective was to describe the long-term health status and quality of life (QoL) after HSCT for childhood leukemia survivors with respect to donor type (SD, MUD or UCB transplantation). Because the patients were transplanted between May 1997 and June 2012 and transplantations involving an HLA haplo-identical family donor were rare in France during this time period, the few patients who underwent such transplantation were not included in this study. METHODS Patients Evaluation of physical health status Medical visits were conducted to detect the occurrence of late effects based on clinical examinations and laboratory tests when required. Clinical follow-up commenced one year after HSCT; these examinations were repeated every two years until the age of 20 and for at least ten years of complete remission; patients were then examined every four years thereafter. Height, weight and body mass index (BMI) were measured at transplantation, study inclusion, and each subsequent medical examination. The measurements were then converted to standard deviation scores (SDS) based on the normal values for the French population [START_REF] Sempé | Auxologie : méthodes et séquences[END_REF]. Growth failure (stunted height) was defined by a cumulative SDS change equal to or lower than -1 (minor failure for a value between -1.0 and -1.9, and major failure for a value equal to or lower than 2). Overweight was defined as a BMI of 25 kg/m² or more for adults (minor: BMI of 25.0-29.9, major: BMI of 30 or more) and a cumulative SDS change of +1 or more for children under 18 (minor: between 1.0 and 1.9, major: equal to or higher than 2). Low weight was defined as a BMI lower than 18.5 kg/m² in adults and a cumulative loss in SDS of -1.0 or more in children under 18. Children were not assessed for gonadal function if they were under 15 years of age and had not experienced menarche (girls) or did not have any pubertal signs (boys). Patients were diagnosed with gonadal dysfunction if they showed signs of precocious puberty or hypergonadotropic hypogonadism (low estradiol levels with high follicle stimulating hormone (FSH) and luteinizing hormone (LH) levels in women; low testosterone with high FSH and LH levels in men). Hypothyroidism was defined as a nontransient increasein thyroid stimulating hormone levels. All second tumors (including basal cell carcinoma) were taken into consideration for this analysis. Cardiac function was considered impaired when any one of the following three conditions was present: the echocardiographic shortening fraction was inferior to 28%, the left ventricular ejection fraction was inferior to 55% or specific treatments were required. Femoral neck and lumbar bone mineral density were measured using dual energy X-ray absorptiometry for all adults. Patients were considered to have low bone mineral density when the Z-score was inferior or equal to -2 in at least one of the two sites examined. Metabolic syndrome was defined according to the NCEP-ATPIII revised in 2005 (metabolic syndrome patients had at least three of the five criteria: [START_REF] Faraci | Non-endocrine late complications in children after allogeneic haematopoietic SCT[END_REF] increased waist circumference (≥102 cm in men, ≥88 cm in women); (2) elevated blood pressure (systolic blood pressure ≥130 mmHg and/or diastolic blood pressure ≥85 mmHg and/or treatment necessitated); (3) reduced high-density lipoprotein cholesterol (≤40 mg/dL in men, ≤50 mg/dL in women); (4) elevated fasting glucose (≥1 g/L or drug treatment needed for elevated glucose levels); and (5) elevated triglycerides (≥ 150 mg/dL or drug treatment required for elevated triglycerides)) [START_REF] Grundy | Diagnosis and Management of the Metabolic Syndrome An American Heart Association/National Heart, Lung, and Blood Institute Scientific Statement[END_REF] The SF-36 (the Medical Outcome Study Short Form 36 Health Survey) is a widely used QoL measure that provides a non-disease-specific assessment of adult functioning and well-being, which enables comparison with a broad range of age-matched norm groups [START_REF] Leplège | The French SF-36 Health Survey[END_REF][START_REF] Reulen | The use of the SF-36 questionnaire in adult survivors of childhood cancer: evaluation of data quality, score reliability, and scaling assumptions[END_REF]. The SF-36 is a generic QoL scale for adults consisting of 36 items describing eight dimensions: physical functioning, social functioning, role limitations due to physical health problems, role limitations due to emotional health, mental health, vitality, bodily pain and general health. Two summary scores are also calculated from the subscales: a physical component score and a mental component score. This is a reliable instrument to assess self-perceived health status in adult survivors of childhood cancer. The French version is well validated. All scores range between 0 and 100, with higher scores indicating better QoL. Statistical analysis Chi-squared, Fisher's exact and ANOVA tests were used to compare demographic and clinical variables between the SD, MUD and UCB transplant groups. ANOVA was used to compare the mean number of late effects experienced per patient in each donor type group. Each of the following complications (as defined above) were considered as one late effect: height growth failure (minor or major), overweight (minor or major), low weight, gonadal dysfunction, hypothyroidism, second tumors, cataracts, alopecia, impaired cardiac function, osteonecrosis, low BMD, diabetes, metabolic syndrome, iron overload and central nervous system complications. To determine the link between each assessed adverse effect and donor type (i.e., SD, MUD or UCB), adjusted logistic regression models were performed. The six following covariates were included in the models: gender, age at diagnosis, age at last visit, history of relapse, conditioning regimen (TBI-versus busulfan-based), and leukemia type (acute myeloid leukemia (AML) versus acute lymphoblastic leukemia (ALL)). GvHD was considered as a potential intervening variable, i.e., a variable that is on the causal pathway between the transplant source and health status. Consequently, GvHD was not included in the model [START_REF] Katz | Multivariable analysis: a practical guide for clinicians[END_REF]. Adjusted odds ratio (OR) and risk of having one type of late effect (including 95% confidence intervals) were estimated. Adjusted multiple linear regression models were generated to explore the link between the long-term QoL scores and donor type with the same covariates. Each model is presented with its standardized β coefficient, which measures the strength of the effect of graft type on the QoL dimension score. The SF-36 mean scores reported by adult patients were compared with those obtained from age-and sex-matched French control subjects, using the paired Student's t-test [START_REF] Berbis | A French cohort of childhood leukemia survivors: impact of hematopoietic stem cell transplantation on health status and quality of life[END_REF]. Statistical significance was defined as p<0.05. RESULTS Patient characteristics A total of 314 patients fulfilled all selection criteria and were included in the analysis. The patient characteristics are summarized in Table 1. One hundred twenty-seven patients had received stem cells from a SD (40.5%), 99 from a MUD (31.5%) and 88 from unrelated UCB (28.0%). The mean follow-up duration from diagnosis and HSCT to last L.E.A. visit were 7.7±0.2 and 6.2±0.2 years, respectively. The mean age at acute leukemia diagnosis was 7.5±0.3 years; UCB recipients were significantly younger at diagnosis (p=0.02). As expected, the percentage of patients who relapsed before HSCT was significantly higher in the MUD and UCB groups than in the SD group (p=0.001). More patients in the SD group (66.1%) were in first hematologic complete remission at the time of transplantation, compared with MUD (51.5 %) and UCB (40.9%); whereas in RC2 or more advance hematologic status, patients more often received an alternative donor type (MUD or UCB) (p=0.009).The incidence of significant GvHD (grade II-IV aGvHD or extensive cGvHD) was lower among UCB recipients (27.3% versus 43.3% for SD; and 62.6% for MUD, p<10 -3 ). A greater proportion of patients in the UCB and MUD groups had received post-transplant corticosteroids (p=0.02); this high percentage in spite of the low GvHD incidence in the UCB group can be explained by the fact that steroids were included in the GvHD prophylaxis regimen of most UCB recipients. The three groups were similar with regard to gender, previous irradiation, age at HSCT, leukemia type, conditioning regimen (TBI-or busulfanbased) and follow-up duration from diagnosis and HSCT to last visit. The patients of the UCB group were younger at last L.E.A. evaluation compared with the other groups, although this difference was not statistically significant (p=0.11). Long-term late effects Overall, 284 of 314 patients (90.4%) were found to have at least one late effect, without any apparent difference between the three groups. Among the SD survivors, 92.1% suffered from at least one late effect compared with the MUD (92.9%) and UCB (85.2%) survivors (p=0.14). The average number of adverse late effects was 2.1±0.1, 2.4±0.2 and 2.4±0.2, respectively (non-significant). Twenty-two percent of the transplanted patients had one late effect, 31% had two late effects and 37% had three or more late effects. As shown in Figure 1, no significant difference was found between the donor cell sources (p=0.52). The occurrence of each side effect for each group is outlined in Table 2. The patients treated using SD transplant were considered the reference group for all comparisons. The multivariate analysis indicated that donor type did not have an impact on most sequelae. The only two significant differences were higher risk of major height growth failure after MUD transplantation (OR[95%CI]=2.42[1.06-5.56], p=0.04) and osteonecrosis following UCB transplantation (OR[95%CI]=4.15 [1.23-14.04], p=0.02). None of the other comparisons revealed significant differences in the multivariate models. Quality of life Adults Adults of the three groups reported very similar QoL (Table 3). The physical composite scores were 52.1±1.6 for the SD group, 50.4±1.8 for the MUD group and 50.3±2.2 for the UCB group (p=0.72). The mental composite scores were 43.4±1.4 for the SD group versus 47.3±1.7 for the MUD group and 43.3±2.6 for the UCB group (p=0.28). Considering SD as the reference group, multivariate linear regression analysis did not show any difference between the donor sources for each dimension. Parents' point of view The QoL of children and adolescents was reported by 204 parents (Table 4). The summary scores were 68.4±1.7 for the SD group, 68.8±2.0 for the MUD group and 69.8±1.9 for the UCB group (p=0.87). Parent-reported scoring of the nine dimensions did not indicate that donor type had an impact on the QoL of children and adolescents. Children and Adolescents The mean scores reported by children (n=35) were comparable for all VSPAe subscales (Table 5). The summary scores were 72.4±3.4, 74.4±3.7 and 70.3±4.2 for the SD, MUD and UCB groups, respectively (p=0.78). Regarding adolescent QoL (Table 6), no significant difference was found between the three groups, with the exception of 'relationship with parents' and 'school work'. In fact, adolescents of the SD group reported a significantly better 'school work' mean score than those of the MUD group (p=0.05) and a lower 'relationship with parents' mean score compared with the UCB group (p=0.03). The summary scores were 64.1±1.7, 67.6±2.0 and 69.2±2.0 for the SD, MUD and UCB groups, respectively (p=0.15). Comparison to French norms The QoL assessed in 84 adults of this cohort was compared to age-and sex-matched French reference scores (Figure 2). Almost all subscales were significantly lower in the L.E.A. cohort. The physical composite (51.2±1.1 versus 55.2±0.1, p<0.001) and mental composite scores (44.3±1.1 versus 47.9±0.3, p=0.001) were both lower in the L.E.A. group. DISCUSSION The main objective of this study was to assess the long-term health status and QoL of a French cohort of childhood leukemia survivors who had received HSCT from three different donor types. HLA-identical sibling transplanted patients were chosen as the reference group and compared with MUD and UCB transplantations. During the immediate post-transplant phase, MUD transplantation patients are at increased risk of GvHD, while UCB transplants are associated with a slower hematologic recovery [START_REF] Eapen | Outcomes of transplantation of unrelated donor umbilical cord blood and bone marrow in children with acute leukaemia: a comparison study[END_REF][START_REF] Gluckman | Resultsof Unrelated Umbilical Cord Blood Hematopoietic Stem Cell Transplantation[END_REF][START_REF] Rocha | Comparison of outcomes of unrelated bone marrow and umbilical cord blood transplants in children with acute leukemia[END_REF]. We aimed to determine whether donor type also had an impact on long-term health status and QoL. With a 6.2-year post-transplant follow-up, this study showed that regardless of the donor type, the development of adverse health outcomes and QoL in long-term survivors were markedly similar. The mean number of late effects experienced per patient was a little more than two for each group; 90.4% of HSCT survivors in this study developed at least one adverse effect. Although the occurrence of late effects in patients transplanted during childhood has been described, the impact of donor type on side effects was seldom taken in consideration. In the study by Bresters et al., among 162 survivors of HSCT, 93.2% had sequelae after a median follow-up time of 7.2 years. Donor type was not found to be a risk factor for increased burden of late effects in a multivariate analysis, although only two patients had received UCB transplantation (1.2%) [START_REF] Bresters | High burden of late effects after haematopoietic stem cell transplantation in childhood: a single-centre study[END_REF]. Armenian et al. have found at least one chronic health condition in 79.3% of childhood HSCT survivors (n=145) after a median follow-up time of 12 years. In a multivariate analysis, compared with conventionally treated cancer survivors, HSCT survivors had a significantly elevated risk of adverse health-related outcomes, and unrelated HSCT recipients were at greatest risk [START_REF] Armenian | Long-term health-related outcomes in survivors of childhood cancer treated with HSCT versus conventional therapy: a report from the Bone Marrow Transplant Survivor Study (BMTSS) and Childhood Cancer Survivor Study (CCSS)[END_REF]. Another study involving a cohort of 463 adults and children has reported a significantly higher cumulative incidence of extensive GvHD, cataracts and bone necrosis at 12 years after MUD, compared with SD transplants [START_REF] Hows | Comparison of long-term outcomes after allogeneic hematopoietic stem cell transplantation from matched sibling and unrelated donors[END_REF]. To our knowledge, the health status of long-term survivors after UCB transplant has never been described. A few studies have reported late complications after HSCT during childhood, in which some patients had received UCB transplantation. However, no comparison between the donor source was performed, and the cohorts included a very limited proportion of UCB recipients: between 1.2% ( 14) and 5% [START_REF] Ferry | Long-term outcomes after allogeneic stem cell transplantation for children with hematological malignancies[END_REF]. The absolute number of late effects per patient is not a sufficient data point to comprehensively describe health status, as the burden of each late effect may markedly vary. Consequently, in this study we described the risks of specific late effects with respect to stem cell sources. Only two late effects were significantly associated with donor type: osteonecrosis was more frequent in the UCB group and major growth failure occurred more often following MUD transplant. Steroids have been shown to play a role in the pathophysiology of post-transplant osteonecrosis; other well-described risk factors include older age, female gender and GvHD [START_REF] Girard | Symptomatic osteonecrosis in childhood leukemia survivors: prevalence, risk factors and impact on quality of life in adulthood[END_REF][START_REF] Li | Avascular necrosis of bone following allogeneic hematopoietic cell transplantation in children and adolescents[END_REF][START_REF] Mcavoy | Corticosteroid Dose as a Risk Factor for Avascular Necrosis of the Bone after Hematopoietic Cell Transplantation[END_REF][START_REF] Mcclune | Bone Loss and Avascular Necrosis of Bone After Hematopoietic Cell Transplantation[END_REF]. In the current study, although GvHD risk was lower following UCB transplant compared with MUD and SD transplant, the use of posttransplant steroids was very common as steroids were included in the GvHD prophylaxis regimen of most UCB recipients. Additionally, the higher proportion of patients with a history of pre-transplant leukemia relapse and ALL in the UCB and MUD groups may have played a role by increasing the pre-transplant cumulative steroid dose. Several studies have reported the impact of TBI conditioning regimens on post-transplant growth [START_REF] Sanders | Growth and development after hematopoietic cell transplant in children[END_REF][START_REF] Bernard | Height growth during adolescence and final height after haematopoietic SCT for childhood acute leukaemia: the impact of a conditioning regimen with BU or TBI[END_REF]. In the present study, the risk of major growth failure was higher in patients who had received MUD, whereas the proportion of patients treated with TBI as a pre-transplant conditioning regimen was not significantly greater. Poor post-transplant growth may be due to many other factors, including GvHD and its treatments [START_REF] Isfan | Growth hormone treatment impact on growth rate and final height of patients who received HSCT with TBI or/and cranial irradiation in childhood: a report from the French Leukaemia Long-Term Follow-Up Study (LEA)[END_REF][START_REF] Majhail | Recommended Screening and Preventive Practices for Long-term Survivors after Hematopoietic Cell Transplantation[END_REF]. Significant GvHD occurred more frequently following MUD transplantation compared with the two other groups. However, our data do not support this explanation as we were unable to demonstrate a significant effect of GvHD on major growth failure in our cohort (data not shown). To evaluate QoL, we used self-reported questionnaires for adults, children and adolescents as well as parent-reported questionnaires for patients less than 18 years of age. We found comparable results among the three study groups for all composite scores. This observation suggests that even if the immediate post-transplant period and burden of early complications experienced by transplanted children may differ with respect to the donor type, this does not explain the QoL reported many years after HSCT. In contrast, the adult QoL scores were significantly lower than sex-and age-matched French norms. Previous L.E.A. reports studying QoL have found similar results regardless of treatment or health condition, thus suggesting that suffering from acute leukemia may also play a role [START_REF] Berbis | A French cohort of childhood leukemia survivors: impact of hematopoietic stem cell transplantation on health status and quality of life[END_REF][START_REF] Michel | Health status and quality of life in long-term survivors of childhood leukaemia: the impact of haematopoietic stem cell transplantation[END_REF]. We acknowledge that the observed differences in the physical and mental composite scores, albeit statistically significant, were relatively small and their clinical relevance must be thus interpreted with caution. Others studies showed that cGvHD is the major contributor to reduced QoL after HSCT [START_REF] Eapen | Outcomes of transplantation of unrelated donor umbilical cord blood and bone marrow in children with acute leukaemia: a comparison study[END_REF][START_REF] Fraser | Impact of chronic graft-versus-host disease on the health status of hematopoietic cell transplantation survivors: a report from the Bone Marrow Transplant Survivor Study[END_REF]. In our study, significant GvHD incidence was statistically higher among recipients of MUD grafts although QoL was similar. This is perhaps due to the fact that QoL scores reported in our study are the most recent measure for each patient and that survivors with resolved cGvHD may have a comparable long-term QoL to those never diagnosed with cGvHD [START_REF] Fraser | Impact of chronic graft-versus-host disease on the health status of hematopoietic cell transplantation survivors: a report from the Bone Marrow Transplant Survivor Study[END_REF][START_REF] Sun | Burden of morbidity in 10+ year survivors of hematopoietic cell transplantation: a report from the Bone Marrow Transplant Survivor Study[END_REF]. Data concerning the impact of donor type on QoL are very scarce. Lof et al. did not identify any difference between patients with a related or unrelated donor [START_REF] Löf | Health-related quality of life in adult survivors after paediatric allo-SCT[END_REF]. Very little is known regarding QoL among long-term survivors following UCB transplant. Routine evaluation of health-related QoL should be an integral part of patient follow-up after childhood leukemia, especially when patients are treated by HSCT regardless of the donor type. As UCB transplant has only recently become available, UCB patients of the L.E.A. cohort had a shorter follow-up duration than SD or MUD patients. More precisely, the date of the first UCB transplant reported to L.E.A. was May 1997. Thus, in the present study, only patients transplanted after that date were included, to both obtain a similar follow-up duration among the three groups and compare patients who had been treated in the same country during the same period of time. As a consequence, the follow-up duration (7.7 years after diagnosis and 6.2 years after HSCT) is shorter than that in other L.E.A. studies. This is a limitation of our study as some late effects may occur after a longer period of time. Some complications such as hypogonadism manifest during adulthood, thus requiring an extended follow-up period for detection. Other studies with a prolonged follow-up period are warranted to confirm our results. It is, however, important to note that more than one third of the patients in our cohort were adults at last assessment. The strengths of this study include cohort size and the large proportion of patients who received UCB transplantation (28%). To our knowledge, this represents the first comprehensive study to describe the long-term late effects and QoL after UCB transplant for childhood leukemia. In conclusion, long-term acute leukemia survivors treated with HSCT during childhood are at risk for treatment-related sequelae, although donor type appears to have a very low impact on long-term outcomes and QoL. This analysis provides additional information for patients and physicians to assist in treatment decisions when a SD is not available and the transplant donor type must be selected between MUD and UCB. To prevent Gonadal dysfunction a (39.4) 26 (37.7) 24 (41.4) 1.17 (0.52 -2.66) 0.71 All patients described here were included in the L.E.A. program. This French multicenter program was established in 2003 to prospectively evaluate the long-term health status, QoL and socioeconomic status of childhood leukemia survivors. Patients were included in L.E.A. program if they met the following criteria: treated for acute leukemia after 1980 in one of the participating centers, were younger than 18 years of age at the time of diagnosis, and agreed (or their parents/legal guardians) to participate in the study. The present L.E.A. study focused on patients who received allogeneic HSCT with HLA-identical SD, MUD or UCB stem cells after a total body irradiation (TBI)-or busulfan-based myeloablative conditioning regimen before June 2012. To avoid potential bias due to different treatment periods and follow-up durations, we only included HSCTs performed after May 1997, the date of the first UCB transplant reported in the L.E.A. cohort. Patients were excluded from the study if they underwent more than one HSCT, if they were treated before May 1997, if they were conditioned with a non-myeloablative regimen, or if they received autologous or HLA mismatched related transplantation. All patients (or their parents) provided written informed consent to participate in the program. The French National Program for Clinical Research and the French National Cancer Institute approved this study. Iron overload was indicated by hyperferritinemia (a serum ferritin dosage ≥ 350ng/ml at least one year after HSCT) in the absence of concomitant high erythrocyte sedimentation rates. Other late effects (cataracts, alopecia, osteonecrosis, diabetes and central nervous system complications) were systematically screened during every medical visit. Evaluation of quality of life (QoL) The VSPAe (Vécu et Santé Perçue de l'Enfant) and VSPA (Vécu et Santé Perçue de l'Adolescent) questionnaires are generic health-related QoL questionnaires specifically designed to evaluate self-reported QoL in 8-to 10-year-old children and 11-to 17-year-old adolescents. VSP-Ap questionnaires (Vécu et Santé Perçue de l'Enfant et de l'Adolescent rapportés par les parents) are used to assess the parental point of view of their child's or adolescent's QoL. These questionnaire responses consider nine dimensions: psychological well-being, body image, vitality, physical well-being, leisure activities, relationship with friends, relationship with parents, relationship with teachers and school work. In addition to specific scores for each subscale, a global health-related QoL score is computed (21-23). Figure 1 : 1 Figure 1: Number of late effects per patient with respect to donor type. Figure 2 : 2 Figure 2: SF-36 results in adults compared with sex-and age-matched French norms. Table 3 :Table 4 : 34 ᵃ:gonadal function was assessable in 221 patients (92 girls and 129 boys/94 SD, 69 MUD and 58 UCB) ᵇ: data available in 67 adults (27 SD, 21 MUD and 19 UCB) c : data of metabolic syndrome was assessable in 78 adults (36 SD, 21 MUD and 21 UCB) ᵈ: iron overload was assessable in 293 patients (121 SD, 90 MUD and 82 UCB) QoL of adults (n=84) using SF-36 questionnaire. -variates: gender, leukemia type, age at diagnosis, age at last visit, relapse and conditioning (TBI/Bu). QoL of children and adolescents reported by their parents (n=204) using VSP-Ap. -variates: gender, leukemia type, age at diagnosis, age at last visit, relapse and conditioning (TBI/Bu). Table 1 : 1 Patient Characteristics (n=314) SD MUD UCB HSCT, hematopoietic stem cell transplantation; ALL, acute lymphoblastic leukemia; AML, acute myeloid leukemia; CNS, central nervous system; TBI, total body irradiation; Bu, Busulphan; CR, complete remission; GvHD, Graft versus host disease *Significant GvHD: comprises acute GvHD grade II-IV and extensive chronic GvHD Table 2 : Occurrence of late effects according to donor type Multivariate analysis SD MUD UCB MUD versus SD UCB versus SD 2 (n=127) (n=99) (n=88) n(%) n(%) n(%) OR (95% CI) p OR (95% CI) p Height growth failure Minor or major (31.5) 44 (44.4) 27 (30.7) 1.68 (0.93 -3.01) 0.08 0.94 (0.50 -1.77) 0.84 Major 12 (9.4) 20 (20.2) 13 (14.8) 2.42 (1.06 -5.56) 0.04 1.60 (0.65 -3.97) 0.30 GH treatment 8 (6.3) 10 (10.1) 6 (6.8) 1.50 (0.54 -4.18) 0.44 0.91 (0.29 -2.86) 0.88 Overweight Minor or major (18.9) 20 (20.2) 18 (20.5) 1.22 (0.61 -2.44) 0.58 1.15 (0.55 -2.41) 0.70 Major 7 (5.5) 8 (8.1) 7 (8.0) 1.57 (0.52 -4.71) 0.42 1.32 (0.41 -4.28) 0.64 Low weight (25.2) 24 (24.2) 20 (22.7) 0.75 (0.39 -1.42) 0.38 0.68 (0.35 -1.35) 0.27 Page 16 of 29 Acknowledgements The study was funded in part by the French National Clinical Research Program, the French National Cancer Institute (InCA), the French National Research Agency (ANR), the Cancéropôle PACA, the Regional Council PACA, the Hérault and the Bouches-du-Rhône departmental comities of the Ligue Contre le Cancer and the French Institute for Public Health Research (IRESP). The authors would like to thank the patients and their family as well as all members of the L.E.A. study group (Supplemental Data S1). Conflict of interest: The authors declare no potential conflict of interest. long follow-up of transplant patients is recommended regardless of the donor type.

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2024/03/05 22:32:07

2017

https://theses.hal.science/tel-01745888/file/44395_BONIFAS_2017_diffusion.pdf

Thèse De Doctorat De L'université Paris-Saclay Préparée À L'école Polytechnique Ecole Doctorale M Nicolas Bonifas Professeur Jean-Charles Billaut Professeur M Jacques Carlier U T Émérite Compiègne M Christian Rapporteur Artigues M Philippe Baptiste Friedrich Eisenbrand Daniel Dadush email: [email protected] Mark Wallace Andreas Schutt Yakov Zinder Leo Liberti Berthe Choueiry François Clautiaux J'ai Emmanuelle Grand Françoise Routon Gérard Vachet Annick Pasini Mariange Ramozzi-Doreau Ronan Ludot-Vlasak Jean-Pierre Lièvre Stéphane Gonnord Florent Jacques Renault Nicolas Bonifas email: [email protected] Marco Di Summa Nicolai Hähnle Martin Niemeier || NNT : 2017SACLX119 Keywords: Closest Vector Problem, Lattice Problems, Convex Geometry Context This work falls in the scope of mathematical optimization, and more precisely in constraint-based scheduling. This involves determining the start and end dates for the execution of tasks (these dates constitute the decision variables of the optimization problem), while satisfying time and resource constraints and optimizing an objective function. In constraint programming, a problem of this nature is solved by a treebased exploration of the domains of decision variables with a branch & bound search. In addition, at each node a propagation phase is carried out: necessary conditions for the different constraints to be satisfied are verified, and values of the domains of the decision variables which do not satisfy these conditions are eliminated. In this framework, the most frequently encountered resource constraint is the cumulative. Indeed, it enables modeling parallel processes, which consume during their execution a shared resource available in finite quantity (such as machines or a budget). Propagating this constraint efficiently is therefore of crucial importance for the practical efficiency of a constraint-based scheduling engine. In this thesis, we study the cumulative constraint with the help of tools rarely used in constraint programming (polyhedral analysis, linear programming duality, projective geometry duality). Using these tools, we propose two contributions for the domain: the Cumulative Strengthening, and the O(n 2 log n) Energy Reasoning. gation, of course without losing feasible solutions. This technique is commonly used in integer linear programming (cutting planes generation), but this is one of its very first examples of a redundant global constraint in constraint programming. Calculating this reformulation is based on a linear program whose size depends only on the capacity of the resource but not on the number of tasks, which makes it possible to precompute the reformulations. We also provide guarantees on the quality of the reformulations thus obtained, showing in particular that all bounds calculated using these reformulations are at least as strong as those that would be obtained by a preemptive relaxation of the scheduling problem. This technique makes it possible to strengthen all propagations of the cumulative constraint based on the calculation of an energy bound, in particular the Edge-Finding and the Energy Reasoning. This work was presented at the ROADEF 2014 conference [BB14] and was published in 2017 in the Discrete Applied Mathematics journal [START_REF] Baptiste | Redundant cumulative constraints to compute preemptive bounds[END_REF]. O(n 2 log n) Energy Reasoning This work consists of an improvement of the algorithmic complexity of one of the most powerful propagations for the cumulative constraint, namely the Energy Reasoning, introduced in 1990 by Erschler and Lopez. In spite of the very strong deductions found by this propagation, it is rarely used in practice because of its cubic complexity. Many approaches have been developed in recent years to attempt to make it usable in practice (machine learning to use it wisely, reducing the constant factor of its algorithmic complexity, etc). We propose an algorithm that computes this propagation with a O(n 2 log n) complexity, which constitutes a significant improvement of this algorithm known for more than 25 years. This new approach relies on new properties of the cumulative constraint and on a geometric study. This work was published in preliminary form at the CP 2014 conference [START_REF] Bonifas | Fast propagation for the Energy Reasoning[END_REF] and was published [START_REF] Bonifas | A O (n^2 log (n)) propagation for the Energy Reasoning[END_REF] at the ROADEF 2016 conference, where it was awarded 2 th prize of the Young Researcher Award. It may be that when we no longer know what to do, we have come to our real work and when we no longer know which way to go, we have begun our real journey. The mind that is not baffled is not employed. The impeded stream is the one that sings. (Wendell Berry) Il est d'usage d'écrire quelques remerciements en introduction d'un manuscrit de thèse, mais j'ai été tellement porté pendant mes études et mes années de doctorat que cela me semble être la chose la plus naturelle du monde. Avant tout, mes remerciements vont à ma famille, et à mes parents extraordinaires. Je mesure la chance que j'ai eu de grandir dans un environnement sécurisant, épanouissant, et aimant ! En ce qui concerne cette thèse, mes remerciements vont à mon directeur de thèse, Philippe Baptiste, qui a su m'indiquer des questions scientifiques dignes d'intérêt et a pris beaucoup de temps dans son emploi du temps chargé pour me guider. Du côté d'IBM, mes encadrants ont été Jérôme Rogerie, que je remercie pour ses conseils, son soutien, et de nombreuses discussions inattendues et éclairantes, ainsi que Philippe Laborie, qui m'a tant appris en optimisation, et qui a su trouver une réponse me permettant d'avancer à chaque question que je lui ai posée. C'était un bonheur de travailler avec vous. Merci aussi à Paul Shaw et Éric Mazeran pour voir rendu cette thèse possible, ainsi qu'à tous mes collègues d'IBM, de l'équipe CP Optimizer et au-delà. Merci à Jacques Carlier et Christian Artigues d'avoir accepté de relire ce manuscrit, ainsi qu'à Jean-Charles Billaut, Catuscia Palamidessi et Peter Stuckey d'avoir bien voulu faire partie de mon jury. Je voudrais aussi remercier toutes les personnes qui, tout autour du monde, m'ont invité à collaborer au cours de cette thèse, et notamment Christian Artigues, Pierre Lopez (que je remercie tout particulièrement pour m'avoir in- Préface Pourquoi optimisons-nous ? Pourquoi faire une thèse en optimisation ? Mon histoire personnelle avec l'optimisation a commencé un jour d'été de 2008 alors que, jeune étudiant dans un master d'informatique enseignant essentiellement la logique formelle, je me suis retrouvé bloqué de nombreuses heures dans un train sans climatisation sous le chaud soleil de la Côte d'Azur. Un incendie sur un appareil de voie obligeait en effet les trains à circuler sur une voie unique, et les stratégies d'alternance du sens de circulation entraînaient une pénible attente. Pour me distraire, j'ai alors réfléchi aux stratégies de reprise du trafic après une telle interruption, et à l'équilibre entre débit de la voie ferrée et temps d'attente maximal des passagers. Rentré chez moi (je ne disposais pas d'internet mobile en ces temps reculés), j'ai appris que tout un domaine scientifique, la recherche opérationnelle, s'attelait déjà à résoudre ces questions. J'ai pris conscience ce jour que l'informatique, bien loin de se cantonner à résoudre des problèmes d'informatique elle-même, pouvait être une discipline ouverte sur le monde, et j'ai développé une passion pour les outils et les applications de l'optimisation, et j'ai alors réorienté mes études vers ce domaine. La victoire des machines sur les humains dans les jeux d'échec ou de go s'est faite non par une meilleure vision stratégique, mais en étendant la vision tactique des dix ou douze demi-coups dont sont capables les meilleurs joueurs humains jusqu'à l'horizon de la partie. De la même façon, l'optimisation permet d'étendre la portée des abstractions d'un système, et d'en réduire le nombre, permettant une maîtrise plus fine du lien entre objectifs et réalisation. Ceci nécessite de la délicatesse pour maintenir l'équilibre subtil entre l'extrême niveau de détail nécessaire à l'aspect tactique, et l'ampleur d'une vision globale. Mais optimiser nécessite de travailler sur une vision idéalisée, mathématisée du monde. Mesurer, c'est projeter une réalité complexe, parfois mal définie, sur l'axe unidimensionnel des certitudes. De la même façon, modéliser, c'est simplifier. C'est ignorer ou déformer certains aspects de la réalité. Introduction Scheduling consists in planning the realization of a project, defined as a set of tasks, by setting execution dates for each of the tasks and allocating scarce resources to the tasks over time. In this introductory chapter, we begin by briefly recalling the history of scheduling and its emergence as a topic of major interest in Computer Science in Section 1.1. In Section 1.2 we underline the essential contribution of mathematical optimization to scheduling. We conclude the chapter by presenting in Section 1.3 our main contributions and an outline of the thesis. The birth of Scheduling within Computer Science Scheduling as a discipline first appeared in the field of work management in the USA during WW1 with the invention of the Gantt chart. Its creator, Henry Gantt, was an American engineer and management consultant who worked with Frederick Taylor on the so-called scientific management. This method was enhanced with the PERT tool in 1955, which was developed as a collaboration between the US Navy, Lockheed and Booz Allen Hamilton. Similar tools were developed independently around the same time, for example CPM (Critical Path Method) at DuPont and MPM (Méthode des Potentiels Métra) by Bernard Roy in France, both in 1958. These tools proved invaluable in managing complex projects of the time, such as the Apollo program. They are still widely used today in project planning thanks to their simplicity, which also imposes severe limitations on the accuracy of the models which can be expressed with these tools. In modern terms, these techniques embed a precedence network only, with no resource constraints. Through operations re-14 CHAPTER 1. INTRODUCTION search think tanks and academic exchanges, many prominent mathematicians and early computer scientists came in contact with these questions. At the same time in the early 1960s, with the advent of parallel computers and time sharing on mainframes, scheduling jobs on computers became a topic of major interest for operating systems designers, and the problem emerged as its own field of research in Computer Science. This is all the more the case since scheduling, perhaps coincidentally, involves very fundamental objects in Computer Science, and was the source of many examples in the early development of graph theory, complexity theory and approximation theory. The widespread, strategic applications of scheduling today justify an everincreased research effort. To mention a few application areas, scheduling is used in organizing the production steps in workshops, both manned and automated. It is used in building rosters and timetables in complex organizations, such as hospitals and airlines. In scientific computing, it can be used to schedule computations on clusters of heterogenous machines [ABED + 15]. Recently, in a beautiful application, the sequence of scientific operations performed by the Philae lander on comet 67P/Churyumov-Gerasimenko (see Figure 1.1) was planned using constraint-based scheduling in a context of severely limited energetic resources, processing power and communication bandwidth [START_REF] Simonin | Scheduling Scientific Experiments on the Rosetta/Philae Mission[END_REF]. Close to half of the optimization problems submitted to IBM optimization consultants have a major scheduling element. The Optimization approach to Scheduling Constraint-based scheduling finds its origins in three scientific domains: Operations Research, Constraint Programming and Mathematical Optimization. We describe the respective contributions of these fields to the state of the art of scheduling engines, and give an overview of the modeling principles when using a modern, Model & Run, solver. Operations Research Starting in the early 1960s, several specific scheduling problems have been extensively studied in the Operations Research and the Algorithms communities. There were originally two distinct lines of research, each with their own technical results and applications: one of them is machine scheduling, which studies problems such as parallel machine scheduling and shop scheduling (these notions will be defined later in section 2.1), for example to plan tasks on the different machines in a factory, and the other is project management, where the RCPSP (defined later in section 3.2) is used to plan the tasks of a project in a large organization. The models which were produced are simple, restricted in expressivity, and each of them relies on an ad-hoc algorithm. In spite of the success of this approach when it is focused on a specific, simplified problem, the solutions are not robust to the addition of most side constraints. These models must reflect the data structures of the underlying algorithms, and thus often lack expressiveness. Not taking into account side constraints results in the production of simplified solutions, which are sometimes difficult to translate back to the original problem, or can be very far from optimal solutions to the complete problem. This limitation significantly restricts the applicability of this approach to practical problems. Moreover, the solving algorithms lack genericity and are not robust to small changes of the problem or to sideconstraints, which requires the development of a brand new algorithm for each particular scheduling problem. Nevertheless, this focus on pure problems motivated a lot of research on complexity results and polynomial-time approximation algorithms for different scheduling problems. This explains why this line of research has been and still is particularly prolific. It is noteworthy that some of these algorithms have been made generic and are heavily used in more recent technologies, notably in propagators and as primal heuristics. CHAPTER 1. INTRODUCTION Another line of research in this community is the use of (mixed-integer) linear programming in conjunction with metaheuristics to solve scheduling problems. In spite of a number of major successes, this technology is often ill-suited since most resource constraints (such as the cumulative constraint) do not fit this model well. On the one hand, these constraints are difficult to linearize (they require at least a quadratic number of variables and have weak linear relaxations) and result in MIP formulations which do not scale. On the other hand, they are slow to evaluate and break the metaheuristics' performance. In spite of the genericity of mixed-integer linear programming and the fact that it is the best choice to tackle many different types of optimization problems, it is often not the technology of choice for scheduling problems. For more information about the history of scheduling, we refer the reader to [START_REF] Herrmann | A History of Production Scheduling[END_REF]. Constraint Programming A different approach was initiated in the early 1990s with the first generation of constraint-programming libraries for scheduling, such as [START_REF] Aggoun | Extending CHIP in Order to Solve Complex Scheduling and Placement Problems[END_REF][START_REF] Wim | Time and Resource Constrained Scheduling: a Constraint Satisfaction Approach[END_REF] and Ilog Scheduler. Constraint programming finds its origins in the 1960s in AI and in Human Computer Interaction research, and was then used from the 1970s in combinatorial optimization, notably with Prolog. These libraries provided a generic language and propagators for each constraint, but the branching strategy was to be written by the final user, and had to be finely tuned to the model. Even though the distinction between model and search has always been clear in theory in CP, the practical necessity and difficulty of writing a search algorithm meant that the separation between the model and the ad-hoc search was sometimes blurred since information about the problem was embedded within the search algorithm. Moreover, most of these languages and solvers were not tuned for scheduling problems, which was treated like any other combinatorial optimization problem. It should be noted that, independently of scheduling, constraint programming is finding even more widespread applications today in different fields of computer science, for example with a recent application to cryptanalysis [START_REF] Gerault | Constraint Programming Models for Chosen Key Differential Cryptanalysis[END_REF]. Mathematical Optimization A further step was taken in 2007 with the release of CP Optimizer 2.0, which incorporates many ideas from the Optimization community, and offers for the first time a model & run platform for scheduling problems. The principle of model & run is to enable the user to solve the problem by modeling it in an appropriate language, without having to write algorithmic code, and by leaving the solving process to the optimization engine. This is for example what is done when using a MIP solver. We explain model & run further in the following subsection. This was achieved for scheduling through the introduction of two new key technologies. First, a generic modeling language for scheduling problems was designed. This language is based on the notion of optional time intervals. These time intervals are used to represent activities of the schedule, as well as other temporal entities: union of activities, availability period of resources, alternative tasks, etc. This language can also natively represent several types of complex resources (temporal relations between tasks, cumulative resources, state resources, etc), which makes it expressive enough to model most common scheduling problems, yet concise. Moreover, this language is algebraic: all quantities can be expressions of decision variables. Thanks to the "unreasonable efficiency of mathematics", this enables very powerful combinations of constraints, which enables to expose less than ten types of constraints to the user, as opposed to classical constraint programming systems, which have hundreds of types of constraints. This language will be explained in more detail in Subsection 2.4.1. The second key technology is the introduction of a generic, automatic search [START_REF] Laborie | Self-Adapting Large Neighborhood Search: Application to Single-Mode Scheduling Problems[END_REF], which had been a feature of mixed-integer linear programming solvers since the end of the 1980s. This automatic search relies on the principles of scheduling, since it is assumed that the problems which will be solved with CP Optimizer are dominated by their temporal structure. Moreover, the search engine exploits the semantics obtained from modeling in a high-level scheduling language, for example to distinguish between temporal and resource variables. The solving engine is based on many technologies in addition to constraint programming, such as local search, linear programming, scheduling heuristics, etc. Its functioning is explained in Section 2.5. A lot of progress in the field of scheduling is due to the OR approach, but optimization and constraint programming made these solutions generic: constraint programming can indeed exploit OR algorithms while staying generic. This is why we can think of constraint programming not as a solving technology, but as an algorithmic integration platform, enabling different algorithms to collaborate, each of them working on partial problems. As far as research in optimization is concerned, scheduling is today a major source of benchmarks to validate new computational techniques, with applications to Operations Research, Artificial Intelligence, AI planning, etc. Model & Run We must say a word here about the modeling principles when using such a Model & Run solver. The progress of Operations Research has resulted in the development of tools which are intended to be simple enough to be used directly by the operational users, such as production engineers. This is due in a large part to the Model & Run approach in constraint programming, in contrary to previous approaches where part of the model was not written explicitely but actually concealed in the solving algorithm. The combinatorial optimization paradigm is particularly fitting to that end. It is based upon an explicit model: clearly identified decision variables, constraints between these variables, and an objective function which must be optimized in accordance with the constraints. This paradigm helps clarify the problem representation. This has several benefits for the user. By separating the modelling and solving processes, the user can focus on the model instead of writing a new search algorithm for each problem as is customary in the Operations Research approach. He thus benefits from the same flexibility as already offered by Mixed-Integer Programming solvers. As we will see below, tuning is done by modifying the model only, instead of having to make changes deep within the search algorithm. To a certain extent, the model is even agnostic to the solving technology. This is the initial idea behind the Optimization Programming Language, which enables the same model to be solved either by a MIP engine or by a CP engine. Nevertheless, using a high-level modeling language, as is possible with CP, has two main advantages. First, it allows to maintain some semantics on the problem, contrary to what can be expressed with a SAT or MIP model, for instance. In the case of constraint-based scheduling, this makes it possible to maintain a distinction between variables representing a temporal quantity and variables representing something else, such as a resource. This semantics can then be exploited by the solving engine through appropriate search techniques. Second, this high-level language also helps the user structure the model by providing her with a language which is designed specifically for scheduling problems instead of letting her write everything in the form of, for example, linear inequalities as is done when using a MIP engine. This allows for a much more compact and natural representation (see Subsection 2.4.1). We now go into the detail of the different questions to keep in mind when writing an optimization model. As a reminder, a model is generally defined as being an instance of a theory (m) which satisfactorily reproduces the characteristics of the world (ϕ). Modeling is thus an art as much as a science, since it requires skillful tradeoffs between the expressiveness of the theory and fidelity to 1.2. THE OPTIMIZATION APPROACH TO SCHEDULING the world characteristics. All models are simplifications and idealizations of the real phenomena they describe and, depending on the purpose, some truths on the world, certain aspects, and effects of different magnitudes are emphasized, approximated, or neglected. The 20 th century statistician George Box famously wrote on this subject: "All models are wrong, but some are useful". The Model & Run approach, with its explicit statement of the model, helps to control these gaps. Having an explicit representation of the problem in the form of a model is of additional interest. Indeed, in contrary to the implicit assumption which we sometimes encounter in the community, that there exists a canonical model which perfectly describes the real-world problem, we believe that there are actually three steps to consider if one considers the question of solving an industrial optimization problem in its globality. After choosing an appropriate modeling language for the problem, we must model the problem in a modeling language, and pass it to an optimization engine. The engine will then find a solution to this model. The final step consists in implementing this solution in the real world, for instance in the full industrial context. A common example of the subtlety of implementing the solution to a model to the real world appears in manpower scheduling, where it is easy to make the mistake of overprescribing the solution. In most manpower scheduling models one will naturally write indeed, each shift will be assigned to a particular worker. Actually, it is often unimportant to the management if two workers prefer to swap their shifts as long as they both have the required skills. In this case, any good implementation of the optimization results into the real world should be flexible enough to provide the possibility for employees to exchange their slots. Thus, the model is not just an encoding of the problem data, but a simplified and modified version. This approach to solving an optimization problem is summarized on Figure 1.2. In this framework, there are three sources of gaps between the world and the solution found by the solver to the model. They reside in the modeling limitations of the language, in the ability of the solving engine to work efficiently with the model provided, and in the gap between the best solution found by the solver and the mathematically optimal solution to the model (optimality gap). Let us detail each of them now. The first source of gap comes from the lack of expressivity of the modeling language. For example, the available constraints may not be an exact match for the real constraints. Non-linearities might have to be linearized, if using a MIP solver. The need to keep the model compact can also be a limit to its precision. The second source of gap, between the language and the solving engine, comes from the need to balance an expressive language with the possibilities and needs of the solver. For instance, there may be different possible formulations with differing properties and efficiency. We might also have to explicitely model (or not, depending on the situation), redundant constraints or symmetry breaking. Of course, the higher the genericity of the modeling language and solver, the closer one can be to reality when modeling. So an engine design goal is to try as much as possible not to limit expressivity because of solving engine limitations, so that the user can deal only with the model instead of fine-tuning the solver to her particular problem. Finally, the last source of gap is the optimality gap, that is the gap between the best solution found by the solving engine in a limited time, and the optimal solution to the problem. Since we are trying to solve NP-hard problems with limited resource, we will always encounter problems which we cannot solve to optimality in a given time (no free lunch theorem). In practice, it is possible to narrow this gap somewhat by giving the solver more time, on a more powerful computer, with more finely tuned engine parameters, and mostly with the continuous improvement of solving engines. Contributions of this thesis We focus in this thesis on offline, deterministic, non-preemptive scheduling problems, solved within a constraint programming framework. Offline means that the instance data is entirely known over the whole time horizon before we start making decisions, contrary to online problems where decisions have to be made without fully knowing what comes ahead. Deterministic means that the instance data is supposed to be perfectly accurate, in contrast to problems involving uncertain data, such as different possible scenarios or stochastic data 1.3. CONTRIBUTIONS OF THIS THESIS (see Subsection 2.4.2). Finally, non-preemptive means that the activities have to be processed without being interrupted once they have started. In other words, the end time of an activity is always equal to its start time plus its processing time. In this context, our topic of interest is the cumulative constraint, in relation with temporal constraints. Our two main contributions are the Cumulative Strengthening and the Fast Energy Reasoning. The Cumulative Strengthening is a way of computing efficient reformulations of the cumulative contraint which strengthen many existing propagations and lower bounds. The Fast Energy Reasoning is a new algorithm for the Energy Reasoning propagation for the cumulative constraint, bringing its algorithmic complexity down from O(n 3 ) to O(n 2 log n). These two contributions rely on duality results and geometric insights. They were both implemented on top of CP Optimizer and evaluated for their practical impact. These contributions are related to the evolution of the field and of solvers technology as outlined in the previous section. Indeed, they reinforce the automatic search even in difficult cases, and the possibility of using the solver as a black box. The reminder of this thesis is organized as follows. In Chapter 2, we review the basic principles of constraint-based scheduling. Since our work was implemented with CP Optimizer, we also present its basic design principles and the basics of automatic search. In Chapter 3, we introduce the cumulative constraint, study some of its properties, give examples of applications in industrial models, and present state of the art propagations and ways of computing lower bounds. Then we introduce the Cumulative Strengthening in Chapter 4. This chapter was already published in a shorter form as [BB14] and as a journal version in [START_REF] Baptiste | Redundant cumulative constraints to compute preemptive bounds[END_REF]. In Chapter 5 we present the Fast Energy Reasoning algorithm. This chapter was already published in a much shorter version as [START_REF] Bonifas | A O (n^2 log (n)) propagation for the Energy Reasoning[END_REF]. Finally, Chapter 6 is a conclusion on this work and an outlook on possible extensions. Chapter 2 Scheduling with constraint programming This chapter introduces the use of constraint programming to model and solve scheduling problems, which is the framework in which our work takes place. Section 2.1 presents scheduling techniques which found their origin in Operations Research. In Section 2.2, we briefly mention the different generic optimization tools which have been applied to scheduling. Section 2.3 is a brief review of constraint programming, and Section 2.4 focuses on its application to scheduling. Finally, the design principles of CP Optimizer and of the automatic search are exposed in Section 2.5. Machine scheduling Most of the early research on scheduling under resource constraints, in the Operations Research community, was done in the context of machine scheduling, where resources correspond to machines. This field is prolific and a large part of the research on scheduling is still performed in this context. We present the most common problems in this field in Subsection 2.1.1, briefly reference general extensions of these problems and show how they are classified to form a theory of machine scheduling in Subsection 2.1.2, and present a selection of algorithms for this field in Subsection 2.1.3. These problems are all special cases of the RCPSP, defined below in Section 3.2, and the constraint-based scheduling approach subsumes these techniques, but they form the algorithmic origin of a part of a constraint-based scheduling engine. 24 CHAPTER 2. SCHEDULING WITH CONSTRAINT PROGRAMMING Machine and shop scheduling problems In machine scheduling problems, we only deal with specific resources, namely machines, which correspond to actual machines in a workshop or factory. The problem consists of n jobs J 1 , . . . , J n which must be scheduled on the machines. These jobs are non-preemptive, meaning that their processing cannot be paused or interrupted once started, the problem is deterministic, meaning that we assume that there is no uncertainty on the problem values, and offline, which means that all information is available before solving the problem. In the simplest cases, the single machine problems, we have only one machine. This machine can only process one job at a time, we must therefore find a total order on the jobs. In parallel machine problems, we now have m machines M 1 , . . . , M m available, which can each process one job at a time. The machines are said to be identical if the jobs can be processed on every machine and the processing time is the same on each machine. They are said to be unrelated if the processing time depends on the machine on which a job is processed. Certain jobs can also be performed only on certain machines. In their most general form, shop scheduling problems consist of m machines M 1 , . . . , M m , and each job J i consists of n(i) operations O i,1 , . . . , O i,n(i) . Each operation O i,j must be processed on a specific machine µ i,j , two operations of the same job can not be processed simultaneously, and a machine can process a single operation at a time. In this most general form, this problem is called open-shop. We mention two important special cases: the job-shop, where the operations of a job are constrained by a chain of precedence constraints, giving a total order on the processing dates of the operations of a job and the flowshop, which is a special job-shop where each job consists of the same operations, which must be performed in the same order for each job. For more information about these models, we refer the reader to Section 1.2 of [START_REF] Brucker | Complex Scheduling[END_REF] and Part I of [START_REF] Michael | Scheduling[END_REF]. Complexity classes The basic problems we defined in the previous subsection accept many variants in terms of additional constraints on the tasks (release dates, due dates and precedences), machines capabilities, objective functions (makespan, lateness, throughput, machine cost, etc.), and an algorithm developed for one scheduling problem P can often be applied to another problem Q, in particular to all special cases of P. When this is the case, we say that Q reduces to P. This provides a hierarchy of problems, solving techniques and complexity results. Formalizing MACHINE SCHEDULING 25 this hierarchy provides the beginning of a theory of scheduling complexity. The most common classification in this context is that of Graham, also called α | β | γ. α corresponds to the machine environment, that is the resources: single machine, several machines which can be identical, uniform, or unrelated, or shop problem: flow shop F, job shop J or open shop O. β represents the job characteristics: precedences between the tasks, release dates r i , deadlines d i and durations p i . γ is the optimality criterion or objective function. Here, C i denotes the completion time of job i, and T i is its tardiness: given a desirable due date δ i for each job i, T i = max(0, C iδ i ) is the lateness of this job with respect to the due date. Among the most common optimality criteria are the makespan max i C i , the total weighted flow time or sum of weighted completion times ∑ i w i C i , the maximum tardiness max i T i and the total weighted tardiness ∑ i w i T i . This classification is restricted to the class of scheduling problems mentioned above (machine and shop scheduling), and most problems considered in this thesis do not belong to the α | β | γ classification. Nevertheless, the study of this field resulted in finding specific algorithms, as well as approximation schemes. We refer the reader to the elegant and exhaustive Scheduling Zoo website [?] for more information on this subfield. Basic algorithms We present here a small selection of classical results, which have been inspirational for further developments in cumulative scheduling. We refer the reader to [START_REF] Michael | Scheduling[END_REF] for more information on this subfield of scheduling. • The first of these results concerns minimizing the sum of completion times of tasks scheduled on one machine, or 1 | | ∑ i C i . This objective is also called the average flow time. In this case, the Shortest Processing Time first (SPT) rule results in an optimal solution: tasks should be run in the order of increasing durations p i . A weighted version of this result exists: if each task has a weight w i and the objective is to minimize ∑ i w i C i , then the tasks should be run in the order of increasing p i w i values. • In the case of 1 | | L max , that is when the tasks are again processed on one machine, have a due date d i , and the objective is to minimize the maximum lateness max i (C i -d i ) where the C i are the actual completion dates, the optimal solution is obtained by using the Earliest Due Date (EDD) CHAPTER 2. SCHEDULING WITH CONSTRAINT PROGRAMMING rule, also called Jackson's rule. The tasks should be run in the order of increasing due dates. • Finally, we present the Pm | prmp | C max problem, that is minimizing the makespan of a set of jobs which can be executed on m identical parallel machines now, and can be preempted (meaning that their execution can be interrupted before it is complete and restarted later, possibly on a different machine). In this case, McNaughton's algorithm (Theorem 3. This schedule is feasible. Indeed, part of a task may appear at the end of the schedule for machine i and at the beginning of the schedule for machine i + 1. Since no task is longer than C * max and preemptions are allowed, this gives a valid schedule. And since this schedule has length C * max which is a lower bound, it is optimal. Combinatorial optimization A way to generalize the approaches seen above and find more generic algorithms to solve scheduling problems is to express scheduling problems in a combinatorial optimization framework. In their most general form, combinatorial optimization algorithms optimize a function c : F → R, defined on a finite set F of feasible solutions. The different techniques we will present apply to different special cases of this general definition, depending on the structure of F . As we will see in the following section, Constraint Programming gives a framework to call on these different techniques (and others!) to solve subproblems of the main problem and integrate the results thus obtained. The following techniques have all been used successfully within a CP framework to solve scheduling problems. Linear Programming Linear programs are expressed as the problem of optimizing (maximizing or minimizing) a linear function defined over a set of continuous (floating-point, typically) variables, which must satisfy a set of linear inequalities over these variables: max n ∑ j=1 c j x j s.t. ∀i ∈ [1, m] n ∑ j=1 a ij x j ≤ b i ∀j ∈ [1, n] x j ≥ 0 There exist different families of algorithms to solve linear programs, the most common among them being simplex algorithms and interior points methods. Even though no one has yet designed a simplex algorithm of subexponential complexity, interior points methods have a polynomial time complexity so the problem of linear programming itself is polynomial. Without going into the details here, we must say that there exists a very rich geometrical theory about the structures linear programming operates on. An active research topic in linear programming consists in the design of simplex algorithms with a provably polynomial complexity. Moreover, the paradigm of linear programming is very broadly used in optimization and operations research, due to its versatility. Indeed, many industrial problems have a natural representation under the form of a linear program, notably problems of assignment of resources to productions: an interpretation of a linear program consists in seeing the variables as production quantities of different items, while the constraints represent limits on the available resources. This versatility has generated an intense industrial interest since the end of the 1940s into the applications and the effective computer solving of linear programs. Very efficient codes are available today, for example CPLEX which is developed by IBM Ilog along with CP Optimizer. A technique which is very commonly used in conjunction with linear programming, notably when we have to represent combinations, patterns, or possible subsets of a certain set, is delayed column generation. Indeed, one of the drawbacks of linear programming is that many problems of interest have a superpolynomial (typically exponential) number of variables in their representation as a linear program, even though only a small number of these variables (no more than the number of constraints when the program is written in standard form, according to duality theory) will have a nonzero value. The principle CHAPTER 2. SCHEDULING WITH CONSTRAINT PROGRAMMING of delayed column generation to solve these very large problems is to consider only a subset of the variables, to solve this partial problem (called the master) and then to check with an adhoc algorithm, depending on the problem in question (called the slave), if setting one of the missing variables to a nonzero value could result in an improved solution. If this is the case, we add this variable (column) to the master problem and repeat the process. In practice, only a small number of variables will be added to the master problem. In spite of the additional complexity of solving many different versions of the master problem and having to solve the slave problems, the full procedure can typically be several orders of magnitude faster than solving the main problem directly (if it fits into memory at all)! Mixed Integer Programming Mixed Integer Linear Programming (typically called MIP) is similar to linear programming, with the difference that a subset J of the variables can only take discrete values (in general Boolean values). max n ∑ j=1 c j x j s.t. ∀i ∈ [1, m] n ∑ j=1 a ij x j ≤ b i ∀j ∈ [1, n] x j ≥ 0 ∀j ∈ J x j ∈ N The main idea behind solving mixed integer linear programs is to use a branch-and-bound technique by recursively splitting the domain into smaller subdomains and solving the continuous relaxation of the problem, i.e. the problem without the constraint that some of the variables can only take integer values (which is thus a linear program). If the optimal solution x * to the linear relaxation is such that all components in J have integer values, then we have an optimal mixed integer solution. Otherwise, a component x * i with i ∈ J of x * is fractional and two subproblems can be generated, respectively with the additional constraints x i ≥ x * i + 1 and x i ≤ x * i , and we can recurse. The optimal value of the continuous relaxation gives a primal bound on the optimal value of a mixed integer solution. In addition, so-called cuts can be generated automatically. They are additional linear inequalities which are generated in such a way that the integer solutions are still feasible, but not necessarily the fractional solutions. The cuts reduce the size of the search space and increase the likelihood that the continuous relaxation will find an integer solution. There has also been a tremendous amount of engineering into making MIP solvers generic and efficient. Moreover, with the addition of integer variables to the linear programming model, all types of combinatorial problems can be modeled in the MIP framework which makes it extremely versatile. Among its drawbacks are the facts that no semantics is kept on the model, and the fact that almost all scheduling problems have MIP representations whose size is exponential in the problem size, making this technology inefficient when compared to constraint programming. We discuss MIP representations of scheduling problems later in Section 3.5. SAT A Boolean formula is defined over Boolean variables, which can only take the values True or False. A literal l is a variable a or its negation ¬a (the negation of a is True whenever a is False and vice versa). A clause c i = (l i,1 ∨ . . . ∨ l i,n i ) is a disjunction of literals (True if any of the literals is True). Finally, a CNF (conjonctive normal form) formula is a conjunction c 1 ∧ . . . ∧ c m of clauses (True if all the literals are True). Of course, the same variable can appear in several literals in one formula. The SAT problem consists in determining whether there exists an assignment of values to the variables of a CNF formula such that that formula is True. This problem is the prototypical NP-complete problem (Cook's theorem, 1971) and it is very generic: many combinatorial optimization problems can be naturally modeled as a SAT problem. For this reason, very efficient algorithms have been developped: first with DPLL-based solvers which perform a recursive search over the possible literal values. These solvers rely heavily on tailored data structures which enable very efficient unit clause propagation (if all literals but one are False in one clause, then set this literal to True everywhere in the formula). A new generation of solvers appeared in the last decade, called clause learning solvers, which make use of the large quantity of RAM available in modern machines to generate a new clause to the formula each time they encounter a failure, so as not to explore this part of the search space again. A lot of engineering has been done in this field, motivated by SAT solver competitions, and very efficient programs are available. Fruitful ideas have also been exchanged with the CP community. CHAPTER 2. SCHEDULING WITH CONSTRAINT PROGRAMMING Dynamic Programming This technique can be used for problems which satisfy the optimal substructure property (sometimes also called Bellman's optimality principle), which states that an optimal solution can be recursively constructed from optimal solutions to subproblems of the same type. When this property applies, we can memorize the solutions to the subproblems in order not to recompute them every time they appear in the resolution of a more general subproblem. A common example in scheduling is the Weighted Interval Scheduling Problem: given a set J of optional jobs j, each with a start time s j , an end time e j and a weight w j , the goal is to compute the maximum sum f (J) of the weights of non-overlapping jobs. In other words, we have only one machine and we must choose a set of compatible tasks which will yield the highest possible profit. We can see that if we assume that job i is executed in an optimal solution, then we can solve the problem separately over jobs which end before s i and over jobs which start after e i , and sum the results to obtain the overall optimal solution, so for any K ⊂ J, f (K) = max i∈K w i + f ( j ∈ K : e j ≤ s i ) + f ( j ∈ K : s j ≥ e i ) and the computation of f (K) is reduced to the computation of f for nonoverlapping subsets of K, making this problem suitable for the dynamic programming framework. Heuristics When complete methods are ineffective on a class of problems, we have to rely on heuristics algorithms. These algorithms make different approximations on the structure of the set of solutions in the hope of finding good enough feasible solutions to very difficult problems, but the solutions obtained and are not guaranteed to be optimal. We generally use the word heuristics for algorithms which are tailored to a very particular problem and metaheuristics for techniques which are more general. There are two general ways of using (meta)heuristics in combinatorial optimization and in scheduling in particular. The first one is local search in which one starts from a feasible solution, possibly of poor quality and which can be obtained via various means, and gradually transform it into a good one via a series of small, incremental steps, from feasible solution to slightly better feasible 2.3. CONSTRAINT PROGRAMMING solution. This approach works well if the space of feasible solutions is densily connected for the heuristic algorithm in use. The second way of using heuristics is to make use of an indirect representation of the problem (which can be a simplified version of the problem, or a very different problem, but which can be effectively transformed into a full solution) and to use the metaheuristics on the indirect representation. If the indirect representation effectively captures the difficult part of the problem (sometimes called problem kernel) and the transformation into a full solution is easy, this can be a very powerful technique. See also Subsections 2.5.1 and 2.5.3.3. Among the most common heuristics and metaheuristics one can find are genetic algorithms, ant colony optimization, hill climbing, simulated annealing, and tabu search. All of them have been used successfully for scheduling. Constraint programming Constraint programming is based on the declaration of a combinatorial problem in the form of decision variables, linked together by constraints, and an objective function of these variables, to be optimized. The basic technique for solving these problems is a tree search of the search space by separation-evaluation, associated at each node of the tree with a stage of constraint propagation. Propagation consists in checking, with the aid of specific algorithms, necessary conditions on the compatibility of the potential values that the variables can take (domains), in order to eliminate some of them and thus reduce the size of the space of search to explore downstream of the current node. As such, constraint programming can be seen as a language, since its directory of decision variables and constraints offers a modeling language, but also, on the solving side, as an algorithms integration framework allowing the problem to be broken down into a set of very orthogonal polynomial subproblems, through search heuristics and propagation algorithms, each working on a well-structured subset of the problem but pooling their results through variables domains (see below) and indirect representations (see Subsection 2.5.1). For example, a typical scheduling problem can be decomposed into a network of precedence constraints, on which we can reason in polynomial time, and into resource constraints on which we can reason using polynomial complexity propagation algorithms (see Section 3.4). Formally, a constraint programming problem consists of a set of decision variables, a set of constraints over these variables, and an objective function. Let us define these terms now. CHAPTER 2. SCHEDULING WITH CONSTRAINT PROGRAMMING The decision variables are the variables which must be determined by the solving procedure. They correspond to the operational decisions that have to be taken. Three types of decision variables are commonly encountered in constraint programming: integer, floating-point and time intervals (in scheduling engines). Two important cases of integer variables are variables representing boolean decisions or the choice of an element in a finite set. In constraint programming, decision variables take their value in a domain, which is a set of potential values. The domains will be reduced as the search goes along and potential values can be eliminated. Constraints are compatibility conditions between the variables. They are typically either binary (they involve two variables) or global (they involve multiple variables, which gives more semantic information on the problem and enables remove inconsistencies more efficiently than by just using binary constraints). In practice, constraints are enforced by propagation algorithms, which remove incompatible values from the domains of variables. Propagations are the keystone of constraint programming. A constraint is said to have a support if there is an assignment to each variable of a value of its domain, such that the constraint is satisfied. If a constraint has no support, the problem is infeasible. We call this a failure. If a value does no belong to a support for any constraint, we say that it is inconsistent and it can be removed from the variable domain. Propagation therefore consists in eliminating values from variable domains that are inconsistent with a constraint. For example, if task B must start after task A ends and task A can not finish before time t = 7, then we know that task B must start at or after time t = 7 and can eliminate smaller values from the domain of its earliest start time. We use different propagation algorithms, which work on different constraints or groups of constraints and check different necessary conditions. Usually we execute them repeatedly until none of them is capable of reducing the domain anymore (we say we have reached the fixed point of propagation), or one of the domains becomes empty ( in which case the problem has no solution). The result of propagation is therefore an equivalent problem, in the sense that it has the same solutions, but more consistent (smaller search space). We distinguish several types of consistency. For example, we say that we have achieved arc consistency if all unsupported values have been removed from all domains. Arc consistency can be very costly to maintain, so we often resort to a weaker form of consistency such as bound consistency. In bound consistency, the domains are intervals, and we only make sure that the minimum and maximum of this domain is not unsupported. However in scheduling the constraints are so complex that even the bound consistency is often too expensive to maintain. We therefore only verify particular cases of BC, differ-ent propagation algorithms for the same constraints having different ratios of computing time versus propagation power and being used in different contexts. In practice, it is essential that propagations be incremental, that is, not all combinations of domains be re-studied from one node to the next, since branching decisions have a local impact. Incremental propagations only re-examine values that could have been impacted by the last branching decision. As for the search algorithm, the tree search starts at each node by calling the propagations. Depending on whether or not a conflict is detected during this step, three outcomes are possible: • If no conflict is detected and each variable has only one value in its domain, we found a feasible solution to the problem. We can compute the objective value for this solution and update the best solution found. • If no conflict is detected but there are variables whose domain still has several values, we call a branching heuristic to divide the search space in two, and call the search recursively on the left node and then on the right node. • If the propagation has detected a conflict (no value left in the domain of a variable), we backtrack, that is we return to the previous node, cancelling the last branching decision. If we are already at the root node and cannot backtrack, we have proven that the problem has no more solutions. Constraint-based scheduling consists in solving in solving scheduling problems using a large part of the conceptual framework of constraint programming, enriched with scheduling-specific notions. Constraint-based scheduling t est i lst i eet i let i p i CHAPTER 2. SCHEDULING WITH CONSTRAINT PROGRAMMING The contraint program thus possesses variables for the execution dates (start and end dates) of each task as well as for the resources use over time. It is crucial to use a scheduling-specific modeling language, instead of simply enriching an integer variables based constraint programming system with scheduling constraints. Indeed, the temporal structure, that is the distinction between the temporal and resource variables and constraints, is essential to the efficiency of the solving algorithms. It is also essential, both for the solver and for the person writing the model, to have a small, compact model. To this end, the modeling language is algebraic, which means that all appearing quantities are themselves decision variables. This enables the engine to understand part of the problem semantics and to exploit scheduling concepts beyond what could be done in pure constraint programming. Language overview Let us now give a very brief overview of the CP Optimizer constraint language. More information can be found in [START_REF] Laborie | Reasoning with Conditional Time-Intervals[END_REF][START_REF] Laborie | Reasoning with Conditional Time-Intervals. Part II: an Algebraical Model for Resources[END_REF]. The language is based on the notion of interval variables. These variables possess a presence status, whose domain contains the two values present and absent, as well as a start date and an end date. The domain of the start date ranges from the earliest start time to the latest start time, and the domain of the end date ranges from the earliest end time to the latest end time (see Figure 2.1). Decisions, made by propagation or during the search, will tend to reduce these ranges. The presence status is essential to facilitate modeling, for example when we have activities whose execution is optional, to model tasks which can be executed on alternative resources or in different modes, or through different processes. Integer variables are also available, if needed, but the scheduling language is expressive enough that they are generally not needed. Several families of constraints are available. Temporal constraints first, such as bounds on the start or end dates of intervals or more complex arithmetic expressions of those. Additionally, precedence constraints state that an interval can start only after another one ended, and optional delays between these two events can be specified. Alternative modes can be modeled using the Alternative constraint, where a master interval is synchronized (same presence status, start and end dates) as one of several slave intervals representing the different modes, the presence status of the other slave intervals being set to absent. A very useful constraint to model hierarchical processes is the Span, which states that a master interval should extend over all of time range covered by slave intervals. The disjunctive is a resource constraint: it models a unique resource used by several intervals, and forces them not to overlap. Since this yields a total order on the execution of these intervals, the sequence that we obtain can also be thought of as a temporal constraint, and constraints are available to manipulate it, such as Prev and Next (constraints on the intervals which should precede or follow another one). The two main other resource constraints are the cumulative, which we will see in great detail in Chapter 3 and is the focus of this thesis, and the state constraint, which is used to model situations where a machine can perform different operations when it is in different states, or industrial ovens where certain tasks can only be performed within a certain temperature range. An objective function can also be specified as a general function of the decision variables. This flexible design allows for very general objective functions, which can be a variant of the makespan or incorporate additional costs such as setups, lateness, as well as non-temporal costs such as resource allocation, non-execution, etc. Sometimes, the real objective is very difficult to express as a function to optimize, since the industrial objective is not to over-optimize every detail of the process at the expense of flexibility, but rather to smooth things out and minimize fits and starts. Modeling these objectives requires a great deal of expertise. Assumption of determinism Of course, not all scheduling problems conceivable can be modeled in this framework even if, in practice, a great breadth of problems fit [START_REF] Laborie | IBM ILOG CP Optimizer for Detailed Scheduling Illustrated on Three Problems[END_REF]. One important limitation to pay attention to is the assumption of determinism, which requires particular care. As first noticed by Fulkerson in [START_REF] Fulkerson | Expected Critical Path Lengths in PERT Networks[END_REF], in the case of uncertainties about the duration of certain tasks in a scheduling problem, using the expectation of the duration as a deterministic value will most of the time lead to a systematic underestimation of the total processing time for the whole project. This is in particular the case if the function f which gives the optimal value of the objective is a convex function of the processing times p. Indeed by Jensen's inequality, f (E(p 1 ), . . . , E(p n )) ≤ E( f (p 1 , ..., p n )) . Notably for us, the RCPSP (see later Section 3.2) falls into that category if the objective function is regular (monotonous with the start and end times of the tasks), since a solution which is feasible with a certain vector of processing times p is obviously still feasible if the processing time of a task is reduced. In practice, we resort in these cases to writing a robust model: different scenarios on the stochastic variables are sampled from the joint probability distribution and evaluated simultaneously, in the sense that the scheduling decisions CHAPTER 2. SCHEDULING WITH CONSTRAINT PROGRAMMING made must be compatible with all scenarios. The objective function also takes the different scenarios into account, for example optimizing for the worst-case. An interesting feature of constraint programming in the context of robust optimization is that we can use global constraints to join decisions taken for the different scenarios at a high-level. An example of such a constraint is the Isomorphism in CP Optimizer, which constrains two set of tasks to be executed in the same order. Automatic search in CP Optimizer CP Optimizer has a powerful automatic search algorithm. It presents several properties: it is generic (the same algorithm is used to solve all problems and does not require to be rewritten when the problem changes), complete (it is guaranteed that the optimal solution will be found or in other terms that all of the search space will eventually be traversed, even if this can take a very long time), deterministic (in spite of the use of randomness during the search and a parallel implementation, it is guaranteed that running the search several times on the same problem with the same parameters will yield the same solution), adaptative (its parameters are automatically adjusted during the solve to the particular instance characteristics) and robust (it is invariant with respect to a number of changes of the instance, for example a dilation of the time scale). This automatic search relies on several techniques on top of Constraint Programming. Among them are Linear Programming, local methods, scheduling heuristics, machine learning, etc. Reaching the objectives stated above requires several new ideas. First, as we have shown above in Section 2.4, we use a scheduling-specific language, notably through the concept of optional interval variables. We also make a strong use of the fact that we work specifically on scheduling problems to guide the search through chronological branching techniques. Second, indirect representations of the problem are used in many parts of the solving algorithm. We will detail these principles in the following subsections. Global variables and indirect representations When working on a difficult problem, it is often advantageous to change its representation, for example as an instance of a problem we can already solve. The main principle behind finding efficient representations is to focus on the parts of the problem that form the core of its difficulty. CP Optimizer makes a heavy use of global and indirect representations of the problem. Some are explicitely given in the model, other are computed or inferred from binary constraints or from deductions made during the search. These indirect representations are used both by the propagations and by the search algorithms, and actually everywhere a decision is taken. They help make the algorithm robust by considering more global structures when taking decisions than just the constraints independently from each other. These representations play a role similar to that of duality and cut generation in MIP engines. Some of these representations are analogous to global constraints, such as the temporal and logical networks, and could even have been expressed as such in the language instead of being aggregated from binary constraints. This choice was made out of convenience for the user. Propagation is performed both on the model variables and on these representations. A few examples of these indirect representations follow. The temporal network is a fundamental concept in the algorithmics of scheduling and comes from AI planning. The principle is to aggregate all precedence constraints in a single graph, which exhibits much more of the temporal structure of the problem than just considering the pairwise relations between intervals. This graph is dynamically updated using pairwise relations discovered during the search. The logical network plays a similar role and aggregates all the constraints between presence statuses of interval variables. It is beneficial to consider these relations globally since the problem of propagating all these statuses globally is equivalent to 2-SAT, for which efficient algorithms exist. These two networks are described in more details in [START_REF] Laborie | Reasoning with Conditional Time-Intervals[END_REF]. Many propagators on global constraints, such as the timetable, the edgefinder, etc, maintain incremental data structures along the search tree, which constitue global and relaxed representations of a part of the problem. Finally, some local search algorithms, such as the Genetic Algorithm, or list scheduling, rely on an indirect representation in the form of a simplified problem which can then be expanded into the full problem. Please see Subsection 2.5.3.3 for more information. Other examples of indirect representations will be shown in the rest of this chapter, such as the linear relaxation, POS in Large Neighborhood Search and partial enumeration of interval variables in FDS. Branching strategies In a tree search, branching strategies are the heuristics which choose the variable whose domain will be split, as well as the partition of that domain that will CHAPTER 2. SCHEDULING WITH CONSTRAINT PROGRAMMING be explored in the left and right branches. They are also called, in scheduling and depending on the context, dates fixing algorithms or completion strategies. They must be worked out in a strategic fashion, since a large fraction of the overall engine performance depends on their ability to drive the search quickly (that is, using mostly left branches, which are explored first) towards a possible solution. Thus, in scheduling, we use chronological branching strategies, which make use of the temporal nature of the problem. The main idea of these strategies is to select as branching variable that whose value will be chosen as small as possible (or as large as possible in the case of a reverse chronology). This combines well with the propagations since it densifies the space occupied locally, at the beginning of the schedule. Set Times The objective function f is said to be regular if for any two schedules S and S such that all dates in S are smaller or equal to the dates in S , f (S) ≤ f (S ). It intuitively means that all other things being equal, we should schedule activities as early as possible. Many common objectives, such as makespan, weighted flow time, maximum lateness, total weighted tardiness are regular but not earliness-tardiness, for example. When the problem has a regular objective, we can use the Set Times branching rule, which is a form of chronological backtracking and makes use of the intuition that tasks should try to be scheduled as early as possible. To the extent of our knowledge, the Set Times rule was first published in [START_REF] Le Pape | Time-versus-Capacity Compromises in Project Scheduling[END_REF]. Other explanations can be found in [START_REF] Godard | Randomized Large Neighborhood Search for Cumulative Scheduling[END_REF][START_REF] Laborie | Temporal Linear Relaxation in IBM ILOG CP Optimizer[END_REF], but many other variants exist. The basic principle of Set Times is to try in the left branch to set the activity with smallest earliest starting time to start at its est. If a failure occurs, we say that the activity is postponed, and we know that we cannot start this activity at its est. We will not select this task anymore until its est has been changed further down in the search tree. Temporal Linear Relaxation When the objective is irregular, such as with earliness costs, variable activities duration or delays between activities, or when it includes non-temporal costs such as setup costs or non-execution costs, scheduling can be extremely difficult, even in the absence of resource constraints. The chronological branching heuristics such as Set Times are poorly suited and do not perform well in this setting. In this case, CP Optimizer uses the solution to a linear programming relaxation of this instance as an oracle, through a branching heuristics, as explained in more details in [START_REF] Laborie | Temporal Linear Relaxation in IBM ILOG CP Optimizer[END_REF]. The idea is to combine the ability of linear programming to solve complex allocation problems, as long as they are linear, with the ability of CP to solve minute details of activities placement and resource consumption. Resources do not need to have a very tight linear relaxation, and since the Temporal Linear Relaxation is used in the context of Large Neighborhood Search, we will make use of the POS (see below in Subsection 2.5.3.2) as a strong and already available linearization of the resource constraints. The Temporal Linear Relaxation is based on a model whose decisions variables are continous: x i ∈ [0, 1] represents the presence value of interval i, s i its start date and e i its end date. Several constraints are already linear, such as logical constraints (logical relations between the presence statuses of several intervals) and precedence constraints. Moreover, since the TLR is tightly integrated within the Large Neighborhood Search, most of the complex resource constraints have already been linearized into precedence constraints in the Partial Order Schedule. Arithmetical expressions are relaxed into convex piecewise linear expressions. Some additional inequalities are added to the linear formulation for additional strength, such as for alternative constraints, or the cumulative constraint for which a global energy bound is computed. We refer the reader to the paper on TLR for more details on these. Finally, the branching heuristics based on the Temporal Linear Relaxation works in a similar fashion to the Set Times heuristics seen above, except that it uses the values computed for s i and e i as references instead of est i . For the presence value of interval i, the probability of setting it present in the left branch is equal to x i . The TLR is also used to compute a global lower bound on the problem, and the reduced costs are used to reduce the variable domains when possible. Using these principles, the Temporal Linear Relaxation gives another example of the possibility to integrate algorithms of a different nature thanks to Constraint Programming. Search strategies As we will see below, for example with the RCPSP in Subsection 3.2.4, scheduling problems are NP-hard in general, so the goal of a commercial solver is to look for as good as possible solutions in a limited computational time. To achieve this goal, the search follows these general principles: • We should avoid spending time exploring parts of the search space where no solution lies. In constraint programming, this is done through constraint propagation (see below in Subsection 2.5.4) and the use of dominance rules when branching (as seen above in Subsection 2.5.2). • When a good solution is found, the search is intensified around it, since this solution probably has properties that are desirable for other good solutions. • There should be no large regions of the search space left unexplored, different techniques are used to diversify the search. • The horizon can be very large in scheduling, so enumerating the temporal variables should be avoided. Reasonings should be performed on events instead. • All searches are regularly restarted. Restart is a very fundamental concept in search space exploration. The goal is to avoid getting trapped in local optima by regularly restarting the search from scratch, hoping to explore a different region of the search space. Some information can be kept from previous searches, such as no-goods and learnt parameters on heuristics and properties of the search tree. More information on restarts can be found in [LSZ93, KHR + 02]. • The search is self-adapting. As many internal parameters as possible are learnt dynamically during the search. • Since CP Optimizer is intended to be used in an industrial setting with limited time available, the emphasis is set on finding good solutions (improving the primal bound). Time is spent improving the dual bound only when it is assumed that we already know the optimal solution or an exhaustive search space exploration is needed. In line with these principles, here are the general steps performed by CP Optimizer's generic search algorithm: 0. A presolve step is run. Different transformations are performed on the model to simplify it, remove some modeling inefficiencies, and make it more suitable for the next steps. 1. A first feasible solution is searched, using a branch & bound tree with aggressive heuristics which emphasize finding a feasible solution over optimizing the solution quality. 2. An initial dual bound is computed by two means. First by running a binary search on its value with a tree search of limited size, second by reusing the value found by the linear relaxation mentioned above. This dual bound will be used in the branch & bound search and to measure the quality of the solutions found. 3. The main search is started. Different strategies are available, and are generally run in parallel, with a different CPU time ratio so as to favor the most promising ones. We detail the available search strategies in the rest of this subsection. Depth-First Search The Depth-First Search method is the most basic search algorithm in CP Optimizer. It consists in a simple branch & bound exploration of the search tree. The branching heuristics is the Set Times if the objective is regular, or the linear relaxation otherwise. This algorithm is complete (all feasible solutions are explored), but the cost of doing so can be tremendous. In practice, it is mostly used for debugging and for counting the number of feasible solutions to a problem. Large Neighborhood Search The Large Neighborhood Search is the main search strategy used by CP Optimizer, once at least one feasible solution to the problem has been found. This strategy was introduced in [START_REF] Shaw | Using Constraint Programming and Local Search Methods to Solve Vehicle Routing Problems[END_REF] and was adapted to scheduling problems and expanded in [START_REF] Laborie | Complete MCS-Based Search: Application to Resource Constrained Project Scheduling[END_REF][START_REF] Laborie | Self-Adapting Large Neighborhood Search: Application to Single-Mode Scheduling Problems[END_REF]. It can be thought of as a local search based metaheuristics on top of the tree search. The principle is to relax a part of the incumbent solution, to reoptimize, and to repeat the process, as shown on Figure 2.2. We will hereunder describe the main ideas of LNS. Since LNS is a local search algorithm, we need to find a feasible solution first. This is done using a tree search, with parameters which are optimized to quickly find a feasible solution, at the expense maybe of its quality. Once that first solution is available, LNS itself can start and will loop over relaxing a part of the current solution, and completing it into a full solution. Relaxing a part of the solution is done by keeping some structure on the rest of the solution and reoptimizing the whole problem with this additional structure. The structure that is kept is called a Partial Order Schedule and was introduced in [START_REF] Policella | Generating robust schedules through temporal flexibility[END_REF]. It consists in a graph whose edges are precedence constraints between activities, with the property that all schedules which are feasible with respect to the POS also satisfy most resource constraints of the problem (except for very particular situations such as a minimum level constraint in a cumul, for which the POS might be underconstrained). Of course, the POS is over-constrained with respect to the original problem since it encodes part of the incumbent solution. Computation of the POS is done resource by resource. For example, for a sequence, the POS contains all precedences from one activity in the sequence to the next. For a state resource, the POS contains all precedences between activities requiring incompatible states, such that one of them is executed fully before the other. For a cumulative resource, the computation is slightly more involved and is described in details in [START_REF] Laborie | Self-Adapting Large Neighborhood Search: Application to Single-Mode Scheduling Problems[END_REF]. The part of the solution which will be relaxed can be chosen by different heuristics, each depending on parameters. For instance, one such heuristics will randomly select activities to be relaxed with a certain probability. Another one will relax all activities which start in a certain time window. The position and size of that time window are parameters of that heuristics. Once we have chosen a fragment to relax, all edges in the POS involving activities from the fragment are removed, and this new problem is solved. The additional constraints provided by the POS will result in a much smaller search space. The LNS is self-adaptating, as are most algorithms in CP Optimizer, in the sense that all its parameters (choice of neighborhood and search heuristics, and the parameters of these heuristics) are learned dynamically. This is done by maintaining weights over the likelihood of choosing each heuristics and over a vector of possible values for the heuristics parameters. The heuristics to be used and its parameters are then randomly chosen according to this distribution. Af-ter each reoptimization, the effect of the heuristics is measured and the weights are updated: if a heuristics resulted in improving the solution, the probability that it is used again in the future will be increased, and vice versa. An overview of LNS is presented on Figure 2.3. CS 1 [q 1 ] CS 1 [q 1 ] LN 1 [p 1 ] Reinforcement Large Neighborhoods portfolio CS 1 [q 1 ] Relax fragment Solve Selection LN i [P i ] CS j [Q j ] First solution Limit reached Reward r LN 1 [p 1 ] No Yes Completion Strategies portfolio Figure 2.3: Self-Adapting LNS overview (from [LG07]) We can already see with this simple description that CP Optimizer relies on concepts which go far beyond "classical" CP: propagation remains a core concept, but the default search uses local search as much as tree search, this local search can itself be guided by a linear relaxation of the problem, learning is used at all stages, etc. Genetic Algorithm The Genetic Algorithm search method is a multipoint local search algorithm (it works with a population of solutions). Since enumerating the time is extremely costly and the solution space is typically not convex, the local search should not be performed directly on the decision variables of the problem (which include starting and ending times), but should instead be performed on an indirect representation of the problem, which is then decoded into an assignement of values to the decision variables. This is what is done in CP Optimizer's genetic algorithm: a genome is maintained and evolved, which encodes mostly precedences between activities. It is then interpreted through a decoding step. In this step, the genome is transformed into additional constraints. A solution is then found using a completion algorithm (the parameters of the completion algorithm are also evolved and decoded from the genome). This algorithm is generally not as robust as the Large Neighborhood Search, but has better performances in some of the benchmark instances and is thus frequently useful on problems with a complex solution landscape. This is mostly due to better diversification. In addition, it is often used in conjunction with other search methods such as LNS to try and improve the solution they found. Failure Directed Search The Failure Directed Search is used when we aim at improving the dual bound on the current solution and hopefully prove that this solution is optimal. It is started under the conditions that the search space is small enough, since FDS is RAM-intensive, and that the LNS is not making progress anymore. In this case indeed, we assume that there is probably no better solution and start working towards a proof. If a better solution exists it is very hard to find with the common techniques and we need to cover the full search space anyway. In order to cover all of the search space rapidly, we now aim at finding failures as quickly as possible. Moreover, no-goods are learned when failures are found so as to avoid exploring the same part of the search space in the future and further reduce the search space. In order to direct the search towards failures, the decisions (branching variable and value) are rated: those which lead to stronger domain reduction, and ideally failure, are preferred and will be used with a greater propability during the next restarts. More information about this technique can be found in [START_REF] Vilím | Failure-Directed Search for Constraint-Based Scheduling[END_REF]. Propagations There are several keys to implementing propagations efficiently in a constraint programming engine. A first good practice is to use a propagation queue, running the least expensive (in terms of computing time) propagators in priority over the heavier ones. In particular, once a heavy propagator manages to change a domain, it should be interrupted in favor of the lighter propagators, instead of trying to reach the fixed point of that propagation. The fixed point of all propagations though must be reached before branching. The goal of incrementality is to maintain information between the nodes during the search, in order to avoid trying the same propagations several times and to restudy only the part of the propagations which may have been affected by the last branching decision. The domain values modified by the last branching decision are called delta domain in this context. A propagation can also be made incremental with the use of supports, which are logical conditions whose truth value must change before it is necessary to reexamine this propagation. Typically, several propagation algorithms are implemented for each constraint. These algorithms have varying properties in terms of algorithmic complexity, deduction power, incrementality, cooperation with other factors such as the presence of other constraints or the search strategy being used. Trade-offs have thus to be considered when choosing a propagation algorithm in a given situation. An instance of such a trade-off is between the time spent propagating and the propagation power. Typically, if we want to search for a feasible solution (that is, improve the primal bound), we will use a search algorithm which will quickly sample the search space in different regions. In this case, we want to move quickly through the search space and need relatively inexpensive propagation algorithms (typically, no more than O(n log n) where n is the number of tasks), at the expense of the propagation power, with the hope that we will not end up in too many dead-ends. On the other hand, if we want to improve the dual bound, we need to cover the search space more extensively, and will benefit from longer but more powerful propagations, which will significantly reduce the size of the search space. For example, the Edge Finder (see below in Subsection 3.4.3), which is an expensive propagation, does not combine well with the LNS when trying to find good feasible solutions, but it works very well in combination with a search strategy aimed at improving the dual bound, such as FDS. Another trade-off concerns the incrementality properties of a propagator. For example, the Timetable propagation can be updated in linear time from a node to one of its successors, thus making it the cornerstone of propagators for the cumulative constraint, in spite of the existence of much stronger propagators, but with poorer incrementality properties. In the next chapter, we will detail some propagations which are used in constraint-based scheduling, notably on the cumulative constraint. Chapter 3 The Cumulative constraint We introduce in this chapter the cumulative constraint, briefly mentioned previously, which is at the heart of this thesis. As we will see, this constraint is used to model very diverse situations and is widely used, since we find a cumulative aspect in most scheduling problems. Besides, there exist several families of powerful algorithms to deal with this constraint. We begin in Section 3.1 by defining this global constraint and the notation which we will use further on. We continue in Section 3.2 by introducing the Resource Constrained Project Scheduling Problem, a fundamental problem in scheduling, which has many uses in and of itself, and has many close ties with the cumulative constraint. Then in Section 3.3, we present numerous varied examples of industrial applications of the cumulative constraint which motivates the extensive study of this constraint on top of its theoretical interest. Finally in Sections 3.4 to 3.8, we present different algorithmic frameworks to solve cumulative schedules. Definition and notations We focus in this chapter on the discrete cumulative resource, an abstraction for the assignment of a fixed quantity of a resource to a task during its execution period. At any point in time, the total resource usage by the tasks being run must not exceed the total resource capacity. This abstraction is heavily used in constraint-based scheduling to model the allocation of a limited resource, such as manpower, budget, parallel machines, reservoirs. In this context, we talk of the cumulative constraint. Definition 1. Given a discrete cumulative resource of capacity C on a set of n tasks with respective length p i and demand c i on the resource, we say that the resource can be satisfied if and only if for each task there exists a start time t i 48 CHAPTER 3. THE CUMULATIVE CONSTRAINT such that: ∀ time t, ∑ i ∈ [1, n] t i ≤ t < t i + p i c i ≤ C In this case, t 1 , ..., t n is called a valid schedule. An example of a common application of this constraint is in the famous Resource Constrained Project Scheduling Problem: this problem can be seen as minimizing the makespan of a set of non-preemptive tasks of given lengths, under a cumulative constraint such as we just introduced, in addition to precedence constraints between tasks. The cumulative constraint was introduced as a constraint in the Constraint Programming framework in [START_REF] Aggoun | Extending CHIP in Order to Solve Complex Scheduling and Placement Problems[END_REF]. Note that this problem is related but clearly different from the two-dimensional packing problems [START_REF] Lodi | Two-Dimensional Packing Problems: a Survey[END_REF], such as strip-packing. One can obtain excellent relaxations of two-dimensional packing problems with cumulative resources though, and all the techniques developed for the cumulative resource apply to two-dimensional packing as well. We will see later in this chapter several algorithmic techniques to deal with the cumulative constraint, but a fundamental tool, used by several of these techniques, is the notion of energy of a set of tasks over a cumulative resource: Definition 2. Given a discrete cumulative resource of capacity C and a set of n tasks with respective lengths p i and demands c i , the energy bound of the tasks on the resource is E = ∑ n i=1 p i c i C . Notice that the energy gives a lower bound for the total time needed to run all the tasks. For example (see Figure 3.1), on a resource of capacity C = 4, if we have n = 5 tasks of lengths p = (3, 4, 9, 1, 4) and respective demands c = (4, 3, 1, 2, 1), then the energy lower bound on the makespan is 3×4+4×3+9×1+1×2+4×1 4 = 9.75 (the actual minimum makespan for this example is 12). RCPSP RCPSP The cumulative constraint is often studied in the context of the Resource Constrained Project Scheduling Problem (RCPSP), as these two notions are intimately linked. The RCPSP is indeed the scheduling problem containing only cumulative constraints (possibly several of them) and precedence constraints between the tasks. The reason for combining the study of these two families of constraints is that they act in an orthogonal fashion on the solution: intuitively, the precedence constraints define the structure of the solution along the temporal axis, while the cumulative constraints act on the behaviour of the problem at a given point in time. Thus, they form the prototypes of constraints acting on these two dimensions and enable the study of their interactions, which have yielded rich and fruitful algorithmic results. In addition to its theoretical interest in relation with the cumulative constraint, the RCPSP, in spite of its simplicity, is also applicable to a broad variety of practical problems. Indeed, it generalizes many practical production problems, such as parallel machines scheduling and shop problems, as we will see below. Of course, even though the cumulative resource is often presented and studied in the context of the RCPSP, it is much more general and can be used in many different settings. This will be illustrated on different examples in the following section. There exists an extensive body of literature about the RCPSP: several hundreds of articles and a dozen of books have been published on this topic. We will only cite one of them here: the recent and excellent [ADN13] which contains surveys of all the sub-fields of the study of the RCPSP and references for the interested reader. Formal definition Let us now give a formal definition of the RCPSP. Given: • R cumulative resources r = 1, . . . , R of respective capacities C r . • n activities i = 1, . . . , n of respective durations p i and demands d i,r on each resource r. • and a set of precedences i → j such that the directed graph whose nodes are the activities and edges are the precedences is acyclic. we must find a start date S i for each activity such that: CHAPTER 3. THE CUMULATIVE CONSTRAINT • at any point in time, for each cumulative resource, the sum of the demands of the tasks being executed does not exceed the capacity of that resource: ∀time t, ∀r ∈ [1, R], ∑ i ∈ [1, n] S i ≤ t < S i + p i d i,r ≤ C r . • for each precedence i → j, the inequality S i + p i ≤ S j is satisfied, so that activity i finishes before activity j begins. Preemption is not allowed. • the makespan max i∈ [1,n] (S i + p i ), that is the latest completion time of any activity, is minimized among all the schedules which satisfy the previous two constraints. Other objectives are possible in variants of the RCPSP. Without loss of generality, all quantities C r , d i,r and p i are assumed to be integer. Example We show here a small example of a project with n = 5 activities and R = 2 cumulative resources of respective capacities C 1 = 3 and C 2 = 5. For each activity i, the lengths p i and demands d i,1 and d i,2 on the two resources are given in the following table: A versatile model The RCPSP, in spite of its simplicity, is a surprisingly expressive model. For example, the problems we introduced above in Section 2.1, such as single machine problems, parallel machine problems and shop scheduling problems, can be modeled as RCPSPs. We give here an outline of these reductions. Single-machine problems, sometimes also called disjunctive problems, where activities have to be scheduled on a resource which accepts one activity at a time, are easily modeled with a cumulative resource of capacity C 1 = 1, on which each activity has a demand d i,1 = 1. Similarly, parallel identical machines problems, where jobs can be scheduled on m identical machines with the same processing speed, can be modeled with an RCPSP with one cumulative resource of capacity C 1 = m and each activity has a demand d i,1 = 1. The open shop scheduling problem with n jobs and m machines can be modeled with an RCPSP with n + m cumulative resources, all of capacity C i = 1. The first n resources correspond to the jobs, and the following m resources correspond to the machines. The activities correspond to the operations O i,j , with a demand of 1 on resource j corresponding to job J i , and a demand of 1 on resource n + µ i,j , corresponding to the use of the machine. The special cases of the shop scheduling problems we mentioned above can also be modeled as RCPSPs. We refer the reader to Chapter 1 of [START_REF] Brucker | Complex Scheduling[END_REF] for more details on modeling with the RCPSP. Algorithmic complexity In spite of its practical relevance, this problem is of formidable difficulty: it is strongly NP-hard, even without precedence constraints and only one cumulative resource of capacity 3 [START_REF] Garey | Complexity Results for Multiprocessor Scheduling under Resource Constraints[END_REF]. The reduction from 3-PARTITION is intuitive. Contrary to many other NP-hard problems, it is also poorly tractable in practice, even for instances of modest sizes. Surprisingly, small instances which contain only 60 tasks and are more than 20 years old as this thesis is written are still unsolved to optimality to this day, in spite of the tremendous recent improvement in algorithms and computing power. The most commonly used benchmark instances for the RCPSP have been collected in the PSPLIB [START_REF] Kolisch | PSPLIB -a Project Scheduling Problem Library[END_REF]. OPL model A model for the RCPSP in the OPL language is presend in Listing 3.1. Let us detail this code: Lines 1-12 correspond to the data input. The RCPSP instance has n tasks and R resources. A data structure containing the informations of each task (length, demand on each of the R cumulative resources, and direct successors in a precedence constraint) is initialized lines 4-9 and filled line 11. The capacity of each of the R cumulative resources is set line 12. Line 14, the decision variables of the model are declared. They are here of type interval which means that they represent interval variables, that is time period over which the corresponding task will be executed. The start and the end of that period are to be determined by the solver. Lines 16-18, the cumulative functions usage[r] are declared. Their value is defined as a sum over the tasks of pulses, that is functions of the time whose value is the demand of the task on that cumulative resource on the execution period of the task and null elsewhere. In this way, the value of the cumulative function at any point in time is the sum of the demands of the tasks being executed. The objective function of the model is declared line 20. It states the makespan objective, that is the minimization of the latest end time of any of the tasks. While widely regarded as quite academic, the makespan is a sensible measure of the throughput of a production system. The constraints of the model appear in the subject to block, lines 21-26. As we mentioned above, there are two families of constraints: Lines 22-23 express the cumulative constraints: for every cumulative resource, the corresponding cumulative function must never exceed the capacity Lines 24-25 correspond to the precedence constraints: for all declared direct successors to a task, we declare an endBeforeStart constraint, which states a precedence betwen the two intervals given as parameters. RCPSP with Multiple Modes An important variant of the RCPSP, which we do not deal with in this thesis but should be mentioned for completeness, is the RCPSP with Multiple Modes (RCPSPMM). In this variant, a set of modes M i is associated with each activity i. To each mode m ∈ M i corresponds a different processing time p i,m and different de-mands d i,r,m on the cumulative resources. This is used to model the frequent situations in which different processes or machines can be used to obtain the same result, albeit with a different processing time. In addition to setting starting dates, the solver must select a mode for each activity. In CP Optimizer, alternatives can be used to model the RCPSPMM. Examples of industrial applications of the cumulative constraint Since this thesis was prepared in an industrial setting, we wish to present some concrete examples of the usage of the cumulative constraint, to show how versatile it is. In our experience, more than half of all scheduling models involve a cumulative constraint. Our goal is to represent the variety of its uses, both from an application and a modelization standpoint. Exclusive zones Many areas in an aircraft under construction are cramped and can only accomodate a limited number of workers, or are supported by scaffolding and have severe weight limits. Exclusive zones are extensively used in aircraft manufacturing to account for these restrictions. The principle is to divide the aircraft into zones, and to limit the number of people working simultaneously in a zone (or a union of zones) using a cumulative constraint. Figure 3.4 shows an example of splitting the forward section of an aircraft into different exclusive zones, and the associated constraints. The cumulative resource is often used to model such volume limitations in the case of processing several items at once, for example in chemical reactors or in ovens, in semiconductor manufacturing. Workforce scheduling A common application of scheduling is the building of rosters. Different tasks have to be assigned to different employees. Each task requires a certain skill, and employees each have a set of skills. The easiest way of modeling this with CP Optimizer is to have an interval for each task t i , and to create optional tasks t ij for each task t i and each employee j which has the skills to complete task t i . Two types of constraints are then used: an alternative from each main task t i to the subtasks t ij , which ensures that the task will be assigned to exactly one employee, and a no-overlap constraint for each worker j on all the tasks t ij It is very helpful here to add a redundant cumulative constraint for each skill. These cumulative constraints will relax the exact assignment of tasks to workers, but they will have a more global view of all the tasks sharing certain workers. This will enable them to make deductions that the no-overlap constraints would have missed. For each skill, we thus define a new cumulative constraint. Its capacity is the number of workers having that skill, and the de-56 Using a cumulative resource as an approximation for identical machines is often a wise and efficient practice for non-critical parts of a model. CHAPTER Long-term capacity planning In supply chain management, capacity planning consists in matching the resource availabilities and customer demands for the products of a company. In general, the long term strategy of the company is at stake. This problem is generally modeled as a linear problem by assimilating the production as a cumul of demands over a small number of fixed time buckets. The expected demands come from strategic forecasts, the time horizon is huge, typically in years, and the buckets duration is long as well, typically one month. The objective is then to choose which resources to use for the production, and to compute an approximate production date. Nevertheless, the long duration time buckets relaxation is not precise enough for some cases. This is for example the case when precedence constraints between demands, or costs in case of early or late delivery, are key factors. It prevents from safely aggregating the demands in buckets and requires a continuous time definition. If we suppose that the relaxation of the production system as cumulative resources is still valid, the core problem is then an RCPSP (eventually an RCPSPMM) with a huge number of tasks, typically up to one million, and a large set of cumulative resources, typically in the hundreds. Berth allocation This example is an original, though not uncommon, application of the cumulative constraint to a geometric problem. A commercial port has a linear quay of length L. Ships can dock anywhere along the quay, but their berthing locations obviously cannot overlap, see Figure 3 In many instances and for all practical purposes, since ships have a large enough length and only a small number of them can fit on the quay at the same time in the real-world problem, this model provides an exact relaxation of the original problem. EXAMPLES OF INDUSTRIAL APPLICATIONS OF THE CUMULATIVE CONSTRAINT 59 The principle of using a cumulative resource to model a strip packing problem with a thin dimension applies to vehicle loading problems such as truck or train loading. The idea consists in using the temporal dimension of the cumulative resource to model the length of the vehicle, as shown on Balancing production load In CP Optimizer, the demand of a task can also be a decision variable, and not only a data of the problem. One example of a use case is in balancing a production load. In many industrial problems, the real objective is to balance the production as evenly as possible, which is not easy to express as an objective function in an optimization model. A way of accomplishing this is to minimize the peak utilization of the resources. In this example, a dummy task of variable demand D is added to the cumulative constraint and the objective is to maximize D, as shown in Figure 3.8 and Listing 3.5. Many variants of this idea can be used to balance a load, such as having a dummy task per work shift. t A 1 A 2 A 3 A 4 A 5 A 6 Batching Another type of constraint, related to the cumulative constraint as defined previously, is the alwaysIn constraint. We do not deal with the algorithmic aspects of the alwaysIn constraint in this thesis, but mention it here for comprehensiveness. alwaysIn( f , a, h min , h max ) states that during the execution interval of activity a, the cumulative function f , representing the sum of the demands of the tasks being executed at a certain point in time, must always be at a level in the range [h min , h max ]. In the following example which is sometimes encountered in process scheduling models (OPL code in Listing 3.6 and illustration in Figure 3.9), the alwaysIn constraint is used to locate production periods, in order to fit maintenance tasks between those periods. The production tasks take place during the A i intervals. We want a W j interval to be present exactly whenever an A i interval is present. To this end, the constraint alwaysIn(CW, A[i], 1, 1) ensures that exactly one W j interval is present whenever an A i interval is present. The constraint alwaysIn(CA, W[i], 1, n) ensures that at least one A i interval is present whenever a W j interval is present, so that no W j interval is present when no A i interval is present. A natural extension of the cumulative resource is the "reservoir", which is a succession both of filling and emptying events, for which we have a minimum level constraint (in general 0) and a maximum level constraint (the reservoir volume). In this thesis, we will not deal with negative demands nor minimum levels in cumulative resources. CUMULATIVE PROPAGATION 61 0 1 n A 4 A 6 A 6 A 5 A 3 A 1 A 2 0 1 W 1 W 2 W 3 Cumulative propagation We describe in this section the most common propagations for the Cumulative constraint. These propagations only take into account the current bounds for the start and end times of the different activities, except for the Precedence Energy which also takes precedence relationships into account. Most of these propagations rely on the notion of Energy Bound, defined above in Section 3.1. CHAPTER 3. THE CUMULATIVE CONSTRAINT The general principle is to select a set of activities, select a time interval where these activities should be scheduled, and move an activity if the total energy of the activities exceeds the available energy in the window. There are dominance relationships between these propagations, in the sense that the deductions made by some of them are subsubmed by the deductions made by another one with a higher algorithmic complexity. In particular, Energy Reasoning is stronger than the Edge-Finders. It is also stronger than the Timetable. It is worth noting though that Energy Reasoning and Not-First, Not-Last are incomparable and can both make deductions that the other cannot. Timetable The timetable propagation was introduced in [START_REF] Aggoun | Extending CHIP in Order to Solve Complex Scheduling and Placement Problems[END_REF]. A lot of improvement has since been made on this topic and exists unpublished in the community. The timetable has a quadratic worst-case complexity but its amortized complexity is linear. In spite of its relative propagation weakness, its low complexity makes it the basis of propagating the cumulative constraint. The timetable amounts to maintaining arc-B-consistency on the equation from Definition 1 which defines a satisfied cumulative resource. The principle is to maintain a so-called resource histogram, recording for each point in time the minimum amount of resource consumption: first we compute for each task its compulsory part, that is M i (t) c i if lst i ≤ t < eet i 0 otherwise We accumulate for each time point the compulsory parts of all tasks: U(t) ∑ i M i (t) = ∑ i∈ 1,n :lst i ≤t<eet i c i Several deductions can be made with this histogram: • If ∃t/U(t) > C, then we know that the current problem has no solution and the search must backtrack. • If for a particular task A i , ∃t 0 / max(lst i , eet i ) ≤ t 0 < let i , U(t 0 ) -M i (t 0 ) + c i > C, then A i cannot end after t 0 , otherwise the resource would be overconsumed. let i can then be updated to t 0 . A symmetric rule exists for updating the start times. In addition to its low complexity, a major advantage of the timetable is that it is easily extendable. It can easily support extensions such as reservoirs (negative demands and minimum level constraints on a cumulative), as well as the joint propagation of other resources such as state functions with transition time, without incurring a performance penalty. The implementation in CP Optimizer is fully incremental with respect to changes to the interval variables domains. Disjunctive If the sum of the demands of two activities A i and A j exceeds the capacity of the cumulative, they cannot overlap in time, so one of them must precede the other. The disjunctive constraint maintains arc-consistency on the formula: c i + c j ≤ C ∨ A i → A j ∨ A j → A i This constraint can be propagated on all activities with a O(n log n) complexity, as explained in [START_REF] Vilím | O(n log n) Filtering Algorithms for Unary Resource Constraint[END_REF]. This propagation is surprisingly powerful in practice, since many instances of cumulative problems exhibits tasks which cannot overlap. Edge Finding and derivatives The Edge-Finding family of propagation techniques reason about the order of execution of activities. Specifically, these algorithms determine if a given activity A i must execute before (or after) a set of activities Ω. Two types of informations can be drawn from this: new precedence relations, sometimes called implicit precedences or "edges", and more importantly new bounds on the execution dates of activities. The general principle is to check if the total energy of a set of activities Ω fits in the available energy for Ω, otherwise a conflict is detected: C(let Ω -est Ω ) < e Ω ⇒ fail. There exists a breadth of Edge-Finder variants, with different propagation power versus algorithmic complexity trade-offs, and research is very active in this area. We present the most common versions. Edge-Finder The original cumulative Edge-Finder was introduced in [START_REF] Wim | Time and Resource Constrained Scheduling: a Constraint Satisfaction Approach[END_REF]. For A i / ∈ Ω, if C(let Ω -est Ω∪{A i } ) < e Ω∪{A i } , then A i must end after lct Ω . Indeed, the previous condition is satisfied if the energy overflows when A i is forced to finish no later than let Ω . The original complexity of this propagation was O(n 2 ) where n is the number of activities, and it was improved in [START_REF] Vilím | Edge Finding Filtering Algorithm for Discrete Cumulative Resources in O(kn log n)[END_REF] to O(kn log n), where k is the number of different demands of the activities, thanks to a new data structure called Θ-tree. Extended Edge-Finder The Extended Edge-Finder was also introduced in [START_REF] Wim | Time and Resource Constrained Scheduling: a Constraint Satisfaction Approach[END_REF]. Instead of considering the [est Ω∪{A i } , let Ω ) window as previously, only the window [est Ω , let Ω ) is now considered and only the part of the energy of A i which necessarily falls into this window is taken into account. The full propagation condition is thus, for A i / ∈ Ω and est i ≤ est Ω < eet i , if C(let Ω -est Ω ) < e Ω + c i (eet i -est Ω ), then A i must end after lct Ω . The original complexity of this propagation was O(n 3 ), and it was improved in [START_REF] Mercier | Edge Finding for Cumulative Scheduling[END_REF] to O(kn 2 ). Timetable Edge-Finder The Timetable Edge-Finder was introduced in [START_REF] Vilím | Timetable Edge Finding Filtering Algorithm for Discrete Cumulative Resources[END_REF], and combines the Timetable propagation with the Edge-Finder, by also taking into account the timetable in the overflow condition. We defined above in Subsection 3.4.1 the U(t) function, which sums the mandatory energy of all tasks at time t. We can thus improve the overload checking rule by adding the energy from the timetable: C(let Ω -est Ω ) < e Ω + let Ω ∑ t=est Ω U(t) ⇒ fail. Depending on the relative positions of A i and Ω, we can add to this equation the amount of energy of A i that necessarily falls in the [est Ω , let Ω ) window and is not yet taken into account in the timetable to obtain an adjustment rule. The complexity of this propagation is O(n 2 ). Not-First, Not-Last The Not-First, Not-Last propagation was originally introduced in [START_REF] Wim | Time and Resource Constrained Scheduling: a Constraint Satisfaction Approach[END_REF]. It gives necessary conditions under which an activity A i has to be scheduled after at least one activity (not-first) or before at least one activity (not-last) of a set Ω of activities. This is complementary to edge-finding. We now describe the Not-First rule. The Not-Last rule is similar. For a set of activities Ω and an activity A i / ∈ Ω, if est Ω ≤ est i < min j∈Ω eet j , and e Ω + c i (min(eet i , let Ω ) -est Ω ) > C(let Ω -est Ω ), then A i must start after min j∈Ω eet j and est i can be updated. Indeed, if we force activity A i to start at its earliest start time, then no activity of Ω can be completed before A i is started, since est i < min j∈Ω eet j . Thus an energy c i (est i -est Ω ) is occupied by no activity before A i starts, as shown in green on Figure 3.10. The sum of the energy of the activities in We can thus deduce that: Ω ∪ {A i } on the window [est Ω , let Ω ) is thus e Ω + c i (min(eet i , let Ω ) -est i ) + c i (est i -est Ω ), est i ≥ max Ω⊆{A 1 ,...,A n }/          A i / ∈ Ω est Ω ≤ est i < min j∈Ω eet j e Ω + c i (min(eet i , let Ω ) -est Ω ) > C(let Ω -est Ω ) min j∈Ω eet j The original complexity of propagating this condition on all tasks was O(n 3 ). It was improved to O(n 2 log n) in [SW10]. Energy Reasoning The energy reasoning was originally introduced in [START_REF] Erschler | Energy-based approach for task scheduling under time and resources constraints[END_REF]. It consists in fixing a time window, and accumulating for all the the tasks the minimum energy within the time window. The propagation is triggered when an overflow is detected. This propagation is extremely powerful and subsumes several other propagations (notably timetabling and edge-finding). But its original computational complexity of O(n 3 ) made it unusable in practice. [START_REF] Erschler | Energy-based approach for task scheduling under time and resources constraints[END_REF][START_REF] Baptiste | Constraint-Based Scheduling: Applying Constraint Programming to Scheduling Problems[END_REF]. The energy reasoning has seen much interest and several improvements in the recent years, notably in a series of two articles [START_REF] Derrien | The Energetic Reasoning Checker Revisited[END_REF][START_REF] Derrien | A New Characterization of Relevant Intervals for Energetic Reasoning[END_REF] in which the constant factor of its complexity was divided by seven, and in techniques aiming at using machine learning to predict cases when running the expensive O(n 3 ) Energy Reasoning propagator will be beneficial, so as to use this propagation sparingly [START_REF] Van Cauwelaert | Supervised learning to control energetic reasoning: Feasibility study[END_REF]. As part of this doctoral work, we developed a new propagation algorithm with a complexity of O(n 2 log n) [START_REF] Bonifas | A O (n^2 log (n)) propagation for the Energy Reasoning[END_REF], making this propagation useful in a higher number of cases. We do not detail this propagation here, since it will be extensively covered in Chapter 5. Energy Precedence The energy precedence propagation, introduced in [Lab03a], is different from the previous propagations in that it combines information coming from the temporal network (precedence constraints) with the information on the cumulative constraint and the current domains of the intervals. This propagation is especially useful when used in combination with a precedence-based search (a branching scheme which adds precedence constraints to the model). The energy precedence ensures that the cumulative resource provides enough energy for all the predecessors of a task to fit before that task can start. More formally, for each subset Ω of predecessors of a task A i , the earliest start time of a task in Ω plus a lower bound on the running time of all the tasks in Ω provides a lower bound on the earliest start time of A i : ∀Ω ⊆ {j ∈ 1, n : A j → A i }, est i ≥ est Ω + e Ω C There exists a symmetric rule with the successors and latest end time of i. Since subsets Ω of the form {j ∈ 1, n : A j → A i , est j ≥ t} dominate the other subsets for this propagation, and there are only p of them where p is the maximal number of predecessors of a given activity in the temporal network (p < n), the energy precedence can be propagated with a worst-time complexity O(n(p + log n)). MIP formulations Numerous MIP formulations have been proposed for the RCPSP. Nevertheless, a major difficulty in coming up with good formulations is that the cumulative constraint is deeply non-linear with respect to the dates of the activities. In other words, it is very difficult to express the temporal and cumulative constraints jointly using linear inequalities. This requires compromises on the size of the formulation and the quality of the linear relaxation, which are done by choosing different variables, representing more or less directly the temporal and cumulative constraints, and thus by different linearizations, more or less precise and more or less compact. Nevertheless, there is no good compact linearization and we have to choose between either a weak approximation or a pseudo-polynomial model. For this reason, only small instances have been solved to optimality, or instances with additional properties that could be exploited in the MIP model. Additionally, the linear relaxations of these formulations are generally weak. The fundamental notion in most of these formulations and attempts at linearizing the cumulative constraint is that of antichains (set of activities which are incomparable in the precedence relation and can a priori be scheduled simultaneously). Following Queyranne and Schulz's [QS94] classification of MIP formulations, we will group them hereafter by the type of decision variables used. Time indexed formulations Time indexed formulations were introduced in [PWW69] . They are, perhaps surprisingly, the most commonly used MIP formulations for the RCPSP, owing to their simplicity, in spite of their poor practical performance on most instances. The basic principle (Discrete Time formulation) is to divide the scheduling horizon H into unit time slots and to use time-indexed binary variables: for any activity i and time t ∈ [0, T], x it = 1 if activity i starts at time t, 0 otherwise. With these variables, the precedence constraint from i to j is expressed as H ∑ t=0 tx jt - H ∑ t=0 tx it ≥ p i and the cumulative constraint as n ∑ i=1 c i t ∑ τ=t-p i +1 x iτ ≤ C for all times t ∈ [0, H -1]. In the Disaggregated Discrete Time formulation, the precedence constraints are desagregated into t-p i ∑ τ=0 x iτ - t ∑ τ=0 x jτ ≥ 0 for all times t ∈ [0, H -1] as introduced in [START_REF] Christofides | Project scheduling with resource constraints: A branch and bound approach[END_REF]. We obtain a stronger continuous relaxation and better lower bounds. Moreover, if the cumulative constraints are dualized with a La-grangian relaxation, the remaining constraint matrix is totally unimodular. This fact was used in [START_REF] Möhring | Solving Project Scheduling Problems by Minimum Cut Computations[END_REF] to solve the remaining problem with a maximum flow algorithm. An even stronger linear relaxation can be obtained by replacing the cumulative constraint as expressed above by its Dantzig-Wolfe decomposition, using antichains. More specifically, we call P the set of antichains P. P i is the set of antichains containing activity i. The decision variable y Pt is equal to 1 if antichain P is being executed at time t, 0 otherwise. The cumulative constraint is now expressed as: ∀i ∈ [1, n], ∑ P∈P i ∑ t∈[0,H-1] y Pt = p i , so that enough antichains are available to execute all of activity i. ∀t ∈ [0, H -1], ∑ P∈P y Pt ≤ 1, to schedule only one antichain per time slot. ∀i ∈ [1, n], ∀t ∈ [0, H -1], x it -∑ P∈P i y Pt -∑ P∈P i y P,t-1 ≥ 0, to synchronize the start date of i with the first use of an element of P i . This formulation was introduced in [START_REF] Mingozzi | An Exact Algorithm for the Resource-Constrained Project Scheduling Problem Based on a New Mathematical Formulation[END_REF]. It was further improved with constraint propagation and specific cutting planes in [START_REF] Brucker | A Linear Programming and Constraint Propagation-Based Lower Bound for the RCPSP[END_REF][START_REF] Baptiste | Tight LP Bounds for Resource Constrained Project Scheduling[END_REF]. All these time indexed formulations are pseudo-polynomial in H. Continous time formulations These formulations are based on two families of variables: n continuous variables S i representing the starting time of each activity and n 2 binary sequencing variables x ij between all pairs of activities, representing precedences between activities: x ij = 1 if activity j does not start before activity i is completed, and 0 otherwise. The basic constraints of these formulations enforce the asymmetry and the triangle inequality of the precedence relation: x ij + x ji ≤ 1 and x ij + x jh -x ih ≤ 1, , and the precedence relationship from i to j with a big-M: S j -S i -Mx ij ≥ p i -M. The first formulation based on these variables [START_REF] Olaguíbel | The project scheduling polyhedron: Dimension, facets and lifting theorems[END_REF] relies on the concept of forbidden sets. A forbidden set F is a set of activities which cannot be scheduled simultaneously because the sum of their demands would exceed the cumulative capacity, so for any forbidden set F, we add a constraint to force a precedence between two activities: ∑ {i,j}⊆F x ij ≥ 1. There is is an exponential number of forbidden sets in general, so this formulation cannot be used in practice. A second formulation based on the concept of resource flow was proposed in [START_REF] Artigues | Insertion techniques for static and dynamic resource-constrained project scheduling[END_REF] to obtain a more compact formulation. The principle is to indicate LP-BASED STRENGTHENING OF THE CUMULATIVE CONSTRAINT69 the quantity of cumulative resource made available for activity j when activity i finishes. Two families of constraints are then added to the model to express that the total incoming and outgoing flow to an activity must match its demand. Two dummy activities are placed at the beginning and end of the schedule to provide and capture all the capacity. This formulation has O(n 2 ) binary variables, O(rn 2 ) continuous variables and O(rn 2 + n 3 ) constraints, so it is reasonably compact. Note moreover that the model size does not depend on the scheduling horizon H. Event-based formulations Event-based formulations, introduced in [KALM11], are based on the fact that an optimal solution to an RCPSP always exists where the start time of any activity coincides with the end time of another activity, or with the scheduling origin (a property which was already exploited implicitly by the flow formulation above), so only n events have to be scheduled. We thus have n continuous variables to represent the events dates, with an ordering constraint to break symmetry. Two families of formulations exist, depending on how the activities are connected to the events. In the Start/End formulations, we have two binary variables for each activity/event pair: x ie = 1 if activity i starts at event e and 0 otherwise, and y ie = 1 if activity i ends at event e and 0 otherwise. In the On/Off formulations, we only use one binary variable for each activity/event pair: z ie = 1 if activity i starts or is still being processed at event e. These formulations both have O(n 2 ) binary variables, O(n) continuous variables and O(n 3 + n(p + r)) constraints where p is the number of precedence constraints. The linear relaxation quality is quite poor, though. LP-based strengthening of the cumulative constraint Many methods have been developed to calculate lower bounds on the makespan of RCPSP. As we will explain in much more detail in Chapter 4, we noticed as part of the work done in preparing this thesis that many of these methods, notably several of these relying on a linear programming formulation, as well as the new cumulative strengthening method that we present in Chapter 4, provide naturally, in addition to a lower bound, a redundant cumulative constraint. Redundant means that we do not lose a feasible solutions but obtain different demands, which can possibly be exploited to compute stronger propagations. This is another example of the integration of various algorithmic methods through Constraint Programming. In these methods, we often first relax the RCPSP into a CuSP, that is we abandon the precedence constraints and keep only one cumulative constraint. The first method known to us to introduce a reformulation of the cumulative constraint was based on Dual Feasible Functions. A good survey was published in [START_REF] Clautiaux | A Survey of Dual-Feasible and Superadditive Functions[END_REF]. DFF are functions that map the original demands to demands in a new cumulative functions, in such a way that no feasible solution is lost. In this line of research, particular classes of functions which satisfied this property, as well as a branch and bound algorithm to generate them all were exhibited. There was no reformulation giving a different demand to each activity, or taking into account special cases, such as very long activities. We are also not aware of DFF being used to strengthen propagation, but only to compute global bounds. As for the methods based on linear programming, those computing a Linear Lower Bound, such as in [START_REF] Carlier | An exact method for solving the multi-processor flow-shop[END_REF], can be used to compute a reformulation as well. Indeed, studying the dual of the linear program immediately yields a redundant cumulative function. Other methods based on linear formulations exist and could be used to compute a form of cumulative strengthening, such as [MMRB98, BK00, CN03], but those formulations are time-indexed and would yield a complex reformulation with a variable demand profile. The computations would probably be too heavy for this to be of any practical use. Note that many other techniques to compute LB which do not yield strenghtening exist, but we do not mention them here. Information about them can be found in the survey [NAB + 06], in Section 3.7 of [START_REF] Brucker | Complex Scheduling[END_REF], or in [START_REF] Artigues | Méthodes exactes pour l'ordonnancement de projets à moyens limités (RCPSP)[END_REF]. The Cumulative Strengthening as we introduce it in Chapter 4 is much stronger than Dual Feasible Functions while being much cheaper to compute than the formulations based on linear programming that we mentioned above. Conflict-based search The most effective way to do this is to look for a conflict as high up as possible in the search tree, and that for two purposes. First, the higher the conflict in the tree, the largest the volume of the search space it cuts. Second, learn a new constraint (no-good) from this conflict, in order not to direct the search anymore in this area of the search space. For very hard feasibility problems or when trying to prove that the solution obtained is optimal (so-called exact method), we must explore the entire search space. Some historical approaches use RCPSP-specific structures, but this is no longer the case in the latter approaches which memorize very simple decisions and operate in a much more general framework than the RCPSP. The first such approach was introduced in [DH97] and relies on cutsets (sets of activities whose predecessors have all been scheduled already). If the search reaches a cutset which has already been processed, we can reuse the result from that earlier search which has been saved. This method was for a decade the best way of exploring the whole search space of an RCPSP and proving lower bounds. Minimal Critical Sets were introduced in [START_REF] Laborie | Complete MCS-Based Search: Application to Resource Constrained Project Scheduling[END_REF]. They are sets of activities which violate a cumulative constraint, and thus cannot be scheduled simultaneously, so there must be a precedence between (at least) two of them. The MCS-based search thus branches on possible precedences which break MCS. Yet another approach was introduced in [SFSW10] and consists in modeling the RCPSP as a SAT problem, using the TTEF propagation to enforce the cumulative constraint. The main idea here is to use the native clause learning mechanism of the SAT solver, using so-called TTEF explanations: when a conflict is detected by the TTEF, a clause is generated and added to the SAT formulation. The basic decisions in this formulation consists of inequalities between start dates of tasks and times. This approach works well for problems with small time horizons, but because the time is enumerated in the formulation, this approach doesn't scale to realistic time horizons. The last approach to date is the Failure Directed Search, which we explain in more detail in Subsection 2.5.3.4. As a general remark, these techniques (and all decision making techniques for scheduling problems, such as branching heuristics, local search techniques, and propagations) fall into two categories, depending on whether they raise a precedence between two activities or they set a temporal bound on the start or end time of an activity. It seems that decisions on precedences have less impact than decisions on times, at least to find a solution, and that decisions on dates scale better as well. This can be seen with the extreme example of assuming that we have a perfect branching heuristic and that we will find the optimal solution by always taking the left branch at each node: with a chronological branching rule such as Set-Times, only n decisions have to be taken in this case, while n 2 decisions have to be taken if we want to raise all precedences. If the goal is to prove optimality, the best approach is less clear. Experiments with the Failure Directed Search algorithm show that the Set-Times rules works a bit better than precedence raising, but the reason is unclear, and this observation might just come from lack of research in this direction. Another reason to prefer taking decisions on dates is that all decision-taking methods in a constraint programming engine have to work well together, and that propagations of the cumulative constraints on precedences have not been developed as much as propagations on dates. As far as we know, no research has been done yet on trying to record precedence disjonctions of small cardinality as no-goods. This research direction might yield interesting results. List scheduling The RCPSP is often referred to as one of the most intractable problems in Operations Research, since even very small instances can not be solved to optimality with the current technology. To circumvent this limitation of exact methods, numerous RCPSP-specific heuristics have been developed. A good survey of the different techniques can be found in [START_REF] Hartmann | Experimental evaluation of state-of-the-art heuristics for the resource-constrained project scheduling problem[END_REF], but detailed experiments and comparisons reveal that all good heuristics and local search techniques boil down to list scheduling. The general principle of list scheduling (sometimes also called Schedule Generation Schemes) consists in representing a solution in the form of a total ordering of the activities (list), which is then transformed into a schedule through a decoding procedure specific to the particular heuristic. An example of a decoding procedure is the Earliest Start Schedule, which consists in taking the first activity in the list, scheduling it at the earliest date possible which does not conflict with an already scheduled activity, and repeating these steps until the list is empty. This particular decoding procedure has the convenient property that for an RCPSP with a regular objective function, a list of all activities always exists such that the procedure Earliest Start Schedule provides an optimal schedule. Many other decoding procedures are available in the literature. On the other hand, this heuristic works poorly when no good decoding procedure exists for the particular constraints or objective of the problem, for example when there are many implicit precedences, or if the objective is irregular. An efficient improvement of list scheduling procedures is called Forward-Backward Improvement. After decoding a given list, a new list is created by ranking the activities in the order of decreasing completion times. This list is decoded into a backward schedule, and the whole procedure is repeated until no new schedules are created. The best schedule which appeared in the course of the process is output. Chapter 4 Cumulative Strengthening As mentioned above in Section 3.6, several methods of computing lower bounds on the makespan of an RCPSP, especially those based on the energy bound, can be extended to provide a redundant cumulative constraint as well. In this chapter, which was already published as [START_REF] Baptiste | Redundant cumulative constraints to compute preemptive bounds[END_REF], we show how to construct redundant cumulative constraints. Introduction The objective of this work is thus not to give a new filtering algorithm, but to improve all the reasonings based on the notion of energy, which will automatically improve the existing filtering algorithms which rely on this notion, notably timetabling and edge-finding. See for example Figure 4.1, which gives a possible reformulation for the instance of Figure 3.1: the energy bound is increased from 9.75 to 11.5, and this will help the filtering algorithms adjust the time bounds for this instance. The reformulation of cumulative constraints that we introduce in this chapter is such that all valid schedules remain valid for the reformulated constraint, but we now have a guarantee that the energy lower bound matches the makespan of the preemptive relaxation. The preemptive relaxation of a cumulative resource CHAPTER 4. CUMULATIVE STRENGTHENING problem is a solution which satisfies the cumulative constraints but enables tasks to be interrupted and restarted later (compared to a non-preemptive schedule where a task of length p must run uninterrupted from its starting time t until t + p). These reformulations are relatively cheap to compute (after a precomputation which does not depend on the instance) and they provide a significant improvement for all algorithms which rely on energy reasonings. In our experiments, we used them within an edge-finding algorithm. Let us start by studying two small examples for a resource of capacity C = 3, to get an idea of the spirit of these reformulations: Since on such a resource a task of demand 3 cannot run in parallel with any other task, it can occupy the whole resource. Moreover, since a task of demand 2 cannot run in parallel with another task of demand 2, a valid bound on the makespan of such an instance is the sum of the lengths of tasks of demand 2 and 3. Hence, a valid reformulation for a resource of capacity C = 3 is to discard tasks of demand 1, and to increase the demand of tasks of original demand 2, so that they now occupy the full capacity of the resource. See Figure 4.2 for an example where this reformulation increases the energy bound from 7.33 to 8. Again for a resource of capacity C = 3, we notice that the longest task of demand 1 cannot run in parallel with any task of demand 3. So a valid lower bound on the makespan is the sum of the lengths of tasks of demand 3, plus the length of the longest task of demand 1. Hence, a valid reformulation for a resource of capacity C = 3 is to discard tasks of demands 1 and 2, except for the longest one of demand 1, and to increase the demand of that task so that it now occupies all of the capacity of the resource. See Fig. 4.3 for an example where this reformulation increases the energy bound from 9 to 10. In the rest of this chapter, we will introduce a new method to compute lower bounds for the cumulative resource problem. We will show that it also yields a new, valid, cumulative constraint which can itself be passed to existing filtering algorithms for further propagation. We will then show that after a precomputation phase which is independent from the instance, computing this redundant 4.2. A COMPACT LP FOR PREEMPTIVE CUMULATIVE SCHEDULING 75 constraint for a given problem can be done very effectively. Then we will provide some experimental results. A similar approach was introduced in [CN07] under the name of dual feasible functions. We will show that our approach generalizes and supersedes the dual feasible functions based redundant constraints. 4 We are given a discrete cumulative resource of capacity C and a set of tasks, each with a fixed length and demand level. We want to compute a lower bound of the makespan of non-preemptive schedules of the set of tasks on the resource. A compact LP for preemptive cumulative scheduling 4     0 0 0 1         1 0 1 0         2 1 0 0         0 2 0 0         4 0 0 0         P 1 P 2 P 3 P 4         0 0 0 1         1 0 1 0         2 1 0 0         P 1 P 2 P 3 P 4     We now introduce tasks configurations: a configuration is an integer vector P of size C such that ∑ C c=1 P c × c = C. P c is the number of tasks of demand c in configuration P. Thus, the configurations are exactly the integer partitions of 76 CHAPTER 4. CUMULATIVE STRENGTHENING C (see Figure 4.4). We denote them P. x P can be interpreted as the time during which configuration P is run. A configuration indicates a number of tasks for each demand level which can run at the same time on the cumulative resource. See Figure 4.5 for an example, where the resource has capacity C = 4: the configuration P = (2, 1, 0, 0) corresponds to the possibility of running 2 tasks of demand 1 and 1 task of demand 2 simultaneously on the resource. Another possible configuration could be P = (4, 0, 0, 0), corresponding to running 4 tasks of demand 1 simultaneously. Note that these are demands configurations, not tasks configurations as often seen with configuration LPs. A key idea in our approach is the use of non-superposition constraints: a task cannot run in parallel of itself, so it can only use one slot on a given configuration. Generalizing this idea, any j tasks of demand c can only occupy up to j slots of demand c on a configuration, otherwise there would be at least a task occupying two slots on a configuration. This remark justifies the second set of constraints in the linear program below, called non-superposition constraints. For each demand c ≤ C, we call s c the sum of the lengths of tasks of demand c, and for 1 ≤ j ≤ C c -1, s c,j the sum of the lengths of the j longest tasks of demand c: ∀1 ≤ c ≤ C, s c = ∑ i:c i =c p i ∀1 ≤ c ≤ C, ∀1 ≤ j ≤ C c -1, s c,j = max I: |I| = j ∀i ∈ I, c i = c ∑ i∈I p i Given the set P of all combinations of the demands of tasks which can be run at the same time, we can show that the optimal value of the following LP is the minimal makespan for a preemptive schedule which satisfies the cumulative constraint: min ∑ P∈P x P s.t. ∀c ∈ [1, C] ∑ P∈P P c x P ≥ s c (4.1) ∀c ∈ [1, C], ∀j ∈ [1, C c -1] ∑ P∈P min(P c , j)x P ≥ s c,j (4.2) ∀P ∈ P x P ≥ 0 The first set of constraints, labelled (4.1), states that for each demand level c, the total time allocated on the resource for tasks of demand c should be greater than the total length of these tasks. Indeed, P c is the number of slots for tasks of 4.2. A COMPACT LP FOR PREEMPTIVE CUMULATIVE SCHEDULING 77 demand c on configuration P and x P is the number of times that configuration P is selected. The sum runs over all the configurations. The second set of constraints, labelled (4.2), are the non-superposition constraints as introduced above. They ensure that whenever we schedule only j tasks of the same demand level, we take into account at most j occurrences of this demand level in each configuration, so that a given task cannot occupy more than its demand in any configuration. Let us illustrate this on a simple example: if we have a resource of capacity 3, and two tasks, both of demand 1, one of length l and the other of length 2l, as illustrated in Figure 4.6, then without the constraints (4.2) the value of the best LP solution would be l. Indeed, with C = 3, the configurations are P 3 = (3, 0, 0), (1, 1, 0), (0, 0, 1). If we label these configurations respectively 0, 1 and 2, the linear program without the non-superposition constraints becomes: min x 0 + x 1 + x 2 s.t. 3x 0 + x 1 ≥ 3l x 0 ≥ 0 x 1 ≥ 0 x 2 ≥ 0 The optimal solution to this program is x 0 = l, x 1 = 0, x 2 = 0, as shown in Figure 4.7. This is a poor solution since the lower bound is not even as long as the longest task. Non-superposition constraints prevent this situation from occuring. Indeed, the non-superposition constraints in this case are: x 0 ≥ 2l 2x 0 ≥ 3l With these additional constraints, the optimal solution is now x 0 = 2l, x 1 = 0, x 2 = 0, and we get a lower bound of 2l on the makespan, as expected. CHAPTER 4. CUMULATIVE STRENGTHENING It is enough for the non-superposition constraints to be satisfied for any j tasks to ensure that the inequality stands with respect to the length of the j longest tasks. These constraints distinguish our reformulation from previous approaches: as we will prove now, these non-superposition constraints are quite strong: they are indeed a necessary and sufficient condition to get a preemptive schedule. Proposition 3. Given a discrete cumulative resource of capacity C and a set of n tasks with respective lengths p i and demands c i , the energy bound E of the best reformulation with non-superposition constraints is equal to the makespan M of the optimal preemptive schedule. Proof. We show that the energy bound E, as defined in definition 2, in our reformulation is equal to the makespan M of the optimal preemptive schedule. First, we show that E ≤ M. Indeed, when we have a preemptive schedule, all tasks have enough time and resources to be completed (so the constraints (4.1) in the primal are satisfied), and the non-superposition constraints are enforced (so the constraints (4.2) are satisfied). All the constraints of the primal are satisfied, and we can derive a feasible solution of the LP. We now show that M ≤ E. To this end, given a solution to the primal LP, that is the duration we will spend in each configuration, we will construct a preemptive schedule. Independently for each demand level c ∈ [1, C], we construct a staircase, representing the slots available to tasks of demand c. The staircase for tasks of demand c is built in the following way: we consider the configurations P ∈ P in the order of decreasing values of P c , and for each of them we add to the staircase a column of demand P c and width x P . We can interpret the staircase as a superposition of C c horizontal lines, ordered by lengths (the shortest lines on the top, the longest at the bottom). We denote l 1 , l 2 , ..., l C c the lengths of the lines, with l 1 ≥ l 2 ≥ ... ≥ l C c . See Figure 4.8 for an example showing different configurations with slots of demand c, shown in white, and slots of other demand levels, shown in dashed lines. On this example, we have a cumulative of capacity 9, and 5 tasks of demand 2 and of respective lengths 15, 13, 9, 8 and 6. The lines can be interpreted as independent machines, and we now assign the tasks of demand c to the configurations in the following manner, similar to McNaughton's method [START_REF] Mcnaughton | Scheduling with Deadlines and Loss Functions[END_REF]: we start with the longest task, and place it on the topmost line, starting from the rightmost free slot. If we fill the whole line, we continue on the next line, starting again from the right, without going further than the rightmost slot occupied by the task on the line above, so that a task does not occupy two slots of the same configuration. If necessary, we repeat until we can place all the task on the staircase. See Figure 4.9 for an example of placing one task. The goal is to preserve a staircase structure after removing the positions covered by the longest task. We then repeat this process with the following tasks, considered in the order of decreasing lengths. See Figure 4.10 for an example, the positions on each line are marked with the number of a task. Let us show that a schedule constructed in this fashion is preemptive. There is enough time for all of the tasks to run (constraints (4.1) of the primal), so we will never have to exceed the capacity of the last line. What remains to prove is that no job has to be scheduled twice on the same configuration. We will prove this by showing that the staircase invariant ∀k ≤ C c , ∑ k j=1 l j ≥ ∑ k j=1 p j (where p 1 ≥ p 2 ≥ ... ≥ p k ) is preserved after placing the longest task as described above. CHAPTER 4. CUMULATIVE STRENGTHENING The case k = 1 is obvious: there is only one line, so there can not be a superposition. Otherwise, assume that the invariant is satisfied for the staircase of lengths l. We will place the longest task according to the rule above and show that the invariant is still satisfied. We define s as the largest integer such that p 1 ∈ [l s+1 , l s [. All the lines of index greater than s will be completely filled when placing the longest task. We call l the length of the staircase lines after placing the task. There are three cases to distinguish: If k < s, then ∑ k j=1 l j = ∑ k j=1 l j ≥ ∑ k j=1 p j ≥ ∑ k j=1 p j -p 1 . (The task p 1 is not placed on line k). If k = s, then ∑ k j=1 l j = ∑ k j=1 l j -(p 1 -l s+1 ) = ∑ k+1 j=1 l j -p 1 ≥ ∑ k+1 j=1 p j -p 1 ≥ ∑ k j=1 p j -p 1 . (The task p 1 occupies the end of line k). If k > s, then ∑ k j=1 l j = ∑ k j=1 l j -(p 1 -l s+1 ) -∑ k j=s+1 (l j -l j+1 ) = ∑ k+1 j=1 l j - p 1 ≥ ∑ k+1 j=1 p j -p 1 ≥ ∑ k j=1 p j -p 1 . (The task p 1 occupies all of line k). This proves that by construction, one can always place the longest task on the staircase with no superposition, the remaining spots are still staircase-shaped, and the total staircase surface is equal to the total task surface. Using this construction for each demand level, we can assign locations on the configurations to tasks independently for the tasks of a given demand. We repeat this construction for every demand c ∈ [1, C]. Since we do not consider time or precedence constraints, changing the order in which configurations are run preserves a preemptive schedule. Reformulation We now consider the dual of the previous linear programming formulation: max C ∑ c=1   s c h c + C c -1 ∑ j=1 s c,j h c,j   s.t. ∀P ∈ P C ∑ c=1   P c h c + C c -1 ∑ j=1 min(P c , j)h c,j   ≤ 1 ∀c ∈ [1, C] (h c ≥ 0 and ∀j ∈ [1, C c -1] h c,j ≥ 0) Here we introduce variables h c for each c ∈ [1, C] and h c,j for each c ∈ [1, C] and 1 ≤ j ≤ C c -1, corresponding to constraints in the primal linear program. The constraints state that all reformulations must remain feasible, and we search for the reformulation which gives the largest energy reasoning bound. REFORMULATION 81 This dual formulation is particularly interesting in that we can interpret the dual variables as corresponding to demands of a new valid cumulative constraint, which can be used with all filtering techniques for cumulative constraints, notably edge-finding. In contrast, previous linear programming approaches to this problem (e.g. [START_REF] Brucker | A Linear Programming and Constraint Propagation-Based Lower Bound for the RCPSP[END_REF][START_REF] Baptiste | Tight LP Bounds for Resource Constrained Project Scheduling[END_REF]) used formulations similar to the primal formulation above, but did not make use of the new information about the cumulative constraint. More formally, we can define a new valid cumulative constraint as follows: Proposition 4. Given a solution to this linear program, if a cumulative resource of capacity C is reformulated with a capacity of 1, and that the tasks are ajusted so that the demand of the i th longest task of initial demand c becomes h c + ∑ j≥i h c,j , then all previously feasible schedules are still feasible with the new cumulative resource. Proof. Schedules which are feasible with the original cumulative resource are such that at any point in time, the tasks I being executed satisfy the constraints ∑ i∈I c i ≤ C, so the set of demands of the tasks being executed is contained in one of the configurations P of P. We can now rewrite the constraint of the linear program above as follows: C ∑ c=1   P c h c + C c -1 ∑ j=1 min(P c , j)h c,j   ≤ 1 ⇔ C ∑ c=1   P c ∑ i=1 h c + C c -1 ∑ j=1 min(P c ,j) ∑ i=1 h c,j   ≤ 1 ⇔ C ∑ c=1   P c ∑ i=1 h c + P c ∑ i=1 C c -1 ∑ j=i h c,j   ≤ 1 ⇔ C ∑ c=1 P c ∑ i=1   h c + C c -1 ∑ j=i h c,j   ≤ 1 This constraint precisely states that for any configuration P ∈ P, if we change the demands of all the tasks in such a way that the demand of the i th longest task of initial demand c becomes h c + ∑ j≥i h c,j , then the new demands of the tasks in P will sum to less than 1. This proves that the schedule is still feasible at any point in time after the reformulation. In constraint-based scheduling, the demands are traditionally integer numbers, while the demands given in the proposition above are fractional and smaller than 1. In an implementation, one can multiply each value by the LCM of the denominators of the new demands, to work with integer values only. This reformulation is stronger than the original constraint, and because the energy lower bound is equal to the minimal makespan for a preemptive schedule, as we have shown previously, the standard propagation algorithms for cumulative constraints should propagate better with this constraint than with the original constraint. Before we can do so, we have to find a way to compute it in an efficient way. Precomputing the vertices Note that since this reformulation is a mapping of the task capacities, the size of the underlying polytope does not depend on the number of tasks involved, but only on the capacity of the resource. Note also that the instance-specific data, the s c and s c,j values, only appear in the objective function but not in the constraints. Thus we can precompute the vertices of the polytope associated with this linear program. A convenient way of doing it is to enumerate the vertices of this polytope using LRS [START_REF] Avis | LRS: A Revised Implementation of the Reverse Search Vertex Enumeration Algorithm[END_REF]. To solve this program we then just have to evaluate its objective function over the vertices which were precomputed. Using a dominance property we show that on some of the vertices, none of the objective functions we consider will reach its maximum, and we eliminate these vertices. Indeed, we are only interested in those vertices that lead to reformulations in which at least one task gets assigned a higher demand than in another formulation. In other words, a reformulation h (solution vector to the LP above) is dominated if there exists a reformulation h , such that ∀i, ∀c, h c + ∑ j≥i h c,j ≤ h c + ∑ j≥i h c,j . We eliminate dominated reformulations. Reciprocally, we noticed in our experiments that all remaining reformulations are useful and can yield the best bound on different instances. After eliminating all the dominated vertices, the number of vertices that remain is relatively small, and is practical up to C = 12 (from about hundred thousands vertices in the polytope, only a few hundreds vertices are not dominated). Discussion As an example, here are the non-dominated solutions for C=3: 4.6. COMPARISON WITH DUAL FEASIBLE FUNCTIONS 83 h 1,1 h 1,2 h 1 h 2 h 3 0 0 1 3 2 3 1 0 0 0 1 1 0 1 2 0 1 2 1 1 0 0 0 1 The first row corresponds to no reformulation. The second row gives a bound which is the sum of the lengths of tasks of demand 2 and 3 (since no two tasks of demand 2 can be executed at the same time), and was used to reformulate the instance in Figure 4.2. The third row is the most interesting, and gives a bound which is the length of tasks of demand 3, plus half the length of tasks of demand 2, plus half the length of the two longest tasks of demand 1. Finally, the fourth row gives a bound which is the length of tasks of demand 3, plus the length of the longest task of demand 1, and was used to reformulate the instance in Figure 4.3. It is interesting to notice that if all tasks are of demand 1, we find Mc-Naughton [START_REF] Mcnaughton | Scheduling with Deadlines and Loss Functions[END_REF] bound for parallel machines. Indeed, in this case the linear program reduces to: max s 1 h 1 + ∑ 1≤j<C s 1,j h 1,j s.t. Ch 1 + ∑ 1≤j≤C-1 jh 1,j ≤ 1 (h 1 ≥ 0 and ∀j ∈ [1, C -1] h 1,j ≥ 0) The optimal solution to this linear program is max( s 1 C , s 1,1 ), which is Mc-Naughton's bound. Comparison with dual feasible functions Dual feasible functions, used in [START_REF] Carlier | Computing Redundant Resources for the Resource Constrained Project Scheduling Problem[END_REF] to get lower bounds for the energy of a set of tasks on a cumulative resource, can also be used to reformulate cumulative resources in a similar fashion to what was done above, by looking at the dual formulation. The valid reformulations obtained using dual feasible functions change the demand of all tasks of original demand c into h c , for h c values which satisfy the ∀P ∈ P C ∑ c=1 P c h c ≤ 1 (4.3) ∀c ∈ [1, C] h c ≥ 0 We can notice that the constraint polytope of this program is the master knapsack polytope. In a similar fashion, the valid reformulations that we introduce change the demand of the i th longest task of initial demand c into h c + ∑ j≥i h c,j , for h c and h c,j values which satisfy the following program: max C ∑ c=1 s c h c + C ∑ c=1 ∑ 1≤j<C/c s c,j h c,j s.t. ∀P ∈ P C ∑ c=1 P c h c + C ∑ c=1 ∑ j<C/c min(P c , j)h c,j ≤ 1 (4.4) ∀c ∈ [1, C] (h c ≥ 0 and ∀j ∈ [1, C/c[ h c,j ≥ 0) Proposition 5. The reformulations induced by dual feasible functions are dominated by the preemptive reformulations. Proof. All solutions of (4.3) are feasible for (4.4), if we set all the h c,j variables to 0, with the same objective value, so the optimal preemptive reformulation dominates the optimal DFF reformulation. Additional constraints We can stightly strenghten this formulation by adding, in the primal program, constraints which state that ∀c ∈ [1, C[, ∑ P∈P min(P c , 1)x P must be greater than the minimum length of a bin in a bin-packing of the tasks of demand c into C c bins. We call these additional constraints bin-packing constraints. They are justified by the fact that each configuration contains at most C c slots for tasks of demand c, and that the assignment of tasks to slots can be viewed as a bin-packing problem. EXPERIMENTS AND RESULTS 85 One possibility to add these contraints is to compute an optimal bin-packing of the tasks of demand c for each c ∈ [1, C[. Another possibility is to use the bound of Marcello and Toth [START_REF] Lodi | Two-Dimensional Packing Problems: a Survey[END_REF], which in this case reduces to ∀c ∈ [1, C[, ∑ P∈P min(P c , 1)x P ≥ p C c + p C c +1 . These two ways of formulating the constraint are both convenient once translated into the dual formulation, which we use to compute the redundant cumulative resource. Experiments and results We ran some experiments by manually adding redundant constraints to RCPSP instances from the PSPLIB [START_REF] Kolisch | PSPLIB -a Project Scheduling Problem Library[END_REF]. In practice, it is too expensive to try all of the reformulations. We have to rely on heuristics to select them, hoping that it will strengthen the propagations enough. The best results were obtained when adding a single redundant constraint per cumulative constraint in the original problem. Choosing the redundant cumulative constraint with the highest global lower bound (highest objective value in the linear program) worked the best also to improve the propagations made by edge-finding. A completely different problem, which we do not address in this Chapter, consists in choosing the best reformulation. We made some progress in this direction, which we report in the Outlook chapter of this thesis, in Section 6.2. We solved the instances with CP Optimizer 12.3 running on a Core i5-2520m processor with a time limit of 10 minutes. We used the default search strategy (Restart), and we set the IloCP::CumulFunctionInferenceLevel parameter to IloCP::Medium, which activates the edge-finding algorithm described in [START_REF] Vilím | Timetable Edge Finding Filtering Algorithm for Discrete Cumulative Resources[END_REF]. Our results show that on average, the energy lower bounds we compute are about 10% better than with the original cumulative resource. This compares to an approximate 6% improvement when using a DFF over the original bounds, but the advantage of our reformulations over DFF is stronger when we consider sets with a small number of tasks, such as the sets on which the edge-finder algorithm typically propagates. These stronger bounds enable stronger propagations, and by using destructive bounds we managed to improve the lower bounds for some of the J60 RCPSP instances of the PSPLIB: inst Conclusion It is remarkable that our LP formulation for computing the makespan of a preemptive schedule on a cumulative resource is independent of the number of tasks, and depends only on the capacity of the resource. This formulation gives a new light on the structure of preemptive relaxations to cumulative problems. Since all parameters specific to the instance can be encoded in the objective function only, this enables us to precompute the vertices of the LP polytope, so that solving the LP in practice is extremely fast. From a constraint programming point of view, this formulation yields redundant cumulative resources which can be exploited by other algorithms for cumulative resources such as edge-finding. Note also that our reformulation subsumes the Dual Feasible Functions reformulation [START_REF] Carlier | Computing Redundant Resources for the Resource Constrained Project Scheduling Problem[END_REF]. Using a very similar method to the one shown in this chapter, other results from the literature, such as [MMRB98, BK00, CN03], could also be turned into a redundant cumulative constraint, but the computations would be much more expensive. We think that our approach of aggregating the activities by similar demands, except for the few longest ones, hits an interesting trade-off between the quality of the bound (notably since we can prove that it is equivalent to a preemptive relaxation of the problem) and the size of the formulation, which stays independent from the number of tasks. Chapter 5 Fast Energy Reasoning In this chapter, we show how to reduce the complexity of the Energy Reasoning propagation, introduced above in Subsection 3.4.5, from O(n 3 ) to O(n 2 log n). This chapter has been published in preliminary form as [START_REF] Bonifas | A O (n^2 log (n)) propagation for the Energy Reasoning[END_REF]. Introduction Energy reasoning has been known for 25 years and is the strongest propagation known for the cumulative constraint (with the exception of not-first, not-last, none of those propagations dominating each other) for the cumulative constraint (full propagation is NP-hard). Improving the propagation of energy reasoning has sparked some interest recently [START_REF] Derrien | A New Characterization of Relevant Intervals for Energetic Reasoning[END_REF], but the best algorithm to date has a complexity of O(n 3 ) for n tasks. This high complexity is the reason why this propagation is seldom used in practice and other, weaker but faster, propagations were introduced. In this chapter, we introduce techniques to propagate energy reasoning in time O(n 2 log n). This is an important theoretical advance to a long-standing open question. Moreover, our experiments suggest that this algorithm should also be of practical interest for difficult problems. Our approach is based on three novel ideas, which form the next sections of the chapter. First, we proved that the so-called additional rule supersedes the other energy reasoning rules in the literature, and we can study this case only. We then show that detecting a propagation with the additional rule reduces to computing the maximum of a set of piecewise affine functions with special properties that we make use of. Finally, we give an algorithm to efficiently compute this maximum by using the point-line duality of projective geometry to reduce the problem to a convex hull computation. We conclude by discussing this new method and giving experimental results. 88 CHAPTER 5. FAST ENERGY REASONING Energy reasoning rules Given a cumulative resource of capacity C, tasks of consumption c i and of length p i , we respectively denote est i and lst i their earliest starting time and latest starting time, in the context of constraint-based scheduling. We say that a task is left-shifted if we try to schedule it as early as possible, that is from est i to est i + p i , and similarly we say that it is right-shifted if we try to schedule it as late as possible, that is from lst i to lst i + p i . We also denote W i (a, b) with b ≥ a the intersection energy of task i in the interval [a, b), that is W i (a, b) = c i • min(b -a, p + i (a), p - i (b)) with p + i (a) being the length of time during which task i executes after a if it is left-shifted, and p - i (b) being the length of time during which task i executes before b if it is right-shifted. The expression of p + i (a) in terms of the est i and lst i variables is Finally, W(a, b) is the sum of the intersection energies of all tasks: p + i (a) = max(0, min(p i , est i + p i -a)) and p - i (b) = max(0, min(p i , b -lst i )). t left shift right shift est i eet i lst i let i a b p + i (a) p - i (b) W(a, b) = ∑ i W i (a, b). In the rest of this chapter, the propagations will be described in the case when we shift the task under consideration as much as possible to the left. There is obviously a symmetric version of all of them in the case when we shift the task to the right. The energy reasoning rule is: If W =i (a, b) + c i • min(b -a, p + i (a)) > C(b -a) then est i ≥ b - C(b -a) -W =i (a, b) c i . One can show that this rule supersedes the other rules in the literature. t p + i (a) a b 0 C c i W =i A i (a) Energy overflow in [a, b[ t a b 0 C W =i A i new est i ( Propagation conditions We will now establish a simple condition to detect when the energy reasoning rule applies. Its application condition is: C • (b -a) -c i • min(p + i (a), b -a) < W =i (a, b) ⇔ C • (b -a) -c i • min(p + i (a), b -a) < W(a, b) -W i (a, b) ⇔ C • (b -a) -W(a, b) < c i • min(p + i (a), b -a) -W i (a, b) The left-hand side of this inequality is constant on a time window, and the right-hand side depends on task i only. For each window, we thus compute max i c i • min(p + i (a), b -a) -W i (a, b) . If this value does not exceed C • (b - a) -W(a, b), there is no excess of intersection energy. If the maximum ex- ceeds C • (b -a) -W(a, b), we know that we should propagate on the task argmax i c i • min(p + i (a), b -a) -W i (a, b ) (and maybe on other tasks, but since our goal is just to detect if there is or not a propagation, we ignore these for now). 90 CHAPTER 5. FAST ENERGY REASONING Efficient detection of intervals with an excess of intersection energy As we saw in the previous section, we can find all intervals on which to propagate if we compute argmax i c i • min(p + i (a), ba) -W i (a, b) for all windows [a, b). We will now show a way of finding the maximum of c i • min(p + i (a), b - a) -W i (a, b) for all dates b in time O(n log n). We define the function f i,a (b More precisely, there are three cases to distinguish, depending on the relative positions of a and the two shifts. In all of these three cases, the function is continuous, piecewise-linear, with a slope of +c i , 0 or -c i depending on the piece, and the coordinates of the slope changes are indicated on Figure 5.4. H is the planning horizon (upper bound on the makespan). There are three cases to distinguish, depending on the relative position of a with est i and lst i : ) := c i • min(p + i (a), b -a) -W i ( • When est i > a, f i,a is constant and equal to 0 from a to est i , has a slope +c i to min(est i + p i , lst i ), is constant to max(est i + p i , lst i ), has a slope -c i to lst i + p i , and is constant to H. • When est i ≤ a < lst i , f i,a has a slope +c i to min(a + p + i (a), lst i ), is constant to max(a + p + i (a), lst i ), has a slope -c i to lst i + p i , and is constant to H. • Finally, when a ≥ lst i , f i,a is constant to a + p + i (a), has a slope -c i to lst i + p i , and is constant to H. b 0 • a • est i • min(est i + p i , lst i ) • max(est i + p i , lst i ) Remember that we want to compute the supremum over i of the f i,a functions. Since these functions have a very special structure, we will resort to geometry instead of analysis to compute their supremum. Specifically, we decompose these functions into line segments, and use the algorithm described in section 15.3.2 of [START_REF] Boissonnat | Algorithmic Geometry[END_REF] to compute the upper envelope of these segments. This will exactly yield the function max i f i,a . • lst i + p i • H +c i -c i (a) f i,a when est i > a. b 0 • a • min(a + p + i (a), lst i ) • max(a + p + i (a), lst i ) • lst i + p i • H +c i -c i (b) f i,a when est i ≤ a < lst i . b 0 • a • a + p + i (a) • lst i + p i • H -c i (c) f i,a when a ≥ lst i . In our context, this algorithm works as follows: the end dates of the O(n) line segments are projected on the x-axis and define O(n) non-overlapping intervals. A balanced binary tree is built, whose leaves are these non-overlapping intervals in ascending order. To a node of the tree corresponds the union of the intervals of the leaves of the subtree. Each line segment is assigned to the node deepest in the tree which contains the projected endpoints of the segment. The upper envelope of all the segments assigned to one node can be computed in time linear with the number of segments at the node. Now, since the upper envelopes of two nodes at the same level in the tree do not overlap (by construction of the tree), we can merge them in linear time, and we can merge all the envelope of one level of the tree in time O(n log n). Finally, we can merge the envelopes of the O(log n) levels of the tree in time O(nα(n) log log n) (cf. section 15.3.2 of [BY98]), which is O(n log n). We now have an algorithm to compute the supremum of O(n) line segments in time O(n log n). Complete algorithm and complexity analysis An important property of energy reasoning, which we make use of, is that when we have n tasks, only O(n) starting dates a have to be considered as candidates for the intervals [a, b) [EL90, BPN01, DP14]. More precisely, when the tasks are left-shifted, the a dates to be considered are the O(n) dates in the set O 1 = {est i , 1 ≤ i ≤ n} ∪ {lst i , 1 ≤ i ≤ n} ∪ {est i + p i , 1 ≤ i ≤ n} (cf. Proposi- tion 19 in chapter 3 of [BPN01] ). With our algorithm, we will study at once all dates b located after a. Actually, even less dates a can be studied by using the stronger characterization from [START_REF] Derrien | A New Characterization of Relevant Intervals for Energetic Reasoning[END_REF]. Therefore there are only O(n) interesting starting dates to consider, on which we might propagate. We start by precomputing all O(n 2 ) values W(a, b) in time O(n 2 ) with the algorithm described in section 3.3.6.2 of [START_REF] Baptiste | Constraint-Based Scheduling: Applying Constraint Programming to Scheduling Problems[END_REF]. Then, for every interesting date a, we use the algorithm of the previous section to detect interesting intervals. Since there are O(n) dates to study and that the study of each of them has complexity O(n log n), the complexity of this step is O(n 2 log n). If we find a propagation, we adjust the earliest starting time of the task according to the additional energy reasoning rule, which is done in constant time. Finally, the total complexity with this algorithm of finding one date adjustment, or finding that there is nothing to propagate, is O(n 2 log n). Moreover, we run the energetic checker (cf. [START_REF] Baptiste | Constraint-Based Scheduling: Applying Constraint Programming to Scheduling Problems[END_REF]) before our algorithm. The energetic checker is a O(n 2 ) test that either guarantees that energy reasoning will not find a propagation, in which case we do not need to run the energy reasoning, or that it will find one without telling which one, in which case we do run the energy reasoning. Input ∈ {est i , 1 ≤ i ≤ n} ∪ {lst i , 1 ≤ i ≤ n} ∪ {est i + p i , 1 ≤ i ≤ n} do for i ← 1 to n do I ← ∅ if est i > a do if C • (b -a) -W(a, b) < f a (b) then i ← label of current segment est i ← b -C•(b-a)-W(a,b)+W i (a,b) c i end end end Algorithm 1: Fast Energy Reasoning Discussion In a similar fashion to what is done with classical algorithms to propagate the energy reasoning, we can then re-apply this algorithm on the adjusted dates until we reach a fixpoint and there is nothing else to propagate. Since this algorithm will detect at least one excess of intersection energy if there is one, and we showed already that energy reasoning propagates exactly when there is an excess of intersection energy, this algorithm will make at least one energy reasoning adjustment if the classical O(n 3 ) algorithm would have done so. The propagation we get is equivalent to the O(n 3 ) algorithm. One difference though is in the rare case where the intersection energy is exceeded for several tasks on the same interval [a, b). In this case this algorithm will only tighten the bound for the task with the highest excess of interval energy, while the original algorithm would have made the adjustments for all the tasks. We believe that in the context of constraint programming, when propagations are rare and we need to run the algorithm iteratively until we reach the fix point anyway, the relevant question is "Given that the O(n 2 ) checker reports that there is something to propagate, how to find one such propagation as quickly as possible?" In this case our algorithm needs time O(n 2 log n) versus O(n 3 ) for the original algorithm. More generally, if there are k ≤ n tasks for which to propagate, our algorithm needs time O(kn 2 log n) versus O(n 3 ) for the original algorithm and thus constitutes an improvement. Indeed, in a complete CP solver, global constraints are called from a propagation queue, and the heaviest propagations are only used when lighter propagations cannot reduce the domains anymore. In practice, once the Energy Reasoning algorithm finds a propagation, it will be stopped there and all lighter propagations will be tried again on the newly reduced instance for further (and less costly) improvements. We implemented this algorithm on top of IBM Ilog CP Optimizer 12.6 and noticed that this improved algorithm gives a performance improvement in practice, on hard instances which require the use of energy reasoning. Chapter 6 Outlook We have presented in the previous chapters two new techniques for dealing with the cumulative constraint. These offer a theoretical contribution by the use of new methods in constraint programming, and they both have a practical interest in solving difficult feasibility problems. This is particularly the case with Failure Directed Search, where the combined use of Cumulative Strengthening and fast Energy Reasoning offer the potential to solve previously inaccessible feasibility problems. More generally, these improvements lay the path for further improvements in cumulative scheduling, of which we give a selection in this section. Static selection of a reformulation In practice, it is far too costly and counterproductive to use all the cumulative reformulations that can be generated by Cumulative Strengthening. Thus, there is still a lot of work to be done in determining which reformulations will or will not help a given propagation algorithm in a given context. Research tracks were laid down during a visit to François Clautiaux in Bordeaux. Indeed, it seems that the gap to the optimal reformulation reduces as an inverse exponential of the number of reformulations considered. It is therefore probable that a small number of reformulations will make it possible to obtain the bulk of the power of this technique at a very low cost. We have two directions to study this phenomenon: one is experimental, by measuring the contribution of a small number of randomly chosen reformulations, and the other is theoretical, aiming at quantifying the energy improvement offered by a given set of reformulations. We have already identified some properties to characterize the cumulative resources for which a given reformulation will improve the energy bound. 95 CHAPTER 6. OUTLOOK Finally, another direction is to try to adapt the work on Vector Packing Dual Feasible Functions (VP-DFF) to quickly compute reformulations which take into account several cumulative constraints on the same intervals. Dynamic computation of a reformulation within propagators A second line of research consists in computing reformulations directly within the propagation algorithms, in order to exploit the full potential of cumulative strengthening. In this context, it is also possible to exploit more information than the cumulative constraints only, as described above. Notably, the state of the timetable as well as the precedence graph can be used. This allows for much stronger reformulations. I directed the M1 internship of Réda Bousseta on this subject and I gave a preliminary presentation on this work at the ROADEF 2017 conference [START_REF] Bonifas | Boosting cumulative propagations with cumulative strengthening[END_REF]. We show in this work how to compute, within the Edge-Finder, reformulations adapted to the set of activities being examined. This makes it possible to obtain much stronger bounds over this set than with a reformulation of the whole instance. The basic Edge-Finder, thus enhanced, gains considerable propagation strength, and we show that it allows for deductions that the unreinforced Timetable Edge-Finder can not find. We also show how to integrate the same idea within Precedence Energy and Energy Reasoning. Integration with the Failure Directed Search Finally, the newly introduced Failure Directed Search technique (see Subsection 2.5.3.4) to improve the dual bounds for very difficult instances, such as those from the PSPLIB, are based on using strong propagations. The two techniques introduced in this thesis contribute to this: the O(n 2 log n) Energy Reasoning can be used instead of the weaker algorithms previously used, and the Cumulative Strengthening can reinforce the deductions made by Energy Reasoning. Using these two tools jointly, within the Failure Directed Search, offers the potential to dramatically improve dual bounds for certain open instances of the PSPLIB which have been long-standing and hitherto unattainable. Introduction One of the fundamental open problems in optimization and discrete geometry is the question whether the diameter of a polyhedron can be bounded by a polynomial in the dimension and the number of its defining inequalities. The problem is readily explained: A polyhedron is a set of the form P = {x ∈ R n : Ax b}, where A ∈ R m×n is a matrix and b ∈ R m is an m-dimensional vector. A vertex of P is a point x * ∈ P such that there exist n linearly independent rows of A whose corresponding inequalities of Ax b are satisfied by x * with equality. Throughout this paper, we assume that the polyhedron P is pointed, i.e. it has vertices, which is equivalent to saying that the matrix A has full column-rank. Two different vertices x * and y * are neighbors if they are the endpoints of an edge of the polyhedron, i.e. there exist n -1 linearly independent rows of A whose corresponding inequalities of Ax b are satisfied with equality both by x * and y * . In this way, we obtain the undirected polyhedral graph with edges being pairs of neighboring vertices of P . This graph is connected. The diameter of P is the smallest natural number that bounds the length of a shortest path between any pair of vertices in this graph. The question is now as follows: Can the diameter of a polyhedron P = {x ∈ R n : Ax b} be bounded by a polynomial in m and n? The belief in a positive answer to this question is called the polynomial Hirsch conjecture. Despite a lot of research effort during the last 50 years, the gap between lower and upper bounds on the diameter remains huge. While, when the dimension n is fixed, the diameter can be bounded by a linear function of m [14,2], for the general case the best upper bound, due to Kalai and Kleitman [11], * An extended abstract of this paper was presented at the 28-th annual ACM symposium on Computational Geometry (SOCG 12) † LIX, École Polytechnique, Palaiseau and IBM, Gentilly (France). [email protected] ‡ Dipartimento di Matematica, Università di Padova (Italy). [email protected] § Ecole Polytechnique Fédérale de Lausanne (Switzerland). [email protected] ¶ Technische Universität Berlin (Germany). [email protected] || Technische Universität Berlin (Germany). [email protected] 106 APPENDIX is m 1+log n . The best lower bound is of the form (1 + ε) • m for some ε > 0 in fixed and sufficiently large dimension n. This is due to a celebrated result of Santos [19] who disproved the, until then longstanding, original Hirsch conjecture for polytopes. The Hirsch conjecture stated that the diameter of a bounded polyhedron1 is at most mn. Interestingly, this huge gap (polynomial versus quasipolynomial) is also not closed in a very simple combinatorial abstraction of polyhedral graphs [6]. However, it was shown by Vershynin [20] that every polyhedron can be perturbed by a small random amount so that the expected diameter of the resulting polyhedron is bounded by a polynomial in m and n. See Kim and Santos [12] for a recent survey. In light of the importance and apparent difficulty of the open question above, many researchers have shown that it can be answered in an affirmative way in some special cases. Naddef [17] proved that the Hirsch conjecture holds true for 0/1-polytopes. Orlin [18] provided a quadratic upper bound for flow-polytopes. Brightwell et al. [3] showed that the diameter of the transportation polytope is linear in m and n, and a similar result holds for the dual of a transportation polytope [1] and the axial 3-way transportation polytope [4]. The results on flow polytopes and classical transportation polytopes concern polyhedra defined by totally unimodular matrices, i.e., integer matrices whose sub-determinants are 0, ±1. For such polyhedra Dyer and Frieze [5] had previously shown that the diameter is bounded by a polynomial in n and m. Their bound is O(m 16 n 3 (log mn) 3 ). Their result is also algorithmic: they show that there exists a randomized simplex-algorithm that solves linear programs defined by totally unimodular matrices in polynomial time. Our main result is a generalization and considerable improvement of the diameter bound of Dyer and Frieze. We show that the diameter of a polyhedron P = {x ∈ R n : Ax b}, with A ∈ Z m×n is bounded by O ∆ 2 n 4 log n∆ . Here, ∆ denotes the largest absolute value of a sub-determinant of A. If P is bounded, i.e., a polytope, then we can show that the diameter of P is at most O ∆ 2 n 3.5 log n∆ . To compare our bound with the one of Dyer and Frieze one has to set ∆ above to one and obtains O n 4 log n and O n 3.5 log n respectively. Notice that our bound is independent of m, i.e., the number of rows of A. The proof method Let u and v be two vertices of P . We estimate the maximum number of iterations of two breadthfirst-search explorations of the polyhedral graph, one initiated at u, the other initiated at v, until a common vertex is discovered. The diameter of P is at most twice this number of iterations. The main idea in the analysis is to reason about the normal cones of vertices of P and to exploit a certain volume expansion property. We can assume that P = {x ∈ R n : Ax b} is non-degenerate, i.e., each vertex has exactly n tight inequalities. This can be achieved by slightly perturbing the right-hand side vector b: in this way the diameter can only grow. We denote the polyhedral graph of P by G P = (V, E ). Let v ∈ V now be a vertex of P . The normal cone C v of v is the set of all vectors c ∈ R n such that v is an optimal solution of the linear program max{c T x : x ∈ R n , Ax b}. The normal cone C v of a vertex of v is a full-dimensional simplicial polyhedral cone. Two vertices v and v are adjacent if and only if C v and C v share a facet. No two distinct normal cones share an interior point. Furthermore, if P is a polytope, then the union of the normal cones of vertices of P is the complete space R n . We now define the volume of a set U ⊆ V of vertices as the volume of the union of the normal cones of U intersected with the unit ball B n = {x ∈ R n : x 2 1}, i.e., vol(U ) := vol v∈U C v ∩ B n . Consider an iteration of breadth-first-search. Let I ⊆ V be the set of vertices that have been discovered so far. Breadth-first-search will next discover the neighborhood of I , which we denote by N (I ). Together with the integrality of A, the bound ∆ on the subdeterminants guarantees that the angle between one facet of a normal cone C v and the opposite ray is not too small. We combine this fact, which we formalize in Lemma 3, with an isoperimetric inequality to show that the volume of N (I ) is large relative to the volume of I . • vol(I ). We provide the proof of this lemma in the next section. Our diameter bound for polytopes is an easy consequence: Theorem 2. Let P = {x ∈ R n : Ax b} be a polytope where all subdeterminants of A ∈ Z m×n are bounded by ∆ in absolute value. The diameter of P is bounded by O ∆ 2 n 3.5 log n∆ . Proof. We estimate the maximum number of iterations of breadth-first-search until the total volume of the discovered vertices exceeds (1/2) • vol(B n ). This is an upper bound on the aforementioned maximum number of iterations of two breadth-first-search explorations until a common vertex is discovered. Suppose we start at vertex v and let I j be the vertices that have been discovered during the first j iterations. We have I o = {v}. If The condition vol( I j ) (1/2) • vol(B n ) implies   1 + 1 π 2 ∆ 2 n 2.5    j vol(I 0 ) 2 n . This is equivalent to j • ln   1 + 1 π 2 ∆ 2 n 2.5    ln(2 n /vol(I 0 )). For 0 x 1 one has ln(1 + x) x/2 and thus the inequality above implies j 2π∆ 2 n 2.5 • ln(2 n /vol(I 0 )). (1) To finish the proof we need a lower bound on vol(I 0 ), i.e., the n-dimensional volume of the set C v ∩ B n . The normal cone C v contains the full-dimensional simplex spanned by 0 and the n row-vectors a i 1 , . . . , a i n of A that correspond to the inequalities of Ax b that are tight at v. Since A is integral, the volume of this simplex is at least 1/n!. Furthermore, if this simplex is scaled by 1/ max{ a i k : k = 1, . . . , n}, then it is contained in the unit ball. Since each component of A is itself a sub-determinant, one has max{ a i k : k = 1, . . . , n} n∆ and thus vol(I 0 ) 1/(n! • n n/2 ∆ n ). It follows that (1) implies j = O ∆ 2 n 3.5 log n∆ . APPENDIX Remarks. The result of Dyer and Frieze [5] is also based on analyzing expansion properties via isoperimetric inequalities. It is our choice of normal cones as the natural geometric representation, and the fact that we only ask for volume expansion instead of expansion of the graph itself, that allows us to get a better bound. Expansion properties of the graphs of general classes of polytopes have also been studied elsewhere in the literature, e.g. [10,9]. Organization of the paper The next section is devoted to a proof of the volume-expansion property, i.e., Lemma 1. The main tool that is used here is a classical isoperimetric inequality that states that among measurable subsets of a sphere with fixed volume, spherical caps have the smallest circumference. Section 3 deals with unbounded polyhedra. Compared to the case of polytopes, the problem that arises here is the fact that the union of the normal cones is not the complete space R n . To tackle this case, we rely on an isoperimetric inequality of Lovász and Simonovits [15]. Finally, we discuss how our bound can be further generalized. In fact, not all sub-determinants of A need to be at most ∆ but merely the entries of A and the (n -1)-dimensional sub-determinants have to be bounded by ∆, which yields a slightly stronger result. Volume expansion This section is devoted to a proof of Lemma 1. Throughout this section, we assume that P = {x ∈ R n : Ax b} is a polytope. We begin with some useful notation. A (not necessarily convex) cone is a subset of R n that is closed under the multiplication with non-negative scalars. The intersection of a cone with the unit ball B n is called a spherical cone. Recall that C v denotes the normal cone of the vertex v of P . We denote the spherical cone C v ∩ B n by S v and, for a subset U ⊆ V , the spherical cone v∈U S v by S U . Our goal is to show that the following inequality holds for each I ⊆ V with vol(S I ) 1 2 vol(B n ): vol(S N (I ) ) 2 π 1 ∆ 2 n 2.5 • vol(S I ). (2) Recall that two vertices are adjacent in G P if and only if their normal cones have a common facet. This means that the neighbors of I are those vertices u for which S u has a facet which is part of the surface of the spherical cone S I . In an iteration of breadth-first-search we thus augment the set of discovered vertices I by those vertices u that can "dock" on S I via a common facet. We call the (n -1)dimensional volume of the surface of a spherical cone S that is not on the sphere, the dockable surface D(S), see Figure 1. The base of S is the intersection of S with the unit sphere. We denote the area of the base by B (S). By area we mean the (n -1)-dimensional measure of some surface. Furthermore, L(S) denotes the length of the relative boundary of the base of S. We use the term length to denote the measure of an (n -2)-dimensional volume, see Figure 1. Given any spherical cone S in the unit ball, the following well-known relations follow from basic integration: vol(S) = B (S) n , D(S) = L(S) n -1 . ( 3 ) To obtain the volume expansion relation (2) we need to bound the dockable surface of a spherical cone from below by its volume and, for a simplicial spherical cone, we need an upper bound on the dockable surface by its volume. More precisely, we show that for every simplicial spherical cone S v one has D(S v ) vol(S v ) ∆ 2 n 3 (4) Once inequalities ( 4) and ( 5) are derived, the bound (2) can be obtained as follows. All of the dockable surface of S I must be "consumed" by the neighbors of I . Using (5) one has thus v∈N (I ) D(S v ) D(S I ) 2n π • vol(S I ). (6) On the other hand, (4) implies v∈N (I ) D(S v ) ∆ 2 n 3 • v∈N (I ) vol(S v ) = ∆ 2 n 3 • vol(S N (I ) ). (7) These last two inequalities imply inequality (2). The remainder of this section is devoted to proving (4) and (5). Area to volume ratio of a spherical simplicial cone We will first derive inequality (4). Lemma 3. Let v be a vertex of P . One has D(S v ) vol(S v ) ∆ 2 n 3 . Proof. Let F be a facet of a spherical cone S v . Let y be the vertex of S v not contained in F . Let Q denote the convex hull of F and y (see Figure 2). We have Q ⊆ S v because S v is convex. Moreover, if h F is the Euclidean distance of y from the hyperplane containing F , then vol(S v ) vol(Q) = area(F ) • h F n . Summing over the facets of S v , we find D(S v ) vol(S v ) = facet F area(F ) vol(S v ) n • facet F 1 h F . ( 8 h F = d (y, H ) 1 n∆ 2 . Plugging this into (8) completes the proof. An isoperimetric inequality for spherical cones We now derive the lower bound (5) on the area to volume ratio for a general spherical cone. To do that, we assume that the spherical cone has the least favorable shape for the area to volume ratio and derive the inequality for cones of this shape. Here one uses classical isoperimetric inequalities. The basic isoperimetric inequality states that the measurable subset of R n with a prescribed volume and minimal area is the ball of this volume. In this paper, we need Lévy's isoperimetric inequality, see e.g. [8, Theorem 2.1], which can be seen as an analogous result for spheres: it states that a measurable subset of the sphere of prescribed area and minimal boundary is a spherical cap. A spherical cone S is a cone of revolution if there exist a vector v and an angle 0 < θ π/2 such that S is the set of vectors in the unit ball that form an angle of at most θ with v: S = x ∈ B n : v T x v x cos θ . Note that a spherical cone is a cone of revolution if and only if its base is a spherical cap. We also observe that two spherical cones of revolution, defined by two different vectors but by the same angle, are always congruent. Therefore, in the following we will only specify the angle of a cone of revolution. inequality for spheres, the optimal shape for such a surface is a spherical cap. As observed above, this corresponds to a cone of revolution. Proof. Using (3), we have to show that L(S) B (S) 2 π n -1 n . (9) This is done in two steps. We first prove that this ratio is minimal for S being the half-ball, i.e., θ = π/2. Then we show that L(S) B (S) 2 π n-1 n holds for the half-ball. Let H be the hyperplane containing the boundary of the base of S. Then H divides S into two parts: a truncated cone K and the convex hull of a spherical cap. The radius r of the base of K is bounded by one. Consider now the half-ball that contains B (S) and that has H ∩ B n as its flat-surface, see Figure 3, and let Σ denote the area of the corresponding half-sphere. One has B (S) Σ and thus L(S) B (S) L(S) Σ . Now Σ and L(S) are the surface of an (n -1)-dimensional half-sphere of radius r and the length of its boundary respectively. If we scale this half-sphere by a factor of 1/r , we obtain the unit half-ball and its length respectively. Since scaling by a factor of 1/r increases areas by a factor of 1/r n-1 and lengths by a factor of 1/r n-2 , we have that L(S) Σ is at least the length of the unit-half-ball divided by the area of the base of the half-ball. Suppose now that S is the half-unit-ball. We show that the inequality L(S)/B (S) 2 π n-1 n holds. The base of S is a half unit sphere and L(S) is the length of the boundary of a unit ball of dimension n -1. Thus B (S) = n 2 π n/2 Γ n 2 + 1 , L(S) = (n -1)π (n-1)/2 Γ n-1 2 + 1 , where Γ is the well-known Gamma function. Using the fact that Γ(x + 1/2)/Γ(x) x -1 4 for all x > 1 4 (see, e.g., [16]), one easily verifies that Γ n 2 + 1 n 2 • Γ n -1 2 + 1 . APPENDIX It follows that L(S) B (S) = 2 π n -1 n Γ n 2 + 1 Γ n-1 2 + 1 2 π • n -1 n . Finally we are now ready to consider the case of an arbitrary spherical cone. This was the final step in the proof of Lemma 1 and thus we have also proved Theorem 2, our main result on polytopes. The next section is devoted to unbounded polyhedra. The case of an unbounded polyhedron If the polyhedron P is unbounded, then the union of the normal cones of all vertices of P forms a proper subset K of R n : namely, K is the set of objective functions c for which the linear program max{c T x : x ∈ P } has finite optimum. Similarly, the set K ∩ B n is a proper subset of B n . Then, given the union of the spherical cones that have already been discovered by the breadth-first-search (we denote this set by S), we should redefine the dockable surface of S as that part of the lateral surface of S that is shared by some neighboring cones. In other words, we should exclude the part lying on the boundary of K ∩ B n . However, this implies that the result of Lemma 6 does not hold in this context. To overcome this difficulty, we make use of the Lovász-Simonovits inequality, which we now recall. Below we use notation d (X , Y ) to indicate the Euclidean distance between two subsets X , Y ⊆ R n , i.e., d (X , Y ) = inf{ xy : x ∈ X , y ∈ Y }. Also, [x, y] denotes the segment connecting two points x, y ∈ R n (see Figure 4). Theorem 7. [15] Let K ⊆ R n be a convex compact set, 0 < ε < 1 and (K 1 , K 2 , K 3 ) be a partition of K into three measurable sets such that ∀x, y ∈ K , d ([x, y] ∩ K 1 , [x, y] ∩ K 2 ) ε • x -y . ( 10 ) Then vol(K 3 ) 2ε 1 -ε min (vol(K 1 ), vol(K 2 )) . We now illustrate how the above result can be used in our context. Let K = K ∩ B n and observe that K is a convex and compact set. Let S ⊆ K be the union of the spherical cones that have already been discovered by the breadth-first-search. We define the dockable surface of S as that part of the lateral surface of S that is disjoint from the boundary of K . We denote by D (S) the area of the dockable surface of S. We can prove the following analogue of Lemma 6: Lemma 8. If vol(S) 1 2 vol(K ), then D (S) vol(S). Proof. Let F denote the dockable surface of S (thus D (S) is the area of F ). For every ε > 0 we define K 3,ε = (F + εB n ) ∩ K , K 1,ε = S \ K 3,ε , K 2,ε = K \ (K 1,ε ∪ K 3,ε ), where X + Y denotes the Minkowski sum of two subsets X , Y ∈ R n , i.e., X + Y = {x + y : x ∈ X , y ∈ Y }. Clearly (K 1,ε , K 2,ε , K 3,ε ) is a partition of K into three measurable sets. Furthermore, condition (10) is satisfied. Thus Theorem 7 implies that vol(K 3,ε ) 2ε 1 1 -ε min vol(K 1,ε ), vol(K 2,ε ) . We observe that vol(K 2,ε ) vol(K \ S) -vol(K 3,ε ) vol(S) -vol(K 3,ε ) vol(K 1,ε ) -vol(K 3,ε ). Combining those two inequalities, we find vol(F + εB n ) 2ε vol(K 3,ε ) 2ε 1 1 -ε (vol(K 1,ε ) -vol(K 3,ε )). ( 11 ) By a well-known result in geometry (see, e.g., [7],) as ε tends to 0 the left-hand side of (11) tends to the area of F , which is precisely the dockable surface D (S). Moreover, as ε tends to 0, vol(K 3,ε ) tends to 0 and vol(K 1,ε ) tends to vol(S). We conclude that D (S) vol(S). Following the same approach as that used for the case of a polytope, one can show the following result for polyhedra. Theorem 9. Let P = {x ∈ R n : Ax b} be a polyhedron, where all sub-determinants of A ∈ Z m×n are bounded by ∆ in absolute value. Then the diameter of P is bounded by O ∆ 2 n 4 log n∆ . In particular, if A is totally unimodular, then the diameter of P is bounded by O(n 4 log n). Introduction An n dimensional lattice L in R n is defined as all integer combinations of some basis B = (b 1 , . . . , b n ) of R n . The most fundamental computational problems on lattices are the Shortest and Closest Vector Problems, which we denote by SVP and CVP respectively. Given a basis B ∈ R n×n of L, the SVP is to compute y ∈ L \ {0} minimizing y 2 , and the CVP is, given an additional target t ∈ R n , to compute a vector y ∈ L minimizing ty 22 . The study of the algorithms and complexity of lattice problems has yielded many fundamental results in Computer Science and other fields over the last three decades. Lattice techniques were introduced to factor polynomials with rational coefficients [20] and to show the polynomial solvability of integer programs with a fixed number of integer variables [20,[START_REF] Lenstra | Integer programming with a fixed number of variables[END_REF]. It has been used as a cryptanalytic tool for breaking the security of knapsack crypto schemes [19], and in coding theory for developing structured codes [8] and asymptotically optimal codes for power-constrained additive white Gaussian noise (AWGN) channels [10]. Most recently, the security of powerful cryptographic primitives such as fully homomorphic encryption [12,13,6] have been based on the worst case hardness of lattice problems. The Closest Vector Problem with Preprocessing. In CVP applications, a common setup is the need to solve many CVP queries over the same lattice but with varying targets. This is the case in the context of coding over a Gaussian noise channel, a fundamental channel model in wireless communication theory. Lattice codes, where the codewords correspond to a subset of lattice points, asymptotically achieve the AWGN channel capacity (for fixed transmission power), and maximum likelihood decoding for a noisy codeword corresponds (almost) exactly to a CVP query on the coding lattice. In the context of lattice based public key encryption, in most cases the decryption routine can be interpreted as solving an approximate (decisional) CVP over a public lattice, where the encrypted bit is 0 if the point is close and 1 if it is far. CVP algorithms in this setting (and in general), often naturally break into a preprocessing phase, where useful information about the lattice is computed (i.e. short lattice vectors, a short basis, important sublattices, etc.), and a query / search phase, where the computed advice is used to answer CVP queries quickly. Since the advice computed during preprocessing is used across all CVP queries, if the number of CVP queries is large the work done in the preprocessing phase can be effectively "amortized out". This motivates the definition of the Closest Vector Problem with Preprocessing (CVPP), where we fix an n dimensional lattice L and measure only the complexity of answering CVP queries on L after the preprocessing phase has been completed (crucially, the preprocessing is done before the CVP queries are known). To avoid trivial solutions to this problem, i.e. not allowing the preprocessing phase to compute a table containing all CVP solutions, we restrict the amount of space (as a function of the encoding size of the input lattice basis) needed to store the preprocessing advice. Complexity. While the ability to preprocess the lattice is very powerful, it was shown in [START_REF] Micciancio | The hardness of the closest vector problem with preprocessing[END_REF] that CVPP is NP-hard when the size of the preprocessing advice is polynomial. Subsequently, approximation hardness for the gap version of CVPP (i.e. approximately deciding the distance of the target) was shown in [11,[START_REF] Regev | Improved inapproximability of lattice and coding problems with preprocessing[END_REF]4], culminating in a hardness factor of 2 log 1-ε n for any ε > 0 [17] under the assumption that NP is not in randomized quasi-polynomial time. On the positive side, polynomial time algorithms for the approximate search version of CVPP were studied (implicitly) in [5,18], where the current best approximation factor O(n/ log n) was recently achieved in [7]. For the gap decisional version of CVPP, the results are better, where the current best approximation factor is O( n/ log n) [1]. v 1 L v 2 v 3 v 6 v 5 v 4 0 Figure 1: The Voronoi cell and its relevant vectors While our algorithm is randomized, it is Las Vegas, and hence the randomness is in the runtime and not the correctness. Furthermore, the amount of randomness we require is polynomial: it corresponds to the randomness needed to generate a nearly-uniform sample from the Voronoi cell, which can be achieved using Monte Carlo Markov Chain (MCMC) methods over convex bodies [9,[START_REF] Lovász | Fast algorithms for logconcave functions: Sampling, rounding, integration and optimization[END_REF]. This requires a polynomial number of calls to a membership oracle. Each membership oracle test requires an enumeration over the O(2 n ) Voronoi-relevant vectors, resulting in a total complexity of O(2 n ). Unfortunately, we do not know how to convert our CVPP improvement to one for CVP. The technical difficulty lies in the fact that computing the Voronoi relevant vectors, using the current approach, is reduced to solving O(2 n ) related lower dimensional CVPs on an n -1 dimensional lattice (for which the Voronoi cell has already been computed). While the MV CVPP algorithm requires O(4 n ) for worst case targets (which we improve to O(2 n )), they are able to use the relations between the preprocessing CVPs to solve each of them in amortized O(2 n ) time per instance. Hence, with the current approach, reducing the running time of CVP to O(2 n ) would require reducing the amortized per instance complexity to polynomial, which seems very challenging. Organization. In the next section, section 2, we explain how to solve CVPP by finding short paths over the Voronoi graph. In particular, we review the iterative slicer [START_REF] Sommer | Finding the closest lattice point by iterative slicing[END_REF] and MV [START_REF] Micciancio | A deterministic single exponential time algorithm for most lattice problems based on voronoi cell computations[END_REF] algorithms for navigating the Voronoi graph, and describe our new randomized straight line procedure for this task. In section 3, we state the guarantees for the randomized straight line procedure and use it to give our expected O(2 n ) time CVPP algorithm (Theorem 8), as well as an optimal relationship between geometric and path distance on the Voronoi graph (Theorem 5). The main geometric estimates underlying the analysis of the randomized straight path algorithm are proved in section 5. Definitions and references for the concepts and prior algorithms used in the paper can be found in section 4. In particular, see subsections 4.1 for basic lattice definitions, and subsection 4.1.1 for precise definitions and fundamental facts about the Voronoi cell and related concepts. Navigating the Voronoi graph In this section, we explain how one can solve CVP using an efficient navigation algorithm over the Voronoi graph of a lattice. We first describe the techniques used by [START_REF] Sommer | Finding the closest lattice point by iterative slicing[END_REF][START_REF] Micciancio | A deterministic single exponential time algorithm for most lattice problems based on voronoi cell computations[END_REF] for finding short paths on this graph, and then give our new (randomized) approach. Paths on the Voronoi graph. Following the strategy of [START_REF] Sommer | Finding the closest lattice point by iterative slicing[END_REF][START_REF] Micciancio | A deterministic single exponential time algorithm for most lattice problems based on voronoi cell computations[END_REF], our search algorithm works on the Voronoi graph G of an n dimensional lattice L. V (L) = {x ∈ R n : x 2 ≤ x -y 2 , ∀y ∈ L \ {0}} = {x ∈ R n : x, y ≤ y, y /2, ∀y ∈ L \ {0}} , the set of points closer to 0 than any other lattice point. When the lattice in question is clear, we simply write V for V (L). It was shown by Voronoi that V is a centrally symmetric polytope with at most 2(2 n -1) facets. We define VR, the set of Voronoi relevant vectors of L, to be the lattice vectors inducing facets of V. The Voronoi graph G is the contact graph induced by the tiling of space by Voronoi cells, that is, two lattice vectors x, y ∈ L are adjacent if their associated Voronoi cells x + V and y + V touch in a shared facet (equivalently xy ∈ VR). We denote the shortest path distance between x, y ∈ L on G by d G (x, y). See Section 4.1.1 for more basic facts about the Voronoi cell. To solve CVP on a target t, the idea of Voronoi cell based methods is to compute a short path on the Voronoi graph G from a "close enough" starting vertex x ∈ L to t (usually, a rounded version of t under some basis), to the center y ∈ L of a Voronoi cell containing t, which we note is a closest lattice vector by definition. (see Figure 2). Iterative slicer. The iterative slicer [START_REF] Sommer | Finding the closest lattice point by iterative slicing[END_REF] was the first CVP algorithm to make use of an explicit description of the Voronoi cell, in the form of the VR vectors. The path steps of the iterative slicer are computed by greedily choosing any Voronoi relevant vector that brings the current iterate z ∈ L closer to the target t. That is, if there exists a VR vector v such that z + vt 2 < zt 2 , then we move to z + v. This procedure is iterated until there is no improving VR vector, at which point we have reached a closest lattice vector to t. This procedure was shown to terminate in a finite number of steps, and currently, no good quantitative bound is known on its convergence time. The Voronoi norm. We now make precise which notion of closeness to the target we use (as well as MV) for the starting lattice vector x to the target t. Notice that for the path finding approach to make sense from the perspective of CVP, we need to start the process from a point x ∈ L that we know is apriori close in graph distance to a closest lattice vector y to t. Given the complexity of G and the fact that we do not know y, we will need a robust proxy for graph distance that we can estimate knowing only x and t. From this perspective, it was shown in [START_REF] Micciancio | A deterministic single exponential time algorithm for most lattice problems based on voronoi cell computations[END_REF] that the Voronoi norm t -x V = inf {s ≥ 0 : t -x ∈ sV } = sup v∈VR 2 v, t -x v, v of tx (i.e. the smallest scaling of V containing tx) can be used to bound the shortest path distance between x and y. Here the quantity tx V is robust in the sense that yx V ≤ tx V + yt V ≤ tx V + 1 by the triangle inequality. Hence from the perspective of the Voronoi norm, t is simply a "noisy" version of y. Furthermore, given that each Voronoi relevant vector has Voronoi norm 2, one can construct a lattice vector x such that tx V ≤ n, by simply expressing t = ∑ n i=1 a i v i , for some linearly independent v 1 , . . . , v n ∈ VR, and letting x = ∑ n i=1 a i v i . The MV Path. We now present the MV path finding approach, and give the relationship they obtain between tx V and the path distance to a closest lattice vector y to t. The base principle of MV [START_REF] Micciancio | A deterministic single exponential time algorithm for most lattice problems based on voronoi cell computations[END_REF] is similar to that of the iterative slicer, but it uses a different strategy to select the next VR vector to follow, resulting in a provably single exponential path length. In MV, a path step consists of tracing the straight line from the current path vertex z ∈ L to the target t, and moving to z + v where v ∈ VR induces a facet (generically unique) of z + V crossed by the line segment [z, t]. It is not hard to check that each step can be computed using O(n|VR|) = O(2 n ) arithmetic operations, and hence the complexity of computing the path is O(n|VR| × path length). The main bound they give on the path length, is that if the start vertex x ∈ 2V + t (i.e. Voronoi distance less than 2), then the path length is bounded by 2 n . To prove the bound, they show that the path always stays inside t + 2V, that the 2 distance to the target monotonically decreases along the path (and hence it is acyclic), and that the number of lattice vectors in the interior of t + 2V is at most 2 n . To build the full path, they run this procedure on the Voronoi graph for decreasing exponential scalings of L 4 , and build a path (on a supergraph of G) of length O(2 n log 2 tx V ). One can also straightforwardly adapt the MV procedure to stay on G, by essentially breaking up the line segment [x, t] in pieces of length at most 2, yielding a path length of O(2 n xt V ). Since we can always achieve a starting distance of xt V ≤ n by straightforward basis rounding, note that the distance term is lower order compared to the proportionality factor 2 n . Randomized Straight Line. Given the 2 n proportionality factor between geometric and path distance achieved by the MV algorithm, the main focus of our work will be to reduce the proportionality factor to polynomial. In fact, will show the existence of paths of length (n/2)( tx V + 1), however the paths we are able to construct will be longer. For our path finding procedure, the base idea is rather straightforward, we simply attempt to follow the sequence of Voronoi cells on the straight line from the start point x to the target t. We dub this procedure the straight line algorithm. As we will show, the complexity of computing this path follows the same pattern as MV (under certain genericity assumptions), and hence the challenge is proving that the number of Voronoi cells the path crosses is polynomial. Unfortunately, we do not know how to analyze this procedure directly. In particular, we are unable to rule out the possibility that a "short" line segment (say of Voronoi length O(1)) may pass through exponentially many Voronoi cells in the worst case (though we do not have any examples of this). To get around the problem of having "unexpectedly many" crossings, we will make use of randomization to perturb the starting point of the line segment. Specifically, we will use a randomized straight line path from x ∈ L to t which proceeds as follows (see Figure 3): (C) Follow the line from t + Z to t. 4 The MV path is in fact built on a supergraph of the Voronoi graph, which has edges corresponding to 2 i VR, i ≥ 0. We briefly outline the analysis bounding the expected number of Voronoi cells this path crosses, which we claim achieves a polynomial proportionality factor with respect to tx V . To begin, note that in phase A, we stay entirely within x + Z, and hence do not cross any Voronoi cells. In phase B, at every time α ∈ [0, 1], the point (1α)x + αt + Z is in a uniformly random coset of R n /L since Z is uniform. Hence the probability that we we cross a boundary between time α and α + ε is identical to the probability that we cross a boundary going from Z to Z + ε(t -x). Taking the limit as ε → 0 and using linearity of expectation, we use the above invariance to show that the expected number of boundaries we cross is bounded by (n/2) tx V , the Voronoi distance between x and t. In essence, we relate the number of crossings to the probability that a uniform sample from V (equivalently, a uniform coset) is close under the Voronoi norm to the boundary ∂V, which is a certain surface area to volume ratio. Interestingly, as a consequence of our bound for phase B, we are able to give an optimal relationship between the Voronoi distance between two lattice points and their shortest path distance on G, which we believe to be independent interest. In particular, for two lattice points x, y ∈ L, we show in Theorem 5 that the shortest path distance on G is at least xy V /2 and at most (n/2) xy V , which is tight for certain pairs of lattice points on Z n . It remains now to bound the expected number of crossings in phase C. Here, the analysis is more difficult than the second step, because the random shift is only on one side of the line segment from t + Z to t. We will still be able to relate the expected number of crossings to "generalized" surface area to volume ratios, however the probability distributions at each time step will no longer be invariant modulo the lattice. In particular, the distributions become more concentrated as we move closer to t, and hence we slowly lose the benefits of the randomness as we get closer to t. Unfortunately, because of this phenomenon, we are unable to show in general that the number of crossings from t + Z to t is polynomial. However, we will be able to bound the number of crossings from t + Z to t + αZ by O(n ln(1/α)), that is, a very slow growing function of α as α → 0. Fortunately, for rational lattices and targets, we can show that for α not too small, in particular ln(1/α) linear in the size of binary encoding of the basis and target suffices, that t + αZ and t lie in the same Voronoi cell. This yields the claimed (weakly) polynomial bound. Analysis and Applications of Randomized Straight Line In this section, we give the formal guarantees for the randomized straight line algorithm and its applications. The analysis here will rely on geometric estimates for the number of crossings, whose proofs are found in Section 5. To begin, we make formal the connection between Voronoi cells crossings, the length of the randomized straight line path, and the complexity of computing it. Lemma 2 (Randomized Straight Line Complexity). Let x ∈ L be the starting point and let t ∈ R n be the target. Then using perturbation Z ∼ Uniform(V ), the expected edge length of the path from x to a closest lattice vector y to t on G induced by the randomized straight line procedure is E[|(L + ∂V ) ∩ [x + Z, t + Z]|] + E[|(L + ∂V ) ∩ [t + Z, t)|] . Furthermore, with probability 1, each edge of the path can be computed using O(n|VR|) arithmetic operations. While rather intuitive, the proof of this Lemma is somewhat tedious, and so we defer it to section 6. Note that (L + ∂V ) ∩ [x + Z, t + Z] corresponds to the phase B crossings, and that (L + ∂V ) ∩ [t + Z, t) corresponds to the phase C crossings. Our bound for the phase B crossings, which is proved in Section 5.1, is as follows. Theorem 3 (Phase B crossing bound). Let L be an n dimensional lattice. Then for x, y ∈ R n and Z ∼ uniform(V ), we have that E Z [|(L + ∂V ) ∩ [x + Z, y + Z]|] ≤ (n/2) y -x V . For phase C, we give a bound on the number crossings for a truncation of the phase C path. That is, instead of going all the way from t + Z to t, we stop at t + αZ, for α ∈ (0, 1]. Its proof is given in Section 5. Using the crossing estimate for phase B, we now show that from the perspective of existence, one can improve the MV proportionality factor between geometric and path distance from exponential to linear in dimension. Theorem 5. For x, y ∈ L, we have that (1/2) x -y V ≤ d G (x, y) ≤ (n/2) x -y V . Furthermore, the above is best possible, even when restricted to L = Z n . Proof. For the lower bound, note that d G (x, y) is the minimum k ∈ Z + such that there exists v 1 , . . . , v k ∈ VR satisfying y = x + ∑ k i=1 v i . Since ∀v ∈ VR, v V = 2, by the triangle inequality Proof. By applying a linear transformation to K and y, we may assume that y = e n . Let π n-1 : R n → R n-1 denote the projection onto the first n -1 coordinates. Define l : π n-1 (K) → R + as l(x) = vol 1 ({(x, x n ) : x n ∈ R, (x, x n ) ∈ K}), i.e. the length of the chord of K passing through (x, 0) in direction e n . y -x V = k ∑ i=1 v i V ≤ k ∑ i=1 v i V = For x ∈ π n-1 (K), let {(x, x n ) : x n ∈ R, (x, x n ) ∈ K} = [(x, a), (x, b)], a ≤ b, denote its associated chord, where we note that |b -a| = l(x). From here, conditioned on Z landing on this chord, note that Z + εe ( ) 3 Let s = 1/ e n K . Since K is centrally symmetric, Re n ∩ K = [-se n , se n ] and hence l(0) = 2s. Note that by central symmetry of K, for all x ∈ π n-1 (K), l(x) = l(-x). Since K is convex, the function l is concave on π n-1 (K), and hence max x∈π n-1 (K) l(x) = max x∈π n-1 (K) 1 2 l(x) + 1 2 l(-x) ≤ max x∈π n-1 (K) l(0) = 2s. Let K = {(x, x n ) : x ∈ π n-1 (K), 0 ≤ x n ≤ l(x)}. By concavity of l, it is easy to see that K is also a convex set. Furthermore, note that K has exactly that same chord lengths as K in direction e n , and hence vol n (K ) = vol n (K). For a ∈ R, let K a = K ∩ (x, a) : x ∈ R n-1 . Here Re n ∩ K = [0, 2se n ], and hence K a = ∅, ∀a ∈ [0, 2s]. Therefore K a = ∅ for a > 2s, since the maximum chord length is l(0) = 2s, as well as for a < 0. By construction of K , we see that K 0 = π n-1 (K) × {0}, and hence vol n-1 (K 0 ) = vol n-1 (π n-1 (K)). Given (3), to prove the Lemma, it now suffices to show that vol n-1 (π n-1 (K)) vol n (K) = vol n-1 (π n-1 (K)) vol n (K ) ≤ (n/2) e n K . For a ∈ [0, 2s], by convexity of K and the Brunn-Minkowski inequality on R n-1 , we have that vol n-1 (K a ) 1 n-1 ≥ vol n-1 ((1 - a 2s )K 0 + a 2s K 2s ) 1 n-1 ≥ (1 - a 2s )vol n-1 (K 0 ) 1 n-1 + a 2s vol n-1 (K 2s ) 1 n-1 ≥ (1 - a 2s )vol n-1 (π n-1 (K)) (1 -a) n-1 da = vol n-1 (π n-1 (K))(2s)/n = 2vol n-1 (π n-1 (K)) n e n , as needed. APPENDIX Taking the limit as ε → 0, we express the infinitessimal probability as a certain weighted surface area integral over s(Lt + ∂V ) (see Lemma 27). In the same spirit as phase B, we will attempt to relate the surface integral to a nicely bounded integral over all of space. To help us in this task, we will rely on a technical trick to "smooth out" the distribution of Z. More precisely, we will replace the perturbation Z ∼ Uniform(V ) by the perturbation X ∼ Laplace(V, θ), for an appropriate choice of θ. For the relationship between both types of perturbatoins, we will use the representation of X as rZ, where r ∼ Γ(n + 1, θ). We will choose θ so that r is concentrated in the interval [1, 1 + 1 n ], which will insure that the number of crossings for X and Z are roughly the same. The benefit of the Laplace perturbation for us will be as follows. Since it varies much more smoothly than the uniform distribution (which has "sharp boundaries"), it will allow us to make the analysis of the surface integral entirely local by using the periodic structure of s(Lt + ∂V ). In particular, we will be able to relate the surface integral over each tile s(yt + ∂V ), y ∈ L, to a specific integral over each cone s(y + conv(0, F v )), ∀v ∈ VR, making up the tile. Under the uniform distribution, the probability density over each tile can be challenging to analyze, since the tile may only be partially contained in V. However, under the Laplace distribution, we know that over s(y + V ) the density can vary by at most e ±s/θ (see Equation 2 in the Preliminaries). The integral over R n we end up using to control the surface integral over s(Lt + ∂V ) turns out the be rather natural. At all scales, we simply use the integral representation of E[ X V ] = nθ (see Lemma 25). In particular, as s → ∞, for the appropriate choice of θ, this will allow us to bound the surface integral over s(Lt + ∂V ) by O(n/s). Integrating this bound from 1 to 1/α yields the claimed O(n ln(1/α)) bound on the number of crossings. This section is organized as follows. In subsection 5.2.1, we relate the number of crossings for uniform and Laplace perturbations. In subsection 5.2.2, we bound the number of crossings for Laplace perturbations. Lastly, in subsection 5.2.3, we combine the estimates from the previous subsections to give the full phase C in Theorem 4. Converting Uniform Perturbations to Laplace Perturbations In this section, we show that the number of crossings induced by uniform perturbations can be controlled by the number of crossings induced by Laplace perturbation. We define θ n = 1 (n+1)- √ 2 (n+1) , γ n = (1 + 2 √ 2 √ n+1- √ 2 ) -1 for use in the rest of this section. The following Lemma shows the Γ(n + 1, θ) distribution is concentrated in a small interval above 1 for the appropriate choice of θ. This will be used in Lemma 21 to relate the number of crossings between the uniform and Laplace perturbations. Lemma 20. For r ∼ Γ(n + 1, θ n ), n ≥ 2, we have that Pr[r ∈ [1, 1 + 2 √ 2 √ n + 1 - √ 2 ]] ≥ 1 2 Proof. Remember that E[r] = (n + where γ n = (1 + 2 √ 2 √ n+1- √ 2 ) -1 . Proof. We shall use the fact that X is identically distributed to rZ where r ∼ Γ(n + 1, θ n ) is sampled independently from Z. Conditioned on any value of Z, the following inclusion holds as needed. Bounding the Number of Crossing for Laplace Perturbations In this section, we bound the number of crossings induced by Laplace perturbations. The expression for the infinitessimal crossing probabilities is given in Lemma 23, the bound on the surface area integral over s(Lt + ∂V ) to E[ X V ] is given in Lemma 25, and the full phase C Laplace crossing bound is given in Theorem 27. For t ∈ R n , and s > 0, the set s(Lt + ∂V ) is a shifted and scaled version of the tiling boundary L + ∂V. For y ∈ L, and v ∈ VR, we will call s(yt + F v ) a facet of s(Lt + ∂V ). Definition 22 (Tiling boundary normals). We define the function η : (L -t + ∂V ) → S n-1 as follows. For x ∈ (L -t + ∂V ), choose the lexicographically minimal v ∈ VR such that ∃y ∈ L satisfying x ∈ (y -t + F v ). Finally, define η(x) = v/ v 2 . Note that for x ∈ s(Lt + ∂V ), η(x/s) is a unit normal to a facet of s(Lt + ∂V ) containing x. Furthermore, the subset of points in s(Lt + F v ) having more than one containing facet has n -1 dimensional measure 0, and hence can be ignored from the perspective of integration over s(Lt + ∂V ). The following lemma gives the expression for the infinitessimal crossing probabilities. From here, we first decompose the expected number of intersections by summing over all facets. This yields , we see that (α ) ∈ w + F e is the correctly computed exiting point (corresponding to (α) at the end of the loop iteration), and that w + F e is the exiting facet. Since the facet w + F e is shared by (w + e) + V, we see that (α ) ∈ (w + e) + ∂V, and hence the invariant is maintained in the next iteration. The line following algorithm is thus correct. E[|(L -t + ∂V ) ∩ [X, αX]|] = 1 Notice that each iteration of the line following procedure clearly requires at most O(n|VR|) arithmetic operations. We note that the conclusions of Claim 28 are only needed to ensure that each iteration of the path finding procedure can be associated with exactly one intersection point in ([x + Z, t + Z] ∪ [t + Z, t)) ∩ (L + ∂V ). In particular, it assures that the minimizer in (18) is unique. This concludes the proof of the Lemma. Proof of Lemma 7 (Bit length bound). Clearly, q ≤ ( ∏ ij q B ij )( ∏ i q t i ) ⇒ log 2 q ≤ ∑ ij log 2 (q B ij ) + ∑ i q t i . ( 19 ) 142 APPENDIX Résumé en français Approches géométriques et duales en ordonnancement cumulatif Problématique Ce travail s'inscrit dans le domaine de l'optimisation mathématique, et plus précisément en ordonnancement à base de programmation par contraintes. Ceci consiste à déterminer les dates de début et de fin d'exécution de tâches (ces dates constituent les variables de décision du problème d'optimisation), tout en satisfaisant à des contraintes temporelles et de ressources, et en optimisant une fonction objectif. En programmation par contraintes, un problème de cette nature est résolu par une exploration arborescente des domaines des variables de décision par séparation-évaluation. De plus, à chaque noeud est effectuée une phase de propagation, c'est-à-dire que l'on élimine des valeurs des domaines des variables de décision en vérifiant des conditions nécessaires pour que les différentes contraintes soient satisfaites. Dans ce cadre, la contrainte de ressource la plus fréquemment rencontrée est la cumulative. Elle permet en effet de modéliser des processus se déroulant de manière parallèle, mais en consommant pendant leur exécution une ressource partagée disponible en quantité finie (par exemple des machines ou un budget). Propager cette contrainte de manière efficace est donc d'une importance cruciale pour l'efficacité pratique d'un moteur d'ordonnancement à base de programmation par contraintes. Nous étudions dans cette thèse la contrainte cumulative en nous servant d'outils rarement utilisés en programmation par contraintes (analyse polyédrale, dualité de la programmation linéaire, dualité de la géométrie projective). À l'aide de ces outils, nous proposons deux contributions pour le domaine : le renforcement cumulatif, et le Raisonnement Énergétique en O(n 2 log n). 147 Figure 1 . 1 : 11 Figure 1.1: Depiction of Philae on comet 67P/C-G (Credits: Deutsches Zentrum für Luft-und Raumfahrt, CC-BY 3.0) Figure 1 . 2 : 12 Figure 1.2: Steps to solving an optimization problem Figure 2 . 1 : 21 Figure 2.1: Bounds of a time interval in constraint-based scheduling Figure 2 . 2 : 22 Figure 2.2: Schedule Improvement with Large Neighborhood Search (Figure courtesy of Philippe Laborie) Figure 3 . 1 : 31 Figure 3.1: Example instance. Energy bound: 9.75. Figure 3 . 2 : 32 Figure 3.2: Precedence graph Figure 3 . 3 : 33 Figure 3.3: An optimal schedule 1: RCPSP model int R = .. . ; // number of resources tuple Task { key int id ; int length ; int demands [0 .. R -1]; { int } succs ; } { Task } Tasks = .. . ; // characteristics of tasks int Capacity [ r in 1 .. R ] = .. . ; // resources capacities dvar interval intervals [ task in Tasks ] size task . length ; cumulFunction usage [r in 1 .. R] = sum ( task in Tasks : task . demands [ r ] > 0) pulse ( intervals [ task ] , task . demands [ r ]) ; minimize max ( t in Tasks ) endOf ( intervals [ t ]) ; subject to { forall ( r in 0 .. R -1) usage [ r ] <= Capacity [ r ]; forall ( task in Tasks , succId in task . succs ) endBeforeStart ( itvs [ task ] , itvs [ < succId > ]) ; } of the cumulative resource. Figure 3 . 4 : 34 Figure 3.4: Exclusive zones in the forward section of an aircraft . 6 .Listing 3 . 2 :Figure 3 . 6 : 63236 Figure 3.6: Berth allocation This problem is a strip-packing problem, if we think of the temporal dimen- Figure 3 Figure 3 . 7 : 337 Figure 3.7: Truck loading Figure 3 . 8 : 38 Figure 3.8: Balancing production load Figure 3 . 9 : 39 Figure 3.9: Intervals synchronization C = 4 tFigure 4 . 1 : 441 Figure 4.1: Reformulation of the instance of Figure 3.1. Energy bound: 11.5. 3 3 Figure 4 . 2 : 3342 Figure 4.2: Energy bound increased from 7.33 to 8. 3 3 Figure 4 . 3 : 3343 Figure 4.3: Energy bound increased from 9 to 10. Figure 4 . 4 : 44 Figure 4.4: The five possible task configurations on a cumulative of capacity C=4. Figure 4 . 5 : 45 Figure 4.5: The three task configurations used to build the reformulation of Figure 3.1. 3 3 Figure 4 . 6 : 46 Figure 4.6: Example schedule 3 3 Figure 4 . 7 : 47 Figure 4.7: Best LP solution without the non-superposition constraints 4. 2 . 2 C = 9 l 1 Figure 4 . 8 : 229148 Figure 4.8: Example of configurations forming a staircase of slots of demand c. Figure 4 . 9 : 2 C = 9 Figure 4 . 10 : 4929410 Figure 4.9: Placing one task on the staircase. Figure 5 . 1 : 51 Figure 5.1: Intersection length W =i (a, b) is the sum of the intersection energies of all tasks different from i: W =i (a, b) = ∑ j =i W j (a, b).Finally, W(a, b) is the sum of the intersection energies of all tasks:W(a, b) = ∑ i W i (a, b).In the rest of this chapter, the propagations will be described in the case when we shift the task under consideration as much as possible to the left. There is obviously a symmetric version of all of them in the case when we shift the task to the right.The energy reasoning rule is: Figure 5 . 2 : 52 Figure 5.2: Energy Reasoning Figure 5 . 3 : 53 Figure 5.3: General shape of f i,a . Figure 5 . 4 : 54 Figure 5.4: Shape of f i,a in different cases. Lemma 1 . 2 π 1 ∆ 2 1212 Let P = {x ∈ R n : Ax b} be a polytope where all sub-determinants of A ∈ Z m×n are bounded by ∆ in absolute value and let I ⊆ V be a set of vertices of G P with vol(I ) (1/2) • vol(B n ). Then the volume of the neighborhood of I is at least vol(N (I )) n2.5 2 π 1 ∆ 2 1 + 2 π 1 ∆ 2 n 2. 5 j 2121215 j 1 and vol(I j -1 ) (1/2) • vol(B n ) we have by Lemma 1 vol(I j ) 1 + n 2.5 vol(I j -1 ) vol(I 0 ). ( a ) a Dockable surface of S. (b) Base of S. (c) Relative boundary of the base of S. Figure 1 : 1 Figure 1: Illustration of D(S), B (S) and L(S). Lemma 4 .Figure 3 : 43 Figure 3: Proof of Lemma 5. Lemma 5 . 5 Let S be a spherical cone of revolution of angle 0 < θ π/2. Then D(S) vol(S) 2n π . Lemma 6 . 6 Let S be a (not necessarily convex) spherical cone with vol(S) 1 2 vol(B n ). Let S * be a spherical cone of revolution with the same volume as S. By Lemma 4, D(S) D(S * ). Now, using Lemma 5 one has D(S) vol(S) D(S * ) vol(S * ) 2n π . 1 K 2 K 3 Figure 4 : 1234 Figure 4: Illustration of the Lovász-Simonovits inequality. Figure 2 : 2 Figure 2: CVP solution is the center of target-containing Voronoi cell ( A) Move to x + Z, where Z ∼ Uniform(V ) is sampled uniformly from the Voronoi cell.(B) Follow the line from x + Z to t + Z. Figure 3 : 3 Figure 3: Randomized Straight Line algorithm 2 . 4 ( 24 Theorem Phase C crossing bound). For α ∈ (0, 1], Z ∼ Uniform(V ), n ≥ 2, we have thatE[|(L + ∂V ) ∩ [Z + t, αZ + t]|] ≤ e 2 √ 2 -1 n(2 + ln(4/α)) . 2k, as needed.For the upper bound, we run the randomized straight line procedure from x to y, i.e. setting t = y. By Lemma 2, the expected path length on G isE[|(L + ∂V ) ∩ [x + Z, y + Z]|] + E[|(L + ∂V ) ∩ [y + Z, y)|] n / ∈ K if and only if Z lies in the half open line segment ((x, bε), (x, b)]. Given this, we have that lim ε→0 Pr[Z + εe n / ∈ K]/ε = lim ε→0 (1/ε) π n-1 (K) min {ε, l(x)} dx vol n (K) = lim ε→0 π n-1 (K) min {1, l(x)/ε} dx vol n (K) = π n-1 (K) dx vol n (K) = vol n-1 (π n-1 (K))vol n (K) . = 1 (K a )da ≥ vol n-1 (π n-1 (K)) vol n-1 (π n-1 (K))(2s) 1 0 Lemma 23 . 23 For α ∈ (0, 1], and X ∼ Laplace(V, θ), we have thatE[|(L + ∂V ) ∩ [X + t, αX + t]|] = t+∂V ) | η(x/s), x/s | f θ V (x)dvol n-1 (x)ds .Proof. Firstly, shifting by -t on both sides, we get thatE[|(L + ∂V ) ∩ [X + t, αX + t]|] = E[|(L -t + ∂V ) ∩ [X, αX]|]. Data: a, b ∈ R n , z ∈ L, a ∈ z + V Result: w ∈ L such that b ∈ w + V w ← z, e ← 0, α ← 0 VR ← {v ∈ VR : v, ba > 0} repeat w ← w + e e ← arg min v∈VR vintersection point (α ), α > α,and the exiting facet w + F e . If α ≥ 1, we know that b ∈ [ (α), (α )] ⊆ w + V, and hence we may return w. Otherwise, we move to the center of the Voronoi cell sharing the facet w + F e opposite w.To verify the correctness, we need only show that the line [a, b] indeed exits w + V through the facet w + F e at the end of each iteration. Note that by our invariant (α) ∈ w + V at the beginning of the iteration, and hencev, (α) -w ≤ 1 2 v, v , ∀v ∈ VR ⇔ v, (1α)a + αb -w ≤ 1 2 v, v , ∀v ∈ VR ⇔ α v, ba ≤ v, v/2 -a + w , ∀v ∈ VR(17)Since we move along the line segment [a, b] by increasing α, i.e. going from a to b, note that the only constraints that can be eventually violated as we increase α are those for which v, ba > 0. Hence, in finding the first violated constraint (i.e. exiting facet), we may restrict our attention to the subset of Voronoi relevant vectors VR = {v ∈ VR : v, ba > 0} as done in the algorithm.From(17), we see that we do not to cross any facet w + F v , v ∈ VR , as long asα ≤ v, v/2 -a + w v, ba , ∀v ∈ VR .Hence the first facet we violate must be induced by e 3. THE CUMULATIVE CONSTRAINT Skills ALT t 1 ALT t 2 ALT t 3 ALT t 4 W 1 W 2 W 3 t Figure 3.5: Workforce scheduling mand of each task t i on the constraint is 1 if it requires that skill, 0 otherwise, as shown in Listing 3.3. which overflows the available energy. A i est Ω est i min j∈Ω eet j eet i let Ω Ω Figure 3.10: Not-First, Not-Last : n: number of tasks p 1 , . . . , p n : length of the tasks c 1 , . . . , c n : consumption of the tasks est 1 , . . . , est n : earliest start dates of the tasks lst 1 , . . . , lst n : latest start dates of the tasks H: planning horizon C: cumulative resource capacity Side Effect: Update a value est i if Energy Reasoning propagates if the energetic checker finds no possible propagation then return end Compute all W(a, b) values using the algorithm of section 3.3.6.2 of [BPN01]. for a then Add to I the 5 segments of figure 2, label them with i end else if est i ≤ a < lst i then Add to I the 4 segments of figure 3, label them with i end else Add to I the 3 segments of figure 4, label them with i end end end Compute the upper envelope of the segments in I using the algorithm of section 15.3.2 of [BY98]. Preserve the segments labels. Call this function f a . for b ∈ endpoints of the segments of f a )It remains to provide a lower bound on h F . Let a 1 , . . . , a n be the row-vectors of A defining the extreme rays of the normal cone of v, and let A v be the non-singular matrix whose rows are a 1 , . . . , a n . Furthermore, suppose that the vertex y lies on the ray generated by a 1 . Let H be the hyperplane generated by a 2 , . . . , a n . The distance d (y, H ) of y to H is equal to d (a 1 , H )/ a 1 . Let b 1 , . . . , b n be the columns of the adjugate of A v . The column-vector b 1 is integral and each component of b 1 is bounded by ∆. Furthermore b 1 is orthogonal to each of a 2 , . . . , a n . Thus d (a 1 , H ) is the length of the projection of a 1 to b 1 , which is |〈a 1 , b 1 〉|/ b 1 1/( n • ∆), since a 1 and b 1 are integral. Thus 110 APPENDIX y a 1 r F Figure 2: Proof of Lemma 3. 1)θ n and that VAR[r] = (n + 1)θ 2 Let L be an n-dimensional lattice, n ≥ 2, and t ∈ R n . Then for α ∈ [0, 1], Z ∼ Uniform(V ) and X ∼ Laplace(V, θ n ), we have thatE Z [|(L + ∂V ) ∩ [Z + t, αZ + t]|] ≤ 2 E X [|(L + ∂V ) ∩ [X + t, γ n αX + t]] 134 APPENDIX Lemma 21. Chebyshev's inequality Pr[|r -E[r]| ≥ √ 2σ] ≤ VAR[r] 2σ 2 = 1 2 n . Letting σ = VAR[r], by The result now follows from the identities E[r] - √ 2 √ n + 1 -√ 2 √ 2 2σ = ((n + 1) -2(n + 1))θ n = 1 E[r] + √ 2σ = ((n + 1) + 2(n + 1))θ n = 1 + [Z + t, αZ + t) ⊆ [rZ + t, γ n αrZ + t) as long as r ∈ [1, γ -1 n ] = [1, 1 + 2 √ 2 ]. By Lemma 20, we get thatE X [|(L + ∂V ) ∩ [X + t, γ n αX + t]|] = E Z [ E r [|(L + ∂V ) ∩ [rZ + t, γ n αrZ + t]|] ] ≥ E Z [|(L + ∂V ) ∩ [Z + t, αZ + t]| Pr[r ∈ [1, γ -1 √ 2 √ n+1- ≥ 1 2 E n ]]] Z [|(L + ∂V ) ∩ [Z + t, αZ + t]|], 3 Proof of Theorem 4 (Phase C crossing bound) Proof. We recall that θ n = -1 . Note that nθ n > 1.By Lemma 21 and Theorem 27, for X ∼ Laplace(V, θ n ), we have thatE[|(L + ∂V ) ∩ [Z + t, αZ + t]|] ≤ 2 E[|(L + ∂V ) ∩ [X + t, γ n αX + t]|] ≤ c 1 1/α max n 2 θ s 2 , n s ds = c 1 s * n 2 θ s 2 ds + c 1/α s * n s ds = cn 2 θ 1/2 - 1 s * + cn ln 1 αs * = cn nθ 2 1 - 1 s * + ln 1 αs * , as needed. 5.2.1 (n+1)-√ 2(n+1) and γ n = 1 + 2 √ n+1-√ 2 √ 2 ≤ 2cn nθ n 2 1 - 1 nθ n + ln 1 αγ n nθ n ≤ e 2 √ 2 -1 n(2 + ln(4/α)) , for n ≥ 2 , as needed. 2 ∑ y∈L,v∈VR E[|(y -t + F v ) ∩ [X, αX]|]. ( 7 ) | η(x/s), x/s | f θ V (x)dvol n-1 (x)ds A counterexample to the same conjecture for unbounded polyhedra was found in 1967 by Klee and Walkup[13].ON SUB-DETERMINANTS AND THE DIAMETER OF POLYHEDRA The SVP and CVP can be defined over any norm, though we restrict our attention here to the Euclidean norm.118APPENDIX The O notation suppresses polylogarithmic factors.SHORT PATHS ON THE VORONOI GRAPH AND CVPP Remerciements 4 Acknowledgements This work was carried out while all authors were at EPFL (École Polytechnique Fédérale de Lausanne), Switzerland. The authors acknowledge support from the DFG Focus Program 1307 within the project "Algorithm Engineering for Real-time Scheduling and Routing" and from the Swiss National Science Foundation within the project "Set-partitioning integer programs and integrality gaps". Appendix: supplementary work In parallel of preparing this PhD thesis in constraint-based scheduling, I continued working on topics I started during my Master's degree, namely on the geometrical structures of linear programming (polyhedra) and of integer linear programming (lattices). This work resulted in two articles which have been prepared and published during my PhD. The article "On Sub-Determinants and the Diameter of Polyhedra" gives a bound on the diameter of the constraint polyhedron of a linear program whose linear inequalities involve only integer coefficients. This bound is a function of the dimension and of the largest subdeterminant appearing in the constraint matrix. This is an extension of results known on the diameter of totally unimodular polyhedra. The article "Short Paths on the Voronoi Graph and Closest Vector Problem with Preprocessing" significantly improves the complexity of the Micciancio-Voulgaris algorithm to solve the Closest Vector Problem with Preprocessing, in the field of Lattice Problems. These two articles are included here for reference. Remark For simplicity, we have assumed that a bound ∆ was given for the absolute value of all sub-determinants of A. However, our proof only uses the fact the the sub-determinants of size 1 (i.e., the entries of the matrix) and n -1 are bounded. Calling ∆ 1 (resp. ∆ n-1 ) the bound on the absolute value of the entries of A (resp. on the sub-determinants of A of size n -1), one easily verifies that all the results discussed above remain essentially unchanged, except that the statement of Lemma 3 becomes and the lower bound on vol(I 0 ) becomes vol(I 0 ) 1 This implies the following strengthened result: Theorem 10. Let P = {x ∈ R n : Ax b} be a polyhedron, where the entries of A (respectively the subdeterminants of A of size n -1) are bounded in absolute value by ∆ 1 (respectively ∆ n-1 ). Then the diameter of P is bounded by O ∆ 1 ∆ n-1 n 4 log n∆ 1 . Moreover, if P is a polytope, its diameter is bounded by O ∆ 1 ∆ n-1 n 3.5 log n∆ 1 . Short Paths on the Voronoi Graph and Closest Vector Problem with Preprocessing Daniel Dadush * Nicolas Bonifas † Abstract Improving on the Voronoi cell based techniques of [START_REF] Sommer | Finding the closest lattice point by iterative slicing[END_REF][START_REF] Micciancio | A deterministic single exponential time algorithm for most lattice problems based on voronoi cell computations[END_REF], we give a Las Vegas O(2 n ) expected time and space algorithm for CVPP (the preprocessing version of the Closest Vector Problem, CVP). This improves on the O(4 n ) deterministic runtime of the Micciancio Voulgaris algorithm [START_REF] Micciancio | A deterministic single exponential time algorithm for most lattice problems based on voronoi cell computations[END_REF] (henceforth MV) for CVPP 1 at the cost of a polynomial amount of randomness (which only affects runtime, not correctness). As in MV, our algorithm proceeds by computing a short path on the Voronoi graph of the lattice, where lattice points are adjacent if their Voronoi cells share a common facet, from the origin to a closest lattice vector. Our main technical contribution is a randomized procedure that given the Voronoi relevant vectors of a lattice -the lattice vectors inducing facets of the Voronoi cell -as preprocessing and any "close enough" lattice point to the target, computes a path to a closest lattice vector of expected polynomial size. This improves on the O(2 n ) path length given by the MV algorithm. Furthermore, as in MV, each edge of the path can be computed using a single iteration over the Voronoi relevant vectors. As a byproduct of our work, we also give an optimal relationship between geometric and path distance on the Voronoi graph, which we believe to be of independent interest. Exact CVPP algorithms. Given the hardness results for polynomial sized preprocessing, we do not expect efficient algorithms for solving exact CVPP for general lattices. For applications in wireless coding however, one has control over the coding lattice, though constructing coding lattices with good error correcting properties (i.e. large minimum distance) for which decoding is "easy" remains an outstanding open problem. In this context, the study of fast algorithms for exact CVPP in general lattices can yield new tools in the context of lattice design, as well as new insights for solving CVP without preprocessing. The extant algorithms for exact CVPP are in fact also algorithms for CVP, that is, the time to compute the preprocessing is bounded by query / search time. There are currently two classes of CVP algorithms which fit the preprocessing / search model (this excludes only the randomized sieving approaches [2,3]). The first class is based on lattice basis reduction [20], which use a "short" lattice basis as preprocessing to solve lattice problems, that is polynomial sized preprocessing. The fastest such algorithm is due to Kannan [16], with subsequent refinements in [15,5,14,[START_REF] Micciancio | Fast lattice point enumeration with minimal overhead[END_REF], which computes a Hermite-Korkine-Zolatoreff basis (HKZ) during the preprocessing phase in O(n n 2e ) 3 time and poly(n) space, and in the query phase uses a search tree to compute the coefficients of the closest vector under the HKZ basis in O(n n 2 ) time and poly(n) space. The second class, which are the most relevant to this work, use the Voronoi cell (see Section 4.1.1 for precise definitions) of the lattice -the centrally symmetric polytope corresponding to the points closer to the origin than to other lattice points -as preprocessing, and were first introduced by Sommer, Feder and Shalvi [START_REF] Sommer | Finding the closest lattice point by iterative slicing[END_REF]. In [START_REF] Sommer | Finding the closest lattice point by iterative slicing[END_REF], they give an iterative procedure that uses the facet inducing lattice vectors of the Voronoi cell (known as the Voronoi relevant vectors) to move closer and closer to the target, and show that this procedure converges to a closest lattice vector in a finite number of steps. The number of Voronoi relevant vectors is 2(2 n -1) in the worst-case (this holds for almost all lattices), and hence Voronoi cell based algorithms often require exponential size preprocessing. Subsequently, Micciancio and Voulgaris [START_REF] Micciancio | A deterministic single exponential time algorithm for most lattice problems based on voronoi cell computations[END_REF] (henceforth MV), showed how to compute the Voronoi relevant vectors during preprocessing and how to implement the search phase such that each phase uses O(4 n ) time and O(2 n ) space (yielding the first 2 O(n) time algorithm for exact CVP!). While Voronoi cell based CVPP algorithms require exponential time and space on general lattices, it was recently shown in [START_REF] Mckilliam | Finding a closest point in lattices of Voronoi's first kind[END_REF] that a variant of [START_REF] Sommer | Finding the closest lattice point by iterative slicing[END_REF] can be implemented in polynomial time for lattices of Voronoi's first kind -lattices which admit a set of n + 1 generators whose Gram matrix is the Laplacian of a non-negatively weighted graph -using these generators as the preprocessing advice. Hence, it is sometimes possible to "scale down" the complexity of exact solvers for interesting classes of lattices. Main Result. Our main result is a randomized O(2 n ) expected time and space algorithm for exact CVPP, improving the O(4 n ) (deterministic) running time of MV. Our preprocessing is the same as MV, that is we use the facet inducing lattice vectors of the Voronoi cell, known as the Voronoi relevant vectors (see Figure 1), as the preprocessing advice, which in the worst case consists of 2(2 n -1) lattice vectors. Our main contribution, is a new search algorithm that requires only an expected polynomial number of iterations over the set of Voronoi relevant vectors to converge to a closest lattice vector, compared to O(2 n ) in MV. One minor caveat to our iteration bound is that unlike that of MV, which only depends on n, ours also depends (at worst linearly) on the binary encoding length of the input lattice basis and target (though the O(2 n ) bound also holds for our procedure). Hence, while the bound is polynomial, it is only "weakly" so. In applications however, it is rather anomalous to encounter n dimensional lattice bases and targets whose individual coefficients require more than say poly(n) bits to represent, and hence the iteration bound will be poly(n) in almost all settings of relevance. Furthermore, it is unclear if this dependence of our algorithm is inherent, or whether it is just an artifact of the analysis. where Z ∼ Uniform(V ). Since y ∈ L and Z ∈ int(V ) with probability 1, note that i.e. the number of steps in phase C is 0. It therefore suffices to bound the number phase B steps. By Theorem 3, we have that as needed. This shows the desired upper bound on the path length. We now show that the above bounds are sharp. For the lower bound, note that it is tight for any two adjacent lattice vectors, since ∀v ∈ VR, v V = 2. For the upper bound, letting L = Z n , V = [-1/2, 1/2] n , VR = {±e 1 , . . . , ±e n }, the shortest path between x = 0 and y = (1, . . . , 1) has length n, while xy Since the Voronoi distance changes by at most 1 when switching from y to t ∈ y + V, we note that the above bound immediately yields a corresponding bound on the path length to a closest lattice vector to any target. As the phase C bound in Theorem 4 only holds for the truncated path, it does yield a bound on the randomized straight line path length for general lattices. However, for rational lattices and targets, we now show that for α small enough, the truncated path in fact suffices. We will derive this result from the following simple Lemmas. Lemma 6 (Rational Lattice Bound). Let L ⊆ Q n , and t ∈ Q n . Let q ∈ N be the smallest number such that qL ⊆ Z n and qt ∈ Z n , and let µ = µ(L) denote the covering radius of L. Proof. Note that t / ∈ y + V iff ty V > 1. From here, we have that for some v ∈ VR. By our assumptions, we note that v, ty = a/q 2 , for a ∈ N. Next, v 2 ≤ 2µ (see the end of section 4.1.1 for details) and v ∈ Z n /q, and hence we can write as needed. The following shows that the relevant quantities in Lemma 6 can be bounded by the binary encoding length of the lattice basis and target. Since it is rather standard, we defer the proof to section 6. Lemma 7 (Bit Length Bound). Let B ∈ Q n×n be a lattice basis matrix for an n dimensional lattice L, with Then for q ∈ N, the smallest number such that qL ⊆ Z n and qt ∈ Z n , we have that log 2 ( qµ(L)) ≤ enc (B) + enc (t) and log 2 (µ(L)/λ 1 (L)) ≤ enc (B). We are now in a position to give our full CVPP algorithm. SHORT PATHS ON THE VORONOI GRAPH AND CVPP 125 Theorem 8 (CVPP Algorithm). Let L be an n-dimensional lattice with basis B ∈ Q n×n , let VR denote the set of Voronoi relevant vectors of L. Given the set VR as preprocessing, for any target t ∈ Q n , a closest lattice vector to t can be computed using an expected poly(n, enc (B) , enc (t))|VR| arithmetic operations. Proof. To start we pick linearly independent v 1 , . . . , v n ∈ VR. We then compute the coefficent representation of t with respect to v 1 , . . . , v n , that is t = ∑ n i=1 a i v i . From here we compute the lattice vector x = ∑ n i=1 a i v i , i.e. the rounding of t. Next, using the convex body sampler (Theorem 18), we compute a (1/4)-uniform sample Z over V. Note that a membership oracle for V can be implemented using O(n|VR|) arithmetic operations. Furthermore, letting λ 1 = λ 1 (L), µ = µ(L), we have that Lemma 15 in the Appendix). Hence, nearly tight sandwiching estimates for V can be easily computed using the set VR. We now run the randomized straight line algorithm starting at lattice point x, perturbation Z, and target t. If the path produced by the algorithm becomes longer than cn(n + (enc (B) + enc (t))) (for some c ≥ 1 large enough), restart the algorithm, and otherwise return the found closest lattice vector. The correctness of the algorithm follows directly from the correctness of the randomized straight line algorithm (Lemma 2), and hence we need only show a bound on the expected runtime. Runtime. We first bound the number of operations performed in a single iteration. Computing v 1 , . . . , v n , t, and the sandwiching estimates for V, requires at most O(n We now show that the algorithm performs at most O(1) iterations on expectation. For this it suffices to show that each iteration succeeds with constant probability. In particular, we will show that with constant probability, the length of the randomized straight line path is bounded by O(n 2 (enc (B) + enc (t))). To do this we will simply show that the expected path length is bounded by O(n 2 (enc (B) + enc (t))) under the assumption that Z is truly uniform. By Markov's inequality, the probability that the length is less than twice the expectation is at least 1/2 for a truly uniform Z, and hence it will be a least 1/4 for a 1/4-uniform Z. To begin, we note that by the triangle inequality Let q be as in Lemma 7, and let α = 1 (4 qµ) 2 , where we have that ln(1/α) = O(enc (B) + enc (t)). Let y ∈ L denote the center of the first Voronoi cell containing t + αZ found by the randomized straight line algorithm. We claim that y is a closest lattice vector to t, or equivalently that t ∈ y + V. Assume not, then by Lemma 6, ty V ≥ 1 + 1 (2 qµ) 2 . On the other hand, since t + αZ ∈ y + V and Z ∈ V, by the triangle inequality 126 APPENDIX a contradiction. Hence y is a closest lattice vector to t. If Z ∼ Uniform(V ), then by Theorems 3 and 4 the expected length of the randomized straight line path up till t + αZ (i.e. till we find y) is bounded by as needed. The theorem thus follows. Preliminaries Basics. For n ≥ 1, we denote R n , Q n , Z n to be the set of n dimensional real / rational / integral vectors respectively. We let N denote the set of natural numbers, and Z + denote the set of non-negative integers. For two sets A, B ⊆ R n , we denote their Minkowski sum We write ∂A to denote the topological boundary of A. For a set A ⊆ R n , its affine hull, affhull(A), is the inclusion wise smallest linear affine space containing A. We denote the interior of A in R n as int(A). We denote the relative interior of A by relint(A), which is the interior of A with the respect to the subspace topology on affhull(A). For two n dimensional vectors x, y ∈ R n , we denote their inner product x, y = ∑ n i=1 x i y i . The 2 (Euclidean) norm of a vector x is denoted x 2 = x, x . We let B n 2 = {x ∈ R n : x 2 ≤ 1} denote the unit Euclidean ball, and let S n-1 = ∂B n 2 denote the unit sphere. For vectors x, y ∈ R n , we denote the closed line segment from x to y by [x, y] and[x, y) the half open line segment not containing y. We denote e 1 , . . . , e n the vectors of the standard basis of R n , that is the vectors such that e i has a 1 in the i th coordinate and 0's elsewhere. Binary encoding. For an integer z ∈ Z, the standard binary encoding for z requires 1 + log 2 (|z| + 1) bits, which we denote enc (z). For a rational number p q ∈ Q, p ∈ Z, q ∈ N, the encoding size of p q is enc p q = enc (p) + enc (q). For an n × m matrix M ∈ Q m×n or vector a ∈ Q n , enc (M), enc (a) denotes the sum of encoding lengths of all the entries. Integration. We denote the k-dimensional Lebesgue measure in R n by vol k (•). Only k = n and k = n -1 will be used in this paper. For k = n -1, we will only apply it to sets which can be written as a disjoint countable union of n -1 dimensional flat pieces. When integrating a function f : R n → R over a set A ⊆ R n using the n dimensional Lebesgue measure, we use the notation Probability. For a random variable X ∈ R, we define its expectation by E[X] and its variance by 2 . For two random variables X, Y ∈ Ω, we define their total variation distance to be Definition 9 (Uniform Distribution). For a set A ⊆ R n , we define the uniform distribution on A, denoted Uniform(A), to have probability density function 1/vol n (A) and 0 elsewhere. That is, for a uniform random variable X ∼ Uniform(A), we have that for any measurable set B ⊆ R n . Complexity. We use the notation O(T(n)) to mean O(T(n)polylog(T(n))). Lattices An n dimensional lattice L ⊆ R n is a discrete subgroup of R n whose linear span is R n . Equivalently, L is generated by all integer combinations of some basis B = (b 1 , . . . , b n ) of R n , i.e. L = BZ n . For k ∈ N, we define the quotient group L/kL = {y + kL : y ∈ L}. It is easy to check that the map The set of cosets of L form a group R n /L under addition, i.e. the torus. We will use the notation A (mod L), for a set A ⊆ R n , to denote the set of cosets L + A. Note that R n /L is isomorphic to [0, 1) n under addition (mod 1) (coordinate wise), under the map x → Bx + L for any basis B of L. We will need to make use of the uniform distribution over R n /L, which we denote Uniform(R n /L). To obtain a sample from Uniform(R n /L), one can take U ∼ Uniform([0, 1) n ) and return BU (mod L). We denote the length of the shortest non-zero vector (or minimum distance) of L as λ 1 (L) = min y∈L\{0} y 2 . We denote the covering radius of L as µ(L) = max t∈R n min y∈L ty 2 to be the farthest distance between any point in space and the lattice. The following standard lemma (see for instance [5]) allows us to bound the covering radius: Voronoi cell, tiling, and relevant vectors For a point t ∈ R n , let CVP(L, t) = arg min x∈L tx 2 , denote the set of closest lattice vectors to t. For y ∈ L, let denote the halfspace defining the set of points closer to 0 than to y. Definition 11 (Voronoi Cell). The Voronoi cell V (L) of L is defined as the set of all points in R n closer or at equal distance to the origin than to any other lattice point. Naturally, V (L) is the set of points of L whose closest lattice vector is 0. We abbreviate V (L) to V when the context is clear. It is easy to check from the definitions that a vector y ∈ L is a closest lattice vector to a target t ∈ R n iff ty ∈ V. The CVP is then equivalent to finding a lattice shift of V containing the target. From this, we see that the Voronoi cell tiles space with respect to R n , that is, the set of shifts L + V cover R n , and shifts x + V and y + V, x, y ∈ L, are interior disjoint if x = y. From the tiling property, we have the useful property that the distribution Uniform(V ) (mod L) is identical to Uniform(R n /L). We note that the problem of separating over the Voronoi cell reduces directly to CVP, since if y ∈ L is closer to a target t than 0, then H y separates t from V. Also, if no such closer lattice vector exists, then t ∈ V. Definition 12 (Voronoi Relevant Vectors). We define VR(L), the Voronoi relevant vectors of L, to be the minimal set of lattice vectors satisfying V (L) = ∩ v∈VR(L) H v , which we abbreviate to VR when the context is clear. APPENDIX Since the Voronoi cell is a full dimensional centrally symmetric polytope, the set VR corresponds exactly to the set of lattice vectors inducing facets of V (i.e. such that V ∩ ∂H v is n -1 dimensional). Definition 13 (Voronoi Cell Facet). For each v ∈ VR, let denote the facet of V induced by v. Here we have that since the intersection of distinct facets has affine dimension at most n -2. Similarly, A central object in this paper will be L + ∂V, the boundary of the lattice tiling. We shall call y + F v , for y ∈ L, v ∈ VR, a facet fo L + ∂V. Here, we see that Note that each facet is counted twice in the above union, i.e. y + An important theorem of Voronoi classifies the set of Voronoi relevant vectors: Theorem 14 (Voronoi). For an n dimensional lattice L, y ∈ L \ {0} is in VR(L) if and only if {±y} = arg min In particular, |VR| ≤ 2(2 n -1). Here the bound on |VR| follows from the fact that the map y → y + 2L from VR to L/(2L) \ {2L} is 2-to-1. Furthermore, note that each Voronoi relevant vector can be recovered from solutions to CVPs over 2L. More precisely, given a basis B for L, each vector in v ∈ VR can be expressed as Bpx, for some p ∈ {0, 1} n \ {0}, and x ∈ CVP(2L, Bp) (we get a Voronoi relevant iff x is unique up to reflection about Bp). We now list several important and standard properties we will need about the Voronoi cell and relevant vectors. We give a proof for completeness. Lemma 15. For an n dimensional lattice L: Proof. We prove each of the above in order: 1. Since each vector y ∈ L \ {0} satisfies y 2 ≥ λ 1 (L), we clearly have that λ 1 (L)/2B n 2 ⊆ H y . The inner containment holds for V since V = ∩ y∈L\{0} H y . For the outer containment, note that for any t ∈ V, that 0 is a closest lattice vector to t. Hence, by definition, t 2 = t -0 2 ≤ µ(L) as needed. 2. Since the set VR ⊆ L \ {0}, the vectors in VR clearly have length greater than or equal to λ 1 (L). Next, let y ∈ L \ {0} denote a shortest non-zero vector of L. We wish to show that y ∈ VR. To do this, by Theorem 14, we need only show that the only vectors of length λ 1 (L) in y + 2L are ±y. Assume not, then there exists z ∈ y + 2L, such that z is not collinear with y having z 2 = λ 1 (L). But then note that (y + z)/2 ∈ L \ {0} and (y + z)/2 2 < λ 1 (L), a contradition. 3. For v ∈ VR, we remember that v = arg min z∈2L+v z 2 . In particular, this implies that v 2 ≤ µ(2L) = 2µ(L) as needed. Since the VR vectors span R n , we can find linearly independent v 1 , . . . , v n ∈ VR. By Lemma 10, we have that as needed. Convex geometry , compact and has non-empty interior. K is symmetric if K = -K. For a symmetric convex body K ⊆ R n , we define the norm (or gauge function) with respect to K by For a set A ⊆ R n , we define its convex hull conv(A) to be the (inclusion wise) smallest convex set containing A. For two sets A, B ⊆ R n , we use the notation conv(A, B) def = conv(A ∪ B). For two non-empty measurable sets A, B ⊆ R n such that A + B is measurable, the Brunn-Minkowski inequality gives the following fundamental lower bound Laplace Distributions. We define the Gamma function, In particular, E[r] = kθ and VAR[r] = kθ 2 . Definition 16 (Laplace Distribution). We define the probability distribution Laplace(K, θ), with probability density function Equivalently, a well known and useful fact (which we state without proof) is: Lemma 17. X ∼ Laplace(K, θ) is identically distributed to rU, where r ∼ Γ(n + 1, θ) and U ∼ Uniform(K) are sampled independently. APPENDIX For our purposes, Laplace(K, θ) will serve as a "smoothed" out version of Uniform(K). In particular, letting f denote the probability density function of Laplace(K, θ), for x, y ∈ R n , by the triangle inequality Hence, the density varies smoothly as a function of • K norm, avoiding the "sharp" boundaries of the uniform measure on K. and O K (x) = 0 otherwise. Most algorithms over convex bodies can be implemented using only a membership oracle with some additional guarantees. In our CVPP algorithm, we will need to sample nearly uniformly from the Voronoi cell. For this purpose, we will utilize the classic geometric random walk method of Dyer, Frieze, and Kannan [9], which allows for polynomial time near uniform sampling over any convex body. Theorem 18 (Convex Body Sampler [9]). Let K ⊆ R n be a convex body, given my a membership oracle O K , satisfying rB n 2 ⊆ K ⊆ RB n 2 . Then for ε > 0, a realisation of a random variable X ∈ K, having total variation distance at most ε from Uniform(K), can be computed using poly(n, log(R/r), log(1/ε)) arithmetic operations and calls to the membership oracle. Bounding the Number of Crossings In this section, we prove bounds on the number of crossings the randomized straight line algorithm induces on the tiling boundary L + ∂V. For a target t, starting point x ∈ L, and perturbation Z ∼ Uniform(V), we need to bound the expected number of crossings in phases B and C, that is The phase B bound is given in Section 5.1, and the phase C is given in Section 5.2. Phase B estimates The high level idea of the phase B bound is as follows. To count the number of crossings, we break the segment [x + Z, y + Z] into k equal chunks (we will let k → ∞), and simply count the number of chunks which cross at least 1 boundary. By our choice of perturbation, we can show that each point on the segment [x + Z, y + Z] is uniformly distributed modulo the lattice, and hence the crossing probability will be indentical on each chunk. In particular, we will get that each crossing probability is exactly the probability that Z "escapes" from V after moving by (y -x)/k. This measures how close Z tends to be to the boundary of V, and hence corresponds to a certain "directional" surface area to volume ratio. In the next lemma, we show that the escape probability is reasonably small for any symmetric convex body, when the size of the shift is measured using the norm induced by the body. We shall use this to prove the full phase B crossing bound in Theorem 3. Lemma 19. Let K ⊆ R n be a centrally symmetric convex body. Then for Z ∼ Uniform(K) and y ∈ R n , we have that lim ε→0 Pr[Z + εy / ∈ K]/ε ≤ (n/2) y K Proof of Theorem 3 (Phase B crossing bound) Proof. Note first that the sets (L + ∂V ) ∩ [x + Z, y + Z] and (L + ∂V ) ∩ [x + Z, y + Z) agree unless y + Z ∈ L + ∂V. Given that this event happens with probability 0 (as L + ∂V has n dimensional Lebesgue measure 0), we get that We now bound the expectation on the right hand side. For s ∈ [0, 1], define the random variable (s) = (1 -s)x + sy + Z. Let A k j , 0 ≤ j < 2 k , denote the event that Clearly, we have that By the monotone convergence theorem, we get that Since L + ∂V is by definition invariant under lattice shifts, we see that Pr[A k j = 1] depends only on the distribution of (j/2 k ) (mod L). Given that Z (mod L) ∼ uniform(R n /L) and that R n /L is shift invariant, we have that (j/2 k ) (mod L) ∼ uniform(R n /L). In particular, this implies that Pr and hence by Lemma 19 as needed. The result follows by combining (4) and (5). Phase C estimates As mentioned previously in the paper, our techniques will not be sufficient to fully bound the number of phase C crossings. However, we will use be able to give bounds for a truncation of the phase C path, that is for α ∈ (0, 1], we will bound We will give a bound of O(n ln 1/α) for the above crossings in Theorem 4. For the proof strategy, we follow the approach as phase B in terms of bounding the crossing probability on infinitessimal chunks of [t + Z, t + αZ]. However, the implementation of this strategy will be completely different here, since the points along the segment no longer have the same distribution modulo the lattice. In particular, as α → 0, the distributions get more concentrated, and hence we lose the effects of the randomness. This loss will be surprisingly mild however, as evidenced by the claimed ln(1/α) dependence. For the infinitessimal probabilities, it will be convenient to parametrize the segment [t + Z, t + αZ] differently than in phase B. In particular, we use t + Z/s, for s ∈ [1, 1/α]. From here, note that The factor 1/2 above accounts for the fact that we count each facet twice, i.e. yt + F v and (y + v)t + F -v . Secondly, note that the intersections we count more than twice in the above decomposition correspond to a countable number of lines passing through at most n -2 dimensional faces, and hence have n dimensional Lebesgue measure 0. The equality in Equation ( 7) thus follows. If we restrict to one facet yt + F v , for some y ∈ L, v ∈ VR, we note that the line segment [X, αX] crosses the facet yt + F v at most once with probability 1. Hence, we get that Let r = v/ v 2 , yt + F v , noting the inner product with v (and hence v) is constant over F v . By possibly switching v to -v and y to y + v (which maintains the facet), we may assume that r ≥ 0. Notice that by construction, for any x in the (relative) interior of s(yt + F v ), we get that r = | η(x/s), x/s |, since then there is a unique facet of s(Lt + ∂V ) containing x. Integrating first in the in the direction v, we get that Note that we use the n -1 dimensional Lebesgue measure to integrate over s(y and hence has measure (and probability) 0. This is still satisfied by the last expression in (9), and hence the identity is still valid in this case. Putting everything together, combining Equation ( 7),( 9), we get that The lower bound given in the following Lemma will be needed in the proof of Lemma 25. Lemma 24. For a, b, c, d ∈ R, c ≤ d, we have that Proof. Firstly, we note that hence it suffices to prove the inequality when c = 0, d = 1. After this reduction, by possibly applying the change of variables h ← 1 -h, we may assume that |a| ≥ |a + b|. Next, by changing the signs of a, b, we may assume that a ≥ 0. Hence, it remains to prove the inequality 136 APPENDIX under the assumption that a ≥ |a + b|, or equivalently a ≥ 0 and -2a ≤ b ≤ 0. Notice that if a = 0 or b = 0, the above inequality is trivially true. If a, b = 0, then dividing inequality 10 by a, we reduce to the case where a = 1, -2 ≤ b < 0. Letting α = -1/b, we have that α ∈ [1/2, ∞). From here, we get that The derivative of the above expression is 1 -1 2α 2 . The expression is thus minimized for α = 1 √ 2 > 1/2, and the result follows by plugging in this value. We now prove the bound on the surface integral in terms of the expectation E[ X V ]. Lemma 25. For s ≥ 1 and X ∼ Laplace(V, θ), we have that Proof. We first prove the equality on the right hand side. We remember that X is identically distributed to rZ where r ∼ Γ(n + 1, θ) and Z ∼ Uniform(V ). From here, we have that We now prove the first inequality. To prove the bound, we write the integral expressing E[ X V ] over the cells of s(Lt + ∂V ), and compare the integral over each cell to the corresponding boundary integral. To begin Fix y ∈ L and v ∈ V in the above sum. Noting that v/ v 2 , sF v = s v 2 /2 by construction, and integrating first in the direction v, we get that s(y-t)+conv(0,sF v ) In the above, we use the n -1 dimensional Lebesgue measure to integrate over 2h embedded in R n (we also do this for ease of notation). Setting β = 2h s v 2 , note that β ∈ [0, 1]. In equation (12), β represents the convex combination between 0 and sF v , that is conv(0, Performing a change of variables, Equation ( 12) simplifies to From here we note that From inequality ( 14), we have that the expression in equation ( 13) is greater than or equal to To compare to the surface integral, we now lower bound the inner integral. Claim 26. For x V ≤ s, we have that Proof. Note that for 0 ≤ h ≤ min 1 n , θ s , we have that . Hence, using the above and Lemma 24, we have that as needed. Given Claim 26, we get that expression (15) is greater than or equal to APPENDIX Putting everything together, combining the above with equation ( 11), we get that where the last equality follows since each facet in s(Lt + ∂V ) is counted twice. The lemma thus follows. The following gives the full phase C bound for Laplace perturbations. Theorem 27. For α ∈ (0, 1] and X ∼ Laplace(V, θ), we have that Proof. Using Lemmas 23 and 25, we have that 6 Missings proofs from Section 3 Proof of Lemma 2 (Randomized Straight Line Complexity). We recall the three phases of the randomized straight line algorithm: (A) Move from x to x + Z. (B) Follow the sequence of Voronoi cells from x + Z to y + Z. (C) Follow the sequence of Voronoi cells from y + Z to y. Characterizing the Path Length: We will show that with probability 1, the length of the path on G induced by the three phases is Firstly, note that since Z ∈ V and x ∈ L, x + Z and x lie in the same Voronoi cell x + Z, and hence phase A corresponds to the trivial path x. Hence, we need only worry about the number of edges induced by phases B and C. The following claim will establish the structure of a generic intersection pattern with the tiling boundary, which will be necessary for establishing the basic properties of the path. Claim 28. With probability 1, the path [x + Z, t + Z] ∪ [t + Z, t) only intersects L + ∂V in the relative interior of its facets. Furthermore, with probability 1, the intersection consists of isolated points, and x + Z, t + Z / ∈ L + ∂V. Proof. We prove the first part. Let C 1 , . . . , C k denote the n -2 dimensional faces of V. Note that the probability of not hitting L + ∂V in the relative interior of it facets, can be expressed as Here the last inequality is valid since L is countable. Analyzing each term separately, we see that To justify the last equality, note that since C i is n -2 dimensional, y + C i -[x, t] is at most n -1 dimensional (since the line segment can only add 1 dimension). Therefore y + C i -[x, t] has n dimensional Lebesgue measure 0, and in particular probability 0 with respect to Uniform(V ). Next, we have that Pr where the last equality follows since ∪ s>1 s(y + C i -t) is at most n -1 dimensional. Hence the probability in ( 16) is 0, as needed. We now prove the second part. Note that if the path [x + Z, t + Z] ∪ [t + Z, t) does not intersect L + ∂V in isolated points (i.e. the intersection contains a non-trivial interval), then either [x + Z, t + Z] or [t + Z, t) must intersect some facet of L + ∂V in a least 2 distinct points. Let F v be the facet of V induced by v ∈ VR. If [t + Z, t) intersects y + F v , for some y ∈ L, in two distinct points then we must have that v, Z = 0. Since Pr[∪ v∈VR { v, Z = 0}] = 0, this event happens with probability 0. Next, note that [x + Z, t + Z] intersects y + F v in two distinct points, if and only if v, xt = 0 and v, Z = v, y + v/2 . But then, the probability of this happening for any facet can be bounded by For the last part, note that since L + ∂V is the union of n -1 dimensional pieces, Pr[x The claim thus follows. Conditioning on the intersection structure given in claim 28, we now describe the associated path on G. Let p 1 , . . . , p k denote the points in ([x + Z, t + Z] ∪ [t + Z, t)) ∩ (L + ∂V ) ordered in order of appearance on the path [x + Z, t + Z] ∪ [t + Z, t) from left to right. Letting p k+1 = t, let y i ∈ L, 1 ≤ i ≤ k, denote the center of the unique Voronoi cell in L + V containing the interval [p i , p i+1 ]. Note that the existence of y i is guaranteed since the Voronoi cells in the tiling L + V are interior disjoint, and the open segment (p i , p i+1 ) lies in the interior of some Voronoi cell by convexity of the cells. Letting y 0 = x, we now claim that y 0 , y 1 , . . . , y k form a valid path in G. To begin, we first establish that y i = y i+1 , 0 ≤ i ≤ k. Firstly, since x + Z / ∈ L + ∂V, we have that Z is in the interior of V, and hence the ray starting at Z in the direction of p 1 exits x + V at p 1 and never returns (by convexity of V). Furthermore, since p 1 = t + Z, the Voronoi cell y 1 + V must contain a non-trivial interval on this ray starting at p 1 , i.e. [p 1 , p 2 ], and hence y 1 = x. Indeed, for the remaining cases, the argument follows in the same way as long as the Voronoi cell y i+1 + V contains a non-trivial interval of the ray exiting y i + V. Note that this is guaranteed by the assumption that t + Z / ∈ ∂V and by the fact that none of the p i s equals t. Hence y i = y i+1 , 0 ≤ i ≤ k, as needed. Next, note that each p i , i ∈ [k], belongs to the relative interior of some facet of L + ∂V. Furthermore, by construction p i ∈ y i-1 + ∂V and p i ∈ y i + ∂V. Since the relative interior of facets of L + ∂V touch exactly two adjacent Voronoi cells, and since y i-1 = y i , we must have that p i ∈ y i-1 + F v , where v = y i -y i-1 ∈ VR. Hence the path y 0 , y 1 , . . . , y k is valid in G as claimed. From here, note that the length of is indeed Since this holds with probability 1, we get that the expected path length is as needed. Computing the Path: We now explain how to compute each edge of the path using O(n|VR|) arithmetic operations, conditioning on the conclusions of Claim 28. In constructing the path, we will in fact compute the intersection points p 1 , . . . , p k as above, and the lattice points y 1 , . . . , y k . As one would expect, this computation is broken up in phase B and C, corresponding to computing the intersection / lattice points for [x + Z, t + Z] in phase B, followed by the intersection / lattice points from [t + Z, t) in Phase C. For each phase, we will use a generic line following procedure that given vectors a, b ∈ R n , and a starting lattice point z ∈ L, such that a ∈ z + V, follows the path of Voronoi cells along the line segment [a, b), and outputs a lattice vector w ∈ L satisfying b ∈ w + V. To implement phase B, we initialize the procedure with x + Z, t + Z and starting point x. For phase C, we give it t + Z, t and the output of phase B as the starting point. We describe the line following procedure. Let (α) = (1α)a + αb, for α ∈ [0, 1], i.e. the parametrization of [a + Z, b + Z] as a function of time. The procedure will have a variable for α, which will be set at its bounds at the beginning and end of the procedure, starting at 0 ending at ≥ 1, and in intermediate steps will correspond to an intersection point. We will also have a variable w ∈ L, corresponding to the current Voronoi cell center. We will maintain the invariant that (α) ∈ w + V, and furthermore that (α) ∈ w + ∂V for α ∈ (0, 1). The line following algorithm is as follows: Described in words, each loop iteration does the following: given the current Voronoi cell w + V, and the entering intersection point (α) of the line segment [a, b] with respect to w + V, we first Hence log 2 q is smaller than the sum of encoding sizes of the denominators of the entries of B and t. Next, it is well known that µ(L) ≤ 1 2 ∑ ij B 2 ij (see for example [5]). From here, we get that Hence log 2 µ(L) is less than the sum of encoding sizes of the numerators of the entries in B. The bound log 2 ( qµ(L)) ≤ enc (B) + enc (t) now follows by adding ( 19), (20). We now bound log 2 (µ(L)/λ 1 (L)). Letting q = ∏ ij q B ij , note that Therefore 1/λ 1 (L) ≤ q. Since log 2 q ≤ ∑ ij log 2 (q B ij ) and log 2 (µ(L)/λ 1 (L)) ≤ log 2 ( qµ(L)), combining with (20) we get that log 2 (µ(L)/λ 1 (L)) ≤ enc (B) as needed. Open Problems Our work here raises a number of natural questions. Firstly, given the improvement for CVPP, it is natural to wonder whether any of the insights developed here can be used to improve the complexity upper bound for CVP. As mentioned previously, this would seem to require new techniques, and we leave this as an open problem. Secondly, while we now have a number of methods to navigate over the Voronoi graph, we have no lower bounds on the lengths of the path they create. In particular, it is entirely possible that either the MV path or the simple deterministic straight line path, also yield short paths on the Voronoi graph. Hence, showing either strong lower bounds for these methods or new upper bounds is an interesting open problem. In this vein, as mentioned previously, we do not know whether the expected number of iterations for the randomized straight line procedure is inherently weakly polynomial. We leave this as an open problem. Contributions Renforcement cumulatif Nous proposons une reformulation de la contrainte cumulative, c'est-à-dire la génération de contraintes redondantes plus serrées, ce qui permet une propagation plus forte, sans perdre bien entendu de solution faisable. Cette technique est couramment utilisée en programmation linéaire entière (génération de coupes), mais il s'agit de l'un des tous premiers exemples d'une contrainte globale redondante en programmation par contraintes. Le calcul de cette reformulation repose sur un programme linéaire dont la taille dépend uniquement de la capacité de la ressource mais pas du nombre de tâches, ce qui permet de précalculer les reformulations. Nous fournissons des garanties sur la qualité des reformulations ainsi obtenues, montrant en particulier que toutes les bornes que l'on calcule en utilisant ces reformulations sont au moins aussi fortes que celles que l'on obtiendrait en faisant une relaxation préemptive du problème d'ordonnancement. Cette technique permet de renforcer toutes les propagations de la contrainte cumulative reposant sur le calcul d'une borne énergétique, notamment l'Edge-Finding et le Raisonnement Énergétique. Ce travail a été présenté lors de la conférence ROADEF 2014 [BB14] et a été publié dans le journal Discrete Applied Mathematics en 2017 [START_REF] Baptiste | Redundant cumulative constraints to compute preemptive bounds[END_REF]. Raisonnement Énergétique en O(n 2 log n) Ce travail consiste en une amélioration de la complexité algorithmique de l'une de des propagations les plus puissantes pour la contrainte cumulative, le Raisonnement Énergétique de Erschler et Lopez, introduit en 1990. Bien que cette propagation permette des déductions fortes, elle est rarement utilisée en pratique en raison de sa complexité cubique. De nombreuses approches ont été développées ces dernières années pour tenter malgré tout de la rendre utilisable (apprentissage automatique pour l'utiliser à bon escient, réduction du facteur constant de sa complexité algorithmique, etc). Nous proposons un algorithme qui calcule cette propagation avec une complexité O(n 2 log n), ce qui constitue une amélioration significative de cet algorithme connu depuis plus de 25 ans. Cette nouvelle approche repose sur de nouvelles propriétés de la contrainte cumulative et sur une étude géométrique. Ce travail a été publié sous une forme préliminaire lors de la conférence CP 2014 [START_REF] Bonifas | Fast propagation for the Energy Reasoning[END_REF] puis a fait l'objet d'une publication [START_REF] Bonifas | A O (n^2 log (n)) propagation for the Energy Reasoning[END_REF] lors de la conférence ROADEF 2016, récompensée par la 2 ème place au Prix du Jeune Chercheur. In this thesis, we study the cumulative constraint with the help of tools rarely used in constraint programming (polyhedral analysis, linear programming duality, projective geometry duality) and propose two contributions for the domain. Cumulative strengthening is a means of generating tighter redundant cumulative constraints, analogous to the generation of cuts in integer linear programming. This is one of the first examples of a redundant global constraint. Energy Reasoning is an extremely powerful propagation for cumulative constraint, with hitherto a high complexity of O(n 3 ). We propose an algorithm that computes this propagation with a O(n 2 log n) complexity, which is a significant improvement of this algorithm known for more than 25 years. Université Paris-Saclay Espace Technologique / Immeuble Discovery Route de l'Orme aux Merisiers RD 128 / 91190 Saint-Aubin, France

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2016

https://univ-rennes.hal.science/hal-01416158/file/Scalp%20tissue%20expansion%20above.pdf

M.D Raphael Carloni email: [email protected] M.D Christian Herlin M.D Benoit Chaput M.D Antoine De Runz M.D E Watier MD Nicolas Bertheuil Scalp tissue expansion above a custom-bone hydroxyapatite cranial implant to correct sequelar alopecia on a transposition flap: a case report Keywords: cranioplasty, tissue expansion, alopecia, transposition flap, calvarial reconstruction Scalp tissue expansion above a custom-made hydroxyapatite cranial implant to correct sequelar alopecia on a transposition flap: a case report INTRODUCTION Tissue expansion of the scalp is a well-codified technique used to improve success before cranioplasty after previous infection, tissue avulsion, and/or radiotherapy, [START_REF] Carloni | Soft tissue expansion and cranioplasty: For which indications? J Cranio-Maxillo-fac[END_REF] and to correct sequelar alopecia during second-step surgery. [START_REF] Bilkay | Alopecia treatment with scalp expansion: some surgical fine points and a simple modification to improve the results[END_REF] The scalp is particularly suitable for expansion because the cranial bone provides a solid basis for the expander. No data are available in the literature regarding the safety of expansion above the implant when an alloplastic material (e.g., hydroxyapatite [START_REF] Stefini | Use of "custom made" porous hydroxyapatite implants for cranioplasty: postoperative analysis of complications in 1549 patients[END_REF] , titanium [START_REF] Cabraja | Long-term results following titanium cranioplasty of large skull defects[END_REF] , methyl metacrylate [START_REF] Greene | Onlay frontal cranioplasty using wire reinforced methyl methacrylate[END_REF] ) is used to replace the cranial bone after cranioplasty. We present the case of a patient who underwent scalp tissue expansion above a custom-made hydroxyapatite implant to correct sequelar alopecia after cranioplasty for dermatofibrosarcoma protuberans. CASE REPORT This 30-year-old man had previously undergone a right temporo-frontal craniotomy for resection of temporo-frontal dermatofibrosarcoma protuberans with bone invasion. Coverage of the dura with a transposition flap and no bone reconstruction was performed at the same time. The donor site of the flap, on the vertex, was covered with a skin graft, leaving a sequelar alopecia zone that measured 9 × 13 cm (Fig. 1). Histological findings showed complete excision of the tumor. Six months later, cranioplasty with a custom-made hydroxyapatite implant was decided on. An incision at the medial edge of the transposition flap provided access to the cranial defect. A rectangular periosteal flap was dissected over the cranial bone defect to expose the dura. Bony edges were sharpened to expose healthy cancellous bone, and the implant was fixed on the bone defect. Dural tenting sutures were performed on the edges of the defect to prevent epidural hemorrhage after the surgery, by fixing dura to the native bone with silk sutures. The postoperative course was free of complications. Although the reconstruction was successful, sequelar alopecia on the vertex remained a major social, psychological, and esthetic concern for the patient. At 7 months after the cranioplasty, we decided to perform soft tissue expansion, with two smooth rectangular expanders placed in the two temporal regions between the galea and the periosteum M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT through incisions placed at the lateral edges of the alopecia area of the vertex; a 120-cc expander was used on the left side and a 240-cc expander was used on the right side (Fig. 2). The expander on the right side was partially placed above the cranial implant and under the previous transposition flap. In both cases, filling reservoirs were internal, placed in the subcutaneous layer. The expanders were filled with normal saline (90 cc on the right side, 40 cc on the left side) at the end of the surgery. Two drains were inserted in the subgaleal pockets and kept in place until the amount of drainage had decreased to 20 cc per day. Antibiotic prophylaxis (cefazolin, 2 g) was administered intraoperatively. Healing was achieved in 15 days. The expanders were filled once per week for 3 months postoperatively. The expanders on the right and left sides were inflated to 355 cc and 130 cc, respectively (Fig. 3). CT was performed before expander placement, 1 day postoperatively, and at the end of the expansion to confirm the absence of fracture or dislocation of the hydroxyapatite implant (Fig. 4). At 1 week after the last inflation, the expanders were removed and two advancement flaps were used to remove the entire alopecic zone of the vertex (Fig. 5). The galea was scored to allow further advancement. No complication occurred during the expansion or postoperatively. The patient was very satisfied with the cosmetic result (Fig. 6). DISCUSSION The choice between autologous bone and alloplastic material for bone reconstruction in cranioplasty remains under debate. [START_REF] Schwarz | Cranioplasty after decompressive craniectomy: is there a rationale for an initial artificial bone-substitute implant? A single-center experience after 631 procedures[END_REF] Debate also exists concerning techniques for soft tissue recruitment when tissue is lacking over the scalp. [START_REF] Carloni | Soft tissue expansion and cranioplasty: For which indications? J Cranio-Maxillo-fac[END_REF][START_REF] Leedy | Reconstruction of acquired scalp defects: an algorithmic approach[END_REF]8 Among techniques, tissue expansion has been demonstrated to be safe when performed over the cranial bone, either before the cranioplasty procedure to increase the rate of success [START_REF] Carloni | Soft tissue expansion and cranioplasty: For which indications? J Cranio-Maxillo-fac[END_REF][START_REF] Merlino | Role of systematic scalp expansion before cranioplasty in patients with craniectomy defects[END_REF] or after cranioplasty, away from the implant, to correct sequelar alopecia. [START_REF] Bilkay | Alopecia treatment with scalp expansion: some surgical fine points and a simple modification to improve the results[END_REF][START_REF] Azzolini | Skin expansion in head and neck reconstructive surgery[END_REF] However, no application of this technique above a cranial implant has been described, due to the supposed risk of the procedure. Through this case report, we aimed to show that safe expansion is possible, even in situations in which the only stretchable tissue is partially situated above a cranial implant. Alopecia of the scalp after cranioplasty is a frequent complaint of patients. It can be created by radiotherapy and infection prior to scalp reconstruction, or as a consequence of soft tissue reconstruction with local or free flaps. Except for the rotational scalp flap, which does not cause alopecia but can be used when only slight tissue recruitment is necessary, 7 these flaps are often unesthetic. For large tissue needs, the choice between flaps 8 and tissue expansion [START_REF] Carloni | Soft tissue expansion and cranioplasty: For which indications? J Cranio-Maxillo-fac[END_REF] Hydroxyapatite implants have been demonstrated to have osteoconductive properties that lead to good osteointegration with the cranial vault. In clinical practice, osteointegration can be checked on cranial scans during patient follow up and is defined as the absence of a radiolucent line at the interface between the living bone and the surface of the implant. [START_REF] Stefini | Use of "custom made" porous hydroxyapatite implants for cranioplasty: postoperative analysis of complications in 1549 patients[END_REF] The porous nature of the implant is supposed to allow ingrowth of osteoprogenitor cells, and increased resistance of the implant. Because hydroxyapatite has the same density as bone on CT scans, this process is difficult to confirm and we cannot say that the implant acquires the same resistance as bone in the months following cranioplasty. However, dislocation or fracture of the implant is very rare. [START_REF] Lindner | Cranioplasty using custom-made hydroxyapatite versus titanium: a randomized clinical trial[END_REF][START_REF] Iaccarino | Preliminary Results of a Prospective Study on Methods of Cranial Reconstruction[END_REF] These properties allowed us to attempt expansion over this kind of implant, with a successful outcome. Indeed, in our patient, the implant provided sufficient strength to support a 355-cc expansion. The main condition to verify before expansion was osteointegration of the implant on the preoperative scan. The risk of infection associated with expansion, which could contaminate the underlying cranial implant, was prevented by respecting some simple rules: (1) intraoperative administration of a prophylactic antibiotic, (2) rapid drain removal, and (3) separation of the implant and the expander by a periosteal flap during cranioplasty. No specific alloplastic material has been shown to be more sensitive to infection in the literature. [START_REF] Yadla | Effect of early surgery, material, and method of flap preservation on cranioplasty infections: a systematic review[END_REF][START_REF] Reddy | Clinical outcomes in cranioplasty: risk factors and choice of reconstructive material[END_REF] Estimated infection rates are 2% for hydroxyapatite implants [START_REF] Stefini | Use of "custom made" porous hydroxyapatite implants for cranioplasty: postoperative analysis of complications in 1549 patients[END_REF] and 24% for expanders. [START_REF] Cunha | Tissue expander complications in plastic surgery: a 10-year experience[END_REF] Tissue expansion must be chosen only in situations in which the scalp has healed completely because wounds could be the port of entry for bacteria, which could contaminate the expander. [START_REF] Carloni | Reconstructive approach to hostile cranioplasty: A review of the University of Chicago experience[END_REF] When the scalp has not yet healed, waiting for complete healing with dressings or the choice of another reconstructive method is preferable, with tissue expansion performed once healing is M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT achieved. Other reported complications of tissue expansion include exposure, rupture of the expander, and hematoma. [START_REF] Cunha | Tissue expander complications in plastic surgery: a 10-year experience[END_REF] The choice of expander size depends on the size of the defect and the location of the alopecia zone to treat. For round expanders, the tissue gain obtained with expansion corresponds approximately to the radius of the expander. [START_REF] Carloni | Soft tissue expansion and cranioplasty: For which indications? J Cranio-Maxillo-fac[END_REF] For rectangular expanders, the gain is more difficult to predict. For the present patient, we chose two rectangular expanders to expand the two temporal regions. No difference in complications according to expander shape has been reported. However, the use of fewer expanders seems be associated with a lower frequency of complications. [START_REF] Leedy | Reconstruction of acquired scalp defects: an algorithmic approach[END_REF][START_REF] Wang | Complications in tissue expansion: an updated retrospective analysis of risk factors. Handchir Mikrochir Plast Chir Organ Deutschsprachigen Arbeitsgemeinschaft Für Handchir Organ Deutschsprachigen Arbeitsgemeinschaft Für Mikrochir Peripher Nerven Gefässe Organ[END_REF] Expanders must be placed as high as possible on the scalp for two reasons: the higher temporal regions and the vertex are easier to stretch, and such placement avoids the patient's sleeping on the expanders during the night, thereby diminishing the risk of exposure. Although the filling reservoir can be placed externally, placement under the skin is more comfortable for the patient. Our expansion protocol did not differ from the protocols described in the literature. In our case, we stopped expander inflation when the filled volume slightly exceeded 150% of its capacity on the right side and 100% of capacity on the left side. Expanders can be overinflated to 150% of their capacity to improve tissue gain. [START_REF] Carloni | Soft tissue expansion and cranioplasty: For which indications? J Cranio-Maxillo-fac[END_REF] Another option to improve tissue gain consists of scoring of the galea. This procedure could compromise the perfusion of the advancement flaps. Thus, we perform it only when necessary. Other techniques described in the literature to correct sequelar alopecia of the scalp include external tissue expansion [START_REF] Reinard | Preoperative external tissue expansion for complex cranial reconstructions[END_REF] and hair grafting in the alopecia zone. 19 In our experience, the outcome of hair grafting is very disappointing in large zones of scar tissue, but this technique can be attempted in small zones, such as enlarged scars. Two possibilities exist in this situation. When possible, a tricophytic technique should be attempted first. 20 When scar enlargement is important, making a tricophytic suture difficult, or when a new suture in the area is risky for healing reasons, a hair graft with follicular unit extraction 19 should be attempted. CONCLUSION Tissue expansion remains the gold standard for the treatment of sequelar alopecia after cranioplasty. This procedure may be of concern to the physician when performed above hydroxyapatite implants because of the risk of infection and rupture of the cranial implant. Precautions to prevent these issues include a preoperative check of the osteointegration of the must be M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT discussed because both options have advantages and disadvantages. Based on our experience, we recommend that scalp expansion be attempted first, except in situations in which an open wound still exists or rapid coverage is necessary. 11 This choice allows the achievement of a more esthetic reconstruction after the first surgery, providing natural hair-bearing skin to correct a tissue defect. When large local flaps of the scalp, such as transposition flaps or bipedicled flaps, have been used in the first surgery, most of the hairy tissue lies above the cranial implant and the only solution for the treatment of alopecia with expansion is to place an expander above the cranial implant. Follicular unit extraction. Facial Plast Surg FPS 2008;24:404-413. 20. Ahmad M. Does the trichophytic technique have any role in facial wound closure? A hypothesis. J Plast Reconstr Aesthetic Surg JPRAS 2009;62:662. Figure 1 . 1 Figure 1. Preoperative picture of the patient presenting with a sequelar alopecia of vertex after a transposition flap of the scalp. Figure 2 . 2 Figure 2. Photo of the two rectangular expanders. Figure 3 . 3 Figure 3. Picture of the patient after maximal inflation of the expanders. Figure 4 . 4 Figure 4. Computed tomographic scan after maximal inflation of the expanders. The scan shows an osteointegration of the hydroxyapatite implant and the absence of fracture or dislocation. Figure 5 . 5 Figure 5. Intraoperative view after removal of the expanders. Figure 6 . 6 Figure 6. Postoperative picture of the patient after correction of the alopecia. M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT implant; a 6-month interval between cranioplasty and expansion; separation of the expander from the implant using periosteum; antibiotic prophylaxis and rapid drain removal. DISCLOSURES The authors have no disclosures to declare. Abbreviations computed tomography (CT)

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https://hal.science/hal-01745901/file/TwoCenturiesDemographicEvolutionSurvey_v1.pdf

Nicole El Karoui email: [email protected] Kaouther Hadji email: [email protected]. Sarah Kaakai email: [email protected]. Inextricable complexity of two centuries of demographic changes: A fascinating modeling challenge demographic transition affected most of European countries and countries with Introduction In over two centuries, the world population has been transformed dramatically, under the effect of considerable changes induced by demographic, economic, technological, medical, epidemiological, political and social revolutions. The age pyramids of ageing developed countries look like "colossus with feet of clay", and the complexity of involved phenomena makes the projection of future developments very difficult, especially since these transitions are unprecedented. The problem does not lie so much in the lack of data or empirical studies. For several years now, a considerable amount of data have been collected at different levels. A number of international organizations 1 have their own open databases, and national statistical institutes 2 have been releasing more and more data. On top of that, more than fifty public reports are produced each year. The private sector is also very active on these issues, especially pension funds and insurance companies which are strongly exposed to the increase in life expectancy at older ages. However, the past few years have been marked by a renewed demand for more efficient models. This demand has been motivated by observations of recent demographic trends which seem to be in contradiction with some firmly established ideas. New available data seem to indicate a paradigm shift over the past decades, toward a more complex and individualized world. Countries which had similar mortality experiences until the 1980s now diverge, and a widening of health and mortality gaps inside countries has been reported by a large number of studies. These new trends have been declared as key public issues by several organizations, including the WHO in its latest World report on ageing and health (World Health Organization (2015)), and the National Institute on Aging in the United States which created in 2008 a panel on Understanding Divergent Trends in Longevity in High-Income Countries, leading to the publication of a comprehensive report in National Research Council and Committee on Population (2011). In the face of the considerable amount of literature, data and points of views concerning the evolution of human longevity and populations in the last two century, we came to the conclusion that it was necessary to highlight a number of key observations to avoid the pitfalls of an overly naive approach. The goal of this cross-disciplinary survey is to help a modeler of human population dynamics to find a coherent way (for instance by taking into account the whole population dynamics and not only old ages) around this mass of multidisciplinary information. Based on numerous surveys from various academic disciplines and many contradictory readings, we offer a subjective selection of what we believe to be the most important ideas or facts, from a mathematical modeling perspective. As we will not be able to devote the necessary time to each point, we try to illustrate some of our points with examples that will support the intuition about mentioned phenomena. It should be emphasized from the start that if the discussion is greatly enriched by the multidisciplinary nature of the field, the presentation of ideas is also made more difficult, especially for matters of vocabulary. It should also be noted that issues related to medical advances and to the biology of human ageing are dealt with in a very cursory way, as we focus mainly on economic and social issues. The survey is composed of three main parts which are summarized in the next subsection. The first part deals with the historic demographic transition. The importance of public health is dealt with, with a specific focus on the cholera epidemic outbreaks that took place in France and in the UK during the nineteenth century. Other features of the historic demographic transition are also considered. In particular, we explore the relationship between the economic growth and mortality improvements experienced during the past century. In the second part of the survey, we examine the implications for population modeling and the key features of this shift in paradigm that have been observed since the 2000s. We first give a brief overview of the so-called demographic transition, and the move toward the description of increased complexity and diverging trends that have been recently observed, based principally on the experience of developed countries. A special attention is paid to socioeconomic differences in health and mortality. In the last part of the survey, we give a short review of microsimulation models and agent based models widely used in social sciences, and in particular in demographic applications. We first describe the main components of a dynamic microsimulation exercise to study heterogeneous individual trajectories in order to obtain macro outcomes by aggregation in the form of a data-driven complex model. Then, we present the agent based models which take into account individual interactions for explaining macroscopic regularities. The historic demographic transition Since the nineteenth century, most countries have experienced a remarkable evolution of their populations, referenced by demographers and economists as the demographic transition [START_REF] Bongaarts | Human Population Growth and the Demographic Transition[END_REF]). The historic demographic transition of the developed world 3 began in the nineteenth century and was completed in over a century (∼ (1850 -1960)). This historical process is mainly referred to as "the secular shift in fertility and mortality from high and sharply fluctuating levels to low and relatively stable ones" [START_REF] Lee | Demographic transition and its consequences[END_REF]). These substantial demographic changes caused life expectancy at birth to grow by more than 40 years over the last 150 ans (for instance, life expectancy at birth rose in the United Kingdom and France from about 40 years in the 1870s 4 , to respectively 81 and over 82 years in 2013 5 ), and the world population to grow from around 1 billion in 1800 to 2.5 billions by 1950 [START_REF] Bloom | The global demography of aging: Facts, explanations, future[END_REF]). The treatment of infectious diseases constituted the vast bulk of the causes that explain the historic fall in mortality. For example, infectious diseases had virtually disappeared by 1971 in England and Wales while they were responsible for 60 percent of deaths in England and Wales in 1848 [START_REF] Cutler | The determinants of mortality[END_REF]). The causes of this reduction have been extensively debated. Among the main causes that have been put forward are economic growth, improvement in living standards, education and most importantly social and public health measures [START_REF] Bloom | The global demography of aging: Facts, explanations, future[END_REF], [START_REF] Cutler | The determinants of mortality[END_REF]). For instance, Cutler and Miller (2005) estimated that the purification of water explained half of the mortality reduction in the US in the first third of the twentieth century. 2.1 The cholera pandemic: a starting point of the demographic transition The cholera pandemic: a starting point of the demographic transition In order to understand the unprecedented rise of life expectancy during the first half of the twentieth century, one has to go back 50 years which foreshadow the demographic transition. At the beginning of the nineteenth century, the Industrial Revolution led to a total upheaval of society, associated with unbridled urban sprawl and unsanitary living conditions. In Paris, the population doubled from 1800 to 1850 to attain over one million inhabitants [START_REF] Jardin | Restoration and Reaction, 1815-1848[END_REF]), while London grew by 2.5 fold during those 50 years, to attain more than 2 million inhabitants [START_REF] Chalklin | The rise of the English town, 1650-1850[END_REF]). In this context, epidemics were frequent and deadly. The cholera pandemic, which struck fear and left indelible marks of blue-black dying faces due to cyanose' on the collective imagination (hence the nickname "blue death"), had the most important social and economic consequences. It is often refered to as an iconic example where medicine was confronted to statistics [START_REF] Dupaquier | Cholera in England during the nineteenth century: medicine as a test of the validity of statistics[END_REF]) and was regarded as "the real spark which lit the tinder of the budding philanthropic movement, culminating in the social reforms and the foundation of the official public health movement seventeen years later" [START_REF] Underwood | The History of Cholera in Great Britain[END_REF]). The cholera pandemics originated in India and spread to Europe in the 1830s. Four subsequent outbreaks (1831, 1848 -1854, 1866 -1867 and 1888 -1889) mainly affected France and England, causing 102.000 deaths in France in 1832 and 143.000 in the 1850s over a population of 36 million [START_REF] Haupt | Histoire sociale de la France depuis 1789. Bibliothèque allemande[END_REF]). In London, 6536 deaths were reported in 1831 and 14137 deaths during the 1848-1849 cholera outbreak [START_REF] Underwood | The History of Cholera in Great Britain[END_REF]). In the following paragraphs, we will focus on the cholera outbreaks in France and England, in order to illustrate the profound changes which occured at different levels (city, state and international), and which still give valuable insight on contemporary challenges. Cholera in England The intensity of the first cholera outbreak in London in 1831, combined with the growing influence of advocates of public health, brought to light the need for public measures to improve sanitation. At that time, a lot of reformers considered that statistics were a prerequisite for any intervention, and the enthusiasm in the field expanded very quickly, which is somehow reminiscent of the current craze for data science. In this context, the General Register Office (G.R.O) was created in 1836, with the aim of centralizing vital statistics. England and Wales were divided in 2193 registration sub districts, administered by qualified registrars (often doctors). In charge 2.1 The cholera pandemic: a starting point of the demographic transition of compiling data from registration districts, W. Farr served as statistical superintendent from 1839 to 1880 and became "the architect of England's national system of vital statistics" [START_REF] Eyler | William Farr on the cholera: the sanitarian's disease theory and the statistician's method[END_REF]). The precise mortality data collected by the G.R.O during cholera outbreaks turned out to be instrumental in the analysis of the disease. In his pioneering Report on the Mortality of Cholera in England, 1848-49 (Farr (1852)), Farr and the G.R.O produced almost four hundred pages of statistics. His main finding, based on the collected data of the 1848-49 outbreak, was the existence of an inverse relationship between cholera mortality rates and the elevation of registration districts above the Thames. Farr was particularly pleased with this statistical law, since it validated his beliefs in the prevailing miasmatic theories, which predicted that the passing on of the disease was airborne. It was actually J. Snow who first claimed that cholera communication was waterborne, with his famous experiment of the Broad street pump [START_REF] Brody | Mapmaking and myth-making in Broad Street: the London cholera epidemic, 1854[END_REF]). However, Farr's statistics were decisive in supporting and validating his theory. Although Snow's theory was not widely accepted, he contributed to raising the issue of water quality. Under the impulsion of the General Board of Health created in 1848, the Metropolis Water Act of 1852 introduced for the first time regulations for (private) water supply companies, to take effect by 1855. At the time of the 1853-54 cholera outbreak, Farr found out that only one company had complied with the new regulations, and that in a number of districts, it was competing with another company drawing water from a highly polluted area. The perfect conditions for a full-scale experiment were brought together, and Farr and Snow joined their investigations to conclude that without doubt, water played an important role in the communication of the disease. In 1866, a smaller outbreak hit London. More specifically, the reintroduction of sewage contaminated water by the East London Water Company caused in just one week 908 over 5596 deaths in London [START_REF] Dupaquier | Cholera in England during the nineteenth century: medicine as a test of the validity of statistics[END_REF], [START_REF] Underwood | The History of Cholera in Great Britain[END_REF]) . Despite the overwhelming amount of evidence, the Medical Officer himself tried to exonerate the company, causing the wrath of Farr. This event, however, was a wake-up call for the English political class to guarantee the supply of clean water. Several public health measures were taken from the 1850s in order to improve public health and water quality. A new administrative network was established in London in 1855, which undertook the development of the city's main drainage system which was completed in 1875. Among other measures were the Rivers Pollution act in 1876 and the carrying out of monthly water reports from the 1860s [START_REF] Hardy | Water and the search for public health in london in the eighteenth and nineteenth centuries[END_REF]). 2.1 The cholera pandemic: a starting point of the demographic transition Cholera in France France's experience with cholera varied from England's, due to its different scientific environment and unstable political situation. The first epidemic reached Paris by the spring of 1832 causing, in four months, the death of almost 2.1% of the 774.338 Parisians (Paillar (1832)). In his remarkable report addressed to the Higher Council of Health (Moreau de Jones (1831) 6 ), the former military A. Moreau de Jonnès (1778 -1870) gave considerable details on the international spread along trade routes of the pandemic that started in India in 1817, including the treatments and precautions taken against the disease. He clearly attested that cholera was "incontestably" contagious. In 1833, he became the first chief of the Statistique Generale de la France (SGF), the nearest equivalent to the G.R.O in England. Like Farr, Moreau de Jonnès published many reports (13 volumes) and contributed to the development of Statistics and its applications in France. Unfortunately, little attention was paid to his findings by the French health care community. In the end of 1831, France was anticipating a cholera epidemic. Health commissions were established in Paris and in other departments in order to control the disease with the help of health councils (conseils de salubrité); but the organization was less systematic and data collection was less reliable than in England. During the first epidemic, social unrest among the lower classes, who saw the disease "as a massive assassination plot by doctors in the service of the state", were the worst fears of the government [START_REF] Kudlick | Cholera in Post-revolutionary Paris: A Cultural History[END_REF]). The government was supported by the Faculty of medicine in its efforts to reduce fear and avoid a population uprising, and the latter stated in 1832 that the disease was not communicable [START_REF] Fabre | Conflits d'imaginaires en temps d'épidémie[END_REF]). In 1848, a public health advisory committee was created and attached to the Ministry of Agriculture and Commerce, in charge of sanitary issues (housing, water and protection of workers) and prophylactic measures to prevent the epidemic from spreading Le Mée (1998). As in 1832, this committee stated that cholera was not contagious [START_REF] Dupaquier | Cholera in England during the nineteenth century: medicine as a test of the validity of statistics[END_REF]). In 1849, the second epidemic broke out after the 1848 revolution. Contrary to the first epidemic characterized by riots and tensions, the reaction to the outbreak was more peaceful, with more efficient collaboration between scientists and the administration. At the same time, the perception of the lower classes also changed with the idea of struggling against destitution in order to prevent revolt [START_REF] Kudlick | Cholera in Post-revolutionary Paris: A Cultural History[END_REF]). As a consequence, the response to the second epidemic was better organized and social laws were passed in 1850 -51. In the following years, hygiene problems and unsanitary living conditions caused by the rapid growth of Paris's population were 2.1 The cholera pandemic: a starting point of the demographic transition addressed to by important public health measures. In particular, the massive public work projects led by Baron Haussman 7 in less than two decades from 1852 to 1870 remains as a symbol of the modernization of Paris at the end of the nineteenth century (Raux (2014)). Cholera Pandemic and International Health Organization The international dimension of the problem raised by cholera, reported in France by Moreau de Jonnès in 1824-31, was widely publicized by The Lancet, which published in 1831 a map on the international progress of cholera 8 (Koch (2014)). This map suggested a relation between human travel and the communication of the disease, accelerated by the industrial revolution in transport, in particular with steamships and railways. Cholera was regarded as an issue transcending national boundaries, which needed international cooperation to control it [START_REF] Huber | The unification of the globe by disease? the international sanitary conferences on cholera, 1851-1894[END_REF]). Europe had succeeded in setting up an efficient protective system against the plague, based on ideas such as quarantine and "cordon sanitaire". But those measures were very restrictive and seemed inefficient against cholera. Moreover, in the second half of the nineteenth century, Western European countries were involved in competitive colonial expansion, and were rather hostile to travel restrictions, even if increased global circulation was a threat to populations. The opening of the Suez Canal in 1869 was an emblematic example of those changes. Under the influence of French hygienist doctors, the first International Sanitary Conference opened in Paris in 1851 [START_REF] Huber | The unification of the globe by disease? the international sanitary conferences on cholera, 1851-1894[END_REF]) gathering European states and Turkey. It was the first international cooperation on the control of global risk to human health, and so the beginning of international health diplomacy. It took more than ten international conferences over a period of over 50 years to produce tangible results. During the first five conferences 9 , the absence of clear scientific explanation on the origin of cholera prevented any agreement. It was only with the formal identification of the V. cholerae bacterium by R. Koch in 1883 10 and the work of L. Pasteur that infectious diseases were clearly identified and efficiently fought against. Indeed, technological progress as evinced through disinfection machines could allow the technological implementation of new measures [START_REF] Huber | The unification of the globe by disease? the international sanitary conferences on cholera, 1851-1894[END_REF]). Furthermore, advances on germ theory "allowed diplomats to shape better informed policies and rules" [START_REF] Fidler | The globalization of public health: the first 100 years of international health diplomacy[END_REF]). At the Seventh Conference (1892), the first maritime regulation treaty was adopted 7 Napoléon III appointed Baron Haussmann as Préfet de la Seine 8 The map was completed in 1832 by Brigham to include Canada and the USA 9 (1851,1859,1866, 1874 and 1881) 10 The bacterium had been isolated before by other scientists such as F.Pacini in 1854, but his work did not had a wide diffusion. 2.1 The cholera pandemic: a starting point of the demographic transition for ship traveling via the Suez Canal. During the ninth conference (1894), sanitary precautions were taken for pilgrims traveling to Mecca. Participants finally agreed that cholera was a waterborne disease in 1903 during the eleventh conference. The International Sanitary Conferences provided a forum for medical administrators and researchers to discuss not only on cholera but also on other communicable diseases, and brought about the first treaties and rules for international health governance. Ultimately, this spirit of international cooperation gave birth in 1948 to the World Health Organization, an agency of the United Nations, conceived to direct and coordinate intergovernmental health activities. Discussion In England, Farr's discoveries could not have been made without the cutting edge organization and the power of the G.R.O. It is worth noting that only a governmental organization such as the G.R.O was able to collect the data fast enough for the 1854 experiment of Farr and Snow [START_REF] Dupaquier | Cholera in England during the nineteenth century: medicine as a test of the validity of statistics[END_REF]) to be possible. The modern organization of the G.R.O undoubtedly contributed to the remarkable quality of today's England vital databases. Across the Channel, France did not manage to create the same kind of centralized authority. On the grounds of their hostility to the communicable disease theory, French doctors did not rely on statistics. On the other hand, the use of statistics made by Farr contributed to a better understanding of the disease. It was only more than a century and a half later that a major breakthrough was made in the understanding of the origins of the disease, with the work of R. Colwell showing that the V. cholerae bacterium appears naturally in the environment. Yet the ambition to find causal factors by the sole analysis of data is not devoid of risks, and thus constitutes a major challenge for the data science era. Farr's elevation law is a textbook case of an unexpected correlation that turns out to have a great influence. Despite claimed impartiality, his choice to highlight the elevation law among all the findings mentioned in his report on the 1848-49 outbreak was clearly biased by his beliefs in the predominant (though false) miasma theory. While he later accepted that epidemics could be waterborne, Farr continued to believe in the prevailing role of elevation, even when deaths due to cholera during the 1854 and 1866 outbreak were not consistent with the elevation law. Rather than allowing the discovery of the causes of cholera, Farr's statistics were actually more useful for testing and validating the relationships predicted by Snow's theory. Another point is that the conditions that made the 1854 experiment possible were quite extraordinary. Testing theories regarding the complex events of health and 2.1 The cholera pandemic: a starting point of the demographic transition mortality in human communities is often nearly impossible. Only a handful of studies can take advantage of natural experiments. More often than not, as stated in National Research Council and Committee on Population (2011), "they are limited ethical opportunities to use randomized controlled trials to study the question at issue". Furthermore, governments failed to come to an agreement during the first international conferences because of the lack of scientific explanations on the origin of cholera. The need of theoretical arguments for public decisions to be made is still an important issue, especially when considering human health and longevity, for which no biological or medical consensus has emerged. As will be developed further in this survey, the use of a mathematical model and simulations can operate as a proxy to real life experiments and help decision making. Even when theories are publicized, there are often important delays (one or two generations) before action is taken. For instance, even if Snow's theory was better known in 1866, and despite the development of germ theory in the early 1880s, political divergences prevented any action before 1892. The example of asbestos, which took 50 years to be banned after the exhibition of its link with cancer, shows us that these delays in public response did not diminish over time [START_REF] Cicolella | Santé et Environnement : la 2e révolution de Santé Publique[END_REF]). More generally speaking, around 30 years elapsed between the first epidemic and the real development of public health policies in England and in France. The example of cholera illustrates the complexity of studying mortality evolution, inseparable from societal and political changes. Although cholera outbreaks occurred at about the same time in France and in England, they were experienced very differently owing to the different political and scientific climates in both countries. This shows that the sole study of mortality data could not be sufficient to understand the future trends of mortality. In particular, the explosion of the London population, whose size was twice as large as that of Paris, brought about social problems on a much greater scale, which played a determining role as a catalyst of public health changes. The cholera outbreaks contributed to the development of important public health measures, which played a major role in the reduction of infectious diseases. For instance, [START_REF] Cutler | The role of public health improvements in health advances: the twentieth-century United States[END_REF] estimated that water purification explained half of the mortality decline in the United States between 1900 and 1930. In comparison, the discoveries of new vaccines for a number of diseases at the beginning of the twentieth century seem to have had little impact on the reduction of mortality from those diseases. For instance, the reduction in mortality due to those diseases (except tuberculosis) following the introduction of those vaccines is estimated to have contributed to the emergence of only 3 percent of total mortality reduction ( A century of economic growth The twentieth century was the century of "the emergence for the first time in history of sustained increases in income per head" (Canning (2011)), and the association of economic growth and mortality improvements have been extensively discussed by economists. During the nineteenth century, individuals in rich and poor countries experienced similar health conditions. The 1870s were a turning point with the improvement of health in rich countries [START_REF] Bloom | Commentary: The preston curve 30 years on: still sparking fires[END_REF]). In his seminal article, [START_REF] Preston | The changing relation between mortality and level of economic development[END_REF] was one of the first economists to examine the relationship between life expectancy at birth and national income per head in different countries 11 , for three different decades: the 1900s, 1930s and 1960s (see Figure 1). In each decade, Preston brought to light a strong positive association between life expectancy and national income. He also stated that the relationship was curvilinear. For instance, the so-called Preston curve of 1960 appeared "to be steeper at incomes under 400$ and flatter at incomes over 600$" [START_REF] Preston | The changing relation between mortality and level of economic development[END_REF]. Preston also noted an upward shift of the curve characterized by a rise of life expectancy over time at all income levels. These empirical results showed that economic growth alone did not explain the remarkable mortality decline. For instance, the income level corresponding to a life expectancy of 60 was about three time higher in 1930 than in 1960. Another example is China which had in 2000 the same income level as the USA in the 1880s, but the life expectancy level of the USA in 1970. Preston (1975) estimated that national income accounted for only 10 to 25 percent of the growth of life expectancy between 1930s and 1960s. [START_REF] Bloom | Commentary: The preston curve 30 years on: still sparking fires[END_REF] also estimated that increases in income between 1938 and 1963 were responsible for about 20% of the increase in the global life expectancy. ). On the one hand, some studies on the causal link between health and wealth suggested that "health can be a powerful instrument of economic development" [START_REF] Bloom | Commentary: The preston curve 30 years on: still sparking fires[END_REF]). On the opposite side, Acemoglu and Johnson (2007) argue that improvements in population health, especially the reduction of children mortality, might have negative impacts on economic growth, due to the increase in the population size. They argue that a positive effect of economic growth on health may be counterbalanced by the negative effect of population growth on health. However, Reher (2011) describes the increase in the proportion of working age poeple in the population that occurred in developed countries during the twentieth century as a situation which had "profound economic implications for society, as long as the economy was able to generate enough jobs to accommodate the growing population of working age". A century of economic growth For a more complete picture, it is thus interesting to go beyond "macro" environmental indicators such as public health and economic growth, and to look at mortality experiences on different scales, by exploring differences between countries, and within countries. 3 A new era of diverging trends 3.1 A second demographic transition? In the early 1970s, many demographers and population scientists had supported for the idea that populations would ultimately reach the last stage of the classical demographic transition, described as an "older stationary population corresponding with replacement fertility (i.e., just over two children on average), zero population growth, and life expectancies higher than 70y" (Lesthaeghe (2014)). More generally speaking, populations were supposed to attain an equilibrium state, characterized by a significant level of homogeneity. For instance, the nuclear family composed of a married couple and their children was expected to become the predominant family model. Yet, in most countries which experienced the historic transition, the baby boom of the 1960s12 was characterized by higher fertility rates, followed by a decline in fertility in the 1970s (baby bust). In response to these fluctuations, attempts were made to modify the original theory. For instance, Easterlin (1980) developed a cyclical fertility theory, linking fertility rates to labor-market conditions. Smaller cohorts would benefit from better living conditions when entering the labor market, leading to earlier marriage and higher fertility rates. On the contrary, larger cohorts would experience worse living conditions, leading to later marriage and lower fertility rates. However, it turned out that this state of equilibrium and homogeneity in populations was never realized. Actually, fertility rates remained too low to ensure the replacement of generations; mortality rates, especially at advanced ages, declined at a faster rate than ever envisaged; and contemporary societies seem to be defined by more and more heterogeneity and diverging trends. The idea of a renewed or sec- Lesthaeghe and Van de Kaa also define the second demographic transition as a shift in the value system. The first phase of the demographic transition was a period of economic growth and aspirations to better material living conditions. In contrast, the past few years have seen a rise of "higher order" needs and individualization. In this new paradigm, individuals are overwhelmingly preoccupied by individual autonomy, self-realization and personal freedom of choice, resulting in the creation of a more heterogeneous world. Even if the framework of the second demographic transition has been criticized, this viewpoint shed an interesting light on recent longevity trends. Indeed, divergences in mortality levels and improvements between and within high income countries are at the heart of numerous debates and research works. As the average life expectancy has been rising unprecedentedly, gaps have also been widening at several scales. What may be somehow surprising is that up until the 1980s, high income countries had roughly similar life expectancy levels. For example, the comparison of the female life expectancy at age 50 in ten high income countries 13 shows that the gap was of less than one year in 1980. By 2007, the gap had risen to more than 5 years, with the United States at the bottom of the panel with Denmark, more than 2 years behind Australia, France, Italy and Japan (National Research Council and Committee on Population (2011)). On another scale, a great amount of evidence shows that socioeconomic differentials have also widened within high income countries. For instance, the gap in male life expectancy at age 65 between higher managerial and professional occupations and routine occupations in England 13 Australia, Canada, Denmark, England and Wales, France, Italy, Japan, Netherlands, Sweden and the United States. 3.2 Diverging trends between high-income countries: the impact of smoking behaviors. and Wales was of 2.4 years in 1982 to 1986, and rose to 3.9 years in 2007 to 2011 14 . The following part focuses on two angles of analysis on these diverging trends: the impact of smoking behaviors and socioeconomic inequalities. The goal of the following discussion is not to detail further the impact of these risk factors, but rather to show the complexity of understanding current longevity trends, which cannot be disentangled from the evolution of the whole population, and which require a multiscale analysis of phenomena while keeping in mind that obtaining comparable and unbiased data is also a challenge in order to explain longevity. 3.2 Diverging trends between high-income countries: the impact of smoking behaviors. In 15 . The evolution is even more striking for women: the ranking for the female life expectancy at age 50 fell from the 13th to the 31th position, with an increase of only about 60 percent of the average increase of high income countries. In addition, the gap with higher achieving countries such as France or Japan grew from less than one year in 1980-85 to more than 3 years in 2010-15 16 . Netherlands and Denmark also show similar patterns of underachievement in life expectancy increases. Although many methodological problems may arise when using cause-of-death statistics, a cause-of-death analysis can provide a powerful tool for understanding divergences in mortality trends. In a commissioned background article for the report, Glei et al. ( 2010) have studied cause-of-death patterns for 10 different countries in order to identify the main causes of death possibly responsible for diverging trends. The 3.2 Diverging trends between high-income countries: the impact of smoking behaviors. case of lung cancer or respiratory diseases, which are relevant indicators concerning smoking is particularly interesting. Age-standardized mortality rates from lung cancer among men aged 50 and older in the U.S decreased from 1980 to 2005 while they increased for women, although they remain higher for men than for women. In addition, the increase of age-standardized mortality rates due to lung cancer for women was much faster in the U.S, Denmark and Netherlands than the average increase of the studied countries and especially than Japan where age-standardized mortality rates remained flat. These findings of [START_REF] Glei | Diverging trends in life expectancy at age 50: A look at causes of death[END_REF] clearly point out to smoking as the main underlying factor explaining those divergences. Over the past 30 years, evolution of mortality due to lung cancer and respiratory diseases has had a positive effect on gains in life expectancy for males, while the effect was negative for females. These gender differences can be linked to the fact that women began to smoke later than men, and have been quitting at a slower pace [START_REF] Cutler | The determinants of mortality[END_REF]). In addition, fifty years ago, people smoked more intensively in the United States, Denmark and the Netherlands than in other European countries or in Japan. These differences can give precious information as for future mortality patterns. Because of its delayed effects on mortality, the impact of smoking behaviors on future trends is somehow predictable. Just as the causes of death of individuals aged 50 and older give some insight on what happened in the past, current behaviors among younger individuals can be a useful indicator of future trends. Thus, life expectancy for males in the United States is likely to increase rather rapidly following reductions in the prevalence of smoking over the past twenty years, while slower life expectancy improvements can be expected for women in the coming years (National Research Council and Committee on Population (2011)). According to a panel of experts, life expectancy in Japan is also expected to increase at a slower pace in the future due to an increase in the prevalence of smoking. Differences in the timing of evolutions in smoking behaviors across gender and countries might also give additional information. The impact of smoking on male life expectancy in the past could help predict future trends for women, and the experience of the United States could shed light on the future impact of smoking in Japan. But smoking is certainly not a sufficient explanation, and other risk factors may have contributed to the underachievement of the United States. In particular, the obesity epidemic may partly account for the slower increase in life expectancy experienced by the United States. Quantifying the impact of the obesity epidemic is much more complicated, since no clear markers are available such as lung cancer and respiratory diseases concerning smoking. According to some researchers, the obesity epidemic in the United States might even offset gains in life expectancy due to the decline of smoking (Stewart et Differences within countries: the impact of social inequalities Discussion The "predictable" effects of smoking could be integrated in a population dynamic framework taking into account the whole age structure of the population. Countries experiencing similar phenomena but with different timings could also be compared in a theoretical framework of population dynamics. Furthermore, a finer-grained model could help to better understand the future impacts of emerging issues such as the obesity epidemic, as well as the potential compensating effect of a decrease in smoking prevalence. Differences within countries: the impact of social inequalities Research on the relationship between socioeconomic status and mortality and health can be traced back as far as the nineteenth century. In France, Villermé (1830)compared mortality rates in Paris' boroughs with the rates of non-taxable households in each borough [START_REF] Mireaux | Un chirurgien sociologue: Louis-René Villermé[END_REF] The persistence and widening of socioeconomic inequalities in longevity has created a new paradigm, in which the increased heterogeneity has brought out even more complexity in understanding longevity evolution, and which has now to be taken into account in mortality predictions. New interlinked problems have arisen on mul-3.3 Differences within countries: the impact of social inequalities tiple scales. On an individual level, underlying factors linking individuals' health to their socioeconomic status are still debated. Another subject of no little interest to us is the critical challenge of understanding the impact of this rising heterogeneity on aggregated variables. In the following part, we will focus on some selected topics which have been discussed by sociologists, demographers, social epidemiologists and other scientists, with the aim of highlighting modeling challenges and solutions hidden beyond these reflections. Measuring the socioeconomic status The concept of socioeconomic status (SES) is broad and can encompass numerous characteristics, observable or not. Translating socioeconomic status into empirical measurements in order to better understand the links between SES and health and mortality, is in itself a challenge. Proxy variables such as educational attainment, occupation, income or wealth usually serve as SES measures, with different practices and habits in different countries [START_REF] Elo | Social class differentials in health and mortality: Patterns and explanations in comparative perspective[END_REF]). However, their ability to model the complexity of the social hierarchy and to produce comparable data through different times and places are often quite limited. Educational systems, even in groups of similar countries, can differ substantially from one country to another and make cross-national comparisons difficult [START_REF] Elo | Social class differentials in health and mortality: Patterns and explanations in comparative perspective[END_REF]). Furthermore, there is a real difficulty in comparing certain groups at different periods in time. Important changes can occur in group sizes and composition. For instance, the proportion of women in France with higher managerial and professional occupations increased from about 2 percent in 1975 to 6 percent in 1999 (10 to 14 percent for males). The evolution in the number of women long term unemployed or not in the labor force is even more striking. Their proportion decreased from 45 percent in 1975 to only 21 percent in 1999. Besides, Blanpain (2011) observed an important widening of mortality inequalities between this subgroup and other occupational subgroups over the period. The widening of these gaps is actually a typical consequence of important changes in the composition of the long term unemployed or not in the labor force subgroup. The major decrease in the size of the subgroup can be explained by the important decrease of the number of housewives over time, leaving only the most precarious in the subgroup. Proxies for the SES can be measured at different periods in the life course of an individual, and can have different causal relations with health or mortality. Education is rather consistent across the lifespan (which allows for an easier dynamic modeling), and permits to assess the stock of human capital accumulated early in life and available throughout the individual life course [START_REF] Elo | Social class differentials in health and mortality: Patterns and explanations in comparative perspective[END_REF]). On the 2006)). Explaining the socioeconomic gradient in health and mortality The difficulty in interpreting results of empirical measurements of the SES gradient in mortality reflects our little understanding of the risk factors that underlie the repercussions of socioeconomic inequalities on health and mortality. Theories explaining the SES gradient are still being debated, and their testing is often not straightforward and not unbiased as far as the measurements are concerned (see below the discussion on absolute versus relative measures). Furthermore, the impact of inequalities on aggregated variables or on the interpretation in terms of public policy can differ substantially according to different theories. The mechanisms through which SES is assumed to generate inequalities in health and longevity are usually grouped in three broad categories: material, behavioral and psychosocial [START_REF] Cutler | The determinants of mortality[END_REF]). Material risk factors Maybe one of the most natural explanation of socioeconomic differences in health is that wealthier individuals have better access to health care, even in countries with national health care coverage where potential two-tiered systems can also create inequalities. Individuals with a higher income can also maintain a healthier lifestyle, being able to buy expensive organic food or pay for gym memberships. However, access to health care or material resources does not appear to be the primary factor explaining the SES gradient (National Research Council Behavioral risk factors The second explanation is that individuals with higher educational attainment are more likely to adopt healthier behaviors and to avoid risks. By accumulating knowledge, skills and ressources, individuals who are higher on the SES ladder should be able to take better advantage of new health knowledge and technological innovations, as well as to turn more rapidly toward healthier behaviors. This behavioral explanation of socioeconomic inequalities is linked to the theory of Link and Phelan of fundamental causes [START_REF] Link | Social conditions as fundamental causes of disease[END_REF] shown that if behavioral differences play a significant role in explaining the SES gradient in mortality, it does not explain everything, and may not even account for the major part of the differentials. For instance, the famous study of Whitehall civil servants [START_REF] Marmot | Social differentials in health within and between populations[END_REF]) showed that health differentials subsisted even when factors such as smoking or drinking were controlled. Psychosocial Factors Another prominent and rather recent theory explaining socioeconomic differentials in mortality is that health is impacted by the SES through pyschosocial factors [START_REF] Cutler | The determinants of mortality[END_REF], [START_REF] Wilkinson | Income inequality and social dysfunction[END_REF]). Among pyschosocial factors are stress, anxiety, depression or anger. Accumulated exposure to stress has received particular attention in literature, due to its pervasive effects on health. Indeed, prolonged exposure to chronic stress affects multiple physiological systems by shifting priorities from systems such as the immune, digestive or cardiovascular systems in favor of systems responding to threat or danger [START_REF] Wilkinson | Income inequality and social dysfunction[END_REF]), and by leading to a state of so-called "allostatic load". The link between low social status and stress has been supported by a number of studies on primates. For instance, [START_REF] Sapolsky | Social status and health in humans and other animals[END_REF][START_REF] Sapolsky | Sick of poverty[END_REF] showed that among wild baboons, subordinate animals presented higher level of glucocorticoids, a hormone with a cen-3.3 Differences within countries: the impact of social inequalities tral role in stress response. The impact on aggregated variables On an aggregated level, socioeconomic inequalities impact national mortality not only through the importance of the SES gradient, but also through the composition of the population and its heterogeneity. From a material point of view, the relationship between income and health or mortality was initially thought of as a curvi-linear relation [START_REF] Preston | The changing relation between mortality and level of economic development[END_REF], [START_REF] Rodgers | Income and inequality as determinants of mortality: an international cross-section analysis[END_REF]). According to this analysis, a reditribution of income from the wealthiest groups to the poorest would result in improving the health of the poor rather than endanger the health of the wealthy. This non-linear relationship shows that the impact of inequality at the aggregated level of a country is not trivial. For instance, if a country experiences a high level of income inequalities, the overall mortality in the country can be higher than in a country with the same average level of income but with a lower level of inequality. But recent studies, based on the pyschosocial explanation of the SES gradient, seem to indicate that the relation between aggregated mortality and inequality is even more complicated. They argue that the presence of inequality itself impacts the health and mortality of individuals. For instance, [START_REF] Wilkinson | The spirit level: Why more equal societies almost always do better[END_REF] studied the association between life expectancy and income inequality 19 among the 50 richest countries of more than 3 million inhabitants, and found out a correlation of 0.44 between life expectancy and the level of inequalities, while no significant association was found between life expectancy and the average income. These results suggest that health and mortality are impacted by the relative social position of individuals, rather than their absolute material living standards [START_REF] Wilkinson | Income inequality and social dysfunction[END_REF], [START_REF] Pickett | Income inequality and health: a causal review[END_REF]). This is closely linked to the theory 18 The RII is an index of inequality which takes into account differences in mortality as well the populations composition, see [START_REF] Regidor | Measures of health inequalities: part 2[END_REF] for details on the computation of the index. 19 Income inequality was measured in each country as the ratio of income of the poorest 20% to the richest 20%. 3.3 Differences within countries: the impact of social inequalities of psychosocial factors, which assumes that it is the relative social ranking which determines the level of exposure to psychosocial problems and the ability to cope with them. [START_REF] Wilkinson | Income inequality and social dysfunction[END_REF] go even further and argue that inequalities affect not only individuals at the bottom of the socioeconomic ladder, but the vast majority of the population. For instance, Wilkinson and Pickett (2008) compared standardized mortality rates in counties of the 25 more equal states in the US and in counties of the 25 less equal states20 (see Figure 3). They found out that for counties with the same median income, mortality rates were higher in counties in the more equal states than in counties in less equal states. The relation held at all levels of median income, with more important differences for counties with lower median income. When measuring the impact of inequality on health, the size of the area appears to be an important variable to take into account. On the one hand, the relationship between health and inequalities, when measured at the level of large areas such as states of big regions, seems to be fairly strong. On the other hand, [START_REF] Wilkinson | Income inequality and social dysfunction[END_REF] note that at the level of smaller areas such as neighborhoods, the average level of income seems to matter more than one's relative social position in the neighborhood. This "neighborhood effect" has been studied by many authors and constitutes a field of research in itself (se e.g [START_REF] Kawachi | Neighborhoods and health[END_REF], Diez Roux (2007), Diez Roux and Mair (2010), Nandi and Kawachi (2011)). Societal inequalities, neighborhood environment and individual socioeconomic characteristics thus impact health and mortality at multiple scales, making the analysis of factors responsible for poorer health highly difficult. It is even more difficult to understand what happens at the aggregated level. 3.3 Differences within countries: the impact of social inequalities Discussion The problems surrounding the measure of the SES is revealing of the issues at hand. The interpretation of data across time and places is a delicate matter. As illustrated by the major changes in the composition of women occupational subgroups, the effects of composition changes have to be carefully addressed to. Besides, it is rather unlikely that a single measure of SES, at only one point in the life course of individuals, could capture accurately the many pathways by which social status can affect health and mortality [START_REF] Elo | Social class differentials in health and mortality: Patterns and explanations in comparative perspective[END_REF]). However, there are many limitations in the ability to obtain reliable data from multiple measures of SES. There are often limited opportunities for the empirical testing of complex theories such as the fundamental cause theory or the theory of psychosocial factors. The design of empirical test is not straightforward, to say the least. Natural experiments, such as the evolution of smoking behavior, or experiments on non-human populations, such as Saplosky's study of baboons, can give valuable insights on theories. However, as stated in the conclusion of the report of National Research Council and Committee on Population (2011), " it is sometimes difficult, expensive, and ethically challenging to alter individual behavior". Pathways involved in translating SES into mortality outcomes can differ substantially according to the theory taken into account. Moreover, the impact of these underlying mechanisms on aggregated variables can also differ a lot, ranging from composition effects due to the curvilinear relation between material resources and longevity, to the global (non-linear) effects of the social stratification on individu- We believe that the dynamic modeling of the evolution of the population may help to address these issues. A fine-grained modeling of the population dynamic could help to evaluate the impact of changes in the composition of socioeconomic subgroups. In addition, modeling the population dynamics can serve as a simulation tool in order to take into account various measures of SES, when empirical data are limited. It can also be used to test hypotheses regarding which aspects of SES are the most important for reducing socioeconomic inequalities in health and mortality. By using population simulation as an experimenting tool when real life experiences are not possible, theories can be tested by comparing the aggregated outcomes produced by the model to what is observed in reality. However, the above paragraph shows us the complexity of the phenomena involved. Socioeconomic inequalities impact health and mortality through complicated pathways. Phenomena are often non-reproducible -risk factors, as well as the economic, social or demographic environment have changed dramatically over the recent years -with effects which are often delayed. Furthermore, findings suggest that the impact of socioeconomic inequalities is highly non-linear. Individual characteristics do not fully explain the longevity of individuals. Mechanisms acting at different scales appear to be equally important. For instance, the neighborhood effect, the relative social position of individuals or the global level of inequality in society are also important factors to take into account. From these examples, it is quite easy to see the modeling challenges brought about by the new paradigm of the second demographic transition. At yet, there is also an urgent need for complex population models, for a better understanding of the observed data, as well as to serve as an alternative when empirical testing is not possible. Modeling complex population evolutions Before the 1980s, demographic models were principally focused on the macro-level, and used aggregate data to produce average indicators. In view of the previous considerations, producing a pertinent modeling directly at the macro-level appears to be a more and more complicated-if not impossible-task. Hence, demographic models have increasingly shifted towards a finer-grained modeling of the population in the last decades. There is thus an intrinsic interest in describing the variability and heterogeneity of the population on a more detailed level, in order to obtain macro-outcomes by aggregation, to be used forecasting/projections and/or policy recommendations, or in a broader sense for the analysis of social economic policies. Over the last two decades, the increase of computing power and improvements in numerical methods have made it possible to study rich heterogeneous individual models. Indeed, a wide variety of models simulating individual behavior have been developed for different purposes and used in different domains. In this section, we give an overview of two types of models widely used in demography: Standard microsimulation models (MSMs) and Agent based models (ABMs) which are derived from the idea of Orcutt (1957) (see [START_REF] Morand | Demographic modelling: the state of the art[END_REF]). 2009)). This model is used for instance to measure the efficiency of reforms on state pension systems, and is based on a representative sample of the national population. Dynamic microsimulation A dynamic microsimulation exercise A demographic micro-model can be viewed as a population database, which stores dynamically information (characteristics) on all members (individuals) of the heterogeneous population [START_REF] Willekens | Biographic forecasting: bridging the micro-macro gap in population forecasting[END_REF]). [START_REF] Zinn | A Continuous-Time Microsimulation and First Steps Towards a Multi-Level Approach in Demography[END_REF] gives the main steps of a microsimulation exercise which consist of: (i) State space and state variables: The state space is composed of all the combinations of the values (attributes) of individuals' characteristics, called state variables. Age, sex, marital status, fertility and mortality status, education or emigration are 21 Since then, updated versions were developed, with for example DYNASIM3 in 2004 (Li and O'Donoghue (2013)). 22 The micmac project is documented in [START_REF] Willekens | Biographic forecasting: bridging the micro-macro gap in population forecasting[END_REF]. Dynamic microsimulation examples of state variables. An example of a state is given by the possible values of state variables: (Female, Married, 1 Child, Alive, Not emigrated, Lower secondary school)23 . (ii) Transition rates: Events occurring during the life course of individuals are characterized by individual hazard functions, or individual transition rates / probabilities. Each of the transition probabilities is related to an event, i.e. a change in one of the state variables of the individual. These probabilities are estimated conditionally on demographic covariates (i.e explanatory variables such as gender, age, educational attainment, children born, ethnicity), and other risk factors that affect the rate of occurrence of some events (environmental covariates that provide external information on the common (random) environment) [START_REF] Spielauer | What is social science microsimulation?[END_REF]). In microsimulation models, the covariates are often estimated by using logit models (see [START_REF] Zinn | A Continuous-Time Microsimulation and First Steps Towards a Multi-Level Approach in Demography[END_REF]). (iii) Dynamic simulation: Dynamic simulation aims at predicting the future state of the population, by making the distinction between events influencing the population itself and those affected by it (population ageing, concentration of wealth, sustainability of social policies...). (iv) Internal consistency: Microsimulation models can handle links between individuals, which can be qualified as "internal consistency" (Van Imhoff and Post (1998)). Individuals can be grouped together in the database into "families" (for instance, if they are married or related). When a state variable of an individual in the group changes, the state variables of the other members are updated if needed. For example, this can be the case when such events as marriage, divorce or a child leaving the parental home take place. (v) Output of microsimulation exercise and representation: The output of a dynamic microsimulation model is a simulated database with longitudinal information, e.g. in the form of individual virtual biographies, viewed as a sequence of state variables. The effects of different factors can be revealed more clearly when grouping individuals with life courses embedded in similar historical context. Usually, individuals are grouped in cohorts (individuals with the same age) or in generations. The aggregation of individual biographies of the same cohort yields a bottom-up estimate of the so-called cohort biography. Nevertheless, in the presence of interactions, all the biographies have to be simulated simultaneously, which is challenging for large populations. Dynamic microsimulation Sources of randomness Microsimulation models are subject to several sources of uncertainty and randomness which have been discussed in detail in the work of Van Imhoff and Post (1998). The so-called "inherent randomness" is due to the nature of Monte Carlo random experiments (different simulations produce variable sets of outcomes). This type of randomness can be diminished when simulating large populations (increasing the number of individuals in the database) or repeating random experiments many times to average the results, which implies important computational cost 24 . In the presence of interactions, one should be careful since the two techniques are not equivalent. Such is the case of agent based models, which are discussed in the next section. The starting population, which is the initial database for the microsimulation model, can be either a sample of the population based on survey data, or a synthetic population created by gathering data from different information sources. This initial population is subject to random variations and sampling errors. Moreover, at the individual level, the state variables and the covariates must be known before starting the simulation, and their joint distribution within the initial database is random. Van Imhoff and Post (1998) note that any deviation of the sample distribution may impact future projections. These previous sources of randomness can be mitigated by increasing the size of the database and are probably less important in comparison with the so-called specification randomness. The outputs of a microsimulation model can be subject to a high degree of randomness when an important number of covariates are included. Indeed, there are calibrating errors resulting from the estimation of empirical data each relationship between probabilities of the model and covariates. Moreover, each additional covariate requires an extra set of Monte Carlo experiments, with a corresponding increase in Monte Carlo randomness. The specification randomness can be reduced by using sorting or alignment methods, a calibration technique that consists in selecting the simulated life course in such a way that the micro model respects some macro properties, including the property of producing the expected values. In destinie 2, this alignment is ensured by adjusting the individual transition rates to obtain the annual number of births, deaths and migration consistent with some macro projections [START_REF] Blanchet | The Destinie 2 microsimulation model: overview and illustrative results[END_REF]). Discussion In many cases, behaviors are more stable or better understood on the micro level than on aggregated levels that are affected by structural changes when the number 24 Various techniques to accelerate Monte Carlo simulation coupled with variance reduction has been developed in many areas. Agent Based Models (ABM) or size of the micro-units in the population changes. Thus, microsimulation models are well suited to explain processes resulting from the actions and interactions of a large number of micro-units. For instance, according to Spielauer (2011), an increase of graduation rates25 at the macro level can lie entirely in the changing composition of the parents' generations, and not necessarily in a change of individuals' behaviors. In order to produce more micro-level explanations for population change, microsimulation models require an increasing amount of high quality data to be collected. [START_REF] Silverman | Feeding the beast: can computational demographic models free us from the tyranny of data?[END_REF] point out the "Over-dependence on potentially immense sets of data" of microsimulation models and the expensive data collection required to provide inputs for those models. Most of the time, only large entities such as national or international institutions are able to complete this demanding task. The size of the samples also has an impact on the run time of the model; the larger the sample's size is, the longer the run speed will be, which will result in a trade-off. [START_REF] Silverman | Feeding the beast: can computational demographic models free us from the tyranny of data?[END_REF] argue in favour of the use of more abstract computational models rather than on highly data-driven research. More recently, Agent Based Models (ABM), which also derived from individual-based models, have been increasingly applied in various areas to analyze macro level phenomena gathered from micro units. These models emphasize interactions between individuals through behavioral rules and individual strategies. In this context, Zinn (2017) stressed the importance of incorporating behavioral rules through ABM models (e.g kinship, mate matching models..) since demographic microsimulation is well suited for population projection, if only the model considers independent entities. Agent Based Models (ABM) What is ABM? The main purpose of the Agent Based Models (ABM) is to explain macroscopic regularities by replicating the behavior of complex, real-world systems with dynamical systems of interacting agents based on the so-called bottom-up approach [START_REF] Tesfatsion | Agent-based computational economics: Growing economies from the bottom up[END_REF], [START_REF] Billari | Agent-Based Computational Modelling: Applications in Demography[END_REF]). ABM consists basically of the simulation of interactions of autonomous agents i.e independent individuals (which can be households, organizations, companies, or nations...depending on the application). As in microsimulation, agents are defined by their attributes. Each single agent is also defined by behavioral rules, which can be simple or complex (e.g utility optimization, complex social patterns...), deterministic or stochastic, on whose basis she/he interacts with other agents and with the simulated environment ( A dynamic exercise of an ABM The key defining feature of an ABM model is the interactions between heterogeneous individuals. Moreover, an agent based model is grounded on a dynamic simulation, which means that agents adapt dynamically to changes in the simulated environment. They act and react with other agents in this environment at different spatial and temporal scales [START_REF] Billari | Agent-Based Computational Modelling: Applications in Demography[END_REF]). This contrasts with Microsimulation models (MSM) which rely on transition rates that are determined a priori (and once). Agent based models are based on some rules, or heuristics, which can be either deterministic or stochastic, and which determine the decision-making process. For example, in an agent based marriage market model, the appropriate partner can be chosen as the one who has the most similar education level to the considered agent, or an ideal age difference [START_REF] Billari | The "Wedding-Ring": An agent-based marriage model based on social interaction[END_REF]). Besides, in comparison with Microsimulation models, which operate on a realistic Conclusion scale (real data), but use very simple matching algorithms (often a Monte Carlo "roll the dice" styled decision rule), agent based models use small and artificial data sets, but show more complexity in modeling how the agents viewed and chose partners. Limitations The design of agent based model needs a certain level of expertise in the determining of behavioral rules. Furthermore, when modeling large systems (large number of agents), computational time rises considerably. Indeed, ABM models are not designed for extensive simulations. The parameters of an agent based model can be either calibrated using accurate data, or consider sensitivity analysis incorporating some level of comparison with actual data. For instance, [START_REF] Hills | Population heterogeneity and individual differences in an assortative agent-based marriage and divorce model (madam) using search with relaxing expectations[END_REF] compare the results of their Agent-Based Marriage and Divorce Model (MADAM) to real age-at-marriage distributions. But the outputs of an agent based model also depend on the "internal" structure of the model, determined by the behavioral rules (Gianluca (2014)). Consequently, the strategies to calibrate parameters, and to overcome the problem of dependency on the model's structure, rely on available empirical information. It is important to note that ABMs are designed to focus on process related factors or on the demonstration of emergent properties, rather than to make projections. Conclusion The interest of dynamic microsimulation is to constitute both a modeling exercise, and an exercise to run the model and experiment with it (Spielauer (2011)). In addition to helping to test theory or to picture the future, the exercise may be used as a simulator by policy makers (or citizens) or for a better assessment of the impact of public policies. The results/outputs of microsimulation models are population projections rather than forecasts, which is what would happen if the assumptions and scenarios chosen were to prove correct on what the future will probably be. The discussion on demographic modeling demonstrated that Microsimulation models (MSM) strongly depend on data [START_REF] Silverman | Feeding the beast: can computational demographic models free us from the tyranny of data?[END_REF]. Then, it faces pragmatic challenges in collecting and cleaning data, in addition to the different sources of randomness discussed above. In parallel with the spread of microsimulation models, there is a growing interest in Agent Based Models, which are suited to model complex systems that take full account of interactions between heterogeneous agents. The major difficulty in using of Agent based model is the absence of theoretical model. Indeed, there is no codified set of recommendations or practices on how to use these models within a program of empirical research. It is essentially based on the cognition and expertise of the developer. In this context, new hybrid applications (combining MSM and ABM models) have been recently proposed in literature. For instance, [START_REF] Grow | Agent-Based Modelling in Population Studies: Concepts, Methods, and Applications[END_REF] present many examples that combine MSM and ABM in demographic models. These new models aim at describing the heterogeneous movements, interactions and behaviors of a large number of individuals within a complex social system at a fine spatial scale. For instance, Zinn (2017) uses a combination of MSM and ABM for modeling individuals and couples life courses by integrating social relations and interactions. The efficiency of these combined and "sophisticated" models to overcome the loopholes of the simple models is an open issue. General conclusion and perspectives Facing all these modeling challenges, we advocate the development of a new mathematical theoretical framework for the modeling of complex population dynamics in demography. As we have seen in Section 3, a number of questions cannot be answered by the sole study of data, and models allow us to generate and experiment with varyous scenarios, so as to test theories or causal links for instance. Theoretical models can help us "to escape from the tyranny of data", as claimed by [START_REF] Silverman | Feeding the beast: can computational demographic models free us from the tyranny of data?[END_REF]. On the other hand, empirical evidence point out a number of key issues which cannot be overlooked, and which demonstrate the "inextricable complexity" of dynamic modeling of realistic human populations. Variables such as mortality or fertility rates are by no means stationary; populations are more and more heterogeneous, with socioeconomic inequality playing an important role at several levels (individual, neighborhood and societal); interactions between individuals and their environment are bidirectional. These are just a few examples illustrating the complexity of modeling. An adapted mathematical framework could contribute significantly to better understand aggregation issues and find out adequate policy recommendations, in concordance with this new paradigm of heterogeneity and non-linearity. More specifically, theoretical models often allow us to reduce complexity by deriving and/or justifying approximations in population dynamics. By changing point of view, data can also be represented differently,and thus permit to go beyond what is usually done. The historical analysis of these two centuries of demographic transitions show that populations have experienced dramatic changes and upheavals. But we can also see, a number of phenomena and timescales present remarkable regularities. These profound regularities, or "fundamental causes", have been noted by several authors, in very different contexts. In our opinion, the identification and understanding of these regularities or cycles is fundamental. Age is also a critical dimension when studying human population dynamics. The age structure of a population generates a lot of complexity in the representation and statistical analysis of data. This so-called Age Period Cohort (APC) problem has been well documented in statistical literature, and should be a main focus in the dynamical modeling of populations. Furthermore, the human life cycle is composed of very different periods, with transition rates of a different order and phenomena of a different nature at each stage. Understanding how to take into account this heterogeneity in age is a critical point. The notion of age itself changes over time. Individuals seem to have rejuvenated, in the sense that today's 65-year-olds are "much younger" than individuals of the same age thirty years ago. Thus, the shift in paradigm observed in recent demographic trends has highlighted a number of new issues which force us to reconsider many aspects of the traditional modeling of human populations. Multiple questions are still open, with difficult challenges ahead, but also exciting perspectives for the future. Figure 1 : 1 Figure 1: Preston curves, 1900, 1930, 1960, reproduced from[START_REF] Preston | The changing relation between mortality and level of economic development[END_REF] Figure 2 : 2 Figure 2: Preston curve in 2000, reproduced from Deaton (2003) al. (2009), Olshansky et al. (2005)). 3. 3 3 Differences within countries: the impact of social inequalities and Committee on Population (2011),[START_REF] Cutler | The determinants of mortality[END_REF]). For instance, the education gradient in the U.S steepened between the sixties and the eighties, even though the Medicare program was enacted in 1965[START_REF] Pappas | The increasing disparity in mortality between socioeconomic groups in the United States, 1960 and 1986[END_REF], cited in[START_REF] Elo | Social class differentials in health and mortality: Patterns and explanations in comparative perspective[END_REF]). Figure 3 : 3 Figure 3: Relationship between median county income and standardized mortality rates among working-age individuals, reproduced from Wilkinson and Pickett (2009) (Figure 11) 3. 3 3 Differences within countries: the impact of social inequalities als. From the fundamental causes point of view, new advances in knowledge and technology related to health will probably increase the SES gradient in health and mortality[START_REF] Phelan | Social conditions as fundamental causes of health inequalities: theory, evidence, and policy implications[END_REF],[START_REF] Cutler | The determinants of mortality[END_REF]). This illustrates how different underlying factors explaining the SES gradient can influence our views on the impact of socioeconomic inequalities on aggregated quantities, and in turn influence choices of public policies. Different types of policy recommendation can be made, according to the underlying factors or measures of SES which are considered to be most prominent. For instance, Phelan et al. (2010) recommend two types of public interventions. The first type focuses on reducing socioeconomic inequality itself in order to redistribute resources and knowledge. This would be consistent with the views of[START_REF] Wilkinson | Income inequality and social dysfunction[END_REF] on the general impact of inequality in a country. The second type of recommendations falls into the domain of public health. Governments should be careful and design interventions which do not increase inequalities, by favouring for instance health interventions which would benefit everyone automatically. But public health should not be underestimated in this new age of "degenerative and man-made diseases"[START_REF] Bongaarts | Trends in causes of death in low-mortality countries: Implications for mortality projections[END_REF]). Public health campaigns against tobacco have played an important role in reducing cardiovascular disease mortality caused by smoking[START_REF] Cutler | The determinants of mortality[END_REF]), although with varying degrees of success depending on countries, gender or social classes. The increase of environmental risks constitutes one of the major challenges faced by contemporary societies, and public action will play a central role in preventing and successfully reducing those risks[START_REF] Cicolella | Santé et Environnement : la 2e révolution de Santé Publique[END_REF]). [START_REF] Cutler | The determinants of mortality[END_REF] ). The second half of the twentieth century was marked by the rise of more intensive medical interventions, and by an epidemiological transition from infectious 2.2 A century of economic growth diseases to chronic diseases. 3.1 A second demographic transition? ond demographic transition, distinct from the classical demographic transition, was originally formulated by[START_REF] Lesthaeghe | Twee demografische transities[END_REF] in an article in Dutch, followed by a series of articles[START_REF] References Lesthaeghe | The unfolding story the second demographic transition[END_REF][START_REF] Lesthaeghe | The second demographic transition: A concise overview of its development[END_REF], Van de Kaa (2010)). In the early 1980s, a number of researchers had already observed that a shift of paradigm (Van de Kaa (2010)) had occured. In particular, the French historian P. Ariès suggested that motivations explaining the decline birth rate in the West had changed[START_REF] Ariès | Two successive motivations for the declining birth rate in the west[END_REF]). During the historic demographic transition, the decline in fertility rates was assumed to originate from an increased parental investment in the child. P. Ariès explained more recent declines in fertility by an increasing interest of individuals in self-realization in which parenthood is only one particular life course choice among many others. More specifically,[START_REF] Lesthaeghe | The second demographic transition: A concise overview of its development[END_REF] characterized the second demographic transition by multiple lifestyle choices and a more flexible life course organization. A striking example can be found in the emerging of multiple types of family arrangements. the comprehensive report of the National Research Council on explaining divergent levels of longevity in high income countries (National Research Council and Committee on Population (2011), a panel of experts have debated on the role of different risk factors for explaining the slower increase of life expectancy in the United States over the last 30 years, in comparison with other high income countries. From 1980 to 2015, the world ranking for life expectancy of the United States kept falling significantly. Furthermore, the gap between the United States and other high income countries widened, due to the slower increase of life expectancy at all ages in the United States. The ranking of the United States for male life expectancy at age 50 fell from 17th in 1980-85 to 28th in 2010-2015, with an increase of life expectancy of 4.58 years, smaller in absolute and relative terms than the average of high income countries ). In England and Wales, a systematic documentation of mortality by occupational class was made by the G.R.O starting from 1851[START_REF] Elo | Social class differentials in health and mortality: Patterns and explanations in comparative perspective[END_REF]). Since then, studies have consistently exhibited a pervasive effect of socioeconomic inequalities on longevity, regardless of the period or country. A recent study based on the French longitudinal survey 17 has found out that males with managerial and higher professional occupations have a life expectancy 6.3 years higher than working-class males (in the mortality conditions of 2000-2008, Blanpain (2011)). Numerous other examples can be found in the review of Elo (2009). Moreover, despite unprecedented rise in life expectancy during the 20th century, evidence shows that socioeconomic inequalities have widened in many developed countries in recent decades (Elo (2009)), or have remained identical at best (Blanpain (2011)). For instance, Meara et al. (2008) (cited in National Research Council and Committee on Population (2011)) argue that the educational gradient in life expectancy at age 25 rose from the eighties to the nineties of about 30 percent. Similarly in England, socioeconomic status measurements using the geographically based Index of Multiple Deprivation (IMD) (see next paragraph for more details) have shown that the average mortality improvement rates at age 65 and older have been about one percent higher in the least deprived quintile that in the most deprived quintile during the period 1982-2006 (Haberman et al. (2014),Lu et al. (2014)). 3.3 Differences within countries: the impact of social inequalities other hand, occupation, income or wealth allow to take into account latter parts of the life course and might allow to capture impacts of public policies better than educational attainment measurements (National Research Council and Committee on Population (2011)). However, the variability of the occupational status through the life course and the difficulty of assigning an occupational group to individuals is important when studying socioeconomic gradients by occupational rankings. For instance, the issue of assigning an occupational group to individuals who are not in the labor force or retirees is classical. Interpretations can also differ significantly depending on the period in the life course at which occupational status is measured. Additional complexity is also generated by the potential bidirectionality of causation, especially concerning economic measures of SES such as income or wealth, for which causal pathways are debated. Evidence from the economic literature has shown that ill health can also lead to a decrease in income or wealth. This is particularly true in countries like the United States with poorer national health care coverage than most Western Europe countries, and where poor health is a significant contributor to bankruptcy (Himmelstein et al. (2009)), retirement or unemployment (Smith (2007), cited in National Research Council and Committee on Population (2011), Case and Deaton (2005), cited in Cutler et al. ( ,[START_REF] Phelan | Social conditions as fundamental causes of health inequalities: theory, evidence, and policy implications[END_REF]). The aim of Link and Phelan's theory is to explain the persistence of pervasive effects of social class inequalities through time, despite dramatic changes in diseases and risk factors. According to the theory, the accumulation among other resources of so-called human capital allows more educated individuals to use re- sources and develop better protective strategies, whenever they can and no matter what the risks are. Let us take the example of smoking, described in [START_REF] Link | Epidemiological sociology and the social shaping of population health[END_REF] ). When first evidence linking smoking to lung cancer emerged in the fifties, smoking was not correlated to SES. But as the knowledge of the harm caused by smoking spread, strong inequalities in smoking behavior appeared, reflected in the fact that more educated individuals quit smoking earlier. However, a number of studies (see National Research Council and Committee on Population (2011) for examples) have For instance, in a study based on the comparison of the United States with 14 European countries,[START_REF] Avendano | Do Americans Have Higher Mortality Than Europeans at All Levels of the Education Distribution?: A Comparison of the United States and 14 European Countries[END_REF] observed that the unusually high educational gradient in mortality in the United States seems to be counterbalanced by an attractive educational distribution. As a consequence, they found out that the Relative Index of Inequality (RII)18 of the United States was not especially high in comparison with other countries with an educational gradient of lower magnitude. The age structure of the population also plays a determining role, and the socioeconomic composition of different age classes can vary a lot (see Chapter 4 for a more detailed discussion on this subject). Examples of microsimulation models The history of microsimulation in social sciences goes back to the work of[START_REF] Orcutt | A new type of socio-economic system[END_REF], who developed so-called data-driven dynamic microsimulation models. Following the original model of Orcutt, the first large-scale dynamic microsimulation model called dynasim21 was developed for the forecasting of the US population up to 2030. This model considered different demographic and economic scenarios, meant to analyse the socioeconomic status and behavior of individuals and families in the US (cost of teenage childbearing for the public sector, unemployment compensations and welfare programs...). Since then, most statistical or demographical government bodies in developed countries have used their own microsimulation models, developed for different purposes. A comprehensive description of various microsimulation models can be found in the surveys of[START_REF] Morand | Demographic modelling: the state of the art[END_REF], Zaidi and Rake (2001), Li and O'Donoghue (2013). For instance dynacan in Canada was designed to model the Canada Pension Plan (CPP) and analyze its contributions and benefits at individual and family level. In Australia, dynamod was developed to carry out a projection of the outlook of Australian population until the year 2050. In Europe, the micmac project 22 was implemented by a consortium of research centers whose objective is to provide demographic projections concerning detailed population categories, that are required for the design of sustainable (elderly) health care and pension systems in the European Union. The specificity of the micmac consists in providing a micro-macro modeling of the population, with micro level projections that are consistent with the projections made by the macro model. In France, the INSEE developed different versions of a microsimulation model, the current version being destinie 2(Blanchet et al. ( 4.1 Dynamic microsimulation 4.1.1 Microsimulation models Microsimulation issues A dynamic microsimulation model provides a simulation tool of individual trajectories in order to obtain macro outcomes by aggregation. It provides a way of combining different processes (biological, cognitive, social) de- scribing the lives of people who evolve over time. One main feature of this class of model is its capacity to interpret macro level changes, represented by macroe- conomic complex quantities (or indicators) (e.g life expectancy, mortality rate,...), resulting from the simulation of the dynamic life courses of individuals, also called micro units. A dynamic microsimulation model, usually relying on an important amount of empirical data, is parametrized with micro-econometrics and statistical methods (Spielauer (2011)). Applications The applications of ABM range from social, economic or political sciences to demography[START_REF] Billari | Agent-Based Computational Modelling: Applications in Demography[END_REF]). For instance,[START_REF] Tesfatsion | Agent-based computational economics: Growing economies from the bottom up[END_REF] used Agent-based Computational economies (ACE) in order to model decentralized economic markets through the interaction of autonomous agents. In demography, ABM are used in[START_REF] Diaz | Transition to Parenthood: The Role of Social Interaction and Endogenous Networks[END_REF] to explain trends in fertility by simple local interactions, in order to solve the difficult problem of age-specific projection of fertility rates.[START_REF] Billari | The "Wedding-Ring": An agent-based marriage model based on social interaction[END_REF] developed an ABM based on the interaction between heterogeneous potential partners, which typically takes place in the marriage market (partnership formation) and which is called "The Wedding Ring model". The purpose of this model is to study the age pattern of marriage using a bottom-up approach. This model was implemented using the software package NetLogo (Wilensky (1999)) which is designed for constructing and exploring multilevel systems 26 .[START_REF] Burke | The Strength of Social Interactions and Obesity among Women[END_REF] suggested the use of an agent based model to explain the differences in obesity rates between women with different educational attainment in the United States. The model integrates biological complex agents (variation of women's metabolism) interacting within a social group, and is able to reproduce the fact that better educated women experience on average lower weights and smaller dispersion of weights. For more examples of agent based models applications, we refer to the work of[START_REF] Morand | Demographic modelling: the state of the art[END_REF] that details different examples of ABM in spatial demography, family demography and historical demography. The book of[START_REF] Billari | Agent-Based Computational Modelling: Applications in Demography[END_REF] also presents various applications of agent-based computational modeling, in particular in demography. 4.2 Agent Based Models (ABM) Morand et al. (2010), Billari (2006)). The report is available on the website of BNF. The national income per head was converted in 1963 U.S dollars. The baby boom affected several countries such as France, the United Kingdom or the United States, although with different timings from the early 1950s to 1970s. Source: Office for National Statistics (ONS). High income country classification based on 2014 GNI per capita from the World Bank. United Nations, Department of Economic and Social Affairs, Population Division (2015). World Population Prospects: The 2015 Revision, custom data acquired via website. description EDP The measure of inequality was based on the Gini coefficient of household income. The state variables Dead and Emigrated are considered as absorbing states, i.e. once they have been entered, they will never be left again. Graduation rate represents the estimated percentage of people who will graduate from a specific level of education over their lifetime. Multi-level agent based models integrate different levels (complementary points of view) of representation of agents with respect to time, space and behavior.

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inferential pragmatism. However, there are also some significant differences that are at center of the dialogical approach to meaning. The present paper does not discuss explicitly phenomenology, however, one might see our proposal as setting the basis for a further study linking phenomenology and the dialogical conception of meaningthe development of such a link is part of several ongoing researches. different fashion, Hintikka's plea for the fruitfulness of game-theoretical semantics in the context of epistemic approaches to logic, semantics, and the foundations of mathematics. 4 From the dialogical point of view, the actions-such as choices-that the particle rules associate with the use of logical constants are crucial elements of their full-fledged (local) meaning: if meaning is conceived as constituted during interaction, then all of the actions involved in the constitution of the meaning of an expression should be made explicit; that is, they should all be part of the object-language. This perspective roots itself in Wittgenstein's remark according to which one cannot position oneself outside language in order to determine the meaning of something and how it is linked to syntax; in other words, language is unavoidable: this is his Unhintergehbarkeit der Sprache. According to this perspective of Wittgensteins, language-games are supposed to accomplish the task of studying language from a perspective that acknowledges its internalized feature. This is what underlies the approach to meaning and syntax of the dialogical framework in which all the speech-acts that are relevant for rendering the meaning and the "formation" of an expression are made explicit. In this respect, the metalogical perspective which is so crucial for model-theoretic conceptions of meaning does not provide a way out. It is in such a context that Lorenz writes: Also propositions of the metalanguage require the understanding of propositions, […] and thus cannot in a sensible way have this same understanding as their proper object. The thesis that a property of a propositional sentence must always be internal, therefore amounts to articulating the insight that in propositions about a propositional sentence this same propositional sentence does not express a meaningful proposition anymore, since in this case it is not the propositional sentence that is asserted but something about it. Thus, if the original assertion (i.e., the proposition of the ground-level) should not be abrogated, then this same proposition should not be the object of a metaproposition […]. 5 was the first to link game-theoretical approaches with CTT. Ranta took Hintikka's (1973) Game-Theoretical Semantics (GTS) as a case study, though his point does not depend on that particular framework: in game-based approaches, a proposition is a set of winning strategies for the player stating the proposition. In game-based approaches, the notion of truth is 4 Cf. Hintikka (1973). 5 Lorenz (1970, p. 75), translated from the German by Shahid Rahman. 6 Lorenz (1970, p. 109), translated from the German by Shahid Rahman. respectively as on one hand must-requests (commitments or obligations) and on the other mayrequests (or entitlements or rights) as follows:1 […] So, let's call them rules of interaction, in addition to inference rules in the usual sense, which of course remain in place as we are used to them. […] Now let's turn to the request mood. And then it's simplest to begin directly with the rules, because the explanation is visible directly from the rules. So, the rules that involve request are these, that if someone has made an assertion, then you may question his assertion, the opponent may question his assertion. (Req1) ⊢ 𝐶 ? ⊢ 𝑚𝑎𝑦 𝐶 Now we have an example of a rule where we have a may. The other rule says that if we have the assertion ⊢ 𝐶, and it has been challenged, then the assertor must execute his knowledge how to do 𝐶. [ … ]. (Req2) ⊢ 𝐶 ? ⊢ 𝐶 ⊢ 𝑚𝑢𝑠𝑡 𝐶′ In relation to the third condition of Brandom, endorsement, it involves the use of assertions brought forward by the interlocutor. In this context Göran Sundholm (2013, p. 17) produced the following proposal that embeds Austin's remark (1946, p. 171) on assertion acts in the context of inference: When I say therefore, I give others my authority for asserting the conclusion, given theirs for asserting the premisses. Herewith, the assertion of one of the interlocutors entitles the other one to endorse it. Moreover, in recent lectures, Per [START_REF] Martin-Löf | Is Logic Part of Normative Ethics? Lecture Held at the research Unity Sciences, Normes, Décisions (FRE 3593)[END_REF] used this dialogical perspective in order to escape a form of circle threatening the explanation of the notions of inference and demonstration. A demonstration may indeed be explained as a chain of (immediate) inferences starting from no premisses at all. That an inference 𝐽 1 . . . 𝐽 𝑛 𝐽 is valid means that one can make the conclusion (judgement 𝐽) evident on the assumption that 𝐽 1 , … , 𝐽 𝑛 are known. Thus the notion of epistemic assumption appears when explaining what a valid inference is. According to this explanation however, we cannot take 'known' in the sense of demonstrated, or else we would be explaining the notion of inference in terms of demonstration when demonstration has been explained in terms of inference. Hence the threatening circle. In this regard Martin-Löf suggests taking 'known' here in the sense of asserted, which yields epistemic assumptions as judgements others have made, judgements whose responsibility others have already assumed. An inference being valid would accordingly mean that, given others have assumed responsibility for the premisses, I can assume responsibility for the conclusion. So, again it is the dialogical take on endorsement which is at stake here that amounts to the following: whatever reason the Opponent has for stating some elementary assertion authorizes the Proponent to use it himself. In other words, whatever reason the Opponent adduces for some elementary assertion the Proponent can take it as rendering the asserted proposition true and therefore can now use the same reason for defending his own assertion of that proposition. In doing so, the Proponent attributes knowledge to the Opponent when he asserts some elementary proposition, but the Opponent does nothe is trying to build a counterargument after all. Thus, the dialogical framework already seems to offer a formal system where the main features of Brandom's epistemological games can be rendered explicit. However, the system so far does not make explicit the reasons behind an assertion. In order to do so we need to incorporate into the dialogical framework expressions standing for those reasons. This requires combining dialogical logic with Per Martin Löfs Constructive Type Theory (1984) in a more thorough way. We call the result of such enrichment of the expressive power of the dialogical framework, dialogues for immanent reasoning precisely because reasons backing a statement, now explicit denizens of the object-language of plays, are internal to the development of the dialogical interaction itselfsee Rahman/McConaugey/Klev/Clerbout (2018). 2However, despite the undeniable links of the dialogical framework to both CTT and Brandom's inferentialist approach to meaning there are also some significant differences that are at the center of the dialogical conception of meaning, namely the identification of a level of meaning, i.e. the play-level, that does not reduce to the proof-theoretical one. We will start by presenting the main features of dialogues for immanent reasoning and then we will come back to the general philosophical discussion on the play-level as the core of what is known as dialoguedefiniteness. The present paper does not discuss explicitly phenomenology, however, Mohammad [START_REF] Shafiei | Intentionnalité et signification: Une approche dialogique[END_REF] developed in his thesis: Intentionnalité et signification: Une approche dialogique, a thorough study of the bearing of the dialogical framework for phenomenology. Nevertheless, his work did not deploy the new development we call immanent reasoning. So, one might see our proposal as setting the basis for a further study linking phenomenology and the dialogical conception of meaning. Local reasons Recent developments in dialogical logic show that the Constructive Type Theory approach to meaning is very natural to the game-theoretical approaches in which (standard) metalogical features are explicitly displayed at the object language-level. 3 This vindicates, albeit in quite a at the level of such winning strategies. Ranta's idea should therefore in principle allow us to apply, safely and directly, instances of game-based methods taken from CTT to the pragmatist approach of the dialogical framework. From the perspective of a general game-theoretical approach to meaning however, reducing a proposition to a set of winning strategies is quite unsatisfactory. This is particularly clear in the dialogical approach in which different levels of meaning are carefully distinguished: there is indeed the level of strategies, but there is also the level of plays in the analysis of meaning which can be further analysed into local, global and material levels. The constitutive role of the play level for developing a meaning explanation has been stressed by Kuno Lorenz in his (2001) paper: Fully spelled out it means that for an entity to be a proposition there must exist a dialogue game associated with this entity, i.e., the proposition A, such that an individual play of the game where A occupies the initial position, i.e., a dialogue D(A) about A, reaches a final position with either win or loss after a finite number of moves according to definite rules: the dialogue game is defined as a finitary open two-person zero-sum game. Thus, propositions will in general be dialogue-definite, and only in special cases be either proofdefinite or refutation-definite or even both which implies their being value-definite. A. 7 Given the distinction between the play level and the strategy level, and deploying within the dialogical framework the CTT-explicitation program, it seems natural to distinguish between local reasons and strategic reasons: only the latter correspond to the notion of proof-object in CTT and to the notion of strategic-object of Ranta. In order to develop such a project we enrich the language of the dialogical framework with statements of the form "𝑝 ∶ 𝐴". In such expressions, what stands on the left-hand side of the colon (here 𝑝) is what we call a local reason; what stands on the right-hand side of the colon (here 𝐴) is a proposition (or set). Within this game-theoretic framework […] truth of A is defined as existence of a winning strategy for A in a dialogue game about A; falsehood of A respectively as existence of a winning strategy against The local meaning of such statements results from the rules describing how to compose (synthesis) within a play the suitable local reasons for the proposition A and how to separate (analysis) a complex local reason into the elements required by the composition rules for A. The synthesis and analysis processes of A are built on the formation rules for A. The most basic contribution of a local reason is its contribution to a dialogue involving an elementary proposition. Informally, we can say that if the Proponent P states the elementary proposition 𝐴, it is because P claims that he can bring forward a reason in defence of his statement, it is this reason that provides content to the proposition. Local meaning and local reasons Statements in dialogues for immanent reasoning Dialogues are games of giving and asking for reasons; yet in the standard dialogical framework, the reasons for each statement are left implicit and do not appear in the notation of the statement: we have statements of the form 𝐗 ! 𝐴 for instance where 𝐴 is an elementary proposition. The framework of dialogues for immanent reasoning allows to have explicitly the reason for making a statement, statements then have the form 𝐗 𝑎 ∶ 𝐴 for instance where 𝑎 is the (local) reason 𝐗 has for stating the proposition 𝐴. But even in dialogues for immanent reasoning, all reasons are not always provided, and sometimes statements have only implicit reasons for bringing the proposition forward, taking then the same form as in the standard dialogical framework: 𝐗 ! 𝐴. Notice that when (local) reasons are not explicit, an exclamation mark is added before the proposition: the statement then has an implicit reason for being made. A statement is thus both a proposition and its local reason, but this reason may be left implicit, requiring then the use of the exclamation mark. Adding concessions In the context of the dialogical conception of CTT we also have statements of the form X ! (x 1 , …, x n ) [x i : A i ] where "" stands for some statement in which (x 1 , …, x n ) ocurs, and where [x i : A i ] stands for some condition under which the statement (x 1 , …, x n ) has been brought forward. Thus, the statement reads: X states that (x 1 , …, x n ) under the condition that the antagonist concedes x i : A i . We call required concessions the statements of the form [x i : A i ] that condition a claim. When the statement is challenged, the antagonist is accepting, through his own challenge, to bring such concessions forward. The concessions of the thesis, if any, are called initial concessions. Initial concessions can include formation statements such as A : prop, B : prop, for the thesis, AB : prop. Formation rules for local reasons: an informal overview It is presupposed in standard dialogical systems that the players use well-formed formulas (wff). The well formation can be checked at will, but only with the usual meta reasoning by which one checks that the formula does indeed observe the definition of a wff. We want to enrich our CTT-based dialogical framework by allowing players themselves to first enquire on the formation of the components of a statement within a play. We thus start with dialogical rules explaining the formation of statements involving logical constants (the formation of elementary propositions is governed by the Socratic rule, see the discussion above on material truth). In this way, the well formation of the thesis can be examined by the Opponent before running the actual dialogue: as soon as she challenges it, she is de facto accepting the thesis to be well formed (the most obvious case being the challenge of the implication, where she has to state the antecedent and thus explicitly endorse it). The Opponent can ask for the formation of the thesis before launching her first challenge; defending the formation of his thesis might for instance bring the Proponent to state that the thesis is a proposition, provided, say, that A is a set is conceded; the Opponent might then concede that A is a set, but only after the constitution of A has been established, though if this were the case, we would be considering the constitution of an elementary statement, which is a material consideration, not a formal one. These rules for the formation of statements with logical constants are also particle rules which are added to the set of particle rules determining the local meaning of logical constants (called synthesis and analysis of local reasons in the framework of dialogues for immanent reasoning). These considerations yield the following condensed presentation of the logical constants (plus falsum), in which "K" in AKB"expresses a connective, and "Q" in "(Qx : A) B(x) " expresses a quantifier. Formation rules, condensed presentation Connective Quantifier Falsum Move X AKB : prop X (Qx : A) B(x) : prop X  : prop Challenge Y ? F K 1 and/or Y ? F K  Y ? F Q 1 and/or Y ? F Q  - Defence X A : prop (resp.) X B : prop X A : set (resp.) X B(x) : prop (x : A) - Synthesis of local reasons The synthesis rules of local reasons determine how to produce a local reason for a statement; they include rules of interaction indicating how to produce the local reason that is required by the proposition (or set) in play, that is, they indicate what kind of dialogic actionwhat kind of move-must be carried out, by whom (challenger or defender), and what reason must be brought forward. Implication For instance, the synthesis rule of a local reason for the implication ABstated by player X indicates: i. that the challenger Y must state the antecedent (while providing a local reason for it): Y p 1 : A 8 ii. that the defender X must respond to the challenge by stating the consequent (with its corresponding local reason): X p 2 : B. In other words, the rules for the synthesis of a local reason for implication are as follows: Synthesis of a local reason for implication Implication Move X ! AB Challenge Y p 1 : A Defence X p 2 : B Notice that the initial statement (X ! AB) does not display a local reason for the claim the the implication holds: player X simply states that he has some reason supporting the claim. We express such kind of move by adding an exclamation mark before the proposition. The further dialogical actions indicate the moves required for producing a local reason in defence of the initial claim. Conjunction The synthesis rule for the conjunction is straightforward: Synthesis of a local reason for conjunction Conjunction Move X ! 𝐴 ∧ 𝐵 Challenge Y ? 𝐿 ∧ or Y ? 𝑅 ∧  Defence X 𝑝 1 : 𝐴 (resp.) X 𝑝 2 : 𝐵 Disjunction For disjunction, as we know from the standard rules, it is the defender who will choose which side he wishes to defend: the challenge consists in requesting of the defender that he chooses which side he will be defending: Synthesis of a local reason for disjunction Disjunction Move X ! 𝐴 ∨ 𝐵 Challenge Y ? ∨  Defence X 𝑝 1 : 𝐴 or X 𝑝 2 : 𝐵 The general structure for the synthesis of local reasons More generally, the rules for the synthesis of a local reason for a constant K is determined by the following triplet: General structure for the synthesis of a local reason for a constant A constant K Implication Conjunction Disjunction Move X ! K X claims that 𝜙 X ! AB X ! 𝐴 ∧ 𝐵 X ! 𝐴 ∨ 𝐵 Analysis of local reasons Apart from the rules for the synthesis of local reasons, we need rules that indicate how to parse a complex local reason into its elements: this is the analysis of local reasons. In order to deal with the complexity of these local reasons and formulate general rules for the analysis of local reasons (at the play level), we introduce certain operators that we call instructions, such as 𝐿 ∨ (𝑝) or 𝑅 ∧ (𝑝). Approaching the analysis rules for local reasons Let us introduce these instructions and the analysis of local reasons with an example: player X states the implication (A∧B)A. According to the rule for the synthesis of local reasons for an implication, we obtain the following: Move X ! (A∧B)B Challenge Y p 1 : A∧B Recall that the synthesis rule prescribes that X must now provide a local reason for the consequent; but instead of defending his implication (with 𝐗 𝑝 2 : 𝐵 for instance), X can choose to parse the reason p 1 provided by Y in order to force Y to provide a local reason for the right-hand side of the conjunction that X will then be able to copy; in other words, X can force Y to provide the local reason for B out of the local reason 𝑝 1 for the antecedent 𝐴 ∧ 𝐵 of the initial implication. The analysis rules prescribe how to carry out such a parsing of the statement by using instructions. The rule for the analysis of a local reason for the conjunction 𝑝 1 : 𝐴 ∧ 𝐵 will thus indicate that its defence includes expressions such as  the left instruction for the conjunction, written 𝐿 ∧ (𝑝 1 ), and  the right instruction for the conjunction, written 𝑅 ∧ (𝑝 1 ). These instructions can be informally understood as carrying out the following step: for the defence of the conjunction 𝑝 1 : 𝐴 ∧ 𝐵 separate the local reason 𝑝 1 in its left (or right) component so that this component can be adduced in defence of the left (or right) side of the conjunction. Here is a play with local reasons for the thesis (𝐴 ∧ 𝐵) ⊃ 𝐵 using instructions: O P ! (𝐴 ∧ 𝐵) ⊃ 𝐵 0 1 𝑚 ≔ 1 𝑛 ≔ 2 2 3 𝑝 1 ∶ 𝐴 ∧ 𝐵 0 𝑅 ∧ (𝑝 1 ) ∶ 𝐵 6 5 𝑅 ∧ (𝑝 1 ) ∶ 𝐵 3 ? 𝑅 ∧ 4 P wins. In this play, P uses the analysis of local reasons for conjunction in order to force O to state 𝑅 ∧ (𝑝 1 ) ∶ 𝐵, that is to provide a local reason9 for the elementary statement 𝐵; P can then copy that local reason in order to back his statement 𝐵, the consequent of his initial implication. With these local reasons, we explicitly have in the object-language the reasons that are given and asked for and which constitute the essence of an argumentative dialogue. The general structure for the analysis rules of local reasons Move Challenge Defence Interaction procedures embedded in instructions Carrying out the prescriptions indicated by instructions require the following three interaction-procedures: 1. Resolution of instructions: this procedure determines how to carry out the instructions prescribed by the rules of analysis and thus provide an actual local reason. 2. Substitution of instructions: this procedure ensures the following; once a given instruction has been carried out through the choice of a local reason, say b, then every time the same instruction occurs, it will always be substituted by the same local reason b. 3. Application of the Socratic rule: the Socratic rule prescribes how to constitute equalities out of the resolution and substitution of instructions, linking synthesis and analysis together. Let us discuss how these rules interact and how they lead to the main thesis of this study, namely that immanent reasoning is equality in action. From Reasons to Equality: a new visit to endorsement One of the most salient features of dialogical logic is the so-called, Socratic rule (or Copycat rule or rule for the formal use of elementary propositions in the standard-that is, non-CTTcontext), establishing that the Proponent can play an elementary proposition only if the Opponent has played it previously. The Socratic rule is a characteristic feature of the dialogical approach: other game-based approaches do not have it and it relates to endorsing condition mentioned in the introduction. With this rule the dialogical framework comes with an internal account of elementary propositions: an account in terms of interaction only, without depending on metalogical meaning explanations for the non-logical vocabulary. The rule has a clear Platonist and Aristotelian origin and sets the terms for what it is to carry out a formal argument: see for instance Plato's Gorgias (472b-c). We can sum up the underlying idea with the following statement: there is no better grounding of an assertion within an argument than indicating that it has been already conceded by the Opponent or that it follows from these concessions. 10 What should be stressed here are the following two points: 1. formality is understood as a kind of interaction; and 2. formal reasoning should not be understood here as devoid of content and reduced to purely syntactic moves. Both points are important in order to understand the criticism often raised against formal reasoning in general, and in logic in particular. It is only quite late in the history of philosophy that formal reasoning has been reduced to syntactic manipulation-presumably the first explicit occurrence of the syntactic view of logic is Leibniz's "pensée aveugle" (though Leibniz's notion was not a reductive one). Plato and Aristotle's notion of formal reasoning is neither "static" nor "empty of meaning". In the Ancient Greek tradition logic emerged from an approach of assertions in which meaning and justification result from what has been brought forward during 10 Recent researches on deploying the dialogical framework for the study of history of logic claim that this rule is central to the interpretation of dialectic as the core of Aristotle's logicsee Crubellier (2014, pp. 11-40) and [START_REF] Marion | Aristotle on universal quantification: a study from the perspective of game semantics[END_REF]. argumentative interaction. According to this view, dialogical interaction is constitutive of meaning. Some former interpretations of standard dialogical logic did understand formal plays in a purely syntactic manner. The reason for this is that the standard version of the framework is not equipped to express meaning at the object-language level: there is no way of asking and giving reasons for elementary propositions. As a consequence, the standard formulation simply relies on a syntactic understanding of Copy-cat moves, that is, moves entitling P to copy the elementary propositions brought forward by O, regardless of its content. The dialogical approach to CTT (dialogues for immanent reasoning) however provides a fine-grain study of the contentual aspects involved in formal plays, much finer than the one provided by the standard dialogical framework. In dialogues for immanent reasoning which we are now presenting, a statement is constituted both by a proposition and by the (local) reason brought forward in defence of the claim that the proposition holds. In formal plays not only is the Proponent allowed to copy an elementary proposition stated by the Opponent, as in the standard framework, but he is also allowed to adduce in defence of that proposition the same local reason brought forward by the Opponent when she defended that same proposition. Thus immanent reasoning and equality in action are intimately linked. In other words, a formal play displays the roots of the content of an elementary proposition, and not a syntactic manipulation of that proposition. Statements of definitional equality emerge precisely at this point. In particular reflexivity statements such as p = p : A express from the dialogical point of view the fact that if O states the elementary proposition A, then P can do the same, that is, play the same move and do it on the same grounds which provide the meaning and justification of A, namely p. These remarks provide an insight only on simple forms of equality and barely touch upon the finer-grain distinctions discussed above; we will be moving to these by means of a concrete example in which we show, rather informally, how the combination of the processes of analysis, synthesis, and resolution of instructions lead to equality statements. Example Assume that the Proponent brings forward the thesis (𝐴 ∧ 𝐵) ⊃ (𝐵 ∧ 𝐴): O P ! (𝐴 ∧ 𝐵) ⊃ (𝐵 ∧ 𝐴) 0 Both players then choose their repetition ranks: O P ! (𝐴 ∧ 𝐵) ⊃ (𝐵 ∧ 𝐴) 0 1 𝑚 ≔ 1 𝑛 ≔ 2 2 O must now challenge the implication if she accepts to enter into the discussion. The rule for the synthesis of a local reason for implication (provided above) stipulates that in order to challenge the thesis, O must state the antecedent and provide a local reason for it: O P ! (𝐴 ∧ 𝐵) ⊃ (𝐵 ∧ 𝐴) 1 𝑚 ≔ 1 𝑛 ≔ 2 Synthesis of a local reason for conjunction 3 𝑝 ∶ 𝐴 ∧ 𝐵 0 According to the same synthesis-rule P must now state the consequent, which he is allowed to do because the consequent is not elementary: O P ! (𝐴 ∧ 𝐵) ⊃ (𝐵 ∧ 𝐴) 0 1 𝑚 ≔ 1 𝑛 ≔ 2 2 3 𝑝 ∶ 𝐴 ∧ 𝐵 0 𝑞 ∶ 𝐵 ∧ 𝐴 4 The Opponent launches her challenge asking for the left component of the local reason 𝑞 provided by P, an application of the rule for the analysis of a local reason for a conjunction described above. O P ! (𝐴 ∧ 𝐵) ⊃ (𝐵 ∧ 𝐴) 1 𝑚 ≔ 1 𝑛 ≔ 2 3 𝑝 ∶ 𝐴 ∧ 𝐵 0 𝑞 ∶ 𝐵 ∧ 𝐴 Analysis of a local reason for conjunction 5 ? 𝐿 ∧ 4 Assume that P responds immediately to this challenge: O P ! (𝐴 ∧ 𝐵) ⊃ (𝐵 ∧ 𝐴) 0 1 𝑚 ≔ 1 𝑛 ≔ 2 2 3 𝑝 ∶ 𝐴 ∧ 𝐵 0 𝑞 ∶ 𝐵 ∧ 𝐴 4 5 ? 𝐿 ∧ 4 𝐿 ∧ (𝑞): 𝐵 6 O will now ask for the resolution of the instruction: O P ! (𝐴 ∧ 𝐵) ⊃ (𝐵 ∧ 𝐴) 0 1 𝑚 ≔ 1 𝑛 ≔ 2 2 3 𝑝 ∶ 𝐴 ∧ 𝐵 0 𝑞 ∶ 𝐵 ∧ 𝐴 4 5 ? 𝐿 ∧ 4 𝐿 ∧ (𝑞): 𝐵 6 Resolution of an instruction 7 ? …/𝐿 ∧ (𝑞) 6 In this move 7, O is asking P to carry out the instruction 𝐿 ∧ (𝑞) by bringing forward the local reason of his choice. The act of choosing such a reason and replacing the instruction for it is called resolving the instruction. In this case, resolving the instruction will lead P to bring forward an elementary statement-that is, a statement in which both the local reason and the proposition are elementary, which falls under the restriction of the Socratic rule. The idea for P then is to postpone his answer to the challenge launched with move 7 and to force O to choose a local reason first so as to copy it in his answer to the challenge. This yields a further application of the analysis rule for the conjunction: Move 11 thus provides P with the information he needed: he can then copy O's choice to answer the challenge she launched at move 7. O P ! (𝐴 ∧ 𝐵) ⊃ (𝐵 ∧ 𝐴) 0 1 𝑚 ≔ 1 𝑛 ≔ 2 2 3 𝑝 ∶ 𝐴 ∧ 𝐵 0 𝑞 ∶ 𝐵 ∧ 𝐴 4 5 ? 𝐿 ∧ 4 𝐿 ∧ ( Note: It should be clear that a similar end will come about if O starts by challenging the right component of the conjunction statement, instead of challenging the left component. Analysis of the example Let us now go deeper in the analysis of the example and make explicit what happened during the play: When O resolves R  (p) with the local reason b (for instance) and P resolves the instruction L  (q) with the same local reason, then P is not only stating b : In other words, the definitional equality 𝑅 ∧ (𝑝) 𝑂 = 𝑏: 𝐵 that provides content to B makes it explicit at the object-language level that an application of the Socratic rule has been initiated and achieved by means of dialogical interaction. B The development of a dialogue determined by immanent reasoning thus includes four distinct stages: 1. applying the rules of synthesis to the thesis; 2. applying the rules of analysis; 3. launching the Resolution and Substitution of instructions; 4. applying the Socratic rule. 5. We can then add a fifth stage: Producing the strategic reason. While the first two steps involve local meaning, step 3 concerns global meaning and step 4 requires describing how to produce a winning strategy. Now that the general idea of local reasons has been provided, we will present in the next chapter all the rules together, according to their level of meaning. The dialogical roots of equality: dialogues for immanent reasoning In this section we will spell out a simplified version of the dialogues for immanent reasoning, that is, the dialogical framework incorporating features of Constructive Type Theory-a dialogical framework making the players' reasons for asserting a proposition explicit. The rules can be divided, just as in the standard framework, into rules determining local meaning and rules determining global meaning. These include: We will be presenting these rules in this order in the next two sections, along with the adaptation of the other structural rules to dialogues for immanent reasoning in the second section. Local meaning in dialogues for immanent reasoning The formation rules The formation rules for logical constants and for falsum are given in the following table. Notice that a statement ' : prop' cannot be challenged; this is the dialogical account for falsum '⊥' being by definition a proposition. Formation rules Move Challenge Defence In the formulation of this rule, "𝜋" is a statement and "𝜏 𝑖 " is a local reason of the form either 𝑎 𝑖 : 𝐴 𝑖 or 𝑥 𝑖 ∶ 𝐴 𝑖 . A particular case of the application of Subst-D is when the challenger simply chooses the same local reasons as those occurring in the concession of the initial statement. This is particularly useful in the case of formation plays: The rules for local reasons: synthesis and analysis Now that the dialogical account of formation rules has been clarified, we may further develop our analysis of plays by introducing local reasons. Let us do so by providing the rules that prescribe the synthesis and analysis of local reasons. For more details on each rule, see section 0. Synthesis rules for local reasons Move Challenge Defence Analysis rules for local reasons Move Anaphoric instructions: dealing with cases of anaphora One of the most salient features of the CTT framework is that it contains the means to deal with cases of anaphora. For example anaphoric expressions are required for formalizing Barbara in CTT. In the following CTT-formalization of Barbara the projection fst(z) can be seen as the tail of the anaphora whose head is 𝑧: (z : (x : D)A)B[fst(z)] true premise 1 (z : (x : D)B)C[fst(z)] true premise 2 -------------- (z : (x : D)A)C[fst(z)] true conclusion In dialogues for immanent reasoning, when a local reason has been made explicit, this kind of anaphoric expression is formalized through instructions, which provides a further reason for introducing them. For example if a is the local reason for the first premise we have P p : (z : (x : D)A(x))B(L  (L  (p) O )) However, since the thesis of a play does not bear an explicit local reason (we use the exclamation mark to indicate there is an implicit one), it is possible for a statement to be bereft of an explicit local reason. When there is no explicit local reason for a statement using anaphora, we cannot bind the instruction L  (p) O to a local reason 𝑝. We thus have something like this, with a blank space instead of the anaphoric local reason: P ! (z : (x : D)A(x))B(L  ( L  ( ) O )) But this blank stage can be circumvented: the challenge on the universal quantifier will yield the required local reason: O will provide 𝑎: (∃𝑥: 𝐷)𝐴(𝑥), which is the local reason for 𝑧. We can therefore bind the instruction on the missing local reason with the corresponding variable-𝑧 in this case-and write P ! (z : (x : D)A(x))B(L  (L  (z) O )) We call this kind of instruction, Anaphoric instructions. For the substitution of Anaphoric instructions the following two cases are to be distinguished: Substitution of Anaphoric Instructions 1 Given some Anaphoric instruction such as L  (z) Y , once the quantifier (∀𝑧: 𝐴)𝐵(… ) has been challenged by the statement a : A the occurrence of L  (z) Y can be substituted by a. The same applies to other instructions. In our example we obtain: P ! (z : (x : D)A(x))B(L  ( L  (z) O )) O a : (x : D)A(x) P b : B(L  (L  (z) O )) O ? a / L  (z) O P b : B(L  (a)) … Substitution of Anaphoric Instructions 2 Given some Anaphoric instruction such as L  (z) Y , once the instruction L  (c)-resulting from an attack on the universal z : has been resolved with a : then any occurrence of L  (z) Y can be substituted by a. The same applies to other instructions. Global Meaning in dialogues for immanent reasoning We here provide the structural rules for dialogues for immanent reasoning, which determine the global meaning in such a framework. They are for the most part similar in principle to the precedent logical framework for dialogues; the rules concerning instructions are an addition for dialogues for immanent reasoning. Structural Rules SR0: Starting rule The start of a formal dialogue of immanent reasoning is a move where P states the thesis. The thesis can be stated under the condition that O commits herself to certain other statements called initial concessions; in this case the thesis has the form ! [  , …,  n ], where 𝐴 is a statement with implicit local reason and 𝐵 1 , … , 𝐵 𝑛 are statements with or without implicit local reasons. A dialogue with a thesis proposed under some conditions starts if and only if O accepts these conditions. O accepts the conditions by stating the initial concessions in moves numbered 0.1, …, 0.n before choosing the repetition ranks. After having stated the thesis (and the initial concessions, if any), each player chooses in turn a positive integer called the repetition rank which determines the upper boundary for the number of attacks and of defences each player can make in reaction to each move during the play. SR1: Development rule The Development rule depends on what kind of logic is chosen: if the game uses intuitionistic logic, then it is SR1i that should be used; but if classical logic is used, then SR1c must be used. SR1i: Intuitionistic Development rule, or Last Duty First Players play one move alternately. Any move after the choice of repetition ranks is either an attack or a defence according to the rules of formation, of synthesis, and of analysis, and in accordance with the rest of the structural rules. Players can answer only against the last non-answered challenge by the adversary. Note: This structural rule is known as the Last Duty First condition, and makes dialogical games suitable for intuitionistic logic, hence the name of this rule. SR1c: Classical Development rule Players play one move alternately. Any move after the choice of repetition ranks is either an attack or a defence according to the rules of formation, of synthesis, and of analysis, and in accordance with the rest of the structural rules. If the logical constant occurring in the thesis is not recorded by the table for local meaning, then either it must be introduced by a nominal definition, or the table for local meaning needs to be enriched with the new expression. Note: The structural rules with SR1c (and not SR1i) produce strategies for classical logic. The point is that since players can answer to a list of challenges in any order (which is not the case with the intuitionistic rule), it might happen that the two options of a P-defence occur in the same play-this is closely related to the classical development rule in sequent calculus allowing more than one formula at the right of the sequent. SR2: Formation rules for formal dialogues SR2i: Starting a formation dialogue A formation-play starts by challenging the thesis with the formation request O ? prop ; P must answer by stating that his thesis is a proposition. SR2ii: Developing a formation dialogue The game then proceeds by applying the formation rules up to the elementary constituents of prop/set. After that O is free to use the other particle rules insofar as the other structural rules allow it. Note: The constituents of the thesis will therefore not be specified before the play but as a result of the structure of the moves (according to the rules recorded by the rules for local meaning). SR3: Resolution of instructions SR4: Substitution of instructions Once the local reason b has been used to resolve the instruction I K (p) X , and if the same instruction occurs again, players have the right to require that the instruction be resolved with 𝑏. The substitution request has the form ?𝑏/I k (p) X . Players cannot choose a different substitution term (in our example, not even X, once the instruction has been resolved). This rule also applies to functions. SR5: Socratic rule and definitional equality The following points are all parts of the Socratic rule, they all apply. SR5.1: Restriction of P statements P cannot make an elementary statement if O has not stated it before, except in the thesis. An elementary statement is either an elementary proposition with implicit local reason, or an elementary proposition and its local reason (not an instruction). SR5.2: Challenging elementary statements in formal dialogues Challenges of elementary statements with implicit local reasons take the form: 𝑿 ! 𝐴 𝒀 ? 𝑟𝑒𝑎𝑠𝑜𝑛 𝑿 𝑎 ∶ 𝐴 Where 𝐴 is an elementary proposition and 𝑎 is a local reason.13 P cannot challenge O's elementary statements, except if O provides an elementary initial concession with implicit local reason, in which case P can ask for a local reason, or in the context of transmission of equality. SR5.3: Definitional equality O may challenge elementary P-statements; P then answers by stating a definitional equality expressing the equality between a local reason and an instruction both introduced by O (for nonreflexive cases, that is when O provided the local reason as a resolution of an instruction), or a reflexive equality of the local reason introduced by O (when the local reason was not introduced by the resolution of an instruction, that is either as such in the initial concessions or as the result of a synthesis of a local reason). We thus distinguish two cases of the Socratic rule: 1. non-reflexive cases; 2. reflexive cases. These rules do not cover cases of transmission of equality. The Socratic rule also applies to the resolution or substitution of functions, even if the formulation mentions only instructions. SR5.3.1: Non-reflexive cases of the Socratic rule We are in the presence of a non-reflexive case of the Socratic rule when P responds to the challenge with the indication that O gave the same local reason for the same proposition when she had to resolve or substitute instruction I. Here are the different challenges and defences determining the meaning of the three following moves: Non-reflexive cases of the Socratic rule The P-statements obtained after defending elementary P-statements cannot be attacked again with the Socratic rule (with the exception of SR5.3.1c), nor with a rule of resolution or substitution of instructions. SR5.3.2: Reflexive cases of the Socratic rule We are in the presence of a reflexive case of the Socratic rule when P responds to the challenge with the indication that O adduced the same local reason for the same proposition, though that local reason in the statement of O is not the result of any resolution or substitution. The attacks have the same form as those prescribed by SR5.3.1. Responses that yield reflexivity presuppose that O has previously stated the same statement or even the same equality. The response obtained cannot be attacked again with the Socratic rule. Definitional Equality transmits by reflexivity, transitivity and symmetry Content and Material Dialogues As pointed out by Krabbe (1985, p. 297), material dialoguesthat is, dialogues in which propositions have content-receive in the writings of Paul Lorenzen and Kuno Lorenz priority over formal dialogues: material dialogues constitute the locus where the logical constants are introduced. However in the standard dialogical framework, since both material and formal dialogues marshal a purely syntactic notion of the formal rule-through which logical validity is defined-, this contentual feature is bypassed,14 with this consequence that Krabbe and others after him considered that, after all, formal dialogues had priority over material ones. As can be gathered from the above discussion, we believe that this conclusion stems from shortcomings of the standard framework, in which local reasons are not expressed at the objectlanguage level. We thus explicitly introduced these local reasons in order to undercut this apparent precedence of a formalistic approach that makes away with the contentual origins of the dialogical project. And yet the Socratic Rule, as defined in the preceding sections of our study entirely leaves the introduction of local reasons to the Opponent (the Proponent only being allowed to endorse what the Opponent introduced). This rule applying to any proposition (or set), it can be considered as a formal rule; so if we are to specify the rules for material reasoningto use Peregrin's (2014, p. 228) apt terminology-, the rules specifying the elementary propositions involved in a dialogue must also be defined: whereas in the structural rules for formal dialogues of immanent reasoning only the Socratic rules dealt with elementary statements, and without providing any specification on that statement beside the simple fact that it must be the Opponent who introduces them in the dialogue, the structural rules for material dialogues of immanent reasoning will have both Socratic rules that are player dependent rules for elementary statements specific to that very statement, but also global rules, that is player independent rules for elementary statements, specific to those statements (thus providing the material level). In fact, in principle; a local reason prefigures a material dialogue displaying the content of the proposition stated. This aspect makes up the ground level of the normative approach to meaning of the dialogical framework, in which use-or dialogical interaction-is to be understood as use prescibed by a rule; such a use is what Peregrin (2014, pp. 2-3) calls the role of a linguistic expression. Dialogical interaction is this use, entirely determined by rules that give it meaning: the linguistic expression of every statement determines this statement by the role it plays, that is by the way it is used, and this use is governed by rules of interaction. The meaning of elementary propositions in dialogical interaction thus amounts to their role in the kind of interaction that is governed by the Socratic and Global rules for material dialogues, that is by the specific formulations of the Socratic and Global rules for precisely those very propositions. It follows that material dialogues are important not only for the general issue on the normativity of logic but also for rendering a language with content. We cannot in this paper develop these kind of dialogues, however we invite the reader to visit the chapter on material dialogues in Rahman/McConaughey/ [START_REF] Rahman | Immanent Reasoning. A plaidoyer for the Play-Level[END_REF], where the main we sketch the main features of material dialogues that include sets of natural numbers and the set 𝐁𝐨𝐨𝐥. The latter allows for expressing classical truth-functions within the dialogical framework, and it has an important role in the CTT-approach to empirical propositions. 15 . The final section of the chapter on Marial dialogues in Rahman/McConaughey/Klev/Clerbout (2018), discusses the epistemological notion of internalization of contes. 16 In this respect, the dialogical framework can be considered as a formal approach to reasoning rooted in the dialogical constitution and "internalization" of content-including empirical content-rather than in the syntactic manipulation of un-interpreted signs. This discussion on material dialogues provides a new perspective on Willfried Sellars ' (1991, pp. 129-194) notion of Space of Reasons: the dialogical framework of immanent reasoning enriched with the material level should show how to integrate world-directed thoughts (displaying empirical content) into an inferentialist approach, thereby suggesting that immanent reasoning can integrate within the same epistemological framework the two conflicting readings of the Space of Reasons brought forward by John McDowell (2009, pp. 221-238) on the one hand, who insists in distinguishing world-direct thought and knowledge gathered by inference, and Robert [START_REF] Brandom | A Study Guide[END_REF] on the other hand, who interprets Sellars' work in a more radical anti-empiricist manner. The point is not only that we can deploy the CTT-distinction between reason as a premise and reason as a piece of evidence justifying a proposition, but also that the dialogical framework allows for distinguishing between the objective justification level targeted by Brandom (1997, p. 129) and the subjective justification level stressed by McDowell. According to our approach the sujective feature corresponds to the play level, where a concrete player brings forward the statement It looks red to me, rather than It is red. The general epistemological upshot from these initial reflections is that, on our view, many of the worries on the interpretation of the Space of Reasons and on the shortcomings of the standard dialogical approach to meaning (beyond the one of logical constants) have their origin in the neglect of the Strategic reasons in dialogues for immanent reasoning The conceptual backbone on which rests the metalogical properties of the dialogical framework is the notion of strategic reason which allows to adopt a global view on all the possible plays that constitute a strategy. However, this global view should not be identified with the perspective common in proof theory: strategic reasons are a kind of recapitulation of what can happen for a given thesis and show the entire history of the play by means of the instructions. Strategic reasons thus yield an overview of the possibilities enclosed in a thesis-what plays can be carried out from it-, but without ever being carried out in an actual play: they are only a perspective on all the possible variants of plays for a thesis and not an actual play. In this way the rules of synthesis and analysis of strategic reasons provided below are not of the same nature as the analysis and synthesis of local reasons, they are not produced through challenges and their defence, but are a recapitulation of the plays that can actually be carried out. The notion of strategic reasons enables us to link dialogical strategies with CTTdemonstrations, since strategic reasons (and not local reasons) are the dialogical counterpart of CTT proof-objects; but it also shows clearly that the strategy level by itself-the only level that proof theory considers-is not enough: a deeper insight is gained when considering, together with the strategy level, the fundamental level of plays; strategic reasons thus bridge these two perspectives, the global view of strategies and the more in-depth and down-to-earth view of actual plays with all the possible variations in logic they allow,17 without sacrificing the one for the other. This vindication of the play level is a key aspect of the dialogical framework and one of the purposes of the present study: other logical frameworks lack this dimension, which besides is not an extra dimension appended to the concern for demonstrations, but actually constitutes it, the heuristical procedure for building strategies out of plays showing the gapless link there is between the play level and the strategy level: strategies (and so demonstrations) stem from plays. Thus the dialogical framework can say at least as much as other logical frameworks, and, additionally, reveals limitations of other frameworks through this level of plays. Introducing strategic reasons Strategic reasons belong to the strategy level, but are elements of the object-language of the play level: they are the reasons brought forward by a player entitling him to his statement. Strategic reasons are a perspective on plays that take into account all the possible variations in the play for a given thesis; they are never actually carried out, since any play is but the actualization of only one of all the possible plays for the thesis: each individual play can be actualized but will be separate from the other individual plays that can be carried out if other choices are made; strategic reasons allow to see together all these possible plays that in fact are always separate. There will never be in any of the plays the complex strategic reason for the thesis as a result of the application of the particle rules, only the local reason for each of the subformulas involved; the strategic reason will put all these separate reasons together as a recapitulation of what can be said from the given thesis. Consider for instance a conjunction: the Proponent claims to have a strategic reason for this conjunction. This means that he claims that whatever the Opponent might play, be it a challenge of the left or of the right conjunct, the Proponent will be able to win the play. But in a single play with repetition rank 1 for the Opponent, there is no way to check if a conjunction is justified, that is if both of the conjuncts can be defended, since a play is precisely the carrying out of only one of the possible O-choices (challenging the left or the right conjunct): to check both sides of a conjunction, two plays are required, one in which the Opponent challenges the left side of the conjunction and another one for the right side. So a strategic reason is never a single play, but refers to the strategy level where all the possible outcomes are taken into account; the winning strategy can then be displayed as a tree showing that both plays (respectively challenging and defending the left conjunct and right conjunct) are won by the Proponent, thus justifying the conjunction. Let us now study what strategic reasons look like, how they are generated and how they are analyzed. A strategic perspective on a statement In the standard framework of dialogues, where we do not explicitly have the reasons for the statements in the object-language, the particle rules simply determine the local meaning of the expressions. In dialogues for immanent reasoning, the reasons entitling one to a statement are explicitely introduced; the particle rules (synthesis and analysis of local reasons) govern both the local reasons and the local meaning of expressions. But when building the core of a winning Pstrategy, local reasons are also linked to the justification of the statements-which is not the case if considering single plays or non-winning strategies, for then only one aspect of the statement may be taken into account during the play, the play providing thus only a partial justification. Take again the example of a P-conjunction, say P 𝑤 ∶ 𝐴 ∧ 𝐵. In providing a strategic reason 𝑤 for the conjunction 𝐴 ∧ 𝐵, P is claiming to have a winning strategy for this conjunction, that is, he is claiming that the conjunction is absolutely justified, that he has a proper reason for asserting it and not simply a local reason for stating it. Assuming that O has a repetition rank of 1 and has stated both 𝐴 and 𝐵 prior to move 𝑖, two different plays can be carried out from this point, which we provide without the strategic reason: Introducing strategic reasons: stating a conjunction O P Concessions Thesis 0 1 𝑚 ≔ 1 𝑛 ≔ 2 2 … … … … … … ! 𝐴 ∧ 𝐵 𝑖 Introducing strategic reasons: left decision option on conjunction O P Concessions Thesis 0 1 𝑚 ≔ 1 𝑛 ≔ 2 2 … … … … … … ! 𝐴 ∧ 𝐵 𝑖 𝑖 + 1 ? ∧ 1 𝑖 ! 𝐴 𝑖 + 2 Introducing strategic reasons: right decision option on conjunction O P Concessions Thesis 0 1 𝑚 ≔ 1 𝑛 ≔ 2 2 … … … … … … ! 𝐴 ∧ 𝐵 𝑖 𝑖 + 1 ? ∧ 2 𝑖 ! 𝐵 𝑖 + 2 So if P brings forward the strategic reason 𝑤 to support his conjunction at move 𝑖, he is claiming to be able to win both Erreur ! Source du renvoi introuvable. and Erreur ! Source du renvoi introuvable., and yet the actual play will follow into only one of the two plays. Strategic reasons are thus a strategic perspective on a statement that is brought forward during actual plays. An anticipation of the play and strategy as recapitulation Since a strategic reason (𝑤 for instance) is brought forward during a play (say at move 𝑖), it is clear that the play has not yet been carried out fully when the player claims to be able to defend his statement against whatever challenge his opponent might launch: bringing forward a strategic reason is thus an anticipation on the outcome of the play. But strategic reasons are not a simple claim to have a winning strategy, they also have a complex internal structure: they can thus be considered as recapitulations of the plays of the winning strategy produced by the heuristic procedure, that is the winning strategy obtained only after running all the relevant plays; this strategy-building process specific to the dialogical framework is a richer process than the one yielding CTT demonstrations-or proof theory in general-, since the strategic reasons will contain traces of choice dependences, which constitute their complexity. Choice dependences link possible moves of a player to the choices made by the other player: a player will play this move if his opponent used this decision-option, that move if the opponent used that decision-option. In the previous example, the Proponent will play move 𝑖 + 2 depending on the Opponent's decision at move 𝑖 + 1, so the strategic object 𝑤 played at move 𝑖 will contain these two possible scenarios with the 𝑖 + 2 P-move depending of the 𝑖 + 1 Odecision. The strategic reason 𝑤 is thus a recapitulation of what would happen if each relevant play was carried out. When the strategic reason makes clearly explicit this choice-dependence of P's moves on those of O, we say that it is in a canonical argumentation form and is a recapitulation of the statement. The rules for strategic reasons do not provide the rules on how to play but rather rules that indicate how a winning strategy has been achieved while applying the relevant rules at the play level. Strategic reasons emerge as the result of considering the optimal moves for a winning strategy: this is what a recapitulation is about. The canonical argumentation form of strategic reasons is closely linked to the synthesis and analysis of local reasons: they provide the recapitulation of all the relevant local reasons that could be generated from a statement. In this respect following the rules for the synthesis and analysis of local reasons, the rules for strategic reasons are divided into synthesis and analysis of strategic reasons, to which we will now turn. In a nutshell, the synthesis of strategic reasons provides a guide for what P needs to be able to defend in order to justify his claim; the analysis of strategic reasons provides a guide for the local reasons P needs to make O state in order to copy these reasons and thus defend his statement. Assertions and statements The difference between local reasons and strategic reasons should now be clear: while local reasons provide a local justification entitling one to his statement, strategic reasons provide an absolute justification of the statement, which thus becomes an assertion. The equalities provided in each of the plays constituting a P-winning strategy, and found in the analysis of strategic objects, convey the information required for P to play in the best possible way by specifying those O-moves necessary for P's victory. This information however is not available at the very beginning of the first play, it is not made explicit at the root of the tree containing all the plays relevant for the P-winning strategy: the root of the tree will not explicitly display the information gathered while developing the plays; this information will be available only once the whole strategy has been developed, and each possible play considered. So when a play starts, the thesis is a simple statement; it is only at the end of the construction process of the strategic reason that P will be able to have the knowledge required to assert the thesis, and thus provide in any new play a strategic reason for backing his thesis. The assertion of the thesis, making explicit the strategic reason resulting from the plays, is in this respect a recapitulation of the result achieved after running the relevant plays, after P's initial simple statement of that thesis. This is what the canonical argumentation form of a strategic object is, and what renders the dialogical formulation of a CTT canonical proof-object. It is in this fashion that dialogical reasons correspond to CTT proof-objects: introduction rules are usually characterized as the right to assert the conclusion from the premises of the inference, that is, as defining what one needs in order to be entitled to assert the conclusion; and the elimination rules are what can be inferred from a given statement. Thus, in the dialogical perspective of P-winning strategies, since we are looking at P's entitlements and duties, what corresponds to proof-object introduction rules would define what P is required to justify in order to assert his statement, which is the synthesis of a P-strategic reason; and what corresponds to proof-object elimination rules would define what P is entitled to ask of O from her previous statements and thus say it himself by copying her statements, which is the analysis of P-strategic reasons. We will thus provide the rules for the synthesis and analysis of strategic reasons (always in the perspective of a P-winning strategy), followed by their corresponding CTT rule. We have in this regard a good justification of Sundholm's idea that inferences can be considered as involving an implicit interlocutor, but here at the strategy level. 3.2 Rules for the synthesis of P-strategic reasons: P-strategic reasons must be built (synthesis of P-strategic reasons); they constitute the justification of a statement by providing certain information-choice-dependences-that are essential to the relevant plays issuing from the statement: strategic reasons are a recapitulation of the building of a winning strategy, directly inserted into a play. Thus a strategic reason for a P-statement can have the form p 2 P ⟦ p 1 O ⟧ and indicates that P's choice of 𝑝 2 is dependent upon O's choice of 𝑝 1 . Strategic reasons for P are the dialogical formulation of CTT proof-objects, and the canonical argumentation form of strategic reasons correspond to canonical proof-objects. Since in this section we are seeking a notion of winning strategy that corresponds to that of a CTTdemonstration, and since these strategies have being identified to be those where P wins, we will only provide the synthesis of strategic reasons for P. 18Synthesis of strategic reasons for P: For negation, we must bear in mind that we are considering P-strategies, that is, plays in which P wins, and we are not providing particle rules with a proper challenge and defence, but we are adopting a strategic perspective on the reason to provide backing a statement; thus the response to an O-challenge on a negation cannot be 𝐏 ! ⊥, which would amount to P losing; this statement "P n O Move Correspondence between the synthesis of strategic reasons and CTT introduction rules and elimination rules Since we are considering a P-winning strategy, we are searching what P needs to justify in order to justify his thesis, which is the point of the synthesis rules for strategic reasons. This search corresponds to the CTT introduction rules, since these determine what one needs in order to carry out an inference. The following table displays the correspondence between the procedures of synthesis of a strategic reason and an introduction rule. Correspondence between synthesis of strategic reasons and introduction rules Synthesis of P-strategic reasons: CTT-introduction rule: Existential quantification 𝐏 ! (∃𝑥 ∶ 𝐴)𝐵(𝑥) (∃𝑥 ∶ 𝐴)𝐵(𝑥) 𝒕𝒓𝒖𝒆 𝑝 1 ∶ 𝐴 𝑝 2 ∶ 𝐵(𝑝 1 ) 〈𝑝 1 , 𝑝 2 〉 ∶ (∃𝑥 ∶ 𝐴)𝐵(𝑥) 𝐎 ? 𝐿 ∃ 𝐏 𝑝 1 ∶ 𝐴 𝐎 ? 𝑅 ∃ 𝐏 𝑝 2 ∶ 𝐵(𝑝 1 ) 𝐏 〈𝑝 1 , 𝑝 2 〉 ∶ (∃𝑥 ∶ 𝐴)𝐵(𝑥) Conjunction 𝑷 ! 𝐴 ∧ 𝐵 𝐴 ∧ 𝐵 Dependences In the case of material implication and universal quantification, a winning P-strategy literally displays the procedure by which the Proponent chooses the local reason for the consequent depending on the local reason chosen by the Opponent for the antecedent. What the canonical argumentation form of a strategic object does is to make explicit the relevant choicedependence by means of a recapitulation of the plays stemming from the thesis. This corresponds to the general description of proof-objects for material implications and universally quantified formulas in CTT: a method which, given a proof-object for the antecedent, yields a proof-object for the consequent. P-strategic reasons as recapitulations of procedures of analysis and record of instructions The analysis of P-strategic reasons focuses on this other essential aspect of P's activity while playing: not determining what he needs in order to justify his statement-that aspect is dealt with by the synthesis of P-strategic reasons-, but determining how he will be able to defend his statement through O's statement and through those alone; that is, the analysis of Pstrategic reasons are a direct consequence of the Socratic rule: since P must defend his thesis using only the elements provided by O, P must be able to analyze O's statements and find the elements he needs for the justification of his own statements, so as to force O to bring these elements forward during the play. In this regard, the analysis of strategic reasons constitute both the analogue of the elimination rules in CTT and the equality rules of a type, to which we now turn. The second presentation on the other hand, allows P to back any proposition 𝐶 with the local reason '𝑦𝑜𝑢 𝑔𝑎𝑣𝑒 𝑢𝑝 (𝑛)' once O has stated ⊥ at move 𝑛. Thus the strategic reason for any proposition 𝐶 is constituted by '𝑦𝑜𝑢 𝑔𝑎𝑣𝑒 𝑢𝑝 (𝑛)', provided that O has provided P with the means for resolving the instruction 𝐿 ⊃ (𝑝). Analysis rules for P-strategic reasons Move Correspondence between the analysis of strategic reasons and CTT equality and elimination rules We will not present here the table of correspondences since they can be reconstructed by the reader emulating the table of correspondence for procedures of synthesis. Let us only indicated that: P 𝑦𝑜𝑢 𝑔𝑎𝑣𝑒 𝑢𝑝 (𝑛) ⟦𝐿 ⊃ (𝑝) 𝑃 = 𝑝 1 𝑷,𝑶 ⟧ : C corresponds to the CTT-elimination-rule for absurdity, that is: ⊥true 𝐶 true interpreted as the fact that we shall never get an element of ⊥ defined as the empty ℕ 0  More precisely, if 𝑐 ∶ ℕ 0 , then the proof-object of 𝐶 is "𝑅 0 " understood as an "aborted programme" 𝑐 ∶ ℕ 0 𝑅 0 (𝑐) ∶ 𝐶(𝑐). In this respect the dialogical reading of the abort-operator is that a player gives up, and the reason for the other player to state C is that the antagonist gave up. A Plaidoyer for the play-level To some extent, the criticisms the dialogical approach to logic has been subject to provides an opportunity for clarifying its basic tenets. We will therefore herewith consider some recent objections raised against the dialogical framework in order to pinpoint some of its fundamental features, whose importance may not have appeared clearly enough through the main body of the paper; namely, dialogue-definiteness, player-independence, and the dialogical conception of proposition. Showing how and why these features have been developed, and specifying their point and the level they operate on, will enable us to vindicate the play level and thus disarm the objections that have been raised against the dialogical framework for having neglected this crucial level. We shall first come back on the central notion of dialogue-definiteness and on the dialogical conception of propositions, which are essential for properly understanding the specific role and importance of the play level. We shall then be able to address three objections to the dialogical framework, due to a misunderstanding of the notion of Built-in Opponent, of the principles of dialogue-definiteness and of player-independence, and of the reflection on normativity that constitutes the philosophical foundation of the framework; all of these misunderstandings can be reduced to a misappraisal of the play level. We shall then go somewhat deeper in the normative aspects of the dialogical framework, according to the principle that logic has its roots in ethics. Dialogue-Definiteness and Propositions The dialogical theory of meaning is structured in three levels, that of the local meaning (determined by the particle rules for the logical constants), of the global meaning (determined by the structural rules), and the strategic level of meaning (determined by what is required for having a winning strategy). The material level of consideration is part of the global meaning, but with particular rules so precise that they determine only one specific expression (through a modified Socratic rule). A characteristic of the local meaning is that the rules are player independent: the meaning is thus defined in the same fashion for each player; they are bound by the same sets of duties and rights when they start a dialogue. This normative aspect is thus constitutive of the play level (which encompasses both the local meaning and the global meaning): it is even what allows one to judge that a dialogue is taking place. In this regard, meaning is immanent to the dialogue: what constitutes the meaning of the statements in a particular dialogue solely rests on rules determining interaction (the local and the global levels of meaning). The strategy level on the other hand is built on the play level, and the notion of demonstration operates on the strategy level (it amounts to having a winning strategy). Two main tenets of the dialogical theory of meaning can be traced back to Wittgenstein, and ground in particular the pivotal notion of dialogue-definiteness: 1. the internal feature of meaning (the Unhintergehbarkeit der Sprache 19 ), and 2. the meaning as mediated by language-games. As for the first Wittgensteinian tenet, the internal feature of meaning, we already mentioned in the introduction that if we relate the notion of internalization of meaning with both languagegames and fully-interpreted languages of CTT, then a salient feature of the dialogical approach to meaning can come to fore: the expressive power of CTT allows all these actions involved in the dialogical constitution of meaning to be incorporated as an explicit part of the object-language of the dialogical framework. In relation to the second tenet, the inceptors of the dialogical framework observed that if language-games are to be conceived as mediators of meaning carried out by social interaction, these language-games must be games actually playable by human beings: it must be the case that we can actually perform them, 20 which is captured in the notion of dialogue-definiteness. 21Dialogue-definiteness is essential for dialogues to be mediators of meaning, but it is also constitutive of what propositions are, as Lorenz clearly puts it: […] for an entity to be a proposition there must exist an individual play, such that this entity occupies the initial position, and the play reaches a final position with either win or loss after a finite number of moves according to definite rules. (Lorenz, 2001, p. 258) A proposition is thus defined in the standard presentation of dialogical logic as a dialoguedefinite expression, that is, an expression 𝐴 such that there is an individual play about 𝐴, that can be said to be lost or won after a finite number of steps, following given rules of dialogical interaction. 22The notion of dialogue-definiteness is in this sense the backbone of the dialogical theory of meaning: it provides the basis for implementing the human-playability requirement and the notion of proposition. 19 See Tractatus Logico-Philosophicus, 5.6. 20 As observed by Marion (2006, p. 245), a lucid formulation of this point is the following remark of Hintikka (1996, p. 158) who shared this tenet (among others) with the dialogical framework: [Finitism] was for Wittgenstein merely one way of defending the need of language-games as the sense that [sic] they had to be actually playable by human beings. […] Wittgenstein shunned infinity because it presupposed constructions that we human beings cannot actually carry out and which therefore cannot be incorporated in any realistic language-game. […] What was important for Wittgenstein was not just the finitude of the operations we perform in our calculi and other language-games, but the fact that we can actually perform them. Otherwise the entire idea of language-games as meaning mediators will lose its meaning. The language-games have to be humanly playable. And that is not possible if they involve infinitary elements. Thus it is the possibility of actually playing the meaning-conferring language-games that is the crucial issue for Wittgenstein, not finitism as such. Dialogue-definiteness sets apart rather decisively the level of strategies from the level of plays, as Lorenz's notion of dialogue-definite proposition does not amount to a set of winning strategies, but rather to an individual play. Indeed, a winning strategy for a player X is a sequences of moves such that X wins independently of the moves of the antagonist. It is crucial to understand that the qualification independently of the moves of the antagonist amounts to the fact that the one claiming 𝐴 has to play under the restriction of the Copy-cat rule: if possessing a winning strategy for player X involves being in possession of a method (leading to the win of X) allowing to choose a move for any move the antagonist might play, then we must assume that the propositions brought forward by the antagonist are justified. There is a winning strategy if X can base his moves leading to a win by endorsing himself those propositions whose justification is rooted on Y's authority. For short, the act of endorsing is what lies behind the so-called Copy-cat rule and structures dialogues for immanent reasoning: it ensures that X can win whatever the contender might bring forward in order to contest 𝐴 (within the limits set by the game). Furthermore, refuting, that is bringing up a strategy against 𝐴, amounts to the dual requirement: that the antagonist Y possess a method that leads to the loss of X ! 𝐴, whatever X is can bring forward, and that she can do it under the Copy-cat restriction: X ! 𝐴 is refuted, if the antagonist Y can bring up a sequence of moves such that she (Y) can win playing under the Copy-cat restriction. Refuting is thus different and stronger than contesting: while contesting only requires that the antagonist Y brings forward at least one counterexample in a kind of play where Y does not need to justify her own propositions, refuting means that Y must be able to lead to the loss of X ! A, whatever X's justification of his propositions might be. In this sense, the assumption that every play is a finitary open two-person zero-sum game does not mean that either there is a winning strategy for 𝐴 or a winning strategy against 𝐴: the play level cannot be reduced to the strategy level. For instance, if we play with the Last-duty first development rule P will lose the individual plays relevant for the constitution of a strategy for ∨ ¬𝐴 . So 𝐴 ∨ ¬𝐴 is dialogue-definite, though there is no winning strategy against 𝐴 ∨ ¬𝐴. The distinction between the play level and the strategy level thus emerges from the combination of dialogue-definiteness and the Copy-cat rule. The classical reduction of strategies against 𝐴 to the falsity of 𝐴 (by means of the saddlepoint theorem) assumes that the win and the loss of a play reduce to the truth or the falsity of the thesis. But we claim that the existence of the play level and a loss in one of the plays introduces a qualification that is not usually present in the purely proof-theoretic approach; to use the previous example, we know that P does not have a winning strategy for ! 𝐴 ∨ ¬𝐴 (playing under the intuitionisitic development rule), but neither will O have one against it if she has to play under the Copy-cat rule herself (notice the switch in the burden of the restriction of the Copy-cat rule when refuting a thesis). Let us identify the player who has to play under the Copy-cat restriction by highlighting her moves: Play against P ! 𝐴 ∨ ¬𝐴 O P ! 𝐴 ∨ ¬𝐴 0 1 𝑛 ≔ 1 𝑚 ≔ 2 2 3 ? ∨ 0 ! 𝐴 4 P wins The distinction between the play and the strategy level can be understood as a consequence of introducing the notion of dialogue-definiteness which amounts to a win or a loss at the play level, though strategically seen, the proposition at stake may be (proof-theoretically) undecidable. Hence, some criticisms to the purported lack of dynamics to dialogical logic are off the mark if they are based on the point that "games" of dialogical logic are deterministic: 23 plays are deterministic in the sense that they are dialogue-definite, but strategies are not deterministic in the sense that for every proposition there would either be a winning strategy for it or a winning strategy against it. Before ending this section let us quote quite extensively [START_REF] Lorenz | Basic Objectives of Dialogue Logic in Historical Perspective[END_REF], who provides a synopsis of the historical background that lead to the introduction of the notion of dialoguedefiniteness and the distinction of the deterministic conception of plays-which obviously operates at the level of plays-from the proof-theoretical undecidable propositions-which operate at the level of strategies: […] It was Alfred Tarski who, in discussions with Lorenzen in 1957/58, when Lorenzen had been invited to the Institute for Advanced Study at Princeton, convinced him of the impossibility to characterize arbitrary (logically compound) propositions by some decidable generalization of having a decidable proof-predicate or a decidable refutation-predicate. […] It became necessary to search for some decidable predicate which may be used to qualify a linguistic entity as a proposition about any domain of objects, be it elementary or logically compound. Decidability is essential here, because the classical characterization of a proposition as an entity which may be true or false, has the awkward consequence that of an undecided proposition it is impossible to know that it is in fact a proposition. This observation gains further weight by L. E. J. Brouwer's discovery that even on the basis of a set of "value-definite", i.e., decidably true or false, elementary propositions, logical composition does not in general preserve value-definiteness. And since neither the property of being proof-definite nor the one of being refutation-definite nor properties which may be defined using these two, are general enough to cover the case of an arbitrary proposition, some other procedure had to be invented which is both characteristic of a proposition and satisfies a decidable concept. The concept looked for and at first erroneously held to be synonymous with argumentation[ 24 ] turned out to be the concept of dialogue about a proposition 23 For such criticismssee Trafford (2017, pp. 86-88). 24 Lorenz identifies argumentation rules with rules at the strategy level and he would like to isolate the interaction displayed by the moves constituting the play levelsee Lorenz (2010a, p.79). We deploy the term argumentationrule for request-answer interaction as defined by the local and structural rules. It is true that nowadays argumentation-rules has even a broader scope including several kinds of communicative interaction and this might produce some confusion on the main goal of the dialogical framework which is in principle, to provide an argumentative understanding of logic rather than the logic of argumentation. However, once this distinction has been drawn nothing prevents to develop the interface dialogical-understanding of logic/logical structure of a dialogue. In fact, it is our claim that in order to study the logical structure of a dialogue, the dialogical conception of logic provides the right venue. A (which had to replace the concept of truth of a proposition A as well as the concepts of proof or of refutation of a proposition A, because neither of them can be made decidable). Fully spelled out it means that for an entity to be a proposition there must exist a dialogue game associated with this entity, i.e., the proposition A, such that an individual play of the game where A occupies the initial position, i.e., a dialogue D(A) about A, reaches a final position with either win or loss after a finite number of moves according to definite rules: the dialogue game is defined as a finitary open twoperson zero-sum game. Thus, propositions will in general be dialogue-definite, and only in special cases be either proof-definite or refutation-definite or even both which implies their being valuedefinite. Within this game-theoretic framework where win or loss of a dialogue D(A) about A is in general not a function of A alone, but is dependent on the moves of the particular play D(A), truth of A is defined as existence of a winning strategy for A in a dialogue game about A; falsehood of A respectively as existence of a winning strategy against A. Winning strategies for A count as proofs of A, and winning strategies against A as refutations of A. The meta-truth of "either 'A is true' or 'A is false' " which is provable only classically by means of the saddlepoint theorem for games of this kind may constructively be reduced to the decidability of win or loss for individual plays about A. The concept of truth of dialogue-definite propositions remains finitary, and it will, as it is to be expected of any adequate definition of truth, in general not be recursively enumerable. The same holds for the concept of falsehood which is conspicuously defined independently of negation. (Lorenz, 2001, pp. 257-258). 4.2 The Built-in Opponent and the Neglect of the Play Level In recent literature Catarina Duthil Novaes (2015) and James Trafford (2017, pp. 102-105) deploy the term internalization for the proposal that natural deduction can be seen as having an internalized Opponent, thereby motivating the inferential steps. This form of internalization is called the built-in Opponent. The origin of this concept is linked to Göran Sundholm who, by 2000, in order to characterize the fundamental links between natural deduction and dialogical logic, introduced in his lectures and talks the term implicit interlocutor. Yet, since the notion of implicit interlocutor was meant to link the strategy level with natural deduction, the concept of built-in Opponent-being the implicit interlocutor's offspring-inherited the same strategic perspective on logical truth. Thus, logical truth can be seen as the encoding of a process through which the Proponent succeeds in defending his assertion against a stubborn ideal interlocutor. 25From the dialogical point of view however, the ideal interlocutor of the strategy level is the result of a process of selecting the relevant moves from the play level. Rahman/Clerbout/ [START_REF] Keiff | Dialogical Logic[END_REF], in a paper dedicated to the Festschrift for Sundholm, designate the process as incarnation, using Jean-Yves Girard's term. Their thorough description of the incarnation process already displays those aspects of the cooperative endeavour, which was formulated by Duthil Novaes (2015) and quoted by Trafford (2017, p. 102) as a criticism of the dialogical framework. Their criticism seems to rest on the idea that the dialogues of the dialogical framework are not truly cooperative, since they are reduced to constituting logical truth. If this is really the point of their criticism, it is simply wrong, for the play level would then be completely neglected: the intersubjective in-built and implicit cooperation of the strategy level (which takes care of inferences) grows out of the explicit interaction of players at the play level in relation to the formation-rules; accepting or contesting a local reason is a process by the means of which players cooperate in order to determine the meaning associated to the action-schema at stake. 26 It is fair to say that the standard dialogical framework, not enriched with the language of CTT, did not have the means to fully develop the so-called material dialogues, that is dialogues that deal with content. Duthil Novaes (2015, p. 602)-but not Trafford (2017, p. 102)-seems to be aware that dialogues are a complex interplay of adversarial and cooperative moves, 27 even in Lorenzen and Lorenz' standard formulation. However, since she understands this interplay as triggered by the built-in implicit Opponent at the strategy level, Duthil Novaes suggestions or corrections motivated by reflections on the Opponent's role cannot be made explicit in the framework, and the way this role contributes in finally constituting a winning strategy cannot be traced back. 28 Duthil Novaes ' (2015, pp. 602-604) approach leads her to suggest that monotonicity is a consequence of the role of the Opponent as a stubborn adversary, which takes care of the nondefeasibility of the demonstration at stake; from this perspective, she contends that the standard presentations of dialogical logic, being mostly adversarial or competitive, are blind to defeasible forms of reasons and are thus […] rather contrived forms of dialogical interaction, and essentially restricted to specific circles of specialists (Duthil Novaes, 2015, p. 602). But this argument is not compelling when considering the strategy level as being built from the play level: setting aside the point on content mentioned above, if we conceive the constitution of a strategy as the end-result of the complementary role of competition and cooperation taking place at the play level, we do not seem to need-at least in many cases-to endow the notion of 26 In fact, when [START_REF] Trafford | Meaning in Dialogue. An Interactive Approach to Logic and Reasoning[END_REF] criticizes dialogical logic in his chapter 4, he surprisingly claims that this form of dialogical interaction does not include the case in which the plays would be open-ended in relation to the logical rules at stake, though it has already been suggested-see for instance in (Rahman & Keiff, 2005, pp. 394-403)-how to develop what we called Structure Seeking Dialogues (SSD). Moreover, [START_REF] Keiff | Le Pluralisme dialogique : Approches dynamiques de l'argumentation formelle[END_REF] PhD-dissertation is mainly about SSD. The idea behind SSD is roughly the following; let us take some inferential practice we would like to formulate as an action-schema, mainly in a teaching-learning situation; we then search for the rules allowing us to make these inferential practices to be put into a schema. For example: we take the third excluded to be in a given context a sound inferential practice; we then might ask what kind of moves P should be allowed to make if he states the third excluded as thesis. It is nonetheless true to say that SSD were studied only in the case of modal logic. 27 To put it in her own words: "the majority of dialogical interactions involving humans appear to be essentially cooperative, i.e., the different speakers share common goals, including mutual understanding and possibly a given practical outcome to be achieved." Duthil Novaes (2015, p. 602). 28 See for instance her discussion of countermoves Duthil Novaes (2015, p. 602) : indefeasibility means that the Opponent has no available countermove: "A countermove in this case is the presentation of one single situation, no matter how far-fetched it is, where the premises are the case and the conclusion is not-a counterexample." The question then would be to know how to show that the Opponent has no countermove available. The whole point of building winning strategies from plays is to actually construct the evidence that there is no possible move for the Opponent that will lead her to win: that is a winning strategy. But when the play level is neglected, the question remains: how does one know the Opponent has no countermove available? It can actually be argued that the mere notion of countermove tends to blurr the distinction between the level of plays and of strategies: a countermove makes sense if it is 'counter' to a winning strategy, as if the players were playing at the strategy level, but that is something we explicitly reject. At the play level, there are only simple moves: these can be challenges, defences, counterattacks, but countermoves do not make any sense. inference with non-monotonic features. The play level is the level were cooperative interaction, either constructive or destructive, can take place until the definitive answer-given the structural and material conditions of the rules of the game-has been reached. 29 The strategy level is a recapitulation that retains the end result. These considerations should also provide an end to Trafford's (2017, pp. 86-88) search for open-ended dialogical settings: open-ended dialogical interaction, to put it bluntly, is a property of the play level. Certainly the point of the objection may be to point out either that this level is underdeveloped in the literature-a fact that we acknowledge with the provisos formulated above-, or that the dialogical approach to meaning does not manage to draw a clean distinction between local and strategic meaning-the section on tonk below intends to make this distinction as clear as possible. At this point of the discussion we can say that the role of the (built-in) Opponent in Lorenzen and Lorenz' dialogical logic has been fully misunderstood. Indeed, the role of both interlocutors (implicit or not) is not about assuring logical truth by checking the non-defeasibility of the demonstration at stake, but their role is about implementing both the dialogical definiteness of the expressions involved and the internalization of meaning.30 Pathological cases and the Neglect of the Play Level The notorious case of [START_REF] Prior | The Runabout Inference-ticket[END_REF] tonk has been several times addressed as a counterargument to inferentialism and also to the "indoor-perspective" of the dialogical framework. This also seems to constitute the background of how Trafford (2017, p. 86) for instance reproduces the circularity objection against the dialogical approach to logical constants. At this point of the discussion, Trafford (2017, pp. 86-88) is clearly aware of the distinction between the rules for local meaning and the rules of the strategy level, though he points out that the local meaning is vitiated by the strategic notion of justification. This is rather surprising as [START_REF] Rahman | On How to be a Dialogician[END_REF], Rahman/Clerbout/ [START_REF] Keiff | Dialogical Logic[END_REF], [START_REF] Rahman | Negation in the Logic of First Degree Entailment and Tonk. A Dialogical Study[END_REF] and [START_REF] Redmond | Armonía Dialógica: tonk Teoría Constructiva de Tipos y Reglas para Jugadores Anónimos[END_REF] have shown it is precisely the case of tonk that provides a definitive answer to the issue. In this respect, three well distinguished levels of meaning are respectively determined by specific rules:  the local meaning of an expression establishes how a statement involving such an expression is to be attacked and defended (through the particle rules);  the global meaning of an expression results from structural rules prescribing how to develop a play having this expression for thesis;  the strategy rules (for P) determine what options P must consider in order to show that he does have a method for winning whatever O may do-in accordance with the local and structural rules. It can in a quite straightforward fashion be shown (see below) that an inferential formulation of rules for tonk correspond to strategic rules that cannot be constituted by the formulation of particle rules. The player-independence of the particle rules-responsible for the branches at the strategy level-do not yield the strategic rules that the inferential rules for tonk are purported to prescribe. For short, the dialogical take on tonk shows precisely how distinguishing rules of local meaning from strategic rules makes the dialogical framework immune to tonk. As this distinction is central to the dialogical framework and illustrates the key feature of player-independence of particle rules, we will now develop the argument; we will then be able to contrast this pathological tonk case to another case, that of the black-bullet operator. The tonk challenge and player-independence of local meaning To show how the dialogical framework is immune to tonk through the importance and priority it gives to the play level, winning strategies are linked to semantic tableaux. According to the dialogical perspective, if tableaux rules (or any other inference system for that matter) are conceived as describing the core of strategic rules for P, then the tableaux rules should be justified by the play level, and not the other way round: the tonk case clearly shows that contravening this order yields pathological situations. We will here only need conjunction and disjunction for dealing with tonk. 31 A systematic description of the winning strategies available for P in the context of the possible choices of O can be obtained from the following considerations: if P is to win against any choice of O, we will have to consider two main different dialogical situations, namely those (a) in which O has uttered a complex formula, and those (b) in which P has uttered a complex formula. We call these main situations the O-cases and the P-cases, respectively. In both of these situations another distinction has to be examined: (i) P wins by choosing i.1. between two possible challenges in the O-cases (a), or i.2. between two possible defences in the P-cases (b), iff he can win with at least one of his choices. (ii) When O can choose ii.1. between two possible defences in the O-cases (a), or ii.2. between two possible challenges in the P-cases (b), P wins iff he can win irrespective of O's choices. The description of the available strategies will yield a version of the semantic tableaux of Beth that became popular after the landmark work on semantic-trees by Raymond Smullyan (1968), where O stands for T (left-side) and P for F (right-side), and where situations of type ii (and not of type i) will lead to a branching-rule. (𝐎)𝐴 ∧ 𝐵 (𝐎)𝐴 ∨ 𝐵 〈𝐏 ?∧ 1 〉 (𝐎)𝐴 〈𝐏?∧ 2 〉 (𝐎)𝐵 〈𝐏? 〉 (𝐎)𝐴  (𝐎)𝐵 However, as mentioned above, semantic tableaux are not dialogues. The main point is that dialogues are built bottom up, from local to global meaning, and from global meaning to validity. This establishes the priority of the play level over the winning strategy level. From the dialogical point of view, Prior's original tonk contravenes this priority. Let us indeed temporarily assume that we can start not by laying down the local meaning of tonk, but by specifying how a winning strategy for tonk would look like with the help of T(left)side and F(right)-side tableaux-rules (or sequent-calculus) for logical constants; in other words, let us assume that the tableaux-rules are necessary and sufficient to set the meaning of tonk. Prior's tonk rules are built for half on the disjunction rules (taking up only its introduction rule), and for half on the conjunction rules (taking up only its elimination rule). This renders the following tableaux version for the undesirable tonk: 32 Tonk is certainly a nuisance: if we apply the cut-rule, it is possible to obtain a closed tableau for T𝐴, F𝐵, for any 𝐴 and 𝐵. Moreover, there are closed tableaux for both {𝐓𝐴, 𝐴𝑡𝑜𝑛𝑘¬𝐴} and {𝐓𝐴, ¬(𝐴𝑡𝑜𝑛𝑘¬𝐴)}. From the dialogical point of view, the rejection of tonk is linked to the fact there is no way to formulate rules for its local meaning that meet the condition of being player-independent: if we try to formulate rules for local meaning matching the ones of the tableaux, the defence yields a different response, namely the tail of tonk if the defender is O, and the head of tonk if the defender is The fact that we need two sets of rules for the challenge and the defence of a tonk move means that the rule that should provide the local meaning of tonk is player-dependent, which should not be the case. Summing up, within the dialogical framework tonk-like operators are rejected because there is no way to formulate player-independent rules for its local meaning that justify the tableaux rules designed for these operators. The mere possibility of writing tableaux rules that cannot be linked to the play level rules shows that the play level rules are not vitiated by strategic rules. This brief reflection on tonk should state our case for both, the importance of distinguishing the rules of the play level from those of the strategy level, and the importance of including in the rules for the local meaning the feature of player-independence: it is the player-independence that provides the meaning explanation of the strategic rules, not the other way round. The black-bullet challenge and dialogue-definiteness Trafford (2017, pp. 37-41) contests the standard inferentialist approach to the meaning of logical constants by recalling the counterexample of Stephen Read, the black-bullet operator. Indeed, [START_REF] Read | Harmony and Modality[END_REF][START_REF] Read | General Elimination Harmony and the Meaning of the Logical Constants[END_REF] introduces a different kind of pathological operator, the black-bullet •, a zero-adic operator that says of itself that it is false. Trafford (2017, p. 39 footnote 35) suggests that the objection also extends to CTT; this claim however is patently wrong, since those counterexamples would not meet the conditions for the constitution of a type. 33 Within the dialogical framework, though player-independent rules for black-bullet can be formulated (as opposed to tonk), they do not satisfy dialogue-definiteness. Let us have the following tableaux rules for the black-bullet, showing that it certainly is pathological: they deliver closed tableaux for both • and ¬ •: (P) • 〈𝐎? 〉 (P)•⊃⊥ (𝐎) • 〈𝐏? 〉 (𝐎) •⊃⊥ We can in this case formulate the following player-independent rules: 33 Klev (2017, p. 12 footnote 7) points out that the introduction rule of such kind of operator fails to be meaninggiving because the postulated canonical set Λ(𝐴) occurs negatively in its premiss, and that the restriction avoiding such kind of operators have been already formulated by Martin-Löf (1971, pp. 182-183), and by [START_REF] Dybjer | Inductive families[END_REF]. Black-bullet player-independent particle rules Move Challenge Defence 𝐗 ! • 𝐘 ? • 𝐗! •⊃⊥ The black-bullet operator seems therefore to meet the dialogical requirement of playerindependent rules, and would thus have local meaning. But if it does indeed have playerindependent rules, the further play on the defence (which is a negation) would require that the challenger concedes the antecedent, that is black-bullet itself: Deploying the black-bullet challenges Y X … … ! • 𝑖 𝑖 + 1 ? • 𝑖 ! •⊃⊥ 𝑖 + 2 𝑖 + 3 ! • 𝑖 + 2 𝑖 + 3 ? • 𝑖 + 4 Obviously, this play sequence can be carried out indefinitely, regardless of which player initially states black-bullet. So the apparently acceptable player-independent rules for playing black-bullet would contravene dialogue-definiteness; and the only way of keeping dialoguedefiniteness would be to give up player-independence!34 Conclusion: the meaning of expressions comes from the play level The two pathological cases we have discussed, the tonk and the black-bullet operators, stress the difference between the play level and the strategy level and how the meaning provided by rules at the strategy level does not carry to the local meaning. Thus, from the dialogical point of view, the rules determining the meaning of any expression are to be rooted at the play level, and at this level what is to be admitted and rejected as a meaningful expression amounts to the formulation of a player-independent rule, that prescribe the constitution of a dialogue-definite proposition (where that expression occurs as a main operator). Notice that if we include material dialogues the distinction between logical operators and non-logical operators is not important any more. If we enrich the dialogical framework with the CTT-language, this feature comes more prominently to the fore. What the dialogical framework adds to the CTT framework is, as pointed out by Martin-Löf (2017a;[START_REF] Martin-Löf | Assertion and Request[END_REF], to set a pragmatic layer where normativity finds its natural place. Let us now discuss the notion of normativity. Normativity and the Dialogical Framework: A New Venue for the Interface Pragmatics-Semantics In his recent book, Jaroslav [START_REF] Peregrin | Inferentialism. Why Rules Matter[END_REF] marshals the distinction between the play level and the strategy level (that he calls tactics) in order to offer another insight, more general, into the issue of normativity mentioned at that start of our volume (Indeed, Peregrin understands the normativity of logic not in the sense of a prescription on how to reason, but rather as providing the material by the means of which we reason. It follows from the conclusion of the previous section that the rules of logic cannot be seen as tactical rules dictating feasible strategies of a game; they are the rules constitutive of the game as such. (MP does not tell us how to handle implication efficiently, but rather what implication is.) This is a crucial point, because it is often taken for granted that the rules of logic tell us how to reason precisely in the tactical sense of the word. But what I maintain is that this is wrong, the rules do not tell us how to reason, they provide us with things with which, or in terms of which, to reason. (Peregrin, 2014, pp. 228-229) Peregrin endorses at this point the dialogical distinction between rules for plays and rules for strategies. In this regard, the prescriptions for developping a play provide the material for reasoning, that is, the material allowing a play to be developped, and without which there would not even be a play; whereas the prescriptions of the tactical level (to use his terminology) prescribe how to win, or how to develop a winning-strategy: This brings us back to our frequently invoked analogy between language and chess. There are two kinds of rules of chess: first, there are rules of the kind that a bishop can move only diagonally and that the king and a rook can castle only when neither of the pieces have previously been moved. These are the rules constitutive of chess; were we not to follow them, we have seen (Section 5.5) we would not be playing chess. In contrast to these, there are tactical rules telling us what to do to increase our chance of winning, rules advising us, e.g., not to exchange a rook for a bishop or to embattle the king by castling. Were we not to follow them, we would still be playing chess, but with little likelihood of winning. (Peregrin, 2014, pp. 228-229) This observation of Peregrin plus his criticism on the standard approach to the dialogical framework, according to which this framework would only focus on logical constants (Peregrin, 2014, pp. 100, 106)-a criticism shared by many others since (Hintikka, 1973, pp. 77-82)naturally leads to the main subject of our book, namely immanent reasoning, or linking CTT with the dialogical framework. The criticism according to which the focus would be on logical constants and not on the meaning of other expressions does indeed fall to some extent on the standard dialogical framework, as little studies have been carried out on material dialogues in this basic framework;35 but the enriched CTT language in material dialogues deals with this shortcoming. Yet this criticism seems to dovetail this other criticism, summoned by Martin-Löf as starting point in his Oslo lecture: I shall take up criticism of logic from another direction, namely the criticism that you may phrase by saying that traditional logic doesn't pay sufficient attention to the social character of language. (Martin-Löf, 2017a, p. 1) The focus on the social character of language not only takes logical constants into account, of course, but it also considers other expressions such as elementary propositions or questions, as well as the acts bringing these expressions forward in a dialogical interaction, like statements, requests, challenges, or defences-to take examples from the dialogical framework-and how these acts made by persons intertwine and call for-or put out of order-other specific responses by that person or by others. In this regard, the social character of language is put at the core of immanent reasoning through the normativity present in dialogues: normativity involves, within immanent reasoning, rules of interaction which allow us to consider assertions as the result of having intertwined rights and duties (or permissions and obligations). This central normative dimension of the dialogical framework at large, which stems from questionning what is actually being done when implementing the rules of this very framework, entails that objections according to which the focus would be only on logical constants will always be, from the dialogical perspective, slightly off the mark. As mentioned in the introduction, in his Oslo and Stockholm lectures, Martin-Löf's (2017a;[START_REF] Martin-Löf | Assertion and Request[END_REF] delves in the structure of the deontic and epistemic layers of statements within his view on dialogical logic. In order to approach this normative aspect which pervades logic up to its technical parts, let us discuss more thoroughly the following extracts of "Assertion and Request": 36 […] we have this distinction, which I just mentioned, between, on the one hand, the social character of language, and on the other side, the non-social […] view of language. But there is a pair of words that fits very well here, namely to speak of the monological conception of logic, or language in general, versus a dialogical one. And here I am showing some special respect for Lorenzen, who is the one who introduced the very term dialogical logic. The first time I was confronted with something of this sort was when reading Aarne Ranta's book Type-Theoretical Grammar in (1994). Ranta there gave two examples, which I will show immediately. The first example is in propositional logic, and moreover, we take it to be constructive propositional logic, because that does matter here, since the rule that I am going to show is valid constructively, but not valid classically. Suppose that someone claims a disjunction to be true, asserts, or judges, a disjunction to be true. Then someone else has the right to come and ask him, Is it the left disjunct or is it the right disjunct that is true? There comes an opponent here, who questions the original assertion, and I could write that in this way: ? ⊢ 𝐴 ∨ 𝐵 𝑡𝑟𝑢𝑒 And by doing that, he obliges the original assertor to answer either that 𝐴 is true that is, to assert either that 𝐴 is true or that 𝐵 is true, so he has a choice, and we need to have some symbol for the choice here. (Dis) ⊢ 𝐴 ∨ 𝐵 𝑡𝑟𝑢𝑒 ? ⊢ 𝐴 ∨ 𝐵 𝑡𝑟𝑢𝑒 ⊢ 𝐴 𝑡𝑟𝑢𝑒 | ⊢ 𝐵 𝑡𝑟𝑢𝑒 Ranta's second example is from predicate logic, but it is of the same kind. Someone asserts an existence statement, ⊢ (∃𝑥 ∶ 𝐴)𝐵(𝑥) 𝑡𝑟𝑢𝑒 and then someone else comes and questions that ? ⊢ (∃𝑥 ∶ 𝐴)𝐵(𝑥) 𝑡𝑟𝑢𝑒 And in that case the original assertor is forced, which is to say, he must come up with an individual from the individual domain and also assert that the predicate 𝐵 is true of that instance. […] So, what are the new things that we are faced with here? Well, first of all, we have a new kind of speech act, which is performed by the| oh, I haven't said that, of course I will use the standard terminology here, either speaker and hearer, or else respondent and opponent, or proponent and opponent, as Lorenzen usually says, so that's terminology but the novelty is that we have a new kind of speech act in addition to assertion. […] So, let's call them rules of interaction, in addition to inference rules in the usual sense, which of course remain in place as we are used to them. […] Now let's turn to the request mood. And then it's simplest to begin directly with the rules, because the explanation is visible directly from the rules. So, the rules that involve request are these, that if someone has made an assertion, then you may question his assertion, the opponent may question his assertion. (Req1) ⊢ 𝐶 ? ⊢ 𝑚𝑎𝑦 𝐶 Now we have an example of a rule where we have a may. The other rule says that if we have the assertion ⊢ 𝐶, and it has been challenged, then the assertor must execute his knowledge how to do 𝐶. And we saw what that amounted too in the two Ranta examples, so I will write this schematically that he will continue by asserting zero, one, or more we have two in the existential case so I will call that schematically by 𝐶0. (Req2) ⊢ 𝐶 ? ⊢ 𝐶 ⊢ 𝑚𝑢𝑠𝑡 𝐶′ The Oslo and the Stockholm lectures of Martin-Löf (2017a; 2017b) contain challenging and deep insights in dialogical logic, and the understanding of defences as duties and challenges as rights is indeed at the core of the deontics underlying the dialogical framework.37 More precisely, the rules Req1 and Req2 do both, they condense the local rules of meaning, and they bring to the fore the normative feature of those rules, which additionally provides a new understanding for Sunholm's notion of implicit interlocutor: once we make explicit the role of the interlocutor, the deontic nature of logic comes out. 38 Moreover, as Martin-Löf points out, and rightly so, they should not be called rules of inference but rules of interaction. Accordingly, a dialogician might wish to add players X and Y to Req2, in order to stress both that the dialogical rules do not involve inference but interaction, and that they constitute a new approach to the action-based background underlying Lorenzen's (1955) Operative Logik. This would yield the following, where we substitute the horizontal bar for an arrow: 39 ⊢ 𝐗 𝐶 ? ⊢ 𝑚𝑎𝑦 𝐘 𝐶 (Req2) ⇓ ? ⊢ 𝑚𝑢𝑠𝑡 𝐗 𝐶 ′ Such a rule does indeed condense the rules of local meaning, but it still does not express the choices while defending or challenging; yet it is the distribution of these choices that determines for example that the meaning of a disjunction is different from that of a conjunction: while in the former case (disjunction) the defender must choose a component, the latter (conjunction) requires of the challenger that, her right to challenge is bounded to her duty to choose the side to be requested (though she might further on request the other side). Hence, the rules for disjunction and conjunction (if we adapt them to Martin-Löf's rules) would be the following: Notice however that these rules only determine the local meaning of disjunction and conjunction, not their global meaning. For example, while classical and constructive disjunction share the same rules of local meaning, they differ at the global level of meaning: in a classical disjunction the defender may come back on the choice he made for defending his disjunction, though in a constructive disjunction this is not allowed, once a player has made a choice he must live with it. What is more, these rules are not rules of inferences (for example rules of introduction and elimination): they become rules of inference only when we focus on the choices P must take into consideration in order to claim that he has a winning strategy for the thesis. Indeed, as mentioned at the start of the present chapter strategy rules (for P) determine what options P must consider in order to show that he has a method for winning whatever O does, in accordance with the rules of local and global meaning. The introduction rules on the one hand establish what P has to bring forward in order to assert it, when O challenges it. Thus in the case of a disjunction, P must choose and assert one of the two components. So, P's obligation lies in the fact that he must choose, and so P's duty to choose yields the introduction rule. Compare this with the conjunction where it is the challenger who has the right to choose (and who does not assert but request his choice). But in both cases, defending a disjunction and defending a conjunction, only one conclusion will be produced, not two: in the case of a conjunction, the challenger will ask one after the other (recall that it is an interaction taking place within a dialogue where each step alternates between moves of each of the players). The elimination rules on the other hand prescribe what moves O must consider when she asserted the proposition at stake. So if O asserted a disjunction, P must be able to win whatever the choices of O be. The case of the universal quantifier adds the interdependence of choices triggered by the may-moves and the must-moves: if the thesis is a universal quantifier of the form (∀𝑥 ∶ 𝐴) 𝐵(𝑥), P must assert 𝐵(𝑎), for whatever 𝑎 O may chose from the domain 𝐴: this is what correspond to the introduction rule. If it is O who asserted the universal quantifier, and if she also conceded that, 𝑎 ∶ 𝐴, then P may challenge the quantifier by choosing 𝑎 ∶ 𝐴, and request of O that she asserts 𝐵(𝑎); this is how the elimination rule for the universal quantifier are introduced in the dialogical framework (for details see chapter 0). These distinctions can be made explicit if we enrich the first-order language of standard dialogical logic with expressions inspired by CTT. The first task is to introduce statements of the form " 𝑝 ∶ 𝐴". On the right-hand side of the colon is the proposition 𝐴, on the left-hand side is the local reason 𝑝 brought forward to back the proposition during a play. The local reason is therefore local if the force of the assertion is limited to the level of plays. But when the assertion "𝑝 ∶ 𝐴" is backed by a winning strategy, the judgement asserted draws its justification precisely from that strategy, thus endowing 𝑝 with the status of a strategic reason that, in the most general cases, encodes an arbitrary choice of O. The rock bottom of the dialogical approach is still the play level notion of dialoguedefiniteness of the proposition, namely For an expression to count as a proposition 𝐴 there must exist an individual play about the statement 𝑿 ! 𝐴, in the course of which 𝑿 is committed to bring forward a local reason to back that proposition, and the play reaches a final position with either win or loss after a finite number of moves according to definite local and structural rules. The deontic feature of logic is here built directly within the dialogical concept of statements about a proposition. More generally, the point is that, as observed by Martin-Löf (2017a, p. 9), according to the dialogical conception, logic belongs to the area of ethics. One way of explaining how this important aspect has been overseen or misunderstood might be that the usual approaches to the layers underlying logic got the order of priority between the deontic notions and the epistemic notions the wrong way round. 40Martin-Löf's lectures propose a fine analysis of the inner and outer structure of the statements of logic from the point of view of speech-act theory, that put the order of priority mentioned above right; in doing so it pushes forward one of the most cherished tenets of the dialogical framework, namely that logic has its roots in ethics. In fact, Martin-Löf's insights on dialogical logic as re-establishing the historical links of ethics and logic provides a clear answer to Wilfried Hodges's (2008) 41 sceptical view in his section 2 as to what the dialogical framework's contribution is. Hodges's criticism seems to target the mathematical interest of a dialogical conception of logic, rather than a philosophical interest which does not seem to attract much of his interest. In lieu of a general plaidoyer for the dialogical framework's philosophical contribution to the foundations of logic and mathematics, which would bring us too far, let us highlight these three points which result from the above discussions: 1) the dialogical interpretation of epistemic assumptions offers a sound venue for the development of inference-based foundations of logic; 2) the dialogical take on the interaction of epistemic and deontic notions in logic, as well as the specification of the play level's role, display new ways of implementing the interface pragmatics-semantics within logic. 3) the introduction of knowing how into the realm of logic is of great import (Martin-Löf, 2017a;[START_REF] Martin-Löf | Assertion and Request[END_REF]. Obviously, formal semantics in the Tarski-style is blind to the first point, misunderstands the nature of the interface involved in the second, and ignores the third. Final Remarks The play level is the level where meaning is forged: it provides the material with which we reason. 42 It reduces neither to the (singular) performances that actualize the interaction-types of the play level, nor to the "tactics" for the constitution of the schema that yields a winning strategy. We call our dialogues involving rational argumentation dialogues for immanent reasoning precisely because reasons backing a statement, that are now explicit denizens of the objectlanguage of plays, are internal to the development of the dialogical interaction itself. More generally, the emergence of concepts, so we claim, are not only games of giving and asking for reasons (games involving Why-questions) they are also games that include moves establishing how is it that the reason brought forward accomplishes the explicative task. Dialogues for immanent reasoning are dialogical games of Why and How. Notice that the notion of dialogue-definiteness is not bound to knowing how to win-this is rather a feature that characterizes winning strategies; to master meaning of an implication, within the dialogical framework, amounts rather to know how to develop an actual play for it. In this context it is worth mentioning that during the Stockholm and Oslo talks on dialogical logic, Martin-Löf (2017a;[START_REF] Martin-Löf | Assertion and Request[END_REF] points out that one of the hallmarks of the dialogical approach is the notion of execution, which-as mentioned in the preface-is close to the requirement of bringing forward a suitable equality while performing an actual play. Indeed from the dialogical point of view, an equality statement comes out as an answer to a question on the local reason 𝑏 of the form how: How do you show the efficiency of 𝑏 as providing a reason for 𝐴? In this sense the how-question presupposes that 𝑏 has been brought-forward as an answer to a why question: Why does A hold? Thus, equalities express the way how to execute or carry out the actions encoded by the local reason; however, the actualization of a play-schema does not require the ability of knowing how to win a play. Thus, while execution, or performance, is indeed important the backbone of the framework lies in the dialogue-definiteness notion of a play. The point of the preceding paragraph is that though actualizing and schematizing are processes at the heart of the dialogical construction of meaning, they should not be understood as performing two separate actions: through these actions we acquire the competence that is associated to the meaning of an expression by learning to play both, the active and the passive role. This feature of Dialogical Constructivism stems from Herder's view 43 that the cultural process is a process of education, in which teaching and learning always occur together: dialogues display this double nature of the cultural process in which concepts emerge from a complex interplay of why and how questions.. In this sense, as pointed out by Lorenz (2010a, pp. 140-147) the dialogical teaching-learning situation is where competition, the I-perspective, and cooperation interact, the You-perspective: both intertwine in collective forms of dialogical interaction that take place at the play level. If the reader allows us to condense our proposal once more, we might say that the perspective we are trying to bring to the fore is rooted in the intimate conviction that meaning and knowledge are something we do together; our perspective is thus an invitation to participate in the open-ended dialogue that is the human pursuit of knowledge and collective understanding, since philosophy's endeavour is immanent to the kind of dialogical interaction that makes reason happen. [START_REF] Brandom | Making it Explicit[END_REF]. Making it Explicit. Cambridge: Harvard University Press. 43 See [START_REF] Herder | Abhandulung über der Ursprung der Sprache[END_REF][START_REF] Herder | Abhandulung über der Ursprung der Sprache[END_REF], Part II). 𝐗 𝑅 ∧ (𝑝) : 𝐵 Existential quantifiation 𝐗 𝑝: (∃𝑥: 𝐴)𝐵(𝑥) 𝐘 ? 𝐿 ∃ or 𝐘 ? 𝑅 ∃ 𝐗 𝐿 ∃ (𝑝) : 𝐴 (resp.) 𝐗 𝑅 ∃ (𝑝) : 𝐵(𝐿 ∃ (𝑝) ) Subset separation 𝐗 𝑝: {𝑥 ∶ 𝐴 |𝐵(𝑥)} 𝐘 ? 𝐿 or 𝐘 ? 𝑅 𝐗 𝐿 {… } (𝑝) : 𝐴 (resp.) 𝐗 𝑅 ∧ (𝑝) : 𝐵(𝐿 {… } (𝑝) ) 𝐘 𝐿 ¬ (𝑝) : 𝐴 𝐗 𝑅 ¬ (𝑝) : ⊥ Also expressed as 𝐗 𝑝: 𝐴 ⊃⊥ 𝐘 𝐿 ⊃ (𝑝) : 𝐴 𝐗 𝑅 ⊃ (𝑝) : ⊥ 1. A player may ask his adversary to carry out the prescribed instruction and thus bring forward a suitable local reason in defence of the proposition at stake. Once the defender has replaced the instruction with the required local reason we say that the instruction has been resolved. 2. The player index of an instruction determines which of the two players has the right to choose the local reason that will resolve the instruction. a. If the instruction I for the logical constant K has the form I K (p) X and it is Y who requests the resolution, then the request has the form Y ?…/ I K (p) X , and it is X who chooses the local reason. b. If the instruction I for the logic constant K has the form I K (p) Y and it is player Y who requests the resolution, then the request has the form Y p i / I K (p) Y , and it is Y who chooses the local reason. P I = a : A SR5.3.1b 𝐏 𝑎 ∶ 𝐴(𝑏) 𝐎 ? = 𝑏 𝐴(𝑏) P I = b : D SR5.3.1c P I = b : D (this statement stems from SR5.3.1b ) 𝐎 ? = 𝐴(𝑏) P A(I) = A(b) : prop Presuppositions: (i) The response prescribed by SR5.3.1a presupposes that O has stated A or a = b : A as the result of the resolution or substitution of instruction I occurring in I : A or in I = b : A. (ii) The response prescribed by SR5.3.1b presupposes that O has stated A and b : D as the result of the resolution or substitution of instruction I occurring in a : A(I). (iii) SR5.3.1c assumes that P I = b : D is the result of the application of SR5.3.1b. The further challenge seeks to verify that the replacement of the instruction produces an equality in prop, that is, that the replacement of the instruction with a local reason yields an equal proposition to the one in which the instruction was not yet replaced. The answer prescribed by this rule presupposes that O has already stated A(b) : prop (or more trivially A(I) = A(b) : prop). 1 〉(𝐏)𝐴  〈𝐏?∧ 2 〉(𝐏)𝐵 〈𝐎? 〉 (𝐏)𝐵The expressions of the form 〈𝐗 … 〉 constitute interrogative utterances.The expressions of the form 〈𝐗 … 〉 constitute interrogative utterances. (O) [or (T)] 𝐴𝑡𝑜𝑛𝑘𝐵 (O) [(T)] 𝐵 (P) [or (F)] 𝐴𝑡𝑜𝑛𝑘𝐵 (P)[(F)] 𝐴 These rules can be considered as inserting in the rules the back and forth movement described byMartin-Löf (2017a, p. 8) with the following diagram: but he is doing this by choosing 𝑏 as local reason for B, that is, by choosing exactly the same local reason as O for the resolution of R  (p).Let us assume that O can ask P to make his choice for a given local reason explicit. P would then answer that his choice for his local reason depends on O's own choice: he simply copied what O considered to be a local reason for 𝐵, that is 𝑅 ∧ (𝑝) 𝑂 = 𝑏: 𝐵. The application of the Socratic rule yields in this respect definitional equality. This rule prescribes the following response to a challenge on an elementary local reason:When O challenges an elementary statement of P such as b : B, P must be able to bring forward a definitional equality such as P R  (p) = b : B. Which reads: P grounds his choice of the local reason b for the proposition B in O's resolution of the instruction R  (p). At the very end P's choice is the same local reason brought forward by O for the same proposition B. .1.2 The substitution rule within dependent statements 𝐗 𝜋(𝑥 1 , … , 𝑥 𝑛 )[𝑥 𝑖 : 𝐴 𝑖 ] 𝐘 𝜏 1 : 𝐴 1 , … , 𝜏 𝑛 : 𝐴 𝑛 𝐗 𝜋(𝜏 1 , … , 𝜏 𝑛 ) Existential quantification X (∃𝑥: 𝐴)𝐵(𝑥): 𝒑𝒓𝒐𝒑 Y ? 𝐹 ∃1 or X 𝐴: 𝒔𝒆𝒕 (resp.) Y ? 𝐹 ∃2 X 𝐵(𝑥): 𝒑𝒓𝒐𝒑[𝑥: 𝐴] Subset separation 𝐗 {𝑥 ∶ 𝐴 |𝐵(𝑥)}: 𝒑𝒓𝒐𝒑 Y ? 𝐹 1 or X 𝐴: 𝒔𝒆𝒕 (resp.) Y ? 𝐹 2 X 𝐵(𝑥): 𝒑𝒓𝒐𝒑[𝑥: 𝐴] Falsum X ⊥: 𝒑𝒓𝒐𝒑 - - 2.2The following rule is not really a formation-rule but is very useful while applying formation rules where one statement is dependent upon the other such as 𝐵(𝑥): 𝒑𝒓𝒐𝒑[𝑥: 𝐴]. 11 Substitution rule within dependent statements (subst-D) Move Challenge Defence Subst-D Y ? 𝐹 ∧1 X 𝐴: 𝒑𝒓𝒐𝒑 Conjunction X 𝐴 ∧ 𝐵: 𝒑𝒓𝒐𝒑 or (resp.) Y ? 𝐹 ∧2 X 𝐵: 𝒑𝒓𝒐𝒑 Y ? 𝐹 ∨1 X 𝐴: 𝒑𝒓𝒐𝒑 Disjunction X 𝐴 ∨ 𝐵: 𝒑𝒓𝒐𝒑 or (resp.) Y ? 𝐹 ∨2 X 𝐵: 𝒑𝒓𝒐𝒑 Y ? 𝐹 ⊃1 X 𝐴: 𝒑𝒓𝒐𝒑 Implication X 𝐴 ⊃ 𝐵: 𝒑𝒓𝒐𝒑 or (resp.) Y ? 𝐹 ⊃2 X 𝐵: 𝒑𝒓𝒐𝒑 Universal quantification X (∀𝑥: 𝐴)𝐵(𝑥): 𝒑𝒓𝒐𝒑 Y ? 𝐹 ∀1 or X 𝐴: 𝒔𝒆𝒕 (resp.) Y ? 𝐹 ∀2 X 𝐵(𝑥): 𝒑𝒓𝒐𝒑[𝑥: 𝐴] ⟦p 1 A" indicates that P's strategic reason for the negation is based on O's move n (where O is forced to state move 𝑛 which is dependent upon O's choice p 1 as local reason for the antecedent of the negation. This yields the following rule for the synthesis of the strategic reason for negation: Synthesis of the strategic reason for negation Move Challenge Defence Strategic reason (synthesis) 𝐎 ! ⊥ Negation 𝐏 ! ¬𝐴 Also expressed as 𝐏 ! 𝐴 ⊃⊥ 𝐎 𝑝 1 : 𝐴 P's successful defence of the negation amounts to a switch such that O must now state that she has a local reason for 𝐴. However this move leads her to give up by bringing forward ⊥ (𝑛) O ⟧ : A The move O p 1 : A, P n O ⟦p 1 allows P to force her to give up in move n, which leads to P's victory. O ⟧ : Note that the analysis of strategic reasons for negation is divided into two presentations of negation, 𝐎 𝑝: ¬𝐴 and 𝐎 𝑝: 𝐴 ⊃⊥, which, at the play level, are governed by SR7 (see p. The first presentation yields O stating ⊥, that is giving up, and therefore the play ends with P winning without further ado. Thus the strategic reason is constituted by the resolution of the instruction for 𝐴 with the means provided by O Analysis of local reasons Analysis of P- Challenge Defence strategic reasons Conjunction O 𝑝: 𝐴 ∧ 𝐵 𝐏 ? 𝐿 ∧ or 𝐏 ? 𝑅 ∧ 𝐎 𝐿 ∧ (𝑝) 𝐎 : 𝐴 (resp.) 𝐎 𝑅 ∧ (𝑝) 𝐎 : 𝐵 𝑷,𝑶 : 𝐴 𝑷,𝑶 : 𝐵 𝐏 𝐿 ∧ (𝑝) 𝑶 = 𝑝 1 (resp.) 𝐏 𝑅 ∧ (𝑝) 𝑶 = 𝑝 2 Existential quantification 𝐎 𝑝: (∃𝑥: 𝐴)𝐵(𝑥) 𝐏 ? 𝐿 ∃ or 𝐏 ? 𝑅 ∃ 𝐎 𝐿 ∃ (𝑝) 𝐎 : 𝐴 (resp.) 𝐎 𝑅 ∃ (𝑝) 𝑂 : 𝐵(𝐿 ∃ (𝑝) 𝐎 ) 𝐏 𝐿 ∃ (𝑝) 𝐎 = 𝑝 1 (resp.) 𝐏 𝑅 ∃ (𝑝) 𝐎 = 𝑝 2 𝐏,𝐎 : 𝐵(𝑝 1 𝐏,𝐎 : 𝐴 𝐏,𝐎 ) Subset separation 𝐎 𝑝: {𝑥 ∶ 𝐴 |𝐵(𝑥)} 𝐏 ? 𝐿 or 𝐏 ? 𝑅 𝐎 𝐿 {… } (𝑝) 𝐎 : 𝐴 (resp.) 𝐎 𝑅 ∧ (𝑝) 𝐎 : 𝐵(𝐿 {… } (𝑝) 𝐎 ) 𝐏 𝐿 {… } (𝑝) 𝐎 = 𝑝 1 (resp.) 𝐏 𝑅 ∧ (𝑝) 𝐎 = 𝑝 2 𝐏,𝐎 : 𝐵(𝑝 1 𝐏,𝐎 : 𝐴 𝐏,𝐎 ) Disjunction 𝐎 𝑝: 𝐴 ∨ 𝐵 𝐏 ? ∨ 𝐎 𝐿 ∨ (𝑝) 𝐎 : 𝐴 or 𝐎 𝑅 ∨ (𝑝) 𝐎 : 𝐵 𝐏 𝐿 ∨ (𝑑) 𝐎 = 𝑑 1 = 𝑑 2 𝐏,𝐎 : 𝐶 𝐏,𝐎 |𝑅 ∨ (𝑑) 𝐎 𝑷 𝑅 ⊃ (𝑝) 𝐎 Implication 𝐎 𝑝: 𝐴 ⊃ 𝐵 𝐏 𝐿 ⊃ (𝑝) 𝐏 : 𝐴 𝐎 𝑅 ⊃ (𝑝) 𝐎 : 𝐵 = 𝑝 2 𝐏,𝐎 ⟦𝐿 ⊃ (𝑝) 𝐏 = 𝑝 1 𝐏,𝐎 ⟧ : 𝐵 Universal quantification 𝐎 𝑝: (∀𝑥: 𝐴)𝐵(𝑥) 𝐏 𝐿 ∀ (𝑝) 𝐏 : 𝐴 𝐎 𝑅 ∀ (𝑝) 𝐎 : 𝐵(𝐿 ∀ (𝑝) 𝐏 ) 𝑷 𝑅 ∀ (𝑝) 𝑶 𝑷,𝑶 ⟦𝐿 ∀ (𝑝) 𝐏 = 𝑝 2 = 𝑝 1 𝑷,𝑶 ⟧: 𝐵(𝑝 1 𝐏,𝐎 ) 𝐎 𝑝: ¬𝐴 𝐏 𝐿 ¬ (𝑝) 𝐏 : 𝐴 𝐎 𝑅 ¬ (𝑝) 𝐎 : ⊥ 𝑷 𝐿 ¬ (𝑝) 𝑷 = 𝑝 1 𝐏,𝐎 : 𝐴 Negation Also expressed as 𝐎 𝑝: 𝐴 ⊃⊥ 𝐏 𝐿 ⊃ (𝑝) 𝐏 : 𝐴 𝐎 𝑅 ⊃ (𝑝) 𝐎 : ⊥ P 𝑦𝑜𝑢 𝑔𝑎𝑣𝑒 𝑢𝑝 (𝑛)⟦𝐿 ⊃ (𝑝) 𝐏 = 𝑝 1 𝑷,𝑶 ⟧ : C (𝐿 ¬ (𝑝) = 𝑝 1 𝑶 ). Erreur ! Signet non défini.). Ansten Klev's transcription of Martin Löf (2017a, pp. 1-3, 7). In fact, the present paper relies on the main technical and philosophical results of Rahman/McConaugey/ Klev/Clerbout (2018). Lorenz (2001, p. 258). Speaking of local reasons is a little premature at this stage, since only instructions are provided and not actual local reasons; but the purpose is here to give the general idea of local reasons, and instructions are meant to be resolved into proper local reasons, which requires only an extra step. This rule is an expression at the level of plays of the rule for the substitution of variables in a hypothetical judgement. SeeMartin-Löf (1984, pp. 9-11). Note that P is allowed to make an elementary statement only as a thesis (Socratic rule); he will be able to respond to the challenge on an elementary statement only if O has provided the required local reason in her initial concessions. Krabbe (1985, p. 297). See[START_REF] Martin-Löf | Truth of Empirical Propositions[END_REF]. By "internalization" we mean that the relevant content is made part of the setting of the game of giving and asking for reasons: any relevant content is the content displayed during the interaction. For a discussion on this conception of internalizationseePeregrin (2014, pp. 36-42). Among these variations can be counted cooperative games, non-monotony, the possibility of player errors or of limited knowledge or resources, to cite but a few options the play level offers, making the dialogical framework very well adapted for history and philosophy of logic. The table which follows is in fact the dialogical analogue to the introduction rules in CTT: dialogically speaking, these rules display the duties required by P's assertions. The fact that these language-games must be finite does not rule out the possibility of a (potentially) infinite number of them. While establishing particle rules the development rules have not been fixed yet, so we might call those expressions propositional schemata. With "ideal" we mean an interlocutor that always make the optimal choices in order to collaborate in the task of testing the thesis. See Rahman (2015) and[START_REF] Rahman | Unfolding parallel reasoning in islamic jurisprudence (I). Epistemic and Dialectical Meaning within Abū Isḥāq al-Shīrāzī's System of Co-Relational Inferences of the Occasioning Factor[END_REF]. Notice that if the role of the Opponent in adversial dialogues is reduced to checking the achievement of logical truth, one would wonder what the role of the Opponent might be in more cooperation-featured dialogues: A soft interlocutor ready to accept weak arguments? We could provide at the local level of meaning a set of player-independent rules, and add some special structural rule in order to force dialogue-definiteness-seeRahman (2012, p. 225); however, such kinds of rules would produce a mismatch in the formation of black-bullet: the formulation of the particle rule would have to assume that black-bullet is an operator, but the structural rule would have to assume it is an elementary proposition. This kind of criticism does not seem to have been aware of[START_REF] Lorenz | Elemente der Sprachkritik. Eine Alternative zum Dogmatismus und Skeptizismus in der Analytischen Philosophie[END_REF] 2009; 2010a; 2010b), carrying out a thorough discussion on predication from a dialogical perspective, which discusses the interaction between perceptual and conceptual knowledge. However, perhaps it is fair to say that this philosophical work has not been integrated into the dialogical logic-we will come back to this subject below. See Lorenz (1981, p. 120), who uses the expressions right to attack and duty to defend. This crucial insight of Martin-Löf on dialogical logic and on the deontic nature of logic seems to underly recent studies on the dialogical framework which are based on Sundholm's notion of the implicit interlocutor, such as Duthil Novaes (2015) and[START_REF] Trafford | Meaning in Dialogue. An Interactive Approach to Logic and Reasoning[END_REF]. See (Martin-Löf, 2017b, p. 9). See also[START_REF] Hodges | Dialogue foundations: A sceptical look[END_REF] andTrafford (2017, pp. 87-88). To usePeregrin's (2014, pp. 228-229) words.

01741824

en

2024/03/05 22:32:07

2012

https://hal.science/hal-01741824/file/Fages-ISSF-SanFrancisco-2012.pdf

Jacques Fages email: [email protected] Elisabeth Rodier Jean-Jacques Letourneau Martial Sauceau Spiro D Alexandratos Polypropylene Grafting in Supercritical Carbon Dioxide Polypropylene (PP), one of the most widely low-cost commodity polymers, has a hydrophobic nature which prevents its use from a large variety of applications. Functionalisation, through free radical grafting of polar vinyl monomers offers an effective route towards higher added value applications especially for environmental remediation, wastewater treatment, and hydrometallurgy. Classic processes for the modification of polypropylene by polar monomers includes reactive extrusion in the molten state and chemical reaction in the solid-state as well as in liquid solvents. Recently, the use of supercritical carbon dioxide (Sc-CO 2 ) has appeared as an innovative and interesting medium because of its environmentally friendly characteristics and its properties of sorption, swelling and foaming of polymers. In this study, PP was put in an autoclave in different forms: pellets, fibres or powders. Vinylbenzyl chloride (VBC) was added along with benzoyl peroxide as initiator of the polymerisation. The autoclave was filled with CO 2 , pressurised and let for a certain amount of time under controlled pressure and temperature. After opening, the swollen polymer was washed with acetone, analysed by infra-red spectroscopy and the chlorine content was determined. When there is sufficient chlorine the samples were used for further modification with a phosphate ligand in order to perform metal ions complexation from water solutions. Microphotographs as well as infra-red spectra show clearly that effective grafting with covalent binding is obtained under sc-CO 2 conditions not only at the surface but also into the pellets. It also appears that the reaction is more complete when polypropylene has previously been pulverised. This may come from a larger surface area on one hand and a different crystalline ratio on the other hand. A two-step process, one aimed at the polymer swelling and the second aimed at the grafting has been successfully compared to the single step process. The influence of several operating parameters was tested: duration of the experiments, pressure and temperature, peroxide content and VBC/PP ratio. After reflux treatment with triethyl phosphite, the best result achieved was a phosphorus elemental analysis of 1.25 mmol/g. This value shows that a high degree of polymerization has been reached and that the polyVBC is grafted on the PP chains, rather than coated on the surface. The next study will deal with the capacity of such grafted PP samples to remove heavy metal ions from wastewater. INTRODUCTION Water is the one commodity on which life on Earth most depends yet it is one that has been taken for granted and used as a waste repository throughout our history. With the advent of the Industrial Revolution, pollution of our waterways has quickened to the extent that it is difficult to find any body of water that has not been impacted by human activity. This now includes the aquifers, rivers, and lakes on which much of humanity relies upon for its drinking water. Purification of water in the environment is thus one of the most pressing problems facing all countries today. The conventional techniques for removing contaminants from water are insufficiently selective for removing the lethal yet low amounts of contaminants such as lead, mercury, and cadmium metal ions. There is thus a need to develop methods to remove contaminants from water and to do so in a cost-effective manner. This paper describes a preliminary study aimed at producing a low-cost and accessible process for producing a plastic filter to be used in the removal of toxic metal ions from water in the environment. The filter will be a modified polypropylene (PP) membrane. Polypropylene is one of the most widely low-cost commodity polymer. However, its hydrophobic nature prevents its use from a large variety of applications. To overcome these limitations free radical grafting of polar vinyl monomers offers an effective route towards higher added value applications. This functionalisation, opens a larger compatibility spectrum of the polyolefin with hydrophilic systems. The modification of PP with polar ligands will permit a new level of applicability to environmental remediation, wastewater treatment, and hydrometallurgy. In these applications, polymers are often applied as beads (ionexchange or chelating resins) requiring a flow of the aqueous stream through a column. The modification of PP will now permit the preparation of a wide range of ion-selective membranes for applications in which flow through membranes is preferred over flow through a column of beads. Classic processes for the modification of polypropylene by polar monomers includes reactive extrusion in the molten state and chemical reaction in the solid-state as well as in liquid solvents. Recently, the use of supercritical carbon dioxide (Sc-CO 2 ) has appeared as an innovative and interesting medium because of its environmentally friendly characteristics and its properties of sorption, swelling and foaming of polymers. This paper describes a method which utilizes supercritical CO 2 in the modification of polypropylene and then further modify this polymer with ion-selective ligands for application to environmental remediation. MATERIALS AND METHODS The experiments were carried out in a multifunctional high-pressure pilot plant (Separex Champigneulles, France). A flowsheet of the apparatus is shown on figure 1. Carbon dioxide is cooled and stored in a liquid CO 2 storage tank (2). It is put in motion and pressurized by a membrane pump (Lewa, Germany) (3). In this study, PP was put in a 500 mL autoclave (5) in different forms: pellets, fibres or powders. Vinylbenzyl chloride (VBC) was added along with benzoyl peroxide as initiator of the VBC polymerisation. The autoclave was then filled with CO 2 , which was previously heated by passing through a heat exchanger (4) to become supercritical until the desired pressure was reached. The pressurised autoclave was let for a certain amount of time under controlled pressure and temperature. Figure 1 -Experimental set-up for the polymer grafting experiments under supercritical carbon dioxide After opening, the swollen polymer was washed with acetone, analysed by FTIR spectroscopy and the chlorine content was determined. When there is sufficient chlorine the samples were used for further modification with a phosphate ligand (triethyl phosphite) in order to perform metal ions complexation from water solutions. Scanning Electron Microscopy (SEM) was also used to visualise the processed polymer. RESULTS SEM microphotographs as well as FTIR spectra show clearly that effective grafting with covalent binding is obtained under sc-CO 2 conditions not only at the surface of PP pellets but also into the pellets. A two-step process, one aimed at the PP swelling and the second aimed at the grafting has been successfully compared to the single step process. In a typical experiment using this approach, polypropylene (PP) pellets (2 mm diameter) were added into a reactor with a volume of 500 mL along with vinylbenzyl chloride (VBC, 10 mL per 2 g PP) and benzoyl peroxide (BPO, 1% based on monomer weight), supercritical CO 2 (scCO 2 ) was introduced at an initial pressure of 12 MPa and contacted with the reactants in two steps: -2 h at 35°C followed by -18 h at 80°C. The reactor was then cooled, the pellets washed extensively with acetone, and dried. The extent of grafting was followed by ATR (the band at 1265 cm -1 is due to C-Cl). Figure 2 shows the spectrum obtained. Figure 2. Polypropylene fiber processed with sc-CO2 The influence of several operating parameters was tested: duration of the experiments, pressure and temperature, peroxide content and VBC/PP ratio. This series of experiments with varying conditions proved the feasibility of the modification. Most importantly, the polyVBC was permanently bound as shown by converting the -CH 2 Cl moieties into phosphonic acid -CH 2 P(O)(OH) 2 by reflux treatment with triethyl phosphite. On figure 3 below, the ATR spectrum shows the broad region around 1103 cm -1 which is representative of the phosphonic acid moiety. The phosphorus capacity of 1.25 mmol/g is indicative of a high level of grafting. This value is quite significant, indicating that the degree of polymerization is high, the polyVBC is grafted on the PP chains, rather than coated on the surface. It also appeared that the reaction is more complete when the PP has previously been pulverised. This may come from a larger surface area on one hand and a different crystallinity ratio on the other hand. CONCLUSION AND PERSPECTIVES These first promising results led us to define what could be the next experiments to achieve. Our specific objective is now to determine the conditions that permit a high degree of modification of polypropylene (PP) with vinylbenzyl chloride (VBC) using supercritical CO 2 as the solvent and benzoyl peroxide (BPO) as the initiator. A unique feature of this research will be the preparation of membranes with higher degrees of modification than previously achieved with BPO by focusing on the swelling time with scCO 2 , its initial pressure, and the ratio of VBC to scCO 2 . The most novel aspect of this research is that, once polypropylene is successfully grafted with polyVBC, the polymer chains will be modified with ion-selective ligands using chemistry that we have established from research on polymer beads, and then the membranes utilized for the detoxification of environmentally critical bodies of water such as sources of drinking water. The general objective of this study is to prepare ion-selective polypropylene membranes in a costeffective manner that can be used for the removal of toxic metal ions from rivers, groundwater, and other sources of drinking water. Figure 3 . 3 Figure 3. Polypropylene fiber modified with phosphonic acid ligands: phosphorus capacity of 1.25 mmol/g ACKNOWLEDGMENT We gratefully acknowledge the Fulbright Scholar Program administered by the Council for International Exchange of Scholars and the Commission Franco-Américaine for support of one of us (SA) while at l'Ecole des Mines d'Albi.

01745950

en

2024/03/05 22:32:07

2018

https://pasteur.hal.science/pasteur-01745950/file/17-1783.pdf

Maria Dolores Fernandez-Garcia Romain Volle Marie-Line Joffret Serge Alain Sadeuh-Mba Ionela Gouandjika-Vasilache Ousmane Kebe Michael R Wiley Manasi Majumdar Etienne Simon-Loriere Anavaj Sakuntabhai Marie-Line Joffret Gustavo Palacios Javier Martin Francis Delpeyroux Kader Ndiaye Maël Bessaud M D Fernandez-Garcia M.-L Joffret Genetic Characterization of Enterovirus A71 Circulating in Africa à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. E nterovirus A71 (EV-A71; species Enterovirus A, genus Enterovirus, family Picornaviridae) is a common etiologic agent of hand, foot and mouth disease in young children. In addition, EV-A71 has been associated with severe and sometimes fatal neurologic diseases, including aseptic meningitis, encephalitis, and poliomyelitis-like acute flaccid paralysis (AFP) [START_REF] Solomon | Virology, epidemiology, pathogenesis, and control of enterovirus 71[END_REF][START_REF] Chang | The current status of the disease caused by enterovirus 71 infections: Epidemiology, pathogenesis, molecular epidemiology, and vaccine development[END_REF]. EV-A71 is classified into 7 genogroups (A-G). Genogroup A includes the prototype strain BrCr that was isolated in the United States in 1969 [START_REF] Solomon | Virology, epidemiology, pathogenesis, and control of enterovirus 71[END_REF][START_REF] Chang | The current status of the disease caused by enterovirus 71 infections: Epidemiology, pathogenesis, molecular epidemiology, and vaccine development[END_REF]. Most EV-A71 isolates belong to genogroups B or C, which are each further divided into subgenogroups [START_REF] Solomon | Virology, epidemiology, pathogenesis, and control of enterovirus 71[END_REF][START_REF] Chang | The current status of the disease caused by enterovirus 71 infections: Epidemiology, pathogenesis, molecular epidemiology, and vaccine development[END_REF]. Subgenogroups B4, B5, and C4 are mainly restricted to countries in Asia, whereas C1 and C2 circulate primarily in Europe and the Asia-Pacific region [START_REF] Solomon | Virology, epidemiology, pathogenesis, and control of enterovirus 71[END_REF]. Genogroup D and the newly proposed genogroup G appear to be indigenous to India, whereas genogroups E and F were recently discovered in Africa and Madagascar, respectively [START_REF] Bessaud | Molecular comparison and evolutionary analyses of VP1 nucleotide sequences of new African human enterovirus 71 isolates reveal a wide genetic diversity[END_REF]. Although EV-A71 has been reported in many parts of the world, its epidemiology remains largely unexplored in Africa. An EV-A71 outbreak was documented in 2000 in Kenya, where HIV-infected orphans were infected by EV-A71 genogroup C (4). Several AFP cases have been associated with EV-A71 infection during 2000-2013 throughout Africa: in Democratic Republic of the Congo (5) (2000, n = 1); Nigeria ( 6 [START_REF] Faleye | Direct detection and identification of enteroviruses from faeces of healthy Nigerian children using a cell-culture independent RT-seminested PCR assay[END_REF]. Molecular identification of all these isolates was based only on the analysis of sequences of the viral protein (VP) 1 capsid protein region. Recombination events may be associated with the emergence and global expansion of new groups of EV-A71 that have induced large outbreaks of hand, foot and mouth disease with high rates of illness and death [START_REF] Leitch | The association of recombination events in the founding and emergence of subgenogroup evolutionary lineages of human enterovirus 71[END_REF]. For EV-A71, genetic exchanges have been described both within a given genogroup and with other types of enterovirus A (EV-A), usually in nonstructural genome regions P2 and P3 [START_REF] Solomon | Virology, epidemiology, pathogenesis, and control of enterovirus 71[END_REF][START_REF] Leitch | The association of recombination events in the founding and emergence of subgenogroup evolutionary lineages of human enterovirus 71[END_REF][START_REF] Yoke-Fun | Phylogenetic evidence for inter-typic recombination in the emergence of human enterovirus 71 subgenotypes[END_REF]. However, before 2017, no complete genome sequence of EV-A71 detected in Africa has been reported, diminishing the power of such analysis. We examined the complete genome of most EV-A71 isolates reported to date in Africa to characterize the evolutionary mechanisms of genetic variability. The Study We sequenced the full genome of 8 EV-A71 isolates obtained from patients with AFP (Table ): isolates 14-157, 14-250, , and 15-355 from West Africa and isolates 08-041, 08-146, and 03-008 from Central Africa. We isolated and typed these isolates as previously described (7-10) and obtained nearly complete genomic sequences using degenerated primers [START_REF] Yoke-Fun | Phylogenetic evidence for inter-typic recombination in the emergence of human enterovirus 71 subgenotypes[END_REF] and additional primers designed for gene-walking (available on request) or unbiased sequencing methods [START_REF] Kugelman | US Army Medical Research Institute of Infectious Diseases; National Institutes of Health; Integrated Research Facility-Frederick Ebola Response Team 2014-2015[END_REF]. We determined the 5′-terminal sequences by means of a RACE kit (Roche, Munich, Germany). We deposited viral genomes in GenBank (accession numbers in Table ) and submitted sequence alignments under BioProject PRJNA422891. We aligned sequences using ClustalW software (http://www.clustal.org). To investigate the genetic relationship between Africa and global EV-A71 isolates, we constructed subgenomic phylogenetic trees based on the P1, P2, and P3 regions of the genome (Figure 1). We identified viral isolates showing related sequences in 1 of these 3 regions by BLAST search (http://www.ncbi.nlm.nih.gov/BLAST) and included them in the corresponding datasets used for analyses. We completed these datasets with a representative global set of EV-A71 sequences available in GenBank and belonging to the different EV-A71 genogroups (https:///wwwnc.cdc.gov/ EID/article/24/4/17-1783-Techapp1.pdf). As expected, in the structural P1 region, the 8 isolates we studied clustered within their respective genogroups (C1, C2, and E), previ-ously established by VP1-based typing (Figure 1, panel A). In particular, the isolates of genogroup E consistently clustered together (bootstrap value 100%), confirming their belonging to the EV-A71 type and their divergence from the other isolates belonging to the common genogroups A, B, and C. Analysis of the nonstructural P2 and P3 genome regions were in agreement with these data. However, the genetic heterogeneity, <12%, observed among the complete genome of genogroup E sequences highly suggested that they have circulated and diverged for years in a large geographic area in Africa. The unique Africa EV-A71-C1 strain clustered with other C1 strains originating worldwide, regardless of which genome region we analyzed. In contrast, the nonstructural sequences of Africa EV-A71 isolates of subgenogroup C2 did not cluster with their non-Africa C2 counterparts or with any of the existing EV-A71 genogroups. The incongruent phylogenetic relationships of Africa C2 strains in the different regions of the genome suggested that recombination events have occurred during evolution. To examine further recombination events, we analyzed EV-A71-C2 study strains by similarity plot against potential parental genomes (Figure 2). This analysis showed that sequences 14-157, 14-250, and 15-355 had high similarity (>95%). By contrast, 13-365 diverged from the other C2 isolates around nt 5600 in the P3 region, suggesting a recombination breakpoint. The analysis showed high sequence similarity (>97%) between the studied EV-A71-C2 isolates and other subgenogroup C2 strains over the P1 capsid region. Conversely, in the noncapsid region, sequence similarity between Africa EV-A71-C2 isolates and classical subgenogroup C2 isolates (e.g., GenBank accession no. HQ647175) was much lower (66%-77%). This finding confirmed a recombination event of the Africa EV-A71 C2 lineage with an unknown enterovirus, the most likely breakpoint being located between nt 3596 and 3740, within the 2A gene. Sequence identity of EV-A71-C2 study strains with their closest related viruses (coxsackievirus A10 [CV-A10], CV-A5, EV-A120, and EV-A71 genogroup E strains) in the 3′ half of the genome was <87.7%. Of note, we found much higher sequence identity with the full-genome sequence of CV-A14 isolate in our database, obtained in 2014 from a patient with AFP in Senegal [START_REF] Fernandez-Garcia | Identification and molecular characterization of non-polio enteroviruses from children with acute flaccid paralysis in West Africa, 2013-2014[END_REF]. This strain features a high similarity value (>97%) with the 3′ half of the genomes of EV-A71-C2 West Africa strains (Figure 2), indicating that their P3 regions share a recent common ancestor. Because these strains belong to 2 different types, this finding strongly suggests that genetic exchanges occurred through intertypic recombination. This result cannot be a result of cross-contamination during the sequencing process because the CV-A14 and EV-A71 isolates were sequenced on 2 different platforms. tensively circulating in Africa. We also suggest that the common ancestor of EV-A71-C2 strains in West Africa has undergone recombination with >1 EV-A circulating in Africa. Genogroup E and recombinant C2 appear to be indigenous to Africa; they have not yet been detected elsewhere. Further exploration of environmental or clinical samples using deep sequencing technology would be of interest to determine the extent of EV-A71 circulation in Africa in the absence of AFP cases. Systematic surveillance based on full-genome sequencing could also serve to monitor these viruses for potential recombinations and to study their role in the emergence of new EV-A71 variants in Africa. Conclusions ) (2004, n = 1, genogroup E); Central African Republic (7) (2003, n = 1, genogroup E); Cameroon (8) (2008, n = 2, genogroup E); Niger (9) (2013, n = 1, genogroup E); and Senegal, Mauritania, and Guinea (9) (2013-2014, n = 3, subgenogroup C2). Four additional EV-A71 strains were obtained from captive gorillas in Cameroon during 2006-2008 (n = 2, genogroup E) (10) and from healthy children in Nigeria in 2014 (n = 2, genogroup E) Figure 1 . 1 Figure 1. Phylogenetic relationships of EV-A71 isolates from patients with acute flaccid paralysis in Africa based on 3 coding regions: A) P1, B) P2, and C) P3. Apart from the studied sequences, subgenomic datasets included their best nucleotide sequence matches identified by NCBI BLAST search (http://www.ncbi.nlm.nih.gov/BLAST) as well as representative sequences of different EV-A71 genogroups and subgenogroups originating worldwide. Trees were constructed from the nucleotide sequence alignment using MEGA 5.0 software (http:// megasoftware.net/) with the neighbor-joining method. Distances were computed using the Kimura 2-parameter model. The robustness of the nodes was tested by 1,000 bootstrap replications. Bootstrap support values >75 are shown in nodes and indicate a strong support for the tree topology. For clarity, CV-A10, CV-A5, and EV-A71 subgenogroups C3, C4, and C5 have been collapsed. Study strains are indicated by laboratory code, country of origin, and year of isolation; previously published strains are indicated by GenBank accession number, isolate code, country of origin, and year of isolation. Black triangles indicate EV-A71 strains from this study; black square indicates the CV-A14 strain from this study. Strains gathered in brackets belong to EV-A71 genogroups or subgenogroups; strains marked in blue color belong to other species of EV-A. Scale bars indicate nucleotide substitutions per site. CV, coxsackievirus; EV, enterovirus. Figure 2 . 2 Figure2. Identification of recombinant sequences in the genome of EV-A71 C2 isolates from patients with acute flaccid paralysis in Africa (14-157, 14-250, 13-365, 15-355) by similarity plot against potential parent genomes (CV-A14 strain 14-254; EV-A71 genogroup E strains 13-194, 08-146, and 03-008) and from GenBank (CV-A10, CV-A5, EV-A120). Similarity plot analysis was performed using SimPlot version 3.5.1 (http://sray.med.som.jhmi.edu/SCRoftware/simplot) on the basis of full-length genomes. For the analysis, we used a window of 600 nt moving in 20-nt steps. Approximate nt positions in the enterovirus genome are indicated. The enterovirus genetic map is shown in the top panel. We used the genome of EV-A71 study strain 14-157 as a query sequence. UTR, untranslated region. Table . . Emerging Infectious Diseases • www.cdc.gov/eid • Vol. 24, No. 4, April 2018 755 Description of enterovirus isolates from patients with acute flaccid paralysis in Africa that were sequenced for characterization of enterovirus A71* Patient age at Genogroup or Strain (reference) Country of isolation diagnosis, y Year Virus subgenogroup Genbank accession no. 14-157 (9) Senegal 3 2014 EV-A71 C2 MG672480 14-250 (9) Mauritania 1.6 2014 EV-A71 C2 MG672481 13-365 (9) Guinea 1.7 2013 EV-A71 C2 MG672479 15-355 (this study) Senegal 2.4 2015 EV-A71 C2 MG013988 13-194 (9) Niger 1.3 2013 EV-A71 E MG672478 03-008 (7) Central African Republic 1.9 2003 EV-A71 E LT719068 08-146 (8) Cameroon 2.6 2008 EV-A71 E LT719066 08-041 (8) Cameroon 1.7 2008 EV-A71 C1 LT719067 14-254 (15) Senegal 3 2014 CV-A14 NA MG672482 *NA, not available. Acknowledgments We thank Karla Prieto and Catherine Pratt, who assisted in obtaining nearly complete genomes of West Africa strains, and Joseph Chitty for analysis of the next-generation sequencing data. The next-generation sequencing equipment at Institut Pasteur of Dakar was provided by the Defense Biological Product Assurance Office under the Targeted Acquisition of Reference Materials Augmenting Capabilities Initiative. This work was supported by the IPD, the Pasteur Institute's Transverse Research Program PTR484, Actions Concertées Inter-Pasteuriennes A22-16, Fondation Total Grant S-CM15010-05B, Roux Howard Cantarini postdoctoral fellowship, and Grant Calmette and Yersin from the International Directorate of the Institut Pasteur. About the Author Dr. Fernandez-Garcia is a scientist with a PhD degree in virology and is involved in research and surveillance of enterovirus infections at Institut Pasteur of Dakar, Senegal. Her research interests include infectious disease epidemiology and public health microbiology.

01718887

en

2024/03/05 22:32:07

2016

https://hal.science/hal-01718887/file/Structural%20and%20magnetic%20properties%20of%20Co2MnSi%20heusler%20alloys%20irradiated%20with%20He%20ions.pdf

Iman Abdallah Nicolas Ratel-Ramond C Magen Béatrice Pécassou Robin Cours Alexandre Arnoult Marc Respaud Jean-François F Bobo Gérard Benassayag Etienne Snoeck B Pecassou N Biziere Structural and magnetic properties of He+ irradiated Co2MnSi Heusler alloys Keywords: Heusler Alloys, Structure, X-ray diffraction, HAADF-STEM niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. average magnetization of the alloy, which is due to D0 3 disorder and local defects induced by irradiation. I. Introduction In the last ten years, full Heusler compounds with general formula X 2 YZ have become a major topic in spintronics, especially for spin torque devices requiring low damping and high spin polarization. Among them, Co 2 MnSi (CMS) is a very promising candidate. It is predicted to be half metallic, with a Curie temperature well above the room temperature [1][2][3][4] and a very low Gilbert damping coefficient as compared to other ferromagnetic metals [5,6]. In all Heusler alloys, the magnetic properties and the atomic order are intimately related. For example, half metallicity has been predicted in CMS for the L2 1 or B2 structure only [7]. The L2 1 structure is the most ordered phase and corresponds to 8 body centered cubic (bcc) sub-lattices having the Co atoms at the corners of the bcc cells and the center sites occupied either by the Mn or Si atoms. A random distribution between Mn and Si or between Mn and Co atoms corresponds to the B2 or D0 3 order respectively while a random distribution of the Co, Mn and Si atoms between the different atomic sites leads to the disordered A2 phase. Ion irradiation with light ions is an efficient technique to improve the local chemical order in different magnetic alloys such as FePt [8] or more recently in Heusler alloy [9]. Indeed, Gaier et al. demonstrated that He + ion irradiation at 30 keV increases the long range order parameter in CMS grown in the B2 phase. Therefore, ion irradiation appears as a very interesting complementary or alternative technique to high temperature annealing, incompatible with microelectronic processes. Before this, further studies about the structural modification induced by irradiation are needed. In this work, we offer to study the structural modification in both the L2 1 and B2 order in Co 2 MnSi irradiated with 150 keV He + ions. Several techniques can be used to characterize the crystal structure such as neutron diffraction, nuclear magnetic resonance, X-ray absorption and circular magnetic dichroism (XAS/XMCD), photoemission spectroscopy (HAXPES) or HAADF-STEM techniques [10][11][12][13]. However, X-ray diffraction is an accessible and commonly used technique for macroscopic characterization of the crystal order [14][15][16]. Quantitative information on the presence of the different phases can be extracted from the measurements of the intensity of different diffraction peaks. For example, superlattice (h,k,l) diffraction peaks for which with h, k and l are odd numbers (e.g. 111) only appear when the L2 1 and/or D0 3 phases are present while diffraction peaks for which h+k+l = 4n+2 (e.g. 002) appear for L2 1 , D0 3 and B2 phases. Finally, the fundamental peaks whose h+k+l = 4n (e.g. 022) appear for all crystal phases. One of the major issues encountered in X-ray diffraction experiments to discriminate between the various ordered phases in CMS is that the Co and Mn scattering factors are very close at the Cu Kα edge, making it almost impossible to distinguish between D0 3 and L2 1 phases. This issue can however be overpassed using a Co-Kα source because, due to anomalous diffraction of the Co at the K-edge, the Co and Mn scattering factors becomes very different and Co and Mn atoms can be differentiated. Thus, the different disorder parameters in CMS can be obtained combining diffraction measurements using Cu and Co Kα sources. The method, based on a model proposed by Niculescu et al. [17], has recently been applied by Takamura et al. [18] for the structural characterization of Co 2 FeSi. In this model α, β and γ are three disorder parameters. α corresponds to the number of Mn atoms located on Si sites and then represents the Mn/Si substitution per CMS unit. Similarly, β and γ corresponds to the number of Co atoms on Si and Mn sites respectively. Then, the structure factors for the different peak of interest are expressed as: F 111 ∝ (1 -2α -β)(f Mn -f Si ) + (γ -β)(f Co -f Mn ) (1) F 002 ∝ (1 -2β)(f Co -f Si ) + (1 -2γ)(f Co -f Mn ) (2) F 022 ∝ 2f Co + f Mn + f Si (3) where f Co , f Mn and f Si are the scattering factors calculated from Ref [START_REF]International Tables for X-ray Crystallography[END_REF]. We can see from eq. 1 and 2 that if f Co and f Mn are close, as for Cu Kα source, the intensity of the diffraction peaks I 111 (∞ │F 111 │ 2 ) and I 002 (∞│F 002 │ 2 ) are not sensitive to D0 3 order while it is for Co-Kα source in anomalous conditions. I. Experimental details. In this study CMS have been grown by magnetron sputtering on MgO (001) single crystals in a Plassys MPU 600 S ultrahigh vacuum (UHV) chamber. Details about the deposition conditions are presented in Ref [START_REF] Ortiz | [END_REF]. The thickness of the CMS layer is 42 +/-1 nm and a 10 nm MgO capping layer is deposited to avoid oxidation. This reference sample is then cut into four pieces, one as a reference and three for He + irradiation at 150 keV performed with a 200A2 Varian ion implanter. The irradiation is performed at room temperature and the fluences for the three samples are 1x10 15 , 5x10 15 and 1x10 16 ions per cm² respectively. The high kinetic energy of the ions prevents from implantation of He + in the CMS film as they stop several hundreds of nm deep in the substrate. X-ray diffraction experiments were performed on a Bruker D8-Discover (Da-Vinci) diffractometer equipped with a Cu Kα 1 source (λ=0.154 nm) to measure the (002) and (004) peaks while a Panalytical Empyrean diffractometer equipped with a Co Kα 1 source (λ=0.179 nm) has been used to measure the (111), ( 002) and (022) diffraction peaks. Note that the (004) and (022) peaks show similar structure factor and are both 'fundamental' peaks. Figure 1 presents examples of different diffraction peaks obtained either with Co or Cu Kα 1 sources. II. Results and discussion. The results for the deduced α, β, γ are presented in Figure 1.d. For all the samples, we observe that the β parameter, i.e. Co/Si disorder, remains almost constant (0.04 +/-0.02) whatever the ion fluence. Additionally α increases from 0.14 +/-0.01 to 0.22 +/-0.01 for the reference sample and the 10 16 irradiated one respectively. While we already observe a small increase of α for 1x10 15 ions/cm², there is a clear step at 5x10 15 . Similarly the γ parameter, i.e. Co/Mn disorder, increases significantly for fluences above 5x10 15 . For lower fluences, the Co-Mn substitution (γ) is not detectable within the uncertainty of the measurements, meaning it is less than 0.02. Using these disorder parameters one can calculate the probability of presence of the different atoms on their original site considering the L2 1 phase as starting structure (see details in Ref 18). We found values of 98, 86 and 83 % for the Co, Mn and Si respectively for the reference sample. These probabilities fall down to 93, 71 and 75% at 10 16 ions/cm². Considering that α = 0.5 for full B2 order, we can estimate that our sample grows as mixture of L2 1 and B2 phases, with ratios of about 25% of B2 and 75% of L2 1 order (neglecting the small amount of Co/Si disorder). We also observe structural modifications of the alloy for fluences above 5.10 15 /cm². The 002 and 004 diffraction peaks of the irradiated sample in Figure 1 are slight shifted toward low angle compared to the peak position of the reference sample, indicating an increase of the out-of-plane lattice parameter from 5.67 to 5.69 Ǻ. In contrary the (111) reflection remains at the same Bragg angle. As the (111) peak is related to the L2 1 and/or D0 3 phases only, we may assume that only regions of the thin film presenting the B2 order have their out-of-plane lattice parameter increasing with the irradiation. While X-ray diffraction gives macroscopic information about the structural order, it does not explain the way the different atomic exchanges get organized in the material. For example, it is difficult to know if the B2 and D0 3 disorders are diluted in the initial matrix or if grains of a particular ordered phase grow from the L2 1 starting structure. To answer this question we performed HAADF-STEM experiments which provide information on the local ordering at the atomic scale. Measurements have been performed at 300 kV on an FEI Titan 60-300 microscope, equipped with a spherical aberration corrector for the probe. Two lamellas have been prepared by FIB, one extracted from the reference sample and the other one prepared from the sample irradiated at 10 16 ions/cm². HAADF-STEM studies were realized in the [-110] zone axis. In this orientation and for L2 1 order, the intensity of each atomic column, which increases with Z, corresponds to only one type of atoms. order. We also note that the difference of intensity between Co and Mn columns is weak as expected from the Co and Mn atomic numbers (Z Co = 27). Similarly to classical X-ray diffraction using a Cu Kα source, HAADF-STEM is not very sensitive to D0 3 disorder. In order to demonstrate that the L2 1 order is the main phase of the sample we performed a statistical analysis of maximum peak intensities of the HAADF-STEM images for the reference sample (Figure 2) and for the sample irradiated at 10 16 ions/cm² (Figure 3). For the first one, three distinct intensity distributions corresponding to the Co, Mn and Si atomic columns are observed (Figure 2.c). The values are normalized by the value of the Co intensity at the center of the Co distribution. The appearance of three different intensities in the HAADF-STEM image is in good agreement with the L2 1 order even if some spreading of the intensity distribution is observed. One source of spreading comes from the slight change of thickness across the lamella prepared for STEM experiment. This effect prevents any statistics over very large areas. In some particular regions of the film other intensity distribution are observed, as the one reported in Figure 2.d which corresponds to statistical analysis performed in the black box in Figure 2.a. The inset in Figure 2.d shows that STEM intensities corresponding to the Mn and Si columns are very similar. Even if the statistical analysis is performed on a small number of atomic columns, it clearly shows that the STEM intensities corresponding to Mn and Si atomic columns converge to a single value, which could be associated to the appearance of the B2 order. This result indicates that very small grains with B2 order are distributed in the L2 1 matrix. While the B2 order in the reference sample is only observed in small regions of similar size as the black box in Figure 2.a, it can be observed in more extended areas for a fluence of 10 16 ions/cm². This is shown in Figure 3.a. The B2 order can be observed either from the intensity profile of the different We also observe for the irradiated sample that the relative STEM intensity of the different atomic columns spread over slightly larger values as compared to the reference sample. The most probable explanation relies on the difference between the two prepared samples, especially the thicknesses which modify the absolute STEM intensity. Another possibility relies on local defects such as vacancies induced by the irradiation. As already explained, it is very challenging to state about any Co/Mn exchange from HAADF-STEM experiments. However one can argue that if Co/Mn substitutions arise in the B2 phase, one should observe a spreading of the STEM intensity associated to the Mn columns that range between Si and Co ones. This is however not what is observed. Moreover, the relative intensity of the Mn columns compared to the Co ones in the L2 1 phase shows similar values (as for the reference sample). Therefore we assume that the D0 3 order most probably arises in the L2 1 matrix. Finally, we measured the magnetization of the four samples (Figure 4) with a QUANTUM design PPMS VSM magnetometer at 300 K. We clearly observe a decrease of the magnetization versus the ion fluence. As demonstrated in Ref [2] the B2 and L2 1 phase have the same magnetic moment (µ 0 Ms ≈ 1.25 T) while it decreases by 10% in the D0 3 phase. Therefore we argue the decrease of the magnetization can be accounted for the local Co-Mn exchange in the L2 1 matrix and vacancies induced by the irradiation. III. Conclusions. In conclusion we demonstrated that He + ion irradiation at 150 keV induces Mn/Si exchange and then favors the B2 structure at the cost of the L2 1 phase. However, for fluences above 5x10 15 ions/cm² , the out of plane lattice parameter of the B2 phase increase and might have an impact upon half metallicity. In addition, for this threshold value, Co/Mn substitution occurs in the L2 1 structure leading to a decreased magnetization and so half metallicity. Below this threshold, no significant improvement of L2 1 order has been observed. Therefore our study demonstrates that ion irradiation is an interesting alternative to annealing only for the B2 order as long as fluence is kept under a threshold value. According to eq 2 and 3, comparing the experimental I 002 /I 004 ratio to the calculated one allows the extraction of the β values. Then α and γ are deduced from the measurements of the I 002 /I 022 and I 111 /I 022 ratios. Figure 1 . 1 Figure 1. (color online) (002) (a) and (111) (b) diffraction peak at Co Kα 1 edge for reference (black) and 10 16 ions/cm² irradiated (red) samples. (c) (002) and (004) diffraction peaks at Cu Kα 1 edge. (d) Disorder parameters obtained from X-ray diffraction experiments. Figure 2 . 2 Figure 2.a shows an example of a HAADF-STEM image obtained on the reference sample. The insert shows a zoom on few atomic columns. Figure 2.b shows the intensity profiles of STEM image taken along different lines reported in Figure2.a. The profiles clearly shows the alternation of high and low peak intensity corresponding to Mn (Z = 25) and Si (Z = 14) columns, demonstrating the L2 1 Figure 2 . 2 Figure 2. (color online) (a) HAADF-STEM image of CMS. The insert at top right is a zoom over 6x6 atomic columns. (b) Intensity profile of the lines denoted by the colored arrows in (a). (c) Statistical analysis of intensity profile obtained from (a) on a region of 11 x 23 atomic columns. (d) statistical analysis of the area denoted by the black box in (a). In insert the intensity profile of the three central lines in the black box. lines (Figure 3 3 .b) or statistically over the selected area (Figure3.c). Similar values for the STEM intensity of the Mn and Si columns are measured while the intensity corresponding to the Co columns remains higher. On other part of the sample we still observe the L2 1 order (Figure3.d). We then suggest that ion irradiation induces a L2 1 to B2 transformation, which occurs around the initial B2 grains. Figure 3 . 3 Figure 3. (color online) (a) HAADF-STEM image of CMS irradiated at 10 16 ions/cm². The black boxes correspond to a zone with B2 or L2 1 character. (b) Intensity profile of the lines denoted by red and black lines in the B2 or L2 1 boxes in (a). Statistical analysis of the B2 (c) and L2 1 (d) regions in (a). Figure 4 . 4 Figure 4. (color online) Evolution of the CMS average magnetization at 300 K as a function of the He + fluence. Acknowledgments The authors thank LPCNO for X-ray diffraction facilities. This work has been supported by the French Agence Nationale pour la Recherche (ANR NASSICS 12-JS10-008 01) and the "French RENATECH network".

01745988

en

2024/03/05 22:32:07

2016

https://pastel.hal.science/tel-01745988/file/TheseLin.pdf

Ph.D students including Ahlem, Yasir, Ovidiu, Hajer, Abdoulaye, Aymen, Hind, Vaggelis, Habib from the previous team, RES Rita Thomas Andrea Leonardo The Léonce Yixi, Christian, Rim, Dalia, Céline Natalya Dang Keywords: Streaming Video, Real-Time Streaming, Adaptive Streaming, Quality of Experience, Video Chunk, Flow-Level Model, Machine Learning. ix Streaming Vidéo, Streaming en Temps Réel, Streaming Adaptatif, Qualité de l First of all, I would like to thank my supervisor Thomas Bonald from Télécom ParisTech for his patience, encouragements and instructions during these three years. Without his helps, I would not be able to complete my thesis. I would also like to thank Dr. Salah Eddine Elayoubi for his Motivation Les objectifs de cette thèse sont la construction de plusieurs modèles de trafic streaming afin d'analyser la mesure objective de la vidéo dans les réseaux sans fils pour différents types de services vidéo. Nous commençons par prendre en compte le service de streaming en temps réel. En supposant que le streaming en temps réel a la plus haute priorité, nous vérifierons la relation entre la charge de trafic et le taux de panne de paquets (PLR). Comme la vidéo à la demande compte pour une plus grande partie du trafic réseau, notre thèse se concentre principalement sur l'étude de ce type de service et en particulier, le streaming adaptatif HTTP, en raison de la maturité de la technologie. La propriété d'adapter le débit binaire vidéo est censée fournir une liberté d'équilibrage entre le débit binaire moyen et la performance du tampon, la fluidité de la vidéo. Cependant, les impacts sur la performance des paramètres ne sont pas clairs, à la fois dans les réseaux sans fils et dans le système de livraison vidéo. Par conséquent, dans cette partie de la recherche, nous nous concentrerons sur les impacts des paramètres suivants du réseau: -Durée du morceau vidéo -Nombre de débits binaires vidéo -Programmes de planification -Mobilité des utilisateurs En appliquant notre modèle de trafic, les opérateurs comprennent comment concevoir correctement les paramètres réseau et vidéo associés pour offrir une meilleure expérience de streaming adaptatif. Dans cette thèse, premièrement nous développerons le modèle de trafic correspondant à l'aide de la dynamique des flux et nous démontrerons ensuite les impacts de performance et proposerons des méthodes de déploiement améliorées. Contributions Dans cette thèse, notre principale contribution est de proposer un modèle analytique basé sur le modèle de niveau de flux pour l'évaluation de la performance vidéo dans différents scénarios. D'autres contributions détaillées serons ensuite précisées: Dans la section 2, nous présenterons quelques connaissances de base pour cette thèse. Dans la section 3, nous contribuerons à développer la distribution des paquets en temps réel des services de streaming en temps réel dans une cellule sans fils. Nous modéliserons une station de base (BS) en appliquant la théorie des files d'attente et en fonction de la propriété quasi-stationnaire, où nous calculerons le délai de paquets en combinant la dynamique des niveaux de paquets et des flux. Dans la section 4, à l'aide d'un modèle de niveau de flux, nous considérons les impacts de la dynamique du trafic sur la performance du streaming adaptatif HTTP. Nous commençons par considérer la durée de fragment vidéo de façon significativement courte. Ensuite, nous étendons notre modèle de trafic au niveau de flux avec la configuration d'une durée de morceau vidéo considérablement importante. Les modèles respectifs représentent une performance extrême liée à toutes les configurations de durées intermédiaires. Ces impacts sur la performance ont été observés en calculant les indicateurs clés de performance (KPI) comme étant le débit binaire moyen de la vidéo (video bitrate) pour la résolution vidéo et le taux moyen de déficit, le temps de service moyen et le surplus moyen de tampon pour la fluidité vidéo. Le modèle adaptatif de circulation en continu est également étendu pour intégrer les effets des conditions radioélectriques hétérogènes, des schémas de programmation et la coexistence avec le trafic élastique en général. Pour compléter les travaux, nous prendrons également en compte la mobilité intra-cellulaire dans notre modèle de niveau de flux. Dans la section 5, nos contributions peuvent être divisées en deux parties. L'une consiste à valider notre modèle de trafic proposé pour le streaming adaptatif par simulation. L'autre consiste à examiner les répercussions sur la performance des différentes configurations de systèmes. Dans la section 6, nous étudions la qualité vidéo de l'expérience par une autre approche, où nous appliquons la technique d'apprentissage automatique pour prédire l'une des principales métriques de qualité d'expérience (QoE), l'assèchement du tampon. Nous démontrons la performance de prédiction de différents flux HTTP et montrons que le streaming statique et adaptatif possède la plus haute précision de prédiction. Nous démontrerons également que différents paramètres de réseau ont des significations importantes et différentes pour prédire l'assèchement. En utilisant la technique d'apprentissage automatique, nous pouvons encore comprendre la relation entre les métriques de performance et les données du système lorsqu'aucun modèle mathématique exact n'est disponible. Cela donne aux opérateurs un accès permettant de comprendre en profondeur la QoE des utilisateurs. Contexte Système sans fils La capacité de canal d'une liaison sans fils entre une paire émetteur-récepteur est limitée par des altérations dues à l'environnement, par exemple. L'affaiblissement de canal vu précédemment et par d'autres transmissions simultanées sur la même bande de fréquence voisine ou adjacente qui génèrent une interférence. Nous utilisons toujours le bruit blanc gaussien (AWGN) pour modéliser une liaison sans fils affectée par le bruit thermique, qui est due à l'agitation thermique des électrons dans les dispositifs électroniques. Avec P u , représentant le signal reçu et I u pour l'interférence globale perçue par un utilisateur spécifique u, la qualité du signal est déterminée par le rapport signal sur interférence et bruit (SINR) donné par: S I N R u = f u P u I u + N 0 , Où f u représente l'effet de chute de canal que nous avons mentionné précédemment, habituellement il est décrit par l'Information de l'Etat du Canal (CSI). Une fois que nous avons obtenu le canal d'évanouissement d'un chemin de signal, nous pouvons calculer la capacité de canal théorique point à point en utilisant la formule de Shannon. Et nous pouvons exprimer la capacité d'un canal comme R = W log(1 + S I N R u ), Où W représente la bande passante du système. Sur la base de la valeur R, l'émetteur s'adaptera à un système de codage de modulation (MCS) approprié pour la transmission. Il est intéressant de mentionner que la formule de Shannon nous offre une borne supérieure pour la capacité du canal, ce qui est un résultat optimiste. Modèle trafic Nous présentons ici la modélisation de base du trafic élastique dans le cas des réseaux mobiles: Nous considérons un ensemble arbitraire de classes d'UE indexées par i ∈ pour refléter les différentes conditions radio, R i (c'est-à-dire les emplacements) dans la cellule considérée. En pratique, le taux de transmission dépend de l'environnement radio et varie dans le temps en raison de la mobilité de l'utilisateur. Sauf indication contraire, nous ignorons les effets de décoloration rapide. Par conséquent, la vitesse de crête R i dépend de la position de l'utilisateur dans la cellule. Nous supposons que le taux de transmission est constant pendant le transfert de données à moins que l'utilisateur ait une grande position change. Dans chaque classe, nous supposons que les flux de données arrivent selon un processus de Poisson avec une intensité λ dans la cellule de référence. Chaque flux reste dans le système tant que les données correspondantes n'ont pas été transmises avec succès à l'UE. On suppose que les tailles flux sont indépendantes et distribuées exponentiellement avec des bits σ moyens, bien que tous nos résultats soient sensiblement insensibles à la distribution. L'intensité du trafic est λ×σ en bit/s. Le taux d'arrivée total λ est composé du taux d'arrivée à chaque classe-i, où λ i = λp i et i∈ p i = 1. X (t) représente le nombre d'utilisateur et suivre un processus de Markov dont le taux de transition dépend du schéma d'ordonnancement, dont nous discuterons dans la section 4. Nous supposons ici que Round Robin (RR) Programme d'ordonnancement. Les métriques de performance considérées sont le débit (en bit/s). Soit τ i est la durée moyenne du flux de classe-i. Selon la formule de Little, E(X i ) = λ i τ i et nous avons γ i = σ τ i = λ i σ E(X i ) . ( 1 ) Il s'agit du rapport entre l'intensité du trafic de la classe-i et le nombre moyen d'flux de classe-i. Cette métrique reflète l'expérience de l'utilisateur, en tenant compte à la fois des conditions radio et de la nature aléatoire du trafic, à travers la distribution stationnaire du processus de Markov X (t). Le débit moyen dans la cellule est donné par: γ = σ τ , (2) où τ est la durée moyenne d'flux de la cellule,τ = i∈ p i τ i . On a γ = i∈ p i γ i -1 . (3) C'est la moyenne harmonique pondérée des débits d'flux par classe, avec des poids donnés par les intensités de trafic par classe. L'idée de la moyenne harmonique des débits a été proposée dans [START_REF] Bonald | Wireless downlink data channels: user performance and cell dimensioning[END_REF]. En appliquant le schéma d'ordonnancement RR, la propriété de la balance est vérifiée et le système de file d'attente peut être considéré comme un réseau de Whittle [START_REF]Network Performance Analysis[END_REF]. Avec la définition de charge ρ i = λ i σ R i , ρ = i∈ ρ i = λσ R , ( 4 ) où R = i∈ p i R i -1 . Par conséquent, la distribution stationnaire du nombre d'flux dans la cellule, x est formulé comme π(x ) = (1 -ρ) |x |! i∈ x i ! i∈ ρ x i i , (5) avec |x | = i x i . Apprentissage automatique L'apprentissage automatique est un outil populaire généralement utilisé pour faire des prédictions, des décisions ou une classification basée sur une grande quantité de données. Il est largement appliqué à la reconnaissance des formes et l'intelligence artificielle, par exemple. Il est étroitement lié aux statistiques informatiques. Comme certaines métriques QoE dans notre modèle de trafic peuvent être trop compliquées pour exprimer sous une forme mathématique exacte, nous essayons d'utiliser l'apprentissage automatique pour découvrir la corrélation entre les caractéristiques du réseaux et les résultats de sortie, plus spécifiquement la QoE des utilisateurs. Dans cette section, nous présentons quelques antécédents d'apprentissage automatique utiles pour la section 6. D'une manière générale, il existe deux types d'apprentissage automatique. L'un est l'apprentissage supervisé. L'autre est l'apprentissage non supervisé. Dans cette thèse, nous nous concentrons sur l'apprentissage supervisé. Un problème général d'apprentissage supervisé est formulé comme Figure 2 . En supposant qu'il existe m paires de données d'apprentissage. Pour les données de formation i-th, nous utilisons le vecteur x i ∈ n pour représenter les variables d'entrée, également appelées caractéristiques d'entrée. Ici, n représente le nombre de caractéristiques dans x i . y i est notée comme la variable de sortie ou cible que nous essayons de prédire. Une paire (x i , y i ) est appelée un exemple d'apprentissage dans l'ensemble de données que nous utilisons pour apprendre est appelé ensemble d'entraînement, = {(x i , y i ) : i = 1, • • • , m}. = {x i } i=1,••• ,m désigne l'espace des valeurs d'entrée, et = {y i } i=1,,m l'espace des valeurs de sortie. Pour décrire le problème d'apprentissage supervisé un peu plus formellement, L'objectif est d'apprendre une fonction h : → de sorte que h(x i ) est un bon prédicteur pour la valeur correspondante de y i . Cette fonction, h est appelée une hypothèse. Lorsque la variable cible que nous essayons de prédire est continue, par ex. x i ∈ n et y ∈ m , nous appelons le problème d'apprentissage un problème de régression. Lorsque y peut prendre un petit nombre de valeurs discrètes, par ex. x i ∈ n et y ∈ {1, -1} m , alors le problème est appelé classification problème. Modèle de trafic streaming en temps réel La qualité des services de données élastiques est principalement évalué sur le débit moyen des utilisateurs en tant que métriques de QoS. Comme nous l'avons mentionné avant, de nombreuses métriques de QoE sont proposées pour le streaming en temps réel. Pour étudier la performance du streaming en temps réel, dans les recherche comme [START_REF] Bonald | On performance bounds for the integration of elastic and adaptive streaming flows[END_REF], [START_REF] Blaszczyszyn | Quality of Real-Time Streaming in Wireless Cellular Networks -Stochastic Modeling and Analysis[END_REF] and [START_REF] Karray | Evaluation and comparison of resource allocation strategies for new streaming services in wireless cellular networks[END_REF], les auteurs ont choisi d'autres paramètres appelés taux de blocage de flux ou taux d'indisponibilité en tant que les principales mesures de performance pour le streaming en temps réel. Sur la base de la métrique, dans [START_REF] Borst | Integration of streaming and elastic traffic in wireless networks[END_REF], la performance des utilisateurs élastiques est évaluée avec la présence d'utilisateurs en streaming utilisant un modèle de niveau d'flux. Nous ne considérons ici que ceux liés à la QoS du réseau, le panne ou la perte de paquets, une métrique de QoS importante pour les services en temps réel mentionnés dans [START_REF] Wu | Transporting real-time video over the internet: challenges and approaches[END_REF] et [START_REF] Mugisha | Packet scheduling for VoIP over LTE-A[END_REF]. Différentes applications en temps réel ont des contraintes de délai de paquet différentes. Les paquets avec un délai supérieur à la contrainte de retard sont considérés comme inutiles. Par conséquent, les opérateurs ont besoin d'un bon modèle pour prédire les performances de retard de paquets sous une intensité de trafic donnée afin de déployer une capacité de système appropriée et de concevoir la politique de contrôle d'admission en conséquence. Notre contribution consiste à développer un modèle de trafic pour les utilisateurs de streaming en temps réel en supposant que les utilisateurs en streaming viennent au système de façon indépendante et que les paquets générés par les utilisateurs sont servis avec une durée de service différente en fonction de leurs propres conditions de canal et leur débit binaire de la vidéo choisi. Pour le service de streaming en temps réel, la station de base ne desservira qu'un seul paquet d'utilisateur à la fois, par rapport aux données vocales, la taille du paquet en continu étant toujours suffisante pour occuper tout le RB dans un TTI. En considérant la contrainte de délai de paquets pour différents types de services de streaming, nous calculons dans la section la capacité maximale du système de streaming en temps réel sous la contrainte que 95% de paquets ont un retard inférieur à un retard d'application spécifique D. Nos autres contributions incluent: -Développement d'un modèle de calcul de la capacité des services de streaming en temps réel compte tenu des retards des paquets. -Proposition d'une méthode de calcul plus simple en utilisant le modèle de fluide. -Extension et validation de notre modèle avec des effets de décoloration rapide. -Nous utilisons la dynamique du niveau d'flux pour décrire la dynamique des utilisateurs dans le système. Modèle avec niveaux flux et paquets Afin d'obtenir la charge maximale du système, nous modélisons le système à deux niveaux, niveau d'flux et niveau de paquet comme Fig. 3. Au lieu d'utiliser le processus de Poisson modulé par MMPP [START_REF] Li | Radio Access Network Dimensioning for 3G UMTS[END_REF] étant la processus d'arrivée, nous supposons que la dynamique du niveau d'flux se produit sur une échelle de temps relativement lente par rapport à la dynamique des paquets, une propriété indiquée dans [START_REF]Performance evaluation of multi-rate streaming traffic by quasi-stationary modelling[END_REF]. En réalité, un service de streaming généré par un utilisateur reste dans une échelle de temps de secondes et le temps de service de paquets est toujours dans une échelle de temps de millisecondes. Par conséquent, la performance de retard au niveau du paquet atteindra approximativement une sorte d'état stationnaire entre les changements dans la population du modèle de niveau d'flux. Comme les paquets sont générés périodiquement et chaque utilisateur va générer indépendamment ses paquets avec un intervalle d'arrivée moyenne, nous modélisons le processus d'arrivée des paquets comme un simple processus d'arrivée de Markov. Dynamique au niveau flux Au niveau du flux, on considère d'abord une classe d'utilisateurs qui ont le même canal et nous modélisons le nombre d'utilisateurs de streaming en temps réel utilisant le chaine de Markov en temps continuous avec le taux d'arrivée, λ f = A -1 f , inverse de l'intervalle d'arrivée du flux et du débit de service, µ f = S -1 f , inverse de l'intervalle de service du flux qui sont indépendants du comportement de départ et de départ de l'autre utilisateur. Sur la base de la formule d'Erlang [START_REF]Network Performance Analysis[END_REF], nous savons que la distribution d'état stationnaire avec des utilisateurs infinis et finis peut être exprimée comme π f (n) =          e -ρ f ρ f n n! , when n ∈ [0, ∞] ρ n f n! 1 + ρ f + • • • + ρ f m m! , when n ∈ [0, m] (6) où ρ f = λ f µ f = S f A f représente la charge de niveau flux pour les utilisateurs streaming en temps réel. Dynamique au niveau paquets Sur la base du régime quasi stationnaire, chaque état n, correspondre à un nombre d'utilisateurs au niveau de flux, correspondant à un régime en niveau de paquet. Dans la file d'attente de paquets, on suppose que chaque utilisateur génèrera périodiquement ses paquets de service à intervalle fixe A p et sera desservie par la station de base à intervalle fixe S p . Comme chaque utilisateur va générer ses paquets de diffusion en continu périodiquement et de nombreux utilisateurs de générer les paquets à des moments différents. L'arrivée des paquets est aléatoire et nous l'approchons d'un processus de Poisson, nous utilisons la file d'attente M/D/1 pour modéliser le système de streaming en temps réel au niveau du paquet. A l'état n, nous modélisons le comportement d'arrivée des paquets en tant que processus de Poisson avec le taux d'arrivée: λ p (n) = n A p (7) Nous considérons que tous les utilisateurs appartiennent à la même condition de canal. La vitesse de départ des paquets à l'état n est indépendante de l'état n: µ p (n) = S p -1 . Avec n utilisateurs dans le système, en utilisant les deux équations précédentes, nous définissons la charge de la file d'attente de paquets comme ρ p (n) = nS p A p = nρ p , où ρ p = S p A p [START_REF]Evolved Universal Terrestial Radio Access (E-UTRA) further advancements for E-UTRA physical layer aspects TR 36.814[END_REF] Avec la dérivation détaillée de la fonction CDF, la distribution du temps d'attente est représentée dans l'équation [START_REF]Evolved Universal Terrestial Radio Access (E-UTRA) radio frequency RF system scenarios TR 36.942[END_REF]. P n (T ≤ x) =      0 , ρ p (n) ≥ 1, (1 -nρ p ) x k=0 (nρ p (k -x )) k k! e nρ p (k-x ) , ρ p (n) < 1, où la function x représente le plus grand entier inférieur ou égal à x variable et x = x S p . Parce que cette équation nous donne la distribution du temps d'attente, pour obtenir la distribution du temps de réponse, il suffit de déplacer la distribution par un S p . Dans l'hypothèse d'un système quasi-stationnaire à deux niveaux et basé sur le théorème bayésien, la distribution globale du retard, P(T ≤ x) est le retard moyen de la distribution du retard de chaque état, n. Par conséquent, avec l'équation (6) and [START_REF]Evolved Universal Terrestial Radio Access (E-UTRA) radio frequency RF system scenarios TR 36.942[END_REF], on obtient P(T ≤ x) = n π f (n)P n (T ≤ x). ( 9 ) Comme tout délai de paquet supérieur à une contrainte de délai donnée, D, est inutile pour le service sensible au retard, nous pouvons obtenir le taux de panne de paquets comme γ(D) = P(T > D) [START_REF] Aalto | On the optimal trade-off between srpt and opportunistic scheduling[END_REF] Compte tenu de ρ p , de la tolérance de paquets ε et d'une certaine contrainte de retard D, nous sommes capables de calculer ρ f maximum, charge système, faisant γ(D) = ε. Extension vers des conditions radio hétérogènes Du point de vue du dimensionnement du système, les utilisateurs peuvent utiliser un taux de codec différent et peuvent avoir des conditions de canal différentes. Par conséquent, nous étendons notre modèle à des utilisateurs de classes multiples avec un modèle M/D/1 modifié et un modèle de fluide. Dans le cas de classes multiples, nous modélisons le système avec plusieurs classes d'utilisateurs ayant des temps de distribution de paquets différents. En outre, en raison de la difficulté d'obtenir la forme fermée de M/D/1 avec plusieurs classes et plusieurs fois de service, le modèle de fluide pourrait devenir un bon modèle pour faciliter le calcul. Dynamique au niveau flux Dans la section précédente, nous utilisons la dynamique du niveau de flux pour modéliser le nombre d'utilisateurs dans le système. En supposant qu'il existe K classes d'utilisateurs qui représentent les utilisateurs avec des canaux différentes et chaque classe k ∈ {1, • • • , K} dispose d'un service de streaming en temps réel avec taux d'arrivée Poissonien λ k et taux de départ µ k . Avec les deux paramètres, on dénote la charge du processeur de classe k par ρ k = λ k /µ k . On dénote le nombre d'appels d'une classe donnée demandant par n(t). Streaming à l'instant t et n(t) = (n 1 (t), • • • , n K (t)) désigne le nombre d'flux dans chaque classe. Sur la base de [START_REF]Network Performance Analysis[END_REF], la distribution stationnaire de l'état π(n) décrivant le nombre d'flux de chaque classe est donnée par π(n) = K k=1 e -ρ k ρ k n k n k ! (11) Dynamique au niveau paquets Correspondant à différentes conditions de canal, chaque classe a son temps de service spécifique S = {S 1 , S 2 , • • • , S K }. Comme plus d'une classe d'utilisateurs coexistent dans le système, nous modifions le taux d'indisponibilité M/D/1 dans l'équation [START_REF] Abhayawardhana | Comparison of empirical propagation path loss models for fixed wireless access systems[END_REF] avec ρ f = (ρ 1 , • • • , ρ K ). γ MD1,m = n π(n)P n (T ≤ D, ρ p ) (12) En considérant plus d'un temps de service, le CDF de délai des paquets peut être calculé par le résultat numérique de la transformée de Laplace inverse obtenue dans l'équation (14), qui est également le modèle M/G/1 montré dans [START_REF] Kleinrock | Queueing Systems: Theory, ser[END_REF] avec multiple temps de service discret, S k et la probabilité correspondante à n k /n. Pn (s) = P n (T ≤ x) = ρ -1 λ -s -λB(s) (13) Où la fonction B(s) est exprimée comme B(s) = K k=1 n k n e -sS k (14) et l'autres variables comme ρ =λ k n k n S k = k n k S k A p , (15) λ = k n k A p , n = k n k . ( 16 ) Résultats de simulations Dans cette section, nous présentons les performances du modèle M/D/1 et du modèle de fluide avec différents services correspondant à différentes configurations de contraintes de paquets délai. Basé sur [START_REF] Perkins | RTP: Audio and Video for the Internet[END_REF], le délai tolérant humain pour le service interactif tel que la vidéo conférence est environ 150ms. Nous configurons les contraintes de délai en tant que 500ms pour le streaming TV en direct. Nous montrons que le modèle de fluide peut être utilisé pour simplifier le streaming TV en direct et qu'il vaut mieux rester avec le modèle M/D/1 dans le dimensionnement de la vidéo conférence. Validation du modèle avec single classe Dans le tableau.1, nous supposons que le temps d'arrivée du flux moyen est S f = 10s qui est cent fois plus grand que le temps moyen d'arrivée des paquets A p = 100ms. Basé sur la distribution du SINR obtenue par [START_REF] Blaszczyszyn | Quality of Real-Time Streaming in Wireless Cellular Networks -Stochastic Modeling and Analysis[END_REF] et sur la configuration de la spécification 3GPP [START_REF]Evolved Universal Terrestial Radio Access (E-UTRA) radio frequency RF system scenarios TR 36.942[END_REF][8], le débit moyen LTE est calculé comme τ = 9.4Mbps. Avec différents codecs, différents S p s'appliquent. Les paramètres sont affichés dans le tableau.1, avec c désignant le codec choisi. S p = c × A p τ ( 17 ) On peut observer que S f , A f >> S p , A p , qui suit le régime quasi-stationnaire nous supposons. Paramètres Symboles Valeur Temps moyen d'arrivé d'un flux (s) A f [4,[START_REF] Balachandran | Developing a predictive model of quality of experience for internet video[END_REF] Temps Validation du modèle avec multiple classes Pour valider l'extension de nos modèles au scénario de plusieurs classes, nous prenons un exemple d'utilisateurs avec deux classes, S = {S c , S e }, représentant respectivement les utilisateurs de bordures des cellules et de centres des cellules. Dans la validation, nous supposons que les utilisateurs utilisent le codec avec un taux de codage de 512kbps. Sur la base de la même distribution SINR et de l'équation [START_REF] Andrew | CS229 lecture notes: Part V Support Vector Machines[END_REF], nous avons calculé le débit moyen et le temps de service d'un paquet en tant que S c = 3.5ms calculé par τ c = 14.63Mbps pour les utilisateurs du cell centre et S e = 13.73ms calculé par τ e = 3.73Mbps pour les utilisateurs de bord de la cellule. Modèle de trafic streaming adaptatif Dans la section précédente, nous avons présenté un modèle de trafic streaming en temps réel. Comme nous savons que la vidéo à la demande compte pour une plus grande proportion de trafic que le streaming en temps réel et le fait que les services comme YouTube et Netflix [4] devient très populaire, un modèle de trafic pour analyser l'impact de la configuration différente de streaming d'HTTP est nécessaire, en particulier Streaming adaptatif d'HTTP (HAS) devient une solution technique mature et populaire selon [START_REF] Mok | QDASH: a QoE-aware DASH system[END_REF][69] [START_REF] Cicco | An experimental investigation of the akamai adaptive video streaming[END_REF][15] [START_REF] De Cicco | Feedback control for adaptive live video streaming[END_REF]. Comme nous l'avons mentionné dans la section introduction, la plus grande différence entre le streaming en temps réel et le streaming d'HTTP est l'existence d'un tampon de lecture vidéo. De plus, TCP est le protocole de couche de transport utilisé pour le streaming d'HTTP. Dans cette section, nous développons un modèle de trafic général qui vise à aider les opérateurs à évaluer la qualité de service perçue par leurs utilisateurs et à dimensionner correctement leurs réseaux. Nous appliquons le modèle de niveau flux bien connu, où un flux peut représenter une session de streaming vidéo ou une session élastique. Il est également appliqué pour le streaming en temps réel. Par exemple, les auteurs de [START_REF] Bonald | On performance bounds for the integration of elastic and adaptive streaming flows[END_REF] ont étudié l'intégration des services élastiques et de streaming en modélisant le streaming adaptatif en temps réel en tant que flux et en fournissant uniquement la performance liée car la propriété d'insensibilité ne tient pas. Toutefois, les services de streaming considérés sont modélisés comme un type spécifique de diffusion en continu, en temps réel adaptatif. Par rapport à la modélisation en temps réel en streaming, la modélisation pour le streaming HTTP est encore à ses débuts. La plupart des travaux existants se concentrent sur l'évaluation du streaming HTTP. Il n'existe pas de modèle de trafic mature pour le streaming adaptatif HTTP et ses compromis de performances. Les modèles de niveau flux mentionnés ci-dessus se sont concentrés sur les services de streaming élastiques et en temps réel classiques et ne tiennent pas compte des impacts du tampon. Le modèle de niveau de flux a été appliqué dans des études de streaming HTTP [START_REF] Xu | Impact of Flow-level Dynamics on QoE of Video Streaming in Wireless Networks[END_REF] [99] avec des tampons vidéo infinis. Les KPI comme la probabilité d'assèchement du tampon ont été calculés en utilisant une analyse tampon détaillée. Cependant, le modèle mentionné n'est pas adapté pour être adapté pour évaluer la performance du streaming adaptatif HTTP en raison du manque de considération pour l'adaptabilité de débit. Les travaux de HAS incluent [START_REF] Ye | Analysis and modelling quality of experience of video streaming under time-varying bandwidth[END_REF], où les auteurs proposent un cadre analytique pour le streaming adaptatif HTTP sous l'hypothèse d'une fréquence d'arrivée de cadres fixes pour différents débits binaires vidéo et [START_REF] Xu | Analytical QoE models for bit-rate switching in dynamic adaptive streaming systems[END_REF]. Comme l'arrivée de fluide modulée par Markov. Les deux ne tiennent pas compte des répercussions de l'autre trafic et des charges globales du système. Pour d'autres études, il est préférable de combiner les impacts de l'autre trafic au débit d'arrivée des paquets, ce qui est la partie la plus difficile.Comment allouer des ressources pour le streaming et les services élastiques devient une question pour les opérateurs. Il est bien connu que la capacité du système sans fils peut être améliorée avec la diversité multi-utilisateurs en utilisant des ordonnanceurs opportunistes de [START_REF] Tse | Multiuser diversity in wireless networks[END_REF] [10] [START_REF] Ayesta | A modeling framework for optimizing the flow-level scheduling with time-varying channels[END_REF]. Toutefois, comme le service de diffusion vidéo en continu comme YouTube et Netflix représente la plus grande partie du trafic système, s'il existe un modèle de flux adaptatif HTTP, il facilitera aux opérateurs de comprendre si ces suggestions sont toujours valables pour les services de diffusion en continu. Organization On commence par introduire le contexte qui explique comment un vidéo était transmis par le système cellulaire et des paramètres systèmes. Et puis on introduit les modèles de trafic de streaming adaptatif d'HTTP avec une configuration de très petit segment de vidéo et très large segment de vidéo. Nous formulerons les KPIs pour mesurer la qualité d'espérance par rapport à la définition de vidéo et la fluidité de la vidéo aussi. Mais les extensions de modèle considérant plusieurs conditions radio, différent ordonnance, différent mobilité et l'intégration des service élastiques ne sont pas démontrés dans la synthèse française mais que dans la thèse anglais. Description du système Cette section présente deux aspects clés qui influent sur la performance des services de streaming adaptatif HTTP fournis dans les réseaux sans fils: la configuration du contenu vidéo et le réseau d'accès sans fils. Selon le mécanisme de streaming adaptatif HTTP que nous avons introduit dans la première section, une vidéo est séparée par plusieurs segments vidéo (segments) et ils sont demandés les uns après les autres par des requêtes d'HTTP. Le débit binaire correspondant est sélectionné au début de chaque téléchargement. Figure . 8 donne un exemple montrant comment un téléchargement vidéo est composé par des tas de morceaux. La durée du bloc, h, est un paramètre système que les fournisseurs de services peuvent contrôler. Configuration de contenu vidéo Intuitivement parlant, en choisissant une durée plus courte, les utilisateurs ont plus de chances d'adapter son débit binaire vidéo. Dans ce section, nous allons étudier les résultats analytiques de deux configurations de durées de blocs extrêmes illustrées dans le tableau 2 suivant. Configurations du segment vidéo Réseau d'accès sans fils Dans cette section, nous nous concentrons sur la performance du streaming adaptatif livré dans une cellule typique comme Fig. 9. Les utilisateurs mobiles de la cellule téléchargent le trafic en flux continu vers leur tampon par les ressources allouées de la cellule. Nous commençons, pour la facilité de compréhension du modèle, par une condition radio homogène où tous les utilisateurs sont supposés voir une capacité R obtenu par utiliser l'état radio moyen sur la cellule. En suite, nous montreront comment étendre ces modèles à de multiples conditions radio et comment intégrer d'autres services comme le trafic élastique. Modèle de petite vidéo segments Pour modéliser la dynamique du nombre de flux, X (t), nous utilisons le modèle de file d'attente avec la propriété, partage de processeur. Avec l'hypothèse d'un état radio homogène et d'une petite durée de blocage, nous modélisons tous les flux dans une classe. Comme X (t) = x, ce qui signifie que x flux sont servis dans le système. Et le taux de départs, µ(x), peut être exprimé comme µ(x) = φ(x) σ(x) , (18) Où φ(x) représente le débit physique alloué à tous les UEs de la cellule à l'état x et σ(x) représente la taille de flux restante à l'état x. Comme nous considérons une taille de tampon infinie du côté des utilisateurs, les utilisateurs peuvent profiter au maximum du débit qui leur est alloué, occupant ainsi le temps de programmation global de la cellule, conduisant à une allocation φ(x) = R dans ce cas. A partir de la taille de flux restante, σ(x), étant donné que nous considérons une petite longueur de bloc conduisant à une adaptation instantanée, le débit binaire vidéo v(x) à l'état x dépend seulement du nombre des flux x et non sur l'historique du système. En outre, comme la durée vidéo est supposée d'être exponentielle, la propriété sans mémoire implique que la taille du flux à l'état x est également exponentiellement distribuée avec sa moyenne σ(x) = v(x)T. ( 19 ) X (t) est donc un processus de Markov dont les vitesses de départ dépendent de la sélection du débit binaire vidéo. Nous montrons dans les sections suivantes comment ce débit est calculé. La vitesse de départ du flux est obtenue comme µ(x) = φ(x) v(x) . Et le système peut être facilement montré avoir une distribution stationnaire de produit-forme calculée comme π(x) = π(0) x n=1 λ µ(n) , (20) Où π(0) = 1 + ∞ x=0 x n=1 λ µ(n) -1 . Sélection du débit binaire vidéo Le débit vidéo choisi par chaque flux est déterminé en fonction du débit instantané, γ(x), que l'utilisateurs observent. Dans le modèle de niveau flux, en appliquant la politique d'ordonnancement round-robin, le débit instantané peut être calculé comme γ(x) = R x . (21) En réalité, les débits binaires vidéo ne sont pas continus. Au lieu de cela, les flux sélectionnent un débit vidéo spécifique à partir d'un ensemble de débits binaires discrets = {v 1 , • • • , v I }, où nous supposons v 1 > • • • > v I . Connaissant le débit, les utilisateurs sélectionnent le débit binaire v(x) = γ(x) , when γ(x) > v I , v I , when γ(x) ≤ v I , (22) Où z représente une fonction qui sélectionne une valeur maximale dans mais inférieure à z. On peut également observer que γ(x) est toujours égal à v(x) ou supérieur à v(x) seulement lorsque γ(x) < v I . Avec le mécanisme de sélection de débit vidéo défini, lorsque γ(x) ≥ v I , le tampon vidéo des utilisateurs augmente ou reste le même. Au lieu de cela, le tampon vidéo ne diminuera que lorsque γ(x) < v I . Modèle de grande vidéo segments Avec la même caractéristique système mentionnée dans la section 1, nous considérons ici le système de streaming avec une durée de blocs infiniment grande, où les flux choisissent leur débit binaire vidéo au début de leur arrivée et gardent celui-ci jusqu'à la fin du téléchargement. Différent du cas de la petite quantité de morceau, où nous modélisons le système avec une classe d'utilisateur qui choisissent le même débit binaire vidéo en même temps. Pour la configuration d'une durée de blocs infinie, plusieurs classes de files d'attente sont nécessaires pour décrire le nombre de débits qui choisissent des débits binaires vidéo différents à un moment donné. Comme dans la section précédente, l'ensemble discret des débits binaires vidéo est noté = {v ∀i, λ i (x ) = λ, if v(x + e i ) = v i , 0, otherwise, (23) Où v(x ) est définie comme Eq. ( 22) et e i représente un vecteur avec une valeur unitaire à la classe i. Appliquer le même concept de l'équation [START_REF] Arsan | Review of bandwidth estimation tools and application to bandwidth adaptive video streaming[END_REF], le débit de sortie du flux et la ressource allouée de la classe i est noté µ i (x ) = φ i (x ) v i T , ( 24 ) φ i (x ) = x i R |x | , (25) Avec le même réglage de Round-Robin ordonnancement. On peut alors calculer la distribution stationnaire π(x ) en utilisant le taux d'arrivée et de départ des deux cas dans les équations (23 -24) et en résolvant les équations d'équilibre notées ∀x , i λ i (x ) + µ i (x ) π(x ) (26) = i λ i (x -e i )π(x -e i ) + µ i (x + e i )π(x + e i ). Avec la distribution stationnaire, π(x ) calculée dans le cas d'une petite quantité de morceaux et d'une grande quantité de morceaux, nous définissons ensuite les principaux indicateurs de performance dans la section suivante. Condition stabilité Le débit d'arrivée du flux doit être inférieur au débit de départ maximum. Dans le cas où v n est la sélection du débit binaire vidéo de n-ième flux, la vitesse de départ du débit maximal implique que ∀n, v n = v I , conduisant à la condition de stabilité suivante pour les deux configurations de durée de fragment et taux d'arrivée maximum, λ max : λ < R v I T ⇒ λ max = R v I T . ( 27 ) Définition des KPIs Pour évaluer le QoE du service de streaming adaptatif, nous proposons quatre indicateurs de performance clés, le débit binaire moyen, le temps de service moyen, le taux de déficit et le surplus de tampon. Tous sont définis sur la base de la distribution stationnaire π(x ). Débit binaire de la vidéo Le débit binaire moyen correspond au débit binaire moyen d'un flux lors de la lecture de la vidéo. Lorsque le débit vidéo moyen est élevé, l'utilisateur a une meilleure expérience vidéo. Nous calculons le débit binaire moyen de la cellule en utilisant le concept suivant, Débit binaire de la vidéo = Resources allouées #Flux servi , [START_REF] Bonald | Inter-cell coordination in wireless data networks[END_REF] où nous divisons toute les ressources allouées sur le nombre de flux multiplié la durée moyenne de la vidéo pour calculer le débit binaire moyen. Ensuite, nous définissons le débit binaire global moyen, v pour une durée de fragment petite et grande, v = x:x>0 π(x)φ(x) λT , v = x :|x |>0 π(x ) i φ i (x ) λT , (29) où |x | = i x i . Un indicateur populaire QoE utilisé pour évaluer la performance en streaming est la probabilité d'assèchement du tampon [START_REF] Xu | Impact of Flow-level Dynamics on QoE of Video Streaming in Wireless Networks[END_REF]. Même si la pause de la vidéo ne se produit que lorsque le débit binaire vidéo est plus grand que le débit instantané, cette dernière condition n'est pas une condition suffisante pour la pause de la vidéo car le tampon peut contrecarrer l'impact de courtes périodes de faible débit. Le calcul de la probabilité de d'assèchement du tampon doit prendre en compte la mémoire du système en introduisant la taille du tampon dans l'analyse markovienne comme dans [START_REF] Xu | Impact of Flow-level Dynamics on QoE of Video Streaming in Wireless Networks[END_REF]. Ici, nous introduisons et examinons trois KPIs appelé temps de service, taux de déficit et excédent de tampon. Temps de service Pour évaluer la probabilité d'assèchement, nous proposons le première KPI, temps de service moyen d'un flux vidéo, qui est calculé par la formule de Little comme S = L λ = x λ , S = L λ = x λ , (30) Taux de déficit Comme [START_REF]Network Performance Analysis[END_REF] mentionné, les flux ont une probabilité plus élevée de rester dans l'état pour le téléchargement v(x) parce que x utilisateurs existent. Par conséquent, en pondérant les métriques correspondantes à l'état x par le nombre de flux, x, également appelé distribution de biais de taille, nous définissons les paramètres suivants. Le taux de déficit est égal à la probabilité qu'un flux voit son débit instantané est inférieur à son débit binaire vidéo choisi. Comme on suppose que l'adaptation du débit binaire vidéo se produit instantanément en réaction aux variations du débit observé, le taux de déficit est défini par la probabilité que le débit instantané, γ(x) = φ(x) x , (31) est inférieur à v(x) dans le cas d'une petite durée de fragment vidéo ou γ i (x) = φ i (x) i x i , (32) Est inférieur à v i dans le cas d'une longue durée de fragment vidéo. Notez que pour la configuration de la petite taille de la section, le déficit ne se produit que lorsque γ(x) < v I , basé sur le mécanisme de sélection, Eq. ( 22). Le taux de déficit global est défini en pondérant la distribution stationnaire à différents états x avec le nombre de flux: D = Pr{γ(x) < v(x)} = x:x>0 xπ(x) x 1 {γ(x)<v(x)} , (33) Où 1 représente la fonction indicateur. Pour le cas de configuration de la durée de gros morceau vidéo, D = Pr{γ i (x ) < v i (x )} = x :|x |>0 π(x ) x i x i 1 {γ i (x )<v i } . ( 34 ) La probabilité de vacuité est aussi positivement liée au taux de déficit. Par conséquent, un taux de déficit plus important entraînera une plus grande probabilité de vacuité. Surplus de tampon Nous introduisons également une autre métrique de performance appelée excédent de tampon, qui représente la variation moyenne de tampon de chaque débit en une seconde. Il est calculé en pondérant toute la variation du tampon γ(x)-v(x) v(x) , à chaque état x as B = x:x>0 xπ(x) x γ(x) -v(x) v(x) , (35) Pour la petite configuration de durée de morceau vidéo. Lorsque γ(x) > v(x), le tampon des utilisateurs accumule une certaine durée de la vidéo. Lorsque γ(x) < v(x), l'utilisateur commence à consommer les paquets vidéo stockés dans la mémoire tampon, ce qui réduit les valeurs de la mémoire tampon moyenne excédentaire. Pour le cas d'une longue durée, le surplus de tampon est calculé comme B = x :|x |>0 π(x ) x i x i γ i (x ) -v i v i . ( 36 ) Un plus grand surplus de tampon diminuera la probabilité de vacuité. Par conséquent, il est négativement lié à la probabilité de vacuité. Simulation de la performance streaming adaptatif Organization Après avoir présenté le modèle de flux adaptatif basé sur HTTP dans la section précédente, nous commencerons par montrer les impacts de différentes configurations de système soit dans le système de distribution vidéo soit dans les réseaux d'accès sans fils, y compris la durée du segment vidéo, le nombre de débits binaires vidéo disponibles et différent ordonnancements. Nous démontrons également l'impact sur le rendement de la mobilité des utilisateurs. Nous ne présentons pas le modèle d'approximation qui peut réduire la complexité du calcul ici dans la synthèse française. Dans la partie anglais, nous appliquons le modèle d'approximatif pour illustrer comment utiliser notre modèle pour le dimensionnement du réseau et étudier les impacts de la limitation du débit binaire vidéo sur la performance en streaming. Impacts de la durée du segments vidéo Selon le rapport technique d'Akamai [1] et la démonstration empirique de [START_REF] Yao | Empirical evaluation of http adaptive streaming under vehicular mobility[END_REF], ils sont démontré que le déploiement d'une durée de segment plus courte offre une meilleure fluidité de la vidéo et plus de chances pour les utilisateurs de sélectionner la bonne débit binaire vidéo. Cependant, le déploiement de segments plus courts générera également un grand nombre de segments vidéo et sa signalisation d'HTTP correspondante. Par conséquent, le rapport suggère que les fournisseurs de services vidéo choisissent leurs configurations de durée de segment d'HTTP adaptatif en continu en 10 secondes. Dans cette section, nous effectuons d'abord des simulations pour vérifier ces résultats en définissant les paramètres comme = {R C , R E } = {10, 4}Mbps, (p C , p E ) = ( 1 2 , 1 2 ) pour les utilisateurs de cell centre et les utilisateurs au bord du cellule. (p e , p s ) = ( 12 , 1 2 ) représente les taux d'arrivés des flux élastiques et streaming. = {v 1 , v 2 } = {2, 0.5}Mbps, T = 10s, σ = 5Mbits et λ max = 1.14 flux/s. Les résultats de simulation dans les figures suivantes montrent la performance de toutes les métriques définies par rapport aux paramètres normalisés de la charge de trafic, (λ/λ max ). Comme la section 4.1 mentionné, les résultats de simulation montrés dans la figure 10 nous donnent deux limites de performance des deux cas extrêmes, h = 0 and h = ∞. Ce Impacts du nombre de débits binaires vidéo Dans la simulation précédente, l'ensemble débit binaire vidéo est configuré comme = {v 1 , v I }. Cependant, une question générale est de savoir quels sont les impacts si plus de débits binaires vidéo sont disponibles pour être choisis dans ? Pour répondre, nous configurons un continu ensemble de débit binaire vidéo cont = {v : v 1 ≤ v ≤ v I } avec la configuration de petite durée de morceau vidéo. Dans ce cas, | cont | = ∞. Différent de l'équation [START_REF] Bolch | Queueing Networks and Markov Chains[END_REF], les flux des utilisateurs à la région de capacité k sont supposés pouvoir sélectionner n'importe quel débit binaire vidéo entre v 1 et v I , exprimé en v k (x ) = max min γ k (x ), v 1 , v I , (37) où γ k (x ) est défini comme l'équation [START_REF] Bonald | On the stability of flow-aware CSMA[END_REF] et [START_REF]Network Performance Analysis[END_REF]. Basé sur la définition de taux de départ, le taux de départ du flux devient |x | est supérieur à v 1 , le tampon commence à remplir sur la mesure que la capacité de téléchargement est plus grande que la vitesse de lecture. Le flux se comporte donc comme un flux élastique car il n'y a pas de limitation sur la taille du tampon. D'autre part, lorsque γ k (x ) est inférieur à v I , le comportement est aussi comme des flux élastique mais avec une faim possible si le tampon se vide en raison du taux de diffusion constant. Entre v 1 et v I , le flux se comporte comme un temps réel et la taille du tampon reste constante. Résoudre les équations du bilan comme Eq. ( 20), on obtient les résultats de simulation représentés sur la Fig. 12 avec trois configurations différentes dont 2 débits binaires vidéo, un nombre infini de débits binaires vidéo et un réglage intermédiaire, 3 débits binaires vidéo. On peut observer que les mesures de performance de 3 débits binaires vidéo sont situées entre celles des deux cas extrêmes. µ k (x ) = φ k (x ) v k (x )T = R max min γ k (x), v 1 , v I T . ( 38 ) De la Fig. 12a, nous observons que le déploiement de débit vidéo continu, ce qui signifie plus de débits binaires vidéo pour approcher le débit instantané réel des utilisateurs, augmentera la résolution vidéo à long terme mais diminuera le débit élastique moyen. De plus, du reste des figures, nous découvrons que le déploiement de plus de débits binaires vidéo augmentera également la faim des tampons à partir des trois mesures liées au buffer. Par conséquent, le déploiement d'un plus grand nombre de débits binaires vidéo générera un compromis qui peut bénéficier à la résolution vidéo moyenne mais diminue le débit élastique et la lisibilité vidéo. Résumé Dans cette section, les résultats de la simulation montrent que la configuration d'une grande longueur de bloc n'améliore pas la résolution vidéo mais diminue largement la lisibilité de la lecture et que la configuration d'un plus grand nombre de débits vidéo fournira également une meilleure résolution vidéo mais dégradera la fluidité vidéo. Dans la synthèse française nous avons montré l'impacts de la durée de segments vidéo et le nombre de débit binaire vidéo. Dans la thèse anglais nous montrons également le compromis de performance de différents schémas d'ordonnancement avec et sans tenir compte de la mobilité des utilisateurs. Prediction de la QoE de streaming video avec technique apprentissage automatique 6.1 Organization Dans les sections précédents, nous avons étudié les performances du streaming en temps réel et du streaming d'HTTP. Pour mesurer le QoE du streaming basé sur HTTP, les métriques, la probabilité de assèchement du tampon ou les événements de tampon sont adoptés populairement. Assèchement du tampon ou les événements de tampon se produisent lorsque le tampon vidéo devient vide et les utilisateurs rencontrent une pause vidéo. Comme il n'est pas facile de développer une forme mathématique et analytique pour les métriques de QoE, en particulier pour le subjectif QoE, des recherches comme [START_REF] Singh | Quality of experience estimation for adaptive http/tcp video streaming using h.264/avc[END_REF] appliquent une analyse statistique pour comprendre la corrélation entre QoE et réseau QoS. Dans cette section, nous nous appuyons sur un simulateur qui génère une grande quantité de paires de données (QoE + caractéristiques réseau) et nous démontrons l'efficacité de la prévision de la QoE, la vacuité de la vidéo, en utilisant les caractéristiques du réseau d'entrée telles que CSI, Le nombre de flux d'utilisateurs et la durée de la vidéo, enregistrés à l'arrivée de chaque utilisateur. La probabilité de vacuité est étudiée en tant que QoE de service vidéo dans de nombreux travaux. Il est modélisé et calculé analytiquement dans [START_REF] Xu | Impact of Flow-level Dynamics on QoE of Video Streaming in Wireless Networks[END_REF]. Cependant, plusieurs contraintes sont nécessaires lors de l'application de ce modèle, par exemple, un débit binaire fixe est requis et le modèle ne peut prendre en compte qu'une seule condition de canal. Dans [START_REF] Xu | Analytical QoE models for bit-rate switching in dynamic adaptive streaming systems[END_REF], la forme analytique de la probabilité de vacuité du streaming adaptatif est proposée sans tenir compte de la dynamique du flux. Nous avons utilisé un modèle de niveau d'flux pour étudier les métriques de performance vidéo dans [START_REF] Bonald | A flow-level performance model for mobile networks carrying adaptive streaming traffic[END_REF] et [START_REF] Lin | Impact of chunk duration on adaptive streaming performance in mobile networks[END_REF]. Cependant, la relation entre la faim vidéo et les mesures de performance proposées ne sont pas claires. L'apprentissage par machine a été largement utilisé pour étudier à la fois subjective et objective QoE. Notre contribution est de démontrer la corrélation entre la faim vidéo de différemment streaming et les fonctionnalités des utilisateurs enregistrés. Nous analysons également l'importance des fonctionnalités de ces utilisateurs pour la prédiction des événements de privation de la vidéo, y compris le nombre d'utilisateurs vidéo existant dans une cellule, les conditions de canal de l'utilisateur vidéo et la durée de la vidéo enregistrée lorsqu'un flux lance son téléchargement vidéo. Description du système Dans cette section, nous présentons d'abord deux types de flux vidéo. Ensuite, nous introduisons le modèle de niveau d'flux utilisé pour calculer la charge maximale du système. Un simulateur piloté par événement est présenté pour générer des données de la vacuité vidéo pour différentes charges. Streaming avec un débit binaire vidéo fixé Les utilisateurs fixent un débit binaire vidéo, v c , depuis le début jusqu'à la fin du téléchargement vidéo, ce qui est la plus simple implémentation. Streaming adaptatif Comme la Fig. 6.1, les services adaptatifs de streaming permettent aux utilisateurs de s'adapter en temps réel à leur débit vidéo. Après avoir fini de télécharger un morceau vidéo avec une durée h, basé sur le débit mesuré, γ j , les utilisateurs peuvent sélectionner un débit binaire vidéo pour le bloc suivant à partir d' ensemble discret = {v 1 , • • • , v I }, où v 1 > • • • > v I . Le débit vidéo sélectionné est donné v j = γ j , when γ j ≥ v I , v I , when γ j ≤ v I , (39) où y est une notation abrégée pour le plus grand débit binaire vidéo en mais pas supérieur à y et γ j représente le débit instantané de l'utilisateur j. Après le téléchargement, les morceaux vidéo seront stockés dans un tampon de lecture. Comme les sections précédentes, nous supposons encore que la mémoire tampon de lecture des utilisateurs est infinie. Une fois qu'un utilisateur entre dans la cellule, il sera programmé une quantité de ressource jusqu'à la fin du téléchargement de la vidéo. Réseau d'accès radio et caractéristiques du trafic Nous considérons un réseau d'accès radio où les utilisateurs ont un débit physique différent en fonction de son emplacement dans la cellule. Nous considérons l'ensemble du débit physique comme -Type I: streaming statique et adaptatif. = {R 1 , • • • , R K } et -Type II: streaming mobile et adaptatif. -Type III: streaming statique et avec fixe débit binaire vidéo. -Type IV: streaming mobile et débit fixe. Dans le réseau réel, le taux d'arrivée du trafic, λ, varie le long des heures, généralement plus élevé en jour et plus bas la nuit. Dans les résultats de simulation suivants, les données de privation de la vidéo sont générées avec des taux d'arrivée de trafic différents. La prévision montrée à chaque charge de trafic correspond à la performance potentielle à chaque heure. Modèle de flux et taux d'arrivée maximal Le concept de modèle de niveau flux a été utilisé pour obtenir la performance du streaming dans le papier [START_REF] Bonald | A flow-level performance model for mobile networks carrying adaptive streaming traffic[END_REF] et [START_REF] Lin | Impact of chunk duration on adaptive streaming performance in mobile networks[END_REF]. Basé sur ce modèle, nous pouvons obtenir le débit maximum d'arrivée de flux qui garantit la stabilité de système. Soit x (t) = (x 1 1 (t), • • • , x i K (t)) le nombre de flux en continu à l'instant t et x i k (t) répresente le nombre de type-i streaming avec R k à l'instant t. Basé sur la méthode d'ordonnancement à tour de rôle, nous avons le débit instantané calculé comme γ i k (x ) = R k |x | , ( 40 ) où |x | = i k x i k est le nombre total de flux en cours. Le débit binaire vidéo du segment vidéo suivant est sélectionné sur la base de l'équation [START_REF] Bolch | Queueing Networks and Markov Chains[END_REF] et [START_REF] Cha | I tube, you tube, everybody tubes: Analyzing the world's largest user generated content video system[END_REF]. Lorsque la charge se rapproche de la capacité du système et des caractéristiques de trafic mentionnées, tous les flux adaptatifs sont forcés de s'adapter à v I , le débit maximum d'arrivée du système peut être calculé en traitant tous les flux comme un trafic élastique avec le volume v T as λ max = w 1 v I T R s + w 2 v I T R m + w 3 v c T R s + w 4 v c T R m -1 . ( 41 ) R s = k p k R k -1 représente le débit radio équivalent pour les utilisateurs statiques. Comme l'indique le travail [START_REF] Nivine Abbas | Mobility-driven scheduler for mobile networks carrying adaptive streaming traffic[END_REF], R m = k q k R k représente le débit radio équivalent pour les utilisateurs mobiles avec q k désigné comme la proportion de temps que les utilisateurs restent avec le débit R k . Simulateur événementiel Dans [START_REF] Bonald | A flow-level performance model for mobile networks carrying adaptive streaming traffic[END_REF] et [START_REF] Lin | Impact of chunk duration on adaptive streaming performance in mobile networks[END_REF], par modèle mathématique de niveau flux, nous ne pouvons obtenir que des métriques objectives de QoS au lieu des informations de tampon réel de j-th utilisateur, b j (t). Par conséquent, nous implémentons un simulateur piloté par événement qui est capable de simuler l'événement de la privation vidéo et d'enregistrer la valeur de la mémoire tampon de tous les utilisateurs. Notre simulateur est implémenté en fonction du concept de niveau d'flux, où chaque session vidéo est considérée comme un flux et chacun d'entre eux peut rencontrer les événements suivants: • Événement d'arrivée: L'arrivée de flux de flux suit la distribution de Poisson. Comme l'utilisateur j arrive à la cellule au moment E a j , plusieurs caractéristiques observées, z j , sont enregistrées et utilisées comme données d'entrée pour prédire la vacuité, y j ∈ {1, -1}. • Événement de départ: Les utilisateurs rencontrent un événement de départ au moment E d j lorsque la vidéo demandée est complètement téléchargée. • Événement Chunk: Les utilisateurs rencontrent un événement chunk au moment E c j lorsque la partie vidéo demandée avec la durée h est téléchargée. • Événement Tampon: Nous classons l'utilisateur de streaming simulé j en trois états, PREFETCH, PLAY et STARVATION(assèchement du tampon), où chacun a un taux de variation de tampon d b j (t) d t = γ j , PREFETCH ou STARVATION, γ j -1, PLAY. Lorsque l'utilisateur commence à demander une vidéo, il restera à PREFETCH et passera à STARVATION jusqu'à b j (t) ≥ B, où B est le tampon initial. Une fois b j (t) = 0, l'utilisateur entre dans STARVATION, où il est reconnu comme un utilisateur connaissant une vacuité de la vidéo, y j = 1, et il attendra que b j (t) ≥ B pour entrer à nouveau PLAY. E b j est le temps des événements de tampon pour l'utilisateur j. • Événement de mobilité: Les utilisateurs avec mobilité changeront le R j pour le débit adjacent lorsque l'événement de mobilité au moment E m j se produit. Dans ce cas, les utilisateurs planifient l'événement de mobilité suivant en fonction de la distribution exponentielle avec le taux ν j . Caractéristiques enregistrées Nous avons mentionné que z j est enregistré à l'arrivée de j-th utilisateur et va être utilisé pour prédire y j . Ici, nous présentons les composants de la fonction de l'utilisateur, z j : R j Condition radio(R) Il est enregistré au début du téléchargement de la vidéo. Si l'utilisateur est statique, R j est fixe. T j Durée vidéo (T ) Il suit une distribution exponentielle. x j Nombre de flux (N ) x j = (x 1 , • • • , x K ) j représente Outil d'apprentissage automatique Comme on a déjà introduit les outils d'apprentissage automatique dans la section 2.3, ici nous allons seulement présenter les indicateurs des performances et les libraries relatives. Indicateurs des performances Afin de vérifier les performances d'apprentissage de la machine, nous définissons les mesures de performance suivantes pour examiner les performances de prédiction parmi les données de test. Cette probabilité représente la précision moyenne de prédiction parmi tous les échantillons testés. P = P { y j =-1} P { ŷ(z j )=-1| y j =-1} + P { y j =1} P { ŷ(z j )=1| y j =1} . Libraries Pour l 'analyse GLM, nous avons utilisé le paquetage R de stats, qui a basé son algorithme sur le GLM proposé par [START_REF] Nelder | Generalized linear models[END_REF]. Pour l'analyse numérique de ce travail, nous appliquons l'une des bibliothèques d'apprentissage machine open source SVM les plus populaires, LIBSVM, proposée par [START_REF] Chang | LIBSVM: A library for support vector machines[END_REF] pour étudier la performance de prédiction En tenant compte des différents paramètres du réseau. Analyse de simulation Dans cette section, nous présentons d'abord les paramètres généraux du système que nous avons configurés pour les simulateurs. Ensuite, nous analysons les performances de prédiction de l'apprentissage machine parmi les différents types de streaming HTTP avec toutes les fonctionnalités enregistrées. Enfin, nous démontrons la performance de prédiction en considérant seulement certaines caractéristiques. Configuration de la simulation Notre simulateur est lancé sur la base des conditions radio réalistes obtenues à partir des données de mesure d'un réseau 4G dans une grande ville européenne, avec un rayon de cellule moyen de 350 mètres. La bande de fréquence concernée est LTE 1800 MHZ. La figure 13 montre la distribution de probabilité mesurée du CQI obtenue à partir de mesures de stations de base collectées à l'aide d'un outil O&M. Chaque CQI est associé à un MCS, en déterminant son efficacité spectrale. Comme nous l'avons mentionné, la charge de trafic peut varier en heures, nous avons démontré 8 taux d'arrivée de flux normalisés par la valeur maximale λ max . Pour chaque λ, le simulateur génère l = 10 6 arrivées en streaming. 80% de données sont sélectionnés au hasard pour la formation et 20% de données pour validation parmi tous les échantillons. Dans la Fig. 6.3, le nombre moyen de privations de la vidéo et de la vacuité de la vidéo utilisées pour la formation est indiqué pour chaque charge. On peut observer que lorsque la charge est faible, la vacuité de la vidéo se produit rarement. Cependant, lorsque la charge approche à λ max , plus de flux vidéo éprouvent la pause de la vidéo. Il est également démontré que les utilisateurs statiques et les utilisateurs de flux adaptatif éprouvent moins de vacuité que les utilisateurs mobiles et fixes. En appliquant les techniques d'apprentissage machine introduites, nous obtenons les deux figures suivantes montrant la performance moyenne de prédiction, P, avec GLM à la Fig. 15a et avec SVM dans la Fig. 15b. On peut observer que les performances de prédiction de GLM et SVM sont similaires. De plus, quel que soit l'outil d'apprentissage que nous utilisons, nous pouvons observer que lorsque la charge augmente, les performances de prédiction diminueront en raison de l'augmentation de l'incertitude. À partir des résultats de la simulation, nous montrons que la QoE des utilisateurs mobiles est beaucoup plus difficile à prévoir, surtout lorsque la charge est importante. Toutefois, les utilisateurs statiques peuvent atteindre plus de 90% de précision. Il n'existe aucune règle générale disant que le débit fixe est plus facile à prévoir que le streaming adaptatif. Cela dépend de la mobilité. Pour les utilisateurs statiques disposant d'une propriété de diffusion en continu adaptative, la prédiction de QoE est plus précise. Pour les utilisateurs statiques, même près de 95% de précision peut être atteint à haute charge. Cependant, pour les utilisateurs mobiles, on peut dire que les informations initiales ne sont pas suffisantes pour la prédiction. Dans cette section, nous vérifions la performance de la prédiction compte tenu de l'accès limité aux fonctionnalités de certains utilisateurs. Comme nous l'avons montré, GLM et SVM ont des résultats similaires à ceux de la Fig. 15a Performances de prédiction de différents flux HTTP Performance de prédiction avec différentes caractéristiques Conclusion Mesurer et améliorer la QoE de la vidéo devient de plus en plus important car la vidéo représente plus de 50% du trafic réseau. Dans cette thèse, nous proposons des modèles pour le dimensionnement de différents types de services de streaming, y compris le streaming en temps réel et le streaming adaptatif HTTP à l'intérieur des réseaux sans fils. Les deux sont développés en appliquant les concepts de dynamique de niveau de flux pour modéliser les arrivées et les départs de la demande de trafic, ce qui est très utilisé pour le trafic élastique et le trafic en temps réel dans la littérature. Dans la section 3, nous développons un modèle de trafic au niveau flux et paquets pour les services de streaming en temps réel. Nous supposons l'existence de propriété quasistationnaire et combinons à la fois le débit et les niveaux de paquets pour calculer le taux de panne de paquets. En utilisant notre modèle, les opérateurs pourraient concevoir l'algorithme de contrôle d'admission correspondant pour les services de streaming en temps réel avec un taux de pannes garanti. Dans la section 4 et 5, nous développons un modèle de trafic au niveau flux pour les services de streaming d'HTTP adaptatif. Le modèle prend en compte la dynamique du niveau flux et vérifie les impacts sur la performance de différents paramètres tels que la durée du morceau vidéo, le nombre de débits binaires vidéo et les schémas d'ordonnancement, etc. Nous abordons ci-dessous les questions potentielles rencontrées par les opérateurs lors de leur déploiement Le service de streaming adaptatif HTTP et souhaitent améliorer la qualité de l'expérience des utilisateurs comme: Impact de la durée du morceau vidéo, Impact du nombre de débits binaires vidéo, propose un nouveau design de la durée du morceau vidéo, Impact des schémas d'ordonnancement sur le streaming adaptatif et aussi impact de la mobilité des utilisateurs. Dans la section finale, comme nous avons constaté qu'il est difficile de trouver une forme analytique exacte pour la QoE vidéo, y compris tous les paramètres possibles du système, nous proposons d'utiliser la technique d'apprentissage automatique pour prédire la qualité vidéo. Les résultats nous aident également à comprendre la corrélation entre la fluidité de la vidéo et les caractéristiques des utilisateurs enregistrées à chaque arrivée. Dans la dernière partie de notre thèse, nous examinons la performance de prédiction en utilisant GLM et SVM avec différentes charges de système. Nous avons constaté que les précisions de prédiction sont plus de 92% pour les utilisateurs statiques à chaque charge et les paramètres de réseau les plus importants incluent le numéro de la condition radio et des flux, ce qui montre que la prédiction de la fluidité vidéo est faisable pour les utilisateurs statiques. Cependant, plus de fonctionnalités des utilisateurs sont nécessaires pour bien prédire les événements de la pause de vidéo. Travaux futurs Les travaux futurs les plus importants, de notre point de vue, sont liés à la section apprentissage machine. En effet, prédire la QoE vidéo sans avoir complètement la solution analytique est une première étape vers l'exploitation des données du réseau pour prédire et améliorer la qualité de vidéo. Par exemple, une première étape pour améliorer l'erreur de prédiction consiste à essayer d'autres modèles d'apprentissage plus aléatoires comme l'arbre de décision, la forêt aléatoire, le réseau neuronal et les voisins K-voisins. De plus, comme nos données d'entraînement sont générées sur la base des simulateurs d'arrivée de Poisson, il est également préférable de mettre en oeuvre un simulateur avec un profil de trafic plus réaliste ou d'utiliser des mesures réelles du réseau. L'ajout de fonctionnalités liées à la mobilité est également important pour la prévision de QoE. Une fois que la méthodologie de prédiction QoE a été améliorée, la deuxième étape sera de l'utiliser pour améliorer QoE, en proposant des algorithmes avancés d'auto-organisation. Comment mettre en oeuvre des algorithmes efficaces mais simples d'auto-organisation pour améliorer QoE de services de streaming pourrait être un sujet intéressant pour une autre thèse de doctorat. List of Acronyms Contents Acknowledgement vii Abstract ix Résumé xi Synthèse en français xiii List of Acronyms xlvii List of Figures Introduction This chapter introduces the subject of this thesis. We first present the technological context of different video services functioning in telecommunication networks. Then we expose the main objectives and contributions of this thesis. We finally list all the publications made during the thesis at the end of this chapter. Context While 4G technology becomes much more mature, broadband services are easily accessible and affordable for every people. With broadband technology, various types of service which are difficult to support in 2G and 3G have large increasing rate in 4G networks. In the report [4], Cisco gave an interesting forecast reported in Table . 1.1 showing that video traffic has already accounted for more than 50% percent of global mobile data traffic in 2015 and that it has relatively larger increasing rate than the other types of traffic. The phenonemen that video traffic is the most important traffic then becomes unavoidable for the internet service providers. Video categories Based on whether the video content are generated at the same time of watching or not, we can simply divide video services into two categories. They are respectively On-Demand Video on Demand (VoD) The on-demand video can be abbreviated to VoD, which means broadcasting programs on the basis of user requirements. Different from the traditional TV broadcasting, users can pause/play video anytime as they wish. The best example of this type of service are provided by YouTube [START_REF] Cha | I tube, you tube, everybody tubes: Analyzing the world's largest user generated content video system[END_REF], Netflix, Hulu and Dailymotion which have an explosive growth and have become one of the most popular research topics since 2005. Simply speaking, the video content of VoD are stored at the cloud side and users can access anytime they want. Real-time video The content of real-time streaming service are generated at the same time of content delivery. Contrary to the VoD streaming service, users of real-time streaming can not replay video as they desired and have to follow the schedule of video content providers. Real-time video can be easily categorized into two parts. One of them has another wellknown name called, Live Streaming. The best examples of this type of service are all the Web TV service like BBC, Orange TV, BFM TV direct, etc. Moreover, real-time video also include the audio conference provided by Skype and other instant messengers like WhatsApp and FB Messengers. In order to support different types of video services, several communication protocols are proposed and we are going to summarize them here. Some of them are open standards and others are only enclosed for the specified usages. HTTP progressive download [14] In progressive download, all the streaming technologies belong to the progressive download, which means that video are downloaded part by part as the word progressive describes. Therefore, users do not need to wait until the end of video download but can start to watch the video while downloading. Hypertext Transfer Protocol (HTTP) is the most popular protocol that supports progressive download and it is currently the most popular technology to deliver on-demand streaming. According to [START_REF] Oyman | Quality of experience for http adaptive streaming services[END_REF], the advantages of using HTTP Web server to deliver video include: • Broad market adoption of HTTP and TCP/IP protocols; they support the majority of the Internet services today. • HTTP-based delivery avoids NAT and firewall traversal issues. • The ability to use standard/existing HTTP servers and caches instead of specialized streaming servers allow reuse of the existing infrastructure. Moreover, progressive download enables users to start watching their video before the whole video are fully downloaded, as video are divided into small segments and are delivered separately to users. Nevertheless, one drawback of HTTP progressive download is that users are limited to choose only one video quality and video format. Real Time Protocol (RTP) RTP is designed for the transport of real-time data including audio and video. It can be used for media-on-demand as well as interactive services such as Internet telephony. RTP usually runs over UDP. Due to the packet loss, it is less reliable compared with HTTP progressive download. RTP is always implemented with Real Time Control Protocol (RTCP) as a control protocol. For on-demand streaming, Real Time Streaming Protocol (RTSP) [3] is an application protocol like HTTP to deliver streaming and it supports the function of pause and return. For some of live streaming RTP is adopted. Most of video conference services like Skpye, are delivered by RTP/RTCP. Real Time Message Protocol (RTMP) RTMP, also known as Flash, is a protocol mainly transmitted based on TCP. More precisely speaking, it is a proprietary protocol for multimedia content transfer between a Flash player and a server. It is also an important video delivery technology but not in the scope of our thesis. HTTP Adaptive Streaming (HAS) HAS is an extended feature of HTTP progressive download and it aims to optimize and adapt the video configurations over time in order to deliver the best possible video quality considering changing link or network conditions. Following the property of HTTP progressive download, video are segmented and stored at the video servers. At the same time, these video segments are encoded in more than one version and hosted along with the Media Presentation Description (MPD) [START_REF] Stockhammer | Dynamic adaptive streaming over http -: Standards and design principles[END_REF] as shown in Fig. 1.1. Based on this MPD metadata information in Fig. 1.2 that describes the relation of the segments and how they form a media presentation, clients use HTTP GET request to access the video segment one after another. started on 2010 and the standard was published in 2012 [6]. In addition, the concept of a Media Presentation is introduced in TS 26.234 [START_REF] Gpp | Transparent end-to-end packet switched streaming service (pss)[END_REF]. • Apple HTTP Live Streaming (HLS): HLS is a communication protocol implemented by Apple as part of QuickTime and iOS. HLS supports both live and video on-demand content. • Microsoft Smooth Streaming [START_REF]Microsoft Corporation: IIS Smooth Streaming Technical Overview[END_REF]: Smooth streaming is an IIS Media services extension. Microsoft is actively involved with 3GPP, MPEG and DECE for standardization. • Adobe HTTP Dynamic Streaming On the client side, the most important decisions are which segments to download, when to start with the download, and how to manage the receiver video buffer. The adaptation algorithm should select the appropriate representation in order to maximize the quality of experience [START_REF] Seufert | A survey on quality of experience of http adaptive streaming[END_REF]. The most common approach is to estimate the instantaneous channel bandwidth and to use it as decision criterion. For channel estimation, authors of [START_REF] Arsan | Review of bandwidth estimation tools and application to bandwidth adaptive video streaming[END_REF] reviewed the available bitrate estimation algorithms. Other decision engines based on Markov Decision Process are described in [START_REF] Jarnikov | Client intelligence for adaptive streaming solutions[END_REF]. In addition to the throughput, there are algorithms considering buffer level, e.g., the authors of [START_REF] Miller | Adaptation algorithm for adaptive streaming over http[END_REF] propose an adaptation engine based on the dynamics of the available throughput in the past and the actual buffer level to select the appropriate representation. Quality of Experience (QoE) Since video services became more popular, how to measure video performance has become a hot topic. Video delivery over wireless network has also been studied for more than 15 years. Ways to measure a video quality can be divided into subjective ones and objective one.s The subjective one is common for all types of video. Instead, corresponding to different types of video services, different types of objective quality measure are defined and proposed. Subjective quality measure Mean Opinion Score (MOS) [5] is a measure, which can be either subjective or objective and is originally used for voice quality measurement. Later it is also applied for the usages of all types of video services. Subjective testing for visual assessment has been formalized in ITU-R Rec. BT.500 [START_REF]Methodology for the subjective assessment of the quality of television pictures[END_REF] and ITU-T Rec. P.910 [START_REF]Subjective video quality assessment methods for multimedia applications[END_REF], which suggest standard viewing conditions, criteria for the selection of observers and test material, assessment procedures, and data analysis methods. MOS quantifies the service qualities into five different levels, from 1, meaning bad quality to 5, meaning excellent quality and the subjective measured method, such as Absolute Category Rating (ACR), Degradation Category Rating (DCR) and Pair Comparison (PC) are standardized in [START_REF]Methods for subjective determination of transmission quality[END_REF]. Several researches also study the video performance by MOS as [START_REF] Singh | Quality of experience estimation for adaptive http/tcp video streaming using h.264/avc[END_REF]. Fig. 1.3 attempts to categorize the objective quality measure of video. In addition, it also tries to clarify the relationship between Quality of Service (QoS) and QoE. For QoS measures, the network QoS community has defined simple metrics to quantify transmission errors, such as Bit Error Rate (BER) and Packet Loss Rate (PLR). None of them take into account the content. Secondly, more approaching to the human visual system, Picture metrics treat the video data by pixel unit. The simplest possible metrics, Mean Square Error (MSE) and Peak Signal to Noise Ratio (PSNR), take into account only signal to noise ratio, although it is also very easy to produce results that deviate form human perception. Packet-or bitstreambased metrics for compressed video delivery over packet networks look at the packet header information and the encoded bitstream directly without fully decoding the video. In paper 1.1. CONTEXT 6 [START_REF] Winkler | The evolution of video quality measurement: From PSNR to hybrid metrics[END_REF], authors focus on MPEG-2. To conclude, it is hard to say which metric is better. Hybrid metrics are proposed. Generally speaking, metrics more at the right side of Fig. 1.3 can approach more to the user perception. Objective quality measure Quality measure for real-time video In this section, we only describe the measures for real-time video transmitted by UDP. The one transmitted by TCP can be measured the performance metrics introduced in the following section. Based on the system mechanism that we have mentioned for the real-time video in the previous section 1.1.1 and due to the UDP property, having possibility to loss video packet, real-time video usually experiences video distortion. Therefore, some popular metrics for measuring the real-time video quality transmitted over unreliable protocol. • PSNR, derived by setting the MSE [START_REF]Survey of objective video quality measurements[END_REF] as M S E = M i=1 N j=1 f (i, j) -F (i, j) 2 M N , ( 1.1) PSN R = 20 log 10 255 M S E , ( 1.2) where f (i, j) is the original signal at pixel (i, j), F (i, j) is the reconstructed signal, and M × N is the picture size. The result is a single number in decibels, ranging from 30 to 40 for medium to high quality video. There are other picture metrics, such as VQM, SSIM, etc. • Packet loss rate (PLR) Packet loss will directly influence the video quality. Therefore, several research works analyzed the impact of packet loss to the real-time video quality such as MPEG-2 [START_REF] Verscheure | User-oriented QoS analysis in MPEG-2 video delivery[END_REF]. In our thesis, when we study the real-time video, we mainly focus on PLR. • Blocking rate In [START_REF] Bonald | On performance bounds for the integration of elastic and adaptive streaming flows[END_REF], blocking rate is used to study the performance of real-time streaming. Blocking rate is a high-level performance metric. Blocking happens when system capacity can not accept any more new video calls. This metrics is highly utilized in the studies of GSM system capacity, e.g. the Erlang models. Quality measure for VoD For the performance metrics of on-demand video, authors in [START_REF] Dobrian | Understanding the impact of video quality on user engagement[END_REF] summarize them into five following terms and quantify the impacts of them on user engagement. In Fig. 1.4, we show a simple illustration of a life time of video session. The state of a video session can be mainly categorized into the following three states: Prefetch state, where user fills its buffer without playing, Playing state, where user start to play its video while download its video and Buffering state, where the video stalls until the buffer is filled up to the STARVATION level. • Join time: Measured in seconds, this metric represents the duration from the beginning of a session connection until the time sufficient playout buffer has filled up. In Fig. 1.4, it is also called start-up delay. • Buffering ratio: Represented as a percentage, this metric is the fraction of the total session time spent in buffering. • Rate of buffering events: Buffer ratio does not capture the frequency of induced interruptions observed by the user. For example, a user may experience video stuttering where each interruption is small but the total number of interruptions is high. • Mean bitrate: Adaptive streaming allows video player to switch between different bitrate streams. Mean bitrate is the sum of played bitrates weighted by the duration each bitrate is played. • Rendering quality: Rendering rate (frames per second) is central to user's visual perception. Rendering rate may drop due to the CPU overload or due to network congestion. • Frequency of bitrate changing: Once the video bit rate is adapted, users will experience a change of video quality. Therefore, it is better not to change the resolution very frequently. There are also another performance metrics that is highly examined in some researches • Rate of bit rate switching: When adaptive streaming is introduced, users are allowed to switch among several video bit rates. This metric measures the frequency of rate switching. Studies suggest users are likely to be sensitive to frequent and significant bitrate switches [44][45]. About the importance of these performance metrics, authors of paper [START_REF] Dobrian | Understanding the impact of video quality on user engagement[END_REF] show that buffering ratio is the most important metric across all content genres and the bitrate is especially critical for Live (sports) content. 1.2. OBJECTIVES 8 Objectives The objectives of this thesis are to build up a traffic model to analyze the objective measure of video inside wireless networks for different types of video services. We begin by taking into account the real-time streaming service. Assuming that real-time streaming has the highest priority, we verify the relationship between • traffic load and Packet Loss Rate (PLR). As on-demand video accounts for larger part of network traffic, our thesis mainly focuses on investigating this type of service and especially, the HTTP adaptive streaming, because of the maturity of the technology. Property of adapting video bit rate is supposed to provide a freedom to balance between mean video bit rate and buffer performance. However, it is not clear the performance impacts of parameters both from the wireless networks and video delivery system. Therefore, in this part of research, we focus on the impacts of following network parameters: • Video chunk duration • Number of video bit rate • Scheduling schemes • Users' mobility By applying our traffic model, operators understand how to well design the related network and video parameters for providing better adaptive streaming experience. In this thesis, we develop the corresponding traffic model using flow-level dynamics, demonstrate the performance impacts and propose the improving deployment methods. Contributions In this thesis, our main contribution is to propose an analytical model based on flow-level model for evaluation of video performance in different scenarios. Other detailed contributions are specified in the following: In chapter 2, we introduce some background knowledges for this thesis. We begin by describing the basics of wireless cellular system and explain how a single-to-single transmission functions and how to model the capacity of this single-to-single link. Then we introduce the corresponding flow-level traffic model, a well-known method established on queueing theory in order to model wireless system. Finally, we present some popular machine learning techniques such as generalized linear model and support vector machine used for the following studies. In chapter 3, our main contribution is to develop the packet delay distribution of realtime streaming services in a wireless cell. We model a Base Station (BS) by applying queueing theory and based on the quasi-stationary property, where we calculate the packet delay by combining both packet-level and flow-level dynamics. Under the obtained packet delay distribution, we can then decide the acceptable flow arrival rate as an admission control policy given a packet delay constraint. We show that with some model extension, fast fading effect can be taken into account in the dimensioning problem. Works mentioned in this chapter are published in [C5]. In chapter 4, we begin by introducing the state of the art of HTTP adaptive streaming modeling. It is shown that using flow-level model, we consider the impacts of traffic dynamics on the performance of HTTP adaptive streaming. We start with considering significantly small video chunk duration. Then we extend our flow-level traffic model with the configuration of significantly large video chunk duration. Respective models stand for an extreme performance bound for any intermediate chunk duration configurations. These performance impacts have been observed by calculating the Key Performance Indicators (KPIs) as the following, mean video bit rate standing for video resolution and mean deficit rate, mean serving time and mean buffer surplus standing for video smoothness. The adaptive streaming traffic model is also extended to integrate the effects of heterogeneous radio conditions, scheduling schemes and coexistence with elastic traffic as a general one. To make the works complete, we also take into account the intra-cell mobility into our flow-level model. This chapter presents all the contributions which have already been published respectively in [C2-4] and [J1]. In chapter 5, our contributions can be divided into two parts. One is to validate our proposed traffic model for the adaptive streaming by simulation. The other is to examine the performance impacts of different system configurations. In order to reduce the complexity of simulation we present an approximation model to simplify the numerical analysis with multiple classes of users flow. Our proposed model can assist service providers to understand the impacts of chunk duration, the impacts of number of video bit rate, the impacts of scheduling schemes and the impacts of mobility. Our results show that smaller chunk duration can offer a better video smoothness with a price to lose little video resolution, vice versa for larger chunk duration configuration. Moreover, our results also show that providing infinite video bit rate may not be a good idea. One of our main contributions is mentioned in this chapter where we propose to deploy the video chunk with same size instead of same duration and we show that this can improve video smoothness. For the publication reference, the results presented in this chapter can be found in [C2-4] and [J1]. In chapter 6, we study the video quality of experience by another approach, where we apply machine learning technique to predict one of the important Quality of Experience (QoE) metrics, video starvation. We demonstrate the prediction performance of different HTTP streaming and show that static and adaptive streaming possess the highest prediction accuracy. We also demonstrate that different network parameters have different importances to predict video starvation. By using machine learning technique, we can still understand the relationship between performance metrics and system statistics when no exact mathematical model is available. This gives an access for operators to understand deeply users' QoE. Contributions of this work are submitted in [C1]. Chapter 2 Background In this chapter, we start by presenting some background knowledges of our thesis, which can be divided into three parties. The background wireless systems will be presented in 2.1. Queueing theory and wireless system modeling are presented in 2.2 and some machine learning techniques are introduced in 2.3. Wireless systems As our thesis is about the video performance in wireless networks, it is important to know the concepts and the characteristics of wireless networks. Wireless channel characteristics Characteristic of the mobile wireless channel is the variations of the channel strength over time and over frequency. The variations are generally composed of slow fading and fast fading, as shown in Fig. 2.1 [START_REF] Tse | Fundamentals of wireless communication[END_REF]. In this thesis, we mainly focus on the slow fading effect for both real-time streaming and HTTP adaptive streaming. In the case of real-time streaming service, we also try to include the fast fading effect. Slow fading Slow-fading is composed of two principle effects, path loss (attenuation) and shadowing, caused by large objects covering such as buildings and hills. The path loss of signal is a function of distance. This occurs as the mobile moves of the order of the cell size, and is typically frequency independent. There are several path loss prediction models that consist in characterizing the propagation medium theoretically and by means of measurements. The empirical models based on statistical analysis over a large number of experimental measures prove to be analytically simple, tractable, and easily extrapolated to other environments with similar propagation conditions as those where measurements were made. The most known examples are HATA and COST-Hata models [START_REF] Abhayawardhana | Comparison of empirical propagation path loss models for fixed wireless access systems[END_REF] [START_REF] Fenton | The sum of log-normal probability distributions in scatter transmission systems[END_REF]. For urban areas, here we present where β is the path loss exponent and l 0 is a fixed term which depends on the system parameters such as the frequency band and the base station height. Moreover, the slow-fading channel is also impacted by shadowing, along the propagation path. This random phenomenon has been described so far by a log-normal distribution [START_REF] Fenton | The sum of log-normal probability distributions in scatter transmission systems[END_REF] and is commonly used for network simulation and performance evaluation. The log-normal distribution with parameters (µ S , σ S ) has the following probabilistic density function: f S (x) = 1 x 2πσ S exp - (log x -µ S ) 2 2σ 2 S , x ≥ 0, (2.2) where µ S and σ S denote the mean and the standard deviation in d B, respectively. A example shows that µ S can be selected as 0 and σ S is ranging from 3d B to 14d B . Fast fading Fast fading is due to the constructive and destructive interference of the multiple signal paths between the transmitter and receiver. This occurs at the scale of the carrier wavelength, and is frequency dependent. Several models are proposed, the Rayleigh propagation model [START_REF] Sklar | Rayleigh fading channels in mobile digital communication systems part ii: Mitigation[END_REF] is applicable to environments where there are many different signal paths, none of which can dominate. As a result, the magnitude of the signal has a Rayleigh distribution as: f R (x) = x σ 2 exp -x 2 2σ 2 , x ≥ 0, (2.3) where 2σ 2 is the average power of the received signal. If there is a dominant line of sight component, Rician fading may be more appropriate [START_REF] Simon | Digital communication over fading channels[END_REF]. Channel capacity The channel capacity of a wireless link between a transmitter-receiver pair is constrained by impairments due to the environment e.g. the channel attenuation seen previously and by other simultaneous transmissions on the same or adjacent frequency band which generate interference. We always use Additive White Gaussian Noise (AWGN) to model a wireless link affected by the thermal noise, which is due to the thermal agitation of electrons in electronic devices. With P u , standing for the received signal and I u standing for the overall interference perceived from a specific user u, the signal quality is determined by the Signal to Interference and Noise Ratio (SINR) ratio given by: S I N R u = f u P u I u + N 0 , (2.4) where f u stands for the channel fading effect we have mentioned before, usually it is described by the Channel State Information (CSI). Once we got the fading channel of a signal path, we can calculate the theoretical point-to-point channel capacity using Shannon's formula. And we can express the capacity of a channel as R = W log(1 + S I N R u ), (2.5) where W denotes the system bandwidth. Based on the R value, transmitter will adapt to a proper Modulation Coding Scheme (MCS) for transmission. It is worthy of mentioning that Shannon formula offers us an upper bound for the channel capacity, which is an optimistic result. Network deployment A simple diagram of a cellular system is shown in Fig. 2.2. A cellular network is a wireless network that provides services by using a large number of Base Station (BS) with limited power, each of which covers a limited area called a cell. These cells provide together the coverage of a wide geographic area. This enables a large number of User Equipments (UE) to communicate with each other and with the fixed infrastructure. When users access video services, the data flow will pass from the servers of video service providers located at Internet through tService Gatway (S-GW) and Packet Data Network (PDN) Gateway to users. When a cellular network is modeled, a cell is usually modeled as an omi-directional cell or a tri-sector cell as shown in Fig. 2.3. In reality, other sectorizations are also possible, i.e. 2, 4, 5 sectors in a cell. The benefit of multiple sectors is to have frequency reuse so as to increase the system capacity. However, the interference between cells will also increase. In the case of tri-sector model, we can regard it as three independent cells with its own BS. In the early 1990's, the Second Generation (2G) starts to emerge using the digital communication and coding technology to guarantee the correctness of data transmission. In terms of services, voice and text message are provided in 2G systems. Although there were several 2G standards, the Global System for Mobile (GSM) is the most successful system and was highly adapted by the operators around the world. This has enabled GSM to be further enhanced and developed so as to support higher data rates. General Packet Radio Services (GPRS) and Enhanced Data Rates for GSM Evolution (EDGE) are the evolutions of GSM with enhanced Adaptive Modulation and Coding (AMC) and some other coding schemes. The evolution of cellular systems continue to third generation which introduced the packet-switched concept coexisting with the circuit-switched method used in the previous system. The Univeral Mobile Telecommunications System (UMTS) developed within the 3rd Generation Partnership Project (3GPP) is one of the candidates that meet the 3G requirement of International Telecommunication Union (ITU) in terms of performance, service and spectrum efficiency. It provides enhanced radio interface called Universal Terrestrial Radio Access (UTRA) network and a core network evolved from the last generation. The Long Term Evolution (LTE) of UMTS system is the 3.9th Generation of cellular networks. It was designed with an Evolve-Univeral Terrestrial Radio Access (E-UTRA) network with a full-IP core network called Evolved Packet Core (EPC). Whole information are supposed to be transmitted in packet-switched network and no circuit-switched service will be offered anymore in 4G. The entire architecture is named Evolved Packet System. Then we have the real 4G system, LTE-Advanced that fulfills the requirement of ITU with the advanced features for both radio and core networks. Nowadays, discussions are launched for 5G. The ITU requirement of 5G has not been defined yet. However, the most recognized system characteristics of 5G are Massive system capacity, Very high data rate everywhere, Very low latency, Ultra-high reliability and availability, Very low device cost and energy consumption, Energy-efficient networks. 5G also includes a lot of specific topics and technology like virtualization and it can support various types of services such as machine-type communication, automatic car, intelligent factory, etc. Radio resource management As radio resources are limited, how to fairly share the wireless resources and how to increase the spectrum efficiency become important issues. In the followings, we introduce two properties concerning the wireless resource management. Duplexing The duplexing aims at defining a transmission technique between the downlink and uplink link. There are two common techniques, Time-Division Duplex (TDD) and Frequency-Division Duplex (FDD) as shown in Fig. 2.4. In TDD, the downlink and uplink transmissions are partitioned over time and on the same frequency band. Generally speaking, TDD provides more freedom to the resource allocation. On the other hand, in FDD, the downlink and uplink are allocated to the separated frequency bands during the whole time axis. In this thesis, we focus on the downlink performance, because the video traffic is usually larger on the downlink. Moreover, only FDD is taken into account. Multiple access Multiple access is the technique which shares the wireless resource among users. The two basic techniques are Time Division Multiple Access (TDMA) and Frequency Division Multiple Access (FDMA). The 2nd Generation family GSM, GPRS, EDGE was based on TDMA and FDMA. However, starting from the 3rd Generation, UMTS utilized a more advanced multiple access technique, called Code Division Multiple Access (CDMA). This technique enables multiple mobile stations to communicate simultaneously on the same frequency and at the same time and more spectrum efficiency are exploited. CDMA is implemented based on the spread spectrum technology, where the original data stream is spread with a code over a longer sequence of transmitted bits. The orthogonality among the transmission channels is ensured by the orthogonality between the spreading codes. In 4G/LTE specifications, Orthogonal Frequency Division Multiple Access (OFDMA) shares the spectrum resource based on OFDM technology, which increases the spectrum efficiency. It consists of splitting each user data stream into several sub-streams, which are sent in parallel on several sub-carriers. These subcarriers provide higher spectrum efficiency because they are orthogonal to each other and less interference are created. In the specification of LTE systems, a Physical Resource Block (RB) is shown in Fig. 2.5, which is composed of 7 × 12 resource elements in a slot (0.5ms). For a LTE with 20MHz spectrum, 100 RBs are available in one slot. Scheduling A resource block is the minimum unit for scheduling. As we have shown in the previous section, an LTE system with 20MHz has 100 RBs able to be allocated. The task of a centralized scheduler is to distribute these wireless resource to a group of users. In our thesis, we assume that we can allocate any fraction of time-frequency block, ϕ u to a certain user u. To fully utilize the wireless resource, we have u∈ ϕ u = 1, (2.6) where denotes as the user set. We assume that the fraction, ϕ u can be any values in [0, 1]. Scheduling algorithm is supposed to improve spectral efficiency of the system. In the thesis, we consider several scheduling schemes starting from Round Robin (RR) scheme, a simple, fair scheduling algorithm that does not exploit fast fading. Other scheduling schemes are introduced in details in section 4.5. Queueing theory and traffic modeling In this section, we introduce the basic knowledge of queueing theory and how to apply queueing theory for traffic modeling. Queueing theory is a popular mathematical tool used to describe a dynamic system with a shared resource. In this section, we only introduce some concepts applied in our thesis. Regarding the details, we refer the readers to [START_REF]Network Performance Analysis[END_REF] [START_REF] Bolch | Queueing Networks and Markov Chains[END_REF] . Queue model We begin by introducing a single station queue as shown in Fig. 2.6. For a queueing network contains more than one station, readers can refer to, [START_REF] Bolch | Queueing Networks and Markov Chains[END_REF]. A queue is characterized There are also some non independent arrival processes including Markov Modulated Poisson Process (MMPP), which is examined in chapter 3 for A. The default discipline for D is typically: • First Come First Served (FCFS): The jobs are served in the order of their arrivals. • Last Come First Served (LCFS): The job that arrived last is going to be served first. • Processor Sharing (PS): This strategy corresponds to round robin with infinitestimally small time slices. It is as if all jobs are served simultaneously and the service time is increased correspondingly. Queue size, n and service discipline, D are not always specified. By default, n is configured as infinite queue size and FCFS is the default discipline for D. M/M/1 queue The simplest example is M/M/1 queue. Recall that in this case, the arrival process is Poisson, the service times are exponentially distributed, and there are a single server and infinite queue size. The system then can be modeled as a birth-death process with arrival rate λ and a constant service rate µ. With the assumption that λ < µ, the underlying Continous-Time Markov Chain (CTMC) is ergodic and hence the queueing system is stable. Instead, if the system is not stable, that is λ > µ, the number of users inside the system will grow to infinity. With the load defined as ρ = λ µ and assuming ρ < 1, the stationary distribution, π(x), having x users inside the system is given by: π(x) = π(0)ρ x , x ∈ , ( 2.8) with π(0) = x ρ x -1 = 1 -ρ. (2.9) We can calculate the mean number of users as E(X ) = x xπ(x) = ρ 1 -ρ , ( 2.10) where we can observe that when ρ → 1, E(X ) → ∞. By Little formula [START_REF]Network Performance Analysis[END_REF], we obtain the mean sojourn time of a user, S, as S = E(X ) λ = 1 µ -λ . ( 2 .11) Whenµ ≈ λ, meaning that ρ → 1, the sojourn time of a user goes to infinity. State dependent queue In order to better model a general system, it is common to generalize the simple M/M/1 queue. The arrival rate and departure rate of each state depends on the state x, φ (x) and φ(x). Therefore, by the reversibility, the stationary distribution π(x) is denoted as π(x) = π(0) λ µ x φ (0)φ (1) • • • φ (x -1) φ(1)φ(2) • • • φ(x) , where x ∈ . (2.12) In the later modeling of HTTP adaptive streaming, we use the concept of state dependent queue for modeling the parameter, chunk duration and in chapter 5, we use the same concept to develop our approximation model. Processor sharing discipline Different from the classical FCFS queues, processor sharing queues provide an interesting property calledinsensitivity, mentioned in [START_REF]Computer Applications[END_REF] and [START_REF] Bonald | Insensitivity in processor-sharing networks[END_REF]. Assuming Poisson arrivals, the stationary distribution of the number of customers does not depend on the distribution of service times, which is not the case with the FCFS discipline. In reality, the service time distribution is not always exponential. Therefore, in practice, this property is very useful and has the practical interest in communication networks of allowing the development of engineering rules independently of precise traffic statistics. Whittle networks We consider a network of n single-server queues coupled by their service rate. Customers arrive to the system according to a Poisson process of intensity λ i at queue i, then leave the network after finishing the service. In Whittle networks, the total service rate of each queue depends on the network state. Customers require an exponential service rate with parameter µ i at queue i is thus equal to µ i φ i (x ) in state x , where x = (x i , • • • , x N ). The network is said to be a Whittle network if the service capacities satisfy the following balance property given by φ i (x )φ j (xe i ) = φ j (x )φ i (xe j ), ∀i, j, ∀x : x i > 0, x j > 0. (2.13) Seeing in Fig. 2.9, let (x , xe i 1 , • • • , xe i 1 • • • -e i n , 0) be a direct path from state x to state 0. A path of length n where n is the number of customers in state x . The balance property implies that the expression Φ(x ) = 1 φ i 1 (x )φ i 2 (x -e i 1 ) • • • φ i n (x -e i 1 • • • -e i n ) , (2.14) is independent of the considered direct path. Therefore, φ i can be uniquely characterized by the function Φ, referred to as the balance function: φ i (x ) = Φ(x -e i ) Φ(x ) , i = 1, • • • , N , x i > 0. (2.15) The Whittle network is stable if and only if: x Φ(x ) N i=1 λ i µ i x i < ∞, (2.16) in which case the stationary distribution is: π(x ) = π(0)Φ(x ) N i=1 λ i µ i x i , ( 2.17) where the proof can be found in [START_REF] Serfozo | Introduction to Stochastic Networks, ser. Stochastic Modelling and Applied Probability[END_REF] if the processor-sharing property holds. This stationary distribution is insensitive to the service requirements at any node. Packet-level modeling v.s. Flow-level modeling Two levels of traffic dynamics can be considered, either at packet-level or at flow-level. Several studies report that examining the dynamics of IP traffic is difficult (e.g. and references therein) since the statistics of packets arriving at and departing from the network exhibit self-similar behavior due to the heavy tailed distribution of document size and all the mechanisms of TCP congestion control [START_REF] Feldmann | Dynamics of ip traffic: A study of the role of variability and the impact of control[END_REF]. As a result, evaluating network performance at packet-level is hardly tractable. In particular, modeling the network using queuing theory with the assumption of Poisson arrival for instance, is not applicable. In [START_REF] Roberts | Quality of service by flow-aware networking[END_REF], the flow-level model have been introduced. The term flow refers to a continuous stream of packets using the same path in a network and characterized by the starting time and the size. In the next section, flow-level modeling and its application for wireless network are introduced. Flow-level modeling Paper [START_REF] Massouliã© | Bandwidth sharing and admission control for elastic traffic[END_REF][80] introduced the concept of flow-level dynamics to model the variations of the resource shares. Each flow represents a service request generated by a client and the departures/arrivals of a flow influence the resource shares. Several researches applied this technique for service operators to investigate the performance of wireless network. An important set of works applying flow level modeling in different networks are summarized in [START_REF]Network Performance Analysis[END_REF] and [START_REF] Bonald | Insensitive traffic models for communication networks[END_REF] focuses on the insensitive property of flow-level modeling in communication networks. In paper [START_REF] Bonald | Wireless downlink data channels: user performance and cell dimensioning[END_REF], a flow level model has been proposed for studying elastic traffic in cellular networks. Works of [START_REF] Bonald | Wireless data performance in multicell scenarios[END_REF] extends to the model considering multiple cells. Note that a flow may be subject to different radio conditions due to the position of the user in the cell and to throughput variations due to the dynamics of arrivals and departures of other users. This model has been extended for taking into consideration mobility in [START_REF] Bonald | Flow-level performance and capacity of wireless networks with user mobility[END_REF] and advanced radio features like intra-cell coordination in [START_REF] Khlass | Flow-level performance of intra-site coordination in cellular networks[END_REF] and inter-cell coordination in [START_REF] Bonald | Inter-cell coordination in wireless data networks[END_REF]. Flowlevel model has also been applied in modeling the traffic in fixed network as [START_REF] Bonald | Dimensioning high speed ip access networks[END_REF] and in modeling WiFi network in [START_REF] Bonald | On the stability of flow-aware CSMA[END_REF]. We present the basic flow-level modeling of elastic traffic in the case of mobile networks here: We consider an arbitrary set of classes of UEs indexed by i ∈ to reflect the different radio conditions, R i (i.e., locations) in the considered cell. In practice, the transmission rate depends on the radio environment and varies over time due to user mobility. Unless otherwise specified, we ignore the fast fading effects. Hence, the peak rate R i depends on the user's position in the cell. We assume that the transmission rate is constant during the data transfer unless user has a large position changes. In each class, we assume that data flows arrive according to a Poisson process with intensity λ in the reference cell. Each flow stays in the system as long as the corresponding data have not been successfully transmitted to UE. Flow sizes are assumed to be independent and exponentially distributed with mean σ bits, although all our results are approximately insensitive to the distribution. The traffic intensity is λ × σ in bit/s. The total arrival rate λ is composed of the arrival rate at each class-i, where λ i = λp i and i∈ p i = 1. (2.18) Let X i (t) be the number of class-i flows at time t. The vector X (t) = (X i ) i∈ is an irreducible Markov process whose transition rates depend on the scheduling scheme, which we will discuss this impact in chapter 4. Here, we assume that Round Robin (RR) scheduling scheme. The performance metrics considered is flow throughput(in bit/s). Let us say τ i is the mean duration of class-i flow. According to the Little's formula, E(X i ) = λ i τ i and we have γ i = σ τ i = λ i σ E(X i ) . (2.19) This is ratio of the traffic intensity of class i to the mean number of class-i flows. This throughput metric reflects user experience, accounting both for the radio conditions and for the random nature of traffic, through the stationary distribution of the Markov process X (t). The mean flow throughput in the cell is given by: γ = σ τ , (2.20) where τ is the mean flow duration of the cell, τ = i∈ p i τ i . We obtain γ = i∈ p i γ i -1 . (2.21) This is the weighted harmonic mean of the per-class flow throughputs, with weights given by the per-class traffic intensities. The idea of the harmonic average of throughputs was proposed in [START_REF] Bonald | Wireless downlink data channels: user performance and cell dimensioning[END_REF]. Applying the RR scheduling scheme, the balance property, Eq. (2.13) is verified and the queueing system can be viewed as a Whittle network [START_REF]Network Performance Analysis[END_REF]. With the load definition of ρ i = λ i σ R i , ρ = i∈ ρ i = λσ R , (2.22) where R = i∈ p i R i -1 . Therefore, the stationary distribution of number of flow in the cell, x is given by: π(x ) = (1 -ρ) |x |! i∈ x i ! i∈ ρ x i i , (2.23) where |x | = i x i . Machine learning Machine learning is a popular tool usually used for making predictions, decisions or classification based on a large amount of data. It is widely applied to pattern recognition and artificial intelligence for instance. It is closely related to the computational statistics. As some QoE metrics in our traffic model might be too complicated to express in an exact mathematical form, we attempt to utilize machine learning to find out the correlation between features and output results, here more specifically the QoE of users. In this section, we present some backgrounds of machine learning useful for chapter 6. Generally speaking, there are two types of machine learning. One is supervised learning. The other is non-supervised learning. In this thesis, we focus on supervised learning. Supervised learning A general supervised learning problem is formulated as Fig. 2.11. Assuming there are m training data pairs. For ith training data, we use vector x i ∈ n to represent the input variables, also called input features. Here n represents the number of features in x i . y i is denoted as the output or target variable that we are trying to predict. A pair (x i , y i ) is called a training example in the dataset that we use to learn is called a training set, {(x i , y i This function, h is called a hypothesis. When the target variable that we are trying to predict is continuous, e.g. x i ∈ n and y ∈ m , we call the learning problem a regression problem. When y can take on a small number of discrete values, e.g. x i ∈ n and y ∈ {1, -1} m , then the problem is called as a classification problem. ) : i = 1, • • • , m}. = {x i } i=1,••• , Cost function and probabilistic interpretation We first demonstrate the simplest hypothesis, h θ (x ), for both regression and classification problem. Then we formulate it to a more generalized form. Regression problem When faced with a regression problem, let us assume that the target variables and the inputs are related via the equation y i = θ T x i + ε i , (2.24) where θ ∈ n and ε i is an error term that captures either unmodeled effects or random noise. Assuming that ε i are distributed IID according to a Gaussian distribution with mean zero and some variance σ 2 . We can write this as ε i ∼ (0, σ 2 ). The density function is given by p(ε i ) = 1 2πσ exp - ε 2 i 2σ 2 (2.25) ⇒p( y i |x i ; ε i ) = 1 2πσ exp - ( y i -θ T x i ) 2 2σ 2 , ( 2.26) where the notation p( y i |x i , θ ) indicates that this is the distribution of y i given x i and parameterized by θ . We define the likelihood function, distribution of y given x , for describing the probability as L(θ ) = L(θ ; X , y) = p(y|X ; θ ). ( 2 .27) Note that by the assumption of independence on the ε i , likelihood function can also be written as L(θ ) = m i=1 p( y i |x i ; θ ) = m i=1 1 2πσ exp - ( y i -θ T x i ) 2 2σ 2 . (2.28) In order to maximize the likelihood function, we can also transform the optimization problem to a log likelihood maximization problem as θ * = arg max θ L(θ ) = arg max θ log L(θ ). (2.29) Hence, maximizing L(θ ) gives the same results as minimizing the cost function, C(θ ): θ * = arg max θ 1 2 m i=1 C(θ ) = arg max θ 1 2 m i=1 ( y i -θ T x i ) 2 . (2.30) Classification problem For the classification problem, the target variable y is confined to a set of values as {1, -1} and the form of hypothesis h θ is chosen as h θ (x) = g(θ T x ) = 1 1 + e -θ T x , (2.31) where g function is called logistic function or sigmoid function. Let us assume that p( y = 1|x; θ ) = h θ (x ). (2.32) With the property of sigmoid function, we have the likelihood function as p( y = -1|x; θ ) = 1 -h θ (x ) = h θ (-x ). (2.33) After summarizing the two previous equations, the likelihood function for classification can be written as p( y|x; θ ) = (h θ ( y x )). (2.34) Assuming that m training samples were generated independently, we can then write down the likelihood function as L(θ ) =p(y|X ; θ ) (2.35) = m i=1 p( y i |x i ; θ ) (2.36) = m i=1 (h θ ( y i x i )). (2.37) Same as the last section, optimal θ * can be obtained by maximizing the likelihood function, which is equivalent to minimize the following cost function, C(θ ), C(θ ) = m i=1 log(h θ ( y i x i )). (2.38) where θ is a vector of scalars corresponding to each element in x i and h θ (x i ) = (1+e θ T x i ) -1 . With the obtained θ * = arg min θ C(θ ), we have ŷ(x i ) = 1, when h θ * (x i ) ≥ 0.5, -1, when h θ * (x i ) < 0.5. (2.39) Generalized Linear Model (GLM) In the regression example, we had y|x ; θ ∼ (µ, σ 2 ), and in the classification one, y|x ; θ ∼ Bernoulli(φ). In this section, both of these methods are special cases of a broader family of models, called Generalized Linear Model (GLM). We say that likelihood function can be written in a more generalized form as p( y; η) = b( y i ) exp(η T T ( y) -α(η)). (2.40) Here, η is called the natural parameter of the distribution; T ( y) is the sufficient statistic and a(η) is the log partition function. For regression problem, choosing η = µ = h θ (x ), T ( y) = y, a(η) = η 2 2 = µ 2 2 , b( y) = 1 2π exp -y 2 2 makes the likelihood function become exactly as Eq. (2.27). Moreover, for classification problem, we can choose the configuration as η = log φ 1 -φ = log h θ (x ) 1 -h θ (x ) , T ( y) = y, a(η) = -log(1 -φ) = log(1 + e η ), b( y) = 1, which makes the likelihood function same as Eq. (2.37). There are many other distributions that are members of the exponential family: The multinomial, the Poisson, the gamma and the exponential, etc. Each member has its corresponding advantages on treating certain type of problems, i.e., Poisson distribution is good for predicting the count-data. Gradient descent algorithm In order to maximize the likelihood function of GLM, Eq. (2.40), gradient descent method starts with an initial θ and repeatedly performs the update as θ j := θ j -α ∂ ∂ θ i C(θ ) = θ j -α m i=1 ( y i -h θ (x ))x j , (2.41) where α represents the learning rate, C(θ ) = m i=1 log p( y i : η(θ , x i )) and two examples are shown in Eq. (2.30,2.38). For both cases, it can be proved that the cost function C(θ ) is convex, therefore gradient descent will converge to a unique θ * . Support Vector Machine (SVM) Support vector machine introduced in [41] [START_REF] Andrew | CS229 lecture notes: Part V Support Vector Machines[END_REF][13] treat the same question as classification problem with the same training set, = {x i ∈ n } and y ∈ {1, -1} m , Margins Assuming there is an hyperplane w T x + b = 0, we want to utilize it for separating a set of data { y i , x i }. Given a x , the distance of this point to the plane is distance = 1 ||w || |w T x + b|. (2.42) Therefore, we formulate an optimization problem as follows: to find out the hyperplane parameters, w , b, that maximize the minimum distance (margin) max b,w min i=1,••• ,m 1 ||w || y i (w T x i + b), s.t. ∀i, y i (w T x i + b) > 0. (2.43) The optimization problem will remain the same if we scale the constraint saying that min i=1,••• ,m y i (w T x i + b) = 1. The optimization problem then becomes max b,w 1 ||w || , s.t. ∀i, y i (w T x i + b) ≥ 1. ⇒ min b,w 1 2 w T w , s.t. ∀i, y i (w T x i + b) ≥ 1. By solving the dual problem of quadratic optimization problem, max all α i ≥0 min b,w 1 2 w T w - M i=1 α i 1 -y i (w T x i + b) , (2.44) the obtained w = m i=1 α i ( y i x i ) and b = y i -w T x i when α i > 0 define the hypothesis hyperplane as g(x ) = sign i α i y i x i x + b , (2.45) where α i is the Lagrange Multiplier. The support vectors on the boundary will satisfy α i > 0. Otherwise, α i = 0. Kernel As we can observe in the previous section, w has the same dimension as x i . If we want to increase more VC-dimension to our learning model, we can introduce a non-linear function z i = φ(x i ), where z i could be any element of d . Therefore, the optimization becomes min b,w 1 2 w T w , s.t. ∀i, y i (w T z i + b) ≥ 1. Like the method that solves the optimization problem in the previous section, we obtain similar results with z i = φ(x i ) and Eq. (2.45) becomes g(x ) = sign i α i y i φ(x i ) φ(x ) + b = sign i α i y i K(x i , x ) + b , (2.46) where K(x i , x ) is the kernel function. The definitions and the physical meaning of kernel are listed as • Linear Kernel: K(x i , x ) = x T i x Linear kernel makes the question back to the original problem, which has the limited VC-dimension to the dimension of x . • Polynomial Kernel: K(x i , x ) = (ζ + γx T i x ) Q Polynomial kernel provides more VC dimension than linear kernel but less than infinity. • Gaussian Kernel: K(x i , x ) = exp(-γ x i -x 2 ) Gaussian kernel uses Radial basis function (RBF) to make VC-dimension to infinite and also less parameters are needed to control compared to the polynomial kernel. Soft-margin SVM Different from the problem formulation introduced in the previous section whose hyperplane can always separate the training set, the soft-margin SVM optimization problem provides the optimal w with a relaxation of margin ξ i . min w ,b,ξ 1 2 w T w + C m i=1 ξ i subject to y i (w T φ(x i ) + b) ≤ 1 -ξ i , ξ i ≥ 0. (2.47) In reality, soft-margin SVM is common in applications, because always finding out a hyperplane to separate all the positive and negative training sets is difficult. Even though using SVM with Gaussian kernel could also realize a well-separated boundary. Some overfitting could be generated. Overviews of machine learning techniques In addition to the machine techniques, GLM and SVM that we applied and introduced in this thesis, we provide an simple overview to some current and popular machine learning techniques, such as Decision Tree, Random Forest, k-Nearest-Neighbor and Neural Network which are not used in this thesis but may be applied in the future works. Decision tree Decision tree is a machine technique that mimics the humans' behavior. Here we present a basic decision tree algorithm with original data set = {(x i , y i )} m i=1 . Four configurations need to be made before launching decision tree algorithm: number of branches, branching criteria, termination criteria and base hypothesis g t (x ). If termination criteria met, hypothesis function g t (x ) will be transmitted back otherwise will be split into several Where C = 2 is the simplest. Branching is forced to terminate when all y n are the same or all x n are the same. For classification, the popular choice of impurity function is c = {(x i , y i ) : b(x i ) = c} impurity( ) = 1 m m i=1 ( y i -ȳ) 2 , and the function for regression is impurity( ) = 1 - K k=1 m i=1 [ y i = k] m 2 . By the concept of divide and conquer, the problem is reduced from G(x ) to several subproblem, G c (x ). G(x ) = C c=1 [b(x ) = c]G c (x ). (2.49) For each G c (x ) the same process above will be executed another time. The learning process stops when the decision tree becomes fully-grown tree and that E in (G) = 0. However, overfitting happens and a regularizer is needed. A decision tree with regularizer is also called pruned decision tree. The general advantages of Decision Tree learning technique are: human-explainable, multiclass easily, categorical features easily, missing features easily and efficient non-linear training (and testing). Several libraries of decision tree are for example C&RT and C4.5. Random forest The idea of random forest is to combine bagging and fully-grown decision tree. ˜ t are generated by taking out several data from . The advantage of Random forest include highly parallel/efficient to learn, inherit pros of decision tree and eliminate cons of fully-grown tree. With these advantages, Random forest is more applicable than only one decision tree. k-nearest-neighbor (KNN) KNN classifier is one of the most basic classifiers for pattern recognition or data classification. The principle of this method is based on the intuitive concept that data instances of the same class should be closer in the feature space. As a result, for a given data point x of unknown class, we can simply compute the distance between x and all the data points in the training data, and assign the class determined by the K nearest points of x . Neural network Neural networks is recently very popular in both research and engineering. Inspired from the mechanism of human neurons. In mathematical model of neural network, the number of layers can be configured depends on the need. Pros of neural network is able to approximate anything complex regression and classification problem if enough neurons and the number of layers are configured. Cons of this technique is more about complexity of calculation and overfitting if too many neurons are considered. Chapter 3 Model of Real-time Streaming Traffic Nowadays Internet provides a wide range of services and applications. Generally speaking, the traffic can be easily separated into two types, elastic data and non-elastic data. Elastic data are those non-delay-sensitive services, such as email, File Transport Protocol (FTP) and web browsing. Non-elastic data includes voice services and video services such as VoD and real-time streaming service as we have introduced in section 1.1.1. Real-time streaming services including live TV streaming and video conference services become more important according to Cisco forecasts [4]. In this chapter, we study the performance of real-time streaming services and establish a traffic model for it. Problem statement and the state of the art Elastic data services quality mainly evaluated users' average throughput as QoS metrics. As we mentioned in section 1.1.2, many QoE metrics are proposed for real-time streaming. In order to study the performance of real-time streaming, in [START_REF] Bonald | On performance bounds for the integration of elastic and adaptive streaming flows[END_REF], [START_REF] Blaszczyszyn | Quality of Real-Time Streaming in Wireless Cellular Networks -Stochastic Modeling and Analysis[END_REF] and [START_REF] Karray | Evaluation and comparison of resource allocation strategies for new streaming services in wireless cellular networks[END_REF], authors chose other metrics called flow blocking rate or outage rate as the main performance metrics for real-time streaming. Based on the metric, in [START_REF] Borst | Integration of streaming and elastic traffic in wireless networks[END_REF], the performance of elastic users is evaluated with the presence of streaming users using flow-level model. Here we only consider the ones related to network QoS, packet outage rate, an important QoS metrics when dealing with real-time services as mentioned in [START_REF] Wu | Transporting real-time video over the internet: challenges and approaches[END_REF] and [START_REF] Mugisha | Packet scheduling for VoIP over LTE-A[END_REF]. Different real-time applications have different packet delay constraints. Packets with delay larger than the delay constraint are regarded as useless. Therefore, operators need a good model to predict the packet delay performance under a given traffic intensity so as to deploy proper system capacity and to design the admission control policy accordingly. Real-time services are specified to generate their packets periodically. Take voice service as an example, each Voice over LTE (VoLTE) user generates their packets every 20ms or longer [START_REF] Poikselkä | Voice over LTE[END_REF]. For the real-time streaming services, packets are generated periodically as voice services but real-time streaming needs larger bandwidth and this presents some challenges for operators on dimensioning. Usually, real-time services are always considered to have higher priority compared to other services. In [START_REF]Performance evaluation of multi-rate streaming traffic by quasi-stationary modelling[END_REF] and [START_REF] Olivier | Internet Data Flow Characterization and Bandwidth Sharing Modelling[END_REF], the packet delay is calculated by the quasi-stationary property. However, it is not applied in the wireless scenario. Contributions Our contribution is to develop a traffic model for real-time streaming users by assuming that streaming users come to the system independently and that the packets generated by the users are served with a different serving time based on their own channel conditions and their chosen video bit rates. For the real-time streaming service, base station will serve only one user's packet at one time because compared with voice data, streaming packet size is always large enough to occupy all the RB in a Transmission Time Interval (TTI). Considering with the packet delay constraint for different type of streaming services, we calculate within chapter, the maximum capacity of the real-time streaming system under the constraint that 95% of packets have a delay lower than a specific application delay, D. Our other contributions include: • Development of a model for calculating the capacity of real-time streaming services considering the packet delays performance. • Proposition a simpler calculation method by using fluid model. • Extension and validation of our model with fast fading effects. Chapter organization Chapter 3, is organized as follows: in section 3.2, we introduce the system model with quasi-stationary regime. We then calculate the packet delay distribution considering the system load described by flow-level dynamics. In section 3.3, we extend the model to multiple classes of users representing users with different channel conditions and video codec usage. In section 3.4, we use the LTE system parameters to simulate the maximum load for different codec configuration and show the applicability of fluid model for LTE real-time services. Finally in section 3.5, model is validated with taking fast fading effects into consideration. It is shown that fast fading can be modeled. Flow-level and packet-level model We utilize flow level dynamics to describe the users dynamics in the system, which has been introduced in section 2.2. In order to obtain the maximum system load, we model the system at two levels, flow level and packet level as Fig. 3.1. Instead of using Markov Modulated Poisson Process (MMPP) [START_REF] Li | Radio Access Network Dimensioning for 3G UMTS[END_REF] as our arrival process, we assume that the flow-level dynamics occur on a relatively slow time scale compared to the packet level dynamics, which is referred to the quasistationary property shown in [START_REF]Performance evaluation of multi-rate streaming traffic by quasi-stationary modelling[END_REF]. In reality, a streaming service generated by a user will stay in a time scale of seconds and packet service time is always in a time scale of milliseconds. Therefore, the packet level delay performance will approximately reach some sort of steady state in between changes in the population of flow level model. As the packets are generated periodically and each user will generate independently its packets with an average arriving interval we model the packet arriving process as a simple Markov arrival process. We validate on the assumption of quasi-stationary property in section 3.4.1. Flow-level dynamics At the flow level, we consider firstly one class of users who have the same channel conditions and we model the number of real-time streaming users n as a continuous-time Markov Chain with arriving rate, λ f = A -1 f , inverse of flow arriving interval and serving rate, µ f = S -1 f , inverse of flow serving interval which are independent from the other user's coming and departure behavior. Based on the Erlang formula [START_REF]Network Performance Analysis[END_REF], we know that the stationary state distribution with infinite and finite users can be expressed like π f (n) =          e -ρ f ρ f n n! , when n ∈ [0, ∞] ρ n f n! 1 + ρ f + • • • + ρ f m m! , when n ∈ [0, m] (3.1) where ρ f = λ f µ f = S f A f represents the flow level load for the real-time streaming user. Packet-level dynamics Based on the quasi-stationary regime, each state n, standing for number of users at flow level will correspond to a packet-level regime In the packet queue, we assume that each user will generate its service packets periodically with fixed interval A p and will be served by the base station with fixed interval S p . As each user will generate its streaming packets periodically and many users generate the packets at different time. The packet arrival is random and we approximate it to a Poisson process, we use M/D/1 queue to model realtime streaming system at the packet level. At state n, we model the packet arriving behavior as a Poisson process with arriving rate: λ p (n) = n A p (3.2) We consider that all the users belong to the same channel condition. The packet departure rate at state n is independent of state n: µ p (n) = S p -1 . With n users in the system, using the two previous equations, we define the load of packet-level queue as ρ p (n) = nS p A p = nρ p , where ρ p = S p A p (3.3) With the detailed derivation of the CDF function in [START_REF] Iversen | Waiting time distribution in M/D/1 queueing systems[END_REF], the waiting time distribution is shown in equation (3.4). P n (T ≤ x) =      0 , ρ p (n) ≥ 1 (1 -nρ p ) x k=0 (nρ p (k -x )) k k! e nρ p (k-x ) , ρ p (n) < 1 where function x represents the largest integer less than or equal to x variable and x = x S p . Because this equation gives us the waiting time distribution, to get the response time distribution we just need to shift the distribution by an S p . Under the assumption of quasistationary two-level system and based on the Bayesian Theorem, the overall delay distribution, P(T ≤ x) is the average delay of the delay distribution of each state, n. Therefore, with equation (3.1) and (3.4), we obtain P(T ≤ x) = n π f (n)P n (T ≤ x). (3.4) Because any packet delay larger than a given delay constraint, D, is useless for the delay sensitive service, we can obtain the packet outage rate as γ(D) = P(T > D) (3.5) Given certain ρ p , tolerated packet outage rate ε and a certain delay contraint D, we are able to calculate maximum ρ f , system load, making γ(D) = ε. Fluid model approximation Seeing the packet traffic as fluid, we propose fluid model approximation to simplify packet level model without considering the delay constraint. Because when system enters the overloaded status, delay of packet will become infinity and both the packets out of delay constraint and dropped packets are seen useless for the service, the overall packet outage rate is composed of the indication function expressed as the following equation, γ fluid = n π f (n)1 {ρ p (n)>1} < ε (3.6) where which shows that fluid model is a lower bound of two-level model and it is independent of value D. ρ p (n) is in equation (3. Extension to heterogeneous radio conditions From the point of view of system dimensioning, users might use different codec rate and might have different channel conditions. Therefore, we extend our model to multiple-class users with modified M/D/1 model and fluid model. In the case of multiple classes, we model the system with multiple classes of users having different packet serving times. In addition because of the difficulty to get the closed form of M/D/1 with multiple classes and multiple serving times, fluid model could become a good model to facilitate the calculation. Flow-level dynamics In the previous section, we use flow level dynamics to model the number of users in the system. Assuming there are K classes of users which represent the users with different channel conditions and each class k ∈ {1, Based on [START_REF]Network Performance Analysis[END_REF], the stationary distribution of the state π(n) describing the number of flows of each class is given by π(n) = K k=1 e -ρ k ρ k n k n k ! (3.8) Packet-level dynamics Corresponding to different channel conditions, each class has its specific service time S = {S 1 , S 2 , • • • S K }. As more than one class of users coexist in the system, we modify the M/D/1 outage rate in equation (3.20) with ρ f = (ρ 1 , • • • , ρ K ). γ MD1,m = n π(n)P n (T ≤ D, ρ p ) (3.9) by considering more than one serving time, the CDF can be calculated by the numerical result of inverse Laplace transform obtained in equation (3.11), which is also the M/G/1 model shown in [START_REF] Kleinrock | Queueing Systems: Theory, ser[END_REF] with multiple discrete serving time, S k and corresponding probability n k n . Pn (s) = P n (T ≤ x) = ρ -1 λ -s -λB(s) (3.10) where B(s) function is expressed as B(s) = K k=1 n k n e -sS k (3.11) and other variables ρ =λ k n k n S k = k n k S k A p (3.12) λ = k n k A p (3.13) n = k n k (3.14) The outage rate of fluid model with multiple class of users, γ fluid,m , is shown as below using the same logic of equation (3.6). γ fluid,m = n π(n)1 {ρ>1} (3.15) Simulation results In this section, we first show the validation of quasi-stationary regime and then we show the performance of M/D/1 model and fluid model with different services corresponding to different delay constraints configuration. Based on [START_REF] Perkins | RTP: Audio and Video for the Internet[END_REF], the human tolerant delay for interactive service such as video conference is about 150ms. We configure the delay constraints as 500ms for live TV streaming. We show that the fluid model can be used for the simplification of live TV streaming and that it is better to stay with M/D/1 model in the dimensioning of video conference. Quasi-stationary regime Single class model validation In the Table . 3.1, we assume that the mean flow arriving time is S f = 10s which is one hundred time larger than the mean packet arriving time A p = 100ms. Based on the signal and interference noise ratio (SINR) distribution obtained in [START_REF] Blaszczyszyn | Quality of Real-Time Streaming in Wireless Cellular Networks -Stochastic Modeling and Analysis[END_REF] and the configuration of 3GPP specification [START_REF]Evolved Universal Terrestial Radio Access (E-UTRA) radio frequency RF system scenarios TR 36.942[END_REF][8], the LTE average throughput is calculated as τ = 9.4Mbps and based on different codec settings, different S p settings are shown in the In Fig. 3.4 and Fig. 3.5, we show that configuration with 2Mbps and 512kbps codec rate and users have the same service time S p = 21.3ms and S p = 5.45ms respectively, the red curve stands for the packet outage rate obtained by fluid model and the other blue curves are the results obtained from M/D/1 model with different delay constraint D = 50ms, 150ms and 500ms. It can be seen that the packet outage rates are lower bounded by fluid model and when delay constraint approaches to 500ms, the performance of fluid model and M/D/1 are the same. Therefore, we say that fluid model is enough to describe the packet-level performance of streaming services for service like live TV streaming. For the interactive services like video conference, it is better to use M/D/1 model. In Table . 3.2, we show the maximum flow-level load obtained by the simulation results for different types of services and different codec configurations which limit the packet outage rate under 5%. Multiple class model validation To validate the extension of our models to the multiple class scenario, we take an example of users with two classes, S = {S c , S e }, representing cell edge and cell center users respectively. In the validation, we assume that users utilize the codec with coding rate 512kbps. Based on the same SINR distribution and equation (3.16) by fluid model has a 2% difference with obtained with delay constraint, 500ms and there is a 25% difference with delay constraint 150ms. Therefore, we say that fluid model is enough to describe the delay performance with multi-class of users for the service like live TV streaming. Validation with fading effect For the completeness of the study, we consider the fast fading effects described by using Rayleigh distribution. Considering a Rayleigh fading with σ = 2/π and coherent time, 1ms, we show the simulation configuration of five different serving times for each user class in Table 3 We then show the outage rate of fluid model as γ fluid,fading = n π(n)1 ρ>1 (3.17) where ρ = k n k i p k,i S k,i A p (3.18) and the outage rate of exact two-level model can be obtained by adjusting the equation (3.11) to Pn (s) = P n (T ≤ x) = (ρ -1) λ -s -λ K k=1 n k n i p k,i e -sS k,i -1 (3.19) γ MD1,m = n π(n)P n (T ≤ D) (3.20) where p k,i stands for the portion of different channel conditions in the same class of users and S k,i stands for the serving time in the Table Summary In this chapter, we include the impact of flow-level dynamics into the calculation of the packet delay of real-time video service. Under the quasi-stationary property, we calculate the packet delay performance with M/D/1 queue at packet level and combine it with the stationary distribution of flow-level dynamics. Chapter 4 Model of Adaptive Streaming Traffic In the previous chapter, we have presented a model of real-time streaming traffic. As we know that on-demand video account for larger proportion of traffic than the real-time streaming and the fact that services like YouTube and Netflix [4] becomes very popular, a traffic model to analyze the impact of different configuration of HTTP streaming is needed, especially HTTP Adaptive Streaming (HAS) becomes a mature and popular technical solution [START_REF] Mok | QDASH: a QoE-aware DASH system[END_REF][69] [START_REF] Cicco | An experimental investigation of the akamai adaptive video streaming[END_REF][15] [START_REF] De Cicco | Feedback control for adaptive live video streaming[END_REF]. As we mentioned in the introduction chapter, the biggest difference between real-time streaming and HTTP streaming is the existence of a video playout buffer. Moreover, TCP is the transport layer protocol used for HTTP streaming. In this chapter, we develop a general traffic model that aims at helping operators to assess the quality-of-service perceived by their users and properly dimension their networks. We apply the well-known flow-level model, where a flow can represent either a video streaming session or a elastic session. Problem statement and state of the art Flow-level model is widely used for evaluating the impacts of traffic for elastic traffic and real-time adaptive streaming. In chapter 2, we introduce how flow-level model can be used for evaluating the performance of elastic data. It is also applied for real-time streaming. For instance, authors of [START_REF] Bonald | On performance bounds for the integration of elastic and adaptive streaming flows[END_REF] investigated the integration of elastic and streaming services by modeling the real-time adaptive streaming as a flow and only provided the performance bound because the insensitivity property does not hold. However, the considered streaming services are modeled as a specific type of streaming, real-time adaptive streaming. Paper [START_REF] Borst | Integration of streaming and elastic traffic in wireless networks[END_REF] examined the video performance with different scheduling methods with the video QoS expressed by a blocking rate with the same assumption of real-time adaptive streaming. Another work also considers the performance of real time streaming services on the flow level [START_REF] Blaszczyszyn | Quality of Real-Time Streaming in Wireless Cellular Networks -Stochastic Modeling and Analysis[END_REF]. Moreover, it also focuses on the same metrics, flow blocking rate. Compared to the real-time streaming modeling, modeling for the HTTP streaming is still at the early stage. Most of the existing works focus on the evaluation of HTTP streaming. There is no mature traffic model for HTTP adaptive streaming and its performance trade-offs. The above mentioned flow level models focused on classical elastic and real-time streaming services and does not consider the impacts of buffer. Moreover, classical performance models developed for real-time adaptive streaming and elastic services are not suitable for HAS as the latter service has similarities with both types of traffic. Flow-level model has been applied in studies of HTTP streaming [100] [START_REF] Xu | Analysis of buffer starvation with application to objective QoE optimization of streaming services[END_REF] with infinite video buffers. The KPIs like starvation probability have been computed using a detailed buffer analysis. However the mentioned model is not suitable to be adapted for evaluating the performance of HTTP adaptive streaming because of the lack of consideration for rate adaptivity. Indeed, the works that considered real-time streaming traffics were limited to real time streaming and works about HTTP streaming did not consider video bit rate adaptivity. Works of HAS include [START_REF] Ye | Analysis and modelling quality of experience of video streaming under time-varying bandwidth[END_REF], where authors propose an analytical framework for HTTP adaptive streaming under the assumption of fix frames arrival rate for different video bit rates and [START_REF] Xu | Analytical QoE models for bit-rate switching in dynamic adaptive streaming systems[END_REF], where authors model the frame arrival as Markov modulated fluid arrival. Both do not consider the impacts of other traffic and overall system loads. For further studies, it is better to combine the impacts of other traffic to the packet arrival rate, which is the most difficult part. How to allocate resources for both streaming and elastic services becomes a question for operators. It has been well-known that wireless system capacity can be enhanced with multi-user diversity using opportunistic schedulers from [START_REF] Tse | Multiuser diversity in wireless networks[END_REF][10] [START_REF] Ayesta | A modeling framework for optimizing the flow-level scheduling with time-varying channels[END_REF]. Slow channel variations due to the mobility can be exploited as well even under a blind fair scheduling strategy like round-robin [START_REF] Abbas | Opportunistic gains of mobility in cellular data networks[END_REF][29] [START_REF] Borst | Capacity of wireless data networks with intra-and inter-cell mobility[END_REF][58] [START_REF] Borst | Mobility-driven scheduling in wireless networks[END_REF]. Applying flow-level dynamics, authors of [START_REF] Abbas | Opportunistic gains of mobility in cellular data networks[END_REF] analyze the performance impacts of mobility in the presence of elastic traffic and they suggest that operators deploy opportunistic schedulers for the elastic data in order to profit from the multi-user diversity generated by the users' mobility. However, as video streaming service like YouTube and Netflix account for larger part of system traffic, if there is a model for HTTP adaptive streaming, it will facilitate operators to understand whether these suggestions are still valid for streaming services. Main contributions Our first contribution is to develop a flow-level model for the adaptive streaming traffic taking into account the main characteristics of this service, the presence of playout buffer, the different configurations of video chunk duration, video bit rates, scheduling schemes and users' mobility. Our second contribution is to extend this model to consider heterogeneous radio conditions and a mixed service between streaming and elastic traffic. As of measures, we believe that users are satisfied if they watch the video with a high video bit rate and if the video play back is smooth, i.e. the playout buffer never gets empty. We look at the KPIs that directly influence the Quality of Experience (QoE) of users such as the average video bit rate observed during the video session and the starvation probability, i.e. the probability that the video buffer becomes empty. Although our flow-level model is able to compute the QoS such as average video bit rate, the computation of starvation probability needs a packet-level analysis as it depends on the behavior of the player in terms of prefetching policy and the detailed buffer state. As our objective is to provide simple models that can be used for mobile network dimensioning purposes, we examine several KPIs related to the starvation probability and they can be computed using the flow-level modeling: the deficit rate expressed as the probability that the instantaneous user throughput is lower than the chosen video bitrate, the buffer surplus representing the average buffer variation during the video download and the mean service time representing the average time to transmit a video session. The model with the consideration of different network parameters are published in three scientific papers. We first introduce a specific flow-level model for adaptive streaming considering with buffer in [START_REF] Bonald | A flow-level performance model for mobile networks carrying adaptive streaming traffic[END_REF], then we examine the impacts of the system parameters, video chunk duration, in [START_REF] Lin | Impact of chunk duration on adaptive streaming performance in mobile networks[END_REF] and in [START_REF] Nivine Abbas | Mobility-driven scheduler for mobile networks carrying adaptive streaming traffic[END_REF], model considering the mobility and different scheduling schemes is introduced. Chapter organization The remainder of this chapter is organized as follows. Section 4.2 describes the key components for deliver HAS and introduces the important configuration, video chunk duration, that we are going to examine. In section 4.3, we develop the flow-level model for adaptive streaming traffic with small and large chunk duration respectively and we define the performance metrics in section 4.4. Then we introduce different heterogeneous radio conditions and different scheduling schemes into our system 4.5 and model are extended to consider the integration of elastic and streaming services in section 4.6. Finally, the users' mobility between different capacity regions are modeled and considered in section 4.7. System description This section introduces two key aspects that influences the performance of HTTP adaptive streaming services delivered in wireless networks: video content configuration and wireless access network. According to the mechanism of HTTP adaptive streaming we have introduced in chapter ??, a video is separated by several video chunks (segments) as Fig. 1.1 and they are requested one after another by HTTP requests. The proper and corresponding video bit rate is selected at the beginning of each chunk download. Fig. 4.1 gives an example showing how a video download is composed by a bunch of chunks. The chunk duration, h, is a system parameters that service providers can control. Video content configuration Intuitively speaking, by selecting a shorter duration, users have more chances to adapt its video bit rate. In this chapter, we are going to study the analytical results of two extreme chunk duration configurations shown in the following Table 4 Wireless access network In this section, we focus on the performance of adaptive streaming delivered in a typical cell as Fig. 4.2. Mobile users in the cell download the streaming traffic to they buffer by the allocated resources of the cell. We begin, for the ease of understanding of the model, by a homogeneous radio condition where all users are supposed to see a capacity R equal to the average radio condition over the cell. Section 4.5 and 4.6 will show how to extend these models to multiple radio conditions and how to integrate other services like elastic traffic. Moreover, to facilitate the flow level modeling, for the streaming system configuration, we assume that each flow has an infinite playout buffer so that each will continuously download until its corresponding video is fully downloaded. During the transmission, each flow requests the sequential video chunks according to the allocated resource. As we mention before, the streaming chunk duration impacts how users adapt their video bit rates. System model with flow-level dynamics In this section, we first consider adaptive streaming services, the mix with other services is considered later. In addition, we make a classical assumption that traffic flows arrive according to a Poisson process with intensity, λ and the video duration of a flow is assumed to be independent and exponentially distributed with mean T . Small chunk duration model • Markov model To model the dynamics of flow number, X (t), we utilize the model of processor-sharing queue. With the assumption of homogeneous radio condition and small chunk duration, we model all the flows in one class. As X (t) = x, meaning that x flows are served in the system, the flow departure rate, µ(x), can be expressed as µ(x) = φ(x) σ(x) , (4.1) where φ(x) stands for the physical throughput allocated to all UEs of the cell at state x and σ(x) stands for the remaining flow size at state x. As we consider an infinite buffer size at users' side, users can profit at the maximum from the throughput allocated to them, occupying thus the whole scheduling time of cell, leading to an allocation φ(x) = R in this case. As of the remaining flow size, σ(x), as we consider a small chunk duration leading to an instantaneous adaptation, the video bit rate, v(x), at state x depends only on the number of flows x and not on the history of the system. Furthermore, as the video duration is assumed to be exponential, the memoryless property implies that the flow size at state x is also exponentially distributed with its mean σ(x) = v(x)T. (4.2) X (t) is thus a Markov process whose departure rates depend on the video bit rate selection. We show in the following sections how this bit rate is computed. With Eq. (4.1) and (4.2), flow departure rate then is obtained as µ(x) = φ(x) v(x)T and the system can be easily shown to have a product-form stationary distribution computed as π(x) = π(0) x n=1 λ µ(n) , (4.3) where π(0) = 1 + ∞ x=0 x n=1 λ µ(n) -1 . • Video bit rate selection Video bit rate chosen by each flow is decided based on the instantaneous throughput, γ(x), that flow observes. In flow-level model, by applying round-robin scheduling policy, the instantaneous throughput can be calculated as γ(x) = R x . ( 4.4) In reality, video bit rates are not continuous. Instead, flows will select a specific video bit rate from a set of discrete video bit rates = {v 1 , • • • , v I }, where we assume v 1 > • • • > v I . Knowing the throughput, users will select the video bit rate as v(x) = γ(x) , when γ(x) > v I , v I , when γ(x) ≤ v I , (4.5) where z stands for a function selecting a maximum value in but lower than z. It can also be observed that γ(x) is always equal to v(x) or larger than v(x) only when γ(x) < v I . With the defined video bit rate selection mechanism, when γ(x) ≥ v I , users' video buffer will increase or remain the same. Instead, video buffer will decrease only when γ(x) < v I . Large chunk duration model With the same system characteristic mentioned in section 1, here, we consider the streaming system with infinitely large chunk duration, where flows choose their video bit rate at the beginning of their arrivals and keep the one until the end of download. Different from the case of small chunk duration, where we model the system with one class of user who select the same video bit rate at the same time. For configuration of infinite chunk duration, multiple classes of queue are needed to describe the number of flows choosing different video bit rates at given time. Same as previous section, the discrete set of video bit rates is denoted as = {v 1 , • • • , v I }, where | | = I. Because flows with different video bit rate possess different arriving and departure rate, we denote the state of system as x = (x 1 , • • • , x I ) ∈ N I , where x i represents the number of flows that chose video bit rate v i at its arrival. The flow arrival rate of queue-i depends on the total number of flows in the system, |x | = i x i . Given a state x , all the flow arrival will only select video bit rate v i . Therefore, we have ∀i, λ i (x ) = λ, if v(x + e i ) = v i , 0, otherwise, (4.6) where v(x ) is defined as Eq.(4.5) and e i represents a vector with an unit value at class i. Applying the same concept of Eq. (4.1), flow departure rate and the allocated resource of class i is denoted as µ i (x ) = φ i (x ) v i T , (4.7 ) φ i (x ) = x i R |x | , ( 4.8) with the same setting of Round-Robin scheduling. We then can calculate the stationary distribution π(x ) by using the arrival and departure rate of both cases in Eqs. (4.6-4.7) and solving the balance equations denoted as ∀x , i λ i (x ) + µ i (x ) π(x ) (4.9) = i λ i (x -e i )π(x -e i ) + µ i (x + e i )π(x + e i ). With the stationary distribution, π(x ) calculated in the case of small chunk duration and large chunk duration, we then define the key performance indicators in section 4.4. Stability condition The flow arrival rate should be smaller than the maximum flow departure rate. In the case that v n is the video bit rate selection of n-th flow, the maximum flow departure rate implies that ∀n, v n = v I , leading to the following stability condition for both chunk duration configuations and maximum flow arrival rate, λ max : λ < R v I T ⇒ λ max = R v I T . ( 4.10) If v I = 0, the system is always stable. However, when v I > 0, the system becomes unstable with a large number of arrivals. KPIs definition To evaluate the QoE of adaptive streaming service, we propose four key performance indicators, mean video bit rate, mean service time, deficit rate and buffer surplus. All of them are defined based on the stationary distribution π(x ). Video bit rate Mean video bit rate stands for the average video bit rate that a flow experiences during playing its video. When mean video bit rate is high, user has a better video experience. We calculate the cell-average video bit rate using the following concept, Video Bit Rate = Allocated Resource #Flows Served , (4.11) where we devide all the allocated resource on the number of flow multiplied the mean video duration to calculate mean video bit rate. Then we define the overall mean video bit rate, v for both small and large chunk duration, v = x:x>0 π(x)φ(x) λT , v = x :|x |>0 π(x ) i φ i (x ) λT , (4.12) where |x | = i x i . A popular QoE indicator used to evaluate streaming performance is the starvation probability [START_REF] Xu | Impact of Flow-level Dynamics on QoE of Video Streaming in Wireless Networks[END_REF]. Even if starvation happens only when the video bit rate is larger than the instantaneous throughput, the latter condition is not a sufficient condition for starvation as the buffer may counteract the impact of short periods of low throughput. The computation of the starvation probability has to take into account the memory of the system by introducing the buffer size in the Markovian analysis as in [START_REF] Xu | Impact of Flow-level Dynamics on QoE of Video Streaming in Wireless Networks[END_REF]. Here, we introduce and examine three KPIs called service time, deficit rate and buffer surplus. Service time To evaluate the starvation probability, we propose another KPIs, mean service time of a video flow, which is calculated by the Little's formula as S = L λ = x λ , S = L λ = x λ , (4.13) with x = x>0 xπ(x), x = |x |>0 |x |π(x ) and that L = x represents the mean number of flow. By observing S, we can imply the starvation probability, which is positively related with S. Saying that smaller S could imply smaller starvation probability. Deficit rate As [START_REF]Network Performance Analysis[END_REF] mentioned, flows have higher probability to stay in the state for downloading v(x) because x users exist. Therefore, by weighting the corresponding metrics at state x by the number of flows, x, also called as size-biased distribution we define the following metrics. The deficit rate is equal to the probability that an ongoing flow sees its instantaneous throughput lower than its chosen video bit rate. As the adaptation of video bit rate is assumed to occur instantaneously in reaction to the variations of the observed throughput, the deficit rate is defined by the probability that the instantaneous throughput, γ(x) = φ(x) x , is smaller than v(x) in the case of small video chunk duration or γ i (x) = φ i (x) i x i , is smaller than v i in the case of large video chunk duration. Note that for small chunk duration configuration deficit happens only when γ(x) < v I , based on the selection mechanism, Eq. (4.5). The overall deficit rate is defined by weighting the stationary distribution at different state x with the number of flows: D = Pr{γ(x) < v(x)} = x:x>0 xπ(x) x 1 {γ(x)<v(x)} , (4.14 D = Pr{γ i (x ) < v i (x )} = x :|x |>0 π(x ) x i x i 1 {γ i (x )<v i } . (4.15) Starvation probability is also positively related to the deficit rate. Therefore, larger deficit rate will cause larger starvation probability. Buffer surplus We also introduce another performance metric called buffer surplus, which represents the average buffer variation of each flow in a second. It is calculated by weighting all the buffer variation, γ(x)-v(x) v(x) , at each state x as B = x:x>0 xπ(x) x γ(x) -v(x) v(x) , (4.16) for the small video chunk duration configuration. When γ(x) > v(x), users' buffer accumulates certain amount of duration of video. When γ(x) < v(x), then user starts to consume the video packets stored in the buffer, which reduces the values of average buffer surplus. For the case of large chunk duration, buffer surplus is calculated as B = x :|x |>0 π(x ) x i x i γ i (x ) -v i v i . (4.17) Larger buffer surplus will decrease the starvation probability. Therefore, it is negatively related to the starvation probability. Performance of Different KPIs v.s. Starvation Probability In order to demonstrate the relationship between desired starvation probability and our introduced KPIs, we simulate the real starvation probability with event-based simulation considering the initial buffer, I, which is a parameter not included in the flow-level model. For the simulation we set up the following configurations, R = 10Mbps, T = 20s, and (v 1 , v 2 ) = (2, 1)Mbps. In Fig. 4.3, we can observe the correlation of starvation probability and our defined KPIs and that we can first observe that three defined KPIs are not dependent on the initial buffer values, I. Moreover, we can also observe some simple correlation between our proposed KPIs and the stationary probability. For example, deficit rate is always the upper bound of starvation probability. Second, when buffer surplus value is equal to zero, starvation probability is around 10 -20%. Third, mean service time should be smaller than the whole video duration. Otherwise, the starvation probability becomes high. In chapter 6, that works that we predict the QoE of video streaming using each users' features is inspired by the insufficiency of correlation between QoE and our proposed KPIs. Scheduling schemes Scheduling is an important topic discussed a lot in the system design of LTE. It is also one of the most important parameters that internet service providers can control. In this section, we first introduce the model considering heterogeneous channel condition and we present several scheduling schemes implemented knowing the channel information. Heterogeneous radio conditions Based on the 3GPP LTE-A standards [START_REF] Dahlman | 4G: LTE/LTE-Advanced for Mobile Broadband[END_REF], users with various positions have different discrete Channel Quality Indicator (CQI), for example from CQI-1 to CQI-15. Therefore, traffic flows can be separated into several classes having a radio condition under round-robin, max C/I [START_REF] Kolding | High speed downlink packet access: Wcdma evolution[END_REF] , max-min [START_REF] Bonald | A score-based opportunistic scheduler for fading radio channels[END_REF] and opportunistic scheduling schemes. As we model each cell by a set of K regions. In each region, radio conditions are supposed to be homogeneous and thus users are served at the same physical data rate on the downlink. In the simple case of two regions illustrated by Figure 4.5, users may be close to the cell center and experience good radio conditions (light gray) or close to the cell edge and suffer from bad radio conditions (dark gray). We model each region by a queue with a specific service rate corresponding to the physical data rate in this region; since all users in the cell share the same radio resources, each cell can be viewed as a set of K parallel queues with coupled processors. The precise coupling depends on the scheduling policy, as explained below. R k , R k ∈ = {R 1 , • • • , R K }, where R 1 ≥ • • • ≥ R K . Each Round-robin scheme Under the round-robin policy, users share the radio resources equally, independently to their radio conditions. Thus users in region k are allocated a fraction of radio resources in state x as ϕ k (x ) = x k j x j . ( 4.18) Then the radio resource allocated to users in region k is calculated as φ k (x ) = ϕ k (x )R k . (4.19) At high load all users in all regions select v min = v I as their video bit rate under the round robin policy. Thus the stability condition follows from (4.10): ρ = k p k λ R k v min T < 1 → λ max = k p k v min T R k -1 , (4.20) where λ max is the maximum arrival rate that the system can handle. Observe that this corresponds to the less restrictive stability condition that can be obtained from (4.10). Max C/I scheme The max C/I policy is an extreme case of opportunistic scheduling strategies that prioritizes those users with the best radio conditions. For two regions for instance, cell-center users are scheduled first and are allocated all the resources whenever active; cell-edge users are served only when there are no active cell-center users. Stability condition approximation: Under the max C/I policy, base station first allocate its resources to the cell-center flows. Then it allocates the rest of resources to users having lower physical throughput. The maximum traffic intensity happens when the rest of resource allocated to the cell-edge users can only support for the selection of v min , which is shown as λ max = k =K p i vk T R k + p K v min T R K -1 , (4.21) where vk is also a function of λ max . Therefore, λ max of max C/I policy can be solved as a fixed-point solution of Eq.(4.21) and vk is calculated as vk = ∞ n=0 π k (n)v k (n), (4.22) with π k (n) = π i (0) n j=1 p k λ max R k v k ( j)T , (4.23) v k (n) = max min R k n , v max , v min , (4.24) where R k = j<k π j (0)R k and π k (0) is denoted as the probability that there is no flows in class-k. Max-min scheme The max-min policy achieves fairness through users throughput equalization. Users in good radio conditions are allocated fewer resources while users in bad radio conditions get the largest share, so that all users get the same throughput: ϕ 1 (x )R 1 x 1 = ϕ 2 (x )R 2 x 2 = . . . ϕ K (x )R K x K . Thus users in region k are allocated a fraction ϕ k (x ) = x k /R k j x j /R j (4.25) of radio resources in state x . Similarly to the round robin policy, all users get v min at high load and the stability condition is given by (4.20). Opportunistic scheduling scheme The capacity shares in equation (4.18) are computed supposing a round robin scheduling. However, we consider a channel aware scheduling. A proportional fair scheduler that operates at the fast fading time scale, as that of [START_REF] Tse | Multiuser diversity in wireless networks[END_REF][64], is assumed. In this case, when there are |x | flows that are active in the LTE cell, the throughput of a user of radio condition R k that gets a proportion ϕ k of the cell resources is equal to ϕ k R k G(|x |), where G(|x |) is the opportunistic scheduling gain that depends on many parameters such as the channel model, the receiver and the Multiple Input Multiple Output (MIMO) scheme [START_REF] Combes | Cross-layer analysis of scheduling gains: Application to lmmse receivers in frequency-selective rayleigh-fading channels[END_REF]. Note that, contrary to real time streaming that does not profit from the opportunistic scheduling gain due to its stringent delay constraints, http streaming with buffering profit from this type of scheduling like elastic traffic. The performance model proposed in 4.3 can thus be extended to the opportunistic scheduling case by introducing G(|x |) into the formulation of allocated resource in Eq. (4.1) and (4.8) and video bit rate selected in either discrete or continuous set. Note that, G(|x |) = 1, when round-robin scheduling is applied. on the stationary distribution π(x ), in the static case, we can compute the mean video bit rate of users in each region i as follows: vk = E (ϕ k (X )R k ) λ k T . ( 4.26) The cell mean video bit rate in both scenarios (without and with mobility) is given by: v = E k ϕ k (X )R k k λ k T . (4.27) Intuitively speaking, the mean video bit rate is obtained by dividing the average wireless resource allocated to different classes by the average number of arriving flows. In the case with mobility, only the cell mean video bit rate v can be calculated. However, in the absence of mobility it is observed that v is nothing then the arithmetic mean of vi weighed by the probabilities p k : v = k p k vk . (4.28) Buffer surplus As we have introduced in section 4.4, the distribution seen by users in region k is the size-biased distribution. In the following discussion, we denote by E i the expectation using corresponding biased distribution: π k (x ) ∝ x k π(x ). In addition to the mean video resolution, the quality of experience is also influenced by the video smoothness measured in terms of buffer surplus. By calculating the buffer surplus as Eq. 4.29, this performance metrics reflects the average relative buffer variation of each flow. B k = E k R k ϕ k (X )/X k -v k (X ) v k (X ) (4.29) = x π k (x ) R i ϕ k x k -v k (x ) v k (x ) = x π(x ) R k ϕ k v k (x ) -x k x x k π(x ) . That is: Bk = E R k ϕ k (X ) -X k v k (X ) v k (X ) /E (X k ) . Similarly, the mean buffer surplus over the cell is: B = E k R i ϕ k (X ) -X k v k (X ) v k (X ) /E k X k . (4.30) Note that B = k p k B k , with p k = E (X k ) j E X j where p k represents the probability that an active user is in region k. When load becomes large, both B and B k values approach -1. Integration of elastic services Section 4.3 proposed performance metrics for adaptive streaming and showed how to compute them when only adaptive streaming flows share the capacity with both cases of small and large chunk duration. Here, we are going to generalize the model to heterogeneous radio conditions and discuss the performance of these flows in the presence of elastic traffic. Here we take round robin scheduling as the example. For other scheduling schemes we can simply change the shared resources in Eq. (4.33). We also consider an extended setting where streaming flows share the cell capacity with elastic flows. Therefore, the system can be represented with two groups of coupled processor-sharing queues. Flows in queue e, k correspond to the elastic traffic with radio condition, R k , and flows in queue s, k, i correspond to the streaming ones with R k and selecting v i . To remind, for small chunk duration, there is only one video bit rate i corresponds to a specific radio condition R k at a given time, therefore, queue s, k, i can be simplified as s, k. However, we show the general case in the following. With the proportion of elastic and streaming denoted as p e , p s and p e + p s = 1, the flow arrival rates at each queue are calculated as λ e k = p e p k λ and µ e k (x ) = φ e k (x ) σ , µ s ki (x ) = φ s ki (x ) v ki (x )T , ( 4.32) where σ is the mean flow size of elastic data. Besides, φ e k (x ) and φ s ki (x ) stand for the allocated wireless resources for each class. These capacity shares can be computed as: φ e k (x ) = x e k R k |x | , φ s ki (x ) = x s ki R k |x | , ( 4.33) where |x | = k (x e k + i x s ki ). Moreover, for small chunk case, the video bit rate, v ki (x ) = v k (x ), is calculated based on the mechanism (4.5) and the instantaneous throughput, γ s ki (x ) = φ s ki (x ) x s ki = R k |x | . ( 4 Stability condition We begin by assessing the stability region of the system. Applying the same implication of section 4.3.3, when system approaches the stability limit, the video bit rate of streaming users decreases to v I . Both streaming flows with small chunk and large chunk behave like elastic traffic. Stability holds only when the sum of offered loads for both services is less than 1, where we obtain the max system traffic intensity, λ max , as k λp e p k σ R k + λp s p k v I T R k ≤ 1 ⇒ λ max = k R k p e p k σ + p s p k v I T -1 . (4.36) KPIs definition for integrating elastic traffic We extend the adaptive streaming related KPIs to general case and define a KPI for the performance of elastic traffic. •Video bit rate We define the overall video bit rate for the general model, v, by summing up all the class and weighting by the flow number of each class at state x as v = x :|x s |>0 π(x ) k i φ ki (x ) λp s T (4.37) and the average video bit rate of class-k with R k is denoted as vk = x :x s ki >0 π(x ) i φ ki (x ) λp s p k T (4.38) where |x s | = k i x s ki . •Deficit rate The overall deficit rate of multiple class model, D and the deficit rate of each class with R k , D k are defined as D = x :|x s |>0 π(x ) x s k i x s ki 1 {γ s ki (x )<v ki (x )} , (4.39) D k = x :x s ki >0 π(x ) xs ki i x s ki 1 {γ s ki (x )<v ki (x )} , (4.40) where 1 is the indicator function equal to 1 when the condition is satisfied, otherwise the indicator function will become 0. In addition, x s = x π(x )|x s |, xs k = x π(x ) i x s ki . •Buffer surplus We then define the overall buffer surplus of multiple class model, B and the buffer surplus of each class with R k , B k as B = x :|x s |>0 π(x ) x s k i x s ki γ s ki (x ) -v ki (x ) v ki (x ) , (4.41) B k = x :x s ki >0 π(x ) xs k i x s ki γ s ki (x ) -v ki (x ) v ki (x ) . ( 4.42) •Service time Same as the previous section, the overall mean service time and mean service time of class-k are denoted as S = x s λ , S k = xs k λ k . ( 4.43) •Average elastic throughput The average elastic throughput is chosen as the performance metric for elastic flows, with γe and γe k standing for the overall metric and mean throughput for each class with R k , γe = Mobility model As Fig. 4.6 shows, we assume that each user in region k moves to region k -1 (for k > 1) and to region k + 1 (for k < N ) after exponential durations, at respective rates ν k,k+1 and ν k,k-1 . The probability that a user is in region i then satisfies: Note that this is not the probability that an active user is in region k, which is given by p k . In addition, based on the mobility rates, the stationary distribution, π(x ), of the Markov process is the solution of the following balance equations: q k ∝ k-1 j=1 ν j, j+1 ν j+1, j . R 2 λ 2 R 1 λ 1 ν 2 ν 1 k (λ k + µ k (x ) + ν k,k+1 + ν k,k-1 )π(x ) = k λ k π(x -e k ) + µ k (x + e k )π(x + e k ) + k x k+1 ν k+1,k π(x -e k + e k+1 ) + x k-1 ν k-1,k π(x -e k + e k-1 ) . Stability condition In this section, we show how to calculate the λ max obtained at the stability condition, which follows from the limiting regime of infinite mobility where ν k,k+1 , ν k+1,k → ∞ and is given by: λ/ μ < 1, where μ is the mean service rate at high load. We shall see that this service rate depends only on the scheduling strategy and is independent from the mobility rate in the presence of mobility. The λ max is defined as λ max = ū. round-robin policy The mean service rate at high load under the round robin strategy is given by: μ = k q k R k v min T . max C/I policy Under the max C/I policy all mobile users are served in the first region (that of the best physical rate R 1 ) at high load. It follows that: μ = R 1 v min T . max-min policy Under the max-min policy, the mean service rate at high load follows from (4.25): μ = 1 v min T k q k j q j /R j . Observe that in the presence of mobility the less restrictive stability condition is obtained under the max C/I policy. However in the absence of mobility this strategy engender the most restrictive stability condition compared to more fair allocation strategies (round robin, max-min). Performance in light traffic The performance in light traffic (that is, when ρ → 0) is independent of the scheduling policy. Indeed, a user when alone in the system is always allocated all radio resources. As Eq. (4.5), the video bit rate in region k can be extended to a continuous set as c k = max (min (R k , v max ) , v min ), that is v max in all regions k where R k > v max . The mean buffer surplus in region k is then: b k = R k -c k c k . Thus the mean buffer surplus in the cell in light traffic is given by: B = k p k b k . When ν k → 0, ∀k ≤ K (no mobility) the mean buffer surplus in the cell in light traffic can be written as: B = k p k /µ k j p j /µ j b k , where µ k = R k c k T . However, in the limiting regime of infinite mobility ν k → ∞ ( ∀k ≤ K) the cell mean buffer surplus is given by: B = k q k b k . For two regions for instance, explicit expressions of the cell buffer surplus in light traffic as a function of the mobility rates can be written as: the case of large chunk duration. Fig. 5.1b tells us that deploying small chunk duration can be beneficial for mean elastic throughput. In terms of buffer performance, we observe that deploying small chunk duration will result in better deficit rate in Fig. 5.2a and that deploying small chunk duration will result in a better buffer surplus in Fig. 5.2b. We also get the similar result in terms of mean service time in Fig. 5.2c. These results show that configuring B = (ν 2 + p 1 µ 2 )b 1 + (ν 1 + p 2 µ 1 )b 2 ν 1 + ν 2 + p 1 µ 2 + p 2 µ 1 . One chunk per HTTP request Assuming that the video chunk duration is configured as h for all video bit rates, v i ∈ and that a general video has duration T , users are going to download their video with a total number of video chunks, N (1) h , calculated as N (1) h = T h , ( 5.1) where function x gives a largest integer value less than x. In this case, all the video bit rates adopt the same value of chunk duration, which has the same configuration as what we have discussed in the previous sections. Here, instead of choosing only one video chunk duration, h, for all the video bit rates, we assume that for different video bit rates, v i , a specific number of chunks a i , where a i ∈ and a i ≥ 1, will be downloaded in an HTTP request. Therefore, it seems that for video bit rate, v i , we have a longer chunk duration h i = a i h as shown in Fig. 5.3. Assuming that each video bit rate is selected equally, we have the number of HTTP requests calculated as Multiple chunks per HTTP request following same size of requests N (2) h = v i ∈ T h i I ≤ T h v i ∈ 1 a i I ≤ T h = N (1) h . (5.2) With the fact that a i ≥ 1, we can show that N (2) h ≤ N (1) h . In order to facilitate the design of {a i }, we propose to select the set, {a i }, as follows: a i = v 1 v i . ( 5.3) This means that for a low video bit rate, more video chunks will be transmitted in an HTTP request. An example is given as Fig. 5.3. Several solution sets of chunk durations, {a i }, are feasible. In this thesis, we only examine the proposed design guideline figuring out the number of video chunks requested in a HTTP request for a certain video bit rate. There might be other ways to design {a i }, for example to design chunk duration proportional to their video bit rate. These alternatives are not studied in this thesis but can be part of the future works. Performance comparison In order to analyze the performance of proposed policies, with the system configuration where = {R 1 , R 2 } = {10, 5}Mbps, (p 1 , p 2 ) = (0.5, 0.5), T = 30s, = {v 1 , v 2 } = {2, 0.5}Mbps, we compare two different ways to download video chunks, one chunk per HTTP request and multiple chunks per HTTP request. We simulate the HTTP adaptive streaming with the mentioned system configuration but with different numbers of chunks requested for two different video bit rates shown in Table 5.1. For the proposed policy, we serve the smaller video bit rate v 2 with a longer chunk duration, a 2 , compared to the first policy. with the first policy, our proposed policy offers slightly worse performance for the mean video bit rate but better performance for the rest of metrics. The degradation of mean video bit rate is not very significant. However, the improvements of other smoothness-related metrics are large. By comparing the simulation results between Fig. 5.1, 5.2 and Fig. 5.4, it can also be concluded that our proposed policy has similar effects as one chunk per HTTP request policy with shorter chunk duration, e.g. h = 9s in this case, which means that the proposed policy needs less video chunks stored at servers while provides the same performance. In addition, adaptive streaming systems that deploy multiple chunk per request policy will also generate less HTTP requests compared to the one chunk per HTTP request policy with the same video chunk duration. Impacts of the number of video bit rates In the previous simulation, video bit rate set is configured as = {v 1 , v I }. However, a general question is what are the impacts if more video bit rates are available to be chosen in ? In order to answer, we configure a continuous video bit rate set cont = {v : v 1 ≤ v ≤ v I } with configuring small video chunk duration. In this case, | cont | = ∞. Different from Eq. (4.5), users' flows at capacity region k are assumed to be able to select any video bit rate between v 1 and v I , expressed as v k (x ) = max min γ k (x ), v 1 , v I , (5.4) where γ k (x ) can be found in Eq. (4.34) and based on Eq. (4.32), the flow departure rate becomes µ k (x ) = φ k (x ) v k (x )T = R max min γ k (x), v 1 , v I T . (5.5) When the available throughput γ k (x ) = R k |x | is larger than v 1 , the buffer starts filling as the download capacity is larger than the playout rate and the flow behaves thus as an elastic one as there is no limitation on the buffer size. On the other hand, when γ k (x ) is smaller than v I , the behavior is also elastic but with possible starvation if the buffer empties due to the constant playout rate. Between v 1 and v I , the flow behaves as a real time one and the buffer size remains constant. Solving the balance equations as Eq. Moreover, in Fig. 5.7, bad buffer surplus shows that the starvation event will strongly happen for the cell-edge users under the max C/I policy. In [START_REF] Abbas | Opportunistic gains of mobility in cellular data networks[END_REF], it is concluded that there is no difference to deploy either round-robin or max C/I policy in the absence of mobility. However, for adaptive streaming services, it is showed that a trade-off exists between the two performance metrics and max C/I policy degrades system stability conditions. With little improvement of mean video bit rate and large degradation of system stability, operators are suggested to deploy round-robin policy for adaptive streaming in the static case. Impacts of intra-cell mobility In this section, we discuss the impacts of intra-cell mobility introduced in Fig. 4.6. We suppose that there is only two regions and users move from the center of the cell to the edge and vice versa. We still assume that there is no fast fading. Considering two capacity regions stand for cell-center and cell-edge with the same traffic and system configurations as before, we suppose that mobility rates are symmetric, that is ν 1 = ν 2 = ν. We consider the static case ν = 0 and a case with a mobility rate ν = 1. The results shown in Figure 5.8 and 5.9 are obtained by the numerical evaluation of the stationary distribution π(x ) of the Markov process X (t). For the video bit rate, we can only obtain the average all over the cell, because when users' mobility is considered, from the flow-level model, no flows belong to only one class. In addition, it can be shown that max C/I policy provides better mean video bit rate and higher system stability. On the other hand, for the mean buffer surplus, max C/I policy elastic data. However, for the adaptive streaming suggested policy depends on the desired optimized performance metric. In the case of mobile users, there is a trade-off to deploy either round-robin or max C/I policy. Therefore, we examine the performance of discriminatory scheduler [START_REF] Altman | Fairness analysis of TCP/IP[END_REF] to provide some intermediate results between the two scheduling policies. The resource allocation of users in region k under the discriminatory scheduler is calculated as follows: Services φ k (x ) = w k x k j w j x j R k , (5.6) where w k is the weight value of users in region k. Impacts of largest video bit rate With the configuration of continuous video bit rate shown in Eq. (5.4), in this section, we demonstrate the impacts of maximum video bit rate, v I = v max on the four performance metrics. Under the same system configuration as previous section but continuous video bit rate set, four performance metrics are shown with different v max configurations in Fig. 5.12. It can be observed that when v max increase, vs and D s increase, but B s and γe decreases, meaning that in the case of v min = 0.5Mbps, decreasing v max can benefit vs , D s and B s regardless of the trade-offs of γe reduction. We can also observe that the deficit rate is not highly influenced when v max is large. Therefore, we believe that compared with the deficit rate buffer surplus has more information than deficit rate which better represents the starvation performance of streaming users. Approximation model Section 4.3 introduces the general model considering the heterogeneous radio conditions with small and large chunk duration. However, the complexity of solving the balance equations mentioned in Eq. (4.35) increases polymonially with the number of classes, making the computation of the stationary distribution very difficult. In this section, we propose an approximation model which can reduce the number of class while keeping the access to the defined KPIs. Here the approximation model is proposed for adaptive streaming with consideration of elastic traffic. Approximation model for significantly small chunk duration Here, we approximate the multiple classes of users to the processor-sharing model carrying only one equivalent class of elastic and adaptive streaming, where the system state is denoted as x = (x e , x s ). The equivalent flow arrivals of these two queues are expressed as: λe (x ) = p e λ k p k η k , λs (x ) = p s λ k p k α k (x ) η k , ( 5.7) where η k = R k R K . ( 5.8) p k and η -1 k weights the contribution of the k-th capacity region to the overall arrived rate. Therefore, classes with large R k have reduced impact. α k (x ) is another factor that impacts the equivalent arrival rate, which represents the video bit rate ratio depending on the definitions of continuous video bit rate, Eq. (5.4) or discrete one, Eq. (4.5): Here, we give an example with continuous video bit rate selection, α k (x ) = v k (x ) v min = max min γ s k (x ), v max , v min v min . ( 5.9) On the other hand, the equivalent flow departure rates with the assumption of round-robin scheduling scheme are formulated as μe = x e |x | R K σ , μs = x s |x | R K v min T . ( 5 .10) With the flow departure rate and the flow arriving rate mentioned above, the approximated stationary distribution of π(x) can be obtained using the same concept of balance equations shown in Eq. (4.35). However, the complexity to solve the stationary distribution with one class is much lower. Here, we succeed to reduce the number of users class impacted by the radio conditions. Note that the stability condition of this approximation is same as Eq.(4.36) and that this approximation also holds when applying opportunistic scheduling; it is thus sufficient to multiply the throughput by the state dependent scheduling gain G(|x |). However, for the other scheduling schemes and consideration of users' mobility, more efforts for extension are needed. Calculation of approximated KPIs of small chunk duration As of the approximated performance metrics, the mean video bit rate, vs , the deficit rate, Ds and the buffer surplus, Bs for adaptive streaming traffic and the mean throughput for elastic traffic, γe , can be computed with the newly calculated stationary distribution π(x ) as vs x e π(x ) xe φ e (x ) = 1 λp s T x :x s >0 x s π(x ) |x | k β k (x ) β(x ) R k , (5.11) Ds = x :x s >0 x s π(x ) xs k β k (x ) β(x ) 1 {γ s k (x )<v min } , (5.12) Bs = x :x s >0 x s π(x ) xs k β k (x ) β(x ) γ s k (x ) -v k (x ) v k (x ) , ( 5 x e = x π(x )φ e (x ) xe , (5.14) where β k (x ) = p k α k (x ) R k , β(x ) = k β k (x ), γ s k (x ) = R k |x | , ( 5.15) and β k (x ) β(x ) represents the fraction of load volume of class-k streaming users when there are in total x users in the system. In addition, xs = x :x s >0 x s π(x ) and xe = x :x e >0 x e π(x ) stand for the average number of streaming and elastic calls in the cell. It is also worth of mentioning that the approximation model can also predict all the metrics for each R k as vs k = 1 λp s p k T x :x s >0 x s π(x ) |x | β k (x ) β(x ) R k , (5.16) Ds k = x :x s >0 x s π(x ) xs β k (x ) β(x ) 1 {γ s k (x )<v min } , (5.17) Bs k = x :x s >0 x s π(x ) xs β k (x ) β(x ) γ s k (x ) -v k (x ) v k (x ) , (5.18) where all the summations are taken off compared to the Eqs. 5.11 and elastic throughput can be obtained by γe k = R k R eq γe , with R eq = k p k R k . Approximation model for significantly large chunk duration In the case of large chunk duration, we propose an approximation model to reduce the system complexity. We can reduce the classes of users from both elastic and streaming services considering different radio conditions and video bit rates into a model classified by different video bit rate as x = (x e , x s v 1 , • • • , x s v i ). We reformulate the arriving rate as λe = p e λ k p k η k , λs i (x ) = k p k λ k,i (x ) η k , ( 5.19) where η k = R k R K has the same meaning as in previous section. Moreover, the corresponding flow departure rates are formulated as μe .20) Based on the arrival and departure rate, we get the stationary distribution π(x ) and we calculate the following KPIs using π(x ). = x e |x | R K σ , μs i (x ) = x s i |x | R K v i T . ( 5 Calculation of approximated KPIs for large chunk duration To calculate the KPIs, we need to know β k,i , the probability that a flow of v i belongs to radio condition R k . Knowing the π(x ), we can obtain the probability that flows of R k will choose v i , calculated as a k,i = x π(x )∞ { γ s k (x +e k,i ) =v i } , (5.21) and we then can calculate the value β k,i as β k,i = p k a k,i /R k k p k a k,i /R k . ( 5.22) With β k,i , we can calculate the performance metrics as the followings: vs = 1 λp s T x π(x ) |x | i k x s i β k,i R k , (5.23) Ds = x π(x ) x s i k x s i β k,i 1 {γ s k (x )<v i } , (5.24) Bs = x π(x ) x s i k x s i β k,i γ s k (x ) -v i v i , (5.25) γe = x π(x ) x e x e γ e (x ). (5.26) To conclude the two previous subsections 5.8.1 and 5.8.2, with the approximation model, we can reduce the number of classes as shown in Table . 5.3, where I is the number of video bit rate and K is the number of capacity regions. We can observe that large chunk duration always needs to have more classes than cases with small chunk duration. Performance of approximation model To validate the performance of approximation model, we follow the previous configurations in section 5.1 with two video bit rates and two radio conditions representing cell center and cell edge respectively. We set the parameters as = {R C , R E } = {10, 4}Mbps, (p C , p E ) = ( 1 2 , 1 2 ) for cell center and cell edge, (p e , p s ) = ( where approximation ratio is frequently used in studying the performance of approximation as [START_REF] Wu | Leveraging the delay-friendliness of tcp with fec coding in real-time video communication[END_REF]. Approximation difference is extended to avoid the case when the exact value of buffer surplus and deficit rate can approach 0. Compared to the maximum value, 2 for buffer surplus and 1 for deficit rate, the difference remains small and acceptable. It can also be concluded that the predictions of approximation model perform well generally. Approximation model performs the best in the case with configuration of small chunk duration and continuous video bit rate. There are some larger differences between the exact and approximation results with discrete video bit rate configured, but they remain acceptable. System dimensioning Based on our models, operators can design their dimensioning algorithm allowing a certain traffic intensity obtained under some QoS constraints, which might be any combination of performance metrics we defined. Here, we demonstrate an example utilizing realistic radio conditions based on measurement data from a 4G network in a large European city, with an average cell radius of 350 meters. The concerned frequency band is LTE 1800 MHZ. Figure 6.2 shows the measured probability distribution function of the Channel Quality Indicator (CQI) obtained from base station measurements collected using an Observation and Measurements (O&M) tool. Each CQI is associate to an MCS, determining its spectral efficiency. Using the CQI-MCS associ- ation figures of [START_REF] Bouguen | LTE et les réseaux 4G[END_REF] and considering a bandwidth of 10 MHZ, the corresponding harmonic capacity of an LTE cell is computed as equal to R e = 16.82 Mbps. As of the opportunistic scheduling gain, we make use of the scheduling gain calculated in [START_REF] Wang | System level analysis of lte-advanced: with emphasis on multi-component carrier management[END_REF] for a MIMO 2×2 LTE system and an AWGN channel and that converges to G(∞) = 1.7 starting from a number of active users in the LTE cell equal to 15. Other traffic-related parameters are configured as T = 20s, σ = 5Mbits. We consider a discrete video bit rate with (v 1 , v 2 ) = (2, 0.5)Mbps. Chapter 6 Predicting QoE of Video Streaming with Machine Learning Technique In the previous chapters, we studied the performance of both real-time streaming and HTTP-based streaming. To measure the QoE of HTTP-based streaming, metrics, the probability of starvation events or so-called buffer events are adopted popularly. Starvation or buffer events occur when the video buffer becomes empty and users encounter a video pause. As it is not easy to develop an analytical form for QoE metrics, especially for the subjective QoE, therefore, researches like [START_REF] Singh | Quality of experience estimation for adaptive http/tcp video streaming using h.264/avc[END_REF] apply statistical analysis to understand the correlation between QoE and network QoS. In this chapter, we rely on a simulator which generates a big amount of data pairs (QoE+network features) and we demonstrate the efficiency of predicting QoE, video starvation, using the input network features such as CSI, the number of users' flows and video duration, recorded at the arrival of each user. Problem statement and state-of-the-art Different from the traditional metrics for measuring the quality of real-time video as we introduced in chapter 1, it is more reasonable to have other metrics for evaluating the Quality of Experience (QoE) [START_REF] Patrick | Qualinet White Paper on Definitions of Quality of Experience[END_REF] of progressive downloaded video. Regarding to the QoE of video users, objective performance metrics, such as starvation probability, rebuffering rate or mean duration of a rebuffering event are summarized in [START_REF] Dobrian | Understanding the impact of video quality on user engagement[END_REF] and they are highly studied in researches like [START_REF] Xu | Impact of Flow-level Dynamics on QoE of Video Streaming in Wireless Networks[END_REF] and [START_REF] Cicco | An experimental investigation of the akamai adaptive video streaming[END_REF]. In fact, the mentioned performance metrics can be easily obtained by setting up a client-server testbed and measuring those video buffer statistics at TCP session level. Authors of [START_REF] Mok | Measuring the quality of experience of http video streaming[END_REF] showed a correlation of users' QoE and network features by collecting those data from a simple testbed. Nevertheless, when it comes to the impact of wireless networks to the video performance, it is a challenging task to correlate the radiorelated parameters like Channel State Information (CSI) to QoE metrics. Can we predict the video starvation using channel state information and other network features? The main difficulty on finding out this correlation lies on the distance (and consequent lack of crosslayer information) between where video application and radio information can be accessed. Information of video services are monitored at higher level like application layer, which is not accessible for operators. Information of transport layer is recorded at the PDN Gateway but not sufficient for studying the QoE, video starvation. On the other hand, radio-related parameters are recorded at lower layer such as data link and physical layer. In order to understand this correlation, we establish a simulator which generate all the correlated data for streaming users to predict their video starvation. Starvation probability is studied as QoE of video service in many works. It is modeled and calculated analytically in [START_REF] Xu | Impact of Flow-level Dynamics on QoE of Video Streaming in Wireless Networks[END_REF]. However, several constraints are needed when applying that model, e.g., fixed video bitrate is required and the model can only take a single channel condition into account. In [START_REF] Xu | Analytical QoE models for bit-rate switching in dynamic adaptive streaming systems[END_REF], analytical form of starvation probability of adaptive streaming is proposed however without consideration of flow dynamics. We utilized flowlevel model to investigate the video performance metrics in [START_REF] Bonald | A flow-level performance model for mobile networks carrying adaptive streaming traffic[END_REF] and [START_REF] Lin | Impact of chunk duration on adaptive streaming performance in mobile networks[END_REF]. However, the relationship between video starvation and the proposed performance metrics are not clear. Machine learning has been widely used to study both subjective and objective QoE. [START_REF] Balachandran | Developing a predictive model of quality of experience for internet video[END_REF] and [START_REF] Zhang | Predicting the quality of experience for internet video with fuzzy decision tree[END_REF] use machine learning to study the correlation between objective users' satisfaction and application metrics such as buffer times. Authors of [START_REF] Singh | Quality of experience estimation for adaptive http/tcp video streaming using h.264/avc[END_REF] studied the QoE with TCP information. However, as aforementioned, the correlation between QoE and users' radio information are not considered. Contributions Our contribution is to demonstrate the correlation between the video starvation of different streaming and the recorded users' features. We also analyze the importance of these users' features to the prediction of video starvation events, including the number of video users existing in a cell, channel conditions of video user and video duration recorded when a flow starts its video download. To do so, we develop a C++ event-driven simulator that generates statistics per video streaming. In order to identify the most important performance features that predict the video starvation event, we apply two typical machine learning models for analysis. We show that generally prediction accuracy decreases as the system load increases and that users' mobility and having fixed bitrate will cause more video starvations. Considering both number of flows and radio condition is sufficient to achieve more than 92% of prediction accuracy for the static users but not sufficient for the mobile users. More features are needed to improve the prediction accuracy. We also show that video duration is not that important for predicting a video starvation and that number of flows and number of flows in starvation are similarly important to the prediction. Organizations The rest of chapter is organized as follows. In Section 6.2 we introduce two types of video streaming delivery and present the flow-level concept used to calculate the maximum flow arrival rate. Moreover, in the same Section, we also describe the structure of our simulator and the users' features that we examine for prediction performance. Section 6.3 introduces the two evaluated machine learning tools, Generalized Linear Model (GLM) and Support Vector Machine (SVM). Prediction performance of four different types of video streaming CHAPTER 6. PREDICTING QOE OF VIDEO STREAMING WITH MACHINE LEARNING TECHNIQUE 87 are shown in Section 6.4 with access of all features and also limited access of all users' features are considered. Section 6.5 concludes the chapter and discusses the future works. System Description In this section, we first present two types of video streamings. Then we introduce the flow-level model used for calculating the maximum system load. Event-driven simulator is presented to generate data of video starvation for different loads. Video streaming According to the mechanism of HTTP streaming services as we mentioned in chapter 1, users request one video chunk after another. Generally, they can be categorized into fixed video bitrate streaming and adaptive streaming as following: Fixed video bitrate streaming Users fix a video bitrate, v c , from the beginning till the end of video download, which is the simplest implementation. Adaptive streaming As Fig. where y is a shorthand notation for the largest video bitrate in but not greater than y and γ j stands for the instantaneous throughput of user j. v I , the maximum system arrival rate can be calculated treating all the flows as an elastic traffic with volume v T as λ max = w 1 v I T R s + w 2 v I T R m + w 3 v c T R s + w 4 v c T R m -1 . (6.3) R s = k p k R k -1 stands for the equivalent radio throughput for static users. As work [START_REF] Nivine Abbas | Mobility-driven scheduler for mobile networks carrying adaptive streaming traffic[END_REF] mentioned, R m = k q k R k stands for the equivalent radio throughput for mobile users with q k denoted as the proportion of time users stay with throughput R k . Event-driven simulator In [START_REF] Bonald | A flow-level performance model for mobile networks carrying adaptive streaming traffic[END_REF] and [START_REF] Lin | Impact of chunk duration on adaptive streaming performance in mobile networks[END_REF], by mathematical flow-level model, we can only obtain some objective QoS metrics instead of the real buffer information of j-th user, b j (t). Therefore, we implement an event-driven simulator which is able to simulate the video starvation event and record the buffer value of all users. Our simulator is implemented based on flow-level concept, where each video session is regarded as a flow and each of them may encounter the following events: • Arrival Event: Flow arrival of streaming follows Poisson distribution. As user j arrives to the cell at time E a j , several observed features, z j , are recorded and used as the input data for predicting starvation, y j ∈ {1, -1}. • Departure Event: Users encounter a departure event at time E d j when its requested video is fully downloaded. • Chunk Event: Users encounter a chunk event at time E c j when its requested video chunk with duration h is downloaded. When user starts to request a video, it will stay at PREFETCH and switch to STARVA-TION until b j (t) ≥ B, where B is the initial buffer. Once b j (t) = 0, user enters into STARVATION, where it is recognized as a user experiencing a video starvation, y j = 1, and it will wait until b j (t) ≥ B to enter again PLAY state. E b j is the time of buffer events for user j. • Mobility Event: Users with mobility will change the R j to the adjacent throughput when mobility event at time E m j occurs. In this event, users schedule the next mobility event based on the exponential distribution with rate ν j . Algorithm 1 summarizes the mechanism of the event-driven simulator with input parameters and output resutls listed in Table 6.1, where is the set of all next event time and is the set of all users in the cell. update new E a j and put it in , put user j in ; / Recorded features In section 6.2.4, we have mentioned that z j is recorded at the arrival of j-th user and is going to be used to predict y j . Here, we introduce the components of user's feature, z j : • R j , Radio condition (R): It is recorded at the beginning of video download. If user is static, R j is fixed. • T j , Video duration (T ): It follows an exponential distribution. • x j , Number of flows (N ): x j = (x 1 , • • • , x K ) j stands for the number of flows in each region. • |x j |, Total number of flows: |x j | = k x k represents the total number of flows in the cell. • x s j , Number of flows in starvation (N s ): x s j = (x s 1 , • • • , x s K ) j stands for the number of flows experiencing video starvation in each region. • |x s j |, Total number of flows in starvation: |x s j | = k x s k represents the total number of flows experiencing starvation. Machine Learning Tool In this section, we introduce two efficient and widely used machine learning tools, GLM and SVM, which were used in the herein presented analysis. The performance metrics and reference the libraries of algorithms are also shown here. Generalized Linear Model (GLM) Given a training set of instance-label pairs (z j , y j ), where j = 1, • • • , l, z j ∈ n and y ∈ {1, -1} l , GLM with logistic regression model tries to minimize the following cost function C(θ ) = 1 l l j=1 log(g(-y j θ T z j )), (6.4) where θ ∈ n is a vector having the same dimention as z j and g( y j θ T z j ) = (1 + e y j θ T z j ) -1 , known as logistic function or sigmoid function. With the obtained θ opt = arg min θ C(θ ), we have the prediction function as ŷ(z) = 1, when g(θ T opt z) ≥ 0.5, -1, when g(θ T opt z) < 0.5. (6.5) Compared with SVM, GLM technique needs less calculation time and it is easier to implement. Support Vector Machine (SVM) In [START_REF] Chang | LIBSVM: A library for support vector machines[END_REF], with the same training set as previous subsection but with y ∈ {1, -1} l , SVM solves the following optimization problem to obtain the optimal w . Based on different data characteristics, SVM provides us a different solution φ(z j ) to choose. If data can not be easily separated by a hyperplane then other types of φ(z j ) need to be considered. By solving the optimization problem using Lagarange multiplier, α * j , we can rewrite the decision function, g(z) as ŷ(z) = g(z) = sign We can choose a linear kernel as K(z, z j ) = x T z j , or other more elaborated kernel such as radial basis function (RBF): K(z, z j ) = exp(-γ zz j 2 ), where we do not optimize γ, but choose a default value γ = 1. Our simulator is launched based on the realistic radio conditions obtained from the measurement data of a 4G network in a large European city, with an average cell radius of 350 meters. The concerned frequency band is LTE 1800 MHZ. Figure 6.2 shows the measured probability distribution function of the Channel Quality Indicator (CQI) obtained from base station measurements collected using an Observation and Measurements (O&M) tool. Each CQI is associate to an Modulation Coding Scheme (MCS), determining its spectral efficiency. As we have mentioned that traffic load might vary along hours, we demonstrated eight flow arrival rate normalized by the maximum value λ max . For each λ, simulator generates l = 10 6 streaming arrivals. 80% of data are randomly selected for training and 20% of data for validation among all the samples. In Fig. 6.3, the average number of video starvation and non-starvation used for training is listed for each load. It can be observed that when load is small, starvation seldomly happens. However, when load approaches to λ max , more video flows experience video starvation. It is also shown that static users and adaptive streaming users experience less video starvation than mobile and fixed bitrate users. Applying the introduced machine learning techniques, we obtain the two following figures showing the average prediction performance, P, with GLM in Fig. 6.4 and with SVM in Fig. 6.5. We can observe that the prediction performance of GLM and SVM are similar. In addition, no matter which machine learning tool we use, we can observe that as load increases, prediction performance will decrease because of the increase of uncertainty. From the simulation results, we show that QoE of mobile users are much more difficult to predict, especially when load is large. However, static users can achieve more than 90% of accuracy. There is no general rule saying that fixed bitrate is easier to predict than adaptive streaming. It depends on mobility. For static users with adaptive streaming property, prediction of QoE is more accurate. For static users, even almost 95% of accuracy can be achieved at high load. However, for mobile users, it can be said that the initial information are not sufficient for prediction. In this section, we verify the prediction performance considering limited access to certain 4. Impact of scheduling schemes to adaptive streaming: In the static case, we suggest to deploy round-robin scheduling or opportunistic scheduling schemes but not max-C/I and max-min scheme. This conclusion is different from the one for elastic traffic saying there is no difference among all scheduling schemes. Prediction performance of different HTTP streaming Prediction performance of different features 5. Impact of users' mobility: When users' mobility is considered inside the cell, deploying RR has better video smoothness but less video resolution. Vice versa for the max C/I scheduling scheme. We show that implementing discriminatory scheduling can achieve an intermediate performance. In addition to the qualitative conclusions given above, our other contribution is to provide the quantitative results, which could assist both the service providers and networks operators to well dimension their systems given certain performance constraints. In chapter 6, as we found that it is difficult to find out an exact analytical form for video QoE including all the possible system parameters, we propose to utilize the machine learning technique to predict the video quality. The results also help us to understand the correlation between the video starvation and the users features recorded at each arrival. In the last part of our thesis, we examine the prediction performance using GLM and SVM with different system loads. We found out that the prediction accuracies are more than 92% for the static users at each load and the most important network parameters include the radio condition and flows number, which shows that prediction video starvation is feasible for static users. However, more users' features are needed to well predict the starvation events. Future works For real-time streaming services, an important future work would be the extension of the model to consider real time streaming with adaptive video bit rate, on both packet and flow levels. As of HTTP adaptive streaming traffic, although we have computed important KPIs for video smoothness that are correlated with one of the common video QoE indicator, starvation probability, obtaining the analytical form of starvation probability is still a challenging future work. We have started a set of works on this that did not lead to concrete results, but that may be good guidance for future works. More in details, in order to obtain an analytical approximation of starvation probability, we have tried to model the metrics by applying the concepts developed in [START_REF] Luan | Impact of network dynamics on user's video quality: analytical framework and qos provision[END_REF] and [START_REF] Hou | Qoe-optimal scheduling for on-demand video streams over unreliable wireless networks[END_REF]: Assuming that elastic traffic dynamics are fast enough, the packet arrival process of streaming service can be modeled as a Brownian motion and important metrics can be computed. However, this assumption for elastic traffic dynamics being unrealistic, the approximation did not seem to work well. More investigations may be needed in the future. Concerning the QoE metrics, future works may study other video QoE metrics as those mentioned in the introduction section, such as join time, buffer ratio, etc [START_REF] Dobrian | Understanding the impact of video quality on user engagement[END_REF], in addition to starvation probability considered in this thesis. The performance impact of buffer configuration is also a good direction for research while we assume that playout buffer is infinite. The most important works, from our perspective, are related to machine learning chapter. Indeed, predicting the video QoE without having to rely completely on analytical solution is a first step towards the exploitation of the network data for predicting and enhancing video QoE. For example, a first step for improving the prediction error is to try other learning models with more randomness like Decision Tree, Random Forest, Neural Network and K-Nearest Neighbors. In addition, because our training data are generated based on the simulators of Poisson arrival, it is also better to implement a simulator with a more realistic traffic profile or to use real network measurements. Adding features related to mobility is also important for QoE prediction. Once the QoE prediction methodology has been improved, the second step would be to use it for enhancing QoE, by proposing advanced selforganizing algorithms. How to implement efficient yet simple self-organizing algorithms for enhancing QoE of streaming services could be an interesting topic for another Ph.D thesis. Figure 1 : 1 Figure 1: Une cellule typique pour des modèles en niveau flux. Figure 2 : 2 Figure 2: Diagramme d'apprentissage supervisé. Figure 3 : 3 Figure 3: Schéma d'arrivée des paquets avec modélisation en deux niveaux. moyen départ d'un flux (s) S f 10 Temps moyen d'arrivé d'un paquet (ms) A p 100 Temps moyen départ d'un paquet (ms) S p 21.3 (2Mbps) et 10.6 (1Mbps) 5.45 (512kbps) et 2.72 (256kbps) Nombre maximum d'utilisateurs m 100 Figure 4 : 4 Figure 4: Taux de paquets en panne avec 2Mbps codec.Figure 5: Taux de paquets en panne avec 512kbps codec. Figure 5 : 5 Figure 4: Taux de paquets en panne avec 2Mbps codec.Figure 5: Taux de paquets en panne avec 512kbps codec. Figure 6 : 6 Figure 6: Taux de panne paquets calculés par modèle fluid avec les utilisateurs de multi-classes. Figure 7 : 7 Figure 7: Taux de panne paquets pour different applications. Figure 8 : 8 Figure 8: Un exemple de délivrer des segments vidéo pour un utilisateur avec T , durée du vidéo, v i , débit binaire vidéo. Figure 9 : 9 Figure 9: Système sans fils avec single région capacité. 1 , • • • , v I }, où | | = I. Parce que les flux avec un débit binaire différent possèdent différents débits d'arrivée et de départ, nous désignons l'état du système comme x = (x 1 , • • • , x I ) ∈ N I , Où x i représente le nombre de flux qui a choisi le débit binaire vidéo v i à son arrivée. Le débit d'arrivée du flux-i dépend du nombre total d'flux dans le système, |x | = i x i . Étant donné un état x , l'arrivée du flux ne sélectionnera que le débit binaire v i . Par conséquent, nous avons Avec x = x>0 xπ(x), x = |x |>0 |x |π(x ) et que L = x représente le nombre moyen de flux. En observant S, nous pouvons impliquer la probabilité de vacuité, qui est positivement liée à S. Dire que le plus petit S pourrait impliquer une plus petite probabilité de vacuité. (a) Débit binaire de la vidéo, v s . (b) Débit moyenne élastiques, γ e . (c) Taux de déficit, D s . (d) Surplus de tampon, B s .(e) Temps de service, S s . Figure 10 : 10 Figure 10: Performances de streaming vidéo obtenues par les deux modèles avec deux durées de chunk. Figure 11 : 11 Figure 11: Un exemple démontrant comment télécharger plusieurs segments dans une requête d'HTTP. (a) v s moyenné dans un cellule. (b) γ e moyenné dans un cellule. (c) D s moyenné dans un cellule. (d) S s moyenné dans un cellule.(e) B s moyenné dans un cellule. Figure 12 : 12 Figure 12: Performances du streaming adaptatif avec un débit binaire vidéo continu et discret. j le nombre de flux dans chaque région. |x j | Nombre total de flux |x j | = k x k représente le nombre total de flux dans la cellule. x s Nombre de flux en pause (N s ) x s j = (x s 1 , • • • , x s K ) répresente le nombre flux qui subissent la pause vidéo dans chaque région. |x s j | Nombre total de flux en pause |x s j | = k x s k représente le nombre total de flux en pause. Figure 13 : 13 Figure 13: La distribution de la probabilité de CQI mesuré dans un réseau LTE. Figure 14 : 14 Figure 14: Formation d'échantillons de quatre types de streaming selon des charges. Figure 15 : 15 Figure 15: Performance moyen de la prédiction Figure 16 : 16 Figure 16: Performance de la prédiction envisageant diffèrent caractéristiques du réseau. Introduction 1 1 . 1 2 11 2 . 1 11221 Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Video categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.2 Quality of Experience (QoE) . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Background Wireless systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . 3 1 . 4 7 2. 1 15 lv 2 . 5 1314711525 v i , débit binaire vidéo. . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvi 9 Système sans fils avec single région capacité. . . . . . . . . . . . . . . . . . . . . xxvii 10 Performances de streaming vidéo obtenues par les deux modèles avec deux durées de chunk. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxiii 11 Un exemple démontrant comment télécharger plusieurs segments dans une requête d'HTTP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxiv 12 Performances du streaming adaptatif avec un débit binaire vidéo continu et discret. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxv 13 La distribution de la probabilité de CQI mesuré dans un réseau LTE. . . . . . . xli 14 Formation d'échantillons de quatre types de streaming selon des charges. . . xli 15 Performance moyen de la prédiction . . . . . . . . . . . . . . . . . . . . . . . . . xlii 16 Performance de la prédiction envisageant diffèrent caractéristiques du réseau. xliii 1.1 A typical HTTP adaptive streaming system [71]. . . . . . . . . . . . . . . . . . . 3 1.2 Media Presentation Data Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Different categories of objective video quality metrics, with QoS added for illustration purposes.[81][95] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 An illustration of a video session life time and associated video player events. Channel quality varies over multiple time-scales. At a slow scale, channel varies due to shadowing, etc. At a fast scale, channel varies due to multi-path effects [91]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2 Cellular system diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3 A typical cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.4 Two ways of duplexing [47] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The definition of a resource block in Orthogonal Frequency Division Multiplexing (OFDM) system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Single station queue [22] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Markov chain model for M/M/1 queue. . . . . . . . . . . . . . . . . . . . . . . . 2.8 Markov chain model for state dependent queue. . . . . . . . . . . . . . . . . . . 2.9 The balance function Φ(x ) is equal to each weight of any path from state x to state 0 [35]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10 A typical flow-level modeling for a typical cell. . . . . . . . . . . . . . . . . . . . 2.11 Diagram of supervised learning. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Packet arriving scheme with two level modeling. . . . . . . . . . . . . . . . . . . 3.2 Performance comparison with λ f λ p ≤ 0.025 . . . . . . . . . . . . . . . . . . . . . . 3.3 Performance comparison with λ f λ p ≤ 0.25 . . . . . . . . . . . . . . . . . . . . . . . 3.4 Packet outage rate with 2Mbps codec. . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Packet outage rate with 512kbps codec. . . . . . . . . . . . . . . . . . . . . . . . 3.6 Packet outage rate calculated by fluid model with multiple class of users. . . . 3.7 Packets outage rate for different applications. . . . . . . . . . . . . . . . . . . . . 3.8 Packets outage rate of fluid model with fast fading channel effect. . . . . . . . 3.9 Packets outage rate for different applications with fast fading channel effect. 4.1 An example of video chunks delivery for a user. . . . . . . . . . . . . . . . . . . 4.2 Wireless system with single capacity region . . . . . . . . . . . . . . . . . . . . . 4.3 Performance comparison between proposed QoS metrics and starvation probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Wireless system with multiple capacity region. . . . . . . . . . . . . . . . . . . . 4.5 A simple model with two cell regions. . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Queuing model for two regions with intra-cell mobility. . . . . . . . . . . . . . . 5.1 Performance of video resolution and elastic traffic obtained by both models with small and large chunk duration. . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Performance of video smoothness obtained by both models with two of chunk durations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 An example showing the mechanism of multiple chunks downloaded in an HTTP request. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Peformance of transmitting multiple video chunks in a HTTP request. . . . . . 5.5 Performance of adaptive streaming with continuous and discrete video bit rate set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Video bit rate performance under different scheduling schemes. . . . . . . . . 5.7 Buffer surplus performance under different scheduling schemes. . . . . . . . . 5.8 Mean video bit rate with and w/o mobility, under round-robin(black), max C/I(blue) and maxmin(green). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9 Buffer surplus with and w/o mobility, under round-robin(black), max C/I(blue) and maxmin(green). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 1 2 Video 2 Streaming and Real-Time Video[START_REF] Simpson | IPTV and Internet Video: Expanding the Reach of Television Broadcasting[END_REF]. Figure 1 . 1 : 11 Figure 1.1: A typical HTTP adaptive streaming system [71]. Figure 1 . 2 : 12 Figure 1.2: Media Presentation Data Model. Figure 1 . 3 : 13 Figure 1.3: Different categories of objective video quality metrics, with QoS added for illustration purposes.[81][95] Figure 1 . 4 : 14 Figure 1.4: An illustration of a video session life time and associated video player events. Figure 2 . 1 : 21 Figure 2.1: Channel quality varies over multiple time-scales. At a slow scale, channel varies due to shadowing, etc. At a fast scale, channel varies due to multi-path effects [91]. Figure 2 . 2 : 22 Figure 2.2: Cellular system diagram Figure 2.3: A typical cell Figure 2 . 4 : 24 Figure 2.4: Two ways of duplexing[START_REF] Erik Dahlman | 4G LTE/LTE-Advanced for Mobile Broadband[END_REF] Figure 2 . 5 : 25 Figure 2.5: The definition of a resource block in OFDM system. Figure 2 . 6 : 26 Figure 2.6: Single station queue[START_REF] Bolch | Queueing Networks and Markov Chains[END_REF] Figure 2 . 7 : 27 Figure 2.7: Markov chain model for M/M/1 queue. Figure 2 . 8 : 28 Figure 2.8: Markov chain model for state dependent queue. Figure 2 . 9 : 29 Figure 2.9: The balance function Φ(x ) is equal to each weight of any path from state x to state 0 [35]. Figure 2 . 2 Figure 2.10: A typical flow-level modeling for a typical cell. m denotes the space of input values, and = { y i } i=1,••• ,m the space of output values. To describe the supervised learning problem slightly more formally, the goal is to learn a function h : → so that h(x i ) is a good predictor for the corresponding value of y i . Figure 2 . 11 : 211 Figure 2.11: Diagram of supervised learning. by learning the branching function, b(x ) which is obtained as b(x ) = arg min h(x ) C c=1 | c with h| × impurity( c with h), (2.48) Figure 3 . 1 : 31 Figure 3.1: Packet arriving scheme with two level modeling. 3 ) 3 and ρ p = S p /A p . Based on the tolerated outage rate, ε, we can calculate the acceptable traffic, ρ f , given a certain resource in the packet level ρ p by verifying the equation (3.6). Seeing the equation (3.5), we get the following relationship between the two packet outage rate models, lim D→∞ γ(D) = γ fluid (3.7) Figure 3 . 2 :Figure 3 . 3 : 3233 Figure 3.2: Performance comparison with λ f λ p ≤ 0.025 Figure 3 . 4 : 34 Figure 3.4: Packet outage rate with 2Mbps codec. Figure 3 . 5 : 35 Figure 3.5: Packet outage rate with 512kbps codec. Figure 3 . 6 :Figure 3 . 7 : 3637 Figure 3.6: Packet outage rate calculated by fluid model with multiple class of users. . 3.4. In Fig. 3.8, we show the packet outage rate obtained by fluid model with consideration of fast fading. It also shows that the fast fading effect will have the same outage rate as the one without fading effect in Fig. 3.6. In Fig. 3.9, we show the packet outage rate of modified M/D/1 model with different D values. Based on the simulation results, we conclude the fluid model is useful for live TV streaming with D ≥ 500ms and when considering service as video conference, it is better to utilize modified M/D/1 model. In addition, we can also show the same results for slow fading with log-normal distribution, σ = 3dB, by the same method. Figure 3 . 8 :Figure 3 . 9 : 3839 Figure 3.8: Packets outage rate of fluid model with fast fading channel effect. To facilitate the calculation, we propose a substitution of M/D/1 packet-level model by fluid model. In the simulation, it is shown that the fluid model approximates well the performance of M/D/1 queue under the LTE networks configuration for the delay tolerant of application like live TV video. On the other hand, for services like video conference which require smaller packet delay, an obvious difference between M/D/1 and fluid model can be observed. Thus we conclude that it is better to use the M/D/1 for packet level modeling. Model extension with multiple classes case are validated both for M/D/1 and fluid model, also considering with the fast fading effects. Figure 4 . 1 : 41 Figure 4.1: An example of video chunks delivery for a user. Figure 4 . 2 : 42 Figure 4.2: Wireless system with single capacity region ) 50 CHAPTER 4 . 51 where 1 504511 MODEL OF ADAPTIVE STREAMING TRAFFIC stands for indicator function. For the case configuring large video chunk duration, class accounts for a proportion of flow arrivals, p k , with K k=1 p k = 1. Then we present the model with corresponding performance results Figure 4 . 4 : 44 Figure 4.4: Wireless system with multiple capacity region. R 2 λ 2 R 1 λ 1 Figure 4 . 5 : 2145 Figure 4.5: A simple model with two cell regions. e = x π(x )|x e |, x e k = x x e k π(x ) and |x e | = k x e k . 4.7. MOBILITY MODEL 60 Figure 4 . 6 : 46 Figure 4.6: Queuing model for two regions with intra-cell mobility. (a) Mean video bit rate v s . (b) Mean elastic throughput γ e . Figure 5 . 1 : 51 Figure 5.1: Performance of video resolution and elastic traffic obtained by both models with small and large chunk duration. Figure 5 . 3 : 53 Figure 5.3: An example showing the mechanism of multiple chunks downloaded in an HTTP request. Figure 5 . 4 : 54 Figure 5.4: Peformance of transmitting multiple video chunks in a HTTP request. (4.35), we obtain the tion of two capacity regions standing for cell-center and cell-edge respectively, the simulation results are shown in Fig. 5.6 and Fig. 5.7 which compare the performance of video bit rate and the buffer surplus with different policies for two regions, where the configurations are set up as p 1 = p 2 = 1/2, R 1 = 25 Mbps, R 2 = 10 Mbps, v min = 0.5 Mbps, v max = 2 Mbps and T = 10s. These results are obtained by the numerical evaluation of the stationary distribution of the Markov process X (t). (a) Cell-average buffer surplus (b) Cell-center buffer surplus (c) Cell-edge buffer surplus. Figure 5 . 7 : 57 Figure 5.7: Buffer surplus performance under different scheduling schemes. Fig. 5 . 5 Fig. 5.6 shows that under the max C/I policy, we can obtain better cell-average video bit rate compared to the other policies. Observe that the cell-average video bit rate is simply v = p 1 v1 + p 2 v2 in accordance with (4.28). However, in terms of stability condition, we observe that max C/I policy provides lower stability condition, λ max , than other two policies. Based on Eq. (4.20) and (4.21), we obtain λ max,RR = 2.85, and λ max,max C/I = 0.54×λ max,RR . It is shown that the analytic calculation give the same results as we obtain in simulation, where λ max,RR and λ max,max C/I are the minimum flow arrival rate that makes vc or ve equal Figure 5 . 8 : 58 Figure 5.8: Mean video bit rate with and w/o mobility, under round-robin(black), max C/I(blue) and maxmin(green). B at medium load max C/I: Better v, stability condition max C/I: Better B at low and high load Depend on the needs Figure 5 . 10 : 510 Figure 5.10: Mean video bit rate with mobility under round-robin (blue), max C/I (black) and discriminatory policy(green). Figure 5 . 11 : 511 Figure 5.11: Mean buffer surplus with mobility under round-robin(blue), max C/I(black) and discriminatory policy(green). Fig. 5 . 5 Fig. 5.10 and 5.11 shows the simulation results using the same system configuration as previous where users are all mobile. Moreover weighting value are configured as (w 1 , w 2 ) = (1000, 1). Simulation results give an intermediate performance between the round-robin, where (w 1 , w 2 ) = (1, 1) and the max C/I policy, where (w 1 , w 2 ) = (∞, 1). (a) Mean streaming video bit rate. (b) Streaming deficit rate (c) Streaming buffer surplus (d) Mean elastic throughput. Figure 5 . 12 : 512 Figure 5.12: Performance with different v max . flows,= {v 1 , v 2 } = {2, 0.5}Mbps, T = 10s, σ = 5Mbits and λ max = 1.14 users/s. We demonstrate the number of classes that we are going to simulate in Table. Figure 5 . 13 : 513 Figure 5.13: Approximation ratio of video bit rate and service time. Figure 5 . 14 : 514 Figure 5.14: Approximation ratio and approximation difference. Figure 5 . 5 Figure 5.15: Measured CQI probability distribution function on a live LTE network. Figure 5 . 16 : 516 Figure 5.16: Dimensioning with real LTE system configuration. 6.1, adaptive streaming services allow users to adapt to their video bitrate in realtime. After finishing downloading a video chunk with duration h, based on the measured throughput, γ j , users can select a video bitrate for the next chunk from the discrete set = {v 1 , • • • , v I }, where v 1 > • • • > v I . The selected video bitrate is given as v j = γ j , when γ j ≥ v I , v I , when γ j ≤ v I , (6.1) Figure 6 . 1 : 61 Figure 6.1: Video chunk selected by the adaptive streaming. • Buffer Event: We classify the simulated streaming user j into three states, PREFETCH, PLAY and STARVATION, where each has a buffer variation rate d b j (t) d t = γ j , PREFETCH or STARVATION, γ j -1, PLAY. Algorithm 1 : 1 Event-driven simulation Input: , p, λ, l, T , B and h. Output: z = (z 1 , • • • , z l ) and y = ( y 1 , • • • , y l ) Initialize y = -1, = {E a 1 } and = ; while j = 1; j ≤ l do E = min e∈ (e); //Arrival Event:if E == E a j then record z j , j = j + 1; to y j (w T φ(z j ) + b) ≤ 1 -ξ j , ξ j ≥ 0. (6.6) j y j α * j φ T (z j )φ(z) + b * = sign j y j α * j K(z j , z) + b * (6.7) Figure 6 . 3 : 94 CHAPTER 6 . 63946 Figure 6.3: Training samples of four types of streaming along loads Figure 6 . 4 :Figure 6 . 5 : 6465 Figure 6.4: Average prediction performance with GLM Figure 6 . 6 : 66 Figure 6.6: Prediction performance considering different network features. Table 1 : 1 Configuration de simulations pour utilisateurs avec une seul class. Table 2 : 2 Deux extrêmes configurations pour la durée de segments vidéo. h Section Transition du débit binaire Petite segments vidéo h ≈ 0 4.2 Plus Grande segments vidéo h ≈ ∞ 4.3 Moins supposons que le débit physique d'un utilisateur statique reste le même pendant le transfert du flux de données entier (La session vidéo). Pour les utilisateurs mobiles, le débit physique varie entre ces valeurs. En ce qui concerne les caractéristiques de trafic, nous ne considérons que les services de streaming dans ce modèle et nous faisons l'hypothèse classique que les flux de données ayant un débit physique R k arrivent comme un processus de Poisson avec le débit λ k = p k λ, où λ est le débit global d'arrivée du flux dans la cellule et p k représente la proportion de trafic avec le débit physique R k , avec k p k = 1. La durée des requêtes est supposée indépendante et distribuée de manière exponentielle avec la moyenne T . Le streaming simulé peut être divisé en quatre types suivants et chacun d'eux compte pour w i proportion d'arrivées avec i w i = 1, où i ∈ {I, II, III, IV}. et 15b. Nous ne démontrons que la performance de prédiction de SVM de quatre types de streaming sur la Fig.16. Il est montré que considérer toutes les caractéristiques (A) peut toujours effectuer la plus haute précision de prédiction et que N et N s fournissent des informations très semblables. Il peut être une bonne nouvelle pour les opérateurs qu'ils n'ont pas besoin de connaître plus d'informations d'application du côté des utilisateurs. De plus, nous pouvons également observer que l'information R devient moins importante lorsque les utilisateurs sont mobiles et T est également moins importante. Cela peut être causé par notre hypothèse de mémoire tampon infinie. Dans la Fig.16, nous démontrons également que considérer à la fois R et N peut fournir de bons résultats dans le cas statique. 3 Model of Real-time Streaming Traffic Machine learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 li 2.3.1 Supervised learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Cost function and probabilistic interpretation . . . . . . . . . . . . . . . 2.3.3 Generalized Linear Model (GLM) . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Gradient descent algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.5 Support Vector Machine (SVM) . . . . . . . . . . . . . . . . . . . . . . . . 2.3.6 Overviews of machine learning techniques . . . . . . . . . . . . . . . . . 3.1 Problem statement and the state of the art . . . . . . . . . . . . . . . . . . . . . . 3.2 Flow-level and packet-level model . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Flow-level dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Packet-level dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Fluid model approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Extension to heterogeneous radio conditions . . . . . . . . . . . . . . . . . . . . Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1.1 Wireless channel characteristics . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1.2 Channel capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1.3 Network deployment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1.4 Cellular system structure and evolution . . . . . . . . . . . . . . . . . . . 14 2.1.5 Radio resource management . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2 Queueing theory and traffic modeling . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2.1 Queue model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2.2 M/M/1 queue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2.3 State dependent queue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2.4 Processor sharing discipline . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2.5 Whittle networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2.6 Packet-level modeling v.s. Flow-level modeling . . . . . . . . . . . . . . 20 2.2.7 Flow-level modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.3 3.3.1 Flow-level dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Packet-level dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 3.4.1 Quasi-stationary regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Single class model validation . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Multiple class model validation . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Validation with fading effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Une cellule typique pour des modèles en niveau flux. . . . . . . . . . . . . . . . xvi 2 Diagramme d'apprentissage supervisé. . . . . . . . . . . . . . . . . . . . . . . . . xviii 3 Schéma d'arrivée des paquets avec modélisation en deux niveaux. . . . . . . . xix 4 Taux de paquets en panne avec 2Mbps codec. . . . . . . . . . . . . . . . . . . . . xxiii 5 Taux de paquets en panne avec 512kbps codec. . . . . . . . . . . . . . . . . . . xxiii 6 Taux de panne paquets calculés par modèle fluid avec les utilisateurs de multiclasses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiv 7 Taux de panne paquets pour different applications. . . . . . . . . . . . . . . . . xxiv 8 Un exemple de délivrer des segments vidéo pour un utilisateur avec T , durée du vidéo, Table 1 . 1 Traffic Type 2015 2016 2017 2018 2019 2020 Web, Data, and VoIP 1,323 1,968 2,779 3,605 4,427 5,158 Video 2,031 3,643 6,232 9,977 15,410 22,963 Audio streaming 279 462 722 1,034 1,398 1,788 File sharing 51 106 196 317 472 653 1: Global Mobile Data Traffic, from 2015 to 2020 (PB per Month) [4] • • • , K} has real-time streaming service with Poisson arrival rate λ k and departure rate µ k . With the two parameters, we denote processor load of class k by ρ k = λ k µ k . We denote by n k (t) the number of calls of a given class requesting streaming at time t and n(t) = (n 1 (t), • • • , n K (t)) denotes the number of flows in each class. Table 3 . 3 Table. 3.1, with c denoting chosen codec rate. It can be observed that S f , A f >> S p , A p , which follows the quasi-stationary regime we assume. S p = c × A p τ (3.16) Parameter Sym Value Mean flow arriving time (s) A f [4, 20] Mean flow serving time (s) S f 10 Mean packet arrival time (ms) A p 100 2Mbps codec → 21.3 Mean packet serving time (ms) S p 1Mbps codec → 10.6 512kbps codec → 5.45 256kbps codec → 2.72 Maximum user number m 100 1: Simulation configuration for single class users Table 3 . 3 in section 3.4, we calculated the 2: Maximum flow-level load with different codec average throughput and one-packet serving time of cell center and cell edge users as Table.3.3. Codec Max Flow-level Load Max Flow-level Load Configuration for Live TV Streaming for Video Conference 2Mbps 1.9 1.75 1Mbps 5.5 5 512kbps 12.4 12 256kpbs 27.6 27.4 Users' Class Rate Serving time Cell center 14.63 Mbps S c = 3.50ms Cell edge 3.73 Mbps S e = 13.73ms Table 3 . 3 33 : Serving time of cell edge and cell center users .4. We use the same average serving time as previous simulation, 3.5ms, 13.73ms. User Class Portion S p User Class Portion S p 3.51% 1.7ms 2.19% 9.7ms 31.89% 2.6ms 20.66% 11.5ms cell center 36.89% 3.5ms cell edge 39.85% 13.3ms 25.39% 4.6ms 26.73% 15.2ms 2.29% 6.4ms 10.57% 17.5ms Table 3 . 3 4: Serving time and probability distribution of two classes of users with fading effect consideration. Table 4 . 4 .1. Configurations Value of h Section Chances of bitrate transition Significantly small chunk duration h ≈ 0 1 Many Significantly large chunk duration h ≈ ∞ 4.3.2 Zero 1: Two extreme video chunk durations. represents the number of flows in each queue. Applying the same concept to formulate the departure rate as equation (4.1), the flow departure rate of elastic and adaptive streaming services can be respectively expressed as Small Chunk, λ s k = p s p k λ, Large Chunk, λ s ki (x ) = 0, p s p k λ, as v k (x + e s otherwise, ki ) = v i , (4.31) where x = (x e 1 , • • • , x e k , x s 11 , • • • , x s ki ) .34) Moreover, for large chunk case, v ki (x ) = v i . We can obtain departure rate values as Eq.(4.32). With the formulation above, note that the balance property introduced in[START_REF] Bonald | On performance bounds for the integration of elastic and adaptive streaming flows[END_REF] is not valid. The Markov chain is not reversible and we have to solve the balance equations (by a matrix inversion) for the stationary distribution π(x ): ∀x , λ j ki + µ j ki (x ) π(x ) (4.35) j=e,s k i = λ j ki (x )π(x -e j ki ) + µ j ki (x + e j ki )π(x + e j ki ) . j=e,s k i Table 5 . 5 1: Chunk Configuration for Two Policies with h = 10s Table 5 . 2 52 : Policies recommended for different cases and services. Table 5 . 5 4: System configuration of examined scenarios. 1 2 , 1 2 ) for the elastic and streaming Table 6 . 6 1: Notations of input parameters and output results. else E c i = E d i ; //Buffer Event: else if E == E b i then if next state is STARVATION then change d b i else Error happens; update all γ i based on new | |; recalculate E d i , E c i , E d i , ∀i ∈ and put them in ; obtain pairs (z, y); /Departure Event: else if E == E d i then remove user i from ; //Chunk Event: else if E == E c i then if user i is adaptive streaming user then user i chooses a v in based on its γ i ; d t = γ i and y i = 1; else if next state is PLAY then change d b i d t = γ i -1; //Mobility Event: else if E == E m i then change R k of user h to R k+1 or R k-1 ; Model of Adaptive Streaming Traffic 4.1 Problem statement and state of the art . . . . . . . . . . . . . . . . . . . . . . . . (a) Performance of Deficit Rate (b) Performance of Buffer Surplus (c) Performance of Mean Service TimeFigure 4.3: Performance comparison between proposed QoS metrics and starvation probability Acknowledgements List of Tables KPIs definition for heterogeneous radio condition Here we extend the KPIs definition with heterogeneous radio conditions and different scheduling schemes, where we only demonstrate the case configured by small chunk duration. Video bit rate Under the assumption that users watch all the downloaded video.The first performance metrics we measure is the mean video bit rate, which is the average video resolution that a user experiences while watching the video. Noting that the expectation is calculated based Summary In this chapter, we present the analytical model for HTTP adaptive streaming considering the impacts of flow-level dynamics. The model takes into account the system parameters like video chunk duration, video bit rate configuration, scheduling schemes and users' mobility. Based on the system stationary distribution, we also define the KPIs or the QoS that we are going to observe in two different senses, video resolution and playback smoothness. Because the complexity of solving balanced equations is too large, there is no exact form for the KPIs that we want to observe. We can only obtain these values by numerical analysis and launching simulation. The performance impacts of these system parameters and the trade-offs of KPIs will be presented and discussed in the next chapter. Chapter 5 Simulation of Adaptive Streaming and Approximation Model After having presented the model for HTTP-based adaptive streaming in the previous chapter, in this chapter, we begin by showing impacts of different system configurations in the video delivery system or wireless access networks including video chunk duration, number of available video bit rates and scheduling schemes. We also demonstrate the performance impact of users' mobility. Then we present an approximation model which can reduce the complexity of calculation. We apply the approximation model for illustrating how to use our model for network dimensioning and studying the impacts of the video bit rate limitation on streaming performance. Impacts of chunk duration According to the technical report of Akamai [1] and the empirical demonstration of [START_REF] Yao | Empirical evaluation of http adaptive streaming under vehicular mobility[END_REF], it is shown that deploying shorter chunk duration provides better video smoothness and more chances for users to select the proper video bit rate. However, deploying shorter chunks will also generate a large number of video chunks and its corresponding HTTP signaling. Therefore, the report suggests that video service providers choose their chunk duration configurations of HTTP adaptive streaming as 10 seconds. In this section, we first perform simulations for checking these findings by setting the parameters as 2 ) for cell center and cell edge, (p e , p s ) = ( 1 2 , 1 2 ) for the elastic and streaming flows, = {v 1 , v 2 } = {2, 0.5}Mbps, T = 10s, σ = 5Mbits and λ max = 1.14 users/s. The simulation results in figs. 5.1 and 5.2 show the performance of all defined metrics with respect to the normalized settings of traffic load, (λ/λ max ). As section 4.2.1 mentioned, the simulation results shown in figs. 5.1 and 5.2 give us two performance bounds of the two extreme cases, h = 0 and h = ∞ which enable us to predict the performance of intermediate chunk duration. In the simulation, we further demonstrates the performance of an intermediate configuration, h = 5s between two extreme cases. In Fig. 5.1a, we find that configuring small chunk has smaller mean video bit rate compared to large chunk duration may increase the users' satisfaction in the sense of resolution but not in the sense of video smoothness. We proposes a method to reduce HTTP signaling while keeping the same performance in the following section. Chunk duration design Based on the analysis of previous section, a shorter chunk duration will slightly deteriorate the mean video bit rate but improve the mean service time, buffer surplus and deficit rate without taking HTTP signaling into consideration. Therefore, in this section, we propose and investigate a new mechanism allowing users to request multiple video chunks in an HTTP requests which can reduce the HTTP request signaling. We show that if we design the number of video chunk transmitted in an HTTP request following the rule to have same data volume in a single HTTP request, we can offer a chance for operators to achieve the system performance as the same aspect of video smoothness but with less number of HTTP requests. simulation results shown in Fig. 5.5 with three different configurations including 2 video bit rates, infinite number of video bit rates and an intermediate setting, 3 video bit rates. It can be observed that performance metrics of 3 video bit rates are located between the ones of the two extreme cases. From Fig. 5.5a, we observe that deploying continuous video bit rate, meaning more video bit rates to approach users' real instantaneous throughput, will increase the long-term video resolution but decrease mean elastic throughput. In addition, from the rest of figures, we discover that deploying more video bit rates will also increase the buffer starvation from the three buffer-related metrics. Therefore, deploying larger number of video bit rates will generate a trade-off which can benefit the mean video resolution but decrease the elastic throughput and video smoothness. performs better at low load and also at the high load and round-robin performs better at medium load. Generally speaking, mobility provides more opportunities for users to exploit diversity gain. However, high video bit rate may degrade the buffer surplus. Impacts of scheduling schemes Discriminatory scheduling scheme In this section, we summarize the suggested scheduling policy for different services in Table . 5.2. In the static case, all scheduling policies have the same performance for elastic traffic. However, for adaptive streaming round-robin policy is suggested as the max C/I degrades the stability condition. In the presence of mobility, max C/I is recommended for the CHAPTER 5. SIMULATION OF ADAPTIVE STREAMING AND APPROXIMATION MODEL 83 In Fig. 5.16, we show the simulation results of two metrics, video bit rate and buffer surplus. Based on the stability condition, we have λ max = 2.189 users/s and take B = 0 as an example of QoS constraint, the traffic intensity should be lower than 0.93 * 2.189 = 2.01 users/s. Otherwise, the mean buffer surplus is smaller than zero, meaning that the starvation events happen very often. With the same concept, more QoS constraints can be considered together such as taking service time constraint as the average video duration as S = T . Summary In this chapter, simulation results show that configuring large chunk duration littlely improves video resolution but largely degrades the playback smoothness and that configuring larger number of video bit rate will also provide better video resolution but degrade the video smoothness. We also show the performance trade-off of different scheduling schemes with and without considering the users' mobility. Moreover, we propose an approximation model in order to reduce the complexity of simulating a system with multiple classes. The numerical analysis validates that the approximation performs well compared to the exact solution. By applying the approximation model, ways of dimensioning is demonstrated. Different metrics, mean codec rate, deficit rate, buffer surplus and elastic mean throughput are taken as the main KPIs. After downloaded, video chunks will be stored at a playout buffer. Like the previous sections, we still assume that the playout buffer of users is infinite. Once a user enters into the cell, there will be scheduled an amount of resource until the end of video download. Radio access network and traffic characteristics We consider a radio access network where users have different physical throughput depending on its location in the cell. We denote the set of physical throughput as = {R 1 , • • • , R K } and assume that the physical throughput of a static user remains the same during the transfer of the whole data flow (i.e., the video session). For mobile users, the physical throughput varies among these values. In terms of traffic characteristics, we only consider streaming services in this model and we make the classical assumption that data flows having physical throughput R k arrive as a Poisson process with rate λ k = p k λ, where λ is the overall flow arrival rate in the cell and p k stands for the traffic proportion with physical throughput R k , with k p k = 1. The duration of the requests are assumed to be independent and exponentially distributed with mean T . The simulated streaming can be divided into four following types and each of them accounts for w i proportion of arrivals with i w i = 1, where i ∈ {I, II, III, IV} • Type I: static and adaptive streaming. • Type II: mobile and adaptive streaming. • Type III: static and fixed bitrate streaming. • Type IV: mobile and fixed bitrate streaming. In the real network, traffic arrival rate, λ, varies along hours, usually higher in day and lower at night. In the following simulation results, video starvation data are generated with different traffic arrival rates. Prediction shown at each traffic load corresponds to the potential performance at each hour. Flow-level model and maximum arrival rate The concept of flow-level model has been utilized to obtain the performance of streaming in paper [START_REF] Bonald | A flow-level performance model for mobile networks carrying adaptive streaming traffic[END_REF] and [START_REF] Lin | Impact of chunk duration on adaptive streaming performance in mobile networks[END_REF]. Based on it, we can obtain the maximum flow arrival rate that guarantees the system stability. Let ) be the number of streaming flows at time t and x i k (t) be the number of type-i streaming with R k at time t. Based on the round-robin scheduling method, we have the instantaneous throughput calculated as where |x | = i k x i k is the total number of ongoing flows. Video bitrate of the next video chunk is selected based on Eq.(6.1) and (6.2). When load approaches to system capacity and the mentioned traffic characteristics, all the adaptive streaming are forced to adapt to Performance metrics In order to verify the machine learning performance, we define the following performance metrics to examine prediction performance among testing data. This probability represents the average prediction accuracy among all testing samples. Libraries For the GLM analysis, we have used R's stats package, which based its algorithm on the GLM's proposed by [START_REF] Nelder | Generalized linear models[END_REF]. For the SVM numerical analysis of this works, we apply one of the most popular open-source SVM machine learning library, LIBSVM, proposed by [START_REF] Chang | LIBSVM: A library for support vector machines[END_REF] to investigate the prediction performance considering different network parameters. Simulation Analysis In this section, we first introduce the general system parameters that we configured for the simulators. Then we analyze the prediction performance of machine learning among different types of HTTP streaming with all recorded features. Finally we demonstrate the prediction performance considering only certain features. users' features. As we have shown that GLM and SVM performs similar results in Fig. 6.4 and 6.5. We only demonstrate the SVM prediction performance of four types of streaming in Fig. 6.6. It is shown that considering all features (A) can always perform the highest prediction accuracy and that N and N s provide very similar information. It may be a good news for operators that they do not need to know more application information from users' side. Moreover, we can also observe that R information becomes less important when users are mobile and T is also less important. This may be caused by our infinite buffer assumption. In Fig. 6.6, we also demonstrate that considering both R and N can provide good results in static case. Simulation configuration Summary In this chapter, by applying the concept of flow-level dynamics, we examine the prediction performance of video starvation of different video streaming by looking at different users' features. We develop an event-driven simulator in C++ and we have used supervised machine learning techniques, GLM and SVM, to obtain our training model along the system loads. We correlate the users' QoE and features, and simulation results show that the prediction performs better when users are static and adaptive streaming. For mobile users, prediction accuracy degrades. In terms of users' features, it can be observed that users' features such as number of flows and radio conditions lead to better prediction of starvation event than others. With the two users' features, we can obtain more than 92% of prediction accuracy in the static cases. In addition, we also observe that GLM provides similar performance with regards to SVM. Future works will consider other QoE metrics like rebuffering time and the evaluation with other machine models such as Random Forests, Neural Networks, Naive Bayes, and K-Nearest Neighbors to widen the view on other techniques which may improve the prediction accuracy. In addition, prediction accuracy of other configurations needs to be checked. Chapter 7 Conclusions and Future Works Measuring and improving the QoE of video become more and more important as video accounts for more than 50% of network traffic. In this thesis, we propose models for dimensioning different types of streaming services, including real-time streaming and HTTP adaptive streaming inside wireless networks. Both of them are developed by applying the concepts of flow-level dynamics for modeling the arrivals and the departures of the traffic demand, which is highly used for both elastic traffic and real-time streaming traffic in the literature. In chapter 3, we develop a flow-level traffic model for real-time streaming services. We assume the existence of quasi-stationary property and combine both the flow and packet levels to calculate the packet outage rate. In addition, we show that fluid approximation can be adapted to apply at the packet level based on different delay constraints for different types of real-time streaming. Using our model, operators could design the corresponding admission control algorithm for real-time streaming services with a guaranteed packet outage rate. In chapter 4, we develop a flow-level traffic model for HTTP streaming services. The model takes the flow-level dynamics into account and verify the performance impacts of different parameters such as video chunk duration, number of video bit rates and scheduling schemes, etc. We address in the following are the potential questions encountered by operators when they deploy the HTTP adaptive streaming service and want to improve the users' quality-of-experience: 1. Impact of video chunk duration: Deploying larger video chunk duration will decrease video smoothness. However, it gives relatively little gain concerning the video resolution. 2. Impact of number of video bit rates: Configuring more options of video bit rates provides better mean video resolution but decreases the video smoothness in the case of small chunk duration. Design of video chunk duration: Designing the number of video chunk transmitted in a HTTP request based on providing the same video volume for each video bit rate could offer a similar video performance as deploying smaller video chunk duration, which offer another option of deployment for video service providers. Analyse de performance des services de vidéo streaming dans les réseaux mobiles Yu-Ting LIN RESUME : Le trafic de vidéo streaming étant en très forte augmentation dans les réseaux mobiles, il devient essentiel pour les opérateurs de tenir compte des spécificités de ce trafic pour bien dimensionner et configurer le réseau. Dans cette thèse, nous nous intéressons à la modélisation du trafic de vidéo streaming dans les réseaux mobiles. Pour le trafic de vidéo streaming en temps-réel, nous obtenons une forme analytique pour une métrique de qualité de service (QoS) importante, le taux de perte de paquets, et utilisons ce modèle à faire du dimensionnement. Pour le trafic de vidéo streaming de type HTTP adaptatif, nous proposons et analysons d'autres métriques de QoS comme le bitrate moyen, le taux de déficit vidéo et le surplus de buffer, afin de trouver le bon compromis entre résolution de la vidéo et fluidité de la diffusion vidéo. Nous étudions par simulation l'impact de quelque paramètres clés du systéme. Nous montrons que l'utilisation de segments de vidéo courts, d'un nombre réduit d'encodages vidéos et de l'ordonnancement de type round robin améliore la fluidité de la vidéo tout en diminuant sa résolution. Nous proposons par ailleurs d'adapter le nombre des segments téléchargés dans une requête HTTP de sorte que chaque requête corresponde au même volume de données. Enfin, nous appliquons les techniques de l'apprentissage automatique pour analyser la corrélation entre les caracteristiques du système et la qualité d'expérience (QoE) des utilisateurs. MOTS-CLEFS: Streaming Vidéo, Streaming Temps Réel, Streaming Adaptatif, Qualité de l'Expérience, Segment Vidéo, Modèle Niveau Flow, L'intelligence Artificielle. ABSTRACT: As the traffic of video streaming increases significantly in mobile networks, it is essential for operators to account for the features of this traffic when dimensioning and configuring the network. The focus of this thesis is on traffic models of video streaming in mobile networks. For real-time video streaming traffic, we derive an analytical form for an important Quality-of-Service (QoS) metric, the packet outage rate, and utilize the model for dimensioning. For HTTP adaptive video streaming traffic, we propose and evaluate other QoS metrics such as the mean video bit rate, the deficit rate and the buffer surplus, so as to find the good trade-off between video resolution and playback smoothness. We study by simulation the impacts of some key parameters of the system. We show that using smaller chunk durations, fewer video coding rates and round-robin scheduling may provide a smoother video playback but decrease the mean video resolution. We also propose to adapt the number of chunks downloaded in an HTTP request so that each HTTP request has the same data volume. Finally, we apply machine learning techniques to analyze the correlation between system characteristics and the quality of experience (QoE) of users. KEY-WORDS: Streaming Video, Real-Time Streaming, Adaptive Streaming, Quality of Experience, Video Chunk, Flow-Level Model, Machine Learning.

01746054

en

2024/03/05 22:32:07

2018

https://inria.hal.science/hal-01746054/file/3DAHM_Puchaud_V3_soumise.pdf

Pierre Puchaud Christophe Sauret Antoine Muller Nicolas Bideau Georges Dumont Helene Pillet Charles Pontonnier Evaluation of geometrically calibrated segment lengths: preliminary results Introduction Geometry of biomechanical models can be personalized through different approaches based on palpated anatomical landmarks, regression equations, reconstructed medical imaging, or optimization relying on optoelectronic data. The latter offers great perspectives since it is faster and lighter than imaging procedures, does not necessitate additional data acquisition and can also correct misplaced markers. In such geometric calibration methods, an osteo-articular model is defined, describing distances and degrees of freedom (DoFs) mimicking joint functions. These quantities can be calibrated to fit optoelectronic data through optimization methods minimizing the reconstruction error1-3. However, validation of the calibrated quantities remains an issue. Research Question Do geometric calibration methods guarantee consistent anatomical lengths compared to medical imaging assessment ? Methods Optoelectronic data (200Hz) of functional motions -activating each DoF of the lower limb -of 7 subjects were recorded. Marker trajectories were smoothed through moving average over five frames. A 33 DoFs musculoskeletal model of lower limbs, trunk and head was used. The lower limb4 exhibited 3 DoFs at hip, 1 DoF at the knee, and 2 intersecting DoFs at the ankle. Firstly, a regression method (RM) linearly scaled with subjects' height initially estimated the segment lengths. Secondly, geometric calibrations (GCn) with RM as an initial guess optimized segment lengths, positions of joint centers, and marker positions1 by minimizing Euclidian distances between measured and reconstructed markers. Different number of frames equally spaced in the motion data -n = {3, 10, 50, 100, 500} -were used to test the efficiency of the method. Reference joint center positions (two hips, two knees, two ankles) were obtained through the EOS® system. For that purpose, the same subjects underwent biplanar radiographs allowing 3D reconstruction of the pelvis, femurs, tibias, fibulas and spine. Reference hip joint centers were defined as the centers of the spheres fitted on the meshes of the femoral heads5. The knee joint centers were defined as the midpoints between the two centers of spheres fitted on both condyles of the femurs6. Finally, the ankle joint centers were defined as the midpoints between distal nodes of tibias and fibulas. Inter-hip, femur, and shank (left and right) lengths were computed for each method (RM, GCn, EOS). Normality of the data was not systematically ensured by Shapiro-Wilk test. Thus, the nonparametric Friedman's test was used to check if the quantities of the seven methods were significantly different (p<0.05). If this hypothesis was ensured, Fisher's LSD tests were applied to detect significant differences among methods (RM, GCn) compared to EOS. Results Abstracts should be no more than two pages in length. Results are presented in Table 1. First, no significant difference among methods was found for the left femur length assessment. Indeed, the mean distance between EOS and RM was 8.5 ±10 mm, and the mean difference between EOS and GCn was between 4.1 ±7.4 mm (GC3) and 1.8 ±7.8 mm (GC50). Right femur and inter-hip lengths estimations were significantly different among methods. Fisher's LSD test revealed significant differences for the right femur length between EOS, RM and GC50,100,500, with mean differences with respect to the EOS reference data ranging from 2.3±6.3 mm (GC3) to 11.3±10.5 mm (RM). EOS inter-hip length was significantly different when assessed from RM and GC3,10 and mean differences to EOS data ranged between -14,5 ±10.6 mm (RM) and -4.47 ±11.4 mm (GC500). Also, significant differences between methods were found for both shanks, Fisher's LSD test revealed significant difference between EOS and each method. The mean errors for these segments ranged between 28.84 ± 5.83 (GC50) and 35.33 ±12.31 (RM) for the left shank and between 33.9 ±12.06 (RM) and 27.46 ± 6.01 (GC500) for the right side. Discussion The results of GCn are promising since it reduced systematically the segments lengths differences compared with the EOS reference data. However, the number of frames used in GCn influenced the results. For inter-hip length and left femur, increasing n tended to minimize significantly the distance to EOS until GC50. For the right femur, the GC3 gave better results but might be a local minimum. This approach seems to give good results compared to regression and functional methods such as sphere fitting to estimate the hip position7. However, GCn didn't matched the measured anatomical length of the shank. Only 7 subjects were involved but the difference was systematic, requiring further investigations to be explained. However, the comparison of the different methods, based on segment length estimation, should be completed by a comparison of joint center position assessment in the segments' local coordinate system.

01746073

en

2024/03/05 22:32:07

2017

https://theses.hal.science/tel-01746073/file/SarahCristophVDpdftk.pdf

Maria Foglia Christophe Cécile Francisco Corinne Léa, Sylvie Thibaud ) Gervaise nécessaire. Merci Francisco de m'avoir fait confiance pour te lancer dans l'aventure de l'encadrement de thèse. Merci de m'avoir accompagnée à travers les problèmes techniques, le bricolage, les discussions théoriques et les résultats intriguant, à travers les manips réussies et (surtout) les manips ratées. Je n'aurais pas pu rêver un meilleur encadrement, merci du fond du coeur pour ta curiosité communicative, ton enthousiasme débordant, ta gentillesse et ta disponibilité sans faille. Je souhaite également adresser tous mes remerciements aux membres de mon jury de thèse Francisco Del Monte et Vanessa Prevot, en qualité de rapporteurs ainsi que Pierre Barré, Bernard Cathala, Sylvie Derenne, et Sylvain Deville en tant qu'examinateurs. Je vous remercie pour le temps que vous avez accordé à la lecture de mon manuscrit. J'ai été ravie d'avoir la chance de pouvoir vous présenter mes travaux vous suis très reconnaissante pour vos commentaires et remarques ainsi que pour la discussion très enrichissante que nous avons pu partager lors de la soutenance. Je tiens également à remercier la directrice du LCMCP Florence Babonneau de m'avoir acceptée au sein d'un laboratoire si propice à l'épanouissement tant scientifique que personnel, dans un environnement où excellence scientifique et convivialité semblent toujours aller de pair. Merci à Christian Bonhomme et à l'ED 397, ainsi qu'au LABEX Matisse pour le financement de cette thèse, qui m'ont permis de me lancer dans le grand bain de la recherche dans des conditions plus que favorables. Merci à Laurence Bonnet-Lericque pour son aide tout au long de la thèse et en particulier pour les aspects logistiques de la soutenance en elle-même. Un immense merci à Corinne Pozzo-Di Borgo et Hélène Gervais pour leur aide tout au long de ces trois ans. Merci pour votre sourire, votre efficacité et votre disponibilité face à toutes sortes d'appels à l'aide. Merci également à Diana Lesueur et Bernard Haye grâce à qui on ne manque de rien pour les manips. Un grand merci également à toutes les personnes qui m'ont aidée pour caractériser mes échantillons et m'ont accordé leur temps et leur savoir-faire. Merci à Isabelle Génois pour m'avoir appris les secrets du MEB, de l'EDX et du BET (et merci pour tout le reste que ce soit lors des catastrophes de lyophilisateur ou pour les joies de l'inventaire et de l'étiquetage des produits chimiques). Merci Bernard Haye pour la préparation des échantillons de TEM et merci Patrick Le Griel et Gervaise Mosser pour l'observation. Merci à Guillaume Laurent pour la RMN (et la gestion du bidon d'azote liquide). Merci à Nora Abdoul-Aribi et Alexandre Bahezre pour l'ATG. Merci à Christophe Hélary et David Pinto pour m'avoir initiée à la microbiologie. Je souhaite également à remercier les différentes personnes hors du LCMCP qui ont contribué à ces travaux. En premier lieu, un très grand merci à Pierre Barré du Laboratoire de Géologie de l'ENS. Merci pour ton aide, tes conseils et ta gentillesse, j'ai été ravie d'avoir l'occasion de travailler avec toi. Merci également à Catherine Garnier et Estelle Bonnin de l'INRA à Nantes, sans qui aucun des matériaux à base de pectine n'auraient pu être étudiés. Merci à Anne-Laure Rollet et Guillaume Meriguet du laboratoire Phenix pour leur expertise pour In the last few decades, the key importance of the human environmental footprint on the various ecosystems of the planet has slowly made its way into the collective consciousness. From this awareness, various initiatives have emerged for pollution monitoring, prevention and remediation, from individual actions to international consensus. Soil pollution is especially concerning due to the various pathways through which it can lead to bioaccumulation (leaching into ground water, accumulation in plants and microorganisms etc…) and ultimately impact all livings beings on Earth. In addition soils are particularly difficult to manage, compared for instance to waters: while ex situ approaches raise important logistical issues and can be highly disturbing for the local ecosystems, in situ treatments face the intrinsic complexity of solid yet highly dynamic media. Bioremediation is Nature's response to pollution of a given ecosystem. It consists in the accumulation or degradation of contaminants by living organisms, including plants, but also animals (such as insects or earthworms) or diverse microorganisms (both eukaryotic, i.e. fungi or prokaryotic, i.e. bacteria). The natural response of a polluted ecosystem (bioattenuation) may however not always be efficient enough to provide total or quick enough restauration of the soil's original condition. In this case two main strategies have been devised to support the natural recovery of the considered environment [START_REF] Adams | Biostimulation and Bioaugmention: A Review[END_REF][START_REF] Juwarkar | A comprehensive overview of elements in bioremediation[END_REF] . Biostimulation requires the amendment of the soil with various nutrients to enhance the endogenous degradation activity. This approach is however limited by the composition of the local biological population. Alternatively, bioaugmentation consists in the introduction of exogenous organisms to provide the soil with the necessary remediation capabilities. This approach may be very efficient to degrade the targeted contaminants, but raises the question of the introduction of exogenous organisms which may responsible for severe unbalances in the original ecosystem. Regardless of the bioremediation approach, a limit to the use of living organisms is their sensitivity to their immediate environment. As a result the physico-chemical properties of the considered soil (for instance concentrations in contaminants or co-contaminants, pH, salinity, temperature etc…) may be deleterious to the viability of species relevant to bioremediation. One way to limit the impact of the microorganisms' direct environment is to design immobilization matrices capable of hosting and protecting living cells. This approach has in particular been used for the design of bioactive materials for applications in environmental science [START_REF] Cassidy | Environmental applications of immobilized microbial cells: A review[END_REF] . Immobilization of organisms within an appropriate matrix can indeed provide valuable protection against detrimental physico-chemical conditions. In addition such entrapment can be a way to limit dispersion of exogenous organisms within a given ecosystem, thus preventing possible biological disequilibria. But the design of an ideal encapsulating matrix for organisms with bioremediation capabilities is a challenge on several levels. The two main objectives of bioremediation approaches based on encapsulated cells are the efficiency of the depollution process and the confinement of the exogenous microorganisms within the matrix. These two apparently simple objectives are actually influenced by a wide range of interrelated parameters (see Figure 1). Some of these are dependent on the characteristics of the polluted site (type of soil, temperature, physico-chemical properties of the soil but also nature of the contaminant) but part of them are related to the cellularized material itself. The device can be seen as the association of a functional unit (the encapsulated metabolically active microorganism) and of a structural part (the encapsulation matrix itself). The efficiency of the depollution process is primarily dependent on the metabolic activity of the entrapped microorganisms, but the structure of the matrix can also be crucial since it is likely to modify the diffusion rates of the substrates and therefore the depollution kinetics. Both functional and structural points of view must therefore be taken in account in the design of an efficient depollution device. Control of these aspects can be achieved through two inter-dependent routes. The composition of the encapsulation matrix must first be selected so as to be non-cytotoxic towards the encapsulated organisms, as well as, on a larger time and space scale, towards the whole considered ecosystem. The nature of the constituents of the matrix must also be chosen to ensure structural stability during the residency of the material in soil, as a way to prevent leaching of the entrapped exogenous organisms. These choices can however not be separated from material engineering aspects and, especially, the matrix shaping process that must also be compatible with encapsulation of living organisms. Regarding the matrix composition, the literature has highlighted the interest of biopolymers as encapsulating matrices [START_REF] Lim | Microencapsulated Islets as Bioartificial Endocrine Pancreas[END_REF][START_REF] Gasperini | Natural polymers for the microencapsulation of cells[END_REF] . These polymers are usually found in organisms as part of the extracellular matrix or directly within the cell walls. As a result most of them are highly cytocompatible, which is a key feature when considering cell encapsulation. This cytocompatibility is however often accompanied by biodegradability, which may be a valuable advantage for biomedical applications, but could be problematic regarding the stability of a matrix supposed to prevent cell leaching. One way to tune the mechanical properties of a matrix, while preserving part or all of its chemical functionalities is the use of hybrid or composites materials. As a result biopolymer-inorganic hybrid and more specifically biopolymer-silica hybrid and composites have been widely used for cell and microorganism encapsulation [START_REF] Coradin | Silica-alginate composites for microencapsulation[END_REF][START_REF] Léonard | Whole-cell based hybrid materials for green energy production, environmental remediation and smart cell-therapy[END_REF] , and could be especially useful for bioremediation approaches [START_REF] Chen | Decolorization of azo dye by immobilized Pseudomonas luteola entrapped in alginate-silicate sol-gel beads[END_REF] . From the structural point of view, matrices can adopt various shapes and structures. In the targeted application, elaboration of a porous material could provide valuable advantages regarding substrate diffusion. Ice-templating can be used for shaping a wide range of compounds (from ceramics to polymers) to yield lots of different pores morphology, including well controlled and tunable oriented pores. Such a porosity could be useful in facilitating substrate transport via capillary phenomena. In addition, the efficiency of freezecasting for physical entrapment of living organisms and preservation of their metabolic activity has been demonstrated in several materials [START_REF] Soltmann | Utilization of sol-gel ceramics for the immobilization of living microorganisms[END_REF][START_REF] Gutiérrez | Hydrogel scaffolds with immobilized bacteria for 3D cultures[END_REF] . Based on a literature survey presented in Chapter I, we hypothesized in this PhD work that cellularized biopolymer-silica hybrid porous materials obtained by freeze-casting, a specific well-controlled and easily-tunable ice-templating technique, could prove very efficient in soil bioremediation processes (see Figure 2). Our approach was based on the preparation of these materials in a two-step process: 1. encapsulation of the chosen microorganisms within a biopolymer porous structure through freeze-casting and subsequent lyophilization. Biopolymer Inorganic moeity Encapsulated bacteria Pollutants Non-toxic by products 2. coating of the cellularized biopolymer scaffold with a silica layer using sol-gel chemistry. Each of these steps had to be adapted to the nature of the compounds to ensure both control over the material morphology and compatibility with cell survival. The strategy we adopted to pinpoint parameters relevant for the control of structural and functional properties of the materials was to first develop the shaping strategy in absence of cells, but keeping in mind its implication for cell encapsulation (see Figure 3). Freeze-casting of pectin foams was first investigated to identify the relevant parameters in the design of unidirectional porous foams (Chapter II). Sol-gel silica deposition using vapor phase precursor was then developed to yield a pectin-silica core-shell structure for the pore walls and gain control over the silica layer characteristics. The stability of such material was then evaluated in a reference soil (Chapter III). Once the key parameters regarding the material's structure were identified, the process was applied to the encapsulation of model microorganisms to explore their influence on the material functionality. Adjustments of the process were implemented to ensure good cell compatibility while keeping control over the structural aspects (Chapter IV). Finally, optimization of the biopolymer nature, silicification process and cell type allowed for the successful evaluation of the depollution capabilities of the cellularized biopolymer-hybrid porous material in a real soil (Chapter V). Soil is a key component in the behavior and equilibrium of various ecosystems. It is at the root of chains and interactions between the various living organisms of these complex systems. Physical or chemical degradation of a given soil may have a wide range of negative consequences [START_REF] Gros | Fonctionnement et qualité des sols soumis à des perturbations physiques et chimiques d'origine anthropique : réponses du sol , de la flore et de la microflore bactérienne tellurique[END_REF] , directly on the local fauna and flora, but also indirectly on human health through different pathways including food and water consumption. As a result monitoring and protection of this precious resource is essential. In several cases however soils may be damaged by anthropic activities but also in some occurrences through natural phenomena. It is then essential to try to restore the considered ecosystem to its original state, concerning the physico-chemical composition of the soil but also regarding the endogenous microbial population. The notion of soil quality may be difficult to restrain to one set of physical or chemical characteristics, since soils are actually a set of interactions, fluxes and processes which may vary both in time (seasons) and space (depth). The soil condition may however be described by chemical (inorganic composition, nutrients, total organic content), physical (texture, erosion) and biological characteristics (microbial population). The notion of pollution can be defined as the presence of an imbalance due to the prevalence of specific compounds. Rehabilitation of a polluted soil usually requires removal or degradation of the considered contaminant and return to the initial ecosystem equilibrium. This notion is however not as simple as it may initially appear, since a wide variety of phenomena may result in the displacement, immobilization or degradation of a target compound. In addition, sources and types of pollutants are extremely diverse and polluted sites often simultaneously contain different types of pollutant. In addition, as was mentioned earlier, each soil is complex and unique ecosystem, which implies that any depollution process must be preceded by a careful analysis and characterization of the considered site. I.1.a.i Common pollutants Sources of contaminations are extremely diverse, but some classes of pollutants are especially widely used. They may be separated based on their chemical structure, their main sources and uses or even depending on their introduction pathways. The different categories however often overlap and one type of classification or the other may be more relevant to discuss certain families of pollutants. The way certain pollutants may end up in a given soil is strongly dependent on their uses. Pollution of a given soil may be the result of single accidental events (oil spills for instance) or of repeated use of certain compounds in human activities as is the case regarding the use of pesticides [START_REF] Shayler | Sources and Impacts of Contaminants in Soils[END_REF] . Some contaminants may also have a non-anthropogenic origin such as wildfires [START_REF] Finlay | Health impacts of wildfires[END_REF] . The contamination vectors are also diverse, since pollutants may end-up in the through a wide range of pathways, for instance by direct diffusion, through liquid infiltration or through the atmosphere (see Several classes of pollutants have been pointed out as especially concerning. In particular pesticides and their fate in soils have attracted a considerable amount of attention [START_REF] Arias-Estévez | The mobility and degradation of pesticides in soils and the pollution of groundwater resources[END_REF][START_REF] Gevao | Bound pesticide residues in soils: A review[END_REF][START_REF] Anjum | Environmental Protection Strategies for Sustainability[END_REF][START_REF] Odukkathil | Toxicity and bioremediation of pesticides in agricultural soil[END_REF] in the last few decades. This type of compounds are somewhat peculiar in the domain of pollutants since they are knowingly and repeatedly introduced in the environment, where they provide valuable advantages [START_REF] Aktar | Impact of pesticides use in agriculture: their benefits and hazards[END_REF] . The ideal pesticide should "be toxic only to the target organism, be biodegradable and undesirable residues should not affect non-target surfaces" [START_REF] Anjum | Environmental Protection Strategies for Sustainability[END_REF] but it is actually estimated that only a very small fraction of the used pesticides reach their intended targets [START_REF] Pimentel | Amounts of pesticides reaching target pests: Environmental impacts and ethics[END_REF][START_REF] Pimentel | Pesticides: Amounts Applied and Amounts Reaching Pests[END_REF] . Even if pesticides are a major and well-advertised source of pollution, it is far from being the sole cause for soil contamination. Polycyclic Aromatic Hydrocarbons (PAH) have been of considerable concern for several decades [START_REF] Migliore | Biodegradation of oxytetracycline by Pleurotus ostreatus mycelium: A mycoremediation technique[END_REF] due to their ubiquity [START_REF] Cerniglia | Biodegradation of polycyclic aromatic hydrocarbons[END_REF] . They are mainly the result of combustion processes ranging from natural fires [START_REF] Freeman | Woodburning as a source of atmospheric polycyclic aromatic hydrocarbons[END_REF] to domestic heating, urban traffic [START_REF] Dubowsky | The contribution of traffic to indoor concentrations of polycyclic aromatic hydrocarbons[END_REF] and industrial incineration. Due to intensive use of coal in the 20 th century industries, considerable amounts of PAH may be found in numerous soils [START_REF] Johnsen | Principles of microbial PAH-degradation in soil[END_REF] . Due to their chemical structure (two or more fused aromatic cycles (see Figure I.3 a) and their hydrophobicity [START_REF] Yu | Natural attenuation, biostimulation and bioaugmentation on biodegradation of polycyclic aromatic hydrocarbons (PAHs) in mangrove sediments[END_REF] these compounds tend to be highly persistent [START_REF] Wilson | Bioremediation of soil contaminated with polynuclear aromatic hydrocarbons (PAHs) : a review[END_REF] . Another family of hydrocarbon derivatives deemed of specific concern is the so called BTEX family, which regroups benzene, toluene, ethylbenzene and xylenes (see Figure I.3 b). These compounds are retrieved from fossil fuels and are widely used as solvents or additives in gasoline. As a result contamination sources include various industrial plants, gas stations but also large traffic axes. These compounds are especially concerning due to their very high toxicity. Diseases associated with BTEX exposure range from respiratory conditions to cardiovascular impediments, endocrine disruption or fertility disorders. Another class of common pollutants is dyes due the very large global demand and use. The main source of dye contamination is the textile industry. It is deemed that 10 to 25% dyes are lost during textile dyeing process, part of which may be directly released in the environment [START_REF] Carmen | Textile Organic Dyes -Characteristics , Polluting Effects and Separation / Elimination Procedures from Industrial Effluents -A Critical Overview[END_REF] . Most dyes are based on aromatic cycles substituted with various chemical functions, among which the azo group is particularly prominent [START_REF] Pereira | Environmental Protection Strategies for Sustainability[END_REF] . As is the case for other organic contaminants, dye can chemically be broken down into various reaction products. Both dyes and by-products may represent risks for the contaminated ecosystems. Beside the visual pollution resulting from dye contamination, these chemical compounds have been reported to cause various conditions ranging from simple skin irritation or allergies to carcinogenic [START_REF] Puvaneswari | Toxicity assessment and microbial degradation of azo dyes[END_REF] and mutagenic effects [START_REF] Robinson | Remediation of dyes in textile effluent: A critical review on current treatment technologies with a proposed alternative[END_REF] . If organic pollutants represent the majority of contaminants, inorganic compounds may also present a risk for environment and health. Metals are ubiquitous and present serious issues regarding the decontamination processes. Contrary to purely organic compounds, metals cannot be broken down into nontoxic compounds such as water or carbon dioxide. Remediation approaches for metals are therefore mainly based on redox modifications to minimize toxicity and mobility or on physical removal of the contaminants. Since these compounds cannot be degraded [START_REF] Valls | Exploiting the genetic and biochemical capacities of bacteria for the remediation of heavy metal pollution[END_REF] , they often bioaccumulate along the food chain [START_REF] Bryan | Bioavailability, accumulation and effects of heavy metals in sediments with special reference to United Kingdom estuaries: a review[END_REF][START_REF] Mclachlan | Bioaccumulation of hydrophobic chemicals in agricultural food chains[END_REF] resulting in possible heavy metal poisoning which may induce wide range of conditions including kidney, pulmonary or cardiac damages, cognitive and neurological impairments as well a cancer [START_REF] Järup | Hazards of heavy metal contamination[END_REF] . Even if sources of contamination are diverse, the fate of the different contaminants share common features. Regardless of the type of pollutant, the issue of bioavailability has a major influence on possible environmental and health risk, as well as remediation possibilities. I. 1.a.ii Bioavailability is key Once a contaminant reaches a given soil, it is susceptible to be subjected to diverse mobility pathways and to biotic or abiotic degradation (see Figure I.4) [START_REF] Stokes | Behaviour and assessment of bioavailability of organic contaminants in soil: relevance for risk assessment and remediation[END_REF] . Pollutants in soil may be partially or totally soluble in water, which can result either in run off phenomena (chemicals are transported mainly at the surface of the soil) or in soil leaching (pollutants are transported in depth in the soil and often down to underground aquifers). Another possible way for the chemicals to be removed from the soil is though volatilization. In both cases, removal of the pollutants from the soil results in the contamination of another part of the ecosystem (water or atmosphere). Depending on the properties of the considered chemical, it may also be retained within the soil by adsorption phenomena (either or organic or inorganic compound of the soil). In addition, contaminants may be internalized and stored by different organisms (from bacteria to animals and plants), resulting in a possible bioaccumulation phenomenon. They may also be modified or degraded by living organisms or abiotic chemical reactions, sometimes resulting in complete decomposition of the contaminants into non-toxic compounds. The fate of pollutants in the soil depends on the type of contaminant, especially on its chemical structure and properties, but the nature of soil itself is also of significant influence. Soils are usually characterized by a wide range of parameters. Soils with different compositions in terms of texture, that is to say contents in sand (particles >50 µm), silt (2 µm < particles < 50 µm) and clay (particles < 2 µm) particles, may have very different behaviors regarding water permeability and adsorption properties. Physico-chemical properties of the soil such as pH, redox potential, salinity, water and oxygen content, temperature or organic content [START_REF] Sijm | Bioavailability in soil or sediment: Exposure of different organisms and approaches to study it[END_REF][START_REF] Chung | Effect of soil properties on bioavailability and extractability of phenanthrene and atrazine sequestered in soil[END_REF] are likely to modify the bioavailability of the pollutants. A distinction is often made between the environmental scientist's view which usually defines bioavailabity as the fraction of a given pollutant that can interact with living organisms (microorganisms, plants or animals) [START_REF] Sijm | Risk Assessment of Chemicals: An introduction[END_REF] (see Figure I.5) and the toxicologist's view which defines bioavailability as the amount of a given compound capable of entering the bloodstream [START_REF] Vermeire | Risk Assessment of Chemicals: An introduction[END_REF] . In this work, the environmental scientist's definition was used as a reference. Figure I.5: The term bioavailability covers a wide range of processes including sorption and desorption and uptake by living organisms. Adapted from Sijm et al. [START_REF] Sijm | Risk Assessment of Chemicals: An introduction[END_REF] Bioavailability of chemicals in soils is often evaluated based on the equilibrium between the adsorbed compounds and the surrounding water (Equilibrium Partition Theory (EPT) [START_REF] Shea | Developing national sediment quality criteria[END_REF][START_REF] Toro | Technical basis for establishing sediment quality criteria for nonionic organic chemicals using equilibrium partitioning[END_REF] ). In this prospect, chemical properties of the considered pollutants are crucial, especially regarding hydrophobicity [START_REF] Cerniglia | Biodegradation of polycyclic aromatic hydrocarbons[END_REF] . As a result, contaminants are often described based on their octanol-water partition coefficient (KOW) [START_REF] Muncke | Estimating the Organic Carbon Partition Coefficient and Its Variability for Hydrophobic Chemicals[END_REF][START_REF] Noble | Partition coefficients (n-octanol-water) for pesticides[END_REF] in order to assess their tendency toward soil adsorption and more largely as an indicator of their bioavailability. Chemical compounds can however be bound to soil particle more or less tightly depending on the type of interactions involved (hydrophobic, ionic and electrostatic, van der Waals, hydrogen bond, ligand exchange etc…) [START_REF] Gevao | Bound pesticide residues in soils: A review[END_REF] . In addition, soils are not a stationary ecosystems. As a result dynamics and mobility of contaminants as well as their bioavailability is variable in time. It is usually accepted that pollutants undergo ageing mechanisms after prolonged residency in soils due to slower, but stronger, adsorption mechanisms [START_REF] Pignatello | Mechanisms of Slow Sorption of Organic Chemicals to Natural Particles[END_REF] . The bioavailable fraction may therefore diminish in function of time [START_REF] Alexander | Aging, bioavaiability, and overestimation of risk from environmental pollutants[END_REF] After residency in soils, contaminants tend to become strongly bound to the soil thus limiting the bioavailable fraction. Adapted from Jones et al. [START_REF] Jones | Persistent organic pollutants (POPs): state of the science[END_REF] Understanding the fate of contaminants and their bioavailability is essential to assess the environmental and health risks [START_REF] Ehlers | Contaminant bioavailability in soil and sediment[END_REF] . A highly toxic compound strongly bound in soil may never induce harmful effects. On the contrary, compounds with low intrinsic toxicity but highly bioavailable may accumulate in living organisms and become concentrated in specific organs resulting in deleterious effects for the concerned organism. One of the main difficulties in assessing the toxicity of specific compounds lies in the fact that the effects are entirely dependent on the dose. The notion of dose itself may be difficult to define, since a given compound may undergo different mechanisms of mobility, concentration or elimination within an organism. As a result the entry dose may not be the effective dose in sensitive target organs. In addition each organism has its own specificities and one compound may be highly toxic to one species and completely harmless or even beneficial to another. It has also been reported that toxicity of a combination of contaminants may induce synergistic toxicity, resulting in higher toxicity of the mixture compared to the sum of effects of each contaminant alone [START_REF] Arnold | Synergistic Activation of Estrogen Receptor with Combinations of V Environmental Chemicals[END_REF] . Finally the term toxicity has been defined as "the capacity of a chemical to cause injury" [START_REF] Vermeire | Risk Assessment of Chemicals: An introduction[END_REF] , which may in fact range from localized and reversible impacts to death of the considered organism. These factors concur to make the task of establishing regulations and standards regarding the threshold levels of hazardous chemical extremely complex. The diversity of pollutants implies a wide range of possible effects on the organisms, including toxicity, but in some cases the contaminant may also be used as part of the organisms' metabolism and therefore be biodegraded. As a result a high bioavailability, which might be detrimental from a toxicity point of view, may be seen as an advantage from the decontamination point of view [START_REF] Megharaj | Bioremediation approaches for organic pollutants: a critical perspective[END_REF] . As mentioned previously, soils may be contaminated by a wide range of chemical compounds. Once in the soil these compounds may be removed through various processes. They may simply migrate to other parts of the ecosystem (for instance leaching into groundwater or volatilize into the atmosphere) but can also be degraded through several mechanisms. These transformations may be abiotic (hydrolysis [START_REF] Han | Study on the Hydrolysis Kinetics of Dimethyl Disulfide[END_REF][START_REF] Venkatesan | Urea hydrolysis of tea soils as influenced by incubation period, soil pH, and nitrification inhibitor[END_REF] , photolysis [START_REF] Sturini | Environmental photochemistry of fluoroquinolones in soil and in aqueous soil suspensions under solar light[END_REF] ) or be part of biological processes (biodegradation). In some cases however these phenomena of natural attenuation may be inefficient or too slow. Active decontamination actions may therefore be needed to prevent toxic effects on the ecosystem and on human health. I.1.b.i Soil depollution The choice of the appropriate cleanup technology must be adapted both to the type of contaminant and to the characteristics of the polluted site. Other parameters such as cost effectiveness, volume of contaminated soil or time scales must also be taken in account when devising a soil remediation strategy. United States Environment Protection Agency provides a list of about twenty different approaches for remediation of soils and groundwater [START_REF]A Citizen's Guide to Cleanup Technologies[END_REF] . Main features of a few of these techniques are compiled in Table I. 1. Each approach has its own specificities, advantages; drawbacks and range of applicability. Complete rehabilitation of a given soil may sometimes require the combined use of two or more different techniques. There are several general approaches to manage risks in the case of a polluted soil. One of the main objectives is usually to prevent health hazards. Toxic compounds tend to enter organisms through transfer from another medium (air or water). One approach to limit risks therefore consists in trying to immobilize the compounds within the soil, or in other terms in limiting bioavailability of the pollutant. Techniques such as solidification or stabilization [START_REF] Tajudin | Stabilization/Solidification Remediation Method for Contaminated Soil: A Review[END_REF] and capping [START_REF] Simon | Standard and alternative landfill capping design in Germany[END_REF] are designed to limit the mobility of the pollutants. These approaches have however the drawback of leaving the pollutant within the soil, which may be problematic in the long run, since soils may undergo significant physical and chemical modifications. These approaches require therefore constant monitoring even after treatment. Pollutants may also be physically removed from the polluted sites. The most straightforward approach is excavation, which is usually combined with other techniques such as thermal desorption [START_REF] Falciglia | Low-temperature thermal desorption of diesel polluted soil: Influence of temperature and soil texture on contaminant removal kinetics[END_REF] , incineration or soil washing [START_REF] Dermont | Soil washing for metal removal: A review of physical/chemical technologies and field applications[END_REF][START_REF] Mulligan | Surfactant-enhanced remediation of contaminated soil: A review[END_REF] . These methods are usually complex from a logistical point of view, especially in the case of large contaminated areas. They are however fast and efficient. It is also possible to treat contaminated soil thanks to in situ methods. In situ thermal treatment [START_REF] Piña | Thermal treatment of soils contaminated with gas oil: Influence of soil composition and treatment temperature[END_REF] allows for extraction of the contaminant without excavation and in some case for direct degradation of the targeted chemicals. The recovered contaminants must then be stored or treated appropriately. Pollutants may also be degraded directly within the soil, for instance using in situ oxidation [START_REF] Tsitonaki | In Situ Chemical Oxidation of Contaminated Soil and Groundwater Using Persulfate: A Review[END_REF] or in situ reduction [START_REF] Ludwig | In situ chemical reduction of Cr(VI) in groundwater using a combination of ferrous sulfate and sodium dithionite: A field investigation[END_REF] . This however requires addition of oxidizing or reducing chemicals to the contaminated sites. soil may be used to refill the excavated site. Removal or volatile or semi volatile compound from soil by heating (between 100°C and 500°C). Vapors are collected to be further treated or recycled. Separation of the contaminants from soil thanks to an appropriate washing liquid. Burning of hazardous materials (soil, sludge but also liquids) at very high temperature (up to 1400°C). Pollutants are usually degraded but residual particles or acid gases may be formed. Residuals ashes can be treated or stored. Use of reducing (ex zero valent iron) or oxidizing agents (ex permanganate, hydrogen peroxide, ozone) to change contaminants into less toxic or less mobile forms. Reactive agents may be directly injected in wells, or introduces as granules. Mobilization of pollutants in soil thanks to heat. Soil is heated and pollutants are directed towards wells for recovery and further treatment. Some types of pollutants may be directly degraded in situ. Cleanup method Solidification and stabilization Capping Excavation Thermal desorption Soil washing Incineration In situ chemical reduction and oxidation In situ thermal treatment A wide range of physical and chemical remediation techniques exist to treat polluted soils. Most of them however require extensive logistic dispositions. Several of these techniques require complementary monitoring measures or treatments because the contaminants are not directly degraded. The in situ chemical cleanup techniques are efficient for the degradation of the pollutants but require the addition of chemical to the contaminated soil. One alternative to these physical and chemical techniques is to take advantage of the degradation capabilities displayed by a wide variety of living organisms. I.1.b.ii Principles of bioremediation Bioremediation can be defined as the use of biological entities for the degradation of contaminants. This apparently simple definition actually covers a wide range of processes since different living organisms (from bacteria and fungi to plants) may be involved in the remediation of various pollutants (hydrocarbons, metals, pesticides, dyes etc…) in a wide variety of environments (wastewaters, sediment, sludge, soil etc…). The term bioremediation also covers different approaches since the goal may simply be to remove the target contaminant from the considered ecosystem or to fully degrade the pollutant into nontoxic chemicals. Bioremediation approaches can be carried out with no or little involvement (e.g. natural in situ attenuation) or with fully engineered processes (e.g. ex situ bioaugmentation). This versatility may explain the increasing interest around bioremediation thematics in the last few decades [START_REF] Juwarkar | A comprehensive overview of elements in bioremediation[END_REF][START_REF] Megharaj | Bioremediation approaches for organic pollutants: a critical perspective[END_REF][START_REF] Hamer | Bioremediation: A response to gross environmental abuse[END_REF][START_REF] Iwamoto | Current bioremediation practice and perspective[END_REF] , as illustrated by the growing number of publications related to soil bioremediation (see Figure I.7). The idea of bioremediation is actually much older since it is the basic principle behind the formation of compost [START_REF] Kulcu | Determination of aeration rate and kinetics of composting some agricultural wastes[END_REF][START_REF] Fogarty | Microbiological degradation of pesticides in yard waste composting[END_REF] . Activated sludges have been used for wastewater treatment since the beginning of the 20 th century [START_REF] Shannon | Science and technology for water purification in the coming decades[END_REF][START_REF] Khan | Aerobic granulation for wastewater bioremediation: A review[END_REF] . Bioremediation in the modern acceptation of the term, that is to say as a way to degrade pollutants, was introduced in the late 60s by George Robinson who used microorganisms to remediate oil spill along the Santa Barbara coast [START_REF] Adams | Biostimulation and Bioaugmention: A Review[END_REF] . Bioremediation approaches have since been used for the depollution of a wide range of ecosystems (freshwater and marine environments, groundwater, sediment, sludge, soil) and with different types of contaminants. Different approaches, both in situ and ex situ [START_REF] Hatzinger | In-Situ and Ex-Situ Bioremediation Options for Treating Perchlorate in Groundwater[END_REF][START_REF] Azubuike | Bioremediation techniquesclassification based on site of application: principles, advantages, limitations and prospects[END_REF] can be used. The choice of the appropriate strategy must take in account the characteristics of the contaminated site but also the type of pollutant itself. Many reviews have been devoted to the bioremediation of specific classes of pollutants such as metals [START_REF] Yao | Review on Remediation Technologies of Soil Contaminated by Heavy Metals[END_REF][START_REF] Salt | Phytoremediation: a novel strategy for the removal of toxic metals from the environment using plants[END_REF] , PAH [START_REF] Cerniglia | Biodegradation of polycyclic aromatic hydrocarbons[END_REF][START_REF] Johnsen | Principles of microbial PAH-degradation in soil[END_REF][START_REF] Yu | Natural attenuation, biostimulation and bioaugmentation on biodegradation of polycyclic aromatic hydrocarbons (PAHs) in mangrove sediments[END_REF][START_REF] Haritash | Biodegradation aspects of polycyclic aromatic hydrocarbons (PAHs): a review[END_REF][START_REF] Samanta | Polycyclic aromatic hydrocarbons: Environmental pollution and bioremediation[END_REF][START_REF] Bamforth | Bioremediation of polycyclic aromatic hydrocarbons: current knowledge and future directions[END_REF] , oil derivatives [START_REF] Ron | Biosurfactants and oil bioremediation[END_REF][START_REF] Mazzeo | BTEX biodegradation by bacteria from effluents of petroleum refinery[END_REF] , pesticides [START_REF] Anjum | Environmental Protection Strategies for Sustainability[END_REF][START_REF] Odukkathil | Toxicity and bioremediation of pesticides in agricultural soil[END_REF] or dyes [START_REF] Kaushik | Fungal dye decolourization: Recent advances and future potential[END_REF][START_REF] Khan | Microbial decolorization and degradation of synthetic dyes: a review[END_REF][START_REF] Saratale | Bacterial decolorization and degradation of azo dyes: A review[END_REF] . Another common way to classify bioremediation approaches is based on the type of organisms involved. This includes a wide variety of organisms from unicellular microorganisms to plants. Some definitions tend to limit the term bioremediation to the use of microorganisms and to consider phytoremediation, the use of plant, as a separate approach [START_REF] Juwarkar | A comprehensive overview of elements in bioremediation[END_REF][START_REF] Singh | Bioremediation: environmental clean-up through pathway engineering[END_REF] . Plants are often favored for metal remediation [START_REF] Salt | Phytoremediation: a novel strategy for the removal of toxic metals from the environment using plants[END_REF][START_REF] Ali | Phytoremediation of heavy metals-Concepts and applications[END_REF][START_REF] Paz-Ferreiro | Use of phytoremediation and biochar to remediate heavy metal polluted soils: A review[END_REF] while microorganisms are often used for the degradation of organic chemicals. Phytoremediation strategies can actually be applied both for bioaccumulation of contaminants (including metals), as a way to immobilize or even extract contaminants [START_REF] Meagher | Phytoremediation of toxic elemental and organic pollutants[END_REF] , and as a way to fully mineralize organic pollutants into non-toxic compounds [START_REF] Cunningham | Phytoremediation of contaminated soils[END_REF] . Animal biodegradation is very scarce compared to plant or microbial activity. Soil animals, especially earthworms, are however important as part as global remediation strategies, since they are in situ indicators of the ecosystem's health. Soil animal may also have a significant importance enhancing other bioremediation processes, especially microbial activity, by improvement of the soil fertility and aeration, as well as increase of pollutants bioavailability [START_REF] Hickman | Earthworm assisted bioremediation of organic contaminants[END_REF] . The generic term microorganism is used to refer to a wide range of potential vectors of bioremediation [START_REF] Iranzo | The use of microorganisms in environmental remediation[END_REF][START_REF] Watanabe | Microorganisms relevant to bioremediation[END_REF] . The three main types of microorganisms relevant to bioremediation are microalgae [START_REF] Lim | Use of Chlorella vulgaris for bioremediation of textile wastewater[END_REF][START_REF] Perales-Vela | Heavy metal detoxification in eukaryotic microalgae[END_REF] , fungi [START_REF] Kaushik | Fungal dye decolourization: Recent advances and future potential[END_REF][START_REF] Perales-Vela | Heavy metal detoxification in eukaryotic microalgae[END_REF] (including yeast [START_REF] Wang | Biosorption of heavy metals by Saccharomyces cerevisiae: A review[END_REF] ) and bacteria [START_REF] Saratale | Bacterial decolorization and degradation of azo dyes: A review[END_REF][START_REF] Lu | Bacteria-mediated PAH degradation in soil and sediment[END_REF] . Bacteria are actually the most represented type of microorganisms used in bioremediation approaches. This might be explained by their versatility and adaptability to a wide range of ecosystems [START_REF] Brim | Engineering Deinococcus radiodurans for metal remediation in radioactive mixed waste environments[END_REF][START_REF] Hao | Crude-oil-degrading thermophilic bacterium isolated from an oil field[END_REF] . As is the case for higher plants, microbial remediation may occur as bioaccumulation and subsequent limitation of leaching phenomena or as transformation of the considered pollutants (modification into as less toxic forms or full break-down of the compounds). Regardless of the organism used, bioremediation mechanisms are often complex and not fully understood due to the large number of parameters involved. These parameters are actually very close to the ones used to describe soil quality. They include pH of the soil, presence of nutrients, presence of electron donors or acceptors, temperature, salinity. Presence of oxygen may also condition the efficiency of bioremediation processes since certain types of organisms require specifically aerobic or anaerobic conditions. Models of degradation of one specific pollutant by one specific strain may be studied in laboratory, under closely controlled conditions, but in situ bioremediation is usually more complex. In the case of in situ bioremediation processes it is common to find multiple contaminants or nutrients as well as different organisms. These different actors may interact and influence each other to create synergetic effects. Consortia of bacteria [START_REF] Dojka | Microbial Diversity in a Hydrocarbon-and Chlorinated-Solvent-Contaminated Aquifer Undergoing Intrinsic Bioremediation[END_REF] , bacteria and fungi [START_REF] Silva | Bioremediation of a polyaromatic hydrocarbon contaminated soil by native soil microbiota and bioaugmentation with isolated microbial consortia[END_REF] , bacteria and algae [START_REF] Fu | Algae and Their Bacterial Consortia for Soil Bioremediation[END_REF] or microbes and animals [START_REF] Hickman | Earthworm assisted bioremediation of organic contaminants[END_REF] have for instance been identified in bioremediation processes [START_REF] Glick | Phytoremediation: Synergistic use of plants and bacteria to clean up the environment[END_REF] . Even if specific degradation mechanisms cannot be fully resolved, a few general processes involved in bioremediation have been identified. A distinction may be applied between processes that occur within the organisms and externally. In the latter case, mechanisms are often enzyme based, where the contaminants are used as substrates or co-substrates. For processes occurring within the organisms, the first step is the internalization of the considered pollutant. This implies bioavailability of the pollutant. Organisms may excrete specific chemical to enhance bioavailability and facilitate internalization of the chemicals. Plant may for instance excrete chelator to make metals more available 100 . Similarly, bacteria have been reported to produce surfactants which may help in desorbing hydrophobic pollutants (for instance hydrocarbons) from soil particles 101 . Chemicals within the various organisms may have different fates. They may be toxic compounds and therefore be sequestrated or degraded as defensive mechanisms. In certain cases, chemical compounds which are considered as toxic compounds from human health perspective can actually be used as part of the metabolic activity of organisms (as primary substrate or in cometabolism) in bioremediation processes 102 . Figure I.8: Various approaches can be devised for bioremediation. Bioattenuation relies on natural degradation of the contaminants, while biostimulation consists in adding various chemical compounds to enhance degradation by endogenous microorganisms. Bioaugmentation relies the addition of microorganisms or microbial consortia in the polluted soil for remediation. Endogenous organisms Soil composition (nutrient, electron donors/acceptors etc…) Contaminants Additives (nutrient, electron donors/acceptors etc…) Exogenous organisms Biostimulation Bioattenuation Bioaugmentation Understanding the mechanisms involved in bioremediation is essential to design organismbased soil depollution strategies. Three main approaches are generally distinguished regarding bioremediation, in particular microbial remediation: bioattenuation, biostimulation and bioaugmentation [START_REF] Adams | Biostimulation and Bioaugmention: A Review[END_REF][START_REF] Megharaj | Bioremediation approaches for organic pollutants: a critical perspective[END_REF]66 (see Figure I.8). One gram of soil is estimated to contain about 100 000 000 bacteria (up to 7000 species) and around 10 000 fungal colonies [START_REF] Anjum | Environmental Protection Strategies for Sustainability[END_REF] . Such a rich biomass has a tremendous remediation potential and indeed one of the most common approach for soil remediation, bioattenuation, consists in exploiting the natural degradation capabilities of a given soil. In this case, the approach consists mainly in monitoring the evolution of the contaminant concentrations. This approach is often successful thanks to the adaptability of microorganisms which can use a wide variety of carbon sources but may also undergo selective evolution where most resistant strain prevail over strains which are not capable of degrading the considered contaminant. As a result efficient strains for the degradation of specific pollutants can often be isolated from the contaminated sites. When the bioattenuation strategy is not efficient enough or too slow, various compound may be added to stimulate the natural metabolism of the endogenous microbial population. This general approach is called biostimulation and may consist in the addition of a wide range of compounds, including nutrients (carbon sources, nitrogen sources or electron donors and acceptors etc...) 103 . In some instances the initial pollutant may only be degraded as a cosubstrate. The soil may therefore need to be amended with a primary substrate to allow efficient remediation, which may itself be a contaminant such as toluene or phenol 104 . When considering aerobic degradation pathways, the oxygen content of the soil may be a limitation to the biodegradation efficiency. In this case, it is possible to inject oxygen in the soil. This specific type of biostimulation is called bioventing. As mentioned previously, bioavailability is essential for efficient biodegradation. Various chemical compounds may therefore be added to contaminated soils in order to enhance bioavailability, such as chelating agents for metals 105 or surfactants for hydrocarbons [START_REF] Mulligan | Surfactant-enhanced remediation of contaminated soil: A review[END_REF] . Even with favorable soil composition, some sites cannot be efficiently depolluted by the endogenous soil microbial population, either because the microorganisms are not intrinsically capable of metabolizing the specific contaminant or because the microbial population is too limited in number. Bioaugmentation approaches aim at introducing microorganisms within the soil to perform bioremediation. The organisms may be strains isolated from the site and cultivated to reach sufficient number or exogenous microorganisms selected for their known efficiency regarding the considered contaminant. These strains may for instance have been isolated from other sites contaminated by similar pollutants, or be genetically engineered strains 106,107 . In some cases microbial consortia may be design to take advantage of cooperative effects. For instance one organism may be chosen to enhance bioavailability of the targeted pollutant, another may metabolize the pollutant in question and another may produce nutrient to promote the metabolic activity or surfactants to enhance bioavailability. Introduction of microbial consortia may also be interesting in the case of co-contaminated sites. Bioremediation is therefore a flexible and widely applicable strategy for depollution of contaminated soils. The approach has however some significant limitations. I.1.b.iii Advantages and limitations of bioremediation approaches As mentioned previously, one the main advantages of bioremediation is its versatility. The vast number of potentially degrading organisms and their uses in different consortia, as well as the possibility to add various nutrients and substrates makes bioremediation an option for the remediation of most contaminated sites. In addition, bioremediation approaches have been estimated to be competitive from the cost point of view, compared for instance to thermal or chemical treatments [START_REF] Juwarkar | A comprehensive overview of elements in bioremediation[END_REF] . Most bioremediation strategies may be conducted both in situ and ex situ. In situ approaches are usually advantageous from the logistical and cost effectiveness perspective and may contribute to limit the environmental impact of the remediation process (no excavation, no destruction of fauna or flora). Ex situ approaches may however be favored when the soil must be amended with nutrients (biostimulation) or when parameters such as oxygen content or temperature must be controlled. The use of living organisms for depollution however presents some fundamental limitations. Bioremediation processes are often slow (from a few month to a few years) compared to physical methods for instance, and may therefore not be used when contamination must be rapidly contained to prevent health risks. While a wide variety of contaminants can be degraded into nontoxic compounds, in some instances the metabolic pathways result in the formation of more toxic by-products [START_REF] Juwarkar | A comprehensive overview of elements in bioremediation[END_REF]108 . Degradation mechanisms must therefore be studied in order to predict as accurately as possible the chemical intermediate likely to be produced during the degradation. Another possible limitation bioremediation is the fact that some types of chemical compounds cannot be degraded (see Table I.2). In the case of metal or radioactive compounds, this fact is inherent to the nature of the chemicals, and bioremediation suffers the same limitations as other remediation techniques. The compounds may be transformed into less toxic or less bioavailable forms and can be extracted from the contaminated sites (with phytoremediation for instance), but they cannot be fully degraded. Another limitation of bioremediation processes is inherent to the use of living organisms. Contrary to chemical or physical processes, bioremediation phenomena are fragile equilibria. Some contaminants easily degraded when present in low concentration may be toxic to the microbial population at high levels. In other case, presence of co-contaminants may inhibit the metabolic activity of the considered metabolically active organisms. Living organisms may also be sensitive to other soil characteristics such as pH, temperature or salinity. One possible option to prevent such deleterious effects is to physically separate the microorganisms from the soil and high toxic compounds concentrations by encapsulating the cells within a compatible matrix [START_REF] Cassidy | Environmental applications of immobilized microbial cells: A review[END_REF]109 . I.2 About encapsulation Immobilization of microorganisms in a matrix could provide valuable protection against deleterious effects of high contaminant concentrations or non-compatible physico-chemical characteristics of the soil. This leads to consider remediation approaches not as the use of biologically species but rather as the use of a full device which is the combination of a functional unit (the metabolically active organism) and of a structural part (the encapsulating matrix). The design of a cellularized material however cannot be considered solely from the material science point of view but rather with an approach integrating both material processing and biology. The main limitation is that structural requirements (for instance in term of mechanical properties or stability) may involve choice of materials or processes which are not compatible with living organisms. When considering encapsulation in a matrix, several aspects must be addressed. The first aspect regards the composition of the matrix, which can be based on organic or inorganic compounds, but also be hybrid. The geometry and structure of the matrix must also be adapted to the targeted application. Whole cells (either eukaryotic or prokaryotic) have for instance been immobilized in different structures such as shells 110 , films 111 , beads [START_REF] Coradin | Silica-alginate composites for microencapsulation[END_REF] , fibers 112 or gels 113 (see Figure I.9). The choice of the composition of the matrices and the desired structure influences the requirements in terms of process, which in turn must be adapted to preserve the integrity of the encapsulate. I.2.a What is encapsulation? To understand the role of material processing in encapsulation approaches, it is first necessary to give a brief definition of the term encapsulation itself. Encapsulation is a type of immobilization, where the encapsulated species is entrapped within a matrix, as opposed to binding where the object is attached on the surface of the material. Binding approaches can be based on a wide range of interactions, either physical (hydrophobic, ionic, Van der Waals) or chemical (covalent bonds). However, species immobilized by encapsulation may be subjected Eukaryotic cells Prokaryotic cells Matrices (organic or inorganic) a) b) d) e) c) to further binding (for instance covalent). The distinction between simple binding and encapsulation is therefore not always clear, for instance when objects are immobilized on the pores surface within a porous material. The distinction may however be made not based on the final location or interactions between the immobilized object and the matrix, but rather based on the processing and shaping of the material. Sheldon 114 defined immobilization as attachment on a prefabricated support, while encapsulation requires formation of the matrix itself in presence of the encapsulate (see Figure I.10). Immobilization and encapsulation techniques may be applied to a wide range of species, besides whole cells, with different goals, advantages and drawbacks. This includes encapsulation of simple molecules such as drugs or more complex ones (small polypeptide, DNA, various protein and enzymes), single cells (for instance animal cells or cells from higher plants) as well as whole organisms (microalgae, fungi, bacteria etc…). Immobilization technologies were initially developed for enzymes with potential in the chemical industry. The need for greener and less solvent-consuming processes highlighted the interest of enzyme-based chemical synthesis 115 . First commercial use of immobilized enzymes was reported in Japan in the late 60s for racemic resolution of L-methionine 116 . Initial goals were to design efficient enzyme-based reactions and immobilization of enzymes proved to be efficient for thermal stabilization 117 of enzymes and long term storage 118 . In addition immobilization provides advantages for handling as well as to separate enzymes from products, resulting in increased reusability 119 . The drawback to immobilization of enzymes is often a drop in intrinsic activity compared to free enzymes, either due to modification in the structure of the enzyme or to diffusion limitations 120 . This may however be compensated by the increase stability and reusability of immobilized enzymes 121 . Since these initial attempts, enzymes have been immobilized within a variety of materials (both organic and inorganic) and for a wide range of syntheses 122 . The knowledge about enzyme immobilization has been extended for encapsulation of a wide range of objects ranging from simple molecules or biomolecules to complex proteins and finally to whole cells and microorganisms. The variety in the type of encapsulates is due to the growing number of applications seeking to take advantage of Nature's efficiency while ensuring stability and reusability. But when considering encapsulation, especially for sensitive entities such as cells and microorganisms, there are also some limits. One key aspect when encapsulating living cells regards their long term viability. It is necessary to design matrices allowing the necessary substrates and nutrients to reach the encapsulated object [START_REF] Léonard | Whole-cell based hybrid materials for green energy production, environmental remediation and smart cell-therapy[END_REF] . When considering entrapped microorganisms, it may also be necessary to address the possibility of cell division since the cells are physically constrained within the matrix 123 . Encapsulation can actually be applied to very simple structures such as single molecules. In this prospect, the most relevant example of application is probably drug delivery 124 . In this case encapsulation in a well-chosen matrix may allow for the protection (for instance from pH variations in the digestive track) of the drug molecule until it reaches the intended delivery site. Another main advantage of drug encapsulation is the control it confers over delivery kinetics. Such delivery systems have also been used for more complex and also more sensitive molecules and biomolecules such as proteins 125,126 (for instance growth factors 127 ). Encapsulation is now also regarded as a possible option for cell delivery 128,129 . In this case encapsulation may provide a valuable protection against immune response, since the encapsulating matrix can be tuned to act as a filtering membrane, allowing small molecules to go through while retaining larger molecules such as immunoglobulins 130,131 . Such approaches also lead to the design of artificial organs. Pancreatic islets have been entrapped in alginate as soon as 1980 [START_REF] Lim | Microencapsulated Islets as Bioartificial Endocrine Pancreas[END_REF] . Encapsulation also allowed the design of artificial single cells, for instance by entrapment of DNA and protein synthesis machinery in alginate-silica-beads 132 . Another aspect of encapsulation relevant to biomedical applications is the design of sensors, one prominent example being the encapsulation of glucose oxidase as glucose sensor 133 . The use of sensors is however not limited to medical applications. Horseradish peroxidase has for instance been use as the functional unit in hydrogen peroxide sensors 134,135 . This system has been widely used as model for enzyme encapsulation, but other enzymes have also been used, for instance as a way to detect potentially harmful chemicals such as phenolic compounds 136 . Encapsulated whole cells, such as photosensitive plant cells or microorganisms sensitive to specific pollutants, have also been used for sensing 137 . Similar as the design of sensors, enzymes have also been used to design electrodes for applications in the domain of sustainable energy 138 , but most popular encapsulates regarding the development of biological energy sources are whole cells. Both microalgae 139,140 and bacteria 141 have been extensively used as biocatalysts for fuel cells. Various entities have also been encapsulated for applications in environmental science. Both enzyme 136 and cells (fungi 142 , algae 143 , bacteria 144,145 ) have been used as sensors due to their high sensitivity and specificity. More than simple pollution monitoring immobilization technology has been used for bioremediation. Materials containing enzymes such as peroxidase have also proven to be of use for remediation of contaminants 146 . But conventional bioremediation is usually performed through the use of living organisms and especially microorganisms. The different cell types may benefit from immobilization from viability and stability point of view. As a result, various cellularized materials have been studied for remediation purposes, using biofunctional units such as bacteria [START_REF] Chen | Decolorization of azo dye by immobilized Pseudomonas luteola entrapped in alginate-silicate sol-gel beads[END_REF]147,148 , fungi 149 , algae 150 . Encapsulation of biomolecules and cells is of use for a wide variety of applications from biomedicine to sustainable energy and environmental application. The main advantages of encapsulation are stabilization and protection of the encapsulated objects. Depending on the application, the matrix may also be tuned for permeability, to act as conductive material or be designed for controlled degradability for instance. Even if the encapsulated molecule or cell can be seen as the functional unit, as opposed to the matrix which is the structural part, both elements have significant influences on each other. As a result, the encapsulating matrix must be designed according to the targeted application and to the encapsulated entity. One key element of such designing process is the choice of the matrix composition. I.2.b Encapsulation in biopolymers Polymers and more specifically natural polymers (also called biopolymers) have been widely used for encapsulation of molecules, proteins (including enzymes) and cells (either eukaryotic or prokaryotic). Although biopolymers present some undeniable advantages from the encapsulation point of view, they also have some shortcomings from the material scientist's perspective. I.2.b.i About biopolymers Polymers are commonly referred to as biopolymer when they can be extracted or recovered from living organisms, whether they are animals (collagen) 151 , plants (cellulose) 152 , algae (agar) 151 and bacteria 153 (xanthan) or less commonly fungi (pullulan) 154 . Most of the commonly used biopolymers can be classified into the polysaccharide or protein categories. However it is worth mentioning that natural rubber (polyisoprene) or lignin (polyphenol) are also biopolymers. The type of polymers present in an organism is dependent of the species, and even of the biological kingdom considered, even though they are no general and absolute rules. Plants and algae are generally composed of polysaccharides, although, as was mentioned earlier, lignin, one of the major components of wood is a polyphenol. On the other hand, most biopolymers in animals are proteins. One exception to this tendency is chitin which is a polysaccharide mainly found in crustacean shells. Table I.3 is an overview of the most common biopolymers, their main sources and applications. Contrary to synthetic polymers or proteins, polysaccharides (extracted from plants for instance) are often polydisperse. If the monomeric units corresponding to the various polysaccharides and the overall structures are generally well known, the fact that these polymers are extracted from natural sources induces variability in the structural composition. In fact biopolymers are usually rather families of macromolecules 151,174 , with variabilities in structure or length depending on the source but also on the extraction methods and possible post treatments. This variability does not however impede the use of biopolymers for a large range of applications. Food industry is a major consumer of biopolymers, especially polysaccharides 175 , due to their wide availability and general low cost. Polysaccharides such as agar, alginate, pectin or xanthan are very widely used in the food industry either as stabilizing, emulsifying, gelling and thickening agents. Proteins are also of interest for the food industry. Gelatin is for instance one of the most commonly used gelling agents. But biopolymers have also been largely used for biomedical applications, from drug delivery to tissue engineering and surgery 168,176 . In this case, the intrinsic biocompatibility of natural polymers is a decisive advantage. Other industrial activities such as textile or paper are also important users of biopolymers such as cellulose and lignin 152,171 . Here the biopolymers are mainly used for their structural features. Use of biopolymers is a general trend in many domains due to the growing demand for alternatives to synthetic polymers (especially petrol based ones). The need for eco-friendly and sustainable alternatives is responsible for a renewed interest in biosourced and natural polymers. I.2.b.ii Advantages and limitations of biopolymer encapsulation Encapsulation technology has also taken advantage of the possibilities given by biopolymers [START_REF] Gasperini | Natural polymers for the microencapsulation of cells[END_REF] . In nature, biopolymers are often found as structural materials 177 . Polysaccharides such as cellulose or pectin are found in plant cell walls and provide mechanical support 178 . In animals cells are usually comprised in an extracellular matrix composed of proteins (i.e. collagen). Biopolymers' initial function is therefore to be an encapsulating matrix. As a result, biopolymers appear as ideal candidates for encapsulation technologies, especially for medical applications, due to their wide availability, low cost as well as their intrinsic bio and cytocompatibility and biodegradability. For several decades, alginate has been one of the most popular biopolymers for encapsulation [179][180][181][182] . In addition to the previously mentioned advantages, alginate can be crosslinked by divalent cations (including calcium) by forming "egg-box" structures 183 as a way to yield self-supporting gel materials. Encapsulation in crosslinked alginate was first reported in 1980 for immobilization of pancreatic islets 4 but has since been used to encapsulate drugs 184 , biomolecules 125 , enzymes 185 , mammalian cells 186 , fungi (including yeast 187 ), algae 188 and bacteria 186 . Even if alginate is by far the most commonly used biopolymer for encapsulation, it is however not the only one. Chitosan 189,190 or pectin 191,192 have for instance been considered for cells 193 or enzyme immobilization as well as drug delivery. The common point to all biopolymers used for encapsulation is their gelling properties. Since the polymer is here intended to be used as a structural unit (most commonly as microcapsules), gelation is necessary to provide self-supporting matrices. The intrinsic softness and toughness of polymers are both an advantage and a drawback from the encapsulation point of view. It provides a suitable environment for living cell and may be advantageous for tissue engineering applications, but might be problematic for applications where the matrix should provide protection against mechanical solicitation. This issue is also linked to the question of degradability and especially biodegradability. For most medical applications biodegradability is a great advantage, especially for controlled delivery over long periods of drug or even cells. But this advantage becomes a drawback for applications in other domains such as energy or environment, where the stability of the matrix can be crucial. One approach to control and tune these aspects has been the use of polymer composites including two or more biopolymers with different properties 194 . From the processing point of view, several biopolymers have the great advantage of being hydrosoluble. This is invaluable for encapsulation processes since cells can be directly dispersed in polymer aqueous solutions prior to shaping of the material. In short, biopolymers appear as ideal candidate for encapsulation due to their bio and cytocompatibility. They can be shaped through a wide variety of approaches, but they suffer from shortcomings in terms of stability and mechanical properties. I.2.c Encapsulation in inorganic matrices Another approach, quite opposite to the use of biopolymers, for the encapsulation of cells and biomolecules is the use of inorganic compounds. Cells and biomolecules have been encapsulated in a wide range of compounds including metal oxides 113,133,195 , carbonates 196 , or layered double hydroxides 148,197,198 . One of the most common medium for encapsulation in inorganic matrices however remains sol-gel silica. I.2.c.i About sol-gel silica Silica (silicon dioxide) is one of the most abundant oxides in soil, but can also be found in mineralized organisms such as diatoms. If silica (for instance fumed silica or fused quartz) can be obtained through high temperature processes, the possibility to obtain silica materials at low temperature from a solution of precursors has attracted attention for more than a century. In 1844 J.J. Ebelmen first reported the phenomenon of hydrolysis of alkoxides in presence of moisture 199 . It is only decades later, around 1930, that materials were actually made by sol gel techniques as thin film by Geffken 200 or aerogels by Kistler 201 . It is only at the end of the 60s that interest in sol-gel approaches actually grew significantly. Sol-gel chemistry has since been used in wide range applications and with several types of metal oxides (for instance based on titanium, zirconium or aluminum). Depending on the targeted application, various types of materials can be obtained through sol-gel. Sol-gel syntheses rely on polymerization of precursors (small molecules) to form a colloidal suspension of silica particle called "sol". The inorganic polymerization can be further controlled to yield precipitation of the silica particles (Stöber process 201 ) or to result in the formation of hydrated gels, where silica forms a percolating network. Such gels can be further processed to remove the solvent. Depending on the drying techniques it is possible to obtain aerogels which retain the initial porous structure of the gel (for instance using supercritical drying) or more condensed xerogels (see Figure I.11). Sol-gel processes are generally based on two main synthetic pathways: the alkoxide-based route and the silicate-based route. Synthesis of sol-gel based materials is commonly carried out using alkoxides, one of the most prominent examples being tetraethyl orthosilicate, also known as tetraethoxysilane (TEOS). Alkoxides can be functionalized by a wide range of reactive groups (including organic groups) as a way to tune the synthetic conditions and the final material properties. The formation of the inorganic polymer occurs in two steps (see Figure I.12). The monomers are first hydrolyzed. Total hydrolysis would result in formation of silicic acid, but in most sol-gel processes, hydrolysis is only partial. Hydrolysis may be catalyzed either by acidic or basic conditions. The hydrolysis is followed by condensation of the silicic acid to form oxygen bond resulting in the formation of oligomers and subsequently silica particles. Depending on the synthetic conditions, especially pH, nucleation or growth of the silica particles may be tuned resulting in different gel morphologies. Another common synthetic pathway to obtain silica gels is the use of metal salts. One of the common precursors source for this approach are sodium silicates (waterglass) of general formula NaxSiyOz. Sodium silicates can be found as solids or basic aqueous solutions, often as a mixture of various silicic oligomers. Gelling is obtained by neutralization of the suspensions resulting in the formation of a percolating network. Both the alkoxide route and the aqueous route can be tuned in terms of gelling times, optical properties or mechanical behvior 137 . Parameters such as pH and temperature may have a significant influence regarding the gelation kinetics and the characteristics of the final material, but they are also of great importance for possible encapsulated biological entities. I.2.c.ii Advantages and limitations of silica encapsulation Given the mild conditions required for the formation of inorganic + 4H 2 O 4 + a) b) Hydrolysis Even though it is mainly part of the mineral world, silica has been used in the world of living organisms as protective shell by diatoms, which can be seen as form of encapsulation. Silica gels can therefore be seen as good candidates in the search of matrices for hosting living organisms. Cytocompatibility of silica has been demonstrated for a wide range of organisms, but the encapsulation process itself must also be designed carefully to prevent any deleterious effect to the encapsulated entity. Sol-gel syntheses have the significant advantage of being conducted at low temperature (for instance at room temperature or 37°C, which is compatible with many cells). But depending on the chosen synthetic pathway, some other parameters are likely to induce toxicity. One of the most used precursors for sol-gel synthesis is TEOS. This precursor is widely available and can easily be hydrolyzed and condensed to form silica gels. However, ethanol is produced during the hydrolysis of TEOS, which is likely to be harmful to most cells. Use of alkoxides in general is subjected to the same limitation with production of the corresponding alcohols (methanol for tetramethyl orthosilicate, propanol for tetrapropyl orthosilicate etc…). Furthermore, hydrolysis of alkoxide is usually conducted under acid or basic condition, which can be highly deleterious for cell viaility. + H 2 O One proposed way to deal with this issue is to resort to a two-step encapsulation. The chosen alkoxide is first hydrolyzed under conventional conditions. The formed alcohol is then removed and the sol is brought to neutral pH before addition of the considered cells 214,215 . Gelation takes then place to entrap the cells within the silica matrix. Alkoxide precursors themselves have also been modified to prevent release of harmful alcohol in order to ensure compatibility of the encapsulation process [216][217][218] . The use of the aqueous sol-gel route may be another alternative since gelation can be carried out at neutral pH and without production of alcohol 207,219,220 . This approach however implies the presence of high salt contents. Well-defined silica colloids (e.g. LUDOX®) have been added to silicates in order to form silica gels while maintaining low salinity. This however results in lower stability of the gels. Such colloidal particles have therefore been associated with silicates to combine stability and low salinity 113 . Immobilization within a silica matrix (gels, films or dry porous materials) of biomolecules 221 or cells 212 has proven to be of interest regarding protection of the encapsulated objects for various applications. This however requires careful design of the encapsulation process since harmful by-product or physico-chemical conditions may be encountered. Immobilization within hard and brittle materials is however not very common in nature, since cells are usually entrapped within polymeric extra cellular matrices, which are often soft and tough materials. One way to combine the durable protection provided by silica matrices with the enhanced compatibility of biopolymer matrices is the design of hybrid materials for encapsulation. I.2.d Encapsulation in hybrid matrices Both organic and inorganic compounds have been used for the encapsulation of cells and microorganisms. Each type of compound possesses its own advantages and drawbacks from the functional and structural points of view. Another approach for encapsulation consists in combining organic and inorganic moieties in order to design materials with tailor-made properties (for instance in terms of mechanical behavior or processing options). I.2.d.i About hybrid materials In general the term hybrid refers to the mixture of two components with different properties. It is used in biology (for instance in botanic) to refer to mixing of two different species, often as a way to yield new or enhanced characteristics. In recent years it has also been used in the automobile domain to refer to cars combining two different energy sources as a way to combine sustainability and performances. Hybridization can therefore been seen as a way to take advantages of two different moieties (in terms of physico-chemical properties for instance) in order to gain new properties or enhanced properties. The term hybrid is often implicitly used to refer to organic-inorganic hybrids. A distinction must be made between hybrid materials and composites materials. The most commonly accepted definition for composite materials is the association of a dispersed phase within a continuous phase (or matrix). The term hybrid is generally used to describe materials where the components are mixed at the nano or molecular scale. This may consist in a homogeneous blend or in a nano-scale dispersion. In this last case, hybrids are sometimes described as nanocomposites. Hybrid materials are commonly separated into class I and class II hybrids. In class I materials interactions between the two moieties are weak (hydrogen, Van der Waals, ionic etc..) while class II materials are characterized by covalent or ionocovalent bonds 222 . Creation of hybrid constructs can also be a way to finely tune physico-chemical properties of the final material such as mechanical and thermal behavior, stability, density, permeability, optical properties, hydrophobicity, biocompatibility etc… 223 . One of the major advantages in the design of hybrid materials is the flexibility it offers through the choice of the components themselves, but also due to the wide range of possible processing techniques. As a result of such versatility, hybrid materials have now become ubiquitous. They have been used in domains as varied as energy, health, housing, microelectronics, micro-optics and environment. In the domains of environmental science and medicine, a type of hybrid in particular has attracted a lot of attention. Hybrid materials based on biopolymers as organic moiety, also called bionanocomposites 224 , present the advantages of conventional hybrids in terms of tunability and enhanced properties, but here the biocompatibility and biodegradability inherent to biopolymers may also be exploited. The addition of an inorganic moiety allows for control over the main limitations of biopolymers (low mechanical strength and low stability) 225 . From the processing point of view one type of inorganic compounds has been especially favored. Metal oxides in general and more specifically sol-gel processes seem especially relevant for controlled synthetic approaches of metal oxide-biopolymer hybrids 226 . Among this family of compounds, silica based materials are especially prominent. From a structural point of view, silica-biopolymer hybrids are distributed between two extreme organizations. On the one end of the spectrum are the materials composed of a biopolymer matrix in which the inorganic moiety is dispersed. On the opposite end of the spectrum are inorganic matrices containing a biopolymer phase. Core-shell particles (either biopolymer@silica or silica@biopolymer) can be seen as specific cases of hybrid compounds, although they do not constitute a material from the macroscopic point of view. Control over the gelation of both the inorganic and organic moieties is key to obtain the desired architecture (see Figure I.14). As was mentioned previously, one of the most widely used biopolymers is alginate, mainly due to its wide availability, non cyto-toxicity and gelling properties. This trend is also found in the design of silica-biopolymer where silica-alginate constructs have been used for numerous applications 228,229 . Silica has however also been associated otherbiopolymers including pectin 230 , chitosan 231 or collagen 232 . The nature of the biopolymer is of paramount importance since it has a direct influence on the interactions between the organic and inorganic moiety, which in turn is determinant for both the structure and function of the final hybrid material 233 . Since both composition and shaping of the hybrid materials have significant impact on their final properties and characteristics, these two aspects represent keys to design tailor-made materials for specific applications. I.2.d.ii Use of hybrid matrices for encapsulation Silica-biopolymer hybrids and nanocomposites are especially favored for applications in medicine or environmental science. The flexibility, tunability and biocompatibility of silicabiopolymer hybrids make them excellent scaffolds for cell growth 234 . In addition the use of nanocomposites is a good way to control the mechanical characteristics of the materials without significant changes of the surface chemistry, which is of great importance for cell growth 235 . Silica-biopolymer bionanocomposites have also been used in depollution applications as adsorbing materials 236 . The use of silica and biopolymers ensures good ecocompatibility since both components are present in nature. Hybrid biopolymer-silica matrices have also largely been used as encapsulation matrices for various biological entities. In this case the system become even more complex since they comprise an inorganic and an organic structural moieties, as well as the biological functional entity. Advantages of encapsulation in silica-biopolymer hybrids are similar to the ones mentioned for the individual moieties concerning stabilization and protection. However combination of the organic and inorganic moiety usually allows for a minimization of the shortcomings while preserving or enhancing the advantages. Such materials may for instance have the advantages of the biopolymers in terms of bio and cytocompatatibility, but with enhanced mechanical properties provided by the silica moiety. Parameters such as the stiffness or the biodegradability of the material, as well as the general geometry (beads, fibers, monoliths etc…) and morphology (porosity or architecture of the different phases) must be tailored depending on the targeted application. In addition to the tuning of the structural part, the functional entity (enzyme, animal cells, plant cells, microorganisms) must also be chosen according to the desired application. First encapsulations in hybrid matrices were performed using enzymes, immobilized within alginate bead and subsequently coated with silica 237 . Such encapsulation provided protection against thermal and chemical denaturation, while the presence of silica ensured mechanical robustness. This type of system has since been used for the encapsulation of various biomolecules and cells [START_REF] Coradin | Silica-alginate composites for microencapsulation[END_REF] . The synthetic pathway itself could be tuned using bioinspired silica deposition 131 . Alginate-silica bead have further been used for immobilization of bacteria 238 , mammalian cells 239 , fungi or microalgae 240 . However depending on the targeted application other biopolymers can be associated to silica as encapsulating matrices (including chitosan 117 , collagen 241 , xanthan 242 or pectin 243 ). Applications for silica-biopolymer-based biohybrids range from protein and cell delivery 241 , energy production 244 or biosensing 245 to environmental science [START_REF] Chen | Decolorization of azo dye by immobilized Pseudomonas luteola entrapped in alginate-silicate sol-gel beads[END_REF]246 . Despite the wide range of available biopolymers and the flexibility of the available processing techniques, most encapsulation routes require gelation of the biopolymer, often as micro beads 247 . The functional entities remain encapsulated within hydrated gels, which may be problematic from storage and handling point of view. In addition, in the case of immobilized cells, the porosity of the matrix is essential to provide the encapsulate with necessary nutrients. This is especially relevant for applications in environmental science, where it is essential that the substrates diffuse to the functional encapsulated species. For such devices, the micro and mesoporosity is important to allow molecular diffusion while preventing cell leaching. I.2.e About encapsulation of living organisms in hybrid materials for soil bioremediation As was demonstrated, encapsulation is a widely applicable strategy. Encapsulation processes can be characterized and classified depending on several criteria including the nature of the encapsulate, the nature of the matrix or the targeted application. These different interrelated aspects must be considered in the design of an encapsulation strategy. This work focuses on the elaboration of cellularized hybrid materials for soil bioremediation. The relevant domain of application (environmental science), the type of encapsulate (metabolically active microorganisms) and the class of matrix (hybrid materials) have all been separately addressed in various works. The combination of these three specific characteristics have however more rarely been reported. Encapsulation processes have been used for environmental science in two main ways: for the development of sensors 143 and for bioremediation strategies. The bioremediation approaches may rely on the encapsulation of enzymes 146 , but the use of encapsulated cells is most commonly reported. A wide variety of organisms (mostly bacteria and algae) relevant to bioremediation have been encapsulated in either organic (including gellan-gum 248 , polyethylene oxide/polycaprolactone/polyethylene glycol composites 147 or polyethylene glycol and poly(vinyl alcohol) fibers 249 ) or inorganic matrices (mainly silica 250,251 , but also layered double hydroxides 148 for instance). Encapsulation in hybrid materials has been widely developed for enzyme immobilization. Examples of microorganisms relevant to bioremediation encapsulated within hybrid materials are however scarce. The bacteria Pseudomonas luteola has been immobilized in alginate-silica beads for the remediation of Reactive Red 22 [START_REF] Chen | Decolorization of azo dye by immobilized Pseudomonas luteola entrapped in alginate-silicate sol-gel beads[END_REF] . Similarly, cyanobacteria Nostoc calcicola was entrapped in silica-coated alginate beads for metal adsorption 246 . An alginate-silica matrix was also used for the immobilization of Stereum hirsutum to remediate malachite green 149 . The efficiency of these functional materials was however assessed for water treatment. Pseudomonas fluorescens encapsulated within alginate-bentonite clay nanocomposite 109 was reported to have enhanced survival rates in soil, but the potential of the cellularized material for bioremediation was not assessed. One of the main difficulties regarding the use of encapsulated microorganisms for soil bioremediation is related to diffusion limitations and substrate transport. The use of a porous encapsulation matrix could be beneficial to favor such phenomena. I.3 Encapsulation by freeze-casting Freeze-casting has been used for a few years to design porous material and more specifically materials with an oriented porosity. Such a structure can be a way to favor capillary mass transport within the material. I.3.a Shaping using ice as a template I.3.a.i About freeze-casting Freeze-casting is a processing technique relying on the use of ice-crystals as templates for the shaping of porous materials. Due to the very low solubility of most compounds in ice, freezing of a solution, suspension or slurry usually results in a segregation phenomenon. As a result the particles or solutes are repelled and concentrated by the growing ice crystal. The phenomenon has for instance been described in the literature through the observation of freezing seawater 252 , but such behavior is actually ubiquitous in nature 253 . While the templating capabilities of growing ice may be deleterious in many cases (for instance damages caused to soils and roads in winter or freezing of cells 254 ), it represents an interesting tool from the material scientist's perspective. First reports of the use of ice crystals as a way to shape materials dates from 1954 255 , but the technique of freeze-casting (which has also been called ice-templating, freeze-gelation or even ice segregation induced self-assembly 254 ) has mainly attracted attention during the last decade (Figure I.15). Freeze-casting is indeed an easy-to-implement way to obtain porous materials. One of the specificities of freeze-casted materials is that oriented porosity can be obtained using the appropriate freezing-setup. Another advantage is that it can be used for processing of a wide variety of materials, from ceramics and metals to polymers. The principle is to freeze a solution, suspension or slurry in order to grow ice-crystals which are subsequently sublimated to free the desired porosity. Due to the low solubility of most compounds in frozen water, segregation occurs during freezing resulting in the formation of ice crystals on the one hand (which will become the pores after drying) and of the pore walls shaped by the growing ice crystals on the other hand (see Figure I.16). One of the major advantages of ice-templating is the control it confers over the pore morphology of the final material. The porous structure can be tuned through a wide variety of parameters which can depend on the setup used (control of the temperature gradient) but also on the composition of the frozen solution or suspension (solvent, particles size, additives etc…). The shape and orientation of the pores can be tuned by controlling the ice nucleation and growth. In order to obtain aligned and oriented porosity a temperature gradient must be established within the sample. A simple way to obtain such gradient is to plunge samples in a liquid nitrogen bath at a chosen speed. In this case however the cooling temperature is set to -196°C. A common freeze-casting setup is therefore composed of a heat conductive material (copper, aluminum etc…) in contact with a cold source (such as liquid nitrogen) 257 Polymer Ice crystals of a heating resistance allows for precise control of the heat-conductive element's temperature. The desired solution or suspension is poured in a mold and put in contact with the cold finger to establish a temperature gradient. As a result ice-crystals will nucleate at the interface with the cold element and grow along the temperature gradient (see Figure I.17). Although such setup is commonly used, it is far from being the only possible configuration. For instance systems with two cold sources have been used for better control of the temperature gradient 258,259 . Figure I.17: Unidirectional porosity can be obtained by applying a chosen temperature gradient to the sample. Use of such a setup allows for a precise control of the temperature both in time (a chosen temperature ramp can be applied) and space (the orientation of the gradient is well defined). The temperature gradient has a direct influence on the freezing-front velocity which is key to the control of the size of the pores 260 . In addition to the influence of the temperature gradient, freezing-front velocity is also largely dependent on the thermal properties of the solidified phase, that is to say on the composition of the initial solution or suspension 257 . Formulation of the initial slurry is key to control the freezing front velocity, but may also influence a wide range of characteristics likely to modify the final porous structure. As mentioned previously, ice-templating can be used with a wide variety of compounds since it mainly relies on physical interactions. It has for instance been used for shaping polymer solutions 261 , or more recently for metals 262 . The method has also been extensively used for shaping of porous ceramics, including silica 263 and alumina 260 . Depending on the interactions between the templated colloids, further sintering or densification may be required after sublimation of the ice-crystals to yield satisfying mechanical properties. Use of a binder (for instance polymer in ceramic slurries) may also be a way to obtain structural integrity of the materials 264 . All the components of such complex suspensions must be carefully selected in order to ensure control over the desired pore morphology. The first element to be selected is the compounds on which the material will be based (for instance the type of polymer or of ceramic). The chemical composition is relevant since it may affect interactions between the colloids, as well as the particle-solvent interactions. The chemical nature of the freeze-casted components is however far from being the only relevant parameter. Figure I.18 illustrates the variety of morphologies which can be obtained for ceramics by changing parameters such as solvent, binders or solid loading. The characteristics of the particles themselves are also critical in the morphology of the final material. Most commonly, such particles are hard (as is the case of ceramics), but it is also possible to apply freeze-casting to soft particles (such as polymer particles or even cells). In the case of ceramic processing, which has been one of the most studied materials, the properties of the slurry are very dependent on the size 266 , shape 254,267 and density of the particles. Usually these parameters are chosen in order to prevent sedimentation phenomena, although these can be taken advantage of to create gradients within the final porous material. Regardless of the nature of the colloids, another crucial parameter is the concentration (or solid loading) within the initial slurry 268 . It may influence the final density (and mechanical properties) of the material, but is also relevant to the freezing process itself since it is likely to modify the viscosity or freezing point of the liquid sample. Water is most often used as the solvent to disperse polymer or ceramic particles. Use of dispersant is often required to obtain stable suspensions. The nature of the solvent used is one of the most important parameter in the design of freeze-casting systems, since it largely influences the crystal nucleation and growth, which is the driving force of ice-templating. In the case of water, one of the most commonly observed morphology is composed of lamellar and well aligned pores, often organized in orientation domains 269 . In some cases, this can be linked to the hexagonal crystallographic structure of ice and to the fact that ice growth is favored along the a axis compared to grow along the c axis 256 (see Figure I.19), however depending on the composition of the initial suspension and freezing conditions, ice growth mechanisms are often much more complex. Depending on the ice growth, morphologies such as microhoneycombs can also be obtained 270 . The presence of additives 272 such as dispersants, common cryoprotectants (sucrose 273 , glycerol 274 ) or binders (such as polymers 275,276 ) is likely to modify the ice nucleation and growth and therefore the porous structure of the material. It is however difficult to predict the effect of such modifications 277 since the additives are likely to impact many parameters simultaneously (ice growth, viscosity, freezing point etc…). The use of a different solvent may also provide access to original shapes of pores. Camphene-based freeze-casting has for instance been used to obtain dendritic pore morphologies 278,279 . Fishbone-like structures have been obtained by freeze-casting in liquid carbon dioxide 280 . The ice growth can therefore be tuned through a variety of approaches, but ice nucleation and the initial stages of growth are also of tremendous importance for the final morphology. Ice undergoes a non-lamellar growth phase before attaining a steady-state growth regime. A transition from an initial planar ice front to a lamellar ice morphology results from destabilization of the solid-liquid interface due to the accumulation of solutes at this interface. As a result, areas with different pore morphologies can be observed close to the interface with the cooling element 260 . Such morphology heterogeneities have been minimized by modifying conventional freeze-casting setup by the introduction of a tilting angle of the cooling element 281 . Importance of the interface with the cooling element has also been demonstrated through modulation of the final pore morphology thanks to patterning of the cold finger 273 . In short, a large number of often interrelated parameters (both from the setup and formulation points of view) are susceptible to influence the porous morphology of the final materials. The comprehension of their influence is key 282 to gain control in the precise design and tuning of materials with a wide variety of structures. I.3.a.ii Range of application As mentioned previously, ice-templating processes can be applied to a wide range of compounds 283 to yield a great diversity of structures. Such versatility has opened the way for the use of freeze-casting in many applications. Each class of materials has its own specificities in terms of shaping even if the principle (use of ice crystals as templates) remains the same. One of the major uses of freeze-casting techniques is for shaping of ceramic compounds 256 (including alumina 260,284,285 , silica 271,286 , titania 287,288 , zirconia 289,290 , clays 291,292 , calcium phosphate 293 and hydroxyapatite [294][295][296] ). Polymers, and water soluble polymers in particular, have also extensively been shaped though ice-templating. In this case, instead of a suspension of hard particles, the frozen medium is a solution of dispersed macromolecules. The interest of the use of macromolecules is that the lyophilized materials are usually self-supporting and do not require further treatment. Examples of freeze-casted polymers include biopolymers (alginate 297 , cellulose 298,299 , chitosan 300 , collagen 301 , gelatin 302 or silk 303 ) but also synthetic polymers like poly(vinyl alcohol) 304,305 or poly(lactide-co-glycolide) 305 . Carbon-based materials have also been shaped by freeze-casting to yield ultralight aerogels with conductive properties, for instance using multi-walled carbon nanotubes 306,307 or graphene 308,309 . Metals have also been shaped through ice-templating, either from metal particles 262 or metal precursors 310 . As is the case for ceramic particles, the choice of the particles and dispersants is crucial for the control of the final porosity of the material in order to avoid sedimentation. In the case of metals however, the question of oxidation must also be considered for metals such as iron 311 , titanium 262 or copper 312 but can be avoided using stable metals (gold 313 , silver 314 or stainless steel 315 for instance). Some of these materials can be shaped individually, but in some cases it may prove interesting to combine them. Polymers have for instance been added to ceramic suspensions 264,275 (as binder or in order to modify the final morphology). But the combined use of different component can also be a strategy for the elaboration of composite materials (either by simultaneous ice-templating of the different moieties 292 or by freeze-casting of a structure and subsequent infiltration 314 ). Such composites may have original architectures 259 or properties 316 (for instance mechanical behavior 317 or conductivity 318 ). Such diversity opens the ways for uses in fields ranging from biomedical application to energy, environmental science or housing materials. Ceramic based materials are for instance of interest when high mechanical, thermic or chemical stability is required. This includes, among many other applications, thermal insulation 290,319 , catalytic supports 287 or biomaterials 296 . Biopolymer 304,320 or biopolymer-based composites 234,321 porous materials have been largely favored for applications in biomedicine. The use of scaffold with an aligned porosity is particularly interesting for cell oriented growth 322 . Ice-templated materials have also be used in the environmental domain, for instance as absorbents [323][324][325] . Freeze-casting has proven to be a widely applicable technique for shaping numerous compounds. The process can be tuned through a variety of parameters (setup, freezing rate, composition, presence of additives, type of solvent etc…). The versatility of this technique which is both straightforward and highly tunable has allowed for the design of materials with tailor-made properties (chemical interactions, mechanical behavior, thermal stability etc…) for specific applications. I.3.b Freezing and drying cells Freeze-casting can be applied to many compounds, including polymers and inorganic moieties, commonly used for biomolecule or cell encapsulation (for instance alginate 297 or silica 326 ). The aligned porous structure of the materials has proven to be interesting for various applications since it may provide oriented support for cell growth or facilitate mass transport in adsorption mechanisms. Furthermore, most freeze-casting approaches use water as solvent which is an advantage from the cytocompatibility point of view. Freeze-casting therefore appears as good candidate for original encapsulation procedures. There are however only few examples of encapsulation of biological entities within freezecasted materials, which may partly be attributed to the challenge that is the freezing of biological entities. I.3.b.i Encapsulating cells by freeze-casting Requirements are different for encapsulation of simple molecules such as drugs, biomolecules (including proteins) or cells (animal cells, plants cells, microorganisms such as yeast or bacteria). Polymer-based freeze-casted materials have been used for entrapment and release of drug 304,327,328 . Control of the porosity and morphology of such structures allows for good control of the release rates (by erosion of the matrix or swelling and dissolution of the drug), which is a key feature of most drug delivery devices. In the case of water soluble drugs entrapment can easily be achieved by dispersing the drug molecules in the initial suspension. Freeze-casting has also been used for the encapsulation of more complex molecules such as enzymes 329 by ice-templating of a PVA-protease mixture. In this case the structure of the matrix can be tuned as a way to maximize mass transport which could be of great interest for the design of high-yield bioreactors. As is the case for drugs, the homogeneous encapsulation of enzymes is facilitated by the fact that the molecules can be efficiently dispersed in the aqueous carrier solution (often polymer) 330 . In the case of enzymes however, the molecules are much more sensitive to their environment and the composition of the matrix must be designed to preserve the structure and enzymatic activity (for instance by using a polymer moiety). The issue of preservation is even more essential when contemplating the encapsulation of more complex structures such as liposomes 331 or cells. Bacteria in particular have be entrapped in freeze-casted ceramic 332 , multiwalled carbon nanotubes 307 or polymer structures [START_REF] Gutiérrez | Hydrogel scaffolds with immobilized bacteria for 3D cultures[END_REF] . From a structural point of view entrapment of cells is not as straightforward as it may seem. Cells can be seen as soft and deformable particles to be entrapped between the growing ice lamellas. Mechanisms of sedimentation, entrapment or rejection by an advancing ice-front have mainly been studied in the case of ceramic particles 257 , but the general mechanisms can be extended to the case of soft particles such as cells or bubbles 333 . Such behavior strongly depends on various parameters such as the ice front velocity, the viscosity of the carrier solution, the size and density of the particles or the interaction between the particles and the ice front Growth of ice-crystals is responsible for shaping and formation of pore walls, but the counterpart to this templating effect is the application of non-negligible mechanical constraints on the cells 334 which may prove highly deleterious to their viability. As a result, cells have been pre-immobilized in alginate beads to ensure better cell survival through the encapsulation process [START_REF] Gutiérrez | Hydrogel scaffolds with immobilized bacteria for 3D cultures[END_REF]307 . Immobilization in such cytocompatible biopolymer may also be of use to prevent damages due to freezing and drying. From a microbiological point of view freezing and lyophilization of living cells is far from being free from consequences for the cells, as can be attested by the extensive efforts invested for several decades in understanding and optimization of cryopreservation 335 . I.3.b.ii Freezing cells Freezing is a highly deleterious phenomenon to most living organisms. Nature has devised strategies to deal with this issue in a few species exposed to extreme temperatures 336 such as insects, and some amphibians or reptiles 337 . If such behavior remains exceptional, the observation and understanding 338 of the underlying phenomena have open the way for the development of advanced cryopreservation strategies. Two main sources of damage have been identified as consequence of freezing: formation of intracellular ice, which induces structural cell damages and rise in extracellular solute concentrations during external ice formation which may result in cell dehydration, significant cell shrinkage and membrane changes. Common cellular protection strategies include the secretion of cryoprotective molecules such as glucose 339 or glycerol 340 . Sufficient concentration of such cryoprotectants in cells may prevent excessive cell shrinkage due to osmotic dehydration during the formation of extracellular ice. Another strategy consists in the limitation or control of ice crystals formation through to the use of proteins (antifreeze proteins 341 ). The action mechanism for these proteins is however not fully understood. In this case, the efficiency does not rely on high solute concentration but rather on specific interactions with ice. Several classes of freeze-protecting proteins have been identified with different behaviors. They may for instance induce controlled ice nucleation to prevent random ice crystal formation or be bound to ice crystals to tailor ice growth. Another effect observed was the presence of a thermal hysteresis, resulting in lower freezing point despite low protein concentrations. The use of cryoprotective agents has been mimicked for the preservation of a whole variety of cells including mammalian cells 342,343 , plant cells 344,345 or microorganisms 346,347 . Even if cryopreservation protocols must be tailored the specific targeted species, the general principles and strategies are similar. A distinction can be made between two main routes 348 . In conventional cryopreservation, focus is put on the control of ice formation and growth. In vitrification approaches, the goals is to completely prevent the formation of ice crystals. In conventional cryopreservation, the most common way to preserve the cells is through the use of cryoprotectants, from which glycerol is the most prominent example since its first use in 1949 349 . The action of glycerol (and other intracellular cryoprotective agents such as DMSO) relies mainly on two effects: the increase of intracellular solute concentration induces a freezing-point depression thus limiting the formation of intracellular ice, and these high concentrations prevent osmotic imbalance during concentration of extracellular solutes during ice formation and subsequent cell shrinkage. It has also been proposed that one possible cause for cell injury during freezing is related to the increase of intracellular salts concentrations up to toxic levels due to dehydration of cells 350 . By preventing excessive dehydration of the cells, glycerol could therefore limit the rise in salt concentration and subsequent toxic effects. The efficiency of such cryoprotective agents therefore relies both on their capacity to efficiently penetrate cells and on the control of their possible toxicity In addition to the presence of cryoprotective agents, another key element in the control of survival in cryoprotective strategies is the freezing rate 351 . Since injuries are highly dependent on osmotic phenomenon, the kinetics of water diffusion through the cell membranes has a heavy influence on the cryoprotection efficiency. At high cooling rates, water cannot permeate through the membrane quickly enough to compensate osmotic unbalances. The remaining intracellular water has higher chemical potential and become increasingly supercooled, ultimately resulting in the formation of intracellular ice. On the other hand, at sufficiently low cooling rates, water diffuses out of the cells and freeze-externally 352 . However if dehydration of the cells is too important, the previously mentioned effects of cell shrinkage and increased salt concentration may result in cell damages (see Figure I.21). Due to these antagonistic effects, an optimal cooling rate must be found to minimize intracellular ice formation, while maintaining non-toxic intracellular solutes concentrations. Such optimum is usually dependent on the type of cells due to the diversity in intracellular composition but also in the nature of the cell membranes. Some cryoprotective agents such as sucrose, trehalose or polyethylene glycol (called nonpenetrating cryoprotectants) are able to protect cells from freezing damages through different mechanisms, since they are not capable of permeating through the cell membrane 346 . Their main pathway of action therefore relies on the control of external ice nucleation (for instance on the formation of smaller ice crystals) to minimize mechanical constraint. Another possible action of such compounds is the modification of water diffusion kinetics 353 which also modifies the ice crystals formation and may help in preventing high osmotic imbalances. A different approach in cryopreservation consists in fully avoiding freezing by vitrification of the intracellular water. Condition for water vitrification are however not easy to obtain. Both high intracellular solute concentrations and very high cooling rates are required (see Figure I .22). A critical cooling rate, which depends on the total solute content of the solution, can be defined as the rate above which ice nucleation kinetics is too slow for freezing to occur 354 . Vitrification has the advantage of completely preventing the formation of intracellular ice crystals which could be damaging to the cells, without need for dehydration and subsequent elevation of the intracellular solutes concentrations 356 . As a result, there is no need to find an optimal cooling rate, which is of great interest for the preservation of multicellular units or whole organs. The downside is however the necessity to use high initial cryoprotectant concentrations which may be concerning both in terms of toxicity and of osmotic pressure. Since freezing is generally used as a tool for long term preservation for further use, it is essential to consider not only the freezing process but also the return of the cells to their initial state. This includes thawing and removal of the cryoprotectants. In both cases the conditions must be carefully chosen. Thawing rate has proven to be just as important as the freezing rate 357 . Cryoprotectant removal is usually performed by washing with a solution with lower concentration in the considered cryoprotective agents. Concentrations must be adjusted carefully in order to prevent osmotic shock. Another approach for the preservation of cell is not the inhibition of cellular activity through the use of very low temperature, but rather by the completed cell dehydration. I.3.b.iii Drying cells Freeze-drying has also been used a way to preserve and store proteins 358 and cells (animals 359 , plants or microorganisms 360,361 ) as well as virus (vaccines 362 ). The main advantage of this approach is the cost efficient storage and facility in transport and handling, since dry cells do not need to be kept at very low temperature such as frozen cells. Freeze-drying consist in the freezing of a cell suspension and subsequent lyophilization under high vacuum. As a result the cells and organisms subjected to freeze-drying must actually face two types of stress: freezing and subsequent drying. The drying step itself is usually composed of primary drying, which is the sublimation of ice crystals under high vacuum and secondary drying which consists in the removal of bound water 363 . Optimization of processes for freeze-drying must therefore take in account both sources of damage. Effects of freezing have been extensively studied for conventional cryopreservation. It is however much more difficult to assess the effects of the drying step alone, since it is always preceded by a freezing step. In addition these effects can be interrelated which means that survival and damages must be considered across the whole process 364 . Optimization of the freeze-drying process however usually consists largely in optimization of the freezing conditions (formulation of the suspension and freezing rate). As a result, cells are usually dispersed in solutions containing common cryoprotective agents. However some compounds also display specific lyoprotective features. Formulations may contain both cryo and lyopreservatives or single compounds which present protective effect during both freezing and drying. Common protective agents used in freeze-drying include trehalose, sucrose or glycerol as well as skim milk. As mentioned previously, some cryoprotectants are capable of permeating through the cell membranes and accumulate in the intracellular medium. Most of these compounds are highly hydrophilic. Such capability to retain water may provide advantages during the drying process 346 . As is the case for conventional cryopreservation, the freeze-dried species must be returned to their original hydration state prior to use. Even if only water has been removed from the cells during ice sublimation, better results are usually obtained with more elaborate rehydration media (for instance saline) 365 . Conclusion Living organisms and microorganisms in particular have proven to be interesting tools for the depollution of soils. Bioremediation approaches have been investigated for the treatment of soils containing various common pollutants such as PAH, pesticides or even heavy metals. The limitation of such strategies is usually the intrinsic sensitivity of living organisms. Encapsulation of cells and microorganisms in various matrices has proved to be an efficient way to design functional materials. Entrapment of metabolically active species usually allows for enhanced stability, reusability or recovery of the cells. Entrapment matrices and encapsulation protocols must be designed in order to satisfy imperatives from a structural as well as structural point of view. These requirements can be met by tuning the composition of the material, as well as the shaping processes. Use of hybrid or composite materials is a good way to combine the cytocompatibility of materials such as biopolymers with the stability of inorganic materials such as silica. Freeze-casting is a shaping process relying on the formation of ice crystals to act as template for the design of porous materials with complex and controlled architectural features. This technique has been of great interest to the material scientists' community due to its versatility and tunability. Morphological control can be gained throughout control of compositional parameters (nature of the compounds, shape of the particles, type of solvent used, presence of additive etc…) or by modifying processing parameters (type of setup, freezing rate etc…). Controlled freeze-casting techniques have been used for the design of complex composites architecture, in particular including various particles in a well-structured matrix. In order to apply such a method to the encapsulation metabolically active microorganisms, the challenge of living organisms must be addressed. Conventional strategies include the addition of cryoprotectants but also tuning of the freezing-rate, which can easily be controlled in icetemplating approaches. As a result freeze-casting could prove to be a valuable tool in for the encapsulation of cells in biopolymer macroporous materials with an oriented porosity and in hybrid macroporous materials. This processing approach could be used both as a way to control the structural features of the material and as way to control the functional aspects of cell encapsulation. Cellularized hybrid materials with an oriented porosity could be of great interest for environmental applications. Entrapment of microorganisms with bioremediation capabilities could be a decisive advantage in depollution strategies of contaminated soils, as a way to introduce efficient exogenous microorganisms while ensuring stability and preservation against potentially harmful conditions for the metabolically active microorganisms, as well as preventing their dispersion. Introduction As mentioned before, the conception of a hybrid material containing microorganisms was considered as a two-step process: formation of a biopolymer scaffold containing the targeted cells and coating with a silica layer. The structural properties of the material were first investigated in absence of cells, while keeping in mind the fact that the designed material is intended to be used as cell-host matrix and that the shaping process must be adapted to cellular encapsulation. The first objective was the elaboration of a dry macroporous biopolymer-base material. The characteristics of the biopolymer structure were investigated using ice-templating techniques. Both the formulation of the initial polymer solution and the freezing setup were explored as possible levers on the pore morphology. The influence of the porous structure was in turn assessed over the macroscopic properties of the material such as mechanical wetting behavior. II.1 A few words about pectin First and foremost, the material must be compatible with living microorganism, which implies the use of a non-toxic compounds, solvent and shaping process. This points towards the use of natural polymers (also known as biopolymers) [START_REF] Gasperini | Natural polymers for the microencapsulation of cells[END_REF] such as polysaccharides (for instance alginate [START_REF] Lim | Microencapsulated Islets as Bioartificial Endocrine Pancreas[END_REF] , chitosan 190 , starch 366 , or pectin 192 ), or proteins 367 (collagen and its derivative gelatin, silk fibroin, keratin ). These types of polymers are especially widely used in the medical domain for applications as varied as drug delivery, wound dressing, scaffolds 156 . The food industry is also one of the main consumers of biopolymers 175 as stabilizing, gelling or thickening agents, but they are actually used in almost every domain including packing, textile or paper industry 368 . Pectin is a polysaccharide found mainly in the plant cell primary wall, especially in young tissues and fruits 369 . The best known, and most commonly used pectin sources are apple pomace and citrus peels 370,371 , but pectin can actually be found in a wide variety of plants including tomato, carrots, beet root, apricots mango or even sunflower 174 . Pectin plays a role in the mechanical behavior of the cell wall 178 and more largely in the texture of the plant or fruit 372 which implies the presence of various pectic substances depending on the location, type and maturity of the cells within the plant. Even in single cells, a distribution of structural domain can be observed 373 . As a result pectin, or rather pectins, are very complex to characterize and no simple or general structure can be given. Oligosaccharide side chains of different length may also be present, containing the following residues : Galacturonic acid ( ), rhamnose ( ), apiose ( ), fucose ( ), aceric acid ( ), galactose ( ), arabinose( ), xylose ( ), glucuronic acid ( ), ketodeoxymanno-octulopyranosylonic acid ( ). Depending on the pectin source different levels of methylation ( ) and acetylation ( ) can be observed. Reproduce from Willats et al. 160 Galacturonic acid Rhamnose HG is a linear polymer composed of a chain of 1,4-linked α-D-galacturonic acid (GalA). This chain may be partially esterified of acetylated depending on the source, extraction method and subsequent treatment. RG-I consists in alternating residues of rhamnose (Rha) and galacturonic acid (GalA) (α-D-GalA-1,2-α-L-Rha-1,4). Neutral side chains can be attached on the Rha residues, most often 1,4-β-D-galactan or 1,5-α-L-arabinan. RG-II, also called substituted homogalacturonan, is the most complex and variable of these three structures 174 . Contrary to what the name rhamnogalacturonan II suggests, the backbone of this structure is made of homogalacturonan 160 but is substituted with a wide variety of side chains. In reality, most pectins present several of these well identified structures 376 and as a results the polymers are commonly accepted as heterogeneous sequence of simple HG regions ("smooth" regions) and more substituted regions 377,378 ("hairy" regions). It has however be proposed that the HG regions might themselves be side chain to the RG-I regions 379 . Since pectin is mainly used in the food industry 371 , especially as a gelling or thickening agent, structural characteristics influencing the gelling properties of the pectin are of great importance. There are however some applications in other domains, which may call upon various other properties of the pectin chains. As mentioned, food industry 380 and medicine or pharmaceutical industry 381,382 are the most prominent users of pectin based materials, but there are also reports in environmental science, for instance as adsorption materials 383,384 . The pectin based materials may adopt various forms such as gel bead 385 , films 386 , three dimensional scaffolds 387 and dry porous materials 388 . Regardless of the targeted application, it is necessary to understand the relationship between the structure of the pectin and its physical properties, in order to choose to best candidate for desired function. The chains of HG or RG may be methylated and/or acetylated which may have and influence on interchain interactions. For this reason, pectin are commonly separated according to their degree of methylation (DM) into highly methoxylated pectins (HM) with a DM above 50% and low methoxy pectins (LM) with DM inferior to 50%. HM pectins are able to form gel in acidic conditions in presence of high sucrose content 389 . This is typically what happens in the confection of jams. LM pectin can also form gels but under different conditions. In this last case, the presence of divalent cations such as Ca 2+ is required. A proposed explanation is the formation of an "egg-box" structure 390 , similar to the one observed upon crosslinking of alginate 183 . It is however necessary to remember that pectins are complex and heterogeneous polymers, which results in complex molecular interactions and macroscopic behaviors. For this reason the two previously cited gelation mechanisms are not completely independent and calcium proved to be able to modulate gelation of HM pectins 391 . In addition, several parameters can influence the gelling properties of pectins, including pH, chain length, temperature and type of divalent cation 392 . The degree of acetylation also plays a major role in the gelling properties of pectin 393 , since the presence of the functions may sterically hinder formation of the gels. This is one of the proposed explanations for the limited gelation capabilities of pectin extracted from beet root compared to citrus or apple pectins 394 . The properties of pectins extracted from beet root pulp 395,396 can be modified depending on the extraction conditions 397 and post-treatments 398 in order to make it compatible with conventional gelling conditions. This however makes the beet root pectin less competitive than citrus or apple pectins. As a result, this type of pectin is not commonly used in the food industry and it largely remains an unexploited by product of the sugar industry. Beet root pectin has however its own specificity, namely the presence of ferulic residues 399 which may confer a possibility to crosslink the pectin 400 in order to obtain gels. In this work we chose to use a shaping technique that does not rely on the gelling properties of pectin, but takes advantage of its polymeric nature and of its solubility in water. II.2 Ice-templating or the use of water to shape materials II.2.a Comparison of different freezing-conditions Ice templating is a simple way to obtain dry porous materials. Besides its simplicity, one of the main advantage of the technique is its versatility since it can be applied to components ranging from polymers 304,320 to ceramics 256 and metal 262 , as well as various composites 234,319,321 . The general idea is to freeze a solution (or suspension) in order to form ice crystals which will serve as templates for the final porosity. As ice crystals grow, solutes are segregated from the solvent (often water), which results in the formation of the future pore walls around the newly grown ice crystals. The latter are then removed by sublimation to free the porosity. This method can be used to shape many different raw materials, since the vast majority of solutes have a very low solubility in ice. The limit then resides in the capacity of the material to retain its shape after removal of the ice crystals. One possibility is to add a sintering step as is the case for the shaping of ceramics. In the case of pectin freeze-casting, the dry foam obtained after lyophilization is self-supporting, but it is important to highlight the fact that this network is not crosslinked, and that any contact with liquid water results in the dissolution of the polymer structure. The method for the formation of a macroporous pectin foams may seem straightforward: freezing of an aqueous solution of the polymer and freeze-drying of the sample. However, many parameters must be taken in account including the formulation of the initial solution (concentration, possible additives), the freezing conditions (freezing-rate, geometry) and the drying conditions. In order to link the processing parameters to the final foam morphology 297 , four different freezing conditions were explored. As a first approach, 4 mL samples were placed in polyethylene cylindrical molds of 19 mm in diameter and subsequently frozen in conventional -20°C and -80°C laboratory freezers. Samples of the same size were obtained by plunging the molds containing the pectin solution into a liquid nitrogen bath. These three approaches were compared with the freeze-casting technique, where a chosen temperature ramp was applied to the sample in contact with a heat conductive element (copper) at a precisely set temperature. Copper rod Material and methods Influence of the freezing conditions was assessed on aqueous solutions of pectin Heating resistance Liquid nitrogen tank Controller To understand the influence of the freezing method over the morphology and properties of the final material, it was necessary to pinpoint the differences between these methods. To characterize each freezing condition, the temperature at the core of the samples were monitored by a thermocouple during the whole freezing process (from a few minutes to four hours). Figure II.3 displays theses temperatures (full line) as well as the target temperature (dotted line), which was either the temperature of the freezer, liquid nitrogen bath, or the temperature at the top of the copper rod (in the case of freeze-casted samples). The first three conditions (freezers at -20°C, -80°C and liquid nitrogen bath) can easily be compared due to the very similar setups (one fixed target temperature, same type of mold and anisotropic source of cold). The main changing parameter is the targeted temperature (-20°C, -80°C or -196°C), but the geometry of the setup was the same. In this last case, the temperature follows a linear gradient along the axis of the cylindrical mold. In addition, the targeted temperature is not fixed, but is a ramp with a chosen slope (in this case 10°C/min). In other terms the temperature can be controlled both spatially and in time. For the samples frozen in laboratory freezer the temperature profiles are similar. After an initial cooling period, the temperature remains constant. This corresponds to the actual phase transition in the sample, and the growth of the ice crystals throughout the sample. After complete freezing of the sample, the temperature at the core of the sample decreases down to the target temperature. At lower temperature (-80°C) the initial cooling rate increases (3,6 °C/min in the -80°C freezer vs 1,2 °C/min in the -20°C freezer) and the phase transition time decreases (around 10 min at -80°C vs 50 min at -20°C). When the samples are plunged in liquid nitrogen, a delay is observed before initial cooling (at 243°C/min). With the freezecasting technique, the temperature at mid height of the sample follows the imposed temperature ramp, but with a slight shift (5 minutes). In this case, the cooling rate is 7.2 °C/min (theoretical ramp was of 5°C/min). To explain these variations, it is necessary to link them to the final morphology of the material, as a way to understand the relation between these temperature profiles and the ice growth. II.2.b Influence on the pores morphology Different morphologies can be observed at the macroscopic scale (see Figure II.4). The sample obtained at -20°C has a homogeneous aspect and no significant difference can be observed between the longitudinal and transversal views. The sample obtained at -80°C looks less homogeneous, with small striations visible on both the longitudinal and transversal views, but no clear orientation is noticeable. Slight iridescence can be seen in samples obtained in a liquid nitrogen bath (oriented in a radial fashion) and by freeze-casting (in the longitudinal direction). It is however noticeable that the pores in the material obtained at -80°C, so presumably with a higher temperature gradient, are slightly elongated. To investigate the influence of a temperature gradient, samples were prepared by immersion in a liquid nitrogen bath. In this case, the temperature at the core of the sample remains stable before dropping sharply (see Figure II.3 c). This might be due to the presence of an important temperature gradient between the core and the sides of the sample. It is probable that ice crystals nucleate almost immediately when the sample is plunged into liquid nitrogen, but only in the outer regions of the sample. Due to the radial temperature gradient, the ice crystals then grow in a radial fashion inside the sample (Figure II.7 b). This organization of the ice crystals results in the observed orientation of the pores. The absence of a plateau at the transition temperature supports the idea that when the center of the sample freezes, the rest of the sample is already frozen. It might be noted that the actual gradient is not strictly radial, since the sample is plunged entirely in liquid nitrogen. A vertical component to the gradient must therefore be present at the top and bottom of the sample, resulting in a slight tilting of the pores. The presence of a temperature gradient inside the liquid sample is responsible for the orientation of the final porosity, however, in the case of freezing with a liquid nitrogen bath, there is no control over this gradient which strongly depends on the mold geometry. To gain better control over the pore morphology, samples were prepared by freeze-casting. This method confers control over the orientation of the temperature gradient, as well the kinetics of the ice growth. By placing the sample in contact with a cooling element at the basis of the cylindrical mold, it is possible to induce a longitudinal temperature gradient. The temperature in the region directly in contact with the cooling element closely follows the imposed temperature ramp. The temperature in the middle region of the sample (where the thermocouple was placed) however follows the cooling ramp with a certain delay (see Figure Both the temperature profiles and morphology observations allowed for the identification of three different types of ice growth, which are summed-up in Figure II.7. The presence of a temperature gradient, as is the case for samples obtained by dipping in liquid nitrogen or freeze-casting, induces the oriented growth of ice crystals and therefore the formation oriented pores. Freeze-casting presents the further advantage of precisely controlling the direction and amplitude of this temperature gradient. II.2.c Influence on mechanical behavior The control of the freezing conditions influences the morphology of the material, which in turn has an influence on the physical properties of the foam. The mechanical behavior of the samples obtained by the four different freezing methods was assessed under compression. [START_REF] Cassidy | Environmental applications of immobilized microbial cells: A review[END_REF] were cut in the different samples. Samples were compressed up to a 50% strain, at a constant displacement rate of 1 mm/min. Charge was measured in function of displacement and corresponding stress and strain were calculated. Five replicates were used for each measurement. Material and methods Mechanical behavior under compression was assessed using an Instron 5965 traction and compression device. Cubes of 1 cm Two orthogonal compression directions were used on 1 cm 3 samples: along the cylinders axes (axial compression) and perpendicularly to the cylinder axes (radial compression). When applicable, these directions were specifically chosen along or orthogonally to the pores of the material. For the -20°C and -80°C samples axial compression (Figure II II.1. An anisotropy ratio was calculated by comparing the Young's modulus measured in the direction of the pores (//) and perpendicularly to the pores (⊥). 𝐴𝑛𝑖𝑠𝑜𝑡𝑟𝑜𝑝𝑦 𝑟𝑎𝑡𝑖𝑜 = 𝐸 ⊥ -𝐸 // 𝐸 ⊥ Compared to the other materials, the sample prepared by freeze-casting displayed a highly anisotropic behavior. When compressed in the direction of the pores, this material has a Young's modulus of 2,8 MPa, which is typical for polymer foams of this density 402 . However under compression orthogonal to the pores, the Young's modulus drops to 125 kPa. This strong mechanical anisotropy is the direct consequence of the structure anisotropy described earlier. Figure II.9: Samples obtained with conventional freezing at -20°C (a) or at -80°C (b) present no significant mechanical anisotropy. Foams obtained with liquid nitrogen (c) present an oriented porosity, but the mechanical behavior is quite similar in both directions. The sample obtained by freeze-casting (d) show both higher Young's modulus and compressive strength when compressed along the pores. The cellular materials with no specific orientation have no or little mechanical anisotropy (see Figure II.9 a and b and Table II.1). The Young's modulus in both directions are in the same order of magnitude (between 1,1 and 1,5 Mpa for the -20°C freezing vs 0,3 Mpa for the -80°C freezing). The observed difference may be attributed to the slightly elongated shape and more lamellar structure of the material prepared at -80°C. The radial structure (sample obtained using a liquid nitrogen bath), despite having a specific pore orientation, displays no mechanical anisotropy. This might be attributed to the fact that the compressive tests are not really performed directly along the pores but rather on a distribution of orientations, due to the radial organization. The apparent mechanical behavior therefore results from a mean over several orientation including compression along the pores but also orthogonal compression (see Figure The morphology can be linked to the mechanical behavior which is a good indication to assess the physical properties of the material. Another relevant information regarding the considered application is the wetting behavior, where oriented or channel like pores seem to be a definitive advantage. II.2.d Influence on the wetting behavior By the use of a controlled version of the ice-templating process, a unidirectional cellular pectin-based material was obtained. This oriented porosity could be useful in soil depollution applications as a way to maximize substrate transport via capillary phenomena. Material and methods Wetting behavior of the foams was assessed by impregnation of the foams with a solution of Disperse Red 1 at 0.2 g/L in absolute ethanol. The impregnation was recorded on camera and images were analyzed the ImageJ software. Wetting profiles were obtained by separating the impregnated regions from the dry ones by the "Threshold" function. The images were then assembled thanks to the "Reslice" function and the profiles were extracted by the "Find Edges" and "Save XY Coordinates" functions. Initial wetting rates were measured from the slope of the profile during the first 1.5 sec of impregnation. To highlight the influence of the porosity orientation, cellular pectin materials with a radial structure (obtained in a liquid nitrogen bath) and with a longitudinal porosity (obtained by freeze-casting) were subjected to the capillary ascension of a solution of Disperse Red 1 in absolute ethanol. The analysis of the images provided the corresponding wetting profiles (Figure II.10). The materials were impregnated both along and orthogonally to the pore orientation as was the case for compression. In the case of radial foams, alignment of the impregnation direction with the porosity could not really be obtained due to the distribution of orientations. Table II.2 compiles the initial wetting rates for radial and freeze-casted foams. As expected, the capillary ascension was much faster along the material's pores. However it is interesting to notice that the radial foam is quickly impregnated up to 50% of its total height, but then the impregnation is drastically reduced. Due to the distribution of orientations, the wetting cannot occur along the porosity throughout the whole sample, and as a result the full wetting of the foam is a combination of longitudinal and orthogonal wetting regimes. In the case of freeze-casted samples, further control of the wetting properties may be gained by taking advantage of the slight pore size gradient usually observed in samples 403 . Table II.2: Initial wetting rates are higher when impregnation occurs along the pores of the foam. Freeze-casted foams have higher initial wetting rate both along and perpendicular to the pores Type of pores Impregnation along the pores Impregnation perpendicular to the pores Axial porosity (a) 2.3 1.4 Radial porosity (b) 1.6 0.8 Both the foams obtained by plunging into liquid nitrogen (radial foams) and the freeze-casted materials (with a longitudinal porosity) may have significant interest from an application point of view based on their mechanical and wetting behavior. They were further investigated in terms of fine tuning of the morphology. II.3 Variations around foams obtain with a liquid nitrogen bath The freezing method has a direct influence on the morphology of the foams and macroscopic properties, but it is also possible to tune these by formulation of the initial solution and variations of the processing parameters. II.3.a Influence of the polymer concentration The morphology of radially oriented foams obtained by plunging polymers solutions into a liquid nitrogen bath can be modified by changing the composition of the initial solution. This may refer to the nature of the polymer itself, to the presence of various additives but also more simply to variations in the concentration of the polymer. Samples were prepared from pectin solutions at various concentrations, which proved to have a direct influence on the pore morphology, as can be seen in SEM cross sections (see Figure II.11). Material and methods Solutions of various pectin concentration (40, 45, 50, 55 The most direct way to characterize the morphology of the pore walls is the measure the pores width, or in other term, the space between two pectin layers. (Figure II.12 a). It is also possible to measure the pore walls thickness (Figure II.12 b). It must however be noted that these values are less statically relevant that pore width due to the limited number of measurement performed. In addition both pore width and pore wall thickness measurements may be subjected to slight perspective errors since SEM pictures are never taken perfectly orthogonally to the sections. They can however give a good order of magnitude for these dimensions and information about general tendencies in size variations. Pore wall thickness and pore width are linked but it is difficult to highlight a simple relation between these two dimensions, because there are many other parameters to take in account, including the pore length, the total polymer content, the pore wall density as well as the total number of pores. Assuming similar pore walls densities and a constant polymer concentration, smaller pores (i.e. smaller ice crystals) would mean more pore and therefore more pore walls, resulting in thinner pore walls. On the other hand, if a constant number of pores is assumed, as well as a constant pore wall density, thicker pore walls may be expected at higher pectin concentration. But in reality it is difficult to modify one of these parameters without changing the others. As a result, there is not clear variation over the concentration range considered (40 to 60 g/L in pectin) for the pore width or pore wall thickness. This may be due to the fact that several parameters are changing at the same time, possibly compensating each other. For a more comprehensive evaluation of the influence of the pectin concentration, it would be interesting to study foams prepared at lower concentrations. However, for too low concentrations (below about 10 g/L), it might be difficult to obtain self-supporting materials due to the lack of a dense polymer network to form the pore walls. Higher pectin concentrations induce higher solution viscosity, which may involve the presence of higher mechanical constraints during the ice growth, thus limiting the formation of ice-crystals. Direct interactions between the polymer and the ice surface may also influence the ice growth. For considered range of concentrations however, no significant pore width change was monitored (see Figure II.12 a). No significant pore wall thickness variation can be observed (see Figure II.12 b). As mentioned earlier, higher concentrations may be expected to result in thicker pore walls. But smaller, and therefore more numerous, pores implies more pore walls, which may therefore be thinner. If the width of the pores or the pore wall thickness does no change dramatically when the concentration changes, the organization of the pores is modified, as can be seen at higher magnification (Figure II.11 a', b', c' and d'). Higher pectin content seem to result in more ordered and more regular pores. Although cutting of the samples for observation in SEM may slightly alter the aspect of the pores, it is clear that the pores are better aligned in the foams obtained at 50 and 60 g/L than the foam at 40 g/L. II.3.b Influence of the addition of a reheating step Beside the pore size, another important aspect of the pore morphology is their interconnectivity. It has been reported that the solvent nature may be able to tune this aspect 280 . In this case however, the aimed application must be taken in account. Since the material is to be used as a host for microorganisms, it is essential to use nontoxic solvent such as water. Instead of modifying the formulation of the initial solution, the tuning of the pore morphology was investigated through processing parameters. Material and methods Foams were prepared in the same way as previously described by dissolving 40 g/L of pectin in deionized water, and stirring overnight at room temperature. 1.8 mL of the solution was poured into 2 mL cryotubes, which were plunged into liquid nitrogen for a few minutes. Half the samples were immediately put to dry in the freeze-drier, and the other half was left 5 minutes in a 0°C bath and subsequently freeze-dried. Transversal slices were cut and sputtered with 20 nm of gold for observation in SEM microscopy. Pore interconnections have been reported in materials obtained by ice templating, but they generally result from the use of solvents yielding highly dendritic structures upon freezing. In presence of specific composition 404 or additive 274,405 , or with different solvents 280 a large array of pore morphologies may be obtained. They result in pore interconnections which are generally finely controlled and well organized. In this case however, ice-formation is strictly similar between the two samples. The protocol only differs after complete freezing of the sample. Some level of reorganization in the ice network upon reheating may occur and the formation of interconnections might be due to the partial thawing of the ice crystal and subsequent local dissolution of the pectin walls. Samples are however put under vacuum before complete melting of the sample. Figure II.14: In samples dried directly after freezing (blue arrow) pressure drops below the triple point and the ice undergoes sublimation which results in lamellar pore walls. When the initial temperature of the sample is higher (red arrow), the sample crosses the melting line. However, since the pressure quickly drop, only small regions have time to melt and dissolve the pectin walls. When samples are placed in the freeze-drier directly after freezing, pressure drops to 0.05 mbar in a few minutes, which is below the pressure of water triple point (6.1 mbar). Although the sample temperature slowly rises again, the phase transition that the sample undergoes is sublimation ( interconnections may be attributed to locally higher solutes concentrations at the surface of the pectin wall, resulting in the local lowering of the ice melting point and subsequent dissolution of small regions of the pectin wall. The change in pore width may however suggests complete thawing and re-freezing of the ice lamellas, resulting in a rearrangements of the pore walls. Other variations of this drying step may be designed. For instance drying of the sample inside the original mold or in a larger container also has an influence on the pore interconnectivity. Such interconnections may present a significant interest regarding the targeted application as a way to ensure good substrate exchanges all throughout the samples. The method however presents the serious drawback of being difficult to precisely control. It relies on a fragile equilibrium between partial thawing of the pore walls and preservation of the general structure. This implies the necessity to control the temperature in the sample, which may prove extremely versatile depending on the sample size and geometry, and results in very low reproducibility. A more controlled way to tune the pore morphology would therefore be to control not the fate of the ice crystal before sublimation, but the growth of ice crystal themselves. II.4 Tuning of freeze-casting conditions Although the method of ice-templating by plunging into liquid nitrogen has the great advantage of being very easy to implement, it does not provide sufficient control over the ice growth itself. As mentioned earlier, the freeze-casting technique confers control over the geometry of the temperature gradient, and therefore over the pore orientation, but it also allows for a control of the freezing-rate as a way to modulate the ice growth, and therefore the pore size and morphology 273 . II.4.a Influence of the solution concentration In order to gain control over the material morphology, key parameters must be identified all along the synthetic path. The first step of the material preparation is the formulation of the aqueous pectin solution. As a result, it may seem logical to start tuning the material properties through the composition of the initial solution. Depending of the type of polymer used, the morphology of the pores may vary significantly 187,261,300,304,320,322 . However, even a change of concentration for a given polymer may have an influence on the aspect of the pores as can be seen in Figure II.15. Material and methods Solutions at different concentrations (20, 30, 40 Cubes of 1x1x1 cm were cut to assess the mechanical behavior under longitudinal compression (compression in the direction of the pores). Each measurement was repeated on 5 different samples. To assess the efficiency of liquid transport within the foams, samples obtained from different concentrations were put in contact with a 0.2 g/L solution of Disperse Red 1 in ethanol. Impregnation was recorded and the images were analyzed using Fiji software to extract the wetting profiles. The orientation domains of well-aligned pore can only be seen for higher polymer content. Increase in polymer concentration induces an increase in the order of the porosity. This might be due to higher mechanical constraints in highly concentrated solutions. The preferential growth of ice in certain directions has a direct influence on the aspect ratio of the pores (see Figure II.16 d). Analysis of the SEM cross section also revealed a significant modification of the pore sizes (see Figure II.16 a and b). Lower pectin densities are responsible for wider (pores range from 8.5 to 15.3 µm) and shorter pores (from 104 to 245 µm). Pore walls are also thinner (from 39 to 208 nm) when the initial concentration in polymer is low. The larger pores, which also means a reduce number of pores since the overall size of the sample does not change, as well as the thinning of pore walls can be linked to the lower amount of polymer available for the formation of the walls when the initial solution has low pectin content. But variation in solutes concentrations also influence the liquid/solid interface, since both the viscosity and freezing points are modified by the solute concentrations. Another parameter to consider is the variation in viscosity of the polymer solution according to concentration (see Figure II.17). Low concentration solutions exhibit lower viscosity, as a result, ice growth is less mechanically constrained which results in larger ice crystals and therefore larger pores. Once again, the tuning of the cellular structure can be directly linked to the control of mechanical behavior. As may be expected, the materials prepared from higher concentration solutions exhibit higher Young's modulus, compressive strength and toughness (see Figure II.18 and Table II.3). The changes in the initial solution concentrations correspond to the variation in the porous materials final density. As was documented by Ashby and coworkers 402 , Young's modulus tends to increase with apparent density, regardless of the type of material. Young's modulus measured for these materials (between 1 and 5 MPa) are typical for polymer foams with density ranging from 20 kg/m 3 to 50 kg/m 3 . The stress-strain profiles show typical behavior for polymer foams, with an initial elastic behavior (up to about 7% strain), followed by a plastic deformation plateau (between 7 and 40% strain). The material then undergoes a densification, which translate in a strain increase beyond 40% deformation. Materials with different densities follow the same general profile, however the yield strength changes significantly (variation between 20 and 186 kPa). This might be linked to the changes in pore walls thickness mentioned previously. The wetting behavior of the foams was assessed by impregnation of 1 cm high foams using a solution of Disperse Red 1 in ethanol (see Annex p 213). Despite differences in the pore size, the wetting profiles were similar for the samples prepared from pectin solutions at different concentrations. All samples were impregnated in less than 0.5 s. As mentioned previously, both the initial solution and the actual processing parameters can be tuned to modulate the pores morphology. One of the major advantages of the freeze-casting technique compared to simple dipping in liquid nitrogen is the possibility to control the cooling rate of the sample as a way to control the structure of the foam. II.4.b Influence of the freezing-rate The freeze-casting technique provides a control of the temperature gradient in terms of kinetics thanks to the use of a controller to set a specific cooling rates. This parameter has a direct influence on the pore morphology (see Figure II.19). Material and methods A solution of pectin at 40 g/L was prepared from beet root pectin and deionized water and stirred overnight at room temperature. Samples of 3 mL were prepared in 15 mm diameter molds. Three cooling rates were used: 1, 5 and 10°C/min. The samples were then lyophilized for 48h. SEM observations were performed on transversal and longitudinal slices, sputtered with a 20 nm gold layer. Mechanical behavior was assessed under longitudinal compression on 5 replicates of 1 cm 3 cubes. Wetting behavior was assessed by impregnation of the foams by a solution of Disperse Red 1 in absolute ethanol. The capillary ascension was filmed and the images were analyzed thanks to Fiji software to recover the wetting profiles. Figure II.21 presents the variations in pore width and length, as well a as pore wall thickness, in function of the freezing-rate. Mean pore length varies from 309 µm to 428 µm when the cooling rate decreases. The large standard deviations for these values must however be underlined. This may be explained by the length variations observed in each orientation domain. A more significant parameter to describe the cellular material's morphology might therefore be the pore width which presents smaller deviations. Young's modulus values are around 3 MPa and compressive strength are about 0,1 MPa while are typical values for polymer foams 402 . A slight drop in Young's modulus can be observed at 10°C/min, but remains within the standard deviations of the measurements at 1°C/min and 5°C/min. The wetting behavior of the foams was assessed with foams prepared at different freezing rates (see Annex p 213). No significant change in the wetting profile could be observed, all the samples were fully impregnated in less than 0.5 s. Modifications of the freezing-rate had a significant influence on the pore morphology and size (smaller and better-organized pores at higher freezing rates). The macroscopic properties of the foam such as the mechanical behavior or the wetting properties were however not impacted. Conclusion Ice templating proved to be an efficient way to shape beet root pectin. Pectin solutions were processed into porous dry foams thank to freeze-drying. Various freezing methods were compared and proved to yield various pores morphology which has a direct influence on the macroscopic properties of the foam such as mechanical or wetting behavior, which are of great importance from the application point of view. Freezing in conventional freezers yielded anisotropic pores, but freezing in liquid nitrogen and freeze-casting resulted in the formation of oriented porosity, respectively organized in a radial or longitudinal fashion. The freeze-casting technique confers control over processing parameters such as the freezing rate. Other parameters like the polymer concentration may also be modified in order to tune the pores morphology. Characteristics such as pore width or pore wall thickness can be modified in order to control the macroscopic properties of the foam. The oriented porosity, which is the key feature of this pectin-based cellular material is however preserved. The shaping of the polymer is the first step toward the elaboration of a hybrid material destined to host living organisms (see Figure II.23). Good comprehension of the various processing parameters which may also have an influence on the viability of the encapsulated species will be essential in the design of the encapsulation protocol in order to accurately evaluate and predict their influence on the foam morphology. Introduction Pectin foams with unidirectional porosity may represent an interesting asset in the design of a device for soil depollution, since they should favor mass transport of the targeted pollutant through capillary phenomenon. Pectin was chosen in particular for its non-cytotoxicity and water solubility, which are essential for hosting living organisms. The water solubility of pectin, though being an advantage from the encapsulation perspective, may be problematic from the structural point of view, since the material is destined to be implanted in soils. The water content of the considered soils is likely to dissolve the pectin structure rendering the device inefficient. One option to prevent such dissolution of the foam is to coat the polymer with an inorganic material capable of ensuring its structural integrity in a hydrated state. Silica appears as a good candidate to play the part of the inorganic moiety in such a hybrid foam. It is both eco-friendly (silica is a major component of soils) and non-cytotoxic to the considered microorganisms (silica gels have been used for bacteria encapsulation 113 ). Another advantage of silica is the fact that it can easily be obtained through sol-gel processes, which can be performed at low temperature and under mild conditions. Such considerations are essential, since the silica coating is meant to be applied on pectin foams containing living bacteria. A vapor phase sol-gel silica deposition method to coat macroporous pectin foam with a silica layer was developed and optimized, as a way to prevent dissolution of the structure upon contact with water. III.1 Silica coating using vapor deposition III.1.a About sol-gel and gas phase deposition This developed vapor phase silica deposition technique is based on sol-gel chemistry of tetraethyl orthosilicate (TEOS) (see paragraph I.2.c.i, p30). TEOS is one of most commonly used sol-gel silica precursors. Alkoxides in general have been used for applications ranging from industrial coatings 406,407 to encapsulation matrix for sensitive biomaterials 111,408 . Typically, TEOS can be hydrolyzed with water in a liquid phase, and can then be condensed to form the inorganic polymer network. Both acidic and basic conditions may be used to catalyze hydrolysis, resulting in different morphologies and properties 409 . In several applications, formation of thin and homogeneous layers of silica is crucial. One way to obtain such materials is to use vapors of silica precursors as a way to precisely control the amount of reactive species involved. Early gas phase deposition techniques have been developed for microelectronic devices 410 and involved use of high temperatures 411 under oxidizing conditions. Alternative methods including use of plasma or ozone 412 have been used to gain better control over the silica layer properties and characteristics. These methods are commonly referred to as chemical vapor deposition (CVD), but many variations have been described regarding deposition conditions (including temperature, pressure, precursors or activating species). However for specific applications such conditions were not suitable, especially when living cells were involved 413 . Classical sol-gel techniques, which have the great advantage of being usable at low temperatures, were therefore adapted to gas phase deposition techniques 414 . Such methods have been especially efficiently used for immobilization of various structures (from lipids 415 to various cells 210,416,417 ) with thin silica layers. Silica precursors (most commonly TEOS or TMOS) have also been deposited on a wide variety of substrates, including polysaccharide hydrogels, in order to yield hybrid matrices for cell encapsulation 418,419 . This approach was therefore chosen for the coating of the pectin-based foams. Materials and methods In a typical coating experiment, pectin macroporous foams were prepared as previously described. 40 As a first approach, mass gain was assumed to correspond to the addition of silica. Weighing was therefore used as a straightforward way to assess silica content of the foams (noted %SiO2). % 𝑆𝑖𝑂2 = 𝑚 𝑓 -𝑚 𝑖 𝑚 𝑓 × 100 where %SiO2 is the weight mass percentage of silica, mi the initial polymer mass, mf the final mass. Pectin foams were put in contact with an atmosphere containing TEOS vapors, HCl and water to ensure hydrolysis and condensation. At 30 °C NaCl saturated aqueous solutions are expected to yield 75.6 % of relative humidity 420 . Presence of HCl at 5 % may however slightly modify this value. Azeotropic point for HCl/H2O mixture is for a 20 % HCl content, which means that solution and atmosphere composition might vary slightly over time. Direct pre-hydrolysis of the liquid TEOS in the initial vials may confer better control over the stoichiometry of TEOS and water, as well as a precise pH control. This would however lead to the rapid condensation or at least formation of oligomers in the vials. The resulting mixture would not be volatile enough to ensure vapor phase deposition of the reactive species on the surface of the foams. In this setup, hydrolysis and condensation occur directly on the polymer surface, resulting in the progressive and controlled formation of a silica layer. The drawback to this precise control is that small amounts of precursors are reacting, resulting in a slow formation of the silica layer. It might be argued that higher processing temperatures may be of interest in speeding the process up (all depositions were performed at 30°C). Use of higher temperatures may allow for higher saturation vapor pressure of TEOS, resulting in larger amounts of available molecules on the pectin pore walls. In addition kinetics of hydrolysis and condensation are temperature dependent. Higher temperatures may however not be compatible with cell survival. Furthermore part of the water and acid contained in the atmosphere of the deposition chamber will eventually be dissolved directly in the liquid TEOS vials, resulting in slow hydrolysis of the liquid precursor. Higher processing temperature would likely result in quicker gelation of the precursor solution. Other parameters may be of importance in the coating kinetics, such as the volume of precursor available, but also the geometry of the vials and more specifically the surface of exchange between the atmosphere of the container and the liquid. A large volume of TEOS with a very small exchange surface would likely not be efficiently deposited. For this reason it was preferred to divide the volume in several vials, to maximize the liquid/gas interface. Similarly, composition of the aqueous acid solution may have a significant influence on the deposition kinetics, just as pH control is essential in classic liquid sol-gel chemistry. In short this silica deposition technique which seems simple and straightforward is in fact very sensitive to a wide range of parameters and must therefore be carefully optimized in order to yield reproducible and controlled silica coating. Several sets of coating conditions were therefore applied to different models of foams. III.1.b Deposition on radial foams Efficiency of the vapor phase deposition process was first assessed on pectin foams prepared by dipping in liquid nitrogen. For these samples the pore morphology was organized in a radial fashion and pores were interconnected. This type of sample has the advantage of being quickly and easily prepared, and could therefore be produced in large batches which allowed for the screening of various processing parameters. III.1.b.i Observation of the silica layer The first step in the validation and optimization of the described coating technique was to assess and characterize the presence of silica in the samples exposed to vapors of TEOS. Material and methods Radial foams were prepared as previously described in Chapter II. An aqueous solution of pectin at 40 g/l was prepared under magnetic stirring at room temperature overnight. About 1.8 mL of solution was poured in cryotubes, which were immerged in liquid nitrogen for a few minutes. The samples were slightly reheated to ensure pore interconnection and subsequently vacuum dried for 24h. Samples were then placed for 14 days in the deposition chamber described in paragraph III.1.a . The samples were left 24 h at 30°C and 24 h at room temperature in a desiccated atmosphere before final weighing. Slices were cut and sputtered with 20 nm of gold for SEM observation. Slices were sputtered with 20 nm of carbon before EDX analysis. Samples were crushed for FT-IR spectroscopy observations on Perkin Elmer Spectrum 400 FT-IR/FT-NIR Spectrometer equipped with Universal ATR sampling accessory. The first and most direct ways to assess the presence of silica after contact with vapors of TEOS was to weigh the samples before and after the treatment. Mass gain may be attributed to the formation of silica species on the surface of the sample. Weighing of the samples confirmed the presence of a mass increase up to about 50 % of the total mass after 14 days. A simple method to identify the nature of the added material is the use of FT-IR spectroscopy (see Figure III.1). Spectra were measured for samples with various mass additions (samples were left for different times in the deposition chamber). A sample of pectin not exposed to vapors of TEOS was used as control. Figure III.1: After foams were exposed to TEOS vapors for 7 days (b) or 14 days (c), new peaks appear in the low wavenumber region compared to samples of pectin alone (a). These peaks are characteristic of silica. An offset was applied to the transmittance curves for clarity and the curves were normalized to the νC=O = 1740 cm -1 peak. The wide but not very intense peak observed in all samples around νO-H = 3400 cm -1 can be attributed to stretching of O-H bonds. Such functions may be present in both pectin and silica and can therefore not be used as a mean of identification of the material corresponding to the mass increase. Two small peaks are common to the three samples (at 1740 cm -1 and 1630 cm -1 ). The peak at νC=O = 1740 cm -1 could be due to the stretching vibration of the C=O bond of acetyl esters in pectin 421 . Since the pectin scaffold is common to the various samples, this peak is likely to remain unchanged after silica deposition. As a result νC=O = 1740 cm -1 was used as reference for normalization of the spectra. Interestingly three defined peaks appear at lower wavenumbers after the deposition step. These peaks can be correlated with typical silica signals. Peaks around ν a Si-O = 1070 cm -1 and ν s Si-O-Si = 800 cm -1 have been attributed to stretching of Si-O-Si bonds, as the asymmetric stretching and symmetric stretching of the oxygen atoms respectively 422 . A peak around νSi-OH = 930-950 cm -1 may be attributed to stretching of Si-OH bonds. The literature values are slightly shifted but small variations have previously been reported depending on the density of the silica network 423 and condensation state of the TEOS silica precursor 424 . Additional shift may be attributed to the type IR spectroscopy detection method (use of an ATR module). This series of bands are nonetheless representative of the presence of silica within the foams exposed to TEOS vapors. The intensities of these specific peaks increased with prolonged deposition time, which is consistent with the measured mass increases. The presence of silica within the samples after the deposition process was confirmed. This analysis was not quantitative, but supported the hypothesis that the mass increase is due to the presence of silica. In further experiments, mass gains were considered as the silica content of the samples. The silica coating was further characterized by SEM. As can be seen in Figure III.2 a and b, the general structure of the materials is not altered by exposition to TEOS vapors, despite acid conditions and about 75% of humidity. At higher magnification (Figure III.2 a' and b') a significant difference between the two samples can be observed. After silica deposition a smooth layer can be seen on both sides of the initial pectin wall, which is likely to correspond to the mass increase previously mentioned. To confirm the composition of the deposited layer, energy dispersive X-Ray spectroscopy (EDX) was performed on a samples exposed 14 days to silica vapors (see Figure III.3). Element analysis showed the presence of silicon and oxygen. The presence of oxygen may not entirely be attributed to the formation of a silica layer since it is present in large amounts in the pectin itself. Silicon is however likely representative of the presence of silica. The homogeneity of the silica layer was confirmed by mapping of the oxygen and silica. The proposed method for addition of silica to pectin foams was therefore validated. This method may however be optimized through various parameters (deposition time, temperature, geometry of the deposition chamber etc…) in order to tune the silica layer itself. III.1.b.ii Tuning of the silica layer To gain further control over the deposition of the inorganic moiety on the polymer foam, samples were exposed for various times to vapors of TEOS as a way to yield various silica contents. O mapping Si mapping SEM observation Material and methods The deposition conditions used were the same as previously described. 1.8 mL freeze-dried pectin foams were prepared and weighted. The samples (about 59 ± 2 mg) were placed in the deposition chamber described in paragraph III.1.a . Samples were removed after various deposition times (between 1 and 15 days). Samples were dried 24h at 30°C and 24h at room temperature in a desiccated atmosphere before final weighing. Samples were cut and sputtered with 20 nm of gold for SEM observation. These thickness values must however be considered with caution. The statistical relevance is questionable since only 5 measurements were taken on only two different pore walls. As a result the values do not take in account possible heterogeneities within the samples. It must also be noted that pictures cannot always be taken strictly perpendicularly to the exposed edge of the pore walls, which may result in some perspective bias in the measurements. The vapor phase deposition method proved efficient in adding the desired inorganic moiety to the pectinbased foams. Various deposition times allowed for tuning of the silica content and thickness of the silica layer. Several other parameters are however likely to influence the deposition process. HCl concentration, presence of NaCl and the volume of introduced TEOS were modified in order to assess the robustness of the addition of silica (see Annex p 214). Silica contents were polydisperse after only 7 days of deposition, but the dispersion was reduced after 14 days of silica deposition. Modification of parameters such as HCl concentration, the presence of NaCl or the modification the TEOS volume did not significantly modify the deposition kinetics. As a result, chemical vapor deposition of TEOS proved to be an efficient way to add silica to pectin foams. The polymer porous scaffolds were coated with a tunable and homogeneous layer of silica and the process proved to be efficiently reproducible and robust. III.1.c Coating of unidirectional foams Preparation of macroporous pectin foams by dipping in liquid nitrogen has the advantage of being quick and adapted to large batches of samples. However, as mentioned in Chapter II, this method implies very high cooling rates, which, as will be discussed in Chapter IV, might be problematic from the encapsulation perspective. As a result the vapor phase was also applied to freeze-casted pectin foams. Material and methods Freeze-casted foams were prepared from 3 mL of 40 g/L pectin solutions, frozen unidirectionally at 10°C/min and subsequently vacuum dried at 0.05 mbar for 24h. The resulting samples were cut to 1.5 cm high cylinders and placed in the deposition chamber previously described. A 5 wt% aqueous solution of HCl, with 400 g/L of NaCl was used to ensure hydrolysis. Four vials containing 10 ml of TEOS were introduced in the closed vessel. Coating was performed up to 16 days at 30 °C/min. After removal from the deposition chamber samples were left 24 h at 30 °C and 24 h at room temperature in a desiccated atmosphere to remove excess humidity before weighing. Silica content on pectin foams with unidirectional porosity (obtained by freeze-casting) was first assessed based on the mass gain. Freeze-casted foams were exposed for various times to TEOS vapors and Figure III.5 present the deposition profile compared to silica coating of radial foams (obtained by dipping in liquid nitrogen). In both cases a plateau can be observed after about a week. In the case of unidirectional foams however the maximum silica content appears much lower (about 15 %SiO2) compared to radial foams (about 40 %SiO2). The maximal mass increases were about 23 mg and 40 mg for freeze-casted and radial foams respectively. Beside the structural differences between the two types of samples (unidirectional porosity vs radial porosity), the considered foams also have different volumes. Since two parameters (morphology of the porosity and mass of the samples) are changed simultaneously, it is however difficult to identify their respective influence. In order to discriminate between these two effects, another series of silica deposition was performed on samples prepared with a single technique (freeze-casting) but cut to different sizes (and therefore different volumes). Material and methods Pectin foams were prepared by freeze-casting at 10°C/min, from a 40 g/L polymer solution. The samples were vacuum-dried and kept under desiccated atmosphere until further use. Samples were cut to 8.2 mm, 4.3 mm or 1.7 mm (mean value on triplicate samples). They were placed in the deposition chamber previously described (see Annex p 210) and left 10 days at 30°C in presence of 4x10 mL of TEOS and 150 mL of 5 wt% HCl with 400 g/L of NaCl. After removal from the closed vessel samples were left 24 h at 30 °C and 24 h at room temperature in a desiccated atmosphere. Final weight of the samples was used to determine the silica content. Silica deposition in vapor phase on freeze-casted foams seemed less efficient in yielding high silica content, or thick silica layers. Thin layers or incomplete layers of silica may not be able to protect efficiently the pectin scaffold against dissolution. In order to investigate the reason of this low efficiency, the influence of the mass of the samples was evaluated. As can be seen from All the samples were prepared with the same method (freeze-casting). As a result all samples have the same diameters and densities. As a consequence the volume of the samples, their mass and their thickness are proportional. The thickness of the foam appears as an especially relevant criterion in the understanding of the silica deposition phenomena due to the oriented porosity of the foams. The concentration in silica precursors (and other reactive species such as water) is assumed to be homogeneous throughout the deposition chamber. The flow of reactive species from the chamber to the inside of the foam is also assumed to be constant and is assumed to occur only through the cross section of the samples. Since the diameter of the foams is independent of their volume (or thickness), the exchange surface is also independent of the volume of the foams (assumed to be equal to the sum of the top and bottom surfaces of the cylinders, which is about 3.5 cm²). With similar flows and exchange surfaces, the total mass of precursors within the foams for a given time is therefore independent of the sample's thickness, which is consistent with the measured silica masses (see Figure III.7). Figure III.7: Silica precursor flow is assumed to be independent of the foam volume. Since all foams have the same diameter, similar masses enter the foams in a given time. As a result the added masses are similar, but the relative mass content (or thickness of the silica layer varies. Silica layer is likely to be more homogeneous in thin foams (b) than in thick samples (a) due to diffusion limitations. In addition, the use of thicker samples is likely to result in less homogeneous silica layer due to diffusion limitations. If a constant free mean path is assumed for precursors within the foams, the probability of presence of precursors diminishes at the center of the foams. As a result, silica layers may be thicker in the upper and lower regions of the foams. In the case of thinner samples, the variations in this probability are likely to be limited resulting in more homogeneous silica layers. It seems that the volume of the sample is also a way to tune the final silica content of the hybrid foam (or in other terms the thickness of the deposited silica layer). In the case of foams with an oriented porosity however, more than the total volume, the diameter of the foams (rather than the total volume) appear to be determinant in setting to total introduced silica mass. III.1.d Characterizations Once the deposition conditions were set, the resulting hybrid pectin-silica foams were characterized with various methods. Samples about 5 mm thick with silica contents between 11 %SiO2 and 52%SiO2 were compressed along the pore direction to 50% of strain at a constant displacement rate of 1 mm/min using Instron 5965 traction and compression device. The silica layers were first observed in microscopy. SEM-FEG images of freeze-casted pectin foams coated with 48 %SiO2 and 60 %SiO2 are presented in Figure III.8 a. Two thin silica layers can be observed on the sides of the pectin pore walls. The measured thicknesses are 23 nm for the sample containing 48%SiO2 and 32 nm for the sample with 60 %SiO2. These values can only give an order of magnitude for the thickness of the silica layer due to the previously mentioned bias of perspective and limited statistical relevance. The aspect of the layers themselves appears slightly different compared to observation made on gold coated samples. Initial observations on gold coated samples revealed very smooth layers. Here however the silica layers appear granular. This difference may be due to the fact that sputtered gold particle are bigger than sputtered platinum particle and may therefore hide some of the topological details of the observed layer. TEM observations were performed on samples embedded in resin. 10 measurements of the thickness of the silica layer where performed on 5 separate pore wall images for each sample. For a 47 %SiO2 silica content, pore walls had a mean thickness of 18 ± 12 nm, while at 34 %SiO2 the silica layers were 8 ± 3 nm thick. The high standard deviations for these values illustrate the inhomogeneities within a single sample. Despite these wide distribution, values are statistically different (at p < 0.05) which is consitant with the layer thickness-silica content dependance described in III.1.b.ii. In presence of silica, this mass loss also starts at 200°C but occurs up to 510°C, which may correspond to a partial stabilization of the pectin structure. After calcination, the residual masses were 3%, 23% and 36% respectively for samples with 0%SiO2, 15%SiO2 and 43%SiO2 apparent silica content. These values were normalized to remove the contribution of water mass. As a result the non-calcined fractions of the dry samples are 3%, 25% and 41% respectively. It seems therefore that the silica contents determined by simple weighing of the samples are slightly shifted compared to the actual inorganic contents. They remain however indicative of the variations between different samples and of the order of magnitude of the inorganic fraction. Even if weighing does not allow for precise determination of the actual silica content, it remains a direct and widely applicable, as well as non-destructive technique. It was therefore kept as a further reference to compare silica content between samples, keeping in mind that the actual mass values might be shifted. Silica was added to the pectin foams primarily to ensure protection against dissolution of the structure. However addition of inorganic material to the polymer foam is likely to modify other properties of the material, including its mechanical behavior. Figure III.10 presents the stress/strain curves as well as Young's modulus and compressive strength depending on the silica content. Up to about 20 %SiO2 the mechanical properties of the material appear similar to the characteristics of the pectin alone samples. As expected, higher mass percentages of the inorganic moiety result in stiffer inorganic material. The addition of silica also results in higher compressive strengths (see Table III.1). The general aspect of the stress/strain curves is also modified by the presence of silica. At low silica content (less than 20%SiO2) the curves are similar to pectin alone foams. This corresponds to the limit silica content necessary for SEM observation of the silica layer and could be attributed to the limit to the formation of a fully percolated silica layer. At higher silica content small irregularities appear. They may be attributed to localized failures of the silica layer, as they strongly resemble the behavior described for porous inorganic materials 316,425 . The samples can be seen as a series of pores walls composed of several layers of materials (silica-pectin-silica) with different mechanical behaviors (see Figure III.11). This corresponds the structure of a lamellar composite, in which case a rule of mixture can be applied 426 . Due to variations in chemical composition in the pectin chains, the surface of the polymer walls is likely to present slight heterogeneities, in particular in terms of hydrophobicity and charge. As a result, silica deposition mechanism may be assumed to start by preferential deposition of the silica where interaction with the pectin substrate is more favorable and then be extended across the durface. At low silica content, the silica is likely to be organized as localized patches rather than a fully percolating layer. As a result the mechanical behavior is governed by the pectin structure. At higher silica content, the applied stress is distributed on materials with different Young's modulus, but same strain is applied. Since Esilica > Epectin most of the stress will be sustained by the silica layers, and the mechanical behavior will strongly resemble behavior of a fully inorganic material. Figure III.11: In samples with no percolating silica layers (a) the applied stress is mainly supported by polymer (Epectin). At higher silica content (b) the same strain is applied to both components. Since Esilica > Epectin the overall stress is mainly representative of the silica layers. This simple model may explain the general differences observed both on the typical mechanical values (Young's modulus and compressive strength) as well as the aspect of the stress/strain curves. By using vapors of silica precursor (TEOS), it was possible to efficiently cover the polymer pore walls of the previously described foams with a homogeneous silica layer. The method proved efficient and reproducible and silica layers could easily be observed and characterized. The thickness of the layer could be tuned either by changing the deposition time, or the geometry of the samples. The goal of this addition of an inorganic moiety to the polymer scaffold was to ensure that the initial pore structure is retained and that the material does not dissolve when used in the conditions of the targeted application. The behavior of the material was therefore assessed in different model conditions. The hybrid porous material is meant to be used in polluted soils as a host matrix for metabolically active bacteria. Non-negligible water contents are expected in such environments, which is likely to result in rapid dissolution of a material solely composed of pectin. The presence of a silica layer all across the polymer pore walls should however limit this phenomenon. III.2.a.i In liquid medium The behavior of the hybrid material was initially assessed in liquid medium, as this represents the "worst case scenario" if the targeted environment is a highly hydrated soil. Material and method Macroporous foams were prepared by freeze-casting at 10°C/min from an aqueous solution of pectin grafted with rhodamine isothiocyanate (RITC) (see Annex p 208). The samples were vacuum dried and various contents of silica (0 %SiO2, 22 %SiO2 and 37 %SiO2) were added to the polymer foams. The samples were plunged in 15 mL of water and left under elliptic agitation at room temperature. Supernatants were regularly sampled and centrifuged to remove any material debris. Pectin content was deduced from UV-vis spectrometry at 558 nm (see calibration in Annex p 208). Silica concentration was assessed by the silico-molybdic method 427 (see Annex p 208). Visual aspect of the supernatants was also monitored. All assays were performed in triplicate. Figure III.12 hybrid foams displaying very different behaviors depending on the silica content. When samples were made of pectin alone (Figure III.12 a), the material rapidly swelled and appeared to be fully dissolved after only 24h. In presence of about 22 %SiO2 (see Figure III.12 b) the material retained its general shape up to 13 days but signs of partial dissolution could be observed after 22 days. Finally the sample containing 37 %SiO2 (see Figure III.12 c) appeared to retain its structure even after 22 days in water. As might be expected, it seemed that higher silica contents are more efficient in preventing pectin dissolution. To characterize more precisely the kinetics of foam ageing in water, the composition of the supernatant was followed over 3 weeks. As can be seen in Figure III.13 a, the pectin content of the supernatant increased sharply for the pectin only sample within the first few hours. Over longer periods (see Figure III.13 a') the pectin concentration in the supernatant appeared to drop slightly, which might be due to sedimentation or aggregation phenomena. It is also possible that some rhodamine bleaching occurred over the 3 weeks considered period. Immersion of hybrid materials (22 %SiO2 and 37 %SiO2) resulted in a small increase of pectin concentration in the supernatant which remains stable over 3 weeks. Higher silica content resulted in lower pectin content in the supernatant. Trace of silica could be detected in the supernatant of the foams composed of pectin alone. This may be explained by the fact that the assay were conducted in glass vials and should therefore be considered as a baseline signal. Another possible explanation could be the presence of a small silica amounts in pectin. In a counter intuitive manner, silica concentrations were higher in the supernatant of samples with lower silica content. The different silica contents were obtained by exposing the samples to vapors of TEOS and HCl for various times. These different deposition times have an influence on the overall silica content but might also be responsible for variations in the condensation state of the silica layer (confirmation might be obtained by solid state NMR). Higher condensation of the silica network may limit the dissolution rate of the layer, which might explain the difference observed between the two ageing profiles for hybrid foams. The considered silicon concentration values must however be considered with caution. The silicomolybdic method used for titration is indeed only efficient for detection of monomeric units of silicic acid. To minimize bias due to this limitation, all supernatant samples were diluted to fit the observation range and kept under stirring for 24h before titration was performed, but it cannot be excluded that part of the silica content may not have been detected. Addition of silica seems efficient in preventing dissolution of the pectin foams. As mentioned previously, it is likely that under a critical silica content (around 20%SiO2) the silica does not form a fully percolating, resulting in exposed pectin areas which can directly be dissolved. In addition thin silica layers are likely to be more prone to defects than thicker layers, possibly resulting in leaching of the pectin core of the walls (see Figure III.14). On the contrary thick silica layers are likely to efficiently protect the pectin structure against dissolution. Thickness of the silica layer is in addition susceptible to modulate the diffusion of water and therefore the rehydration kinetics of the pectin foams. Effects of rehydration may however not be the only possible source of material ageing in soil. For instance, these assays do not take in account possible degradation by endogenous microorganisms, which is likely to occur in soil. To assess such behavior, hybrid foams were introduced in reference soil samples and monitored over several weeks. III.2.a.ii In soil Behavior of pectin-silica hybrid foams was assessed in a reference soil (upper horizon silt loam Luvisol, see details in Annex p 204) over 5 weeks in order to identify possible biodegradation mechanisms. Material and methods Samples were prepared as previously described by freeze-casting and lyophilization of a pectin solution and subsequent deposition of silica by exposition to vapors of TEOS. After only 24h, the sample appears much smaller and after one week it can barely be distinguished from the surrounding soil. Addition of 25 %SiO2 seems to slow down the degradation process, however significant contractions can be observed after 20 days and 37 days. Higher silica content (34 %SiO2) seems to protect slightly more efficiently the structure, but the general evolution of the material is similar to the one observed for samples at 25 %SiO2. For 39 %SiO2, the sample appears to remain stable over more than 5 weeks. From the application point of view, preservation of the macroscopic structure is important to prevent leaching of the encapsulated bacteria. However preservation of the pore morphology may also be necessary in order to insure capillary mass transport of the pollutant within the material. Evolution of the porous structure was therefore followed through SEM observation (see Figure III.16). As might be expected, changes in the porous structure follow the macroscopic features described previously. Foams with no silica rapidly lose their porous structure. No significant morphology can be observed, even after only one day in soil. This might be due to rapid rehydration and subsequent dissolution of the pectin pore walls. With 20 %SiO2, the shape of the foam can still be distinguished, but the porous pattern seems however largely disrupted. At higher silica contents (34 %SiO2 and 39 %SiO2) the oriented and aligned pores can still clearly be observed, even after 5 weeks in soil. Assays in both liquid medium and in soil seem to indicate that a minimal silica content or thickness of the silica layer is necessary for efficient protection of the pectin structure. This might be linked to the fact that no clear silica layer could be observed in SEM below 20%SiO2. Similarly no significant changes of mechanical properties were observed for samples containing less than 20%SiO2. This may indicate that at low silica content, silica does not form a full layer, which is not sufficient to prevent the dissolution of the pectin pore walls. The hybrid materials obtained through a two-step process (freeze-casting to obtain a pectin scaffold and silica deposition) appear to be able to sustain prolonged stays in soil. The targeted application is the use of these materials as host matrix for metabolically active microorganisms as a way to degrade pollutants in soil. Interactions between the soil, the contaminants and the matrix, as well as diffusion aspects are keys to the efficiency of the functional material. Foams were therefore introduced in soil containing dye as model pollutant. III.2.b Behavior in polluted soil The final goal is to introduce metabolically active species within the pectin pore walls as functional units. The matrix itself may however have a contribution to the depollution process though adsorption of the pollutant. Pectin has been widely used for adsorption of metallic species. In this case however the targeted pollutants are rather organic species such as hydrocarbons, pesticides or dyes. Dyes represent a good laboratory scale model due to the fact that they are both easy to handle and easy to detect and characterize. In addition some dyes species have been proven to be efficiently adsorbed on both pectin-based 428,429 and silicabased 430,431 materials. III.2.b.i In liquid medium Adsorption properties of the hybrid foams were first assessed in aqueous solution and compared to the adsorption properties of soil. Reactive Black 5 (hereafter noted RB5), an anionic dye mostly used in textile industry, was used as a model pollutant. and 62 ± 5 % of discoloration for foams with 10 %SiO2 and 20 %SiO2 respectively). Dye concentrations for assays with the foams at 10 %SiO2 and 20 %SiO2 were not statistically different. The silica content of the hybrid foam does therefore not seem to have a significant effect on the adsorption properties of the foam. Both samples have however relatively low silica contents due to their thickness (which limits the maximal relative silica content as was demonstrated in III.1.c p 100. Higher silica contents may have more pronounced effect. It might be assumed that the discoloration can entirely be attributed to adsorption phenomena. Since the assays were performed at 25°C, in deionized water and over 24h only, degradation of the dye by microorganisms from the soil or water may be neglected. Dye mass loading on the foams was therefore directly calculated from the concentration difference between the initial and final supernatants: Material and methods 𝑞 = ([𝑅𝐵5] 𝑖 -[𝑅𝐵5] 𝑓 ) * 𝑉 𝑅𝐵5 𝑚 𝑠𝑎𝑚𝑝𝑙𝑒 Where q is the dye loading, [RB5]i the initial RB5 concentration, [RB5]f the final RB5 concentration, VRB5 the added volume of aqueous RB5 solution and msample the mass of the foam sample. Dye loadings were found to be 1.2 ± 0.1 mgdye/gfoam for both hybrid samples Soil itself is likely to have some adsorption properties toward the RB5. However in this case, both use of low dye concentrations and of small quantities of soil for adsorption may have prevented detection of this phenomenon. It is however likely that bigger amounts of soil may adsorb significant dye masses. III.2.b.ii In soil Efficiency of the pectin-silica porous material for discoloration of a RB5 loaded soil was assessed by introducing 10 %SiO2 foams in 5 g of soil saturated by a RB5 solution. Material and methods Hybrid foams (about 35 mg) were prepared accordingly to the previously described process (freeze-casting and drying of pectin and silica deposition). Mean silica content on the triplicate samples was 12 ± 1 %SiO2. Samples were placed in 5 g soil saturated by 3.5 mL of 0.1 g/L of RB5 aqueous solution. The total mass of dye represents about 1 % of the foam mass and 0.007 % of the soil mass. Controls were performed in triplicate with an aqueous solution of RB5 at the same concentration without soil and with soil saturated by the same solution but with no foam. Soil was also put in contact with water containing no dye to establish a baseline. After 24h at 25°C under static conditions, 3.5 mL of PBS 2X were added to each sample to simulate rinsing water and 2 mL of supernatant were centrifuged 10 min at 5000 rpm and UV-vis spectra were measured. Concentration variations were assessed by UV-vis spectroscopy. Absorbance was measured at 598 nm. Contribution of the yellow coloration due to the soil was removed based on the control sample. Both concentrations in rinsing water from soil with and without foam where significantly lower than the initial concentration. Discoloration rates were found to be 82 ± 1 % and 57 ± 1 % respectively. Dye adsorption was therefore significantly more efficient in presence of pectin-silica hybrid foams. In this case, due the presence of larger amounts of soil, it was possible to measure the absorbed mass of dye on the soil, which was found to be 0.038 mgdye/gsoil. In comparison the dye loading of the foam, which weighed only about 0.35 mg, was estimated to be 2.28 mgdye/gfoam. This value was calculated assuming that the amount of dye adsorbed on soil was the same in presence and in absence of the hybrid foams. The pectin-silica material seems in itself to be efficient in depolluting a real soil-based system. A large amount of pollutant is adsorbed on the foam which results in a diminution of the apparent concentration in the rinsing water. Even if it is difficult to assess with this set-up, the foam is also likely to diminished the amount of dye adsorb on the soil particles by displacement of equilibrium. Conclusion Macroporous pectin foams were coated with silica through sol-gel chemistry. The specificity of the process relied on the use of vapors phase silica. This method allowed for efficient and controlled deposition of a silica layer, without any modification of the porous structure of the pectin scaffold. The silica appeared homogeneously distributed on the pectin pore wall surface and silica thickness could be controlled through two main processing parameters (deposition time and sample morphology). Such a pectin-silica core shell structure could be of interest for various applications besides soil depollution. Fine control over the silica layer could prove useful in tuning the diffusion properties of the material. This could for instance be useful for the design of controlled drug delivery vehicles (control of the diffusion of drugs encapsulated within the pectin structure through the silica layer). The resulting hybrid foam proved to have enhanced stability both in liquid medium and in a typical soil example, which is crucial from the application perspective (see Figure More importantly the material proved efficient in adsorbing a model pollutant (Reactive Black 5 dye) up to about 2 mgdye/gfoam. These values are low compared to materials conventionally used for adsorption processes such as activated carbon 432,433 . The hybrid pectin-silica materials are however not meant to be used as simple adsorption devices, but as host matrix for microorganisms with pollutant-degrading capabilities. The fact that the dye could efficiently diffuse from a model environment and be adsorbed within the porous material is very encouraging regarding the depollution capabilities of this hybrid structure. Introduction Microorganisms [START_REF] Adams | Biostimulation and Bioaugmention: A Review[END_REF][START_REF] Watanabe | Microorganisms relevant to bioremediation[END_REF] such as yeast [START_REF] Wang | Biosorption of heavy metals by Saccharomyces cerevisiae: A review[END_REF] , but also and more significantly bacteria [START_REF] Saratale | Bacterial decolorization and degradation of azo dyes: A review[END_REF][START_REF] Lu | Bacteria-mediated PAH degradation in soil and sediment[END_REF] , have been successfully used for the degradation of various types of pollutants [START_REF] Anjum | Environmental Protection Strategies for Sustainability[END_REF][START_REF] Haritash | Biodegradation aspects of polycyclic aromatic hydrocarbons (PAHs): a review[END_REF][START_REF] Valls | Exploiting the genetic and biochemical capacities of bacteria for the remediation of heavy metal pollution[END_REF] . One of the main limitations to the use of microorganisms in soils is their sensitivity to various parameters such as temperature, pH or ionic strength, which are much more difficult to control in soil than in a liquid suspension. In addition introduction and dispersion of exogenous microorganisms in a given ecosystem may result in significant disturbance of the local biodiversity. Immobilization and more specifically encapsulation of the considered microorganisms within a solid have been contemplated as efficient ways to limit these possible drawbacks. Such matrix must be designed as a way to allow and facilitate diffusion and mass transport of the targeted contaminants while preventing leaching of the exogenous microorganisms within the soil (for instance by designing appropriate macro, meso and micro porosities). As a result the encapsulation process should allow for efficient entrapment of the living cells, preservation of their metabolic activity from the functional point of view, but also for shaping of the matrix itself from the structural point of view. Freeze-casting has proven to be an efficient way to elaborate macroporous polymer-based materials and to gain control over their oriented porous structure. Freeze-casting of a suspension of microorganisms in the same biopolymer could be used as an encapsulation process to prepare cellularized porous bioactive materials. Ice-templating parameters such as the freezing rate are usually considered as levers from the structural point of view, for instance to control the final pore size. In this case they could also prove valuable in order to control physical entrapment and viability of functional units within the porous material. In order to evaluate the feasibility of cell encapsulation in the previously described macroporous pectin foams, Saccharomyces cerevisiae was used as a model. This proof of concept was then extended by using Pseudomonas aeruginosa, which is a more relevant model from the point of view of bioremediation. IV.1 Saccharomyces cerevisiae as a proof of concept IV.1.a About Saccharomyces cerevisiae Saccharomyces cerevisiae is part of everyday life for many people, since it is commonly used in the bakery and brewing industry. Besides these applications, S. cerevisiae is also part of the life of many microbiologists. It is often referred as the eukaryotic equivalent to Escherichia coli 434 , being widely used as cell model. This yeast has the advantage of growing in various media, both in aerobic and anaerobic conditions, with rapid doubling time (90 min in rich medium) 435 . A common culture medium is yeast peptone dextrose (YPD), but cells suspensions can also be obtained by simple rehydration of lyophilized cells in water. In order to evaluate the viability of yeasts in suspension several techniques can be used. Plate counting on LB-agar gel is an option, though not very commonly used for yeasts. Metabolic activity assays are generally preferred, though the information obtained by that way is not exactly the same. Indeed the notion of cell viability can be difficult to define since some cells may still have a metabolic activity without being capable of replication (viable but nonculturable state) 436 . It is for instance possible to take advantage of the degrading capabilities of S. cerevisiae toward glucose, which is key in the brewing industry 437 . As a result, titration of the evolution of the glucose concentration, for instance by mean of a hexokinase assay kit, may provide information on the metabolic activity of the cells. Another possible way to investigate the metabolic activity of S. cerevisiae is to take advantage of its enzymatic reducing properties. Methylene blue has been used to monitor cell metabolic activity thanks to the Methylene Blue dye Reduction Test (MBRT). It has been used both for prokaryotic 438 and eukaryotic 439 cells. Under oxidizing conditions, methylene blue exhibits a characteristic blue color, but its reduced form is colorless, which makes it an easy-to-use visual probe for reducing activity Material and methods S. cerevisiae type II was obtained from Sigma-Aldrich under dry form. Yeasts were rehydrated in water at 35 °C with chosen mass loadings and added to pectin solutions (final pectin concentration is 40 g/L and final yeast concentration ranged from 0 to 133.3 g/L). About 1.8 mL of suspension was poured into 2 mL cryotubes and the samples were plunged into liquid nitrogen for 5 minutes. The samples were then vacuum-dried overnight. Dry foams similar to those, obtained in absence of cells were prepared as controls. For SEM observation, transversal slices were cut with a scalpel and sputtered with 20 nm of gold. It is also possible to include higher cell contents in the initial suspension. This still results in a self-supporting macroporous material after drying, but the pores morphology is modified (see Figure Assuming a similar density for the pectin powder and dry cells, an initial 33.3 g/L of cell content and 40 g/L of pectin concentration would account for a cell volume percentage of about 45% of the total dry wall volume. For a higher initial cell content (133.3 g/L) the cell volume fraction goes up to 77% of the pore walls. The cellularized materials with highest cell loadings could therefore be compared to ice-templated ceramic materials where solid particles can be structured into porous materials. The structural integrity can be obtained through sintering (which is not applicable in the case of heat sensitive cells) or by using a polymer as a binder. In this case the cells could be considered as soft and deformable particle and pectin as the binder. For the targeted application, physical encapsulation of the cell is however not sufficient. The metabolic activity of the entrapped microorganism must be preserved in order to produce a depolluting device. Material and methods Yeast-containing foams were prepared as previously described and dispersed in water. Methylene blue was added. Volumes were adjusted to yield 33.3 g/l of yeast, 40 g/L in pectin Even though discoloration is slightly quicker with suspensions containing fresh yeasts, the samples obtained from the yeast-containing foams are efficiently discolored after 72h. This suggests a drop in the amount of metabolic active cells due to the freezing and drying but a portion of the cells seem to remain active. One must however be careful in analyzing these results, since the discoloration mechanism is enzymatic. As a result enzymes may be trapped in the foams even though cells are dead, resulting in discoloration of the methylene blue. These result should therefore be correlated with other analytic methods, for instance fluorescence microscopy to confirm the cell viability. S. cerevisiae is a common laboratory model since it is an easily cultivated, non-pathogenic, easy-to-observe and robust microorganism. It has been used for biosorption in bioremediation processes. It is however not commonly used for biodegradation, contrary to bacteria. Encapsulation of cells in ice-templated pectin matrices was therefore extended to P. aeruginosa. Such an organism is more relevant from a bioremediation point of view, but represents new challenges regarding encapsulation due to its sensitivity. IV.2 Encapsulation of Pseudomonas aeruginosa in freeze-casted pectin foams IV.2.a About Pseudomonas aeruginosa Pseudomonas aeruginosa is gram-negative bacteria, capable of growing various environments (including water, soil, sediments) or hosts (plants, animals and humans). As a result of this adaptability, P. aeruginosa can be grown in a wide variety of conditions, in different media and at various temperatures. A commonly used growth medium is LB broth at 37°C 441 , but this species can also be grow in minimal medium, in order to pinpoint the effect of specific carbon or nitrogen sources for instance. The versatility of Pseudomonas genus in general, and of P. aeruginosa in particular, regarding nutrient sources make them ideal candidates for bioremediation applications. These bacteria have been used for the degradation of a wide range of organic compounds (see Table IV.1). In brief, P. aeruginosa is a bacteria commonly used in microbiology laboratories. It is easy to handle due to its adaptability to various growth conditions and this same adaptability results in the common use of these bacteria in different bioremediation processes. Both these advantages make P. aeruginosa a good candidate for encapsulation in a soil depollution device. IV.2.a.i Growth curves Before any encapsulation attempt can be made, it is crucial to precisely characterize the growth of the specific considered strain. Growth of P. aeruginosa was monitored in LB medium at 30°C to identify its different growth stages. Figure IV.4 presents the evolution of OD for a culture of P. aeruginosa in LB medium at 30°C and 150 rpm. A lag time of about 2h can be observed. The bacteria then enter the exponential growth phase. After about 12 h the OD reaches a plateau around 2, which corresponds to the stationary phase of the bacterial culture. OD was correlated to the number of colony forming units (CFU) in the suspension. Similar cell concentrations were observed at 9h30 and 24h despite different OD. This might be explained by the presence non culturable cells in in the stationary phase suspension which may contribute to the OD. Once culture conditions for P. aeruginosa were set, assays could be designed towards the encapsulation in the hybrid macroporous foams. Material and methods IV.2.a.ii Viability in pectin solutions Before freezing, it is necessary to obtain a stable suspension of bacteria in the polymer solution. For practical and efficiency reasons, bacteria were not cultivated directly in the polymer solution, but rather grown separately and added to a pre-mixed polymer solution. Material and methods P. aeruginosa was cultivated as previously described in 75 cm² culture flasks at 30°C and 150 rpm, from a 1/50 dilution in LB medium of a pre-culture, itself inoculated from a single colony of P. aeruginosa. In typical experiments, culture was maintained for 5h until 0.5 OD was reached. The 60 mL of culture medium were then centrifuged for 10 min at 5000 rpm and 20°C. The bacteria were then suspended in 3 mL of 0.22 µm filtered water (one twentieth of the original volume). The 3 mL of concentrated suspension were then added to 12 mL of a 50 g/L pectin solution. Solutions of pectin were prepared in advance at 50 g/L to accommodate the addition of a small volume of cell suspension (one fifth of the final volume) and result in a final concentration of 40 g/L. Typically 12 mL of 50 g/L pectin solution were prepared and 3 ml of cell suspension were added to yield 15 mL of bacterial suspension in 40 g/L of pectin. Solutions were prepared by adding a chosen mass of dry pectin powder in 0.22 µm filtered water. The mixture was left 24h at room temperature under elliptic agitation, but with not magnetic stirring. Alternatively, pectin was dispersed in a 125 mM piperazine-N,N′-bis (2-ethanesulfonic acid) buffer (hereafter called PIPES) to stabilize pH closer to 7.. This yielded after addition of the cell suspensions a solution at 40 g/L in pectin and 100 mM in PIPES. The PIPES buffer solution was prepared by dissolving the desired amount of PIPES in a small volume of deionized water (typically 3.78 g for 100 mL of final solution), adjusting pH to 7 with 1M NaOH and adding water to reach desired volume and concentration (typically 125 mM). In order to evaluate the cytocompatibility of pectin solutions towards P. aeruginosa, the number of colony forming units (CFU/mL) was compared for cells dispersed in water, in a 40g/l solution of pectin, in a 100 mM aqueous solution of PIPES and in a solution containing both 40g/L of pectin and 100 mM of PIPES. Plate counting was performed after 5 min and 24 h of contact (solutions were kept at 4°C in meantime) (see Figure IV.5). The pH of the solutions was measured in all solutions and was stable over 24 hours (pH = 5). A significant difference between the pectin samples with and without PIPES buffer could be observed. The drop in the number of CFU in pectin solutions might be attributed to the low pH of the pectin solution (around pH 3), which is due to the acidic properties of the pectin, including carboxylic acid residues. The addition of PIPES buffer at 100 mM stabilizes the pH around 5. Increase of pH up to 7 might be beneficial for cell survival, but it would require addition of more concentrated buffer solutions. High solute contents may be problematic due to the sensitivity of bacteria to high osmotic pressures. It is all the more concerning, since the solutes introduced in the suspension tend to be concentrated by the freezing step of the shaping process. Since no significant changes in viable cell concentrations were monitored between the suspension in pH 5 pectin solution and pH 7 PIPES solution, compared to controls in water and aqueous solution of PIPES, the 100 mM buffer concentration was used in all further experiments. As a result slight modification in the composition of the initial solution was efficient in preserving the viability of the cells in suspension. It was however mentioned in Chapter II, that a modification of the formulation of the initial solution has a direct influence on the morphology of the encapsulating matrix. Samples with no cells but in presence of PIPES buffer were prepared for SEM observation. IV.2). In presence of PIPES, the pore width almost doubles. The pore length is however smaller, which results in a large change in the pore aspect ratio. Another significant change is noticeable in the pore wall thickness. The addition of the solute at high concentration significantly modifies the pore morphology, which may be due to interaction with ice crystals thus modifying ice growth 273 . A 100 mM concentration in PIPES corresponds to a mass concentration about 30 g/L, which must be compared to the 40 g/L concentration in pectin. Added solutes may indeed modify the ice growth by adsorbing preferentially on specific crystallographic surfaces and hinder the growth in one direction. But the PIPES also modify the pH of the initial solution, which changes from about 3 to 5. This has a direct influence on the pectin chains themselves, and is especially likely to influence the carboxylic acid functions which may be at least partially deprotonated. As a result the interaction between the pectin chains could be modified which may result in variations in the physico-chemical properties of the initial solution such as viscosity. As a consequence the ice growth during the freezing step can be modified, resulting in changes in the pore morphology, but also the pectin wall density and thickness. A stable bacteria suspension in pectin is only the first step toward the encapsulation of living organisms in macroporous foams. This suspension must then be shaped, by the use of icetemplating as previously described. The resulting material must then be coated by silica in order to obtain a suitable material for the targeted use in soils. IV.2.b Encapsulation of Pseudomonas aeruginosa in freeze-casted pectin foams Bacteria cells can be entrapped in macroporous pectin foams by freezing of a cells suspension in 40 g/L of pectin. The ice growth results in the formation of pores after vacuum drying. The bacteria cells are incorporated within the cell walls during the formation of the ice crystals. This encapsulation is only possible due to a combination of various factors. Phenomena of particles sedimentation, rejection or entrapment have mainly been studied regarding solid ceramic particles. Such consideration can however be adapted and extended to soft particles such as bubbles or cells 333 . The fate of a particle in front of a freezing front is influenced by a wide variety of parameters. The first risk is sedimentation in the initial suspension. In the case of cells, such phenomenon is prevented by the fact that the cell density is close to the density of the suspending medium (the polymer solution). During the progression of the ice front, the fate of the particles (or cells) is mainly governed by two phenomena : the repulsive force due the different interactions energies between the particle, the solid and the liquid and the viscous drag exerted by the liquid displaced by the advancing front that pushes the particle towards the ice front 257 . As a result any parameter modifying one of these two aspects is likely to influence the encapsulation efficiency. For instance the chemical nature of the particle or the presence of additives in the liquid phase may significantly modify the surface interactions. On the other hand modifications of the solution viscosity or of the speed of the ice-front are likely to modify the regime and characteristics of the flow around the particles. In the case of cell encapsulation, the particles (i.e. the cells) are about 1 µm in size and are dispersed in a pectin solution at 40 g/L, which has a viscosity around 1 Pa.s. These conditions are compatible with the immobilization of the cells directly inside the pectin walls. The shape of the cells under the pectin layer may be observed in SEM (see Figure IV.7), providing that the pore walls are thin enough. In some instances, the bacteria may be directly seen within the pore wall section, depending on the way the samples are cut. As pointed out earlier, physical encapsulation of the cell is essential but in order to be efficient, the shaping process must be compatible with the survival of the bacteria. IV.2.b.i Influence of the freezing-rate The ice-templating shaping process involves freezing of living cells. If freezing of living cells as a mean of encapsulation in a polymeric matrix has not been widely documented, there are extensive studies both regarding freezing of cells for preservation in the frozen state and freeze-drying of cells for preservation in a dry state 335 (see I. 3.b , p45). Two main deleterious effects of freezing on living cells have been identified. Formation of intracellular ice may result in disruption or destruction of the cellular membranes. The formation of extracellular ice is usually deemed as less deleterious to the cells 348 , but may however induce significant mechanical stress on the cells 334 . The formation of extracellular ice also results in the concentration of the extracellular solutes, which in turn may induce dehydration of the cells. This may result in strong shrinking and deformation of the cells, but also in a rise of intracellular solutes to toxic levels. Common cryopreservation strategies include the use of cryoprotectants such as glycerol and control of the applied cooling rates. In both cases the goal is to adjust the osmotic balance between the intra and extracellular medium in order to prevent formation of intracellular ice while maintaining non-toxic solutes concentrations. Introduction of molecules (such as glycerol) capable of penetrating the cell membrane may indeed help to balance the rise in external solute concentration. Influence of the freezing rate has also proven to be crucial. High cooling rate limit the diffusion of water through the membrane and may result in supercooling of the internal wall which is responsible for the formation of intracellular ice. On the other hand if cooling rates are too low, water permeation may result in high intracellular solutes concentration which could be damaging to the cells 352 . As a result an optimal cooling rate can usually be found to maximize cell survival. Such optimum is however cell dependent since it may be influenced by different types of membrane or intracellular solutes concentrations. It is also likely to be modified by the composition of the freezing medium 348 . a) b) In this work however the freezing conditions are quite different from conventional cryopreservation. It must first be pointed out that no common cryopreservative is used during freezing. The freezing medium however contain a polysaccharide, which may act in a similar way as non-penetrating cryoprotective agents by tuning the formation of extracellular ice 346 . The presence of the polymer as an encapsulating layer may also provide protection against mechanical solicitation during freezing. In addition to the stress of the freezing step, the encapsulated cells are also exposed to high vacuum (0.05 mbar) during lyophilization. A few studies 360 have been published regarding the freeze-drying of microorganisms, but generally they are freeze-dried from concentrated cell suspensions in order to obtain a powder for cell storage and not for encapsulation in a matrix, which may induce significantly behavior, particularly in terms of mechanical solicitation. In order to assess and optimize survival rates during encapsulation in pectin through freezecasting, P. aeruginosa cells were frozen at various speeds and the numbers of CFU were compared. Material and methods P. aeruginosa was cultivated in LB medium up to 0.5 OD (about mid exponential phase) at 150 rpm and 30°C. The cell culture (typically 60 mL) was centrifuged 10 min at 5000 rpm and dispersed in one twentieth of the initial culture volume in water (typically 3 mL). This suspension was introduced in a pectin and PIPES solution to yield a 40 g/L concentration in pectin, 100 mM in PIPES, and a cell concentration equivalent to 4 times the initial culture (about 2.10 8 CFU/mL). The suspension was then frozen and subsequently vacuum dried. Part of the samples were directly thawed after freezing (30 mins at 30°C) in order to dissociate the effect of the freezing step from the drying step. Cooling rates of 1°C/min, 5°C/min and 10°C/min were applied by freeze-casting. As a comparison point, samples were prepared by direct dipping in liquid nitrogen to yield a 250°C/min cooling rate. The survival rates were compared between the samples frozen and thawed and the sample frozen and dried overnight. Thawed samples were diluted and spread on LB-agar plates and incubated at 37°C for 24h. Dry samples were dispersed in five times their volume of water and successive dilutions were prepared. Three samples were prepared for each cooling-rate and each dilution was plated in triplicate. The initial number of cells introduced was measured and slight variations were measured (between 1.10 8 and 1.10 9 CFU/mL). To facilitate comparison between the cooling rates the number of CFU in frozen and thawed samples, and frozen and dried samples, the number of surviving cells were normalized to an initial number of cells of 1.10 9 CFU/mL. Figure IV.8 presents the number of CFU in samples frozen at different cooling rates. The number of CFU is only one type of indication regarding the viability of the encapsulated bacteria. The notion of survival regarding single cell organisms is complex 436 , since some cells may be incapable of multiplying while retaining a metabolic activity. Other complementary technique may be applied to investigate the viability of encapsulated cells, but plate counting remains a simple and widely applicable technique which will further be used as reference. As a result, the expressions survival rate or viability will be used as the number of CFU. The samples obtained at 1°C/min, 5°C/min and 10°C/min were all processed by freezecasting, with the same setup and are therefore strictly comparable. The device used is however unable to yield higher cooling rates and in order to assess the effect of very high cooling rates (around 250°C/min) samples were prepared by dipping in liquid nitrogen. These samples are therefore only an indication of the survival rates at theses freezing rates, but it must be kept in mind that the geometries are not strictly comparable, which might also have an influence on the survival of the cells since the mechanical constraints are different. Samples frozen and thawed were compared to samples frozen and dried, as way to dissociate the effects of freezing and drying. However it is impossible to fully separate these two effects, or rather, it is impossible to assess the effects of freezing alone, since samples must necessarily be thawed before counting the number of CFU. As a result it is only possible to compare the combined effects of freezing and thawing to the combined effects of freezing, drying and rehydration. Figure IV.8: Freezing and thawing of the sample result in higher survival rates than freezing and subsequent vacuum drying. Optimal cell survival is obtained for a 5°C/min freezing-rate, which is slow enough to allow cell dehydration and prevent formation of intracellular ice, but fast enough to prevent limit high solutes concentration. Freezing and thawing results in viability losses between 1 and 3 logarithmic units. Survival rates are similar at 5°C/min and 10°C/min (drop of 1 logarithmic unit). At 1°C/min a loss about 2 logarithmic units can be observed. At very high cooling rate, the lowest survival rate was observed (3 logarithmic units). As a result it seems that intermediary cooling rates yield higher survival rate which is in agreement with observations performed on cells suspension. It is however difficult to predict this optimal cooling rate, since there are large variations depending on the type of cell 352 .The survival rate after freezing and thawing represents only an indication of the influence of the freezing rate, but the most relevant information is the number of viable cells encapsulated in the dry macroporous matrix. Regardless of the cooling rate, freezing and drying resulted in lower survival rates than freezing and thawing. After vacuum drying, the effects of the freezing rate are even more noticeable. Here again, the viability loss cannot be strictly attributed to the freezing and drying steps, but also to the rehydration conditions. It has indeed been shown that the medium used for cell rehydration has a significant influence on survival rates 365 . At 1°C/min about 10 5 cells are capable of forming colonies per milliliter of the initial suspension; at 5°C/min this value increases to 10 7 CFU/mL, but decreases to 10 6 CFU/mL at 10°C/min and even down to less than 10 3 CFU/mL at very high cooling rate. After drying, the presence of an optimal freezing rate is clearly visible. The highest number of CFU is observed for freezing at 5°C/min. Even if the drying conditions are strictly identical, the prior freezing conditions have an influence on the survival during the drying phase. As the temperature rises again, presence of internal ice created during freezing at high cooling rate may induce further cell damage. Another source of possible damages is the fact that cells are subjected to mechanical constrains, both during freezing 334 and under a 0.05 mbar vacuum, which is likely to induce sharp drops in survival rates. The applied vacuum is identical for all freezing-rates, but the cells might have different sensitivities due to the different freezing conditions. In addition, freezing rates have an influence of the pore walls themselves, including their thickness. These structural variations may have a direct influence on the cell survival since the pectin wall provides a protection against mechanical damage. In all further encapsulations assays, cooling rate was set at 5°C/min even though changing other parameter, such as the bacterial growth phase, may in turn slightly shift the optimal cooling rate due to possible changes in the cells properties. It was however assumed that this variation was limited. Once this critical processing parameter was set, the state of the cells themselves was modified to evaluate the growth conditions best suited to the encapsulation process. IV.2.b.ii Influence of the cell growth phase and concentration The physiological state of the cells (and therefore the growth phase considered) has a significant influence on their resilience to the encapsulation process. The effect of the growth phase must however be distinguished from the influence of the initial cell concentration. In order to assess the influence of these two different parameters, cells were grown up to exponential and stationary phase and concentrated or diluted to different levels prior to encapsulation. Material and methods A culture of P. aeruginosa was grown in LB medium at 150 rpm and 30°C. Bacteria were cultivated either for 5 h (up to a 0.5 OD, exponential growth phase) or 24 h (up to a 2 OD, stationary growth phase). The culture media were centrifuged and dispersed in water in order to yield suspensions at two distinct concentrations for each growth phase. The cells were then suspended in previously prepared pectin and PIPES solutions (final concentrations of 40 g/L and 100 mM respectively). 3 mL of each bacteria suspension in pectin were then freeze-casted at 5°C/min and vacuum dried at 0.05 mbar for 24 h. The dry foams were then dispersed in five times their volume of water, and successive dilutions were prepared for plate counting. Three samples were prepared for each growth stage and concentration, and each dilution was plated in triplicate. Figure IV.9 shows survival rates for encapsulated P. aeruginosa at different growth stages and different initial concentrations. At similar initial concentration (5.10 8 CFU/mL), cells encapsulated in stationary phase (Figure IV.9 c) have higher survival rates (5.10 7 CFU/mL) than cells encapsulated in exponential growth phase (5.10 4 CFU/mL, see Figure IV.9 a). This observation is consistent with previous studies performed on other bacterial and yeast species 347,464 . In order to ensure the highest efficiency for the final biomaterial, high initial cell concentrations may be of help. Cell cultures were concentrated to various levels to investigate the influence of cell concentration on survival. When larger concentrations of cell were introduced in the pectin solution, the final number of CFU (after freeze-casting and drying) were actually similar or lower than sample prepared from lower initial concentrations (see Figure IV.9 b compared to a and d compared to c). As a result introduction of less cells in the initial suspension actually yield higher rates of viable encapsulated cells. IV.2.b.iii Long term survival For encapsulated cells to be used in soil depolluting applications, the macroporous pectin foams must be coated with silica (see Chapter III). The silica coating provides protection against rapid degradation of the foam. The deposition process however requires contact with acidic atmosphere at 30°C for several days. In order to ensure complete encapsulation in the hybrid matrix, long term survival of the encapsulated cells must be assessed. Material and methods P. aeruginosa-loaded pectin foams were prepared as previously described. Bacteria were cultivated in LB medium and centrifuged, suspended in water and added to a pectin and PIPES solution. 3 mL of suspension were then freeze-casted at 5 °C/min and vacuum dried for 24 h. Cell loaded foams were cut into 4 equal parts and maintained 24 h in various conditions: at 4 °C and room humidity, 30 °C and room humidity and 30°C and a controlled 75 % humidity. One fourth of the foams were dispersed in water directly after drying and plated. Foams were also prepared and dried for 48h instead of 24 h. As observed previously, several logarithmic units are lost in the encapsulation process (see Figure IV.10). After 24 h of drying the number of CFU drops from 10 9 CFU/mL to 10 6 CFU/mL. After only 24 h at 30 °C, regardless of the humidity level, no colonies could be observed after redissolution of the foams and plating. Slightly better results were obtained when the cell-loaded foams were kept at 4°C, but the number of CFU still dropped down to 10 4 CFU/mL after 24 h. However when the samples were left for 24 h supplementary hours under vacuum about 10 6 CFU/mL were counted, which is similar to the cell concentrations after only 24 h of drying. These observations highlight the crucial role of cell hydration in the long term survival. It seems that partial rehydration due to room humidity results in loss of viability, while complete rehydrated (when foams are immediately dispersed in water) present better survival rates. When cells are maintained under vacuum (completely dehydrated state) for 48h, the survival rates are similar to those obtained for shorter lyophilization times, which seems to indicate that once the initial drop in pressure has occurred, cells remain in a stable state. Higher survival rates were obtained when cells were maintained at 4 °C in a conventional fridge. This may be explained by a slowing of the cells metabolism, and maybe by a variation of humidity levels between the interior of the fridge and the 30 °C oven. Regardless of the storage conditions, survival rates in pectin foams at long terms were not satisfactory. This is problematic regarding the global encapsulation strategy which includes silica deposition. The silica deposition method using TEOS vapors previously developed allowed for the formation of a homogeneous and well controlled silica layer on the pectin pore walls. This proved efficient in preventing the foam dissolution. The technique however required prolonged (several weeks) exposition of the foam to TEOS vapors at 30°C, which is not compatible with the survival of the encapsulated bacteria. IV.2.c Influence of the presence of cryoprotective additives A common strategy for optimization of cell survival in the frozen or dry state is the addition of cryoprotective 346 and lyoprotective agents 360 . These types of additive may be of help in the preservation of encapsulated cells over several days. IV.2.c.i Influence on the encapsulation efficiency Glycerol and trehalose were selected as possible protective agents during the freeze-casting and drying process. Influence of the presence of each additive or a combination of both was assessed regarding the survival rate of encapsulated P. aeruginosa, directly after drying and after 24h in a desiccated atmosphere. Glycerol was chosen as a typical cryoprotective agent, while trehalose has been used as protectant for preservation of bacteria during vacuum drying. Even if glycerol is expected to be especially efficient for protection during the freezing and trehalose might have a more significant action regarding lyophilization, it should not be excluded that each of these component may have an effect during the different stages of encapsulation. Additives were used separately and in combination in order to uncover potential synergic effects. Figure IV.12 shows the number of CFU encapsulated cells at different stages of the shaping process. After freezing at 5°C/min and thawing, only a small loss of viability was monitored (less than 1 LU). After freezing and drying however, the presence of glycerol proved efficient in enhancing the cell survival rate. Samples prepared with only pectin or pectin and trehalose (Figure IV.12 a and b) contained both around 7.10 5 CFU/mL, while samples with glycerol or a combination of glycerol and trehalose (Figure IV.12 c and d) were loaded with 4.10 6 CFU/mL and 2.10 6 CFU/mL respectively. The effect was even more noticeable after 24 h of storage. The samples were kept at room temperature in a desiccated atmosphere to prevent partial rehydration of the samples. Sample with no additives presented no detectable viable encapsulated cells after 24h. The presence of trehalose resulted in the survival of 2.10 3 CFU/mL, while addition of glycerol resulted in the presence of 6.10 4 CFU/mL after 24h. The combined effect of trehalose and glycerol yielded a 4.10 5 CFU/mL cell concentration. Material and methods a) b) Figure IV.12: The number of CFU was assessed in various samples in presence of additives at different stages of the shaping process. No significant effect of the additives could be observed after freezing and thawing. After drying, similar survival rates were observed for samples containing no additives (a) and only trehalose (b). In the presence of glycerol the number of CFU was higher (c and d). After 24h at room temperature in a desiccated atmosphere, no cells capable of multiplying were observed in foams only composed of pectin. However the presence of additives greatly enhanced the survival rate, the best results being obtained in presence of both glycerol and trehalose. Values marked with different sets of symbols were compared for statistical significance. Values with the same number and type of symbol are not statistically different (at p<0.05). After freezing and thawing, the addition of common cryoprotectants did not seem to modify the survival rate for encapsulate P. aeruginosa. The presence of pectin may be sufficient to provide protection against the main deleterious effects of freezing. After drying however, the presence of additives greatly enhanced the long term survival. Highest survival rates are observed for samples containing glycerol or glycerol and trehalose. Vacuum drying is usually composed of two separate steps: sublimation of the ice crystals and desorption of bound water. This second step usually requires gentle heating of the samples (up the about 20°C) 364 , which was not performed in this case. In addition, intracellular water of cells immobilized within the pectin matrix is likely to be harder to desorb due to diffusion limitations. As a result, bound water might not be removed after the vacuum drying step. The presence of hydrophilic cryoprotectants (glycerol and trehalose) within the cells may be responsible for the protective effect towards drying 346 , since higher amounts of bound water are retained. This could prove decisive to maintain viable cells within the pectin foams throughout the silica coating process by exposure to vapors of TEOS, which requires long deposition times. IV.2.c.ii Influence on the foam morphology The presence of additives provides protection against viability loss, especially on the long run. These solutes are therefore beneficial to the functional part of the targeted depolluting device. Their influence on the structural part of the device must however not be overlooked, since presence of additives during freeze-casting usually results in morphological changes 272 . Material and methods Foams with various additive contents (no additive, trehalose, glycerol or both glycerol and trehalose) were prepared as previously described, but with no addition of bacteria. Longitudinal and transversal slices were cut and sputtered with 20 nm of gold for SEM observation. Samples were maintained in a desiccated atmosphere between the end of drying and any further use. Addition In both cases, some level of anisotropy can still be observed. Longitudinal slices show a lamellar pore wall arrangement, with well aligned pores. Transversal sections show a change in the pore aspect, which are not elongated but rather rounded. Pores are larger compared to samples with no glycerol, and more polydisperse in size. Foams prepared in presence of glycerol tend to be easily deformed during cutting of the samples. As a result the morphology observed in SEM may be slightly modified compared to bulk foams. In addition, foams containing glycerol tend to partly lose their structure when aged at ambient humidity. The pore walls tend to aggregate, resulting in a loss of porosity. This may be explained by a rapid adsorption of air humidity resulting in the rehydration of pore walls and subsequent loss of mechanical stability. As a result samples were maintained in a desiccated atmosphere until observation to prevent loss of structural features. Trehalose and glycerol are both used as cryoprotective agents. Despite non negligible trehalose contents, no significant effect on the morphology could be seen. This might be attributed to the nature of trehalose which is a disaccharide (see Figure IV.11 a), and may therefore present common chemical properties with pectin which is a polysaccharide. Addition of glycerol however significantly changes the foam morphology. Glycerol presents high water affinity, which may explain the modification of the pore aspect. Pore present no specific elongation, but are rather rounded, which could be cause by interactions with glycerol preventing ice growth in specific directions. The pore walls themselves undergo changes in aspect and apparent mechanical behavior, which may be attributed to the presence of glycerol. Due to the high affinity of water and glycerol, dehydration of the polymer and glycerol mixture may be more difficult both during the formation of ice crystals and the vacuum drying. The resulting pore walls might have higher water content which may explain their tendency toward easy deformation and wrinkled aspect under SEM observation. Addition of trehalose has no significant impact on the structure of the matrix, but used alone it has a limited impact on viability preservation. The best survival rates were observed in presence of glycerol, but this additive completely changed the structure and properties of the encapsulating matrix. Conclusion Encapsulation of cells in pectin foams may at first seem easy, as freezing and subsequent lyophilization of a simple cell suspension in a pectin solution results in the formation of a macroporous foam, with cells clearly visible within the pore walls. But this apparent simplicity uncovers a wide range of crucial parameters in order to yield a viable biohybrid material. Physical entrapment is only possible due to a combination of cells and suspending solution characteristics This encapsulation of cells is however not meant as a structural feature, but rather as the incorporation of functional units. This means that maintaining the cell metabolic activity is crucial. Assays with a robust and easy to handle model such as S. cerevisiae yielded encouraging results. This model was an efficient proof of concept, but regarding the targeted application, that is to say soil depollution, bacterial model are more relevant. Unfortunately, bacteria are also more sensitive. For this reason each step of the encapsulation process must be tailored to ensure optimal survival rate of the encapsulated species, in order to yield the highest possible metabolic rates in the final device. The initial formulation of the suspension media was composed of pectin and PIPES in order to efficiently control the pH and prevent cytotoxic effects after suspension of the cells in the biopolymer solution. During the freezing step, the cooling rate proved to have a significant influence on the survival rate. Damages caused by freezing have been observed in the domain of cell cryopreservation and are commonly attributed to two main effects: the formation of intracellular ice and the increase in solutes concentrations. The cooling rate has an influence on both these phenomena, since slow cooling yields dehydration of the cells, thus limiting the formation of intracellular ice. The counterpart of the dehydration is an elevation of the solute concentrations, sometimes up to toxic levels. As a result, an optimal freezing rate may be found, in order to prevent formation of intracellular ice, while maintaining reasonable solute concentrations. In the case of the encapsulation of P. aeruginosa in a 40 g/L solution of pectin in presence of PIPES the optimal freezing-rate was 5°C/min. But encapsulation in a biopolymer foam is only the first step toward the design of a depolluting device. As previously stated, foams need to be coated with a silica layer in order to withstand a prolonged stay in soil. The silica deposition process in vapor phase requires several days to obtain a coating thick enough to prevent the material degradation. It is therefore necessary to ensure survival at long term in the desiccated state. Unfortunately, cells entrapped in a simple pectin foam were not able to withstand a 24h storage. Addition of a common cryoprotective agent (glycerol) and of trehalose, which is known to preserve cells during lyophiliation, proved effective in enhancing the cell survival after 24h of storage. But this significant gain from the functional point of view was accompanied by major drawbacks from the structural point of view, since the porous structure was altered, and more importantly, foams became sensitive to aging at room humidity. As a result, even though the encapsulation in a biopolymer foam was successful, obtaining a hybrid pectin-silica foam containing living cells of P. aeruginosa proved difficult (see Figure IV.14). The crucial role of water regarding cell survival was however demonstrated. Use of pectin alone resulted in interesting structural features, but poor water binding properties, and therefore in low survival rates at long term. Addition of cryoprotective agents was efficient from a functional point of view since it enhanced cell viability, but resulted in unwanted structural changes. In order to combine, water binding and structural properties, encapsulation was attempted in another biopolymer. Introduction Encapsulation in pectin-based porous materials was achieved by means of ice-templating. Storage of the cell-loaded materials however resulted in cellular death after short periods of time, which was incompatible with further processing of the foams into hybrid materials. The addition of cryoprotective agents was efficient from a functional point of view, since viable cells could be recovered after storage. From a structural point of view however, foams were significantly modified. Both structural and functional aspects are essential for the targeted application. The material must retain its integrity and oriented porosity to ensure mass transport of the pollutants while preventing dissemination of exogenous microorganisms in the soil. However if the encapsulated species are not metabolically active, the device will have limited remediation efficiency. In order to combine long-term cell survival and structural properties, encapsulation in another type of polymer was investigated. Alginate was chosen as an alternative to pectin. It is also a polysaccharide, commonly extracted from brown algae. It has however also been reported as part of bacterial biofilms, including in P. aeruginosa 465 , which could be an indication of possible cytocompatibility. Alginate has the advantage of being easily crosslinked in presence of divalent cations, which has made this biopolymer a very popular option for many encapsulation approaches [START_REF] Lim | Microencapsulated Islets as Bioartificial Endocrine Pancreas[END_REF]181 . Crosslinking of freeze-casted alginate foams may be a great advantage for depositionof silica through aqueous solution while preventing dissolution of the alginate structure. The optimization of the encapsulation in a biopolymer foam and subsequent silica deposition should result in a bionanocomposite where both the structure of the matrix and the encapsulated functional units are supporting the depolluting activity of the device. Such activity must be assessed on a model of pollutant. Investigations in the domain of bioremediation have highlighted a wide range of pollutants susceptible of being degraded by given microorganism species and specific strains [START_REF] Adams | Biostimulation and Bioaugmention: A Review[END_REF]466 . Application of the device for remediation of actual polluted sites would require in depth analysis and characterization of the site to determine the best suited organism or consortium for optimal remediation 467 . In this work the approach is however reversed since one specific microorganism was chosen to assess feasibility and efficiency of encapsulation. A model pollutant should therefore be chosen according to the encapsulated microorganism. The first and most crucial criteria to choose such model should be the capacity of P. aeruginosa to degrade it, but other parameters such as the methods of detection, availability or solubility of the pollutant for instance, should be taken in account. This pollutant is meant to work as a proof of concept for the targeted application. It should therefore be relevant in the context of bioremediation, but it must be remembered that laboratory conditions cannot fully represent natural polluted sites, and some experimental conditions may be chosen as a way to highlight remediation phenomena rather than to fully reproduce field conditions. V.1 Alginate as an alternative biopolymer V.1.a About alginate Alginate is a polysaccharide usually extracted from brown algae (Phaeophyceae) 175 , but a few bacterial species have been found to produce alginate polymers, including P. aeruginosa 468,469 . The vast majority of world production is used in food industry as stabilizers, emulsifiers and thickening or gelling agents 470 . However alginate has also been widely used in biomedical and pharmaceutical application for encapsulation of a wide range of molecules and biological species from drugs and enzymes to whole cells 125,179,184,188,419,[471][472][473] . The high biocompatibility of alginate-based materials also make them ideal biomaterials 156,474 . It As is the case for many biosourced polymers, the structure of the alginate may vary considerably depending on the source and extraction conditions. This results in a wide range of possible physical and chemical properties. One of the most interesting and exploited properties of alginate is its capability to form gels in presence of divalent cations. The commonly accepted model regarding the ion binding properties of alginate is the so called "eggbox" structure proposed in 1973 183 (see the formation beads or capsules by crosslinking of alginate in a CaCl2 bath 247 . But alginate gels have also been used as films or coating [476][477][478] , sponges 479 and even fibers 480 , sometimes in combination with other polymer. As mentioned earlier, use of alginate is very widely spread for many applications. Depending on the targeted application some of its properties, including the mechanical behavior or rheological properties, may not be entirely adapted. For this reason alginate has been associated with all sorts of materials from other natural 366,[481][482][483][484] or synthetic polymers 485 to various inorganic compounds 228,229,238,486 . If gelling properties of alginate are extensively exploited for processing of this biopolymer, the material can also be shaped without gelation. Alginate has for instance been shaped thanks to ice templating processes in order to form porous materials 234,297,322 . V.1.b Freeze-casted alginate foams Similarly to the approach used for the design of pectin-based foams, the structural aspects of alginate-based materials were first explored. Some parameters were however directly set according to the observations made in the case of pectin. Polymer concentration was set at 40 g/L and the materials were shaped exclusively using freeze-casting. Material and methods A solution at 40 g/L was prepared by dissolution of alginic acid sodium salt (15-25 cP at 1% in water, procured by Sigma-Aldrich) in deionized water. The viscous solution was then freeze-casted at different cooling rates. The resulting samples were vacuum dried at 0.05 mbar for 24h. Cross sections and longitudinal slices were sputtered with 20 nm of gold for SEM observation. Freeze-casted alginate foams appeared very similar at the macroscopic scale to pectin-based foams (see Figure V.2). These similarities can also be found in the morphology of the pores (see Figure V.3). As is the case for pectin (see Chapter II), freeze-casting of an aqueous alginate solution results in the formation of a macroporous material, with oriented and well aligned pores. Such similarities a) Pectin b) Alginate may be expected from the fact that both pectin and alginate are polysaccharide, with close chemical structures and functions. In both cases, polymers were used as 4 wt% aqueous solutions and the freezing setup and freezing rate were the same. The pore width is of the same order of magnitude as for pectin foams (mainly between 10 µm and 50 µm). Like in pectin foams, the pore morphology can be modified thanks to processing parameters such as the freezing rate. As can be seen from Figure V.4, higher cooling rates result in smaller pores in alginate foams. This variation is consistent with observations made on pectin foams (see Chapter II). In the case of alginate, the pores obtained at low freezingrates also appear less ordered. In other terms, the orientation domains observed in cross sections seem less extended. As expected, alginate-based freeze-casted materials strongly resemble pectin-based foams. This confirms that alginate may be used as a suitable replacement for pectin in the design of a bionanocomposite for soil depollution. V.1.c Encapsulation of Saccharomyces cerevisiae Prior to encapsulation of P. aeruginosa, alginate was used as a matrix for encapsulation of S. cerevisiae 187 . The large size of the yeast cells allows for easy observation in microscopy and the robustness of the species makes for a good first step in the evaluation the functional aspects of encapsulation in alginate foams. Material and methods S. cerevisiae Type II was procured in the dry state from Sigma-Aldrich. Cells were dispersed in PBS (typically 1 mL) and added to Yeast extract Peptone Dextrose (YPD) medium (typically 20 mL). S. cerevisiae was grown 24h at 32°C under static conditions. The culture medium was then centrifuged (5 min at 1000 rpm) and the pellets were dispersed in PBS (typically 2 mL) and added to a 40 g/L alginate solution. The suspension was the freezecasted at chosen cooling rates (1°C/min, 2°C/min and 5°C/min). Viability was monitored by means of a glucose hexokinase assay kit. Briefly dry foams or thawed samples were dispersed in YPD and incubated 48h under static conditions at 32°C. Glucose concentration was then measured by means of glucose hexokinase assay kit and UVvis spectroscopy. Positive controls were performed for suspensions of S. cerevisiae in PBS, alginate solution and negative control with blank alginate foams. Each assay was made in triplicate. From the structural point of view, S cerevisiae cells were easily entrapped in freeze-casted alginate foams (see Figure V.5). The general structure of the foam was not altered by introduction of the cells, but at higher magnification, cells can clearly be seen within the polymer pore walls. The structural integrity of the cells themselves seems to be efficiently preserved. The cell viability was not monitored by plate counting as is common for bacteria, but rather by a metabolic assay. S. cerevisiae is capable of metabolizing glucose as is the case the fermentation process of beer 437 . As a result monitoring glucose content is indicative of cellular activity. Initial glucose concentration in YPD medium was 20 g/L. The negative control indicates that this concentration remains stable after 48 h of incubation. Positive controls for cells suspensions in PBS and alginate are not significantly different (2.6 g/L and 2.5 g/L respectively which represent degradation of 87% of the initial glucose content). This illustrates the fact that alginate presents no cytotoxic effects towards S. cerevisiae. Glucose concentrations, after incubation of cells dispersed from freeze-casted and vacuum dried foams, are higher than the ones obtained with positive controls, but are lower than the initial 20 g/L, which denotes significant metabolic activity after encapsulation in alginate. Samples prepared at 1 °C/min, 2 °C/min and 5 °C/min cooling rates resulting in degradation of 62 %, 53 % and 72 % of the initial glucose content respectively. However due to large standard deviations on the triplicate assays, these values are not statistically different from each other. This may indicate that freezing-rates in the considered range, which remains limited, have no significant influence on viability of encapsulated S. cerevisiae in alginate. Encapsulation of S. cerevisiae confirmed that alginate could be used as a suitable material for encapsulation of microorganisms in a self-supporting, no cytotoxic, porous matrix. Encapsulation of microorganisms more relevant from the bioremediation point of view was then investigated, using P. aeruginosa as a model. V.1.d Encapsulation of Pseudomonas aeruginosa P. aeruginosa was successfully encapsulated in pectin biopolymer, however no viable encapsulated cell could be observed after 24 h of storage (see Chapter IV). Addition of common cryoprotective agents was efficient in preserving the viability of encapsulated cells, but the structural features of the matrix were greatly degraded. In order to prevent these issues encapsulation efficiency was assessed in different polymers. V.1.d.i Encapsulation in various polymers Material and methods P. aeruginosa was cultivated as previously described. One colony was pre-cultivated in 10 ml of LB medium 24h at 30°C and 150 rpm. This pre-culture was then diluted by a factor 50 in LB medium and cultivated for 5h30 up to a 0.5 OD. The culture medium was centrifuged and pellets were dispersed in water. This suspension was then added to various polymer solutions in presence PIPES buffer. The final concentrations were 40 g/l in polymer and 100 mM in PIPES buffer. The polymer solutions were prepared in advance by dispersion of a chosen mass in 125mM aqueous solution of PIPES buffer and magnetic stirring overnight at room temperature (except for gelatin which was heated at 35°C). The following polymers were used: beet root pectin, sodium alginate, bovine gelatin and PVA Solutions were maintained at 35°C to prevent thickening and gelling of the gelatin solution. 3 mL of cell suspension were freeze-casted at 5°C/min and vacuum dried for 24h. The resulting dry foams were immediately dispersed in water and successive dilutions were prepared and plated on LB-agar gels. The plates were incubated 24h at 37°C and the number of developed colonies was counted. Three foams were prepared for each polymer and each dilution was plated in triplicate. Longitudinal and transversal slices were prepared and sputtered with 20 nm of gold for SEM observation. Different polymers were chosen as possible alternatives to beet root pectin. Sodium alginate, like pectin, is a polysaccharide. Its structure and reactive functions are however different from pectin since it is composed of mannuronic acid and guluronic acid (while pectin is mainly composed of galacturonic acid and rhamnose). Gelatin was chose as another bio-sourced polymer, however it is not a polysaccharide like pectin and alginate, but a protein derived from collagen. PVA was chosen as a synthetic polymer, with known properties of biodegradability 487,488 and biocompatibility 489,490 . Solutions of these polymers at a concentration of 40 g/L in presence of 100 mM of PIPES were prepared and freeze-casted at the same rate (5°C/min) and subsequently lyophilized. Regardless of the polymer, selfsupporting macroporous materials were obtained, but the pore morphology varied depending on the polymer as can be seen in Figure V.7. Morphology of beet root pectin based sample was described in Chapter II. Pores are oriented according to the applied temperature gradient. Pore walls have a lamellar aspect and are organized in orientation domains. Samples obtained from an alginate solution have a very similar morphology, but the pores are much shorter, or in other words more transversal bridges may be observed between the lamellar pore walls. All these samples were obtained in solutions containing PIPES buffer, which is likely to modify the ice growth. Non-polysaccharide based foams displayed different morphologies. Materials obtained from bovine gelatin also displayed some level of anisotropy, however the pore alignment and order was greatly diminished compared to pectin or alginate based materials. Some higher level of Material and method 10 9 CFU/mL were introduced in initial solutions of pectin and alginate matrices, in presence of PIPES buffer to prepared cellularized biopolymer foams as previously described. Cell were cultivated up to a 0.5 OD, centrifuged and dispersed in solution of 40 g/L in polymer and 10 mM in PIPES buffer. 3 mL of the suspensions were freeze-dried at 5 °C/min and vacuum dried. The cell suspensions in polymers were maintained 24 h at 4 °C to assess possible cytotoxicity. After 24 h of lyophilization the samples were cut in two along the ice growth direction. One half was immediately dispersed in water, diluted and plated. The second half was maintained 24 h at room temperature in a desiccated atmosphere. The stored samples were then dispersed in water and plated. All LB-agar plates were incubated 24 h at room temperature before counting the number of colonies. Cells were encapsulated in three separates samples of pectin and alginate and each dilution was plated in triplicate. The number of CFU/mL was similar in both pectin and alginate solution (with PIPES buffer) after 24 h of contact, which confirms that neither polymer have a cytotoxic effect at the considered pH. After encapsulation by freeze-drying at 5°C/min and vacuum drying, the number of viable cells encapsulated in the alginate matrix is significantly superior to the number of CFU/mL in pectin (6.10 6 CFU/mL and 2.10 6 CFU/mL respectively). The difference was even more noticeable after 24 h of storage under a dry atmosphere and room temperature. In pectin only about 500 CFU/mL were counted, with high variability between the triplicates, while alginate foam still contained around 4.10 5 CFU/mL. Figure V.9: Survival rates are similar for P. aeruginosa after 24h in suspension in solutions of 40 g/L of alginate or pectin in presence of 100 mM of PIPES buffer (b). After encapsulation in dry foams (c) survival was slightly higher when alginate was used as the encapsulating matrix. After 24h of storage in a desiccated atmosphere, the survival rate was very low in pectin (d). In alginate however, as much as 4.10 5 CFU/mL were still encapsulated after 24h of storage. Values marked by the same number of symbols are not significantly different (at p < 0.05). * * A comparison of the survival rates in pectin and alginate highlighted the interest of the latter polymer for long term encapsulation, which could be an advantage during the silica deposition process. In order to further increase the potential final number of viable cells, several parameters were investigated in order to optimize the encapsulation process. V.1.d.ii Optimization of the encapsulation conditions In order to compare encapsulation in pectin and alginate, both polymers were dissolved in 100 mM PIPES. The presence of buffer is mandatory in the case of pectin (as was demonstrated in chapter IV, p130) due to the acidic properties of the pectin solution. An aqueous solution of alginate has however a pH close to 7 which should not be deleterious to P. aeruginosa viability. The influence of the growth phase was also highlighted during encapsulation in pectin (see chapter IV, p137). The influence of these two parameters (presence of PIPES and growth phase) on the encapsulation efficiency in alginate was therefore investigated. Material and methods P. aeruginosa was grown in LB medium up to 0.5 OD (exponential growth phase) and 1.9 OD (stationary growth phase). Cultures were centrifuged and pellets were suspended in appropriate amounts of water to yield about 5.10 9 CFU/mL. These suspensions were then added to solutions of alginate (final concentration 40 g/L) with or without PIPES buffer (final concentration100 mM). The introduced amount of cells was therefore 10 9 CFU/mL. 3 mL samples were prepared by freeze-casting at 5°C/min and subsequently lyophilized 24h at 0.05 mbar. Dry foams were immediately dispersed in water. The suspension were diluted and plated on LB-agar gels. As can be seen in Figure V.10, successful encapsulation rates are higher for cells in stationary phase than for cells in exponential phase in alginate, regardless of the presence of PIPES. This is consistent with the observation made for this strain of P. aeruginosa encapsulated in pectin, as well as observations made for various bacteria species during freezing for preservation 347,464 . PIPES buffer was added to pectin solutions to regulate pH, since acidic conditions resulted in cell loss even before freezing. In the case alginate the aqueous polymer solution is neutral, which should render the use of buffer useless. Plate counting however reveals higher survival rates when cells are encapsulated in the presence of PIPES. PIPES is not used as a common cyoprotectant. Part of the efficiency of cryoprotective agents is however based on modification of the osmotic equilibrium between the intra and extracellular medium 348 . In this propect, introduction of PIPES could help in balancing osmotic pressures to prevent cell damages during freezing. As a result the best encapsulation conditions for P. aeruginosa in alginate seem to be the use of stationary phase cells in an alginate/PIPES solution. Figure V.11 shows the amount of viable cells encapsulated in these conditions at various stages. No cytotoxicity of the solution can be observed after 24h of contact at 4°C. Freezing and thawing result in about 2.10 8 CFU/mL (with 1.10 9 CFU/mL introduced in the initial polymer solution), which is similar to the combined effects of freezing, drying and rehydration in water. When samples are maintained at room temperature in a desiccated atmosphere over 3 days, the number of viable cells drops to 5.10 6 CFU/mL. It is difficult to evaluate a minimal cell loading to predict functional efficiency of a cellularized material. Maintaining the highest number of viable cells throughout the various steps of the encapsulation however appears a good strategy to ensure maximal efficiency. Figure V.11: 10 9 CFU/mL were dispersed in the initial alginate and PIPES solution (a). After 24h of contact (at 4°C) no loss of viability could be observed (b). After freezing and thawing (30 min at 30°C), about 20% of the cell were still capable of replicating (c). Similar survival rates were observed after freezing, drying and rehydration (d). Even after 3 days of storage, the alginate foam still contained about 5.10 6 CFU/mL (e). The silica deposition through vapors of TEOS requires several days (up to two weeks) to yield a thick enough silica layer (see chapter III, p100), in the case of pectin foams. Deposition kinetics on alginate may however be modified, at least in the early stages, due to different interactions between the silica precursors and the polymer. V.2 Encapsulation in a hybrid material In order to yield fully functional bio-hybrid materials, usable in soils, the alginate foams must be modified to prevent dissolution in contact with water. As mentioned earlier, one possible way to ensure the durability of the material in soil is to coat the polymer scaffold with a thin layer of silica. V.2.a Alginate-silica hybrid materials V.2.a.i Vapor phase deposition Pectin foams were coated by silica through a vapor phase deposition of TEOS. This method proved to be efficient in obtaining a thin and homogeneous controlled layer of silica (see Chapter III, p100). Vapor phase deposition has also been reported as a way to coat alginate beads 418 . The same technique was therefore assessed for silica coating of alginate porous scaffolds. Material and methods Polymer foams were obtained by freeze-casting of 40 g/L solutions of biopolymers (either beet root pectin or sodium alginate). The solutions were prepared by dispersing the polymer powder into deionized water and left under magnetic stirring overnight to yield homogeneous viscous solutions. 3 mL of solutions were then freeze-casted at 10 °C/min and vacuum dried 24 h at 0.05 mbar. Dry foams were cut to 1 mm thick discs and maintained 24 h in a desiccated atmosphere before weighting of the initial mass. The samples were then placed in a silica deposition chamber (see scheme in Chapter III) in presence of a 5 wt% HCl in water mixture saturated by NaCl and four vials containing 10 mL of tetraethoxysilane (also known as tetraethyl orthosilicate or TEOS). The deposition chamber was maintained at 30 °C and samples were removed after 4 or 10 days. The samples were left 24h at 30 °C and ambient humidity and 24 h at room temperature in a desiccated atmosphere. The samples were then weighted again and the relative mass gain was obtained by the following equation. 𝑤𝑡% 𝑔𝑎𝑖𝑛𝑒𝑑 = 𝑚 𝑓 -𝑚 𝑖 𝑚 𝑓 where mi is the initial polymer mass and mf is the final mass after various deposition times. For SEM observations, samples were sputtered with 20 nm of gold. For energy-dispersive X-Ray spectroscopy (EDX) samples were sputtered with 20 nm of carbon. Silica deposition was compared on alginate and pectin foams. The samples were placed simultaneously in the same deposition chamber. As was demonstrated in Chapter III, silica deposition tends to be faster on thinner discs. Deposition was therefore compared on 1 mm thick discs of pectin and alginate. As can be observed in Figure V.12, mass gain was much higher on pectin-based foams, after the same deposition time. After 4 days, the amount of mass gained in the case of pectin was around 72 % of the final mass, while in the case of alginate it was only 12 %. After 10 days the mass gains were 82 % and 31 % for pectin and alginate respectively. This may indicate that in the case of pectin an almost saturated phase was already attained after 4 days, contrary to the alginate samples. The foams were observed in SEM after silica deposition. In the case of pectin a homogeneous and smooth layer of silica could be observed (see Figure V.13 a and a'). In the case of alginate however, heterogeneous square-like structures could be observed across the surface (see Figure V.13 b and b'). The shape of these structures points towards a crystalline material. In this case however we expect to obtain amorphous silica. In order to characterize the coatings on both pectin and alginate foams, EDX analysis was performed (see Annex p 219). The observed structures were sodium chloride, which was likely formed by interaction of the sodium contained in the alginate and the chlorine from the acidified atmosphere. While a stay in an atmosphere saturated in silica precursors in presence of water and acid resulted in the rapid formation of a homogeneous layer of silica on pectin foam, the same process yielded very different result for alginate-based material. The mass gain was much slower in the case of alginate, which may be problematic with the view of coating a material containing living cells. Furthermore the added mass cannot be solely attributed to formation of a silica layer. These differences may be attributed to different interactions between the polymer and the silica precursors. The constitutive monomers of pectin and alginate have similar pKa values (3.5 for galacturonic acid 491 , 3.4 for mannuronic acid and 3.7 for guluronic acid 491 ). pH values of both aqueous polymer solutions are different (about 3 for pectin and 7 for alginate), which may result in higher negative charge in alginate. Assuming the ionization state does not significantly change after drying, ionic interactions with the negatively charged silica and the polymer surfaces may therefore be different between pectin and alginate. The presence of sodium cations in alginate might be expected to have a screening effect on the negative charges of alginate, but as was mentioned previously at least part of this sodium is associated to chlorine in surface crystals of sodium chloride. Repulsive electrostatic interactions may explain the slower deposition rate of silica on the surface of alginate. Longer exposure time would therefore be required for the formation of a percolating silica layer on the alginate scaffolds. Amounts of viable encapsulated P. aeruginosa tend to sharply decrease during long storage, especially in non-desiccated atmospheres. The addition of hydrochloric acid to the atmosphere may present further challenges, even though the fact that the cells are embedded within the polymer wall may provide some degree of protection. As a result, it seems that vapor phase silica deposition is not the best suited technique for coating of alginate based material. Other sol-gel silica approaches have therefore been explored. V.2.a.ii Sodium silicates-based sol-gel chemistry The main alternative to vapor phase sol-gel silica deposition is the use of a liquid phase solgel process. Two main routes are commonly used: the alkoxide route, where silica precursors such as TEOS are hydrolyzed and subsequently condensed and the silicate route, where basic solutions of commonly named waterglass are acidified to yield formation of a silica gel. Combination of LUDOX colloidal particles suspensions and sodium silicate solutions allows the formation of tailor-made gels, with various gelling times, mechanical and optical properties 492 . One drawback regarding liquid phase routes is the necessity to use water as a solvent to ensure non cytotoxicity. Immersion of the biopolymer foam in aqueous solutions results in rapid dissolution of the material. It is however possible to crosslink alginate by using divalent cations such as Ca 2+ to yield the formation of "egg box" crosslinking point, resulting in the formation of a non-water soluble material. Use of a liquid silica deposition phase therefore requires a prior step of crosslinking. Silica formation from alkoxides precursors (such as TEOS) is easy to handle and presents tuning possibilities due to the wide variety of available functionalized precursors. The main drawback to the use of TEOS is the formation of ethanol during the hydrolysis of the precursor. The LUDOX/silicate route has previously be proven to be more efficient than the alkoxide route for the direct encapsulation of bacteria in silica gels 220 . It was therefore chosen for the addition of silica on alginate porous foams. Material and methods Alginate foams were prepared as previously described by freeze-casting at 10°C/min of a 40 g/L solution of alginate and subsequent vacuum drying. Dry foams were then immerged in a 0.5 M solution of CaCl2 during 24h at 4°C. Foams were then rinsed with water and briefly deposited on absorbing paper to removed excess water from the pores. The samples were then immerged in a premixed solution of LUDOX and silicates. The LUDOX/silicate mixture was prepared as follows. A 0.2 M in Si solution of sodium silicates and a 7.8 M in Si solution of LUDOX were prepared separately. Sodium silicate at 27 wt% (waterglass) was obtained from Sigma-Aldrich. LUDOX TM-50 (50 wt% suspension colloidal suspension of silica particles in water) was purchased also from Sigma-Aldrich. A 50:50 mixture was prepared from these two solutions to yield a total Si concentration of 4M (0.1 M from silicates and 3.9 M from LUDOX). The mixture was then acidified to pH = 5 by addition of HCl 4M. The gelling time for this mixture was 1 h at room temperature. Crosslinked alginate foams were added to the LUDOX/silicate solution and maintained 45 min under elliptic agitation. Foams were removed from the LUDOX/silicate solution before gelling and rinsed with deionized water. Samples were then left at ambient temperature and humidity for 45 min. Samples were then stored in water for further uses. For SEM observation, samples were dried. Samples were immerged in successive bath of increasing ethanol content (20%, 40%, 60%, 80% and 100%). Samples were left at least 2 h hours in each bath and 48 h in 100% ethanol. Samples were then left to dry at room temperature. Samples were then cut and sputtered with 20 nm of gold. Samples were weighed before addition of silica and after complete drying in order to evaluate mass gain. After the silica deposition process, samples gained between 35% and 55% of weight mass, which is comparable to the mass gain for vapor phase silica deposition on typical pectin samples. With the view of encapsulating cells in these matrices, it seems preferable to remain in a hydrated environment after formation of the silica gel to prevent further cell damage from drying. However, in order to easily characterize the silica layer it necessary to dry the samples, which may result in significant changes at the macro scale (shrinkage) but also on the microscopic scale (modification of the silica condensation state). Figure V.14 illustrate the shrinking of foams alginate crosslinked foams (with and without silica) after dehydration in ethanol and drying. The main advantage of the liquid phase silica deposition is its capability to introduce large amounts of silica in a short time. This is however accompanied by a major drawback, since it becomes much more difficult to control the repartition and homogeneity of the silica layer. It can be seen at the macroscopic scale since the LUDOX/silica mixture only penetrates in the foam to a limited depth. The center of the foam is not fully impregnated, which results in an inhomogeneous repartition of the silica throughout the foam. This inhomogeneity also results in the partial obstruction of the porosity (see Both crosslinked and dried alginate foams (a, a' and a'') and alginate foams crosslinked and coated with silica (b, b' and b'') before drying present very similar aspects. Oriented pores can still be observed, but significant distortion of the pore walls results irregularities in the shape of the pore themselves (a' and b'). Upon closer observation a granulose-looking surface can be seen on the silica-coated samples (b'') which is not visible on samples simply crosslinked (a''). This layer is however very inhomogeneous in thickness. Scale bars: 500 µm. The silica layer itself appears not very homogeneous in the alginate pore walls (see Figure V.15 b''). The granular aspect can be explained by the chosen synthetic route since it is mainly composed of silica particles from the LUDOX suspensions, aggregated by the use of sodium silicates. The deposited silica does not appear to form a full percolated layer, but rather small aggregates on the surface of the polymer, which may result in inefficient protection against degradation in soils. V.2.a.iii Assessment of the behavior of the hybrid foams in soil The behavior of several alginate-based macroporous foams was assessed in real soil samples over two months. In-soil assays were performed at the Laboratoire de Geologie de l'ENS, in collaboration with Pierre Barre. Compared to pectin, alginate foams (directly after freeze-drying, no crosslinking or silica deposition) seem less prone to immediate degradation in soil. Pectin samples were immediately rehydrated by the moisture content in the soil (see Chapter III p 112), resulting in the disappearance of the sample after one week in soil. In case of alginate however, even though significant contraction may be observed, the samples are still recognizable after 3 weeks (see Figure V.16 a). Shrinkage may be attributed to impregnation by water from the surrounding soil, resulting in a filling of the pores and the presence of capillary forces likely to shrink the material. Furthermore, hydration of the pore walls may significantly modify their mechanical properties and even rheological behavior. Material and methods The crosslinked samples (with or without silica) show no clear change at the macroscopic scale. Samples that were introduced in the hydrated state in the soil (see Upon SEM observation, it can clearly be seen that the pore morphology of non-crosslinked foam has been completely altered (see Figure V.17 b) compared to the initial foam structure (Figure V.17 a). This might once again be attributed to the rehydration of the alginate pore walls, resulting in partial aggregation and remodeling of the pore walls. It must however be highlighted that samples are dried before SEM observation, which may be responsible for slight morphology modifications, especially further contraction of the pore walls due to capillary forces. SEM observation of the foams retrieved after 3 weeks in soil (see Figure V.18) show no significant modification in the pore morphology. Contractions due to drying can be observed, but except for this, the oriented pores can still clearly be observed. The silica layers do not seem to be significantly modified by the 3-week stay in the soil (see Figure V.18 b'' and d''). From a structural point of view, silica does not seem to enhance to durability of the material after 2 month in soil. Simple crosslinking seems efficient in preventing the degradation of the material. Both hydrated and dried materials have similar behavior. This might be explained by rapid water exchanges between the soil and the foam, as suggested by the shrinking of the hydrated foams. The conservation of the aligned and oriented porosity is expected to favor water mass exchanges through capillarity. From the application point of view, these exchanges should to be advantageous, since they are essential to the efficient diffusion of pollutants toward the encapsulated metabolically active species. V.2.b Cell survival in silica coated foams P. aeruginosa cells have successfully been encapsulated in alginate freeze-casted matrix. Survival rates were satisfactory (2.10 8 CFU/mL) after freezing and vacuum drying and even after 3 days of storage foams still contained about 5.10 6 CFU/mL. Silica deposition by contact with vapors of TEOS did however not prove very efficient from a processing point of view. V.2.b.i Silica coating by aqueous sol-gel chemistry Silica coating was therefore performed by sol-gel chemistry of LUDOX and silicates. This proved efficient in quickly adding high amounts of inorganic moiety to the polymer foams. The final goals was however not simply to design a hybrid alginate/silica foam, but to create a matrix destined to host bacteria. It must therefore be verified that the process used to add the inorganic moiety to the cell-loaded polymer foam is compatible with cell survival. TEM observations confirm that silica does not form a homogeneous layer on the polymer wall, on the contrary silica scattered particles aggregates are only present in small amounts. Cells encapsulated in silica-coated alginate have a similar aspect compared to P. aeruginosa embedded in simply crosslinked matrices. Cell themselves appear more or less rounded, even if P. aeruginosa is a bacillus. This is due to the various angles at which the bacteria were cut. Since P. aeruginosa is elongated, transversal cut yield apparent rounded bacteria, while longitudinal cuts yield elongated shapes. Statistically however due to the aspect ratio of the cell shapes, it is more likely to observed bacteria cut in a transversal fashion. To combine structural and functional analysis of the encapsulated cells, foams were stained with LIVE/DEAD® viability kits. The kit is composed of Syto 9® dye capable of staining in fluorescent green both living and dead cells. The second component, Propidium Iodide can only permeate through the membrane of dead cells, resulting in red staining and diminution of the green fluorescence if Syto 9® is also present 493 . Depending on the experimental conditions, it has however been reported that stained P. aeruginosa dead cell may still exhibit green fluorescence resulting in green staining of living cells and yellow staining of dead cells 494 . The samples were included in fluorescence mounting medium for observation in confocal microscopy to prevent bleaching of the dyes. At low magnification stained cells underline the structure of the foams. The advantage here is that the material is mounted directly in the hydrated state. This means that the pore wall distortion and pore contraction observed in SEM due to the drying step are mainly avoided. Thin slices of soft, hydrated matrix must however be cut for observation in confocal microscopy, which may result in slight morphology deformations due to the blade, especially in bending of the structure. Samples coated with silica appear less distorted (see Figure V A few cells are however stained in green, which means that cells are intact and therefore considered as viable. This low viability rate is due to a number of factors. The encapsulation process itself, as it was previously mentioned, is not innocuous. Both freezing and drying of P. aeruginosa may be responsible for cell death. The silica deposition process (both crosslinking and immersion in LUDOX and silicates) may also induce cell death. 16% of living cells can be observed on the confocal microscopy images. This value must however be considered with precaution, since it was only measure on a single image (on 400 cells). Material and methods µm 10 µm Possible heterogeneities of the samples are therefore no taken in account. In addition counting was performed on projections in the z axis of 40 µm of sample resulting in possible cell superposition. No significant difference in cell repartition and green/red ratio can be observed between crosslinked samples and crosslinked and silica-coated samples, which may indicate that most viability loss occurs during freezing and drying and possibly during rehydration in the crosslinking media. V.2.b.ii Cell viability at various encapsulation stages Cell survival to the freezing and drying steps has been previously assessed, resulting in about 2.10 8 CFU/mL in dry foams. Silica deposition was then performed through aqueous sol-gel synthesis to provide enhanced matrix stability. Even if sol-gel synthesis under mild conditions, cell is may happened during these supplementary processing step. Plate counting was performed to monitor cell viability along the encapsulation process. Material and methods Estimation of the cell viability was performed by plate counting. Part of the samples was dispersed in water immediately after vacuum drying as a control. Part of the samples was dispersed in water after 24h at 4°C in the crosslinking medium. Samples were cut into pieces of about 1 mm and vigorously shaken in water. Part of the samples were coated with silica and dispersed in water right after silica gelation (similarly, samples were cut down to 1 mm pieces and vigorously shaken in water). Silica coated samples were placed in water or LB at 30°C under static conditions for 3 days, rinsed in water, cut and dispersed in water. Cell suspensions obtained from samples dispersion at different stages were diluted and plates on LB-Agar gels, and subsequently incubated 24h at 37°C before counting the number of colonies formed. Samples were prepared in triplicate and each dilution was plated in triplicate. In order to dissociate the influence of the various encapsulation steps, several plate countings were performed (see Figure V.22). Plate counting presents no difficulty in the case of simple freeze-dried matrices, since alginate alone is soluble in water. However after the crosslinking step, the materials do not dissolve in water anymore (which is the goal from a structural point of view in the final application, but a problem for characterization of the encapsulated cells). As a result, the best way to performed plate counting was to shred the foams in pieces about 1 mm and to suspend them in water. This suspension was then vigorously shaken in order to disperse the encapsulated cells in the supernatant. This method however includes several aspects which may not be very reproducible, especially regarding the cell extraction process. The results of the plate counting can therefore only give an idea about the general variation of the viable population, but this value may be underestimated due to the fact that part of the encapsulated cells may not be efficiently extracted from the matrix. Another possibility would be to modify the nature of the supernatant in order to reverse the crosslinking of the alginate network. It is for instance possible to use non-gelling ions (Na + ) or chelating agents (EDTA) 179,495 to disrupt the "eggbox" reticulation points and solubilize the alginate gel. This would however introduce a number of supplementary parameters and steps, each of which may modify the survival rate of the cells. As a result it would be difficult to dissociate the effects of the crosslinking itself from the effects the reversion of the crosslinking. These limitations must be kept in mind when comparing plate counting from alginate foams alone which dissolve completely, and crosslinked materials (or crosslinked and silica coated materials). As can be seen in Figure V.22 a small drop in the apparent number of CFU/mL can be observed after 24h of crosslinking. Besides the fact that this value may be underestimated due to the dispersion method, a portion of the encapsulated cells may actually be lost during this processing step. The crosslinking medium is a 0.5 M of CaCl2, which may be responsible for osmotic damages. It must however be remembered that the cells are not in suspension, but immobilized within the alginate pore walls. As a result effects of a possible osmotic shock are difficult to predict. This crosslinking step must also be seen as a rehydration step. As was mentioned in Chapter IV, control of humidity conditions for storage of dry, cell-loaded polysaccharide foams is a key feature in survival rates. Good survival rates were obtained in desiccated atmospheres, however ambient humidity results in very low or zero survival rates. Complete rehydration was obtained when foams were immerged in water for plate counting, resulting however in the dissolution of the polysaccharide foam. Although influence of the rehydration medium on survival of freeze-dried bacteria has previously been investigated 365,496 , use of water yielded satisfactory survival rate. Immersion in the aqueous crosslinking medium may be considered as complete rehydration of the cells, assuming a rapid diffusion of water molecules through the polysaccharide walls. Influence of the crosslinking step is therefore expected to have a limited impact on cell viability, which seems confirmed by plate counting. Crosslinked foams were then coated with silica, by gelation of a LUDOX/silicate solution. Various cells have previously been encapsulated in silica gels, either by the use of alkoxide precursors 209,497 or LUDOX and silicates as precursors 113,203 . The chosen silicification method may therefore be expected to be compatible with cell survival, all the more since cells are protected by a layer of crosslinked alginate. A slight diminution of viability can be observed in silica coated foams compared to simply crosslinked foams (see Figure V.22 d and c respectively), but this variation may be due to previously mentioned incertitude regarding the dispersion method which are difficult to precisely quantify. In order to evaluate the capability of the cells to recover from the various encapsulation steps, silica-coated foams were placed at 30°C under static conditions for three days, both in sterile water and liquid broth. Samples were washed three times with sterile water and subsequently cut down before dispersion in water for plate counting. For material stored in water about 3.10 7 CFU/mL were counted and 1.10 8 CFU/mL were monitored in foams maintained in LB medium (see Figure V.22 e and f). This seem to indicate that cells are capable of efficiently recover from encapsulation in appropriate culture conditions. These results must however once again be considered with caution. In addition to the difficulties regarding complete extraction of the encapsulated cells prior to plate counting, which could lead to an underestimation of the number of viable cells, possible cell leaching must be considered. When foams are stored in a liquid medium, a small fraction of the immobilized cells may leach from the matrix, even if the matrix is crosslinked and coated with silica. Even very small amounts of cells in the initial storage solution may however result in large cells concentrations in the supernatant after three days at 30°C especially in the case of LB medium. Even if the samples were thoroughly washed before plate counting, it is possible that some non-encapsulated from the supernatant remained on the surface of the pore walls, resulting in an overestimation of the number of CFU/mL encapsulated in the material. Figure V.22: Freeze-casting at 5°C/min of suspension of P.aeruginosa (a) at a concentration of 10 9 CFU/mL in alginate and PIPES buffer and subsequent vacuum drying (b) resulted in the encapsulation of 7.10 7 CFU/mL. After crosslinking of the alginate matrix (c), 1.10 6 CFU/mL were monitored and the viability rate was of 3.10 5 CFU/mL immediately after silica deposition (d). When cell-loaded foam were stored at 30°C in water (e) or LB medium (f), cell counting revealed the presence of 2.10 7 CFU/mL and 1.10 8 CFU/mL respectively. All values are significantly different (at p < 0.05). The cell leaching issue is crucial when considering the targeted application. Introduction of an exogenous microorganism in a polluted soil can be necessary for efficient remediation, but leaching of the encapsulated bacteria may result in disturbance of the ecosystem's balance, which should be prevented. V.2.b.iii Assessment of cell leaching Assessment of leaching directly in soil represents a major challenge due to the presence of numerous endogenous microorganisms. It would therefore be difficult to distinguish by simple means such as plate counting the endogenous microorganisms from the soil and the microorganisms introduced by the foam. Leaching in liquid medium may be easier to assess but may not be representative of the behavior of the device in soil. As a compromise, leaching was assessed at the surface of LB-agar gels. Material and methods P. aeruginosa was encapsulated as previously described in alginate/silica hybrid foams. After culture in LB medium, cells were collected and dispersed in alginate and PIPES buffer. The suspension was freeze-casted at 5°C/min and vacuum-dried. The resulting dry alginate foam was crosslinked in a 0.5 M CaCl2 solution and coated with silica (gelation of a LUDOX/silicate solution). The resulting foams were place on LB agar plates and left at room temperature for several days. In order to compare the leaching possibilities in a matrix simply crosslinked and in a matrix crosslinked and coated with silica, samples were placed on LB-agar gels (Figure V.23). After three days, a halo of bacteria could be observed around both samples with or without silica (see Figure V.23). The halo of bacteria however appears more dense and regular around foams that were not coated with silica, which could indicate that the silica layer prevents cells leaching. Interestingly, the bacterial colonies do not seem to expend further after 5 days. The non-silica coated foams also appear to shrink slightly, while the silica coated ones remain identical. Shrinking might partially be explained by partial drying of the foams, which would result in high capillary solicitation on the pore walls. The presence of silica may provide enhance mechanical resistance against such contraction. The foams are however also likely to be degraded by bacterial activity. Degradation of the polymer matrix could result in enhanced leaching. If wall have polymer-silica core-shell structure, leaching could be prevented even if the polymer core is degraded and the general structure can be retained. After 5 days, foams are colored in green (the color is more intense in the silica coated sample). This is likely due to the secretion of pyoverdine and pyocyanine by P. aeruginosa, which is indicative of metabolic activity. As mentioned earlier the silica coating process from a mixture of LUDOX and silicates has both advantages and drawbacks from a structural point of view. The method allows for rapid deposition of large amounts of silica. The counterpart to this efficiency is the low level of control over the thickness, the morphology and the homogeneity of the silica layer itself, as well as the necessity to crosslink the polymer structure prior to silica deposition. The fact that foams with or without silica seem to age in similar way in soil (when no cells are encapsulated) may lead to question the necessity of addition of a silica layer. It seems however that this silica layer, even if it is not homogenous across the alginate pore walls, allows for a limitation of cell leaching, which is a considerable advantage from the application standpoint. V.3 Bioremediation assays P. aeruginosa was encapsulated in hybrid macroporous foams and survival rates were monitored at various stages of the encapsulation process. After freezing, drying, crosslinking and silica deposition, apparent number viable of encapsulated cells was still around 3.10 5 CFU/mL. Presence of living encapsulated cells was confirmed by confocal microscopy. These evaluations of cell viability (either plate counting or staining) were based on the capability of the cells to replicate or on physical integrity of the cells, but the crucial information from the application standpoint is the metabolic activity of the cells. In order to assess the efficiency of the bionanocomposite, the degradation capabilities towards several pollutants were investigated. V.3.a Choice of a target pollutant As mentioned in Chapter IV, P. aeruginosa has been used in a wide range of bioremediation processes. It must however be underlined that in most cases, the strains used are directly collected and cultivated from polluted sites. It was therefore essential to identify a pollutant that could be degraded by free cells of the specific available strain, before assessing the efficiency of the encapsulated bacteria. A first group of targeted pollutant was polyaromatic hydrocarbons (PAH) which have been documented as potentially degraded by P. aeruginosa [447][448][449] . PAH are of major concern as soil pollutants due the wide range of possible sources and potentially high levels of toxicity [START_REF] Johnsen | Principles of microbial PAH-degradation in soil[END_REF][START_REF] Haritash | Biodegradation aspects of polycyclic aromatic hydrocarbons (PAHs): a review[END_REF] . These compounds presented the additional interest of being detectable by fluorescence spectroscopy at low concentrations. They have however low solubility in aqueous media. Unfortunately, degradation assays for anthracene and pyrene in water were not conclusive. Azo dyes were then chosen as potential models of pollutant 251,458 . These compounds are less ubiquitous than PAH since they are mainly localized in industrial effluents from the textile industry [START_REF] Pereira | Environmental Protection Strategies for Sustainability[END_REF] . The large volume of contaminated waste however represent a serious concern [START_REF] Carmen | Textile Organic Dyes -Characteristics , Polluting Effects and Separation / Elimination Procedures from Industrial Effluents -A Critical Overview[END_REF] . Concentrations of Reactive Black 5 498 and Methyl Orange were measured by UV-vis spectroscopy with or without suspensions of P. aeruginosa, but no significant difference could be observed. In order to illustrate bioremediation capabilities of the device, a different approach was then selected. Reactive Black 5 was set as the model pollutant and a bacterial strain capable of degrading this particular contaminant was then chosen. V. Shewanella Oneidensis is, like Pseudomonas aeruginosa, a gram-negative bacteria found in a wide range of habitats. Similarities between the two species are not anecdotic, since the first isolated Shewanella species were initially classified as Pseudomonas, before being renamed a few years later 501 . S. Oneidensis is capable of both aerobic and anaerobic metabolisms and is well-known for its capability to reduce heavy metals 502,503 in anaerobic conditions. These properties of oxydo-reduction have also been used in the design of microbial fuel cells 504 . The versatility of S. Oneidensis allows for growth in various conditions. Optimal growth is monitored at 30°C, but S.Oneidensis is capable of growing at much lower temperatures (3°C) 505 . In terms of growth medium, S. Oneidensis can be grown in LB and plated on LBagar gels 506 . More specific growth medium with different carbon and nitrogen sources can be designed, resulting in various metabolic activities 500 . Efficiency of S. Oneidensis for remediation of Remazol Black 5 was first assessed in "ideal" conditions in order to validate the microorganism/pollutant model. Discoloration was monitored in a MR1 medium adapted to S. Oneidensis to confirm the bioactivity of this strain, prior to the addition of parameters such as the presence of the encapsulating matrix or the use of soil as a medium. Material and methods MR1 strain of Shewanella Oneidensis was stored in 30 vol% glycerol aliquots at -80°C. About 10 µL of aliquot was dispersed in 10 mL of LB medium in a glass tube and precultivated for 24h at 30°C and 150 rpm up to a 0.9 OD. The pre-culture was then diluted by a factor 50 in the culture medium (see Annex p 204 for composition) (typically 0.8 mL were diluted in 39.2 mL of medium) and placed at 30°C and 150 rpm. After 20h of culture part of the culture suspension was autoclaved. Reactive Black 5 degradation assays were then performed in the same medium at 37°C in static conditions over 24h. Three concentrations of Reactive Black 5 (0 mg/L, 10 mg/L and 100 mg/L) were investigated. Samples were prepared with no bacteria and autoclaved bacteria for control and with bacteria directly transferred from the culture medium. Each degradation assay was performed in triplicate. For determination of dye concentrations, 1.5 mL of suspension was centrifuged 10 min at 5000 rpm. The supernatant was mixed with 1.5 mL of PBS 2X before acquisition of the absorbance spectra. If needed, samples were diluted in order for the concentration to be within the linearity range determined by the calibration curve (see Annex p 207). Absorbance maximum for Reactive Black 5 was found to be 598 nm. Plate counting was performed on samples containing fresh bacteria in the initial suspensions and after 24h. Successive dilutions of the cells suspensions were plated in triplicate and incubated 24h at 37°C. For quantification of the discoloration efficiency, samples were centrifuged to remove the spectroscopic contribution due to diffusion by suspended cells, and absorbance was measured at 598 nm. Samples containing 0 mg/L of RB5 but same cell contents were used as baselines. As can be seen from Figure V.26 no significant difference can be seen between samples containing no cells and autoclaved cells, either at 10 mg/L RB5 or 100 mg/L in RB5. After 24h about 10% of discoloration can be observed in the case of 10 mg/L solutions and 3% of discoloration in the case of 100 mg/L solutions, in absence of viable cells. This may reflect the instability of the compound in the considered incubation conditions (mineral medium at 37°C). When the medium was inoculated with 10 7 CFU/mL of S. Oneidensis significant discoloration could be observed both in 10 mg/L and 100 mg/L solutions (see absence of RB5 and with both 10 mg/L and 100 mg/L of dye. Final cell concentrations were not significantly different, which seems to indicate that there is no significant toxicity. S. Oneidensis proved to be an efficient model for decolorization of Remazol Black 5, in suspension in an adapted medium. This is however only a pre-requisite toward the actual proof of concept, since the degradation process of interest is not in solution but in soil, with cells encapsulated in the previously described macroporous hybrid foam. V.3.c Soil depollution Moving from a model of biodegradation of cells in suspension in a liquid media to a model of encapsulated cells for depollution of soil is likely to induce several changes in the assessment of degradation efficiency. The presence of soil as a medium makes the monitoring of dye content more difficult. In addition the presence of soil introduces the issue of adsorption of the dye on soil particles and of diffusion inside the soil itself, as well as endogenous bioremediation phenomena. Encapsulation of the functional units in a matrix is likely to significantly change remediation kinetics due limitations in terms of substrate diffusion inside the material itself. Material and methods S. Oneidensis was encapsulated according to the protocol previously described for encapsulation of P.aeruginosa. S. Oneidensis was pre-cultivated in LB medium for 24 h (up to 0.9 OD). This pre-culture was then diluted by a factor 50 in fresh LB medium and cultivated 7 h at 30°C and 150 rpm (up to 2.2 OD). The medium was centrifuged and the pellets were dispersed in alginate and PIPES buffer to yield the following concentrations: 10 Bioremediation efficiency was assessed at small scale in upper horizon silt loam Luvisol kindly provided by the Laboratoire de Géologie de l'ENS as a typical soil sample from the Parisian region. Concentrations of pollutant in leaching water were monitored in presence or in absence of encapsulated S. Oneidensis. Cell-loaded foams were left in contact with a soil soaked by a solution of RB5, which may be seen as a model of industrial spill. Samples were incubated at 37°C to ensure rapid diffusion and degradation kinetics, but such temperature is mostly not representative of actual polluted sites. The soil samples were then briefly washed to simulate water infiltrations, and dye concentration in this supernatant was monitored. Visual observation (see Figure V.27) shows that RB5 concentration appears to be much lower than the initial 0.5 g/L in all rinsing solutions. This is to be expected from dye adsorption on soil particles. The discoloration may also be due to the presence of endogenous microbial species capable of degrading RB5. To further characterize the dye concentrations in rinsing solutions, UV-vis spectra were measured. A slight discoloration (4.5%) of discoloration was observed on the control solution of RB5 in the simplified mineral medium after 42h at 37°C, which is consistent with stability observations reported in paragraph V.3.b . Contact with the soil alone resulted in significant reduction of the dye concentration in the rising water, since final measured concentration was 0.048 g/L (see Figure V.28 a), which corresponds to a 90.4% decolorization rate. This discoloration is likely due to a combination of adsorption and endogenous microbial degradation, since the soil used is not sterile. Adsorption for an initial dye concentration of 0.1 g/L in water at 25°C was found to be 0.038 mgdye/gsoil after 24h (see Chapter III, p118). Such a dye loading could account for a drop of concentration to 0.47 g/L. Even considering the changes in adsorption kinetics due to the higher temperature (37°C vs 25°C), the longer exposure time (42 h vs 24 h) and the change in medium (mineral medium vs water), it is unlikely that the observed discoloration is solely due to adsorption. Since the soil used is a field example of soil which has not been sterilized, the system should contain various endogenous microbial species. The use of mineral medium and incubation temperature are likely to have stimulated the biodegradation phenomena in the soil, resulting in the high discoloration rate observed. The final concentration in the system containing only soil was therefore used as a control value. Introduction of blank foams does not induce a statistical difference in the final RB5 concentration of the washing medium (see Figure V.28 b). The main contribution of such foam would be dye adsorption on the foam itself. Such effect may however be difficult to observe due to the low mass/mass ratio between the foam and the soil. Rinsing water from the soil sample treated with the hybrid cell loaded foam proved to have a final RB5 concentration significantly compared to the control with soil alone (see Figure V.28 a and c). Final RB5 concentrations were 0.042 g/L and 0.048 g/L respectively, which represents a 12.5% discoloration for the cell loaded foam compared to the control. This difference is not very big, though statistically significant (nsamples=3, p<0.05). The difference in concentration between assays in presence of blank foams and cell-loaded foams were however not statistically different (see Figure V.28 b and c). This means that the observed efficiency of the cell loaded foam may be attributed to a combination of dye adsorption on the foam and biological activity. As may be expected, dye concentration in rinsing water for soil directly inoculated with free bacteria is much lower (0.026 g/L or 46.8% of discoloration compared to the soil control) compared to the other assays. Free cells are indeed less likely to be subjected to diffusion limitations, which may help their efficiency compared to encapsulated cell. In this case however the main reason for the difference observed and the relatively low efficiency of the depollution device may be explained by a simple parameter that was not yet discussed in this paragraph: the amount of viable encapsulated cells. The encapsulation process has been developed and optimized using P. aeruginosa strain as a model. S. Oneidensis was however chosen for its superior dye degradation capabilities. The same process and conditions were used for the encapsulation of this second bacteria strain. Unfortunately, under these conditions only 300 CFU/mL were counted in freeze-dried foams, and this number is likely to be even lower after silica deposition. As a result it is difficult to compare efficiency of free and entrapped cells due to the very large difference in the number of introduced cells (less than 900 encapsulated CFU vs 2.5.10 8 free CFU). The bionanocomposite device composed metabolically active bacteria encapsulated in an alginate/silica porous matrix proved to be efficient in the discoloration of rinsing solution from a RB5 polluted soil. The effect was however very limited due to the fact that a very small amount of CFU S. Oneidensis was immobilized in the hybrid foam. A large number of cells was initially introduced in the material, and it cannot be excluded that part of the nonreplicating cell may still have a metabolic activity toward RB5. It seems however that the remediation process could greatly benefit from the optimization of the encapsulation conditions. The immobilization process was initially optimized for P.aeruginosa, which allowed identification of several key parameters in the encapsulation efficiency, including formulation of the matrix (type of polymer, addition of buffer or cryoprotective agents), growth phase of the bacteria or freezing-rate. These parameters would have to be adapted to S. Oneidensis MR1 to ensure optimal encapsulation efficiency. Use of a different bacterial species may even uncover other relevant encapsulation parameters, to which P. aeruginosa is not especially sensitive. Conclusion P. aeruginosa was successfully encapsulated in alginate freeze-casted foams. Alginate proved efficient in maintaining cell viability for several days in desiccated conditions, which could be of use for vapor phase deposition of silica. In the case of alginate-based materials however, these silica depositions conditions resulted in the formation of large amounts of sodium chloride rather than a smooth silica layer. The silica deposition process was therefore modified. Use of a liquid phase silica deposition required crosslinking of the alginate foams, but presented the advantage of rapid deposition and immediate rehydration of the encapsulated cells. Cell survival was monitored throughout the encapsulation and silica deposition processes. Despite a small viability drop during encapsulation, cells were still capable of growth. The drawback to this deposition technique in liquid phase was the relative lack of control regarding the amount of deposited silica and the uniformity of the silica layer, compared to the vapor phase deposition. The hybrid materials however proved to be very stable in soil. Despite efficient encapsulation, the P. aeruginosa strain used proved inefficient for remediation of the tested pollutants (mainly PAHs and dyes). On the other hand S. Oneidensis was very efficient in degrading Reactive Black 5. The latter bacteria was therefore chosen for remediation assays in soil. Lower dye concentrations were monitored in rinsing water when the cellularized hybrid foam was introduced in the soil. The differences observed were however very small, which may be attributed to the limited amount of viable S. Oneidensis cells encapsulated in the macroporous hybrid foam. These low survival rates can be explained by the fact that the encapsulation protocol was optimized for P. aeruginosa. Despite the similarities between the two bacteria species, behavior regarding freezing and drying may be significantly different. Key parameters such as culture medium, freezing temperature, presence of additives or composition of the rehydration medium should allow for the optimization encapsulation and increase of the viable cell content, which should in turn allow for better characterization of the bioremediation potential of the device. General conclusions and perspectives The goal of this work was the design of a new type of cellularized materials for soil bioremediation. Several key requirements were identified both in terms of structure and functionality to maximize the efficiency for the targeted application. Since encapsulated microorganisms (more specifically bacteria) were used as the biofunctional units of the device, it was essential to preserve their metabolic activity. The depollution efficiency not only depends on the activity of the encapsulated cells but also on the accessibility of the targeted substrates (in this case, soil contaminants). One limitation was however the need to prevent dissemination of the chosen metabolically active exogenous organisms within the soil, in order to avoid possible disturbance of the endogenous ecosystem. The matrix had therefore to be designed keeping these constraints in mind. Two main pathways were explored to design and control the encapsulating matrix. First the composition of the matrix was chosen to ensure maximal compatibility with the functional units while providing sufficient structural integrity and stability to withstand prolonged use in soils. In addition the shaping process needed to be carefully engineered to provide both an oriented porosity to favor substrate diffusion and encapsulation conditions compatible with maximal bacterial survival rates. These two aspects could however not be considered separately since the nature of the matrix components and the possibilities in terms of processing are mutually dependent. Regarding the composition of the matrix, biopolymers were selected for their intrinsic biocompatibility as well as their versatility in terms of shaping possibilities. In particular, biopolymer solutions could be processed by freeze-casting. Such a method allowed simultaneous encapsulation of the functional unit and shaping of the matrix into a porous foam. The main limit to the use of cellularized freeze-casted biopolymer foams as soil remediation devices was however the instability of biopolymer-based materials. To prevent rapid degradation of the matrix and subsequent leaching of the encapsulated bacteria, an inorganic moiety was associated to the biopolymer structure. Sol-gel silica appeared as especially relevant thanks to the mild synthetic conditions required. Design of cellularized macroporous hybrid materials by freeze-casting required the development of a multi-step encapsulation process, taking into account the structural and functional requirements of the targeted application. The general strategy of this work was not to fully engineer each aspect of the process individually but rather to identify the key parameters at each stage of the encapsulation, their interdependence and their influence on the other steps of the process. The possibilities in terms of general structure and morphologies were first evaluated by icetemplating of beet-root pectin aqueous solutions. Various freezing setups were investigated (freezing in conventional freezers at -20°C and -80°C, use of a liquid nitrogen bath, use of a freeze-casting setup) and revealed the crucial importance of the presence and orientation of a temperature gradient to obtained well controlled and aligned porosity. Freeze-casting offers a wide range of possibilities in terms of morphological control. Parameters regarding the formulation of the initial solution (type of solvent, type of polymer, presence of additives etc…) or the freezing setup (number of cold fingers, geometry of the cooling element, patterning of the cold surface etc…) might be explored to gain further control over the pore structure and yield original morphologies. The pectin-based macroporous foams were then modified by addition of a silica layer on the pore-wall surface. The silica was deposited through exposure to tetraethoxysilane vapors in presence of an acid aqueous atmosphere to promote hydrolysis and condensation at the surface of the pectin pore walls while preventing their dissolution. The method proved efficient for the deposition of a fully percolated and homogeneous silica layer, without any pore obstruction. The deposition kinetics allowed for fine tuning of the final silica thickness, which in turn granted control of the mechanical properties of the foam. The final structure could be described as a macroporous hybrid material composed of pectin-silica core-shell walls. The presence of the silica layer proved efficient in preventing dissolution of the foam in aqueous medium. Ageing of the foams was also monitored in silt loam Luvisol soil and hybrid materials were stable over 5 weeks. These hybrid materials were also used for adsorption of a model contaminant (Reactive Black 5 (RB5), which is a common dye, mainly used in the textile industry) in liquid medium. Interestingly, the device was also efficient when placed with a real soil sample impregnated with a dye solution. This could be an indication of good substrate diffusion from the soil to the material. Such materials could be of interest in themselves. From a material scientists' perspective the original hybrid structure could be tuned to yield a wide variety of morphologies and properties. This could be used for various applications including biomedical scaffolds or vehicles for controlled drug delivery, but also in environment science for materials combining adsorption properties, oriented porosity and mechanical stability. In this case however the structure was used as a matrix for the encapsulation of metabolically active microorganisms. The components and processing techniques were chosen keeping in mind the constraints imposed by the presence of microorganisms (use of water as a solvent, no exposure to high temperature, use of biopolymers). Efficiency of freeze-casting for direct encapsulation in the previously described matrix however still needed to be confirmed. S. cerevisiae was used as a first model for encapsulation. The density of living cells, viscosity of the pectin suspension and considered ice-front velocities were compatible with entrapment of the cells within the pectin walls. The cell metabolic activity was preserved after freezing and drying. Yeast cells are a good laboratory model but are not especially relevant from the bioremediation perspective. A second model organism, P. aeruginosa, was therefore investigated. Freeze-casting also proved compatible with survival of this bacterial species, despite the biological stress generated by freezing and drying. Although freezing and drying protocols have been developed to prevent cell damages for cell cryopreservation, they usually require introduction of cryoprotective compounds such as glycerol or lyoprotective compounds such as trehalose. In the freeze-casting encapsulation process, no common cryopreservative is added. The presence of a biopolymer such as pectin may however provide some degree of protection against deleterious effects of freezing and mechanical solicitations. In addition to the presence of cryoprotectant, another crucial parameter in conventional cryopreservation is the control of the cooling rate. The freeze-casting technique confers a good control over this parameter which allowed for further optimization of the encapsulated cell survival rate. This encapsulation matrix however proved limited regarding long term survival of the entrapped bacteria. This was problematic regarding the silica coating step, which requires several days to several weeks to yield a silica layer thick enough to provide long term structural stability. The composition of the matrix had therefore to be modified and encapsulation was performed in a different polysaccharide. Bacteria encapsulated in alginate displayed higher survival rates compared to cells in pectin matrices, especially after 24 h of storage. The sol-gel vapor phase deposition was however far less efficient in the case of alginate foams. Sol-gel deposition was therefore performed in liquid phase through the aqueous route (mixture of sodium silicates and commercial colloidal particles), by taking advantage of the crosslinking properties of alginate in presence of divalent cations (in this case Ca 2+) . The silica deposition method proved to be an efficient and quick alternative to vapor phase silica deposition. The hybrid materials obtained by this method showed good stability in the reference soil previously mentioned over 2 months. From a functional perspective, viable cells were observed within the hybrid macroporous foam at the different steps of the encapsulation process and cell growth could be obtained after freezing, drying, crosslinking and silica coating. The silica layer obtained by the liquid aqueous sol-gel route seemed to have a beneficial influence to prevent cell leaching. The assessment of this effect was however only performed on agar gels as a model of solid environment. Evaluation of leaching within the actual reference soil could be of great interest but would require extensive analytical resources (for instance using 16S rRNA sequencing). Control of the silica layer porosity (either obtained by vapor phase deposition or through the liquid aqueous sol-gel route) could provide control over the material's diffusion properties. Such a controlled membrane could be of great interest in various applications such as drug or cell delivery. Tuning of the barrier properties of the silica layer may also be decisive for soil depollution, since the contaminant must diffuse into the polymer layer. To assess the efficiency of cellularized macroporous hybrid materials as depollution devices, various model pollutants were investigated. The specific P. aeruginosa strain used for the development of the encapsulation protocol was unfortunately inefficient for the degradation of the tested contaminants. As a result a different approach was pursued. A contaminant model was set (RB5) and a bacterial species with bioremediation capabilities regarding this specific contaminant was then selected. Shewanella oneidensis displayed high RB5 discoloration capabilities and was therefore further used as the functional unit of the cellularized alginate-silica porous material. The efficiency of the material was evaluated in the reference soil previously mentioned. The soil was oversaturated with a concentrated RB5 solution and dye concentration of rinsing water was monitored after incubation. The addition of the cellularized hybrid porous material in the soil resulted in a slight but statistically significant drop in the contaminant concentration. This system therefore proved to be an efficient proof of concept. Several pathways can be considered in order to confirm these results and enhance the efficiency of the depolluting material. The main limitation of the specific S. oneidensis-alginate-silica structure was the very low viable cell loading. This can be explained by the fact that the various encapsulation parameters were optimized for maximal survival of P. aeruginosa. As a result several conditions may be optimized to increase the concentration of viable S. Oneidensis in the porous matrix. One of the most efficient ways to increase the survival rate may be by tuning the cooling rate during freeze-casting as it was demonstrated that an optimum freezing-rate, which is cell dependent, can be found. Other parameters such as the growth phase, growth medium or initial cell concentration may also be modified to adjust the cell survival rate. The depollution model in itself may also be improved in order to be closer to field conditions. In this proof of concept, the soil was oversaturated with a mineral medium containing high dye concentrations and incubated at 37°C to maximize substrate diffusion and bacterial metabolic activity. These conditions were however likely to be responsible for biostimulation of the endogenous microbial population since the used soil was a non-sterilized field sample. Efficiency of the cell-loaded material may therefore be monitored with different soil hydration levels, various contaminants and nutrient contents as well as different incubation temperature in order to fully dissociate the effect of the endogenous biostimulation from the bioaugmentation. This proof of concept nonetheless appears has as a very encouraging step towards the elaboration of efficient materials for in situ soil depollution. Thanks to its adaptability, the described encapsulation process may be used for the entrapment of a wide variety of microorganisms (bacteria but also fungi or algae for instance). Through the choice of the appropriate microorganisms (or consortia) and tuning of the encapsulating matrix properties (in terms of diffusivity, mechanical behavior, stability etc...), tailor-made depollution materials may be designed for specific contaminated soils. Visual summary: Bacteria were encapsulated in biopolymer-silica macroporous foams. Entrapment of the cells and unidirectional porosity were obtained by freeze-casting and silica coating was performed by sol-gel chemistry. The Shewanella oneidensis-loaded alginate-silica porous material proved efficient in the discoloration of a soil containing Reactive Black 5. Alginate Silica Shewanella oneidensis Reactive Black 5 For EDX analysis, the samples were cut in the same way and coated with 20 nm of carbon. Analysis was typically performed under 10 kV acceleration and 30 µA probe current. Titanium was used as reference. -SEM-FEG SEM observations were also performed using a Hitachi SU-70 equipped with a Field Emission Gun. The samples were cut as previously described and coated with 5 nm of platinum by metal sputtering. The acceleration voltage was typically 1 kV and the emission current was 44 µA. -TEM Transmission Electron Microscopy was performed on a Cryomicroscope Tecnai spirit G2 equipped with a Gatan Orius camera. Prior to observation, samples were embedded in epoxy resin. The samples were stabilized 24h in a 8% paraformaldehyde solution and fixated in in a glutaraldehyde solution (8% glutaraldehyde in a 0.05M cacodylate buffer). After rising with a 0.1M cacodylate buffer and 0.6 M saccharose solution, the samples were fixated by osmium tetraoxyde. After rinsing with 50% overnight, the samples were dehydrated with successive bathes of ethanol (50, 70, 95 and 100%) and a bath of propylene oxide. The samples were then embedded in epoxy resin (mixture of Araldite, E812, dodecenylsuccinic anhydride (DDSA), N,N-dimethylbenzylamine (BDMA) and N-methylacetamide (NMA)). After drying of the embedding resin (3 days at 60°C), slices (between 50 nm and 80 nm) were cut on a Leica EM UC7 microtome. Samples were contrasted with uranyle acetate one day before TEM observation. - Confocal microscopy Confocal microscopy was performed at the Center for Interdisciplinary Research in Biology (CIRB) at College de France. Observations were performed on Leica DMI6000 inverted microscope. Acquisitions were performed on about 40 µm along the z axis with 0.3 µm steps. Samples were incubated 30 min with Live/Dead® dye (Propidium Iodide at a 0.3 mM concentration and Syto 9 at a 0.05 mM concentration) and rinsed with sterile water before observations. Images were analyzed with Fiji software, using the "3D projection" function.  Spectroscopy -IR Infra Red Spectroscopy was performed on a Perkin Elmer Spectrum 400 FT-IR/FT-NIR Spectrometer equipped with Universal ATR sampling accessory. Samples were crushed or shredded into less than 1 mm large fragments. A few milligrams of sample were placed on the diamond detector. -UV-vis UV-visible spectroscopy was performed on CARY 5000 from Agilent Technologies. Typically, water was used as reference and baseline was adjusted on the dispersing medium (for instance phosphate buffer saline). Wavelength range was adapted to the considered species to detect. Scan rate was typically 600 nm/min, data interval was 1 nm, average time was 0.1 s and spectral band width was 2 nm. Calibration curves for the different components observed are provide in the Experimental section, p207.  TGA Thermogravimetric analysis was performed on Netzsche STA 409 PC Luxx thermal analyzer. Samples were crushed or shredded into less than 1 mm large fragments. Between 10 and 20 mg of sample were place in the crucible. Thermal analysis was performed under air between 25 °C and 1200 °C, at 5 °C/min.  Rheology Rheology measurements were performed on a MCR 302 Anton Paar rheometer under planecone geometry. The cone diameter was 24.969 mm with a 1.0110° angle and a 50 µm truncation. Aqueous pectin solutions at various concentrations were prepared by magnetic stirring overnight at room temperature. Measurements were performed with shear rates between 1 and 100 s -1 .  Mechanical compression Stress/strain curves were acquired on an Instron 5965 universal testing machine equipped with a 100 N load cell. Samples were cut into 1 cm 3 cubic and compressed up to 50% strain at constant displacement rate of 1mm/min. The stress/strain curves were typically acquired for 5 replicates per type of sample.  Data treatment -Directional analysis SEM images were analyzed using the Fiji software and the Orientation J plugin (OrientationJ, java plugin for Fiji/ImageJ, written by Daniel Sage at the Biomedical Image Group, EPFL, Switzerland 401 ). Color orientation maps were obtained using orientation as hue value, constant saturation and original image for brightness. Cubic spline gradient was used with a 8 px Gaussian window. - Statistical analysis Data presented are typically the mean value on triplicate samples (mechanical compression of performed on 5 replicates, plate counting were usually performed on triplicate sample, and each sample was plate in triplicate). The error bars are standard deviation values. For pore size analysis 150 measurements were typically performed on SEM images (100 measurements on a x50 magnification image and 50 measurements on a x100 magnification image). Data were presented in box-and-whisker plots, presenting median, value, mean value as well as 25th-75th and 5th-95th percentiles (see Statistical significance was tested using a Student test at two samples, using a level at the level of significance p < 0.05. to 20 colonies on average (which corresponds to 1.10 9 to 2.10 9 CFU/mL). This plate was stored at 4°C up to 3 weeks and individual colonies were used to prepare fresh cultures before each experiment. -Bacteria culture Preparation of fresh cultures was always preceded by a pre-culture step. One colony of the previously mentioned storage plate was introduced in 10 mL of fresh LB medium in 30 mL glass tube and incubated 24h at 30°C and 150 rpm. The OD reached values around 0.8 on average. The culture itself was then made in 75 cm² culture flasks with 1.2 mL of the 0.8 OD pre-culture and 58.8 mL of fresh LB medium (dilution of the pre-culture by a 50 factor). Cultures were incubated at 30°C and 150 rpm for various times until the desired OD was obtained. -Bacteria encapsulation Bacteria were cultivated as previously mentioned, typically 5h up to 0.5 OD (viability was also investigated after 24h of culture). The culture was centrifuged 10 min at 5000 rpm and pellets were dispersed in water (typically 3 mL of water for 60 mL of culture). This suspension was the mixed with a solution at 50 g/L in pectin and 125 mM in PIPES (typically 1 mL of bacteria suspension for 4 mL of PIPES/pectin solution). The final concentrations were 40 g/L in pectin, 100 mM in PIPES and about 10 9 CFU/mL. -Plate counting Plate counting was generally performed in 12 well plates (see Figure A.2). Bacteria suspensions were obtained directly from culture or from dissolution of cellularized biopolymer foams in sterile water. In the case of cellularized hybrid foam, the matrix was previously shredded into less than 1 mm fragments and suspended in water before vigorous agitation. The bacteria suspensions were diluted in water (logarithmic successive dilutions) and four dilutions were plated in triplicate for each sample. Plates were incubated 24 h at 37°C before counting. The number of CFU/mL was estimated using the dilutions yielding between 5 and 50 colonies. -Alternative freezing methods Samples were frozen in different conditions for comparison with the freeze-casting technique. Conventional freezers at -20°C and -80°C as well as a liquid nitrogen bath were used. Typically 4 mL of solution was poured in a Ø = 19 mm polyethylene mold or 1.8 mL was poured in a 2 mL cryotube. The sample was then placed overnight into the freezer or 5 min in a liquid nitrogen bath. -Drying Frozen samples were lyophilized in a Christ Alpha 2-4 LD freeze-dryer under a 0.05 mbar vacuum. Samples were dried at 24 h, either directly in the freeze-dryer chamber or in side vials.  Sol-gel chemistry -Vapor phase deposition Biopolymer foams were kept at least 24 h in a desiccated atmosphere and weighed before the beginning of the silica deposition. Cylindrical samples were cut down to the desired thickness and place in the deposition chamber presented in Figure A.7. The chamber diameter approximate volume was 1 L. The setup was composed of a saturated NaCl acid aqueous solution (typically 17 mL of 37 % HCl, 133 mL of water and 60 g of NaCl) and of 4 vials (Ø = 29 mm) containing 10 mL of tetraethyl orthosilicate (TEOS). The deposition chamber was sealed a kept at 30°C typically between 1 and 14 days. The samples were then kept 24 h at 30°C in an open container and 24 h at room temperature in a desiccated atmosphere before final weighing. -Behavior in simulated polluted soils Influence of macroporous hybrid foams (with or without encapsulated bacteria) on polluted soil was assessed using Reactive Black 5 (RB5) as a model contaminant. 10 g of soil was saturated with 7 mL of RB5 solution (typically 100 mg/L in water or 500 mg/L in mineral medium). The soil was incubated (either 24 h at 25 °C or 42 h at 37 °C) and washed with 7 mL of PBS 2X. The supernatant was then centrifuged 10 min at 5000 rpm and absorbance at 598 nm was measured to calculate the residual RB5 concentration (see calibration curve p 207). Complementary results  Foam wetting behavior Material and methods Pectin foam were prepared by freeze-casting as previously described. Foams were prepared from 20 g/L, 30 g/L, 40 g/L and 50 g/L aqueous pectin solutions at 10 °C/min and from 40 g/L pectin solution at 1°C/min, 5 °C/min and 10°C/min. Samples were vacuum dried 48 h at 0.05 mbar and cut to about 0.8 cm height. Samples were put in contact with the surface of a solution of Disperse Red 1 (0.2 g/L) in ethanol. The wetting was recorded and the wetting profile was extracted from the videos using the Fiji software. Only a limited number of points were recorder for each wetting profile due to the rapid foams impregnation. All samples (regardless of the initial polymer concentration or freezing rate) had similar impregnation profiles, despite variations in the size and shape of the pore (see II.4 ).  Assessment of the reproducibility and robustness of the vapor phase silica deposition process Material and method Pectin samples were prepared as previously mentioned by plunging 1.8 mL of 40 g/L pectin solution in liquid nitrogen and subsequently freeze-drying the samples. Samples were placed in the silica deposition chamber with either 5 wt% or 20 wt% HCl solution, with or without NaCl (400 g/L). Either 4 vials with either 10 mL or 20 mL of TEOS were introduced in the chamber. Reproducibility of the method was also assessed by adding silica to a large number of samples at the same time (18 samples, half were removed after 7 days of exposition and the rest was left 14 days in the chamber). HCl concentration was 5 wt% and 400 g/L of NaCl were added to the acidic aqueous solution. 4 vials with 10 mL of TEOS were introduced in chamber. Figure A.11 a presents silica contents variations depending on the deposition time for a wide range of conditions (different initial TEOS amounts, HCl concentrations and presence of NaCl). No clear influence of the tested parameters can be observed. The general deposition kinetics appears to be independent of the three tested parameters. Each set of conditions was however only tested on a limited number of samples, so that these assays do not take in account possible variations in a series of sample exposed to the same conditions. To assess distribution of the added silica content on samples place simultaneously in the deposition chamber 18 samples were exposed to TEOS vapors. Two deposition times were investigated (7 days for 9 samples and 14 days for 9 samples). When a large number of samples are silicified at the same time, a wide distribution of silica content can be observed. The histogram of the silica contents is presented in Mean values are 35 ± 13 %SiO2 and 50 ± 7 %SiO2 for samples left 7 days and 14 days respectively. After only 7 days of silica depositions silica content range from 16 %SiO2 to 53%SiO2. After 14 days, the mean silica content increases, but most interestingly the values distribution is much smaller. This may be explained by the fact that TEOS content may not be homogeneously distributed throughout the sealed vessel, as was demonstrated in the case of TMOS vapors 415 . Samples closer to TEOS vials may have higher silica contents. The diminution of the distribution width at long deposition times may be explained by the saturation phenomenon previously described. Samples closer to the precursor source may be coated faster, but final silica content are similar. Material and methods Pectin foams containing encapsulated S. cerevisiae cells were prepared as previously described. The foams were then immerged for 10h in pre-hydrolyzed TEOS at room temperature. The sample were then rinsed with a 50/50 water/ethanol mixture and dried at room temperature. The cell loaded hybrid materials were observed in SEM microscopy, after sputtering with 20 nm of gold. The dry hybrid foams were immerged in a methylene blue solution and kept under static conditions at room temperature until complete discoloration of the dye. Non-encapsulated yeast cells and foam with no encapsulated cells were used as controls. The silica deposition protocol results in strong contraction and deformation of the foams (Figure A.12 a, b and c). This is mainly due to the rinsing and drying step, where the material is subjected to high capillary forced. The use of several successive bath of increasing ethanol content may diminish the contraction of the foams. This may represent an advantage from the structural point of view, but contact with absolute ethanol is cytotoxic for most cells. S. cerevisiae are however use in brewery and have a certain degree of tolerance towards the presence of ethanol. Furthermore, the cells are embedded in a pectin layer which may provide additional protection against deleterious effects of ethanol. As a result a 50/50 water/ethanol mixture seemed to be an acceptable compromise between the structural properties and the viability of the encapsulated species. After silica deposition, the cells can still be seen within the pectin pore walls. On top of the pectin layer, a smooth looking layer of silica can be observed (Figure A.12 a', b' and c'). Discoloration efficiency towards methylene blue of the encapsulated yeast was assessed in an aqueous solution of methylene blue dye at room temperature. The foams containing no yeast were able to adsorb the methylene blue, but this resulted in the coloration of the foams themselves (Figure A.13 b). On the contrary, the samples containing yeast remained colorless, which means that the dyes was adsorbed but also efficiently reduced into its colorless form. It is however to be noticed that the discoloration occurs significantly slower compared to a suspension of free yeast cells. This might be attributed to a loss of viable cells during the various steps of encapsulation since freezing, drying under vacuum, silica deposition and rinsing with a 50% ethanol solution may all be deleterious to the survival and metabolic activity of S. cerevisiae. In addition diffusion issues must also be taken in account since the substrates must go through both the silica and pectin layers, resulting in significantly slower kinetics. The S. cerevisiae / methylene blue system has been used as a model to show the feasibility of a depolluting device based on microorganisms entrapped in a pectin-silica hybrid matrix. This model is however limited since methylene blue is only reduced to a colorless state. In addition S. cerevisiae, though being a common and useful model in microbiology, has no significant uses in bioremediation. This encapsulation assay with a simple and sturdy microorganism however works as a good proof of concept toward the encapsulation of more relevant, but also more sensitive organisms such as bacteria.  EDX analysis of alginate and pectin foam after vapor phase silica deposition process Material and methods Polymer foams were obtained by freeze-casting of 40 g/L solutions of polymers (either beet root pectin or sodium alginate) at 10 °C/min and vacuum dried during 24 h.. Dry foams were cut to 1 mm thick discs and maintained 24h in a desiccated atmosphere before weighting of the initial mass. The samples were then placed in a vapor phase silica deposition chamber in presence of a 5 wt% HCl in water mixture saturated by NaCl and four vials containing 10 mL of TEOS. The samples were removed after 4 or 10 days. The samples were left 24h at 30°C and ambient humidity and 24h at room temperature in a desiccated atmosphere before final weighing. For energy-dispersive X-Ray spectroscopy (EDX) samples were sputtered with 20 nm of carbon. Atomic analysis of the pectin-based samples revealed the presence of high amounts of silicon and oxygen (see Figure A.14 a). The theoretical oxygen: silicon ratio in silica is 2 : 1. However in silica obtained through sol-gel at ambient temperature defects are to be expected. In this case however, the polymer itself contains non-negligible amounts of oxygen which limits the quantitative aspect of this analysis. It must also be pointed out that the EDX technique usually requires the use of plane and smooth surfaces for quantitative analysis, which is not the case here. This analysis however remains a good source of qualitative information regarding the nature of the observed layer. Mapping of the identified elements (see Figure A.14 a') confirms that the silicon and oxygen are homogeneously distributed on the pectin pore wall. EDX analysis on alginate-based foam highlights the presence of high amount of chlorine and sodium, and lower amounts of silicon and oxygen (see Figure A.14 b). The presence of silicon may indicate the partial deposition of silica on the alginate surface. Once again, the presence of oxygen can largely be attributed to the polymer itself, but part of it may be due to the presence of small silica contents. The high chlorine and sodium amounts can easily be traced to the nature of the polymer and the conditions used for silica deposition. In this case the polymer used is alginate, or more precisely alginic acid sodium salt. This means that non-negligible amounts of sodium are introduced in the initial solution and remain in the solid foam after drying. The silica deposition chamber is saturated by a HCl/water atmosphere. As a result it is not surprising to detect traces of chlorine on the samples. Part of the chlorine may be deposited on the pore wall and in presence of the sodium form crystals of sodium chloride, which can be observed as geometric structure on the polymer surface. This theory is supported by the element mapping (see Figure A.14 b'), which shows that oxygen and small amounts of silicon are homogeneously distributed on the pore wall, while chlorine and sodium are concentrated on the square-shaped structures observed in SEM. Le contrôle de ces aspects peut être modulé par deux voies interdépendantes. La composition de la matrice d'encapsulation doit être sélectionnée de façon à assurer la non-toxicité vis-à-vis des organismes encapsulés, mais également vis-vis de l'ensemble de l'écosystème considéré. Le choix des composants de la matrice doit aussi tenir compte des exigences en matière de stabilité dans les sols, afin de minimiser la dissémination des organismes exogènes immobilisés. Le choix de la composition ne peut cependant pas être fait indépendamment des considération d'ingénierie des matériaux et plus particulièrement des aspects de mise en forme des matériaux qui doivent être compatibles avec la survie des microorganismes immobilisés. Concernant la composition de la matrice, la littérature met en lumière l'intérêt des biopolymères comme matrices d'encapsulation. Ces polymères peuvent être trouvés dans les organismes vivants notamment dans les matrices extracellulaires ou les parois cellulaires des plantes. En conséquence la majorité de ces biopolymères présentent une bonne cytocompatibilité, ce qui est essentiel pour l'encapsulation de cellules. Cette cytocompatibilité va cependant souvent de pair avec une biodegradabilité, ce qui peut être un avantage important pour des applications biomédicales, mais peut se révéler problématique du point de vue de la stabilité de la matrice. Une approche permettant la modulation des propriétés de la matrice, tout en conservant une partie de la composition chimique du matériau, consiste en l'utilisation de structures hybrides ou composites. De fait l'utilisation d'une structure hybride biopolymère-inorganique et plus particulièrement l'utilisation de matériaux hybride biopolymères-silice pourrait s'avérer utile dans l'élaboration de matériaux pour la bioremédiation des sols. Du point de vue structural, les matrices d'encapsulation peuvent adopter de nombreuses formes. Concernant l'application ciblée, l'utilisation d'un matériau macroporeux pourrait être un avantage notable du point de vue de de la diffusion des substrats. La méthode du freezecasting peut être utilisée pour mettre en forme de nombreux matériaux (des céramiques aux polymères) de façon à obtenir une large gamme de matériaux à porosité contrôlée et notamment des matériaux à porosité orientée. Ce type de porosité pourrait représenter un avantage décisif pour faciliter la mobilité des substrats ciblés par transport capillaire. Ces travaux portent donc sur l'élaboration de matériaux hybrides contenant des microorganismes encapsulés via un procédé de freeze-casting. L'approche pour la préparation de ces matériaux est basée sur un procédé en deux étapes : Les mousses macroporeuses de pectine ainsi obtenues ont ensuite été modifiées par l'addition d'une couche de silice à la surface des parois des pores. La silice a été déposée par exposition des mousses de pectine à des vapeurs de tétraéthyl orthosilicate en présence d'une atmosphère acide à humidité contrôlée afin de permettre l'hydrolyse et la condensation à la surface des parois des pores, tout en limitant leur dissolution. Cette méthode a permis l'obtention d'une couche de silice entièrement percolée et homogène, sans obstruction des pores. La cinétique de dépôt a pu être contrôlée pour maîtriser l'épaisseur de la couche déposée de façon à influencer les propriétés mécaniques du matériau. La structure hybride peut ainsi être décrite comme un matériau hybride macroporeux avec des parois présentant une structure de type coeur-coquille. La présence de silice a permis de limiter la dissolution des mousses dans des conditions hydratées. Le vieillissement des mousses a notamment été observé dans un sol de type Luvisol, où les matériaux ont présenté une bonne stabilité durant 5 semaines. Ces matériaux hybrides ont également été utilisés pour l'adsorption en milieu liquide d'un polluant modèle (Reactive Black 5, un colorant fréquemment utilisé dans l'industrie textile). Les matériaux ont également été testés dans un sol imprégné par une solution du même colorant. La diminution de la concentration apparente en colorant dans le sol pourrait être une indication de la capacité des substrats à diffuser du sol vers le matériau. La densité de cellules viables, la viscosité de la solution de polymère utilisée ainsi que la vitesse de front de glace employés ont notamment permis d'assurer l'immobilisation des cellules à l'intérieur des parois des pores du matériau. L'activité métabolique des cellules après congélation et séchage a pu être partiellement conservée. Les levures sont de on modèles de laboratoire mais n'ont pas d'intérêt spécifique en matière de bioremédiation. Un deuxième modèle, la bactérie Pseudomonas aeruginosa, a donc été étudié. La méthode du freeze-casting s'est également avérée compatible avec la survie des bactéries, en dépit du stress biologique induit par la congélation et la lyophilisation. Les procédés de cryoprotection et lyoprotection conventionnels nécessitent généralement l'utilisation d'additifs comme le glycérol ou le tréhalose. Dans le cas de l'encapsulation par le biais du freeze-casting aucun cryoprotecteur n'a été utilisé. La présence de biopolymère semble cependant fournir un certain degré de protection vis-à-vis des dommages causés par les contraintes osmotiques et mécaniques lors de la congélation. La vitesse de congélation est également un paramètre crucial dans les protocoles conventionnels de cryoprotection. La technique du freeze-casting a l'avantage de coférer un bon contrôle sur les vitesses de refroidissement, ce qui a permis l'optimisation des taux de survie cellulaire lors de l'encapsulation. L'encapsulation dans une matrice de pectine de betterave s'est cependant avérée limitée du point de vue de la survie à long terme. Cette limitation est problématique compte-tenu des durées importantes (plusieurs jours à plusieurs semaines) nécessaires à l'obtention d'une couche de silice d'épaisseur suffisante par la méthode du dépôt en phase vapeur. La composition de la matrice a donc été modifiée par l'usage de différents polysaccharides. Un meilleur taux de survie des bactéries encapsulées a été observé dans les matrices d'alginate comparées aux matrices de pectine, en particulier après 24h de stockage. Le dépôt de silice en phase vapeur était cependant fortement ralenti dans le cas de l'alginate. Le dépôt de silice a donc été effectué par le biais de chimie-sol gel en phase aqueuse (utilisation d'un mélange de silicates de sodium et d'un mélange commercial de particules de silice colloïdale), en utilisant des possibilités de réticulation de l'alginate en présence d'ions divalents (ici Ca 2+ ). Les matériaux hybrides obtenus par ce procédé sont stable jusqu'à 2 mois dans le sol de référence utilisé pour des essais de vieillissement. Du point de vue fonctionnel, des bactéries viables ont pu être observées aux différents stades du procédé d'encapsulation et après dépôt de silice. La couche de silice déposée semble également permettre la limitation des phénomènes de dissémination des bactéries encapsulée. L'efficacité de la matrice cellularisée en tant que matériau pour la dépollution a ensuite été évaluée en présence de différents polluants modèles. La souche de Pseudomonas aeruginosa utilisée pour l'optimisation du procédé d'encapsulation n'a malheureusement pas présenté d'activité significative pour la dégradation des contaminants modèles étudiés. Una pproche différente a donc été adoptée, en fixant un contaminant cible (le Reactive Black 5) et une souche bactérienne capable de dégrader ce polluant an ensuite été sélectionnée. La bactérie Shewanella oneidensis a donc été utilisée en tant qu'unité fonctionnelle du matériau alginatesilice. L'efficacité du matériau cellularisé a été étudiée dans le sol de référence mentionnée précédemment. Le sol a été saturé par une solution concentrée de RB5 et la concentration en colorant dans le surnageant a été mesurée après incubation. L'ajout des matériaux dans le sol a permis une faible diminution de la concentration apparente en colorant. L'effet n'est cependant que peu marqué, du faible nombre de bactéries Shewanella oneidensis viable encapsulées, ce qui s'explique par le fait que le procédé d'encapsulation a été optimisé pour un autre type de bactéries. Plusieurs options sont envisageables pour confirmer ces observations et améliorer l'efficacité du matériau dépolluant. La principale voie d'amélioration consisterait à optimiser les paramètres d'encapsulation de Shewanella oneidensis, par exemple en ajustant la vitesse de refroidissement lors du freeze-casting, en ajustant la composition de la matrice (présence d'additifs par exemple) ou encore en modifiant les conditions de culture (milieu de culture, phase de croissance). Le modèle pour les essais de dépollution pourrait également être amélioré pour se rapprocher des conditions de terrain (en termes de température ou de taux d'hydratation par exemple). Ce matériau reste néanmoins une preuve de concept encourageante en vue de l'élaboration de matériaux pour la dépollution des sols in situ. Grace à son adaptabilité, le procédé d'encapsulation décrit pourrait être utilisé pour l'immobilisation d'une large gamme de microorganismes (bactéries, champignons, algues etc…). Ainsi, par le choix de microorganismes appropriés (ou de consortia) et la modulation des caractéristiques de la matrice (en termes de propriétés de diffusion, mécaniques, stabilité etc…), des matériaux sur mesure pourraient être créés pour la dépollution de sites contaminés spécifiques. Figure 1 : 1 Figure 1: Bioremediations processes using encapsulated cells as biofunctional units are influenced by a wide variety of interdependent parameters. Figure 2 : 2 Figure 2: The goal of this work is to design a macroporous hybrid material for encapsulation of metabolically active bacteria. The biopolymer structure should provide biocompatibility while the silica layer should ensure structural stability of the material after prolonged stays in soil. Figure 3 : 3 Figure 3:The shaping process was first developed on a material containing no cells to pinpoint relevant parameter to the structural aspects. The process was then applied and adjusted for encapsulation of living organisms. Efficiency of the cellularized hybrid porous material was then assessed in a soil contaminated with a model pollutant. Figure I.1). Most contaminants can be traced to human industrial or commercial activities, among which agriculture and petrochemistry have often been pointed out as major actors. However non negligible contaminations can occur from ill disposal of common consumers goods and domestic materials (see Figure I.2 a). According to European Environment Agency (EEA) heavy metals as well as hydrocarbon derivatives (oil minerals and Polycyclic Aromatic Hydrocarbons (PAH)) are especially widely spread in the surveyed European soils(see Figure I.2 b). Figure I. 1 : 1 Figure I.1: Contaminants can reach soil through several different pathways. Pollution might for instance occur through isolated incidents or repeated exposures. Figure I. 2 : 2 Figure I.2: Overview of economic activities causing soil pollution reveals predominant impact of industrial production and oil industry (a). Sources of contaminations have direct influence on the type of pollutants most commonly found in soil and groundwater (b). Data from European Environmental Agency published in 2009. Retrieved from http://www.eea.europa.eu in July 2017. Figure I. 3 : 3 Figure I.3: (a) Common Polycyclic Aromatic Hydrocarbons. Adapted from Haritash et al. 28 (b) Chemical structure of BTEX (benzene, toluene, ethylbenzene, xylenes. Adapted from Bolden et al. 29 . Figure I. 4 : 4 Figure I.4: Pollutants in soil may undergo a wide range of movements or transformations. Figure I.6:After residency in soils, contaminants tend to become strongly bound to the soil thus limiting the bioavailable fraction. Adapted from Jones et al.[START_REF] Jones | Persistent organic pollutants (POPs): state of the science[END_REF] Contaminant loss by a combination of leaching, volatilization, biodegradation, photolysis etc… Bioavailable contaminant (labile) Irreversibly bound (non-extractable) Recalcitrant but extractable as intact parent compound in solvent C = Concentration at time t, C 0 = Initial concentration I.1.b Bioremediation Figure I. 7 : 7 Figure I.7: Number of publications corresponding to various keywords. Results collected from Web of Science database (retrieved in July 2017). Figure I. 9 : 9 Figure I.9: Cells can be immobilized within matrices with varied geometries such as shells 110 (a), films 111 (b), beads 6 (c), fibers 112 (d) and gels 113 (e). Figure I. 10 : 10 Figure I.10:In encapsulation approaches the immobilization matrix is formed directly around the object to immobilize (b), contrary to surface binding where the object is attached to a pre-existing support (a). Figure I. 11 : 11 Figure I.11: Different types of silica-based materials can be obtained by the sol-gel process. Adapted from Owens et al 202 . Figure I. 12 : 12 Figure I.12: TEOS (a) hydrolysis results in the formation of silicic acid (b). Silicic acid can then be condensed to form a silica network (c and d). Scheme d is reproduced from Wang et al. 203 . Figure I.13: a) SEM image of Arabidopsis thaliana plant cell in matrix prepared from sodium silicate and organosiloxanes (reproduced from Meunier et al. 208 ), b) SEM image of Chlorella vulgaris algae TMOS-based network (reproduced from Darder et al. 209 ), c) SEM image of Pichia pastoris yeast cell in a sodium silicate based matrix (reproduced fromGuan et al. 213 ), d) TEM image of Escherichia coli in a gel prepared from LUDOX and sodium silicate in presence of glycerol (reproduced from Nassif et al. 113 ). Figure I. 14 : 14 Figure I.14: A wide range of architectures can be obtained for silica-biopolymer hybrids. Reproduced from Christoph et al. 227 . Figure I. 15 : 15 Figure I.15: Number of publication related to the keywords freeze-gelation, ice-templating and freeze-casting. Results collected from Web of Science database (retrieved in July 2017). Figure I. 16 : 16 Figure I.16: Polymeric porous materials can be obtained by unidirectional freezing and subsequent lyophilization of and aqueous polymer solution. Adapted from Deville 256 . Figure I. 18 : 18 Figure I.18: A wide variety of morphologies can be obtained for freeze-casted porous ceramic by tuning the formulation of the initial slurry (solvent, binders, dispersants, solid loading and pH). Reproduced from Deville 265 Figure I. 19 : 19 Figure I.19: (a) Hexagonal ice crystals grow preferentially along the a axis resulting in honeycomb (b) or lamellar (c) pore morphologies. (a) reproduced from Deville 256 , (b) reproduced from Mukai et al. 271 , (c) reproduced from Christoph et al. 187 . (see Figure I.20). Figure I. 20 : 20 Figure I.20: Interaction between the freezing front and particles are influences by a wide variety of parameters. Reproduced from Deville 267 . Figure I. 21 : 21 Figure I.21: High cooling rates may induce formation of intracellular ice while too slow rates may result in cytotoxic intracellular solutes concentration (a). A cell dependent optimal cooling rate can be found to minimize these effects (b). (a) adapted from Mazur 352 , (b) adapted from Pegg 348 . Figure I. 22 : 22 Figure I.22: Vitrification requires high cryoprotectant concentrations and very high cooling rates. Adapted from Fahy et al. 355 . It is however possible to identify specific recurrent regions and structures, from which the three most common are homogalacturonan (HG), rhamnogalacturonan I (RG-I) 374 and rhamnogalacturonan II (RG-II) 375 (see Figure II.1). Figure II. 1 : 1 Figure II.1: Pectins are a family of mainly linear polymer based on covalently linked galacturonic acid.Oligosaccharide side chains of different length may also be present, containing the following residues : Galacturonic acid ( ), rhamnose ( ), apiose ( ), fucose ( ), aceric acid ( ), galactose ( ), arabinose( ), xylose ( ), glucuronic acid ( ), ketodeoxymanno-octulopyranosylonic acid ( ). Depending on the pectin source different levels of methylation ( ) and acetylation ( ) can be observed. Reproduce fromWillats et al. 160 at 40 g/L. The pectin used was kindly supplied by Estelle Bonnin and Catherine Garnier at the Biopolymères Interaction Assemblages laboratory at INRA Nantes and was extracted from sugar beet pulp. The dry powder was used without further treatment. Pectin solutions were prepared by dissolution of the powder in deionized water and stirred overnight at room temperature. The resulting viscous solutions were then frozen by various methods described hereafter and subsequently freeze-dried in a Christ Alpha 2-4 LD freezedryer for 48 hours, under 0.05 mbar vacuum. The samples frozen in laboratory freezer were typically 4 mL samples of 40 g/L pectin solution placed in polyethylene cylindrical molds of 19 mm in diameter. The samples were left overnight in the freezer either at -20°C or -80°C before drying. To obtain samples frozen at -196°C, similar samples were directly plunged into a liquid nitrogen bath for around 5 minutes (until no ebullition of the liquid nitrogen was visible). Various samples were obtained by the freeze-casting technique. As previously described, the freeze-casting method aims at providing a controlled temperature gradient, in order to freeze the sample at a controlled rate and with a specific orientation. The homemade setup (see Figure II.2 and Annex p 209) is composed of three copper rods (Ø = 15 mm), plunging in a liquid nitrogen tank for half their length. Each rod is equipped with a heating resistance linked to a PID thermocontroller. A thermocouple is place about one centimeter bellow the top of the central copper rod. Typical cooling profiles include a 3 min equilibration step at 20°C and a cooling ramp at a rate between 1°C/min and 10°C/min. When the temperature reaches -60°C, temperature remains constant until the frozen sample is removed. Cylindrical polypropylene mold can be adjusted on top of the rods to pour the pectin solutions. Scanning Electron Microscopy (MEB) observations were performed on Hitachi S-3400N SEM. The samples were sputtered with 20 nm of gold and observed under 3 to 4 kV acceleration and 30 µA probe current. The OrientationJ plugin to the Fiji software was used to obtain orientation distributions and mapping. (OrientationJ, java plugin for Fiji/ImageJ, written by Daniel Sage at the Biomedical Image Group, EPFL, Switzerland 401 ). Figure II. 2 : 2 Figure II.2:The homemade freeze-casting device is composed of three copper rods plunging in a liquid nitrogen tank to provide cooling and equipped with heating resistances. The temperature is set through a PID thermocontroller. A polypropylene cylindrical mold can be affixed on top of the copper rod. Figure II. 3 : 3 Figure II.3: The temperature profiles obtained in -20°C (a) and -80°C (b) conventional freezers are similar. A slight delay in cooling can be observed in the sample obtained in liquid nitrogen (c). The temperature profile for the freeze-casted sample (d) is very different since the targeted temperature is set to follows a ramp. Figure II. 4 : 4 Figure II.4: The sample obtained in a -20°C conventional freezer (a and a') looks homogeneous. The sample obtained at -80°C (b and b') displays non-aligned striations. The sample obtained in a liquid nitrogen bath (c and c') shows radially arranged structures. The freeze-casted sample (d and d') shows longitudinal iridescence. Scale bars: 5 mm. Figure II. 5 : 5 Figure II.5: SEM observation reveals differences between longitudinal (a, b, c and d) and transversal slices (a', b', c', and d'). Pore are anisotropic in samples obtained at -20°C (a and a'). Pores seem slightly elongated in samples obtained at -80°C (b and b'). Both samples obtained in liquid nitrogen (c and c') and freeze-casted samples (d and d') show well aligned and oriented pores. Scale bars: 1mm. Figure II. 6 : 6 Figure II.6: Mapping of the pore orientation in SEM images of samples obtained at -20°C (a) and -80°C (b) in conventional freezer reveal no specific orientation. Cross sections of samples obtained in a liquid nitrogen bath (c) reveal a radial organization. Longitudinal cuts freeze-casted samples (d) also reveal oriented porosity, but all pores are aligned. Pore orientation distribution confirms these observations (e). The pore orientation distributions were centered on 0° for clarity. Scale bars: 500µm. .3 d). As soon as the temperature of the cooling element reaches the transition temperature, ice crystals nucleate in the bottom region of the sample. The crystals then grow along the temperature gradient, which is to say along the cylinder axis(Figure II.7 c). SEM observation e) of a cross section (Figure II.5 d) reveals the presence of a lamellar structure, where the pores are organized in orientation domains. This lamellar structure is characteristic of materials obtained by freeze-casting of aqueous solutions or suspensions and is the direct consequence of the crystallographic properties of ice 256 . The observation of the longitudinal section (seeFigure II.5 d') reveals that all the pores follow a common direction along the direction of the temperature gradient. This translates into the presence of a single well defined peak in the pore orientation distribution (see FigureII.6). Figure II. 7 : 7 Figure II.7:Ice grows in different ways according to the freezing technique. In conventional freezers (-20°C and -80°C) nucleates homogeneously and ice grows in an anisotropic fashion (a). When plunged into liquid nitrogen ice nucleates in the outer region of the sample and grows radially towards the center of the sample (b). With freeze-casting process, ice nucleates at the interface between the copper and the sample and grows upward in a channel-like manner (c). . 8 a) was to considered orthogonal to the pores and radial compression (Figure II.8 b) was considered along the pores by comparison with the liquid nitrogen samples. In the case of the freezecasted samples however, axial compression (Figure II.8 c) was along the pores and radial compression (Figure II.8 d) was orthogonal to the pores. Figure II. 8 : 8 Figure II.8: Samples were cut down to 1 cm 3 cubes and the mechanical behavior was assessed under compressive strength. Samples were solicited under axial (a and c) and radial (b and d).compression. Figure Figure II.9 shows the stress/strain behavior for materials with various pore organizations and the values for Young's modulus and compressive strength are summed up in TableII.1. An anisotropy ratio was calculated by comparing the Young's modulus measured in the direction of the pores (//) and perpendicularly to the pores (⊥). .8 b). Figure II. 10 : 10 Figure II.10: Foams obtained by freeze-casting (a) show very different wetting behavior compared to radial foams obtained by ice-templating in liquid nitrogen (b). Mass transport seems to be more efficient in freezecasted foams. Figure II. 11 : 11 Figure II.11: Foams prepared at 40 g/L (a and a') display wider and less ordered pores than foams prepared at 50 g/L (b and b') and 60 g/L (c and c'). Scale bars: 500 µm for a, b and c, 100 µm for a', b' and c'. Figure II. 12 : 12 Figure II.12 : Pore width (a) and pore wall thickness (b) are not significantly different when the polymer concentration changes. Figure Figure II.13 presents the aspect of foams obtained by plunging a pectin solution in liquid nitrogen and drying the obtained samples. A reheating step up to 0°C was added to study the influence of the initial temperature of drying. Figure II. 13 : 13 Figure II.13: When sample are reheated prior to drying (a and a') pores are bigger and more interconnected than when sample are dried immediately after freezing (b and b'). Scale bars: 500 µm for a and b, 100 µm for a' and b'. Figure II.14, blue arrow) and not thawing. When the sample is reheated slightly below 0°C, the sample is subjected to partial melting before the pressure reaches a value sufficiently low to ensure ice sublimation.(Figure II.14, red arrow). The repartition of the Figure II. 15 : 15 Figure II.15: Pores in foams obtained from solutions at 20 g/L (a and a') and 30 g/L (b and b') are wider and shorter, as well as less organized than pores in materials freeze-casted from 40 g/L (c and c') and 50 g/L (d and d') pectin solutions. Scale bars: 500 µm. Figure II. 16 : 16 Figure II.16: At higher concentration, pore exhibit narrower (a) and longer (b) pores, as well as thicker pores (c). Variations in width and length of the pore results in a significant shift in the pore aspect ratio (d). Figure II. 17 : 17 Figure II.17 : Higher polymer concentrations result in higher viscosity (a). Above a critical concentration (40 g/L) the polymer chains are percolating resulting in sharp viscosity increase (b). Figure II. 19 : 19 Figure II.19: Transversal (a, b and c) and longitudinal (a', b' and c') SEM observations of foams obtained by freeze-casting of 40 g/L pectin solutions at different freezing rates show common oriented and lamellar porosities. The general morphology is similar for foams obtained at 1°C/min (a and a'), 5°C/min (b and b') and 10°C/min (c and c'), however pore size varies. Scale bars: 500 µm. Figure II. 20 : 20 Figure II.20: Foams obtained at 1°C (a and a'), 5°c/min (b and b') and 10°C/min (c and c') have similar, well aligned morphologies. Observation of transversal slices (a to c) show several orientation domains resulting in the presence of multiple peaks in the orientation distribution. Orientations domains appear larger at low cooling rates. (d). Mapping of the pores orientation on longitudinal slices (a' to c') reveals perfectly aligned pores which translates into single well defined peaks in the orientation distribution (e). Scale bars: 500µm. Orientation distribution curves were normalized, centered on zero and an offset was applied for clarity. Figure II. 21 : 21 Figure II.21: Faster cooling results in narrower (a) and shorter (b) pores, but the aspect ratio does not change dramatically (d). The pore wall thickness decreases (c) at higher freezing rate. Figure II. 23 : 23 Figure II.23: Pectin macroporous foams with unidirectional porosity and tunable pore size were obtained by freeze-casting. Figure III. 2 : 2 Figure III.2: Samples with no silica (a) and with 58 %SiO2 (b) present identical general morphologies. At higher magnification a smooth layer can be observed on both sides of the polymer pore wall (b') for the sample containing silica. Scale bars: 1 mm for a and b, 3 µm for a' and b'. Figure III. 3 : 3 Figure III.3: Element analysis confirms the presence of both oxygen and silicon within the sample (a). Element mapping seems to indicate homogeneous repartition of the silica layer on the pectin pore walls (b). Scale bars: 10 µm. Figure III. 4 g 4 Figure III.4 g present the mass gain depending on the deposition time. Figure III. 4 :Figure 4 Figure III.4: Silica content can be increased by prolonged exposition to TEOS vapors (g), a plateau is reached after about one week. SEM observation of samples with different silica contents reveals the presence of silica layers of different thicknesses (a to f) which vary according the silica content. Scale bars: 1µm. Figure III. 5 : 5 Figure III.5:Plateau in the silica content can be observed on both freeze-casted and radial foams (a). The final deposited mass is higher for radial foams (b). Figure III. 6 : 6 Figure III.6: Silica mass percentage decreases with sample thickness (a). Initial and final mass are proportional to the sample thickness, but the added silica mass is similar for all samples (b). Figure III. 8 : 8 Figure III.8: SEM-FEG observation of sample coated with 5 nm platinum reveal the presence of silica layers with a granulose aspect on the pectin pore walls (a and a'). Measurement of silica thicknesses on TEM images (b and b') confirms the links between silica content and layer thickness. Scale bars: 1 µm for a and a' and 200 nm for b and b'. Figure III. 9 : 9 Figure III.9: TGA of samples with 0 %SiO2 (a), 15 %SiO2 (b) and 43 %SiO2 (c) (as measured by weighing). Loss of mass due to water desorption can be observed in all samples, but residual mass percentage after calcination of pectin vary according to the apparent silica content. Figure III. 10 : 10 Figure III.10:Young's modulus (a) and compressive strength (b) are similar up to 20 %SiO2 and increase for higher silica content. Similarly stress/strain curves for foams with silica contents up to 20 %SiO2 (c) have similar general aspects. At higher silica content, signs of silica failures appear (black arrows). 2 2 Behavior of the hybrid material in a typical soil III.2.a Influence of the silica content on ageing Figure III. 12 : 12 Figure III.12: Pectin foams are rapidly dissolved in water (a). Addition of silica is efficient in preventing the foam dissolution. Samples with high silica content (c) seem to retain their structure over longer period than materials with lower silica content (b). Figure III. 13 : 13 Figure III.13: Pectin concentration increases sharply for samples with no silica in the first few hours (a) and remains stable afterwards (a'). Silica concentration increases faster for supernatant of samples with lows silica content (b'). Figure III. 14 : 14 Figure III.14: Non-percolating silica layers (a) or defects in the silica layer may result in dissolution and leaching of the pectin pore wall. Scale bars: 2 µm for a and 200 nm for b. Figure III. 15 : 15 Figure III.15: High silica contents appear to efficiently protect hybrid porous foams against macroscopic loss of structure. Scale bars: 2cm. Figure III. 16 : 16 Figure III.16: In absence of silica (a), no porosity can be observed in the remaining material. Low silica content (b) allows for better retention of the general foam structure, but the characteristic oriented porosity appears to be damaged. A high silica contents (c and d), the initial porosity appears to be efficiently preserved. Scale bars: 500 µm. Figure III. 17 : 17 Figure III.17: Chemical structure of Reactive Black 5. Figure III. 18 : 18 Figure III.18: No discoloration of the RB5 solution can be observed from contact with 50 mg of soil (b) compared to the aqueous control solution (a).The addition of hybrid foams with 10%SiO2 (c) or 20 %SiO2 (d) results in the apparent discoloration of the supernatant and in the coloration of the foams themselves. Figure III. 19 : 19 Figure III.19: Control RB5 solution (a) and solution in contact with 50 mg of soil (b) do not present any significant discoloration. In presence of foams containing either 10 %SiO2 (c) or 20 %SiO2 (d), about 60 % of discoloration can be observed. Visual observation of the supernatants (see FigureIII.20), which can be seen as the equivalents of rinsing water in an actual polluted sites, show a significant difference between the system containing only soil (Figure III.20 c) and the system containing a foam within the soil(Figure III.20 d). The efficiency of the device seems to be confirmed by the fact that the initially white foam is blue after 24h in the dye loaded soil. Figure III. 20 : 20 Figure III.20: Apparent color of the control sample (a) does not seem to change after 24h at 25°C. The control performed by putting water containing no dye in contact with the reference soil results in a slight yellowish coloration of the supernatant (b). Supernatant of the soil where a pectin-silica foams was introduce (d) appear lighter than the sample containing only soil (c). This is confirmed by the blue coloration of the foam itself (d'). a) RB5 control in water [RB5] i = 100 mg/L b) Water + soil c) RB5 in water + soil d) RB5 in water + soil + foam (10 % SiO2 ) d') Foam after adsorption Figure III. 21 : 21 Figure III.21:The control solution is stable in water over 24h (a). Part of the dye content is adsorbed on the soil (b). Discoloration of the rinsing water is significantly improved in presence of a hybrid foam (c). .22). Figure III. 22 : 22 Figure III.22: Pectin foams where efficiently coated by a silica layer. Structural integrity of the materials was retained after several weeks in soil. Figure IV. 1 : 1 Figure IV.1: S. cerevisiae cells can easily be observed under optical microscope (b). The use of Live/Dead® staining kits (a) allows for easy discrimination between metabolically active cells with intravacuolar red staining and non-active cells with diffuse green staining (green arrows). Scale bars: 20 µm. Due to their size (about 10 µm), S. cerevisiae cells can easily be observed by conventional optical microscopy (seeFigure IV.1 b). This allows for the use of epifluorescence techniques to assess the viability of a yeast suspension, thanks to the use of Live/Dead® assay kits440 . The fluorescent probe FUN1 penetrates the membrane of yeast cells regardless of their viability. Only metabolically active cells are capable of converting the dye from a diffuse green staining, to a red intravacuolar marking (see Figure IV.1 a). IV.1.b Yeast encapsulation in a pectin foam S. cerevisiae was encapsulated in a pectin matrix by simple freezing of a suspension of cells in a 40 g/L biopolymer solution plunged in a liquid nitrogen bath. Figure IV. 2 2 Figure IV.2 gathers SEM images for pectin foams with three different cell contents (no cells, 33.3 g/L and 133.3 g/L). The thickness of the pore wall in absence of cells is inferior to 100 nm, which is smaller than the cell diameter (about 3µm). As a result the shape of the cell can easily be observed within the pore walls. With low cell contents individuals cells can be seen embedded in the pectin pore walls (see Figure IV.2 b'). The overall structure is slightly modified compared to non-cellularized foams, but elongated pores can still be seen (see Figure IV.2 b).It is also possible to include higher cell contents in the initial suspension. This still results in a self-supporting macroporous material after drying, but the pores morphology is modified (see FigureIV.2 c). No clear pore orientation can be observed, and the pores appear much larger. Upon closer look on the pore walls (see Figure IV.2 c') it is noticeable that they are actually mainly composed of cells. At intermediate concentrations, cells appear to be slightly more concentrated around the edges of the pores walls, which may be explained by the presence of different segregation rate during freezing for the cells and the polymer. Assuming a similar density for the pectin powder and dry cells, an initial 33.3 g/L of cell content and 40 g/L of pectin concentration would account for a cell volume percentage of about 45% of the total dry wall volume. For a higher initial cell content (133.3 g/L) the cell volume fraction goes up to 77% of the pore walls. The cellularized materials with highest cell loadings could therefore be compared to ice-templated ceramic materials where solid particles Figure IV.2 gathers SEM images for pectin foams with three different cell contents (no cells, 33.3 g/L and 133.3 g/L). The thickness of the pore wall in absence of cells is inferior to 100 nm, which is smaller than the cell diameter (about 3µm). As a result the shape of the cell can easily be observed within the pore walls. With low cell contents individuals cells can be seen embedded in the pectin pore walls (see Figure IV.2 b'). The overall structure is slightly modified compared to non-cellularized foams, but elongated pores can still be seen (see Figure IV.2 b).It is also possible to include higher cell contents in the initial suspension. This still results in a self-supporting macroporous material after drying, but the pores morphology is modified (see FigureIV.2 c). No clear pore orientation can be observed, and the pores appear much larger. Upon closer look on the pore walls (see Figure IV.2 c') it is noticeable that they are actually mainly composed of cells. At intermediate concentrations, cells appear to be slightly more concentrated around the edges of the pores walls, which may be explained by the presence of different segregation rate during freezing for the cells and the polymer. Assuming a similar density for the pectin powder and dry cells, an initial 33.3 g/L of cell content and 40 g/L of pectin concentration would account for a cell volume percentage of about 45% of the total dry wall volume. For a higher initial cell content (133.3 g/L) the cell volume fraction goes up to 77% of the pore walls. The cellularized materials with highest cell loadings could therefore be compared to ice-templated ceramic materials where solid particles Figure IV. 2 : 2 Figure IV.2: SEM observation of S. cerevisiae cells embedded in pectin foams. Foams with no yeasts (a and a') display an oriented porosity and lamellar pectin walls. The introduction of 33.3 g/L of yeast cells (b and b') in the initial suspension results in clearly visible cells embedded in the pore walls. At a 133.3g/L cell concentration (c and c') the general morphology of the foam is modified and the pore walls are mainly composed of cells. Scale bars: 500 µm in a, b and c, 20 µm in a', b' and c'. .3 g/L of S. cerevisiae b') 33.3 g/L of S. cerevisiae c) 133.3 g/L of S. cerevisiae c') 133.3 g/L of S. cerevisiae and 1.67.10 -2 g/L in methylene blue. The suspensions were then maintained at 35°C until reduction of the methylene blue.Discoloration rates were compared with samples prepared directly from fresh yeast, pectin and methylene blue at the same concentrations. Figure IV. 3 : 3 Figure IV.3:Yeast cells that were encapsulated in pectin foams (a) were able to discolor methylene blue when the foams were dispersed in water. The discoloration is however slower compared to samples prepared from fresh yeast suspension (b). Figure IV. 4 : 4 Figure IV.4: Growth curve for P. aeruginosa in LB medium at 30°C and 150 rpm (a) and corresponding number of CFU per mL at various incubation times (b). Figure IV. 5 : 5 Figure IV.5:Bacteria in a 40 g/l solution of pectin undergo a drop in the number of CFU/mL. This may be explained by the acidic properties of the pectin which yield solutions at a pH around 3. The use of a PIPES buffer results in a number of CFU/mL similar to the control samples in water of aqueous solution of PIPES. Figure IV. 6 : 6 Figure IV.6: SEM observations of foams prepared from different pectin solutions. Samples prepared from a solution composed of 40 g/L of pectin and 100 mM of PIPES buffer (b and b') present larger pores and thicker pore walls compared to foam prepared from 40 g/L pectin solutions (a and a'). Scale bars: 500 µm for a and b, 20 µm for a' and b'. Figure IV. 7 : 7 Figure IV.7: SEM observation of cellularized pectin foams. The shape of the bacteria can be seen under a thin pectin layer (a) and some cells be directly observed within the pore wall (b). Scale bars: 3 µm. Figure IV. 9 : 9 Figure IV.9:Cell in stationary growth phase (c and d) have higher survival rate than cells in exponential growth phase (a and b). When samples were prepared from the same cell culture but concentrated to different initial cell loadings, better survival rate were obtained from less concentrated suspensions. Cell losses are indicated in logarithmic units. Figure IV. 10 : 10 Figure IV.10: Dramatic losses in cell viability are monitored 24 h after the end of drying. At 30 °C (e and f) no CFU was observed, at 4°C (d) 10 4 CFU/mL were counted. Whether vacuum drying was carried out for 24 h or 48 h, similar survival rates were obtained (a and b). Values marqued with the same number of symbols are not significantly different (at p<0.05) Figure IV. 11 : 11 Figure IV.11: Trehalose (a) is a disaccharide commonly used in cryoprotection and as additive in preservation during drying. Glycerol (b) is one of the most commonly used cryoprotectant during freezing of several types of cells. of trehalose at 20 g/L (Figure IV.13 c and c') does not induce significant morphology changes compared to foams prepared from pectin and PIPES alone (Figure IV.13 a an a'). Well oriented and aligned pore can be seen on longitudinal slices of both samples. The transversal section present typical elongated pores arranged in orientations domains. The addition of glycerol at 38 g/L (3 vol%) however significantly modifies the pore morphology. Samples containing pectin, PIPES and glycerol (Figure IV.13 b and b') are similar to samples composed of pectin, PIPES, trehalose and glycerol (Figure IV.13 d and d'). Figure IV. 13 : 13 Figure IV.13: SEM observation of longitudinal (a', b', c' and d') and transversal (a, b, c and d) slices of pectin foams prepared in the presence of cryprotective agents. Foams containing 20 g/L of trehalose (c and c') have similar morphologies to foams containing no additives (a and a'). The addition of 3 vol% of glycerol (b and b') or a 30%vol glycerol and 20 g/L trehalose (d and d') mix result in a significant morphology change. Scale bars: 500 µm. Figure IV. 14 : 14 Figure IV.14: Encapsulation in pectin foam was achieved with about 10 7 CFU/mL after vacuum drying. However, viability at long term proved insufficient to allow vapor phase sol-gel silica deposition. would however be impossible to give an exhaustive overview of the different uses of alginate since it has been one of the most popular biopolymers since it was first extracted by E. C. C. Standford in 1881. Alginate is mainly composed of β-D-mannuronic acid and α-L-guluronic acid (see Figure V.1 a). Figure V. 1 : 1 Figure V.1: Structure of β-D-mannuronic and α-L-guluronic acid, two main components of alginate (a). Alginate is capable of forming gel by binding of divalent cation in an "eggbox" configuration (b). (b) is reproduced from Leick et al. 475 . Figure V.1 b). Gelation properties of alginate have been used to yield materials in a wide variety of shapes. The most common geometry is Figure V. 2 : 2 Figure V.2: Pectin-based freeze casted foams (a) and alginate-based freeze-casted foams (b) have very similar macroscopic aspects. Scale bars: 1 cm. Figure V. 3 : 3 Figure V.3: Freeze casting of 40 g/L alginate solution result in macroporous foams with a pore morphology very close to the one observed in the case of pectin. When freeze-casting is performed at 1°C/min (a and a') pores are wider and shorter than with a 2°C/min freezing-rate (b and b'). At 5°C/min, (c and c') pores are narrower and better organized. Scale bars: 500 µm. Figure V. 4 : 4 Figure V.4: Pore width decreases when higher cooling-rates are used during freeze-casting of alginate solutions. All values are significantly different (at p < 0.05). Figure V. 5 : 5 Figure V.5: Cell-loaded foams have the same general aspect as blank alginate foams (a). At higher magnification (b) cells can be observed in the pectin walls. Cells do not appear to lose structural integrity (c). Figure V. 6 : 6 Figure V.6: Control cells in PBS (a) or alginate (b) are capable of degrading about 87% of the initial glucose content. Dispersion of alginate foams containing no cells (c) does not modify the glucose content. Dispersion of cells encapsulated in alginate foams obtained at either 1°C/min (d), 2°C/min (e) and 5°C/min (f) induces significant glucose consumption. Values marked by the same number of symbols are not statistically different (p < 0.05). Figure V. 7 : 7 Figure V.7: Various solution of polymer freeze-casted at 5°C/min yield different pore morphologies. Beet root pectin (a and a') results well aligned and elongated pores. Alginate based materials (b and b') have a comparable morphologies, but with narrower and shorter pores. The porosity of gelatin samples (c and c') is not very well ordered. A degree of anisotropy can however be observed by comparison of cross-sections (c) and longitudinal slices (c'). The morphology of the PVA foams (d and d') is completely different with almost square-like and very regular pores. The alignment of the pores can be observed in the longitudinal section (d'). Scale bars: 500 µM for full images and 50 µm for inserts. and alginate foams in presence of PIPES buffer and the number of CFU was monitored at different stages (see Figure V.9). Figure V. 10 : 10 Figure V.10: Higher survival rates are observed for cell in stationary phase (c and d) compared to cells in exponential phase (a and b). The presence of PIPES (a and c) significantly enhances the survival rates. Values marked with different numbers of symbols are significantly different (at p < 0.05). Figure V. 12 : 12 Figure V.12: After 4 days of silica deposition, mass gain represented around 70 % of the final mass in pectin samples, but only about 10 % in alginate samples. After 10 days of deposition the mass gain represented 80 % of pectin samples and 30 % of alginate samples. Figure V. 13 : 13 Figure V.13: After treatment by vapors of silica precursor (TEOS) in an acidic atmosphere (HCl), pectin samples are coated by a smooth layer of silica (a and a'). Alginate samples (b and b') present square-like structures on top of the pore wall surface. Scale bars: 10 µm. Figure V. 14 : 14 Figure V.14: Freeze-casted alginate foams and crosslinked alginate foams (b) shrink significantly after drying compared to alginate foams directly after lyophilization (a). Foams of crosslinked alginate coated with silica undergo similar shrinkage. Figure V. 15 : 15 Figure V.15: Significant foam shrinkage can be observed due to the drying step necessary for SEM observation.Both crosslinked and dried alginate foams (a, a' and a'') and alginate foams crosslinked and coated with silica (b, b' and b'') before drying present very similar aspects. Oriented pores can still be observed, but significant distortion of the pore walls results irregularities in the shape of the pore themselves (a' and b'). Upon closer observation a granulose-looking surface can be seen on the silica-coated samples (b'') which is not visible on samples simply crosslinked (a''). This layer is however very inhomogeneous in thickness. Scale bars: 500 µm. Figure V.16 d and e) tend to slightly shrink due to probable partial dehydration. On the other hand, samples that were introduced in a dry state (Figure V.16 b and c) do not appear to undergo any change in aspect even after almost two month. Figure V. 16 : 16 Figure V.16: Foams obtained by simple freeze-casting of alginate (a) undergo significant aspect modifications.After one day the foam appears to have shrunk and after one week there seem to be a degradation of the material. Alginate foams which were crosslinked and dried (b) or crosslinked, silica coated and dried (c) do no change in aspect, even after two months in soil. Similar samples introduced in a hydrated state (only crosslinked (d) or crosslinked and silica-coated (e)), seem to shrink slightly after one day, but remain stable afterward. Figure V. 17 : 17 Figure V.17: The oriented and organized porosity of freeze-casted alginate foams (a) is completely lost after a 3-week stay in a hydrated soil and subsequent drying for SEM observation (b). Scale bars: 500 µm. Figure V. 18 . 18 Figure V.18. The general morphology of crosslinked (a and c) and crosslinked and silica coated (b and d) foams is not altered by a 3-week stay in hydrated soil. Oriented channel-like pore can still be observed after a stay in soil and subsequent drying for SEM observation (a',b',c' and d'). Initial dry (a and b) or hydrated state (c and ) of the samples does not seem to have a significance on the ageing behavior. At higher magnification (a'',b'',c'',d'') the silica layers do not see to be significantly modified. Scale bars: 500 µm for a,a',b,b',c,c',d, and d', 10 µm for a'',b'',c'' and d''. Figure V. 19 : 19 Figure V.19: P. aeruginosa cells could be observed within the alginate pore walls. Figure V. 20 : 20 Figure V.20: Bacteria (light blue arrows) can be observed within the walls of crosslinked alginate foams (a and a') and crosslinked and silica coated alginate foams (b and b'). Cells are entrapped in small cavities within the polymer matrix. Silica can be seen as small particle aggregates (dark blue arrows) but does not form a continuous or homogeneous layer. Scale bars: 1 µm for a and b, 500 nm for a' and b'. .21 b) compared to samples simply crosslinked (Figure V.21 a), which might be due to modification of the mechanical properties in presence of silica gels. At higher magnification individual cells may be observed (see Figure V.21 a and b). A large majority of them appear stained in red, which means that cells are structurally damaged. a) Bacteria in crosslinked alginate a') Bacteria in crosslinked alginate b) Bacteria in crosslinked and silica coated alginate b') Bacteria in crosslinked and silica coated alginate Figure V. 21 : 21 Figure V.21: Living encapsulated cells are stained in green by Syto 9® dye and dead cells are stained in red by propidium iodide. The high concentration of cells within the pore walls allows the observation of the porous morphology of the foam at low magnification. Crosslinked and silica coated samples (b) seem to undergo less deformation during sample preparation than simply crosslinked foams (a). At higher magnification, no significant difference in cell repartition or live and dead ratio is noticeable between crosslinked materials (a') and crosslinked and silica-coated material (b'). Scale bars: 200 µm for a and b, 50 µm for a' and b'. Figure V. 23 : 23 Figure V.23:When alginate samples are simply crosslinked (a) a larger halo of bacteria can be observed compared to samples that were both crosslinked and coated with silica (b). Samples with no silica appear to shrink slightly. After 5 days at room temperature, a greenish coloration can be observed due to the production of pyoverdine and pyocyanine by encapsulated P. aeruginosa. Scale bars: 1 cm. Figure V. 24 : 24 Figure V.24: Chemical structure of Reactive Black 5. Figure V. 25 25 Figure V.25 present the visual aspect of the various degradation assays after 24h. Efficiency of S. Oneidensis for decolorization of Reactive Black 5 was visually confirmed, both at a concentration of 10 mg/L (Figure V.25 a) and at 100 mg/L (see Figure V.25 b). These concentrations are not representative of actual pollutant concentrations on contaminated sites, but they work as efficient models for a proof of concept, within the limitations of easy and efficient detection in laboratory conditions. Figure V. 25 : 25 Figure V.25: Control samples of RB5 solutions at 10 mg/L (a) and 100 mg/L (b) after 24h of incubation at 37°C and samples containing about 10 7 U/mL of autoclaved S. Oneidensis (a' and b') appear identical. Samples inoculated with 10 7 CFU/mL of fresh S. Oneidensis appear either discolored when the initial dye concentration was 100 mg/L (b'') or completely colorless when the initial dye concentration was 10 mg/L (a''). Figure V.26 a and b). A slight shift and deformation in the absorbance maxima (see Figure V.26 a' and b') could be observed (to 594 nm and 588 nm for [RB5] = 10 mg/L and [RB5] = 100 mg/L respectively). RB5] = 10 mg/L -No bacteria a') [RB5] = 10 mg/L -Autoclaved bacteria a'') [RB5] = 10 mg/L -Fresh bacteria b) [RB5] = 100 mg/L -No bacteria b') [RB5] = 100 mg/L -Autoclaved bacteria b'') [RB5] = 100 mg/L -Fresh bacteria Figure V. 27 :Figure V. 28 : 2728 Figure V.27: Soil soaked in [RB5] = 0.5 g/L solutions in simplified mineral medium were incubated 42h at 37°C. (see Figure V.29). Figure V. 29 : 29 Figure V.29: Efficient encapsulation of P. aeruginosa was achieved and viability was maintained throughout the silica deposition process. S. Oneidensis was encapsulated and efficiency regarding remediation of Reactive Black 5 was assessed. Figure A.1). Figure A. 1 : 1 Figure A.1: Box-and-whiskers plots display the mean and median values as well as the 25 th -75 th and 5 th -95 th percentiles. Figure A. 2 : 2 Figure A.2: For plate counting, 50 µL of 4 successive dilutions were spread in triplicate, in 12 well-plates filled with LB-agar gel. Figure A. 3 :Figure A. 5 : 35 Figure A.3: Calibration curve for RB5 solutions at 598 nm. Figure A. 6 : 6 Figure A.6: Freeze-casting setup. Figure A. 7 :- 7 Figure A.7: Freeze-dried macroporous pectin foams (a) were place in a closed vessel in presence of an acidic atmosphere (b) and vapors of TEOS (c). The deposition chamber was maintained at 30°C throughout the deposition process. Figure A. 8 : 8 Figure A.8: Ageing behavior of macroporous hybrid foams was assessed in a upper horizon silt loam Luvisol. Figure A. 9 : 9 Figure A.9: Influence of macroporous hybrid foams on simulated polluted soils was assessed using RB5 as a model contaminant. Figure A. 10 : 10 Figure A.10: Pore size variations induced by different initial pectin concentration or freezing rates have no significant effect on the wetting profiles of the foams. Figure A.11 a presents silica contents variations depending on the deposition time for a wide range of conditions (different initial TEOS amounts, HCl concentrations and presence of NaCl). No clear influence of the tested parameters can be observed. The general deposition kinetics appears to be independent of the three tested parameters. Each set of conditions was however only tested on a limited number of samples, so that these assays do not take in account possible variations in a series of sample exposed to the same conditions. To assess distribution of the added silica content on samples place simultaneously in the deposition chamber 18 samples were exposed to TEOS vapors. Two deposition times were investigated (7 days for 9 samples and 14 days for 9 samples). When a large number of samples are silicified at the same time, a wide distribution of silica content can be observed. The histogram of the silica contents is presented in Figure A.11 b. Mean values are 35 ± 13 %SiO2 and 50 ± 7 %SiO2 for samples left 7 days and 14 days respectively. After only 7 days of silica depositions silica content range from 16 %SiO2 to 53%SiO2. After 14 days, the mean silica content increases, but most interestingly the values distribution is much smaller. This may be explained by the fact that TEOS content may not be homogeneously distributed throughout the sealed vessel, as was demonstrated in the case of TMOS vapors 415 . Samples closer to TEOS vials may have higher silica contents. The diminution of the distribution width at long deposition times may be explained by the saturation phenomenon previously described. Samples closer to the precursor source may be coated faster, but final silica content are similar. Figure A. 11 : 11 Figure A.11: Silica content evolution does not seem to depend strongly on the HCl and NaCl content or on the introduced volume of TEOS (a). A wide distribution of silica content may be observed when a large number of samples are coated simultaneously, however this distribution is narrower after longer deposition times (b). Figure A. 12 : 12 Figure A.12: SEM observation of S. cerevisiae cells in pectin foams. Foams containing 0 g/L (a and a'), 33.3 g/L (b and b') and 133 g/L (c and c') of yeast cells are slightly contracted by the rinsing and drying treatment. Cells can still be seen within the silica coated pectin wall. Scale bars: 1 mm in a, b and c, 20 µm in a', b' and c'. .3 g/L of S. cerevisiae b') 33.3 g/L of S. cerevisiae c) 133.3 g/L of S. cerevisiae c') 133.3 g/L of S. cerevisiae Figure A. 13 : 13 Figure A.13: Foams containing encapsulated yeasts (d) were able to efficiently discolor a methylene blue solution (a). Non-encapsulated yeast (c) were able to discolor the dye solution quicker, but control foams with no encapsulated yeast (b) were only able to adsorb the model of pollutant. Figure A. 14 : 14 Figure A.14: Pectin-based foams present high contents of oxygen and silicon (a) which may be assumed to be part of a silica layer. These two elements are homogeneously distributed on the whole pore wall surface (a'). In alginate-based material, the two main detected elements are sodium and chloride (b). They are mainly concentrated in the square-like structures observed on the surface of the pore wall (b') which are likely to be NaCl crystals. Scale bars: 10 µm. Figure A. 15 : 15 Figure A.15: Les procédés de bioremédiation reposant sur l'utilisation de microorganismes encapsulés comme unités fonctionnelles sont influencés par une large gamme de paramètres interdépendants. 1 . 1 encapsulation des microorganismes choisis dans une matrice poreuse de biopolymère par freeze-casting 2. dépôt d'une couche de silice par chimie du sol-gel Chacune de ces étapes doit être adaptée à la nature des composés utilisés et contrôlée pour garantir la morphologie souhaitée et la compatibilité avec la survie cellulaire. La stratégie adoptée pour identifier les paramètres d'intérêt pour le contrôle structural et fonctionnel des matériaux a été de commencer par l'élaboration séquentielle du matériau hybride en l'absence de microorganismes (obtention d'une structure de biopolymère à porosité orientée et dépôt de silice par. Le procédé ainsi mis au point a ensuite été utilisé pour l'obtention du matériau cellularisé, les différentes étapes étant modulées pour garantir une survie optimale des microorganismes (voir Figure A.16). Les possibilités en terme de structure et de morphologie ont dans un premier temps été évaluées par congélation de pectine de betterave en solution aqueuse. Plusieurs dispositifs de congélation ont été étudiés (congélation dans des congélateurs conventionnels à -20°C et -80°C, utilisation d'un bain d'azote liquide et utilisation d'un montage de freeze-casting). Ces essais ont permis de mettre en évidence le rôle de la présence d'un gradient de température dans l'obtention de de matériaux avec une porosité orientée. Figure A. 16 : 16 Figure A.16: Le procédé de mise en forme a d'abord été mis au point en l'absence de microorganismes de façon à identifier les paramètres pertinents du point de vue structural. Le procédé a ensuite été adapté pour l'encapsulation de microorganismes. L'efficacité du matériau pour des applications de dépollution a enfin été évaluée dans un sol de référence contenant un polluant modèle. La structure ainsi mise au point a ensuite été utilisée comme matrice d'encapsulation pour des organismes avec des capacités de bioremédiation. Les composants et les méthodes de mise en forme ont été choisis initialement en tenant compte des contraintes imposées par la présence finale des microorganismes (utilisation d'eau comme solvant, températures modérées, utilisation de biopolymères). L'utilisation du freeze-casting comme méthode d'encapsulation a d'abord été confirmée sur la levure Saccharomyces cerevisiae comme organisme modèle. Figure A. 17 : 17 Figure A.17: Des bactéries bioactives ont été encapsulées dans un matériau hybride poreux. La technique du freeze-casting a été utilisée comme un moyen d'obtenir à la fois une encapsulation efficace des cellules et une morphologie contrôlée. La chimie du sol-gel a permis d'obtenir une couche de silice sur les parois des pores. Les matériaux poreux hybrides à base d'alginate et silice contenant la bactérie Shewanella oneidensis ont ainsi pu être utilisés pour la dégradation du Reactive Black 5 dans un sol de référence. Table I . 1 : I1 Comparison of the main cleanup techniques for soil remediation. Based on data from EPA 56 . Limitations -Contaminants are still in the soil -Not applicable for large areas -Contaminant is not removed from the site -Requires use of complementary technique -Requires precise mapping of contaminated area -Requires excavation -Not applicable for all types of soil (like soils with high clay or organic contents) -Requires excavation -Wash solutions must be treated appropriately -Requires excavation -Not efficient for metals -Requires a lot of fuel -Addition of chemicals to the polluted site -Diffusion of the reactive agents may be slow -Requires secondary treatment -Not very efficient on soil with high silt or clay contents Advantages -Efficient for metals and radioactive contaminants -Quick and cost effective -Quick and cost effective -Can be applied to large contaminated sites -Often quickest cleanup technique -Prevents propagation of the contamination -Efficient for volatile or semi- volatile compounds -Possibility to condensate vapors for recycling -Efficient for a wide range of pollutants (from fuel residues and organic compounds to metals -Efficient reduction of the contaminated volume -Efficient on a wide range of pollutant -Low residuals amounts of polluted materials -In situ technique, no need for excavation -Can be used for deep contaminations -In situ technique -Can be used for deep contaminations Description Solidification is the formation of a solid block through addition of a binding agent (cement, asphalt, clay etc…) where compounds are immobilized. Stabilization is the chemical modification of contaminant to prevent leaching. Installation of a cover (asphalt or concrete, vegetative layer, geomembrane, clay) over contaminated materials to prevent contact of the contaminated areas with human or animals. Physical removal of contaminated soil, sediments or sludge by digging. Excavated materials are usually treated with a complementary cleanup method. Cleaned Table I . I Chemical class Examples Biodegradability Aromatic hydrocarbons Benzene, tolune Aerobic and anaerobic Ketones and esters Acetone, LEK Aerobic and anaerobic Petroleum hydrocarbons Fuel oil Aerobic Chlorinated solvents TCE, PCE Aerobic (methanotrophs), anaerobic (reductive dechlorination) Polyaromatic hydrocarbons Anthracene, benzo(a)pyrene, Aerobic creosote Polychlorinated biphenyls Arochlors Some evidence, not readily degradable Organic cyanides Aerobic Metals Cadmium Not degradable, experimental biosorption Radioactive materials Uranium, plutonium Not biodegradable Corrosives Inorganic acids, caustics Not biodegradable Asbestos Not biodegradable 2: Biodegradability of different classes of chemical. Reproduced from Iwamoto et al. 2 Table I . 3 : I3 Sources and applications of most common biopolymers. Type Polymer Main source Applications Ref. Polysaccharides Anionic Alginate Brown seaweed Food, biomaterials 155,156 Agar Red seaweed Food, microbiology 157 Carrageenan Red seaweed Food, cosmetics 158 Pectin Fruits (citrus, apple, beetroot…) Food, beverage, biomedecine 159,160 Xanthan Bacteria Food, cosmetics, pharmaceutic 161,162 Cationic Chitin Crustacean Food, biomedecine 163,164 Chitosan Chitin derivative Food, cosmetics, biomedecine 163,164 Neutral Cellulose Higher plant cell wall Paper industry, biomedecine 152,165 Starch Corn, wheat, potato Food, pharmaceutic 166167 Proteins Collagen Animals (cattle, porcine) Biomedecine 168 Gelatin Collagen derivative Food, biomedecine 168 Fibroin Silk (spider, moth) Biomedical 169 Zein Corn Textile, coatings 170 Polyphenol Lignin Wood Paper, chemistry 171,172 Polyisoprene Rubber Hevea Automobile, consumers goods 173 gels in sol-gel techniques, such materials have been used for immobilization of sensitive molecules such as dyes in 1955 204 or enzymes in 1971 205 . Silica gels have since been used for the encapsulation of various biological species including proteins 206 , mammalian cells 207 , plant cells 208 , yeast 111 , bacteria 113 or algae 209 (see Figure I.13). Fields of application include medicine (drug, protein and cell delivery 203,210 ), biosensors 211 or environmental science 212 . d) Monomer Dimer Trimer Silica particle Table II . 1 : II1 Young's modulus and compressive strength for foams obtained from freezing of 40 g/l aqueous pectin solutions under various freezing conditions. Conditions Freezing rate (°C/min) Anistropy ratio Young's modulus (MPa) Std dev (MPa) Compressive strength (kPa) Std dev (kPa) -20°C freezer 1.2 28.9 1.07 0.50 65 17 Compression -80°C freezer 3.6 19.4 0.27 0.10 NA NA orthogonal to the pores (⊥) Liquid nitrogen 243.3 21.0 0.49 0.09 NA NA (-196°C) Freeze casting at 7.2 95.2 0.13 0.03 NA NA 10°C/min -20°C freezer 1.2 28.9 1.50 0.47 67 34 Compression -80°C freezer 3.6 19.4 0.34 0.15 NA NA along the pores (//) Liquid nitrogen 243.3 21.0 0.62 0.78 NA NA (-196°C) Freeze casting at 7.2 95.2 2.61 0.65 92 2 10°C/min Table II .3 : Mechanical II Concentration (g/L) Density (kg/m 3 ) Std dev (kg/m 3 ) Young's modulus (MPa) Std dev (MPa) Compressive strength (kPa) Std dev (kPa) 20 23.7 1.3 0.97 0.17 20 2 30 31.3 1.5 1.13 0.25 58 1 40 40.7 1.2 2.91 0.76 115 6 50 48.5 1.9 4.96 2.05 185 8 characteristics of foams obtained by freeze-casting at 10°C/min from pectin solutions at various concentrations under compression along the pores direction. Figure II.18: Higher pectin concentrations result in higher apparent density which results in higher Young's modulus and compressive strength The pore width mean values range from 10,4 µm at 10°C/min to 18,7 µm at 1°C/min. Slower freezing is responsible for the formation of larger ice crystal. The resulting pores after freeze-drying are therefore wider. A decrease in the pore wall width can also be observed at high cooling rates (see Figure II.21 c) (200 nm at 10°C/min vs 312 nm at 1°C/min). All these samples have a similar density, so that when the number of pores per cm² increases (i.e. when pores are smaller), the amount of polymer available for each pore wall decreases, resulting in thinner pore walls. Contrary to samples prepared at different pectin concentrations, in this case the apparent density of the foams is similar although the morphology of the pore and the pore wall thickness are different. Freezing-rate (°C/min) Young's modulus (MPa) Std dev (MPa) Compressive strength (kPa) Std dev (kPa) 10 1.21 0.32 91 18 5 2.98 1.70 87 7 1 2.84 1.84 87 18 These small structural changes have no clear influence on the mechanical properties of the foams. Values for Young's modulus and compressive strength are presented in Figure II.22 and Table II.4. Figure II.22: No significant changes in mechanical properties can be seen in samples prepared at various freezing rates. Table II .4: Mechanical characteristics of foams obtained from pectin solutions at various freezing rates under compression along the pores direction. ranging from a few days to several weeks), samples were removed from the closed vessel and maintained 24h at 30°C in an open container and subsequently at least 24h in desiccated atmosphere at room temperature. Samples were weighted again and this value was considered as the final mass (note mf). g/L beet root pectin solutions were frozen either by dipping in liquid nitrogen (hereafter noted radial foams) or by freeze-casting at 10°C/min (hereafter noted freeze-casted foams), and subsequently vacuum dried at least 24h at 0.05 mbar. Foams were maintained in a desiccated atmosphere until further use. Before silica deposition, pectin foams were weighted (initial mass was noted mi). Silica deposition in vapor phase was performed in a closed vessel (see Annex p 210) containing an aqueous solution of HCl at 5 wt%, saturated with NaCl (typically 17 mL of 37 wt% HCl was introduced in 133 mL of deionized water and 60 g of NaCl was added). Samples were placed on a perforated plate. Vials of tetraethyl orthosilicate (TEOS) were introduced (typically 4 vials containing 10 mL of TEOS). The deposition chamber was then sealed and maintained at 30°C. After chosen deposition times ( Table III . 1 : III1 Young's modulus, compressive strength and densities for foams with various silica contents. 200 C 490 C 440 C Silica content Apparent density Young's Modulus Compressive (%SiO2) (mg/cm 3 ) (MPa) strength (kPa) 0 41.0 1.3 145 12 50.7 1.1 140 17 48.8 1.0 147 24 58.9 1.3 179 28 59.2 2.4 186 32 51.9 2.1 204 43 76.2 3.4 287 52 85.4 4.8 336 Table IV . 1 : IV1 Bioremediation capabilities of various P. aeruginosa strains. (*) corresponds to compounds mentioned in the REACH restrictions list (Annex XVII of REACH as retrieved from https://echa.europa.eu in July 2017) Pollutants Reference Hydrocarbons Hexadecane 442,443 Crude oil 444-446 Anthracene (*) 447 Pyrene 448 Phenanthrene 449,450 BTEX Mixture 451 Benzene, toluene, xylene 452 Benzene 453 Toluene (*) 454 Toluene derivatives Dinitrotoluene 455 Trinitrotoluene 456,457 Azo dyes Direct orange 39 458 Remazol black, methyl orange, 251 benzyl orange Phenol derivatives Phenol 459 Bisphenol A (*) 460 PCB PCB 461 Pesticides Tetramethylthiuramdisulfide 462 Fenpropathrin 463 Table IV . 2 : IV2 Pore morphology with and without PIPES buffer in the initial pectin solution. Samples Pore width (µm) Std dev (µm) Pore length(µm) Std dev (µm) Aspect ratio Std dev (µm) Pore wall thickness (µm) Std dev (µm) Pectin at 40 g/L, 5°C/min 12 4 366 258 30 13 0.2 0.1 Pectin at 40 g/L PIPES at 100 mM, 5°C/min g/L(15 vol%) solution of glycerol or a 100 g/L trehalose and 189 g/L(15 vol%) glycerol solution. The cells were left 30 min at room temperature in theses suspensions and subsequently dispersed in a solution of 50 g/L of pectin and 125 mM of PIPES. The final suspension contained about 1.10 9 CFU/mL, 40 g/L of pectin and 100 mM of PIPES. The final additives concentrations were 20 g/L for trehalose and 38 g/L (3 vol%) for glycerol. Cell suspensions were freeze-casted at 5°C/min and subsequently vacuum dried for 24 h. Viability was monitored by plate counting. The number of CFU was controlled in samples with various additives at three different stages: in samples frozen and thawed (30 min at 30°C), samples dispersed in water directly after 24h of vacuum drying and in samples kept 24h at room temperature in a desiccated atmosphere after drying. Three samples were prepared for each initial suspension composition and each dilution of dissolved foam was plated in triplicate. P. aeruginosa cells were encapsulated in pectin foams in presence of glycerol and/or trehalose (see Figure IV.11) . Bacteria were grown in LB medium at 30°C and 150 rpm up to 0.5 OD. The culture was then centrifuged and dispersed in water, a 100 g/L solution of trehalose, a 189 Syto 9 at a 0.05 mM concentration). The foams were then washed thrice with water to remove excess dye and embedded in Dako fluorescence mounting medium for confocal microscopy observation. Confocal microscopy (see Annex p201) acquisitions were performed on about 40 µm of thickness with a 0.3 µm step. A z projection of all images was made for cell counting. P aeruginosa cells were embedded in alginate foams. SEM microscopy allowed for observation of the cells, either under a polymer layer, or directly when pore walls were cut appropriately (see Figure V.19). P. aeruginosa was encapsulated in alginate foams as previously described by freeze-casting at 5°C/min of a stationary phase cell suspension in alginate and PIPES buffer and subsequent vacuum drying. Samples were then immerged in a 0.5 M solution of CaCl2 (filtered at 0.2 µm) and kept 24h at 4°C to ensure homogeneous crosslinking of the sample. The foams were then rinsed in sterile water and the excess water in the pores was removed by briefly putting the samples in contact with absorbing paper. The samples were then plunged in a mixture of 50 vol% of LUDOX ([Si]=7.8 M) and 50 vol% of sodium silicates ([Si]=0.2 M) acidified to pH=5 by HCl 4M. Total concentration in Si was therefore 4M. Samples were soaked 45 min in this mixture and removed before gelation. Samples were briefly rinsed with sterile water and left 30 min at room temperature and ambient humidity. Samples were then typically kept in sterile water at 4°C. Samples were included in epoxy embedding medium (see Annex p 200) and 50 nm to 80 nm slices were cut for TEM observation. For confocal microscopy, samples were cut to thin slices (less than 1 mm) and place on glass slide before addition of Live/Dead® dye (Propidium Iodide at a 0.3 mM concentration and 3.b Decolorization of Reactive Black 5 by Shewanella OneidensisShewanella oneidensis, has been reported to have degrading abilities towards dyes499,500 . Degradation proved very efficient in mineral medium, both for Methyl Orange and Reactive Black 5. Reactive Black 5 (hereafter noted RB5) (see FigureV.24) was used for further assays due to its higher solubility in water. .5.10 7 CFU/mL of free S. Oneidensis. 7 mL of the [RB5] = 0.5 g/L in simplified mineral medium solution was also incubated 42h at 37°C with no soil or bacteria to assess the stability of the dye in the considered medium. 9 CFU/mL of S. Oneidensis, 40 g/L of alginate, 100 mM of PIPES buffer. The suspension was then freeze- casted at 5 °C/min and vacuum dried. Blank samples were prepared in the same manner but without introduction of cells. Foams were then crosslinked 6 h in CaCl2 (0.5 M) and subsequently impregnated with LUDOX/silicate at pH=5 ([Si]LUDOX = 3.9 M and [Si]silicates = 0.1 M). Foams were rinsed and left at room temperature until formation of silica gel before being introduced in soil for remediation assays. Samples were prepared in triplicate. Efficiency the encapsulation was controlled by dissolution of part of the cell loaded foams directly after vacuum drying in water and plate counting. In typical remediation assays, 10 g of soil (silt loam Luvisol sampled at the Versilles INRA station) was soaked with 7 mL of simplified mineral medium (see composition in Annex p 204) containing 0.5 g/L in RB5. After 42h under static conditions at 37 °C, 7 mL of PBS 2X was added and the mixture was vigorously shaken for. 2 mL of supernatant was centrifuged 10 min at 5000 rpm and the UV-vis absorbance spectrum was measured (samples were diluted by an appropriate factor to fit in the linearity range of the calibration if needed). Assays were made in triplicate with cell-loaded foams and blank foams. Control were performed in triplicate with soil in contact with the [RB5] = 0.5 g/L solution, soil in contact with the mineral medium only ([RB5] = 0 g/L) and soil in contact with [RB5] = 0.5 g/L inoculated with 3 -max module could be equipped for Energy Dispersive X-Ray spectroscopy. For SEM observation were samples were usually cut with a scalpel into slices thinner than 1 mm and coated with 20 nm of gold by metal sputtering. Observations were typically conducted under 3 to 4 kV acceleration and 30 µA probe current. If required, the samples were previously dehydrated by successive bathes of increasing ethanol content. The samples were plunged at least 30 mins (up to several hours depending on the sample size) in ethanol solutions at 20, 40, 60, 80 and 100%. The ethanol impregnated samples were then left to dry at room temperature. Publications Characterizations -Journal articles  Microscopy  Christoph, S.; Barré, P.; Haye, B.; Coradin, T.; Fernandes, F. M., Ice-templated hybrid -SEM/EDX biofoams: bacterial encapsulation, viability and biodegradation activity in soil, Submitted Scanning Electron Microscopy was performed on Hitachi S-3400N microscope. An Oxford 2017  Christoph, S.; Hamraouia, A.; Bonnin, E.; Garnier, C.; Coradin, T., Fernandes, F. M., Ice-Instruments -X templating pectin: towards texture, mechanics and capillary properties control in fully biodegradable foams, In preparation 2017  Christoph, S., Kwiatoszynski, J., Coradin, T. and Fernandes, F. M. Cellularized Cellular Solids via Freeze-Casting, Macromol. Biosci., 2016, 16 (2), 182-187  Christoph, S.; Fernandes, F. M.; Coradin, T., Immobilization of Proteins in Biopolymer- Silica Hybrid Materials: Functional Properties and Applications, Curr. Org. Chem., 2015, 19 (17), 1669-1676 -Book section  Bionanocomposites: Integrating Biological Processes for Bio-inspired Nanotechnologies Chapitre 5.4 : Bionanocomposite materials for biocatalytic applications Editors : Carole Aime, Thibaud Coradin Wiley, 2017 Communications -Poster presentation  HINT Training School : Bottom-up Approaches of Hybrid Materials: Preparation and Design, Ljubljana, Slovenia; 2015  International Symposium on Macroporous Materials: From Novel Preparation Techniques to Advanced Applications, Paris, France; 2016 -Oral presentations  EMRS 2016, Lille, France; 2016 Degradation products  CellMAT 2016, Dresden, Germany; 2016 chemicals were usually purchased from Sigma-Aldrich and use as received. Solvents at reagent grade (ethanol, N,N-dimethylformamide) were purchased from VWR. Beet root pectin was kindly provided by Estelle Bonnin and Catherine Garnier at the Biopolymères Interaction Assemblages laboratory at INRA Nantes The reference soil was kindly provided by Pierre Barré at the Laboratoire de Géologie de l'ENS. The soil is upper horizon (0-30 cm) of silt loam Luvisol 507 , developed on loess deposits. The texture of the samples was characterized by 18 % clay (particles < 2 µm), 57 % silt (2 µm < particles < 50 µm) and 25 % sand (particles > 50 µm). The cation exchange capacity (11 cmol(+)/kg) is saturated mainly by calcium. The pH is 6.1 and the total organic carbon content of the upper horizon is 13 gcarbon/kgsoil. The soil was typically rehydrated with 0.16 mLwater/gsoil (18 mL of water for 110 g of soil). Water use was typically deionized water (9.2 MΩ.cm). For microbiology, water was filtered at 0.2 µm. Experimental section  Material  Microbiology -Culture media LB medium (lysogen broth, also known as Luria-Bertani medium) was prepared from deionized water and 20 g/L of commercial LB powder (Sigma-Aldrich Lennox broth, 10 g/L tryptone, 5 g/L yeast extract, 5 g/L NaCl). The solutions were autoclaved 2h at 100°C. YPD (Yeast Peptone Dextrose) medium was purchased from Gibco. Mineral media were prepared with the compositions presented in Table A.1 to Table A.6. The 95th percentile medium was sterilized by ultrafiltration at 0.2 µm. Simplified mineral medium was used for assessment of soil depollution (V.3.c ) and MR1 medium was used for decolorization assays 75th percentile in liquid phase (V.3.b ). Median value Mean value 25th percentile 5th percentile Commercials Table A . 1 : A1 Simplified mineral medium composition (AfterYang et al. 500 ). Compound Concentration (g/L) KH2PO4 1.5 Yeast Extract 1 NaCl 0.5 NH4Cl 0.1 Lactate 2 Table A.2: MR1 medium composition. Compound Concentration (mM) Concentration (mL/L) Remerciements Je tiens en tout premier lieu à remercier Thibaud Coradin et Francisco Fernandes qui m'ont accueillie au LCMCP pour ce qui devait être un stage de Master de six mois et avec qui j'ai finalement eu la joie de partager trois années de thèse. Merci Thibaud pour ton enthousiasme, ta bonne humeur, tes bon conseils et ton oeil neuf qui m'a souvent aidée à prendre un recul I Bibliography II III trois années, autour d'un thé ou d'une Chapter II : Pectin as a supporting structure Chapter III : Design of a biopolymer-silica hybrid porous structure Chapter IV : Encapsulation of microorganisms in macroporous foams Chapter V : From cell encapsulation to depollution order have however been reported for ice-templated gelatin solution 268,320 . This low degree of orientation may be attributed to the presence of the PIPES buffer. In the case of PVA foams, the morphology is very different compared to the foams obtained from biopolymers. The pores are well aligned and oriented, but cross section uncovers almost square-like pores. Longitudinal section reveals the presence of numerous and regularly-spaced transversal walls. Such morphology is consistent with previously described PVA ice-templated macroporous materials 304 . All the observed structures appeared compatible with the targeted application and were used for encapsulation of P. aeruginosa. Highest survival rates were obtained in alginate foams. About 7.10 6 CFU/mL were encapsulated in these foams which represents a loss of 2 LU. As a point of comparison 3 LU were lost during encapsulation in beet root pectin. Bovine gelatin also yielded high survival rates (around 3.10 6 CFU/mL). PVA and Citrus pectin resulted in significantly lower survival rates (10 4 CFU/mL and 10 3 CFU/mL respectively). Both alginate and gelatin appeared as suitable alternative to beet root pectin in terms of cell survival. Alginate however presents two major advantages compared to gelatin. First of all, alginate solutions are easier to handle from a processing point of view, since they do not gel at room temperature contrary to gelatin solution, which must be maintained at 35°C. From a structural point of view, alginate freezecasted foams display a more controlled and organized porosity. As a result, alginate was selected for further investigation. The main drawback for the use of pectin as the main component of the encapsulating matrix was the low survival rate after storage. P. aeruginosa cells were encapsulated in both pectin Results presented are for absorbance measured at 598 nm. One possible explanation could be the formation of intermediary degradation products, such as aromatic compounds. Discoloration rate was higher for an initial RB5 concentration of 10 mg/L (86%) compared to initial concentration of 100 mg/L (75%). Higher dye content may be responsible for toxicity toward cells as has been observed in other decolorization processes 251 . Suspensions inoculated with the same amount of cells but with different dye contents were plated after 24h of incubation at 37°C, in static conditions. Cell growth was observed in -Preparation of agar plates LB-Agar gels for plate counting were prepared from commercial LB-Agar powder (Sigma-Aldrich) dissolved at 35 g/L in deionized water and autoclaved 2h at 100 °C. Plates were prepared either with 20 mL of solution for Petri dishes (Ø=90 mm), or with 2 mL of solution per well in 12 well plates (Ø = 22 mm). -Bacteria storage Pseudomonas aeruginosa (ATCC® 27853™ strain) was stored at -80°C in 30% glycerol aliquots. About 10 µL of aliquot was added to 10 mL of LB medium in 30 mL glass tubes. The cell suspension was incubated 24h at 30°C and 150 rpm. The optical density (OD) after this pre-culture was comprised between 0.8 and 0.9. This cell suspension was then diluted to 10 -7 and 100 µL of this dilution was spread on a LB-agar plate in Ø=90 mm petri dish. This plate was incubated 24h at 37°C, resulting in the formation of 10 -RITC grafted pectin Beet root pectin was grafted with rhodamine isothiocyanate (RITC). A 10 g/L pectin solution in carbonate buffer (pH = 9.3) was prepared. 10 mL of 1 g/L solution of RITC in N,N-Dimethylformamide was added and the solution was left under magnetic agitation at room temperature overnight. The solution was dialysed using a MWCO 3500 membrane and freezedryed. Samples (foams and calibration solutions) were typically prepared using 10 wt% of RITCgrafted pectin and 90 wt% of non-grafted pectin. Uv-vis absorbance was measured at 558 nm and pectin concentration was calculated using the calibration curve presented in -Silica Silica concentration was measured using the blue silicomolybdic titration method 427 . Typically 75µL of solution A (20 g/L ammonium molybdate tetrahydrate, 60 mL/L hydrochloric acid) was added to 800 µL of appropriately diluted samples (samples were left 24h under agitation at room temperature after dilution) to yield the formation of silicomolybdic acid. After 30 min at room temperature, 375 µL of solution B (20 g/L oxalic acid, 6.67 g/L 4-methylaminophenol sulphate, 4 g/L anhydrous sodium sulfite, 100 mL/L sulphuric acid) was added and the solution was left 2h at room temperature before the absorbance at 810 nm was measured.

00000546

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https://inria.hal.science/inria-00000546v2/file/eabloat.pdf

Sylvain Gelly Olivier Teytaud Nicolas Bredeche Marc Schoenauer Apprentissage statistique et programmation génétique: la croissance du code est-elle inévitable ? programmation genetique : la croissance du code est-elle inevitable ? pp163-178. Proceedings of CAP'2005. Universal Consistency, the convergence to the minimum possible error rate in learning through genetic programming (GP), and Code bloat, the excessive increase of code size, are important issues in GP. This paper proposes a theoretical analysis of universal consistency and code bloat in the framework of symbolic regression in GP, from the viewpoint of Statistical Learning Theory, a well grounded mathematical toolbox for Machine Learning. Two kinds of bloat must be distinguished in that context, depending whether the target function has finite description length or not. Then, the Vapnik-Chervonenkis dimension of programs is computed, and we prove that a parsimonious fitness ensures Universal Consistency (i.e. the fact that the solution minimizing the empirical error does converge to the best possible error when the number of examples goes to infinity). However, it is proved that the standard method consisting in choosing a maximal program size depending on the number of examples might still result in programs of infinitely increasing size with their accuracy; a fitness biased by parsimony pressure is proposed. This fitness avoids unnecessary bloat while nevertheless preserving the Universal Consistency. Introduction Universal Consistency denotes the convergence of the error rate, in expectation on the unknown distribution of examples, to the optimal one. Despite it's a fundamental element of learning, it has not been widely studied yet in Genetic Programming (GP). Its restricted version, consistency, i.e. convergence to the optimum when the optimum lies in the search space, has not been more studied. Code bloat (or code growth) denotes the growth of program size during the course of Genetic Programming runs. It has been identified as a key problem in GP from the very beginning [START_REF] Koza | Genetic Programming: On the Programming of Computers by Means of Natural Selection[END_REF], and to any variable length representations based learning algorithm [START_REF] Langdon | The evolution of size in variable length representations[END_REF]. It is today a well studied phenomenon, and empirical solutions have been proposed to address the issues of code bloat (see section 2). However, very few theoretical studies have addressed the issue of bloat. The purpose of this paper is to provide some theoretical insights into the bloat phenomenon and its link with universal consistency, in the context of symbolic regression by GP, from the Statistical Learning Theory viewpoint [START_REF] Vapnik | The nature of statistical learning theory[END_REF]. Statistical Learning Theory is a recent, yet mature, area of Machine Learning that provides efficient theoretical tools to analyse aspects of learning accuracy and algorithm complexity. Our goal is both to perform an in-depth analysis of bloat and to provide appropriate solutions to avoid it. The paper is organized as follows : in the section below, we briefly survey some explanations for code bloat that have been proposed in the literature, and provide an informal description of our results from a GP perspective before discussing their interest for the GP practitioner. Section 2 gives a brief overview of the basic results of Learning Theory that will be used in Section 3 to formally prove all the advertised results. Finally, section 5 discusses the consequences of those theoretical results for GP practitioners and gives some perspectives about this work. The several theories that intend to explain code bloat are : -the introns theory states that code bloat acts as a protective mechanism in order to avoid the destructive effects of operators once relevant solutions have been found [START_REF] Nordin | Complexity compression and evolution[END_REF][START_REF] Mcphee | Accurate replication in genetic programming[END_REF][START_REF] Blickle | Genetic programming and redundancy[END_REF]. Introns are pieces of code that have no influence on the fitness: either subprograms that are never executed, or sub-programs which have no effect; -the fitness causes bloat theory relies on the assumption that there is a greater probability to find a bigger program with the same behavior (i.e. semantically equivalent) than to find a shorter one. Thus, once a good solution is found, programs naturally tends to grow because of fitness pressure [START_REF] Langdon | Fitness causes bloat: Mutation[END_REF]. This theory states that code bloat is operatorindependent and may happen for any variable length representation-based algorithm. As a consequence, code bloat is not to be limited to population-based stochastic algorithm (such as GP), but may be extended to many algorithms using variable length representation [START_REF] Langdon | The evolution of size in variable length representations[END_REF]; -the removal bias theory states that removing longer sub-programs is more tacky than removing shorter ones (because of possible destructive consequence), so there is a natural bias that benefits to the preservation of longer programs [START_REF] Soule | Exons and code growth in genetic programming[END_REF]. While it is now considered that each of these theories somewhat captures part of the problem [START_REF] Banzhaf | Some considerations on the reason for bloat[END_REF], there has not been any definitive global explanation of the bloat phenomenon. At the same time, no definitive practical solution has been proposed that would avoid the drawbacks of bloat (increasing evaluation time of large trees) while maintaining the good performances of GP on difficult problems. Some common solutions rely either on specific operators (e.g. size-fair crossover [START_REF] Langdon | Size fair and homologous tree genetic programming crossovers[END_REF], or different Fair Mutation [START_REF] Langdon | The evolution of size and shape[END_REF]), on some parsimony-based penalization of the fitness [START_REF] Soule | Effects of code growth and parsimony pressure on populations in genetic programming[END_REF] or on abrupt limitation of the program size such as the one originally used by Koza [START_REF] Koza | Genetic Programming: On the Programming of Computers by Means of Natural Selection[END_REF]. Some other more particular solutions have been proposed but are not widely used yet [START_REF] Ratle | Avoiding the bloat with probabilistic grammar-guided genetic programming[END_REF][START_REF] Silva | Dynamic maximum tree depth : A simple technique for avoiding bloat in tree-based gp[END_REF][START_REF] Luke | Lexicographic parsimony pressure[END_REF]. In this paper, we prove, under some sufficient conditions, that the solution given by GP actually converges, when the number of examples goes to infinity, toward the actual function used to generate the examples. This property is known in Statistical Learning as Universal Consistency. Note that this notion is a slightly different from that of Universal Approximation, that people usually refer to when doing symbolic regression in GP: because polynomial for instance are known to be able to approximate any continuous function, GP search using operators {+, * } is also assumed to be able to approximate any continuous function. However, Universal Consistency is concerned with the behavior of the algorithm when the number of examples goes to infinity: being able to find a polynomial that approximates a given function at any arbitrary precision does not imply that any interpolation polynomial built from an arbitrary set of sample points will converge to that given function when the number of points goes to infinity. But going back to bloat, and sticking to the polynomial example, it is also clear that the degree of the interpolation polynomial of a set of examples increases linearly with the number of examples. This leads us to start our bloat analysis by defining two kinds of bloat. On the one hand, we define the structural bloat as the code bloat that unavoidably takes place when no optimal solution (i.e. no function that exactly matches all possible examples) is approximated by the search space. In such a situation, optimal solutions of increasing accuracy will also exhibit an increasing complexity, as larger and larger code will be generated in order to better approximate the target function. The extreme case of structural bloat has also been demonstrated in [START_REF] Gustafson | Problem difficulty and code growth in Genetic Programming[END_REF]. The authors use some polynomial functions of increasing difficulty, and demonstrate that a precise fit can only be obtained through an increased bloat (see also [START_REF] Daida | What makes a problem gp-hard? analysis of a tunably difficult problem in genetic programming[END_REF] for related issues about problem complexity in GP). On the other hand, we define the functional bloat as the bloat that takes place when programs length keeps on growing even though an optimal solution (of known complexity) does lie in the search space. In order to clarify this point, let us use a simple symbolic regression problem defined as follow : given a set S of examples, the goal is to find a function f (here, a GP-tree) that minimized the Least Square Error (or LSE). If we intend to approximate a polynomial (ex. : 14 * x 2 ), we may observe code bloat since it is possible to find arbitrarily long polynomials that gives the exact solution (ex. : 14 * x 2 + 0 * x 3 + ...). Most of the works cited in section 1 are in fact concerned with functional bloat which is the simplest, yet already problematic, kind of bloat. Overview of results. In section 3, we shall investigate the Universal Consistency of Genetic Programming, and study in detail structural and functional bloat that might take place when searching program spaces using GP. A formal and detailed definition of the program space in GP is given in Lemma 1, section 3, and two types of results will then be derived: i) Universal Consistency results, i.e. does the probability of misclassification of the solution given by GP converges to the optimal probability of misclassification when the number of examples goes to infinity? ii) Bloat-related results, first regarding structural bloat, and second with respect to functional bloat in front of various types of fitness penalization and/or bounds on the complexity of the programs. Let us now state precisely, yet informally, our main results. First, as already mentioned, we will precisely define the set of programs under examination, and prove that such a search space fulfills the conditions of the standard theorems of Statistical Learning Theory listed in Section 2. Second, applying those theorems will immediately lead to a first Universal Consistency result for GP, provided that some penalization for complexity is added to the fitness (Theorem 3). Third: the first bloat-related result, Proposition 4, unsurprisingly proves that if no optimal function belongs to the search space, then converging to the optimal error implies an infinite increase of bloat. Fourth, theorem 5 is also a negative result about bloat, as it proves that even if the optimal function belongs to the search space, minimizing the LSE alone might lead to bloat (i.e. the com-plexity of the empirical solutions goes to infinity with the sample size). Finally, the last two theorems (5' and 6) are the best positive results one could expect considering the previous findings: it is possible to carefully adjust the parsimony pressure so as to obtain both Universal Consistency and bounds on the complexity of the empirical solution (i.e. no bloat). Section 4 discuss some properties of alternate solutions for complexity penalization : cross-validation or hold out, with various pairing of data sets. Note that, though all proofs in Section 3 will be stated and proved in the context of classification (i.e. find a function from R d into {0, 1}), their generalization to regression (i.e. find a function from R d into R) is straightforward. Discussion The first limit of our work is the fact that all these results consider that GP finds a program which is empirically the best, in the sense that given a set of examples and a fitness function based on the Least Square Error (and possibly including some parsimony penalization), it will be assumed that GP does find one program in that search space that minimizes this fitness -and it is the behavior of this ideal solution, which is a random function of the number of examples, that is theoretically studied. Of course, we all know that GP is not such an ideal search procedure, and hence such results might look rather far away from GP practice, where the user desperately tries to find a program that gives a reasonably low empirical approximation error. Nevertheless, Universal Consistency is vital for the practitioner too: indeed, it would be totally pointless to fight to approximate an empirically optimal function without any guarantee that this empirical optimum is anywhere close to the ideal optimal solution we are in fact looking for. Furthermore, the bloat-related results give some useful hints about the type of parsimony that has a chance to efficiently fight the unwanted bloat, while maintaining the Universal Consistency property -though some actual experiments will have to be run to confirm the usefulness of those theoretical hints. Elements of Learning theory In the frameworks of regression and classification, Statistical Learning Theory [START_REF] Vapnik | The nature of statistical learning theory[END_REF] is concerned with giving some bounds on the generalization error (i.e. the error on yet unseen data points) in terms of the actual empirical error (the LSE error above) and some fixed quantity depending only on the search space. More precisely, we will use here the notion of Vapnik-Chervonenkis dimension (in short, VCdim) of a space of functions. Roughly, VC-dim provides bounds on the difference between the empirical error and the generalization error. Consider a set of s examples (x i , y i ) i∈{1,...,s} . These examples are drawn from a distribution P on the couple (X, Y ). They are independent identically distributed, Y = {0, 1} (classification problem), and typically X = R d for some dimension d. For any function f , define the loss L(f ) to be the expectation of |f (X) -Y |. Similarly, define the empirical loss L(f ) as the loss observed on the examples: L(f ) = Theorem A [5, Th. 12.8, p206] : Consider F a family of functions from a domain X to {0, 1} and V its VC-dimension. Then, for any ǫ > 0 P ( sup P ∈F |L(P ) -L(P )| ≥ ǫ) ≤ 4 exp(4ǫ + 4ǫ 2 )s 2V exp(-2sǫ 2 ) and for any δ ∈]0, 1]P ( sup P ∈F |L(P ) -L(P )| ≥ ǫ(s, V, δ)) ≤ δ where ǫ(s, V, δ) = 4-log(δ/(4s 2V )) 2s-4 . Interpretation : In a family of finite VC-dimension, the empirical errors and the generalization errors are probably closely related. Other forms of this theorem have no log(n) factor ; they are known as Alexander's bound, but the constant is so large that this result is not better than the result above unless s is huge ([5, p207]): if s ≥ 64/ǫ 2 , P ( sup P ∈F |L(P ) -L(P )| ≥ ǫ) ≤ 16( √ sǫ) 4096V exp(-2sǫ 2 ) We classically derive the following result from theorem A: Theorem A' : Consider F s for s ≥ 0 a family of functions from a domain X to {0, 1} and V s its VC-dimension. Then, sup P ∈Fs |L(P ) -L(P )| → 0 as s → ∞ almost surely whenever V s = o(s/ log(s)). Interpretation : The maximal difference between the empirical error and the generalization error goes almost surely to 0 if the VC-dimension is finite. Proof : We use the classical Borell-Cantelli lemma 1 , for any ǫ ∈ [0, 1] : s≥64/ǫ 2 P (|L(P ) -L(P )| > ǫ) ≤ 16 s≥64/ǫ 2 ( √ sǫ) 4096Vs exp(-2sǫ 2 ) ≤ 16 s≥64/ǫ 2 exp(4096V s (log( √ s) + log(ǫ)) -2sǫ 2 ) which is finite as soon as V s = o(s/ log(s)). Theorem B in [5, Th. 18.2, p290] : Let F 1 , . . . , F k . . . with finite VC-dimensions V 1 , . . . , V k , . . . Let F = ∪ n F n . Then, being given s examples, consider P ∈ F s minimizing the empirical risk L among F s . Then, if V s = o(s/log(s)) and V s → ∞, P (L( P ) ≤ L( P ) + ǫ(s, V s , δ)) ≥ 1 -δ P (L( P ) ≤ inf P ∈Fs L(P ) + 2ǫ(s, V s , δ)) ≥ 1 -δ and L( P ) → inf P ∈F L(P ) a.s. Note that for a well chosen family of functions (typically, programs), inf P ∈F L(P ) = L * for any distribution ; so, theorem B leads to universal consistency (i.e. ∀P ; L( P ) → L * ), for a well-chosen family of functions. 1 If P n P (Xn > ǫ) is finite for any ǫ > 0 andXn > 0, then Xn → 0 almost surely. Interpretation : If the VC-dimension increases slowly enough as a function of the number of examples, then the generalization error goes to the optimal one. If the family of functions is well-chosen, this slow increase of VC-dimension leads to universal consistency. In the following theorem, we use d ′ , t ′ , q ′ instead of d, t, q for the sake of notations in a corollary below. Theorem C (8.14 and 8.4 in [START_REF] Antony | Neural network learning : Theoretical foundations[END_REF]) : Let H = {x → h(a, x); a ∈ R d ′ } where h can be computed with at most t ′ operations among α → exp(α) ; +, -, ×, / ; jumps conditioned on >, ≥, =, ≤, = ; output 0 ; output 1. Then : V Cdim(H) ≤ t ′2 d ′ (d ′ + 19 log 2 (9d ′ )) . Furthermore, if exp(.) is used at most q ′ times, and if there are at most t ′ operations executed among arithmetic operators, conditional jumps, exponentials, π(H, m) ≤ 2 (d ′ (q ′ +1))2/2 (9d ′ (q ′ + 1)2 t ) 5d ′ (q ′ +1) (em(2 t ′ -2)/d ′ ) d ′ where π(H, m) is the m th shattering coefficient of H, and hence V Cdim(H) ≤ (d ′ (q ′ + 1)) 2 + 11d ′ (q ′ + 1)(t ′ + log 2 (9d ′ (q ′ + 1))) Finally, if q ′ = 0 then V Cdim(H) ≤ 4d ′ (t ′ + 2 . . . with finite VC-dimensions V 1 , . . . , V k , . . . Let F = ∪ n F n . Assume that all distribution lead to L F = L * where L * is the optimal possible error (spaces of functions ensuring this exist). Then, given s examples, consider f ∈ F minimizing L(f ) + 32 s V (f ) log(e × s), where V (f ) is V k with k minimal such that f ∈ F k . Then : • if additionally one optimal function belongs to F k , then for any s and ǫ such that V k log(e × s) ≤ sǫ 2 /512, the generalization error is lower than ǫ with probability at most ∆ exp(-sǫ 2 /128) + 8s V k × exp(-sǫ 2 /512) where ∆ = ∞ j=1 exp(-V j ) is assumed finite. • the generalization error, with probability 1, converges to L * . Interpretation : The optimization of a compromise between empirical accuracy and regularization lead to the same properties as in theorem B, plus a stronger convergence rate property. Results This section presents in details results surveyed above. They make an intensive use of the results of Statistical Learning Theory presented in the previous section. More precisely, Lemma 1 defines precisely the space of programs considered here, and carefully shows that it satisfies the hypotheses of Theorems A-C. This allows us to evaluate the VC-dimension of sets of programs, stated in Theorem 2. Then, announced results are derived. Finally, next we propose a new approach combining an a priori limit on VC-dimension (i.e. size limit) and a complexity penalization (i.e. parsimony pressure) and state in theorem 6 that this leads to both universal consistency and convergence to an optimal complexity of the program (i.e. no bloat). We first prove the following Lemma 1 : Let F be the set of functions which can be computed with at most t operations among : • operations α → exp(α) (at most q times); • operations +, -, ×, / ; • jumps conditioned on >, ≥, =, ≤, = ; and • output 0 ; • output 1 ; • labels for jumps ; • at most m constants ; • at most z variables by a program with at most n lines. We note log 2 (x) the integer part (ceil) of log(x)/ log [START_REF] Banzhaf | Some considerations on the reason for bloat[END_REF]. Then F is included in H as defined in theorem C, for a given P with t ′ = t + t max(3 + log 2 (n) + log 2 (z), 7 + 3 log 2 (z)) + n(11 + max(9log 2 (z), 0) + max(3log 2 (z) -3, 0)), q ′ = q, d ′ = 1 + m. Interpretation : This lemma states that a family of programs as defined above is included in the parametrizations of one well-chosen program. This replaces a family of programs by one parametric program, and it will be useful for the computation of VC-dimension of a family of programs by theorem C. Proof : In order to prove this result, we define below a program as in theorem above that can emulate any of these programs, with at most t ′ = t + t max(3 + log 2 (n) + log 2 (z), 7 + 3 log 2 (z)) + n(11 + max(9log 2 (z), 0) + max(3log 2 (z) -3, 0)), q ′ = q, d ′ = 1 + m. The program is as follows : • label "inputs" • initialize variable(1) at value x(1) • label "output 0" • output 0 • label "output 1" • output 1 "operation decode c" can be developed as follows. Indeed, we need m real numbers, for parameters, and 4n integers c(., .), that we will encode as only one real number in [0, 1] as follows : • initialize variable(2) at value x(2) • . . . • initialize variable(dim(x)) at value x(dim(x)) • label "constants" • initialize variable(dim(x) + 1) at value a 1 • initialize variable(dim(x) + 2) at value a 2 • . . . • initialize variable(dim(x) + m) at value a m • label "Decode the program into c" • operation decode c • label "Line 1" • operation c(1, 1. let y ∈ [0, 1] 2. for each i ∈ [1, . . . n] : • c(i, 1) = 0 • y = y * 2 • if (y > 1) then { c(i, 1) = 1 ; y = y -1 } • y = y * 2 • if (y > 1) then { c(i, 1) = c(i, 1) + 2 ; y = y -1 } • y = y * 2 • if (y > 1) then { c(i, 1) = c(i, 1) + 4 ; y = y -1 } 3. for each j ∈ [2, 4] and i ∈ [1, . . . n] : • c(i, j) = 0 • y = y * 2 • if (y > 1) then { c(i, j) = 1 ; y = y -1 } • y = y * 2 • if (y > 1) then { c(i, j) = c(i, j) + 2 ; y = y -1 } • y = y * 2 • if (y > 1) then { c(i, j) = c(i, j) + 4 ; y = y -1 } • . . . • y = y * 2 • if (y > 1) then { c(i, j) = c(i, j) + 2 log2(z)-1 ; y = y -1 } The cost of this is n × (3 + max(3 × log 2 (z), 0)) "if then", and n × (3 + max(3 × log 2 (z), 0)) operators ×, and n(2 + max(3(log 2 (z) -1), 0)) operators +, and n × (3 + max(3 × log 2 (z), 0)) operators -. The overall sum is bounded by n(11 + max(9 log 2 (z), 0) + max(3log 2 (z) -3, 0)). The result then derives from the rewriting of "operation c(i, 1) with variables c(i,2) and c(i,3)". This expression can be developed as follows: • if c(i, 1) == 0 then goto "output1" • if c(i, 1) == 1 then goto "output 0" • if c(i, 2) == 1 then c = variable(1) • if c(i, 2) == 2 then c = variable(2) • . . . • if c(i, 2) == z then c = variable(z) • if c(i, 1) == 7 then goto "Line c" (must be encoded by dichotomy with log 2 (n) lines) • if c(i, 1) == 6 then goto "exponential(i)" • if c(i, 3) == 1 then b = variable(1) • if c(i, 3) == 2 then b = variable(2) • . . . • if c(i, 3) == z then b = variable(z) • if c(i, 1) == 2 then a = c + b • if c(i, 1) == 3 then a = c -b • if c(i, 1) == 4 then a = c × b • if c(i, 1) == 5 then a = c/b • if c(i, 4) == 1 then variable(1) = a • if c(i, 4) == 2 then variable(2) = a • . . . • if c(i, 4) == z then variable(z) = a • label "endOfInstruction(i)" For each such instruction, at the end of the program, we add three lines of the following form : • label "exponential(i)" • a = exp(c) • goto "endOfInstruction(i)" Each sequence of the form "if x=... then" (p times) can be encoded by dichotomy with log 2 (p) tests "if ... then goto". Hence, the expected result. Theorem 2 : Let F be the set of programs as in lemma 1, where q ′ ≥ q, t ′ ≥ t + t max(3 + log 2 (n) + log 2 (z), 7 + 3 log 2 (z)) + n(11 + max(9log 2 (z), 0) + max(3log 2 (z) -3, 0)), d ′ ≥ 1 + m. V Cdim(H) ≤ t ′2 d ′ (d ′ + 19 log 2 (9d ′ )) V Cdim(H) ≤ (d ′ (q ′ + 1)) 2 + 11d ′ (q ′ + 1)(t ′ + log 2 (9d ′ (q ′ + 1))) If q = 0 (no exponential) then V Cdim(H) ≤ 4d ′ (t ′ + 2). Interpretation : interesting and natural families of programs have finite VCdimension. Effective methods can associate a VC-dimension to these families of programs. Proof : Just plug Lemma 1 in Theorem C. We now consider how to use such results in order to ensure universal consistency. First, we show why simple empirical minimization (consisting in choosing one function such that L is minimum) does not ensure consistency. Precisely, we state that, for some distribution of examples, and some i.i.d sequence of examples (x 1 , y 1 ), . . . , (x n , y n ), there exists P 1 , . . . , P n , . . . such that ∀i ∈ [[1, n]]P n (x i ) = y i and however ∀n ∈ NP (f (x) = y) = 0. The proof is as follows. Consider the distribution with x uniformly drawn in [0, 1] and y constant equal to 1. Consider P n the program that compares its entry to x 1 , x 2 , . . . , x n , and outputs 1 if the entry is equal to x j for some j ≤ n, and 0 otherwise else. With probability 1, this program output 0, whereas y = 1 with probability 1. We therefore conclude that minimizing the empirical risk is not enough for ensuring any satisfactory form of consistency. Let's now show that structural risk minimization, i.e. taking into accound a penalization for complex structures, can do the job, i.e. ensure universal consistency, and fast convergence when the solution can be written within finite complexity. Theorem 3 : Consider q f , t f , m f , n f and z f integer sequences, non-decreasing functions of f . Define V f = V Cdim(H f ), where H f is the set of programs with at most t f lines executed, with z f variables, n f lines, q f exponentials, and m f constants. Then with q ′ f = q f , t ′ f = t f + t f max(3 + log 2 (n f ) + log 2 (z f ), 7 + 3 log 2 (z f )) + n f (11 + max(9log 2 (z f ), 0) + max(3log 2 (z f ) -3, 0)), d ′ f = 1 + m f , V f = (d ′ f (q ′ f + 1)) 2 + 11d ′ f (q ′ f + 1)(t ′ f + log 2 (9d ′ f (q ′ f + 1))) or, if ∀f q f = 0 then define V f = 4d ′ f (t ′ f + 2). Then, being given s examples, consider f ∈ F minimizing L(f ) + 32 s V (f ) log(e × s), where V (f ) is the min of all k such that f ∈ F k . Then, if ∆ = ∞ j=1 exp(-V j ) is finite, -the generalization error, with probability 1, converges to L * . -if one optimal rule belongs to F k , then for any s and ǫ such that V k log(e × s) ≤ sǫ 2 /512, the generalization error is lower than L * + ǫ with probability at most ∆ exp(-sǫ 2 /128) + 8s V k × exp(-sǫ 2 /512) where ∆ = ∞ j=1 exp(-V j ) is as- sumed finite. Interpretation : Genetic programming for bi-class classification, provided that structural risk minimization is performed, is universally consistent and verifies some convergence rate properties. Proof : Just plug theorem D in theorem 2. We now prove the non-surprising fact that if it is possible to approximate the optimal function (the Bayesian classifier) without reaching it exactly, then the "complexity" of the program runs to infinity as soon as there is convergence of the generalization error to the optimal one. Proposition 4: Consider P s a sequence of functions such that P s ∈ F V (s) , with F 1 ⊂ F 2 ⊂ F 3 ⊂ . . . , where F V is a set of functions from X to {0, 1} with VC-dimension bounded by V . Define L V = inf P ∈FV L(P ) and V (P ) = inf{V ; P ∈ F V } and suppose that ∀V L V > L * . Then, (L(P s ) s→∞ -→ L * ) =⇒ (V (P s ) s→∞ -→ ∞). Interpretation : This is structural bloat : if your space of programs approximates but does not contain the optimal function, then bloat occurs. Proof: Define ǫ(V ) = L V -L * . Assume that ∀V ǫ(V ) > 0. ǫ is necessarily non-increasing. Consider V 0 a positive integer ; let us prove that if s is large enough, then V (P s ) ≥ V 0 . There exists ǫ 0 such that ǫ(V 0 ) > ǫ 0 > 0. For s large enough, L(P s ) ≤ L * + ǫ 0 , hence L Vs ≤ L * + ǫ 0 , hence L * + ǫ(V s ) ≤ L * + ǫ 0 , hence ǫ(V s ) ≤ ǫ 0 , hence V s > V 0 . We now show that the usual procedure defined below, consisting in defining a maximum VC-dimension depending upon the sample size (as usually done in practice and as recommended by theorem B) and then using a moderate family of functions, leads to bloat. With the same hypotheses as in theorem B, we can state Theorem 5 (bloat theorem for empirical risk minimization with relevant VCdimension): Let F 1 , . . . , F k . . . non-empty sets of functions with finite VC-dimensions V 1 , . . . , V k , . . . Let F = ∪ n F n . Then, given s examples, consider P ∈ F s minimizing the empirical risk L in F s . ¿From Theorem B we already know that if V s = o(s/log(s)) and V s → ∞, then P (L( P ) ≤ L( P ) + ǫ(s, V s , δ)) ≥ 1 -δ, and L( P ) → inf P ∈F L(P ) a.s.. We will now state that if V s → ∞, and noting V (f ) = min{V k ; f ∈ F k }, then ∀V 0 , P 0 > 0, ∃P , distribution of probability on X and Y , such that ∃g ∈ F 1 such that L(g) = L * and for s sufficiently large P (V ( P ) ≤ V 0 ) ≤ P 0 . Interpretation : The result in particular implies that for any V 0 , there is a distribution of examples such that ∃g; V (g) = V 1 and L(g) = L * , with probability 1, V ( f ) ≥ V 0 infinitely often as s increases. This shows that bloat can occur if we use only an abrupt limit on code size, even if this limit depends upon the number of examples (a fortiori if there's no limit). Proof (of the part which is not theorem B) : See figure 3 for a figure illustrating the proof. Consider V 0 > 0 and P 0 > 0. Consider α such that (eα/2 α ) V0 ≤ P 0 /2. Consider s such that V s ≥ αV 0 . Let d = αV 0 . Consider x 1 , . . . , x d d points shattered by F d ; such a family of d points exist, by definition of F d . Define the probability measure P by the fact that X and Y are independent and P (Y = 1) = 1 2 and P (X = x i ) = 1 d . Then, the following holds, with Q the empirical distribution (the average of Dirac masses on the x i 's) : 1. no empty x i 's : P (E 1 ) → 0 where E 1 is the fact that ∃i; Q(X = x i ) = 0, as s → ∞. 2. no equality : P (E 2 ) → 0 where E 2 is the fact that E 1 occurs or ∃i; We now only have to use classical results. It is well known in VC-theory that S(a, b) ≤ (ea/b) b (see for example [5, chap.13]), hence S(d, d/α) ≤ (ed/(d/α)) d/α and P (E 3 |E 2 does not hold) ≤ (eα) d/α /2 d ≤ P 0 /2. If n is sufficiently large to ensure that P (E 2 ) ≤ P 0 /2 (we have proved above that P (E 2 ) → 0 as s → ∞) then Smallest Q(Y = 1|X = x i ) = 1 2 . 3. P (E 3 ) ≤ P (E 3 |¬E 2 )×P (¬E 2 )+P (E 2 ) ≤ P (E 3 |¬E 2 )+P (E 2 ) ≤ P 0 /2+P 0 /2 ≤ P 0 We now show that, on the other hand, it is possible to optimize a compromise between optimality and complexity in an explicit manner (e.g., replacing 1 % precision with 10 lines of programs or 10 minutes of CPU) : Theorem 5' (bloat-control theorem for regularized empirical risk minimization with relevant VC-dimension): Let F 1 , . . . , F k . . . be non-empty sets of functions with finite VC-dimensions V 1 , . . . , V k , . . . Let F = ∪ n F n . Consider W a user-defined complexity penalization term. Then, being given s examples, consider P ∈ F s minimizing the regularized empirical risk L(P ) = L(P ) + W (P ) among F s . If V s = o(s/log(s)) and V s → ∞, then L( P ) → inf P ∈F L(P ) a.s. where L(P ) = L(P ) + W (P ). Interpretation : Theorem 5' shows that, using a relevant a priori bound on the complexity of the program and adding a user-defined complexity penalization to the fitness, can lead to convergence toward a user-defined compromise ( [START_REF] Zhang | Balancing accuracy and parsimony in genetic programming[END_REF][START_REF] Zhang | Evolutionary induction of sparse neural trees[END_REF]) between classification rate and program complexity (i.e. we ensure almost sure convergence to a compromise of the form "λ 1 CPU time + λ 2 misclassification rate + λ 3 number of lines", where the λ i are user-defined). Remark : the drawback of this approach is that we have lost universal consistency and consistency (in the general case, the misclassification rate in generalization will not converge to the Bayes error, and whenever an optimal program exists, we will not necessarily converge to its efficiency). Proof : See figure 3 We now turn our attention to a more complicated case where we want to ensure universal consistency, but we want to avoid a non-necessary bloat ; e.g., we require that if an optimal program exists in our family of functions, then we want to converge to its error rate, without increasing the complexity of the program. We consider a merge between regularization and bounding of the VC-dimension ; we penalize the complexity (e.g., length) of programs by a penalty term R(s, P ) = R(s)R ′ (P ) depending upon the sample size and upon the program ; R(., .) is user-defined and the algorithm will look for a classifier with a small value of both R ′ and L. We study both the universal consistency of this algorithm (i.e. L → L * ) and the no-bloat theorem (i.e. R ′ → R ′ (P * ) when P * exists). Theorem 6 : Let F 1 , . . . , F k . . . with finite VC-dimensions V 1 , . . . , V k , . . . Let F = ∪ n F n . Define V (P ) = V k with k = inf{t|P ∈ F t }. Define L V = inf P ∈FV L(P ). Consider V s = o(log(s)) and V s → ∞. Consider P minimiz- ing L(P ) = L(P ) + R(s, P ) in F s and assume that R(s, .) ≥ 0. Then (consistency), whenever sup P ∈FV s R(s, P ) = o(1), L( P ) → inf P ∈F L(P ) almost surely (note that for well chosen family of functions, inf P ∈F L(P ) = L * ). Moreover, assume that ∃P * ∈ F V * L(P * ) = L * . Then with R(s, P ) = R(s)R ′ (P ) and with R ′ (s) = sup P ∈FV s R ′ (P ) : 1. non-asymptotic no-bloat theorem : R ′ ( P ) ≤ R ′ (P * ) + (1/R(s))2ǫ(s, V s , δ) with probability at least 1 -δ (this result is in particular interesting for ǫ(s, V s , δ)/R(s) → 0, what is possible for usual regularization terms as in theorem D, 2. almost-sure no-bloat theorem : if R(s)s (1-α)/2 = O(1), then almost surely R ′ ( P ) → R ′ (P * ) and if R ′ (P ) has discrete values (such as the number of instructions in P or many complexity measures for programs) then for s sufficiently large, R ′ ( P ) = R ′ (P * ). 3. convergence rate : with probability at least 1 -δ, Interpretation : Combining a code limitation and a penalization leads to universal consistency without bloat. L( P ) ≤ inf P ∈FV s L(P ) + R(s)R ′ (s) Remarks : The usual R(s, P ) as used in theorem D or theorem 3 provides consistency and non-asymptotic no-bloat. A stronger regularization leads to the same results, plus almost sure no-bloat. The asymptotic convergence rate depends upon the regularization. The result is not limited to genetic programming and could be used in other areas. As shown in proposition 4, the no-bloat results require the fact that ∃V * ∃P * ∈ F V * L(P * ) = L * . Interestingly, the convergence rate is reduced when the regularization is increased in order to get the almost sure no-bloat theorem. Proof : Define ǫ(s, V ) = sup f ∈FV | L(f )-L(f )|. Let us prove the consistency. For any P , L( P )+R(s, P ) ≤ L(P )+R(s, P ). On the other hand, L( P ) ≤ L( P )+ǫ(s, V s ). So : L( P ) ≤ ( inf P ∈FV s ( L(P ) + R(s, P ))) -R(s, P ) + ǫ(s, V s ) ≤ ( inf P ∈FV s (L(P ) + ǫ(s, V s ) + R(s, P ))) -R(s, P ) + ǫ(s, V s ) ≤ ( inf P ∈FV s (L(P ) + R(s, P ))) + 2ǫ(s, V s ) as ǫ(s, V s ) → 0 almost surely2 and (inf P ∈FV s (L(P ) + R(s, P ))) → inf P ∈F L(P ), we conclude that L( P ) → inf P ∈F L(P ) a.s. We now focus on the proof of the "no bloat" result : By definition of the algorithm, for s sufficiently large to ensure P * ∈ F Vs , L( P ) + R(s, P ) ≤ L(P * ) + R(s, P * ) hence with probability at least 1 -δ, R ′ ( P ) ≤ R ′ (P * ) + (1/R(s))(L * + ǫ(s, V s , δ) -L( P ) + ǫ(s, V s , δ)) hence R ′ ( P ) ≤ R ′ (V * ) + (1/R(s))(L * -L( P ) + 2ǫ(s, V s , δ)) As L * ≤ L( P ), this leads to the non-asymptotic version of the no-bloat theorem. The almost sure no-bloat theorem is derived as follows. R ′ ( P ) ≤ R ′ (P * ) + 1/R(s)(L * + ǫ(s, V s ) -L( P ) + ǫ(s, V s )) hence R ′ ( P ) ≤ R ′ (P * ) + 1/R(s)(L * -L( P ) + 2ǫ(s, V s )) R ′ ( P ) ≤ R ′ (P * ) + 1/R(s)2ǫ(s, V s ) All we need is the fact that ǫ(s, V s )/R(s) → 0 a.s. For any ǫ > 0, we consider the probability of ǫ(s, V s )/R(s) > ǫ, and we sum over s > 0. By the Borell-Cantelli lemma, the finiteness of this sum is sufficient for the almost sure convergence to 0. The probability of ǫ(s, V s )/R(s) > ǫ is the probability of ǫ(s, V s ) > ǫR(s). By theorem A, this is bounded above by O(exp(2V s log(s) -2sǫ 2 R(s) 2 )). This has finite sum for R(s) = Ω(s -(1-α)/2 ). Let us now consider the convergence rate. Consider s sufficiently large to ensure L Vs = L * . As shown above during the proof of the consistency, L( P ) ≤ ( inf so with probability at least 1 -δ, ≤ inf P ∈FV s L(P ) + R(s)R ′ (s) + 2ǫ(s, V s , δ) P Extensions We have studied above : -the method consisting in minimizing the empirical error, i.e. the error observed on examples (leading to bloat (this is an a fortiori consequence of theorem 5) without universal consistency (see remark before theorem 3)) ; -the method consisting in minimizing the empirical error, i.e. the error observed on examples, with a hard bound on the complexity (leading to universal consistency but bloat, see theorem 5) ; -the method, inspired from (but slightly adapted against bloat) structural risk minimization, consisting in minimizing a compromize between the empirical error and a complexity bound including size and computation-time (see theorem 6). We study the following other cases now : -the case in which the level of complexity is chosen through resamplings, i.e. crossvalidation or hold out ; -the case in which the complexity penalization does not include any time bound but only size bounds ; We mainly conclude that penalization is necessary, cannot be replaced by crossvalidation, cannot be replaced by hold-out, and must include time-penalization. About the use of cross-validation or hold-out for avoiding bloat and choosing the complexity level Note UC for universal consistency and ERM for empirical risk minimization. We considered above different cases : -evolutionary programming with only "ERM" fitness ; -evolutionary programming with ERM+bound (leading to UC + bloat) ; -evolutionary programming with ERM+penalization+bound (leading to UC without bloat). One can now consider some other cases : 1 ) 1 with variables c(1, 2) and c(1, 3) and c(1, 4) • label "Line 2" • operation c(2, 1) with variables c(2, 2) and c(2, 3) and c(2, 4) • . . . • label "Line n" • operation c(n, 1) with variables c(n, 2)and c(n, 3) and c(n, 4) Fig. 1 . 1 Fig. 1. Illustration of the proof. With a larger k, F k has a smaller best error. for a figure illustrating the proof. sup P ∈Fs | L(P ) -L(P )| ≤ sup P ∈Fs | L(P ) -L(P )| ≤ ǫ(s, V s ) → 0, almost surely, by theorem A'. Hence the expected result. =o( 1 ) 1 by hypothesis+2ǫ(s, V s , δ) where ǫ(s, V, δ) = 4-log(δ/(4s 2V )) 2s-4is an upper bound on ǫ(s, V ) = sup f ∈FV | L(f ) -L(f )| (given by theorem A), true with probability at least 1 -δ. ∈FV s (L(P ) + R(s, P ))) + 2ǫ(s, V s ) ≤ ( infP ∈FV s (L(P ) + R(s)R ′ (P ))) + 2ǫ(s, V s ) ≤ inf P ∈FV s L(P ) + R(s)R ′ (s) + 2ǫ(s, V s ) the best function is not in F V0 : P (E 3 |E 2 does not hold) ≤ S(d, d/α)/2 d where E 3 is the fact that ∃g ∈ F d/α=V0 ; L(g) = inf F d L, with S(d, d/α) the relevant shattering coefficient, i.e. the cardinal of F d/α restricted to {x 1 , . . . , x d }. Illustration of the proof. With a larger k, F k has a smaller best error, but the penalization is stronger than the difference of error. Smallest Error in Fk HatLTilde HatL LTilde L Complexity Vk of family Fk Fig. 2. s i |f (x i ) -y i |.Finally, define L * , the Bayes error, as the smallest possible generalization error for any mapping from X to {0, 1}.The following 4 theorems are well-known in the Statistical Learning community: See theorem A' Acknowledgements This work was supported in part by the PASCAL Network of Excellence. We thank Bill Langdon for very helpful comments. -hold out in order to choose between different complexity classes (i.e., in the Paretofront corresponding to the compromise between ERM and complexity, choose the function by hold out) ; -idem through cross-validation. This section is devoted to these cases. First, let's consider hold-out for choosing the complexity level. Consider that the function can be chosen in many complexity levels, F 0 ⊂ F 1 ⊂ F 2 ⊂ F 3 ⊂ . . . , where F i = F i+1 . Note L(f, X) the error rate of the function f in the set X of examples: where l(f, X i ) = 1 if f fails on X i and 0 otherwise. Define f k = arg min F k L(., X k ). In hold-out, f = f k * where k * = arg min k l k where In all the sequel, we assume that We consider that all X k 's and Y k 's have the same size n. There are different cases : X k = Y k and ∀k, X k = X 0 is the naive case (studied above). The case with hold out leads to different cases also : -Greedy case: all X k 's and Y k 's are independent. -Case with pairing: Case of greedy hold-out. -consider the case of an output y independent of the input x, and P (y = 1) = k * is therefore a Poisson law with parameter 1/2 n . Its expectation is TODO and its standard deviation is TODO -therefore, almost surely, k * → ∞ as n → ∞. This is shown with one distribution, which does not depend upon the number of examples. This happens whereas an optimal function lies in F 0 . Case of hold-out with pairing. - with v minimal realizing this condition. -Consider A = {a 1 , . . . , a V }, a set of points shattered by F v . -Consider a distribution of examples with x uniform on A and y independent of x with P (y = 1) = P (y = 0) = 1 2 . -Consider PX the empirical distribution associated to X and PY the empirical distribution associated to Y . -There is at least one function on A which does not belong in F k-1 . -With probability at least (1 -P (E Y ))/2 V , this function is optimal for L(., Y 0 ). -With probability at least (1 -P (E X ))/2 V , f k is equal to this function. -Combining the two probabilities above, as the events are independent, we see that with probability at least p -this implies the first result : P (k * ≥ v) does not go to 0, whereas a function in F 0 is optimal. -Now, let's consider that we can change the distribution as n moves. -For n sufficiently large, choose v maximal such that p(v, n) =≥ 1/n and F v has VC-dimension greater than the VC-dimension of F v-1 . Consider the distribution associated to v as above (uniform on A, a set of shattered point). We have therefore shown, with a distribution dependent on n, that k * → ∞. And for a distribution that does not depend upon n, that P (k * < v) is lower bounded. In both cases, an optimal function lies in F 0 . We now turn our attention to the case of cross-validation. We formalize N-crossvalidation as follows : Greedy cross-validation could be considered as in the case of hold out above. This leads to the same result (for some distribution, k * → ∞). We therefore only consider cross-validation with pairing : X i k = X i We only consider cross-validation as a method for computing k * , and not for computing the classifier. We note Pi the empirical law associated to X i . We consider A a set of points shattered by F v , |A| = V , A not shattered by F v-1 . We consider f ∈ F v realizing a dichotomy of A that is not realized by F v-1 . We define E i the event {∀a ∈ A; Pi (y = f (a)|x = a) > 1 2 }. We assume that the distribution of examples is, for x, uniform on A, and for y, independently of x, uniform on {0, 1}. The probability of E i goes to We therefore have the following result : Theorem : one can not avoid bloat with only hold-out or cross-validation. Consider greedy hold-out, hold out with pairing and cross-validation with pairing. Then, -for some well-chosen distribution of examples, greedy hold-out almost surely leads to k * → ∞ whereas an optimal function lies in F 0 . -whatever may be V = V C -dimension(F v ), for some well-chosen distribution, hold-out with pairing almost surely leads to k * > V infinitely often whereas an optimal function lies in F 0 . -whatever may be V = V C -dimension(F v ), for some well-chosen distribution, cross-validation with pairing almost surely leads to k * > V infinitely often whereas an optimal function lies in F 0 . Is time-complexity required ? Consider any learning algorithm working on a sequence of i.i.d examples (x 1 , y 1 ), . . . , (x n , y n ) and outputting a program. We formalize as follows the fact that this algorithm does not take into account any form of time-complexity but only the size of programs : If the learning program outputs P , then there is not program P ′ with the same length as P that has a better empirical error rate. We show in the sequel that such a learning program can not verify convergence rates as shown in theorem 6, i.e. a guaranteed convergence rate in O(1/ √ n) when an optimal function has bounded complexity. In the sequel, we assume that the reader is familiar with statistical learning theory and shattering properties ; the interested reader is referred to [START_REF] Devroye | A probabilistic theory of pattern recognition[END_REF]. The main element is that theorem C does not hold without bounded time. The following program has bounded length, only one parameter α, but generates as α ∈ R a family of functions which shatters an infinite set : -consider x the entry in R and α ∈ R a parameter ; -if x ≤ 0 then go to FINISH. -if α ≥ 0.5, output 1 and stop. -output 0 and stop. This program shifts α and x to the left until x > 1 2 . It then replies 1 if and only if α, after shift, has its first digit equal to 1. Therefore, this program can realize any dichotomy of { 1 2 , 1 4 , 1 8 , . . . }. This is exactly the definition of the fact that this set is shattered. So, we have shown that an family of functions shattering an infinite set was included in the set of programs with bounded length. Now, consider a learning program which has a guaranteed convergence rate in a family of functions including the family of functions computed by the program above. Ie, we assume that Theorem : fitnesses without time-complexity-pressure do not ensure consistency. What ever may be the sequence a 1 , . . . , a n , . . . decreasing to 0, there's no learning program ensuring that for any distribution of examples such that P (y = f (x)) = 1 for some f with bounded length, the expectation of P (P n (x) = y) is O(a n ). Conclusion In this paper, we have proposed a theoretical study of two important issues in Genetic Programming known as universal consistency and code bloat. We have shown that GP trees used in symbolic regression (involving the four arithmetic operations, the exponential function, and ephemeral constants, as well as test and jump instructions) could benefit from classical results from Statistical Learning Theory (thanks to Theorem C and Lemma 1). This has led to two kinds of original outcomes : i) some results about Universal Consistency of GP, i.e. almost sure asymptotic convergence to the optimal error rate, ii) results about the bloat. Both the unavoidable structural bloat in case the ideal target function does not have a finite description, and the functional bloat, for which we prove that it can be avoided by simultaneously bounding the length of the programs with some ad hoc bound and using some parsimony pressure in the fitness function. Some negative results have been obtained, too, such as the fact though structural bloat was known to be unavoidable, functional bloat might indeed happen even when the target function does lie in the search space, but no parsimony pressure is used. Interestingly, all those results (both positive and negative) about bloat are also valid in different contexts, such as for instance that of Neural Networks (the number of neurons replaces the complexity of GP programs). Moreover, results presented here are not limited to the scope of regression problems, but may be applied to variable length representation algorithms in different contexts such as control or identification tasks. Finally, going back to the debate about the causes of bloat in practice, it is clear that our results can only partly explain the actual cause of bloat in a real GP run -and tends to give arguments to the "fitness causes bloat" explanation [START_REF] Langdon | Fitness causes bloat: Mutation[END_REF]. It might be possible to study the impact of size-preserving mechanisms (e.g. specific variation operators, like size-fair crossover [START_REF] Langdon | Size fair and homologous tree genetic programming crossovers[END_REF] or fair mutations [START_REF] Langdon | The evolution of size and shape[END_REF]) as somehow contributing to the regularization term in our final result ensuring both Universal Consistency and no-bloat.

01746090

en

2024/03/05 22:32:07

2017

https://hal.science/hal-01746090/file/manuscript.pdf

We-Hyo Soe Yasuhiro Shirai Corentin Durand Yusuke Yonamine Kosuke Minami Xavier Bouju Marek Kolmer Katsuhiko Ariga Christian Joachim Waka Nakanishi Conformation Manipulation and Motion of a Double Paddle Molecule on an Au(111) Surface The molecular conformation of a bisbinaphthyldurene (BBD) molecule is manipulated using a lowtemperature ultrahigh-vacuum scanning tunneling microscope (LT-UHV STM) on an Au(111) surface. BBD has two binaphthyl groups at both ends connected to a central durene leading to anti/syn/flat conformers. In solution, dynamic nuclear magnetic resonance indicated the fast interexchange between the anti and syn conformers as confirmed by density functional theory calculations. After deposition in a submonolayer on an Au(111) surface, only the syn conformers were observed forming small islands of self-assembled syn dimers. The syn dimers can be separated into syn monomers by STM molecular manipulations. A flat conformer can also be prepared by using a peculiar mechanical unfolding of a syn monomer by STM manipulations. The experimental STM dI/dV and theoretical elastic scattering quantum chemistry maps of the low-lying tunneling resonances confirmed the flat conformer BBD molecule STM production. The key BBD electronic states for a step-by-step STM inelastic excitation lateral motion on the Au(111) are presented requiring no mechanical interactions between the STM tip apex and the BBD. On the BBD molecular board, selected STM tip apex positions for this inelastic tunneling excitation enable the flat BBD to move controllably on Au(111) by a step of 0.29 nm per bias voltage ramp. With the tip of a scanning tunneling microscope (STM), atomic-scale manipulation protocols are well-known since the pioneering work of D. Eigler, [START_REF] Frisch | Revision B.01[END_REF] and precise studies have described the various mechanisms of single atom (a small molecule) mechanical manipulations. [START_REF] Becke | Density-Functional Exchange-Energy Approximation with Correct Asymptotic Behavior[END_REF][START_REF] Becke | Density-Functional Thermochemistry. III. The Role of Exact Exchange[END_REF] Pushing a single large molecule on a surface with the tip of the STM [START_REF] Lee | Development of the Colle-Salvetti Correlation-Energy Formula into a Functional of the Electron Density[END_REF][START_REF] Ditchfield | Self-Consistent Molecular-Orbital Methods. IX. An Extended Gaussian-Type Basis for Molecular-Orbital Studies of Organic Molecules[END_REF] is now a standard procedure to position precisely functioning molecules on a surface for single molecule mechanics experiments [START_REF] Hehre | Self-Consistent Molecular Orbital Methods. XII. Further Extensions of Gaussian-Type Basis Sets for Use in Molecular Orbital Studies of Organic Molecules[END_REF] and also for single molecule electronic measurements. [START_REF] Hariharan | Accuracy of AH n Equilibrium Geometries by Single Determinant Molecular Orbital Theory[END_REF] To perform an atomically precise lateral manipulation of a molecule on a metallic surface with no mechanical interactions between the STM tip apex and the molecule, the bias tip must be able to feed up energy to the molecule with a few picometer lateral precision. [START_REF] Hariharan | The Influence of Polarization Functions on Molecular Orbital Hydrogenation Energies[END_REF] This excitation can be either inelastic from the tunneling current itself or originate from the enhanced electric field located in the biased tip/surface junction when the molecule carries a local dipolar moment. 9 For inelastic tunneling excitations, the energy entry port is generally the low-lying reduced electronic states of the molecule. [START_REF] Hariharan | Accuracy of AH n Equilibrium Geometries by Single Determinant Molecular Orbital Theory[END_REF]10 Here, a precise design of the molecule is required to avoid the energy provided by the tunneling current passing through the molecule from being equally distributed among the many mechanical degrees of freedom of this molecule. If not, a conformation change of the molecule may happen but with no lateral displacements. The molecule can also be broken in small chemical groups by the applied bias voltage pulse 11 instead of moving step-by-step on the supporting surface by steps, generally the commensurable surface atomic lattice constant. To also avoid energy redistribution toward the supporting surface, different leg and wheel molecular groups have been early identified. They can efficiently maintain a space (van der Waals distances) between the planar molecular chassis and the supporting surface. 12-14 Due, for example, to steric crowding, lateral chemical groups not having the shape of a leg or a wheel, mounted on the chassis in a symmetric way and holding it at van der Waals distances from the surface, are also interesting for molecular design as presented in this paper. [START_REF] Soe | Mapping the Excited States of Single Hexa-Peri-Benzocoronene Oligomers[END_REF] The light-driven molecular motor of the Feringa group the first switchable chemical group to be mounted by the Tour group on a chassis equipped with four wheels 16 in an attempt to leave space for this molecular group to change its conformation/configuration using an optical excitation. 17 A similar molecular switch was used by the Feringa group to obtain a molecule with four of those, used as switchable legs under a tunneling inelastic excitation. 18 Other switchable chemical groups are also available for equipping a molecular chassis. For example, molecules carrying a photoisomerizable double bond, such as stilbene, azobenzene, or diarylethene, have been used as molecular switches. 19 Their photoisomerization is usually studied in the gas phase or in solution. Conformation/configuration change triggered by tunneling electrons has also been observed in STM single molecule experiments, like with azobenzene. [20][21][22] Other molecules are also available which can twist around a single bond by photoirradiation. Twisted intramolecular charge transfer (TICT) molecules provide a nice example of such a light-activated conformation change. 23 Binaphthyl molecules or their derivatives (Scheme ) belong to another group of photosensitive compounds which are also known to change conformation under UV irradiation. 24,25 In this paper, we present the design and synthesis of a bisbinaphthyldurene (BBD) molecule (Scheme ) for STM imaging, single molecule manipulation, and step-by-step lateral motions. This molecule is equipped with two binaphthyl paddles mounted laterally on a very simple central phenyl chassis. On a planar BBD, we demonstrate here how to use the low amplitude vibration modes of its 1,1'-binaphthyl lateral paddles 26,27 for manipulating the BBD along a Au(111) surface using STM inelastic tunneling effects. Not existing in solution, this planar conformation is stabilized by the Au(111) surface. On Au(111), it enters in competition with its native in solution nonplanar conformation which can also be reached by the same excitation on a metallic surface as presented below. In the initial subsections of the Results and Discussion, the design, the synthesis, and the structural analysis of the BBD molecules in solution are provided together with a detailed DFT theoretical study of the different possible conformations of a BBD molecule. In the following subsections, STM images of the BBD molecules on the Au(111) surface acquired at low-temperature (LT) and in ultrahigh-vacuum environment (UHV) are provided. We demonstrate how to prepare the BBD molecule in a planar conformation on the Au(111) surface using a very specific STM tip lateral molecular manipulation protocol. In this planar conformation, the BBD electronic probability density map of its electronic states around the Au(111) surface Fermi level can be recorded to prepare the BBD inelastic manipulation. In the final subsection, the entry ports for tunneling electron energy transfer to the BBD molecule are identified. It is shown how to STM manipulate step-by-step the BBD molecule by step of 0.29 nm on the Au(111) surface. (TICT) molecules provide a nice example of such a lightactivated conformation change. 23 Binaphthyl molecules or their derivatives (Scheme 1) belong to another group of photosensitive compounds which are also known to change conformation under UV irradiation. 24,25 In this paper, we present the design and synthesis of a bisbinaphthyldurene (BBD) molecule (Scheme 1) for STM imaging, single molecule manipulation, and step-by-step lateral motions. This molecule is equipped with two binaphthyl paddles mounted laterally on a very simple central phenyl chassis. On a planar BBD, we demonstrate here how to use the low amplitude vibration modes of its 1,1′-binaphthyl lateral paddles 26,27 for manipulating the BBD along a Au(111) surface using STM inelastic tunneling effects. Not existing in solution, this planar conformation is stabilized by the Au(111) surface. On Au(111), it enters in competition with its native in solution nonplanar conformation which can also be reached by the same excitation on a metallic surface as presented below. around this Coriginating from 360°naphthylbond. 24,25,[28][29][30] T are classified usin their absolute co binaphthyls and between -180° and those betw configurations, r and as a function conformation an mol of energy. conformation ch 90°saddle point at |θ| ∼ 70°and | chiral binaphthy (CD) spectra, co was demonstrate mechanical force binaphthyl relaxe singlet state con internal mechani the S 1 (θ) relativ curve minima. 26,[START_REF] Becke | Density-Functional Exchange-Energy Approximation with Correct Asymptotic Behavior[END_REF] (|θ| > 90°), the S discussion, and th BBD design. For theory (TD-DFT naphthol) molec synthesis; Sche conformation be that triggers an a optically from Information). Entering now molecules are use reversal of the re and S 0 and bec Scheme 1. Synthetic Route of BBD 10358 Scheme 1. Synthetic Route of BBD. RESULTS AND DISCUSSIONS A. Design and Chemical Synthesis A 1,1'-binaphthyl molecule consists of two naphthalene moieties with one single phenyl per moiety connected via a C-C single bond. The distinct characteristics of a 1,1'-binaphthyl are (1) its flexibility around this C-C bond 24,25,28-30 and (2) the axial chirality originating from the inhibition by steric crowding of a complete 360 • naphthyl-naphthyl rotation around its joint C-C bond. 24,25,[28][29][30] The enantiomers of axially chiral compounds are classified using the stereochemical labels R and S based on their absolute configuration around a stereocenter. Chiral 1,1'-binaphthyls and derivatives having a naphthyl torsion angle between -180 • < θ < 0 • correspond to the R configurations and those between 0 • < θ < 180 • correspond to the S configurations, respectively. In its S 0 electronic ground state and as a function of the torsion angle |θ|, the 1,1'-binaphthyl conformation angle can vary from 60 • to 120 • within < 1 kcal/ mol of energy. The potential energy curve along this conformation change is a flat-bottomed well where the |θ| ∼ 90 • saddle point separates two shallow wells whose minima are at |θ| ∼ 70 • and |θ| ∼ 110 • . 24,25,[28][29][30] Since the conformation of chiral binaphthyls can be monitored by circular dichroism (CD) spectra, conformation controllability in this ground state was demonstrated at the air-water interface by applying a small mechanical force. 31,32 The difference between the S 0 1,1'-binaphthyl relaxed conformations and the S 1 lowest excited singlet state conformations 26,27 is at the origin of our BBD internal mechanical vibrations because of the reversal in θ of the S 1 (θ) relative to the S 0 (θ) double well potential energy curve minima. 26,27 Although for cisoid (|θ| < 90 • ) and transoid (|θ| > 90 • ), the S 1 and S 0 relaxed conformations are still under discussion, and this difference was important to preserve in the BBD design. For example, time-dependent density functional theory (TD-DFT) calculations show that the (R)-1,1'-bi (2-naphthol) molecule (the starting compound for the BBD synthesis; Scheme ) still preserves a different relaxed conformation between S 0 (θ = -91 • ) and S 1 (θ = -119 • ) that triggers an almost 30 • paddle effect going back and forth optically from S 0 to S 1 (Figure S16 in the Supporting Information). Entering now in the design of our molecule, two binaphthyl molecules are used in the BBD as lateral paddles because of this reversal of the relative minimum energy |θ| value between S 1 and S 0 and because of the low 1 kcal/mol energy barrier between the two minima in the S 0 ground state. The two binaphthyls are connected laterally to a very small chassis made simply of a central phenyl (BBD in Scheme 1). This covalent binding of each binaphthyl via the methylene oxy bridges modifies the paddle switch ability with, for example, the suppression of the S 1 (θ = -119 • ) torsion angle energy minimum. What is important here is that S 0 keeps its awaited mechanical characteristics, that is, the possibility of its vibrational oscillations around its new (θ = -61 • ) ground-state minimum (TD-DFT calculated) reachable, for example, by optical excitation and relaxation via its new S 1 (θ = -58 • ) relaxed conformation for the (R,R) isomer (Figure S17 in the Supporting Information). On a metallic surface and in a planar conformation, the two BBD binaphthyl groups permit to space the BBD chassis away from this surface at a distance compatible with a physisorption state. To drive the BBD molecule step-by-step along an fcc track of the Au(111) surface using the inelastic effect of the STM tunneling current, one has to first virtually prepare this molecule in its instantaneous virtual reduced electronic state well described for its mechanics by considering in first approximation the BBD S 1 excited state. Afterward, relaxation to the S 0 ground state will result in a small amplitude and noncoherent binaphthyl oscillations. As a function on the tip apex location on BBD, this will generate a surface lateral motion over the lateral diffusion barrier of the Au(111) fcc portion of the herringbone surface reconstruction. The BBD molecule was synthesized by a one-pot reaction from commercially available (R)-1,1'-bi(2-naphthol) and α,α ,α ,α -tetrabromodurene (Scheme 1). Before its evaporation in the STM preparation chamber, it was further ultrapurified by sublimation to produce a colorless powder with no crystallinity. BBD UV-vis absorption spectrum is similar to the one of binaphthyl molecules 33 and shows an absorption peak maximum at 334 nm (Figure S1 in the Supporting Information), demonstrating a good electronic separation between the two BBD lateral paddles. Recorded conventional CD spectra of chiral binaphthyls (Figure S2 in the Supporting Information) confirmed that the chirality of the binaphthyl groups remained after purification. B. The Native BBD Molecule Conformation in Solution. Variable-temperature (VT) analysis with NMR spectroscopy (Figure S13 in the Supporting Information) revealed the dynamic fluctuations between two BBD conformers with the same equilibrium population in solution. The two sets of 1 H NMR peaks that originate from those two conformers were also observed at low temperature from 218 to 223 K showing no sign of a favored conformer (Figure S13 in the Supporting Information). The analysis of those VT NMR spectra by a line-shape-fitting provides the experimental energetics for the interexchange processes between the two BBD conformers. Using an Eyring plot, the parameters of this interexchange were estimated to be ∆H = 9.8 kcal/mol and ∆S = -17 cal/(mol K) (Figures S14 and S15 in the Supporting Information), supporting the possibility of a fast interconversion of the two conformers at ambient temperature and in solution. As presented in Figure 1, three BBD conformers were identified using DFT calculations (B3LYP/6-31G(d,p)) depending on the location of the two binaphthyl paddles relative to the central phenyl. They correspond to the flat, syn, and anti conformations of a BBD molecule. The syn and anti conformers are expected to be the principal BBD isomers in solution since after molecular structure optimization, the flat, syn, and anti relative conformation energies are 20, 0.4, and 0.0 kcal/mol, respectively. Experimental ROESY peaks in NMR spectroscopy are also consistent with the existence of the anti-syn conformer in solution since a weak correlation was observed between the central aromatic CH and the side binaphthyl aromatic CH proton (Figures S10-S12 in the Supporting Information). the S 0 ground state. The two rally to a very small chassis made BD in Scheme 1). This covalent via the methylene oxy bridges ability with, for example, the -119°) torsion angle energy here is that S 0 keeps its awaited that is, the possibility of its d its new (θ = -61°) groundlculated) reachable, for example, ation via its new S 1 (θ = -58°) (R,R) isomer (Figure S17 in the n a planar conformation, the two it to space the BBD chassis away compatible with a physisorption lecule step-by-step along an fcc using the inelastic effect of the has to first virtually prepare this virtual reduced electronic state hanics by considering in first cited state. Afterward, relaxation result in a small amplitude and lations. As a function on the tip will generate a surface lateral sion barrier of the Au(111) fcc rface reconstruction. nthesized by a one-pot reaction e (R)-1,1′-bi(2-naphthol) and (Scheme 1). Before its evapoation chamber, it was further to produce a colorless powder V-vis absorption spectrum is thyl molecules 33 and shows an at 334 nm (Figure S1 in the monstrating a good electronic BBD lateral paddles. Recorded iral binaphthyls (Figure S2 in the firmed that the chirality of the fter purification. le Conformation in Solution. nalysis with NMR spectroscopy ting Information) revealed the two BBD conformers with the in solution. The two sets of 1 H those two conformers were also from 218 to 223 K showing no (Figure S13 in the Supporting those VT NMR spectra by a linexperimental energetics for the een the two BBD conformers. meters of this interexchange were /mol and ΔS = -17 cal/(mol K) the Supporting Information), syn, and anti relative conformation energies are 20, 0.4, and 0.0 kcal/mol, respectively. Experimental ROESY peaks in NMR spectroscopy are also consistent with the existence of the antisyn conformer in solution since a weak correlation was observed between the central aromatic CH and the side binaphthyl aromatic CH proton (Figures S10-S12 in the Supporting Information). A f lat BBD conformer is supposed by design to render accessible the different entry ports on its board for local STM excitations. In the following section, we will demonstrate how this f lat conformer can be produced molecule per molecule by STM single molecule mechanical manipulations. When obtained, this flat conformer turns out to be quite stable on the Au(111) surface. Native BBD Conformation and 2D Organization on the Au(111) Surface. Two typical constant current STM images obtained after BBD molecules deposition on the Au(111) reconstructed surface are presented in Figure 2. They were mainly found self-assembled in small 2D islands (Figure 2a). In some place, single BBD molecule lines can also be observed. This pseudo-1D growth along the Au(111) herringbone track is usually stopped at both ends of the line by a different surface BBD molecular ordering (Figure 2b). In all those observed pseudo-1D and 2D surface molecular orderings, the BBD molecules appear having the shape of a curve letter Two typical constant current STM images obtained after BBD molecules deposition on the Au(111) reconstructed surface are presented in Figure 2. They were mainly found self-assembled in small 2D islands (Figure 2a). In some place, single BBD molecule lines can also be observed. This pseudo-1D growth along the Au(111) herringbone track is usually stopped at both ends of the line by a different surface BBD molecular ordering (Figure 2b). In all those observed pseudo-1D and 2D surface molecular orderings, the BBD molecules appear having the shape of a curve letter " ". Those " " BBD molecules have three possible adsorption directions on the Au(111) surface (see the Figure 2a insert). As presented in Figure 2b, the observed single " " BBD lines confirm that the BBD molecules are sensitive to the lateral ridges of the herringbone reconstruction (in average 0.03 nm in height). [START_REF] Hehre | Self-Consistent Molecular Orbital Methods. XII. Further Extensions of Gaussian-Type Basis Sets for Use in Molecular Orbital Studies of Organic Molecules[END_REF] As certified experimentally by STM single molecule manipulations in the next subsection, each " " STM molecular feature is a BBD dimer consisting of two syn conformers oriented perpendicular to the surface plane. They are coupled by a pair along one of the three As discussed in the previous subsection, the BBD molecules are found equally in the syn and anti conformations in solution. Molecular dynamics (MD) simulations were performed to simulate a hot adsorption process of the BBD molecules on a Au(111) surface (see Supporting Information). When the BBD molecules are annealed on the surface up to 500 K, the syn and anti conformers are deformed but remain on the surface with no transformation in the flat conformer. At this temperature and during their 2D diffusion around the Au(111) surface, the BBD molecules have enough kinetic energy to mutually transform between the syn and anti as they certainly performed in solution and at room temperature. Upon cooling down the surface to room temperature, the BBD molecules thermalize toward the syn conformers since syn is 9.0 kcal/mol lower in energy as compared to anti on the Au(111) surface. Furthermore, the syn thermalize with still their central phenyl perpendicular to the Au(111) surface because laterally stabilized by their two paddles. During a slow thermalization process, the syn will also continue to diffuse on the surface. While in this perpendicular adsorption conformation, they can pair via a central phenyl π-stacking interactions as also confirmed by MD calculation (see Supporting Information). It results in the " " As discussed in the previous subsection, the BBD molecules are found equally in the syn and anti conformations in solution. Molecular dynamics (MD) simulations were performed to simulate a hot adsorption process of the BBD molecules on a Au( 111) surface (see Supporting Information). When the BBD molecules are annealed on the surface up to 500 K, the syn and anti conformers are deformed but remain on the surface with no transformation in the f lat conformer. At this temperature and during their 2D diffusion around the Au(111) surface, the confirmed by MD calcula It results in the "f " STM Figure 2 assembled in th islands. According to MD calcul conformers would be mo surface syn and anti confor conformer from the nativ Au(111) surface would r about 1500 K. Therefore, the surface directly afterwa conformers. At such a h molecules will break and/ the gold melting point is l below, f lat conformers can on the Au( 111 According to MD calculations and if accessible, the flat BBD conformers would be more stable than the orthogonal to the surface syn and anti conformers. However, getting directly a flat conformer from the native syn and anti conformers on the Au(111) surface would require an annealing temperature of about 1500 K. Therefore, during deposition or by heating up the surface directly afterward, it will be difficult to produce flat conformers. At such a high temperature, most of the BBD molecules will break and/or desorb from the surface. Anyhow, the gold melting point is lower than 1500 K. As demonstrated below, flat conformers can be produced molecule per molecule on the Au(111) surface starting from the orthogonal to the surface syn conformers using a very specific STM single molecule mechanical manipulation protocol. D. STM Single Molecule Mechanical Manipulation for Preparing flat BBD Conformers. To produce a flat BBD conformer, a selected syn conformer dimer adsorbed perpendicular to the Au( 111) surface (one of the " " molecular units imaged in Figure 2) must be first separated into independent syn monomers. For this purpose and starting from a 2D island of the sort imaged in Figure 2a, STM lateral BBD molecule mechanical manipulation has been first performed as presented in Figure 3. Here, the threshold STM tunneling resistance for molecule manipulation is around R T = 270 MΩ. In most cases, a " " dimer can readily be separated out of the 2D island but only as a single " " dimer entity with no monomer separation as presented in the Figure 3a,b. Then, a " " dimer can be step-by-step displaced over quite long distances over the surface in such STM manipulation conditions. When they are sometimes disassembled into two syn monomers during this process, one syn of the " " pair is generally transferred to the STM tip, and the other one remains in the island as shown in the sequence Figure 3b,c. Notice also that after the breaking of a " " pair at the 2D island border and in its orthogonal to the surface adsorption configuration, the syn monomer left in the island (as obtained in Figure 3c) can be further extracted from this island byr a further lateral STM manipulation. In this case, it has also a high probability to be captured by the tip apex, confirming how this syn monomer orthogonal configuration is not very stable on an Au(111) surface. We have succeeded to manipulate a few of those syn monomers toward specific Au(111) surface areas like the herringbone kinks where generally the surface atomic order is not regular and can stabilize them. They can also be dragged e. They are coupled 11] crystallographic this surface (see the cation of this pairing the only molecular BDs molecular row. , the BBD molecules rmations in solution. were performed to BD molecules on a ion). When the BBD o 500 K, the syn and on the surface with At this temperature Au(111) surface, the energy to mutually certainly performed n cooling down the olecules thermalize 0 kcal/mol lower in Au( 111) surface. their central phenyl because laterally slow thermalization use on the surface. formation, they can nteractions as also conformers. At such a high temperature, most of the BBD molecules will break and/or desorb from the surface. Anyhow, the gold melting point is lower than 1500 K. As demonstrated below, f lat conformers can be produced molecule per molecule on the Au(111) surface starting from the orthogonal to the surface syn conformers using a very specific STM single molecule mechanical manipulation protocol. STM Single Molecule Mechanical Manipulation for Preparing f lat BBD Conformers. To produce a f lat BBD conformer, a selected syn conformer dimer adsorbed perpendicular to the Au( 111) surface (one of the "f " molecular units imaged in Figure 2) must be first separated into independent syn monomers. For this purpose and starting from a 2D island of the sort imaged in Figure 2a, STM lateral BBD molecule mechanical manipulation has been first performed as presented in Figure 3. Here, the threshold STM tunneling resistance for molecule manipulation is around R T = 270 MΩ. In most cases, a "f " dimer can readily be separated out of the 2D island but only as a single "f " dimer entity with no monomer separation as presented in the Figure 3a,b. Then, a "f " dimer can be step-by-step displaced over quite long distances over the surface in such STM manipulation conditions. When they are sometimes disassembled into two syn monomers during this process, one syn of the "f " pair is generally transferred to the STM tip, and the other one remains in the island as shown in the sequence Figure 3b,c. Notice also that after the breaking of a "f " pair at the 2D island border and in its orthogonal to the surface adsorption configuration, the syn monomer left in the island (as obtained in Figure 3c) can be further extracted from this island by a further lateral STM manipulation. In this case, it has also a high probability to be captured by the tip apex, confirming how this syn monomer orthogonal configuration is not very stable on an along the surface during standard imaging conditions, that is, for STM R T around 10 GΩ and a tunneling current below 10 pA. At the border of a 2D-island and using R T ∼ 120 MΩ, a new specific BBD dimer molecule manipulation protocol can bring a different manipulation outcome. When in a " " pair located at the border of a 2D island, a syn BBD molecule is manipulated without trying to extract it directly from this island border; however, following the appropriate manipulation tip trajectory presented in Figure 4, a flat monomer can be produced with its two-fold paddles now fully open. Figure 4 presents an example of such an outcome with a reasonable 10% probability of success. There are two essential conditions for this specific protocol to produce successfully a flat conformer and to open the BBD two paddles. First, the syn targeted BBD molecule must be paired with a syn BBD anchored at the edge of a 2D island but not at a corner. This anchoring will serve as a pivot for the opening following a classical molecular mechanical motion of the molecule, as if it was a solid and rigid body pivoting around a fix point. Second, the targeted syn BBD must be "rubbed" laterally on another " " dimer of the island during the manipulation, that is, the manipulation trajectory must maintain a lateral interaction with the other " " dimer for the flat flipping of the manipulated BBD molecule to be complete. As a consequence, the manipulated BBD molecule performs a 90 • flip down to the surface to reach a planar central phenyl configuration with the two paddles opened flat on the Au(111) surfacem as illustrated in Figure 4a,b (more examples in Supporting Information section 9). In this case, the lateral required interactions between the manipulated BBD and the border 2D island BBDs seem to be attractive according to the recorded manipulation signal, but this requires a more detailed interpretation in the future. When produced, a flat conformer is very stable on the Au(111) surface as theoretically predicted by MD calculations. After its production, this is confirmed by the experimental RT threshold value for a flat conformer STM molecular manipulation in a pushing mode along the Au(111) surface, the lowest (∼ 66 MΩ) of all the R T values used for the different BBD molecule configurations met on this surface. A flat BBD monomer can be truly and reproducibly STM manipulated mechanically over long distances as presented in Figure 4b,c. E. Tunneling Spectroscopy and States Mapping of the Flat BBD Conformer. A flat BBD conformer on the Au(111) surface opens access to the detail STM dI/dV mapping of its low-lying molecular electronic states around the Au(111) surface Fermi energy. Such a the herringbone kinks where generally the surface atomic order is not regular and can stabilize them. They can also be dragged along the surface during standard imaging conditions, that is, for STM R T around 10 GΩ and a tunneling current below 10 pA. At the border of a 2D-island and using R T ∼ 120 MΩ, a new specific BBD dimer molecule manipulation protocol can bring a different manipulation outcome. When in a "f " pair located at the border of a 2D island, a syn BBD molecule is manipulated without trying to extract it directly from this island border; however, following the appropriate manipulation tip trajectory presented in Figure 4, a f lat monomer can be produced with its two-fold paddles now fully open. Figure 4 presents an example of such an outcome with a reasonable 10% probability of success. There are two essential conditions for this specific protocol to produce successfully a f lat conformer and to open the BBD two paddles. First, the syn targeted BBD molecule must be paired with a syn BBD anchored at the edge of a 2D island but not at a corner. This anchoring will serve as a pivot for the opening following a classical molecular mechanical motion of the molecule, as if it was a solid and rigid body pivoting around a fix point. Second, the targeted syn BBD must be "rubbed" laterally on another "f " dimer of the island during the manipulation, that is, the manipulation trajectory must maintain a lateral interaction with the other "f " dimer for the flat flipping of the manipulated BBD molecule to be complete. As a After its production, threshold value fo manipulation in a pu the lowest (∼66 MΩ) BBD molecule config monomer can be tru mechanically over lon Tunneling Spect Flat BBD Conforme surface opens access low-lying molecular surface Fermi energy confirm the f lat confo specific manipulation determining the loca weight of its reduced BBD molecular struct principal port for intra tunneling electrons in structure to move on push. Figure 5a presents f lat BBD molecule ad STM tip apex was po lobes identified on the identifying the Au(11 V) in this spectrum, observed, one at -1. +2.7 V. This gives an for a f lat BBD mo compared to the 3.7 e the syn and anti conf Performed exactly precise dI/dV STM m molecular orbital el reduced and oxidized two tunneling resonan dV elastic scattering calculations were pe imental dI/dV maps. adapted to provide a molecules. 35,36 Thes contributor to theoccupied molecular unoccupied molecu resonance. Those im optimized f lat BBD co confirms how the B conformation by STM to Figure 5 images, electronically decoupl Step-by-Step Ma mapping is of importance to confirm the flat conformation interpretation after the Figure 4 specific manipulation protocol. It is also very appropriate for determining the location of the maximum molecular orbital weight of its reduced and oxidized electronic states along the BBD molecular structure. Those maxima are known to be the principal port for intramolecular inelastic excitations induced by tunneling electrons in a way to bring energy to the molecular structure to move on a metallic surface with no mechanical push. Figure 5a presents a typical dI/dV spectrum recorded on a flat BBD molecule adsorbed on the Au(111) surface. Here, the STM tip apex was positioned at the center of one of the two lobes identified on the topographic image (see Figure 4c). After identifying the Au(111) surface states energy location (∼ -0.5 V) in this spectrum, two differential conductance peaks are observed, one at -1.6 V and a small bump centered around +2.7 V. This gives an apparent electronic gap of about To manipulate a BBD molecule by inelastic electron tunneling effects along a fcc portion of the Au(111) surface, the single molecule must capture enough energy from the tunneling current to pass over the fcc surface lateral diffusion barrier. One way to trigger the inelastic energy release on the BBD vibronic mode is to increase the tunneling current intensity through the molecule reaching its first low-lying reduced states. According to the dI/dV spectrum in Figure 5a, this can be achieved with a bias voltage applied to the tunnel junction greater than about 2 V to reach at least the tail of the +2.7 V BBD tunneling electronic resonance. Notice here that the energy captured by the vibronic modes of the BBD molecule will be a small fraction of the 2 eV. 38 To maximize the tunneling inelastic excitations, it is also usually taken for granted to position the STM tip apex at the locations at the highest electron density of the targeted in energy molecular electronic states. This strategy also helps to minimize the STM bias voltage range in a way not to destroy the molecule. 11 The BBD dI/dV images presented in Figure 5 are mapping those maxima and minima in its flat surface conformation. This mapping results from the electronic coupling between the tip apex and the molecular orbitals entering in the composition of resonating BBD electronic states when considering that those electronic states can be well described by a superposition of Slater determinants constructed using a molecular orbital basis set. 37,39 The pixel-by-pixel construction of the Figure 5b,c maps results from the local measurement of the conductance of the BBD molecule at each pixel. In effect, this measurement projects the total BBD electronic probability density on a 2D plane. This is a very convenient way to identify where to position the tip for triggering a tunneling inelastic effect. For BBD, the highest electronic probability density (the highest dI/dV) sites are located on the BBD binaphthyl paddles as observed in Figure 5c. ACS Nano As indicated in Figure 5c, when positioning the tip apex at location 1 on the BBD molecule and then ramping up the bias voltage further than +2.3 V (but without reaching +2.7 V), the BBD changes its conformation with one naphthyl paddle going up the surface in a conformation similar to syn (syn/flat-like conformation, see Figure S21 in the Supporting Information). This conformational change occurs systematically opposite to the paddle been excited. On the corresponding reduced state potential energy surface, this indicates that the energy captured by the BBD molecule from the tunneling current is initiating a conformation change trajectory at the onset of the +2.7 V resonance. Starting from the flat ground-state conformation, this trajectory certainly reaches a minimum on this reduced state potential energy manifold corresponding to the rotating up of a paddle to form a "syn/flat-like" conformation. Then, the BBD molecule relaxes in its ground state in this new stable "syn/flat-like" conformation not observable natively event after an STM mechanical lateral manipulation procedure. Notice that in the first approximation, the reduced state potential energy manifold can be explored using the potential energy surface of the BBD S 1 vertically accessible excited electronic state. After having tried to induce the paddle vibrations directly, we have selected the electronic probability density maximum 2 as indicated in Figure 5c. For this new excitation location and as presented in left column of Figure 6, when the bias voltage reaches about +2.3 V and the tunneling current several hundred pico-amperes, the current intensity through the BBD suddenly jumps up due to the molecule one step lateral translation. This very reproducible behavior shows how a BBD molecule can be step-by-step driven by steps of 0.29 nm on an fcc flat area on the Au(111) surface. To be more precise on the inelastic manipulation direction, an atomic resolved image of the Au(111) surface recorded using molecule terminated tip is inserted in Figure 6a to certify the accurate moving direction: exactly one of 110 orientations of the Au(111) surface inits fcc portion. The interatomic gold atom distance along those orientations is 0.288 nm, in complete agreement with the experimentally observed 0.29 nm long step motion per voltage ramp. As presented in Figure 6, a controllable inelastic motion is only possible when the molecule lies parallel to and in between two herringbones of the reconstruction. When the molecule sits on a herringbone (even when only one paddle end is laying on it), it is stuck and difficult to manipulate inelastically, as also recently observed with a windmill molecule. 40 For a BBD molecule laying parallel to the herringbones and located at the fcc portion of the Au(111) surface, the probability of a controllable stepwise motion is about 50% after a single shot bias voltage ramp. Incidentally, the probability of molecule motion regardless the adsorption site is around 5% and the probability of a breaking or a conformation changes is around 3%. As a consequence and after a three consecutive lateral step-by-step motions, the BBD molecule is stopped because of herringbone lateral diffusion barrier and must be reprepared for a new run like with the windmill molecule. 40 Here, the BBD molecule must be manipulated with care not to open a conformation change path on its reduced state potential energy surface nor a chemical reaction path breaking some of its chemical bonds leading to the final destruction of the molecule since a molecule is often very unstable under high positive STM bias voltage pulses. [START_REF] Hariharan | Accuracy of AH n Equilibrium Geometries by Single Determinant Molecular Orbital Theory[END_REF]41 While exciting the BBD molecule at two different spatial locations of the same resonance maxima, the difference of mechanical response is a nice indication of how the electronic coupling between the tip apex and the electronic states of a molecule can give rise to different mechanical responses. Here and during an STM excitation (or imaging), the effective lateral extension of the tunneling electrons inelastic excitation is much narrower than the BBD molecular orbitals spatial lateral extension. As a consequence, the electronic coupling between the tip apex and the BBD molecule is very local. For each tip positioning on the molecule, this brings out a very specific is often very unstable under high positive STM bias voltage pulses. [START_REF] Hariharan | Accuracy of AH n Equilibrium Geometries by Single Determinant Molecular Orbital Theory[END_REF]41 While exciting the BBD molecule at two different spatial locations of the same resonance maxima, the difference of mechanical response is a nice indication of how the electronic coupling between the tip apex and the electronic states of a molecule can give rise to different mechanical responses. Here and during an STM excitation (or imaging), the effective lateral extension of the tunneling electrons inelastic excitation is much narrower than the BBD molecular orbitals spatial lateral extension. As a consequence, the electronic coupling between the tip apex and the BBD molecule is very local. For each tip positioning on the molecule, this brings out a very specific superposition of BBD molecular orbitals to contribute to the motion as also observed for STM imaging in the case, for example, of an hexabenzocoronene (HBC) molecule. 42 This can trigger a large conformation of the paddle for one tip apex location or a gentle stepwise lateral motion for another location supposing that the effective potential energy surface built up from this superposition is different in the two cases. For negative applied bias voltage, we have also tried the same strategy by locating the tip apex at one of the many maxima indicated in Figure 5b. No movement of the BBD molecule was observed down to a bias voltage ramp reaching a maximum of -2.0 V with several nA of tunneling current intensity. We do not yet have a detailed explanation of this observation. CONCLUSION A bisbinaphthyldurene (BBD) molecule was designed, synthesized, and deposited on an Au(111) surface, mechan- superposition of BBD molecular orbitals to contribute to the motion as also observed for STM imaging in the case, for example, of an hexabenzocoronene (HBC) molecule. 42 This can trigger a large conformation of the paddle for one tip apex location or a gentle stepwise lateral motion for another location supposing that the effective potential energy surface built up from this superposition is different in the two cases. For negative applied bias voltage, we have also tried the same strategy by locating the tip apex at one of the many maxima indicated in Figure 5b. No movement of the BBD molecule was observed down to a bias voltage ramp reaching a maximum of -2.0 V with several nA of tunneling current intensity. We do not yet have a detailed explanation of this observation. 111) is a perpendicular dimer conformation. Once a BBD molecule was prepared in its flat conformation, dI/dV molecular orbital mapping was performed to determine the best tip apex location to free up this molecule inelastically for a manipulation on an Au(111) fcc flat terrace. The intuitive on-paddle excitation is not a good entry port for an inelastic tunnel manipulation since it leads to a drastic conformation change of the BBD molecule entering in competition with its step-by-step motion on the Au(111) surface. We have demonstrated that on the molecule and nearby the paddle location, there exists another energy entry port where the BBD molecule remains flat on the surface and can be laterally manipulated step-by-step with a step of about 0.29 nm per excitation. The BBD molecule was used by the MANA-NIMS Japanese team during the first international nanocar race in Toulouse. 43 MATERIALS AND METHODS The synthetic procedure of BBD is described in the Supporting Information. The STM experiments were conducted as follows. The BBD molecules were deposited on a Au(111) single crystal surface previously cleaned by standard metal surface UHV preparation methods consisting of several cycles of ion sputtering and subsequent annealing. [START_REF] Hehre | Self-Consistent Molecular Orbital Methods. XII. Further Extensions of Gaussian-Type Basis Sets for Use in Molecular Orbital Studies of Organic Molecules[END_REF]10 The BBD molecules were sublimated from about 3 mg of the colorless BBD molecular powder by heating a Kentax quartz crucible at 563 K during 30 s. The gold substrate temperature was kept below 323 K during this deposition. The evaporation parameters were selected in a way to deposit a minute amount of BBD molecules to produce a submonolayer coverage in order to leave enough large molecule-free areas on the clean Au(111) surface to be able to use the STM single molecule manipulation protocol. The Au(111) sample was then loaded on the STM sample stage kept at cryogenic temperature (LT) and rapidly cooled down to ∼ 5 K. All low-temperature ultrahigh-vacuum scanning tunneling microscopy (LT-UHV STM) experiments presented, that is, constant current imaging, molecule manipulations, tunneling spectroscopic measurements, and intramolecular dI/dV mapping were performed on one of the four STM heads of our new ScientaOmicron LT-UHV 4 independent STM instrument. 44 * E-mail: [email protected] † Present Address: Department of Chemical Engineering, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan. ‡ E-mail: [email protected]. Photophysical properties See Fig. S1 and Fig. S2. NMR spectra See Fig. S3 to Fig. S15. Theoretical calculations (DFT) All calculations were performed using the Gaussian 09 program, 1 and the results were analyzed and visualized on GaussView 5.0.9. Calculations were performed at the density functional theory (DFT) level with the B3LYP functional, the gradient correction of the exchange functional by Table S16. Potential energy (S 0 and S 1 ) of (a) 1,1'-binaphthyl and (b) BINOL, which were modified from the reported one for 1,1'-binaphthyl. 9 As reported, two sets of optimized structure of 1,1'-binaphthyl were found both in S 0 and S 1 by DFT and TD-DFT (B3LYP/6-31G(d,p)) calculations. On the other hand, only one sets of optimized structure were found for BINOL by the same procedures. Theoretical calculations (MD simulations) Structures The simulated systems consisted of one or two molecules on Au(111) ------------------------------------------------------------------- ------------------------------------------------------------------ ------------------------------------------------------------------- ------------------------------------------------------------------- ------------------------------------------------------------------ ------------------------------------------------------------------- ------------------------------------------------------------------- ------------------------------------------------------------------ ------------------------------------------------------------------- ------------------------------------------------------------------- ------------------------------------------------------------------ .661329 ---------------------------------------------------------------------Table S5. Cartesian coordinate anti-conformer SCF Done: E(RB3LYP) = -2226.74970717 A.U. after 14 cycles ---------------------------------------------------------------------Center Atomic Atomic Coordinates (Angstroms) Number Number Type X Y Z --------------------------------------------------------------------- 1.490691 ---------------------------------------------------------------------Table S5. deformed. On the other hand, the stability of flat-conformer mainly originate from the large surface interaction energy (E surfaceinteraction_flat -E sur f aceinteraction_syn = -29.3 kcal/mol more favorable than syn-conformer) rather than molecular deformation energy (E flat -E syn = +6.9 kcal/mol higher than syn-conformer). Total energy of syn-syn dimer on Au(111) surface (E total ) was -42 912.6 kcal/mol. The following energies were obtained as a single point energy calculation with the same force field. The energy of Au ( 111) (E surface ) was -43 945.7 kcal/mol, that of syn-syn dimer (E syn-syn-dimer ) is 1 438.2 kcal/mol, those of each syn-conformers (E syn-conformer-1 and E syn-conformer-2 ) are 724.3 kcal/mol and 724.3, respectively. The interaction energy between Au(111) and syn-syn dimers (E surfaceinteraction ) was calculated to be -405.1 kcal/mol and that between syn-syn dimers (E molecularinteraction ) was to be -10.4 kcal/mol based on the following equations: E surfaceinteraction = E total -(E surface + E syn-syn-dimer ) E molecularinteraction = E syn-syn-dimer -(E syn-conformer-1 + E syn-conformer-2 ) The formation of syn-syn dimer is more favorable than being two syn-monomers, by the energy of E = (E total -E surface )/2 -(E total s yn -E surface m onomer ) = -2.8 kcal/mol. The formation of syn-syn dimer (in vacuum) is estimated to be E syn-syn-dimer /2 -Esyn = +4.8 kcal/mol more unfavorable compared with being monomers. On the other hand, surface interaction energy became E surfaceinteraction /2 -E surfaceinteraction s yn = -7.6 kcal/mol more favorable for formation of dimers. As a result, each syn-conformers are E = -7.6 + 4.8 = -2.8 kcal/mol more favorable as a dimer compared with being monomers. Conformation transformations by forcible molecule manipulation Here we provide more details and examples of lateral single molecule manipulation protocol in a mechanical mode to succeed to prepare one by one single planar BBD molecules are presented and used in the manuscript. See Fig. S20. Inelastic electronic tunneling effect After some of the inelastic electronic tunneling excitation of a BBD molecule by locating the STM tip apex on the highest electronic probability density site of a BBD molecule, we have S18 conformer on Au(111) surface was also obtained with higher energy cycle temperature was raised to 1500 K, flat-conformer was found wit compared with the syn-conformer. For calculations of a dimer on Au syn-conformers were used as initial structures, and syn-syn-dimer on A a stable conformer. Figure 1 .Figure 1 . 11 Figure 1. Various possible conformers of BBD molecule. (a) Flat, (b) syn, and (c) anti conformers obtained from DFT calculations (B3LYP/6-31G(d,p)) with relative energies of +20, +0.4, and 0.0 kcal/mol, respectively. Each structure was optimized with its D 2 , C 2 , and C 2 symmetry, respectively. The calculated torsion angles of the binaphthyl are θ = -61°, -64°and -64°, respectively. In solution, only the anti and syn conformers have been identified. Physisorbed on an Au(111) surface, STM molecular manipulations of BBD lead to the production of the flat conformer. C . Native BBD Conformation and 2D Organization on the Au(111) Surface. [211] crystallographic orientations of the Au(111) fcc portion of this surface (see the Figure 2a insert). A first experimental indication of this pairing is evidenced in Figure 2a by analyzing the only molecular alignment defect at the top of the last left BBDs molecular row. oriented perpendicular to the surface plane. They are coupled by a pair along one of the three [211] crystallographic orientations of the Au(111) fcc portion of this surface (see the Figure 2a insert). A first experimental indication of this pairing is evidenced in Figure 2a by analyzing the only molecular alignment defect at the top of the last left BBDs molecular row. ) surface s surface syn conformers molecule mechanical man STM Single Molecul Preparing f lat BBD Co conformer, a selected perpendicular to the Au(1 units imaged in Figure independent syn monom from a 2D island of the so BBD molecule mechan performed as presented STM tunneling resistance R T = 270 MΩ. In most separated out of the 2D i entity with no monomer s 3a,b. Then, a "f " dimer can Figure 2 .Figure 3 .Figure 2 . 232 Figure 2. (a) A typical STM topographic image of BBD molecules forming a small 2D island on the Au(111) reconstructed surface.Each building block of the imaged island is a dimer of syn-syn conformers having the shape of a curved "f " letter. Insert a: the three possible adsorption orientations of the syn dimers. The atomic resolved image was also recorded using a molecule terminated STM tip to confirm the molecular orientation. (b) A good example of supramolecular assembly with a line of 10 BBD molecular dimers formed along an Au(111) herringbone whose growth was stopped at both ends of the BBD line by 2 "f " dimers of a different surface orientation. (All LT-UHV STM constant current image images were generally recorded at I = 20 pA, V = 0.5 V.) Figure 3 . 3 Figure 3. An example of the molecule manipulation experiments using the tunneling condition near the threshold R T = 270 MΩ. (a) 2D island of self-assembled syn-syn dimers on the Au(111) surface observed just after Au(111) sample preparation. (b) A "f " BBD molecule was step-by-step manipulated and extracted from its island while maintaining its dimer structure. (c) A monomer was detached from a down right corner "f " of the island and adsorbed to the tip apex and remains of the second syn of this dimer initially perpendicular to the surface conformation (same STM image conditions as in Figure 2. Image size: 15 nm × 15 nm). Figure 3 . 3 Figure 3. An example of the molecule manipulation experiments using the tunneling condition near the threshold R T = 270 MΩ. (a) 2D island of self-assembled syn-syn dimers on the Au(111) surface observed just after Au(111) sample preparation. (b) A " " BBD molecule was step-by-step manipulated and extracted from its island while maintaining its dimer structure. (c) A monomer was detached from a down right corner " " of the island and adsorbed to the tip apex and remains of the second syn of this dimer initially perpendicular to the surface conformation (same STM image conditions as in Figure 2. Image size: 15 nm × 15 nm). Figure 4 . 4 Figure 4. An example of the forcible molecule manipulation experiments using tunneling conditions less than 1/2 R T in Figure 3. Topographic images (a) before and (b) after R T = 120 MΩ manipulation. Using this condition, a structural transformation from syn to flat conformer occurs. (c) The so-produced flat conformer was manipulated with the STM tip away from its original four "f " dimer line with R T = 66 MΩ in order to isolate it to prevent the influence from other molecules during dI/dV STM spectroscopic measurements. The specific tip trajectories during the manipulation to produce a flat conformer are indicated by the arrows in (a) and (b). The tip location selected during the Figure 5 spectrum recording on this flat conformer is indicated by a dot in (c). Same STM image conditions as in Figure 2. Image size: 12 nm × 15 nm.Figure 4. An example of the forcible molecule manipulation experiments using tunneling Figure 4 . 4 Figure 4. An example of the forcible molecule manipulation experiments using tunneling conditions less than 1/2 R T in Figure 3. Topographic images (a) before and (b) after R T = 120 MΩ manipulation. Using this condition, a structural transformation from syn to flat conformer occurs. (c) The so-produced flat conformer was manipulated with the STM tip away from its original four "f " dimer line with R T = 66 MΩ in order to isolate it to prevent the influence from other molecules during dI/dV STM spectroscopic measurements. The specific tip trajectories during the manipulation to produce a flat conformer are indicated by the arrows in (a) and (b). The tip location selected during the Figure 5 spectrum recording on this flat conformer is indicated by a dot in (c). Same STM image conditions as in Figure 2. Image size: 12 nm × 15 nm.Figure 4. An example of the forcible molecule manipulation experiments using tunneling conditions less than 1/2 R T in Figure 3. Topographic images (a) before and (b) after R T = 120 MΩ manipulation. Using this condition, a structural transformation from syn to flat conformer occurs. (c) The so-produced flat conformer was manipulated with the STM tip away from its original four " " dimer line with R T = 66 MÎl' in order to isolate it to prevent the influence from other molecules during dI/dV STM spectroscopic mΩ. The specific tip trajectories during the manipulation to produce a flat conformer are indicated by the arrows in (a) and (b). The tip location selected during the Figure 5 spectrum recording on this flat conformer is indicated by a dot in (c). Same STM image conditions as in Figure 2. Image size: 12 nm × 15 nm. 14 4. 3 3 eV for a flat BBD molecule on the Au(111) surface to be compared to the 3.7 eV (334 nm) UV optical gap observed for the syn and anti conformers in solution.Performed exactly at these resonances, the Figure5very precise dI/dV STM mapping permits to determine the spatial molecular orbital electronic distribution of the flat BBD reduced and oxidized electronic states at the origin of those two tunneling resonances. Monoelectronic constant current dI/dV elastic scattering quantum chemistry (ESQC) 34 image calculations were performed to compare with those experimental dI/dV maps. The ESQC calculation is particularly welladapted to provide accurate STM images for large adsorbed molecules.35,36 These calculations confirm that the main contributor to the -1.6 V resonance is the BBD highest occupied molecular orbital (HOMO) and the BBD lowest unoccupied molecular orbital (LUMO) to the +2.7 V resonance. Those images were calculated starting from an optimized flat BBD conformation on the Au(111) surface. This confirms how the BBD molecule can be prepared in a flat conformation by STM single molecule manipulation. According to Figure5images, the central phenyl chassis is relatively electronically decoupled for the Au(111) surface. F . Step-by-Step Manipulation for Moving a flat BBD Conformer on the Au(111) Surface. Figure 5 . 5 Figure 5. (a) The dI/dV spectrum and (b and c) dI/dV maps recorded on flat BBD produced in Figure 4. The first tunneling resonances appear at -1.6 V and +2.7 V bias voltage (sample grounded on the LT-UHV 4-STM). A dI/dV map captures the spatial distribution of the electron density of the corresponding molecular electronic states contributing to the resonance. 37 (d and e) At energies corresponding to HOMO and LUMO of the flat BBD monomer, monoelectronic ESQC STM calculated images are also presented for comparison. Resonance at -1.6 V appears to be mainly coming from the HOMO component of the imaged ground state and at +2.7 V from the LUMO contribution of BBD reduced (image size: 3.6 nm × 2.4 nm). Figure 5 . 5 Figure 5. (a) The dI/dV spectrum and (b and c) dI/dV maps recorded on flat BBD produced in Figure 4 . 4 Figure 4. The first tunneling resonances appear at -1.6 V and +2.7 V bias voltage (sample grounded on the LT-UHV 4-STM). A dI/dV map captures the spatial distribution of the electron density of the corresponding molecular electronic states contributing to the resonance. 37 (d and e) At energies corresponding to HOMO and LUMO of the flat BBD monomer, monoelectronic ESQC STM calculated images are also presented for comparison. Resonance at -1.6 V appears to be mainly coming from the HOMO component of the imaged ground state and at +2.7 V from the LUMO contribution of BBD reduced (image size: 3.6 nm × 2.4 nm). Figure 6 . 6 Figure 6. Manipulation of a BBD molecule on the Au(111) surface using inelastic electrons tunneling excitations. (a-d) A series of images demonstrating the BBD molecule motion along the timeline. An atomic resolved image and its significant crystallographic orientation [101̅ ] is also presented. The red dots on each image indicate the tip apex location for applying the bias voltage ramp, resulting in the measured tunneling current. The moving direction, of a BBD molecule is always the [101̅ ] orientation and its average lateral motion per voltage ramping is around 0.29 nm, which perfectly matches with the 0.288 nm interatomic distance on Au(111) in the ⟨11̅ 0⟩ surface directions. The voltage ramp duration from +1.0 to +2.3 V was 20 s in each case. Left column shows I-V characteristics for each inelastic event. Right column: 12.0 nm × 5.6 nm constant current STM images recorded at V = +0.5 V and I = 10 pA. In each image, top left, two dimers and one BBD monomer in a syn conformation have been imaged at the same time to provide a clear measurement of the BBD molecule motion per excitation. Figure 6 . 6 Figure 6. Manipulation of a BBD molecule on the Au(111) surface using inelastic electrons tunneling excitations. (a-d) A series of images demonstrating the BBD molecule motion along the timeline. An atomic resolved image and its significant crystallographic orientation [101] is also presented. The red dots on each image indicate the tip apex location for applying the bias voltage ramp, resulting in the measured tunneling current. The moving direction, of a BBD molecule is always the [101] orientation and its average lateral motion per voltage ramping is around 0.29 nm, which perfectly matches with the 0.288 nm interatomic distance on Au(111) in the 110 surface directions. The voltage ramp duration from +1.0 to +2.3 V was 20 s in each case. Left column shows I -V characteristics for each inelastic event. Right column: 12.0 nm × 5.6 nm constant current STM images recorded at V = +0.5 V and I = 10 pA. In each image, top left, two dimers and one BBD monomer in a syn conformation have been imaged at the same time to provide a clear measurement of the BBD molecule motion per excitation. CONCLUSIONA bisbinaphthyldurene (BBD) molecule was designed, synthesized, and deposited on an Au(111) surface, mechanically manipulated with a STM tip, and then with the STM inelastic contribution of the tunneling current passing through this molecule. The BBD molecule is equipped with two lateral binaphthyl paddles mounted on a very simple central phenyl chassis to separate it from the supporting surface. Single molecule STM lateral mechanical manipulation must be first performed for this molecule to reach a flat conformation on the Au(111) surface since its native surface conformation on Au( S3 4 . 4 Figure S1. UV-vis absorption spectra of BBD in THF (5.6 × 10 -6 M). Figure S2 . S2 Figure S2. CD spectra of BBD in THF (1.0 × 10 -5 M). Figure S1 . S1 Figure S1. UV-vis absorption spectra of BBD in THF (5.6 × 10 -6 M). S3Figure S1 . S1 Figure S1. UV-vis absorption spectra of BBD in THF (5.6 × 10 -6 M). Figure S2 .Figure S2 . S2S2 Figure S2. CD spectra of BBD in THF (1.0 × 10 -5 M). Becke 2 , 2 [START_REF] Becke | Density-Functional Thermochemistry. III. The Role of Exact Exchange[END_REF] and the correlation functional by Lee, Yang and Parr,[START_REF] Lee | Development of the Colle-Salvetti Correlation-Energy Formula into a Functional of the Electron Density[END_REF] and the 6-31G(d,p) split valence plus polarization basis set[START_REF] Ditchfield | Self-Consistent Molecular-Orbital Methods. IX. An Extended Gaussian-Type Basis for Molecular-Orbital Studies of Organic Molecules[END_REF][START_REF] Hehre | Self-Consistent Molecular Orbital Methods. XII. Further Extensions of Gaussian-Type Basis Sets for Use in Molecular Orbital Studies of Organic Molecules[END_REF][START_REF] Hariharan | Accuracy of AH n Equilibrium Geometries by Single Determinant Molecular Orbital Theory[END_REF][START_REF] Hariharan | The Influence of Polarization Functions on Molecular Orbital Hydrogenation Energies[END_REF] was used. Relaxed S 1 structures (= optimized structures in S 1 state) were calculated by time-dependent (TD) DFT.See Tab. S1, Tab. S2, Fig.S16, Fig.S17 Figure S5 . S5 Figure S5. DEPT 90 of BBD. Figure S5 .S5Figure S5 . S5S5 Figure S5. DEPT 90 of BBD. Figure S6 . S6 Figure S6. DEPT 135 of BBD. Figure S6 . S6 Figure S6. DEPT 135 of BBD. Figure S7 . S7 Figure S7. COSY of BBD. Figure S7 .S6Figure S7 . S7S7 Figure S7. COSY of BBD. Figure S8 . S8 Figure S8. HMQC of BBD. Figure S8 . S8 Figure S8. HMQC of BBD. Figure S9 . S9 Figure S9. HMBC of BBD. Figure S9 .S7Figure S9 . S9S9 Figure S9. HMBC of BBD. Figure S10 . S10 Figure S10. NOESY of BBD. Figure S10 . S10 Figure S10. NOESY of BBD. Figure S11 .Figure S12 . S11S12 Figure S11. ROESY of BBD. Figure S11 . S11 Figure S11. ROESY of BBD. Figure S11 . S11 Figure S11. ROESY of BBD. Figure S12 .S9Figure S13 . S12S13 Figure S12. Selected correlation in NOESY (dashed lines) and Figure S13 .S10Figure S14 . S13S14 Figure S13. Temperature dependent 1 H NMR (400 MHz, CDCl 3 ) spectra of BBD. The temperature was changed from 313 to 218 K. White and black dots at 218 and 223 K correspond to signals from two conformers. Proton resonances of aromatic region at the lower magnetic field were further analyzed for energetics, since they are comparably isolated. Two doublet peaks from one isomer and one overlapped doublet peaks from another isomer were observed at 218 K, which are merged to form two doublet peaks at 313 K. Figure S15 .FigureFigure S15 .Figure S16 . S15S15S16 Figure S15. The Eyring plot for BBD. The energetics parameters are shown at the bottom of the plots. H = 9.8 kcal/mol, S = -17 cal/mol K. Figure S18 . S18 Figure S18. (a, d) anti-, (b, e) syn-, (c, f) flat-BBD on Au(111) layers simu Fig. S18 . S18 Fig. S18. (a, d) anti-, (b, e) syn-, (c, f) flat-BBD on Au(111) layers simulated by MD calculations. Bartels, L.; Meyer, G.; Rieder, K.-H. Basic Steps of Lateral Manipulation of Single Atoms and Diatomic Clusters with a Scanning Tunneling Microscope Tip. Phys. Rev. Lett. 1997, 79, 697-700. Bouju, X.; Joachim, C.; Girard, C.; Tang, H. Mechanics of (Xe) N Atomic Chains under STM Manipulation. Phys. Rev. B: Condens. Matter Mater. Phys. 2001, 63, 085415. 1 Eigler, D. M.; Schweizer, E. K. Positioning Single Atoms with a Scanning Tunnelling Microscope. Nature 1990, 344, 524-526. 2 3 6-layer, with a vacuum layer. The size of unit cell for a single molecule on Au(111) was 19.9795 × 23.0703 × 51.7730 Å and that for two molecules was 29.9693 × 34.6055 × 61.7730 Å. All simulations were performed 7. Cartesian coordinates Table S3. Cartesian coordinate flat-conformer SCF Done: E(RB3LYP) = -2226.71856349 A.U. after 14 cycles - S14 Table S4 . S4 -Cartesian coordinate syn-conformer SCF Done: E(RB3LYP) = -2226.74911677 A.U. after 7 cycles - Table S4 . S4 -Cartesian coordinate syn-conformer SCF Done: E(RB3LYP) = -2226.74911677 A.U. after 7 cycles - Table S5 . S5 -Cartesian coordinate anti-conformer SCF Done: E(RB3LYP) = -2226.74970717 A.U. after 14 cycles - Jung, T. A.; Schlittler, R. R.; Gimzewski, J. K.; Tang, H.; Joachim, C. Controlled Room-Temperature Positioning of Individual Molecules: Molecular Flexure and Motion. Science 1996, 271, 181-184. Hochstrasser, R. M. The Effect of Intramolecular Twisting on the Emission Spectra of Hindered Aromatic Molecules. Can. J.Chem. 1960, 39, 459-470. ACKNOWLEDGMENTS We thank Dr. Y. Okawa, Dr. T. Uchihashi, and Dr. K. Sagisaka in NIMS for prescreening of STM conditions and helpful discussions and Prof. M. Aono for its continuous support during this work. Funding WPI MANA MEXT program and by JSPS KAKENHI (16H07436, JP16H06518, 26790003). M.K. acknowledges financial support received from the Foundation for Polish Science (FNP). We thank TOYOTA as an official sponsor for our NIMS MANA team in the nanocar race. 43 Notes The authors declare no competing financial interest. SUPPORTING INFORMATION 1-General Analytical thin-layer chromatography (TLC) was performed on a glass plate coated with silica gel (230-400 mesh, 0.25 mm thickness) containing a fluorescent indicator (silica gel 60F254, Merck). Flash silica gel column chromatography was performed on silica gel 60N (spherical and neutral gel, 40-50 µm, Kanto). Infrared (IR) spectra were recorded on Thermo Scientific Nicolet NEXUS 670 FT-IR and were reported as wavenumbers (ν) in cm -1 . Proton ( 1 H) and carbon ( 13 C) nuclear magnetic resonance (NMR) spectra were recorded on a JEOL JNM-ECA400 spectrometer. Mass spectra were obtained on an Applied Biosystems Voyager DE STR SI-3 instrument (MALDI-TOF MS). UV-Vis absorption spectra were obtained on JASCO V-670. Circular dichroism (CD) spectra were obtained on JASCO, J-820. Materials Solvents and materials were purchased from Aldrich, Tokyo Kasei Chemical Co. or Wako Chemical Co., and were used without further purification. Synthesis To a mixture of (R)-(+)-1,1'-bi (2-naphthol) (1.00 g, 3.49 mmol) and CsCO 3 (2.84 g, 8.73 mmol) in dry acetone (100 mL) was added 1,2,4,5-tetrakis(bromomethyl)benzene (715 mg, 1.59 mmol) and the mixture was refluxed 48 h. The mixture was extracted with CH 2 Cl 2 (2 × 200 mL) and concentrated in vacuo. The crude material was purified by silica gel column chromatography (eluent: CH 2 Cl 2 /Hexane) to give pure desired compound, bisbinaphthyldurene (BBD) (410 mg, 37%). The compound was further purified by sublimation (< 300 • C) for the STM experiments. Mp > 250 • C; FT-IR (KBr, cm 1 ) 3047, 2934, 2880, 1918, 1593, 1472, 1321, 1244, 1147, 1079, 1009, 893, 805, 749; 1 H NMR (400 MHz, CDCl 3 ) δ 5. 15 (d, J = 11.2 Hz, 4H), 5.21 (d, J = 11.2 Hz, 4H), 7.16 (ddd, J = 7.6, 7.6, 0.8 Hz, 4H), 7.18 (dd, J = 7.6, 7.2 Hz, 4H), 7.27 (ddd, J = 7.2, 7.2, 1.4 Hz, 4H), 7.32 (s, 2H), 7.46 (d, J = 9.0 Hz, 4H), 7.75 (d, J = 7.6 Hz, 4H), 7.80 (d, J = 9.0 Hz, 4H); 13 C NMR (400 MHz, CDCl 3 ): δ 71. 4, 117.0, 121.6, 123.9, 126.1, 126.3, 128.1, 129.3, 129.8, 133.4, 134.5, 136.6, 154.4 Theoretical calculations (DFT) All calculations were performed using the Gaussian 09 program, 1 and the results were analyzed and visualized on GaussView 5.0.9. Calculations were performed at the density functional theory (DFT) level with the B3LYP functional, the gradient correction of the exchange functional by Becke 2,[START_REF] Becke | Density-Functional Thermochemistry. III. The Role of Exact Exchange[END_REF] and the correlation functional by Lee, Yang and Parr, [START_REF] Lee | Development of the Colle-Salvetti Correlation-Energy Formula into a Functional of the Electron Density[END_REF] and the 6-31G(d,p) split valence plus polarization basis set [START_REF] Ditchfield | Self-Consistent Molecular-Orbital Methods. IX. An Extended Gaussian-Type Basis for Molecular-Orbital Studies of Organic Molecules[END_REF][START_REF] Hehre | Self-Consistent Molecular Orbital Methods. XII. Further Extensions of Gaussian-Type Basis Sets for Use in Molecular Orbital Studies of Organic Molecules[END_REF][START_REF] Hariharan | Accuracy of AH n Equilibrium Geometries by Single Determinant Molecular Orbital Theory[END_REF][START_REF] Hariharan | The Influence of Polarization Functions on Molecular Orbital Hydrogenation Energies[END_REF] was used. Relaxed S1 structures (= optimized structures in S1 state) were calculated by time-dependent (TD) DFT. Table S1. Torsion angle of flat-, syn-, and anti-BBD optimized by DFT (B3LYP/6-31G(d,p)). Torsion angle Flat Syn Anti C2-C1-C1'-C2' ( ) -60.5 -63.9 -63.9 C2'-C1'-C1''-C2'' ( ) -60.5 -63.9 -63.9 C2-O1-C3-C4 -158.9 -61.6 -61.7 C2'-O1'-C3'-C4' -158.9 -144.1 -144.6 C2''-O1''-C3''-C4'' -158.9 -144.1 -61.7 C2'''-O1'''-C3'''-C4'' ' -158.9 -61.6 -144.6 Table S2. Table S1. Table S1. Torsion angle of flat-, syn-, and anti-BBD optimized by DFT (B3LYP/6-31G(d,p)). Torsion angle Flat Syn Anti C2-C1-C1'-C2' ( ) -60.5 -63.9 -63.9 C2'-C1'-C1''-C2'' ( ) -60.5 -63.9 -63.9 C2-O1-C3-C4 -158.9 -61.6 -61.7 C2'-O1'-C3'-C4' -158.9 -144.1 -144.6 C2''-O1''-C3''-C4'' -158.9 -144.1 -61.7 C2'''-O1'''-C3'''-C4'' ' -158.9 -61.6 -144.6 Table S2. Table S2. S13 in the NVT, and controlled using a NHL thermostat, with a decay constant of 1 ps. To find several conformations of one or two molecules on Au(111) surface, anneal dynamics were used. Anneal dynamics consists of a dynamics simulation where the temperature is periodically increased from an initial temperature (4 K) to a mid-cycle temperature (500 K) and back again. All MD calculations were performed with Forcite in Materials Studio, and force field COMPASS was used. All the structures obtained were optimized with the same force field. For calculations of a monomer on Au(111), the DFT optimized anti-structure was used as an initial structure, and syn-conformer on Au(111) surface was obtained as a stable conformer. The anti-conformer on Au(111) surface was also obtained with higher energy (+9.0 kcal/mol). When a mid-cycle temperature was raised to 1500 K, flat-conformer was found with lower energy, -22.5 kcal/mol compared with the syn-conformer. For calculations of a dimer on Au(111), the DFT optimized two syn-conformers were used as initial structures, and syn-syn-dimer on Au(111) surface was obtained as a stable conformer. See Fig. S18 and Fig. S19. Interaction energies Each total energies of syn-, anti-, and flat-conformers on Au( 111 The interaction energy between Au(111) and syn--conformer (E surfaceinteraction_syn ) was calculated to be -194.92 kcal/mol based on the following equation: The interaction energy between Au(111) and anti-and flat-conformer (E surfaceinteraction_anti , E surfaceinteraction_flat ) was calculated with the same method to be -194.3 and -244.2 kcal/mol, respectively. The surface interaction energy for syn-or anti-conformers were at the same level (E surfaceinteraction_syn = -194.9 kcal/mol, E surfaceinteraction_anti = -194.3 kcal/mol). The difference in stability of synand anti-conformers on Au(111) originates from the difference in molecular deformation energy (E anti -E syn = +8.4 kcal/mol higher than syn-conformer) rather than interaction energy (E surfaceinteraction_anti -E surfaceinteraction_syn = +0.6 kcal/mol more unfavorable for the anti-conformer). In another words, to obtain similar interaction with Au(111) surface, anti-conformer needed to be Interaction energies Each total energies of syn-, anti

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2024/03/05 22:32:07

2015

https://hal.science/hal-01746106/file/Piazolo_Montagnat2015_postRevVersion.pdf

S Piazolo M Montagnat F Grennerat H Moulinec J Wheeler Effect of local stress heterogeneities on dislocation fields: Examples from transient creep in polycrystalline ice Keywords: stress heterogeneities, dislocation field, electron backscatter diffraction, full-field modeling, kink bands, viscoplastic anisotropy This work presents a coupled experimental and modeling approach to better understand the role of stress field heterogeneities on deformation behavior in material with a high viscoplastic anisotropy e.g. polycrystalline ice. Full-field elasto-viscoplastic modeling is used to predict the local stress and strain field during transient creep in a polycrystalline ice sample. Modeling input includes the experimental starting microstructure and a validated slip system dependent flow law. EBSD measurements on selected areas are used to estimate the local dislocation field utilizing the Weighted Burgers Vector (WBV) analysis. Areas of local stress concentration correlate with triple junctions and grain boundaries, originating from strain incompatibilities between differently oriented grains. In these areas of highly heterogeneous stress patterns, (a) kink bands are formed and (b) WBV analysis shows a non-negligible c-axis component of the WBV. The correlation between this defect structure and presence of kink bands suggest that kink band formation is an efficient accommodation deformation mode. Introduction When polycrystalline material is plastically deforming, stress and strain heterogeneity fields are developing due to strain incompatibilities between grains of different crystallographic orientations. Depending on the level of viscoplastic anisotropy of the material, the heterogeneity amplitude can be high. The viscoplastic anisotropy of ice is known to be very strong, with dislocations gliding mostly on the basal plane with three equivalent < 11 20 > Burgers vector directions [START_REF] Hondoh | Physics of Ice Core Records[END_REF]. This results in strong kinematic hardening at grain boundaries and triple junctions [START_REF] Duval | [END_REF]. As such ice is a good model for materials with high viscoplastic anisotropy, such as magnesium [3,4], quartz [START_REF] Hobbs | Preferred orientation in deformed metals and rocks, 405 chapter The geological significance of microfabric analyses[END_REF] and olivine [START_REF] Nicolas | Crystalline Plasticity and Solid State Flow 408 in Metamorphic Rocks[END_REF]. In ice, strong heterogeneity fields were measured by Digital Image Correlation on polycrystalline samples deformed by compression creep with local strain amplitude as high as ten times 18 the macroscopic strain [START_REF] Grennerat | [END_REF]. The strain heterogeneities 19 were also indirectly observed through lattice misori-20 entation measurements via EBSD [8,[START_REF] Montagnat | Lebensohn 414 RA[END_REF] and were 21 simulated using full-field viscoplastic approaches based 22 on Fast Fourier Transform formulation [10,[START_REF] Montagnat | Lebensohn 414 RA[END_REF][START_REF] Grennerat | [END_REF]. In [10,[START_REF] Montagnat | Lebensohn 414 RA[END_REF]. The same numerical constraint was derived for 34 polycrystalline Mg in conditions where twinning was not activated [11]. However, up to now there exists no unequivocal evidence for macroscopic strain resulting 37 from non-basal slip in ice [12,13]. 38 The presence of heterogeneous dislocation fields in experimentally and naturally deformed ice was mostly 40 observed indirectly from substructures observations (Xray diffraction, optical analyses, EBSD...) [8,[START_REF] Montagnat | Lebensohn 414 RA[END_REF]14,[START_REF] Donges | Nuclear Instru-424 ments and Methods in Physics Research Section B: Beam Inter-425 actions with[END_REF]. 42 In particular kink bands and double kink bands has been 43 commonly observed in polycrystalline ice deformed 44 in the laboratory. The origin of these kink bands was following [16,17]. Accuracy of EBSD data is within figures with both the sample coordinate system and the 125 crystal coordinate system (see Fig. 2). 83 0.3 • -0.4 • [18]. 126 Table 1 presents the integral WBV calculated over se-127 lected areas in the analyzed samples (Fig. 2 and Fig. 3), 1 shows that the integral WBV values vary significantly depending on microstructure type and location. However, for each selected area, we obtained several WBV values and chose to present here values that are representative for the selected area. The accuracy of the integral WBV is dependent on the angular resolution of the EBSD data. Using EBSD data with high angular resolution (here within 0.3 degrees), we consider an integral WBV ratio of one specific Burger vector over the maximum WBV value of 0.5 significant. Local stress field estimation Full-field numerical simulations were performed using the CraFT code as presented in [START_REF] Suquet | [END_REF]. The code is based on the FFT method initially proposed in [22,23], extended to elasto-viscoplastic composites using a step-by-step integration in time in [24] (see also the numerical details in [START_REF] Suquet | [END_REF]). The method used in the CraFT code finds a strain rate field associated with a kinematically admissible velocity field that minimizes the average local work rate under the compatibility and equilibrium constraints. An iterative scheme is used following a fixed point approach. It is numerically more efficient than the finite element method [25], but is limited to simulations with microstructures with periodic boundary conditions. The exact experimental boundary conditions (stress free lateral surface) could not be reproduced due to the numerical periodicity constraint. therefore, Accordingly, stress and strain fields predicted close to the specimen edges are not expected to be very accurate, however Grennerat et al. [START_REF] Grennerat | [END_REF] showed a limited impact on the macroscopic response and estimated fields, especially in the center of the modeled microstructure, where the 185 areas of interest here are located. 186 The elasto-viscoplastic response of ice was modeled 187 following the law and hypothesis discussed in details in 188 [START_REF] Suquet | [END_REF]. To model ice, the used crystal plasticity formula- This is illustrated by the decomposition of the integral 261 WBV data (Table 1) and represented in the inverse pole 262 figure plots in Fig. 3 (see for instance mapG1-3 areas A 263 and B and mapG4-1-5 area B, Table 1). 264 As illustrated by the inverse pole figure of subarea D 265 from map G4-1-5, there is a systematic presence of a 266 minor but significant c-axis component close to triple 267 junctions. Such a c-axis component is considered as 268 non-negligible when the ratio K c /max(K ai ) is higher 269 than 0.5 (see Table 1). In contrast, most of the areas 270 analyzed in a grain interior show a c-axis component of 271 the WBV which is very close to the detection limit (see 272 Table 1). that cannot be resolved by the easy slip system i.e. 297 basal slip in ice [3,4,29]. In metals, kink bands 298 can occur at large deformation as strain localization 299 modes, as predicted by the bifurcation analysis of 300 [30], but also at the early stage of deformation in the 301 presence of obstacles [31]. Chang et al. [31] used 373 -The approach presented is applicable to materials 374 with significant viscoplastic anisotropy. Analyses areas Integral WBV (µm) - 2: Relative CRSS for the three slip system families at the beginning of the transient creep, τ iniR , and at the end, τ statR . n s is the stress exponent that was adjusted by [START_REF] Suquet | [END_REF] for each family. The absolute value of τ ini for the basal slip systems was fitted at 0.1 by [START_REF] Suquet | [END_REF], and the absolute value of τ stat for the basal slip systems was fitted at 0.022. 23 these 23 modelling approaches, plasticity is simulated 24 by the activation of several slip systems (e.g. basal, 25 prismatic and pyramidal slip in the case of ice), where 26 each slip system is assigned a relative critical resolved 27 shear stress for slip activation. In this frame, these 28 last works have shown, among other results, that 29 observed high stress and strain heterogeneities could 30 only be correctly simulated providing a significant 31 amount of non-basal slip activity in the corresponding 32 areas, while the global non-basal activity remained low 33 3 . 3 Fig. 1 presents selected EBSD maps 84 showing the local variations in orientations. 85 86 Weighted Burgers Vector analysis 87 To quantitatively analyze the EBSD point grid data 88 we utilized the Weighted Burgers Vector (WBV) analy-89 sis explained in details in [19]. The WBV is a recently 90 developed new quantity to constrain dislocation densi-91 ties and dislocation types using EBSD data on two di-92 mensional sections through crystalline materials. The 93 WBV is defined as the sum, over all dislocation types, 94 of [(density of intersections of dislocation lines with a 95 map) × (Burgers vector)] and as such can be calculated 96 from a planar set of orientation measurements such as 97 in an EBSD orientation map. There is no assumption 98 about the orientation gradient in the third dimension. 99 The magnitude of the WBV gives a lower bound of 100 the magnitude of the dislocation density tensor. The di-101 rection of the WBV can be used to constrain the types 102 of Burgers vectors of the geometrically necessary dis-103 locations present in the microstructure and their geo-104 metric relationship to intra-grain structures, for exam-105 ple subgrain walls. The WBV can then be decomposed 106 in terms of the 3 main lattice vectors [20]. In the case 107 of ice, the two a-axis lattice vectors are further decom-108 posed into the 3 equivalent a-axis lattice vectors. We 109 can calculate the net Burgers vector content of disloca-110 tions intersecting a given area of a map by an integra-111 tion around the edge of this area. This integral WBV 112 method is fast, complements point-by-point WBV cal-113 culations and, thanks to this integration, reduces the ef-114 fect of noise on the analysis. A lower bound of the 115 density of geometrically necessary dislocations can be 116 estimated from this calculation. This estimation is not 117 absolute but can be used for comparison purposes. It 118 should be noted that the net Burgers vector value de-119 rived using the integral WBV method is sensitive to the 120 chosen area both in terms of size and location. 121 WBV analysis data is represented in color coded maps 122 showing the WBV magnitude, WBV directions as pro-123 jected arrows onto the color coded maps and in pole 124 128 decomposed into the three a-axes and the c-axis of the 129 ice crystallographic structure. Areas were selected to 130 represent the observed spectrum of different microstruc-131 tures present within the analyzed sample. Such mi-132 crostructures include triple junctions, grain boundaries 133 with or without asperities, areas in close vicinity or at distance to grain boundaries. Data from Table The specimen undeformed microstructure was discretized into 512 × 512 Fourier points with a single layer of Fourier points in the third (Z) direction, assuming thus infinite column length. To represent the microstructure, each Fourier point is allocated a c-axis orientation according to the measured orientation of the underformed experimental sample. Consequently grain boundary are not specifically defined as discrete objects with specific physical characteristics other than a change in crystal orientation. Throughout the numerical simulations, no crystallographic orientation changes are imposed, thus the microstructure (orientation, grain boundaries) does not evolve. Creep conditions equivalent to the experimental ones were applied (constant 0.5 MPa stress in the vertical direction, transient creep up to 1% macroscopic strain). 225 189 tion accounts for three different families of slip systems, 190 namely the basal, prismatic and pyramidal systems, 191 the latter two being taken stiffer than basal slip. A 192 power law is considered for the evolution of the shear 193 strain rate as a function of the resolved shear stress on 194 each slip system that integrates kinematic hardening, 195 slip system interactions and their evolution with time 196 during transient creep. The material parameters of the 197 law were determined by adjusting them according to 198 experimental data available for single crystals as well 199 as for polycrystals, as detailed in [21]. The Critical 200 Resolved Shear Stress (CRSS) on each slip system 201 was allowed to evolve with strain between initial and 202 stationary values. Table 2 provides the relative CRSS 203 for each slip system family, and the value of the stress 204 exponent attributed to each family. It is worth noting 205 that the relative CRSS is much higher for the pyramidal 206 family, and that there is a relative softening of basal 207 systems during transient creep [21]. 208 The local heterogeneities of the strain and stress fields 209 are illustrated in Fig. 4a) and b) where the spatial 210 variation of the equivalent stress (σ eq = 3 2 σ i j σ i j ) 211 and equivalent strain (ε eq = 3 2 ε i j ε i j ) are repre-212 sented, respectively. For each simulated pixel of 213 the microstructure, the deviatoric part of the stress 214 tensor was decomposed into its eigenvector frame. 215 From this decomposition, we extracted the eigenvector 216 corresponding to the largest absolute eigenvalue. We 217 built a composite vector representation based on the 218 in-plane projection of a vector having the direction of 219 this eigenvector, and the amplitude of the associated 220 eigenvalue. This representation is given in Fig. 4c) for 221 the area of interest, superimposed on the equivalent 222 stress contour plot. Numerical modeling shows that local stress concen-226 trations and stress field heterogeneities occur close to 227 triple junctions and grain boundaries. The higher the 228 mismatch between crystallographic grain orientations, 229 the higher the concentration and heterogeneities (Fig. 230 Fig. 3 3 Fig.3where the heterogenous stress band crossing 253 260 273 Close to triple junctions, we observe near straight 274 subgrain boundaries. Their traces are parallel to the 275 c-axis, and the WBV are predominantly oriented along 276 the a-axis perpendicular to the subgrain boundary trace 277 (Fig. 1 and Fig.3). The other a-axis contribution is 278 related to a continuous crystal bending perpendicular to 279 the c-axis (see Fig.5with a sketch of the bending axis). 280 Similar observations were made in[8,[START_REF] Montagnat | Lebensohn 414 RA[END_REF]. 281 These only slightly curved subgrain boundaries com-282 monly occur in parallel pairs with the main WBV 283 pointing in opposite directions (Fig. 3 mapG1-3 284 and mapG4-1-5, Table 1). The parallelism of the 285 subgrain boundary traces, their orientation relative to 286 the crystallographic axes of the host grain together 287 with the orientation of the main WBV a-axis compo-288 nent is consistent with a strongly crystallographically 289 determined boundary development. The opposing 290 directions of WBV in pairs of subgrain boundaries 291 (Fig. 3) is consistent with kink bands with alternating 292 opposing dislocation structures [26, 27, 28]. These 293 crystallographically well defined kink boundaries could 294 appear similar to the twinning modes in magnesium 295 in the way that they both accommodate shear stress 296 302 Dislocation Dynamic simulation to illustrate the stable 303 position of an edge dislocation within the stress field 304 of an alignment of edge dislocations forming the kink 305 band. Consequently, in contrast to what was initially 306 thought [9], climb or cross-slip do not seem to be 307 necessary mechanisms to explain kink band formation. 308 In summary, observations suggest that kink bands form 309 in areas of very high local stress fields originating either 310 from strain incompatibilities developed in materials 311 with high viscoplastic anistropies or from high strain 312 rates. 313 It should be noted that the used numerical model does 314 not account for specific grain boundary properties. The 315 question of the strain continuity at an interface such as 316 grain boundary is therefore not addressed. In contrast, 317 recent work [32] showed that stress accumulation at 318 boundaries can explain grain boundary delamination 319 in alloys. In the case of ice deforming during tran-320 sient creep under low applied stress, no cracking was 321 observed at grain boundaries, but a better account for 322 continuity conditions of the plastic distortion at GB 323 might be necessary to evaluate the impact of strain 324 heterogeneities on recrystallization mechanisms for 325 instance. 326 Our data show that in high stress concentration areas, 327 close to triple junctions, a Burgers vector component 328 parallel to c-axis is present. Cross slip of dislocations 329 with Burgers vector lying in the basal plane cannot 330 explain this; instead it could be interpreted by a local 331 activity of a non-basal slip system. Conceptually, 332 such a non-basal slip activity is expected only in such 333 high stress areas, as non-basal activity requires high 334 level of local stress in order to overcome the high 335 critical resolved shear stress required for non-basal slip 336 dislocation activity in ice were performed in conditions 338 with very few dislocations activated, that could hardly 339 be extrapolated. Indeed, they were performed using 340 X-ray diffraction topography which enable individual 341 dislocation observation at the very early stage of 342 deformation [33, 34]. 343 The observed crystal bending parallel as well as per-344 pendicular to the kink bands which is accommodated 345 by the shown presence of Burgers vectors along not 346 one but two a-axes in high stress areas, may results in 347 the development of subgrain boundaries both parallel 348 and perpendicular to c-axis. This then will result in 349 the formation of a subgrain with near perpendicular 350 subgrain boundaries, one parallel to c-axis and one 351 perpendicular to c-axis. Such a developing subgrain is 352 shown in Fig. 5 and has been described in [8]. This 353 subgrain boundary formation represents one of the re-354 crystallization processes relaxing strain heterogeneities 355 at macroscopic strain exceeding the transient creep 356 regime [2, 35]. - Coupled full field elasto-viscoplastic modelling 360 and detailed EBSD analysis show the effect of 361 stress heterogeneities (magnitude and orientation) 362 on the dislocation field. 363 -The WBV c-axis component measured in high 364 stress areas could be consistent with local activa-365 tion of non-basal slip. 366 -At low strain, formation of kink bands with a well 367 defined crystallographic character appears as an ef-368 ficient accommodation deformation mode, similar 369 to twinning in Mg. 370 -Distinct misorientations across and perpendicular 371 to kink boundaries form substructures which act as 372 precursors for grain nucleation. Figure 1 :Figure 2 : 12 Figure1: Microstructure of a selected area of the columnar ice sample (in the plane perpendicular to the columns) after compressive strain of 10% along y direction. The central microstructure is color coded according to the measured c-axis orientation as represented by the color wheel (inset). EBSD maps of selected areas (marked by white rectangles) are shown with color code for change in orientation according to color scheme provided on the left side of the central microstructure. Insets of 3D crystal orientations (grey hexagons) are provided by EBSD analyses. Grains are given numbers for ease of reference (e.g. gr 1). Note labeling of the selected areas for which EBSD maps were obtained corresponds to the grain numbers of the grains present in the respective map. Figure 3 : 3 Figure 3: Comparison between stress field and dislocation field; deviatoric stress eigenvectors (left) zoomed in the areas selected for the WBV analyses (right maps). WBV analyses are shown with WBV magnitude (color range), WBV directions shown as white arrows on map (over a threshold value), and on inverse pole figures for selected areas (labeled A to D). Only upper limits are shown for these areas. Integral WBV of subareas are given in Table1. Black lines represent grain boundaries. Due to limitation in the simulation configuration (see text), the correspondence between modeled and observed microstructures such as the position of grain boundaries is approximate. Figure 4 : 4 Figure 4: Maps of equivalent stress (in MPa) (a) and of equivalent strain (b) predicted by the full-field simulation using CraFT on the area of interest after 0.01 macroscopic strain under compression creep. c) provides a map of the deviatoric stress eigenvectors projected on the plane, superimposed on the equivalent stress contour plot (same scale as in a)).Grain boundary traces are superimposed for clarity but they are not physical entities in the code. Figure 5 : 5 Figure 5: Planes of crystal bending (white stippled lines) for subgrain boundaries and associated rotation directions. The nearly horizontal subgrain boundaries are associated with abrupt misorientations, while the vertical ones correspond to more progressive misorientations. Both sub-structures were quantified in term of WBV, see Fig.3. 3 3 Figure 5: Planes of crystal bending (white stippled lines) for subgrain boundaries and associated rotation directions. The nearly horizontal subgrain boundaries are associated with abrupt misorientations, while the vertical ones correspond to more progressive misorientations. Both sub-structures were quantified in term of WBV, see Fig.3. Figure 5: Planes of crystal bending (white stippled lines) for subgrain boundaries and associated rotation directions. The nearly horizontal subgrain boundaries are associated with abrupt misorientations, while the vertical ones correspond to more progressive misorientations. Both sub-structures were quantified in term of WBV, see Fig.3. Table 1 : 1 Integral WBV decomposed onto the three crystallographic axis of the ice crystal, for specific areas in EBSD maps shown in Figs.1, 2 and3. The two Ka vector directions are symmetrically equivalent, they were decomposed into the 3 a-directions following[START_REF] Wenk | Minerals: Their Constitution and Origin[END_REF]. The relative activity of K c /max(K ai ) is also given. ρ provides a lower bound estimate of the geometrically necessary dislocation density in the area. Step size is 15µm for all EBSD maps. * minimum value of the net Burgers vector magnitude. "gr" signifies grain number (cf. Fig.1) and "area", subarea as shown in Figs.2 and 3. 2 ρ Table Table Acknowledgments: Financial support by French ANR is 377 project DREAM #ANR-13-BS09-0001). Together with 378 support from institutes INSIS and INSU of CNRS, 379 France. The authors gratefully acknowledge the ESF support of the European Science 384 Foundation under the EUROCORES Programme, 385 EuroMinSci, MinSubStrDyn, No. ERAS-CT-2003-386 980409 of the European Commission, DG Research, 387 FP6. SP acknowledges additional funding from ARC 388 DP120102060 and FT1101100070. Funding by ARC 389 Centre of Excellence for Core to Crust Fluid Systems 390 (www.CCFS.mq.edu.au) allowed a Research visit of 391 M.M. to Macquarie University. This is contribution 392 534 from the ARC Centre of Excellence for Core to 393 Crust Fluid Systems (http://www.ccfs.mq.edu.au) and 394 975 in the GEMOC Key Centre. Funding from the 395 Australian Research Council (ARC) through the CCFS 396 Visiting Researcher scheme is gratefully acknowledged

01617193

en

2024/03/05 22:32:07

2017

https://inserm.hal.science/inserm-01617193/file/RambeaudIntJCardiol2017.pdf

Pierre Rambeau Emilie Faure Alexis Théron Jean-François Avierinos Chris Jopling Stéphane Zaffran Adèle Faucherre Reduced aggrecan expression affects cardiac outflow tract development in zebrafish and is associated with bicuspid aortic valve disease in humans Keywords: Hemodynamic forces have been known for a long time to regulate cardiogenic processes such as cardiac valve development. During embryonic development in vertebrates, the outflow tract (OFT) adjacent to the ventricle comes under increasing hemodynamic load as cardiogenesis proceeds. Consequently, extracellular matrix components are produced in this region as the cardiac cushions form which will eventually give rise to the aortic valves. The proteoglycan AGGRECAN is a key component of the aortic valves and is frequently found to be deregulated in a variety of aortic valve diseases. Here we demonstrate that aggrecan expression in the OFT of developing zebrafish embryos is hemodynamically dependent, a process presumably mediated by mechanosensitive channels. Furthermore, knockdown or knockout of aggrecan leads to failure of the OFT to develop resulting in stenosis. Based on these findings we analysed the expression of AGGRECAN in human bicuspid aortic valves (BAV). We found that in type 0 BAV there was a significant reduction in the expression of AGGRECAN. Our data indicate that aggrecan is required for OFT development and when its expression is reduced this is associated with BAV in humans. Introduction Arterial valve leaflets are composed of three distinct layers of Extra-Cellular Matrix (ECM) called fibrosa, spongiosa and ventricularis. The spongiosa layer is particularly rich in proteoglycans which will provide compressive properties to the tissue and allow the leaflet to change shape during the cardiac cycle. Indeed, valves are submitted to extreme hemodynamic forces such as shear stress and cyclic strain that will subsequently regulate both their development and function. Studies in chick and zebrafish have shown that disruption of the hemodynamic forces results in cardiac valves defects including defects of the outflow tract (OFT) cushion formation and severe heart defects associated with a total absence of valves [START_REF] Menon | Altered hemodynamics in the embryonic heart affects outflow valve development[END_REF][START_REF] Hove | Intracardiac fluid forces are an essential epigenetic factor for embryonic cardiogenesis[END_REF]. Among valve diseases, bicuspid aortic valve (BAV) is one of the most common pathologies found in patients, occurring in 1-2% of living births and are frequently associated with aortic stenosis, regurgitation, endocarditis and calcified valves [START_REF] Padang | Genetic basis of familial valvular heart disease[END_REF]. AGGRECAN is one of the major members of the large proteoglycans found in cartilage and provides the ability to resist compressive loads [START_REF] Lincoln | Hearts and bones: shared regulatory mechanisms in heart valve, cartilage, tendon, and bone development[END_REF]. A recent transcriptomic study of human BAV has shown that AGGRECAN expression is decreased in BAV patients with mild calcification compared with calcified tricuspid aortic valve (TAV) [START_REF] Padang | Comparative transcriptome profiling in human bicuspid aortic valve disease using RNA sequencing[END_REF]. Zebrafish possess 2 aggrecan paralogues, aggrecanA (acana) and aggrecanB (acanb). Here we show that acana expression in the zebrafish OFT is dependent on hemodynamic forces and that knockdown or knockout of acana during development induces cardiovascular defects. Moreover, we observe that in humans, AGGRECAN (ACAN) expression is reduced in type 0 BAV compared to normal TAV. Materials and methods Zebrafish strains and husbandry Zebrafish were maintained under standardized conditions [START_REF] Westerfield | The zebrafish book, A Guide for the Laboratory Use of Zebrafish (Danio rerio)[END_REF] and experiments were conducted in accordance with the European Communities council directive 2010/63. The Tg(fli1a:GFP)y1 line was provided by the CMR[B]. ISH were performed as described previously [START_REF] Brend | Zebrafish whole mount high-resolution double fluorescent in situ hybridization[END_REF][START_REF] Thisse | High-resolution in situ hybridization to whole-mount zebrafish embryos[END_REF] (see Supplementary information). For RR treated fish, the proteinase K treatment was reduced to 15 min. Ruthenium Red treatment Morpholinos and injections Morpholino oligonucleotides were obtained from Gene Tools (Philomath, OR, USA) and injected into one-cell stage embryos (see Supplementary information). CRISPR/Cas9 Acana target sequences were identified using ZiFiT online software [START_REF] Sander | Zinc Finger Targeter (ZiFiT): an engineered zinc finger/target site design tool[END_REF]. 150 pg of acana gRNA was co-injected with nls-Cas9 protein (N.E.B) (see Supplementary information). Cardiovascular parameters analysis Cardiovascular parameters were determined using the MicroZebraLab™ software from ViewPoint [START_REF] Parker | A multi-endpoint in vivo larval zebrafish (Danio rerio) model for the assessment of integrated cardiovascular function[END_REF] (see Supplementary information). Real-time quantitative Reverse Transcription Polymerase Chain Reaction (qRT-PCR) Human aortic valve tissues were collected after surgery by the Department of cardiac surgery at "La Timone" Hospital, Marseille, France. The protocol was evaluated and authorised by the "CPP Sud Méditerranée" n°: 13.061. and by the "Agence de la biomedicine" n°PFS14-011. (see Supplementary information). Results AggrecanA is expressed in the zebrafish OFT and is dependent on hemodynamic forces To determine the expression pattern of aggrecan, we performed in situ hybridization (ISH) on 4 days post-fertilization (dpf) embryos using antisense acana and acanb probes. Acana showed a high expression in craniofacial cartilage as previously described [START_REF] Kang | Molecular cloning and developmental expression of a hyaluronan and proteoglycan link protein gene, crtl1/hapln1, in zebrafish[END_REF], however we were also able to observe a clear expression of acana in the OFT (Fig. 1.A). In contrast, we could not detect any acanb expression in this cardiac structure (Fig. 1.B). In order to confirm that acana expression was localised to the OFT, we performed double fluorescent ISH on 4dpf larvae using antisense probes targeting acana and elastin b (elnb), an abundant component of the zebrafish OFT [START_REF] Miao | Differential expression of two tropoelastin genes in zebrafish[END_REF]. In this manner we were able to observe that acana expression co-localised with elnb expression in the OFT (Fig. 1.C-H). Because ECM composition can be modulated by hemodynamic forces, we sought to determine whether this was the case for acana expression in the OFT. To achieve this, we used a previously described morpholino targeting tnnt2 which effectively stops the heart from beating [START_REF] Sehnert | Cardiac troponin T is essential in sarcomere assembly and cardiac contractility[END_REF]. In this manner we found that tnnt2 morphants display an apparent lack of acana expression in the OFT (Fig. 1.I). Previous research has indicated that mechanosensitive ion channels (MSC) can detect hemodynamic forces and trigger valve formation [START_REF] Heckel | Oscillatory flow modulates mechanosensitive klf2a expression through trpv4 and trpp2 during heart valve development[END_REF]. To determine whether MSC could mediate acana expression in response to hemodynamic forces, we employed the nonselective MSC blocker Ruthenium Red (RR). In this manner we could observe that larvae incubated with RR showed a dose responsive decrease of acana expression in the OFT (Fig. 1.J-L). AggrecanA is involved in cardiac development Due to its expression in the OFT, we speculated that acana may play a role in cardiac development. To answer this question we adopted a morpholino (MO) mediated approach targeting an internal splice site. Injection of this MO produced a phenotype characterized by defective cardiogenesis including a larger atrium and associated edema at 3dpf (Fig. 1.M-N, P-Q). To confirm the specificity of the phenotype, we performed several control experiments (Suppl. Fig. 1 and Suppl. information). We also implemented a previously described transient CRISPR/ Cas9 knockout strategy [START_REF] Willems | The Wnt Co-receptor Lrp5 is required for cranial neural crest cell migration in zebrafish[END_REF][START_REF] Moriyama | Evolution of the fish heart by sub/neofunctionalization of an elastin gene[END_REF]. In this manner we were able to observe zebrafish embryos displaying a similar cardiac phenotype to that observed when acana was knocked-down with a MO (Fig. 1.O,R). Although this approach will result in a mosaic knockout of acana, all embryos which displayed a cardiac phenotype tested positive for KO of acana by T7 assay (n = 6/44) (Suppl. Fig. 2). Together these results indicate that loss of acana either by knockdown or knockout leads to perturbed cardiogenesis. To better understand how the OFT has been affected, we analysed this structure in Tg(fli1a:GFP)y1 larvae which express GFP in all endothelial cells. At 3dpf, the wild-type OFT displays a typical "pear-shape" structure (Fig. 2.A). By contrast, in the acana morphants this structure has failed to develop (Fig. 2.B). In parallel, we also analysed a number of cardiovascular parameters such as heart rate and stroke volume. No differences were observed in the heart rate between the wild-type and acana morphants (Suppl. Fig. 3). However, the cardiac output and stroke volume were significantly decreased in acana morphants when compared to controls (Fig. 2.C and Suppl. Fig. 3). We also recorded high-speed movies of the beating embryonic heart and observed that in acana morphants there was a significant amount of blood regurgitation between the ventricle and the atrium, most likely caused by failed OFT development, forcing blood back through the AV canal (Suppl. movies M1 and M2). ACAN expression is reduced in human BAV patients Based on our findings in zebrafish, we hypothesised that reduced AGGRECAN (ACAN) expression may also be associated with aortic valve diseases such as BAV. Firstly, we determined the relative abundance of ACAN in the aortic valve by qPCR and were able to detect ACAN expression during foetal valve development in human (Suppl. Fig. 4). By 13 weeks of gestation, the ACAN expression greatly increased and this expression was even more abundant in adult valves. To assess the possibility that ACAN expression could be reduced in BAV patients, we performed RT-qPCR analysis using RNA extracted from aortic valves surgically removed from patients diagnosed type 0 "pure" BAV. As a control we used RNA extracted from normal TAV. We observed a significant (16 fold) decrease in the expression of ACAN in the type 0 BAV samples when compared to the control TAV samples (Fig. 2.D). Discussion Here we have shown that hemodynamically dependent acana expression is required for OFT development in zebrafish. Moreover, we showed that a MSC could be the sensor of these hemodynamic forces. It has recently been shown that the Trpv4 MSC, a target of RR, is involved in atrioventricular valve development in response to hemodynamic forces [START_REF] Heckel | Oscillatory flow modulates mechanosensitive klf2a expression through trpv4 and trpp2 during heart valve development[END_REF]. However, because RR is non-specific, we cannot determine the true identity of the MSC at this juncture. Because the analogous region in humans will give rise to the aortic valves, we also analysed ACAN expression during human aortic valve development and in patients who suffer from type 0 BAV we found a significant reduction when compared to TAV. Because of the rarity of this condition, we were only able to analyse relatively few patients, however the difference was in excess of 16 fold. It will therefore be necessary to expand this cohort to determine fully the reduction in the expression of AGGRECAN associated with BAV type 0. At present little is known about the genetic causes of BAV, with only a handful of genes thus far identified and, as one can imagine, there is even less known about what causes the different types of BAV. Although we cannot categorically state that reduced AGGRECAN expression is the root cause of type 0 BAV, its association with this condition does appear to be significant. Decreased AGGRECAN expression with BAV type 0 may not be so surprising considering it is required to strengthen and provide rigidity to the developing valves, and when this is lost the developmental process will malfunction. Why AGGRECAN expression is reduced in BAV type 0 remains unclear at this juncture, however it is possible that defective hemodynamics or defective mechanosensation of these forces during OFT development could be involved with this condition. Supplementary data to this article can be found online at https://doi. org/10.1016/j.ijcard.2017.09.174. Fig. 1 . 1 Fig. 1. Acana is hemodynamically expressed in the OFT and is involved in heart development. (A-B) ISH of 4dpf larvae using antisense acana (A) and acanb (B) probes. Acana is expressed in the OFT (A, yellow arrowhead). Acanb expression was not detected in the OFT (B, yellow arrowhead). (C-E) Confocal maximal projection images of double fluorescent ISH on 4dpf WT larvae using antisense acana (C, red) and elnb (D, green) probes (Blue: DAPI, yellow arrowheads indicate the OFT). (F-H) Same larvae at higher magnification (dashed rectangle in E). The merged images (E,H) indicate that acana is expressed in the OFT labelled by elnb. Scale bars E: 100 μm; H: 20 μm. (I) ISH of 4dpf tnnt2-MO injected larvae using an antisense acana probe. Acana expression was not detected in the OFT (yellow arrowhead). (J-L) ISH of 4dpf larvae using an antisense acana probe on nontreated (J, control, n = 19), or treated specimen with 10 μM (K, n = 15) and 20 μM (L, n = 6) of Ruthenium Red (RR). Black arrowheads indicate the OFT. (M-O) Bright field images of non-injected (M), acana-MO (N) and CRISPR/Cas9 injected (O) 3dpf larvae. Both morphant and transient knockout display heart edema (black arrowheads). (P-R) myl7 ISH showing the morphology of a wild-type heart (P) or of hearts from acana-MO (Q) and CRISPR/Cas9 injected (R) 3dpf larvae. Both morphant and transient knockout display a larger atrium (v: ventricle; a: atrium). Fig. 2 . 2 Fig. 2. Knockdown of acana disrupts OFT development and AGGRECAN expression is modified in human BAV patients. (A,B) Confocal maximal projection image of 3dpf Tg(fli1a:GFP)y1 OFT from WT (A) and acana morphant (B) larvae. White dashed lines indicate the OFT. (C) Graph showing the difference in stroke volume between noninjected (NI, n = 13) and acana morphants (n = 12). ***p b 0.001 (Student's t-test) (D) Graph showing relative mRNA expression of ACAN between TAV controls (n = 4) and BAV Type 0 (n = 2). **p b 0.05 (nonparametric Mann and Withney test). Error bars represent mean ± SEM in all histograms. Acknowledgements A.F is currently supported by a Labex ISCT postdoctoral fellowship with previous support provided by a Fondation Lefoulon-Delalande postdoctoral fellowship. P.R is supported by the Labex ISCT PhD program. C.J is supported by an INSERM ATIP-AVENIR grant and a Marie Curie CIG (PCIG12-GA-2012-332772). A.F, P.R and C.J are members of the Laboratory of Excellence «Ion Channel Science and Therapeutics» supported by a grant from the ANR. Work in C.J lab is supported by a grant from the Fondation Leducq and by the Fédération pour la Recherche sur le Cerveau (FRC). Work in S.Z lab is supported by INSERM, the Fédération Française de Cardiologie, and the Association Française contre les Myopathies (AFM-Telethon). Conflict of interest The authors report no relationships that could be construed as a conflict of interest.

01746129

en

2024/03/05 22:32:07

2017

https://hal.sorbonne-universite.fr/hal-01746129/file/ARACHNIDA-16-Leiurus-hoggarensis-compressed_sans%20marque.pdf

Wilson R Lourenço email: [email protected] Mohamed Lamine Kourim Salah Eddine Sadine Vachon Una nuova specie africana del genere Leiurus Ehrenberg, 1828 (Scorpiones: Buthidae) Keywords: Scorpion, new species, Leiurus hoggarensis sp. n., Buthidae, Algeria, Hoggar. Riassunto Scorpione, nuova specie, Leiurus hoggarensis sp. n., Buthidae, Algeria, Hoggar A new species of buthid scorpion belonging to the genus Leiurus Ehrenberg 1828 is described on the basis of four males and six females collected in the region of Amesmessa Tamanrasset in the south of Algeria. The new species, Leiurus hoggarensis sp. n., most certainly corresponds to the Leiurus population previously cited by Vachon from both the Hoggar and the Tassili N'Ajjer as Leiurus quinquestriatus. Several characteristics, however, attest that this population is unquestionable distinct from these found in Egypt, and both species can be distinguished by a distinct coloration pattern, different morphometric values and different number of teeth on the pectines. The type locality of the new species represents the most westerly record of the genus Leiurus in Africa, and the new species also inhabit a more mesic zone when compared to the central compartment of the Saharan desert. Leiurus hoggarensis sp. n., apparently does not present characteristics of a psamophilic species and may be considered as a lithophilic species. This is the 12 th species to be described for this buthid genus. Introduction As already outlined in several previous publications [START_REF] Lourenço | Description of a new species of Leiurus Ehrenberg, 1828 (Scorpiones, Buthidae) from the South of Jordan[END_REF][START_REF] Lourenço | The African species of the genus Leiurus Ehrenberg, 1828 (Scorpiones: Buthidae) with the description of a new species[END_REF][START_REF] Lourenço | One more African species of the genus Leiurus Ehrenberg, 1828 (Scorpiones: Buthidae) from Somalia[END_REF], the genus Leiurus Ehrenberg, 1828 was represented over many decades by a single species, Leiurus quinquestriatus, containing two subspecies, L. quinquestriatus quinquestriatus (Ehrenberg, 1828) and L. quinquestriatus hebraeus (Birula, 1908). Leiurus quinquestriatus seems to be a common species of desert faunas in certain regions of Egypt, Sinai and Sudan, although the precise identity of some regional populations from these areas requires yet further investigations [START_REF] Lowe | A review of the genus Leiurus Ehrenberg, 1828 (Scorpiones: Buthidae) with description of four new species from the Arabian Peninsula[END_REF]. Contrarily, L. hebraeus Birula, 1908 (now recognized as a valid species) is largely distributed in Israel and nearby countries [START_REF] Levy | Leiurus quinquestriatus hebraeus (Birula, 1908) (Scorpiones; Buthidae) and its systematic position[END_REF]. Leiurus species are globally infamous since they secrete one of the most noxious venoms among buthid scorpions in general, and are responsible for very serious human incidents (for details refer to [START_REF] Lourenço | One more African species of the genus Leiurus Ehrenberg, 1828 (Scorpiones: Buthidae) from Somalia[END_REF]. Because of its infamous reputation as a very dangerous scorpion, the toxins of both L. quinquestriatus and L. hebraeus have been the subject of numerous biochemical studies (for references see [START_REF] Simard | Venoms and Toxins[END_REF][START_REF] Hammock | Structure and neurotoxicity of venoms[END_REF]. Neverthless, many aspects of the taxonomy of the genus Leiurus remained confused for many decades (for more details see [START_REF] Lourenço | The African species of the genus Leiurus Ehrenberg, 1828 (Scorpiones: Buthidae) with the description of a new species[END_REF][START_REF] Lourenço | One more African species of the genus Leiurus Ehrenberg, 1828 (Scorpiones: Buthidae) from Somalia[END_REF]. Only in recent years, totally new species were finally described for the genus Leiurus. The description which really changed most conservative views about this group of scorpions was that of Leiurus jordanensis Lourenço, Modry et Amr, 2002 described from Jordan [START_REF] Lourenço | Description of a new species of Leiurus Ehrenberg, 1828 (Scorpiones, Buthidae) from the South of Jordan[END_REF]. Just a few years later, Leiurus savanicola Lourenço, Qi et Cloudsley Thompson, 2006[START_REF] Lourenço | The African species of the genus Leiurus Ehrenberg, 1828 (Scorpiones: Buthidae) with the description of a new species[END_REF] was described from Cameroon, representing the second confirmed species from Africa. In a more recent contribution, [START_REF] Lowe | A review of the genus Leiurus Ehrenberg, 1828 (Scorpiones: Buthidae) with description of four new species from the Arabian Peninsula[END_REF] proposed, in a very extensive article, a full revision of the genus Leiurus, but dealing mainly with the populations from the Arabian Peninsula. The status of some old species was revalidated, one recently described species was placed in synonymy, one subspecies was raised to species and four new species were described. This raised the total number of species in the genus Leiurus to ten. The characters used by these authors to define the species, as well as the proposed dichotomic key are rather difficult to be used. Nevertheless, we globally agree with these authors and, in particular with their opinion about the African species, stated as follows: "Our findings show that, like many other scorpion genera, Leiurus is comprised of an assemblage of allopatric or parapatric species spread across different regions separated by physiographic barriers, each adapted to local environments and substrates. Additional species diversity may emerge when other local populations are analysed in more detail, for example those in southern Sinai and in more central parts of North Africa". Moreover, after the study of the original syntypes used in the description of Leiurus quinquestriatus, these same authors [START_REF] Lowe | A review of the genus Leiurus Ehrenberg, 1828 (Scorpiones: Buthidae) with description of four new species from the Arabian Peninsula[END_REF] suggest as follows: "The syntypes include assorted material from the Sinai, the Nile Valley in Egypt and Sudan, and the desert region of Egypt east of the Nile. These could represent more than one species if the populations in the Sinai are distinct from those of the Nile Valley". Once again we globally agree with this suggestion. It is most obvious that the African populations of Leiurus have been largely neglected and still require intensive further studies. If [START_REF] Vachon | Etude sur les Scorpions[END_REF] associated the few specimens he studied from Hoggar to L. quinquestriatus this can be attributed to both the typical incertitude of this author, but also to the very limited material (mainly fragments) he disposed. Again [START_REF] Vachon | Scorpions, Mission scientifique au Tassili des Ajjer[END_REF] in a synopsis of the scorpions from the Tassili N'Ajjer, another mountain range in southern Algeria, cited Leiurus quinquestriatus from this locality, but based on the study of two very young juveniles. None of these specimens was located in the collections of the Museum in Paris and probably were deposited in other collections such as that of the Institute Pasteur of Algeria. In his monograph on the scorpions of North Africa, [START_REF] Vachon | Etude sur les Scorpions[END_REF] also referred to several specimens collected in the Fezzan (Libya) as L. quinquestriatus. It is quite possible however, that this population corresponds to Buthus quinquestriatus libycus Birula, 1908 (= Leiurus quinquestriatus libycus). Nevertheless, only the study of more fresh material from Libya will allow a confirmation to this suggestion. In the present contribution we describe a new species based on material collected in the region of the Hoggar Massif in the south of Algeria. This raises the number of Leiurus species to twelve. Methods Illustrations and measurements were obtained using a Wild M5 stereomicroscope with a drawing tube and ocular micrometer. Measurements follow [START_REF] Stahnke | Scorpion nomenclature and mensuration[END_REF] and are given in mm. Trichobothrial notations follow [START_REF] Vachon | Etude des caractères utilisés pour classer les familles et les genres de Scorpions (Arachnides)[END_REF] and morphological terminology mostly follows [START_REF] Hjelle | Anatomy and morphology[END_REF]. Etymology: specific name makes reference to the Hoggar, the region where the new species was found. Diagnosis. Scorpion of large size when compared with the other species of the genus, having a maximum total length of 77.7 mm for male and 94.6 mm for female. Ground colour yellow to orangeyellow with the body and pedipalps almost totally orangeyellow. Male carapace with a brownish which covers the ocular tubercle; metasomal segment V only slightly infuscate, including in juvenile specimens; other metasomal segments orangeyellow. Ocular tubercle strongly prominent. Pectines with 32 to 34 and 26 to 29 teeth for males and females respectively. Median carinae on sternites IIIIV moderately to strongly marked; sternite VII with mediate intercarinal surface presenting a thin granulation. Pedipalp fingers with 1112 or 1212 rows of granules for both males and females. Description based on male holotype and paratypes. (Morphometric measurements in Table I). Coloration. Ground colour yellow to orangeyellow; body and pedipalps almost totally orangeyellow; legs yellow. Male carapace orangeyellow with a brownish spot which covers the ocular tubercle. Mesosoma tergites with some infuscations in male, absent from female. Metasoma orangeyellow on segments I to IV; segment V slightly infuscate, including on juveniles. Vesicle yellow with reddish tonalities on lateral sides; aculeus yellow at the base and dark red at its extremity. Venter yellow to slightly orange yellow without spots. Chelicerae yellow without any dark reticulated spots; teeth dark red. Pedipalps yellow to orangeyellow overall except for the rows of granules on chela fingers which are dark red. Legs yellow with some zones slightly orangeyellow. Morphology. Prosoma: Anterior margin of carapace with a weak concavity. Carapace carinae moderately to strongly developed; central median and posterior median carinae moderate to strong; anterior median carinae strong; central lateral carinae moderate to strong; posterior median carinae moderate to strong, terminating distally in a small spinoid process that extends very slightly beyond the posterior margin of the carapace. All carinae better marked on males. Intercarinal spaces with very few irregular granules, and the reminder of the surface almost smooth, in particular laterally and distally. Median ocular tubercle in a central position and strongly prominent; median eyes large in size and separated by about two ocular diameters. Four/five pairs of lateral eyes; the fourth and fifth are vestigial. Mesosomal tergites III pentacarinate; IIIVI tricarinate. All carinae strong, granular, better marked on male; each carina terminating distally in a spinoid process that extends slightly beyond the posterior margin of the tergite. Median carinae on I moderate, on IIVI strong, crenulated. Tergite VII pentacarinate, with lateral pairs of carinae strong and fused; median carinae present on the proximal half in female and on the 2/3 on male, moderate to strong. Intercarinal spaces weakly to moderately granular. Lateral carinae absent from sternite III; moderate to strong on sternites IVVI; strong, crenulate on VII; median carinae on sternites IIIIV moderate to strong. Pectines long; pectinal tooth count 3434 on male holotype and 2928 for female paratype (see diagnosis for variation). Metasomal segments IIII with ten carinae, moderately crenulate; lateral inframedian carinae on I moderate; on II present on the posterior third; on III limited to a few posterior granules; IV with eight carinae. Dorsal and dorsolateral carinae moderate, without any enlarged denticles distally. All the other carinae moderate to weak on segments IIV. Segment V with five carinae; ventromedian carinae with several slightly spinoid granules distally; anal arch with three slightly spinoid lobes, better marked in female. Dorsal furrows of all segments weakly developed and smooth; intercarinal spaces almost smooth, with only a few granules on the ventral surface of segment V. Telson smooth; subaculear tubercle absent; aculeus as long as vesicle. Chelicerae with two reduced denticles at the base of the movable finger [START_REF] Vachon | De l'utilité, en systématique, d'une nomenclature des dents des chélicères chez les Scorpions[END_REF]. Pedipalps: Trichobothrial pattern orthobothriotaxic, type A [START_REF] Vachon | Etude des caractères utilisés pour classer les familles et les genres de Scorpions (Arachnides)[END_REF]; dorsal trichobothria of femur in configuration [START_REF] Vachon | Sur l'utilisation de la trichobothriotaxie du bras des pédipalpes des Scorpions (Arachnides) dans le classement des genres de la famille des Buthidae Simon[END_REF]. Femur pentacarinate; all carinae moderately β crenulate. Patella with seven carinae; all carinae moderately to weakly crenulate; dorsointernal carinae with 23 spinoid granules distally. Chelae slender, with elongated fingers; all carinae weakly marked, almost vestigial. Dentate margins of fixed and movable fingers composed of 1112 or 1212 almost linear rows of granules in both sexes. Legs: Ventral aspect of tarsi with short spiniform setae more or less arranged in two rows. Tibial spurs present on legs III and IV, moderately marked. Pedal spurs present on all legs, strongly marked. Relationships. Based on the key supplied by [START_REF] Lowe | A review of the genus Leiurus Ehrenberg, 1828 (Scorpiones: Buthidae) with description of four new species from the Arabian Peninsula[END_REF], the new species seems to present affinities with L. quinquestriatus 'typicus' normally only distributed in Egypt, mainly Sinai, and perhaps also in Sudan. Nevertheless the two species differs by a number of characters: I) distinct patterns of pigmentation, with the population from Hoggar showing a more orangeyellow colour; II) quite distinct morphometric values for specimens of a similar global size; III) lower numbers of pectinal teeth counts. Moreover, the geographic distributions of the African populations are not continuous. The future examination of material from Libya should confirm the existence of an intermediate population between Egypt and Algeria. Ecology As already outlined in a recent publication [START_REF] Lourenço | Scorpions from the region of Tamanrasset, Algeria. Part I. A new species of Buthacus Birula, 1908 (Scorpiones: Buthidae)[END_REF], the Region of El Ahaggar (Tamanrasset) which corresponds to the El Ahaggar National Park is very important in surface. It is located in the Central Massif of the southeastern Algerian region (Fig. 15) and covers a total area of ca. 450,000 km 2 . The main locality in this area is Tamanrasset [START_REF] Wacher | lnventaire de la faune sahelosaharienn[END_REF]. The very diverse geomorphological features are constituted by the Regs, Ergs, Stone Plateaux (Figs. 1617) but also very high summits such as the Tahat with more than 3000 meters, constituting the highest mountain in Algeria [START_REF] Sahki | Guide des principaux arbres et arbustes du Sahara central (Ahaggar et Tassili)[END_REF]. The region of El Ahaggar, as all the others regions around Tamanrasset is characterized by a typical arid climate with mild winters but also important thermal amplitudes between the day and the night [START_REF] Kourim | Biodiversité faunistique dans le Parc National de l'Ahaggar[END_REF]. The hottest months range from June to August. Rain fall is extremely rare in the region of El Ahaggar and the average values can vary extremely accordingly to the year; very important dry periods can be observed over more than three years. Maximum rain is generally observed during the hot period from June to August [START_REF] Hamdine | Conservation du Guépard (Acinonyx jubatus Schreber, 1776) de la région de l'Ahaggar et du Tassili n'Adjjer en Algérie[END_REF]. The new species described here was collected in the region of Amesmessa which is located about 450 Km NW of Tamanrasset. This site is located in a region of mountains with quite many sand deposits which are the consequence of gold mining. The new Leiurus species appears as the most common scorpion in the region of study, representing up to 76.24% in the region of Tamanrasset and up to 84.72% in the region of Amesmessa. Leiurus hoggarensis sp. n., apparently does not present characteristics of a psamophilic species and may be considered as a lithophilic species (Fig. 18). Figs. 14 . 14 Figs. 14. Leiurus hoggarensis sp. n. Habitus. 12. Male holotype. 34. Female paratype. 59. Leiurus hoggarensis sp. n. Male holotype (58). Female paratype (9). 5. Chelicera, dorsal aspect. 6. Cutting edge of movable finger, showing rows of granules. 7. Idem, detail of the extremity. 89. Metasomal segment V and telson, lateral aspect.Taxonomic treatmentFamily Buthidae C. L. Koch, 1837 Genus Leiurus Ehrenberg, 1828Leiurus hoggarensis sp. n. (Figs. 114)Type material: Algeria, AmesmessaTamanrasset (21°03' N -02°28' E), 326/X/2015 (M. L. Kourim). Male holotype, 2 males and 3 females paratypes deposited in the Muséum national d'Histoire naturelle, Paris, France. 1 male and 3 female paratypes deposited in the University of Ghardaïa, Algeria. Figs. 1014 . 1014 Figs. 1014. Leiurus hoggarensis sp. n. Male holotype. Trichobothrial pattern. 1011. Chela, dorsoexternal and ventral aspects. 1213. Patella, dorsal and external aspects. 14. Femur, dorsal aspect. Fig. 15 . 15 Fig. 15. Map of Algeria showing in detail the region of El Ahaggar (Tamanrasset), with the type locality of Leiurus hoggarensis sp. n. (black star). Figs. 1617 . 1617 Figs. 1617. Aspects of the biotopes found in the region of El Ahaggar with typical sites in Amesmessa where the Leiurus hoggarensis sp. n. was collected. Fig. 18 . 18 Fig. 18. A preadult male of Leiurus hoggarensis sp. n. alive in its natural habitat. Table I . I Morphometric values (in mm) of the male holotype and female paratype of Leiurus hoggarensis sp. n. Male holotype Female paratype Total length 77.7 94.6 Carapace: length 8.4 10.5 anterior width 5.8 7.2 posterior width 9.5 12.5 Mesosoma length 17.8 20.7 Metasomal segment I: length 6.7 8.2 width 5.6 6.2 Metasomal segment II: length 8.2 9.8 width 5.2 5.3 Metasomal segment III: length 8.3 10.3 width 4.7 4.9 Metasomal segment IV: length 9.1 11.4 width 4.3 4.6 Metasomal segment V: length 10.5 12.5 width 4.0 4.6 depth 3.6 3.9 Telson length 8.7 11.2 Vesicle: width 3.4 4.2 depth 3.2 3.8 Pedipalp: Femur length 8.9 11.1 Femur width 2.2 2.7 Patella length 9.8 12.3 Patella width 2.8 3.2 Chela length 15.8 19.9 Chela width 2.5 3.2 Chela depth 2.6 3.3 Movable finger: length 11.2 14.4 Acknowledgements We are most grateful to Mohamed Kraimat (University of Ghardaïa) for his assistance both during the field work and the preparation of this article, and in special to EliseAnne Leguin (MNHN, Paris) for her assistance in the preparation of the photos and plates.

01637751

en

2024/03/05 22:32:07

2018

https://hal.science/hal-01637751/file/Anthraco-typology%20as%20a%20key%20approach_revised.pdf

Alexa Dufraisse email: [email protected] Sylvie Coubray Olivier Girardclos Noémie Nocus Michel Lemoine Jean-Luc Dupouey Dominique Marguerie approach to past firewood exploitation and woodland management reconstructions. Dendrological reference dataset modelling with dendro-anthracological tools Keywords: firewood management, dendro-anthracological tools, anthraco-typology, anthraco-group, deciduous oak scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Introduction Anthracology and past woodland reconstruction The questions raised by relationships between people and the environment in time and space can be explored by archaeological, ethnographic or environmental approaches. The management of the environment for (plant or animal) food strategies reflects, to some extent, human societies and their organization, their lifestyles, their perception of the environment and the landscape in which they operate. Forest exploitation in order to produce wood material for multiple needs, is perceptible at different scales: the tree, the woodland and the landscape (Michon, 2005(Michon, , 2015)); Humans R e v i s e d m a n u s c r i p t "domesticate" the tree by modifying its architecture, its growth cycle, its production and its reproductive functions. This domestication also concerns the forest ecosystem transformed by practices. Wood stands are shaped by and for societies living in them as a result of the installation of fields, herds and villages in forest areas. Variable spatial patterns results from this articulation between forest and agriculture, the landscapes. Firewood management contributes to this "domestication". It is part of a complex system closely related to social organization, technical and economic systems, and the environment itself (Chabal, 1997;[START_REF] Picornell Gelabert | People, trees and charcoal: somes reflections about the use of ethnoarchaeology in archaeological charcoal analysis[END_REF][START_REF] Dufraisse | Firewood and woodland management in their social, economic and ecological dimensions[END_REF][START_REF] Salavert | Understanding the impact of socio-economic activities on archaeological charcoal assemblages in temperate areas: comparative analysis of firewood management of two Neolithic societies of Western Europe (Belgium, France)[END_REF]. Thus, archaeological charcoal fragments, residues of firewood selected and transported by humans, are valuable ecofacts reflecting use, techniques and woodland management, and are themselves conditioned by environmental resources (available wood resources, i.e. biodiversity and biomass). In forest science, the criteria characterizing wood stands are the composition (dominant and secondary species), stand density (number of stems per hectare), structure (distribution of age and diameter classes of trees) and modes of regeneration (seeded or vegetative renewal) [START_REF] Rondeux | La mesure des arbres et des peuplements forestiers[END_REF]. The methods usually employed in dendrochronology to extract this information are not suited to anthracological material. In dendrochronology, samples usually come from timber wood and generally from trunks and/or branches or roots, the wood is not charred, the methods are based on statistical tools that require at least 50 consecutive rings and it is possible to individualize the signals (study of distinct elements). In anthracology, fragments derive from trunks and/or branches or roots, the wood is charred, fragmented and incomplete as it is partially reduced to ashes, the fragments present on average less than five rings and result from the exploitation of many indistinguishable individuals [START_REF] Dufraisse | Charcoal anatomy potential, wood diameter and radial growth[END_REF][START_REF] Marguerie | Short Tree ring series: the study materials of the dendro-anthracologist[END_REF]. Consequently, in the absence of adequate tools, charcoal analysis is most often limited to the study of a list of taxa and their relative frequency without exploiting the information contained in the wood anatomy. The identification of the morphological characteristics of harvested firewood (part of the tree, age, R e v i s e d m a n u s c r i p t shape, etc.) still raises methodological problems even though it is a fundamental element for characterizing firewood exploitation techniques and reconstructing the populational and environmental parameters of wood stands. In order to address this need to learn more about forest exploitation and practices, the ANR DENDRAC project "Development of dendrometrical tools used in anthracology: study of the interactions between Man, resources and environments" aims to convert dendroecological data measured on fresh wood material from modern-day oak wood standscorresponding to different types of historical woodland practices -into parameters adapted to charcoal analysis, using a method similar to that developed by A. Billamboz, termed dendrotypology [START_REF] Billamboz | Applying dendro-typology to large timber series[END_REF][START_REF] Billamboz | Regional patterns of settlement and woodland developments: Dendroarchaeology in the Neolithic pile dwellings on Lake Constance (Germany)[END_REF]. His method consists in establishing a typological classification of tree-ring series according to their growth patterns. The application of a similar method in anthracology involves associating the identification of the taxa with the examination of dendrological and anatomical parameters; a concept that leads to the notion of dendro-anthracology (Marguerie et al., 2010). Deciduous European oak (Quercus petraea/robur) was chosen for its abundance in temperate forests, its anatomy with clearly identifiable growth rings and its representativity in anthracological spectra. In the present study, we postulate that the characteristics of an assemblage of treerings can be exploited, without taking into account tree-ring series in terms of time series like in dendrochronology. The first step consisted in developing dendro-anthracological tools based on morpho-anatomical features. The second step was to convert dendroecological data to form an anthraco-typological grid, which could then be used as a key approach for the interpretation of archaeological charcoal assemblages. This approach was applied to three modern-day wood stands: a coppice under standard, a high forest and a young stand formed by a mixture of seeded and coppice trees. Analysis was conducted at different levels: the whole tree, and trunks and branches separately, in order to model different modes of wood exploitation. R e v i s e d m a n u s c r i p t The dendro-anthracological tools Growth rate is a widely used dendro-anthracological parameter, but the successive tree-rings width series of each charcoal fragment must be localized as precisely as possible on the stem cross-section. In that aim, different dendro-anthracological tools are proposed in order i) to distinguish sapwood from heartwood which provides information about the minimal age of the wood (heartwood formation i.e. duraminisation starts when deciduous oak is around 25 years old) ii) to localize the tree-ring series in respect to the center of the stem, iii) to model dendroecological data from modern wood stands into dendro-anthracological parameters adapted to charcoal analysis. The Heartwood-Sapwood discriminating tool In some species the coloration of heartwood due to the deposition of lignins and polyphenols makes heartwood recognizable, but the charcoalification process that occurs during carbonization obliterates the colour difference, making this feature unusable in anthracology. Fortunately, in some Angiospermae, such as deciduous European oak (Quercus petraea/robur), the formation of tyloses (cellulose walls expansions) in earlywood vessels is an important feature of the changeover of sapwood to heartwood. However, tylosis formation also occurs in sapwood and increases with the formation of heartwood, from 0% of tyloses in the cambial region and close to 100% in the heartwood. Thus, we quantify the number of vessels sealed by tylosis in order to establish discriminating thresholds between sapwood and heartwood (Fig. 1a) [START_REF] Dufraisse | Contribution of tyloses quantification in early wood oak vessels to archaeological charcoal analyses: estimation of a minimum age and influences of physiological and environmental factors[END_REF]. Trunks and branches of ten deciduous oak trees from 15 to 60 years old were sampled in three stations in order to evaluate the number of earlywood vessels with tylosis in sapwood and heartwood. For an application to archaeological charcoal (tyloses are preserved until 800°C), at least one tree ring and 15 vessels must be counted. The best strategy is to count 50 vessels spread over 3-4 tree rings. Thresholds of less than 65% for sapwood and up to 85% for heartwood are R e v i s e d m a n u s c r i p t significant. Besides the discrimination of sapwood and heartwood, the process of heartwood formation starts when deciduous oak is about 25 years old. The absence of heartwood is thus an indication of the exploitation of young wood (trunks or branches). The pith estimation tool The pith estimation tool is used to measure the distance between the charcoal fragment and the center of the stem (or the missing pith), named the "charcoal-pith distance". This measurement is taken with the trigonometric pith estimation tool based on measurements of the angle and the distance between two ligneous rays (Fig. 1b) [START_REF] Dufraisse | Mesurer les diamètres du bois de feu en anthracologie. Outils dendrométriques et interprétation des données[END_REF]Paradis-Grenouillet et al., 2013). This tool was evaluated on fresh and carbonized oak wood discs with different angle values and distances between ligneous rays. This work enables us i) to propose exclusive criteria (angle < 2° and distance < 2 mm) for reducing the margin of error and improving results in archaeological applications, ii) to establish correction factors linked to the trigonometric tool itself (underestimation of distance values between 5 and 10 cm) and the shrinkage which occurs during charcoalification, iii) to highlight that there are no reliable measurements for charcoal-pith distances beyond 12.5 cm, i.e. diameter of 25 cm [START_REF] Dufraisse | Mesurer les diamètres du bois de feu en anthracologie. Outils dendrométriques et interprétation des données[END_REF][START_REF] Martinez | Correction factors on archaeological wood diameter estimation[END_REF]. The values were ordered into diameter classes chosen to be compatible with standards used in dendrometrical plans by foresters [START_REF] Gaudin | Dendrométrie des peuplements[END_REF][START_REF] Deleuze | Estimer le volume total d'un arbre, quelles que soient l'essence, la taille, la sylviculture[END_REF]. For Angiospermae the conventional wood cuts are 4 cm, 7 cm, 20 cm, etc. Two cuts were added at 2 and 10 cm for more accurate interpretation of charcoal diameters, that is to say ]0-2] cm, ]2-4] cm, ]4-7] cm, ]7-10] cm, ]10-20] and >20 cm. For Gymnospermae it is more appropriate to add a cut at 14 cm, namely ]0-2] cm, ]2-4] cm, ]4-7] cm, ]7-10] cm, ]10-14] cm, ]14-20] cm and >20 cm. Therefore, an Analysis Diameter model (ADmodel) was developed, based on the fact that a trunk is biologically considered to be a stack of cones (Fig. 1c) [START_REF] Dufraisse | Les Habitats littoraux néolithiques des lacs de Chalain et Clairvaux (Jura, France): collecte du bois de feu, gestion de[END_REF][START_REF] Dufraisse | Charcoal anatomy potential, wood diameter and radial growth[END_REF][START_REF] Dufraisse | Mesurer les diamètres du bois de feu en anthracologie. Outils dendrométriques et interprétation des données[END_REF]. These cones are hollow and their thickness corresponds to the amplitude of the diameter classes. It is based on a calculation table that provides the respective distribution of these cones in terms of volume. The ADmodel breaks down unburnt wood diameter into an expected distribution of charcoal-pith distances. In return, the ADmodel is a helpful tool to interpret the distribution of charcoal-pith distances from a charcoal assemblage as unburnt wood diameter (UWD). However, this model does not reconstruct the initially quantity of burnt wood [START_REF] Théry-Parisot | Charcoal analysis and wood diameter: inductive and deductive methodological approaches for the study of firewood collecting practices[END_REF]. In the present study, only the ADmodel running into UWD decomposition is used. In the present study, each cone thickness was also characterized by a growth rate (cumulated tree-ring width divided by the number of tree rings) and its sapwood/heartwood affiliation. Material and Method The general analytical protocol consists in sampling modern-day oak woodlands corresponding to specific archaeological questions, removing logs from felled trees, cutting wood discs from logs and producing experimental charcoal assemblages (Fig. 2). Various kinds of datasets were produced: i) dendrometrical plans to characterize tree morphology and wood stands (composition, structure stand density, regeneration modes), ii) dendrochronological data from wood discs, iii) anthracological data modelled with the dendro-anthracological tools. With respect to historical woodland practices and to answer to specific archaeological issues such as the distinction branch/ trunk or coppice /high forest three "contrasted" deciduous oak stands managed by the National Forestry Office (ONF) in France were chosen (Fig. 3). The first one is " Les Cagouillères ", located in the Vienne department, on a limestone plateau (altitude: 115m). It is in an old abandoned coppice woodland, about 62 years old, currently undergoing conversion to high forest. The second stand is "Bogny-sur-Meuse" located in the Ardennes department. This is a coppice-under-standard growing on an acidic brown soil on schists, about 68 years old. The third stand is "Le Bois de l'Or", also located in the Ardennes department, near Bogny-sur-Meuse. This is a young stand, about 15 years old, formed by a mixture of even aged seeded and coppice trees (altitude: 350m). Stand analysis In order to characterize the wood stands, forest inventory and dendrometric data were compiled. The basal area increment (m²/ha), stand density (number of stems per hectare), dominant height of the trees and the average square diameter were recorded distinguishing trees with diameters of more and less than 7.5 cm (table 1, Fig. 4). Dendrometry and tree ring analysis The dendrological information for each tree, such as diameter, age, growth rate and radial growth trend, was defined at breast height on the field and from disc located at 1.30 m above ground, as is usual in dendroecology. However, the nature and representativeness of archaeological samples are different in dendroecology and anthracology. Consequently, for R e v i s e d m a n u s c r i p t the conversion in anthracological data according to anthracological constraints, the dendrological data were measured in the whole tree. For the study of tree ring-climate relations in sessile oak, six is the number of optimal trees to sample. For our purpose, and taking into account our archaeological questions, one to five trees were felled and registered meter by meter, one dominant tree in the coppiceunder-standard at "Bogny-sur-Meuse", four dominant stems from distinct multi-stem trees at "Les Cagouillères" and five coppice shoots and five seeded trees at "Le Bois de l'Or". For each tree, the total height, the height of the first large branch insertion on the stem and the height of the crown base were recorded as well as the diameter at breast height and regeneration modes. The set of tree morphology indicators is presented in table 1 and figure 4. In order to estimate the relative proportion of trunk and branches for each tree and each stand, each tree was cut into logs of 1-metre-long including branches with a diameter of more than 4 cm. A code was attributed to each log according to its position in the tree (height, number of branches, location in the branch). Length and circumference (at three points) of each log were measured to calculate the mean diameter and the volume. Branches with a diameter of less than 4 cm were packed into bundles according to two diameter classes; 0-2 cm and 2-4 cm. Each bundle was weighed. Sub-samples of wood were collected from each bundle to estimate the density of the wood and then to calculate the volume of each bundle. In order to characterize each tree and then each wood stand at different levels (whole tree, and trunks and branches separately), one disc was removed from the extremity of each log. In the present study, a subsample of the set of discs was taken by selecting discs at different heights in the trunk and in the crown (23 discs for the four trees at "Les R e v i s e d m a n u s c r i p t Cagouillères", 14 discs for the tree at "Bogny-sur-Meuse" and 77 discs for the 10 trees at "Le Bois de l'Or" (Table 2a,2b). The tree-ring widths (discriminating earlywood and latewood) of each disc were measured to the nearest 0.01 mm using a LINTAB measurement device and associated TSAP software (Frank Rinn, Heidelberg, Germany) along 5 radii and averaged in order to reduce intra-tree variability. Each tree-ring was then associated with a diameter class (calculated by the cumulated ring widths) and sapwood/heartwood. Thus, the proportion of sapwood and heartwood was characterized by averaging tree-ring numbers, tree-ring width and wood volume. The usual dendro-anthracological parameters were first independently considered to obtain a "whole tree" estimation, and then the trunks and branches were separated. i) the distribution of growth-ring width, ii) the proportion of sapwood/heartwood, iii) the distribution of the decomposed unburnt wood diameters (UWD) were recorded. Results 3.1. Dendrological features of the three wood stands (Fig. 4). For the four sampled 62 years-old trees from "Les Cagouillères", the average diameter at breast height is 20.75 cm. The average tree height is 17.7 m and 90,4% of the wood volume is from to the trunk. The low proportion of branches, with a diameter of less than 7 cm, reflects an undeveloped crown (probably due to competition, a consequence of the abandonment of forest management). The dominant tree at "Bogny-sur-Meuse" is 68 years old with a diameter of 33 cm at breast height. The height is 20.3 m with a first large branch at 7.7 m and a more developed crown; branches represent 37.4% of the tree volume and can reach a diameter of 20 cm. The trees at "Le Bois de l'Or" are 14 years old, their diameters average 10.21 cm, 8.6 m high, the trunk forms 78.38% of the volume and the diameter of branches less than 7 cm in diameter. Thus, the tree at "Bogny-sur-Meuse" is less R e v i s e d m a n u s c r i p t slender than trees at "Les Cagouillères" and "Le Bois de l'Or" (see the height/diameter ratio, table 1). In the three sampled stands, trunk volume is always predominant and branches are poorly represented. The diameter 20-40 cm class is the best represented at "Bogny-sur- Meuse" whereas the 10-20 cm diameter class characterizes "Les Cagouillères". The main volume at "Le Bois de l'Or" is distributed between 7-10 diameter but a few trees reach 11 cm and thus belong to the 10-20 cm diameter class. Radial growth rate and growth trends are different in each stand. Tree-ring widths average 1.23 mm/year at "Les Cagouillères" coppice, and the growth trend has been decreasing over the past 20 years due to strong competition between shoots, intra-tree and between stools. At "Bogny-sur-Meuse", growth-ring widths average 1.35 mm/year and the growth trend has been decreasing slightly over the past 20 years. At "Le Bois de l'Or", growth-ring widths average 2.99 mm/year and are marked during the 1 to 10 first years by a steady increase in the coppice trees while seeded trees are characterized by narrowest rings than coppice from around the pith to 6-7 years, followed by an intensive growth period before a relatively sudden decrease (for more details, see [START_REF] Girardclos | Improving identification of coppiced and seeded tress in past woodland management by comparing growth and wood anatomy of living sessile oaks (Quercus petraea)[END_REF]. Simple dendro-anthracologial parameters Growth rate The distribution of the growth rates indicates differences at stand and tree levels (Fig. 5a). First, the difference in growth rate observed in § 3.1 and only based on one disc localized at 1.30 m in the trunk, is conserved when the whole tree is taken into account, what is more realistic for anthracology. The growth rate at "Les Bois de l'Or" is the highest, followed by "Bogny-sur-Meuse" and "Les Cagouillères". For a same stand, we also note a significant difference between trunks and branches, the latter being characterized by a lower rate. Moreover, considering the different parts of the trunk (base, top, upper part in the crown), we note that the annual ring-width in the top of the bole is wider than in the lower R e v i s e d m a n u s c r i p t part, and that the growth rate of the trunk localized in the crown is comparable with branches (Fig. 5b). However, this latter observation is less clear at "Le Bois de l'Or". Sapwood/heartwood The trees at "Le bois de l'Or", less than 15 years old, are characterized by the absence of heartwood, contrasting with "Bogny-sur-Meuse" and "Les Cagouillères" (Fig. 6). However, at "Les Cagouillères", heartwood formation is not yet initiated in branches. Conversely, the trunk and branches of the dominant tree at « Bogny-sur-Meuse » are characterized by heartwood and sapwood. The relative proportion of sapwood in trunk is less important at "Bogny-sur-Meuse" than at "Les Cagouillères". Likewise, the average number of sapwood tree-rings is less important at "Bogny-sur-Meuse". Nevertheless the average sapwood ring width is higher at "Bogny-sur-Meuse" reflecting more vigorous growth. Diameters The unburnt wood diameters (UWD) were decomposed with the ADmodel, according to the relative volume of each hollow cone composing the logs (Fig. 1c,7). The raw dendrological data indicate that there is little overlap between the diameters of branches and trunks. In fact, the low proportion of the trunk represented in the smallest diameter classes corresponds to the upper part of the trunk localized in the crown. Therefore, for each wood stand, the distribution of the decomposed UWD of branches is clearly distinct from the trunk. Besides, as the volume of branches is weak, the wood diameter pattern for whole trees does not show clear differences with that of the trunk. The first combination consisted in assessing growth trends by characterizing each wood stand. In dendroecology, growth trends are obtained by combining tree-ring width and cambial age. Given that i) the analysis of tree-ring patterns in segment of cambial age is considered relevant for studying forest dynamics and development [START_REF] Haneca | Growth trends reveal the forest structure during Roman and Medieval times in Western Europe: a comparison between archaeological and actual oak ring series (Quercus robur and Quercus petraea)[END_REF] ii) the distance of the charcoal from the pith can be estimated by the charcoal-pith tool, we combined tree-ring width with diameter classes. For the dendrological data, average tree-ring width was calculated for each cambial age. For the modelled anthracological data, average tree-ring width was calculated for each diameter class (Fig. 8). The radial growth trends of the three wood stands are different and the modelled anthracological data correspond well to their dendrological characteristics. Even though the anthracological data are smoother because of the calculation of average ring width per diameter class, the radial growth trend is consisting of i) a strong increase in the radial growth of trees at "Le Bois de l'Or", reflecting a free juvenile growth, ii) the increase followed by a decrease at "Les Cagouillères" due to the high density of trees over a long period of time, iii) a slight decrease in the life of the tree at "Bogny-sur-Meuse" due to a managed coppice-under-standards. However, the differences observed between seeded and coppice trees at "Le Bois de l'Or" are no longer evident. The branches at "Les Cagouillères" and "Bogny-sur-Meuse" are characterized by a lower growth rate than in the corresponding trunks (cf. § 3.2.1.) and by a downward growth. In contrast, the young seeded and coppice trunks at "Le Bois de l'Or", with diameters comparable to the branches, are characterized by a clearly higher growth rate and a more upward growth. The radial growth rate of whole trees is lower in the first diameter classes than in the trunk considered separately, as it includes the low growth rates of branches. Then, radial growth increases from the boundary of the step between diameters of trunks and branches. R e v i s e d m a n u s c r i p t Diameter classes versus heartwood/sapwood The second combination aims to improve the interpretation of the distribution of the decomposed UWD by associating them with the presence or absence of heartwood, and the sapwood/heartwood ratio in each diameter class, as decomposed by the ADmodel (Fig. 9). The distribution of heartwood/sapwood according to the diameter classes shows specific patterns for the different wood stands and possible exploitation modes (whole trees, trunks/branches separately). At "Les Cagouillères", where branches are characterized by the absence of heartwood, the volume of the trunk is mainly distributed in the penultimate diameter class. The pattern of the whole trees is similar to that of the trunk, as branches only represent 9.58% of the volume. At "Bogny-sur-Meuse", the same pattern is observable but the main volume is represented in the last two diameter classes. However, regarding the whole tree, sapwood is better represented in the small diameter classes than at "Les Cagouillères", as branches account for 37.4% of the tree volume. While the mature trees contain a central heartwood core (reflected by heartwood in the small diameter classes) and peripheral sapwood (reflected by sapwood in the largest diameter classes), the absence of heartwood in trunks from "Le Bois de l'Or" and in branches from "Les Cagouillères" is in agreement with young trunks and young branches respectively (less than 25 years old for oak). They are characterized by small diameters with sapwood, and the absence of heartwood and of large diameters. The biggest branches of the tree from "Bogny-sur-Meuse", i.e. 10-20 cm, contain small amounts of heartwood, and traces of heartwood in the smaller classes. The third combination consists in combining tree-ring width with the decomposed UWD and their respective affiliation to sapwood or heartwood (Fig. 10). Globally, the pattern between whole trees and trunks from a same stand is similar. This is less obvious at "Bogny-sur-Meuse" where no disc from the upper part of the trunk without heartwood has yet been studied. However, we can expect the same pattern, characterized by sapwood and heartwood in all the diameter classes, and a lower average tree-ring width in sapwood corresponding to the external rings, which is coherent with the growth dynamic of trees [START_REF] Fritts | Tree Rings and Climate[END_REF]. The exploitation of branches only is clearly distinct, with a low growth rate and the absence of heartwood in the case of young branches, as at "Les Cagouillères". If branches are a little older as at "Bogny-sur-Meuse", heartwood is absent in the largest classes of diameter. Lastly, the young vigorous seeded and coppice trees are characterized by a high growth rate in sapwood, while heartwood is absent. Discussion and application to charcoal assemblages The dendrological characteristics of each wood stand, discriminating branches, trunks and whole trees, were defined with the help of the dendro-anthracological tools. The dendroanthracological parameters (growth rate, heartwood/sapwood, diameters) were recorded independently of each other and then combined, forming anthraco-types (Fig. 11). First of all, annual ring width was considered individually. Considering the whole tree and the trunk, ring width distribution is significantly different among stands. However, the distribution between seeded and coppice trees at "Le Bois de l'Or" is not significantly different. Likewise, the distribution between branches at "Les Cagouillères" and "Bogny-sur-Meuse" is not different. In each stand, branches are characterized by a lower growth rate than in the trunk. This observation is in agreement with the study of the variation of annual tree-ring width along the stem marked by a slight increase from the base to the top of the At the scale of a charcoal assemblage, these data can be obtained by measuring each tree ring of each charcoal fragment and averaging them (per fragment). However, their interpretation remains problematic at this stage as they may come from different wood stands, trunks and/or branches, and it is not possible to distinguish them. In addition, it is difficult to interpret growth rate without contemporary, diachronic or modern-day reference standards. The presence or the absence of heartwood and the proportion of sapwood/heartwood are good indicators of the maturity of the wood. In anthracology, sapwood and heartwood can be distinguished using the proportion of vessels sealed by tylosis [START_REF] Dufraisse | Contribution of tyloses quantification in early wood oak vessels to archaeological charcoal analyses: estimation of a minimum age and influences of physiological and environmental factors[END_REF]. Then each fragment can be affiliated to sapwood or heartwood. However, if although the absence of heartwood reflects the exploitation of young trees, it is difficult to interpret sapwood and heartwood proportions as external and internal sapwood are not differentiated. Unburnt wood diameter (UWD) was decomposed using ADmodel. In a charcoal assemblage, charcoal diameters are obtained by measuring the charcoal-pith distance. The results indicate a diameter limit between branches and trunks for each wood stand, with almost no overlap, which is in agreement with the literature [START_REF] Deleuze | Estimer le volume total d'un arbre, quelles que soient l'essence, la taille, la sylviculture[END_REF]. However, the exploitation of whole trees is difficult to distinguish from the exploitation of trunks on account of the low branch volume. Consequently, if we hypothesize the exploitation of whole trees, the proportion of branches will be inconspicuous and difficult to distinguish from the exploitation of trunks. In addition, as regards the exploitation of different wood stands, it is R e v i s e d m a n u s c r i p t problematic to differentiate branches and young trunks solely on the basis of diameter distribution. Thus, growth rate, heartwood/sapwood and wood diameters are three parameters that can be applied to charcoal assemblages. However, their use independently of each other is somewhat limited and sometimes difficult to interpret despite their information potential. A first combination consisted in associating heartwood/sapwood and diameter parameters in order i) to differentiate the two kinds of sapwood: external sapwood in mature woods, and internal sapwood (absence of heartwood) in young woods ii) to improve the interpretation of the distribution of the decomposed UWD. Specific patterns were recorded according to wood stands and the exploitation modes (whole trees, trunks/branches separately). Young woods (trunk or branches) are characterized by absence of heartwood and small diameter classes, whereas mature wood is characterized by heartwood in small diameter classes and sapwood in the largest ones. In the scope of application to charcoal assemblages, this first combination yields four groups of charcoal fragments depending on their position in the wood: i) small diameter associated with sapwood corresponding to young woods, ii) small diameter associated with heartwood corresponding to the internal part of mature woods, iii) large diameter associated with heartwood corresponding to the middle part of mature woods and iv) large diameter associated with sapwood corresponding to the external part of mature woods. The association of growth rates with the sapwood/heartwood ratio can provide information about the vigour of wood stands and tree morphology. For example, the proportion of sapwood is higher in trunks from "Les Cagouillères" (high forest) than in the trunk of the dominant tree at "Bogny-sur-Meuse" (coppice-under-standard). However, average sapwood ring-width and sapwood width are higher at "Bogny-sur-Meuse" than in "Les Cagouillères" (Fig. 7). This observation shows that i) for a same age (Bogny: 68 years R e v i s e d m a n u s c r i p t old, Cagouillères: 62 years old), the most vigorous trees have a more extensive sapwood surface [START_REF] Lebourgeois | Les chênes sessile et pédonculé (Quercus petraea Liebl. et Quercus robur L.) dans le réseau RENECOFOR : rythme de croissance radiale, anatomie du bois, de l'aubier et de l'écorce[END_REF] ii) sapwood width is higher in coppice-under-standard than in high forest [START_REF] Dhôte | Profil de la tige et géométrie de l'aubier chez le Chêne sessile (Quercus petraea Liebl.)[END_REF]. Thus, the under-representation of sapwood in the trunk of the tree in "Bogny-sur-Meuse" is probably due to a larger tree diameter, 33 cm as opposed to 20.75 cm. The third combination consists in associating tree-ring width and diameters (distribution of the decomposed unburnt wood diameters). For an application to charcoal assemblage, each tree ring is associated with a charcoal-pith distance, then to a diameter class and finally an average tree-ring width is calculated for each diameter class. Radial growth trends appear to be preserved keeping with dendrological radial growth. An original pattern marked by a low growth rate along the smallest diameter classes followed by a higher rate in the largest diameter classes may be a characteristic of the exploitation of whole trees. However, as it is often the case in dendroecology, one pattern may correspond to several scenarios. Here for example, a partial clearing of the wood stand could lead to a comparable growth trend. Thus interpretations have to be associated with the results established by other disciplines. In addition, an initial distinction between young trunks (coppice) and young branches becomes possible as their growth rate and growth trend differ (high rate and upward trend for coppice, low rate and downward trend for branches). However, no further distinction is visible between coppice and seeded trees at "Le Bois de l'Or". In fact, only the proportion of earlywood is only significant when radius is up to 1,6 cm [START_REF] Girardclos | Improving identification of coppiced and seeded tress in past woodland management by comparing growth and wood anatomy of living sessile oaks (Quercus petraea)[END_REF]. The last combination is the association of all the dendro-anthracological parameters: heartwood/sapwood, tree-ring width and diameters. Besides the distinction between young and mature woods based on the association between heartwood/sapwood and diameters, it becomes possible to discriminate branches from trunks among young woods. Indeed, branches are characterized by sapwood, a low growth rate and rather downward growth, R e v i s e d m a n u s c r i p t whereas young trunks (coppice and seeded trees) are characterized by sapwood, a high growth rate and rather upward growth. Specific patterns appear depending on the stand and the potential types of wood exploitation (trunks and/or branches). Thus, anthracological types could be defined forming an interpretative grid which can act as a useful key for the interpretation of archaeological charcoal assemblages. Moreover, the recorded dendrological information is not the same depending on the position in the tree. For example, the information recorded in tree-ring width depends on the position of the charcoal fragment; tree-ring width and growth trend in young woods may be a good indicator of the origin of the wood in the tree (crown or bole) whereas stand characteristics (stand density according to strong or low competition between trees) are more perceptible in trunk, i.e. large diameter of mature wood (Marguerie and Hunot, 2007). These results entail a new approach to anthracological material. Charcoal fragments have to be sorted according to their position in the stem cross-section and in the tree. For that purpose, an anthracological key based on dendro-anthracological parameters and forming anthraco-groups is proposed (Fig. 12). Each oak fragment is characterized by a charcoal-pith distance, sapwood/heartwood affiliation and annual tree-ring width. The first division at the threshold of a diameter of 7 cm is often used by foresters and corresponds to the diameter limit between branches and trunks in deciduous oak forest. Concerning tree-ring width, charcoal fragments with regular and irregular tree-ring width series are taken into account separately. For example in northern France, according to V. Bernard (1998, p. 96), narrow rings are less than 0.7 mm/year and large rings are between 0.7 and 3 mm/year for deciduous oak. Very large rings, up to 3 mm, can be also considered (i.e. 12 groups). The use of this anthracological key enables us to sort charcoal fragments according to their position in the tree. Then, measurements of each batch can be processed separately. R e v i s e d m a n u s c r i p t To close, it is important to make several remarks concerning the dendroanthracological tools and their applications. i) The application of dendro-anthracological tools requires a minimum transversal plane size of about 4 mm x 4 mm and at least one whole growthring. The optimal number of fragments to analyze is around 100 per sampling unit (structure, layer, etc. according to the problematic). ii) The choice of the diameter classes, chosen to be compatible with standards used in dendrometrical plans by foresters, seems to be relevant. However, a charcoal fragment can be classified in a class or the other when the value of the charcoal-pith distance is close to a limit but usually the interpretation is not affected. iii) Given that it exists a boundary between the diameters of trunks and branches within a wood stand and that the part of the trunk located in the crown presents the same dendrological characteristics as a branch, it is more relevant and accurate for charcoal analysis to distinguish bole from crown than trunk from branch when considering oak and probably more generally Angiospermea. However, by Gymnospermea, the trunk can be easily followed until the apex with a clear separation of the branch material. Thus this distinction bole/crown or trunk/branch has to be adapted according to the architecture of the tree. In addition, variations in growth rates are often considered and interpreted in terms of environmental (light, soil or climate) and human factors (clearings or woodland management). However, we have to keep in mind that they can also result from a change in exploitation techniques (whole trees, trunks, branches). The use of the anthracological key may allow for the classification of the growth-ring width signal and thus bring more accurate information. R e v i s e d m a n u s c r i p t iv) Shrinkage during charcoalification leads to lower tree-ring width. This process is not consistent, depending on sapwood/heartwood and charcoal-pith distance. A preliminary study on shrinkage offers promising results in order to propose correction factors (Garcia [START_REF] Martinez | Correction factors on archaeological wood diameter estimation[END_REF]. v) The relative frequency of the different taxa in charcoal assemblages is representative of the used biomass (wood volume). In the same way, the use of the dendro-anthracological parameters is based on the assumption that charcoal fragments represent the different parts of trees proportionally to their volume, with their dendrological characteristics (growth, ratio sapwood/heartwood, diameter). That is why the ADmodel is based on wood volume (and not on the number of fragments). However, we stress that, this model cannot reconstruct the quantity of initially burnt wood. vi) As for the interpretation of tree-ring width (Marguerie, 1992, p. 72;Marguerie and Hunot, 2007), several conditions are required to interpret the dendroanthracological parameters: charcoal assemblages must come from numerous trees, tree-ring series are randomly distributed in the transversal sections of charcoal fragments, ring series must be numerous enough and with a homogeneous width, acquisitions areas are subjected to the same climatic influences and the geological substratum must be homogeneous. Conclusion In line with the work of D. Marguerie (Marguerie, 1992, Marguerie and Hunot, 2007[START_REF] Marguerie | Short Tree ring series: the study materials of the dendro-anthracologist[END_REF], Marguerie et al., 2010), combining charcoal identification and dendrological examination, the aim of this study was to improve methods to assess whether it was pertinent to develop quantitative measurements, such as estimating pith-charcoal distance, and whether the combination of dendro-anthracological parameters provides new information on wood exploitation and forest management. R e v i s e d m a n u s c r i p t Besides the measurement of tree-ring width, the present study is based on the development of three anthracological tools consisting in i) measuring charcoal-pith distance, ii) discriminating heartwood/sapwood and iii) modelling dendrological data to make them compatible with charcoal analysis. Three dendro-anthracological parameters i.e. growth ring width, charcoal-pith distances and heartwood/sapwood, modelled with ADmodel, were tested on modern-day oak wood stands chosen with respect to historical woodland practices: a coppice-under-standard, an old coppice undergoing conversion to high forest and a young stand formed by a mixture of seeded and coppice trees. For a more realistic representation of dendrological data according to anthracological constraints, different levels of analysis were considered: the whole tree, and trunks and branches separately, allowing us to further consider various modes of wood exploitation. The dendro-anthracological parameters taken into account independently of each other provide interesting results but rather limited interpretation, especially for tree-ring width or sapwood/heartwood. Indeed the dendrological information cannot be interpreted in the same way depending on its position in the tree. For example, growth conditions and thus paleo-environmental information are essentially recorded in the trunk. In contrast, the combination of the dendro-anthracological parameters highlights specific patterns between organs, stands and regeneration modes, and enables us to establish an anthraco-typology forming an interpretative grid. A major result here is the identification of the position of the charcoal fragment belonging to young woods or internal/middle/external parts of mature woods and the distinction between branches and young trunks when associated with the tree-ring width. These results lead to the establishment of an anthracological key aiming to sort charcoal fragments into anthraco-groups according to their position in the tree and their growth rate. Finally, these results offer new opportunities for the interpretation of archaeological charcoal assemblages as well as the development of new dendro-anthracological tools adapted to species other than deciduous oak. R e v i s e d m a n u s c r i p t also grateful to two anonymous reviewers for their valuable remarks and suggestions which helped to improve this publication. Captions Table 1 Dendrological characteristics of each wood stand and sampled trees. undergoes both mass loss and charcoal fragmentation. Consequently, the distribution of the charcoal-pith distances does not indicate unburnt wood diameter. 3.3. Combination of dendro-anthracological parameters 3.3.1. Decomposed UWD versus tree-ring width R e v i s e d m a n u s c r i p t 3.3.3. Tree-ring width versus diameter classes versus sapwood/heartwood s c r i p t trunk and a strong decreasing in the upper part of the trunk (in the crown). These results are similar to those of[START_REF] Dhôte | Profil de la tige et géométrie de l'aubier chez le Chêne sessile (Quercus petraea Liebl.)[END_REF], based on 82 Quercus petraea distributed in five regions in France. Consequently, growth conditions are mainly recorded in the trunk and branches should be avoided for palaeo-environmental reconstruction. This result fits with the method of D.Marguerie and J.-Y. Hunot (2007) whose the principle is to keep only tree-ring width measurements based on charcoal with large charcoal-pith distance. Fig. 1 . 1 Fig. 1. Dendro-anthracological tools. Fig. 2 . 2 Fig. 2. General analytical protocol developed in the ANR DENDRAC program. Experimental charcoal assemblages are not considered in this paper. Fig. 4 . 4 Fig.4. Main dendrological characteristics of the wood stands: modes of regeneration, average age, average diameter at breast height, relative proportion of trunks and branches (expressed according to volume), distribution of the diameters of trunks and branches (each log and its volume was attributed to an unburnt wood diameter class), average growth rate and growth trends (tree-ring width measurements were taken on each disc at a height of 1.30 m, along 5 radii and averaged). Fig. 5 . 5 Fig. 5. Annual ring-width was averaged from 5 radii in each disc. (a) Distribution of annual ring-width (maximum and minimum values, 1st and 3rd quartiles and median) considering whole trees (white), trunks (brown) and branches (green). (b) Distribution of the annual ringwidth along the trunks and in branches. Fig. 6 . 6 Fig.6. Relative proportion of heartwood (brown) and sapwood (yellow) for each stand, considering whole trees, trunks and branches. The volume proportion of sapwood and heartwood was estimated for each disc, then each log and tree, and averaged for each wood stand. The average number of sapwood tree-rings, average sapwood width (cm) and average sapwood growth rate (mm/year) are indicated in boxes when heartwood is present. Fig. 8 . 8 Fig. 8. Dendrological data (simple line): average tree-ring width calculated by cambial age. Modelled anthracological data (solid line): average tree-ring width calculated for each diameter class. Fig. 10 . 10 Fig. 10. Average tree-ring width according to the diameter classes (decomposed UWD) and their respective affiliation to sapwood (yellow) or heartwood (brown). Fig. 11 .Fig. 12 . 1112 Fig. 11. Anthraco-typology for deciduous oak: an interpretative grid for archaeological charcoal assemblages. Table 2a 2a Analyzed wood discs and dendrological characteristics: Bogny-sur-Meuse; Les Cagouillères Table2bAnalyzed wood discs and dendrological characteristics: Bois de l'Or. Acknowledgment The authors thank the Agence Nationale de la Recherche (ANR JCJC 200101 DENDRAC, dir. A. Dufraisse) for financing this study and Louise Byrne for English correction. They are

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2011

https://hal.univ-lorraine.fr/tel-01746228/file/SCD_T_2011_0061_WANG.pdf

Keywords: Tetramethoxysilane APTES: Aminopropyltriethoxysilane GPS: 3-Glycidoxypropyl-trimethoxysilane PDDA: Poly(dimethyldiallylammonium chloride) PAA: Poly (allylamine Ferrocene functionalized organoalkoxysilane MWCNTs: Multiwalled carbon nanotubes Abbreviations 2 SWCNTs: Single-walled carbon nanotubes Cyclic voltammetry UV: UV-vis spectrophotometry IR: infrared spectrophotometry GPES: General Purpose Electrochemical System SECM: Scanning electrochemical microscopy SEM: Scanning electron microscopy Silice, sol-gel, déshydrogénase, cofacteur NAD + /NADH, médiateur, polyélectrolyte, bioencapsulation, électrogénération, film, électrodes macroporeuses, nanotubes de carbone DSDH: D-sorbitol dehydrogenase Preface Considerable interests have been drawn in the development of electrochemical reactors for the manufacture of fine chemicals with dehydrogenases as a process with almost zero waste emission. The system requires that all active compounds like cofactor, mediator and dehydrogenase can be functionally immobilized on the working electrode surface in such a way that dehydrogenases are durably immobilized and active, their cofactor is durably immobilized close to the enzyme, and the mediator can reduce the overpotential without leaching. However, it is still a challenge to construct this kind of functional layer with long term stability. This is the goal of the present thesis prepared in the frame of an European program (ERUDESP, enantioselective……). Stable immobilization of active enzyme on electrodes is a prerequisite for bio-electrochemical applications. Sol-gel-derived silica-based materials have proven to provide a rather suitable environment for biomolecules entrapment, ensuring conformation changes similar to their water solution and even enhanced stability. The quite recent development of the electrochemicallyassisted deposition of sol-gel silica thin films opens new opportunities for enzyme encapsulation into sol-gel thin film onto electrode surfaces [1]. In this method, a sufficiently negative potential was applied to the electrode surface to generate hydroxyl ions, which play the role of the catalyst for the polycondensation [2]. In comparison with the traditional sol-gel methods which involve deposition by evaporation (dip-, spin-, spray-coating) and can only be applied to basically flate surfaces, the combination of electrochemistry with the sol-gel process makes possible the well controlled modification of complex electrode surfaces, for example, macroporous gold electrodes. This approach is promising for the electrochemical biocatalysis application. Although there have been a long history of studying and using sol-gel material to encapsulate enzymes, little has been reported regarding the encapsulation of enzyme in sol-gel silica films in the course of their electrodeposition, especially for dehydrogenase encapsulation. The realization of cofactor regeneration with both the dehydrogenase and its cofactor immobilized is the biggest challenge in construction of the functional layer. One obstacle for this approach is that the water soluble cofactor is a relatively small molecule, so it is likely to diffuse away from the electrode surface into solution, thus limiting the long-time durability of the Preface 4 modified electrode. Another obstacle is that high overpotential at electrode surfaces leading to undesired side reactions producing enzymatically-inactive dimers and isomers of cofactors. As far as biocatalysis with electrochemical regeneration of cofactor is concerned, the effectiveness of electron transfer is a key parameter affecting the performance of the process so that the resort to charge transfer mediators has often been proposed to improve the turnover of the overall reaction. However, further difficulties may arise with many mediators due to their poor stability or due to electrode modification procedures. Taking into account the above problems that are likely to exist in the construction of such functional films, this research developed a series of sol-gel immobilization matrixes to improve the perfomance and long-term stability of the biocomposite films. The details are summarized as follows: In Chapter I, the first part is devoted to a brief introduction to the ERUDESP project and to a definition of the subject. Then, we describe the main immobilization methods of enzyme, cofactor and mediator reported sofar in the literature. In the experimental section (Chapter II), we describe the physico-chemical properties of the studied compounds, various sol-gel preparation methods, electrode modifications and the experimental techniques used in this work. In Chapter III, we show the feasibility of dehydrogenase encapsulation in sol-gel matrices by both drop-coating and electrodeposition. DSDH was found to be very sensitive to the silica gel environment, first, the influence of polyelectrolyte additives on the sol-gel encapsulation of dehydrogenases has been evaluated by drop-coating. Then, we report that the electrochemicallyassisted deposition of silica thin films can be a good strategy for DSDH immobilization as well as DSDH and diaphorase co-immobilization in an active form (Diaphorase is an useful enzyme to catalyze cofactor regeneration in a smooth way). In Chapter IV, one has compared various strategies for cofactor immobilization in sol-gel matrices, i.e. the simple encapsulation of the native cofactor, the encapsulation via attachment to a high molecular weight compound (NAD-Dextran), the adsorption on carbon nanotubes introduced in the sol-gel matrix and finally the use of glycidoxypropylsilane (GPS) as additive. In Chapter V, the immobilization of mediators (ferrocene species and osmium polymer) in the sol-gel matrix is first studied. The influence of GPS as additives for the mediator immobilization is also presented. Then, the feasibility of co-immobilization base on sol-gel film is evaluated by one step drop-coating and electrodeposition. In Chapter VI, different strategies for mediator immobilization on CNTs are developed. Here, the co-immobilization strategies base on CNTs/sol-gel matrix are used to develop a reagent free device because of the problems encountered with mediator immobilization through one step electrodepostion in chapter V. The first layer of CNTs functionalized with mediator is covered with an additional drop-coated or electrodeposited sol-gel layer containing the dehydrogenase (and eventually diaphorase) and the cofactor covalently bond to GPS. Chapitre I. Introduction Ce chapitre introductif présente tout d'abord le projet ERUDESP dans le cadre duquel s'est déroulé ce travail de thèse. L'objectif de cette étude était la co-immobilisation au sein d'une couche mince sol-gel d'une déshydrogénase, du cofacteur enzymatique NAD + et du médiateur électrochimique permettant de catalyser la régénération électrochimique de ce cofacteur. Cette couche mince devant ensuite être déposée sur la surface interne d'une électrode d'or macroporeuse et être intégrée dans le réacteur pour application en électrosynthèse enzymatique (ces derniers travaux étant menés dans le projet ERUDESP, mais hors du cadre de cette thèse). L'état de l'art sur les différents aspects de ce projet est ensuite donné (chap. I). Les méthodes couramment utilisée pour obtenir l'immobilisation d'une protéine sous une forme active à la surface d'une électrode sont présentés. Le procédé sol-gel et son application en bioencapsulation sont décrits. Enfin la génération de couche minces sol-gel par assistance électrochimique est présentée ainsi que son utilisation pour l'immobilisation de protéines. Nous discutons ensuite des difficultés et des besoins concernant l'immobilisation du cofacteur enzymatique NAD + et de sa régénération électrochimique à l'aide de médiateurs électrochimiques dans le cadre particuliers de l'électrosynthèse enzymatique. Les méthodes conventionnelles pour la régénération du cofacteur sont alors présentées. Finalement, une revue des travaux existant sur la co-immobilisation de déshydrogénases, du cofacteur enzymatique et de médiateurs est donnée. La présentation de cette étude expérimentale a été organisée en différents chapitres décrivant les étapes successives de cette recherche. L'encapsulation de déshydrogénase au sein de la matrice sol-gel a tout d'abord été décrite, par les méthodes de drop-coating et d'électrogénération. Bien que l'encapsulation de protéines, voire de déshydrogénase au sein de couches minces sol-gel ait été décrite, il est vite apparu que la matrice de silice perturbait fortement l'activité enzymatique de la protéine. L'addition de charges positive au sein du matériau sol-gel, par introduction de polyélectrolyte chargés positivement dans le sol, a alors permis l'encapsulation de déshydrogénases sous leur forme active. Le matériau sol-gel peut alors être déposé sur l'électrode sous forme de couche mince par évaporation ou par électrogénération (chap. III). La co-immobilisation de la déshydrogénase avec une diaphorase a également été étudiée. La diaphorase catalyse alors l'oxydation de NADH en présence d'un médiateur électrochimique en solution, par exemple le ferrocenedimethanol. La déshydrogénase catalyse les réactions d'oxydation ou de réduction du substrat enzymatique en présence du cofacteur NAD + /NADH. Contrairement à d'autres cofacteurs enzymatiques qui sont liés à la protéine (par exemple FAD), NAD + est libre de diffusé en solution. Il y a alors un grand intérêt à immobiliser cette molécule pour limiter le coût du procédé (en diminuant la quantité de molécules utilisées). Différentes stratégies d'immobilisation ont été comparés dans le chapitre IV, la simple encapsulation dans le sol-gel, l'adsorption sur des nanotubes de carbone immobilisés, l'encapsulation du NAD + chimiquement lié à la macromolécule dextran pour limiter sa diffusion et enfin la condensation avec le groupement epoxy du glycidoxypropylsilane (GPS). Ce dernier système a ensuite été utilisé pour élaborer une couche mince sol-gel dans laquelle sont coimmobilisées la déshydrogénase (et éventuellement la diaphorase), le cofacteur enzymatique NAD + et le médiateur électrochimique. Plusieurs stratégies ont été mises en oeuvre, en incorporant le médiateur dans la matrice sol-gel (Chap. V) ou en utilisant des nanotubes de carbone fonctionnalisés (Chap. VI). Une attention particulière a été donnée à la préparation de la couche mince sol-gel par électrogénération. Nous verrons ainsi que si de nombreux systèmes peuvent fonctionner lorsqu'ils sont préparés par évaporation, il est beaucoup plus difficile d'atteindre la co-immobilisation de tous les éléments autorisant leur communication lorsque le matériau est généré électrochimiquement. Bien que la plupart des travaux aient été menés pour obtenir un système bioélectrocatalytique fonctionnant en oxydation, nous avons également étudié l'immobilisation sur nanotubes de carbone d'un médiateur électrochimique à base de rhodium permettant de catalyser la réduction de NAD + . Chapter I. Introduction In the chapter, we start by the introduction of the ERUDESP project and the definition of the PhD subject. We then discuss the immobilization of enzymes on electrode surfaces. We recall some of the conventional enzyme immobilization methods, in particular, we introduce the sol-gel materials for enzyme encapsulation and electrochemically-assisted generation of silica films for bioencapsulation. After that, we also discuss the need for cofactor immobilization and regeneration as well as the immobilization of charge transfer catalyst on electrode surfaces. We explain the necessity and difficulties of cofactor immobilization and regeneration, and introduce the conventional methods to modify the electrode and to regenerate the cofactor. At last, we describe the functional films for co-immobilization of dehydrogenase, cofactor and mediator, and review the work of the literature dealing with such reagentless system. ERUDESP project The project full title is "Development of Electrochemical Reactors Using Dehydrogenases for Enantiopure Synthon Preparations" (ERUDESP website: http://www.erudesp.eu). The main objective of the project is the development of electrochemical reactors for the manufacture of fine chemicals with dehydrogenases as a process with almost zero waste. The production of enantiopure compounds with high enantiomeric excess (EE) can be achieved by using dehydrogenases as biocatalysts, because they express high enantioselectivity in ketone reduction combined with broad substrate spectra by some of these enzymes. As these dehydrogenases typically require co-substrate regeneration by aid of a second enzymatic reaction, we are looking for alternative solutions for cofactor regeneration to avoid the contamination of the reaction fluid by other proteins and chemical compounds. In this project we will use an electrochemical approach for the regeneration of cofactors. Hydrogen gas is oxidized into a mixture of hydrogen ions and electrons on the anode side of the cell, then, hydrogen ions diffusion through a nafion membrane to the cathode side of the cell. If all active compounds like the mediator, the cofactor and the dehydrogenase can be functionally immobilized on the working electrode surface (on the cathode of the cell), educt in the input flow will be reduced into the product in the output flow avoiding any contaminations. The interesting thing is that the same electrochemical reactor can also be used for oxidation reaction. In this case, all active compounds will be immobilized on the anode side of the cell. Oxygen continuously passes over the cathode, which react with the electrons and protons (coming from the oxidation reaction at the anode side of the cell) to form water. As counter electrode a gas diffusion electrode will be employed; it delivers clean protons (no liquid anolyte!) to the catholyte and simultaneously reduces the cell voltage by about 1 V; hence undesired side reactions/degradation processes will be suppressed and thus the longterm stability of the whole electroenzymatic system will be improved. There are six participants in this project. Participant 1 (Saarland University, Germany) In this project, we are Participant 3, my PhD research was focused on designing functional layers based on silica sol-gel thin films to co-immobilize enzyme, cofactor and electron mediator to get the active systems and such modifications of electrode surfaces should be adaptable to the macroporous electrodes. Initially, the reduction of prochiral ketones to enantiopure hydroxylated products was the most desirable reaction. However, a major obstacle was encountered with the loss of reduction mediator activity upon immobilization on electrode surface. The analyses of the actual market by the industrial partner demonstrated that the production of rare sugars was also interesting. Therefore, the oxidation of sugar alcohols to rare sugars with electrochemical cofactor oxidation has also come into focus of the project. includes Immobilization of enzymes on electrode surfaces The development of a simple and effective strategy for immobilizing enzymes on or into an electrode is a crucial step in the design and fabrication of electrochemical biosensors, bioreactors or biofuel cells. Ideally, this immobilization should be totally irreversible and stable with time, without deactivation of the biomolecule, while managing excellent accessibility to this biomolecule and ensuring a certain conformational mobility [ 1 ]. Adsorption [ 2 , 3 ], covalent grafting [ 4 , 5 ], and entrapment [ 6 , 7 ] are conventional immobilization methods. Adsorption is a rather soft and the simplest immobilization method, but the disadvantage of the method is the low mechanical strength of the assembly. Desorption phenomena are regularly observed when environmental conditions (pH, ionic strength) change. Covalent grafting is a chemical immobilization technique consisting of creating a covalent bond between the sensitive element and the support. This technique permits the orientation of the grafted molecules and thus optimization of the recognition probability. However, deactivation and the irreversible immobilization of the enzyme components restrict the performance of these types of enzyme immobilization. Entrapment of the receptor in a host matrix can be done by simply mixing the various components and depositing the mixture onto a suitable support. This method is the most widespread. A large diversity of materials is thus used: inorganic materials, natural or synthetic organic polymers. However, steric stresses and interactions with the matrix may denature the species. Also diffusional limitations may occur when the receptor is not sufficiently accessible. Some other immobilized schemes and advanced materials that can improve the analytical capacities of sensor devices are highly desired. The last decade has seen a revolution in the area of sol-gel-derived biomaterials since the demonstration that these materials can be used to encapsulate biological species such as enzymes, antibodies and other proteins in a functional state [8]. Upon encapsulation, the biomolecules retain their spectroscopic properties and biological activity [9,10]. Silica Solgel could offer some advantages such as improved mechanical strength and chemical stability. It does not swell in aqueous or organic solvents, preventing leaching of entrapped biomolecules. Silica sol-gel materials are particularly interesting, because they can be synthesized with a large variety of organic functionalities, such as hydrophobic or hydrophilic ones [ 11 , 12 , 13 ]. They can be used to retain metallic complexes (e.g., mediators) by covalently grafting, electrostatic pH dependent interaction or simply as adsorption by the intermediate of weak physical bonds [11]. The trapped biomolecules reside in an interconnected mesoporous network and become part of the entire material [13], and they usually exhibit better activity and longer life times than free enzymes. During encapsulation, they remain trapped within a silica cage tailored to their size, and it provides a chemical surrounding that favours the activity (Figure I-2)). Sol-gel immobilization is characterized by physical entrapment without chemical modification. This approach also permits the biomolecules to be isolated and stabilized against aggressive chemical and thermal environments [10,13]. While the relatively large biomolecules are immobilized within the silica network, small ions or molecules can be easily transported into the interior of the matrix, which has been largely exploited in the field of biosensors [9,14,15]. Figure I-2. The enzymes entrapped in sol-gel matrices [8]. [16] A sol is a stable dispersion of colloidal particles or polymers in a liquid. Colloids are solid particles with diameters of 1-100 nm. A gel is an interconnected, rigid network with pores of submicrometer dimensions and polymeric chains whose average length is greater than a micrometer. A silica gel may be obtained by formation of an interconnected 3-D network by the simultaneous hydrolysis and polycondensation of a precursor. When the pore liquid is removed as a gas phase from the interconnected solid gel network under hypercritical conditions, the network does not collapse and a low density aerogel is produced. The sol-gel process generally involves the use of alkoxide precursors, which undergo hydrolysis, condensation, aging and drying to give gels or xerogel. Sol-gel process Hydrolysis The preparation of a silica glass begins with an appropriate alkoxide, such as Si(OR) 4 , where R is mainly CH 3 , C 2 H 5 , or C 3 H 7 , which is mixed with water and a mutual solvent to form a solution. Hydrolysis leads to the formation of silanol groups (SiOH). It has been well established that the presence of H 3 O + in the solution increases the rate of the hydrolysis reaction. Condensation In a condensation reaction, two partially hydrolyzed molecules can link together through forming siloxane bonds (Si-O-Si). This type of reaction can continue to build larger and larger silicon-containing molecules (linkage of additional Si-OH) and eventually results in a SiO 2 network. The H 2 O (or alcohol) expelled from the reaction remains in the pores of the network. When sufficient interconnected Si-O-Si bonds are formed in a region, they respond cooperatively as colloidal (submicrometer) particles or a sol. The gel morphology is influenced by temperature, the concentrations of each species (attention focuses on R ratio, R = [H 2 O]/[Si(OR) 4 ), and especially acidity: • Acid catalysis generally produces weakly-crosslinked gels which easily compact under drying conditions, yielding low-porosity microporous (smaller than 2 nm) xerogel structures • Under some conditions, base-catalyzed and two-step acid-base catalyzed gels (initial polymerization under acidic conditions and further gelation under basic conditions [17,18]) exhibit hierarchical structure and complex network topology (Figure I-3c). Aging Gel aging is an extension of the gelation step in which the gel network is reinforced through further polymerization, possibly at different temperature and solvent conditions. During aging, polycondensation continues along with localized solution and reprecipitation of the gel network, which increases the thickness of interparticle necks and decreases the porosity. The strength of the gel thereby increases with aging. An aged gel must develop sufficient strength to resist cracking during drying. Drying The gel drying process consists in removal of water from the interconnected pore network, with simultaneous collapse of the gel structure, under conditions of constant temperature, pressure, and humidity. Large capillary stresses can develop during drying when the pores are small (<20 nm). These stresses will cause the gels to crack catastrophically unless the drying process is controlled by decreasing the liquid surface energy by addition of surfactants or elimination of very small pores, by hypercritical evaporation, which avoids the solid-liquid interface, or by obtaining monodisperse pore sizes by controlling the rates of hydrolysis and condensation. Optimization of the sol-gel process for bioencapsulation Sol-gel is known to be a suitable matrix for bioencapsulation. However, sol-gel conditions are sometimes not mild enough for biomolecules. Recent efforts have been made to optimize the process by controlling the porosity of bioencapsulates and the chemical environment of trapped species [8]. This notably involved the use of biocompatible silane precursors, sugars and amino acid N-methylglycine or polymers. Biocompatible silane precursors The release of alcohol during the hydrolysis-condensation of silicon alkoxides has been considered an obstacle, due to its potential denaturing activity on the entrapped biological moiety. Biocompatible silane precursors have to be used for avoiding enzyme denaturation due to alcohol release. TMOS, Si(OMe) 4 , is therefore currently used instead of TEOS, Si(OEt) 4 , as methanol is less harmful than ethanol. However some enzymes are specially sensitive to traces of alcohol so that the usual two-step alkoxide route has to be modified. One way to overcome this drawback would be to remove the alcohol via evaporation under vacuum in order to get a fully hydrolyzed solution before adding enzymes [20]. Aqueous solgel processes have been developed in order to avoid any trace of alcohol, for example a sodium silicate solution [21], or a mixture of sodium silicate and Ludox suspension [22]. Another way to avoid denaturation by alcohol is to use biocompatible alcohols such as polyolbased silanes as hydrolyzable groups that can be hydrolyzed under mild pH conditions [23]. A biocompatible reagent, glycerol, is produced so that the catalytic efficiency and long-term stability of enzymes are significantly improved [24]. Sugar and amino acid N-methylglycine additives Sugar and amino acid N-methylglycine additives can be used to stabilize enzymes within sol-gel matrices. Chymotrypsin and ribonuclease T1 have been trapped in the presence of sorbitol and N-methylglycine. Both osmolytes significantly increase the thermal stability and biological activity of the proteins by altering their hydration and increasing the pore size of the silica matrix [25]. D-glucolactone and D-maltolactone have also been covalently bound via a coupling reagent aminopropyltriethoxysilane (APTES) to the silica network giving nonhydrolyzable sugar moieties. Firefly Luciferase, trapped in such matrices, has been used for the ultra sensitive detection of ATP via bioluminescent reactions [26]. Polymer additives The silica matrix forms around the trapped biomolecule, but some shrinkage always occurs during the condensation process and the drying of the gel. Stresses can then lead to some partial denaturation of the enzymes. Polymers have been used as additives to form hybrid organic-inorganic gels in order to reduce shrinkage via a 'pore filling' effect. Some polymer additives such as polyethylene glycol (PEG) or polyvinyl alcohol (PVA) have been introduce in sol-gel matrix to increase the catalytic activity of entrapped enzyme [27,28,29]. Electrostatic interactions may also occur between silicate sites and specific residues on the protein surface. Silica surfaces are negatively charged above the point of zero charge (pH =3) and electrostatic interactions mainly depend on the isoelectric point of the protein. Sometimes, electrostatic interactions decrease the catalytic activity of the enzyme. However, the detrimental effects of these electrostatic interactions can be reduced by complexing the enzyme with a polyelectrolyte that shields the critical charged sites [30,31]. Electrochemically-assisted generation of silica films Principle and significant Usually, sol-gel silica films are prepared by polycondensation of silicon alkoxides (alone or in the presence of organosilanes) as induced by evaporation of a sol solution via spincoating, dip-coating or spraying. When the sol is doped with biomolecules, the silica framework is formed around them, leading to the desired composite films containing physically encapsulated biomolecules [ 32 ]. Such "conventional" sol-gel film formation procedures are however restricted to flat surfaces, which are unsuited to porous substrates. At the end of the nineties, a novel method to prepare silica-based thin films on electrode surfaces has been proposed by Shacham et al [ 33 ], involving an elegant combination of electrochemistry with the sol-gel process. The principle (Figure I-4) is based on the electrochemical manipulation of pH on the electrode surface affecting thereby the kinetics associated to the sol-gel process. The electrode is immersed in a stable silica sol (mild acidic medium: pH 3-4) where a negative potential is applied to increase the pH locally at the electrode/solution interface, inducing therefore polycondensation of the silica precursors only on the electrode surface which makes the process applicable to deposit thin films on nonplanar surfaces [34]. The electrochemically-assisted generation of sol-gel thin films have been applied to produce porous silica deposits [35,36], zirconia or titania thin films [37,38], as well as protective layers against corrosion [ 39 ]. The versatility of this novel process was also exploited to produce functionalized silica coatings [40] or molecularly imprinted silica films [41], which can be used for electrochemical sensing purposes [41]. It is noteworthy beyond these considerations that such electrochemically-assisted deposition can be advantageously combined with the surfactant templating process to generate highly ordered mesoporous silica films with unique mesopore orientation normal to the underlying support [42,43] and this was also exploited to prepare vertically aligned silica mesochannels bearing organo-functional groups [44,45]. Application in bioencapsulation The developed electrochemically-assisted generation of sol gel thin films is interesting for biomolecules immobilization. Figure I-5 shows the process of protein enscapsulated in thin sol-gel film through electrodeposition. In the presence of a biomolecule in the starting sol, this electrodeposition process leads to its encapsulation within the thin film, after washing and drying, a thin sol-gel film containing biomolecule modified electrode is obtained. The possible application was shown recently by some research groups [46,47,48]. Nadzhafova, et al. point out that electrogeneration of silica gel (SG) films on glassy carbon electrodes (GCEs) can be applied to immobilize biomolecules -hemoglobin (Hb) or glucose oxidase (GOD) or both of them in mixture [46]. After encapsulation, Hb was found to keep its peroxidase properties and GOD its enzymatic activity, and the biomolecules are very close to the electrode surface as direct electrochemistry has been pointed out. At the same time, Xia, et al. also present a strategy for the one-step immobilization of GOD in a 3D porous silica matrix using an electrochemically promoted sol gel process and bubble template [47], and the formed silica structure was proven to reduce mass transport resistance greatly because the porous structure existed. Tian, et al. report the development of microelectrode biosensors based on a Ruthenium Purple (RP)-coated gold electrode [48]. A desired gel layer is formed on the RP modified electrode using controllable sol-gel deposition technique to fabricate ATP and hypoxanthine amperometric biosensors. The formation of the gel film is not affected by the inner RP layer and the bioactivity of enzymes entrapped in gel film is well retained. Despite these different examples, the application of electrochemically-assisted generation of sol gel thin films in bioencapsulation has been little exploited. To date, and to our knowledge, no attempt has been made to use this synthetic approach to entrap dehydrogenase in sol-gel silica films in the course of their electrodeposition, moreover, such modifications has not been applied to non-planar supports to functionalize the internal surfaces of macroporous (pore size, 0.5 -1 µm) gold electrodes. Glassy carbon Dehydrogenase-based electrochemical reactor: the need for immobilization and cofactor regeneration Enzymes and cofactor Electroenzymatic reaction can provide a promising process to synthesize chiral compounds. Furthermore, in most cases no side reactions occur and therefore downstream processing can be simplified. It is reported that there are more than 250 NAD-dependent dehydrogenases and 150 NADP dependent dehydrogenases which can catalyze the oxidation/reduction of a variety of substrates. The advantage of dehydrogenase enzymes over the oxidases is that they are oxygen-independent, more abundant, and more substrate specific. [49]. The dehydrogenase needs a small molecule, called a cofactor, in order to be active [50]. The cofactor acts an acceptor or donor of small groups or atoms or electrons and provides the driving force for the oxidation or reduction of the substrate. However, the applications of dehydrogenase in electrosynthesis and biosensor are not yet a success story. This can be attributed to some restrictions in the use of dehydrogenases. The primary limitation is that, in contrast to oxidases, which have redox cofactors tightly bound within their molecules, the NAD(P) cofactor is not bound to dehydrogenase molecules. Therefore, unlike the case of oxidase-based biosensors, dehydrogenase-based systems cannot be readily incorporated into reagentless devices. The latter systems require that both a dehydrogenase and its cofactor are immobilized in such a way that a cofactor has an easy access to the enzyme. Another limitation in using dehydrogenases is the fact that their cofactors are recycled only at high potentials at most electrodes, which leads to interferences from redox active species and usually yield enzymatically inactive NAD-dimers [49]. Cofactor immobilization Physical entrapment The easiest solution to immobilize the cofactor is to bulk-modify carbon paste like materials and this approach is quickly adopted for the majority of studies involving NAD-dependent dehydrogenase electrode. In this case the cofactor is mixed with carbon power and a binder (typically paraffin oil) before being packed into a cavity to form the electrode [51]. Noguer et al. designed amperometric acetaldehyde biosensor based on sol-gel immobilization of aldehyde dehydrogenase (AlDH) and NAD + on screen-printed electrodes [52]. However, the simple encapsulation displays a major drawback, as it leads to rapid leaching of the cofactor in the solution during the electrochemical operation. Covalent bonding One possibility to improve the stability of the immobilization is the chemical attachment of the cofactor to a macromolecule that can be encapsulated or immobilized on the electrode without leaching. Mobility of the cofactor is vital for its efficient interaction with enzymes. The spacer is usually linked to the adenine moieties of the NAD(P) + molecule, and should provide some flexibility for the bioactive part of the cofactors, allowing them to be associated with the enzyme molecules. One approach of cofactor immobilization involves the covalent coupling of the NAD(P) + derivatives to some water-soluble polymers, dextran [START_REF] Leca | Reusable ethanol sensor based on a NAD + -dependent dehydrogenase without coenzyme addition[END_REF][START_REF] Leca | Reagentless ethanol sensor based on a NAD-dependent dehydrogenase[END_REF][START_REF] Montagnk | Bi-enzyme amperometric D-lactate sensor using macromolecular NAD +[END_REF], poly(ethylenimine) (PEI) [START_REF] Zheng | Electrical communication between electrode and dehydrogenase by a ferrocene-labeled high molecular-weight cofactor derivative: application to a reagentless biosensor[END_REF], poly(ethyleneglycol) (PEG) [START_REF] Mak | An amperometric bienzyme sensor for determination of formate using cofactor regeneration[END_REF] or chitosan [49]. For example, a reagentless ethanol biosensor was developed by incorporating alcohol dehydrogenase, NADH-oxidase and NAD + -dextran in a poly(vinylalcohol) (PVA) matrix on an electrode surface [START_REF] Leca | Reagentless ethanol sensor based on a NAD-dependent dehydrogenase[END_REF]. Zheng et al [START_REF] Zheng | Electrical communication between electrode and dehydrogenase by a ferrocene-labeled high molecular-weight cofactor derivative: application to a reagentless biosensor[END_REF] explored a reagentless biosensor by using a ferrecene-labled high meolecular weight cofactor derivative (PEI-Fc-NAD + ). Mak et al [START_REF] Mak | An amperometric bienzyme sensor for determination of formate using cofactor regeneration[END_REF] presents an amperometric formate biosensor using FDH and SHL coimmobilized with PEG-NAD + (MW=20 000) in a photopolymerized PVA matrix in front of a Clark-electrode. Zhang et al [49] developed an electrochemical biosensor through the co-immobilization of glucose dehydrogenase (GDH) and its cofactor NAD + on the scaffolds of a biopolymer chitosan, in this method, the dehydrogenase and the cofactor are covalently attached to the polymeric chains on the surface of the electrode. This electrode displays a good operational stability during continuous 25-h long experiments. Of related interest is the immobilization of the cofactor on particles. Liu et al. reported that the cofactor NAD + covalently attached to silica nanoparticles could be successfully coordinated with particle-immobilized enzymes and enabled multistep biotransformations [START_REF] Liu | Nanoparticle-supported multi-enzyme biocatalysis with in situ cofactor regeneration[END_REF]. In this method, NAD + was immobilized onto the silica particles by forming covalent bonds through the epoxide group on the surface of Glycidoxypropyl trimethoxysilane (GPS) functionalized silica nanoparticles. One limitation of covalent coupling comes from the rather complex modification, which results in a substantial decrease of cofactor efficiency. However, the reaction between epoxide group and the adenine moieties of the cofactor is rather simple, which allows good activity of cofactor to be detected with high stability. In the present work, we will consider GPS for direct attachment of NAD + in sol-gel thin film. π-π stacking The soft immobilization of NAD + on carbon nanotubes by π-π-stacking is also a promising avenue to avoid decrease of their efficiency. Zhou et al [START_REF] Zhou | Noncovalent attachment of NAD + cofactor onto carbon nanotubes for preparation of integrated dehydrogenase-based electrochemical biosensors[END_REF] describes a facile approach to the preparation of integrated dehydrogenase-based electrochemical biosensors through noncovalent attachment of NAD + onto carbon nanotubes with the interaction between the adenine subunit in NAD + molecules and multiwalled carbon nanotubes (MWCNTs). Compared with the existing methods for surface confinement of NAD + cofactor, this method is simple and is thus envisaged to be useful for general development of integrated dehydrogenase-based bioelectrochemical devices. To date, however, this method was not estimated with the use of additional electron transfer mediator. Electrochemical cofactor regeneration The electrochemistry of NAD(P) + /NAD(P)H is highly irreversible, and the oxidation of NADH or the reduction of NAD + at bare electrode only occurs at high overpotential. Several efficient methods have been developed for cofactor regeneration. Syntheses where the cofactor NAD(P) + / NAD(P)H has to be regenerated to its oxidized/reduced state can be carried out with direct, indirect and enzyme-coupled electrochemical cofactor regeneration Indirect electrochemical regeneration Indirect electrochemical regeneration is widely used to overcome the problems of the high overpotential and the formation of inactive NAD-dimers. A mediator meeting the following criteria has to be applied [START_REF] Steckhan | Electroenzymic synthesis[END_REF]: (1) The mediator must be stable. (2) The electrochemical activation of the mediator must be possible at suitable potentials. (3) The mediator must not transfer the electrons to the substrate. (4) Only enzymatically active cofactor must be formed. NAD(P) + -dependent oxidation reactions For the efficient electrochemical oxidation of NAD(P)H, mediated electrocatalysis is necessary, and a wide range of mediators has been studied. Organic compounds that undergo two-electron reduction-oxidation processes and also function as proton acceptors-donors upon their redox transformations have been found to be ideal for the mediation of NAD(P)H oxidation. Some redox mediators such as quinones [ 62 , 63 , 64 ], diamines [ 65 , 66 ] and phenazine and phenoxazine dyes [START_REF] Wu | Patternable Artificial Flavin: Phenazine, Phenothiazine, and Phenoxazine[END_REF][START_REF] Lawrence | Chemical adsorption of phenothiazine dyes onto carbon nanotubes: Toward the low potential detection of NADH[END_REF][START_REF] Zhang | Electrochemical sensing platform based on the carbon nanotubes/redox mediators-biopolymer system[END_REF] have been used to recycle the NADH back to the enzymatically active NAD + . The activity of these mediators toward the NADH oxidation has been explained in terms of a hydride transfer mechanism in which the mediator accepts the hydride and has the ability to delocalize the electrons. The oxidation of NADH has also been investigated using ferrocene derivatives [START_REF] Kwon | An electrochemical immunosensor using paminophenol redox cycling by NADH on a self-assembled monolayer and ferrocenemodified Au electrodes[END_REF], transition-metal complexes [START_REF] Wu | Electrocatalytic Oxidation of NADH at Glassy Carbon Electrodes Modified with Transition Metal Complexes Containing 1,10-Phenanthroline-5,6-dione Ligands[END_REF], conductive polymers [START_REF] Manesh | Electrocatalytic oxidation of NADH at gold nanoparticles loaded poly (3,4-ethylenedioxythiophene)-poly(styrene sulfonic acid) film modified electrode and integration of alcohol dehydrogenase for alcohol sensing[END_REF][START_REF] Valentini | Chemical reversibility and stable lowpotential NADH detection with nonconventional conducting polymer nanotubule modified glassy carbon electrodes[END_REF], nitrofluorenone derivatives [START_REF] Munteanu | NADH electrooxidation using carbon paste electrodes modified with nitro-fluorenone derivatives immobilized on zirconium phosphate[END_REF][START_REF] Mano | Electrodes modified with nitrofluorenone derivatives as a basis for new biosensors[END_REF], dichloroindophenol [START_REF] Forrow | Development of a commercial amperometric biosensor electrode for the ketone D-3-hydroxybutyrate[END_REF], and tetracyanoquinodimethane-tetrathiafulvalene [START_REF] Pandey | Ethanol biosensors and electrochemical oxidation of NADH[END_REF]. Some compounds demonstrate very high rates for the mediated oxidation of NAD(P)H in aqueous solutions. However, such mediator-based electrodes have displayed intrinsic difficulties, which were related to the limited stability of mediators and their leaching from the electrodes. Another approach to the facilitated oxidation of NADH included the use of electrodes based on different forms of carbon, e.g., carbon nanotubes (CNTs) [START_REF] Zhou | The characteristics of highly ordered mesoporous carbons as electrode material for electrochemical sensing as compared with carbon nanotubes[END_REF][START_REF] Wang | Rapidly Functionalized, Water-Dispersed Carbon Nanotubes at High Concentration[END_REF][START_REF] Musameh | Electrochemical activation of carbon nanotubes[END_REF] and pyrolytic graphite [START_REF] Moore | Basal plane pyrolytic graphite modified electrodes: comparison of carbon nanotubes and graphite powder as electrocatalysts[END_REF]. Such materials significantly decreased the NADH overpotential, which was ascribed to the edge-plane sites/defects present in the pyrolytic graphite and suspected in CNTs [START_REF] Banks | Edge plane pyrolytic graphite electrodes in electroanalysis: An overview[END_REF]. Recently, one interesting paper demonstrates that further decrease in the NADH overpotential can be achieved at CNTs that were activated by microwaving CNTs in concentrated nitric acid [START_REF] Wooten | Facilitation of NADH Electro-oxidation at Treated Carbon Nanotubes[END_REF], as indicated by a shift in the anodic peak potential of NADH (E NADH ) from 0.4 V to 0.0 V. NAD(P)H-dependent reduction reactions Contrary to the wide range of mediators available for NADH oxidation, the number of mediators for the regeneration of reduced cofactors is relatively small. The mediator for the regeneration of reduced cofactors must indeed operate at potentials less cathodic than -0.9 V (otherwise direct electrochemical reduction of NAD(P) + will lead to dimmer formation) and more cathodic than the standard potential of the cofactor redox couple ( i.e., -0.59V vs. SCE for NAD + /NADH [START_REF] Karyakin | Equilibrium (NAD + /NADH) potential on poly(Neutral Red) modified electrode[END_REF]) to make the reduction reaction thermodynamically feasible. The systems to date fulfilling these requirments are (2, 2'-bipyridyl)rhodium complexes, Rh(bpy)). The electrocatalytic process includes the regioselective transfer of two electrons and a proton to NAD(P) + . In these systems, hydrido-rhodium species are assumed to be the active catalytic moiety. Tris(bipyridine)rhodium (III) [START_REF] Wienkamp | Indirect electrochemical regeneration of NADH by a bipyridinerhodium(I) complex as electron-transfer agent[END_REF], (pentamethylcyclopentadienyl-2, 2'-bipyridine-chloro) rhodium (III) [START_REF] Ruppert | Efficient indirect electrochemical in situ regeneration of NADH: electrochemically driven enzymatic reduction of pyruvate catalyzed by D-LDH[END_REF][START_REF] Koelle | The effect of chloride on the electroreduction of NAD + in the presence of [Cp*RhIII] 2+ species[END_REF], and chlorotris[diphenyl(m-sulfonatophenyl)phosphine] rhodium (I) [START_REF] Willner | Thermal and photochemical regeneration of nicotinamide cofactors and a nicotinamide model compound using a water-soluble rhodium phosphine catalyst[END_REF] have been used as homogeneous mediation of electrons to NAD(P) + . The catalytic efficiency of a series of Rh-complexes has been studied [START_REF] Steckhan | Analytical study of a series of substituted (2,2'-bipyridyl) (pentamethylcyclopentadienyl) rhodium and -iridium complexes with regard to their effectiveness as redox catalysts for the indirect electrochemical and chemical reduction of NAD(P) +[END_REF] and it was shown that the catalyst activity decreases in the presence of electron-withdrawing substituents in the 2, 2'-bipyridine ligand and increases by electron-donating substituents. Substituents in the 6-position of the ligand slow the catalytic reaction because of steric effects. Structure-activity relationships were found in the mechanism of the regioselective reduction of NAD + by Rh-complexes [START_REF] Lo | Bioorganometallic chemistry part 11. regioselective reduction of NAD + models with [Cp*Rh(bpy)H] + : structure-activity relationships and mechanistic aspects in the formation of the 1,4-NADH derivatives[END_REF]. Enzyme-coupled electrochemical regeneration When considering electrosynthesis applications, it is essential to perform a very smooth regeneration of the cofactor, especially if the cofactor is successfully immobilized with the protein(s), to prevent non-controlled oxidation of the expensive NADH (or reduction of NAD + ). The regeneration of NAD(P)H with the participation of mediator-contacted enzymes ensures that the regeneration of cofactor proceeds selectively, and only enzymatically active cofactor is produced. NAD(P) + -dependent oxidation reactions It is possible to combine indirect regeneration of NAD(P) + with an enzymatic regeneration step. For example, diaphorase has been applied to oxidize NADH using a variety of quinone compounds [START_REF] Ogino | Reactions between diaphorase and quinone compounds in bioelectrocatalytic redox reactions of NADH and NAD +[END_REF], ferrocene derivatives [START_REF] Kashivagi | Electrocatalytic oxidation of NADH on thin poly(acrylic acid) film coated graphite felt electrode coimmobilizing ferrocene and diaphorase[END_REF][START_REF] Osa | Electroenzymatic oxidation of alcohols on a poly(acrylic acid) film coated graphite felt electrode terimmobilizing ferrocene, diaphorase and alcohol dehydrogenase[END_REF], or osmium redox polymers [START_REF] Nikitina | Bi-enzyme biosensor based on NAD + -and glutathione-dependent recombinant formaldehyde dehydrogenase and diaphorase for formaldehyde assay[END_REF][START_REF] Antiochia | Development of a carbon nanotube paste electrode osmium polymer-mediated biosensor for determination of glucose in alcoholic beverages[END_REF] as mediators between the enzyme and electrode. Ferrocene/diaphorase is commonly employed for NAD + regeneration. Kashivagi et al. report the study on the characteristics of poly(acrylic acid) coated electrode coimmobilizing Fc and diaphorase and the application of the electrode to macro-electrocatalytic oxidation of NADH [START_REF] Kashivagi | Electrocatalytic oxidation of NADH on thin poly(acrylic acid) film coated graphite felt electrode coimmobilizing ferrocene and diaphorase[END_REF]. To continue the studies, this group has carried out further immobilization of ADH into poly(acrylic acid) layer of the above modified electrode and achieved a smooth electrocatalytic oxidation of alcohol [START_REF] Osa | Electroenzymatic oxidation of alcohols on a poly(acrylic acid) film coated graphite felt electrode terimmobilizing ferrocene, diaphorase and alcohol dehydrogenase[END_REF]. Osmium/diaphorase is also an efficient system for NAD + regeneration. Nikitina et al. report on the development of a bi-enzyme biosensor using diaphorase and formaldehyde dehydrogenase (FDH) as bio-recognition elements. The sensor architecture comprises a first layer containing diaphorase cross-linked with an osmium complex-modified redox polymer. On its top, a second layer was formed by additional cross-linking of FDH with poly(ethylene glycol)(400)diglycidyl ether [START_REF] Nikitina | Bi-enzyme biosensor based on NAD + -and glutathione-dependent recombinant formaldehyde dehydrogenase and diaphorase for formaldehyde assay[END_REF]. Antiochia et al. develop an amperometric biosensor for glucose monitoring. Glucose dehydrogenase(GDH) and diaphorase (DI) were co-immobilized with NAD + into a carbon nanotube paste (CNTP) electrode modified with an osmium functionalized polymer [START_REF] Antiochia | Development of a carbon nanotube paste electrode osmium polymer-mediated biosensor for determination of glucose in alcoholic beverages[END_REF]. NADH oxidase is also immobilized on the electrode surface for NAD + regeneration. Its stability and its range of useful pH are better than those of diaphorase. A series of dehydrogenase biosensors without cofactor addition have been developed by Marty's group, which depended on the immobilized NADH oxidase to regenerate NAD + [START_REF] Leca | Reusable ethanol sensor based on a NAD + -dependent dehydrogenase without coenzyme addition[END_REF][START_REF] Leca | Reagentless ethanol sensor based on a NAD-dependent dehydrogenase[END_REF][START_REF] Montagnk | Bi-enzyme amperometric D-lactate sensor using macromolecular NAD +[END_REF]. The biosensors were developed through the combination of alcohol dehydrogenase, NADH oxidase and NAD-dextran with addition of a mediator hexacyanoferrate (III). The detection was based on the oxidation of the mediator hexacyanoferrate (III) by applying a potential difference of 100 mV between two platinum electrodes in the presence of an excess of hexacyanoferrate (III), corresponding to 250 mV versus SCE [START_REF] Leca | Reusable ethanol sensor based on a NAD + -dependent dehydrogenase without coenzyme addition[END_REF][START_REF] Montagnk | Bi-enzyme amperometric D-lactate sensor using macromolecular NAD +[END_REF]. Contrary to diaphorase, NADH oxidase also accepts oxygen as an electron acceptor. A reagentless sensor without addition of cofactor and mediator can thus be designed since oxygen is always present in the working medium [START_REF] Leca | Reagentless ethanol sensor based on a NAD-dependent dehydrogenase[END_REF]. NADH oxidase catalyses the reaction of NADH oxidation in the presence of oxygen to generate hydrogen peroxide. Hydrogen peroxide can be detected by applying a potential difference of 550 mV between two platinum electrodes, equivalent to ca. 600 mV versus SCE. NAD(P)H-dependent reduction reactions As it is very difficult to find an electrochemical redox catalyst that fulfils all requirements for regenerating NAD(P)H effectively, the interest of this second enzyme is to extend the choice of mediators that are likely to regenerate NAD(P)H. Many enzymes have been used in this context to provide the bioelectrocatalytic reduction of NAD(P) + . ferredoxin-NADP + reductase (FNR) [START_REF] Kano | Quinone-mediated bioelectrochemical reduction of NAD(P) + catalyzed by flavoproteins[END_REF], lipoamide dehydrogenase (LipDH) [START_REF] Delecouls-Servat | Designing membrane electrochemical reactors for oxidoreductase-catalysed synthesis[END_REF], diaphorase [START_REF] Kashiwagi | Preparative, electroenzymic reduction of ketones on an all componentsimmobilized graphite felt electrode[END_REF][START_REF] Kang | Optimization of the mediated electrocatalytic reduction of NAD + by cyclic voltammetry and construction of electrochemically driven enzyme bioreactor[END_REF], alcohol dehydrogenase [START_REF] Yuan | Fabrication of novel electrochemical reduction systems using alcohol dehydrogenase as a bifunctional electrocatalyst[END_REF], and hydrogenase [START_REF] Cantet | Bioelectrocatalysis of NAD + reduction[END_REF][START_REF] Delecouls | Mechanism of the catalysis by Alcaligenes eutrophus H16 hydrogenase of direct electrochemical reduction of NAD +[END_REF]. A variety of low potential electron-transfer mediators have been used to activate the reductive enzymes, for instance viologen derivatives [START_REF] Delecouls-Servat | Designing membrane electrochemical reactors for oxidoreductase-catalysed synthesis[END_REF][START_REF] Kashiwagi | Preparative, electroenzymic reduction of ketones on an all componentsimmobilized graphite felt electrode[END_REF][START_REF] Kang | Optimization of the mediated electrocatalytic reduction of NAD + by cyclic voltammetry and construction of electrochemically driven enzyme bioreactor[END_REF], flavins [START_REF] Cantet | Bioelectrocatalysis of NAD + reduction[END_REF], quinones [START_REF] Kano | Quinone-mediated bioelectrochemical reduction of NAD(P) + catalyzed by flavoproteins[END_REF], or the redox protein ferredoxin [START_REF] Nishiyama | Aminosilane modified indium oxide electrodes for direct electron transfer of ferredoxin[END_REF]. Some redox enzymes can directly communicate with electrode supports and thus stimulate the regeneration of the NAD(P)H cofactor. For example, hydrogenases (from Rhodococcus opacus and Atcaligenes eutrophus H16) have been successfully applied for the bioelectrocatalytic regeneration of NADH without the application of a redox-mediator [START_REF] Gros | Direct electrochemistry of Rhodococcus opacus hydrogenase for the catalysis of NAD + reduction[END_REF]. However, the electrocatalytic rates of these systems are generally too slow to produce observable catalytic current on the cyclic voltammetric time scale. Among the electrontransfer mediators, viologen is often used in combination with enzyme for NADH regeneration. The viologen together with LipDH has been tested in a continuous process. Bergel et al. applied this regeneration system in their Dialysis-Membrane Electrochemistry Reactor (D-MER) together with an alcohol dehydrogenase for the synthesis of cyclohexanol from cyclohexanone [START_REF] Delecouls-Servat | Designing membrane electrochemical reactors for oxidoreductase-catalysed synthesis[END_REF]. Kashiwagi, et al present a poly(acrylic acid) coated graphite felt electrode immobilizing all the components of viologen, diaphorase, NAD + , and alcohol dehydrogenase for the enzymatic reaction. NADH was regenerated with viologen together with diaphorase [START_REF] Kashiwagi | Preparative, electroenzymic reduction of ketones on an all componentsimmobilized graphite felt electrode[END_REF]. The regeneration system viologen/diaphorase is also used incombination with NAD + and D-lactate dehydrogenase, The optimal concentration of diaphorase, viologen and NAD + in the mediated electrocatalytic reduction of NAD + were studied by applying cyclic voltammetry [START_REF] Kang | Optimization of the mediated electrocatalytic reduction of NAD + by cyclic voltammetry and construction of electrochemically driven enzyme bioreactor[END_REF]. Among the enzymes for cofactor regeneration, diaphorase is the most interesting one. The same diaphorase can be used for both NADH oxidation and NAD + reduction. The coimmobilization in active forms of both dehydrogenase and diaphorase would allow the reactor to perform alternatively oxidation or reduction reaction with simply changing the mediator system and the applied potential. For this reason the preparation of such active bio-composite layers would be of great value for electrosynthesis application. Immobilization of charge transfer catalyst on electrode surface For a technologyically useful configuration, the mediator must be immobilized on the electrode surface. A successful system should show good stability of the immobilized mediator, regenerate NAD + faster than it can be consumed by the enzyme, to appropriately shift the dehydrogenase equilibrium towards the product side so that high current densities can be obtained. A wide variety of ways to immobilize mediator species at the electrode surfaces have been described in the literature. They are briefly summarized hereafter. Carbon paste electrode The carbon paste (CP) electrodes provide a straightforward way to immobilize the mediators. In several cases mediators have been directly incorporated in CP electrodes [START_REF] Dominguez | A carbon paste electrode chemically modified with a phenothiazine polymer derivative for electrocatalytic oxidation of NADH. Preliminary study[END_REF][START_REF] Koyuncu | A new amperometric carbon paste enzyme electrode for ethanol determination[END_REF][START_REF] Weiss | Dehydrogenase based reagentless biosensor for monitoring phenylketonuria[END_REF] to produce NADH and dehydrogenase substrate electrodes. Weiss et al. present a base-stable electron mediator, 3,4-dihydroxybenzaldehyde (3,4-DHB), modified electrode with the mediator mixed directly into the carbon paste [START_REF] Weiss | Dehydrogenase based reagentless biosensor for monitoring phenylketonuria[END_REF]. The kinetics of the catalytic oxidation of NADH is studied at these modified carbon paste electrodes. However, the voltammograms were unstable and change upon continued scanning. The key element for such configurations is that the mediator has to have a higher affinity for the hydrophobic carbon paste phase than for the aqueous analytical matrix. This has not always been achieved, and mediator leaching could be the limiting stability factor in this case. To solve the problem, Yao, et al synthesized an oil-soluble mediator, 7-dimethylamine-2-methyl-3-b-naphtamidophenothiazinium chloride (3-NTB) and apply the mediator to the CP/dehydrogenase electrode [START_REF] Yao | Preparation of a carbon paste /alcohol dehydrogenase electrode using polyethylene glycol-modified enzyme and oil-soluble mediator[END_REF]. In the case of the 3-NTB modified electrode, 3-NTB was not soluble in an aqueous solution, so that the magnitude of the current response did not change for 2 weeks. Thus, the usage of an oil-soluble mediator can improve the long term stability of the CP electrode. Surface activation Surface activation approaches have also been reported in the last few years. As mentioned earlier, these electrodes cause electrocatalytic oxidation at lower potentials but not necessarily at those mediators. For example, electrochemical anodization [START_REF] Kuhr | Dehydrogenasemodified carbon-fiber microelectrodes for the measurement of neurotransmitter dynamics. 1. NADH voltammetry[END_REF][START_REF] Nowall | Electrocatalytic Surface for the Oxidation of NADH and Other Anionic Molecules of Biological Significance[END_REF], microwave plasma [START_REF] Wooten | Facilitation of NADH Electro-oxidation at Treated Carbon Nanotubes[END_REF], vacuum heat treatments [START_REF] Fagan | Vacuum heat-treatment for activation of glassy carbon electrodes[END_REF] are used to improve the heterogenenous electron transfer rate of selected redox couples at various forms of carbon electrode. Remarkable stability and retention of the electrocatalytic activity were observed when carbon fibers were electroactivated, and this carbon fiber was used for NADH detection and discrimination from interferents with fast scan voltammetry [START_REF] Kuhr | Dehydrogenasemodified carbon-fiber microelectrodes for the measurement of neurotransmitter dynamics. 1. NADH voltammetry[END_REF][START_REF] Nowall | Electrocatalytic Surface for the Oxidation of NADH and Other Anionic Molecules of Biological Significance[END_REF]. In a recent publication, CNTs were activated by microwaving in concentrated nitric acid, the shift in E NADH was due to the redox mediation of NADH oxidation by traces of quinone species that were formed on the surface of treated CNTs [START_REF] Wooten | Facilitation of NADH Electro-oxidation at Treated Carbon Nanotubes[END_REF]. Precipitation One simple method included "precipitation" on electrode surfaces of various transition metal hexacyanoferrate [START_REF] Chen | Preparation, characterization, and electrocatalytic properties of copper hexacyanoferrate film and bilayer film modified electrodes[END_REF]. Among the transition metal hexacyanoferrates, cobalt hexacyanoferrate is considered as attractive material to modify the electrode surfaces for NADH oxidation due to its excellent reversible redox behavior [START_REF] Cai | Cobalt hexacyanoferrate modified microband gold electrode and its electrocatalytic activity for oxidation of NADH[END_REF][START_REF] Chen | Preparation, Characterization, and Electrocatalytic Properties of Cobalt Oxide and Cobalt Hexacyanoferrate Hybrid Films[END_REF]. The simple electrochemical deposition process can lead to the formation of the metal hexacyanoferrate on the electrode surface. In general, this type of modified electrodes, showed low detection limits, well behaved electrochemistry, fast response times, and high sensitivities to NADH, while being amenable to careful mechanistic studies and valid conclusions for mediator design for NADH electrochemical oxidation. However, their stability was generally low. Monolayers Various ways to immobilize mediator monolayers are available. Self assembled monolayers could be formed if the mediator contains groups that absorb strongly, such as thiol groups on gold [START_REF] Lorenzo | Thermodynamics and kinetics of adsorption and electrocatalysis of NADH oxidation with a self-assembling quinone derivative[END_REF][START_REF] Raj | Facilitated electrochemical oxidation of NADH and its model compound at gold electrode modified with terminally substituted electroinactive selfassembled monolayers[END_REF]. Another way to form monolayers is to covalently attach the mediator to the electrode surface [ 117 , 118 ]. To do this the electrode surface is first fuctionalized by generation of groups that will permit the subsequent covalent attachment of the mediators, for example, the surface of electrode could be functionalized by a strongly absorbed species, such as thiol compound on gold, which has an amino or carboxylic group at the other end. In this case, the sulfur strongly absorbs onto the gold surface, giving an electrode functionalized with NH 2 or COOH groups. The mediator species themselves can then be covalently attached to the electrode by forming a chemical bond to the NH 2 or COOH groups. Recently, one interesting application of the monolayer is developed. For the first time the inner surface of highly organized macroporous electrodes was modified with monlayer catalyst [START_REF] Ben-Ali | Electrocatalysis with monolayer modified highly organized macroporous electrodes[END_REF] and also a model bioelectrocatalytical system containg a redox mediator, a cofactor, and dehydrogenase [START_REF] Ben-Ali | Bioelectrocatalysis with modified highly ordered macroporous electrodes[END_REF][START_REF] Szamocki | Macroporous Ultramicroelectrodes for Improved Electroanalytical Measurements[END_REF]. Such monolayer could provide a means to control the distribution and orientation of immobilized species, however, the concentration of mediator groups at the electrode surface is limited by steric packing constraints. Electropolymerization Conducting polymers are a natural choice for preparing arrays of voltammetric sensors because they have a rich electrochemical behavior and their electrochemical properties can be modulated by introducing chemical modifications in the sensitive materials [START_REF] Macdiarmid | Synthetic Metals: A Novel Role for Organic Polymers (Nobel Lecture)[END_REF]. Electropolymerization is a good approach to prepare polymer modified electrodes (PMEs) as adjusting electrochemical parameters can control film thickness, permeation and charge transport characteristics. PMEs have many advantages in the detection of analytes because of its selectivity, sensitivity and homogeneity in electrochemical deposition, strong adherence to electrode surface and chemical stability of the film [START_REF] Kumar | Poly(4-amino-1-1'-azobenzene-3, 4'-disulfonic acid) coated electrode for selective detection of dopamine from its interferences[END_REF]. Various dyes like Meldola blue [START_REF] Vasilescu | Strategies for developing NADH detectors based on Meldola Blue and screen-printed electrodes: a comparative study[END_REF], phenothiazine [START_REF] Gao | Electro-oxidative polymerization of phenothiazine dyes into a multilayer-containing carbon nanotube on a glassy carbon electrode for the sensitive and low-potential detection of NADH[END_REF], thionine [START_REF] Gao | Preparation of poly(thionine) modified screen-printed carbon electrode and its application to determine NADH in flow injection analysis system[END_REF], and methylene green [START_REF] Dai | Electrocatalytic detection of NADH and ethanol at glassy carbon electrode modified with electropolymerized films from methylene green[END_REF] have been immobilized on electrode by polymerization for detection of NADH at low potential. These films typically have a 10 -6 -10 -8 mol/cm 2 surface coverage and they present more important swelling problems in aqueous solutions than the previously mentioned adsorbed monolayers. These problems are reflected in longer response time, higher detection limit, and lower, in general, sensitivity for NADH detection. Dehydrogenase, cofactor and mediator coimmobilization Dehydrogenases have attracted considerable attention because they are of increasing interest for electroenzymatic synthesis, biosensor and bio-fuel-cells or biobattery. The main technological barrier in the fabrication these reagent free devices is the development of functional film allowing the stable immobilization of all component of the electrochemical detection like cofactor, mediator and dehydrogenase at the surface of the transducer. Ideally, the immobilization should be done in such a way (Figure I-9) that dehydrogenases are durably immobilized and active, the cofactor is durably immobilized close to the enzyme, and mediator can reduce the overpotential without leaching. Moreover, in this system, it is essential that the produced NADH (or NAD + ) is instantaneously consumed by the mediator (or another enzyme on the electrode surface for cofactor regeneration), otherwise the equilibrium will be reached and the further production of NADH (or NAD + ) will cease. The reduced (or oxidised) mediator in turn must also be rapidly reoxidised (or reduced) to recreate its active form. In essence, this means that all three reaction steps (enzymatic, mediated and electrochemical) need to occur very close in space for a successful approach. To achieve the co-immobilization, a suitable matrix should be found, which should have good balance between the permeability of the substrate or products and the retention of the enzyme, cofactor and mediator. Consequently, and predictably, the development of this kind of functional film has been rather slow, only few examples in the application of biosensor are available. Typically, such biosensors have been designed using redox mediators to recycle enzyme cofactors and immobilizing the dehydrogenases and their cofactors by entrapping them in carbon pastes, membranes, composite materials, macroporous electrodes, and assembled layers. Table I-1 shows the comparison of amperometric reagentless biosensors based on dehydrogenase/cofactor system. We can observe from the table, some of them show low sensitivity, some of them have the problem of the stability. So it is still a challenge to construct this kind of functional layer. Figure I-9 A scheme of functional film. Up to now, sol-gel chemistry was not considered for durable cofactor immobilization, which is however mandatory for the elaboration of reagentless devices. The focus of the thesis is on the comparison of different strategies in order to get the stable immobilization of a bienzymatic system (a dehydrogenase and a diaphorase), the cofactor NAD + /NADH and an electron mediator in a sol-gel matrix deposited as a thin film on an electrode surface. This layer can be applied in electro-enzymatic synthesis, but the finding of this work can also be applied in reagent free amperometric sensors and bio-fuel-cells or biobattery. cofactor Mediator Substrate Enzyme Objective beyond the state-of-the-art Based on the works reported in the literature, we have first investigated in detail the preparation of different kinds of sol-gel routes for dehydrogenase encapsulation. It was known that protein and enzyme encapsulation into electrogenerated sol-gel thin films was possible [46,47,48]. However, first attempt to use similar approach for bioencapsulation of dehydrogenase was not successful. DSDH was found to be very sensitive to the silica gel environment. The addition of a positively-charged polyelectrolyte was necessary to ensure effective operational behavior of the biomolecules, which allow the successful encapsulation of dehydrogenase inside sol-gel matrix by using both drop-coating and electrodeposition (chapter III). Dehydrogenase is dependent on free diffusing cofactor in stoichiometric amounts to shuttle the redox equivalents from the enzyme to the substrate. We have investigated different strategies allowing the stable immobilization of this cofactor in the sol-gel matrix (chapter IV). In order to overcome the inherent difficulties of cofactor regeneration, a common approach is to confine the mediator at the electrode surface to facilitate the interfacial electron transfer reaction. Several strategies for mediator immobilization have been developed and used for the elaboration of reagentless device with co-immobilized dehydrogenase and cofactor (chapter V and chapter VI). A special attention was given to the preparation of the sol-gel layer by electrochemically-assisted deposition. While several systems operated well when the sol-gel layer was prepared by drop-coating, it was much more difficult to co-immobiliz all the elements of the biocomposite by electrochemical deposition...... Most of the studies presented in this thesis have been performed in the oxidation reaction. In addition, we also studied the immobilization of Rh(III) mediator on carbon nanotubes for the electrocatalytic reduction of NAD + . Chapitre II. Partie expérimentale Chapter II. Experimental part To conduct the work presented in this thesis, several chemicals and techniques have been used to prepare and characterize the active film. This chapter presents the description of the sol-gel precursors, enzymes, cofactors, mediators, Polymer additives and the protocols used for the cofactor and mediator synthesis. A series of methods used to prepare the starting sols and the working electrodes in this thesis are also described. At last, the analytical techniques used in the studies are described. Chemicals and biomolecules Enzymes In the production of chiral compounds, the oxidation or reduction of prochiral substrates by dehydrogenases is the preferred reaction. Two kinds of dehydrogenase have been cloned and used in this project, D-sorbitol dehydrogenase (DSDH) and galactitol dehydrogenase (GatDH). DSDH and GatDH have been applied in both anodic and cathodic modes, using respectively D-sorbitol and fructose (for DSDH) and hexanediol and hydroxyacetone (for GatDH). D-sorbitol dehydrogenase solution (DSDH, 10 mg/mL, 100 units/mg) and Galactitol dehydrogenase (GatDH, 10 mg/mL, 14 units/mg ) have been provided by Pro. G. W. Kohring (The group of microbiology of Saarland University, Germany), which have been prepared by overproduction of the His(6) tagged protein in Escherichia coli BL21GOLD (DE3) and purification of the enzyme with Histrap columns (GE Healthcare). The activity of the protein suspension was measured as NADH production by oxidation of D-glucitol in a photometric assay at 340 nm. Diaphorase (DI, 1020 units/mg) was obtained from Unitika, Japan. Cofactors Commercial cofactors The Nicotinamide redox cofactors (NAD + and NADH) play important roles in biological electron carriers, which are based on the transfer of two electrons and one proton. β-Nicotinamide adenine dinucleotide (NAD + , ~98 %), β-Nicotinamide adenine dinucleotide, reduced dipotassium salt (NADH, ~97 %) and β-Nicotinamide adenine dinucleotide-dextran (NAD + -dextran, attached through C8 to soluble dextran, D-4133, with a 6 carbon spacer) were supplied by Sigma. NAD-derivatives Synthesis PEI-NAD PEI-NAD was prepared according to the reference [1]. NAD + (200 mg) was dissolved in 40 mL dimethylsulfoxide containing 8 g succinic anhydride. After 24 h at room temperature, the NAD + components (mixture of unreacted NAD + and succinyl-NAD) were precipitated with 60 mL acetone. The precipitate was washed 3 times with acetone and recovered by centrifugation. After centrifugation, 20 mL buffer containing 700 mg EDC was added, and pH was adjusted to 4.7 for activating the carboxylic groups of succinyl-NAD for 1 h. After activation, 300 mg PEI was added and reacted for another 12 h at 4 °C. The reaction mixture was dialysis against 50 mM phosphate buffer (pH 7.0) for 12 h at 4 °C. NAD-GPS The method involves the direct in situ functionalization of NAD + with a glycidopropylsilane, GPS, precursor [2]. NAD + and GPS educts were typically prepared by mixing together 25 mg NAD + and 37.5 mg GPS in 400 µL Tris-HCl buffer solution (pH 7.5) at 4°C under shaking for 12 h. Different kinds of MWCNTs were treated by microwaving based on the procedure reported in the previous article [ 5 ]. Table II-3 shows their information. The MWCNTs were microwaved in a concentrated nitric acid (70%) for 10-20 min. The microwaving was performed at 50 °C, 20 psi, and 100% power using the Discover Labmate single-mode microwave oven (CEM, 300 W). After microwaving, the MWCNTs were subjected to at least three centrifugation/decanting cycles in fresh aliquots of deionized water to remove any remaining impurities. In addition, the acid-treated MWCNTs suspensions were neutralized with a sodium hydroxide solution and washed extensively with water to a neutral pH. The final rinsing was performed with ethanol. The MWCNTs were dried in an oven at 85 °C overnight and stored in closed vials at room temperature. Mediators • Polymerized methylene green(MG) on the MWCNTs modified GCE (GCE/MWCNTs-PMG) Chitosan/MWCNTs suspension was prepared by dispersing 1.0 mg MWCNTs (MWCNTs, Aldrich) in 1mL of chitosan solution (0.2 % in 0.05 M acetic acid solution). An aliquot (5 µL) of this resulting solution was deposited onto the surface of the GCE. The solution was then allowed to dry at room temperature to get MWCNTs/GC electrode. Electropolymerization of MG on MWCNTs/GC electrode was carried out using cyclic voltammograms in 0.1 M pH 7.0 PBS containing 0.5 mM MG and 0.1 M KCl in a potential range from -0.5 to 1.2V at a scan rate of 50 mV/s. After successively cycling for 10 cycles, the electrodes were rinsed with doubly distilled water thoroughly and kept at room temperature drying for further use [6,7]. • MWCNTs-Osmium (MWCNTs-Os) Polymer additives A series of polymer additives with different charges have used in this work. The effects of the introduction of these polymer additives into sol-gel for bioencapsualtion are also investigated. Table II-4 shows the information of used polymer additives. The other chemicals and solutions All other reagents are of analytical grade. They include K 2 HPO 4 (99%, Prolabo), KH 2 PO 4 (99%, Prolabo), ethanol (Merck) and HCl (36%, Prolabo). Chitosan (medium molecular weight) was supplied by Aldrich. D-Sorbitol (98%), D-fructose (99 %), ferrocene carboxyaldehyde (FcCHO), succinic anhydride, 1-ethyl-3-(3-dimethylaminopropyl) carbodiimide hydrochloride (EDC), sodium borohydride (NaBH 4 ) and tris(hydroxymethyl) amino-methane (Tris) were obtained from Sigma. Tris-HCl (pH 9.0) buffers were prepared by adding suitable amounts of HCl to 0.1 M Tris solutions, which was used to investigate the oxidation reaction. Phosphate buffer solution (PBS, 0.1M, pH 6.5) is used to investigate the reduction reaction. All solutions were prepared with high-purity water (18 M cm -1 ) from a Millipore milliQ water purification system. Electrodes In this work, a glassy carbon electrode (GCE, 3mm in diameter), a gold electrode (Au, 4 mm in diameter) or a macroporous gold electrode served as workiong electrode. Prior each measurement, glassy carbon electrodes (GCE) or gold electrodes were first polished on wet emery paper 4000, using Al 2 O 3 powder (0.05 mm, Buehler), then rinsed thoroughly with water and ultrasonicated in water and alcohol bath to remove the embedded alumina particles. Macroporous gold electrodes (Figure II-4) were provided by partner 2 (ENSCPB, Bordeaux), which were obtained by electrodeposition of gold through a silica bead colloidal crystal followed by the dissolution of the silica template. The Langmuir Blodgett technique is used to transfer successive layers of monodisperse beads on gold-coated glass substrates, previously treated by cysteamine in order to make the sample surface hydrophilic [14,15]. The gold electrodeposition was operated at -0.66 V vs. saturated Ag/AgCl after 10 minutes dipping in the commercial gold plating bath in order to let the solution infiltrate the template. After the deposition step the samples were rinsed with distilled water and placed 10 minutes in 5% HF in order to remove the silica colloidal crystal. The pore diameter and the thickness of the porous material were controlled as described in previous work [16]. Preparation of sol-gel for bioencapsulation We have explored various starting sol compositions for bioencapsulation (see Table II-5). At that stage of the project, it has thus been decided to initiate studies with chitosan as an alternative encapsulation matrix, to evaluate the interest of polymeric additives and, depending on the obtained results, to exploit the conclusions to improve the response of solgel biocomposites. Sol G was developed for dehydrogenase enzymes encapsulation. All investigations performed using drop-coating or spin-coating to deposit sol-gel films have been found to lead to electrochemically detectable bioactivity. However, no detectable electrochemical signal was observed by electrodeposition. The first experiments performed The release of alcohol during the hydrolysis of with Sol H and DSDH showed the feasibility of encapsulation of active enzymes in electrodeposited sol-gel thin films. However, this kind of sol-gel can not work for enzyme and cofactor co-immobilization. At this stage, Sol I was developed for enzyme and cofactor coimmobilization. Finally, Sol J and K are developed for co-encapsulate enzyme, cofactor and mediator. Table II-5. Information of different methods used to prepare the starting sols Materials Procedures used to prepare the starting sols for films formation Sol A [17] The sol was prepared by dissolving 2.125 g TEOS, 2 mL H 2 O and 2.5 mL HCl (0.01 M), which were mixed for 12 h using a magnetic stirrer, then NaOH was added in the medium to increase pH at a value of about 4 and electrodeposition was applied. Sol B Same system as Sol-Gel A, but using TMOS instead of TMOS Sol C [18] The sol was prepared by stirring 2.56 g TMOS with 0.6 mL H 2 O and 0.06 mL HCl (0.62 M) for 20 min. Phosphate buffer (1.0 mL, 0.01M, pH=8.2) was added to the sol (1.0 mL) which is shaken vigorously. Sol D [19] The sol was prepared by stirring of 4.46 mL TEOS, 1.44 mL H 2 O and 0.04 mL HCl (0.62 M) for 1 h. Then 1 mL of the resulting sol was mixed with 1 mL of deionised water, and evaporated for a weight loss of 0.62 g. Sol E [20] 11.5 g Sodium silicate (3.25 SiO 2 /Na 2 O) was combined with 34 mL DI water. To this aqueous solution is added 15.4 g of strongly acidic cation-exchange resin with stirring to bring the pH of the solution to a value of 4. The resin is then filtrated. 0.3 mL of 2 M Hydrochloric acid was added to the sol to adjust the pH 2.0. A phosphate buffer (1 M, pH 7) containing enzyme was added to the sol solution in a 1:5 (volume) ratio. Sol F [21] 0.61 g Sodium silicate was combined with 50 mL de-ionized water. To this aqueous solution was added 1.6 mL 37% HCl to decrease the pH to 0.84. 2 mL of resulting 0.1 M sodium silicate was added to 3 mL ludox (SM-30), which was shaken vigorously. Sol G The sol was prepared by dissolving 0.04 g TEOS, 800 µL ethanol and 1 mL HCL (0.01M), which were mixed for 2.5 h using a magnetic stirrer. Sol H Same system as Sol-Gel A, but diluted 3 times with water for further use. Sol I Same system as Sol-Gel A, but diluted 6 times with water for further use. Sol J The sol was prepared by dissolving 0.18g TEOS, 0.13g GPS, 0.5 mL H 2 O and 0.625 mL HCl (0.01M), which were mixed for 12 h using a magnetic stirrer. Then diluted 2 times with water for further use. Sol K The sol was prepared by dissolving 0.18 g TEOS, 0.13 g GPS, 0.02 g Fc-silane, 0.5 mL H 2 O and 0.625 mL HCl (0.01 M), which were mixed for 12 h using a magnetic stirrer. Then diluted 2 times with water for further use. where electrochemically-assisted deposition was performed at -1.3 V at room temperature for 60 s. The electrodes were immediately rinsed with water, and dried overnight at 4°C. MWCNTs-PMG &sol-gel matrix The preparation of GCE/MWCNT-PMG has been described in chapter II 1.4.1.2. Chitosan/CNTs/Rh solution and allowed to evaporate at the room temperature. ① Methods of analysis 5.1 Electrochemical measurements [ 24] Cyclic voltammetry (CV) Cyclic voltammetry is the most widely used technique for acquiring qualitative information about electrochemical reactions. It is often the first experiment performed in an electroanalytical study. In particular, it offers a rapid location of redox potential of the electroactive species, and convenient evaluation of the effect of various parameters on the redox process. This technique is based on varying the applied potential at a working electrode in both forward and reverse directions (at selected scan rates) while monitoring the resulting current. The corresponding plot of current versus potential is termed a cyclic voltammogram. Chronoamperometry in hydrodynamic mode The basis of chronoamperometry techniques is the measurement of the current response to an applied potential. A stationary working electrode and stirred solution are used. The resulting current-time dependence is minitored. As mass transport under these conditions is solely by convection (steady state diffusion), the current-time curve reflects the change in the concentration gradient in the vicinity of the surface, which is directly related to concentration in solution. In this work, all cyclic voltammetry (CV) and chrono amperometry experiments are conducted using a conventional three-electrode cell. The working electrode was a glassy carbon electrode (GCE, 3mm in diameter), a gold electrode (Au, 4 mm in diameter) or a macroporous gold electrode, a gold wire served as auxiliary electrode, and an Ag/AgCl electrode (saturate KCl internal electrolyte) was used as the reference. All electrochemical experiments have been performed at room temperature and carried out using an Autolab PGSTAT-12 potentiostat (Eco Chemie) monitored by the GPES (General Purpose Electrochemical System) software. UV-VIS Spectroscopy (UV) The principles of UV centre on the fact that molecules have the ability to absorb ultraviolet or visible light. This absorption corresponds to the excitation of outer electron in the molecules concerned. When a molecule absorbs energy an electron is promoted from the Highest Occupied Molecular Orbital (HOMO) to the lowest Unoccupied Molecular Orbital (LUMO). As with any UV-Vis spectrometer, three of the main elements are a UV-light source, a monochromator and a detector. The monochromator works as a diffraction grating to dispense the beam of light into various wavelengths. The detectors role is to record the intensity of the light which has been transmitted. Before the samples are run, a reference must first be taken. This calibrates the spectra to screen out any spectral interference. In the case of liquid samples the solvent which has been used to dissolve the sample is used. However, there are certain criteria that solvent must pass before they can be deemed as suitable solvents. The main criterion is that the solvent should not absorb ultraviolet radiation in the same region as the sample being analysed. The apparatus used in this study is a UV-vis spectrophotometer Beckman Du 7500, which was used to measure NADH concentrations in solution at 340 nm. Attenuated Total Reflectance-Fourier Transform Infrared Spectroscopy (ATR-FTIR spectroscopy) Figure II-6. A multiple reflection ATR system. Infrared spectroscopy is a widely used technique that for many years has been an important tool for investigating chemical processes and structure [25]. The combination of infrared spectroscopy with the theories of reflection has made advances in surface analysis possible. Specific IR reflectance techniques may be divided into the areas of specular reflectance, diffuse reflectance, and internal reflectance. An innovative technique (ATR-FTIR spectroscopy) for monitoring the transport process of low molecular weight species was established based upon internal reflectance. This enabled the monitoring of individual species in-situ, while providing additional chemical information on any changes that may be occurring during the transport process. Figure II-6 shows the principles of ATR-FTIR. An attenuated total reflection accessory operates by measuring the changes that occur in a totally internally reflected infrared beam when the beam comes into contact with a sample. An infrared beam is directed onto an optically dense crystal with a high refractive index at a certain angle. This internal reflectance creates an evanescent wave that extends beyond the surface of the crystal into the sample held in contact with the crystal. It can be easier to think of this evanescent wave as a bubble of infrared that sits on the surface of the crystal.This evanescent wave protrudes only a few microns (0.5 µ -5 µ) beyond the crystal surface and into the sample. Consequently, there must be good contact between the sample and the crystal surface. In regions of the infrared spectrum where the sample absorbs energy, the evanescent wave will be attenuated or altered. The attenuated energy from each evanescent wave is passed back to the IR beam, which then exits the opposite end of the crystal and is passed to the detector in the IR spectrometer. The system then generates an infrared spectrum. The apparatus used in this study is a Bruker Vector 22 spectrometer equipped with a KBr beam splitter and a deuterated triglycine sulfate (DTGS) thermal detector. FTIR spectra were recorded between 4000 and 700 cm -1 . Recording of spectra, data storage and data processing were performed using the Bruker OPUS 3.1 software. The resolution of the single beam spectra was 4 cm -1 . The number of bidirectional double-sided interferogram scans was 200, which corresponds to a 2 min accumulation. All interferograms were Fourier processed using the Mertz phase correction mode and a Blackman-Harris three-term apodization function. Measurements were performed at 22 ± 1°C in an air-conditioned room. A nine-reflection diamond ATR accessory (DurasamplIR ™, SensIR Technologies) was used for acquiring spectra. The incidence angle was 45° and the refraction index of the crystal was 2.4. No ATR correction was performed. 80 µL of the studied solution were put on the crystal. Appropriate spectra were used to remove spectral background: an air-reference, a water-reference, or a Tris-HCl buffer solution-reference. Water vapor subtraction was performed when necessary. In the course of reaction monitoring experiments, ATR-FTIR spectra were recorded every 5 or 15 min.. Scanning electron microscopy (SEM) The scanning electron microscope (SEM) is a type of electron microscope that images the sample surface by scanning it with a high-energy beam of electrons in a raster scan pattern. The electrons interact with the atoms that make up the sample producing signals that contain information about the sample's surface topography, composition and other properties such as electrical conductivity. Due to the very narrow electron beam, SEM micrographs have a large depth of field yielding a characteristic three-dimensional appearance useful for understanding the surface structure of a sample. A wide range of magnifications is possible, from about 10 times (about equivalent to that of a powerful hand-lens) to more than 500,000 times, about 250 times the magnification limit of the best light microscopes. Thus we are able to observe the particle morphology closely on a very fine scale. For conventional imaging in the SEM, specimens must be electrically conductive, at least at the surface, and electrically grounded to prevent the accumulation of electrostatic charge at the surface. Nonconductive specimens are therefore usually coated with an ultrathin coating of electrically-conducting material, commonly gold or graphite, deposited on the sample either by low vacuum sputter coating or by high vacuum evaporation. The technique of SEM is to focus on a surface of specimen by lens using a condensed electron beam. The interaction between electrons and the material leads to the backscattered electron emission, X-rays, secondary electrons and so on. These electrons are collected by a detector, converted to a voltage and finally amplified. In this work, The morphologies and structures of the membranes of DSDH-encapsulated electrogenerated sol-gel silica on a GCE were examined on a scanning electron microscope (SEM, Hitachi X-650, Japan). [ 26, 27] Scanning electrochemical microscopy (SECM) is a technique in which the current that flows through a very small electrode tip (generally an ultramicroelectrode with a tip diameter of 10 pm or less) near a conductive, semiconductive, or insulating substrate immersed in solution is used to characterize processes and structural features at the substrate as the tip is moved near the surface. The tip can be moved normal to the surface (the z direction) to probe the diffusion layer, or the tip can be scanned at constant z across the surface (the x and y directions). The tip and substrate are part of an electrochemical cell that usually also contains other (e.g., auxiliary and reference) electrodes. The device for carrying out such studies involves means of moving the tip with a resolution down to the A region, for example, by means of piezoelectric elements or stepping motors driving differential springs, and is called a scanning electrochemical microscope. The abbreviation SECM is used interchangeably for both the technique and the instrument. In SECM the current is carried by redox processes at tip and substrate and is controlled by electron transfer kinetics at the interfaces and mass transfer processes in solution, so that measurements at large spacings, e.g., the range of 1 nm to 10 pm, can be made. In addition to the electrochemical measurement, the machine can be equipped by a sheaforce detection module that helps in positioning the electrode before electrochemical measurement. Here, only the sheaforce detection was used in order to determine the film thickness on conductive surface. The apparatus has been developed in the lab on the base of the SECM instrument from Senslytics (Ruhr-Universität, Bochum, Germany). Scanning electrochemical microscopy (SECM) Chapitre Chapter III. Feasibility of dehydrogenase encapsulation in sol-gel matrix In this chapter, the work focuses on the immobilization of the active dehydrogenase on the electrode surface. Here, sol-gel is chosen as the matrix for the D-sorbitol dehydrogenase (DSDH) encapsulation, which is expected to be likely to provide suitable environment for bioencapsulation. First of all, the feasibility of DSDH encapsulation in sol-gel film is evaluated by drop-coating (see section 2). DSDH encapsulation in pure silica thin films resulted in undetectable electrochemical signal. Then, the influence of polyelectrolyte (PE) additives on the sol-gel encapsulation of dehydrogenases has been evaluated by drop-coating. DSDH was found to be very sensitive to the silica gel environment and the addition of a positively-charged polyelectrolyte was necessary to ensure effective operational behavior of the biomolecules. Since the suitable sol-gel environment for DSDH encapsulation has been found by drop-coating, we then investigate the electrochemically-assisted deposition of solgel thin films for DSDH encapsulation (see section 3). This was achieved via the electrolysis of a hydrolyzed sol containing the biomolecules to initiate the polycondensation of silica precursors upon electrochemically-induced pH increase at the electrode/solution interface. The composition of the sol and the conditions for electrolysis have been optimized with respect to the intensity and the stability of the electrochemical response to D-sorbitol oxidation. The electrochemically-assisted deposition of silica thin films was found to be a good strategy for DSDH immobilization as well as DSDH and diaphorase co-immobilization. At the end, this process has been extended to macroporous electrodes exhibiting a much bigger electroactive surface area. Introduction Dehydrogenases are interesting enzymes for electrosynthesis applications, especially for the production of rare sugars as building blocks for pharmaceutical and food industry [1]. One of the main requirement for electrosynthesis is the stable immobilization of a large amount of active proteins on the electrode surface of the reactor [2,3]. The encapsulation of proteins in a silica matrix using the sol-gel process is known to prevent their denaturation and to keep these proteins active for a long time, in some case longer than in solution, and can also allow the use of some enzymes in rather harsh conditions of pH and temperature [4 ]. When tetraethoxysilane (TEOS) is used as silica precursors, the hydrolysis of the ethoxy groups leads to a significant amount of ethanol potentially harmful for the enzyme. Ethoxy groups can be replaced by methoxy groups, to produce methanol that was reported to be less toxic for the enzyme [5]. Alcohol molecules can also be removed by solvent evaporation [6] and other sol-gel routes can also be involved in order to prevent the presence of any alcohol molecules in the starting sol [7]. A good example of this last strategy was reported recently for the encapsulation of horseradish peroxidase (HRP) in silica thin films obtained by electrochemically-assisted deposition with a sol based on ammonium hexafluorosilicate [8]. A favorable environment can also be obtained by the introduction of additives into the sol [9]. Recent efforts have been made to optimize the process by controlling the porosity of the material and the chemical environment of trapped species. This notably involved the use of biocompatible silane precursors, protein-stabilizing additives, charged polymers, or sugars and aminopeptides [4,10,11]. For electrochemical applications, the silica thin films can be effectively deposited on the surface of a flat electrode by the conventional sol-gel method using the controlled evaporation of the solvent by dip-coating, drop-coating, etc [12]. This has been largely exploited to design amperometric biosensor devices [ 13 ]. In the particular case of electro-enzymatic synthesis, the electroactive surface area of the reactor has to be increased in order to allow high flux of substrate to be oxidized or reduced. This can be achieved, e.g., by the use of macroporous electrodes exhibiting very high surface areas [14,15]. However, coating such porous electrodes in a controlled way with an enzyme-doped silica gel is somehow difficult with the evaporation methods cited above because they are mostly restricted to flat surfaces [16]. As an alternative, the silica layer can be produced at porous electrode surfaces by a local modulation of pH [17]. This is achieved by a controlled electrolysis of the pre-hydrolyzed sol that induces the rapid gelification of the thin film. The possible application of the electrochemically-assisted sol-gel deposition to bio-encapsulation was shown recently by some research groups, using glucose oxidase as a model enzyme [18,19]. It is also essential for electrosynthesis application to ensure a very smooth regeneration of the cofactor, especially if the cofactor is successfully immobilized with the protein(s), to prevent non-controlled oxidation of the expensive NADH (or reduction of NAD + ). Diaphorase catalyzes this cofactor regeneration in the presence of a molecular mediator, for example ferrocene species for oxidation or methylviologen for reduction [ 20 ]. The coimmobilization of both dehydrogenase and diaphorase in active forms into a thin silica film would allow the reactor to perform alternatively oxidation or reduction reactions by "simply" changing the mediator system and the applied potential [21,22]. For this reason the wellcontrolled deposition of such active bio-composite layers in macroporous electrodes would be of great value for bioelectrocatalytic application. In this work, we used DSDH as model enzyme to evaluate the feasibility of dehydrogenase encapsulation in sol-gel film. First, we have evaluated the interest of various polyelectrolytes in combination to silica films for DSDH encapsulation (section 2). We have chosen positively-charged polyelectrolytes because of expected favorable interactions with the negatively-charged enzyme surface [23]. Comparing the electrochemical responses observed in the presence and absence of these additives allowed to evidence the critical role played by the polyelectrolyte in enhancing cofactor regeneration. Then, we show the electrochemicallyassisted deposition of silica-based thin films for DSDH immobilization as well as DSDH and diaphorase co-immobilization (section 3). The process and the sol composition have been optimized on flat glassy carbon electrodes before being applied to macroporous gold electrodes. The electrodeposited bio-composite containing both DSDH and diaphorase has been tested for electrochemical oxidation of D-sorbitol and a comparison was made between flat and macroporous gold electrodes. Critical effect of polyelectrolytes on the electrochemical response of dehydrogenases entrapped in sol-gel thin films 2.1 Preliminary observations D-sorbitol dehydrogenase (DSDH) needs electron transfer cofactors (NADH/NAD + ) for its enzymatic activity. A large amount of work has been done over the past years in order to decrease the overpotentials for the electrochemical detection of this cofactor, especially when operating in oxidation mode [24]. For simplicity, we first considered the direct oxidation of NADH (i.e., without mediator) to evaluate the activity of the enzyme encapsulated into the sol-gel matrix to be tested. The overall reaction scheme is shown on Enzyme encapsulation in pure silica films The encapsulation of DSDH was first evaluated with using pure silica thin films (i.e., without any additive). UV monitoring of the enzyme activity in gel monoliths In order to distinguish between the above hypotheses, DSDH was encapsulated in sol-gelderived monoliths and its biological activity was monitored by UV spectroscopy via NADH generation upon addition of D-sorbitol in the medium. The biomaterial was prepared according to a protocol reported by Miller et al. [25], requiring notably a rather long aging period (6 days at 4°C). Significant shrinkage of the monolith occurred during gelification. The solid was then washed three times in 3 mL phosphate buffer solution in order to remove weakly encapsulated enzymes. It was then introduced into a solution containing 0.36 mM NAD + and 5 mM D-sorbitol. The activity of encapsulated DSDH can be evidenced by UV monitoring of the solution phase at 340 nm (maximum absorbance for NADH detection). The pH of the sol used for the enzyme encapsulation was comprised in between 5 and 7. In these conditions, negative charges are present on the silica surface due to silanol deprotonation (the point of zero charge of silica is reported to be in the range 2-3 [26]). The isoelectric point of DSDH is 4.3 [27]. During the protein encapsulation, the interaction between the silica matrix and the protein seems to be not so favorable in the pH conditions used here. The above sol-gel entrapment and UV monitoring experiments have thus been repeated on the basis of monoliths prepared in the presence of a positively-charged polyelectrolyte (PDDA) likely to act as a stabilizing intermediate between the enzyme and the silica surface. Results presented in Figure III-3 (curve "c") show that this is indeed the case as a much higher activity was observed with the gel containing PDDA for which almost 90 % of NAD + have reacted after the same period of time (6 h). It is noteworthy that the process is still much slower than for the free enzyme in solution, but the presence of PDDA in the solgel encapsulation matrix provides definite advantage in comparison to undoped silica. Interest of additives for DSDH encapsulation onto electrode surfaces 2.2.1 Sol-gel matrices The above results suggest that the presence of positively-charged moieties in the silica/DSDH biocomposite would be helpful to improve the enzyme activity and thereby to enhance the electrochemical response of GCE covered with such biocomposite films. We have evaluated two ways to introduce positive charges in the material: (1) the resort to a positively-charged organosilane (i.e., protonated aminopropyltriethoxysilane, APTES) and ( 2) the addition of positively-charged polyelectrolytes. 4A). This suggests that the presence of the silica film induces some resistance to charge transfer kinetics. One can conclude from this first series of experiments that favorable electrostatic interactions between DSDH and the aminopropyl-functionalized silica matrix (point of zero charge = 9.8 [28]) is beneficial for getting electrochemically detectable enzymatic activity. The positive charges held by the aminopropyl groups could also interact with NAD + (which is also negatively charged), bringing DSDH and NAD + together and allowing a higher net production of NADH [29,30]. Figure III-4B shows that even more impressive behavior can be obtained when using a polyelectrolyte (i.e., 5% PDDA) as additive in the starting sol. A well-defined electrochemical response was observed, increasing regularly by increasing the D-sorbitol concentration from 1 to 9 mM, in the same conditions as those applied for films prepared in the absence of PDDA Extension to chitosan and chitosan sol/gel composites Chitosan is produced by deacetylation of chitin bio-polymer. This reaction allows generating a certain fraction of amine functions that makes the polymer soluble and suitable for bio-encapsulation [32]. This property has been advantageously used for electroanalytical purposes [33]. One observed in the previous section that doping silica sol-gel films with polymers bearing protonated amine or ammonium groups was essential for improving the enzymatic activity of DSDH immobilized on GCE. We have thus evaluated if chitosan, in combination with silica gel, could provide this favorable environment because of the amine groups it holds. Factors affecting the electrode response 2.3.1 Sol composition The sol composition has been optimized in order to define conditions leading to the highest electrochemical response. Both polyelectrolyte (PDDA has been selected because it gave rise to best results among others, see Fig. III-5) and precursor (TEOS) concentrations have been found to affect significantly the film electrode response (Figure III-8). Figure III-8A shows the influence of PDDA content into the starting sol. As already discussed above, the absence of PDDA led to inactive films since no NADH was detected in the presence of D-sorbitol (FigureIII-2). Addition of PDDA, even in few amounts (e.g., 1.7 %) resulted in significant electrochemical signals, the intensity of which increasing up to 5 % and decreasing then quite regularly for higher polyelectrolyte contents (up to 10 %). All electrodes displayed well-defined voltammetric peaks for NADH oxidation. This trend leading to an optimal value of 5 % can be explained by the role played by PDDA, acting somewhat as "macromolecular glue" between the enzyme and silica surfaces, too low or too high contents of this additive contributing to unbalance the stabilizing effect. Another explanation can be found in the concentration-dependent effect of polycationic macromolecules on biosilication, in providing a favorable interaction with silica precursors during the sol-gel process that contributes to the gelification [7]. They can also induce modification in the texture of the final material [34], which would affect mass transport processes in the biocomposite and, therefore, the rate of cofactor generation/regeneration. No noticeable response can be detected when using TEOS concentrations above 1 M. The electrochemical response is not only related to the enzyme activity but also strongly dependent on the diffusion of both D-sorbitol substrate and NAD + cofactor inside the film. While a low concentration of silica precursor induced the formation of a rather porous matrix, suitable for both encapsulation and diffusion, increasing TEOS concentrations led to densification of the silica matrix and thicker films [35], which became less suitable for fast diffusion of species from the solution to the enzyme and from the enzyme to the electrode surface. Stability with time Both composition of the film and type of additives have been found to affect significantly the operational and long-term stability of the biocomposite electrodes. This has been studied With the Silica/PDDA/DSDH composite (curve "c") an increase in peak current intensity ca. 65% was first observed, which can be due to changes originating from the film hydration (dissolution/precipitation of silica can occur and the presence of PDDA can influence strongly this silication process [7]). After this initial step, the current reached more stable values, yet continuing to increase slowly with time, varying from 1.65 to 1.8 for the last 3 h of the experiment. Immobilization of DSDH in a pure PDDA film, in the absence of silica precursor, also allowed measuring a current response when the electrode was introduced in solution but this was not stable in successive measurements (increase in the first 30 min of experiment and then continuous decrease down to the initial value). This can be due to lack of mechanical stability of the film, with progressive leaching of the enzyme in solution. Despite Chapter III. Feasibility of dehydrogenase encapsulation in sol-gel matrix 91 chitosan is known to be a suitable matrix for bio-encapsulation [33], the last system Chitosan/PDDA/DSDH gave rise to even more variable response, starting with a sharp increase in peak currents and following with dramatic decrease in the signal intensity after 30 min of use. For all electrodes, a reorganization/modification of the film resulted in an enhancement of the electrochemical response during the first minutes. This probably arises from easier diffusion of the substrate and cofactor species into the composite matrix. However, after this first step, only the sol-gel-derived film resulted in steady-state values of peak currents, all other systems underwent significant degradation of the electrochemical response. This can be ascribed to the rigid character of the inorganic network likely to ensure more durable bioencapsulation, which appears promising for bio-electrochemical applications. The long-term stability of the best systems (sol-gel based biocomposite film electrodes with added polyelectrolytes) has been also considered. PDDA on the DSDH activity is confirmed as no polyelectrolyte in the silica film means no detectable enzymatic activity. In the conditions of electrochemically-assisted deposition, negative charges are present on the silica surface due to silanol deprotonation (the point of zero charge of silica is reported to be in the range 2-3 [26]). The isoelectric point of DSDH is 4.3 [27]. During the protein encapsulation, the interaction between the silica matrix and the protein seems to be not so favourable. The positively charged polyelectrolyte (PDDA) is likely to act as a stabilizing intermediate between the enzyme and the silica surface. to the silica network (~1000-1200 cm -1 ) and the surface silanol groups Si-OH (~900-1000 cm - 1 ), the amide I and II bands of the proteins (1653 and 1538 cm -1 ), the band from the ammonium groups of PDDA (1474 cm -1 ), and the C-H stretching bands of both PDDA and proteins (~2800-3050 cm -1 ) were distinctively observed on the same spectrum. So the FTIR measurement supports the observation made by cyclic voltammetry and indicates that PDDA and proteins are indeed co-encapsulated in the electrodeposited silica network. Optimization of the electrode response The effect of sol composition has been thoroughly studied and the three major parameters The quantity of protein introduced into the sol has a strong influence on the electrode response, a rapid increase was observed from 0.3 to 2.5 mg/mL, which started to level off for 3.3 mg protein/mL of sol. Higher concentrations of protein cannot be used for this process due to a rapid gelification of the sol (proteins or buffer of the suspension facilitate the sol-gel transition). PDDA concentration has also a dramatic influence on the electrode response. In the absence of PDDA, no signal could be measured (see Figure III-12B). The electrode became active in the presence of 1.8 % polyelectrolyte and the electrode response increased then regularly with increasing the polyelectrolyte concentration up to 6.7 %. As for the protein content, a higher quantity of PDDA is difficult to handle as it facilitates also the gelification of the sol before application of the electrolysis potential. The influence of TEOS concentration follows a different trend as it induces first an increase in the electrode response, passes through a maximum and then decreases. The optimal signal was observed for 0.17 M TEOS in the sol. We can assume that the deposition rate is too slow for low TEOS concentrations [35] inducing thereby less efficient protein encapsulation. On the other hand, a higher TEOS concentration can lead to hindered mass transport (see the next section) as a result of thicker films that limit the efficiency of the bio-electrode (restricted diffusion of the reactants). Deposition time has also a strong influence on the electrode response. Thickness / µm Deposition time / s by the PDDA. The optimal sol composition that has been determined using only electrochemical measurements leads also to the more homogeneous sol-gel biocomposite, showing a good incorporation of both protein and polyelectrolyte. Relationship between the reactivity, the permeability and the film stability All the optimization steps reported in the previous section have been made in order to obtain the highest electrochemical response as possible. When considering electrosynthesis applications it is critical to have an intense electrochemical response, but it is also critical to get sufficient long-term stability of the electrode, at least at a time scale compatible with industrial processes. The quantity of TEOS in the starting sol has in principle an influence on the permeability of the layer [35]. Linear sweep voltammetry for ferrocenedimethanol (Fc) oxidation has been performed at a rotating disc electrode with a bare glassy carbon electrode (GC) and GC covered by thin silica films prepared with sols containing 0.08, 0.17 Comparison of the enzyme activity into the film and in solution The variation of the electrochemical response for increasing concentrations of D-sorbitol into the solution has been studied (see Figure III-16). The measurements have been done during the one hour for which the electrode gave a stable electrochemical response. In these conditions, the peak current intensity increases regularly with the D-sorbitol concentration up to 10 mM and starts to level off for higher concentrations. We can extract from these data a K m value of about 3 mM, slightly lower than the K m of about 6 mM observed for the same protein in solution [36]. The electrochemically-assisted deposition allows thus maintaining an enzymatic activity similar as in solution, with a small improvement due to the encapsulation in the silica gel. pH is known to affect strongly the enzymatic activity of DSDH [36]. (phosphate buffer), followed by a sharp decrease above this optimum value. With the oxidation of D-sorbitol, the electrode response increases more regularly from pH 6 to 9 and decreases slightly at pH 10. The behavior of DSDH in the silica layer is very comparable with the data coming from protein activity in solution with an optimal pH of 6.5 for the reduction of fructose and an optimal oxidation of D-sorbitol at pH 9, suggesting again that the electrogenerated silica layer offers a good environment for DSDH encapsulation onto the electrode surface. The immobilized protein exhibits good activity, similar as the free enzyme in solution concerning both the enzymatic kinetics and the sensitivity to pH. Figure III-17. Evolution of the peak current response versus the pH for (A) reduction of 6 mM fructose in 0.1 M Tris-HCl buffer with 1 mM NADH and (B) oxidation of 6 mM Dsorbitol in 0.1 M tris-HCl buffer with 1 mM NAD + . The modified glassy carbon electrode was prepared with a sol containing 0.17 mM TEOS, 6.7 % PDDA and 3.3 mg/mL DSDH. The electrochemically-assisted deposition was done by applying -1.3 V for 60 s. All cyclic voltammograms have been performed 50 mV/s potential scan rate. Co-immobilization of DSDH and diaphorase The direct electrochemical detection of NADH, occurring at high potential compared to E 0 , induces uncontrolled oxidation and rapid deactivation of the bio-molecule. A large number of studies have been and are still devoted to the electro-catalytic oxidation of NADH that allows Chapter III. Feasibility of dehydrogenase encapsulation in sol-gel matrix 102 decreasing this over-potential. An elegant and very versatile regeneration can be obtained with using diaphorase. An additional electron mediator is then used to transfer the electrons from this protein to the electrode. Interestingly, the same diaphorase can be used for both NADH oxidation and NAD + reduction. Among others, ferrocene species can be used for mediating electron transfer for the oxidation of NADH and methylviologen can be used for the reduction of NAD + [20]. giving rise to a similar trend as for DSDH alone, with an optimal pH value located around pH 9. Note that the bi-enzyme system is much more sensitive to small pH variations, which is illustrated by a current decrease by 75 % when passing from pH 9 to pH 9.5 or pH 8. This influence of pH is mainly due to the strong effect of pH on the diaphorase activity (Figure III-19B). The co-immobilization of DSDH and diaphorase does not prevent the efficient communication between the two proteins. NADH can diffuse from one enzymatic center to the other for efficient bioelectrocatalysis. The electrochemically-assisted silica gel deposition is thus an effective method for the elaboration of complex enzymatic layers on electrode surfaces. Extension to the particular case of macroporous electrodes Macroporous electrodes display pores of about 440 nm with well defined interconnections allowing good mass transport [38]. Figure III-20. Preparation of the macroporous gold electrode by electrodeposition through nanospheres assembly. To further point out the interest of the electrochemically-assisted deposition method for bioelectrocatalysis purposes, the above approach was extended to macroporous electrodes The electrochemical response increases significantly from one half layer to three half layers when detecting 1 mM D-sorbitol. This result is consistent with recent observations made with macroporous electrodes modified by a thin silica film containing hemoglobin [17] in which it was shown that the electrochemical signal of hemoglobin as well as the catalytic current for H 2 O 2 detection is increasing significantly with increasing the number of half-layer from 3 to 9. However, the direct transposition of the results from this previous study to the present work is not possible because the compositions of the sols are very different. The sol used for hemoglobin encapsulation contained much less silane precursors (13.6 mM TEOS) than the sol used here for DSDH and diaphorase encapsulation (0.17M). In addition, the sol we developed for dehydrogenase is complex to handle with the macroporous electrodes because of the presence of the polyelectrolyte. These experiments point out the interest and the complexity of surface modification of such porous material with an elaborated sol-gel biocomposite. Additional optimization will be necessary to carefully control the film deposition inside the macropores. The optimal thickness of the macroporous electrode will also have to be defined with respect to the application in electrosynthesis. Conclusion The first part of the work has pointed out the importance of adding positively-charged polyelectrolytes into sol-gel-derived films doped with dehydrogenase enzymes for providing a good environment for encapsulation of the biomolecules in an active form. Among the tested additives, PDDA offered the best results. The improved behavior in the presence of polyelectrolyte was also observed for other kinds of thin films (i.e., based on chitosan). Then, Chapter IV. Co-immobilization of dehydrogenase and cofactor in sol-gel matrix In this chapter, successful strategies for dehydrogenase and cofactor co-immobilization in sol-gel films have been developed by both drop-coating and electrochemically-assisted deposition. First of all, we compare various strategies directed to the durable immobilization of NAD + /NADH cofactors in biocompatible sol-gel matrices encapsulating a bi-enzymatic system (a dehydrogenase and a diaphorase, this latter being useful to the safe regeneration of the cofactor), which were deposited by drop-coating as thin films onto glassy carbon electrode surfaces. These strategies are (1) the "simple" entrapment of NAD + in the sol-gel matrix, alone or in the presence of carbon nanotubes; (2) the formation of interpenetrated organicinorganic networks using a high molecular weight NAD derivative (NAD-dextran); (3) the chemical attachment of NAD + to the silica matrix using glycidoxypropylsilane in the course of the sol-gel process (in smooth chemical conditions). The third approach based on chemical bonding of the cofactor (which was checked by infrared spectroscopy) led to much better performance in terms of long-term stability of the electrochemical response. The coimmobilization of DSDH, diaphorase (DI) and NAD + was then obtained by electrochemically-assisted deposition. Finally, the functional layer has been successfully deposited in macroporous gold electrodes and applied for the oxidation of D-sorbitol. Introduction The main difficulty to fabricate reagentless dehydrogenase-based bioelectrodes is that the cofactor must be immobilized and regenerated in a stable and active way. In most studies, the native cofactor is added to the starting electrolyte before the enzymatic reaction. Despite the good performance of these methods, a maybe drawback exists: the operation is not only complicated but also involves high cost because the expensive cofactor cannot be reused. One way to overcome this problem is to co-immobilize cofactor and dehydrogenase in the sensing layer, but the difficulty is that the water soluble cofactor is a relatively small molecule, so is likely to diffuse away from the electrode surface into the solution, thus limiting the long-time durability of the modified electrode. Different strategies have been proposed for cofactor immobilization on electrode surfaces. The simple encapsulation leads to rapid leaching of the cofactor in the solution during the electrochemical operation and can only be considered for disposable sensors [1]. One possibility to improve the stability of the immobilization is the chemical attachment of the cofactor to a macromolecule that can be encapsulated or immobilized on the electrode without leaching. Dextran [2,3,4], PEG [5], Chitosan [6], and PEI [7], have been reported to allow this cofactor immobilization for biosensor applications. The stability of some systems has been studied and the bio-electrode can in some cases be operating for several days. One limitation of this approach comes from the rather complex modification, especially if proteins [6] or mediator [7] are also immobilized on the same macromolecule. This induces a significant cost and limits the number of functionalized groups in the layer; More recently the adsorption of NAD + on carbon nanotubes was also proposed as a new strategy for the biomolecules immobilization [8]. Up to now, sol-gel chemistry was only used for enzyme and mediator immobilization [1, 9, 10], and was not considered for durable cofactor immobilization. During operation of bioelectrodes containing coimmobilized dehydrogenase and cofactor, the cofactor must be detected or regenerated electrochemically and this operation has to be done with using a mediator. The demand for electrocatalytic detection of NAD + cofactor comes from the nature of this molecule, free diffusing in the living cell and for which a high electrochemical overpotential is observed for both oxidation and reduction reactions. The molecule is protected from side reaction, but the direct electrochemical detection at high overpotential can lead to the irreversible degradation of the compound and to the simultaneous detection of interfering species. Many strategies for electrocatalytic detection of the cofactor have been developed. Organic mediators [11,12,13,14,15] , carbon nanotubes (CNTs) [10,16,17,18,19] or even gold nanoparticles [20] have been proposed to recycle the NADH back to the enzymatically active NAD + . The cofactor can also be efficiently regenerated by the use of diaphorase in the presence of several molecular mediators, metal complexes [21 , 22 ], quinones and flavins [ 23,24], and also viologens [25 ]. This later approach being very appealing when smooth cofactor regeneration is needed for improving the long term stability of the device, notably if NAD + is immobilized for reagentless device. In this study, we will use diaphorase (DI) in combination with ferrocenedimethanol for cofactor regeneration. We have compared here different strategies for cofactor immobilization in sol-gel matrix, i.e. simple encapsulation of the native cofactor, encapsulation of NAD-Dextran, adsorption on carbon nanotubes introduced in the sol-gel matrix and finally the use of glycidoxypropylsilane (GPS) as additive. This later molecule displays an epoxide ring susceptible to react with the adenine moieties of the cofactor. According to the literature, such coupling has to be prepared in basic solution in order to get the most active biomolecules [ 26 ], but sol-gel bioencapsulation can only be obtained in neutral conditions. We will show here that despite this limitation, the confinement of the linked cofactor in the sol-gel matrix with using GPS allows good activity to be detected with high stability. Evidence of reaction between cofactor and GPS were obtained by FTIR. The co-encapsulation of DSDH and NAD was then evaluated by electrochemically-assisted deposition. The process and the sol composition have been optimized on flat glassy carbon and gold electrodes before being applied to macroporous gold electrodes. At the end, the electrodeposited bio-composite containing DSDH, diaphorase and cofactor has been tested for the electrochemical oxidation of D-sorbitol and a comparison was made between flat and macroporous gold electrodes. Co-immobilization of dehydrogenase and cofactor in solgel matrix by drop-coating "Simple" physical entrapment of the cofactor in the sol-gel film A straightforward way to associate the cofactor to the biocomposite layer is its addition to the starting sol so that, after gelification, it would be physically entrapped in the silica matrix. In Figure IV-2A shows the electrochemical response obtained with the modified electrode prepared with PEI as polyelectrolyte additive. Before addition of D-sorbitol, a well defined electrochemical signal due to ferrocenedimethanol could be measured (plain line). The addition of D-sorbitol in the solution from 2 to 16 mM led to a significant increase in the current response (dashed lines) and typical S-shape curves expected for a bio-catalytic process. The same experiment was performed with PDDA or PAA as additive, but no electrochemical activities were observed (see Figure IV-2B&C). PEI was already reported in the literature to allow efficient immobilization of several dehydrogenase, alcohol [27], D-lactate [28], or glucose dehydrogenase [29]. The effect of PEI for co-immobilization could be explained by the formation of "conjugates" by electrostatic attraction among the cationic polymer and the negatively charged dehydrogenase and NAD + . These "conjugates" could make the enzyme more rigid and stable against unfolding, presenting a more stable conformation, which also could enrich the cofactor in the vicinity of to ensure cofactor entrapment in an active form in the films whereas other polyelectrolytes did not. A possible explanation can be related to the fact that PEI is a branched polyelectrolyte while the others are linear macromolecules. for the biocatalytic event. This explanation is also supported by the faster signal decrease when operating under convective conditions (response vanishing in less than 2 hours). So, in spite of exhibiting a rather good bioelectrocatalytic response, the system based on the simple encapsulation of NAD + into the sol-gel biocomposite (GCE/TEOS/PEI/(DSDH+DI)/NAD + ) Figure IV-3. Evolution of the peak current intensity recorded for successive analyses of 10 mM D-sorbitol solutions at distinct periods of time, using GCE modified with the same sol as does not allow to get long-term stability. Effect of carbon nanotubes on the electrode stability One knows from a recent report by Zhou et al. [8] that noncovalent attachment of NAD + to carbon nanotubes is possible by taking advantage of the strong π-π stacking interaction between the adenine moiety in the NAD + molecule and the nanotube surface. We have thus evaluated if such interaction could contribute to improve the long-term stability of cofactor immobilization in our sol-gel biocomposites. In the present case, however, it was necessary to use carboxylate-functionalized SWCNTs because crude carbon nanotubes cannot be easily dispersed in the water-based sols utilized here. The first straightforward attempt was to incorporate SWCNTs in the starting sol containing all other ingredients (TEOS, PEI, enzymes, NAD + ) so that they are expected to be dispersed Encapsulation of NAD-Dextran Chemical attachment of the cofactor to a macromolecule is the most usual protocol for their immobilization on electrode surface [5,6,7]. Cofactor immobilization via the formation of interpenetrated organic-inorganic networks using a high molecular weight NAD + derivative (NAD-dextran) was also tested. The commercially available NAD-Dextran compound is indeed known to be active when associated to dehydrogenases [2,3,4]. The macromolecule was introduced in the sol-gel matrix following a similar protocol as previous experiments. The electrode response increased during the first hour of experiment and was then stable for more than 6 hours. This experiment was performed with using ferrocenedimethanol in solution as electron mediator between DI and the electrode surface for recycling the immobilized cofactor. (A) (B) Figure IV-5. (A) Cyclic voltammograms obtained with GCE/TEOS/PEI/(DSDH+DI)/NADdextran in the absence of D-sorbitol (solid lines) and in the presence of D-sorbitol from 2 to 16 mM (dashed line). (B) Evolution of the peak current intensity recorded for successive analyses of 10 mM D-sorbitol solutions at distinct periods of time, using GCE modified with the same sol as (A). Cyclic voltammograms have been performed in Tris-HCl buffer (pH 9) containing 0.1 mM FDM. Potential scan rate was 50 mV/s. Covalent attachment to the silica matrix with Glycidoxypropylsilane Glycidoxypropyl-trimethoxysilane (GPS) is a well known compound in sol-gel chemistry, widely used as adhesive layer in composite material, and can find application as part of protective coatings. This compound was recently proposed separately for chemical attachment of cofactor of silica nanoparticles [30] or protein immobilization into macroporous silica monolith [31]. Both approaches used first chemical grafting of the GPS on the silica substrate before taking advantage of the epoxide ring for further functionalization with the enzyme or the cofactor. GPS was also successfully used as silica precursor for the encapsulation of enzymes [32,33]. In these previous reports, however, no attempt was made to get in situ According to the literature, coupling an epoxy ring with the adenine moieties of the cofactor should be made in basic medium in order to get the most active biomolecules [26], but such conditions are prevented here as sol-gel bio-encapsulation can only be obtained in neutral conditions. We will show hereafter that despite this limitation, the confinement of the linked cofactor in the sol-gel matrix with using GPS allows good activity to be detected with high stability. Evidence of reaction between NAD + and GPS can be obtained indirectly by electrochemistry and directly by infrared spectroscopy. Electrochemical evidences of NAD + immobilization Here GPS was first let to react with NAD + for at least 12 hours, before to be introduced as NAD-GPS educt into the sol that was deposited on the electrode surface along with other components (TEOS, PEI, enzymes). First of all, GPS provides to the film a very good adhesion to the glassy carbon electrode surface, because of the adhesive properties of this organosilane [ 34 ]. Typical amperometric and voltammetric responses of ATR-FTIR spectroscopy ATR-FTIR monitoring of GPS hydrolysis Because the methoxy groups of GPS can easily hydrolyze in aqueous media, it was important to identify the characteristic spectral features occurring during the hydrolysis reaction before the analysis of the spectra in the presence of NAD + . All spectra are discussed in the more useful 1800-700 cm -1 region. Figure IV-7. (a)-(j) Time evolution of ATR-FTIR spectra during 18 hours GPS hydrolysis in Tris-HCl buffer (pH=7.5) on a diamond ATR crystal (one spectrum every 2 hours). (k) ATR-FTIR spectrum of GPS in water after 1 h 40 min. hydrolysis on a diamond crystal. Offsets of spectra are used for clarity. One interesting band absorbs at 911 cm -1 . It is characteristic of the epoxide group and it was assigned to C-O, C-C stretchings and C-O torsion modes of this little cycle. One can note that the wavenumber of this band do not change in the course of hydrolysis. However, one can observe a quite big intensity increase with the increase of hydrolysis time. This is probably due a drastic change in the polarity of GPS with hydrolysis that leads to higher transition moments for these vibrational modes. This led us to conclude that the epoxide group did not open in the Tris-HCl buffer used in the study. The hydrolysis reaction in Tris-HCl buffer is also slower than in pure water, since after 18 hours of reaction the spectrum shows a mixture of the spectra of pure GPS and hydrolyzed GPS (Figure j,k)). It was probably because of some protecting interactions between the Si(OCH 3 ) 3 group of GPS and the amino or hydroxide groups of the Tris molecule. Figure IV-8. ATR-FTIR spectra on a diamond ATR crystal of (a) pure GPS, (b) GPS and PEI(5%) after about 15 h in Tris-HCl buffer (pH=7.5), (c) GPS after about 14 hours in Tris- HCl buffer, (d) GPS after 1h40 in pure water and (e) and 5 % PEI in Tris-HCl buffer at pH 7.5. Offsets of spectra are used for clarity. ATR-FTIR spectrum of NAD + Figure IV-9 shows the spectrum of NAD + in Tris-HCl buffer with main group assignments determined according to the literature [37,38,39,40,41] . Wavenumber (cm -1 ) Figure IV-9. ATR-FTIR spectrum of NAD + (0.3 M) in Tris-HCl buffer (pH=7.5) on a diamond ATR crystal. Evidence of reaction between cofactor and GPS Figure IV-10 shows the time-evolution of the ATR-FTIR spectra during 18 hours reaction of GPS with NAD + (1:1 molar ratio) in Tris-HCl buffer. Here, C=O stretching band at 1696 cm -1 from the nicotinamide of NAD + does not change during the course of the reaction. It can be concluded that this group do not react with GPS (such reaction would have prevented electrocatalytic activity). Weak bands that absorb between 1370 and 1310 cm -1 are characteristic of the adenine part of NAD + [39]. The profile of these weak, poorly resolved bands change with increasing time. This suggested that the adenine group react with GPS, but it is not possible to identify precisely the reacting bonds. Figure IV-10. Time evolution of ATR-FTIR spectra during 18 hours reaction of GPS (0.3 M) with NAD + (0.3 M) in Tris-HCl buffer (pH=7.5) on a diamond ATR crystal (one spectrum every 5 min. during 30 min. then at 1 h, 2 h, 3 h, 5 h, 8 h, 11 h, 14 h, 18 h of reaction). Inset plot shows a detail of this spectrum between 870 and 940 cm -1 . The scheme shows the possible reaction between N-1 of NAD + and the epoxide ring of GPS. The intensity of the bands between 1300 and 1000 cm -1 increases with increasing time. Even though GPS was already partly hydrolyzed when FTIR monitoring started (bands around 1100 and at 1015 cm -1 ), it shows the continuing hydrolysis and rearrangement of hydrolyzed species during the time of the reaction monitoring (note that PO 2 elongation from phosphate groups can also be found in this region). One interesting spectral feature is the intensity decrease of the band at 911 cm -1 that is assigned to the epoxide group of GPS. It shows that this group is opened when in the presence of NAD + while no opening of this epoxide ring was observed in Tris-HCl buffer alone (Figure IV-7.). The intensity of this band decreases during 8 hours of reaction, and then stays almost constant showing the end of the reaction. After 8 hours, a residual weak band stays with constant intensity on the spectrum. It is due to a ribose νC-C mode from NAD + (see Figure IV-9). According to previous work from Fuller et al [26], the conditions that were used here should lead to the alkylation of NAD + at position N-1 (see scheme on Figure IV-10). This compound displays very low activity with dehydrogenase [42]. For optimal enzymatic activity, the cofactor should better be attached through the C-6 amino group rather than the N-1 ring nitrogen. This is usually obtained in basic medium using hours of reaction. But such condition cannot be used here as it would induce a rapid gelification and would make impossible the further encapsulation of proteins. Despite this possible limitation, the high concentration of Figure IV-12. Time evolution of the ATR-FTIR spectra during the 18 hours reaction of GPS (0.3 M) with NADH (0.3 M) in Tris-HCl buffer (pH=7.5) on a diamond ATR crystal (one spectrum at 0, 15, 30 min and at 1, 2, 5, 8, 11, 14 and 18 h of reaction). Inset plot shows a detail of the same spectra between 940 and 870 cm -1 . A final control experiment was made by replacing NAD + by NADH for the reaction with GPS following the same protocol. Indeed, the electrochemical biosensor/bioreactor is expected to operate independently on the initial form of the cofactor (because the required form is continuously regenerated via the electron mediator). This time, no decrease was observed at 911 cm -1 (Figure IV-12). At the opposite, the band ascribed to GPS was slightly increasing due to the hydrolysis of alkoxysilane moieties. A similar effect was observed during the simple hydrolysis of GPS in Tris-HCl buffer (Figure IV-7). The reduced form of the cofactor did not react with the epoxide ring in the smooth condition of the sol-gel protocol. This absence of alkylation was confirmed by electrochemical measurements. The electrode was responding to D-sorbitol additions, but the electrochemical response was not stable and disappeared in less than 3 hours (Figure IV-11). This trend is comparable to the one observed with the film prepared in the absence of GPS (Figure IV-6B, curve b). In addition to the attachment of the cofactor, the different components participate to the good stability of the sol-gel film. By comparison, PEI was here replaced by PDDA. As mentioned in the first section of the discussion, this later polyelectrolyte did not lead to electrocatalytic activity when NAD + is present in the film. In addition, it was observed that PDDA led to swelling films with limited stability (few hours). Only the association between PEI and GPS permitted to obtain good film stability with good catalytic activity. FTIR was also used to monitor an eventual reaction between PEI and GPS but no evidence of coupling between them could be observed by following the intensity of the signal at 911 cm -1 (Figure IV-8). Contrarily to PAA or PDDA that are linear polymer, PEI is a branched polymer and we suppose it can be better distributed in the sol-gel layer. This would be the main explanation of the good stability of this bio-organic-inorganic hybrid sol-gel layer, but reaction between PEI and GPS can not be totally exclued during drying and aging that were not monitored by FTIR. Co-immobilization dehydrogenase and cofactor in electrodeposited sol-gel thin film It was demonstrated in section 2 of this chapter that dehydrogenase and cofactor can be readily co-immobilized in silica sol-gel films on electrode surfaces, showing effective bioelectrochemical activity if using PEI and GPS as additive in the sol, but those works were performed using drop-coating to deposit sol-gel films. In the present section, this approach has been extended to electrochemically-assisted deposition of the sol-gel layer. The feasibility on GCE First of all, the co-immobilization of DSDH, DI and NAD-GPS by electrochemicallyassisted deposition is evaluated on glassy carbon electrode. Before addition of D-sorbitol, a well defined electrochemical signal due to ferrocenedimethanol could be observed (plain line). The addition of D-sorbitol in the solution from 2 to 6 mM led to a significant increase in the current response (dashed lines). The electrochemically-assisted silica gel deposition is thus an effective method for the coimmobilization of DSDH and NAD on electrode surfaces. Note that PEI was here used as polyelectrolyte for stabilizing the protein in the sol-gel film. Figure IV-13. Cyclic voltammograms obtained using GCE modified by TEOS/NAD-GPS/PEI/(DSDH+DI) film in the absence and presence of D-sorbitol. Films have been deposited by electrolysis at -1.3 V for 60 s with a sol containing 0.15M TEOS, 14 mM NAD-GPS, 2.3 mg/mL DSDH, 0.76 mg/mL DI and 1.5% PEI. Cyclic voltammograms have been performed in 0.1 M Tris-HCl buffer, in the presence of 0.1m M FDM. Potential scan rate was 50 mV/s. The feasibility on flat gold electrode The co-immobilization of DSDH, DI and NAD-GPS by electrochemically-assisted deposition was then evaluated on flat gold electrode. We first started with the same conditions as used for the experiment on glassy carbon (Figure IV-13), but these conditions did not allow to be extended to gold electrode without changing. It was in fact necessary to introduce a small amount of PDDA in the sol for improving the adhesion of the film on the gold electrode. Cofactor immobilization together with DSDH and diaphorase was successfully achieved in sol-gel film prepared by electrodeposition. As for the proteins, the appropriate choice of polyelectrolyte and concentration in the sol was critical for getting the optimal film deposition and electrocatalytic activity. A combination of PEI and PDDA was used for electrochemically-assisted deposition of this sol-gel biocomposite with co-immobilized DSDH, diaphorase and cofactor on gold surface. Such process has been also applied to the functionalization of macroporous electrodes. Figure IV-14. Cyclic voltammograms obtained using gold electrode modified by TEOS/NAD-GPS/PEI/PDDA/(DSDH+DI) film in the absence and presence of D-sorbitol. Films have been deposited by electrolysis at -1.3 V for 60 s with a sol containing 0.15M TEOS, 14 mM NAD-GPS, 2.3 mg/mL DSDH, 0.76 mg/mL DI and 1 % PEI and 0.5% PDDA. Cyclic voltammograms have been performed in 0.1 M Tris-HCl buffer, in the presence of 0.1m M FDM. Potential scan rate was 50 mV/s. Extension to the macroporous gold electrode The above approach was extended to macroporous gold electrode displaying a much larger electroactive surface area in comparison to the geometric one. A deposition condition -1.3 V, 60 s (versus the Ag/AgCl reference electrode) was used in previous experiment. Despite the good catalytic response was obtained on the flat gold electrode, the deposited sol-gel film was rather thick. In macroporous electrodes the situation is even more complex as the pore interconnections can be rapidly blocked by the electrogenerated silica gel layer. Low potential and deposition times are here needed to prevent the rapid macropore clogging. Here, films have been deposited by electrolysis at -1.1 V for 30 s. polyetheleneimine additive (PEI). All the operations are done in smooth conditions compatible with the sol-gel bio-encapsulation process. By comparison with the simple encapsulation of NAD + or NAD-dextran or adsorption of NAD + on carbon nanotubes, the strategy with GPS is cheaper, simpler to implement and leads to more stable sol-gel films. Conclusion The electrode shows good stability under stirring for more than 14 hours. The sol-gel biocomposite can be deposited on the electrode surface either by evaporation of the sol or by sol electrolysis (i.e. electrochemically-assisted deposition). Finally, the functional layer has been successfully deposited in macroporous gold electrodes and applied for the oxidation of D-sorbitol. The macroporous texture of the gold electrode improves significantly the catalytic efficiency of the sol-gel biocomposite in comparison with a flat gold electrode. By comparison with the methods previously employed, the strategy that is described here offers a simple, cheap and clean way but effective approach to the development of integrated dehydrogenase-based bio-electrochemical devices. Chapter V. Mediator immobilization in sol-gel matrix and co-immobilization with dehydrogenase and cofactor In this chapter, diaphorase has been used in addition to the dehydrogenase in order to ensure the safe regeneration of the NAD + cofactor, using ferrocene species or osmium polymer as electron shuttles between the diaphorase and the electrode. First of all, the immobilization of ferrocene species and osmium polymer in sol-gel matrix was studied. The influence of GPS as additives for the mediator immobilization was described. Then, the feasibility of coimmobilization base on sol-gel film was evaluated by one step drop-coating. In addition, electrochemically-assisted deposition of sol-gel thin film to co-immobilize dehydrogenase, diaphorase, the cofactor NAD + and an electron mediator was also investigated. Introduction Near 300 dehydrogenases are known to catalyze the oxidation of a variety of substrates. However, there are some difficulties in the development of dehydrogenase-based reagentless devices because of the high over-potential for the direct oxidation of NADH that results in the formation of non-active form of cofactors. Some of these difficulties can be minimized through the use of redox mediators that shuttle electrons between the cofactor and the electrode surface [ 1]. The oxidation of NADH can also be achieved using the enzyme diaphorase (DI). The association of diaphorase with mediators to regenerate cofactor at the electrode surface has already been reported [2,3,4]. The regeneration of cofactor with the participation of diaphorase ensures that only enzymatically active cofactors are produced selectively. Mediator is necessary because the diaphorase shows only very slow rates of electron transfer to electrode surfaces [5]. Ferrocene species [3] are by far the most common among the mediators used in combination with diaphorase for cofactor regeneration. Osmium redox polymers have also been proposed recently for such applications [4]. The difficulty to elaborate reagent free device comes from the development of a suitable matrix in which all component of the electrochemical detection are immobilized in a stable form, i.e. the enzyme(s), the cofactor and the electrocatalytic system for cofactor detection and regeneration. Sol-gel materials are known to have very promising features as immobilization matrices, because they can be prepared at room temperature and can retain the catalytic activity of the biomolecules [6,7,8]. By changing the experimental conditions of the sol-gel process, gel structures with related characteristics can be obtained. A very large number of enzymes have been trapped within sol-gel films, showing that they usually retain their catalytic activity and can even be protected against degradation [9,10,11]. Moreover, during the sol-gel process, additional substances such as redox mediators, e.g. Ruthenium complex [ 12 ], toluidine blue [ 13 ], thionin [ 14 ], or ferrocene [ 15 ], can also be easily incorporated into the final structure. Despite the long history of using sol-gel material for enzymes immobilization and mediator immobilization, no work has been reported regarding the co-immobilization of enzyme, cofactor and mediator by using a sol-gel material to construct this reagentless device. In this work, a series of strategies allowing dehydrogenase, cofactor and electron mediator coimmobilization in sol-gel thin films have been investigated. As illustrated on Scheme V-1, oxidation of the enzymatic substrate by the immobilized dehydrogenase induces NAD + reduction to NADH. Diaphorase catalyses then the oxidation of the immobilized NADH back to NAD + and the electron transfer from the diaphorase to the glassy carbon electrode surface is carried out by the immobilized mediators. These mediator species can be ferrocene species (ferrocene linked to a poly(ethylenimine) or a ferrocene-silane) or an osmium polymer. First of all, the immobilization ferrocene species and osmium polymer in the sol-gel matrix are shown. The influence of glycidoxypropylsilane (GPS) as additive to improve the long term stability of the mediator immobilization is studied. Then, the feasibility of co-immobilization of dehydrogenase, diaphorase and cofactor in the sol-gel film is evaluated by one step drop-coating. Attempt to perform this co-immobilization by electrochemically-assisted sol-gel deposition is finally presented. Scheme V-1. Illustration of the electrochemical pathway used for the detection of the dehydrogenase enzymatic substrate. Fc-PEI as co-immobilized mediator Ferrocene (Fc) species are commonly used in combination with diaphorase for cofactor regeneration. The great interest in ferrocene derivatives originates from their fast electron transfer properties, their pH-independent redox potentials, and their efficient electrochemical reversibility [16]. In this process, NADH reacts with diaphorase and the electrochemically generated ferricinium ions to produce NAD + and ferrocene species that can be re-oxidized in the electrocatalytic scheme (see Scheme V-1). Co-immobilization in drop-coated sol-gel film 2.1.1 Effect of GPS on Fc-PEI immobilization The ferricinium ion is much more soluble than ferrocene itself [17]. Therefore, the main problem in using Fc is a gradual leakage, from the electrode surface, of the mediator via its oxidized form. One way to overcome the above problem is attaching the Fc to a polymer. Poly(ethylenimine) (PEI) is an attractive candidate to serve as a redox polymer backbone for a high degree of functional density on the polymer to facilitate modification and high segmental mobility. Liu et al. coupled ferrocene carboxaldehyde to PEI and incorporated these redox polymers into polyelectrolyte multilayer films via a layer-by-layer deposition technique [18]. Hodak et al. reported the redox mediation of glucose oxidase (GOx) in a self-assembled structure of cationic protonated poly-(allylamine) modified by ferrocene (PAA-Fc) and anionic GOx deposited electrostatically layer-by-layer on negatively charged alkanethiolmodified Au surfaces [19]. Zheng et al. prepared PEI with attaching both the redox mediator (ferrocene) and the native cofactor (NAD + ), and then the modified polymer was immobilized with the NAD-dependent dehydrogenase to construct reagentless amperometric biosensor [20]. Here, the synthesized Fc-PEI is introduced into the sol-gel matrix to construct the mediator modified electrode with the idea to form an interpenetrated organic-inorganic hybrid ensuring durable immobilization of the mediator. Figure V-1A shows cyclic voltammograms recorded with a GCE modified by TEOS sol-gel film containing Fc-PEI. A well-defined CV signal can be observed, but it is not stable on multiple potential scan due to the rapid leaching of the mediator. Unfortunately, the leakage of Fc from the electrode can not be prevented by the chemical attachement of Fc to PEI. The chemical structure of the alkoxysilane precursors and the composition of the sol-gel mixture influenced the roughness, the size and the distribution of pores in the sol-gel films, which are important criteria for both efficient enzyme and mediator encapsulation. It is reported that the formation of sol-gel from GPS precursor comparing with the other (organol)alkoxyoxysilane precursors lead to the formation of a more uniform thin film with smaller pores [21,22], which would allow stable enzyme and mediator immobilization. Here, we try to introduce GPS inside the sol-gel matrix to improve the stability of mediator immobilization with the idea to encapsulate more effectively Fc-PEI to the structure via favourable interaction between amino group and epoxide function of GPS. Co-immobilization of Fc-PEI, DSDH, DI and cofactor Here, the co-immobilization of Fc-PEI, cofactor, DSDH and diaphorase inside the silica gel has been investigated. Fc-PEI, GPS functionalized cofactor (NAD-GPS), DSDH and 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0, I / µA E versus Ag/AgCl / V 0,0 0,1 0,2 0,3 0,4 An increasing electrocatalytic response is obtained upon the addition of 0.2 mM D-sorbitol. The electrocatalytic response is about three times lower with the mediator inside the film by comparison with the response with the mediator in solution (Figure IV-6). Figure V-2C shows the corresponding calibration plot, the current intensity increases regularly with the Dsorbitol concentration up to 2.2 mM and starts to level off for higher concentrations. We have here studied the electrode stability (Figure V-3). The response to 2 mM Dsorbitol of electrodes prepared the same way, with (curve a) and without GPS (curve b) are monitored during 14 hours. In the absence of GPS, the electrode current decreases dramatically during the first 500 s (before the addition of D-sorbitol), due to the rapide loss of Ferrocene and NAD + in the solution. After the addition of D-sorbitol, only faintly visible current increase is obtained, that reach quickly a current value close to zero. At the opposite, the electrode prepared with GPS display a very good stability, and just a limited (few percents) decrease in current intensity was observed, possibly due to loss of enzymatic activity during the long operation. GPS has here two functions, it allows to chemically attach NAD + to the silica matrix and it stabilizes the overall assembly for improved long-term stability. Co-immobilization in electrogenerated sol-gel films 2.2. Immobilization of Fc-PEI in electrodeposited sol-gel thin films The electrochemically-assisted deposition of silica thin film involves the local increase of the pH that induces rapid gelification at the electrode surface. In order to develop a mediator immobilization method compatible to macroporous electrodes, we try to extend the previous droping/evaporation approach to the electrochemically-assisted deposition of sol-gel film. ferrocene was observed in the cyclic voltammograms. At the contrary to drop-coating, the electrodeposition involves a significant increase of pH during the gelification. It seems that these conditions limit the homogeneous incorporation of the ferrocene species in the sol-gel film, and limit strongly its electrochemical detection. Figure V-4. Cyclic voltammograms recorded with a GCE modified by electrodeposited TEOS/GPS/PEI/Fc-PEI in the 0.1 M Tris-HCl buffer (pH 9) at a scan rate of 50 mV/s, scan cycle, 10. Films have been deposited by electrolysis at -1.3 V for 60 s with a TEOS/GPS sol containing Fc-PEI. Co-immobilization in electrodeposited sol-gel thin film It has been demonstrated in chapter IV that dehydrogenase and cofactor can be readily encapsulated in electrochemically-assisted deposition of sol-gel films on electrode surfaces. Although the immobilization of Fc-PEI by electrodepostion was not successful, we here try to co-immobilize DSDH, NAD-GPS and Fc-PEI in electrochemically-assisted deposition sol-gel films. Indeed, the introduction of enzyme and cofactor in the starting sol solution may bring some changes of the film properties. However, only faintly visible electrochemical signal of ferrocene could be observed in the absence of D-sorbitol (Figure V-5). The addition of Dsorbitol did not induce any increase of the anodic current. Oppositely, the electrochemical signal of ferrocene decreased strongly. Obviously the differences in gel texture/structure between the film obtained by dropcoating and electrodeposition lead to different electrochemical behaviour. In order to improve the incorporation of the ferrocene species in the electrogenerated sol-gel film, another strategy involving Fc-silane has been tested and is presented in the next section. Fc-silane as co-immobilized mediator The synthesis of sol-gel silica material [23,24,25] has become a vast area of research during the last few years. The silica framework can be synthesized in part from alkoxide precursors containing a nonhydrolyzable Si-C bond, i.e. R 4-x Si(OR') x , where R represents the desired reagent or functional group. One of the possible applications of such materials in the development of sensors is the attachment of the redox material to the surface of electrode. Audebert et al. [25] developed a modified electrode from organic-inorganic hybrid gels formed by hydrolysis-polycondensation of some trimethoxysilylferrocenes. In this work, the used organic-inorganic hybrid gels contain ferrocene units covalently bonded inside a silica network. There is a great potential to study such ferrocene linked sol-gel silica material for mediated biosensor applications. Some works on ferrocene based sol-gel sensors are available [26,27,28]. Here, Fc-silane (see the top of Figure V-6) comes to our consideration due to the problems of mediator immobilization through electrodepostion as demonstrated above. We expect Ferrocene functionalized with silane could improve mediator immobilization through electrodeposition. As previously, the first tests have been performed on the basis of dropcoated sol-gel film before to consider electrodeposition. Co-immobilization of Fc-silane, DSDH, DI and cofactor The ferrocene-silane compound has been then associated with the other components of the reagentless device. First, a sol solution with Fc-silane as co-condensation precursors was prepared, then, GPS functionalized cofactor, DSDH and diaphorase were introduced into this Co-immobilization in electrogenerated sol-gel film Immobilization of Fc-silane in electrodeposited sol-gel thin film Although several reports on silylated ferrocene derivatives immobilized on electrode surfaces are available [25,26,27,28], most of them were prepared by drop-coating or spincoating. To our knowledge, no work has been reported regarding ferrocene-silane derivatives 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 no.sorbitol_C A7mMsorbitol_B I / µA E versus Ag/AgCl / V 0 100 200 300 400 500 0,0 Good catalytic characteristic to D-sorbitol oxidation has been indicated in previous experiments with both proteins and cofactor immobilized in the electrodeposited sol-gel layer (Chapter IV section 3). In order to analyse this negative result, a similar film has been prepared without immobilized NAD + (TEOS/GPS/Fc-silane/PEI/DSDH/DI/NAD-GPS) (Figure V-10). In this experiment, the enzyme and the mediator were co-immobilized on the electrode surface, and the cofactor was introduced into the solution before electrochemical experiments. In the absence of D-sorbitol, only the reversible electrochemical signal of ferrocene was observed. The addition of D-sorbitol into the solution does not lead to noticeable modification of the current response and no increase of peak current can be observed at the potential of NADH oxidation. Obviously, the co-immobilization of the enzymes and mediator in the sol-gel film by electrodepostion does not exhibit electrochemically detectable activity. The communication between of immobilized ferrocene and diaphorase was not sufficient to allow electro-catalysis. The different gel texture expected for film obtained by evaporation (drop-coating) or electrodeposition could explain the difference observed between these two kinds of electrode. The co-immobilization of dehydrogenase, cofactor and electron mediator (Fc-PEI or Fcsilane) in sol-gel matrix by drop-coating was successful. However, such co-immobilization by by using electrogenerated sol-gel thin films was not possible. For Fc-PEI, no electrochemical signal of ferrocen was observed after the immobilization by electrodepostion. The incorporation of ferrocene in the electrogenerated sol-gel film could be improved by using Fcsilane, but no electrocatalysis was observed. These negative results obtained with ferrocene species in sol-gel electrodeposition led us to use different redox polymer for improving the connection between diaphorase and the electrode surface. Due to electrochemical reversibility, high electron transfer rate constant and stability of the Os-complexes, Osmium polymer has been recently proposed for such application and has been tested here. Co-immobilization in electrodeposited sol-gel thin film Os-polymer as co-immobilized mediator Flexible osmium redox polymers attracted the attention of a number of researchers due to the efficient electron shuttling properties combined with the polymeric structure, promoting a stable adsorption, as well as a possibility to immobilize the enzyme into multiple layers [29,30] on the electrode surface. Osmium polymers can serve as mediator for a wide range of oxidases to fabricate biosensor, such as, glucose oxidase [31], lactate oxidase [32] and alcohol oxidase [33]. Osmium polymer as mediator can also be used to fabricate dehydrogenase 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 -0,4 -0,2 0,0 0,2 developed reagentless amperometric formaldehyde-selective biosensors based on the recombinant yeast formaldehyde dehydrogenase. In this method, the polymer layers simultaneously served as a matrix for keeping the negatively charged cofactors and glutathione in the bioactive layer [36]. Here, we tested four kinds of synthesized Os-polymer (see Immobilization of DI The catalytic characteristic of the immobilized osmium was evaluated through the simple co-encapsulation of osmium and diaphorase. -0,1 0,0 0,1 0,2 0,3 0,4 0,5 -0,1 0,0 0,1 0,2 0,3 0,4 0,5 -1 The conclusion for Os-polymer is the same as for ferrocene species. Co-immobilization of all components (dehydrogenase, cofactor and electron mediator) in the sol-gel films can be successfully achieved by using drop-coating. However, the co-immobilization by using electrogenerated sol-gel thin films was not so successful due to the same limitation with the mediator immobilization. Conclusion A series of strategies for dehydrogenase, cofactor and electron mediator co-immobilization in sol-gel thin films have been developed. First of all, ferrocene species (Fc-PEI and Fc-silane) or osmium polymers as mediators between diaphorase and the electrode have been successfully immobilized on the electrode within the sol-gel film, which allows the smooth regeneration of the cofactor. The importance of introducing GPS as an additive into the TEOS sol-gel-derived films has been pointed out with respect to mediator immobilization. GPS can greatly enhance the stability of the electrochemical response. Finally, successful co- and co-immobilization of the NAD + cofactor. Introduction The direct oxidation or reduction of cofactor on a bare electrode requires high overpotential, and usually leads to enzymatically inactive NAD-dimers and serious side reactions. However, only reversible cofactors recycling can guarantee the reagentless aspect of the bioelectrocatalytic devices. In order to overcome these inherent difficulties, a common approach is to confine the mediator at the electrode surface to facilitate the interfacial electron transfer kinetics. The mediators for the regeneration of NAD + are diverse. Several mediators such as quinones [ 1 , 2 ], oxometalates [ 3 ], ruthenium complexes [ 4 ], phenazines [ 5 , 6 ] and phenoxazines [ 7 , 8 ] quinonoid redox dyes have been proposed for NAD + regeneration. Traditionally, the mediators were directly adsorbed, electropolymerized or covalently bound onto the electrode surface [9,10,11]. The mediators for the regeneration of NADH are relatively few. The best systems to date fulfilling these requirements are tris(2,2'-bipyridyl) rhodium complexes [ 12 , 13 , 14 ] and substituted or non substituted (2,2'-bipyridyl) (pentamethylcyclopentadienyl)-rhodium complexes [ 15 ], and some others [ 16 ]. The mechanism of this electrocatalytic process has been largely studied and the effect of various parameters (e.g., solution composition, temperature) has been discussed in the literature [17,18]. However, it might be surprising that not much effort was made to immobilize these mediators in an electrocatalytically-active form and further use it with immobilized dehydrogenase. During the last decades, carbon nanotubes (single-or multiwalled) have emerged as attractive materials in electroanalysis [ 19 , 20 ]. Indeed they display attractive chemical stability, strong absorptive properties and excellent biocompatibility [21,22]. CNTs-based electrodes are known to decrease the overpotential for the oxidation of NADH, however, the extent of decrease is not sufficient for the selective detection and regeneration of cofactor [23]. Recently, Wooten, et al demonstrates that further decrease in the NADH overpotential can be achieved at CNTs that were activated by microwaving in concentrated nitric acid [24]. An alternative method of incorporation the mediators onto carbon nanotubes (CNTs) for NAD + regeneration have attracted considerable study. A number of mediators such as toluidin blue [25], nile blue [26,27], meladola blue [28], methylene green (MG) [29] and an osmium polymer [30] have been immobilized by adsorption onto CNTs, resulting in a remarkable improvement of electocatalysis toward NADH oxidation. Another approach to form stable films of mediators on electrode surfaces, is to use electropolymerization which has many advantages including selectivity, sensitivity and homogeneity in electrochemical deposition, strong adherence to electrode surface and chemical stability of the film [31,32,33]. CNTs/mediator composite as electrode materials has been already explored for the construction of dehydrogenase-based biosensors [34]. Of particular interest is the report by Yan et al which described the assembly of integrated, electrically contacted NAD(P) +dependent enzyme-SWCNT electrodes [27]. The SWCNTs were functionalized with Nile Blue, and the affinity complexes of dehydrogenase with cofactor were crosslinked with glutaric dialdehyde and the biomolecule-functionalized SWCNT materials were deposited on glassy carbon electrodes. This is the unique example of reagentless device based on the combination of dehydrogenase and carbon nanotubes. The combination of sol-gel material and carbon nanotubes has also been considered for electroanalytical applications [35]. Recently, co-immobilization of lactate dehydrogenase and functionalized carbon nanotubes in sol-gel has been developed for biosensor [ 36 ]. The nanocomposite was prepared by the sol-gel process incorporating a redox mediator and carbon nanotubes, which was mixed with enzyme solution in a certain ratio for enzyme encapsulation. To date, and to our knowledge, no attempt was made to use sol-gel thin film to co-immobilized dehydrogenase and NAD + cofactor on the electrode surface of CNTs/mediator composite. Moreover, electrochemically-assisted deposition of sol-gel biocomposite on carbon nanotube assembly is also a new approach. We have investigated here various strategies for the elaboration of a reagentless sensor based on NAD-dependant dehydrogenase using the electrochemically-assisted deposition of the sol-gel biocomposite on carbon nanotubes (CNTs). CNTs have been functionalized by three different protocols in order to provide them catalytic properties for NADH detection. These protocols are (1) micro-wave treatment (MWCNTs-µW), ( 2 complex on SWCNTs has also been studied. Deposition sol-gel film at microwaved MWCNTs (GCE/MWCNTs-µW) A recent report by Wooten et al. described that microwave treatment of MWCNTs resulted in a dramatic shift of the oxidative peak potential of NADH (E NADH ) to a lower value, from +0.4 V to about 0 V [24]. The efficient system could be interesting to be further used in combination with immobilized dehydrogenase and cofactor to develop the reagentless device. Electrocatalytic oxidation of NADH at GCE/MWCNTs-µW Q + NADH + H + → QH 2 + NAD + (1) which is followed by the recycling of quinone species on the surface of treated MWCNTs. QH 2 → Q + 2e -+ 2H + (2) Because reactions 1 and 2 are faster than the direct electrooxidation of NADH to NAD + , the mediated process (1)-( 2) allows conversion of the NADH to NAD + at less-positive potentials close to the formal potential of the Q/QH 2 redox couple (~ 0 V). Importance of bilayers (drop-coated sol-gel film) The efficient system was further used in combination with the sol-gel biocomposite. V (see curve a). Faintly visible electrocatalytic response started to be observed at +0.5V, and the optimal potential was found to be +0.8V. Co-immobilization of DSDH and cofactor in electrogenerated sol-gel thin film This efficient approach has been further used to develop a reagentless system (coimmobilization of DSDH and cofactor with MWCNTs-µW as mediator). The presence of MWCNTs-µW on the GCE surface greatly improved the electrocatalytic efficiency of the bioelectrode and decreased to a certain extent the overpotential of NADH detection (from +0.8V to +0.4V). But the benefit of the microwave treatment was lost when MWCNTs-µW was covered with an additional sol-gel layer. Moreover, the system failed to regenerate the immobilized cofactor. One alternative modification of the multiwalled carbon nanotube is the electrodeposition of methylene green. We expected that this functionalization could be less sensitive than the surface quinones to the sol-gel deposition, and could be more suitable to regeneration of the immobilized cofactor. Deposition of sol-gel film at MWCNTs modified by poly (methylene green) (GCE/MWCNTs-PMG) Electrocatalytic oxidation of NADH at GCE/MWCNTs-PMG The methylene green (MG) has shown to be a good NADH oxidation electrocatalysts [37,38] and was employed in this study. The principle of this mediator can be schematized as responded rapidly to the changes of the concentration and displayed a higher sensitivity and wider detection range at +0.2 V. If a higher potential was applied, the direct uncatalysed oxidation of the enzymatically generated NADH may happen and lead to enzymatically inactive NAD-dimers and serious side reactions. In this experiment, +0.2 V was selected as the working potential which guarantees both good selectivity and sensitivity. This potential was very comparable with peak potential observed in cyclic voltammograms. Encapsulation of DSDH and cofactor in sol-gel film drop-coated onto GCE/MWCNTs-PMG As aforementioned, co-immobilizing all the necessary enzyme, cofactor and mediator onto the electrode surface without any additional reagents in solution is an advantage to reagentless device. As disscussed in chapter IV, Glycidoxypropyl-trimethoxysilan (GPS) [39,40] provides a promising approach for NAD + immobilization. Electrodeposition of sol-gel thin film at GCE/MWCNTs-PMG Electrodeposition of sol-gel thin film containing DSDH at GCE/MWCNTs-PMG (cofactor in solution) The above approach has been extended to electrochemically-assisted deposition. First of all, the encapsulation of DSDH inside the electrodeposited silica gel has been investigated. Then 1 mM NAD + was added in the solution and a significant increase in the current response was observed, indicating a good enzymatic activity of the immobilized protein. DSDH encapsulated in the electrodeposited sol-gel film was active, but the immobilized cofactor did not interact with the carbon nanotubes modified with poly(methylene-green). -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 -80 The strategy based on MWCNTs-PMG can decrease the oxidation overpotential to be 0.2V for the smooth regeneration of cofactor and the mediator was now stable in the presence of sol-gel layer. Co-immobilization of DSDH and cofactor on MWCNTs-PMG could be obtained by drop-coating. However, this strategy did not allow the extension to electrochemicallyassisted deposition. A bad communication was observed between the immobilized cofactor and the poly(methylene-green) deposited on MWCNTs. A system base MWCNT wrapped by Osmium(III) polymer has finally been tested. This functionalization is expected to give sufficient flexibility to mediator to interact with the sol-gel material electrochemically deposited on its surface. Deposition of sol-gel film at MWCNT wrapped by Osmium(III) polymer (GCE/MWCNTs-Os) Carbon nanotubes as Osmium immobilization support The final system considered in this study was the carbon nanotubes modified with an osmium polymer. At first, the osmium polymer was simply mixed with the carbon nanotube in 0.2wt% chitosan. As display in Figure VI-13A, this protocol led to a well-defined CV. But Co-immobilization of DSDH, DI and cofactor at GCE/MWCNTs-Os The co-encapsulation of GPS functionalized cofactor, DSDH and DI on GCE/MWCNT-Os was then evaluated. The protocol for the electrode preparation was rather simple; cofactor, DSDH and diaphorase are mixed together inside the sol and are casted on GCE/MWCNT-Os. It is shown in Electrodepostion of sol-gel film at GCE/MWCNTs-Os Electrocatalytic oxidation of NADH This approach has been extended to electrochemically-assisted deposition. First of all, the encapsulation of DI inside the electrodeposited silica gel has been investigated. Co-immobilization of DSDH, DI and cofactor at GCE/MWCNTs-Os The co-encapsulation of DSDH, DI and cofactor inside the electrodeposited silica gel has been finally investigated. Up to now, carbon nanotubes wrapped by osmium has been the only system that was successfully combined with sol-gel electrodeposition for bioencapsulation of dehydrogenase and cofactor. With MWCNT-µW and MWCNT-PMG, the functionalization was confined at the surface of the nanotube as part of the material (MWCNT-µW) or as a thin surface layer (MWCNT-PMG). The NADH cofactor had to come close the carbon nanotube surface in order to be regenerated in the enzymatically active NAD + . When cofactor was free diffusing in the solution, the regeneration was always successful and carbon nanotubes provided higher surface area resulting in higher sensitivity of the bio-electrode to D-sorbitol with comparison to the bare GCE. However the immobilization of the cofactor in the electrochemically deposited sol-gel thin film did not provide a sufficient mobility to the cofactor and the molecule did not interact with the functionalized MWCNT (MWCNT-µW or MWCNT-PMG). Osmium polymer is composed by a polyacrylate backbone with the osmium complex attached by 5 atoms linkers. While the polymer is linked to the MWCNT by wrapping, the linker gives sufficient flexibility to the mediator to interact with the sol-gel material electrochemically deposited on its surface. The same polymer was first introduced in the sol-gel matrix. But the electrochemical properties of the polymer were lost during the electrodeposition process (see Chapter V). A first immobilization on the carbon nanotube surface before sol-gel electrodeposition gave the unique opportunity to conserve the electrochemical activity of the osmium complexes and to display good flexibility for effective interaction with the encapsulated diaphorase that catalyses the NADH oxidation. The direct electrochemical reduction of NAD(P) + requires high overpotentials and usually leads to enzymatically inactive NAD-dimers generated due to the one-electron transfer reaction [41,42]. The mediators for the regeneration of NADH should react with NAD + and not transferring directly the electrons (or hydride ion) to the substrate, and the potential window for electrochemical activation of the catalysts is rather narrow (-0.59 ~ -0.9 V vs. SCE) [43]. (2,2'-bipyridyl) rhodium complexes are the best systems for the regeneration of NAD + . As illustrated on Figure VI-18, their electrocatalytic behaviour is rather complicated, involving first a two-electron electrochemical reduction of Rh III (M ox ) into transient Rh I species (M red1 ; this reaction occurring itself in several steps [44]) that can be transformed upon protonation into a rhodium hydride complex (M red2 ), which is then likely to transfer the hydride to NAD(P) + under formation of only 1,4-NAD(P)H [45,46]. The interest of Substituent effects and mediator immobilization attempts Derivatives functionalized with thiol groups are attractive for immobilization of reagents in the form of self-assembled monolayers (SAMs) on gold electrodes [47,48], and the aminefunctionalized ones are good precursors to form organosilane reagents likely to be grafted onto metal oxides or incorporated within sol-gel matrices then deposited onto electrode surfaces [49,50,51]. We have examined the behaviour of several functionalized mediators of this family bearing various organic groups, which could be used as precursors to immobilize such compounds onto electrode surfaces. Here, we just give one example. Compounds 11 is a thiol-functionalized Rh complex. Conclusion The goal of the study was the implementation of a reagentless device displaying an efficient interaction between the immobilized dehydrogenase and cofactor in the electrogenerated sol-gel matrix and the functionalized multi-walled carbon nanotubes (MWCNTs). In this work, three different protocols have been developed to functionalize The catalytic property of macrowaved MWCNTs was significantly disturbed by sol-gel material and failed to regenerate the cofactor at 0V. The modification of the MWCNTs with electrodeposited poly(methylene-green) and wrapped Osmium(III) polymer were not so sensitive to the sol-gel material, and allowed the smooth regeneration of the immobilized cofactor in drop-coated sol-gel film. However, when the reagentless devices obtained by drop-coating were extended to electrochemically-assisted deposition, a direct interaction between NAD + immobilized in the sol-gel matrix and the functionalized MWCNTs was not possible with MWCNTs-µw and MWCNTs-PMG. Sol-gel deposition by electrogenerated limited the interaction of NAD + with the mediator confined on MWCNTs. Only MWCNT wrapped with Osmium(III) polymer and in the presence of diaphorase allowed to observe the electrochemical detection of D-sorbitol in a reagentless configuration. This is probably due to high mobility of the osmium complexes immobilized on MWCNTs. For the reduction reaction, we studied the immobilization of functionalized mediators of [Cp*Rh(bpy)Cl] + family. The presence of substituents bearing nucleophilic moieties such as S-or N-containing groups, on the bipyridine ligand, was proven to be harmful to the electrocatalytic properties of the [Cp*Rh(bpy)Cl] + mediator, limiting therefore most immobilization strategies based on covalent bonding onto electrode surfaces. A way to circumvent this problem is the soft immobilization of such [Cp*Rh(bpy)Cl] + mediator by π-πstacking onto CNTs. However, this strategy does not show good operational behaviour when associated to a dehydrogenase enzyme with silica sol-gel for protein encapsulation. Conclusion et perspectives Conclusion and outlook The focus of the research work carried out in this thesis is on the development of different strategies allowing the stable immobilization of a dehydrogenase, the cofactor NAD + /NADH and an electron mediator in a sol-gel matrix deposited (either by evaporation or by electrogeneration) as a thin film on an electrode surface. This layer is intended to be applied in electro-enzymatic synthesis for the production of fine chemicals. To achieve the objective of the project, we have divided the work into three steps: (1) dehydrogenase immobilization, to the epoxide group of glycidoxypropylsilane (GPS) before co-condensation of the organoalkoxisilane with tetraethoxysilane in the presence of the proteins (dehydrogenase and diaphorase) and a poly(etheleneimine) additive (PEI). All the operations were performed in smooth conditions compatible with the sol-gel bio-encapsulation process. By comparison with the simple encapsulation of NAD + or NAD-dextran or adsorption of NAD + on carbon nanotubes, the strategy with GPS is either cheaper or simpler to implement, and leads by far to more stable sol-gel films and durable bioelectrocatalytic responses. At the end, the efficient drop-coating method has been extended to the electrochemically-assisted deposition of sol-gel film with encapsulated enzymes and cofactor on macroporous electrodes. Finally, the last part of this work has been devoted to the development of different strategies for mediator immobilization which could used for the elaboration of reagentless device with co-immobilized dehydrogenase and cofactor. First of all, a series of successful strategies for co-immobilization of all components (dehydrogenase, cofactor and electron mediator) in sol-gel films have been developed by using one step drop-coating. here DSDH was chosen as model enzyme, NAD + functionalized with GPS was used as mediator, and ferrocene species or osmium polymers were introduced inside the matrix as co-immobilized mediators. The importance of introducing GPS as an additive into the TEOS sol-gel-derived films has been pointed out with respect to the stable mediator immobilization. However, such co-immobilization applied to electrochemically-assisted deposition of sol-gel thin films was not successful due to some problems to keep mediators in an active form. To overcome this problem, we have developed different strategies for the elaboration of a reagentless devices based on deposition of the sol-gel biocomposite on mediators functionalized multi-walled carbon nanotubes (MWCNT). Surface modification of the carbon nanotube with quinone moieties by microwave treatment or the electrodeposition of poly(methylene-green) on the MWCNT resulted in good electrocatalytic detections of the free diffusing NADH produced by the immobilized dehydrogenase. However these systems failed to regenerate the cofactor immobilized in electrogenerated sol-gel film as the mediators immobilized on carbon nanotubes did not display enough mobility to react with NAD + linked to the sol-gel matrix. Finally, the sol-gel thin film with co-immobilized dehydrogenase, diaphorase and cofactor was deposited on MWCNT wrapped by osmium polymer. The flexibility of the osmium complexes allowed the smooth regeneration of the immobilized Conclusion and outlook 207 cofactor. All the components are able to communicate inside the silica gel layer for efficient electro-catalytic oxidation of D-sorbitol. The combination of the carbon nanotubes with the Osmium(III) polymer was a suitable electrode material for further electrogeneration of sol-gel materials with co-immobilized proteins and cofactor. The layer developed in this study can be applied in electro-enzymatic synthesis for the manufacture of chiral fine chemicals. Because all reactive agents have been immobilized on the electrode surface, it represents an environmentally friendly process by avoiding organic solvents and reducing purification steps to a minimum. This concept meets the standards of green chemistry and leads to processes which come close to zero waste emissions. However, it will still take some time to improve the reaction productivity before electroenzymatic processes can be applied on an industrial scale. The finding of this work can also be applied in the development of dehydrogenase-based reagentless biosensors. It is reported there are more than 300 kinds of dehydrogenases. These enzymes catalyze the oxidation of a variety of substrates including alcohols, aldehydes, glucose and etc., which are of great interests from the analytical point of view because of the practical application on food industry, environment, and clinical chemistry. This study also offers a facile and versatile approach to the development of some other integrated dehydrogenase-based electrochemical devices, such as biofuel cells and biobattery. [ 1 ] 1 Shacham, R.; Avnir, D.; Mandler, D., Adv. Mater. 1999, 11, 384 [2] Walcarius, A.; Mandler, D.; Cox, J. A.; Collinson, M. M.; Lev, O., J. Mater. Chem. 2005, 15, 3663. Figure I- 1 . 1 Figure I-1. Model of an electrochemical reactor for enantiopure synthon preparation. Since all active compounds (mediator, cofactor, dehydrogenase) are immobilized only the educt and the product are components of the reaction buffer. The gas diffusion counter electrode provides clean protons and improves the long-term stability. Figure I- 1 1 Figure I-1 shows the scheme of an electrochemical reactor for enantiopure synthon preparation. Two main electrochemical reactions occur in the electrochemical reactor. Participant 4 ( 6 ( 46 two groups: Physical Chemistry and Applied Microbiology. The Physical Chemistry group designed of electrochemical multicell with 16 individual cells and electrochemical reactor with a macroporous working electrode and a proton conducting membranes/gas diffusion counterelectrode. The Applied Microbiology group provided several dehydrogenases with enhanced stability and activity in the environment of the developed electrode surfaces. Participant 2 (Ecole Nationale Supérieure de Chimie et de Physique de Bordeaux, Molecular Sciences, France) developed macroporous metal electrodes with high surface area support to immobilize mediators and enzymes in functional internal surface layers using a sol-gel matrix prepared and optimized by partner 3. Participant 3 (CNRS, Laboratory of Physical Chemistry and Microbiology for the Environment, France) developed electrode surface layers for functional immobilization of enzymes, cofactors and mediators. University of Copenhagen, Denmark) included two groups: Biophysical Chemistry Group and Dept. of Chemistry, Bioinformatics. The biophysical group crystallized the dehydrogenases produced by partner 1b and determined their three-dimensional atomic structures by crystallographic methods. The bioinformatics group used computer modeling to provide a molecular level description of how the enzyme activities and stabilities can be enhanced. Participant 5 (Middle East Technical University, Turkey) This group developed mediators for electron transfer to the cofactor NAD + in the described systems. Participant IEP GmbH Wiesbaden IEP, Germany) supported the project from an industrial point of view. (• Conditions of neutral to basic pH result in relatively mesoporous xerogels after drying, as rigid clusters a few nanometers across pack to form mesopores. The clusters themselves may be microporous (Figure I-3b). Figure I- 3 . 3 Figure I-3. Schematic wet and dry gel morphologies and representative transmission electron micrographs. (Adapted from Brinker and Scherer, Sol Gel Science, chapter 9, figures 2a-2c. [19].) Figure I- 4 . 4 Figure I-4. The principle of electrochemically-assisted generation of silica film on the electrode surface. Figure I- 6 . 6 Figure I-6. The structures of cofactors NAD(P) + and NAD(P)H. [ 60 ] 60 Figure I-7. NAD(P) + /NAD(P)H-dependent reactions: (A) direct electrochemical regeneration, (B) indirect electrochemical regeneration, (C) enzyme-coupled electrochemical regeneration[START_REF] Kohlmann | Electroenzymatic synthesis[END_REF]. Figure I- 8 . 8 Figure I-8. Structures of some typical mediators. 12 Figure II- 3 . 123 Figure II-3. Functionalized Rh(III) mediators. Figure II- 4 . 4 Figure II-4. The 3-dimensional structure of macroporous gold electrode. sol A has been considered an obstacle, due to its potential denaturing activity on the entrapped protein. Methanol being less harmful than ethanol, TEOS can be replaced by TMOS (Sol B and C). Alcohol can also be removed from the sol by evaporation (Sol D). Aqueous sol-gel routes (Sol E and F) have been tested in order to avoid any trace of alcohol. A natural polymer (chitosan), or a polyelectrolyte (PDDA and PEI) was used as additive in Sol G, H, I, J and K which can be also advantageous in providing a better environment for biosencapsualtion. Sol A and B have been mainly used during the first series of electrodeposition and Sol C, D, E, and F have been widely used in control experiments and tested, when possible, for spincoating deposition, drop-coating or electrodeposition. Actually, none of them were satisfactory when used as thin biocomposite films on glassy carbon as no voltammetric signals can be detected in the presence of the enzymatic substrates, independently on the used enzyme (DSDH, GatDH), contrarily to what is observed for haemoglobin (which is found electrochemically and electrocatalytically active in all gels). 4 . 1 . 2 4 . 2 Preparation of electrodes for chapter IV 4 . 2 . 1 ③ 41242421 Sol G was mixed with 15 µL PDDA solution (20 wt. %) and 20 µL of the enzyme solution (10 mg/mL). An aliquot (5 µL) of this resulting sol was deposited onto the surface of the GCE. The solution was then allowed to dry at 4 °C overnight. The prepared electrodes were rinsed thoroughly with water and stored in Tris-HCl buffer solution for 15 min prior to the electrochemical measurement. In attempting to optimize the film composition, the electrodes containing various amounts of TEOS and additives (PDDA, PEI, PAA, Nafion, Chitosan) were prepared by adjusting the concentrations of each component as desired and applying the same protocol as above to form the biocomposite films. Composites made of both silica and chitosan were typically prepared by mixing together 20 µL of the starting sol G with 20 µL of a chitosan solution (0.5 % in 0.05 M acetic acid solution), 20 µL PDDA solution and 20 µL DSDH suspension. The pure chitosan film was obtained by mixing 20 µL chitosan, 20 µL PDDA and 20 µL DSDH suspension. When PDDA was not introduced, it was replaced by the same volume of water. An aminopropyl-functionalized material was also prepared by introducing aminopropyltriethoxysilane (APTES) into the sol in addition to TEOS (1:1 molar ratio). Some biocomposite materials were also prepared in the form of monoliths for UV-vis monitoring of the enzymatic activity. They were prepared according to a protocol from the literature[22]. Sol 1 was prepared by mixing 125 µL of the starting Sol C with 125 µL of phosphate buffer (5 mM, pH 8.0) and 25 µL of the DSDH suspension. Sol 2 was prepared as sol 1 and 25 µL of PDDA solution was also added. The gels were allowed to age at 4°C for 6 days. Before use, the gels have been washed three times with 3 mL of 0.1M Tris-HCl (pH 9.0) to remove weakly adsorbed DSDH. Electrodepostion 100 µL of DSDH (10 mg/mL) and 100µL of PDDA were added to 100µL of the above hydrolyzed Sol H. The mixture was put into the electrochemical cell where electrochemicallyassisted deposition was performed at -1.3 V at room temperature for several tens of s. The electrodes were immediately rinsed with water, and dried overnight in a fridge at 4 °C. The prepared electrodes were rinsed thoroughly with water and stored in Tris-HCl buffer solution for 15 min prior to the electrochemical measurements. In attempting to optimize the film composition, the electrodes containing various amounts of TEOS, PDDA and DSDH were prepared by adjusting the concentrations of each component as desired and applying the same protocol as above to form the bio-composite films. Co-immobilization of DSDH and NAD + in sol-gel matrix prepared by drop-coating ① Preparation of GCE/TEOS/PEI/(DSDH+DI)/NAD + 20 µL of sol I was mixed with 10 µl PEI solution (20 wt%, pH 9.0), 10 µl NAD + solution (90 mM), 15 µl DSDH solution (10 mg/ml) and 10 µl DI solution (5 mg/ml). The biocomposite film was then formed by drop-coating, by depositing an aliquot (5 µl) of this resulting sol onto the GCE surface and allowing the solvent to evaporate by drying overnight at 4 °C. ② GCE/TEOS/PDDA/(DSDH+DI)/NAD + and GCE/TEOS/PAA/(DSDH+DI)/NAD + have been prepared as above by replacing the PEI solution by PDDA and PAA solutions (20 wt. % in water), respectively. Electrodes doped with carbon nanotubes, GCE/TEOS/PEI/SWCNT/(DSDH+DI)/NAD + and GCE/TEOS/PEI/(DSDH+DI)/NAD-SWCNT were obtained by suspending 1.0 mg SWCNT or NAD-SWCNT in 1.0 ml of sol I prior to mixing with PEI, DSDH and DI solutions and following the same drop-coating procedure afterwards. The NAD-SWCNT sample (carbon nanotubes with adsorbed/noncovalently attached NAD + ) was previously prepared according to a protocol reported in the literature [23], by mixing 2 mg SWCNT with 50 mg NAD + in 2 mL water under stirring for 48 h and then recovering the solid phase by filtration (0.45 µm, Millipore).④ GCE/TEOS/PEI/(DSDH+DI)/NAD-dextran and GCE/TEOS/PEI/(DSDH+DI)/NAD-GPShave been also prepared as above except that the 10 µl aliquot of NAD + solution was replaced by respectively 10 µl of NAD-dextran solution (62.5 mg/ml in Tris-HCl buffer at pH 7.5) or 10 µl of NAD-GPS solution (typically prepared by mixing together 25 mg NAD + and 37.5 mg GPS in 400 µl Tris-HCl buffer solution (pH 7.5) at 4°C under shaking for 12 h. 4. 2 . 1 4 . 3 . 2 4 . 4 2143244 Co-immobilization of DSDH and NAD + in electrodeposited sol-gel thin film 70 µL DSDH (10 mg/mL), 30 µL PDDA, 40 µL PEI, 40 µL DI and 50 µL NAD-GPS composite were added to 70 µL of the above hydrolyzed Sol H. The mixture was put into the electrochemical cell where electrochemically-assisted deposition was performed at -1.3 V at room temperature for 60 s. The electrodes were immediately rinsed with water, and dried overnight in a fridge at 4 °C. The prepared electrodes were rinsed thoroughly with water and stored in Tris-HCl buffer solution for 15 min prior to the electrochemical measurements. In attempting to optimize the film composition, the electrodes containing only PDDA or PEI were prepared by adjusting the concentrations of each component as desired and applying the same protocol as above to form the bio-composite films.4.3 Preparation of electrodes for chapter V 4.3.1 Fc-PEI and Os-polymer as co-immobilized mediator 20 µL Sol J was mixed with 10 µL PEI solution (10 wt. %, pH 9.0), 15 µL the Fc-PEI or Os-polymer solution, 10µL DI (5mg/mL), 15µL DSDH and 10µL NAD-GPS. An aliquot (5 µL) of this resulting sol was deposited onto the surface of the GCE. The solution was then allowed to dry at 4 °C overnight. The prepared electrodes were rinsed thoroughly with water and stored in Tris-HCl buffer solution for 15 min prior to the electrochemical measurement. The electrodes prepared in the absence of GPS or the enzyme and cofactor were obtained by adjusting the concentrations of each component as desired and applying the same protocol as above to form the biocomposite films. 50 µL PEI, 60 µL Fc-PEI or Os-polymer solution, 40 µL DI, 60 µL DSDH and 40 µL NAD-GPS were added to 100 µL of the Sol J. The mixture was put into the electrochemical cell where electrochemically-assisted deposition was performed at -1.3 V at room temperature for 60 s. The electrodes were immediately rinsed with water, and dried overnight at 4°C. Fc-silane as co-immobilized mediator Drop-coated sol-gel film modified electrodes were prepared with the same protocol as above by mixing 20 µL Sol K, 10 µL PEI solution (10 wt. %, pH 9.0), 15 µL water, 10 µL DI (5 mg/mL), 15 µL DSDH and 10 µL NAD-GPS. Electrodeposited so-gel film modified electrodes are prepared with the same protocol as above by mixing 50 µL PEI, 60µL water 40µL DI, 60µL DSDH, 40µL NAD-GPS and 100µL of the Sol K. Preparation of electrodes for chapter VI 4.4.1 MWCNTs-µW & sol-gel matrix GCE/MWCNTs-µW (GCE/MWCNTs) were prepared by casting 5 µL of suspension of 1.0 mg of microwaved MWCNTs (see chapter II 1.4.1.2) (untreated CNTs) in 1.0 mL of 0.1wt % chitosan solution on the surface of GC electrode and dried for 2 h at room temperature. The film electrodes were rinsed repeatedly with water and soaked in a pH 7.40 phosphate buffer solution while stirring to remove any loose materials. The electrodes were stored at room temperature when not in use. ① ① ① ① GCE/MWCNT-µWs&TEOS/PEI/DSDH 20 µL of Sol I was mixed with 20 µL PEI solution (10 wt %, pH 9.0) and 20 µL DSDH solution (10 mg/mL). An aliquot (5 µL) of this resulting sol was deposited onto the surface of the GCE/MWCNT-µWs. The solution was then allowed to dry at 4°C overnight. ② ② ② ② GCE/MWCNT-µWs &electrodepositedTEOS/PEI/DSDH 70 µL PEI, 80 µL DSDH and 50 µL water were added to 100 µL of the Sol H. The mixture was put into the electrochemical cell with GCE/MWCNT-µWs as working electrode where electrochemically-assisted deposition was performed at -1.3 V at room temperature for 60 s. The electrodes were immediately rinsed with water, and dried overnight at 4°C. ③ ③ ③ ③GCE/MWCNT-µWs&electrodepositedTEOS/PEI/DSDH/NAD-GPS film modified electrode 70 µL PEI, 80 µL DSDH and 50 µL NAD-GPS were added to 100 µL of the Sol H. The mixture was put into the electrochemical cell with GCE/MWCNT-µWs as working electrode 4 . 4 . 3 4 . 1 . 2 ) 4 . 4 . 4 4 . 2 . 3 ) 443412444423 ① ① ① GCE/MWCNT-PMG &TEOS/PEI/DSDH film modified electrode 20 µL of Sol I was mixed with 10 µL PEI solution (10 wt %, pH 9.0), 10 µL water and 20 µL DSDH solution. An aliquot (5 µL) of this resulting sol was deposited onto the surface of the GCE/MWCNT-PMG. The solution was then allowed to dry at 4°C overnight. The prepared electrodes were rinsed thoroughly with water and stored in Tris-HCl buffer solution for 15 min prior to the electrochemical measurement. ② ② ② ② GCE/MWCNT-PMG &TEOS/PEI/DSDH/NAD-GPS film modified electrode 20 µL of Sol I was mixed with 10 µL PEI solution (10 wt %, pH 9.0), 10 µL NAD (or NAD-GPS composite) solution and 20 µL DSDH solution. An aliquot (5 µL) of this resulting sol was deposited onto the surface of the GCE/MWCNT-PMG. The solution was then allowed to dry at 4°C overnight. The prepared electrodes were rinsed thoroughly with water and stored in Tris-HCl buffer solution for 15 min prior to the electrochemical measurement. ③ ③ ③ ③ GCE/MWCNT-PMG&electrodepositedTEOS/PEI/DSDH/NAD-GPS film modified electrode 70 µL PEI, 80 µL DSDH and 50 µL NAD-GPS are added to 100 µL the Sol H. The mixture is put into the electrochemical cell with GCE/MWCNT-PMG as working electrode where electrochemically-assisted deposition was performed at -1.3 V at room temperature for 60 s. The electrodes are immediately rinsed with water, and dried overnight at 4°C. MWCNTs-Os&sol-gel matrix Chitosan/MWCNTs-Os solution was prepared by mixing in 1:1 volume the resulting MWCNTs-Os solution (see chapter II 1.to 0.2 % chitosan solution (0.2 % chitosan solution in 1 % acetic acid). GCE/MWCNT-Os was prepared by depositing 5 µL Chitosan/MWCNTs-Os solution and allowed to evaporate at the room temperature. ① ① ① ① GCE/MWCNT-Os &TEOS/PEI/DSDH/NAD-GPS/DI 20 µL of Sol I was mixed with 10 µL PEI solution (10 wt %, pH 9.0), 10 µL NAD-GPS solution, 10 µL DI and 15 µL DSDH solution. An aliquot (5 µL) of this resulting sol was deposited onto the surface of the GCE/MWCNT-Os. The solution was then allowed to dry at 4°C overnight. The prepared electrodes were rinsed thoroughly with water and stored in Tris-HCl buffer solution for 15 min prior to the electrochemical measurement. ② ② ② ② GCE/MWCNT-Os &electrodepositedTEOS/PEI/DSDH/NAD-GPS/DI 50 µL NAD-GPS, 40 µL DI (5mg/mL), 70 µL DSDH (10 mg/mL) and 60 µL PEI were added to 80 µL of the above Sol H. The mixture was put into the electrochemical cell with GCE/MWCNT-Os as working electrode where electrochemically-assisted deposition was performed at -1.3 V at room temperature for 60 s. The electrodes were immediately rinsed with water, and dried overnight at 4°C. CNTs/Rh(III)&sol-gel matrix Chitosan/CNTs/Rh solution was prepared by mixing in 1:1 volume the resulting CNTs/Rh suspension (see chapter II 1.to 0.5 % chitosan solution (0.5 % chitosan solution in 1 % acetic acid). The CNTs-Rh modified electrode was prepared by depositing 5 µL Figure II- 5 5 Figure II-5 shows the response of a reversible redox couple during a single potential cycle. It is assumed that only the oxidized form O is present initially. Thus, a negative-going potential scan is chosen for the first half-cycle, starting from a value where no reduction occurs. As the Figure II- 5 5 Figure II-5 Typical cyclic voltammogram for a reversible O + ne-→ R and R → O + ne-Redox process. III. Etude de faisabilité de l'encapsulation d'une déshydrogénase dans une matrice sol-gel Ce chapitre montre les études qui ont été menées pour immobiliser sous une forme active la D-sorbitol déshydrogénase (DSDH) dans une couche mince sol-gel à la surface d'une électrode. Les études ont été menées sur électrodes de carbone vitreux avant d'être appliquées à la modification d'électrodes d'or macroporeuses. Dans un premier temps, l'étude a été menée en utilisant le dépôt par évaporation à partir d'un sol à base seulement de TEOS conduisant à l'encapsulation de la DSDH dans une matrice de silice pure. Ces conditions d'immobilisation ne permettent pas de mesurer une activité catalytique (par oxydation du NADH devant être produit par la protéine pendant l'oxydation du sorbitol). L'influence de l'introduction de différents additifs dans le sol sur l'activité enzymatique a ensuite été étudiée. Il a ainsi été montré que l'introduction de polyélectrolyte positivement chargés au sein de la couche mince sol-gel permettait d'observer une bonne activité de la DSDH vis-à-vis de l'oxydation du D-sorbitol. La bioencapsulation sol-gel a ensuite été obtenue par électrochimie. L'électrodépôt sol-gel est basé sur la modulation électrochimique du pH à la surface de l'électrode qui conduit à une transition sol-gel rapide seulement à proximité de l'électrode et à la formation de la couche mince. Le rôle des polyélectrolytes mis précédemment en évidence a été confirmé et la composition du sol ainsi que les paramètres d'électrogénération (temps et potentiel d'électrolyse) ont été optimisés. Il a été observé que ce procédé d'électrogénération sol-gel était parfaitement adapté à l'encapsulation de la DSDH et à la co-immobilisation avec la diaphorase (cette dernière protéine catalysant la régénération du cofacteur enzymatique en présence d'un médiateur redox). Enfin, le protocole optimal d'électrogénération a été appliqué à la fonctionnalisation contrôlée d'électrodes d'or macroporeuses présentant une grande surface électroactive. Figure III- 1 . 1 The sol-gel film contains encapsulated DSDH and the cofactor is present in solution (typically at a concentration of 1 mM) to which various concentrations of D-sorbitol were added. DSDH catalyses the oxidation of D-sorbitol into fructose and simultaneously NAD + is reduced into NADH. The equilibrium of this enzymatic reactions strongly favors the substrate (D-sorbitol) rather than the product (fructose) side because of the very low formal potential of the NAD + /NADH redox couple (-560 mV versus saturated calomel electrode, pH 7.0, 25°C). Here, the electrochemical oxidation of NADH at the electrode surface pushes the reaction to the product side. The reduced form of the cofactor can be detected electrochemically on glassy carbon. In our experimental conditions, a well-defined oxidation peak was observed on bare glassy carbon electrode (GCE) around +0.7 V (versus Ag/AgCl reference electrode). Figure III- 1 . 1 Figure III-1. Enzymatic and electrochemical reactions occurring on the gel-enzyme modified Figure III- 2 2 reports the electrochemical characterization of the film deposited by drop-coating on GCE. It is shown that the addition of D-sorbitol into the solution does not lead to noticeable modification of the current response and no peak current can be observed at the potential of NADH oxidation. Obviously, the enzyme does not exhibit electrochemically detectable activity when encapsulated into the pure silica layer. The actual sol contains a certain amount of alcohol that could be harmful for the enzyme (possible denaturation). Other sol-gel protocols have thus been tested: alcohol evaporation before enzyme encapsulation, use of other silica precursors (aqueous silicates or TMOS instead of TEOS). In all cases, the biocomposite films deposited on GCE did not reveal any measurable electroactivity. Several hypotheses can be proposed to explain the absence of response: (1) DSDH was not successfully incorporated into the sol-gel matrix; (2) enzyme entrapment was successful but led to loss in its biological activity; (3) DSDH was successfully entrapped in an active form but lack of effective electrochemical transduction made the detection not visible. Figure III- 2 . 2 Figure III-2. Cyclic voltammograms obtained using GCE modified with a silica/DSDH film in the absence (dashed line) and in the presence (plain line) of 12 mM D-sorbitol. The measurements have been performed in 0.1 M Tris-HCl buffer solution (pH 9) containing 1 mM NAD + . Potential scan rate was 50 mV/s. Figure III- 3 3 Figure III-3 shows the variation of generated NADH concentrations versus the contact time between the D-sorbitol solution and the enzyme-entrapped gel. As shown, DSDH was still active when encapsulated into the sol-gel silica material (see curve "b" in Figure III-3) but the response was much slower and much lower than that of free enzyme in solution (compare with inset in Figure III-3) as only 20 % of NAD + have been transformed after 6 hours of reaction. A blank experiment (curve "a" in Figure III-3) confirms that UV response was indeed due to NADH generation originating from the activity of encapsulated DSDH. Figure III- 3 . 3 Figure III-3. Evolution of the NADH concentration in solution (a) in the absence of enzyme encapsulated silica gel and in the presence of a gel monolith containing (b) only DSDH and (c) both DSDH and PDDA (1.7 % w/w) . Experiments were performed in 0.1M pH 9.0 Tris-HCl buffer solution containing 0.36 mM NAD + and 5 mM D-sorbitol. NADH concentration was determined by UV absorbance at 340 nm. Inset: free enzyme in solution. Figure Figure III-4A shows the typical response of a bio-composite film deposited on GCE from a sol made of 50 mol% APTES relative to the total content of precursor (TEOS+APTES), for increasing D-sorbitol concentrations in the solution. The modified electrode is now sensitive to the addition of the substrate in the 2 to 12 mM concentration range. The signals correspond to the oxidation of NADH produced by the enzyme encapsulated into the film. They are however less well-defined and positively-shifted (by ca. 100-150 mV) in comparison to NADH oxidation on bare GCE (inset in Figure III-4A). This suggests that the presence of the Figure III- 4 . 4 Figure III-4. Influence of additives (A, 50 mol % APTES; B, 5 % w/w PDDA) introduced into the synthesis sol on the electrochemical response of DSDH-silica composite films on GCE, as measured by cyclic voltammetry in the presence of increasing concentrations of D-sorbitol. Cyclic voltammogram for solution-phase NADH (curve "a" without NADH and curve "b" with 5mM NADH) at bare GCE has been added as inset in part A of the figure. Potential scan rate 50 mV/s. Others conditions as in Figure III-2. ( Figure III-2.). The signal corresponds to the oxidation of NADH produced by the enzyme encapsulated into the film. The measured peak currents (~30 µA for 9 mM D-sorbitol, Figure III-4B), were largely superior to those reported when using APTES (~4µA for 10 mM Dsorbitol, Figure III-4A). The interpenetrating PDDA-silica network thus provides a suitable microenvironment for DSDH encapsulation; it may also contribute to generate a more open structure than in pure silica, which would accelerate mass transport of the various reagents (substrate and cofactors). The key parameter seems however to be the favorable electrostatic interactions originating from the positive charges of the polymer additive, as confirmed by comparing the results obtained for various polyelectrolytes, i.e., three positively-charged compounds in the condition of encapsulation (PDDA, PAA and PEI) and one displaying negative charges (Nafion). This is illustrated in Figure III-5, showing the variation of peak currents versus the concentration of D-sorbitol for the different polyelectrolytes. The encapsulation of alcohol dehydrogenase in Nafion membrane has been reported[31], but here the introduction of this negatively charged polyelectrolyte into the sol, led to inactive electrode as no NADH can be detected when increasing the substrate concentration in the solution (curve "a"). At the opposite, all polyelectrolytes bearing positive charges gave rise to good sensitivity of the modified electrode to the enzymatic substrate, the use of PDDA (curve "d") being somewhat more efficient than PEI (curve "b") or PAA (curve "c"). These results support the aforementioned assumption concerning the need of positive charges into the gel for ensuring good encapsulation of DSDH in an active form. Figure III- 5 . 5 Figure III-5. Variation in peak currents for NADH oxidation versus the d-sorbitol concentration measured with films prepared with TEOS as silica precursor and (a) Nafion, (b) PEI, (c) PAA and (d) PDDA as polyelectrolyte (5% w/w). Other conditions as inFigure III-2. Figure III-6A shows the typical response observed with a chitosan-silica gel composite with encapsulated DSDH. The electrode was first tested with cyclic voltammetry in a Tris-HCl buffer solution containing 1 mM NAD + (dashed line) and, as expected, no anodic peak was observed. More surprising was the absence of signal upon addition of D-sorbitol from 2 to 8 mM in the medium, suggesting that no NADH was produced by the DSDH enzyme (only a decrease in background currents was observed). It was necessary to introduce PDDA in addition to chitosan in the biocomposite film to observe the production of NADH in the presence of D-sorbitol (Figure III-6B). The measured currents are in the same magnitude as those observed with silica gel/polyelectrolyte composites (Figure III-6B). Figure III- 6 . 6 Figure III-6. Cyclic voltammograms obtained at GCE modified with (A) a chitosan/silica/DSDH film and (B) a chitosan/silica/PDDA/DSDH film. Measurements have been performed in the absence (dashed line) and in the presence (plain line) of D-sorbitol, up to 12 mM. Potential scan rate 50 mV/s. Other conditions as in Figure III-2. Figure III- 7 . 7 Figure III-7. Cyclic voltammograms obtained at GCE modified with (A) a chitosan/DSDH film and (B) a chitosan/PDDA/DSDH film. Measurements have been performed in the absence (dashed line) and in the presence (plain line) of D-sorbitol, up to 10 mM. Potential scan rate 50 mV/s. Other conditions as in Figure III-2. Figure III- 8 . 2 . 82 Figure III-8. Variation of peak currents sampled at DSDH-doped sol-gel modified GCE in the presence of 8 mM D-sorbitol, as a function of (A) the amount of PDDA in the starting sol prepared with 42 mM TEOS and (B) the TEOS concentration initially introduced in the synthesis medium, in the presence of 5 % PDDA. Other conditions are the same as in Figure III-2. for three typical cases (PDDA/DSDH, Chitosan/PDDA/DSDH, and Silica/PDDA/DSDH films on GCE) and variations in the relative peak currents of the modified electrodes to 6 mM D-sorbitol in the presence of 1 mM NAD + are shown in Figure III-9. The relative current values are given versus the peak height measured during the first cycle, about 1 min after the electrode was placed in solution. Let's first consider the short-term operational stability (Figure III-9A). Figure III- 9 . 9 Figure III-9. Evolution of the relative peak currents recorded for successive analyses of 6 mM D-sorbitol solutions (in the presence of 1 mM NAD + ) at distinct periods of time, using various DSDH-doped film electrodes: (A) short-term measurements have been made with GCE modified with (a) PDDA/DSDH film, (b) chitosan/PDDA/DSDH film, and (c) silica/PDDA/DSDH film; (B) long-term measurements have been made with GCE modified with (a) silica/PAA/DSDH film, (b) silica/PEI/DSDH film, and (c) silica/PDDA/DSDH film. Other conditions as in Figure III-2. 3 . 1 . 31 Figure III-9Bshows the evolution of the electrode response during almost 1 month for three electrodes prepared with combining positively charged polyelectrolytes (PAA, PEI and PDDA) with silica precursors for the enzyme encapsulation. The worst stability was observed with PAA (curve "a") for which current response of the electrode was found to drop by 50 % within three weeks. PEI (curve "b") displayed a rather stable electrochemical response as the measured peak currents did not change significantly during the first 15 days of investigation and then decreased by less than 15 % in the second half of the month. Finally, the behavior of the electrode prepared with PDDA (curve "c") resembles to that prepared with PEI, being slightly less stable, as its response was maximum after 15 days and decreased somewhat afterwards, suggesting some lack of long-term stability.3. Electrochemically-assisted deposition of sol-gel biocomposite with co-immobilized dehydrogenase and diaphorase Feasibility of the electrochemically-assisted depositionThe electrochemically-assisted deposition of silica thin films involves the local perturbation of the pH at the electrode solution interface. Starting from a stable sol, slightly acidic, the electrochemically-induced pH increase allows a fast gelification only at the electrode surface. The electrolysis of the sol at -1.3 V leads to the rapid formation of the thin sol-gel biocomposite layer. All films that are displayed in Figure III-10 have been prepared using 60 s electrolysis. Figure III-10A shows the electrochemical response of the DSDH-modified electrode to successive addition of D-sorbitol in the solution from 1 to 8 mM. Before addition of this enzymatic substrate, no electrochemical signal could be observed between 0 and 1 V. A well defined voltammetric signal with a peak potential located between 0.7 and 0.8 V (versus Ag/AgCl reference electrode) appears when D-sorbitol is added to the solution and this signal increases with the D-sorbitol concentration. It corresponds to the electrochemical oxidation of the NADH cofactor produced by the encapsulated DSDH while oxidizing Dsorbitol. Here the enzymatic cofactor is directly detected at the glassy carbon electrode without using electron mediator. Figure III-10B confirms the results observed in Figure III-10A, with the same enzyme, but for the reduction of fructose. No electrochemical signal could be observed in the absence of fructose. The addition of fructose from 1 to 8 mM produces well defined electrochemical responses with a peak potential close to -1.1 V versus Ag/AgCl reference electrode. This signal corresponds to the reduction of the NAD + cofactor produced by DSDH in the presence of fructose. The electrode was here sensitive to the concentration of fructose from 1 to 8 mM. DSDH immobilized in electrogenerated silica film is thus active in both oxidation and reduction sides. Figure III- 10 . 10 Figure III-10. Electrochemical responses to D-sorbitol (A) and to fructose (B) measured at GCE modified by DSDH with using sol-gel E-AD. E-AD was done at -1.3 V for 60 s from a sol containing 0.17 M TEOS, 3.3 mg/mL DSDH and 6.7% PDDA. (A) Responses in the absence of D-sorbitol (dashed line) and in the presence of D-sorbitol from 1 to 8 mM (solid lines). CVs were done in Tris-HCl buffer (pH 9) containing 1 mM NAD + (B) Responses in the absence of fructose (dashed line) and in the presence of fructose from 1 to 8 mM (solid lines).CVs were done in 0.1 M phosphate buffer (pH 6.5) containing 1 mM NADH. (C) Blank experiments have been performed in tris-HCl buffer (pH 9.0) containing 1 mM NAD + and 6mM D-sorbitol with GCE modified by a film prepared with the same procedure as (A) but in the absence of the protein (solid line) or PDDA (dotted line) or prepared with the same sol (with protein and PDDA) but without applying the electrolysis potential for the EAD (dashed line). Figure Figure III-10C shows several blank experiments (CV in the presence of 6 mM D-sorbitol) with films prepared in the absence of DSDH (plain line), in the absence of polyelectrolyte (dotted line), or by using the same sol and the same protocol as for the electrode presented in Figure III-10A, but without applying the electrolysis step (dashed line). In all three cases, no electrochemical signal was observed. The later blank experiment (no electrolysis) demonstrates that the electrode response observed in Figure III-10A is only due to the DSDH protein encapsulated in a silica film that has been produced by electrochemistry and neither to any non specific protein adsorption nor gel deposition by evaporation that might have occurred in the course of the electrode preparation. Electrode modification is totally controlled by the electrochemically-assisted sol-gel deposition. Moreover the blank experiment performed in the absence of protein confirms that the signal observed around 0.7- Figure III- 11 . 11 Figure III-11. FTIR spectrum measured on a thin biocomposite film deposited on indium tin oxide electrode by sol-gel E-AD at -1.3 V for 10 s from a sol containing 0.17 M TEOS, 3.3 mg/mL DSDH and 6.7% PDDA. Figure III- 11 11 Figure III-11 shows the FTIR spectrum, which has been obtained on a thin film deposited on indium tin oxide electrode by sol-gel electrochemically-assisted deposition at -1.3 V for 10 s from the same sol as reported in Figure III-10 (60 s electrochemically-assisted deposition led to a signal saturation, so shorter electrolysis time was necessary). Background measurement was done on unmodified ITO electrode. As expected the different bands related of the sol, i.e. the DSDH concentration (Figure III-12A), the PDDA concentration (Figure III-12B) and the TEOS concentration (Figure III-12C) have been varied and their influence on the electrochemical response of the resulting film to 6 mM D-sorbitol has been studied. Figure III-12D shows the effect of the time of electrolysis on the electrode response for sols containing (a) 1.67, (b) 3.35 and (c) 6.7 % PDDA. For high polyelectrolyte concentration, the optimal deposition time was found to be 60 s (3.35 and 6.7 %, curves a&b) and a higher time was observed with lower PDDA content (about 120 s for 1.67 %, curve c). It can be assumed that the quantity of encapsulated protein increases with increasing the deposition time, but film thickness probably becomes a significant limitation for too long deposition times. Figure III- 12 .Figure III- 13 . 1213 Figure III-12. Influence of DSDH concentration (A), PDDA (B), deposition time (C) and TEOS concentration (D) on the peak current response measured at glassy carbon electrodes modified by an electrodeposited silica films with encapsulated DSDH. (A) The electrodes were prepared with 0.17 M TEOS sol, 6.7% PDDA and different concentrations of DSDH from 0.33 mg/mL to 3.3 mg/mL. The electrochemically-assisted deposition was done at -1.3 V for 60 s. (B) The electrodes were prepared with 0.17 M TEOS sol, 3.3 mg/mL DSDH and different concentrations of PDDA from 0 to 6.7 %. The electrochemically-assisted deposition was done at -1.3 V for 60 s. (C) The electrodes were prepared with a sol containing 3.3 mg/mL DSDH, 6.7 % PDDA and different concentrations of TEOS. The electrochemicallyassisted deposition was done at -1.3 V for 60 s. (D) The electrodes were prepared with 0.17 M TEOS sol, 3.3 mg/mL DSDH and different concentrations of PDDA from 1.67 to 6.7 %. The electrochemically-assisted deposition was performed at -1.3 V for different deposition time from 10 to 150 s. For all experiments, cyclic voltammetry has been performed in 0.1 M Tris-HCl buffer (pH 9.0) containing 1 mM NAD + and 6 mM D-sorbitol. Potential scan rate was 50 mV/s. Figure III- 14 14 Figure III-14 reports some SEM pictures of films prepared with the optimal sol composition (A) or containing less protein (B) or less PDDA (C). Optimal composition leads to a film with a homogenous texture (Figure III-14A). Only few filaments about 1-2 µm long, probably due to PDDA can be observed on the top of the film. Changing the composition of the sol does not change significantly the texture of the silica gel layer (the surface of allelectrodes is covered by a sol-gel film). However, using a lower protein concentration leads to the presence of more filaments on the surface (FigureIII-14B). In the presence of lower PDDA concentration, these filaments disappeared and some aggregates can be observed (FigureIII-14C). We suppose that these aggregates are due to proteins that are not protected Figure III- 14 . 14 Figure III-14. SEM image of different films prepared by sol-gel electrochemically-assisted deposition at -1.3 V for 60 s from a sol containing (A) 0.17 M TEOS, 3.3 mg/mL DSDH and 6.7% PDDA; (B) 0.17 M TEOS, 3.3 mg/mL DSDH and 3.35 % PDDA; (C) 0.17 M TEOS, 1.65 mg/mL DSDH and 6.7% PDDA. and 0. 33 M 33 TEOS. This experiment allows the estimation of the effective kinetic of the heterogeneous electron transfer reaction (k eff ) for the chosen electrochemical reaction. This constant can be affected by the film permeability, the eventual defects in the layer and the thickness of the electrodeposited films. Here, k eff decreases regularly from 0.039 to 0.033 cm.s -1 when increasing the TEOS concentration in the starting sol from 0.08 to 0.33 M (see Figure III-15A). Figure III- 15 . 15 Figure III-15. (A) 1/I versus 1/w 2 measured in the presence of 0.05 mM Ferrocenedimethanol with (a, squares) bare glassy carbon electrode and electrodes modified by a thin film prepared from a sol containing various TEOS concentration: (b, circles) 0.08 M, (c, triangles) 0.17 M and (d, stars) 0.33 M. Steady state current was measured by linear sweep voltammetry with using a potential scan rate of 20 mV/s. (B) Evolution of the peak current response of glassy carbon electrode modified by film (b), (c) and (d) in the presence of 1mM NAD + and 6 mM D-sorbitol in 0.1 M Tris-HCl buffer (pH 9). All electrodes have been prepared by electrochemically-assisted deposition at -1.3 V for 60 s from a sol containing 6.7% PDDA, 3.3 mg/mL DSDH and various concentrations of TEOS (see above). Potential scan rate 50 mV/s. Figure III- 16 . 16 Figure III-16. Peak current response versus D-sorbitol concentration measured with a glassy carbon electrode that has been modified by a thin silica film with encapsulated DSDH. The film was prepared with a sol containing 0.17 mM TEOS, 6.7 % PDDA and 3.3 mg/mL DSDH. The electrochemically-assisted deposition was done by applying -1.3 V for 60 s. All cyclic voltammograms have been performed at 50 mV/s, in the presence of 1 mM NAD + and different D-sorbitol concentrations in Tris-HCl buffer (pH 9). Figure III- 17 17 shows the influence of pH on the electrochemical response for both reduction of fructose (Figure III-17A) and oxidation of D-sorbitol (Figure III-17B), from measurements carried out with glassy carbon electrodes covered by the thin silica gel layer with encapsulated DSDH. A sharp increase in the electrode response is observed for fructose reduction around pH 6.5 Figure III- 18 . 18 Figure III-18. Electrochemical behavior of glassy carbon electrode modified by a silica thin film with encapsulated diaphorase. (A) Electrode response in the absence (a) and in the presence of 1 mM (b) and 2 mM (c) NADH. (B) Evolution of the electrode response to 0.4 mM NADH at pH 7 (a), 8 (b) and 9 (c). Films have been deposited by electrolysis at -1.3 V for 60 s with a sol containing 6.7 % PDDA, 0.17 M TEOS sol and 0.83 mg/mL diaphorase. Cyclic voltammograms have been performed in 0.1 M Tris-HCl buffer, in the presence of 0.1 M ferrocenedimethanol (and different NADH concentrations). Potential scan rate was 50 mV/s. Figure III- 19 . 19 Figure III-19. Electrochemical behavior of glassy carbon electrodes modified by a silica film with co-encapsulated DSDH and diaphorase. (A) Evolution of the electrode response to increasing concentration of D-sorbitol from 1 to 9 mM. The measurement was done at pH 9. (B) Evolution of the peak current response to 6 mM D-sorbitol from pH 6 to 10. Films have been deposited by electrolysis at -1.3 V for 60 s with a sol containing 6.7 % PDDA, 0.17 M TEOS sol and 0.83 mg/mL diaphorase. The cyclic voltammograms have been performed in 0.1 M Tris-HCl buffer in the presence of 1 mM NAD + and 0.1 mM ferrocendimethanol. Potential scan rate was 50 mV/s. They were obtained by gold electrodeposition inside an opal network of silica beads. The height of the different gold macroporous electrodes were 220 nm for one half layer of gold and 660 nm for three half layers (Figure III-20). displaying a much larger electroactive surface area in comparison to the geometric one (Figure III-21B to 21C) and compared to flat gold electrode (Figure III-21A). The electrolysis of the sol does not occur on gold at the same potential as for glassy carbon. A potential of -1.3 V versus the Ag/AgCl reference electrode was used in previous optimization on glassy carbon but only -1.1 V was necessary for macroporous electrodes in order to trigger the sol-gel deposition. Figure III- 21 . 21 Figure III-21. Electrochemical behavior of flat and macroporous gold electrodes modified by silica films with co-encapsulated DSDH and diaphorase. (A) Evolution of the flat gold electrode response to increasing concentrations of D-sorbitol from 1 to 7 mM. Dashed curves show the electrode response in the absence of D-sorbitol. (B) Evolution of the macroporous gold electrode response to increasing concentrations of D-sorbitol from 0.5 to 2.5 mM. Dashed curves show the electrode response in the absence of D-sorbitol. The measurement was done with a macroporous electrode of 660 nm thickness (three half layers) modified with using 30 s electrolysis. (C) Evolution of the peak current response to 0.5 mM D-sorbitol with modified electrodes prepared when using different electrolysis times from 10 s to 60 s (curves a to c). The measurements were done with a macroporous electrode of 660 nm thickness (three half layers). (D) Evolution of the peak current response to 0.5 mM D-sorbitol with macroporous electrodes displaying one half layer (a) and three half layers (b). Films were obtained when using 30 s electrolysis. (A-D) All films have been deposited by electrolysis at -1.1 V with a sol containing 6.7 % PDDA, 0.17 M TEOS sol and 0.83 mg/mL diaphorase. The cyclic voltammograms have been performed in 0.1 M Tris-HCl buffer at pH 9 in the presence of 1 mM NAD + and 0.1 mM ferrocendimethanol. Potential scan rate was 50 mV/s. Figure III - III Figure III-21A displays the response to D-sorbitol obtained with a flat gold electrode modified by the electrogenerated silica film with encapsulated DSDH and diaphorase. The electrode was found to be sensitive to successive additions of D-sorbitol, as shown by the increase in peak current intensity for the oxidation of ferrocenedimethanol. The addition of 1 mM D-sorbitol led here to 16 % peak current increase. By comparison, a much higher response was observed when doing the same experiment with a macroporous electrode (three half layers) modified in the same conditions by the bio-composite layer (Figure III-21B), and the peak current intensity was increasing by 1000 % when adding 1 mM D-sorbitol in the solution. The macroporous texture of the gold electrode improves thus significantly the catalytic efficiency of the sol-gel biocomposite. electrochemically-assisted deposition of silica gel layer has been successfully adapted to the encapsulation of DSDH. The encapsulated DSDH enzyme is likely to either oxidize Dsorbitol and produce simultaneously NADH inside the silica gel layer or reduce fructose and generate NAD + . The functional layer has been successfully deposited in macroporous gold electrodes and applied for the oxidation of D-sorbitol. When deposited in the same conditions and in the presence of the same concentration of D-sorbitol, the catalytic current increased much more with a macroporous electrode (1000 %) in comparison with a flat gold electrode(16 %). The macroporous texture of the gold electrode improves thus significantly the catalytic efficiency of the sol-gel biocomposite. The next step(s) of this work will be the immobilization of the electron-mediator and the enzymatic cofactor inside the silica gel matrix for application in zero-waste electrosynthesis. Chapitre IV. Co-immobilisation d'une déshydrogénase et du cofacteur enzymatique NAD + au sein de la matrice sol-gel L'immobilisation du cofactor enzymatique NAD + au sein de la matrice sol-gel a été étudiée en présence de DSDH et de diaphorase co-encapsulés au sein de cette même matrice. Dans cette configuration, le cofacteur doit avoir suffisamment de mobilité pour se déplacer du centre enzymatique de la DSDH au centre enzymatique de la diaphorase. Un médiateur électrochimique, le ferrocenedimethanol, permet alors la communication électronique entre la diaphorase et la surface de l'électrode. La faisabilité de cette étude a tout d'abord été menée en préparant la couche mince sol-gel par évaporation du sol initial. Différentes stratégies d'immobilisation du cofacteur ont alors été étudiées : (1) la simple encapsulation du NAD + dans la matrice sol-gel, seul ou en présence de nanotubes de carbone ; (2) l'encapsulation d'un dérivé du NAD + à haut poids moléculaire (NAD-dextran) ; (3) la fixation chimique du NAD + à la matrice silicatée via sa condensation sur le groupement époxy du glycidoxypropylsilane au cours du processus sol-gel (cette réaction a alors été suivie par spectroscopie infrarouge). Cette dernière approche s'est révélée être la plus intéressante en terme de réponse électrochimique et de stabilité du signal bioélectrocatalytique. Cette même approche a ensuite été étendue pour la génération électrochimique de couches minces sol-gel pour la co-immobilisation de la DSDH, de la diaphorase et du cofacteur NAD + . Cette expérience est alors applicable aux électrodes planes (carbone vitreux ou or) et aux électrodes macroporeuses. Figure IV- 1 1 Figure IV-1 describes the reaction pathway used in this work. Oxidation of the enzymatic substrate by the immobilized dehydrogenase induces NAD + reduction to NADH. Diaphorase catalyses then the oxidation of NADH back to NAD + . Electron transfer from the diaphorase to the glassy carbon electrode surface is carried out by ferrocene species that are introduced in the solution. The main technological barrier is the durable immobilization of the cofactor inan active form, the comparison of the various approaches proposed here to overcome this limit will be made with the redox mediator in solution. Figure IV- 1 . 1 Figure IV-1. Illustration of the electrochemical pathway used for the detection of the dehydrogenase enzymatic substrate. Figure IV- 2 . 2 Figure IV-2. Cyclic voltammograms obtained with (A) GCE/TEOS/PEI/(DSDH+DI)/NAD + ; (B) GCE/TEOS/PDDA/(DSDH+DI)/NAD + and (C) GCE/TEOS/PAA/(DSDH+DI)/NAD + in the absence of D-sorbitol (solid lines) and in the presence of D-sorbitol from 2 to 16 mM. All cyclic voltammograms have been performed in Tris-HCl buffer (pH 9) containing 0.1 mM FDM. Potential scan rate was 50 mV/s. dehydrogenase. In principle, various polyelectrolytes can be used to ensure the biological activity of DSDH and DI enzymes, such as PDDA, PAA or PEI, which gave rise to good catalytic responses when NAD + was introduced in solution. In the present case, the cofactor was part of the sol-gel film and only PEI gave rise to well-defined bioelectrocatalytic responses (Figure IV-2A) whereas no response to D-sorbitol could be observed when using PDDA or PAA additives (Figure IV-2B&C). The reason for such behavior/difference is not clear because favorable electrostatic interactions are expected in all cases, but comparison between Figure IV-2A and Figure IV-2B and 2C clearly demonstrates the crucial role of PEI Figure IV-3. Evolution of the peak current intensity recorded for successive analyses of 10 mM D-sorbitol solutions at distinct periods of time, using GCE modified with the same sol as Figure IV-2A, (a) in the absence and (b) in the presence of SWCNTs. All cyclic voltammograms have been performed in Tris-HCl buffer (pH 9) containing 0.1 mM FDM. Potential scan rate was 50 mV/s. in the final sol-gel biocomposite film deposited on GCE and likely to hold the cofactor by adsorption. Typical response of GCE/TEOS/PEI/SWCNT/(DSDH+DI)/NAD + to successive detections of 10 mM D-sorbitol is illustrated in Figure IV-3 (curve b). As shown, the voltammetric signals were slightly larger (by ca. 30 %) than in the absence of SWCNT, as well as some improved stability (almost constant response for 4 hours) but not at long time (> 50 % lost in sensitivity between 4 and 6 hours of use). The larger signal intensities can be explained by the increase in the electroactive surface area of the bio-electrode as a result of the introduction of SWCNT, but the fact that such increase remains rather small also suggests poor electrical interconnection, possibly due to the deposition of insulating sol-gel material on the surface of the individual nanotubes during the film formation.In a second step, experiments have been performed by adsorbing first the cofactor to carbon nanotubes (to get NAD-SWCNT) and introducing them afterwards in the biocomposite solgel matrix without any additional "free" NAD + . The resulting system (GCE/TEOS/PEI/(DSDH+DI)/NAD-SWCNT) led however to a limited electrochemical activity (about 10 times lower electrochemical response) (FigureIV-4), probably because the quantity of cofactor introduced by this route was rather low, and the stability of the cofactor immobilization was not significantly improved by comparison with the previous route. So, even if carbon nanotubes provides some improvements, the long-term operational stability was not yet satisfactory. Figure IV- 4 . 4 Figure IV-4. (A) Chronoamperogram recorded at 0.4 V with GCE modified with GCE/TEOS/PEI/(DSDH+DI)/SWCNT-NAD film to successive additions of D-sorbitol from 0.5 to 14.5 mM; (B) Cyclic voltammograms measured with the same electrode as (A) in the absence of D-sorbitol and in the presence of 14.5 mM D-sorbitol. Measurements have been performed in Tris-HCl buffer (pH 9) containing 0.1mM FDM. Figure IV - IV Figure IV-5A reports the typical response when increasing concentrations of D-sorbitol were introduced in the solution. The catalytic current increased regularly up to 16 mM, confirming the good behavior of this system. The stability of the electrode response was again evaluated by successive cyclic voltammograms in the presence of 10 mM D-sorbitol (Figure IV-5B). Chapter IV. Co-immobilization of dehydrogenase and cofactor in sol-gel matrix 123 functionalization in a one-pot reaction and no discussion was provided on the question if, or how, the epoxy ring participates effectively to the immobilization process. GCE/TEOS/PEI/(DSDH+DI)/NAD-GPS are reported inFigure IV-6A. Note that in this experiment the electron mediator (ferrocenedimethanol) was not immobilized on the electrode surface but simply introduced in solution. In these conditions a good electrocatalysis could be observed as shown by the increase in current upon successive additions of D-sorbitol from 0.5 to 15.5 mM. CV curves in the presence of 15.5 mM D-sorbitol also display the typical curve of electrocatalytic detection (inset of Figure IV-6A). FigureFigure IV- 6 . 6 Figure IV-6B compares the response of bioelectrodes prepared with and without GPS (i.e., GCE/TEOS/PEI/(DSDH+DI)/NAD-GPS and GCE/TEOS/PEI/(DSDH+DI)/NAD + , respectively) to 2 mM D-sorbitol. Both electrodes showed comparable current response around 6 µA at the beginning of the experiment but, while the amperometric signal of the bioelectrode prepared with GPS kept a rather stable value for ca. 14 hours of continuous reaction (curve a), that prepared without GPS showed very low stability as the bioelectrocatalytic response decreased dramatically during the first hour of the experiment and totally vanished after 5 hours (curve b). The current value of ca. 2.5 µA remaining at that time corresponds only to that of ferrocene (FDM) species in solution (which is even lower than that recorded in a control experiment made on bare GCE (about 3µA)). Figure IV- 7 7 Figure IV-7 shows the time-evolution of the ATR-FTIR spectra during 18 hours GPS hydrolysis in Tris-HCl buffer at pH=7.5. This figure shows also the ATR-FTIR spectrum of GPS in water after 1 h 40 min hydrolysis. The bands assignment was made according to the literature [35, 36]. The more interesting bands against hydrolysis of GPS are discussed below. In the spectrum of pure GPS (Figure IV-8), broad, poorly resolved bands are centered at 1076 cm -1 . They are mainly assigned to C-O, Si-O stretching modes of glycidoxy and methoxy groups. Their NAD + and the confinement inside the gel at close distance to the proteins (dehydrogenase and diaphorase) allows measuring good enzymatic activity with DSDH (Figure IV-6.). Figure IV- 11 . 11 Figure IV-11. Experiment of chronoamperometry recorded at 0.4 V with GCE/TEOS/PEI/(DSDH+DI)/NADH+GPS (i.e. replacing NAD + by NADH and following the same protocol). Measurements have been performed for 14 hours under convective conditions in 0.1 M Tris-HCl buffer (pH 9) containing 0.1 mM FDM and 2 mM D-sorbitol. Figure IV- 13 13 displays the response to D-sorbitol obtained with a GCE electrode modified by the electrogenerated silica film with encapsulated DSDH, diaphorase and GPS functionalized cofactor (NAD-GPS). Figure IV-14 displays the response to D-sorbitol obtained with a flat gold electrode modified by the electrogenerated silica film with encapsulated DSDH, diaphorase and NAD-GPS. Figure IV- 14 14 shows the electrochemical response obtained with the modified electrode prepared with both PEI and PDDA as polyelectrolyte additive. Before addition of D-sorbitol, a well defined electrochemical signal due to ferrocenedimethanol could be observed. The addition of Dsorbitol in the solution from 2 to 10 mM led to a significant increase in the current response.The co-immobilization of DSDH and GPS-NAD on gold electrode surfaces was properly achieved by using both PEI and PDDA as polyelectrolyte additive. Figure IV- 15 . 15 Figure IV-15. Cyclic voltammograms obtained using (A) flat and (B) macroporous (660 nm, three half layers) gold electrode modified by TEOS/NAD-GPS/PEI/PDDA/(DSDH+DI) film in the absence and presence of D-sorbitol. Films have been deposited by electrolysis at -1.1 Vfor 30 s with a sol containing 0.15M TEOS, 14 mM NAD-GPS, 2.3 mg/mL DSDH, 0.76 mg/mL DI and 1 % PEI and 0.5%. Cyclic voltammograms have been performed in 0.1 M Tris-HCl buffer, in the presence of 0.1m M FDM. Potential scan rate was 50 mV/s. A successful strategy for dehydrogenase, diaphorase and cofactor co-immobilization in solgel films has been developed. It involves the chemical bonding of NAD + to the epoxide group of glycidoxypropylsilane (GPS) before co-condensation of the organoalkoxisilane with tetraethoxysilane in the presence of the proteins (dehydrogenase and diaphorase) and a Figure V- 1 . 1 Figure V-1. Cyclic voltammograms recorded with a GCE modified by drop-coated (A) TEOS/PEI/Fc-PEI;(B) TEOS/GPS/PEI/Fc-PEI film in the 0.1 M Tris-HCl buffer (pH 9) at a scan rate of 50 mV/s, scan cycle, 5. Figure V- 2 . 2 Figure V-2. (A) Cyclic voltammograms recorded at GCE modified with TEOS/GPS/Fc-PEI/NAD-GPS/DSDH/DI film in the absence of D-sorbitol and in the presence of 2.8 mM Dsorbitol. (The film was prepared with a sol containing 0.077 M TEOS, 0.0385MGPS, 14 mM NAD-GPS, 2.3 mg/mL DSDH, 0.76 mg/mL DI and Fc-PEI by drop-coating); (B) Amperometric responses recorded at an applied potential of 0.4 V to successive additions of 0.2mM D-sorbitol in stirred solution. (C) Corresponding calibration plot. All measurements were performed in the 0.1 M Tris-HCl buffer (pH 9). Figure V - V Figure V-2A shows the electrochemical response of the film to the addition of D-sorbitol in solution. In the absence of D-sorbitol, only the electrochemical signal of ferrocene is observed. The addition of D-sorbitol induces an increase of the anodic current and a decrease of the cathodic current. It corresponds to the electrochemical oxidation of the NADH cofactor produced by the encapsulated DSDH while oxidizing D-sorbitol. Here the enzymatic cofactor is detected at the diaphorase modified GCE by using ferrocene as electron mediator. NADH reacts with diaphorase and the electrochemically generated ferricinium ions to produce NAD + and ferrocene species that can be re-oxidized in the electrocatalytic scheme.Figure V-2B Figure V-2Bshows amperometric responses of the modified electrode to D-sorbitol in a stirred solution. Figure V- 3 . 3 Figure V-3. Chronoamperograms recorded at 0.4 V with GCE modified by (a) TEOS/GPS/Fc-PEI/NAD-GPS/DSDH/DI; (b) TEOS/Fc-PEI/NAD-GPS/DSDH/DI film. Measurements have been performed for 14 hours oxidation under convective conditions in 0.1 M Tris-HCl buffer (pH 9) containing 2 mM D-sorbitol. Figure V- 4 4 Figure V-4 shows typical cyclic voltammograms recorded with a GCE modified by electrodeposited TEOS/GPS sol-gel film containing Fc-PEI. No electrochemical signal of Figure V- 5 . 5 Figure V-5. Electrochemical response of glassy carbon electrode modified by TEOS/GPS/PEI/Fc-PEI/NAD-GPS/DSDH/DI film in the absence and in the presence of Dsorbitol from 2mM to 4mM. Films have been deposited by electrolysis at -1.3 V for 60 s with a TEOS/GPS sol containing DSDH, DI, NAD-GPS and Fc-PEI. Cyclic voltammograms have been performed at 50 mV/s in 0.1 M Tris-HCl buffer. 1 Figure V- 6 . 16 Figure V-6. Cyclic voltammograms recorded with a GCE modified by drop-coated (A) TEOS/Fc-silane/PEI;(B) TEOS/GPS /Fc-silane/PEI film in the 0.1 M Tris-HClO 4 buffer (pH 9) at a scan rate of 50 mV/s, scan cycle, 10. /AgCl / V sol and drop-coated on GCE. Figure V-7A shows that it worked quite nicely with a well defined reversible electrochemical signal of ferrocene (plain line). It was found that the oxidation peak increase in the presence of 7 mM D-sorbitol and the reduction peak disappeared at the same time (dashed line). Figure V- 7 . 7 Figure V-7. Cyclic voltammograms recorded with a GCE modified by drop-coated TEOS/GPS /Fc-silane/PEI/DSDH/DI/NAD-GPS film in the absence and in the presence of 7 mM D-sorbitol. Potential scan rate: 50 mV/s. (B) Amperometric responses recorded at an applied potential of 0.4 V to successive additions of different concentration of D-sorbitol in 0.1 M Tris-HClO 4 buffer (pH 9). Figure V- 8 . 8 Figure V-8. Cyclic voltammograms recorded with a GCE modified by electrodeposited TEOS/GPS/PEI/Fc-silane film in the 0.1 M Tris-HClO 4 buffer (pH 9) at a scan rate of 50 mV/s, scan cycle, 10. Films have been deposited by electrolysis at -1.3 V for 60 s with a TEOS/GPS sol with Fc-silane as co-condensation precursors. Figure V- 9 Figure V- 9 . 99 Figure V-9 shows cyclic voltammograms recorded with a GCE modified by electrodeposited sol-gel film prepared with TEOS, GPS and Fc-silane and containing DSDH, DI and NAD-GPS. In the absence of D-sorbitol, the well reversible electrochemical signal of ferrocene was observed. However, the addition of D-sorbitol in the solution did not induce any increase of the anodic current. Figure V- 10 . 10 Figure V-10. Electrochemical response of glassy carbon electrode modified by TEOS /GPS /Fc-silane/PEI/DSDH/DI film in the absence and in the presence of D-sorbitol from 2 to 4mM. Films have been deposited by electrolysis at -1.3 V for 60 s. Cyclic voltammograms have been performed at 50 mV/s in 0.1 M Tris-HClO 4 buffer containing 1mM NAD + . 4. 1 1 Co-immobilization in drop-coated sol-gel film4.1.1 Effect of GPS on Os-polymer immobilizationOne knows from literatures, osmium polymers as mediators have been immobilized on electrode surface to develop reagentless dehydrogenase biosensors. Riccarda et al. reported a biosensor based on glucose dehydrogenase (GDH) and diaphorase (DI) co-immobilized with NAD + into a carbon nanotube paste (CNTP) electrode modified with an osmium functionalized polymer[34]. The carbon nanotubes could be wrapped up in the Os-polymer molecules when they are mixed together in the paste and this combination could improve the transfer of electrons between the mediator and the electrode material itself. Olha et al. Figure II- 2 )Figure V- 11 . 211 Figure V-11. Cyclic voltammograms recorded with a GCE modified by drop-coated (A) TEOS/PEI/Os-polymer;(B) TEOS/GPS/PEI/Os-polymer film in the 0.1 M Tris-HCl buffer (pH 9) at a scan rate of 50 mV/s, scan cycle, 10. Figure V- 12 12 shows the typical electrocatalytic response of the TEOS/GPS/PEI/Os-polymer/DI film to NADH. In the absence of NADH, the reversible electrochemical signal of osmium was observed. The addition of 0.3 mM NADH induced a modification of the electrochemical response, the anodic current increased while the cathodic signal disappeared, and the anodic peak current continued to increase with the NADH concentration. Diaphorase kept good catalytic characteristic toward NADH inside the sol-gel film, and the immobilized osmium can efficiently transfer the electron between the electrode and the diaphorase. Figure V- 12 . 4 . 1 . 3 12413 Figure V-12. Cyclic voltammograms recorded with a GCE modified by drop-coated TEOS/GPS/PEI/Os-polymer/DI film in the absence and presence NADH from 0.3 to 1.2mM. All cyclic voltammograms have been performed in Tris-HCl buffer (pH 9) at a scan rate of 50 mV/s. 2 2 Co-immobilization in electrodeposited sol-gel thin film The encapsulation of DSDH, DI, cofactor and Os redox polymer inside the electrodeposited silica gel led to a similar result and no visible electrochemical signals of Os neither before nor after the addition of D-sorbitol was observed (Figure V-15). Figure V- 15 . 15 Figure V-15. Electrochemical response of glassy carbon electrode modified by TEOS/GPS/PEI/Os/NAD-GPS/DSDH/DI film in the absence and in the presence of D-sorbitol from 2mM to 4mM. Films have been deposited by electrolysis at -1.3 V for 60 s with a TEOS/GPS sol containing DSDH, DI, NAD-GPS and Os-polymer. Cyclic voltammograms have been performed at 50 mV/s in 0.1 M Tris-HCl buffer. /AgCl / V immobilization of all components (dehydrogenase, cofactor and electron mediator) in sol-gel films was achieved by using drop-coating. All the components were able to communicate inside the silica gel layer for efficient electro-catalytic oxidation of D-sorbitol. The electrode displayed good stability under stirring for more than 14 hours. However, such coimmobilization applied to electrochemically-assisted deposition of sol-gel thin films was not successful, revealing limitation with the mediator immobilization. An adequate distribution of the mediator in the sol-gel film could be achieved by drop-coating, allowing an efficient communication between the electrode and the protein. However, using the same sol, film prepared by electrodeposition did not display this property, and the electrocatalysis (for Fcsilane) or even the oxidation/reduction of the mediator (for Fc-PEI and osmium polymer) was not observed. Chapitre VI. Immobilisation du médiateur électrochimique sur les nanotubes de carbone et co-immobillisation avec une déshydrogénase et le cofacteur NAD + Dans ce chapitre, différentes stratégies ont été développées pour élaborer un composite solgel/nanotubes de carbone permettant la co-immobilisation de la D-sorbitol déshydrogénase (DSDH), du cofacteur NAD + et du médiateur (et éventuellement de la diaphorase). Une configuration en bicouche a été utilisée consistant à immobiliser dans un premier temps les nanotubes de carbone fonctionnalisés et à déposer ensuite la couche mince sol-gel par évaporation du sol ou par électrogénération. Une attention particulière a été donnée à la coimmobilisation par électrochimie des différents éléments du système bioélectrocatalytique dans la mesure où cet objectif n'avait pu être atteint par électrogénération en une étape (chapitre V). Les nanotubes de carbones à parois simples (SWCNT) ou multiples (MWCNT) ont été fonctionnalisés par quatre protocoles différents pour leur donner des propriétés catalytiques intéressantes pour l'oxydation de NADH (ou la réduction du NAD + ) ; Ces protocoles sont (1) le traitement des nanotubes par micro-ondes (MWCNT-µW), (2) l'électropolymérisation du vert de méthylène (MWCNT-PMG), (3) le recouvrement des nanotubes par un polymère de type polyacrylate portant des complexes d'osmium(III) (MWCNT-Os), et (4) adsorption des complexes d'un complexe de rhodium (III) à la surface de nanotubes de carbone à paroi simple. Les électrodes de carbone vitreux fonctionnalisées par ces nanotubes de carbone présentes de bonnes propriétés catalytiques pour l'oxydation de NADH (1 et 2) ou la réduction de NAD + (4), ou l'oxydation de NADH en présence de diaphorase (3). Les films sol-gel ont ensuite été déposés à la surface de ces nanotubes de carbone by évaporation du sol ou par électrogénération afin d'obtenir la co-immobilisation de la DSDH (et de la diaphorase quand nécessaire) et du cofacteur NAD + .Chapter VI. Mediator immobilization on carbon nanotubes and co-immobilization with dehydrogenase and cofactorIn this chapter, various bilayers strategies based on CNTs/sol-gel matrix are developed for the fabrication of reagent free devices in attempting to overcome the problems encountered with mediator immobilization through one step electrodepostion (chapter V). Carbon nanotubes (CNTs) have been functionalized by four different protocols in order to provide them catalytic properties for NADH (or NAD + ) detection; they include (1) micro-wave treatment (MWCNTs-µW), (2) electrochemical deposition of poly(methylene green) (MWCNTs-PMG), (3) wrapping with a polyacrylate polymer holding osmium(III) complexes (MWCNTs-Os), and (4) adsorption a Rh (III) complex on SWCNTs. GCE electrodes modified with these functionalized CNTs show good electrochemical properties and allow the direct electrocatalytic detection of NADH (1 and 2) or NAD + (4), or the detection of NADH in the presence of diaphorase (3). To the last configuration, a sol-gel thin film has been further deposited on the carbon nanotube layer by drop-coating or by electrochemically-assisted deposition for encapsulation of D-sorbitol dehydrogenase (and diaphorase when necessary) ) electrochemical deposition of poly(methylene green) (MWCNTs-PMG), and (3) wrapping with a polyacrylate polymer holding osmium(III) complexes (MWCNTs-Os). The catalytic properties of the functionalized carbon nanotubes have been first checked by covering them with an additional drop-coated sol-gel layer before use as support for electrodeposited sol-gel films. The electrochemical response of the biocomposite containing the immobilized protein has been compared when NAD + was simply introduced in the solution or when it was chemically attached to the sol-gel matrix (i.e., reagentless device). The study has been developed with the enzyme D-sorbitol dehydrogenase and D-sorbitol was used as a model analyte. Immobilization of Rh (III) Figure Figure VI-1A shows the cyclic voltammograms recorded at a glassy carbon electrode modified with the original (i.e., not treated) MWCNTs. Before addition of NADH, no electrochemical signal could be observed between -0.4 and +0.6 V. A well defined voltammetric signal with a peak potential located at ~ +0.4 V (versus Ag/AgCl reference electrode) appears when NADH was added to the solution. It corresponds to the electrochemical oxidation of the NADH cofactor on MWCNTs modified electrode. Figure VI-1B shows the cyclic voltammograms recorded at the glassy carbon electrode modified with acid-microwaved MWCNTs. A well defined voltammetric signal with a peak potential located at ~ 0 V appears when NADH was added to the solution and this signal increased with the NADH concentration. The microwaving of MWCNTs in acidic solution for 20 min resulted in a dramatic shift of the oxidative peak potential of NADH (E NADH ) to a lower value, from ~ +0.4 V to ~ 0 V (in agreement with previous observations [24]). The shift in E NADH illustrated can be attributed to the redox mediation of NADH oxidation by surface quinones (Q) (formed during the microwave treatment). Figure VI- 1 . 1 Figure VI-1. Cyclic voltammograms recorded at (A) GCE/MWCNTs, (B) GCE/MWCNTs-µW in deoxygenated solutions with and without NADH. Measurements have been performed in 0.1 M Tris-HCl buffer (pH 9), Scan rate, 5 mV/ s. FigureVI-Figure VI- 2 . 2 . 3 2 . 3 . 1 223231 Figure VI-2. Electrochemical response of (A) GCE/chitosan/TEOS/MWCNTs-µW/PEI/DSDH; (B) GCE/MWCNTs-µW&TEOS/PEI/DSDH in the absence and in the presence of D-sorbitol from 2 to 8 mM. Cyclic voltammograms have been performed at 5 mV/s in deoxygenated 0.1 M Tris-HCl buffer containing 1mM NAD + . Figure VI- 3 B 3 Figure VI-3 B shows its amperometric responses at different applied potential. No electrocatalytic response was obtained upon addition of D-sorbitol in stirred solution at +0.4 Figure VI- 3 CFigureVI- 3 DFigure VI- 3 . 333 Scheme of (A)&(B)Scheme of (C)&(D) Figure VI- 4 AFigure VI- 4 . 44 Figure VI-4. Electrochemical response of GCE/MWCNTs-µW&TEOS/PEI/DSDH/NAD-GPS film. (A) Cyclic voltammograms in the absence and in the presence of D-sorbitol from 2 to 4mM. Potential scan rate: 5 mV/s. (B) Amperometric responses recorded at an applied potential of 0.4 V to successive additions of D-sorbitol in the solution. Figure VI- 5 . 5 Figure VI-5A shows the electropolymerization process of MG on the MWCNTs modified GC electrode. It is shown that the peak current increased with the number of cycles, which means more and more MG are deposited on the surface of the electrode. After a few cycles the deposition gradually reaches equilibrium. Figure VI-5B shows a comparative amperometic response of PMG modified GCE in the absence and presence of MWCNTs upon successive addition of 0.2 mM NADH to 0.1M pH 9.0 tri-HCl buffer at an applied potential of +0.2V. A catalytic response is obtained in the absence of MWCNTs, but current intensity were very low. A significant improvement in the current response was obtained when MG was electrodeposited on MWCNT modified GCE. An increasing electrocatalytic response was observed due to the addition of NADH. So with carbon nanotubes, PMG film induces a stable and significantly improved electrochemical response for the mediated oxidation of NADH. 3. 2 3 . 2 . 1 2321 Drop-coating of sol-gel film at GCE/MWCNTs-PMG Encapsulation of DSDH in sol-gel film drop-coated onto GCE/MWCNTs-PMGThe efficient system (GCE/MWCNTs-PMG) was first evaluated in combination with Dsorbitol dehydrogenase. Figure VI-6 shows the amperometic response of different films to D-sorbitol in the Tris-HCl buffer containing 1mM NAD + at an applied potential of 0.2V. The background current was allowed to decay to a steady value before the addition of D-sorbitol. Obvious current responses were observed due to the addition of D-sorbitol on the GCE/MWCNTs-PMG/&TEOS/PEI/DSDH, and this signal increased with the D-sorbitol concentration (curve a). It corresponds to the electrochemical oxidation of the NADH cofactor produced by the encapsulated DSDH while oxidizing D-sorbitol. Curve b and c shows the comparison of the response obtained with films prepared in the absence of MWCNTs and in the absence of PMG. In both cases, only small electrochemical signals were observed. So only the system including both MWCNTs and PMG display a good sensitivity to D-sorbitol. The effects can be ascribed to the increased electroactive surface area obtained with the introduction of MWCNTs for PMG deposition. Figure VI- 6 . 6 Figure VI-6. A comparative view of the amperometric response obtained by using (a) GCE/MWCNTs-PMG&TEOS/PEI/DSDH; (b) GCE/PMG&TEOS/PEI/DSDH; (c) GCE/MWCNTs& TEOS/PEI/DSDH to successive additions of 0.5, 0.5, 1, 1, 2, 2, 3, 3mM Dsorbitol (Eappl = +0.2 V vs. Ag/AgCl). Measurements have been performed in 0.1 M Tris-HCl buffer (pH 9) containing 1mM NAD + . Figure VI- 7 . 7 Figure VI-7. (A) Cyclic voltammograms obtained by using GCE/MWCNTs-PMG/&TEOS/PEI/DSDH in the absence of D-sorbitol (solid line) and in the presence of Dsorbitol from 2 to 6mM(dashed lines), scan rate: 50mV/s. (B) A comparative view of the amperometric response at different applied potential to successive additions of 0.5, 0.5, 1, 1, 2, 2, 3, 3mM D-sorbitol. All measurements have been performed in 0.1 M Tris-HCl buffer (pH 9) containing 1 mM NAD + . Figure /DSDH/NAD + to D-sorbitol. The current increased regularly with D-sorbitol concentration during the first measurement. However, the second measurement made after /AgCl / V electrolyte medium exchange led to complete vanishing of the response. Possible reason is that NAD + is a small molecule, diffusing easily away from the electrode surface into solution. Figure VI- 8 . 8 Figure VI-8. Amperometric response recorded at +0.2 V with GCE/MWCNTs-PMG&TEOS/PEI/DSDH/NAD + in 0.1 M Tris-HCl buffer (pH 9), different concentration of D-sorbitol 0.5,0.5, 1, 1, 2, 2mM were added after 2 min of current recording. Figure VI- 9 9 compares the electrode response to 1 mM D-sorbitol of bioelectrodes obtained with (curve a) and without GPS (curve b) in the starting sol. Both electrodes showed comparable response around 3 µA at the beginning of the experiment. But the electrode prepared with GPS displayed a good operational stability, more than 75% current response was kept after continuous 14-h long experiments in a stirred solution (curve a). At the opposite, the bioelectrode prepared without GPS showed very low stability and the electrode response decreased dramatically during the first hours of experiments (curve b). Most of the catalytic activity was lost during the first 6 hours, as the electrode response almost reached null current. The immobilized cofactor in the sol-gel matrix can be regenerated by PMG deposited on MWCNTs. And as it was previously observed, this functionalization of NAD + with GPS can greatly enhance the stablilty of electrochemical response of the reagentless device. Figure VI- 9 . 9 Figure VI-9. Chronoamperograms recorded at 0.2 V with (a) GCE/MWCNTs-PMG&TEOS/PEI/DSDH/NAD-GPS; (b) GCE/MWCNTs-PMG&TEOS/PEI/DSDH/NAD + . Measurements have been performed for 14 hours oxidation under convective conditions in 0.1 M Tris-HCl buffer (pH 9) containing 1 mM D-sorbitol. Figure VI- 10 AFigure VI- 10 . 3 . 3 . 2 1010332 Figure VI-10 A shows the electrochemical response obtained with GCE/MWCNTs-PMG covered with an additional electrodeposited sol-gel layer with encapsulated DSDH, which has been deposited by electrolysis at -1.3 V for 60 s with a sol containing DSDH (GCE/MWCNTs-PMG/&TEOS/PEI/DSDH), here the cofactor was not attached to the silica matrix. The addition of D-sorbitol induced the modification of oxidation peak current at 0.1-0.2 V, and this signal increased with the D-sorbitol concentration from 2 to 6 mM. Figure VI-10 B shows its amperometric responses at an applied potential of +0.2V. An increasing and stable electrocatalytic response was obtained upon addition of D-sorbitol in the stirred solution. The electrochemically-assisted deposition of silica thin films can be a good strategy to immobilize DSDH in an active form on GCE/MWCNTs-PMG. Figure VI- 11 . 11 Figure VI-11. Electrochemical response of GCE/MWCNTs-PMG/&TEOS/PEI/DSDH/NAD-GPS. (A) Cyclic voltammograms in the absence and in the presence of 8mM D-sorbitol. Potential scan rate: 50 mV /s. (B) Amperometric response at an applied potential of +0.2 V to successive additions of 2mM D-sorbitol in the solution. All measurements have been performed in 0.1 M Tris-HCl buffer (pH 9). Figure VI- 12 . 12 Figure VI-12. Electrochemical response of GCE/MWCNTs-PMG/&TEOS/PEI/DSDH/NAD-GPS. (A) Amperometric response at an applied potential of 0.2 V to successive additions of 2mM D-sorbitol and 0.2 mM NADH in the solution (B) Amperometric response at an applied potential of 0.2 V to additions of 1mM D-sorbitol and 1mM NAD + in the solution. All measurements have been performed in 0.1 M Tris-HCl buffer (pH 9). this electrochemical signal was not stable with time and decreased when multiple potential scan were performed. So, simply mixing MWCNTs and Os-polymer solution in the presence of chitosan did not result in the stable immobilization of osmium polymer in the MWCNT layer. A second protocol for immobilization of this polymer with carbon nanotube was tested.MWCNTs have been first sonicated for 1h and then incubated in osmium-polymer solution for 12h. In these conditions, the positively-charged polymer can wrap on the sidewall surface of MWCNT (MWCNT-Os). Figure VI- 13 . 4 . 2 4 . 2 . 1 1342421 Figure VI-13. Cyclic voltammograms recorded with (A) GCE/MWCNTs/Os (B) GCE/MWCNTs-Os. 10 successive scans in the 0.1 M Tris-HCl buffer (pH 9), potential scan rate: 50 mV/s. Figure VI- 14 . 14 Figure VI-14. Amperometric response obtained at GCE/MWCNT-Os&TEOS/PEI/DI to successive additions of 0.2 mM NADH (Eappl = + 0.3 V vs. Ag/AgCl). Measurements have been performed in 0.1 M Tris-HCl buffer (pH 9). Figure V- 15 . 15 Figure V-15. Cyclic voltammograms recorded with GCE/MWCNT-Os&TEOS/PEI/DI/NAD-GPS in the absence and in the presence of 3 mM D-sorbitol. Potential scan rate: 50 mV/s. (B) Amperometric responses recorded at an applied potential of 0.3 V to successive additions of 0.5 mM D-sorbitol in 0.1 M Tris-HCl buffer (pH 9). Figure VI- 16 shows 16 the amperometric responses at an applied potential of 0.3V. An increasing and stable electrocatalytic response was obtained upon addition of NADH in stirred solution. The cyclic /AgCl / V voltammograms recorded before addition of NADH (Inset of Figure shows that the electrochemical signal from the osmium complexes has been significantly decreased in the presence of the sol-gel material. Nethertheless, the addition of NADH leads to a significant increase in the current intensity. The shape of the catalytic signal was poorly resolved and could be a mix between the detection of NADH through osmium complex and diaphorase and the simultaneous detection of NADH at the carbon nanotube surface, as reported in Figure VI-1A. Figure VI- 16 . 16 Figure VI-16. Amperometric response obtained at GCE/MWCNT-Os&TEOS/PEI/DI to successive additions of 0.2 mM NADH (Eappl = + 0.3 V vs. Ag/AgCl). Inset plot shows cyclic voltammograms recorded before and after addition of 1.4mM NADH. Films have been deposited by electrolysis at -1.3 V for 60 s. Measurements have been performed in 0.1 M Tris-HCl buffer (pH 9). Figure VI-17. (A) Cyclic voltammograms recorded with GCE/MWCNT-Os&TEOS/PEI/DI/NAD-GPS in the absence and in the presence of D-sorbitol from 2 to 8mM. Potential scan rate: 50 mV/s. (B) Amperometric responses recorded at an applied potential of 0.3 V to successive additions of 0.4mM D-sorbitol in stirred solution. Inset plot shows the corresponding calibration plot. (C) Amperometric response obtained in the same conditions. measurements have been performed for 14 hours under convective conditions in 0.1 M Tris-HCl buffer (pH 9) containing 2 mM D-sorbitol. SWCNTs for [Cp*Rh(bpy)Cl] + immobilization: towards a device operating in reduction 5.1 [Cp*Rh(bpy)] 2+ as suitable mediator for NADH regeneration Figure VI- 18 . 18 Figure VI-18. The mechanism of [Cp*Rh(bpy)] 2+ electrocatalytic process. Figure VI- 19 Figure VI- 19 . 1919 Figure VI-19. Cyclic voltammograms recorded with compounds 11 immobilized as a selfassembled monolayer on gold electrode: (A) effect of potential scan rate; (B) effect multisweep; (C) electrode response in the absence and presence of increasing NAD + concentrations. Potential scan rate: 100 mV/ s. Figure VI- 20 .Figure VI- 21 . 2021 Figure VI-20. [Cp*Rh(bpy)] + -type mediators functionalized with siloxy-function. MWCNTs. They include (1) the simple microwave treatment of MWCNTs, (2) electrochemical deposition of poly(methylene green) on MWCNTs, and (3) wrapping of MWCNTs with an Osmium(III) polymer. All the developed strategies show significantly decreased overpotentials for NADH oxidation. A sol-gel thin film has been further deposited functionalized carbon nanotubes by drop-coating for encapsulation of D-sorbitol dehydrogenase (and diaphorase when necessary) and co-immobilization of the NAD + cofactor. Le but des travaux menés dans cette thèse était l'immobilisation sous une forme stable et active de protéines de type déshydrogénase, du cofacteur NAD + /NADH et d'un médiateur électrochimique au sein d'une matrice sol-gel déposée sous forme de couche mince à la surface d'une électrode par évaporation du sol ou électrogénération. Cette électrode ainsi modifiée doit alors être utilisée en électrosynthèse enzymatique. Afin d'atteindre les objectifs posés au commencement du projet, ce travail a été divisé en trois grandes étapes :(1) immobilisation au sien du film sol-gel de protéines de type déshydrogénase sous une forme active, (2) immobilisation du cofacteur et(3) immobilisation du médiateur électrochimique et co-immobilisation avec la déshydrogénase et le cofacteur. Les besoins identifiés pour l'électrosynthèse enzymatique sont l'immobilisation stable d'une quantité importante de déshydrogénase actives à la surface de l'électrode du réacteur. Les études de faisabilité sur l'encapsulation de déshydrogénase au sein de la matrice sol-gel ont été menés en utilisant la D-sorbitol déshydrogénase (DSDH) comme enzyme modèle et en déposant la couche mince par évaporation du sol. Il est rapidement apparu que la DSDH était très sensible à l'environnement sol-gel et que son encapsulation dans une matrice de silice pure conduisait à une absence d'activité électrocatalytique. L'intérêt d'additifs dans le sol initial a ainsi été évalué et il a été montré que l'ajout de polyélectrolytes positivement chargés permettait d'augmenter de façon importante l'activité catalytique de la protéine encapsulée. Les charges positive du polymère (par exemple, Poly(dimethyldiallylammonium chloride), PDDA) ont en effet une interaction favorable avec la protéine négativement chargée pendant le processus d'encapsulation. Ce protocole a ensuite été adapté et optimisé pour le dépôt électrochimiquement assisté de cette couche mince sol-gel. Il a également été montré que la diaphorase pouvait être co-encapsulée avec la DSDH et permettre la régénération du cofacteur NADH en présence du médiateur ferrocènedimethanol. Cette procédure de bioencapsulation sol-gel électrochimique a ensuite été appliquée à la modification d'électrodes macroporeuses et il a été montré que la plus grande surface électroactive apportée par cette macroporosité permettait d'augmenter de façon significative la réponse bioélectrocatalytique vis-à-vis de l'oxydation du D-sorbitol. Conclusion and outlook 202 Un des plus gros défis de ce travail concernait l'immobilisation du cofacteur NAD + / NADH tout en permettant de conserver son activité pendant un temps le plus long possible. Une stratégie efficace pour répondre à cet objectif a ainsi été développée dans la seconde partie de ce travail. Elle met en oeuvre la réaction chimique entre NAD + et le groupement epoxy du composé glycidoxypropylsilane (GPS) avant sa co-condensation avec le tetraethoxysilane (TEOS) en présence des protéines (déshydrogénase et diaphorase) et de poly(ethyleneimine) (PEI). Toutes ces opérations se font dans des conditions douces compatibles avec la bioencapsulation sol-gel. Par comparaison avec l'encapsulation simple de NAD + avec ou sans nanotubes de carbone ou l'encapsulation de NAD-dextran, la procédure utilisant GPS est moins couteuse, et plus simple à mettre en oeuvre. Enfin, ce protocole conduit à une activité catalytique très stable, pendant plus de 12h en solution en présence de convection. Finalement cette méthode développée pour le dépôt du film par évaporation du sol a été adaptée pour l'électrogénération sur électrode d'or macroporeuse. Dans toutes ces études le médiateur électrochimique ferrocènediméthanol se trouvait dans la solution. Finalement, la dernière partie de ce travail a été consacrée à l'immobilisation du médiateur électrochimique à la surface de l'électrode de façon à ce qu'il puisse communiquer efficacement avec l'électrode et avec la diaphorase ou directement avec le cofacteur et ainsi obtenir la régénération électrochimique du cofacteur. Différentes stratégies ont ainsi été étudiées pour obtenir le système final contenant l'ensemble des éléments de cette chaîne bioélectrocatalytique co-immobilisés dans la matrice sol-gel. Ces études ont d'abord été développées en déposant le film sol-gel par évaporation du sol. Comme pour les études précédentes la DSDH est utilisée comme protéine modèle, le cofacteur est immobilisé par couplage avec le GPS et le médiateur est introduit dans la matrice sol-gel à l'aide de polymère portant des espèces ferrocène ou osmium ou à l'aide d'un ferrocène fonctionnalisé par un groupement silane pouvant être co-condensé avec les autres silanes du sol (TEOS et GPS). Il a tout d'abord été montré que GPS permettait d'augmenter fortement la stabilité de l'immobilisation du médiateur en plus de son rôle dans l'immobilisation du cofacteur. Ceci est du à la plus grande stabilité mécanique des films préparés avec GPS. La co-immobilisation de ces médiateurs avec le cofacteur et les protéines (DSDH et diaphorase) a ainsi permis d'obtenir pour la première fois dans une matrice sol-gel une activité catalytique stable. Finalement, les tentatives pour transposer ce résultat obtenu par évaporation du sol à l'électrogénération ont échoués. La distribution du médiateur électrochimique dans les Conclusion and outlook 203 matrices sol-gel obtenus par électrochimie semble ne pas être suffisamment homogène ou leur mobilité ne pas être suffisante pour permettre le transport électronique et une interaction avec l'électrode ou la diaphorase. Afin de résoudre le problème posé par l'absence d'activité catalytique avec les films préparés par électrogénération, nous avons développé une stratégie différente, basée sur le dépôt de la couche mince sol-gel à la surface de nanotubes de carbone fonctionnalisés par différents médiateurs. Cette fonctionnalisation des nanotubes de carbone a été obtenue par la formation de fonctions quinones par traitement micro-ondes, par électropolymérisation du vert de méthylène et par recouvrement des nanotubes par un polymère de type acrylate portant des complexes d'osmium(III). Seul ce dernier système a alors permis d'observer, en présence de diaphorase, une régénération du cofacteur immobilisé dans la matrice sol-gel obtenue par électrogénération. La flexibilité du polymère d'osmium et l'intermédiaire de la diaphorase permettent alors la régénération douce du cofacteur enymatique. Tous les composants immobilisés communiquent efficacement à l'intérieur du gel de silice pour permettre l'oxydation électrocatalytique du D-sorbiltol. Les nanotubes de carbone fonctionnalisés par un médiateur preséntant une certaine faisabilité est un bon substrat d'électrode pour l'électrogénération sol-gel pour la co-immobilisation du cofacteur et des protéines. La couche mince sol-gel qui a été développée au cours de cette thèse peut être appliquée à la préparation de composés chiraux par électrosynthèse enzymatique. Tous les éléments participant à cette synthèse étant immobilisés sur l'électrode, cette approche est intéressante autant d'un point de vue économique que environnemental, en évitant l'utilisation de solvant et en réduisant es étapes de purification au minimum. Un tel concept correspond aux standards de la chimie verte avec un procédé induisant très peu de déchets. Cependant, il est encore nécessaire d'améliorer l'efficacité du procédé avant application à une échelle industriel, ceci passe notamment par le dépôt de ces couches minces sol-gel sur des électrodes macroporeuses de grandes dimensions et leur intégration dans le réacteur. Les découvertes faites dans cette thèse peuvent également être utilisées pour le développement de biocapteurs à base de déshydrogénase ne nécessitant l'ajout d'aucun réactif supplémentaire dans le milieu à analyser. Il existe en effet plus de 300 sortes de déshydrogénase catalysant l'oxydation d'une grande variété de substrats (éthanol, glucose, Conclusion and outlook 204 lactate, etc ) qui peuvent être d'un grand intérêt d'un point de vue analytique du fait de leur présence ou utilisation dans l'industrie alimentaire, l'environnement ou pour suivre certaines pathologies. Les résultats de cette étude sont également d'un grand intérêt pour le développement futur de systèmes électrochimiques à base de déshydrogénase, tel que les biopiles à combustible et biobatteries. ( 3 ) 3 cofactor immobilization and (3) mediator immobilization and co-immobilization with dehydrogenase and cofactor. The first requirements for electrosynthesis involves a stable immobilization of a large amount of active dehydrogenase on the electrode surface of the reactor. Thus, the feasibility of dehydrogenase encapsulation in sol-gel film is first evaluated using D-sorbitol dehydrogenase (DSDH) as a model enzyme by drop-coating. However, DSDH is sensitive to the sol composition during the encapsulation in a silica gel. The direct encapsulation of DSDH in a pure silica gel leads to total deactivation of the protein. We have thus evaluated the interest of various polyelectrolytes in combination with sol-gel deposition of silica films with encapsulated DSDH. We have chosen positively-charged polyelectrolytes because of the expected favorable interactions with the negatively-charged enzyme surface. Then, electrochemically-assisted deposition of silica gel layer has been successfully adapted to the encapsulation of DSDH as well as the co-encapsulation of DSDH and diaphorase in the presence of Poly(dimethyldiallylammonium chloride) (PDDA). The process and the sol composition have been optimized on flat glassy carbon electrodes before being applied to gold macroporous electrodes. At the end, the electrochemically-assisted deposition of the solgel bio-composite has been extended to macroporous electrodes displaying a much bigger electroactive surface area. The macroporous texture of the gold electrode improves thus significantly the catalytic efficiency of the sol-gel biocomposite. The bioelectrocatalytic response looks promising for electro-enzymatic applications. The durable attachment of NAD + /NADH cofactors with long-term activity was the biggest challenge in the project. A successful strategy for cofactor immobilization in sol-gel thin films has been developed in the second part of our work. It involves the chemical bonding of NAD + Conclusion and outlook 206 Si (O Et) 4 Si(OEt) 4 Si(OEt) 4 Si(OEt) 4 S i( O E t ) 4 Si (O Et) 4 Si (O Et) 4 Si(OEt) 4 H + Si(OEt) 4 S i( O E t ) 4 S i( O E t ) 4 H + S i( O E t) 4 S i( O E t) 4 H 2 protein Si(OEt) 4 Si(OEt) 4 Si(OEt) 4 Si(OEt) 4 S i( O E t) 4 S i( O E t) 4 S i( O E t) 4 S i( O E t) 4 H 2 protein S i( O E t) 4 S i(O E t) 4 4 Si( OE t) OH -protein S i( O E t) 4 S i( O E t) 4 S i(O E t) 4 S i(O E t) 4 Si( OE t) 4 4 Si( OE t) OH -protein Si(OEt) 4 S i( O E t) 4 S i( O E t) 4 H 2 O Si(OEt) 4 Si(OEt) 4 S i( O E t) 4 S i( O E t) 4 4 S i( O E t) H 2 O 2. Washing and drying and drying 2. Washing Si OH O Si O O Si O O Si O O O OH protein Si O OH OH O O Si O Si O O Si O O protein Si O Si O O O H Si O Si Si O HO OH OH O Si O protein Si OH OH Si O O O Si Si O H O O Si O Si O HO Si O Si OH OH OH O Si OH Si O -O Si O Si OH O OH OH O Si Si -O OH O Si O O O O Si O Glassy carbon Glassy carbon Glassy carbon 1. Electrodeposition procedure 1. Electrodeposition procedure 3. Thin SG-protein film 3. Thin SG-protein film protein Figure I-5.The process of protein enscapsulated in thin sol-gel film through electrodeposition. Table I -1. Comparison of Amperometric Reagentless Biosensors Based on Dehydrogenase /Cofactor System I Sensor assembly a Sensitivity (mA M -1 cm-2 ) Stability b Linear range (mM) Ref GC/CNT/GDI/CHIT/Nafion (GDH) 1.8 (24 h, 100%)/ (1.5 months, 64%) 0.02-2.0 49 GC/ TFC/Ca 2+ /PL(GDH) 0.67 ? ? 128 GC/ MB/Nafion(GDH) 0.49 ? ? 127 CP/PS-TBO(GDH) 4.0 ? 0.1-5.0 129 CP/PMA(GDH) 0.014 ?/(4 months, ?) 5-36 130 CP/PMA/Nafion(GDH) 0.002 ? 10-330 130 CP/osphendione(GDH) ? (8 h, 92%)/(1 month, 92%) ? 131 CP/Ru complex(GDH) ? ?/(7 days, 60%) ? 132 CP/MB(GDH) ? (1 day, 10%) ?-20 133 Au/Ca 2+ /Nafion(GDH) ? (10 cycles, ?)/? ? 134 GC/NAD + /MWCNT(GDH) ? ? 0.01-0.30 59 Au/PEI-Fc-NAD (ADH) ? ? ?-30 56 Pt/PVA-SbQ/NADH oxidase /NAD-dextran/ cellophane 2 (80 assays)/(6 days) 0.0003-0.1 54 membrane (ADH) a GC, glassy carbon; TFC, trinitrofluorenonecarboxylic acid; PL, polylysine; MB, Meldola Blue; CP, carbon paste; PS, modified polystyrene; TBO, toluidine blue O; PMA, polymethacrylate; CHIT, chitosan; GDI, glutaric dialdehyde;; PVA-SbQ, Poly(vinylalcohol) bearing styrylpyridinium groups; GDH, Glucose Dehydrogenase; ADH, Alcohol dehydrogenase; CNT, carbon nanotubes; PEI-Fc-NAD, A ferrocene-labeled high molecular weight cofactor derivative;?, not reported. b Operational stability (hours, h)/long-term stability (days or months). 1.1. Sol-gel reagents This work focuses on designing functional layers based on silica sol-gel thin films to co- immobilize dehydrogenase, cofactor and mediator. A series of precursor with different properties are used (Table II-1). Tetraethoxysilane and Tetramethoxysilane are the most common precursors used to prepare sol-gel film. 3-Glycidoxypropyl-trimethoxysilane has three OCH 3 groups and a glycidoxy group. The epoxy ring at the end of the glycidoxy group displays chemical activity and can react with other active groups. Aminopropyltriethoxysilane is an aminopropyl-functionalized silane precursor. The positive charges held by the protonated aminopropyl groups could provide suitable environment for bioencapsulation. sodium metasilicate, Ludox® HS-40 colloidal and Sodium silicate solution have been used for protein encapsulation in order to avoid any trace of alcohol (i.e. aqueous sol-gel route) . Table II -1. Information of used precursors II Chemicals Formula Grade MW (g.mol -1 ) Suppliers Tetraethoxysilane (TEOS) Si(OC 2 H 5 ) 4 98 % 208.33 Alfa Aesar Tetramethoxysilane (TMOS) Si(OCH 3 ) 4 99 % 152.22 Aldrich 3-Glycidoxypropyl-trimethoxysilane (GPS) C 9 H 20 O 5 Si 98 % 236.34 Aldrich Aminopropyltriethoxysilane (APTES) H 2 N(CH 2 ) 3 Si(OC 2 H 5 ) 3 99 % 221.37 Aldrich Sodium metasilicate Na 2 SiO 3 95 % 122.06 Aldrich Ludox® HS-40 colloidal SiO 2 40 % 60.08 Sigma-Aldrich Sodium silicate solution Na 2 O SiO 2 10.6 % 26.5 % Sigma-Aldrich Sodium silicate solution (water glass) Na 2 O SiO 2 8.7 % 28.4 % Molekula 1.4.1 Electron mediator for oxidation of NADH 1.4.1.1 Commercial electron mediator The use of an electron-transfer mediator can help to overcome the problems observed during the direct electrochemical oxidation of NADH. In this work, we have tested a series of mediators ( Table II-2). Table II-2. Information of used electron mediator for NADH oxidation Chemicals Formula MW (g.mol -1 ) Suppliers Methylene Green (MG) C 16 H 17 ClN 4 O 2 S.0.5ZnCl 2 433.01 Sigma Meldola's blue (MB) C 18 H 15 ClN 2 O.xZnCl 2 370.78 Acros organics Nile blue chloride (Nb) C 20 H 20 ClN 3 O 85% 353.85 fluka Ferrocenedimethanol (FDM) C 12 H 14 FeO 2 98 % 246.09 Aldrich Ferrocenemethanol (FM) C 11 H 12 FeO 97 % 216.07 Aldrich 1 .4.1.2 Synthesis of non commercially-available electron mediator another 2 h. Finally, the mixture was dried under vacuum condition and the residue was extracted with distilled water. The aqueous solution was purified by membrane dialysis against water for 12 h and dried. The polymer so obtained was referred as PEI-Fc. ② Ferrocene functionalized organoalkoxysilane: ferrocene-alkyl-silane (provided by the group of A. Demir, METU, Ankara, Turky) H N 4 Si(OMe) 3 Fe 1 CHO + H 2 N H 2 N 2 or 3 N H Si(OMe) 3 Si(OMe) 3 1. THF, MS 4 o A 2. NaBH 4 , EtOH Fe Fe H N or 5 H N Si(OMe) 3 Figure II-1. Functionalization strategies of Ferrocene-type mediators with siloxy-functions. Synthesis of siloxane end-group mediators was also considered for possible immobilization in sol-gel thin films. For the synthesis of ferrocene-siloxane compounds with different chain lengths (Figure II-1), two methoxysilane derivatives 2 and 3 was used. 1 mM of 1 (214 mg) and 1 mM of 2 or 3 were dissolved in 10 ml of dry THF in the presence of 100 mg of MS 4 o A under argon atmosphere. The mixture was stirred overnight and filtered off to remove molecular sieves. The filtrate was concentrated and dissolved in 10 ml of absolute ethanol. 2 mmole of NaBH 4 was added portionwise to the solution in a ice-bath. After 3 h, the ethanol was removed. The remaining mixture was dissolved in 10 ml of DCM and extracted with 10 ml of water. The aq. phase was washed with 10 ml of DCM. The combined organic phases was dried over MgSO 4 and concentrated to give 4 or 5. ③ Synthesis of Os-containing redox polymers ① PEI-Fc Os-containing redox polymers were kindly provided by group of Prof. Wolfgang Ferrocene was tethered to poly(ethyleneimine) (PEI) based on the procedure reported in the Schuhmann, and synthesised by Sascha Pöller (Analytische Chemie -Elektroanalytik & literature [3, 4]. FcCHO (90 mg) was dissolved in 15 mL ethanol and added within 1 h to 30 Sensorik & Center for Electrochemical Sciences -CES; Ruhr-Universität Bochum, Bochum, mL of ethanol solution containing 400 mg PEI. The mixture was stirred for 1 h at room Germany). Molecular structures of the synthesized Os-containing redox polymers are shown temperature, then NaBH 4 was carefully added in portions at 0 °C, and stirred continually for in Figure II-2. Table II-3. Characteristic and origin of carbon nanotubes used in this thesis Chemicals Diameter (nm) Length (µm) Purity Suppliers Multi-walled carbon nanotubes (MWCNTs) 5.5 5 95 % Aldrich Carboxylic acid functionalized Single-walled 4-5 0.5-1.5 90 % Aldrich carbon nanotubes (SWCNTs-COOH) Carboxylic acid functionalized Muti-walled 15±5 1-5 95 % Nanolab carbon nanotubes (MWCNTs-COOH) Multi-walled carbon nanotubes (MWCNTs) 15±5 1-5 95 % Nanolab • Microwaved MWCNTs (MWCNTs-µW) .2 Electron mediators for reduction of NAD + 1.4.2.1 Syhthesis of [Cp*Rh(bpy)Cl] + derivatives 2 mg MWCNTs (MWCNTs-COOH, Nanolab) were dispersed in 2 mL distilled water by sonication. Then, 250µL the resulting solution and 150 µL osmium-polymer solution were mixed together. The mixture was first sonicated for 1 h and then stirred for 24 h at room temperature. 1.4A series of substituted (2,2'-bipyridyl) (pentamethylcyclopentadienyl)-rhodium complexes ([Cp*Rh(bpy)Cl] + ) derivatives have been synthesized by by the group of A. Demir (METU, Ankara, Turky) (Figure II-3) according to literature [8, 9, 10, 11, 12, 13], which could be used afterwards as precursors to immobilize such compounds onto electrode surfaces for the reduction of NAD + . Basically, the "simple" mediators (1-10, Figure II-3) have been prepared by reaction of the corresponding unsubstituted and suitably 4,4'-derivatized 2,2'-bipyridine derivatives with (RhCp*Cl 2 ) 2 . 2,2'-bipyridine, 4,4'-dimethoxy-2,2'-bipyridine, 4,4'-dimethyl- 2,2'-bipyridine, 4,4'-diamino-2,2'-bipyridine,4,4'-di-t-butyl-2,2'-bipyridine and (RhCp*Cl 2 ) 2 are commercially available and purchased from Aldrich. Other derivatives of 2,2'-bipyridine were synthesized from 4,4'-dimethyl-2,2'-bipyridine. The preparation of the more "sophisticated" derivatives 11 & 12, which are likely to be used as precursor reagents for immobilization on electrode surfaces, has required more elaborated synthetic procedures. 1.4.2.2 CNTs-Rh H 2 N NH 2 Cp* = N N Cl - N N Cl - N N Cl - Rh Rh Rh Cl Cp* Cl Cp* Cl Cp* Br OH COH SH Cl - Cl - Cl - Cl - N N N N N N N N Rh Rh Rh Rh Cl Cp* Cl Cp* Cl Cp* Cl Cp* MeO OMe MeO 2 C CO 2 Me Cl - Cl - Cl - N N N N N N Rh Rh Rh Cl Cp* Cl Cp* Cl Cp* The non functionalized [Cp*Rh(bpy)Cl] + complex can be efficiently immobilized on the CNT surface by π-π-stacking interaction. 5 mg [Cp*Rh(bpy)Cl] + and 2 mg CNTs (SWCNTs- COOH, nanolab) were dispersed in 2 mL distilled water. The mixture was first sonicated for 1 h and then stirred for 24 h at room temperature. Table II-4. Information of used polymer additives Chemicals Short name Concentration (wt %) Suppliers Poly(dimethyldiallylammonium chloride) PDDA 20 % Aldrich Poly(ethyleneimine) PEI Water free Aldrich Poly(allylamine) PAA 20 % Aldrich Nafion perfluorinated ion-exchange resin NF 5 % Aldrich. Co-immobilization in electrogenerated sol-gel film 4.2.1 Immobilization Os-polymer in electrodeposited sol-gel thin film 8 7 buffer_B A0.3mMNADH_B 6 5 A0.6mMNADH_B A0.9mMNADH_B A1.2mMNADH_B I / µA 2 3 4 1 0 -1 Figure V-13. Cyclic voltammograms recorded with a GCE modified by drop-coated TEOS/GPS/PEI/Os-polymer/DSDH/DI/NAD-GPS film in the absence and presence D-sorbitol E versus Ag/AgCl / V from 2 to 6mM. All cyclic voltammograms have been performed in Tris-HCl buffer (pH 9) at a scan rate of 50 mV/s. 4.2 The encapsulation of Os redox polymer inside the electrodeposited silica gel has finally been investigated. Figure V-14 shows cyclic voltammograms recorded with a GCE modified by electrodeposited TEOS/GPS sol-gel film containing Os-polymer. No electrochemical signal of osmium can be observed in the cyclic voltammograms, so that this way of sol-gel film formation does not seem to be an adequate strategy for Os-polymer immobilization. Figure V-14. Cyclic voltammograms recorded with a GCE modified by electrodeposited TEOS/GPS/PEI/Os-polymer in the 0.1 M Tris-HCl buffer (pH 9) at a scan rate of 50 mV/s, scan cycle, 10. Films have been deposited by electrolysis at -1.3 V for 60 s with a TEOS/GPS sol containing Os-polymer. PEI-NAD + : Poly (ethylenimine)-NAD + NAD-GPS: NAD + -Glycidoxypropylsilane Compound1 Compound 2 properties toward NADH regeneration. We find some alternatives "soft" immobilization procedures not requiring the resort to functional groups harmful to the mediator functioning. Compound 3 Compound Interest of carbon nanotubes as immobilization support The noncovalent functionalization of single-walled carbon nanotubes (SWCNTs) with molecular metal-containing compounds via π-π-stacking starts to become a versatile alternative to covalent bonding [ 52 ]. An approach has thus been tested here for the immobilization of the non functionalized [Cp*Rh(bpy)Cl] + complex, with the aim to overcome the aforementioned deactivation of the electrocatalytic processes when using derivatized [Cp*Rh(bpy)Cl] + derivatives. Conferences participations International Conferences In this thesis, the research work was focused on designing functional layers based on silica sol-gel thin films to co-immobilize dehydrogenase, cofactor and electron mediator to get the most highly active systems and such modifications of electrode surfaces should be adaptable to the macroporous electrodes. Immobilization of dehydrogenase in an active form in a sol-gel matrix was obtained with using a positively-charged polyelectrolyte as additive in the starting sol. This polymer provides a good environment for the protein in the sol-gel. The optimal sol can be deposited by evaporation or by electrodeposition and was successfully deposited in macroporous electrodes. Diaphorase was also successfully co-immobilized with dehydrogenase for the electroenzymatic regeneration of the NAD + cofactor. The immobilization of the cofactor was investigated by simple entrapment, adsorption to carbon nanotube, encapsulation of NAD + chemically attached to dextran (NAD-dextran), and by in-situ coupling with glycidoxypropyltrimethoxysilane (GPS). The last approach allowed stable immobilization of the cofactor, and was extended to electrodeposition and applied to macroporous electrodes. Keywords: Silica, sol-gel, dehydrogenase, NAD + /NADH cofactor, electron mediator, polyelectrolyte, bioencapsulation, electrochemically-assisted deposition, thin films, reagentless device, porous electrodes, carbon nanotubes. La recherché menée dans cette thèse concerne l'élaboration de couches minces sol-gel permettant la co-immobilisation de déshydrogénase, du cofacteur NAD + /NADH et d'un médiateur électrochimique afin d'obtenir le système présentant une activité électrocatalytique optimale et pouvant être déposé au sein d'électrodes macroporeuses. L'immobilisation de la D-sorbitol déshydrogénase (DSDH, l'enzyme modèle de cette étude) sous une forme active dans la matrice sol-gel a été obtenue en utilisant un polyélectrolyte positivement chargé comme additif dans le sol de départ. La présence de ce polymère dans le sol de silice procure un environnement favorable à l'activité enzymatique de la déshydrogénase. Le film peut être déposé par évaporation du sol optimal ou électrogénéré par électrolyse de ce même sol, ce dernier procédé ayant été appliqué à la fonctionnalisation d'électrodes d'or macroporeuses. La diaphorase a également pu être co-encapsule avec la DSDH pour la régénération électroenzymatique du cofacteur NAD + . L'immobilisation du cofacteur dans cette matrice sol-gel a ensuite été étudiée. Le cofacteur a tout d'abord été simplement encapsulé dans la matrice sol-gel en présence ou non de nanotubes de carbone. L'encapsulation d'une forme macromoléculaire du NAD + (NADdextran) a également été étudiée et finalement une voie alternative a été étudiée, utilisant le couplage chimique du NAD + avec le groupement époxy du glycidoxypropylsilane (NAD-GPS). Cette dernière approche s'est montrée être la plus intéressante, notamment en ce qui concerne la stabilité du signal électrocatalytique. Les études de faisabilité ont été menées en utilisant le dépôt sol-gel par évaporation du sol sur électrode plane et la méthode a ensuite été transposée aux électrodes macroporeuses pour dépôt par électrogénération. Plusieurs stratégies d'immobilisation du médiateur électrochimique ont alors été étudiées. Les espèces de type ferrocène ou des complexes d'osmium(III) peuvent être incorporées dans la matrice sol-gel par encapsulation de polymères portant ces médiateurs (Fc-PEI et polymère d'osmium) ou par co-condensation avec un ferrocène fonctionnalisé par un groupement silane. Ces trois systèmes se sont montrés opérationnels lorsque la couche mince sol-gel était déposées par évaporation du sol contenant l'ensemble des éléments de la co-immobilisation (DSDH, diaphorase, NAD-GPS et médiateur). Par contre le dépôt par électrogénération ne permet aux médiateurs de transférer les électrons entre la diaphorase et l'électrode, empêchant toute activité catalytique. Finalement d'autres stratégies basées sur la fonctionnalisation de nanotubes de carbone par différents médiateurs électrochimiques ont alors été étudiées pour dépasser le problème rencontré avec les films déposés par électrogénération (perte de la fonction de médiateur). Les nanotubes de carbones ont été fonctionnalisés par des fonctions quinone grâce à un traitement micro-onde, par électropolymérisation du vert de méthylène, ou par recouvrement par un polymère de type acrylate portant des complexes d'osmium(III). Il alors été possible de coimmobiliser l'ensemble des éléments de ce processus électrocatalytique en utilisant l'électrogénération d'une couche mince sol-gel servant à immobiliser les protéines (DSDH et diaphorase) et le cofacteur (NAD-GPS) à la surface des nanotubes fonctionnalisés par le polymère d'osmium(III). Enfin, les nanotubes de carbone ont permis l'immobilisation sous une forme active de complexes de Rh(III) permettant la régénération du cofacteur NADH.

01430833

en

2024/03/05 22:32:07

2017

https://inserm.hal.science/inserm-01430833/file/Stefanoic%26ZaffranMechDev2017-1.pdf

Sonia Stefanovic email: [email protected] Stéphane Zaffran email: [email protected] Mechanisms of retinoic acid signaling during cardiogenesis Keywords: Substantial experimental and epidemiological data have highlighted the interplay between nutritional and genetic factors in the development of congenital heart defects. Retinoic acid (RA), a derivative of vitamin A, plays a key role during vertebrate development including the formation of the heart. Retinoids bind to RA and retinoid X receptors (RARs and RXRs) which then regulate tissue-specific genes. Here, we will focus on the roles of RA signaling and receptors in gene regulation during cardiogenesis, and the consequence of deregulated retinoid signaling on heart formation and congenital heart defects. Introduction The heart is the first organ to function and is essential for the distribution of nutrients and oxygen in the growing mammalian embryo. Normal cardiac morphogenesis is thus vital for embryonic survival. Heart development is a complex process that requires the precise and coordinate interactions between multiple cardiac and extra-cardiac cell types. Any perturbation in the cells that contribute to heart formation leads to cardiac defects. Congenital heart defects affect 1-2% of live births, and are found in up to one-tenth of spontaneously aborted fetuses [START_REF] Bruneau | The developmental genetics of congenital heart disease[END_REF][START_REF] Fahed | Genetics of congenital heart disease: the glass half empty[END_REF]. Studies in the invertebrate Drosophila melanogaster have defined numerous regulators that determine cardiac cell specification and differentiation, revealing that the cardiac regulatory network is remarkably conserved during evolution. More recently, genetic studies have identified mutations in genes encoding components of signaling pathways as well as proteins organizing chromatin structure that are responsible for congenital heart defects [START_REF] Miyake | KDM6A point mutations cause Kabuki syndrome[END_REF][START_REF] Vissers | Mutations in a new member of the chromodomain gene family cause CHARGE syndrome[END_REF][START_REF] Zaidi | De novo mutations in histone-modifying genes in congenital heart disease[END_REF]. The specification of multipotent heart progenitor cells and their differentiation into different cell lineages is under tight spatial and temporal transcriptional control. Defining the transcriptional networks underlying normal heart development is a prerequisite for understanding the molecular basis of congenital heart malformation. Vitamin A (or provitamin A carotenoid) deficiency is a major public health problem in underdeveloped countries [START_REF] Zile | Vitamin A-not for your eyes only: requirement for heart formation begins early in embryogenesis[END_REF]. Young children, pregnant and breast feeding women are the main groups affected because their requirements for Vitamin A are higher and the impact of deficiency more severe than the other population subgroups. Malformations following maternal vitamin A deficiency were first reported by [START_REF] Hale | The relation of vitamin A to anophthalmos in pigs[END_REF] (F., 1935). The mammalian embryo is strongly dependent on the maternal delivery of retinol (carotenoids and retinyl esters) through transplacental transfer. The fetus needs vitamin A throughout pregnancy [START_REF] Comptour | Nuclear retinoid receptors and pregnancy: placental transfer, functions, and pharmacological aspects[END_REF]. Consequently, both deficiency and excess of vitamin A cause severe damage during prenatal and postnatal development. Nutritional and clinical studies on animals and humans have shown that maternal vitamin A insufficiency can result in fetal death, or a broad range of abnormalities including cardiac malformations (D'Aniello and Waxman, 2015;[START_REF] Wilson | An analysis of the syndrome of malformations induced by maternal vitamin A deficiency. Effects of restoration of vitamin A at various times during gestation[END_REF]. Moreover, it is suggested that the elevated incidence of heart malformations in developing countries could be partly explained by a low availability of retinol due to vitamin A deficiency in the diet [START_REF] Sommer | Impact of vitamin A supplementation on childhood mortality. A randomised controlled community trial[END_REF][START_REF] Underwood | Vitamin A deficiency disorders: international efforts to control a preventable "pox[END_REF]. Conversely, a high level of retinol during pregnancy leads to toxicity of many organs including the heart. For example, maternal intake of isotretinoin has been shown to cause congenital cardiac defects in addition to other malformations [START_REF] Guillonneau | Teratogenic effects of vitamin A and its derivates[END_REF]. Importantly, genetic alterations reducing retinol uptake [START_REF] Golzio | Matthew-Wood syndrome is caused by truncating mutations in the retinol-binding protein receptor gene STRA6[END_REF][START_REF] Kawaguchi | A membrane receptor for retinol binding protein mediates cellular uptake of vitamin A[END_REF][START_REF] Pasutto | Mutations in STRA6 cause a broad spectrum of malformations including anophthalmia, congenital heart defects, diaphragmatic hernia, alveolar capillary dysplasia, lung hypoplasia, and mental retardation[END_REF] or retinoic acid (RA) production [START_REF] Pavan | ALDH1A2 (RALDH2) genetic variation in human congenital heart disease[END_REF][START_REF] Roberts | Cyp26 genes a1, b1 and c1 are down-regulated in Tbx1 null mice and inhibition of Cyp26 enzyme function produces a phenocopy of DiGeorge Syndrome in the chick[END_REF] have been implicated in human congenital heart disease. Altered RA signaling either genetically or nutritionally could be a predominant risk factor, increasing the frequency of congenital heart diseases in humans [START_REF] Huk | Increased dietary intake of vitamin A promotes aortic valve calcification in vivo[END_REF][START_REF] Jenkins | Noninherited risk factors and congenital cardiovascular defects: current knowledge: a scientific statement from the American Heart Association Council on Cardiovascular Disease in the Young: endorsed by the American Academy of Pediatrics[END_REF][START_REF] Underwood | Vitamin A deficiency disorders: international efforts to control a preventable "pox[END_REF]. In this review, we will discuss the role of retinoids in cardiac gene regulation and congenital heart defects. Early heart development The mammalian heart has four chambers and is composed of a variety of cell types. Distinct sets of cardiac progenitors differentiate to form the different parts of the heart. It develops from cardiac progenitors that can be traced back to the early gastrulating embryo (embryonic day (E) 6.5 in the mouse). The earliest progenitors originate from the primitive streak and migrate toward the anterior lateral region to form the cardiac crescent, defined as the first heart field (E7-7.5). By E7.5-8.0, during folding of the embryo and formation of the foregut, the two sides of the cardiac crescent are brought together to form the primary heart tube (Fig. 1). The embryonic myocardium of the tube is characterized by a primitive phenotype, i.e. lower proliferation, a poorly developed contractile apparatus and slow conduction [START_REF] Christoffels | Development of the pacemaker tissues of the heart[END_REF][START_REF] Moorman | Cardiac chamber formation: development, genes, and evolution[END_REF]. Growth of the heart tube depends on the addition of progenitor cells from adjacent pharyngeal mesoderm to the arterial and venous poles. This cell population, named the second heart field, was first identified in the mouse and the chick models [START_REF] Buckingham | Building the mammalian heart from two sources of myocardial cells[END_REF][START_REF] Zaffran | New developments in the second heart field[END_REF]. These progenitor cells, located in a dorsal/medial position relative to the linear heart tube, are kept in an undifferentiated and rapidly proliferating state. These cells ultimately contribute to the outflow tract, right ventricle and a major part of the atria, while the linear heart tube gives rises mainly to the left ventricle [START_REF] Buckingham | Building the mammalian heart from two sources of myocardial cells[END_REF][START_REF] Kelly | The arterial pole of the mouse heart forms from Fgf10-expressing cells in pharyngeal mesoderm[END_REF][START_REF] Zaffran | New developments in the second heart field[END_REF][START_REF] Zaffran | Right ventricular myocardium derives from the anterior heart field[END_REF]. Specific regions in the embryonic heart tube subsequently acquire a chamber-specific gene program, differentiate further and expand, or "balloon" by rapid proliferation to form the ventricular and atrial chamber myocardium (NE8.5) (Fig. 1). In contrast, the regions in between these differentiating chambers, the sinus venosus, the atrioventricular canal and the outflow tract, do not differentiate or expand, and consequently form constrictions. The inflow tract cells of the heart tube develop into atrial cells, pulmonary myocardial cells and myocardial cells of the superior caval veins. Expression of the LIM homeobox 1 Islet1 (Isl1) in second heart field cells led to an appreciation of the full contribution of these progenitors to the venous, as well as the arterial pole of the heart [START_REF] Cai | Isl1 identifies a cardiac progenitor population that proliferates prior to differentiation and contributes a majority of cells to the heart[END_REF]. However, differences in gene expression between progenitors of the venous and arterial poles revealed that the second heart field is pre-patterned [START_REF] Galli | Atrial myocardium derives from the posterior region of the second heart field, which acquires left-right identity as Pitx2c is expressed[END_REF][START_REF] Snarr | Isl1 expression at the venous pole identifies a novel role for the second heart field in cardiac development[END_REF]. Recent genetic lineage analysis in the mouse has shown that anterior Homeobox (Hox) genes Hoxa1, Hoxa3 and Hoxb1 expression define distinct sub-domains within the posterior domain of the second heart field that contribute to a large part of the atrial and sub-pulmonary myocardium [START_REF] Bertrand | Hox genes define distinct progenitor sub-domains within the second heart field[END_REF][START_REF] Diman | A retinoic acid responsive Hoxa3 transgene expressed in embryonic pharyngeal endoderm, cardiac neural crest and a subdomain of the second heart field[END_REF]. This suggests that Hox-expressing progenitor cells in the posterior domain of the second heart field contribute to both poles of the heart tube. Indeed, fate mapping and clonal analysis experiments have confirmed that posterior second heart field cells contribute to outflow tract, and that sub-pulmonary and inflow tract myocardial cells are clonally related [START_REF] Dominguez | Asymmetric fate of the posterior part of the second heart field results in unexpected left/ right contributions to both poles of the heart[END_REF][START_REF] Laforest | Genetic lineage tracing analysis of anterior Hox expressing cells[END_REF][START_REF] Lescroart | Lineage tree for the venous pole of the heart: clonal analysis clarifies controversial genealogy based on genetic tracing[END_REF]. Genetic tracing of Hoxb1 lineages in deficient embryos for the transcription factor T-box1 (Tbx1) showed that the deployment of Hoxb1-positive cell during the formation of the heart is regulated by Tbx1 [START_REF] Rana | Tbx1 coordinates addition of posterior second heart field progenitor cells to the arterial and venous poles of the heart[END_REF]. RA signaling functions during heart development Many studies have demonstrated that the formation of the heart depends on the vitamin A metabolite RA, which serves as a ligand for nuclear receptors (Fig. 2) [START_REF] Niederreither | Embryonic retinoic acid synthesis is essential for early mouse post-implantation development[END_REF][START_REF] Niederreither | Embryonic retinoic acid synthesis is essential for heart morphogenesis in the mouse[END_REF]. RA metabolic pathways have been the subject of some excellent recent reviews [START_REF] Niederreither | Retinoic acid in development: towards an integrated view[END_REF][START_REF] Rhinn | Retinoic acid signalling during development[END_REF]. Excess exposure in humans to vitamin A or its analogs, the retinoids, can cause embryonic defects and congenital heart disease, including conotruncal and aortic arch artery malformations such as transposition of the great vessels, double outlet right ventricle, and tetralogy of Fallot [START_REF] Lammer | Retinoic acid embryopathy[END_REF][START_REF] Mark | Function of retinoid nuclear receptors: lessons from genetic and pharmacological dissections of the retinoic acid signaling pathway during mouse embryogenesis[END_REF]. In rodents, treatment with RA was one of the earliest teratogenic models of heart defects [START_REF] Wilson | Congenital anomalies of heart and great vessels in offspring of vitamin A-deficient rats[END_REF]. Retinoic exposure produces transposition of the great arteries and a wide spectrum of great artery patterning defects [START_REF] Ratajska | Coronary artery embryogenesis in cardiac defects induced by retinoic acid in mice[END_REF][START_REF] Yasui | Morphological observations on the pathogenetic process of transposition of the great arteries induced by retinoic acid in mice[END_REF]. The variability, low penetrance, lack of molecular and electrophysiological data of any particular defect makes these RA-induced teratogenic defects difficult to identify in human patients. Indeed, early disturbances of RA signaling may lead to severe CHDs associated with embryonic death and thus be only rarely observed as a cause of congenital heart disease in humans. The canonical RA synthetic pathway has been elucidated over the last two decades (Fig. 2), mainly via gene targeting studies of several enzymes in the mouse [START_REF] Niederreither | Retinoic acid in development: towards an integrated view[END_REF]. Two sequential reactions are required to transform retinol, the major source of retinoids, into retinaldehyde and RA. The first reversible oxidation is catalyzed by cytosolic alcohol dehydrogenases (ADHs) and microsomal retinol dehydrogenase (RDH), and retinaldehyde is then irreversibly oxidized to RA by retinaldehyde dehydrogenase (RALDHs also known as ALDHs). There are three members of the RALDH family, each with a unique developmental expression patterns [START_REF] Mic | Novel retinoic acid generating activities in the neural tube and heart identified by conditional rescue of Raldh2 null mutant mice[END_REF][START_REF] Mic | RALDH3, a retinaldehyde dehydrogenase that generates retinoic acid, is expressed in the ventral retina, otic vesicle and olfactory pit during mouse development[END_REF]Niederreither et al., 2002b) (Fig. 2). Analysis of knockout mice demonstrated that RALDH2 (ALDH1A2) is responsible for almost all RA production during early development. Studies in mouse and avian embryos have shown that RA deficiency is associated with anomalies of anteroposterior patterning of the primitive heart [START_REF] Hochgreb | A caudorostral wave of RALDH2 conveys anteroposterior information to the cardiac field[END_REF][START_REF] Osmond | The effects of retinoic acid on heart formation in the early chick embryo[END_REF][START_REF] Yutzey | Expression of the atrial-specific myosin heavy chain AMHC1 and the establishment of anteroposterior polarity in the developing chicken heart[END_REF]. Using in situ hybridization experiments, [START_REF] Hochgreb | A caudorostral wave of RALDH2 conveys anteroposterior information to the cardiac field[END_REF] have described two phases of Raldh2 expression [START_REF] Hochgreb | A caudorostral wave of RALDH2 conveys anteroposterior information to the cardiac field[END_REF][START_REF] Moss | Dynamic patterns of retinoic acid synthesis and response in the developing mammalian heart[END_REF]. The first phase is characterized by a large expression domain in lateral mesoderm in proximity with posterior cardiac precursors. The second phase is characterized by progressive encircling of cardiac precursors [START_REF] Hochgreb | A caudorostral wave of RALDH2 conveys anteroposterior information to the cardiac field[END_REF]. Furthermore, treatment of chick embryos with a pan-antagonist of RA signaling at stages HH4-7 causes changes in inflow architecture, indicating that a caudal to rostral wave of Raldh2 conveys anteroposterior information to the forming heart tube. The phenotype of Raldh2-null mice supports this notion (Niederreither et al., 2002a;[START_REF] Niederreither | Embryonic retinoic acid synthesis is essential for early mouse post-implantation development[END_REF][START_REF] Niederreither | Embryonic retinoic acid synthesis is essential for heart morphogenesis in the mouse[END_REF]. In the mouse, deletion of Raldh2 causes heart defects with poor development of the atria and sinus venosus (Niederreither et al., 2002a;[START_REF] Niederreither | Embryonic retinoic acid synthesis is essential for early mouse post-implantation development[END_REF][START_REF] Niederreither | Embryonic retinoic acid synthesis is essential for heart morphogenesis in the mouse[END_REF]. Interestingly, some of these abnormalities can be rescued by transient maternal RA supplementation from E7.5 to E8.5-9.5, suggesting that cardiac precursors commit to their fate early during cardiogenesis [START_REF] Mic | Novel retinoic acid generating activities in the neural tube and heart identified by conditional rescue of Raldh2 null mutant mice[END_REF][START_REF] Niederreither | The regional pattern of retinoic acid synthesis by RALDH2 is essential for the development of posterior pharyngeal arches and the enteric nervous system[END_REF]. Our investigation of the role of Raldh2 revealed that RA signaling plays a role in establishing the boundary of the second heart field in the embryo [START_REF] Ryckebusch | Retinoic acid deficiency alters second heart field formation[END_REF][START_REF] Sirbu | Retinoic acid controls heart anteroposterior patterning by down-regulating Isl1 through the Fgf8 pathway[END_REF]. Analysis of markers of the second heart field, including Isl1, Tbx1, Fgf8 and Fgf10 in Raldh2 mutant embryos has shown abnormal expansion of the expression domains of these genes in posterior lateral mesoderm, suggesting that RA signaling is required to define the posterior boundary of the second heart field [START_REF] Ryckebusch | Retinoic acid deficiency alters second heart field formation[END_REF][START_REF] Sirbu | Retinoic acid controls heart anteroposterior patterning by down-regulating Isl1 through the Fgf8 pathway[END_REF] (Fig. 3). Similarly, the zebrafish mutation neckless (nls), which disrupts function of raldh2, causes formation of large hearts [START_REF] Keegan | Retinoic acid signaling restricts the cardiac progenitor pool[END_REF]. This excess of cardiomyocytes results from an increase number of cardiac progenitor cells as revealed by increased number of nkx2.5-expressing cells [START_REF] Keegan | Retinoic acid signaling restricts the cardiac progenitor pool[END_REF]. The other RADLH enzymes do not play major roles during heart development since deletion of Raldh1 does not result in any observable phenotype [START_REF] Fan | Targeted disruption of Aldh1a1 (Raldh1) provides evidence for a complex mechanism of retinoic acid synthesis in the developing retina[END_REF], and Raldh3-null mice have only defects in ocular and nasal regions as well as neuronal differentiation in the brain [START_REF] Dupe | A newborn lethal defect due to inactivation of retinaldehyde dehydrogenase type 3 is prevented by maternal retinoic acid treatment[END_REF][START_REF] Molotkova | Role of retinoic acid during forebrain development begins late when Raldh3 generates retinoic acid in the ventral subventricular zone[END_REF]. Another important enzyme for RA synthesis and for early embryogenesis is RDH10. Rdh10 expression is localized in the lateral plate mesoderm of the cardiac crescent and later in the venous pole of the heart tube [START_REF] Sandell | RDH10 is essential for synthesis of embryonic retinoic acid and is required for limb, craniofacial, and organ development[END_REF]. Using the RARE-hsp68-lacZ reporter transgene, it has been shown that RA activity in Rdh10 null embryos is almost completely eliminated at the critical E8.0-E8.5 stage of development [START_REF] Rhinn | Involvement of retinol dehydrogenase 10 in embryonic patterning and rescue of its loss of function by maternal retinaldehyde treatment[END_REF][START_REF] Sandell | RDH10 oxidation of Vitamin A is a critical control step in synthesis of retinoic acid during mouse embryogenesis[END_REF][START_REF] Sandell | RDH10 is essential for synthesis of embryonic retinoic acid and is required for limb, craniofacial, and organ development[END_REF]. RDH10 loss-of-function is lethal between E10.5 and E14.5 [START_REF] Cammas | Expression of the murine retinol dehydrogenase 10 (Rdh10) gene correlates with many sites of retinoid signalling during embryogenesis and organ differentiation[END_REF][START_REF] Romand | Dynamic expression of the retinoic acid-synthesizing enzyme retinol dehydrogenase 10 (rdh10) in the developing mouse brain and sensory organs[END_REF][START_REF] Sandell | RDH10 is essential for synthesis of embryonic retinoic acid and is required for limb, craniofacial, and organ development[END_REF]. Rdh10 mutant embryos exhibit abnormalities characteristic of RA deficiency. Some severally affected mutant (b 10%) fail to undergo normal looping and chamber formation, remaining, instead, simple tubes, which can be partly rescued by maternal RA supplementation [START_REF] Rhinn | Involvement of retinol dehydrogenase 10 in embryonic patterning and rescue of its loss of function by maternal retinaldehyde treatment[END_REF][START_REF] Sandell | RDH10 oxidation of Vitamin A is a critical control step in synthesis of retinoic acid during mouse embryogenesis[END_REF]. Rdh10 mutants obtained at E12.5-E14.5 have poor myocardial trabeculation. Zebrafish Rdh10a deficient embryos have enlarged hearts with increased cardiomyocyte number (D'Aniello et al., 2015). The retinaldehyde reductase DHRS3 regulates retinoic acid biosynthesis through a feedback inhibition mechanism and the interaction between RDH10 and DHRS3. Dhrs3 mutant embryos die late in gestation and display defects in cardiac outflow tract formation, atrial and ventricular septation [START_REF] Adams | The retinaldehyde reductase activity of DHRS3 is reciprocally activated by retinol dehydrogenase 10 to control retinoid homeostasis[END_REF][START_REF] Billings | The retinaldehyde reductase DHRS3 is essential for preventing the formation of excess retinoic acid during embryonic development[END_REF][START_REF] Feng | Dhrs3a regulates retinoic acid biosynthesis through a feedback inhibition mechanism[END_REF]. The transport of vitamin A appears to be mediated by STRA6, a membrane bound protein that can interact with cellular retinol binding proteins (CRABPs and CRBP), which bind retinol in the serum (Fig. 2). Human mutations in STRA6 underlie Matthew-Wood syndrome, associated with multiple developmental defects including, occasionally, outflow tract, atrial and ventricular septal defects [START_REF] Golzio | Matthew-Wood syndrome is caused by truncating mutations in the retinol-binding protein receptor gene STRA6[END_REF][START_REF] Pasutto | Mutations in STRA6 cause a broad spectrum of malformations including anophthalmia, congenital heart defects, diaphragmatic hernia, alveolar capillary dysplasia, lung hypoplasia, and mental retardation[END_REF]. Surprisingly, deletion of Stra6 in the mouse has only a modest effect on the levels of RA signaling in most tissues, with the exception of the eye [START_REF] Amengual | STRA6 is critical for cellular vitamin A uptake and homeostasis[END_REF]. Cyp26A1 is a RA degrading enzyme that belongs to the p450 family (Fig. 2). Interestingly, Cyp26A1 expression is spatially restricted in the cardiac crescent and later at the poles of the E8.0 heart tube [START_REF] Maclean | Cloning of a novel retinoic-acid metabolizing cytochrome P450, Cyp26B1, and comparative expression analysis with Cyp26A1 during early murine development[END_REF][START_REF] Rydeen | Cyp26 enzymes are required to balance the cardiac and vascular lineages within the anterior lateral plate mesoderm[END_REF]. Loss of Cyp26 enzymes in zebrafish and mice results in severe phenotypes with embryonic lethality which include smaller atria, looping defects and outflow tract defects [START_REF] Abu-Abed | The retinoic acid-metabolizing enzyme, CYP26A1, is essential for normal hindbrain patterning, vertebral identity, and development of posterior structures[END_REF][START_REF] Emoto | Retinoic acid-metabolizing enzyme Cyp26a1 is essential for determining territories of hindbrain and spinal cord in zebrafish[END_REF][START_REF] Hernandez | Cyp26 enzymes generate the retinoic acid response pattern necessary for hindbrain development[END_REF]Niederreither et al., 2002a;[START_REF] Sakai | The retinoic acid-inactivating enzyme CYP26 is essential for establishing an uneven distribution of retinoic acid along the anterio-posterior axis within the mouse embryo[END_REF]. Although Cyp26c1 knock-out mice do not have significant defects, double Cyp26A1 and Cyp26C1 mutants have more severe looping defects [START_REF] Uehara | CYP26A1 and CYP26C1 cooperatively regulate anterior-posterior patterning of the developing brain and the production of migratory cranial neural crest cells in the mouse[END_REF]. Loss of CYP26 enzymes in humans is associated with numerous developmental syndromes [START_REF] Rydeen | Cyp26 enzymes are required to balance the cardiac and vascular lineages within the anterior lateral plate mesoderm[END_REF]. Inhibition of CYP26A1 is associated with DiGeorge syndrome-like phenotypes that causes heart defects such as conotruncal malformations (interrupted aortic arch, persistent truncus arteriosus, tetralogy of Fallot, and ventricular septal defects [START_REF] Roberts | Cyp26 genes a1, b1 and c1 are down-regulated in Tbx1 null mice and inhibition of Cyp26 enzyme function produces a phenocopy of DiGeorge Syndrome in the chick[END_REF]). The role for CRABPs and CRBP is less clear. However, studies using knock-out mouse suggest that these proteins appear not to be essential for heart development [START_REF] Lampron | Mice deficient in cellular retinoic acid binding protein II (CRABPII) or in both CRABPI and CRABPII are essentially normal[END_REF]. RA regulates development by acting as a diffusible signaling molecule that controls the activity of retinoic acid receptors (RARs). A total of six receptors (RARα, -β, -γ, RXRα, -β, and -γ) transduce the activities of RA [START_REF] Metzger | Contribution of targeted conditional somatic mutagenesis to deciphering retinoid X receptor functions and to generating mouse models of human diseases[END_REF]. Unlike the specific and restricted pattern of Raldh2 or the spatial and temporal availability of RA during development, RARα, RXRα, and RXRβ are ubiquitously expressed in embryonic and adult tissues, whereas RARβ, RARγ, and RXRγ expression is more restricted [START_REF] Dolle | Developmental expression of murine retinoid X receptor (RXR) genes[END_REF][START_REF] Dolle | Retinoic acid receptors and cellular retinoid binding proteins. I. A systematic study of their differential pattern of transcription during mouse organogenesis[END_REF][START_REF] Ruberte | Retinoic acid receptors and cellular retinoid binding proteins. II. Their differential pattern of transcription during early morphogenesis in mouse embryos[END_REF][START_REF] Ruberte | Specific spatial and temporal distribution of retinoic acid receptor gamma transcripts during mouse embryogenesis[END_REF]. Studies with knockout strategies for RARs and RXRs in mutant mice, have demonstrated their crucial role in many developmental processes [START_REF] Kastner | Genetic analysis of RXR alpha developmental function: convergence of RXR and RAR signaling pathways in heart and eye morphogenesis[END_REF][START_REF] Li | Normal development and growth of mice carrying a targeted disruption of the alpha 1 retinoic acid receptor gene[END_REF][START_REF] Lohnes | Function of retinoic acid receptor gamma in the mouse[END_REF][START_REF] Lufkin | High postnatal lethality and testis degeneration in retinoic acid receptor alpha mutant mice[END_REF][START_REF] Mendelsohn | Function of the retinoic acid receptors (RARs) during development (II). Multiple abnormalities at various stages of organogenesis in RAR double mutants[END_REF][START_REF] Sucov | RXR alpha mutant mice establish a genetic basis for vitamin A signaling in heart morphogenesis[END_REF]. As in vitamin A deficient syndrome, fetal or postnatal damages were found in RAR or RXR single-mutant mice, but the defects are less severe, suggesting functional redundancy among these receptors. Whereas RXRα-null mice exhibit embryonic lethality, functional redundancy between the RAR and other RXR isotypes has been demonstrated [START_REF] Mendelsohn | Function of the retinoic acid receptors (RARs) during development (II). Multiple abnormalities at various stages of organogenesis in RAR double mutants[END_REF]. Mutation of either Raldh2 or RXRα results in similar phenotypes characterized by profound embryonic lethality with prominent myocardial defects suggesting a role of RXRα in myocardial growth [START_REF] Dyson | Atrial-like phenotype is associated with embryonic ventricular failure in retinoid X receptor alpha-/-mice[END_REF][START_REF] Gruber | RXR alpha deficiency confers genetic susceptibility for aortic sac, conotruncal, atrioventricular cushion, and ventricular muscle defects in mice[END_REF][START_REF] Kastner | Genetic analysis of RXR alpha developmental function: convergence of RXR and RAR signaling pathways in heart and eye morphogenesis[END_REF][START_REF] Li | Normal development and growth of mice carrying a targeted disruption of the alpha 1 retinoic acid receptor gene[END_REF][START_REF] Sucov | RXR alpha mutant mice establish a genetic basis for vitamin A signaling in heart morphogenesis[END_REF]. On the other hand, other defects in double RXR-RAR mutants are not observed in Raldh2 mutants. RA signaling through RARα1/RXRα regulates differentiation of second heart field cells and outflow tract formation [START_REF] Li | Retinoic acid regulates differentiation of the secondary heart field and TGFbeta-mediated outflow tract septation[END_REF]. Mechanisms of transcriptional regulation RA has been characterized as a diffusible morphogen that acts directly on cells in a concentration-dependent manner to assign positional identities [START_REF] Briscoe | Morphogen rules: design principles of gradient-mediated embryo patterning[END_REF]. RA has a non-cell-autonomous (paracrine) effect on neighboring cells but there is also evidence for it acts in an intracrine manner in cells that synthesize it [START_REF] Azambuja | Retinoic acid and VEGF delay smooth muscle relative to endothelial differentiation to coordinate inner and outer coronary vessel wall morphogenesis[END_REF]. RA signaling is dependent on cells that have the ability to metabolize retinol to RA. RA can form gradients capable of inducing sharp boundaries of target gene expression. The underlying mechanisms include activities of RA-degrading enzymes [START_REF] White | How degrading: Cyp26s in hindbrain development[END_REF]. Several enzymatic activities such as RDHs and CYP26s are required in addition to RALDHs to control RA distribution within the embryo. In zebrafish, it has been demonstrated that RA degradation by CYP26 enzymes progressively determines the limits of RA-dependent gene expression [START_REF] Hernandez | Cyp26 enzymes generate the retinoic acid response pattern necessary for hindbrain development[END_REF]. CYP26s enzymes would thus function to establish boundaries in RA responsiveness. RA gradients induce sharply defined domains of gene expression also through tight feedback regulation of RA synthesis and interactions with other localized morphogens [START_REF] Schilling | Dynamics and precision in retinoic acid morphogen gradients[END_REF][START_REF] Shimozono | Visualization of an endogenous retinoic acid gradient across embryonic development[END_REF]. This has been explored mainly in the context of brain development. The basic mechanism for transcriptional regulation by RARs relies on DNA binding to specific sequence elements, the RA response elements (RAREs). RARs and RXRs are highly conserved among mammals. Unlike RARs, RXRs are not specific to the retinoic pathway, and can be involved in other signaling by binding vitamin D receptors, liver X receptors, thyroid receptors, and peroxisome proliferator-activated receptors. RXR can act as either a homodimer or heterodimer with RARs. In the latter case, regulation of gene transcription is achieved by the binding of the heterodimer RAR/RXR to a specific sequence composed classically of two direct repeats of a hexameric motif. In the classical view, functional RAREs near genes that require RA for normal expression during development typically consist of hexameric direct repeats (DRs) (A/G)G(T/ G) TCA with interspacing of 5 bp (DR5 elements) or 2 bp (DR2 elements), unlike vitamin D and thyroid hormone response elements, which typically exhibit DR3 and DR4 configurations, respectively. Even if spacing is required, the specificity of RAR and RXR binding seems also regulated by interactions of other transcription factors and the epigenetic landscape around the RARE. The cell specificity of the response to RA signaling is probably due to interactions with different regulatory proteins. Recently a two-hybrid assay in yeast demonstrated that RXRα interacts with the cardiac transcription factor Nkx2.5 [START_REF] Waardenberg | Prediction and validation of protein-protein interactors from genome-wide DNA-binding data using a knowledge-based machine-learning approach[END_REF]. In humans, mutations of NKX2-5 result in congenital heart defects such as atrial septal defects and conduction block [START_REF] Prendiville | Insights into the genetic structure of congenital heart disease from human and murine studies on monogenic disorders[END_REF]. Mutations of Nkx2.5 alters this interaction suggesting that defects seen in patients carrying Nkx2.5 mutations may in part due to disrupted protein partner interaction between Nkx2.5 and RXRα. Local chromatin environment, nearest neighboring factor binding motifs are likely important parameters underlying the RARE recognition code. Identifying RARs and RXRs co-factors has the potential to shed light on the complex gene regulatory processes underlying normal development and is likely critical for better differentiation protocols used to drive stem cells into specific cardiac cell types. The use of chromatin immunoprecipitation (ChIP), with antibodies against RARs has demonstrated a greater diversity of RAREs than previously appreciated [START_REF] Boergesen | Genome-wide profiling of liver X receptor, retinoid X receptor, and peroxisome proliferator-activated receptor alpha in mouse liver reveals extensive sharing of binding sites[END_REF][START_REF] Chatagnon | RAR/RXR binding dynamics distinguish pluripotency from differentiation associated cis-regulatory elements[END_REF][START_REF] He | The role of retinoic acid in hepatic lipid homeostasis defined by genomic binding and transcriptome profiling[END_REF][START_REF] Lalevee | Genome-wide in silico identification of new conserved and functional retinoic acid receptor response elements (direct repeats separated by 5 bp)[END_REF][START_REF] Mendoza-Parra | Dissecting the retinoid-induced differentiation of F9 embryonal stem cells by integrative genomics[END_REF][START_REF] Moutier | Retinoic acid receptors recognize the mouse genome through binding elements with diverse spacing and topology[END_REF]. Other hexameric repeat configurations have been found to bind to RARs in cell line studies involving ChIP-seq, but there in vivo importance is unknown. A recent ChIP study coupled to sequencing and performed in ES cells suggested that the presence of RA might also induce de novo RAR/RXR binding to numerous RAREs that are not bound by unliganded receptors [START_REF] Mahony | Ligand-dependent dynamics of retinoic acid receptor binding during early neurogenesis[END_REF]. There is also evidence that inverted repeats with no spacer can also be targets for RARs. RAR ChIP studies and in silico analyses have discovered 13,000-15,000 potential RAREs. Many of these RAREs have not been attributed to endogenous RA signaling and seem to be off-targets due to treatment with high amounts of RA or RAR antagonists. RXR ChIP-seq analyses also revealed that a large fraction of genomic regions occupied by RXR are not associated with a recognizable DR binding site, indicating indirect binding via DNA looping and interaction with co-factors [START_REF] Delacroix | Cell-specific interaction of retinoic acid receptors with target genes in mouse embryonic fibroblasts and embryonic stem cells[END_REF]. Consistent with this, in vitro reporter assays suggest that the transcriptional activities of RARs and RXRs do not necessarily require direct DNA binding [START_REF] Clabby | Retinoid X receptor alpha represses GATA-4-mediated transcription via a retinoid-dependent interaction with the cardiac-enriched repressor FOG-2[END_REF][START_REF] Molkentin | Transcription factor GATA-4 regulates cardiac muscle-specific expression of the alpha-myosin heavy-chain gene[END_REF]. Since ChIP assesses protein-DNA proximity by cross linking, and not direct binding, it will be necessary to verify RAR and RXR binding using in vivo foot printing. Importantly, the presence of an RAR or RXR does not conclusively show that RA will bind to the receptor and regulate gene expression in an RA-dependent manner. RA acts as a ligand for RAR and RXR nuclear receptors, switching them from potential repressors to transcriptional activators. Whether a change of RA gradient concentrations guide this function is unknown. Since the spatial organization of the nucleus may impact on activation or repression of gene expression and interaction with co-factors, assessment of the localization of RA receptors within the nucleus might be relevant. When RA is absent, RAR/RXR heterodimers bind RAREs where they recruit repressive complexes that inhibit transcription. In the presence of RA, however, the repressive complex bound to the receptor is exchanged for an activating complex and transcription at the target site is activated. As the receptors are already present on many target genes, this makes RA the limiting factor in deciding whether or not target genes are activated. The main determinant that drives RA signaling is RA availability rather than nuclear-receptor abundance, which is likely to be secondary. Mechanisms underlying the function of governing the decision of whether RARs and RXRs function as activators or active repressors of a targets gene have been studied in depth using in vitro systems and in the context of the several developmental processes [START_REF] Gillespie | Retinoid regulated association of transcriptional co-regulators and the polycomb group protein SUZ12 with the retinoic acid response elements of Hoxa1, RARbeta(2), and Cyp26A1 in F9 embryonal carcinoma cells[END_REF][START_REF] Janesick | Active repression by RARgamma signaling is required for vertebrate axial elongation[END_REF][START_REF] Kashyap | Epigenetic regulatory mechanisms distinguish retinoic acidmediated transcriptional responses in stem cells and fibroblasts[END_REF][START_REF] Kumar | Retinoic acid controls body axis extension by directly repressing Fgf8 transcription[END_REF][START_REF] Nagy | Nuclear receptor repression mediated by a complex containing SMRT, mSin3A, and histone deacetylase[END_REF]. HDAC inhibitors increase RA sensitivity by promoting dissociation of repressive complexes from RAR [START_REF] Lee | High histone acetylation and decreased polycomb repressive complex 2 member levels regulate gene specific transcriptional changes during early embryonic stem cell differentiation induced by retinoic acid[END_REF]. Indeed, RARs associate with histones acetylases (HATs) and histone deacetylases (HDACs) to modulate gene activity and dictate cell fate [START_REF] Weston | Active repression by unliganded retinoid receptors in development: less is sometimes more[END_REF]. In the repressive unliganded state, the RAR-RXR heterodimer recruits co-repressors such as histone deacetylase (HDAC) protein complexes and Polycomb repressive complex 2 (PRC2). This results in histone H3 lysine 27 trimethylation, chromatin condensation and gene silencing. RA binding to RAR-RXR induces a conformational change in the heterodimer, which promotes the replacement of repressive factors by co-activators such as histone acetylase (HAT) complexes and Trithorax proteins, which mediate H3K4me3, chromatin relaxation and gene activation. These epigenetic factors thus act as mediators or partners in the action of cardiac RARs and RXRs on chromatin structure. In summary, in the presence of RA, RARs bind RA response elements (RAREs) and recruit HATs. In the absence of RA, RARs can actively repress gene transcription by recruiting HDACs that promote chromatin compaction and gene repression. Surprisingly, there are exceptions to this classical model: during neurogenesis RARE sequences upstream of Fgf8 and Hoxb1 mediate gene repression, rather than activation, because RA binding to RAR leads to the recruitment of PRC2 and HDACs, and triggers H3K27me3 [START_REF] Boudadi | The histone deacetylase inhibitor sodium valproate causes limited transcriptional change in mouse embryonic stem cells but selectively overrides Polycomb-mediated Hoxb silencing[END_REF][START_REF] Kumar | Retinoic acid controls body axis extension by directly repressing Fgf8 transcription[END_REF][START_REF] Studer | Role of a conserved retinoic acid response element in rhombomere restriction of Hoxb-1[END_REF]. Transcriptional activities of retinoic receptors in mammalian heart development Activities during early cardiogenesis Several members of the Hox gene family, including Hoxa1 and Hoxb1, are regulated by RAREs, which has been demonstrated in vitro and in vivo [START_REF] Dupe | In vivo functional analysis of the Hoxa-1 3′ retinoic acid response element (3′RARE)[END_REF][START_REF] Huang | A conserved retinoic acid responsive element in the murine Hoxb-1 gene is required for expression in the developing gut[END_REF][START_REF] Langston | Retinoic acid-responsive enhancers located 3′ of the Hox A and Hox B homeobox gene clusters. Functional analysis[END_REF][START_REF] Marshall | A conserved retinoic acid response element required for early expression of the homeobox gene Hoxb-1[END_REF][START_REF] Oosterveen | The direct context of a hox retinoic acid response element is crucial for its activity[END_REF]. The transcription factor homeobox gene Hoxa1 (LaRosa and [START_REF] Larosa | Early retinoic acid-induced F9 teratocarcinoma stem cell gene ERA-1: alternate splicing creates transcripts for a homeobox-containing protein and one lacking the homeobox[END_REF], is a direct target of RA and possesses an enhancer containing a RARE. Consistent with such regulation, reduction or increase of RA signaling causes defects in the contribution of Hoxa1-; Hoxa3-and Hoxb1-expressing progenitor cells to the heart [START_REF] Bertrand | Hox genes define distinct progenitor sub-domains within the second heart field[END_REF]. Our study has demonstrated that RA is required to activate Hoxa1 expression in the posterior second heart field, a subpopulation of cardiac progenitor cells that will later give rise to atrial and sub-pulmonary myocardium [START_REF] Bertrand | Hox genes define distinct progenitor sub-domains within the second heart field[END_REF][START_REF] Ryckebusch | Retinoic acid deficiency alters second heart field formation[END_REF]. Reduction or excess of RA signaling causes abnormalities in the cardiac contribution of Hoxa1 and Hoxb1 expressing progenitors [START_REF] Bertrand | Hox genes define distinct progenitor sub-domains within the second heart field[END_REF]. Enhancers for Hoxa3 and Hoxb1 genes driving expression in cardiac progenitors have been identified [START_REF] Diman | A retinoic acid responsive Hoxa3 transgene expressed in embryonic pharyngeal endoderm, cardiac neural crest and a subdomain of the second heart field[END_REF][START_REF] Nolte | Shadow enhancers flanking the HoxB cluster direct dynamic Hox expression in early heart and endoderm development[END_REF]. It has been reported that enhancers for Hoxb1 gene mediate reporter expression in the second heart field and the proepicardium. These cardiac enhancers have RAREs and may be the direct targets of RA signaling. RA signaling is maintained by an autoregulatory mechanism via Hox genes. Indeed, in the context of brain development, Raldh2 expression is under the direct transcriptional control of HOX, PBX and MEIS complex [START_REF] Vitobello | Hox and Pbx factors control retinoic acid synthesis during hindbrain segmentation[END_REF]. HOXA1-PBX1/2-MEIS2 binds a regulatory element required to maintain normal Raldh2 expression [START_REF] Vitobello | Hox and Pbx factors control retinoic acid synthesis during hindbrain segmentation[END_REF]. The expression profile of Pbx and Meis factors overlaps Hox genes in the second heart field [START_REF] Chang | Pbx1 functions in distinct regulatory networks to pattern the great arteries and cardiac outflow tract[END_REF][START_REF] Stankunas | Pbx/Meis deficiencies demonstrate multigenetic origins of congenital heart disease[END_REF][START_REF] Wamstad | Dynamic and coordinated epigenetic regulation of developmental transitions in the cardiac lineage[END_REF]. Mice deficient for Pbx1, Meis1 and Hox genes have similar cardiac phenotypes [START_REF] Makki | Cardiovascular defects in a mouse model of HOXA1 syndrome[END_REF][START_REF] Paige | A temporal chromatin signature in human embryonic stem cells identifies regulators of cardiac development[END_REF]Stankunas et al., 2008). Together it suggests that PBX/MEIS and HOX proteins may cooperatively regulate Raldh2 gene expression in cardiac progenitors. RA signaling activates another marker of cardiac progenitor cells, the Tbox transcription factor Tbx5 [START_REF] Liberatore | Ventricular expression of tbx5 inhibits normal heart chamber development[END_REF][START_REF] Niederreither | Embryonic retinoic acid synthesis is essential for heart morphogenesis in the mouse[END_REF][START_REF] Sirbu | Retinoic acid controls heart anteroposterior patterning by down-regulating Isl1 through the Fgf8 pathway[END_REF]. Tbx5 is expressed in the posterior domain of the second heart field as well as the first heart field and is required to activate chamber-specific genes, such as atrial natriuretic factor (Nppa) and atrial natriuretic factor (Nppb) [START_REF] Bruneau | A murine model of Holt-Oram syndrome defines roles of the T-box transcription factor Tbx5 in cardiogenesis and disease[END_REF][START_REF] Mori | Tbx5dependent rheostatic control of cardiac gene expression and morphogenesis[END_REF]. RA signaling specifies Tbx5 expressing cells, progenitors of the first heart field to a venous and atrial cell fate [START_REF] Xavier-Neto | A retinoic acid-inducible transgenic marker of sino-atrial development in the mouse heart[END_REF]. DNA elements conferring tissue-type specific gene expression are ideal to analyse the molecular mechanisms that underlie the localized cardiac gene expression. For example, in the context of somite development a site-directed mutagenesis study demonstrated that a RARE upstream of Fgf8 is required for RA repression of Fgf8 in transgenic mouse embryos, thereby showing that RA directly represses Fgf8 transcription in vivo [START_REF] Kumar | Retinoic acid controls body axis extension by directly repressing Fgf8 transcription[END_REF]. Studies of Raldh2 null embryos showed that RA restricts the size of the second heart field by repressing Fgf8 expression in the second heart field [START_REF] Ryckebusch | Retinoic acid deficiency alters second heart field formation[END_REF][START_REF] Sirbu | Retinoic acid controls heart anteroposterior patterning by down-regulating Isl1 through the Fgf8 pathway[END_REF], and zebrafish heart development also requires FGF8 repression by RA [START_REF] Sorrell | Restraint of Fgf8 signaling by retinoic acid signaling is required for proper heart and forelimb formation[END_REF]; however, the of use CRISPR/Cas9-mediated genomic deletion of the Fgf8 RARE showed no defect in heart development or cardiac Fgf8 expression [START_REF] Kumar | Nuclear receptor corepressors Ncor1 and Ncor2 (Smrt) are required for retinoic acid-dependent repression of Fgf8 during somitogenesis[END_REF]. Several RXR/RAR target genes have been identified, including genes within the retinoid pathway, such as the cardiac expressed genes Rarα [START_REF] Dolle | Retinoic acid receptors and cellular retinoid binding proteins. I. A systematic study of their differential pattern of transcription during mouse organogenesis[END_REF][START_REF] Leroy | Mouse retinoic acid receptor alpha 2 isoform is transcribed from a promoter that contains a retinoic acid response element[END_REF][START_REF] Ruberte | Retinoic acid receptors and cellular retinoid binding proteins. II. Their differential pattern of transcription during early morphogenesis in mouse embryos[END_REF]), Rarβ2 (de The et al., 1990), Cyp26a1 [START_REF] Loudig | Transcriptional cooperativity between distant retinoic acid response elements in regulation of Cyp26A1 inducibility[END_REF][START_REF] Maclean | Cloning of a novel retinoic-acid metabolizing cytochrome P450, Cyp26B1, and comparative expression analysis with Cyp26A1 during early murine development[END_REF] Crbp1 [START_REF] Smith | A retinoic acid response element is present in the mouse cellular retinol binding protein I (mCRBPI) promoter[END_REF] and Crabp2 [START_REF] Durand | All-trans and 9-cis retinoic acid induction of CRABPII transcription is mediated by RAR-RXR heterodimers bound to DR1 and DR2 repeated motifs[END_REF]. RA represses the expression of the major embryonic production enzymes RDH10 and RALDH2 [START_REF] D'aniello | Depletion of retinoic acid receptors initiates a novel positive feedback mechanism that promotes teratogenic increases in retinoic acid[END_REF][START_REF] Niederreither | Restricted expression and retinoic acid-induced downregulation of the retinaldehyde dehydrogenase type 2 (RALDH-2) gene during mouse development[END_REF][START_REF] Strate | Retinol dehydrogenase 10 is a feedback regulator of retinoic acid signalling during axis formation and patterning of the central nervous system[END_REF]. Whether this feedback mechanism implies a direct transcriptional mechanism is currently unknown. Ectopic RA signaling affects outflow tract cushion development through the direct repression of a functional RARE in the promoter region of the myocardial Tbx2 gene [START_REF] Sakabe | Ectopic retinoic acid signaling affects outflow tract cushion development through suppression of the myocardial Tbx2-Tgfbeta2 pathway[END_REF]. The chicken slow MyHC3 promoter (slow myosin heavy chain 3) directs transgene expression in the cardiac venous pole at E8.5 and in the atrium at E9.5 with a persistent expression at later stages. This 168 bp regulatory element contains a RARE, suggesting that atrial specific gene expression is controlled directly by the localized synthesis of RA. An inhibitory protein complex composed of RXRα and IRX4 that binds this RARE to inhibit slow MyHC3 expression in primary cultures of embryonic atrial and ventricular quail cardiomyocytes [START_REF] Wang | Irx4 forms an inhibitory complex with the vitamin D and retinoic X receptors to regulate cardiac chamber-specific slow MyHC3 expression[END_REF]. GATA factors also bind the slow MyHC3 regulatory element in vitro, suggesting a cooperative effect between RAR/RXR and GATA factors [START_REF] Wang | A positive GATA element and a negative vitamin D receptor-like element control atrial chamber-specific expression of a slow myosin heavy-chain gene during cardiac morphogenesis[END_REF]. Similarly, RA receptors regulate the chamber-specific genes Nppa (Anf, atrial natriuretic factor) and Nppb (Bnf, brain natriuretic factor) via direct interaction with Gata4 and its co-repressor, Fog2 [START_REF] Clabby | Retinoid X receptor alpha represses GATA-4-mediated transcription via a retinoid-dependent interaction with the cardiac-enriched repressor FOG-2[END_REF][START_REF] Wu | 1,25(OH)2 vitamin D3, and retinoic acid antagonize endothelin-stimulated hypertrophy of neonatal rat cardiac myocytes[END_REF]. Fog factors facilitate the chromatin occupancy of Gata factors and interact with the repressive nucleosome remodeling and deacetylase (NuRD) complex [START_REF] Chlon | Combinatorial regulation of tissue specification by GATA and FOG factors[END_REF][START_REF] Stefanovic | GATA-dependent transcriptional and epigenetic control of cardiac lineage specification and differentiation[END_REF][START_REF] Vakoc | Proximity among distant regulatory elements at the beta-globin locus requires GATA-1 and FOG-1[END_REF]. Thus, RARE elements could act as a platform to recruit these cardiac co-factors and drive chamber specific gene programs [START_REF] Prendiville | Insights into the genetic structure of congenital heart disease from human and murine studies on monogenic disorders[END_REF]. RA activities and cardiac stem cells biology Unlike many other adult tissues, the myocardium of mammals has a limited ability to compensate for the loss of cells after cardiac damage. The ability of RA to stimulate cellular differentiation has been exploited in regenerative medicine differentiation of atrial and ventricular myocytes from human embryonic stem cells [START_REF] Devalla | Atrial-like cardiomyocytes from human pluripotent stem cells are a robust preclinical model for assessing atrial-selective pharmacology[END_REF][START_REF] Gassanov | Retinoid acid-induced effects on atrial and pacemaker cell differentiation and expression of cardiac ion channels[END_REF][START_REF] Wobus | Retinoic acid accelerates embryonic stem cellderived cardiac differentiation and enhances development of ventricular cardiomyocytes[END_REF][START_REF] Zhang | Direct differentiation of atrial and ventricular myocytes from human embryonic stem cells by alternating retinoid signals[END_REF]. Treatment of differentiating embryonic stem cells with RA promotes atrial specification [START_REF] Devalla | Atrial-like cardiomyocytes from human pluripotent stem cells are a robust preclinical model for assessing atrial-selective pharmacology[END_REF]. Several studies have indicated the involvement of COUP-TFs in RA signaling [START_REF] Jonk | Isolation and developmental expression of retinoic-acid-induced genes[END_REF][START_REF] Van Der Wees | Developmental expression and differential regulation by retinoic acid of Xenopus COUP-TF-A and COUP-TF-B[END_REF]. Atrial identity is determined by a COUP-TFII regulatory network [START_REF] Pereira | The orphan nuclear receptor COUP-TFII is required for angiogenesis and heart development[END_REF][START_REF] Wu | Atrial identity is determined by a COUP-TfiI regulatory network[END_REF]. Furthermore COUP-TFI and COUP-TFII are upregulated in differentiated cardiomyocytes in response to RA [START_REF] Devalla | Atrial-like cardiomyocytes from human pluripotent stem cells are a robust preclinical model for assessing atrial-selective pharmacology[END_REF], indicating that a RA-COUP-TF network module participates in the normal control of atrial specification. COUP-TFII is also present in a complex with the HAT p300 and RXR/RAR at the RAREs and enhances RA's actions on target genes during development [START_REF] Vilhais-Neto | Rere controls retinoic acid signalling and somite bilateral symmetry[END_REF]. COUP-TFI and COUP-TFII receptors bind DR elements used by RARs [START_REF] Kliewer | Retinoid X receptor-COUP-TF interactions modulate retinoic acid signaling[END_REF]. Further studies will reveal whether COUP-TfiI is also a RXR/RAR co-factor or a direct target of RA signaling during cardiogenesis. In cell culture, RA can also promote epicardial lineage specification. The combined action of RA, BMP and WNT signaling is required in specifying an epicardium-like lineage from human embryonic stem cells under chemically defined conditions [START_REF] Iyer | Robust derivation of epicardium and its differentiated smooth muscle cell progeny from human pluripotent stem cells[END_REF]. Varying concentration of RA may be responsible for generating in vitro these different cardiac subtypes of cells. Cardiac specific differentiation is also certainly influenced by the use of temporal and combined morphogens. During embryonic stem cell differentiation RA treatment does not affect Hcn4 expression [START_REF] Wobus | Retinoic acid accelerates embryonic stem cellderived cardiac differentiation and enhances development of ventricular cardiomyocytes[END_REF], a major gene in the cardiac conduction tissue [START_REF] Liang | Insights into cardiac conduction system formation provided by HCN4 expression[END_REF], indicating that RA signaling may not be implicated in the differentiation of pacemaker-like cells. In line with this, there is currently interest in reprogramming cells to pacemaker cells by transducing transcription factor genes [START_REF] Boink | The past, present, and future of pacemaker therapies[END_REF]. Through such a reprogramming attempt, it has been shown that the use of RARγ or RXRα together with the cardiac transcription factors Gata6 and Tbx3 does not generate cells with spontaneous beating activity, a key feature of pacemaker cells [START_REF] Nam | Induction of diverse cardiac cell types by reprogramming fibroblasts with cardiac transcription factors[END_REF]. It also remains to be determined whether manipulating the RARs-RXRs-HAT/RARs-RXRs-HDAC complexes could target many silent cardiac-specific sites, open the chromatin for active transcription and enhance reprogramming toward human atrial cells. Activities during late stages of cardiogenesis RA is also involved in processes taking place during late cardiac development (reviewed in Xavier-Neto et al., (2015)). RA signaling acts on neural crest cells orientation and positioning, myocardial specification, and the endothelial-to-mesenchymal transition of endocardial cells process to allow proper endocardial cushion fusion and complete outflow tract septation [START_REF] El Robrini | Cardiac outflow morphogenesis depends on effects of retinoic acid signaling on multiple cell lineages[END_REF][START_REF] Niederreither | Embryonic retinoic acid synthesis is essential for heart morphogenesis in the mouse[END_REF]. RA signaling is involved in the formation of the epicardium [START_REF] Braitsch | Pod1/Tcf21 is regulated by retinoic acid signaling and inhibits differentiation of epicardium-derived cells into smooth muscle in the developing heart[END_REF][START_REF] Moss | Dynamic patterns of retinoic acid synthesis and response in the developing mammalian heart[END_REF][START_REF] Von Gise | WT1 regulates epicardial epithelial to mesenchymal transition through beta-catenin and retinoic acid signaling pathways[END_REF]. Indeed both Raldh2 and RARE-hsp68-lacZ transgene are expressed in the epicardium from stage E11.5 [START_REF] Moss | Dynamic patterns of retinoic acid synthesis and response in the developing mammalian heart[END_REF][START_REF] Xavier-Neto | Sequential programs of retinoic acid synthesis in the myocardial and epicardial layers of the developing avian heart[END_REF]. The heart phenotypes of Raldh2 and RXRα deficient embryos are very similar and are characterized by a severe hypoplasia of the ventricular myocardium, a phenotype mimicking other mutants with defective epicardial function [START_REF] Brade | Retinoic acid stimulates myocardial expansion by induction of hepatic erythropoietin which activates epicardial Igf2[END_REF][START_REF] Merki | Epicardial retinoid X receptor alpha is required for myocardial growth and coronary artery formation[END_REF]. Raldh2 is a direct target of Wt1 in epicardial cells [START_REF] Guadix | Wt1 controls retinoic acid signalling in embryonic epicardium through transcriptional activation of Raldh2[END_REF]. Wt1 regulates epicardial epithelial to mesenchymal transition through β-catenin and RA signaling pathways [START_REF] Von Gise | WT1 regulates epicardial epithelial to mesenchymal transition through beta-catenin and retinoic acid signaling pathways[END_REF]. The transcription factor Tcf21 is regulated by RA signaling and inhibits differentiation of epicardium-derived cells into smooth muscle in the developing heart [START_REF] Braitsch | Pod1/Tcf21 is regulated by retinoic acid signaling and inhibits differentiation of epicardium-derived cells into smooth muscle in the developing heart[END_REF]. It was found that epicardium-derived cells that maintain the expression of Wt1 and Raldh2 initially populate the subepicardial space and subsequently invade the ventricular myocardium. As epicardiumderived cells differentiate into the smooth muscle and endothelial cell lineage of the coronary vessels, the expression of Wt1 and Raldh2 becomes downregulated [START_REF] Perez-Pomares | Experimental studies on the spatiotemporal expression of WT1 and RALDH2 in the embryonic avian heart: a model for the regulation of myocardial and valvuloseptal development by epicardially derived cells (EPDCs)[END_REF]. RA stimulates myocardial expansion by induction of hepatic erythropoietin which activates epicardial Igf2 [START_REF] Brade | Retinoic acid stimulates myocardial expansion by induction of hepatic erythropoietin which activates epicardial Igf2[END_REF]. Erythropoietin and RA, secreted from the epicardium, are required for cardiac myocyte proliferation [START_REF] Stuckmann | Erythropoietin and retinoic acid, secreted from the epicardium, are required for cardiac myocyte proliferation[END_REF]. The RA pathway regulates myocardial growth signals such as Pi3k/ERK, Fgf2-9 [START_REF] Kang | Convergent proliferative response and divergent morphogenic pathways induced by epicardial and endocardial signaling in fetal heart development[END_REF][START_REF] Lin | Endogenous retinoic acid regulates cardiac progenitor differentiation[END_REF][START_REF] Merki | Epicardial retinoid X receptor alpha is required for myocardial growth and coronary artery formation[END_REF] and Wnts [START_REF] Merki | Epicardial retinoid X receptor alpha is required for myocardial growth and coronary artery formation[END_REF]. RA and VEGF delay smooth muscle relative to endothelial differentiation to coordinate inner and outer coronary vessel wall morphogenesis [START_REF] Azambuja | Retinoic acid and VEGF delay smooth muscle relative to endothelial differentiation to coordinate inner and outer coronary vessel wall morphogenesis[END_REF]. RA deficiency reduces expression of Sonic Hedgehog targets and the factors required in the coronary vasculature [START_REF] Lavine | Endocardial and epicardial derived FGF signals regulate myocardial proliferation and differentiation in vivo[END_REF]. Wt1 and RA signaling in the subcoelomic mesenchyme control the development of the pleuropericardial membranes and the sinus horns [START_REF] Norden | Wt1 and retinoic acid signaling in the subcoelomic mesenchyme control the development of the pleuropericardial membranes and the sinus horns[END_REF]. RA signaling is activated in the postischemic heart suggesting that it may play a role in regulation of damage and repair during remodeling. [START_REF] Bilbija | Retinoic acid signalling is activated in the postischemic heart and may influence remodelling[END_REF], Tables 1 and2). Future directions Although there has been progress in characterizing the function of RA signaling, many gaps remain with respect to the underlying mechanisms of RA-mediated gene regulation. Interpretation of experimental data are complicated by the fact that exposure to RA (in cultured cells, whole embryos or explants) may have different, sometimes opposite, effects depending on the concentration, stage or duration of exposure. Strategies that interfere with endogenous retinoid signaling through genetic loss-of-function appear more reliable than approaches using exogenous retinoids, including RAR/RXR antagonists that may lead to the forced repression of target gene loci. Given the ability of RA to signal across cells, understanding the site of action of RA receptors remains difficult. Recent studies have led to novel insights into the interplay between retinoid and other transcription factors in several developing systems. Our knowledge of the relationship between RA signaling and other signaling pathways also remains rudimentary. In the context of heart development, our understanding of the transcriptional targets of RA signaling is also limited. As previously mentioned several cardiac genes have been identified as regulatory targets of RA. In a few cases this regulation is direct, driven by a heterodimer of retinoid receptors bound to a DNA response element; in others, it has either not been investigated in depth or it is indirect, reflecting the actions of intermediate factors. However, our understanding of the role of retinoids will be enhanced if such a distinction can be made for each regulated target gene. Treated epicardium-derived cells Activation [START_REF] Braitsch | Pod1/Tcf21 is regulated by retinoic acid signaling and inhibits differentiation of epicardium-derived cells into smooth muscle in the developing heart[END_REF] Table 2 Some known co-factors associated proteins that regulates RAR/RXR function. Coactivator proteins Function References NcoA-1 Histone acetylation [START_REF] Kashyap | Epigenetic regulatory mechanisms distinguish retinoic acidmediated transcriptional responses in stem cells and fibroblasts[END_REF][START_REF] Kumar | Nuclear receptor corepressors Ncor1 and Ncor2 (Smrt) are required for retinoic acid-dependent repression of Fgf8 during somitogenesis[END_REF] HAT p300 Histone acetylation [START_REF] Gillespie | Retinoid regulated association of transcriptional co-regulators and the polycomb group protein SUZ12 with the retinoic acid response elements of Hoxa1, RARbeta(2), and Cyp26A1 in F9 embryonal carcinoma cells[END_REF][START_REF] Kashyap | Epigenetic regulatory mechanisms distinguish retinoic acidmediated transcriptional responses in stem cells and fibroblasts[END_REF][START_REF] Vilhais-Neto | Rere controls retinoic acid signalling and somite bilateral symmetry[END_REF] FOG2 Transcription factor [START_REF] Clabby | Retinoid X receptor alpha represses GATA-4-mediated transcription via a retinoid-dependent interaction with the cardiac-enriched repressor FOG-2[END_REF] Baf60a/c Recruit SWI/SNF complex [START_REF] Chiba | Two human homologues of Saccharomyces cerevisiae SWI2/SNF2 and Drosophila brahma are transcriptional coactivators cooperating with the estrogen receptor and the retinoic acid receptor[END_REF][START_REF] Flajollet | The core component of the mammalian SWI/SNF complex SMARCD3/BAF60c is a coactivator for the nuclear retinoic acid receptor[END_REF] GATA4 Transcription factor [START_REF] Clabby | Retinoid X receptor alpha represses GATA-4-mediated transcription via a retinoid-dependent interaction with the cardiac-enriched repressor FOG-2[END_REF] Nkx2.5 Transcription factor [START_REF] Waardenberg | Prediction and validation of protein-protein interactors from genome-wide DNA-binding data using a knowledge-based machine-learning approach[END_REF] COUP-TFI-II Transcription factor [START_REF] Vilhais-Neto | Rere controls retinoic acid signalling and somite bilateral symmetry[END_REF] HDACs Histone deacetylation [START_REF] Kashyap | Epigenetic regulatory mechanisms distinguish retinoic acidmediated transcriptional responses in stem cells and fibroblasts[END_REF] There is no doubt that emerging molecular technologies will help in understanding the function of retinoic receptors in cardiac lineage specification. Studies using ChIP-chip and ChIP-seq against RARs and RXRs are available for cultured cell lines and adult tissues [START_REF] Boergesen | Genome-wide profiling of liver X receptor, retinoid X receptor, and peroxisome proliferator-activated receptor alpha in mouse liver reveals extensive sharing of binding sites[END_REF][START_REF] Chatagnon | RAR/RXR binding dynamics distinguish pluripotency from differentiation associated cis-regulatory elements[END_REF][START_REF] He | The role of retinoic acid in hepatic lipid homeostasis defined by genomic binding and transcriptome profiling[END_REF][START_REF] Lalevee | Genome-wide in silico identification of new conserved and functional retinoic acid receptor response elements (direct repeats separated by 5 bp)[END_REF][START_REF] Mendoza-Parra | Dissecting the retinoid-induced differentiation of F9 embryonal stem cells by integrative genomics[END_REF][START_REF] Moutier | Retinoic acid receptors recognize the mouse genome through binding elements with diverse spacing and topology[END_REF]. The exploitation of ChIP-seq technologies from embryonic tissue will enhance the ability to distinguish direct and indirect regulation of cardiac gene expression. For most targets, RA receptors will be present on cardiac regulatory regions regardless of whether the associated gene is transcriptionally active or not and thus interpreting the information will require additional epigenetic data at those sites to determine the likely transcriptional status associated with specific RAREs. The ENCODE project has provided access to valuable data on genome-wide chromatin occupancy of transcription factors, chromatin modifying and remodeling enzymes and histone modifications in heart tissues [START_REF] Bernstein | An integrated encyclopedia of DNA elements in the human genome[END_REF]. Retinoic receptors can mediate looping of distant DNA sequences, enabling transcriptional regulation by far-upstream enhancers [START_REF] Yasmin | DNAlooping by RXR tetramers permits transcriptional regulation "at a distance[END_REF]. Whether this is the case in the context of heart development is unknown. Chromosome conformation capture technologies (e.g. 3-5C, Hi-C) were developed to identify long-range chromatin interactions [START_REF] De Laat | Topology of mammalian developmental enhancers and their regulatory landscapes[END_REF]. Merging these data sets can further facilitate identification of RA regulatory elements. The functional importance of RAREs can be assessed in vivo using recent genome editing technologies [START_REF] Harrison | A CRISPR view of development[END_REF]. Another technical issue is obtaining the starting material from small, localized populations of cardiac progenitor cells. This represents a significant challenge but will be necessary to determine the specificity of the cardiac gene programs. Overcoming these technical challenges will provide important new data in our understanding of RA signaling and its role in cardiac development. Addressing this question is critical for understanding the origin of congenital heart defects. Finally, defining how RA signaling and its interacting factors act to enable epigenetic regulatory events will provide insight into the biology of cardiac progenitor cells leading to methods for increasing the efficiency of directed differentiation of pluripotent cells and cellular reprogramming into cardiac subtypes. Fig. 1 . 1 Fig. 1. Heart fields and their contributions to the developing heart. (A) The second heart field (light grey) is located dorsally from the forming heart derived from the first heart field (dark grey). The second heart field is added at the venous and arterial poles of the definitive heart. Ballooning model of cardiac chamber formation (B). The early heart tube has an embryonic phenotype (dark grey). Chamber myocardium (light grey) expands from the outer curvature, whereas non-chamber myocardium (grey) of the inflow tract, atrioventricular canal, outflow tract and inner curvature does not expand. a indicates atrium; ift, inflow tract; la, left atrium; Iv, left ventricle; ra, right atrium; rv, right ventricle; sv, sinus venosus. Fig. 2 . 2 Fig. 2. Metabolism of vitamin A. Retinyl esters, retinol, and β-carotene are taken into the body from the diet. Both retinol and β-carotene may be converted into the transcriptionally active vitamin A forms after first being converted to retinaldehyde. RA then regulates transcription of vitamin A-responsive genes. When RA is no longer needed, it is catabolized by cytochrome enzymes (CYP26 enzymes). Fig. 3 . 3 Fig. 3. Retonic acid is required to define the posterior limits of the second heart field. Fgf10 is a molecular marker of the murine SHF. The use of the Mlc1v-nlacZ-24 reporter line, in which a lacZ transgene has been integrated upstream of Fgf10 gene, shows the SHF. Ventral views of wild-type (A) or Raldh2-/-(B) embryos at embryonic day 8.5 showing posterior expansion of Fgf10-lacZ transgene expression (arrowhead). The heart has been removed to allow observation of the X-gal staining, compare Raldh2-/-(B) with WT (A) embryos. Table 1 1 Retinoid-responsive cardiac genes. Gene Source Response to RA Mode References MHCα Neonatal rat cardiomyocytes Activation (Rohrer et al., 1991) Cardiac α actin Chicken cardiac mesoderm Repression (Wiens et al., 1992) MHC1a Treated chicken embryo Activation (Yutzey et al., 1994) MLC2a Treated mouse embryos Repression (Dyson et al., 1995) a-Actinin Treated chicken embryos Repression (Dickman and Smith, 1996) SERCA Neonatal rat cardiomyocytes Activation (Rohrer et al., 1991) Na/K/ATP1A3 subunit Neonatal rat cardiomyocytes Repression (He et al., 1996) Large chloride conducting channel Sheep ventricular cells Activation (Rousseau et al., 1996) G protein-coupled endothelin signaling Neonatal rat cardiomyocytes Repression (Zhou et al., 1995) G protein-coupled a-adrenergic signals Neonatal rat cardiomyocytes Repression (Zhou et al., 1995) ANF Neonatal rat cardiomyocytes Activation Direct (Clabby et al., 2003; Wu et al., 1996; Zhou et al., 1995) GLUT4 Adult mouse heart Activation (Castello et al., 1994) GATA4 Cell culture Activation (Arceci et al., 1993; Clabby et al., 2003) FGF8 Raldh2-/-embryos Repression (Ryckebusch et al., 2008; Sirbu et al., 2008) ISL1 Raldh2-/-embryos Repression (Ryckebusch et al., 2008; Sirbu et al., 2008) TBX2 Treated mouse embryos, C2C12 Repression Direct (Sakabe et al., 2012) TBX5 Raldh2-/-, chicken embryos Activation (Liberatore et al., 2000; Niederreither et al., 2001; Sirbu et al., 2008) RARα P19 cells Activation Direct (Ruberte et al., 1991) RARβ2 Treated chicken embryos Activation (Kostetskii et al., 1998) Cyp26α1 P19 cells Activation (MacLean et al., 2001) Raldh2 Treated mouse embryos Activation (Niederreither et al., 1997) Beta-integrin Treated mouse embryos Activation (Hierck et al., 1996) Flectin Treated chicken embryos Activation (Tsuda et al., 1996) Heart lectin-associated matrix protein Treated chicken embryos Activation (Smith et al., 1997) JB4/fibrillin-related protein Treated chicken embryos Repression (Smith et al., 1997) EPO3-IGF2 Raldh2-/-embryos Activation Direct (Brade et al., 2011) TGF2 Treated mouse embryos Activation (Mahmood et al., 1992) Tcf21 Acknowledgments We thank Dr. R. Kelly who critically read this manuscript and offered valuable suggestions. S. Stefanovic is supported by post-doctoral awards from l'Institut de France Lefoulon-Delalande and H2020-MSCA-IF-2014. S. Zaffran is an INSERM research fellow. Work in S. Zaffran's laboratory is supported by the INSERM, the Agence Nationale pour la Recherche (ANR-13-BSV2-0003-01) and the Association Française contre les Myopathies (AFM-Telethon).

01746435

en

2024/03/05 22:32:07

2018

https://hal.science/hal-01746435/file/sampling_switching_systems_V8.pdf

Antonio Ventosa Cutillas Carolina Albea Alexandre Seuret Francesco Gordillo Relaxed periodic switching controllers of high-frequency DC-DC converters using the δ-operator formulation This paper deals with the design of new periodic switching control laws for high frequency DC-DC converters. The contributions are twofolds. On a first hand, the DC-DC converter model are rewritten as a periodic switched affine systems thanks to a δ-operator formulation, which represent an efficient framework for the numerical discretization at high frequencies. On a second hand, three different control laws are provided, the first one being the usual Lyapunov-based control law and the two others being relaxed versions of this first solution. The benefits of these two new control laws over the usual Lyapunovbased one are demonstrated on an simple example. More particularly, it is showed that the selection of sampling period and of the control law strongly influence the size of the region of attraction. I. Introduction Nowadays, there is a relevant interest for DC-DC converters due to their numerous applications in the industry, as for example in computer power supply, cell phones, appliances, automotive, aircraft, etc. These systems can be modeled as switched affine systems (SASs), which represent a particular nonlinear class of switched systems. They correspond to a class of hybrid dynamical systems consisting of several operating modes represented by continuous-time subsystems and a rule that selects between these modes [START_REF] Liberzon | Basic problems in stability and design of switched systems[END_REF]. Compared to the linear case, the affine structure of these systems imposes a set of operating points defined for an averaged dynamic, leading to solutions in the generalized sense of Krasovskii. Many works found in the literature in continuoustime control the SASs by a min-projection strategy [START_REF] Albea | Hybrid dynamic modeling and control of switched affine systems: application to DC-DC converters[END_REF], [START_REF] Deaecto | Switched affine systems control design with application to DC-DC converters[END_REF], [START_REF] Pettersson | Stabilization of hybrid systems using a min-projection strategy[END_REF], even for systems with a general nonlinear form [START_REF] Liu | On the (h0, h)-stabilization of switched nonlinear systems via state-dependent switching rule[END_REF], [START_REF] Lu | A piecewise smooth control-lyapunov function framework for switching stabilization[END_REF]. In these works the provided controllers are good, but may lead to arbitrarily fast switching control. Some solutions to this problem can be found in the literature, as [START_REF] Buisson | On the stabilisation of switching electrical power converters[END_REF], [START_REF] Senesky | Hybrid modelling and control of power electronics[END_REF], [START_REF] Theunisse | Robust global stabilization of the dc-dc boost converter via hybrid control[END_REF], where, the authors aim at ensuring a dwelltime associated with an admissible chattering around the operating point. Nevertheless, [START_REF] Buisson | On the stabilisation of switching electrical power converters[END_REF] does not prove a minimum time associated to the spacial regularization. In [START_REF] Theunisse | Robust global stabilization of the dc-dc boost converter via hybrid control[END_REF], a focus on specific electronic architecture related to boost converters is proposed. In addition, the contributions of [START_REF] Senesky | Hybrid modelling and control of power electronics[END_REF] do not provide a complete stability proof. On the other hand, in [START_REF] Colaneri | Stabilization of continuous-time switched nonlinear systems[END_REF], the authors present an open-loop stabilization strategy based on dwell-time computation, Antonio Ventosa Cutillas and Francisco Gordillo are with University of Seville. Ad. Camino de los Descubrimientos s/n. 41092, Sevilla, Spain. aventosa,[email protected] C. Albea and A. Seuret are with LAAS-CNRS, Univ. de Toulouse, UPS, LAAS, 7 avenue du colonel Roche, F-31400 Toulouse, France. calbea, [email protected] [START_REF] Albea | Practical stabilisation of switched an systems with dwell-time guarantees[END_REF] proposes a minimum dwell-time with a space and time regularization, [START_REF] Hauroigne | Switched affine systems using sampled-data controllers: Robust and guaranteed stabilisation[END_REF] guarantees a minimum and maximum dwell-time by solving optimization problems. These solutions presents a common characteristic: systems are controlled by aperiodic switching. In many occasions, it is necessary to control this class of systems with periodic switching, due to physical constraints. In order to deal with this issue, a solution consisting in the discretization of the continuous-time model with a fixed periodic sampling time was provided in [START_REF] Deaecto | Discrete-time switched linear systems state feedback design with application to networked control[END_REF], [START_REF] Hetel | Robust sampled-data control of switched affine systems[END_REF]. The authors of [START_REF] Deaecto | Discrete-time switched linear systems state feedback design with application to networked control[END_REF] present a controller based on a Lyapunov function synthesized by solving an optimization problem, whose objective is to minimize the area around the equilibrium, where the solutions converge. On the other hand, in [START_REF] Hetel | Robust sampled-data control of switched affine systems[END_REF], the authors design a sampleddata switching control with an upper-bound designed to ensure robustness in continuous-time systems. Through Linear Matrix Inequalities (LMI) based conditions, the upper-bound of the length of the inter-sampling interval can be directly related to the size of the asymptotic stability set around the considered equilibrium. Practical stability is obtained using Lyapunov-Krasovskii functional and the Jensen's inequality. Both solutions do not consider a high-frequency sampling, beside the fact that they are conservative because the controller are based on a Lyapunov-function. In some applications when discretizing these systems, if the sample time is very low, several problems may appear to assess stability, because of numerical issues. Several solutions have been considered in the literature of automatic control. Among them, the δ-operator has been introduced in [START_REF] Middleton | Improved Finite Word Length Characteristics in Digital Control Using Delta Operators[END_REF]. It consists in a discretization method that becomes sufficiently close to the continuoustime model, ensuring continuity of the conditions for high-frequency samplings. In [START_REF] Eidson | Proc. IEEE South East conference[END_REF], a comparison is made between the operator q and the operator δ to perform this discretization. Here, it is possible to observe as the operator δ presents as advantages the natural convergence to a continuous system, while avoiding numerical problems when the sampling period is very short. In [START_REF] Viji | Improved Delta Operator based Discrete Sliding Mode Fuzzy Controller for Buck Converter[END_REF] it is possible to observe that the delta operator is used to perform the sliding mode fuzzy controller of a DC-DC buck converter due to the need for very fast sampling. Therefore, the use of the δ-operator presents a great advantage in the design of controllers with very fast sampling times like the one presented in this paper. In this paper, we model the DC-DC converters in discrete-time by using the δ-operator and we control the system with a well-known min-projection strategy, based on a Lyapunov function. Then, we propose a periodic-sampling relaxed controller for these systems, allowing to obtain results less conservatives, even for highfrequency systems. Moreover, this approach presents a tradeoff between the sampling period and the size of the chattering effects. Some simulations in Matlab valid our contribution. The paper is organized as follow: the problem formulation is stated in Section II. Then, a classical controller is presented in Section III. From this, Section IV proposes some relaxed controllers. An optimization of the controllers is given in Section V. Section VI illustrates the potential of this method on a particular DC-DC converter. The paper ends with a conclusion section. Notation: Throughout the paper N and R denote the set of natural and real numbers, respectively. R n the ndimensional Euclidean space and R n×m the set of all real n×m matrices. The set composed by the first N positive integers, namely {1, 2, ..., N }, is denoted by K N . I is the identity matrix of suited dimension. The Euclidean norm of vector x ∈ R n is denoted by |x|. For any symmetric matrix M of R n×n , the notation M 0 (M ≺ 0) means that the eigenvalues of M are strictly positive (negative). II. Problem formulation A. System data Inspired by the work in [START_REF] Deaecto | Discrete-time switched linear systems state feedback design with application to networked control[END_REF], we focus on the following class of switched affine systems, which is relevant in the context of DC-DC converters ż = A σ z + a σ , (1) where z ∈ R n is the state and it is accessible, A σ and a σ present suited dimensions. The control action is performed through the high frequency switching signal σ ∈ K N := {1, 2, ..., N }, which may be only modified at sampling instants t k , with k ∈ N. In this paper, the length of the sampling interval t k+1 -t k = T is assumed to be constant, known and small enough. This paper focuses on the design problem of a feedback law for the high frequency periodic switching signal σ, in such a way to ensure suitable practical convergence properties of the plant state z to a operating equilibrium z e , which is not necessarily an equilibrium for the continuous-time dynamics in ( 1), but can be obtained as an equilibrium for the switching system with arbitrary switching. A necessary and sufficient condition characterizing this equilibrium is then represented by the following standard assumption (see [START_REF] Deaecto | Switched affine systems control design with application to DC-DC converters[END_REF], [START_REF] Liberzon | Basic problems in stability and design of switched systems[END_REF]). Assumption 1: There exists λ = [λ 1 , λ 2 , ..., λ N ] satisfying i∈K λ i = 1, such that the following convex combination holds: i∈K λ i (A i z e + a i ) = A(λ)z e + a(λ) = 0. ( 2 ) Remark 1: It is emphasized that Assumption 1 is both necessary and sufficient for the existence of a suitable switching signal ensuring forward invariance of the point z e (namely inducing an equilibrium at z e ) when understanding solutions in the generalized sense of Krasovskii or Filippov. Indeed, under (2), we can conclude that the error equation of (1): ẋ = A σ x + B σ , (3) where the error vector is denoted by x := z-z e and where the matrices B σ are defined by B σ := A σ x e + a σ and verify the following convex combination i∈K λ i B i = 0. The objective is to ensure that the error state x converges to the equilibrium x = 0 in the Filippov sense. In addition, the following property is assumed. Assumption 2: The matrices A i , for i ∈ K N are nonsingular and A(λ) is Hurwitz. B. δ-operator for high frequency switching function In this paper, we will propose a discrete-time model based on the δ-operator [START_REF] Middleton | Improved Finite Word Length Characteristics in Digital Control Using Delta Operators[END_REF], which is suitable for high switching frequencies. The δ-operator has been widely used in the literature to avoid numerical problems in the computation of discrete-time dynamics. This is based on the continuous ones in the situation where the sampling period T is potentially very small. The definition of the δ-operator is as follows. For any function ξ from R + to R n , the vector δξ k , at any sampling instant t k ∈ R + , is defined as follows δξ k := 1 T (ξ k+1 -ξ k ), ∀k ≥ 0, where we used the convention ξ k = ξ(t k ) and δξ k = δξ(t k ), for all integer k ≥ 0. Hence the dynamics of system (1) can be rewritten in the framework of the δoperator, which yields the following dynamics δx k = E σ x k + F σ (4) where the matrices that defines the system dynamics are given by E σ = 1 T (e AσT -I), F σ = 1 T T 0 e Aσ(T -s) dsB σ . (5) The interest of this formulation compared to the usual discrete-time formulation comes from the fact that, when T goes to zero, matrices E σ and F σ converge to A σ and B σ , respectively. Another important issue is that matrices E σ and F σ depend explicitly on the switching period T . Indeed, considering small values of T may lead to several numerical problems when discretizing (3). Remark 2: Note that if matrix A σ is non singular, then a simple expression of F σ is provided by F σ = e AσT -I T A -1 σ B σ . It is worth noting that model (4) does not account for the continuous evolution of (1) during the intersampling time. It is however possible to characterize the continuous solution by integrating the solution over a sampling interval, leading, for all t ∈ [t k , t k + T ] and for all k ∈ N, to x(t) = e Aσ(t-t k ) x(t k ) + t t k e Aσ(τ -t k ) dτ a σ Since, the matrices A σ are assumed to be Hurwitz, and since t belongs to the bound interval [t k , t k + T ], the solutions to the system are obviously bounded during the inter sampling time. C. Control objectives When considering such switching affine systems, asymptotic stability to zero is in general not possible. Therefore one has to relax the control objectives and to consider attractor sets, which are not necessarily reduced to the equilibrium set. In this paper, we will consider an estimation of the attractive set, which is of the following quadratic form E := x ∈ R n , x Sx ≤ 1 . (6) with S being a symmetric positive definite matrix to be optimized. This formulation is quite usual and has been used in other contexts as in [START_REF] Albea | Practical stabilisation of switched an systems with dwell-time guarantees[END_REF], [START_REF] Deaecto | Discrete-time switched linear systems state feedback design with application to networked control[END_REF], [START_REF] Hetel | Robust sampled-data control of switched affine systems[END_REF]. This paper focuses on the design problem of a feedback law for the high frequency periodic switching signal σ, in such a way to ensure suitable practical convergence properties of the plant state x to 0, which is not necessarily an equilibrium for the continuous-time dynamics in ( 1), but can be obtained as an equilibrium for the switching system with arbitrary switching. A necessary and sufficient condition characterizing this equilibrium is then represented by the following standard Assumption 2 (see [START_REF] Deaecto | Switched affine systems control design with application to DC-DC converters[END_REF], [START_REF] Liberzon | Basic problems in stability and design of switched systems[END_REF]). The problem can be summarized as follows Problem 1: For any small sampling period T , the problem is to find a switching control law that selects, at each sampling time, the mode or subsystem among all possibilities that stabilizes system (1) with certain performance guarantees at its equilibrium. III. Lyapunov-based switching control Looking at the literature on switched affine systems, one can find the well-known min-projection control law for such a class of systems [START_REF] Deaecto | Discrete-time switched linear systems state feedback design with application to networked control[END_REF], [START_REF] Hetel | Robust sampled-data control of switched affine systems[END_REF], [START_REF] Pettersson | Stabilization of hybrid systems using a min-projection strategy[END_REF]. The underlying idea of this control law is to select the mode of the system which minimizes the decrease of a quadratic Lyapunov function given by V (x) := x P x, ∀x ∈ R n (7) where P 0 is a positive definite matrix of R n×n . This idea is formalized in the following theorem. Theorem 1: Consider Assumption 1 and Property 2 and matrices P 0 and S 0 of suited dimension that are solution to the feasibility problem Γ (1) ij ≺ 0, ∀i, j ∈ K (8) for any pair (i, j) in K 2 . Γ (1) ij = Ψ i (P ) + µ i S 0 0 -1 + γ i [Ψ j (P ) -Ψ i (P )] , Ψ i (P ) = P E i + E i P + T E i P E i P F i + T E i P F i F i P + T F i P E i T F i P F i , (9) for some given parameters γ i > 0 and µ i > 0. Then the switching control (C1) law defined by (C1) σ(x k ) = argmin i∈K N x k 1 Ψ i (P ) x k 1 (10) guarantees ∆V <0 outside of set E. Proof: Consider the Lyapunov function given in [START_REF] Eidson | Proc. IEEE South East conference[END_REF], where P 0 ∈ R n×n . Let us first compute the expression of δV k as follows δV (x k ) = 1 T (V (x k+1 ) -V (x k )) = 1 T ((x k + T δx k ) P (x k + T δx k ) -x k P x k ) = 2δx k P x k + T δx k P δx k . (11) Replacing δx k by its expression given in (4), and using the definition of the matrix Ψ i (P ) provided in (9) yields δV (x k ) = x k 1 Ψ σ (P ) x k 1 . Using ( 9), the previous expression can be rewritten as follows, for any j in K δV (x k ) = x k 1 Γ (1) σj -µ σ S 0 0 -1 -γ σ [Ψ j (P ) -Ψ σ (P )] x k 1 Since the matrix inequalities Γ i,j ≺ 0 holds for any pair (i, j) in K 2 , we have δV (x k ) < µ σ (1 -x k Sx k ) -γ σ x k 1 [Ψ j (P ) -Ψ σ (P )] x k 1 Note that the switching control law [START_REF] Hauroigne | Switched affine systems using sampled-data controllers: Robust and guaranteed stabilisation[END_REF] ensures that the last term of the right hand side of the previous inequality is negative, which guarantees that δV (x k ) < µ σ (1 -x k Sx k ) The previous inequality finally guarantees that for any values of x k outside of E (i.e. 1 -x k Sx k < 0), the quantity δV (x k ) is strictly negative, which concludes the proof. IV. Relaxed switching controller In the previous section, a Lyapunov-based switching control was presented. This control was clearly inspired from the existing literature on switched affine systems such as [START_REF] Deaecto | Discrete-time switched linear systems state feedback design with application to networked control[END_REF], [START_REF] Hetel | Robust sampled-data control of switched affine systems[END_REF] but adapted to the δ-operator modelling of DC-DC converters. The motivation of this section is to present a relaxed version of the previous control law. This relaxation considered here is related to the fact that the a priori intuition behind this Lyapunov-based control law might be too restrictive in the sense that the selection of the Lyapunov matrix P is done to verify two distinct purposes, namely, the definition of the Lyapunov function and the construction of the switching signal. Based on this comment, a relaxed control law can be provided by decoupling the two problems of selecting a Lyapunov function and of designing the switching control law. The relaxed control law is based on the simple idea consisting of keeping the same structure of control law presented in [START_REF] Hauroigne | Switched affine systems using sampled-data controllers: Robust and guaranteed stabilisation[END_REF]. However, instead of using the Lyapunov matrix P , a new unconstrained matrix is introduced to define the switching law. This is formalized in the following theorem. Theorem 2: Consider Assumption 1 and Property 2 and matrices P 0 and S 0 of suited dimension, a new matrix N ∈ R n×n that are solution to the feasibility problem Γ (2) ij ≺ 0, ∀i, j ∈ K (12) where, for any pair (i, j) in K 2 , Γ (2) ij = Ψ i (P ) + µ i S 0 0 -1 + γ i [Ψ j (N ) -Ψ i (N )] , (13) where the matrix Ψ i (P ) and Ψ i (N ) are given in [START_REF] Grant | CVX: Matlab software for disciplined convex programming[END_REF] and, again, for some given parameters γ i > 0 and µ i > 0. Then the switching control (C2) law defined by (C2) σ(x k ) = argmin i∈K N x k 1 Ψ i (N ) x k 1 (14) guarantees ∆V <0 outside of set E. Proof: The proof strictly follows the proof of Theorem 1, except that, now, the switching control law is not characterized by the Lyapunov matrix P but by an arbitrary matrix N , which only has to be a solution to the feasibility problem [START_REF] Liberzon | Basic problems in stability and design of switched systems[END_REF]. Remark 3: Compared to Theorem 1, there are two main advantages. The first one relies on the fact that matrix N is not required to be symmetric nor positive. The second one consists in the fact that the switching law is now completely decoupled from the definition of the Lyapunov function. Moreover, one can see that selection N = P in Theorem 2 leads to the same statement as in Theorem 1. This ensures that the set of feasible solutions of ( 12) is greater than the ones of [START_REF] Grant | Graph implementations for nonsmooth convex programs[END_REF]. It is then expected to derive relaxed solutions that will be presented in the example section where the optimization procedure presented later on in Section V is included. V. Optimisation procedure The feasibility problems proposed in Theorems 1 and 2, only ensure that there exists a switching control law that stabilizes the system to a bounded region around the equilibrium. Without an optimization process, the resulting regions might be too large to be relevant from the physical point of view. Indeed, considering a too large set E can possibly increase the chattering effects which are the main phenomena to avoid or limit in the control design of DC-DC converters. This chattering behavior can damage or even break the devices. Therefore, it is necessary to include an optimization procedure in these theorems, whose objective is to minimize the size of the set E. Since this set is fully characterized by the symmetric positive definite matrix S, minimizing the size of E can be achieved by maximizing the determinant of S. Based on the discussion above, the following proposition of stated dealing with the optimization of the solutions to the conditions of Theorems 1 and 2 is stated. Proposition 1: For any a priori fixed scalar parameters µ j and γ i , the optimisation problem max S,P det(S), s.t. Γ (c) ij ≺ 0, ∀i, j ∈ K (15) for any c = {1, 2}, minimizes the size set [START_REF] Deaecto | Discrete-time switched linear systems state feedback design with application to networked control[END_REF] guaranteeing ∆V <0 outside of set E. Remark 4: Note that we do not optimize the chattering region, because it is constrained to an ellipse form and, the controller is also constrained for a given structure. Thus, it is expected that set E will be relaxed with the control given in Theorem 2, with respect to the control law C1.. Remark 5: This optimization process strongly depends on the selection of the scalar parameters γ i and µ j . Hence it is expected that an iterative procedure to selected the best parameters needs to be included. Remark 6: Other optimization objectives can be considered such as the maximization of the eigenvalues of S, which can be done by maximizing a scalar τ such that τ I ≤ S. VI. Application to a boost converter The control laws introduced above are evaluated on a classical boost converter system. This converter switches at high frequency between two modes (N = 2) corresponding to two affine subsystems. The state variable is defined by x = [i L v c ] , where i L denotes the inductor current and v c the capacitor voltage. We take the parameters given in [START_REF] Deaecto | Switched affine systems control design with application to DC-DC converters[END_REF] for comparison with the switched control algorithm presented therein, which switches with arbitrary aperiodic switching in the steady-state. This type of switching tends to be complicated in physical applications. The considered nominal values are: V in = 100V , R = 2Ω, L = 500µH, C o = 470µF and R o = 50Ω. The switched system state space model ( 1) is defined by the following matrices for i = 1, 2: A i = -R L (i-1) L (i-1) C0 -1 R0C0 , a i = 1 L 0 . The chosen simulation parameters are given by z e = 3 120 , λ = [0.22 0.78] for which simple calculations ensure the satisfaction of Assumption 2. The optimization problems given in Proposition 1 is solved using the CVX solver [START_REF] Grant | Graph implementations for nonsmooth convex programs[END_REF], [START_REF] Grant | CVX: Matlab software for disciplined convex programming[END_REF]. The results obtained with this software illustrate our the efficiency of the new control law presented in Theorem 2 with respect to the Lyapunov-based controller employed in the literature, presented in Theorem 1. As pointed out in Remark 5, the optimization scheme presented in Proposition 1 delivers different results for different values of γ i and µ j . Therefore, a random algorithm has been considered to obtain the best tuple of parameters. The values of these parameters µ 1 , µ γ 2 , obtained for several values of the sampling period T s are provided in Table I. Figure 1 shows the state trajectories, set E and δV > 0 surface in the state-plane for the different controllers given in Proposition 1, as well as for different sampling periods. Note that the δV > 0 surface is in the interior of the set E as is expected from Theorem 1 and 2. Remark also as E is reduced, as T s decreases. We can see that set E is reduced with control law C2 w.r.t. C1, showing as the control law C2 provide reduced region of attraction with respect to C1. This is consistent with Remark 3. Another important remark concerns the fact that the region where δV > 0 is concentrated in a smaller area in control law C2 with respect to C1. These simulations demonstrate the advantages of the relaxed control law over the existing Lyapunov-based one. Indeed, our proposed controller C2 allow to control efficiently the DC-DC converters under study even with relatively high sampling period with a notable reduction of the switches of these systems, which ensures an increase of the lifespan and the reduction of the dissipated energy. VII. Conclusions and future work In this paper, we have presented two main contributions. The first one deals with the definition of accurate discrete-time model for high frequency sampling DC-DC converters. The second contributions consists in the extension of the usual Lyapunov-based control laws employed in this domain to a less restrictive control law that encompass this first formulation. This new control law deliver notable improvements with respect to the Lyapunov-based control law in terms when comparing the estimate of the set E of the switched affine system. (a) C1 with T = 10 -4 . (b) C1 with T = 10 -5 . C1 with T = 10 -6 ; (d) C2 with T = 10 -4 .(e) C2 with T = 10 -5 . C2 with T = 10 -6 . Fig. 1 : 1 Fig. 1: Numerical results of Proposition 1. Time trajectories in green, set E in yellow and δV > 0 in red. 2 , γ 1 and Parameters Ts µ 1 µ 2 γ 1 γ 2 Th. 1 10 -4 9.84 9.86 0.948 0.0515 10 -5 0.16 0.157 0.777 0.223 10 -6 0.907 0.907 0.782 0.218 Th. 2 10 -4 3.16 3.16 0.916 0.327 10 -5 1 1 0.349 0.1 10 -6 0.313 0.313 10 2.793 TABLE I : I Numerical values of µ 1 , µ 2 , γ 1 and γ 2 .

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https://ineris.hal.science/ineris-00174700/file/ARTICLE_TOURS.pdf

V Renaud F Lahaie G Armand T Verdel P Bigarré Conception, numerical prediction and optimization of geomechanical measurements related to a vertical Mine-by-Test at the Meuse/Haute-Marne URL Keywords: instrumentation, conception, design, under-excavation technique, numerical modelling, field stresses, measurement Andra is conducting scientific experiments in the Meuse/Haute-Marne Underground Laboratory among which REP experiment is a vertical mine-by-test focusing on short and long term hydromechanical response of the argilite to the main shaft sinking. Displacements, strains, and pore pressures will be monitored while the shaft is passing down. Andra and INERIS intend to back-analyse most recorded geomechanical data based on under-excavation numerical technique in order to estimate pre-existing field stresses. The under-excavation interpretative technique consists in determining the pre-existing stress tensor related to a quite large volume of rock based on generalized inversion of geomechanical measurements recorded during the disturbance of the host rock (typically the excavation of an underground opening). In the framework a numerical study aiming to test accurately the sensitivity and numerical stability of this interpretative technique, 3D modelling of a step-by-step vertical mine-by-test, based on REP design, has been undertaken. One major step of the numerical INTRODUCTION 1.1 OBJECT/CONTEXT This study is part of the research related to REP experiment in the underground laboratory of Meuse/Haute-Marne (located at Bure, France). Main objective of REP experiment is to study the short and long-term response of argillite to the sinking a vertical shaft (REP). The experiment will allow to record a great number of geomechanical measurements (strains, displacements, tilts.) around the excavated works. It is then possible to estimate, through a generalized numerical inversion, the pre existing stress field in the virgin rock mass. Background literature and field experimentations show both this technique, known as the "under-excavation" or "undercoring" technique, as very promising. The under-excavation technique was first described and proposed by Wiles and Kaiser [START_REF] Wiles | In Situ Stress Determination Using the Underexcavation Technique -I[END_REF][START_REF] Wiles | In Situ Stress Determination Using the Underexcavation Technique -II[END_REF], which applied it quite successfully in the granitic context of the underground laboratory of the AECL (Atomic Energy off Canada Ltd, [START_REF] Read | Technical summary of AECL's Mine-by Experiment Phase 1: Excavation response[END_REF][START_REF] Thompson | In situ rock stress determinations in deep boreholes at the Underground Research Laboratory[END_REF][START_REF] Tonon | Stresses in anisotropic rock masses: an engineering perspective building on geological knowledge[END_REF]). This technique was evaluated by Andra and INERIS in a marl formation (potash mining in Alsace: MDPA, France [START_REF] Bigarré | Casamance, Mesures de déformations, interprétation des données[END_REF]) and clays type (Mont Terri, [START_REF] Bigarré | Mt. Terri Project, Phase III, Stress Measurement Experiment[END_REF][START_REF] Martin | Measurement of in-situ stress in weak rocks at Mont Terri Rock Laboratory, Switzerland[END_REF]). These experiments allowed to confirm both all the interest of this technique [START_REF] Galybin | A measuring scheme for determining in situ stresses and moduli at large scale[END_REF] (even if it is not yet largely used [START_REF] Martin | Stress, instability and design of underground excavations[END_REF]) while raising up some issues concerning its sensitivity and stability as regards both uncertainties affecting the important amount of input data handled and moreover the model used. Research has then been undertaken based on numerical simulation of a standard, synthetic 3D experiment, aiming to a careful, detailed evaluation of this technique. DESCRIPTION OF THE UNDER-EXCAVATION TECHNIQUE The under-excavation technique requires several assumptions, whose principal one is the linear elastic behaviour of the rock mass. If several types of instruments are to be set up around the work (CSIRO cells, extensometers, inclinometers, convergence meters, clinometers, etc. figure 1), a linear relationship between strains/displacements and stresses can be expressed in the following matrix form:   ... M       (this implies an elastic behaviour). [M] is the influence matrix.                                                          N N N N N N N P Q Q a a a                                                        X Y P P P P P P Z YZ XZ Q Q Q Q Q Q XY R R R R R R b b b b b b b b b b b c c c c c c c c c c c c d d d d d d d d d d d d      (1)   M where:    is the initial stress tensor and the a ij , b ij and c ij are the influence coefficients respectively connecting the measurement variations of the sensors, with the components of the initial stress tensor    on the assumption of a linear elastic behaviour of the rock mass. Determining the six components of the initial stress tensor amount thus to solve the system of N+P+Q+… linear equations with 6 unknown. The difficulty of the under-excavation method lies in the direct problem: i.e. the determination of the coefficients of the matrix [M] which requires the numerical modelling of the experiment with 6 canonical loading schemes. For each one of these 6 simulations and with each stage of work excavation, the method consists in recovering, at the location of each virtual sensor, the local value of the stress shift (for CSIRO cells), the new position (for extensometers and inclinometers), the new angle (clinometer), i.e. all induced perturbations due to the mining work progress. These values are then transformed into virtual measurements of strain (CSIRO cells), relative displacement (extensometer) and so on. GENERAL CONSIDERATIONS The synthetic experiment run consisted on the "virtual" monitoring of the progressive excavation of a 6.25 m in diameter vertical shaft around which were laid out beforehand different sensors,. Simulations have been carried out starting from the computer code FLAC 3D (transverse isotropic elastic behaviour + continuous and homogeneous medium). 3D mesh has been designed in order to fit as much as possible location of sensors, made in this case of 3 CSIRO cells, 3 multipoint extensometers and 1 multipoint inclinometer. An additional calculation was carried out using results of triaxial loading tests in accordance with the one already estimated on the field. This was needed in order to check numerically several supplementary functionalities implemented in SYTGEOmath interpretative tool developed by INERIS. DESCRIPTION OF THE NUMERICAL MODEL MODEL GEOMETRY The model geometry is presented on figure 1. Dimensions of the model are 75 m x 75 m x 95 m: the lower and higher dimensions being respectively -515 m and -420 m. A meshing made up of 114009 elements, that is to say 120870 nodes was generated. The model is defined in the coordinates system of principal stresses (X, Y, Z), which corresponds to a rotation of 25° of the general East-North (x, y, z) coordinates system. The meshing has been adapted to take into account the theoretical location of some measurement points (figure 1). MECHANICAL PROPERTIES -EXCAVATION PROCESS Input mechanical properties data are those presented in chapter VI of Andra report "Geological Referential of the site of East" [START_REF] Collectif | Référentiel Géologique du site de Meuse/Haute-Marne -Tome 4 Le Callovo-Oxfordien[END_REF]. They are summarized for each main geological facies in table 1. The shear modulus G 13 of the transverse isotropic elastic law was calculated with the relation of Lekhnitskii [START_REF] Lekhnitskii | Theory of elasticity of an anisotropic elastic body[END_REF], based on laboratory tests:   13 13 13 1 3 12 EE G EE    (2) The boundary conditions correspond to a null normal displacement on the vertical faces and the lower horizontal face. The upper horizontal face is loaded with stress components. All simulations have been carried out following two principal phases:  first phase corresponds to the calculation of the initial state of equilibrium before shaft sinking onset ;  second phase corresponds to the shaft sinking simulation in 31 successive stages. MODEL VALIDATION The numerical model was validated based on 3 x 3 x 4 calculated stress profiles compared with the analytical solutions of the stress field in elastic homogeneous medium around an infinite cylindrical shaft. Error acceptance has been set as very low in order to minimize errors due only to numerical modelling artefacts in the overall procedure. Table 2 recapitulates the maximum absolute and relative error (between numerical and analytical results) made near the various measurement points. ANALYSIS OF THE MEASUREMENTS OBTAINED ON THE VARIOUS INSTRUMENTS OF THE VIRTUAL EXPERIMENTAL DEVICE Measurements obtained in the canonical loading simulations Measurements obtained correspond to the influence coefficients relating these measurements to the corresponding components of the initial stress tensor. Basic analysis of these measurements allows to evaluate amplitude versus time fonction of each sensor (maximum, signal to noise ratio, gradients, etc.). Then analysis of these data (of the 42 graphs like those presented on figure 4.), instrument by instrument, offers a unique mean to identify quantitatively numerous singular conditions as, for example:  redundancy of sensors inside a same instrument (extensometer) can not be justified in terms of quantitative improvement of the overall instrumentation set up;   a sensor may show a narrow predicted useful data range to be back analysed; this can be anticipated by further considerations on the front face working progress of the opening;  best numerical conditioning is to be obtained by combining instruments bringing of additional information, for example CSIRO cells (information on all the components of [ 0 ] except  0 ZZ ) and axial extensometer 3 (information only on  0 ZZ ). Numerical results of the shaft sinking The graphs of evolution of the numerically simulated measurements obtained on each instrument in the triaxial case (loaded with an estimated stress state representative of the experiment depth) are presented on figure 4. On can note some of the relevant points below:  the order of amplitude of the measurement variations obtained on CSIRO 1 cell (more than 4000 µm/m in extension) is relatively high taking into account the range of recommended use for CSIRO cells, i.e. 2500-3000 µm/m. This remark thus encourages to recommend the taking of this cell away the shaft side wall;  the variations of maximum displacement obtained on the sensors of extensometers 1 and 2 lie between 500 and 2400 µm (4000 to 8500 µm for extensometer 3). These variations are at the same time sufficient with respect to the precision of these instruments ( 50 µm) and remain quite lower than their measurement range (105 µm);  the variations of displacement obtained on extensometer 3 are at the same time sufficient with respect to their precision ( 50 µm) and lower the tolerance range considered for this instrument (100000 µm);  the variations of measurements obtained by the inclinometer lie between 100 and 1000 µm. These variations are weak, but remain sufficient for the points closest to the shaft (points n°3 to 5). On the other hand, they become insufficient, with respect to the device precision ( 100 µm/m) for the points furthest away from the shaft (points n°1 and 2);  the comparison (figure 3) of the displacements obtained at the position of the reference points of extensometers 1, 2 and of the inclinometer with those obtained at the position of their first measurement point (point n°1) shows well that displacements of the reference positions are significant and do not have to be neglected. Those simulations have been completed with elastoplastic modelling of the shaft sinking in order to finalize the layout of the experiment. As this article is focus on the under excavation method which requires elastic model, the result of elastoplastic are not shown here. DATA INVERSION STRATEGY PRESENTATION Increasing number of measurements of varied types make the data to be selected and to be inversed quite difficult. Because there are a too great number of possible choices which relate at the same time to:  number of sensors (data sub sets) to be inversed;  temporality of the considered measurement intervals compared to the face advance;  number and width of measurement intervals;  relative temporal shift between the considered intervals, etc… At the same time, as it was mentioned for the under-excavation technique, by Wiles & Kaiser [START_REF] Wiles | In Situ Stress Determination Using the Underexcavation Technique -II[END_REF] who proposed a methodology of selection of the data to be taken into account in the inversion (figure 5). Wiles and Kaiser propose for the stage B three different strategies (figure 6):  "simple interval" (SI);  "multiple intervals with shifted origins" (MISO);  "multiple intervals with common origins" (MICO). In this study, we explored a large set of possible choices for combination of instruments at stage A (12 combinations of instruments: C1, C2, C3, C1-E3, C2-E3, C3-E3, C1-C2-C3, C1-C2-C3-E3, E1-E2-I1, E1-E2-I1-E3, C1-C2-C3-E1-E2-I1 and C1-C2-C3-E1-E2-I1-E3) and all possible combinations for stages B and C. Moreover, for stage B, we tested a fourth procedure (called TOT), which consists in taking into account all the possible intervals of measurement in the inversion. The aim of this last approach is to be able to give a more complete answer on the influence of each assumption on the estimated initial stresses. It is thus possible to establish the type of instrument or the intervals of measurement which it is necessary to consider in the inversion to be able to limit to the maximum the influence of the assumptions that one wishes to test. APPLICATION TO THE TRIAXIAL LOADING SIMULATION The triaxial tests simulation allows to select the most favourable inversion methods. The application of the methodology of inversion led to 15660 inversions to be run through an automated function implemented in the interpretative tool (1305 by combination of instruments). All possible inversions are studied in order to have a better estimation of stresses and their linked error. The assumptions used in this method being the same as those supposed to build the matrix [M], we have to find, after inversion, the initial stress tensor imposed on the model boundaries if the matrix [M] is well conditioned (the matrix conditioning is the ratio of its greater eigenvalue on its smaller eigenvalue). It is allowed that a conditioning ranging between 0 and 10 is "very good", "good" between 10 and 20, "acceptable" between 20 and 30, and that beyond 30, the inversion of measurements presents a significant amplification risk of numerical errors. Figure 7 shows the cumulative distribution (P<(x), percentage of case where x is lower than a given value) of the conditioning value according to the combination of instruments taken into account in the inversion. The conditioning value of the influence matrix (figure 7) is very variable according to the combination of instruments considered in the inversion. Best conditioning is not obtained by considering all the instruments, but only the CSIRO cells and the extensometer located in the shaft axis (E3). For these two combinations of instruments, almost all the inversion methods lead to an acceptable conditioning (< 30). On the other hand, the inversions only carried out on the extensometers except the one placed axially inside the front face of the shaft and inclinometer (E1-E2-I1) never lead to an acceptable conditioning. As for conditioning, we note that the difference between the back calculated and prescribed stresses is very variable according to the combination of instruments considered in the inversion. The combinations of instruments giving place to a bad conditioning tend overall to generate a more important error on the estimated stresses. This global correlation between the conditioning (COND) of the influence matrix and the made error DEVMAX on the estimated stresses is shown on figure 9. However, one can note certain differences between figures 7 and 8. For example, the combination of instruments giving place to best conditioning (C1-C2-C3-E3) is not that which produces less error on the estimated stresses (even if it remains among the best). A limited number of inversion cases were selected. These favourable inversion cases are those which lead at the same time to best conditioning and the weakest error on the prescribed stresses (COND < 30, DEVMAX < 1%). The number of cases thus selected for the study continuation is 2151, that is to say 13.7 % of the number of initial inversions. CONCLUSION Within the framework of REP experiment in Bure (France), Andra and INERIS intend to develop an under-excavation interpretative technique in order to reduce uncertainty on the insitu stress state in the Callovo-Oxfordian argillite formation. A 3D synthetic numerical study has then been completed in order to assess quantitatively the overall reliability and performance of the under-excavation technique. The methodology has been extended in the way that:  intermediate results needed as calculated influence coefficients and total predicted measurements of all varied sensors to be implemented could be back analysed in terms of operational recommendations, regarding a given sensor or a subset of sensors;  inversion numerical strategies are explored based on calculated indicators able to quantify comparatively different instrumentation schemes versus excavation front face overall lay outs, aiming to minimize computational undesired artefacts. This study is one of the studies necessary to the REP experiment design, nevertheless, the calculations presented in this article allow to improve the experimental device by:  moving away CSIRO 1 from the shaft side wall of approximately 3 m;  relocating the inclinometer sensors by reducing the bars length and by increasing the number of measurement points. Moreover, this study shows that the reference points of the extensometers out of shaft and the inclinometer will move significantly in the shaft passage (comparison of figure 3a with figure 3b). This problem can be avoided by not inversing measurements out of shaft that starting from the stage -460 m. 3D numerical conception of an under-excavation interpretative experiment, e.g. aiming to back estimate an unique, overall quantitative result as field stresses, appears to be of major interest for optimizing data quality to be recorded, first for a specific sensor considered, secondly and moreover for the quality of the overall instrumentation scheme, or whatever can be called "information wealth" of data set to be recorded. This "optimal 3D design" approach includes complex factors usually difficult to handle at the same time for the rock mechanics engineer as: 1-best instrumentation coverage of the 3D complex geometry of the advancing excavation inside the instrumented volume of rock; 2-geomechanical properties of host rock to be monitored; 3-correct spreading of the varied instruments to be set up. no measuring instrument considered individually shows a satisfactory sensitivity to the whole components of [  ]. This result is clearly accentuated by the 2D final geometry of the experiment, once the shaft has been completely passing by the monitored volume of rock; Figure 8 8 Figure 8 shows the distribution of the maximum relative difference (noted DEVMAX) between the values of stresses imposed on the model  0 i imp and those estimated by inversion Figure 10 shows 10 Figure10shows the distribution of the favourable inversion cases according to the 2 C 2 o m è tr e I1 E xte n s o m è tr e E X 1 E xte n s o m è tr e E X 2 C ellu le C S IR O 1 C ellu le C S IR O Figure 2 : 2 Figure 2 : 3D overview of the measurement device around the shaft Figure 3 : 3 Figure 4 :Figure 5 : 3345 Figure 3 : Comparison between displacements of the n°1measurement and reference points of extensometer 2 Figure 6 :Figure 7 : 67 Figure 6 : Selection procedures of intervals of measurements taken into account in the inversion = stage B (adapted of Wiles & Kaiser [15]) Figure 8 :Figure 9 :Figure 10 : 8910 Figure 8 : Distribution of the maximum relative difference (DEVMAX) between the estimated stresses and the prescribed stresses in the triaxial case Table 1 : 1 Mechanical characteristics of facies A, B and C Parameter Facies A Facies B & C Wet density  h = 2420 kg/m 3  Young modulus  to the plan of transverse isotropy E 3 = 5200 MPa E 3 = 5200 MPa Young modulus  to the plan of transverse isotropy E 1 = 6300 MPa E 1 = 6300 MPa Poisson's ratio   =   = 0,30 Shear modulus G 13 G 13 = 2144 MPa G 13 = 2144 MPa  rr      zz   r  Relative error 1.58% 1.01% 1.21% 0.62% Absolute error (MPa) 6.71E-02 1.00E-03 1.00E-03 2.13E-03 Table 2 : 2 Absolute and relative maximum errors between numerical and analytical resultsFigure 1 : Detail of the horizontal meshing and localisation of measurement points

00174722

en

2024/03/05 22:32:07

2005

https://ineris.hal.science/ineris-00174722/file/article_GISOS_2005_english_last.pdf

Renaud Vincent email: [email protected] Tritsch Jean-Jacques email: [email protected] Franck Christian email: [email protected] MODELING AND ASSESSMENT FOR SUBSIDENCE HAZARD IN INCLINED IRON MINING Keywords: hazard, subsidence, inclined seams, iron mine, modeling aléa, affaissement, gisement penté, mine de fer, modélisation The old iron mines of the North-West of France have geometrical and exploitation configurations appreciably similar, with dips varying between 30° and 90°. Within the framework of the establishment of risk maps related to these exploitations, the observation of subsidence in certain basins leds us to try to better know the conditions of occurrence and the consequences on the surface of these phenomena, and in particular the influence of the dip on their relevance. A modeling was thus undertaken, consisting initially of back-analysis of a subsidence trough observed and studied, in order to seek the initiating mechanism within mining work and to appreciate the influence and the degree of reliability of the parameters, and in the second time the parameterised analysis of the zones of potential failure according to the dip, the opening of the mine seam, the extraction ratio and the thickness of the overburden. The contribution of this modeling and the experience feedback of other mining basins allowed to fix the principles of evaluation of the subsidence alea, in terms of intensity and occurrence, of these deposits. Introduction In the framework of the assessment and the prevention of mining hazards, the establishment of risk maps related to the movements above the iron deposits exploited in the North-West of France (figure 1) highlighted their relative homogeneity and singularity. These basins indeed have geological and exploitation characteristics of formation relatively homogeneous. In addition, and mainly because these exploitations have an important dip (between 30° and 90°), it quickly appeared during the information collection and the first observations of disorders on the surface that the risk evaluation of ground movement, and especially of subsidence occurrence, had to take into account the singularity of these deposits and could not be completed with the analyses made for horizontal mining works. This article initially describes the characteristics specific to these exploitations. Then, the objectives are presented, steps and results of the modeling made on the basis of back-analysis of an observed subsidence phenomenon. Finally the transcription of these results and the evaluation of the subsidence risk are discussed. Characteristics of North Western iron mining exploitations The risk analysis is developed for studies at the scale of a whole basin of risk, even several risk basins, if they present strong analogies. It is the case of the iron deposits of the synclinals of Soumont, May/Orne, La Ferrière-aux-Etangs (Normandy) and Segré (Pays-de-Loire). Figure 1. Localisation of the iron-bearing basins of the North-West of France (Varoquaux and Gerard, 1980). The various basins present much analogies on the geological and exploitation aspects. These deposits fit in the dissymmetrical synclinal whose periods of deposit (Ordovician or Silurian) and of crumpling are near on a geological scale. They are fairly to strongly slopes (figure 2), located at very close depths (between 10 and 600 m) and hold one or two veins of low or average thickness (overall 2 to 4 m, locally more). [START_REF] Maury | Aménagement de la mine de May-sur-Orne en stockage souterrain d'hydrocarbures[END_REF]. The nature and the strength of the iron ore are relatively variable. The ore is mainly constituted of haematite at May-sur-Orne and at a shallow depth under the calcareous overthrust of Soumont. The ore is carbonated in-depth in others basins, like at Segré and La Ferrière-aux-Etangs. The compressive strength of the ore is about 100 MPa at May-sur-Orne and 200 MPa (perpendicular to the bedding plane) at La Ferrière-aux-Etangs. The mining methods used in those several basins are appreciably similar (figure 3). The oldest mining sites were exploited by short dip faces also called stops, then by dip strike faces. Thereafter, one systematically applied the method of the rise faces or the mechanised strike faces for the mining sites with low slopes (dip lower than 50°) and the shrinkage method for the mining sites slopes to high slopes (dip higher than 50°). These works are connected by level galleries connected to the works of ore extraction, and spaced of 30 m to 75-80 m in altitude, according to the basins and the methods used. The observed disorders (table 1) in these various basins are similar (primarily some localized sinkholes by crown section rupture, shaft or raise clearings, or collapses of galleries). One notes however the existence of collapses of important districts at the bottom, in production run, in general without repercussions on the surface, except for Soumont and La Ferrière-aux-Etangs. The observed disorders on surface are traditional depressions with spread out board, with opened cracks but without frank breaks of shearing, which can be connected with subsidence troughs. On the other hand, the documentary analyses do not identify any accident of huge collapse type: the only events known in the western French basins are exclusively the fact of slate exploitations whose common factors are their complex geometry, very different from iron mining works, and the presence of important residual voids [START_REF] Tritsch | Assistance technique à l'élaboration d'un dossier de demande d'abandon, carrières de Misengrain -site de Noyant[END_REF]. Modeling of the inclined deposits of the West of France Modeling approach The study of modeling, using code UDEC (2D calculations in discontinuous medium), was organised in two stages:  the back-analysis of a subsidence trough in Soumont developed in 1966 above a well delimited underground collapse;  parametric analysis of the zones of potential failure (extension, amplitude) according to the dip (30 to 65°), the extraction ratio (70 to 90 %), the layer thickness (1.5 to 5 m), and thickness of the formations of overburden in discordance (0 to 50 m). Back-analysis of the subsidence phenomenon of Soumont appeared in 1966 Several collapses occurred in the mine of Soumont between 1929 and 1966. They mainly induced subsidence troughs on the surface (figure 4). The latest one is the most documented for underground visits and analyses of the causes were carried out in close connexion with this collapse. Thus this event has been selected for the back-analysis. Collapse occurred between the levels -120 and -250 m, 40 years after the exploitation of this sector of mining works. The dip of the layer is 30° and the extraction ratio is high (80-85 %).The maximum subsidence measured at that time was 65 cm. The back-analysis was based on a modeling on the collapsed district scale. The objective was to specify the conditions which were at the origin of collapse and the most relevant mechanisms. This work was carried out by respecting the three following checking points:  A subsidence amplitude of 0.65 m was measured on surface;  The absence of collapse in work of depth higher than 220 m;  The stability of the mining works exploited with the same depth by shrinkage, to the west of the collapsed districts. Description of the mining conditions The extraction ratio of the lower stages decreases with depth. The section of the way is trapezoidal and the opening of these levels is 4.5 m. The pillar width of the lower stages varies between 4.5 m and 6 m. Above the ore layer, one meets massive and resistant schist beds for a total thickness of 120 m, then sandstone beds over 95 m, then a schist alternation and sandstone of weak thickness (10 m) and finally again sandstone. In lower part of the ore layer, there is a 10 m thick schist bed then a series of sandstones. The whole of these formations is covered by a calcareous slab whose thickness can be evaluated to 30 m at the location of the concerned sector [START_REF] Tincelin | Mine de Soumont -Mesures à entreprendre pour prévoir l'imminence d'un risque d'effondrement survenant à l'aplomb des routes nationales n°158 ou départementales n°43[END_REF]. Geomechanical characteristics The geomechanical characterisation of various materials of the southern side of the mine of Soumont is not complete. Only the iron ore and its immediate roof and floor have been tested in laboratory. Hence we estimated the data according to various sources:  values obtained in laboratory: bibliographical study carried out at the time of the preliminary phase [START_REF] Delaunay | Phase préliminaire à la réalisation d'une modélisation numérique sur les gisements pentés des bassins ferrifères de Soumont[END_REF];  data resulting from a study on the slate mine of Misengrain [START_REF] Tritsch | Assistance technique à l'élaboration d'un dossier de demande d'abandon, carrières de Misengrain -site de Noyant[END_REF];  data from the database by [START_REF] Fine | Le soutènement des galeries minières[END_REF];  data (for the sandstone) resulting from the synthesis of the mechanical characterisations for the HBL [START_REF] Mery | Synthèse des caractérisations géomécaniques[END_REF];  data from the geological map of Mézidon (BRGM); The values of strengths (tensile and compressive) were then degraded while taking into account:  the scale effect being estimated at 0.47: according to [START_REF] Bieniawski | The significance of in-situ tests on large rock specimens[END_REF], referring to a curve obtained on unconfined iron ore samples;  the 2D aspect of modeling by preserving the strength/stress ratio for the pillar in 2D and 3D by decreasing the compressive strength of the seam (equation 1): Let us recall that for square rooms and pillars:  the time influence on the material (by estimating that the coefficient of reduction of strengths is founded on a ratio elastic strength/ peak strength). Table 2 shows a synthesis of all strengths values obtained according to the various effects taken into account. The behaviour law retained takes into account a hardening then softening post-failure behaviour. 1981) showed that  h / v = 0.5. However, many stress measurements carried out in the West of France, within synclinal structures, show that the horizontal stress is always higher or equal to the vertical stress. For the sites of Grais (May/Orne) and St-Sigismond (Maine-et-Loire), the ratio  h / v varies between 1 and 1.5 [START_REF] Burlet | Détermination du champ de contrainte régional à partir de tests hydrauliques en forages, résultats de neuf expérimentations in-situ réalisées en France[END_REF]. The stress tensor being of doubtful validity, we considered three values for the ratio  h / v : 0.5, 1 and 2. Results The solutions being able to explain the collapse of 1966 being plural, some certain data input were regarded sure (extraction ratio, dimensions of the various stages, width of the stage pillars, exploitation thickness, geology, geomechanical characterisation of materials other than the iron ore) and others were regarded as variables of the study. Each method of calculation of this study was analysed in terms of subsidence on surface, distribution of plasticity, displacements, principal stresses and plastic deformation in five tested pillars (figure 5). To sum up, the different methods of calculation carried out made it possible to study the influence of the:  the length of the plastic zone cannot exceed 200 m. The mechanism highlighted cannot thus be repeated in the lower stages;  the verification of the 2 nd checking point showed the importance of the strength value of the barrier pillars. The pillar strength must be higher (51 MPa) than that of the pillars of the higher stage (34 MPa) for the mechanism to occur. That is compatible with the fact that the iron ore of the lower stages is more carbonated (so more resistant);  for  h / v = 0.5 and 1, we notice that the value of maximum subsidence on the surface is close to that measured in 1966: 65 cm;  the state of the initial stresses has a relatively weak influence on the thresholds of pillar strength of the lower stages;  the characteristics (spacing and friction angle) of the stratification network parallel to the dip are essential parameters in the mechanism which we highlighted: the increase in spacing between the joints inhibits the mechanism of collapse. It is the same for the friction angle;  the values of strengths which we introduced into our models are compatible with the intervals of variation of the in-situ characteristics [instantaneous value ; value in the long term integrating the effect of time].  Parametric analysis The second part of this study consisted in carrying out a numerical modeling on the scale of the mine in order to develop the back-analysis collapse (of Soumont in 1966) and to evaluate the criteria to specify the risk by carrying out a parametric study (allowing a valorization on the whole inclined deposits of the same type). We thus studied the sensitivity of four parameters by carrying out twenty calculations:  the dip (between 30 and 65°);  the extraction ratio (between 70 and 90 %);  the exploitation thickness (between 1.5 and 5 m);  the height of overburden (between 0 and 50 m). The analysis of these twenty calculations was focused on the extension of the zones of potential failures (plasticity), on the value of maximum displacement in the pillars and on the value of the maximum subsidence on the surface. This reveals that the mechanism identified at the time of the back-analysis can be reproduced under the geometrical conditions synthesised in table 4. In addition, we noticed that the subvertical faults can inhibit or amplify the mechanism of failure by shearing. Moreover, the reduction of the height/width ratio of the pillars (or thickness reduction) has a very significant positive role on the exploitation stability. Evaluation of the "subsidence" hazard The hazard assessment is classically made by combining the awaited intensity of the phenomenon with its probability of occurrence, this being the predisposition of the site with respect to the dreaded phenomenon. Qualification of the intensity It is recognised that the characteristics of depression which materialise the most severe damage for the goods located on surface are the horizontal differential strains and movements of ground inclined setting rather than maximum vertical subsidence in itself. Table 3 gives indicative values of the strains and slopes which make it possible to evaluate the phenomenon intensity. Negligible  < 1  < 0.2 Very low 1 <  < 5 0.2 <  < 1 Low 5 <  < 10 1 <  < 2 Medium 10 <  < 30 2 <  < 6 High  > 30  > 6 The value of these two parameters can be appreciably influenced by different factors studied before. It appears so that value of maximum subsidence is in the form: A max = 0.3. w., with:  A max = maximum subsidence;  w = exploited thickness (in the districts exploited by shrinkage);   = extraction ratio (or recovery factor). It can be easily deduced from them the values from the strains ( max ) and slopes ( max ) starting from the following traditional relations:  max =  . A max / P  max =  . A max / P Where:  P is the average depth of the panel;   and  of the coefficients estimated respectively at 1.5 and 5 in the western iron basin. The values of the coefficients  and  are deduced from the studies in experience feedback carried out on the Iron Mines of Lorraine and adopted for their drastic security character. Qualification of the occurrence probability In the inclined exploitations of the iron deposits of the West of France, it is mainly the stability of the barrier pillars, the slabs or the pillars left in place to ensure the behaviour of the immediate strata which controls the subsidence predisposition. To evaluate the long-term stability of the undermined surface, main factors that have to be take into account are:  dimensions of the panels;  dip of the layers;  extraction ratio;  opening (height exploited between immediate strata);  strength of pillars. In a more precise way, the parametric analysis described previously provides fundamental indications on the configurations of layer and exploitation for which the occurrence of a subsidence can be excluded (table 4, below). > 55° ≤ 90% ≤ 4 m ≤ 85% ≤ 5 m 45° to 55° ≤ 90% ≤ 3 m ≤ 80% ≤ 5 m 30° to 45° ≤ 80% ≤ 3 m ≤ 70% ≤ 5 m The influence of an increase in the dip appears by a displacement of the zones of failure toward the surface (or of the outcrop): the greater the dip value, the more one affects the grounds close to surface (plastic points). In addition to these configurations of exploitation, other conditions must be taken into account for a reduction of the hazard level, like:  condition n° 1: for a subsidence to occur entirely, it is necessary that dimensions of the mining sites (width L) reach or exceed the depth (H) (that is: L ≥ H), which represents, in the context of these exploitations, a width along the dip from 250 to 290 m (depth lower than 220-250 m). In lower part (L < H), subsidence is all the more, the hazard level is lower;  condition n° 2: it is considered that there are no repercussions on surface (non perceptible subsidence) if the mining site has a width L < 0.4 H;  condition n° 3: if the minimum depth of mine working is higher than 250-300 m (according to the geometry of the mining sites), it is considered that the failure zones are not likely to reach surface. Hazard zoning The limits materialising on surface the zone influenced by subsidence are established, taking in account an angle called "influence angle", measured from the vertical, which connects the end of the panel, at the bottom, to the points of surface where subsidence, strains or slopes are regarded as unperceivable or null. Although an single influence angle () value of 30° to 35° is retained for flat veins, three angle limit values are defined for inclined layers (exploitations) These are:  the limiting angle value (), in the direction of drivage which is equal to the limiting angle in flat vein;  the "upstream" angle value, lower than the angle ;  the "downstream" angle value, always greater than the angle ; Looking at the data obtained in Soumont, it can be noticed (table 6) that the values of failure angles measured upstream and downstream (on average respectively about 7° and 30°), for a dip ranging between 30° and 40°, are very close to the corresponding values of the abacuses of the Lorraine coalfields or the Nord Pas-de-Calais region [START_REF] Proust | Etude sur les affaissements miniers dans le Bassin du Nord et du Pas-de-Calais[END_REF]. Hence, it can be deduced that the influence angles must be also very close and take for the layer of Segré some values of influence angle equal to 30° (upstream side) and 45° (downstream side). Let us specify that the downstream influence angle is taken at the base of the exploited panels, and the upstream influence angle at the higher part of the panels. Figure 2 . 2 Figure 2. Example of the mine configuration of May-sur-Orne (according to Maury, 1972). Figure 3 . 3 Figure 3. Mining methods of the basin of Soumont (according to Perrotte and Lidou, 1983). Figure 4 . 4 Figure 4. Plane view of the seam worked by rooms and pillars at Soumont in the collapse zone of 1966. stress field with the ratio  h / v (3 values: 0.5, 1 and 2);  density of the roof stratification;  friction angle of the bedding planes;  effects of faults on the collapse mechanism;  joint behaviour law;  strength of the pillars;  panel width;  opening effect. The various calculations allow to reproduce a mechanism and explain the subsidence observed on surface in 1966. It is due initially to the relative compressions of pillars and then to the deflection of the roof. These two zones are the place of strong shear mechanisms which imply a potential failure by shearing up to surface. The three checking points (collapse of 1966 in the upper stage, stability of the lower stages and shrinkage stability) were checked. Figure 5 : 5 Figure 5: Distribution of plasticity: joint spacing of 10 m + variation of subsidence on the surface Figure 6 : 6 Figure 6: Diagram showing the dissymmetry of the upstream and downstream influence angles in inclined deposit Table 1 : 1 Comparative analysis of various Western iron- bearing basins Basin MAY-SUR-ORNE SOUMONT LA FERRIERE-AUX- ETANGS SEGRE Dates of exploitation 1896 -1968 1907 -1989 1905 -1970 1907 -1984 Maximum depth 450 m 650 m 400 m 490 m Mining methods stoping, dip faces, shrinkage Rise faces, shrinkage, strike faces or "stoping" Stopings, rise faces, retreating workings, shrinkage shrinkage Dip 45° to 90° 30° to 60° 25° to 45° 60° to 90° Number of worked seams 1 (very locally 2) 1 1 2 (intercalated bed of 40 to 50 m thickness) Haematite under Dominant nature of the iron ore Facies haematite calcareous overthrust, Carbonated and siliceous Chlorito-carbonated. Little haematite Carbonated in-depth Content of iron 35-50% 36-50% 35-50% Average 52% Compressive 80 MPa parallel to the bedding strength of the 100 MPa 115 MPa plane, 200 MPa perpendicular ??? ore to the bedding plane Table 2 . 2 Synthesis of the compressive strengths for various materials of the study. Iron ore Table 3 : 3 Classes of intensity of the risk "subsidence" (purely indicative values) Classify intensity Horizontal differential strains  (in mm/m) Surface inclination  (in %) Table 4 : 4 conditions of exclusion of the process of subsidence (according to[START_REF] Renaud | Contribution à l'analyse des conditions d'effondrement des gisements pentés des bassins ferrifères de Soumont[END_REF] Dip Extraction ratio ( %) Thickness (W) Table 5 : 5 Classes of predisposition of the site for the risk "subsidence" Site predisposition Ratio L/H Depth (H) Very sensitive L < 250 m sensitive L/H # 1 < 250 m Not very sensitive 0.4 < L/H <1 < 250 m negligible L/H < 0.4 L < 250 m > 250-300 m Table 6 : 6 Values given in the subsidence abacuses of Nord/Pas-de-Calais, Saar and Lorraine basins Dip values 0° 15° 25° 30° 40° 50° 60° Angles of rupture giving the limits of Upstream angle 18 14 12 11 9 7 6 fracturing on the surface Downstream angle 18 22 25 27 30 33 36 Angles of influence giving the limits of null subsidence Upstream angle Downstream angle 35 35 32 38 30 40 30 43 30 45 28 47 27 48

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2024/03/05 22:32:07

2006

https://ineris.hal.science/ineris-00174736/file/Sea_to_sky_vancouver_2006_p481-487.pdf

Yannick Wileveau Vincent Renaud Jean-Bernard Kazmierczak RHEOLOGICAL CHARACTERIZATION OF A CLAY FORMATION FROM DRIFTS EXCAVATION : ELASTIC AND ELASTOPLASTIC APPROACH An extensive scientific programme has been carried out by Andra (French Agency in charge of radioactive waste management) for investigating feasibility of High Level Activity Waste disposal in deep geological formation. An Underground Research Laboratory (URL) is currently being constructed in North-eastern France to assess the adequacy of a hard-clay argillite layer (Callovo-Oxfordian formation) situated between 420 m and 550 m of depth. Geotechnical measurements have been carried out during the shafts and drifts excavation and particularly upon the main level of the laboratory (-490 m). The drifts are "horseshoe section" type with about 17 m² in area mainly supported by metallic ribs and rock bolts. The digging has been performed with classical pneumatic hammer. Measurement sections have been instrumented very close to the front face using convergencemeters and radial extensometers. This paper presents a comparison between in situ measurements and numerical modelling. Elastic calculations are not in agreement with the measured deformations. An elastoplastic constitutive model considering damage and using Hoek & Brown criteria has been developed and implemented in the FLAC 3D numerical code. Mechanical parameters came from lab tests performed on core samples. For the first meters, model provides consistent displacements. Beyond 4 meters, a time dependent convergence takes place and has to be integrated in the model to take into account creep and/or hydromechanical behaviour. RÉSUMÉ Un important programme scientifique a été conduit par l'Andra (l'agence française de la gestion des déchets radioactifs) afin d'étudier la faisabilité d'un stockage de déchets de haute activité et à vie longue dans une formation géologique profonde. Un laboratoire de recherche souterrain est actuellement en cours de construction dans le Nord-Est de la France pour évaluer les propriétés d'une roche indurée argileuse (argillite du callovo-oxfordien). Cette roche se situe entre 420 m et 550 m de profondeur. Des expérimentations géomécaniques ont été mises en oeuvre lors de la construction des puits et des galeries en particulier au niveau principal (-490 m). Une méthode classique de creusement au marteau piqueur a été utilisée pour excaver les galeries de type 'section fer à cheval' d'une surface de 17 m 2 environ avec un soutènement se composant de cintres métalliques et de boulons aciers. Des sections de mesures ont été équipées au plus tôt près du front avec des plots de convergence et des extensomètres radiaux. Dans cet article, une comparaison entre modèles mécaniques et mesures in situ est présentée. Les résultats d'une approche purement élastique ne sont pas en accord avec les déformations observées. Une loi rhéologique élastoplastique utilisant les critères Hoek & Brown a été développée et implémentée dans le code numérique FLAC 3D . Les paramètres du modèle ont été ajustés sur des essais sur échantillons. Le modèle prédit bien les déplacements mesurés lors des 4 premiers mètres d'excavation. Après, la convergence est dominée par les effets différés, nécessitant l'intégration d'une loi de fluage ou d'un comportement hydromécanique dans le modèle. . 1. INTRODUCTION In November 1999, having completed the preliminary work phase, Andra started construction work of an underground research laboratory (figure 1) in the district of Bure (Meuse département), located in the Northeastern of France. From 2000 to 2005, the construction of the experimental site has allowed to study radioactive waste storage possibilities in deep geological formation [START_REF] Delay | The French Underground Research Laboratory at Bure as a model precursor for deep geological repositories -IGC symposium -Florence[END_REF]Andra, 2005a). The target horizon for the laboratory is a 130 m thick layer of argillaceous rocks that lies between about 420 and 550 meters below the surface at the URL site. From a lithological view point, the depositional period straddles the Callovian and Oxfordian subdivisions of the middle to upper Jurassic. Argillaceous rocks contain a mixture of clay minerals and clay-sized fractions of other compositions. The clays, which constitute 40 % -45 % on average of the Callovo-Oxfordian argillaceous rocks, offer groundwater isolation and radionuclides retention. Silica and carbonate-rich sedimentary components strengthen the rock to contribute to stability of the underground construction. The stratigraphy of the URL is one of alternating limestone-rich and clay-rich units. On the upper part, the Oxfordian limestones lie from about 150 to 400 meters depth. Between the surface and the Oxfordian limestones is a 150 m thick sequence of mixed argillaceous rocks, [START_REF] Vigneron | Apport des investigations multi échelles pour la construction d'un modèle conceptuel des plateformes carbonatées de l'Oxfordien moyen et supérieur de l'est du Bassin de Paris[END_REF]. The state of in situ stress at the Meuse/Haute-Marne site has been measured by comprehensive combined methods. The vertical stress profile is presently well known on the site. The orientation of the σH stress (N155°E) is consistent with the regional stress field. The horizontal stress anisotropy is estimated between 1.1 < KH = σH/σh < 1.3 and the vertical stress and minimum horizontal stress have been directly measured close to the main level (-500 m depth) respectively equal to 12.7 MPa and 12.4 MPa [START_REF] Wileveau | Complete in situ stress determination in an argillite sedimentary formation[END_REF]. The main purpose of the geomechanical in situ investigation is to understand the rock response to the excavation of underground engineered structures and to the development of the damaged zone. The damaged zone characterization during shafts and drifts excavations will not be developed in this paper. Mechanical measurements are grouped in drift sections and within specific shaft excavation monitoring experiments socalled "mine by tests". These geomechanical experiments include a set of boreholes or convergence sections designed to monitor the behaviour of rock when openings are restarted. Figure 2 presents the overall layout of the underground network of excavated rock in the Meuse/Haute Marne laboratory and the location of the experimental drifts constructed in the clay formation. The first geomechanical mine by experiment is located in the -445 m experimental drift, corresponding to the upper layer of the Callovo-Oxfordian layer, where the mechanical response is mainly elastic [START_REF] Wileveau | Complete in situ stress determination in an argillite sedimentary formation[END_REF]. After an important instrumentation carried out from the -445 m level, the vertical mine by test has been monitored between -465 m and -480 m during the main shaft sinking [START_REF] Souley | Hydromechanical response to a mine by test experiment in a deep claystone -SGC Sea to Sky conference -Vancouver -Oct[END_REF]. The third geomechanical experiment, dealing with this paper, is the drift excavation tests at the main level of the URL, which is around -490 m beneath the surface. SMR1.1 and SMR1. EXPERIMENTAL RESULTS The two instrumented sections have been put in place very close to the front face (around 1.5 meter) in order to investigate the maximal deconfinement from drift excavation. The sections are composed of radial extensometers for which the end point is fixed at 20 m far from the wall, and convergence measurements with 6 points on the section (see figure 3). One notice that the feature of the drifts is "horseshoe section" with about 17 m 2 in area mainly supported by metallic sliding arches composed of three parts and rock bolts of 2.4 m length. The floor is also covered by a concrete slab of 0.7 m of thickness. Classical pneumatic hammer has been used to dig the galleries. The convergences have been measured manually using a system of invar wire along 9 directions of bases. The accuracy of this method is +/-0.2 mm. The reading frequency has been adapted to the excavation advance rate in order to obtain a high density of measurements within a distance of 12 meters from the section, corresponding approximately to 3 times the excavation diameter. The convergence measurements are given on figure 4 for the two sections SMR1.1 and SMR1.3 until early October 2005. The convergence is still monitored. It is not presented in this paper. Obviously, the behaviour of these two perpendicular sections is very different and strongly linked with the in situ stress anisotropy. The evolution of convergence in the SMR1.3 is very similar in the vertical and horizontal direction (respectively, measured on bases 6-3 and 1-5) (Fig. 4b). On the SMR1.1 section (Fig. 4a), the vertical convergence is much higher than the horizontal one what is in good agreement with the stress concentration due to the maximum horizontal stress σH acting on the walls. These values are given below (the convergence is reset to zero just before the excavation starting). Figure 5 shows the measurements for the cases of vertical downward and horizontal extensometers. Several names are used (e.g. GMR, GLE, GKE, GNI) for the drifts dug in the experimental area. PM indicates the distance between the workface and the axis of the previous gallery. Only 4 curves by extensometer are presented (0 m, 2 m, 5 m, 10 m relatively to the anchor installed at 20 m which is considered as a fixed point. One observes the deformation rate reacts gradually with the progress of the face. Moreover, in the particular case of SMR1.3 section where the history of excavation is more complex, the effect of the others openings is clearly measured. Such interaction between drifts is mainly due to the general layout of the URL designed to have a rapid access to the facilities for the time schedule constraint. The smallest distance between parallel galleries is equal to times the diameter of the gallery between the GLE and GKE drifts. This effect has not been identified on the SMR1.1 section. The magnitude of extension at the wall relatively to the reference point taken at the 20 m anchor is comparable to the convergence measurements, even if the starting date of measurement are differed of few days. The table 1 shows the value for both instrumentations. The values obtained by convergence and extension make up a consistent set of data, even though the deformations obtained by the convergence method give in most of the case a larger deconfinement. This difference can be explained by the delay to install the extensometer compared to the convergence section or also by the non measurable part of deformation in place up to 20 m from the wall of the openings. In the following chapter, one takes the convergence values to compare the results obtained by modelling. INTERPRETATION OF CONVERGENCES Results of elastic model The first analysis has been made using a classical approach in the framework of linear elasticity assuming a plane strain approach developed by [START_REF] Panet | Contribution à l'étude du soutènement derrière le front de taille[END_REF]. More complex calculations including a 3D simulation of the drift excavation is presented in the next chapter. The results are not in agreement with the observations as it is shown in figure 6 for the case of SMR1.3 where the in situ stresses are nearly isotropic around the gallery. A better agreement is obtained while reducing Young modulus up to one order of magnitude. This assumption is inconsistent with the mechanical behaviour observed usually on samples subjected to lab tests and can not be validated. Moreover, the significant elastic deconfinement in the first meters from section predicted by plane strain model is not reproduced. To better represent the complex behaviour of the argillite, one also needs to consider a damageable rock. The fracture network has clearly detected within the first 2 meters from the gallery wall by several direct methods (geological survey on cores samples, resin injection within the fracture network followed by overcoring, borehole camera) and indirect methods (velocity measurement, tomography). The application of elastic model, combined with in situ observations, leads us to consider (as it was previously forecasted) an elastoplastic approach for the argillite lying at this depth. Results of elastoplastic model Numerical calculations were carried out in 3D to simulate the drift progress (in 5 phases) of a horseshoe gallery at 490 m of depth (figures 3 and 7) according to the two orientations of the galleries with respect to the stress tensor : σh and σ Η . A first phase allows to reach the fine grid zone. Then, 3 phases of drift advancing (1 meter by 1 meter) are simulated. Finally, the drift advancing is continued to have a complete deconfinement until the end meshed zone. The geometrical model extends on 49.4 m in X direction, 25.3 m in the direction Y and on 50.6 m height (Z direction). It consists of 627183 gridpoints and 605784 zones. The used mesh is sufficiently fine for highlighting the characteristics of the zones exceeding the damage or failure criteria. In the same way, nodal points were selected in various directions (in front of the face, at the walls side, in vault and under floor) to follow the evolution of displacements (figure 8) and stresses during the phases of drift advancing. These points also allowed easy confrontation with in situ measurements on the two sections SMR1.1 and SMR1.3. Modelling assumptions The commercial computation software used for this study is delivered by Itasca ( 2002 The state of natural stresses (at -490 m depth) is as follows: σv = 12.7 MPa, σh = 12.4 MPa and σH = 1.3 xσh. From the assumptions on natural stress field, the worst case has been considered (KH = 1.3). Geomechanical properties Numerical calculations are carried out with a damageable elastoplastic model with hardening (as developed by Hoek and Brown). Table 2 indicates the parameters of this model used as reference parameters for studied zone (named B&C zones) of argillite [START_REF]Dossier 2005 -Référentiel du site Meuse/Haute-Marne[END_REF]. Figure 8 illustrates the isovalues of total displacement obtained by the numerical calculation at the end of the 3 steps of one meter excavation. The higher extrusion on the workface is clearly represented, as well as the corner effect. In order to compare the SMR1.1 and SMR1.3 sections to the modelling results, we have put the tracked point at one meter before the initial face. The zero displacement is then considered when initial face is reached. The calculation results corresponding to 3 m of excavation with respect to the initial face show a value of vertical convergence of about 10 mm in the direction of the major stress and of 25 mm in minor stress direction, that is to say a ratio of 2.5 between these two directions. We notice that this tendency is also observed for the measurements, even if the ratio of displacements is not exactly the same. The differences between the displacements calculated in the four configurations depicted on Figure 9 show a relative good agreement of modelling with the measurements on the 4 first meters of face progress. In spite of the fact that the results of displacement calculation are rather well correlated with the measurements in the case of drift advancing for a gallery oriented according to σH, the model predicts displacements higher than the measured values for the other direction (σh). This is natural, since the model does not integrate neither creep, nor hydraulic coupling. Therefore, long-term convergences cannot be reproduced by the model. CONCLUSION The main purpose of this paper is to assess the classical approach of tools used in geomechanics to interpret the in situ displacements measured on experimental mine by test conducted by Andra in the Meuse/Haute-Marne URL. For such hard clay lying at 490 m of depth in the Callovo-Oxfordian formation, a complex behaviour is observed, including different phenomena as elastoplastic behaviour, time dependent effect, and hydro-mechanical coupling. On one hand, the elastic approach has not successfully provided reasonable comparison when the global elastic mechanical response of the clay has been confirmed at the 445 m of depth [START_REF] Wileveau | Complete in situ stress determination in an argillite sedimentary formation[END_REF]. On the other hand, a damageable elastoplastic constitutive model using Hoek & Brown criteria has been used. Main conclusion of this analysis is that for the first meters only, displacements observed are consistent with the elastoplastic model. After, the time dependent mechanisms take effect as a predominant part of deformations. One notices that the elastoplastic modelling presented here has been carried out in the framework of studies on 3D complexity of URL (real geometry of drifts, anisotropy of horizontal stress, and working phases). Thus, we voluntarily simplified the argillite behaviour by neglecting the effects related on creep and hydraulic couplings. Some other numerical modelling are in progress to better understand the strong hydro-mechanical coupling observed in this clay, the real geometry and possible interaction between drifts, the role and link between plasticity and creep. Figure 1 . 1 Figure 1. Localization of the underground research laboratory at the Meuse/Haute-Marne site marls, and limestones of the Kimmeridgian. Underlying the Callovo-Oxfordian argillaceous rocks are the Bathonian and Bajocian-age Dogger limestones and dolomitic limestones[START_REF] Vigneron | Apport des investigations multi échelles pour la construction d'un modèle conceptuel des plateformes carbonatées de l'Oxfordien moyen et supérieur de l'est du Bassin de Paris[END_REF]. Figure 2 . 2 Figure2presents the overall layout of the underground network of excavated rock in the Meuse/Haute Marne laboratory and the location of the experimental drifts constructed in the clay formation. The first geomechanical mine by experiment is located in the -445 m experimental drift, corresponding to the upper layer of the Callovo-Oxfordian layer, where the mechanical response is mainly elastic[START_REF] Wileveau | Complete in situ stress determination in an argillite sedimentary formation[END_REF]. After an important instrumentation carried out from the -445 m level, the vertical mine by test has been monitored between -465 m and -480 m during the main shaft sinking[START_REF] Souley | Hydromechanical response to a mine by test experiment in a deep claystone -SGC Sea to Sky conference -Vancouver -Oct[END_REF]. The third geomechanical experiment, dealing with this paper, is the drift excavation tests at the main level of the URL, which is around -490 m beneath the surface. SMR1.1 and SMR1.3 sections have been respectively installed in april 2005 and august 2005 during the works Figure 3 . 3 Figure 3. Example of instrumented section SMR1.1 with combined extensometer and convergence measurement at the same location. Base "n" indicates the reference number of convergence point. The extensometers, on the same section than the convergence measurements, have monitored the deformations of the rock mass automatically (frequency : 4 data/hour) during the excavation advance of the drifts.Figure5shows the measurements for the cases of vertical downward and horizontal extensometers. Several names are used (e.g. GMR, GLE, GKE, GNI) for the drifts dug in the experimental area. PM indicates the distance between the workface and the axis of the previous gallery. Only 4 curves by extensometer are presented (0 m, 2 m, 5 m, 10 m relatively to the anchor installed at 20 m which is considered as a fixed point. Convergence measurements on sections SMR1.1 (a) and SMR1.3 (b) Figure 5 . 5 Figure 5. Extension of the rock mass versus time around the drift during the drifts progress -5a) and 5b) SMR1.1 section -5c) and 5d) SMR1.3 section -the values are calculated considering the fixed point at 20 m. Figure 6 . 6 Figure 6. Comparison between the results of convergence measurements of SMR1.3 and plane strain modelling Considering isotropic mechanical behaviour of the argillite, the elastic parameters Young modulus (E = 4.0 GPa) and Poisson's ratio (ν = 0.3) are taken from the Andra's labtests on samples coming from deep boreholes drilled on the Meuse/Haute-Marne site[START_REF]Dossier 2005 -Référentiel du site Meuse/Haute-Marne[END_REF]. are as follows: • null normal displacements are prescribed on the lower face of the model (Z = -515.3), on the face corresponding to the gallery symmetry plane (Y = 0) and on the "left" face of the model (X = -23.0); • stress conditions are prescribed on the upper face of the model (Z = -464.7), on the "back" face of the model (Y = 25.3) and on the right face of the model (X = 26.4). Figure 7 . 7 Figure 7. Mesh of the model including horseshoe shape and refined mesh around the drift (see zoom) Figure 9 9 Figure 8. Variation of total displacement for a 3 m advance 3.2.3Comparison between convergence measurements SMR 1.1 and SMR 1.3 and modelling Figure9illustrates the comparison between the results of numerical modelling and the convergence measurements at sections SUG1350, SUG1360 and SUG1170 (whose axis is parallel to σH) and sections SUG1150, SUG1160 and SUG1180 (whose axis is parallel to σh) according to the face advance. Figure 9 . 9 Figure 9. Measurements comparison between horizontal and vertical convergences and modelling results Table 1 . 1 Comparison between convergence and extension measurements (SMR1.1 and SMR1.3). Ref. number Drift axis Period (days) 1 type 2 (mm) Value SUG1350 σH 163 base 1-5 horiz. base 6-3 vertical 35.8 39.7 SMR1.3 SUG1360 σH 163 base 1-5 horiz. base 6-3 vertical 42.7 39.8 SUG1301 σH 163 horiz. extens. 33.8 SUG1303 σH 163 vertical extens. 27.0 SUG1150 σh 45 base 1-5 horiz. base 6-3 vertical 8.6 33.2 SMR1.1 SUG1160 SUG1103 σh σh 45 45 base 1-5 horiz. base 6-3 vertical extens. horiz. 8.9 53.0 5.7 SUG1105 σh 45 vertical extens. 37.2 SUG1118 σh 45 vertical extens. 23.6 1 Period of observation calculated until the 1 2005 2 The values given are the half-convergence measured st October between two bases Table 2 . 2 Reference values of B & C zones (used in the modelling (S and m: parameters of the Hoek & Brown criterion, α and β: parameters characterising the residual strength evolution) Elastic parameters Young's modulus (MPa) Poisson's ratio 4000 0.3 Criteria of the Hoek & Brown model Failure criterion S (rup) 0.128 m (rup) 2 σc (rup) in MPa 33.5 Damage initiation S (dam) 1 m (dam) 1.5 σc (dam) in MPa 9.6 Residual strength α β in MPa 2.8 3

01747433

en

2024/03/05 22:32:07

2018

https://hal.science/hal-01747433/file/ECC18_0293_FI.pdf

Maxime Thieffry Alexandre Kruszewski Thierry-Marie Guerra Christian Duriez Reduced Order Control of Soft Robots with Guaranteed Stability This work offers the ability to design a closedloop strategy to control the dynamics of soft robots. A numerical model of a robot is obtained using the Finite Element Method, which leads to work with large-scale systems that are difficult to control. The main contribution is a reduced order model-based control law, that consists in two main features: a reduced state feedback tunes the performance while a Lyapunov function guarantees the stability of the large-scale closed-loop systems. The method is generic and usable for any soft robot, as long as a FEM model is obtained. Simulation results show that we can control and reduce the settling time of the soft robot and make it converge faster without oscillations to a desired position. I. INTRODUCTION Soft robots -robots made of deformable materialspromise disruptive advances in many areas and bring transversal challenges, among which are dynamical modelling and control, see [START_REF] Rus | Design, fabrication and control of soft robots[END_REF], [START_REF] Majidi | Soft robotics: A perspective-current trends and prospects for the future[END_REF] and [START_REF] Kim | Soft robotics: A bioinspired evolution in robotics[END_REF]. Being lighter and more compliant than rigid ones, vibrations issues arise when dealing with soft robots dynamics. We propose a feedback control design to handle these issues. Designing such a feedback law brings different challenges such as dynamical modelling, large scale control and stability preservation. Modelling soft robots analytically is hard to achieve as this new type of robot has a theoretical infinite number of degrees of freedoms. Several approaches has been proposed so far: piece-wise constant curvature (PCC), Cosserat rod theory and finite element method (FEM). Depending on the robot geometry, the constant curvature is not always valid. Moreover, the equations coming from the Cosserat model are not often suitable for controller design. Recent work has been done to deal with these issues in [START_REF] Renda | Discrete Cosserat approach for multi-section soft robots dynamics[END_REF] and [START_REF] Falkenhahn | Dynamic modeling of bellows-actuated continuum robots using the Euler-Lagrange formalism[END_REF] to study continuum manipulators or beam-like soft robots. The goal of the present method is to be as generic as possible concerning the geometry of the robot, therefore, the finite element method is used. This spatial discretization gives rise to large scale systems, which are abounding in many fields of research, such as control theory. Standard control theory tools, like Linear Matrix Inequalities (LMI) or Lyapunov equation solvers, can not deal with a too large number of decision variables. Numerical efficiency for control applications is also an active field of research, see [START_REF] Benner | Numerical solution of large and sparse continuous time algebraic matrix Riccati and Lyapunov equations: a state of the art survey[END_REF], [START_REF] Koo | Decentralized fuzzy observerbased output-feedback control for nonlinear large-scale systems: an LMI approach[END_REF] or [START_REF] Chang | H∞ fuzzy control synthesis for a largescale system with a reduced number of LMIs[END_REF]. If classical tools of automatic control have to be applied, model order reduction must be considered. The methods used in this work are rather standard and we refer the reader to [START_REF] Benner | Model Reduction and Approximation : Theory and Algorithms[END_REF] for more details. Recent work has been done on soft robots control: [START_REF] Marchese | Dynamics and trajectory optimization for a soft spatial fluidic elastomer manipulator[END_REF] provides an open-loop stategy for dynamics and trajectory optimization, [START_REF] Lismonde | Trajectory planning of soft link robots with improved intrinsic safety[END_REF] provides an open-loop trajectory planning approach based on FEM model and [START_REF] Zhang | Kinematic modeling and observer based control of soft robot using real-time finite element method[END_REF] provides a closedloop controller based on FEM model but restricted to kinematics. Recently, space reduction has also been used to study continuum manipulators in [START_REF] Sadati | Control space reduction and real-time accurate modeling of continuum manipulators using Ritz and Ritz-Galerkin methods[END_REF]. This paper aims at providing a closed-loop controller that fixes the dynamics of a reduced order system while guaranteeing the stability of the full order model using Lyapunov framework. The remainder of this document is organised as follows. Section II presents a dynamic model of the robot while the main contribution of this work is presented in section III and section IV presents tracks to improve this result. Simulation results are provided along with the theoretical results to illustrate and show the effectiveness of the methodology. II. SOFT ROBOT MODELLING A. FEM model Modelling soft robots relies on both continuum mechanics theory and numerical approaches to solve the underlying equations. Here, a corotated FEM allows us to define position and velocity vectors, respectively q(t) ∈ R n and v(t) ∈ R n whose dimension n is proportional to the size of the FEM mesh used to model the robot. The more nodes the mesh has, the more the model tends to be accurate and, for soft robots application, the size N of the FEM mesh goes from hundreds to thousands of variables. The dimension n of the previous vectors is made of 3 × N variables, as the position and velocity vectors are given in the 3 dimension of space. Using the SOFA framework, the method proposed in [START_REF] Coevoet | Software toolkit for modeling, simulation, and control of soft robots[END_REF] describes concretely how to use FEM to model soft robots, in the particular context of real-time simulation. The non-linear equation of motion of the robots is given by the second law of Newton: M (q) v = P (q) -F (q, v) + H T (q)u(t) (1) where M (q) is the mass matrix, H T (q)u is the actuators contribution : H T (q) contains the direction of the actuators forces and u their amplitude. The matrix F (q, v) represents the internal forces and P (q) gathers all the known external forces. As we consider only the gravity field, P (q) is constant and P (q) = P . Let q 0 ∈ R n be a stable equilibrium point induced by P and u(t) = u 0 , i.e. q 0 is solution to 0 = P -F (q 0 , 0) + H T (q 0 )u 0 (2) Equation ( 1) can also be written as: M (q) v = P -F (q, v) + H T (q)u(t) -P + F (q0, 0) -H T (q0)u0 ⇔ M (q) v = F (q0, 0) -F (q, v) + H T (q)u(t) -H T (q0)u0 (3) We can approximate the internal forces F with a first order Taylor expansion around this equilibrium point: F (q, v) ≈f (q0, 0) + ∂F (q, v) ∂q q=q 0 (q -q0) + ∂F (q, v) ∂v v=0 v (4) where ∂F (q,v) ∂q = K(q, v) is the compliance matrix, and ∂F (q,v) ∂v = D(q, v) is the damping matrix. By definition, mass, compliance and damping matrices are positive definite. With these notations, equation (3) becomes: M (q) v ≈ -K(q0, 0)(q-q0)-D(q0, 0)v+H T (q)u(t)-H T (q0)u0 (5) Let d be the displacement vector defined by: d = q -q 0 (6) The equation of motion around an equilibrium point q 0 is thus given by the following relation: M (q) v ≈ -K(q 0 , 0)d-D(q 0 , 0)v +H T (q)u(t)-H T (q 0 )u 0 (7) B. State-space Equation Without loss of generality, considering u 0 = 0 allows us to define the following non-linear state-space equation:          ẋ = -M (x) -1 D(x) -M (x) -1 K(x) I 0 A(x) x + M (x) -1 H T (x) 0 B(x) u y = Cx (8) where x = v d , x ∈ R 2n and where system matrices are large-scale sparse non-linear matrices, i.e. A(x) ∈ R 2n×2n , B(x) ∈ R 2n×m , C ∈ R p×2n , where m is the number of actuators and p the number of outputs. The results showed in this paper are obtained on simulation experiments, where the non-linear model is used to simulate the robot. For control application, it is more direct to work on a linear model to design the control law and around the equilibrium point q 0 , system (8) can be approximated by the following linear representation:    ẋ = -M -1 D -M -1 K I 0 x + M -1 H T 0 u y = Cx (9) where M = M (q 0 , 0), D = D(q 0 , 0) and K = K(q 0 , 0). Remark 1: The FEM model of the robot allows us to compute its energy in real-time. The kinetic energy of a soft robot is defined as: E k (x) = 1 2 v T M v (10) and its potential energy: E p (x) = 1 2 d T Kd (11) The total energy of the robot is then: E(x) = 1 2 v d T M 0 0 K v d (12) As M and K are positive definite, i.e. M > 0 and K > 0, this energy function is positive definite: E(x) > 0 (13) and E(x) = 0 ⇔ (q, v) = (0, 0) (14) III. LARGE SCALE CONTROL DESIGN The general objective of this paper is to compute a state feedback control law u = Lx (15) that guarantees the performance of the system (9) in closedloop. Yet, using pole placement or LMI approaches, the dimension of x implies that the computation of the matrix L cannot be done with numerical efficiency. Moreover, this controller could guarantee the closed-loop stability of the large scale model, but to be usable it would require the measurement of the whole state, which is of course impossible in practice due to the large number of variables to measure. The objective of this work is also to reduce the number of parameters in the controller. The use of Lyapunov stability and LMI constraints to study the stability and design the controller permits to optimize the performance of the system. In many cases, energy functions can be used as Lyapunov functions. Without actuation the studied soft robot converges to a natural equilibrium point where its energy is zero, this energy function is also a Lyapunov function for the system in open-loop. This allows us to design a Lyapunov function based on the system matrices and limits the complexity in the choice of the Lyapunov function. However, this does not reduce the complexity of the controller design, as the matrix L still contains a lot of variables to tune. The contribution of this paper is a method to deal with this issue: relying on model reduction techniques, a reduced order state feedback control law u = L r ξ r (16) is computed and a proof of stability for the original large scale system is given, using large-scale Lyapunov functions. A. Model Order Reduction Before developping the control part, this subsection presents notion of model order reduction that are required to develop the main results. There are two main branches of model order reduction methods: the first one based on optimisation [START_REF] Vuillemin | Poles residues descent algorithm for optimal frequency-limited H 2 model approximation[END_REF] and the other based on projection [START_REF] Gallivan | Sylvester equations and projection-based model reduction[END_REF]. As the implementation of optimisation based model reduction is still challenging for very large scale systems, only projection methods are used here . These methods aim at computing two projectors W r ∈ R n×r and V r ∈ R n×r , with W T r V r = I r , to approximate a large scale system ẋ = f (x, u) , x ∈ R n (17) with a reduced order one: ξr = W T r f (V r ξ r , u), ξ r ∈ R r , r n (18) where ξ r = W T r x (19) is the reduced order state and ξ r is the neglected one: ξ r = W T r x, ξ r ∈ R r , r = n -r (20) such that: x = V r ξ r + V r ξ r (21) For soft robotics applications, to compute an approximation of the full-order state x = v d , it is interesting to use structure preserving model order reduction. The reduced and neglected states will also keep their initial structure: ξ r = ξ rv ξ rd ; ξ r = ξ rv ξ rd ; (22) Concretely, this requires to find projectors for the velocity and the displacement vectors such that equation (21) writes: v d = V rv 0 V rv 0 0 V rd 0 V rd     ξ rv ξ rd ξ rv ξ rd     ⇔ x = VΞ (23) where Ξ ∈ R n is the projected state. To find the projectors V and W, state-of-the-art model reduction methods are available, such as Improved Balanced Truncation [START_REF] Benner | An improved numerical method for balanced truncation for symmetric second-order systems[END_REF], Iterative Interpolation Methods [START_REF] Gugercin | An iterative SVD-krylov based method for model reduction of large-scale dynamical systems[END_REF] and Proper Orthogonal Decomposition (POD). The first two methods are for now restricted to the linear case whereas the POD is a projection based method adapted to non-linear systems. B. Reduced Order Model-Based Controller To get rid of the whole state measurement to control the large scale system using equation [START_REF] Vuillemin | Poles residues descent algorithm for optimal frequency-limited H 2 model approximation[END_REF], we aim at computing a reduced order state feedback controller. Starting from the Lyapunov function defined previously: E(x) = x T M 0 0 K x = Ξ T V T M 0 0 K VΞ (24) In open-loop, with ẋ defined in eq [START_REF] Benner | Model Reduction and Approximation : Theory and Algorithms[END_REF], the derivative of this function is: Ė(x) = x T -D 0 0 0 x + ( * ) = -2v T Dv (25) where ( * ) represents the transpose of the matrix preceding it. The reduced order state ξ r is of reasonable dimension and we can design a reduced order feedback u = L r ξ r = L rv L rd ξ rv ξ rd (26) that fixes the performance of the closed loop. With this control law, the derivative of the Lyapunov function becomes: Ė(x) = x T -D 0 0 0 x + x T H T 0 u + ( * ) (27) which, in the projected space, is equivalent to: Ė(Ξ) = Ξ T V T -D 0 0 0 VΞ + Ξ T V T H T 0 u + ( * ) = Ξ T QΞ + ξ r ξ r T V T H T 0 L r 0 ξ r ξ r + ( * ) = Ξ T (Q + R)Ξ (28) where Q =     -V T rv DV rv 0 -V T rv DV rv 0 0 0 0 0 -V T rv DV rv 0 -V T rv DV rv 0 0 0 0 0     + ( * ) (29) and R =     V T rv H T L v V T rv H T L d 0 0 0 0 0 0 V T rv H T L v V T rv H T L d 0 0 0 0 0 0     + ( * ) (30) with Q, R ∈ R 2n×2n . The computation of the matrix L r is done using the framework of LMI constraints. The LMI is defined as: Ė(x) < -λE(x) ⇔ (Q + R) < -λ M 0 0 K (31) where λ is a positive scalar that fixes the decay rate of the function E(x). The stability is proven for the large-scale system, even if the number of variables in the LMI is equal to m×r, where m is the number of input and r is the reduced dimension. C. Simulation Results The method is tested on a simulated model of a soft robot made of silicone, shown on the following picture: This robot is actuated with 8 cables mounted on the structure as shown on figure 1 so that the robot can deform in any direction of space. Friction between the robot and its cables is neglectable thanks to flexible tubes that guide the cables and that are added in the simulation. SOFA integrates the CGAL library to compute a FEM mesh from a visual model. In this case, the mesh is made of 1557 nodes and 5157 tetrahedron elements, as shown on figure 2. The size n of both velocity and position vectors is equal to n = 3 × 1557 = 4671, and thus system ( 9) is of dimension 9342. Computing a full-order state feedback, as in [START_REF] Vuillemin | Poles residues descent algorithm for optimal frequency-limited H 2 model approximation[END_REF] would have implied the computation of a matrix L of dimension 8 × 9342, with 74 736 variables to compute. Instead of that, the model reduction step provide us with a reduced system of dimension 6, leading to a feedback matrix L r ∈ R m×r where m = 8 and r = 6. Solving LMI defined in equation ( 31) needs the computation of 48 decision variables. In simulation experiments the state vector, i.e. both velocity and position vectors, is directly available and we use POD reduction method to obtain the reduced order state. The simulation results are given for the following example: the robot starts from its initial shape (left of figure 2) and converges to its rest position (right of figure 2), where the model has been linearized. The behaviour of the robot in open-loop is illustrated on figure 3 and4, it shows that the robot converges to its equilibrium point after some oscillations. The goal of the proposed control method is to reduce, or suppress, this oscillations and make the robot converge faster to the desired position. The control law defined in ( 26) is directly usable in practice, as it only requires the measurement of the state of the robot, i.e. displacement and velocity vectors. In simulation, they are directly available and we reconstruct the reduced state by applying equation ( 19) at each time step. (To test on real robots, an observer should be used to compute the reduced order state from the sensor measurement, this work is not conducted in this paper.) With this feedback controller, the oscillations vanish, as shown on figure 5 and6. This control law allows us to suppress the oscillations in the robot behaviour and the convergence time of the robot is decreased. However, the gain L rd is near zero and does not seem to have an impact on the closed-loop performance. Looking at the definition of matrices Q and R in ( 29) and (30), one can see zero entries on the diagonal corresponding to the displacement quadratic term. This brings conservatism in the choice of matrix L rd , next section proposes an extension of the Lyapunov function (24) to handle this issue. IV. REDUCTION OF CONSERVATISM A. Choice of Lyapunov function Adding parameters in the Lyapunov function reduces the conservatism of the results obtained using the previous Lyapunov function. The following result holds: Theorem 1: The following functions are Lyapunov functions for system (9): V(x) = x T (1 + )M M M (1 + )K + D x (32) for all scalar such that: 0 < < α 1 -α ( 33 ) where α is the mass-damping coefficient of the material (see appendix for details). In open-loop, the derivative of V(x) becomes: V(x) = 2x T -(1 + )D + M 0 0 -K x ( 34 ) By adding parameter in the Lyapunov function, we add a non-zero entry on the diagonal of its derivative. Fig. 5. Results in closed-loop using feedback gains computed thanks to eq (31). Left : Norm of the displacement (cm); Right: Norm of velocity (cm/s 2 ). Fig. 6. Reduced order state in closed-loop using eq (31) B. Closed-loop algorithm The derivative of the Lyapunov function according to the trajectories of the closed-loop writes: V(x) = x T -( 1 + )D + M 0 0 -K + (1 + )H T H T L x+( * ) (35) which in the projected space is equivalent to: Ξ T V T -( 1 + )D + M 0 0 -K + (1 + )H T H T L VΞ + ( * ) (36) from which the design of the controller is made possible thanks to the following theorem: Theorem 2: System (9) with feedback (26) is stable with a decay rate λ if: (S + T) < -λ (1 + )M M M (1 + )K + D (37) with S = V T -(1 + )D + M 0 0 -K V + ( * ) (38) which also writes S =     -(1 + )D rv rv + M rv rv 0 -(1 + )D rv rv + M rv rv 0 0 -K rv rv 0 -K rv rv -(1 + )D rv rv + M rv rv 0 -(1 + )D rv rv + M rv rv 0 0 -K rv rv 0 -K rv rv     where D rv rv = V T rv DV rv , D rv rv = V T rv DV rv and T = V T (1 + )H T H T LV + ( * ) =     (1 + )V T rv H T Lrv (1 + )V T rv H T L rd 0 0 V T rd H T Lrv V T rd H T L rd 0 0 (1 + )V T rv H T Lrv (1 + )V T rv H T L rd 0 0 V T rd H T Lrv V T rd H T L rd 0 0     + ( * ) (39) Remark 2: Sketch of the proof. V (x) is a Lyapunov function defined in Theorem 1 and it holds: (37) ⇔ V(x) < -λV(x) (40) Moreover, L r = 0 is solution of the previous LMI for λ = 0 which makes it possible to use optimization algorithms. Using the Lyapunov function V(x), more flexibility is given in the choice of the matrix L r than in (31) with the same number of decision variables. C. Simulation Experiments The same experiment is done with the robot presented on figure 1 but with the control method of this section, i.e. Theorem 2. The objective remains the same, the oscillations of the open-loop behaviour of figure 3 and4 should be attenuated by the controller. The closed-loop results are presented on figures 7 and 8. It is clear that the oscillations are removed with this control law, and in this case the tuning of the decay rate of the Lypunov function is easier than with the controller defined in (31). During firsts experiments whose results are shown on figure 5, the displacement vector converges after t = 11s whereas on figure 7, it converges at t = 6s. A more complete control of the robots dynamics is also made possible using the method presented in this section. The large scale Lyapunov function V(x) guarantees the stability of the large scale model but here both reduced velocity and displacement have a direct impact on the closed-loop performance. We also have more flexibility in the tuning of the controller. Remark 3: The resolution of the LMI (37), with 48 variables and 9342×9342 constraints took 75 minutes on a Intel Core i7 CPU. V. CONCLUSIONS We presented a generic method that offers the possibility to control the dynamical behaviour of soft robots. The first benefit of this work is the use of model order reduction methods to provide the user with a reduced order system that models a large-scale accurate model of the robot. Thanks to the reduction of the state variable, the design of a reduced order state feedback is made possible. The second advantage of our work is the use of Lyapunov function to prove the stability of the large-scale closed-loop system. Simulation results provided in this paper show the performance of our approach, which is generic in the sense that it is applicable to any robot with a stable equilibrium point, as long as a FEM mesh of the robot is obtained. A few steps are still required before testing on real prototypes, as the one presented on figure 1. Our next move is the integration of an observer in the control design to make it possible to reconstruct the reduced state from measurements. For now, the control design allows the user to control a studied soft robot from any initial shape to a desired position, where the model has been linearized. An extension of the approach would be interesting to be able to control the robot from any initial shape to any desired position. Further research could focus on the linearization assumption required in the method, removing the linearization step to design a controller for the initial non-linear system (8) could also lead to a controller where the region of convergence is guaranteed. APPENDIX : PROOF OF THEOREM 1 The continuous function V(x) is a Lyapunov function in open-loop for system (9) if: 1) V (x) > 0 2) V (x) is radially unbounded 3) V (x) < 0 • Proof of 1) V(x) > 0 ⇔ (1 + )M M M (1 + )K + D > 0 Using Schur complement, it is equivalent to: (1 + )M > 0 ; (1 + )K + D -      > - 1 1 + β ⇒ (1 + + β)K > 0 0 < < α 1 -α ⇒ (α - 1 + ) > 0 The following condition is also a sufficient condition for V(x) to be positive definite in open-loop for system (9): 0 < < α 1 -α (41) • Proof of 2) is trivial. • Proof of 3) V (x) < 0 ⇔ -2 (1 + )D -M 0 0 K < 0 With > 0, it directly follows K > 0. One just need (1 + )D -M > 0, for which condition (41) is also a sufficient condition. Fig. 1 . 1 Fig. 1. Top: Visual model of the robot Bottom: Design of the robot: slice view (left) and side view (right). The robot is actuated with 8 cables: 4 short cables in red, and 4 long ones in green. Fig. 2 . 2 Fig. 2. FEM mesh of the Trunk-like robot, made of 1557 nodes. Left : Deformed position; Right : Rest position. Fig. 3 . 3 Fig. 3. Left : Norm of the displacement (cm) in open-loop Right: Norm of velocity (cm/s 2 ) in open-loop. Fig. 4 . 4 Fig. 4. Reduced order state in open-loop. Fig. 7 . 7 Fig. 7. Results in closed-loop using feedback gains computed thanks to eq (37). Left : Norm of the displacement (cm); Right: Norm of velocity (cm/s 2 ). Fig. 8 . 8 Fig.8. Reduced order state in closed-loop using eq (37) ⇔ > - 1 ; 1 By definition, matrices M, K and D are positive definite. The damping matrix is defined using Rayleigh definition:D = αM + βKwhere α and β are respectively the mass-proportional and the stiffness-proportional damping coefficients of the material. Both are positive scalars lower than one. It holds:V(x) > 0 ⇔ > -1; (1 + )K + (αM + βK) -(1 + + β)K + (α -1 + )M > 0Sufficient conditions are: Univ. Valenciennes, UMR 8201 -LAMIH F-59313 Valenciennes, France Univ. Lille, CNRS, Centrale Lille, Inria, UMR 9189 -CRIStAL -Centre de Recherche en Informatique Signal et Automatique de Lille, F-59000 Lille, France

00174770

en

2024/03/05 22:32:07

2002

https://shs.hal.science/halshs-00174770/file/tallon.pdf

Alain Chateauneuf email: [email protected] Rose-Anne Dana email: [email protected] Jean-Marc Tallon email: [email protected] Optimal Risk-Sharing Rules and Equilibria with Choquet-Expected-Utility Keywords: Choquet expected utility, comonotonicity, risk-sharing, equilibrium This paper explores risk-sharing and equilibrium in a general equilibrium set-up wherein agents are non-additive expected utility maximizers. We show that when agents have the same convex capacity, the set of Pareto-optima is independent of it and identical to the set of optima of an economy in which agents are expected utility maximizers and have same probability. Hence, optimal allocations are comonotone. This enables us to study the equilibrium set. When agents have different capacities, matters are much more complex (as in the vNM case). We give a general characterization and show how it simplifies when Pareto-optima are comonotone. We use this result to characterize Pareto-optima when agents have capacities that are the convex transform of some probability distribution. comonotonicity of Paretooptima is also shown to be true in the two-state case if the intersection of the core of agents' capacities is non-empty; Pareto-optima may then be fully characterized in the two-agent, two-state case. This comonotonicity result does not generalize to more than two states as we show with a counter-example. Finally, if there is no-aggregate risk, we show that non-empty core intersection is enough to guarantee that optimal allocations are full-insurance allocation. This result does not require convexity of preferences. Introduction In this paper, we explore the consequences of Choquet-expected-utility on risk-sharing and equilibrium in a general equilibrium set-up. There has been over the last fifteen years an extensive research on new decision-theoretic models [START_REF] Karni | Utility theory with uncertainty[END_REF] for a survey), and a large part of this research has been devoted to the Choquet-expected-utility model introduced by [START_REF] Schmeidler | Subjective probability and expected utility without additivity[END_REF]. However, applications to an economy-wide set-up have been relatively scarce. In this paper, we derive the implications of assuming such preference representation at the individuals level on the characteristics of Pareto optimal allocations. This, in turn, allows us to (partly) characterize equilibrium allocations under that assumption. Choquet-expected-utility (CEU henceforth) is a model that deals with situations in which objective probabilities are not given and individuals are a priori not able to derive (additive) subjective probabilities over the state space. It is well-suited to represent agents' preferences in situation where "ambiguity"(as observed in the Ellsberg experiments) is a pervasive phenomenon1 . This model departs from expected-utility models in that it relaxes the sure-thing principle. Formally, the (subjective) expected-utility model is a particular subclass of the CEU of model. Our paper can then be seen as an exploration of how results established in the von Neumann-Morgenstern (vNM henceforth) case are modified when allowing for more general preferences, whose form rests on sound axiomatic basis as well. Indeed, since CEU can be thought of as representing situations in which agents are faced with "ambiguous events", it is interesting to study how the optimal social risk-sharing rule in the economy is affected by this ambiguity and its perception by agents. We focus on a pure-exchange economy in which agents are uncertain about future endowments and consume after uncertainty is resolved. Agents are CEU maximizers and characterized by a capacity and a utility index (assumed to be strictly concave). When agents are vNM maximizers and have the same probability on the state space, it is well-known since [START_REF] Borch | Equilibrium in a reinsurance market[END_REF] that agents' optimal consumptions depend only on aggregate risk, and is a non-decreasing function of aggregate resources : at an optimum, an agent bears only (some of) the aggregate risk. It is easy to fully characterize such Pareto Optima (see Eeck-houdt and Gollier [1995]). More generally, in the case of probabilized risk, [START_REF] Landsberger | Co-monotone allocations, Bickel-Lehmann dispersion and the Arrow-Pratt measure of risk aversion[END_REF] and [START_REF] Chateauneuf | A review of some results related to comonotonicity[END_REF] have shown that Pareto Optima (P.O. henceforth) are comonotone if agents' preferences satisfy second-order stochastic dominance. This, in particular, is true in the rank-dependent-expected-utility case. The first goal of this paper is to provide a characterization of the set of P.O. and equilibria in the rank-dependent-expected-utility case. Our second and main aim is to assess whether the results obtained in the case of risk are robust when one moves to a situation of non-probabilized uncertainty with Choquet-expected-utility, in which there is some consensus. We first study the case where all agents have the same capacity. We show that if this capacity is convex, the set of P.O. is the same as that of an economy with vNM agents whose beliefs are described by a common probability. Furthermore, it is independent of that capacity. As a consequence, P.O. are easily characterized in this set-up, and depend only on aggregate risk (and utility index). Thus, if uncertainty is perceived by all agents in the same way, the optimal risk-bearing is not affected (compared to the standard vNM case) by this ambiguity. The equivalence proof relies heavily on the fact that, if agents are vNM maximizers with identical beliefs, optimal allocations are comonotone and independent of these beliefs : each agent's consumption moves in the same direction as aggregate endowments. This equivalence result is in the line of a result on aggregation in appendix C of [START_REF] Epstein | Intertemporal asset pricing under Knightian uncertainty[END_REF]. Finally, the information given by the optimality analysis is used to study the equilibrium set. A qualitative analysis of the equilibrium correspondence may be found in [START_REF] Dana | Pricing rules when agents have non-additive expected utility and homogeneous expectations[END_REF]. When agents have different capacities, matters are much more complex. To begin with, in the vNM case, we don't know of any conditions ensuring that P.O. are comonotone in that case. However, in the CEU model, intuition might suggest that if agents have capacities whose cores have some probability distribution in common, P.O. are then comonotone. This intuition is unfortunately not correct in general, as we show with a counterexample. As a result, when agents have different capacities, whether P.O. allocations are comonotone depends on the specific characteristics of the economy. On the other hand, if P.O. are comonotone, they can be further characterized, although not fully. It is also in general non-trivial to use that information to infer properties of equilibrium. This leads us to study cases for which it is possible to prove that P.O. allocations are comonotone. A first case is when the agents' capacities are the convex transform of some probability distribution. We then know from [START_REF] Chateauneuf | A review of some results related to comonotonicity[END_REF] and [START_REF] Landsberger | Co-monotone allocations, Bickel-Lehmann dispersion and the Arrow-Pratt measure of risk aversion[END_REF] that P.O. are comonotone. Our analysis then enables us to be more specific than they are about the optimal risk-bearing arrangements and equilibrium of such an economy. The second case is the simple case in which there are only two states (as in simple insurance models à la [START_REF] Mossin | Aspects of rational insurance purchasing[END_REF]). The non-emptiness of the cores' intersection is then enough to prove that P.O. allocations are comonotone, although it is not clear what the actual optimal risk-sharing arrangement looks like. If we specify the model further and assume there are only two agents, the risk-sharing arrangement can be fully characterized. Depending on the specifics of the agents' characteristics, it is either a subset of the P.O. of the economy in which agents each have the probability that minimizes, among the probability distributions in the core, the expected value of aggregate endowments, or the less pessimistic agent insures the other. (This last risk-sharing arrangement typically cannot occur in a vNM setup with different beliefs and strictly concave utility functions.) The equilibrium allocation in this economy can also be characterized. Finally, we consider the situation in which there exists only individual risk, a case first studied by [START_REF] Malinvaud | The allocation of individual risks in large markets[END_REF][START_REF] Malinvaud | Markets for an exchange economy with individual risks[END_REF]. comonotonicity is then equivalent to full-insurance. We show that a condition for optimal allocations to be full-insurance allocations is that the intersection of the core of the agents' capacities is non-empty, a condition that can be intuitively interpreted as minimum consensus. This full-insurance result easily generalizes to the multi-dimensional set-up. Using this result, we show that establish that any equilibrium of particular vNM economies is an equilibrium of the CEU economy. These vNM economies are those in which agents have the same characteristics as in the CEU economy and have common beliefs given by a probability in the intersection of the cores of the capacities of the CEU economy. When the capacities are convex, any equilibrium of the CEU economy is of that type. This equivalence result between equilibrium of the CEU economy and associated vNM economies suggests that equilibrium is indeterminate, an idea further explored in [START_REF] Tallon | Risque microéconomique, aversion à l'incertitude et indétermination de l'équilibre[END_REF] and [START_REF] Dana | Pricing rules when agents have non-additive expected utility and homogeneous expectations[END_REF]. The rest of the paper is organized as follows. Section 2 establishes notation and define the characteristics of the pure exchange economy that we deal with in the rest of the paper. In particular, we recall properties of the Choquet integral. We also recall there some useful information on optimal risk-sharing in vNM economies. Section 3 is the heart of the paper and deals with the general case of convex capacities. In a first sub-section, we assume that agents have identical capacities, while the second sub-section deals with the case where agents have different capacities. Section 4 is devoted to the study of two particular cases of interest, namely the case where agents' capacities are the convex transform of a common probability distribution and the two-state case. The case of no-aggregate risk in a multi-dimensional set-up is studied in section 5. Notation, definitions and useful results We consider an economy in which agents make decisions before uncertainty is resolved. The economy is a standard two-period pure-exchange economy, but for agents' preferences. There are k possible states of the world, indexed by superscript j. Let S be the set of states of the world and A the set of subsets of S. There are n agents indexed by subscript i. We assume there is only one good2 . C j i is the consumption by agent i in state j and C i = (C 1 i , . . . , C k i ). Initial endowments are denoted w i = (w 1 i , . . . , w k i ). w = n i=1 w i is the aggregate endowment. We will focus on Choquet-Expected-Utility. We assume the existence of a utility index U i : IR + → IR that is cardinal, i.e. defined up to a positive affine transformation. Throughout the paper U i is taken to be strictly increasing and strictly concave. When needed, we will assume differentiability together with the usual Inada condition: Assumption U1: ∀i, U i is C 1 and U ′ i (0) = ∞. Before defining CEU (the Choquet integral of U with respect to a capacity), we recall some properties of capacities and their core. Capacities and the core A capacity is a set function ν : A → [0, 1] such that ν(∅) = 0, ν(S) = 1, and, for all A, B ∈ A, A ⊂ B ⇒ ν(A) ≤ ν(B). We will assume throughout that the capacities we deal with are such that 1 > ν(A) > 0 for all A ∈ A, A = S, A = ∅. A capacity ν is convex if for all A, B ∈ A, ν(A ∪ B) + ν(A ∩ B) ≥ ν(A) + ν(B). The core of a capacity ν is defined as follows core(ν) =    π ∈ IR k + | j π j = 1 and π(A) ≥ ν(A), ∀A ∈ A    where π(A) = j∈A π j . Core(ν) is a compact, convex set which may be empty. Since 1 > ν(A) > 0 ∀A ∈ A, A = S, A = ∅, any π ∈ core(ν) is such that π ≫ 0, (i.e., π j > 0 for all j). It is well-known that when ν is convex, its core is non-empty. It is equally well-known that non-emptiness of the core does not require convexity of the capacity. If there are only two states however, it is easy to show that core(ν) = ∅ if and only if ν is convex. We shall provide an alternative definition of the core in the following sub-section. Choquet-expected-utility We now turn to the definition of the Choquet integral of f ∈ IR S : f dν ≡ E ν (f ) = 0 -∞ (ν(f ≥ t) -1)dt + ∞ 0 ν(f ≥ t)dt Hence, if f j = f (j) is such that f 1 ≤ f 2 ≤ . . . ≤ f k : f dν = k-1 j=1 [ν({j, . . . , k}) -ν({j + 1, . . . , k})] f j + ν({k})f k As a consequence, if we assume that an agent consumes C j in state j, and that C 1 ≤ . . . ≤ C k , then his preferences are represented by: v (C) = [1 -ν({2, .., k})] U (C 1 )+... [ν({j, .., k}) -ν({j + 1, .., k})] U (C j )+...ν({k})U (C k ) Observe that, if we keep the same ranking of the states as above, then v (C) = E π U (C), where C is here the random variable giving C j in state j, and the probability π is defined by: π j = ν({j, . . . , k})ν({j + 1, . . . , k}), j = 1, . . . , k -1 and π k = ν({k}). If U is differentiable and ν is convex, the function v : IR k + → IR defined above is continuous, strictly concave and subdifferentiable. Let ∂v(C) = {a ∈ IR k | v(C) -v(C ′ ) ≥ a(C -C ′ ), ∀C ′ ∈ IR k + } denote the subgradient of the function v at C. In the open set C ∈ IR k + | 0 < C 1 < C 2 < . . . < C k , v is differentiable. If 0 < C 1 = C 2 = . . . = C k then, ∂v(C) is proportional to core(ν). The following proposition gives an alternative representation of core(ν) that will be useful in section 5. Proposition 2.1 core(ν) = π ∈ IR k + | k j=1 π j = 1 and v(C) ≤ E π (U (C)) , ∀ C ∈ IR k + Proof: Let π ∈ core(ν) and assume C 1 ≤ C 2 ≤ . . . ≤ C k . Then, v(C) = U (C 1 )+ν({2, . . . , k})(U (C 2 )-U (C 1 ))+. . .+ν({k})(U (C k )-U (C k-1 )) Hence, since π ∈ core(ν), and therefore ν(A) ≤ π(A) for all events A: v(C) ≤ U (C 1 )+ k j=2 π j (U (C 2 )-U (C 1 ))+. . .+π k (U (C k )-U (C k-1 )) = E π (U (C)) which proves one inclusion. To prove the other inclusion, let π ∈ π ∈ IR k + | k j=1 π j = 1, v(C) ≤ E π (U (C)) , ∀ C . Normalize U so that U (C) = 0 and U ( C) = 1 for some C and C. Let A ∈ A and C A = C1 A c + C1 A . Since v(C A ) = ν(A) ≤ E π (U (C A )) = π(A), one gets π ∈ core(ν). A corollary is that if core(ν) = ∅, then v(C) ≤ min π∈core(ν) E π U (C). 3 comonotonicity We finally define comonotonicity of a class of random variables ( C i ) i=1,...,n . This notion, which has a natural interpretation in terms of mutualization of risks, will be crucial in the rest of the analysis. Definition 1 A family ( C i ) i=1,...,n of random variables on S is a class of comonotone functions if for all i, i ′ , and for all j, j ′ , C j i -C j ′ i C j i ′ -C j ′ i ′ ≥ 0. An alternative characterization is given in the following proposition (see [START_REF] Denneberg | Non-additive measures and integral[END_REF]): Proposition 2.2 A family ( C i ) i=1,...,n of non-negative random variables on S is a class of comonotone functions if and only if for all i, there exists a function g i : IR + → IR + , non-decreasing and continuous, such that for all x ∈ IR + , n i=1 g i (x) = x and C j i = g i n m=1 C j m for all j. The family ( C i ) i=1,...,n is comonotone if they all vary in the same direction as their sum. Optimal risk-sharing with vNM agents We briefly recall here some well-known results on optimal risk-sharing in the traditional vNM case (see e.g. [START_REF] Eeckhoudt | Risk, evaluation, management and sharing[END_REF] or [START_REF] Magill | Theory of incomplete markets[END_REF]). Consider first the case of identical vNM beliefs. Agents have the same probability π = (π 1 , . . . , π k ), π j > 0 for all j, over the states of the world and a utility function defined by v i (C i ) = k j=1 π j U i (C j i ), i = 1, . . . , n. The following proposition recalls that the P.O. allocations of this economy are independent of the (common) probability, depend only on aggregate risk (and utility indices), and are comonotone4 . Proposition 2.3 Let (C i ) n i=1 ∈ IR kn + be a P.O. allocation of an economy in which agents have vNM utility index and identical additive beliefs π. Then, it is a P.O. of an economy with additive beliefs π ′ (and same vNM utility index). Furthermore, (C i ) n i=1 is comonotone. As a consequence of propositions 2.2 and 2.3, it is easily seen that, at a P.O. allocation, agent i's consumption C i is a non-decreasing function of w. If agents have different probabilities π j i , j = 1, . . . , k, i = 1, . . . , n, over the states of the world, it is easily seen that P.O. now depend on the probabilities and on aggregate risk. It is actually easy to find examples in which P.O. are not necessarily comonotone (take for instance a model without aggregate risk in which agents have different beliefs : the P.O. allocations are not state-independent and therefore are not comonotone). 3 Optimal risk-sharing and equilibrium with CEU agents: the general convex case In this section we deal first with optimal risk-sharing and equilibrium analysis when agents have identical convex capacities and then move on to different convex capacities. Optimal risk-sharing and equilibrium with identical capacities Assume here that all agents have the same capacity ν over the states of the world and that this capacity is convex. We denote by E 1 the exchange economy in which agents are CEU with capacity ν and utility index U i , i = 1, . . . , n. Define D ν (w) as follows: D ν (w) = {π ∈ core(ν) | E π w = E ν w} The set D ν (w) is constituted of the probabilities that "minimize the expected value of the aggregate endowment". In particular, if w 1 < w 2 . . . < w k , D ν (w) contains only π = (π 1 , . . . , π k ) with π j = ν({j, j + 1, . . . , k})ν({j + 1, . . . , k}) for all j < k and π k = ν({k}). If w 1 = . . . = w k , the set D ν (w) is equal to core(ν). It is important to note that the Choquet integral of any random variable that is non-decreasing with w is actually the integral of that random variable with respect to a probability distribution in D ν (w). In particular, we have the following lemma. Lemma 3.1 Let ν be a convex capacity, U an increasing function and C ∈ IR k + be a non-decreasing function of w. Then, v(C) = E π U (C) for any π ∈ D ν (w). Proof: Since C is non-decreasing in w, if w 1 ≤ . . . ≤ w k , then C 1 ≤ . . . ≤ C k . Furthermore, w j = w j ′ implies C j = C j ′ . The same relationship holds between w j k j=1 and U (C j ) k j=1 , U being increasing. It is then simply a matter of writing down the expression of the Choquet integral to see that v(C) = E π U (C) for any π ∈ D ν (w). Proposition 3.1 The allocation (C i ) n i=1 ∈ IR kn + is a P.O. of E 1 if and only if it is a P.O. of an economy in which agents have vNM utility index U i , i = 1, . . . , n and identical probability over the set of states of the world. In particular, P.O. are comonotone. Proof: Since the P.O. of an economy with vNM agents with same probability are independent of the probability, we can choose w.l.o.g. this probability to be π ∈ D ν (w). Let (C i ) n i=1 be a P.O. of the vNM economy. Being a P.O., this allocation is comonotone. By proposition 2.2, C i is a non-decreasing function of w. Hence, applying lemma 3.1, v i (C i ) = E π [U i (C i )], i = 1, . . . , n. If it were not a P.O. of E 1 , there would exist an allocation (C ′ 1 , C ′ 2 . . . C ′ n ) such that v i (C ′ i ) = E ν [U i (C ′ i )] ≥ v i (C i ) = E π [U i (C i )] for all i, and with at least one strict inequality. Since E π [U i (C ′ i )] ≥ E ν [U i (C ′ i )] for all i, this contradicts the fact that (C i ) n i=1 is a P.O. of ′ i ) n i=1 such that E π [U i (C ′ i )] ≥ E π [U i (C i )] ≥ v i (C i ) for all i, and with a strict inequality for at least an agent. (C ′ i ) n i=1 being Pareto optimal, it is comonotone and it follows by proposition 2.2 that C ′ i is a non-decreasing function of w. Hence, applying lemma 3.1, v i (C ′ i ) = E π [U i (C ′ i )], i = 1, . . . , n. This contradicts the fact that (C i ) n i=1 is a P.O. of E 1 . Note that this proposition not only shows that P.O. allocations are comonotone in the CEU economy, but also completely characterizes them. We may now also fully characterize the equilibria of E 1 . Proposition 3.2 (i) Let (p ⋆ , C ⋆ ) be an equilibrium of a vNM economy in which all agents have utility index U i and beliefs given by π ∈ D ν (w), then (p ⋆ , C ⋆ ) is an equilibrium of E 1 . (ii) Conversely, assume U1. If (p ⋆ , C ⋆ ) is an equilibrium of E 1 , then there exists π ∈ D ν (w) such that (p ⋆ , C ⋆ ) is an equilibrium of the vNM economy with utility index U i and probability π ∈ D ν (w). Proof: See [START_REF] Dana | Pricing rules when agents have non-additive expected utility and homogeneous expectations[END_REF]. Corollary 3.1 If U1 is fulfilled and w 1 < w 2 < . . . < w k , then the equilibria of E 1 are identical to those of a vNM economy in which agents have utility index U i , i = 1, . . . , n and same probabilities over states π j = ν{j, j + 1, . . . , k} -ν{j + 1, . . . , k}, j < k and π k = ν({k}). Hence, (w, C ⋆ i , i = 1, ..., n) are comonotone. To conclude this sub-section, observe that P.O. allocations in the CEU economy inherits all the nice properties of P.O. allocations in a vNM economy with identical beliefs. In particular, P.O. allocations are independent of the capacity. However, the equilibrium allocations in the vNM economy do depend on beliefs, and it is not trivial to assess the relationship between the equilibrium set of a vNM economy with identical beliefs and the equilibrium set of the CEU economy E 1 . Note for instance that E 1 has "as many equilibria" as there are probability distributions in the set D ν (w). If D ν (w) consists of a unique probability distribution, equilibria of E 1 are the equilibria of the vNM economy with beliefs equal to that probability distribution. On the other hand, if D ν (w) is not a singleton, it is a priori not possible to assimilate all the equilibria of E 1 with equilibria of a given vNM economy. Optimal risk-sharing and equilibrium with different capacities We next consider an economy in which agents have different convex capacities. Denote the economy in which agents are CEU with capacity ν i and utility index U i , i = 1, . . . , n by E 2 . We first give a general characterization of the set of P.O., when no further restrictions are imposed on the economy. We then show that this general characterization can be most usefully applied when one knows that P.O. are comonotone. Proposition 3.3 (i) Let (C i ) n i=1 ∈ IR kn + be a P.O. of E 2 such that for all i, C j i = C ℓ i , j = ℓ. Let π i ∈ core(ν i ) be such that E ν i U i (C i ) = E π i U i (C i ) for all i. Then (C i ) n i=1 is a P.O. of an economy in which agents have vNM utility index U i and probabilities π i , i = 1, . . . , n. (ii) Let π i ∈ core(ν i ), i = 1, . . . , n and (C i ) n i=1 be a P.O. of the vNM economy with utility index U i and probabilities π i , i = 1, . . . , n. If E ν i U i (C i ) = E π i U i (C i ) for all i, then (C i ) n i=1 is a P.O. of E 2 . Proof: (i) If (C i ) n i=1 is not a P.O. of a vNM economy, then there exists (C ′ i ) n i=1 such that E π i U i (C ′ i ) ≥ E ν i U i (C i ) with a strict inequality for some i. Since t i C ′ i + (1 -t i )C i C i , ∀t i ∈ [0, 1] , by choosing t i small, one may assume w.l.o.g. that C ′ i is ranked in the same order as C i . Hence, E π i U i (C ′ i ) = E ν i U i (C ′ i ) for all i, which contradicts the fact that (C i ) n i=1 is a P.O. of E 2 . (ii) Assume there exists a feasible allocation ( C ′ i ) n i=1 such that E ν i U i (C ′ i ) ≥ E ν i U i (C i ) with a strict inequality for at least some i. Then, E π i U i (C ′ i ) ≥ E π i U i (C i ) with a strict inequality for at least some i, which leads to a contradiction. We now illustrate the implications of this proposition on a simple example. Example 3.1 Consider an economy with two agents, two states and one good, that thus can be represented in an Edgeworth box. Divide the latter into three zones : • zone (1), where C 1 1 > C 2 1 and C 1 2 < C 2 2 • zone (2), where C 1 1 < C 2 1 and C 1 2 < C 2 2 • zone (3), where C 1 1 < C 2 1 and C 1 2 > C 2 2 In zone (1), everything is as if agent 1 had probability (ν 1 1 , 1ν 1 1 ) and agent 2, probability (1ν 2 2 , ν 2 2 ). In zone (2), agent 1 uses (1ν 2 1 , ν 2 1 ) and agent 2, (1ν 2 2 , ν 2 2 ), while in zone (3), agent 1 uses (1ν 2 1 , ν 2 1 ) and agent 2, (ν 1 2 , 1ν 1 2 ). In order to use (ii) of proposition 3.3, we draw the three contract curves, corresponding to the P.O. in the vNM economies in which agents have the same utility index U i and the three possible couples of probability. Label (a), (b) and (c) these curves. One notices that curve (a), which is the P.O. of the vNM economy for agents having beliefs (ν 1 1 , 1ν 1 1 ) and (1ν 2 2 , ν 2 2 ) respectively, does not intersect zone (1), which is the zone where CEU agents do use these probability distributions as well. Hence, no points are at the same time P.O. of that vNM economy and such that E ν i [U i (C i )] = E π i [U i (C i )], i = 1, 2. a c b (3) (1) (2) ✻ ✲ ❄ ✛ C 1 1 C 2 2 C 1 2 C 2 1 1 2 ✻ ✲ ❄ ✛ C 1 1 C 2 2 C 1 2 C 2 1 1 2 contained in zone (2) . That part constitutes a subset of the set of P.O. that we are looking for. We will show later on that, in order to get the full set of P.O. of the CEU economy, one has to replace the part of curve (b) that lies in zone (3) by the segment along the diagonal of agent 2. ♦ It follows from proposition 3.3 that, without any knowledge on the set of P.O., one has to compute the P.O. of (k!) n -1 economies (if there are k! extremal points in core(ν i ) for all i). Thus, the actual characterization of the set of P.O. of E 2 might be somewhat tedious without further information. In the comonotone case however, the characterization of P.O. is simpler, even though it remains partial. Corollary 3.2 Assume w 1 ≤ w 2 ≤ . . . ≤ w k . (i) Let U1 hold and (C i ) n i=1 ∈ IR kn + be a comonotone P.O. of E 2 such that C 1 i < C 2 i < . . . < C k i for all i = 1, . . . , n. Then, (C i ) n i=1 is a P.O. allocation of the economy in which agents are vNM maximizers with utility index U i and probability π j i = ν i ({j, . . . , k})ν i ({j + 1, . . . , k}) for j < k and π k i = ν i ({k}). (ii) Let (C i ) n i=1 ∈ IR kn + be a P.O. of the economy in which agents are vNM maximizers with utility index U i and probability π j i = ν i ({j, . . . , k})ν i ({j + 1, . . . , k}) for j < k and π k i = ν i ({k}). If (C i ) n i=1 is comonotone, then it is a P.O. of E 2 . These results may now be used for equilibrium analysis as follows. Proposition 3.4 Assume w 1 ≤ w 2 ≤ . . . ≤ w k . (i) Let (p ⋆ , C ⋆ ) be an equilibrium of E 2 . If 0 < C ⋆1 i < . . . < C ⋆k i for all i, then (p ⋆ , C ⋆ ) is an equilibrium of the economy in which agents are vNM maximizers with utility index U i and probability π j i = ν i ({j, . . . , k})ν i ({j + 1, . . . , k}) for j < k and π k i = ν i ({k}). (ii) Let (p ⋆ , C ⋆ ) be an equilibrium of the economy in which agents are vNM maximizers with utility index U i and probability π j i = ν i ({j, . . . , k})ν i ({j + 1, . . . , k}) for j < k and π k i = ν i ({k}). If C ⋆ is comonotone, then (p ⋆ , C ⋆ ) is an equilibrium of E 2 . Proof: (i) Since (p ⋆ , C ⋆ ) is an equilibrium of E 2 , and since v i is differentiable at C ⋆ i for every i, there exists a multiplier λ i such that p ⋆ = λ i U ′ i (C ⋆1 i )π 1 i , . . . , U ′ i (C ⋆k i )π k i , where π j i = ν i ({j, . . . , k})ν i ({j + 1, . . . , k}) for j < k and π k i = ν i ({k}) for all i. Hence, (p ⋆ , C ⋆ ) is an equilibrium of the economy in which agents are vNM maximizers with probability π j i for all i, j. (ii) Let (p ⋆ , C ⋆ ) be an equilibrium of the economy in which agents are vNM maximizers with probability π j i for all i, j. Assume C ⋆ is comonotone. We thus have p ⋆ C ′ i ≤ p ⋆ w i ⇒ E π i U i (C ′ i ) ≤ E π i U i (C ⋆ i ) Since E ν i U i (C ′ i ) ≤ E π i U i (C ′ i ) and E ν i U i (C ⋆ i ) = E π i U i (C ⋆ i ) , we get E ν i U i (C ′ i ) ≤ E ν i U i (C ⋆ i ) for all i, which implies that (C ⋆ , p ⋆ ) is an equi- librium of E 2 . Observe that, even though the characterization of P.O. allocations is made simpler when we know that these allocations are comonotone, the above proposition does not give a complete characterization. comonotonicity of the P.O. allocations is also useful for equilibrium analysis. This leads us to look for conditions on the economy under which P.O. are comonotone. Optimal risk-sharing and equilibrium in some particular cases In this section, we focus on two particular cases in which we can prove directly that P.O. allocations are comonotone. Convex transform of a probability distribution In this sub-section, we show how one can use the previous results when agents' capacities are the convex transform of a given probability distribution. In this case, one can directly apply corollary 3.2 and proposition 3.4 to get a characterization of P.O. and equilibrium. Let π = (π 1 , . . . , π k ) be a probability distribution on S, with π j > 0 for all j. Proposition 4.1 Assume w 1 ≤ w 2 ≤ . . . ≤ w k . Assume that, for all i, U i is differentiable and ν i = f i • π, where f i is a strictly increasing and convex function from [0, 1] to [0, 1] with f i (0) = 0, f i (1) = 1. Then, at a P.O., C 1 i ≤ C 2 i ≤ . . . ≤ C k i for all i. Proof: Since U i is differentiable, strictly increasing and strictly concave, and f i is a strictly increasing, convex function for all i, it results from corollary 2 in Chew, [START_REF] Chew | Risk aversion in the theory of expected utility with rank dependent preferences[END_REF] that every agent strictly respects second order stochastic dominance. Therefore it remains to show that if every agent strictly respects second order stochastic dominance, then, at a P.O., C 1 i ≤ C 2 i ≤ . . . ≤ C k i for all i. We do so using proposition 4.1 in [START_REF] Chateauneuf | A review of some results related to comonotonicity[END_REF]. Assume (C i ) n i=1 is not comonotone. W.l.o.g., assume that andC ′ 2 be determined by the feasibility condition C 1 1 > C 2 1 , C 1 2 < C 2 2 , and C 1 1 + C 1 2 ≤ C 2 1 + C 2 2 . Let C ′ be such that: C 1′ 1 = C 2′ 1 = π 1 C 1 1 + π 2 C 2 1 π 1 + π 2 and C j′ 1 = C j 1 , j > 2 Let C ′ i = C i for all i > 2, C 1 + C 2 = C ′ 1 + C ′ 2 . Hence, C 1′ 2 = C 1 2 + π 2 π 1 + π 2 (C 1 1 -C 2 1 ), C 2′ 2 = C 2 2 - π 1 π 1 + π 2 (C 1 1 -C 2 1 ) and C j′ 2 = C j 2 , j > 2 It may easily be checked that C 2 1 < C 1′ 1 = C 2′ 1 < C 1 1 , and C 1 2 < C 1′ 2 ≤ C 2′ 2 < C 2 2 . Furthermore, π 1 C 1′ 1 + π 2 C 2′ 1 = π 1 C 1 1 + π 2 C 2 1 , and π 1 C 1′ 2 + π 2 C 2′ 2 = π 1 C 1 2 + π 2 C 2 2 . Therefore, C ′ i i = 1 , 2 is a strictly less risky allocation than C i i = 1, 2, with respect to mean preserving increases in risk. It follows that agents 1 and 2 are strictly better off with C ′ , while other agents' utilities are unaffected. Hence, C ′ Pareto dominates C. Thus, any P.O. C must be comonotone, i.e., C 1 i ≤ C 2 i ≤ . . . ≤ C k i for all i. Using corollary 3.2, we can then provide a partial characterization of the set of P.O. Note that such a characterization was not provided by the analysis in [START_REF] Chateauneuf | A review of some results related to comonotonicity[END_REF] or [START_REF] Landsberger | Co-monotone allocations, Bickel-Lehmann dispersion and the Arrow-Pratt measure of risk aversion[END_REF]. Proposition 4.2 Assume w 1 ≤ . . . ≤ w k and that agents are CEU maximizers with ν i = f i • π, f i convex, strictly increasing and such that f i (0) = 0 and f i (1) = 1. Then, (i) Let (C i ) n i=1 ∈ IR kn + be a P.O. of this economy such that C 1 i < C 2 i < . . . < C k i for all i = 1, . . . , n. Then, (C i ) n i=1 is a P.O. allocation of the economy in which agents are vNM maximizers with utility index U i and probability π j i = f i k s=j π s -f i k s=j+1 π s for j = 1, . . . , k -1, and π k i = f i π k . (ii) Let (C i ) n i=1 ∈ IR kn + be a P.O. of the economy in which agents are vNM maximizers with utility index U i and probability π j i = f i k s=j π s -f i k s=j+1 π s for j = 1, . . . , k -1, and π k i = f i π k . If (C i ) n i=1 is comonotone, then it is a P.O. of the CEU economy with ν i = f i • π. Proof: See corollary 3.2. The same type of result can be deduced for equilibrium analysis from proposition 3.4, and we omit its formal statement here. The previous characterization formally includes the Rank-Dependent-Expected-Utility model introduced by [START_REF] Quiggin | A theory of anticipated utility[END_REF] in the case of (probabilized) risk. It also applies to so-called "simple capacities" (see e.g. [START_REF] Dow | Uncertainty aversion, risk aversion, and the optimal choice of portfolio[END_REF]), which are particularly easy to deal with in applications. Indeed, let agents have the following simple capacities: ν i (A) = (1ξ i )π(A) for all A ∈ A, A = S, and ν i (S) = 1, where π is a given probability measure with 0 < π j < 1 for all j, and 0 ≤ ξ < 1. These capacities can be written ν i = f i •π where f i is such that f i (0) = 0, f i (1) = 1, is strictly increasing, continuous and convex, with: f i (p) = (1 -ξ i )p if 0 ≤ p ≤ max {π(A)<1} π(A) f i (1) = 1 Hence, ν i is a convex transformation of π, and we can apply the results of this sub-section to characterize the set of P.O. in an economy where all agents have such simple capacities. The two-state case We restrict our attention here to the case S = {1, 2}. Agent i has a capacity ν i characterized by two numbers ν i ({1}), ν i ({2}) such that ν i ({1}) ≤ 1ν i ({2}). To simplify notation, we'll denote ν i ({s}) = ν s i . In this particular case, core (ν i ) = {(π, 1 -π) | π ∈ [ν 1 i , 1 -ν 2 i ]}. Call E 3 the two-state exchange economy in which agents are CEU maximizers with capacity ν i and utility index U i , i = 1, . . . , n. Assumption C: ∩ i core(ν i ) = ∅ This assumption is equivalent to ν 1 i + ν 2 j ≤ 1, i, j = 1, . . . , n, or stated differently, to ∩ i [ν 1 i , 1 -ν 2 i ] = ∅. Recall that in the two-state case, under C, agents' capacities are convex. We now proceed to show that this "minimal consensus" assumption is enough to show that P.O. are comonotone. Proposition 4.3 Let C hold. Then, P.O. are comonotone. Proof: Assume w 1 ≤ w 2 and C not comonotone. W.l.o.g., assume that C 1 1 > C 2 1 , C 1 2 < C 2 2 . Let (π, 1 -π) ∈ ∩ i core(ν i ) and C ′ be the feasible allocation defined by C 1′ 1 = C 2′ 1 = πC 1 1 + (1 -π)C 2 1 and C 1′ 2 and C 2′ 2 are such that C j′ 1 + C j′ 2 = C j 1 + C j 2 , j = 1, 2, i.e. C 1′ 2 = C 1 2 + (1 -π)(C 1 1 -C 2 1 ), C 2′ 2 = C 2 2 -π(C 1 1 -C 2 1 ) One obviously has C 1 2 < C 1′ 2 ≤ C 2′ 2 < C 2 2 . Finally, let C j′ i = C j i , ∀i > 2, j = 1, 2. We now prove that C ′ Pareto dominates C. v 1 (C ′ 1 ) -v 1 (C 1 ) = U 1 (πC 1 1 + (1 -π)C 2 1 ) -ν 1 1 U 1 (C 1 1 ) -(1 -ν 1 1 )U 1 (C 2 1 ) > (π -ν 1 1 ) U 1 (C 1 1 ) -U 1 (C 2 1 ) ≥ 0 since U 1 is strictly concave and π ≥ ν 1 i . Now, consider agent 2's utility: v 2 (C ′ 2 ) -v 2 (C 2 ) = (1 -ν 2 2 ) U 2 (C 1′ 2 ) -U 2 (C 1 2 ) + ν 2 2 U 2 (C 2′ 2 ) -U 2 (C 2 2 ) Since U 2 is strictly concave and C 1 2 < C 1′ 2 ≤ C 2′ 2 < C 2 2 , we have: U 2 (C 1′ 2 ) -U 2 (C 1 2 ) C 1′ 2 -C 1 2 > U 2 (C 2 2 ) -U 2 (C 2′ 2 ) C 2 2 -C 2′ 2 and hence, U 2 (C 1′ 2 ) -U 2 (C 1 2 ) 1 -π > U 2 (C 2 2 ) -U 2 (C 2′ 2 ) π . Therefore, v 2 (C ′ 2 ) -v 2 (C 2 ) > (1 -ν 2 2 ) 1 -π π -ν 2 2 U 2 (C 2 2 ) -U 2 (C 2′ 2 ) ≥ 0 since (1 -ν 2 2 )(1 -π) -πν 2 2 = 1 -ν 2 2 -π ≥ 0 and U 2 (C 2 2 ) -U 2 (C 2′ 2 ) > 0. Hence, C ′ Pareto dominates C. Remark: If ν 1 i + ν 2 j < 1, i, j = 1, . . . , n , which is equivalent to the assumption that ∩ i core(ν i ) contains more than one element, then one can extend proposition 4.3 to linear utilities. Remark: Although convex capacities can, in the two-state case, be expressed as simple capacities, the analysis of sub-section 4.1 (and in particular proposition 4.1), cannot be used here. Indeed, assumption C does not require that agents' capacities are all a convex transform of the same probability distribution as example 4.1 shows. Example 4.1 There are two agents with capacity ν 1 1 = 1/3, ν 2 1 = 2/3, and ν 1 2 = 1/6, ν 2 2 = 2/3 respectively. Assumption C is satisfied since π = (1/3, 2/3) is in the intersection of the cores. The only way ν 1 and ν 2 could be a convex transform of the same probability distribution is ν 1 = π and ν 2 = f 2 • π with f 2 (1/3) = 1/6 and f 2 (2/3) = 2/3. But f 2 then fails to be convex. ♦ Intuition derived from proposition 4.3 might suggest that some minimal consensus assumption might be enough to prove comonotonicity of the P.O. However, that intuition is not valid in general, as can be seen in the following example, in which the intersection of the cores of the capacities is non-empty, but where (some) P.O. allocations are not comonotone. Example 4.2 There are two agents, with the same utility index U i (C) = 2C 1/2 , but different beliefs. The latter are represented by two convex capacities defined as follows: ν 1 ({1}) = 3 9 ν 1 ({2}) = 3 9 ν 1 ({3}) = 1 9 ν 1 ({1, 2}) = 6 9 ν 1 ({1, 3}) = 6 9 ν 1 ({2, 3}) = 4 The intersection of the cores of these two capacities is non-empty since the probability defined by π j = 1/3, j = 1, 2, 3 belongs to both cores. The endowment in each state is respectively w 1 = 1, w 2 = 12, and w 3 = 13. We consider the optimal allocation associated to the weights (1/2, 1/2) and show it cannot be comonotone. In order to do that, we show that the maximum of v 1 (C 1 ) + v 2 (C 2 ) subject to the constraints C j 1 + C j 2 = w j , j = 1, 2, 3 and C j i ≥ 0 for all i and j, does not obtain for C 1 i ≤ C 2 i ≤ C 3 i , i = 1, 2. Observe first that if C 1 i ≤ C 2 i ≤ C 3 i , i = 1, 2, then: v 1 (C 1 )+v 2 (C 2 ) = 2 5 9 C 1 1 + 3 9 C 2 1 + 1 9 C 3 1 + 4 9 C 1 2 + 2 9 C 2 2 + 3 9 C 3 2 Call g(C 1 1 , C 2 1 , C 3 1 , C 1 2 , C 2 2 , C 3 2 ) the above expression. Note that v 1 (C 1 ) + v 2 (C 2 ) takes the exact same form if C 1 1 < C 3 1 < C 2 1 and C 1 2 < C 2 2 < C 3 2 . The optimal solution to the maximization problem: max g(C 1 1 , C 2 1 , C 3 1 , C 1 2 , C 2 2 , C 3 2 ) s.t. C j 1 + C j 2 = w j j = 1, 2, 3 C j i ≥ 0 j = 1, 2, 3 i = 1, 2 is C 1 1 , C 2 1 , C 0 < C 1 1 < C 3 1 < C 2 1 and 0 < C 1 2 < C 2 2 < C 3 2 . Therefore: v 1 ( C 1 )+v 2 ( C 2 ) > v 1 (C 1 )+v 2 (C 2 ) for all C such that C 1 i ≤ C 2 i ≤ C 3 i , i = 1, 2 and hence the P.O. associated to equal weights for each agent is not comonotone.♦ One may expect that it follows from proposition 4.3 that P.O. of E 3 are the P.O. of the vNM economy in which agents have probability π i = 1-ν 2 i , i = 1, . . . , n. However, it is not so, since as recalled in sub-section 2.4, P.O. of a vNM economy with different beliefs are not in general comonotone. We can nevertheless use proposition 3.3 to provide a partial characterization of the set of P.O. In this particular case of only two states, we can obtain a full characterization of the set of P.O. if there are only two agents in the economy. This should then be interpreted as a characterization of the optimal risk-sharing arrangement among two parties to a contract (arrangement that has been widely studied in the vNM case). Figure 2: ✻ ✲ ❄ ✛ C 1 1 C 2 2 C 1 2 C 2 1 1 2 (a) ✻ ✲ ❄ ✛ C 1 1 C 2 2 C 1 2 C 2 1 1 2 (b) 3 (a), the thin line represents P.O. of the vNM economy in which agent i uses probability (1ν 2 i , ν 2 i ) that are not P.O. of the CEU economy, as they are not comonotone. If agent 2 has a utility index of the DARA type, it is easy to show that the set of P.O. of the economy in which agents have utility index U i and probability (1ν 2 i , ν 2 i ) crosses agent 1's diagonal at most once, hence preventing the kind of situation represented on figure 3. When there are only two (types of) agents, we can also go further in the characterization of the set of equilibria. Proposition 4.5 Assume k = 2, n = 2, w 1 ≤ w 2 , C and U1 hold and ν 2 1 < ν 2 2 . Let (p ⋆ , C ⋆ ) be an equilibrium of E. Then there are only two cases: (i) Either C 1⋆ i < C 2⋆ i , i = 1, 2 and (p ⋆ , C ⋆ ) is an equilibrium of a vNM economy in which agents have utility index U i and beliefs given by π i = 1 -ν 2 i , i = 1, 2. (ii) Or, C 1⋆ 1 = C 2⋆ 1 = C ⋆⋆ and C ⋆⋆ satisfies the following : (a) (1 -ν 2 2 )(w 1 1 -C ⋆⋆ )U ′ 2 (w 1 -C ⋆⋆ ) + ν 2 2 (w 2 1 -C ⋆⋆ )U ′ 2 (w 2 -C ⋆⋆ ) = 0 (b) ν 2 1 1-ν 2 1 ≤ -(w 1 1 -C ⋆⋆ ) (w 2 1 -C ⋆⋆ ) Proof: It follows from proposition 4.4 that either C 1⋆ i < C 2⋆ i , i = 1, 2 or C 1⋆ 1 = C 2⋆ 1 = C ⋆⋆ . The first case follows from proposition 3.4. In the Figure 3: ✻ ✲ ❄ ✛ C 1 1 C 2 2 C 1 2 C 2 1 1 2 (a) ✻ ✲ ❄ ✛ C 1 1 C 2 2 C 1 2 C 2 1 1 2 (b) ❛ w ❵ ❵ ❵ ❵ ❵ ❇ ❇ ❇ ❇ ❇ ❇ second case, the P.O. allocation is supported by the price (1 -ν 2 2 )U ′ 2 (w 1 - C ⋆⋆ ), ν 2 2 U ′ 2 (w 2 -C ⋆⋆ ) , hence the tangent to agent two's indifference curve at (w 1 -C ⋆⋆ , w 2 -C ⋆⋆ ) has the following equation in the (C 1 , C 2 ) plane: (1 -ν 2 2 )U ′ 2 (w 1 -C ⋆⋆ )(C 1 -C ⋆⋆ ) + ν 2 2 U ′ 2 (w 2 -C ⋆⋆ )(C 2 -C ⋆⋆ ) = 0 Now C ⋆ is an equilibrium allocation iff (w 1 1 , w 2 1 ) fulfills that equation. Condition (b) follows from condition (a) and condition (ii) from proposition 4.4. There might therefore exist, for a range of initial endowments, equilibria at which agent 1 is perfectly insured even though agent 2 has strictly convex preferences. Observe also that nothing excludes a priori the possibility of having different kind of equilibria for the same initial endowment (see figure 3 (b)). Optimal risk-sharing and equilibrium without aggregate risk We now turn to the study of economies without aggregate uncertainty5 . This corresponds to the case of individual risk first analyzed by [START_REF] Malinvaud | The allocation of individual risks in large markets[END_REF] and [1973]. A particular case is the one of a sunspot economy, in which uncertainty is purely extrinsic and does not affect the fundamentals, i.e., each agent's endowment is independent of the state of the world (see [START_REF] Tallon | Do sunspots matter when agents are Choquet-expectedutility maximizers[END_REF] for a study of sunspot economies with CEU agents). Our analysis of the case of purely individual risk might also yield further insights as to which type of financial contracting (e.g. mutual insurance rather than trade on Arrow securities defined on individual states) is necessary in such economies to decentralize an optimal allocation. It turns out that the economy under consideration possesses remarkable properties: P.O. are comonotone and coincide with full insurance allocations, under the relatively weak condition C. Furthermore, this condition, which is weaker than convexity of preferences, is enough to prove existence of an equilibrium. Finally, the case of purely individual risk lends itself to the introduction of several goods. We thus move to an economy with m goods, indexed by subscript ℓ. C j iℓ is the consumption of good ℓ by agent i in state j. We have, C j i = (C j i1 , . . . , C j im ), and C i = (C 1 i , . . . , C k i ). If C j i = C j ′ i for all j, j ′ , then C i will denote both this constant bundle (i.e. C j i ≡ C i ) and the vector composed of k such vectors, the context making it clear which meaning is intended. Let p j ℓ be the price of good ℓ available in state j, p j = (p j 1 , . . . , p j m ), and p = (p 1 , . . . , p k ). The utility index U i is now defined on IR m + , and is still assumed to be strictly concave and strictly increasing. We will also need a generalization of assumption U1 to the multi-good case, that ensures that the solution to the agent's maximization program is interior. Assumption Um: ∀i, {x ′ ∈ IR m + | U i (x ′ ) ≥ U i (x)} ⊂ IR m ++ , ∀x ∈ IR m ++ . Aggregate endowment is the same across states, although its distribution among households might differ in each state. Therefore, we consider a pure exchange economy E 4 with n agents and m goods described by the list: E 4 = v i : IR km + → IR, w i ∈ IR km + , i = 1, . . . , n . We will denote aggregate endowments w, i.e., w = i w i . Before dealing with non-additive beliefs, we first recall some known results in the vNM case. Proposition 5.1 Assume all agents have identical vNM beliefs, π = (π 1 , . . . , π k ). Then, (i) At a Pareto optimum, C j i = C j ′ i for all i, j, j ′ and C 1 i n i=1 is a P.O. of the static economy (U i , wi = E π (w i ) , i = 1, . . . , n). (ii) Let Um hold. (p ⋆ , C ⋆ ) is an equilibrium of the vNM economy if and only if there exists q ⋆ ∈ IR m + such that p ⋆ = q ⋆ π 1 , q ⋆ π 2 , . . . , q ⋆ π k and (q ⋆ , C ⋆ ) is an equilibrium of the static economy (U i , wi = E π (w i ) , i = 1, . . . , n). It is also easy to see that if agents' beliefs are different, agents will consume state-dependent bundles at an optimum. We now examine to what extent these results, obtained in the vNM case, generalize to the CEU setup, assuming that condition C holds. Using proposition 2.1 this assumption is equivalent6 to P = ∅, where P =    π ∈ IR k ++ | k j=1 π j = 1 and v i (C 1 i , . . . , C k i ) ≤ E π (U i (C i )) , ∀ i, ∀ C i    Recall that assumption C was not enough to prove comonotonicity of P.O. in the general case (though it was sufficient in the two-state case). We now proceed to fully characterize the set of P.O. Proof: (i) Assume, to the contrary, that there exist an agent (say agent 1) and states j and j ′ such that C j 1 = C j ′ 1 . Let Cj i ≡ Ci = E π (C i ) for all j and all i where π ∈ P. This allocation is feasible: i Ci = E π ( i C i ) = w. By definition of P, v i C 1 i , . . . , C k i ≤ E π (U i (C i )) for all i. Now, E π (U i (C i )) ≤ U i (E π (C i )) = U i Ci = v i C1 i , . . . , Ck i for all i, since U i is concave. This last inequality is strict for agent 1, since C j 1 = C j ′ 1 , π ≫ 0, and U 1 is strictly concave. Therefore, v i C 1 i , . . . , C k i ≤ v i C1 1 , . . . , Ck i for all i, with a strict inequality for agent one, a contradiction to the fact that (C i ) n i=1 is an optimum of E 4 . (ii) We skip the proof for this part of the proposition for it relies on the same type of argument as that of proposition 3.3. Thus, even with different "beliefs" (in the sense of different capacities), agents might still find it optimal to fully insure themselves : differences in beliefs do not necessarily lead agents to optimally bear some risk as in the vNM case. We now proceed to study the equilibrium set. Proposition 5.3 Let C hold. (i) Let (p ⋆ , C ⋆ ) be an equilibrium of a vNM economy in which all agents have utility index U i and beliefs given by π ∈ P, then (p ⋆ , C ⋆ ) is an equilibrium of E 4 . (ii) Conversely, assume ν i is convex and U i satisfies Um for all i. Let (p ⋆ , C ⋆ ) be an equilibrium of E 4 , then there exists π ∈ P such that (p ⋆ , C ⋆ ) is an equilibrium of the vNM economy in which all agents have utility index U i and probability π. Furthermore, p ⋆ = q ⋆ π 1 , q ⋆ π 2 , . . . , q ⋆ π k with π ∈ P and q ⋆ = λ i ∇U i (C ⋆ i ), λ i ∈ IR + for all i. Proof: (i) Let (p ⋆ , C ⋆ ) be an equilibrium of a vNM economy in which all agents have beliefs given by π. We have, C j⋆ i = C j ′ ⋆ i for all j, j ′ and all i. By definition of an equilibrium, i C j⋆ i = i w j i for all j and, for all i: C ′ i ≥ 0, p ⋆ C ′ i ≤ p ⋆ w i ⇒ E π (U i (C ′ i )) ≤ E π (U i (C ⋆ i )) Now, since π ∈ P, v i (C ′ i ) ≤ E π (U i (C ′ i )). Notice that v i (C ⋆ i ) = E π (U i (C ⋆ i )). Hence, v i (C ′ i ) ≤ v i (C ⋆ i ) , and (p ⋆ , C ⋆ ) is an equilibrium of E 4 . (ii) Let (p ⋆ , C ⋆ ) be an equilibrium of E 4 . Assume ν i is convex and U i satisfy Um for all i. Then C ⋆ i ≫ 0 for all i. From proposition 5.2 and the first theorem of welfare, C j⋆ i = C j ′ ⋆ i for all j, j ′ and all i. From first-order conditions and Um, there exists λ i ∈ IR + for all i, such that p ⋆ ∈ λ i ∂v i (C ⋆ i , . . . , C ⋆ i ). Therefore, p ⋆ = λ i ∇U i (C ⋆ i ) π 1 i , . . . , λ i ∇U i (C ⋆ i ) π k i where π i ∈ core(ν i ). Summing over p ⋆ 's components, one gets: j λ i ∇U i (C ⋆ i ) π j i = j p j⋆ = j λ i ′ ∇U i ′ (C ⋆ i ′ ) π j i ′ that is: λ i ∇U i (C ⋆ i ) = λ i ′ ∇U i ′ (C ⋆ i ′ ) ∀ i, i ′ Hence, π j i is independent of i for all j, i.e. π i ∈ ∩ i core(ν i ) = P. Let q ⋆ = λ i ∇U i (C ⋆ i ) and π = π i . One gets p ⋆ = q ⋆ π 1 , q ⋆ π 2 , . . . , q ⋆ π k with π ∈ P. It follows from proposition 5.1 that (p ⋆ , C ⋆ ) is an equilibrium of the vNM economy with utility index U i and probability π. This proposition suggests equilibrium indeterminacy if P contains more than one probability distribution. This road is explored further in Tallon [1997] and [START_REF] Dana | Pricing rules when agents have non-additive expected utility and homogeneous expectations[END_REF]. A direct corollary concerns existence: Corollary 5.1 Under C, there exists an equilibrium of E 4 . Hence, since capacities satisfying assumption C need not be convex, convexity of the preferences (which is equivalent, in the CEU setup, to the convexity of the capacity and concavity of the utility index, see [START_REF] Chateauneuf | Diversification, convex preferences and non-empty core[END_REF]) is not necessary to prove that an equilibrium exists in this setup. [START_REF] Malinvaud | Markets for an exchange economy with individual risks[END_REF] noticed that P.O. allocations could be decentralized through insurance contract in a large economy. [START_REF] Cass | Individual risk and mutual insurance[END_REF] showed, in an expected utility framework, how this decentralization can be done in a finite economy: agents of the same type share their risk through (actuarially fair) mutual insurance contract. The same type of argument could be used here in the Choquet expected utility case. It is an open issue whether P.O. allocations can be decentralized with mutual insurance contract where agents in the same pool have different capacities. The same is true for curve (c) and zone (3). On the other hand, part of curve (b) is Figure 1: b 4 if and only if it is a P.O. of a vNM economy with utility index U i and identical probability over the set of states of the world. Hence, P.O. are independent of the capacities. Proposition 5.2 Let C hold. Then, (i) Any P.O. (C i ) n i=1 of E 4 is such that C j i is independent of j for all i and C 1 i n i=1 is a P.O. of the static economy in which agents have utility function (U i ) n i=1 . (ii) The allocation (C i ) n i=1 ∈ IR kmn + is a P.O. of E See Schmeidler [1989],[START_REF] Ghirardato | Coping with ignorance: unforeseen contingencies and nonadditive uncertainty[END_REF],[START_REF] Mukerji | Understanding the nonadditive probability decision model[END_REF]. In section 5, we will deal with several goods and will introduce the appropriate notation at that time. It is well-known (see[START_REF] Schmeidler | Integral representation without additivity[END_REF]) that when ν is convex, the Choquet integral of any random variable f is given by f dν = min π∈core(ν) Eπf . [START_REF] Borch | Equilibrium in a reinsurance market[END_REF] noted that, in a reinsurance market, at a P.O., "the amount which company i has to pay will depend only on (...) the total amount of claims made against the industry. Hence any Pareto optimal set of treaties is equivalent to a pool arrangement." Note that this corresponds to the characterization of comonotone variables as stated in proposition 2.2. For a study of optimal risk-sharing without aggregate uncertainty and an infinite state space, see[START_REF] Billot | Sharing beliefs: between agreeing and disagreeing[END_REF]. Recall that we are dealing with capacities such that 1 > ν(A) > 0 for all A = ∅ and A = S. Proposition 4.4 Assume n = 2, w 1 < w 2 and that agents have capacities ν i , i = 1, 2 which fulfill C, and such that ν 2 1 < ν 2 2 . Assume finally U1 and let (C i ) i=1,2 be a P.O. of E 3 . Then, there are only two cases: ,2 is a P.O. of the vNM economy with utility index U i and probabilities (1) Proof: It follows from proposition 4.3 that there are three cases : ; hence the right-hand side of (2) is fulfilled and (1) is equivalent to (2). Lastly, the case 0 < C 1 2 = C 2 2 is symmetric. The first-order corresponding conditions imply ν 2 1 ≥ ν 2 2 which contradicts our hypothesis. We can illustrate the optimal risk-sharing arrangement just derived in an Edgeworth box. Figure 2 (a) represents case (i) and the optimal contract is the same as the one in the associated vNM setup. However, figure 2 (b) gives a different risk-sharing rule, that can interpreted as follows. The assumption ν 2 1 < ν 2 2 is equivalent to E ν 1 (x) ≤ E ν 2 (x) for all x comonotone with w. But we have just shown that we could restrict our attention to allocations that are comonotone with w. Hence, the assumption ν 2 1 < ν 2 2 can be interpreted as a form of pessimism of agent 1. Under that assumption, agent 2 never insures himself completely, whereas agent 1 might insure do so. This is incompatible with a vNM economy with strictly concave and differentiable utility indices. Finally, risk-sharing arrangements such as the one represented on figure 3 (a) cannot be excluded a priori, i.e., there is no reason that the contract curve in the vNM economy crosses agent 1's diagonal only once. On figure

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https://hal.univ-lorraine.fr/tel-01748278/file/SCD_T_2007_0124_CATAK.pdf

Nh Saron Catak Gérald Monard Viktorya Aviyente Manuel F Ruiz-López I. DEAMIDATION IN PEPTIDES AND PROTEINS -BIOLOGICAL RELEVANCE Asparagine (Asn) and glutamine (Gln), two of the 20 amino acid residues that ordinarily occur in proteins, are inherently unstable under physiological solvent conditions. Asn and Gln were shown to be normal constituents of proteins [1, 2] and first studied with respect to deamidation [3] in 1932. The chemistry of the free amino acids was thoroughly understood by 1961 [4] and, by 1974 a partial understanding of peptide deamidation had been gained [5]. Biological Relevance The deamidation of Asn and Gln in peptides and proteins is of significant biological interest, because it can often produce substantial structural changes. At neutral pH, a negative charge is added to the molecule, and, in some cases, the resulting Asp is isomerized. It has been hypothesized that deamidation serves as a molecular clock for the timing of biological processes [12,13]. The timed processes of protein turnover, development, and aging have been suggested as possible roles for deamidation. The biological turnover rates of rat cytochrome c [14,15] and rabbit muscle aldolase [16,17] have been shown to be controlled by deamidation. Increased amounts of deamidated proteins have been found in some aged and diseased tissues, such as human eye lens cataracts [18] and Alzheimer's plaques [19]. About 1,700 research papers on various aspects of the deamidation of peptides and proteins have been published since the biological importance of deamidation was first emphasized [7,12]. This literature is, however, mostly fragmented into special studies of individual peptides and proteins in a wide variety of conditions. Most of this work has been hindered by the fact that experimental studies of protein deamidation with available techniques are laborious, time-consuming and lack a means of reliably estimating the instability of a particular peptide or protein with respect to non-enzymatic deamidation of the amide residues. Since the original suggestion in 1970 [12] that deamidation plays a positive biological role, especially as a molecular timer, some evidence has accumulated to support this hypothesis. It was found that deamidation rates could be varied over a wide range by changing primary sequence [5,20]. The distribution functions of naturally occurring sequences around amide residues were found to be non-random as were the amide compositions of proteins [12,21]. Specific roles for some deamidations were found [22] and, in two cases, it was shown that deamidation regulates the rate of protein turnover [14][15][16][17]. Studies of the occurrence of deamidation in a wide variety of proteins have been reported [23][24][25][26]. The question remains, however, as to whether or not deamidation is an interesting property of proteins that occasionally has biological usefulness, or is it of widespread biological importance. The estimated deamidation rates of proteins in the Brookhaven Protein Data Bank show that deamidation may be expected to occur in a substantial percentage of proteins under physiological conditions and within biologically interesting time intervals. These estimates agree well with the actual protein deamidation experiments that have been reported. Moreover, since instances are known wherein deamidation increases protein susceptibility to biological degradation, even more deamidation may be occurring than is ordinarily seen in biological preparations. Most Asn residues in proteins do not have fast deamidation rates. The amides that are most interesting biologically consist of only a few percent of the entire set. Therefore, biologically stable amides could easily be genetically provided. Unstable amides need not be present in proteins unless their instability is biologically functional. The change in charge that accompanies deamidation has a substantial effect on protein structure, as does the isomerization that sometimes occurs. There are many reports that these changes markedly affect protein function or stability or both. Amide residues genetically programmed for the proteins of biological systems would, therefore, be expected to be stable with respect to deamidation unless the shorter-lived amides have positive biological uses. To summarize, one can recall the conclusions from a recent work of Robinson [27]: "Proteins contain amide residue clocks. These residues are found in almost all proteins and amide residue clocks are found to be set to timed intervals of biological importance, even though settings to longer times are not only available, but also make up most of the genetically available settings. Deamidation changes protein structures in fundamentally important ways. If deamidation were not of pervasive and positive biological importance, these clocks would be set to time intervals that are long with respect to the lifetimes of living things The fact that they are found to be set instead to biologically relevant time intervals strongly supports the original hypothesis that amides play, through deamidation, a special biologically important role". Deamidation is obviously being used for some widespread and fundamental biological purpose. If this were not so, it would be genetically suppressed since it is otherwise very disruptive to protein structures and would not be tolerated in living systems. CHAPTER II OBJECTIVE and SCOPE II. OBJECTIVE and SCOPE The previous chapter was meant to provide a brief overview on deamidation in peptides and proteins and emphasize its biological importance. As mentioned earlier, mechanistic aspects of deamidation have been previously investigated both experimentally and theoretically, but many aspects have not been elucidated yet. The present study aims to get a deeper insight on the mechanisms leading to deamidation of asparaginyl residues using computational techniques to investigate the feasibility of alternative reaction mechanisms. Some of these mechanisms have been proposed in the literature. Others are put forward here for the first time. The following chapter (Chapter III -Theoretical Background) presents the basic principles of the theoretical approaches employed throughout the course of this study; pertinent methodological details as well as practical aspects of the computational methods utilized are presented. This study aims to tackle four different topics (Chapters IV -VII) related to deamidation of Asn in peptides and proteins. These chapters were intended to be full articles, hence they were written in a standalone, independent format, where each chapter includes a detailed introduction and methodology section. The first topic of interest (Chapter IV) is the effect of solvent molecules on deamidation mechanism, where the difference in mechanistic details and energetics of water-catalyzed deamidation is investigated. The second topic (Chapter V) is based on a comparative study of alternative routes for deamidation, more specifically a comparative analysis of direct hydrolysis of Asn versus the cyclic imide-mediated route. A possible side reaction -nonenzymatic peptide backbone cleavage at Asn and Asp residues-is studied and the relative feasibility of peptide fragmentation near these residues is discussed (Chapter VI). Finally, the influence of protein primary structure on deamidation rates is investigated (Chapter VII), where the effect of the Asn carboxyl-side residue's identity on deamidation rates is explored. Important conclusions drawn from each chapter are summarized (Chapter VIII) and specific suggestions for future work are discussed (Chapter IX). CHAPTER III THEORETICAL BACKGROUND III. THEORETICAL BACKGROUND Molecular Quantum Mechanics All electronic structure methods seek an approximate solution to the Schrödinger equation. For small systems consisting of a few particles, a very accurate solution for the Schrödinger equation can be found but, unfortunately, there is no simple way of obtaining exact solutions for many electron atoms and molecules. Therefore some assumptions have to be made. The molecular orbital (MO) programs build a set of MO's to be occupied by the electrons assigned to the molecule. The MO calculation then simply involves finding the combination of the atomic orbitals that have the proper symmetries and that give the lowest electronic energy. This is known as the linear combination of atomic orbitals (LCAO) [1]. Many simple MO methods are based on one-electron treatment, in that the electron is considered not to interact with others in the molecule. The Hückel and Extended Hückel theory is based on this approach. The SCF method takes electron-electron terms into account by considering the interaction between an electron in a given orbital and the mean field of the other electrons in the molecule. It involves an iterative process in which the orbitals are improved from cycle to cycle until the electronic energy reaches a constant minimum value and the orbitals no longer change. This situation is described as self-consistent. At the SCF level the electron-electron interaction is actually overestimated. The theory does not allow the electrons to avoid each other but assumes that their instantaneous positions are independent of one another. However, the error is reasonably consistent, so that its effects can be made to cancel by the use of proper comparison. The SCF method is also known as the Hartree-Fock or single determinant theory. Both ab initio and semi-empirical calculations treat the linear combination of orbitals by iterative computations, which establish a self-consistent electric field (SCF) and minimize the energy of the system. In ab initio calculations, electron-electron repulsion is specifically taken into account, while the semi-empirical method appears as an intermediate approach. Semi-empirical Methods Semi-empirical methods are characterized by their use of parameters derived from experimental data in order to simplify the approximation to the Schrödinger equation. As such, they are relatively inexpensive and can be practically applied to very, very large molecules. The semi-empirical quantum-mechanical methods developed by Dewar and coworkers [2][3][4][5] have been successful at reproducing molecular energies, replicating molecular structures and interpreting chemical reactions [6,7]. To overcome some of the computational difficulties, approximations are made in which several of the integrals involving core orbitals are replaced by parameters. The number of two-electron integrals calculated is reduced, by simply ignoring them or calculating them in an approximate fashion. Three levels of approximation have been defined by Pople and Beveridge, in which certain two-electron integrals are neglected [8]. The first is known as the complete neglect of differential overlap (CNDO) [9]. It assumes the atomic orbitals to be spherically symmetrical when evaluating electron repulsion integrals. The directionality of p-orbitals was included only via the one-electron resonance integrals, the size of which depend on the orientations and distances of the orbitals and on a constant assigned to each type of bond. The second, known as intermediate neglect of differential overlap (INDO) [10], contains all terms that CNDO contains and includes all one-center two-electron integrals. The third is known as neglect of diatomic differential overlap (NDDO) [11] in which all two-electron two-center integrals involving charge clouds arising from pairs of orbitals on an atom were retained. In 1975, Dewar and coworkers published the MINDO/3 method, which is a modified version of the INDO method. MINDO/3 uses a set of parameters in approximation. These parameters, along with the constants used to evaluate the resonance integrals, allow the results to be fitted as closely as possible to experimental data. The first practical NDDO method was introduced by Dewar and Thiel in 1977 [12] called modified neglect of diatomic overlap (MNDO), the model was parameterized on experimental molecular geometries, heats of formation, dipole moments and ionization potentials. The orbital exponents and the core integral were again treated as empirical parameters to be determined in the fitting procedure. The inability of MNDO has lead to a reexamination of the model, leading to the Austin Model 1 (AM1) [13]. In this model a term was added to MNDO to correct for the excessive repulsions at van der Waals distances. For this purpose, each atom was assigned a number of spherical gaussians, which were intended to mimic long-range correlation effects. The third parameterization of MNDO is the Parametric Method Number 3 (PM3), AMl being the second. In PM3, the parameters were optimized using a large set of reference molecular data. This allowed 12 elements to be optimized simultaneously [14]. The PM3 method has been used in this study for the semi-empirical level calculations. Semi-empirical and ab initio methods differ in the trade-off made between computational cost and accuracy of the result. Semi-empirical calculations are relatively inexpensive and provide reasonable qualitative descriptions of molecular systems and fairly accurate quantitative prediction of energies and structures for systems where good parameter sets exist. Semiempirical methods may only be used for systems where parameters have been developed for all of their component atoms. In addition to this, semi-empirical models have a number of well-known limitations. Types of problems on which they do not perform well include hydrogen bonding, transition structures, and molecules containing atoms for which they are poorly parameterized. Ab Initio Methods The main difficulty in solving the Schrödinger equation is the presence of the electronelectron interactions terms. It is very difficult to find analytical solutions to the Schrödinger equation that has this as part of its potential energy term, but computational techniques are available that give very detailed and reliable numerical solutions for the wavefunctions and energies [15,16]. The time-independent Schrödinger equation is: ψ ψ E H = ˆ (3.1) where E is the electronic energy, ψ is the wave function describing the system and H ˆis the Hamiltonian operator. Nuclei are much heavier than electrons. Hence the electrons move much faster than the nuclei, and to a good approximation, one can regard the nuclei as fixed while the electrons carry out their motion. The Born-Oppenheimer approximation neglects the kinetic energy of nuclei and the Hamiltonian operator in atomic units is: ) 2 1 ( ˆ1 2 ∑ = ∇ - = N i i H -∑ = N i i r 1 1 + ∑∑ = > N i N i j ij r 1 1 + ∑ ∑ = > N N r 1 1 α α β αβ (3.2) where the first term is the kinetic energy operator for electrons, the second term is the operator for the attraction between electrons and nuclei, the third term is the interelectronic repulsion operator and the final term is the internuclear repulsion operator for an N electron system. Then approximating ψ as an antisymmetrized product of n orthonormal orbitals ) (x i r Ψ with a Slater determinant, x r denoting both spin and spatial coordinates of an electron, the electronic energy (called Hartree-Fock (HF) energy at this level of approximation) may be written as: ) 2 ( 2 ˆ1 , 1 ij n j i ij n i i HF HF HF K J H H E - + = = ∑ ∑ = = ψ ψ (3.3) x d x x v x H i i i r r r r ) ( ) ( 2 1 ) ( 2 * Ψ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + ∇ - Ψ = ∫ (3.4) where ) (x v r is the external potential due to the nuclei or other sources, ij J is the Coulomb integral and ij K is the exchange integral: 2 1 2 2 * 12 1 * 1 ) ( ) ( 1 ) ( ) ( x d x d x x r x x J j j i i ij r r r r r r Ψ Ψ Ψ Ψ = ∫∫ (3.5) 2 1 2 * 2 12 1 1 * ) ( ) ( 1 ) ( ) ( x d x d x x r x x K j i j i ij r r r r r r Ψ Ψ Ψ Ψ = ∫∫ (3.6) The average value of the energy, including the normality conditions, is given by equation 3.7. Then, using the variation principle, one finds the wave function that minimizes the energy. ψ ψ ψ ψ H E = (3.7) Hartree-Fock theory is very useful for providing initial, first-level predictions for many systems. It is also reasonably good at computing the structures and vibrational frequencies of stable molecules and some transition states. As such, it is a good base-level theory. However, the Hartree-Fock method does not take into account the correlation energy resulting from the instantaneous interactions of electrons. Its neglect of electron correlation makes it unsuitable for the accurate modeling of the energetics of reactions and bond dissociation. Since electrons are correlated with each other the instantaneous electron correlation should be included into the wavefunction. Methods that include correlation energy such as configuration interaction methods (CI), coupled cluster (CC) and perturbation theory are called post-Hartree-Fock methods [17]. There are other correlated methods [18], such as multiconfiguration SCF (MCSCF) and multi-reference configuration interaction (MRCI). The Møller-Plesset perturbation theory (MP) treats the effect of electronic interactions as a perturbation to a system consisting of non-interacting electrons [19]. The first order perturbation introduces the interaction between the electrons in the ground state and is equal to the Hartree-Fock theory. The second order perturbation (MP2) takes into account the interaction of the doubly excited configurations with the ground state configuration. The third order perturbation (MP3) adds the contribution of doubly excited configurations interacting with each other. The fourth order perturbation (MP4) brings in interactions involving single, double and quadruple excitations. Although the MP4 method is very accurate it is very expensive computationally. Post-SCF methods are not suitable for this project, since there are many structures to optimize; high computational expenses demand an alternative method. Fortunately, the density functional theory (DFT) provides a cheaper method to treat correlation energy. Density Functional Theory The DFT method computes electron correlation via general functionals of the electron density. DFT functionals partition the electronic energy into several components which are computed separately: the kinetic energy, the electron-nuclear interaction, the Coulomb repulsion, and an exchange-correlation term accounting for the remainder of the electronelectron interaction (which is itself divided into separate exchange and correlation components in most actual DFT formulations.) The density functional theory is based on the Kohn-Hohenberg theorems proposed in 1964 [20][21][22][23]. The first theorem states that the electron density ρ(r) determines the external potential v(r), i. e. the potential due to the nuclei. The second theorem introduces the variational principle. Hence, the electron density can be computed variationally and the position of nuclei, energy, wave function and other related parameters can be calculated. The electron density is defined as: ( ) ∫ ∫ Ψ = n n dx dx dx x x x N x ... ) ,... , ( ... 2 1 2 2 1 ρ (3.8) where x represents both spin and spatial coordinates of electrons. The electronic energy can be expressed as a functional of the electron density: [ ] ( ) ( ) [ ] [ ] ρ ρ ρ ρ ee V T dr r r v E + + = ∫ (3.9) where T[ρ] is the kinetic energy of the interacting electrons and V ee [ρ] is the interelectronic interaction energy. The electronic energy may be rewritten as: In Kohn-Sham density functional theory, a reference system of independent non-interacting electrons in a common, one-body potential V KS yielding the same density as the real fullyinteracting system is considered. More specifically, a set of independent reference orbitals ψ i satisfying the following independent particle Schrödinger equation are imagined: [ ] ( ) ( ) [ ] [ ] [ ] ρ ρ ρ ρ ρ xc s E J T dr r r v E + + + = ∫ (3. i i i KS V ψ ε ψ = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + ∇ - 2 2 1 (3.11) with the one-body potential V KS defined as: ( ) [ ] ( ) [ ] ( ) r E r J r v V xc KS ρ ρ ρ ρ ∂ ∂ + ∂ ∂ + = (3.12) ( ) ( ) r v dr r r r r v V xc KS + - + = ' ' ' ) ( ρ (3.13) where v xc (r) is the exchange-correlation potential. The independent orbitals ψ i are known as Kohn-Sham orbitals and give the exact density by: ( ) ∑ = N i i r 2 ψ ρ (3.14) if the exact form of the exchange-correlation functional is known. However, the exact form of this functional is not known and approximate forms are developed starting with the local density approximation (LDA). This approximation gives the energy of a uniform electron gas, i. e. a large number of electrons uniformly spread out in a cube accompanied with a uniform distribution of the positive charge to make system neutral. The energy expression is: [ ] [ ] ( ) ( ) [ ] [ ] ∫ + + + + = b xc s E E J dr r v r T E ρ ρ ρ ρ ρ (3.15) where E b is the electrostatic energy of the positive background. Since the positive charge density is the negative of the electron density due to uniform distribution of particles, the energy expression is reduced to: [ ] [ ] [ ] ρ ρ ρ xc s E T E + = (3.16) [ ] [ ] [ ] [ ] ρ ρ ρ ρ c x s E E T E + + = (3.17) The kinetic energy functional can be written as: [ ] ( ) dr r C T F s ∫ = 3 5 ρ ρ (3.18) where C F is a constant equal to 2.8712. The exchange functional is given by: [ ] ( ) ∫ - = dr r C E x x 3 4 ρ ρ (3.19) with C x being a constant equal to 0.7386. The correlation energy, E c [ρ],for a homogeneous electron gas comes from the parametrization of the results of a set of quantum Monte Carlo calculations. The LDA method underestimates the exchange energy by about 10 per cent and does not have the correct asymptotic behavior. The exact asymptotic behavior of the exchange energy density of any finite many-electron system is given by: r U x x 1 lim - = ∞ → σ (3.20) σ x U being related to E x [ρ] by: [ ] ∑ ∫ = σ σ σ ρ ρ dr U E x x 2 1 (3.21) A gradient-corrected functional is proposed by Becke: ∑ ∫ - + - = σ σ σ σ σ β ρ β dr x x x E E LDA x x 1 2 3 4 sinh 6 1 (3.22) where σ denotes the electron spin, 3 4 σ σ σ ρ ρ ∇ = x and β is an empirical constant (β=0.0042). This functional is known as Becke88 (B88) functional [24]. The adiabatic connection formula connects the non-interacting Kohn-Sham reference system (λ=0) to the fully-interacting real system (λ=1) and is given by: ∫ = 1 0 λ λ d U E xc xc (3.23) where λ is interelectronic coupling-strength parameter and λ The closed shell Lee-Yang-Parr (LYP) correlation functional [26] is given by: The mixing of LDA, B88, exact x E and the gradient-corrected correlation functionals to give the hybrid functionals [27] involves three parameters: dr e t t C b d a E c w w F c ∫ ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∇ + + - + + - = - - - - 3 ( ) local non c c B x x LDA x exact x LDA xc xc E a E a E E a E E - ∆ + ∆ + - + = 88 0 (3.27) where 88 B x E ∆ is the Becke's gradient correction to the exchange functional. In the B3LYP functional, which was utilized in this study, the gradient-correction ( local non c E - ∆ ) to the correlation functional is included in LYP. However, LYP contains also a local correlation term which must be subtracted to yield the correction term only: VWN c LYP c local non c E E E - = ∆ - (3.28) where VWN c E is the Vosko-Wilk-Nusair correlation functional, a parametrized form of the LDA correlation energy based on Monte Carlo calculations. The empirical coefficients are a 0 =0.20, a x =0.72 and a c =0.81 [28]. Basis Sets A basis set is the mathematical description of the orbitals within a system, (which in turn combine to approximate the total electronic wavefunction) used to perform the theoretical calculation. Larger basis sets more accurately approximate the orbitals by imposing fewer restrictions on the locations of the electrons in space. In 1951, Roothaan proposed the Hartree-Fock orbitals as linear combinations of a complete set of known functions, called basis functions. There are two types of set of basis functions for atomic Hartree-Fock calculations, Slater-type functions and Gaussian-type functions. In the simplest Hartree-Fock model, the number of basis functions on each atom is as small as possible that is only large enough to accommodate all the electrons and still maintain spherical symmetry. As a consequence, the molecular orbitals have only limited flexibility. If larger basis sets are used, the number of adjustable coefficients in the variational procedure increases, and an improved description of the molecular orbitals is obtained. Very large basis sets will result in nearly complete flexibility. The limit of such an approach termed the Hartree-Fock limit represents the best that can be done with a single electron configuration. There are numerous different Gaussian basis sets with which SCF calculations can be carried out [28]. The most widely used are those developed by Pople and co-workers [30]. The simplest and lowest basis set is called STO-3G. This means that the Slater-type orbitals are represented by three gaussian functions. It is a minimal basis set which means that it has only as many orbitals as are necessary to accomodate the electrons of the neutral atom. The next level of basis sets developed by Pople is referred to as the split-valence basis sets. The problem of any minimal basis set is its inability to expand or contract its orbitals to fit the molecular environment. One solution to the problem is to use split valence or double zeta basis sets in which the basis are split into two parts, an inner, compact orbital and an outer, more diffuse one. Thus the size of the atomic orbital that contributes to the molecular orbital can be varied within the limits set by the inner and outer basis functions. Split-valence basis set splits only the valence orbitals in this way, whereas double zeta basis also have split core orbitals. For greater flexibility the split-valence basis set can be augmented with polarization functions. In polarization basis sets, which are the next level of improvement in basis set, d orbitals are added to all heavy atoms is designated with a * or (d). Polarization can be added to hydrogen atoms as well, this would be done by **. Diffuse functions are large-size versions of s-and p-type functions (as opposed to the standard valence-size functions). They allow orbitals to occupy a larger region of space. Basis sets with diffuse functions are important for systems where electrons are relatively far from the nucleus: molecules with lone pairs, anions and other systems with significant negative charge, systems in their excited states, and so on. The 6-31+G (d) basis set is the 6-31G (d) basis set with diffuse functions added to heavy atoms. The double plus version, 6-31++G (d), adds diffuse functions to the hydrogen atoms as well. The 6-31+G** split valence basis set has been used for gas phase calculations in this study, with the addition of polarization and diffused functions on heavy atoms as well as polarization functions on hydrogens. Continuum Solvation Models In continuum solvation models [31,32], the solvent is represented as a uniform polarizable medium characterized by its static dielectric constant ε. In basic continuum solvation models, the solute is described at a homogenous quantum mechanical (QM) level and the solutesolvent interactions are limited to those of electrostatic terms. The total solvation free energy may be written as ∆G solvation = ∆G cavity + ∆G dispersion + ∆G electrostatic (3.29) In this representation, ∆G cavity is the energetic cost of creating a cavity in the medium producing a destabilization effect. Dispersion interactions between solvent and solute add stabilization to solvation free energy term expressed as ∆G dispersion . The latter electrostatic term, ∆G electrostatic , has a stabilization effect and furthermore it appears to be responsible for the main structural changes of the solute. The solute charge distribution within the cavity induces a polarization of the surrounding medium, which in turn induces an electric field within the cavity called the reaction field. This field then interacts with solute charges, providing additional stabilization. The effect of the reaction field may be modeled by an appropriately distributed set of induced polarization charges on the surface S of the dielectric. The charge density on the surface of the cavity, ( ) The potential from the surface charge V σ is given by the molecular charge distribution but also enters the Hamiltonian and thus influences the molecular wave function, the procedure is therefore iterative. In the Polarized Continuum solvation Models (PCM), the solute is embedded in a cavity defined by a set of spheres centered on atoms (sometimes only on heavy atoms), having radii defined by the van der Waals radius of the atoms multiplied by a predefined factor (usually 1.2). The cavity surface is then subdivided into small domains (called tesserae), where the polarization charges are placed. Among the many solutions to the electrostatic problem is the Integrated Equation Formalism (IEF) originally formulated by Cancés and Menucci [33][34][35]. The IEF-PCM method is a recent development in the polarized continuum models and has been utilized for solvent calculations in this study. It is based on the use of operators largely exploited in the theory of integral equations. The concept of cavity and of its tessellation is conserved. The IEF formalism is in fact able to treat a larger class of electrostatic problems. Apart from the apparent surface charge (ASC) methods, other solutions for calculating the electrostatic solute-solvent interactions in continuum models have been proposed such as, multipole expansion (MPE) methods, generalized Born approximation (GBA), image charge (IMC) methods, finite element methods (FEM) and finite difference methods (FDM). Molecular Dynamics Molecular dynamics (MD) simulations is one of the principal methods in the theoretical investigation of biological molecules that provides information on the time dependent behavior of molecular systems. MD simulations are routinely used to examine the structure, dynamics and thermodynamics of biological molecules and their complexes. Molecular dynamics is a deterministic method; the state of the system at any future time can be predicted from its current state. Successive configurations of the system are generated by integrating Newton's law of motion. The result is a trajectory that specifies how the positions and velocities of the particles in the system vary with time. The Force Field At the heart of molecular mechanics lies the force field which describes the potential energy surface of the system. The force field is composed of various contributions like bonded or valence terms (bond stretching, angle bending and torsion angle) and non bonded terms (van der Waals and Coulomb forces) all of which contain empirical parameters fitted to results of experimental studies or high level calculations. In the light of these contributions the potential energy V(R) is defined as: (3.39) The specific contributions to the potential energy can be described as follows: (3.40) where k b is specific force constant, l is bond length and l 0 is equilibrium bond distance (3.41) where k θ is specific force constant, θ is bond angle and θ 0 is equilibrium bond angle (3.42) where V n is the amplitude, n is the number of minima on the potential energy surface, ω is the torsion angle and γ is the phase factor (3.43) where van der Waals interaction between two atoms i and j separated by distance r ij is described by Lennard Jones potential with parameters A ij and B ij , and Coulomb potential is described by electrostatic interaction between a pair of atoms i and j using q i and q j as charges on atom and Є as the dielectric constant of medium. Theory of Molecular Dynamics The state of any classical system can be completely described by means of specifying the positions and momenta of the all particles: (3.44) Since a phase point is defined by the positions and momenta of all particles, it determines the location of the next phase point in the absence of outside forces acting upon the system. Therefore, the relationship between two positions in any time interval is given by: (3.45) where; (3.46) Similarly, the relationship between any two momentum vectors is: (3.47) using Newton's Second Law of Motion: (3.48) MD simulations generate information (atomic positions and velocities) at the microscopic level. The conversion of this microscopic information to macroscopic observables such as pressure, energy, heat capacities, etc., requires statistical mechanics. In statistical mechanics, time averages are defined as ensemble averages. The average value of any property during time evolution is: (3.49) where M is the number of times the property is sampled. If the system is allowed to evolve in time indefinitely, it will eventually pass through all possible states and the above equation becomes [18,36]: (3.50) assuming Ergodic hypothesis to be valid and independent of choice of t 0 . IV. EFFECT OF SOLVENT MOLECULES ON MECHANISM OF ASPARAGINE DEAMIDATION In this chapter, the effect of solvent molecules on the deamidation mechanism will be investigated. A cyclic imide-mediated route has been previously suggested, as the pathway for deamidation of Asn residues in relatively unrestrained peptides; previous computational INTRODUCTION The deamidation of proteins may occur under physiological conditions and is known to limit the lifetime of proteins [1]. Glutamyl residue H 2 C O NH 2 H 2 C O O - Scheme -1. Deamidation of Asparaginyl and Glutaminyl Residues The deamidation of Asn and Gln in peptides and proteins is of significant biological interest. Over 1,700 research papers on various aspects of the deamidation of peptides and proteins have been published [3] since the biological importance of deamidation was first emphasized. Deamidation of asparaginyl and glutaminyl residues causes time-dependent changes in charge and conformation of peptides and proteins [4][5][6]. Deamidation rates of 1371 asparaginyl residues in a representative collection of 126 human proteins have been calculated [7]. Deamidation half-times for these proteins were shown to range from 1 -1000 days. These rates have suggested that deamidation is a biologically relevant phenomenon in a remarkably large percentage of human proteins. The timed processes of protein turnover, development, and aging have been suggested as possible roles for deamidation [8]. It has been hypothesized by Robinson et al. that deamidation serves as a molecular clock for the timing of biological processes [9]. The fact that static properties of asparaginyl and glutaminyl residues are not unique and can be easily duplicated by some of the other 18 commonly occurring amino acid residues suggestively indicates the essence of their disruptive effect on peptide and protein structure by deamidation reactions. Furthermore, it has been proposed by Robinson et al. [9] that the instability of asparaginyl and glutaminyl residues is their primary biological function, and that they serve as easily programmable molecular clocks. Scheme -2. Deamidation Mechanism Suggested by Capasso et al [10]. The deamidation of Asn residues via succinimide intermediates has been previously investigated theoretically [14] using the Density Functional Theory (B3LYP/6-31G*). The cyclization, deamination (loss of NH 3 ) and hydrolysis reactions that lead to the deamidation of a model peptide have been studied and the succinimide intermediate suggested by Capasso et al. [10] has been confirmed. The formation of the succinimde intermediate has been proposed to be a multi-step process [14], in which the initial cyclization step forms a tetrahedral intermediate (Scheme-3). This is followed by the deamination step to produce the succinimide ring. Finally, hydrolysis of the succinimide ring leads to L-Asp or L-iso-Asp residues [15,16]. The rate determining step in neutral media has been proposed to be the formation of the tetrahedral intermediate [14]. Gly-Xxx-Gln-Yyy-Gly, where Xxx and Yyy are any of the 20 normally occurring amino acid residues [17]. This study has indicated that deamidation rate is controlled primarily by the carboxyl side residue (Yyy), also commonly referred to as the n+1 residue, with smaller effects from the amino side residue (Xxx). This is consistent with the succinimide reaction mechanism that was originally proposed [10] to explain the deamidation rates of Asn-Gly sequences and iso-Asp formation. In the present study, alternative pathways, namely water-assisted deamidation mechanisms have been investigated. The effect of explicit H 2 O molecules on the mechanism and energetics of deamidation has been explored using computational techniques. Both concerted and stepwise reaction mechanisms leading to the succinimide intermediate have been taken into account. As an alternative mechanism the tautomerization of the Asn side chain amide functionality has been explored. Activation barriers for these pathways have been used for comparative purposes and to identify the most probable mechanism for deamidation of peptides in solution. COMPUTATIONAL METHODOLOGY Preliminary analysis of the potential energy surfaces (PES) for all proposed mechanisms were carried out at a semi-empirical level (PM3) [18]. Further geometry optimizations were performed using the density functional theory (DFT) [19][20][21] at the B3LYP/6-31+G** [22][23][24] level of theory. Geometries of stationary points were optimized without any constraints. All stationary points have been characterized by a frequency analysis from which zero-point energy and thermal corrections have also been attained using the ideal gas approximation and standard procedures. Local minima and first order saddle points were identified by the number of imaginary vibrational frequencies. The intrinsic reaction coordinate (IRC) approach [25,26], followed by full geometry optimization, has been used to determine the species reached by each transition structure. Note that the resulting energy minima do not necessarily correspond to the lowest conformation of the system (particularly for solute-water complexes), though differences are not expected to be large. Free energies of activation (∆G ‡ ) are calculated as the difference of free energies between transition states and reactive conformers reached by IRC calculations. All energy values for gas phase optimized structures listed throughout the discussion include thermal free energy corrections at 298 K and 1 atm. The effect of a polar environment on the reaction path has been taken into account by use of the self-consistent reaction field (SCRF) theory. Single-point energies in water (ε = 78.5) utilizing the integral equation formalism-polarizable continuum (IEF-PCM) model [27][28][29][30] at the B3LYP/6-31++G** level were calculated on gas phase B3LYP/6-31+G** optimized structures. Bondi radii [31] scaled by a factor of 1.2 have been used for solvent calculations. All solvent energies reported include thermal corrections to free energies, obtained from gas phase optimizations and non-electrostatic corrections. All gas phase optimizations and single point solvent calculations have been carried out using the Gaussian 03 program package [32]. Reaction mechanisms shown in figures throughout the text contain gas phase optimized geometries (B3LYP/6-31+G**) of reactive conformers, transition states and products, respectively. All distances shown in the figures are in Angstroms (Å). RESULTS and DISCUSSION: In the first part of this study, the succinimide formation mechanism proposed by Capasso et al. [10] and previously modeled by Konuklar et al. [14] has been computationally revisited with a slightly larger model compound (Scheme-4), in order to more efficiently mimic Asn and its neighboring residues. More specifically, instead of mimicking the backbone NH with an NH 2 group as in the previous model, an acetyl (CH 3 C=O) group has been added to the Asn backbone NH, to help prevent unrealistic intramolecular H-bonding that can be formed by a less restricted NH 2 . In addition, the basis set has been improved by the addition of diffuse functions on heavy atoms and polarization functions on hydrogen atoms, while the method was preserved. The previously suggested concerted mechanism [14] will serve as a benchmark for comparison with energetics of the newly proposed mechanisms in this study. H 3 C N H H N CH 3 O O Model Compound H 2 C H 2 N O Scheme -4. Model Peptide with L-Asparaginyl Residue The previously suggested concerted mechanism has been employed on the new model compound using B3LYP/6-31+G** (Figure -1). This is a four-centered asynchronous concerted mechanism, where the hydrogen on the backbone NH, which belongs to the n+1 residue, is transferred to the Asn side chain carbonyl group early in the reaction as seen in the transition state. Later in the concerted step, the backbone nitrogen attacks the carbonyl carbon and the five-membered tetrahedral intermediate forms. The free energy of activation (∆G ‡ ) for this concerted reaction is quite high, 49.7 kcal/mol in gas phase (40.0 kcal/mol in solution), for a reaction that spontaneously and nonenzymatically occurs under physiological conditions. The same concerted reaction was shown to have an activation barrier of 50.3 kcal/mol in gas phase (47.5 kcal/mol in solution) in the previous study where both a smaller model compound and basis set were used [14]. Stepwise -Water Assisted -Cyclization The second pathway explored was the water assisted stepwise cyclization, which consists of two consecutive steps. The first of these steps is the simultaneous deprotonation of the backbone n+1 The backbone rotation mentioned has not been modeled in this study and it has been assumed that it will not affect the overall barrier of the water assisted stepwise cyclization appreciably, since the two intermediates are very close in energy, the latter being a little more stable. The second step in the water assisted cyclization is the ring closure. This step is expected to have a low barrier, since a negatively charged nitrogen atom is attacking a positive carbon, hence the barrier for ring closure alone is 8 kcal/mol in gas phase (12.3 kcal/mol in solution). There is an effective H-bond network between the peptide and the three peripheral water molecules throughout the reaction, with H-bond distances ranging from 1.420 -1.965 Å. Figure -4. Potential Free Energy Profile for the Water Assisted Stepwise Cyclization Reaction The overall free energy of activation for the water assisted stepwise cyclization is shown to be 33.4 kcal/mol in gas phase (33.9 kcal/mol in solution). This energy barrier is more than 15 kcal/mol lower than the free energy of activation for the gas phase concerted reaction previously reported [14]. It is also lower in energy than the concerted water assisted cyclization mechanisms previously discussed in this text. a. Asparagine Side Chain Tautomerization The asparagine side chain, like any other amide functionality, can tautomerize into an amidic acid tautomer, with the transfer of a proton from the side chain NH 2 group to the side chain carbonyl oxygen. The tautomerization reaction is depicted together with the ring closure that leads to the tetrahedral intermediate (Scheme-6). This is an alternative pathway leading to the tetrahedral intermediate, which can then be converted to the succinimide intermediate through expulsion of NH 3 . Scheme -6. Asparagine Side Chain Tautomerization followed by Cyclization to Tetrahedral Intermediate Side chain tautomerization may occur with or without the assistance of water molecules. Asn side chain tautomerization with no help from surrounding water molecules has been modeled (Figure -5). This is a four-centered concerted reaction with an expected high ∆G ‡ , which was calculated to be approximately 45 kcal/mol in both gas phase and solution. The tautomerization of the amide functional group to the amidic acid tautomer is not likely to proceed through a concerted four-centered reaction but rather through a water-assisted concerted one, where the proton transfer occurs via peripheral water molecules in the vicinity of the amide functionality. Amide tautomerization has been subject to some recently published theoretical studies on smaller amides [33][34][35]. Energetics for the water assisted tautomerization reaction for Asn calculated in this study are in good agreement with these studies, which showed formamide tautomerization to be approximately 20 kcal/mol with water assistance. Relative Free Energies 3b. Ring Closure of Amidic Acid Tautomer The amidic acid tautomer formed through tautomerization of the Asn side chain may undergo ring closure to yield the tetrahedral intermediate (Scheme-6). The energy required for a concerted ring closure starting from the amidic acid tautomer is approximately 27 kcal/mol in gas phase, but the effect of a polar environment significantly reduces this barrier to 17.5 kcal/mol in solution (Figure -8). A variant of the concerted ring closure mechanism with two peripheral water molecules was also modeled (Figure -9). These solvent molecules are not actively involved in the reaction, however However, since the difference in activation energies are small, the most prominent outcome of these results is that all water assisted mechanisms are more than 10 kcal/mol lower in energy than the previously proposed waterless concerted cyclization mechanism [14]. DEAMINATION The Succinimide Intermediate In this part of the study, the deamination of the tetrahedral intermediate with no H 2 O assistance, as previously proposed [14], has been re-evaluated with the new model compound (Figure -12) and a higher basis set, similar to the case for the cyclization reaction. This is a concerted fourcentered step with a barrier of 31.5 kcal/mol in gas phase. If deamination is considered to go through a four-centered mechanism [14], comparison of energetics between cyclization and deamination suggests that deamination may as well be the rate determining step in neutral media rather than the cyclization step. Once the succinimide ring forms the reversal to the Asn residue is considered not feasible. NH 3 attack on the succinimide ring is not foreseen, due to the negligibly small amount of NH 3 produced during the reaction. At this point, hydrolysis is inevitable and either an Asp or an iso-Asp residue will form. Relative Free Energies CONCLUSION The aim of this study was to reinvestigate the energetics of the deamidation mechanism in peptides with the effect of solvent molecules, suggesting different pathways involving the assistance of explicit H 2 O molecules. Three different mechanisms were suggested for the cyclization step of the deamidation reaction, which was previously proposed to be the rate determining step [14]. All water assisted cyclization reactions investigated in this study were shown to have a significantly lower barrier than the previously proposed concerted waterless mechanism. Three water assisted mechanisms with identical molecularity (Figures-3, This study has established the effect of water molecules on the deamination step of the deamidation reaction, verifying that the cyclization step, with a substantially higher barrier for activation, is the rate determining step. It is also noteworthy that the involvement of water molecules in the deamination step has lowered the barrier to half. This study has shown that water molecules in the vicinity of asparaginyl residues serve as a catalyst in deamidation reactions. Deamidation in proteins may as well be enhanced by other residues, which are capable of accepting or donating protons. However, experimental results have shown that the three-dimensional structure of the protein accelerates deamidation in only 6% of the cases [7,36]. Therefore, one may conclude that deamidation in proteins or enzymes will be more probable for those potential deamidating sites that exhibit the largest accessibility by solvent molecules. This investigation also suggests that a quantitative description of the process may require carrying out a detailed statistical treatment of the solvent effect. This will be done in future studies using molecular dynamics simulations and combined QM/MM potentials. Further investigations will also include examination of 1) solvent assisted mechanisms in enzymes having sites with different deamidation rates and 2) the effect of the identity of the n+1 residue on deamidation, which is known to have different half-times for different amino acids. CHAPTER V DIRECT HYDROLYSIS VERSUS SUCCINIMIDE-MEDIATED DEAMIDATION MECHANISMS Chapter V 63 V. DIRECT HYDROLYSIS VERSUS SUCCINIMIDE-MEDIATED DEAMIDATION MECHANISMS The previous chapter has introduced important insight on the catalytic effect of solvent molecules on the mechanism as well as energetics of the steps leading up to the succinimide intermediate. Previous computational studies on succinimide formation and succinimide hydrolysis have suggested that the rate determining step for the overall deamidation process is the cyclization step leading to the tetrahedral intermediate. However, since these studies did not take into account the effect of water molecules and activation barriers were calculated for the waterless reaction, it is imperative to re-analyze these barriers using explicit solvent molecules and to check whether water-assistance modifies the rate determining step for deamidation process. The following study, therefore, takes into account all steps of deamidation including succinimide hydrolysis. This chapter also includes the analysis of another potential mechanism for deamidation, which could compete with the succinimide-mediated pathway. Thus the direct hydrolysis of the Asn side chain amide has been investigated, in order to allow a comparative analysis between both deamidation mechanisms. The following article has recently been submitted for publication in Chemistry -A European Journal. INTRODUCTION Asparagine (Asn) and glutamine (Gln) residues are known to undergo spontaneous nonenzymatic deamidation to form aspartic acid (Asp) and glutamic acid (Glu) residues under physiological conditions [1][2][3][4][5]. The conversion of the neutral amide side chain to the negatively Chapter V 65 charged carboxylate causes time-dependent changes in conformation and limits the lifetime of peptides and proteins [1,[6][7][8]. Deamidation half-times in human proteins were shown to occur over a wide range of biologically relevant time intervals [9], and have been associated with the timed process of protein turnover, development and ageing [10]. Robinson has proposed the molecular clock hypothesis, which suggests that deamidation is a biological molecular timing mechanism that could be set to any desired time interval by genetic control of the primary, secondary, and tertiary structure surrounding the amide [11]. Recent experiments [12][13][14][15][16][17][18][19] and computations [20][21][22][23][24][25] have been in accord with this hypothesis and provided compelling evidence of its significance. The deamidation reaction mechanism was initially believed to be an acid or base catalyzed direct hydrolysis, with a minimum reaction rate near neutral pH. Deamidation products for L-Asn and L-Gln were expected to be L-Asp and L-Glu with little racemization to D-Asp and D-Glu at basic pH. However, the pH minimum for deamidation was actually observed to be around 5 for both peptides and proteins and an L-iso-Asp product was observed in addition to the L-Asp [13]. The succinimide-mediated deamidation mechanism (Scheme 1) is suggested to be responsible for shifting the minimum to pH 5 and for the variety of reaction products observed. [12,14]. The cyclic imide then hydrolyzes at either one of the two carbonyls to give Asp and iso-Asp. The ratio of L-Asp to L-iso-Asp was experimentally found to be 3:1 [13]. Chapter V 66 Experimental evidence indicates that, at low pH, hydrolysis of the side chain amide functionality occurs with ease and imide intermediates are not observed, contrary to neutral and basic pH conditions where, succinimide derivatives have almost always been identified [12][13][14][15][16][17][18][19]. As pH decreases below 5, direct hydrolysis via acid catalysis takes place at an increasing rate. The fact that direct hydrolysis is most prevalent is indicated by the marked drop of the iso-Asp:Asp ratio [5]. Ordinary base catalysis also occurs at high pH, but the rapidity of the imide mechanism at high pH usually obscures this. However, since reaction rates are often related to transient chemical species that are difficult to observe and are subject to influences of solvent and other factors, there is often uncertainty concerning a proposed mechanism, as is the case with deamidation. Nevertheless, the succinimide-mediated mechanism is supported by many experimental observations [12][13][14][15][16][17][18][19]. Previous computational studies on deamidation of Asn include modeling the formation of the succinimide as a two-step process; cyclization followed by deamination [20]. Subsequent computations on this mechanism established the fact that water molecules in the vicinity of Asn residues catalyze deamidation [26]. Radkiewicz et al. have computationally explored the racemization of Asp and Asn via succinimide intermediates [23] and have studied the effect of neighboring side-chains on backbone NH acidity [23]. Peters et al. explored pH dependence of the deamidation mechanism [25]. Experimental findings have shown that the rate of deamidation of Asn residues is primarily controlled by the carboxyl side residue (n +1) with smaller effects from the amino side residue (n -1) [16]; this is also consistent with the succinimide reaction mechanism. However, the relationship between the size and/or charge of the n +1 residue and rate of deamidation is not clear. In peptides with substantial freedom of movement, sequence dependence has also been detected for other residues further along the peptide chain in both directions [16]. The effects of more distant residues are probably largely suppressed in proteins. usually inhibits deamidation, there are many instances in which protein structure near the amide allows deamidation to occur at its primary sequence controlled rate [16]. There are also relatively rare instances in which protein structure actually increases the deamidation rate [19]. This study aims to get a deeper insight on plausible mechanisms leading to deamidation of Asn residues and comparatively discuss relative energetics and feasibilities of these pathways at physiological conditions. For this purpose, several succinimide-mediated deamidation paths together with a direct hydrolysis mechanism will be considered. Some of these mechanisms are described here for the first time. Catalysis of these reactions by different number of explicit solvent molecules will be investigated. COMPUTATIONAL METHODOLOGY All gas-phase geometry optimizations were performed using the density functional theory (DFT) [27][28][29][30] at the B3LYP/6-31+G** level [31][32][33]. Diffuse and polarization functions are included on heavy atoms, since utilization of diffuse functions is especially necessary in the optimization of anionic systems; polarization functions were also added on hydrogen atoms in order to account for the presence of hydrogen-bonds. Geometries of stationary points were optimized without any constraints. All stationary points were characterized by a frequency analysis from which zero-point energy and thermal corrections were attained using the ideal gas approximation and standard procedures. Local minima and first order saddle points were identified by the Chapter V 68 number of imaginary vibrational frequencies. The intrinsic reaction coordinate (IRC) approach [34,35], followed by full geometry optimization, was used to determine the species connected by each transition structure. Energy values for gas phase optimizations listed throughout the discussion include thermal free energy corrections at 298 K and 1 atm. The energetics of pathways leading to deamidation investigated herein will be comparatively discussed with Asn deamidation mechanisms previously studied by the authors [26]. All structures whose energy values are reproduced from previous deamidation studies are labeled with an asterisk (*). The same model peptide, level of theory and basis set are used for a legitimate comparison with aforementioned studies. The catalytic effect of water molecules on deamidation was previously established [26] and has been taken into account in this study. For clarity, comparative discussion of energetics between different mechanisms is always made among species with identical molecularity, i.e. for species that are associated with the same n in the initial peptide model -(H 2 0) n complex. This choice minimizes the errors in the estimation of entropic contributions [36]. Relative free energies of activation (∆G ‡ ) are calculated as the difference of free energies between transition states and the initial reactant. Following previous studies on amide hydrolysis [36] in water-assisted mechanisms, the initial reactant is taken as the solute-water complex with the relevant number of water molecules. Nomenclature of each transition state corresponds to the name indicated in the relevant scheme and the explicit number of water molecules used in that particular step. All calculations were carried out using the Gaussian 03 program package [37]. Structures shown throughout the text are gas-phase optimized geometries (B3LYP/6-31+G**). Distances and free energies listed in the discussion are in angstroms (Å) and kcal/mol, respectively. RESULTS and DISCUSSION In the first part of this study, we describe different mechanisms for deamidation. The main characteristics of the molecular structures will be commented. The direct hydrolysis of the Asn side A. DIRECT HYDROLYSIS OF ASPARAGINE TO ASPARTATE (asn → asp) Amide hydrolysis has been extensively studied by computational methods, formamide hydrolysis in particular [36]. The hydrolysis of the Asn side chain to the carboxylic acid via a concerted mechanism was modeled with one or two water molecules. In the two-water case, the reaction involves a proton relay mechanism where solvent molecules serve as a conduit; amide hydrolysis involves the expulsion of an NH 3 group and deamidation takes place in a single step (Scheme 2). Optimized structures are shown in Figure 2. Given that the water molecule is a reactant, TS1-1H 2 O is in fact an "unassisted" case, while TS1-2H 2 O shows one-water assistance. Transition b. Two Step (asn → tet → suc) This mechanism has been previously described [26]. Hydrolysis of the Succinimide Intermediate (suc → asp) Deamidation is completed when the succinimide intermediate undergoes hydrolysis and an Asp residue forms. It is noteworthy to indicate that the hydrolysis may take place at either one of the carbonyl groups on the succinimide. However products will be different, Asp and iso-Asp may both form; when iso-Asp forms, the peptide backbone is altered (Scheme 1). An extra atom on the peptide chain will be an unstabilizing factor in a protein three-dimensional structure, as it might disrupt intermolecular interactions within the chain. The formation of iso-Asp has been previously investigated computationally [22] and will not be discussed in this study, since the aim is to find the most plausible pathway for the complete deamidation of Asn into Asp. Hydrolysis of the succinimide intermediate can take place via a concerted ring opening (Scheme 6) or through a gemdiol intermediate (Scheme 7); both will give the same product. Previous studies on amide hydrolysis have shown that a stepwise mechanism going through a gemdiol intermediate has a considerably lower barrier than a concerted reaction [36]. These studies have also revealed that water molecules catalyze hydrolysis of an amide, as mentioned earlier. In the concerted amide hydrolysis reaction, a water molecule attacks the ring carbonyl, the N-C bond breaks, as a proton is transferred from the water molecule to the ring NH. As a result, a carboxylic acid forms (Scheme 6). However, in the gemdiol-mediated stepwise mechanism (Scheme 7), the initial step is the addition of a water molecule to the ring carbonyl, forming a gemdiol intermediate, which consequently undergoes ring opening in the second step to reveal the same product. In this way, the transformation of the Asn side chain amide into a carboxylate group is complete. Stepwise succinimide hydrolysis was modeled in the same manner, with one and two water molecules (Figure 5). In the unassisted stepwise mechanism, a water molecule is added to the ring in step 1 (TS8-1H 2 O), the second step (TS9) is the ring-opening. In the two-water case, one of the solvent molecules is added to the ring in step 1 (TS8- 79 C. DIRECT HYDROLYSIS VERSUS SUCCINIMIDE-MEDIATED DEAMIDATION: COMPARISON OF ENERGETICS AND MECHANISMS In this part, imide-mediated routes will be energetically compared amongst each other as well as against the direct hydrolysis pathway. Relative barriers and feasibilities will be discussed. Competing mechanisms with identical molecularity (same number of atoms, depending on the number of water molecules in the initial complex, 0, 1 or 2) are grouped and presented in Figures 6 (no water), 7 (one-water) and 8 (two-water) along with energetics. All structures whose energy values are reproduced from previous deamidation studies [26] are labeled with an asterisk (*). Although previously explored by Konuklar et al. [20][21][22], the energetics for 1) the formation of the succinimide intermediate and 2) its decomposition into aspartic acid by means of water hydrolysis has not been analyzed with the same number of water molecules and within the same energetic scale. These reactions have been previously explored separately [20,22] and barrier heights have been calculated with respect to the initial structure of each individual reaction. Therefore, activation energies are not relative to one another, but to the reactant of each step. In this study, we have evaluated all steps of the deamidation process with respect to a single reference point, the model peptide (Figure 1). Activation barriers calculated in earlier studies [26] for waterless, one-and two-water cases were re-calculated with respect to this reference point for a legitimate comparison; slight changes in free energies of activation have been observed for some steps, but differences are within the range of 2 kcal/mol. The free energy of a single ammonia molecule was added to each component in the succinimide hydrolysis mechanism for scaling purposes. The waterless mechanism (Figure 6 However, the hydrolysis of the succinimide also shows relatively high barriers (Figure 7). The concerted hydrolysis (suc → asp) was expected to have a higher barrier (58.2 kcal/mol) than the gemdiol-mediated stepwise route (suc → gem → asp); the barrier difference between the two mechanisms is approximately 5 kcal/mol. The ring-opening step (gem → asp) is ratedetermining for the stepwise pathway (53.9 kcal/mol as opposed to 48.5 kcal/mol in suc → gem). When imide-mediated deamidation is considered, the most plausible pathways for succinimide formation and succinimide hydrolysis are the tautomerization (asn → taut → tet → suc) and gemdiol (suc → gem → asp) mechanisms, respectively. The rate-determining step for the complete imide-mediated deamidation process seems to be the ring opening step (gem → asp) in succinimide hydrolysis with a barrier much higher (53.9 kcal/mol) than the cyclization step. However, it should be noted that the succinimide hydrolysis barriers involve mechanisms with only one water molecule, i.e. the hydrolysis step depicted in Figure 7 is unassisted, unlike the succinimide formation steps. The water molecule is a reactant in hydrolysis, as mentioned earlier (Figure 4 and5); therefore the one-water case does not constitute for a fair comparison of energetics between the formation and hydrolysis of the succinimide. The two-water case (Figure 8) will provide more realistic grounds for comparison. Chapter V 83 The direct hydrolysis mechanism (asn → asp) with one-water molecule is relatively slower (47.9 kcal/mol) than the succinimide formation steps, but like the imide hydrolysis steps, direct hydrolysis with one-water is unassisted, and therefore a legitimate comparison cannot be made. In the two-water case (Figure 8), all steps of the imide-mediated deamidation reaction and direct hydrolysis are water-assisted; activation barriers of all steps are lower than their uncatalyzed counterparts (Figure 6 The activation barriers for the stepwise hydrolysis of the succinimide with two-water molecules have much lower barriers than the previous one-water -unassisted-case (Figure 7). Again the concerted hydrolysis (suc → asp) has a higher barrier (58.4 kcal/mol) than the gemdiol-mediated stepwise route (suc → gem → asp); the barrier difference between the two mechanisms is approximately 10 kcal/mol. The ring-opening step (gem → asp) is rate-determining for the stepwise pathway (46.3 kcal/mol as opposed to 36.9 kcal/mol in suc → gem). In imide-mediated deamidation, the most plausible pathways for succinimide formation and succinimide hydrolysis are the tautomerization (asn → taut → tet → suc) and gemdiol (suc → gem → asp) mechanisms, respectively. The rate-determining step for the complete imidemediated deamidation process seems to be the ring-opening step (gem → asp) in succinimide hydrolysis with a barrier (46.3 kcal/mol as opposed to 39.7 kcal/mol) almost 6 kcal/mol higher than the cyclization step (taut → tet) in succinimide formation. For all mechanisms (waterless and water-assisted) the tautomerization route (asn → taut → tet → suc) has the lowest barrier for the formation of the succinimide intermediate. The cyclization step is the rate-determining step for succinimide formation in all water assisted mechanisms. The deamination step is faster than cyclization, except for the waterless deamination case (Figure 6). Succinimide hydrolysis barriers are higher than cyclization barriers, indicating that the bottleneck of the deamidation process is in fact the hydrolysis step. This may in fact explain the observation and isolation of succinimide intermediates during deamidation experiments. Previous computational studies on succinimide formation and succinimide hydrolysis have indicated that the rate determining step for the overall deamidation process is the cyclization step leading to the tetrahedral intermediate [20,22], however, these studies did not take into account the effect of water molecules and activation barriers were calculated for the waterless reactions. Nevertheless, the identity of neighboring residues or backbone orientation may change the position of the bottleneck in the imide-mediated deamidation process. In addition, direct hydrolysis seems to be a competitive reaction to the imide-mediated route even in the absence of acid or base catalysis. The activation barrier for direct hydrolysis (asn → asp) with two-water molecules (42.9 kcal/mol) is slightly lower than the rate-determining step (gem → asp) in the imide mechanism (46.3 kcal/mol). Although, it is well-known that direct hydrolysis is the dominant mechanism for deamidation of Asn under acidic conditions, it has been suggested that at neutral pH the succinimide mechanism is in effect [12][13][14][15][16][17][18][19]. The fact that these barriers are only 3 kcal/mol apart suggests that they are rather competitive and neither mechanism can be ruled out for the deamidation of Asn. Availability of solvent molecules, position of water bridges, intramolecular interactions within the peptide or protein, as well as spatial distribution of neighboring residues and three-dimensional aspects, such as hindrance by secondary structure may effect which mechanism will be at play. The absence of iso-Asp as a product of deamidation in some cases [5] may be a result of secondary structure inhibiting cyclization; in that case direct hydrolysis is destined to be the major pathway for deamidation. CONCLUSION Main conclusions that can be drawn from this study are as follows: 1) water assistance increases the rate of deamidation, a fact already established [26]; 2) the tautomerization route has the lowest barrier for the formation of the succinimide intermediate regardless of the number of water molecules that assist the reaction, including the waterless mechanism; 3) cyclization is the rate-determining step for succinimide formation in all water assisted mechanisms; 4) hydrolysis of the succinimide intermediate is likely to go through a stepwise mechanism, where a gemdiol intermediate is formed; 5) more importantly, succinimide hydrolysis barriers are higher than those for succinimide formation. These conclusions contrast in part with previous calculations that had shown that cyclization is the rate determining step for the formation of the succinimide intermediate [20]. Our study suggests that when the entire deamidation process is considered, the hydrolysis step is the actual rate determining step. As shown in Figure 8, the stepwise hydrolysis barrier is the rate determining step for the overall water-assisted deamidation process, which is likely to proceed through the tautomerization route. This also explains the isolation of succinimide intermediates during deamidation reactions. The bottleneck of this process is therefore, proposed to be the hydrolysis of the succinimide intermediate. Another important finding is the relative ease of direct hydrolysis. In all mechanisms involving the use of explicit solvent molecules (Figures 7 and8) direct hydrolysis seems to be a competitive reaction to the imide-mediated route even in the absence of acid or base catalysis. VI. NON-ENZYMATIC PEPTIDE BOND CLEAVAGE AT ASPARAGINE AND ASPARTIC ACID The previous chapter showed that, as long as water-assistance is possible, the cyclization step is not likely to be the rate-determining step for the overall deamidation process. The bottleneck of the imide-mediated route was instead suggested to be the hydrolysis of the succinimide intermediate; this also explains the isolation and observation of succinimide derivatives in some deamidation reactions. In addition, the direct hydrolysis mechanism was shown to be as feasible as the imide-mediated mechanism, indicating why iso-aspartate is not always observed as a result of deamidation under physiological conditions. However, deamidation INTRODUCTION Asparagine (Asn) and glutamine (Gln), two of the 20 most common natural amino acids, are known to be uniquely unstable under physiological conditions; they spontaneously and nonenzymatically deamidate into aspartic acid (Asp) and glutamic acid (Glu), respectively [1][2][3] However, deamidation is not the only possible fate for these two amino acids; non-enzymatic peptide bond cleavage at the carboxyl-side of Asn and Gln residues has also been experimentally observed [4][5][6]. The multitude of products observed is most likely due to the tendency of Asn and Gln to form rings. Deamidation of Asn at acidic pH is known to occur exclusively through a direct hydrolysis mechanism, where the neutral side chain amide is transformed into a carboxylic acid, forming an Asp residue [7]. However, Asn deamidation at neutral pH has been suggested to take place via a cyclic imide, further experimental [8][9][10][11][12][13][14] and computational [15][16][17][18][19][20] studies have supported this idea. In the succinimide-mediated deamidation mechanism, a nucleophilic attack of the carboxyl-side backbone NH to the Asn side chain carbonyl occurs, forming a cyclic tetrahedral intermediate (Scheme 1). The cyclization step is then followed by deamination, where an ammonia molecule is ejected, to form an even more stable intermediate, the succinimide. Subsequent hydrolysis of the amide bonds on the succinimide ring forms the Asp and iso-Asp products. The rate determining step of this mechanism has been suggested to be the initial cyclization step [8,15], where the backbone amide acts as a nucleophile. However, a recent theoretical study on the deamidation mechanism of a small model peptide has shown that the barrier for the cyclization step is much lower with waterassistance and succinimide hydrolysis is the actual rate determining step of the overall deamidation reaction [21], noting that results may vary for different n +1 residues. On the other hand, it was also shown that for the same system deamidation through direct hydrolysis of the Asn side chain at neutral pH is at least as feasible as deamidation through a cyclic imide [22]. 93 where succinimide formation cannot occur [4,[25][26][27][28], while Asn-Ile gave both deamidation and peptide fragmentation products. Cleavage products are formed in most peptide deamidation experiments, but the reaction is usually much slower than deamidation. Asn peptide deamidation half-times range from about 1 to 400 days, Asn cleavage rates range from about 200 to more than 10,000 days [14]. Cleavage of Asn-Pro is the fastest sequence and, since its backbone nitrogen lacks a proton, Asn-Pro deamidates by slow hydrolysis; hence cleavage is the principal degradative pathway for Asn-Pro sequences [4,26]. Peptide bond cleavage at Asp and Glu residues has also been reported in peptides and proteins [29][30][31][32] and the occurrence rate is higher than that for Asn and Gln. The side-chain carboxylic acid group is suggested to play a key role in the cleavage process; this is referred to as the "aspartic acid side chain effect" [33][34]. Since Asn deamidation occurs more readily than backbone cleavage, it is important to investigate the probability of deamidation preceding cleavage, in which case, mechanism and energetics of Asp cleavage becomes essential. The aim of this study is to computationally explore the mechanistic differences between deamidation and peptide bond cleavage at Asn and investigate backbone cleavage at Asp. In order to account for the rate difference between cleavage and deamidation, energetics of peptide fragmentation will be compared with previous calculations [21] on deamidation of Asn. Moreover, mechanistic and energetic information on Asp backbone cleavage will enable a reasonable prediction for the likelihood of Asn deamidation preceding cleavage. Instead of Asn itself undergoing cleavage, it may convert to Asp via deamidation and cleavage may take place henceforth. Knowledge on the mechanism and energetics of Asn cleavage may also be useful in cases where deamidation is hindered by protein tertiary structure, in which case cleavage may become a competing reaction. COMPUTATIONAL METHODOLOGY As mentioned earlier, Asn peptide bond cleavage mechanism studied herein will be compared with the Asn deamidation mechanism previously studied by the authors [21] and therefore the same methodology was employed in this study. Moreover, the same model peptide (Scheme 2) was used for a fair analysis. All structures and energy values reproduced from previous deamidation studies are indicated by an asterisk (*). Previous computational studies have shown that water molecules have a catalytic effect on deamidation [21], whereby they serve as efficient proton conduits. The backbone cleavage mechanism was investigated in light of this information. SCHEME 2: Model Peptide with Asn Residue. H 3 C N H H N CH 3 O O Model Compound H 2 C H 2 N O Full geometry optimizations were performed in gas-phase -without any constraints-using the density functional theory (DFT) [35][36][37][38] at the B3LYP/6-31+G** level [39][40][41]. The use of this basis set and method in similar peptide systems is well established [20,21]. Stationary points were characterized by a frequency analysis. Zero-point energy and thermal corrections were attained using the ideal gas approximation and standard procedures. Local minima and first-order saddle points were identified by the number of imaginary vibrational frequencies. The species reached by each transition structure was determined by intrinsic reaction coordinate (IRC) calculations [42,43]. Relative free energies of activation (∆G ‡ ) are calculated as the difference of free energies between transition states and reactants (reactantwater complex where applicable) of each step. Energy values for gas-phase optimizations listed throughout the discussion include thermal free energy corrections at 298 K and 1 atm. The self-consistent reaction field (SCRF) theory, utilizing the integral equation formalismpolarizable continuum (IEF-PCM) model [44][45][46][47] in H 2 O (ε = 78.0) at the B3LYP/6-31++G** level was used to account for the effect of a polar environment. Bondi radii [48] scaled by a factor of 1.2 were used for all solvent calculations. Single point energies for solvent calculations include nonelectrostatic and thermal free energy corrections obtained from gas-phase optimizations. All calculations were carried out using the Gaussian 03 program package [49]. All distances and free energies listed in the discussion are in angstroms (Å) and kcal/mol, respectively. Single point solvent energies are given in parenthesis. RESULTS and DISCUSSION The Previous studies on the deamidation mechanism have revealed the catalytic effect of water molecules [21]. In this particular reaction step, the assistance of the water molecule has a favorable effect, however it is not substantial, as may be seen in the barrier heights reported (Figure 1). Single point solvent energies in water indicate the stabilizing effect of a polar environment. Both transition states have a somewhat zwitterionic structure, and therefore benefit from electronic interaction with the dielectric continuum. This is indicated by approximately 8 kcal/mol decrease in barrier height for both ring closure reactions involving TS1 and TS1-H 2 O (Figure 1). Although, the attacking species and the products are different, the free energy of activation for the concerted non-assisted ring closure in deamidation and cleavage pathways is rather close, with less than 1 kcal/mol free energy difference (Figure 2). Energy values in gas phase (normal) and solution (italics) are given in kcal/mol. Fates of Intermediate I While the deamidation pathway proceeds with a water assisted deamination [21] (expulsion of NH 3 ), Intermediate I does not have a free NH 2 group for this to occur. However, it may undergo a water-assisted cleavage of the peptide backbone and form fragmentation products. Besides the ring amide may be hydrolyzed to give a non-cyclic intermediate. Both reactions are studied below. a. Peptide Bond Cleavage at Intermediate I Water molecules assist the fragmentation of the peptide bond on the carboxyl-side (Scheme 4) by allowing a proton relay from the -OH on the ring to the carboxyl-side backbone NH group. In this way, the alcohol functionality will be converted into a carbonyl group enhancing conjugation in the cyclic structure, while the backbone of the peptide breaks. This will lead to two fragments, one of which bears a free NH 2 group and the other will be carrying a succinimide ring at its tail end. 3). The transition state geometries for the backbone cleavage step reveal some prerequisites for peptide fragmentation (Figure 4). Considering backbone rigidity in a peptide or protein, the proton transfer from the ring -OH to the backbone NH must take place via a solvent molecule. In addition, proper alignment of these two entities is crucial; the -OH should not be involved in any other intermolecular interactions, the NH group should be in the proper conformation to accept the proton and above all, a solvent molecule must form a water bridge between these two entities in order to facilitate proton transfer. These requirements are not too challenging for a peptide chain with considerable flexibility and access to solvent molecules. However, inside the three-dimensional structure of a protein the possibility of proper alignment and water bridge availability may be limited. Å in the ground state). The nucleophilic attack distance is 2.104 Å. The cleavage of the ring bearing the gemdiol is depicted in TS5-H O (Figure 6). In this step, a proton is transferred from the gemdiol to the adjacent nitrogen via a water molecule. TS5-H 2 O may be described as an early transition state (reactant-like), since the proton transfer is not complete and the cleavage of the ring is not quite advanced, as indicated by the relatively short C-N bond (1.562 Å compared to the ground state C-N distance of 1.466 Å). Activation energies for the stepwise amide hydrolysis (Figure 7) shows that both steps are energetically rather close and the rate-determining step of the two-step process is the ring opening step. Previous studies on amide hydrolysis have suggested that a stepwise mechanism going through a gemdiol intermediate has a considerably lower barrier than a concerted reaction [50]. Energetics of the water-assisted amide hydrolysis modeled herein is in accord with this expectation; the stepwise (gemdiol-mediated) route is more favorable than the concerted pathway by almost 10 kcal/mol. The energetic comparison of the possible fates of Intermediate I reveals a barrier difference of approximately 25 kcal/mol between carboxyl-side peptide bond cleavage (Figure 4, ∆G ‡ =14.7 kcal/mol) and ring amide hydrolysis (Figure 7, ∆G ‡ =38.2 kcal/mol). It is apparent that fragmentation is much more favorable than amide hydrolysis at the Intermediate I stage. TS4-H 2 O However, backbone cleavage may be prohibited inside a protein, where the conformational space around the Asn residue is severely restricted by spatial distribution of neighboring residues and/or secondary structure, such as intramolecular H-bonds that form β-sheets or αhelices. In that case, at Intermediate I, ring amide hydrolysis may take place to form Intermediate II which in a subsequent step could undergo deamidation to reveal Asp and/or cleavage at the carboxyl-side peptide bond. This is discussed in the next section. Energy values in gas phase (normal) and solution (italics) are given in kcal/mol. Fates of Intermediate II Intermediate II may go through a water-assisted deamination step (Scheme 6) that will reform the backbone peptide bond and an Asp residue is formed. This route may be suggested as an alternative pathway to the traditional deamidation mechanism (Scheme 1). Intermediate II may as well undergo peptide backbone fragmentation to reveal cleavage products, given that proper orientation of the backbone is feasible for this reaction (Scheme 7). Cleavage products are similar to those at the Intermediate I stage (Scheme 4), however, in this case there are no rings at any of the tail ends. The availability of a water bridge between these two entities is essential and depends on the position and proper alignment of the backbone NH group. Energy values in gas phase (normal) and solution (italics) are given in kcal/mol. Deamidation versus Peptide Backbone Cleavage at Asn The overall energetics of peptide bond cleavage at Asn is depicted in Figure 9. The initial cyclization step, with an activation barrier of 47.6 kcal/mol, is incidentally the ratedetermining step for peptide bond cleavage at Asn. Deamidation may proceed through different pathways, as suggested by the large range of experimental reaction rates even for the same primary sequence of amino acids [51]. In general, it has been assumed that the first step in deamidation consists of a concerted cyclization. We have shown that the corresponding activation barrier is comparable to that found for the equivalent concerted cyclization step in backbone cleavage (either in nonassisted or water-assisted mechanisms, see Figures 2 and3 respectively). Thus, if deamidation involves concerted cyclization as the first reaction step, deamidation and backbone cleavage would display comparable reaction rates, in contrast with known experimental facts [14]. Actually, we have reported previously that deamidation may proceed through alternative pathways, specifically direct hydrolysis of the Asn side chain [22] or cyclic imide formation via a tautomerization route [21]. Both mechanisms involve lower activation barriers than concerted cyclization provided water assistance is accessible. Comparison between theoretical and experimental data for deamidation versus backbone cleavage suggests therefore that deamidation proceeds through one of these water-assisted processes. Note that in case backbone cleavage at Intermediate I is inhibited, the hydrolysis of the ring amide may eventually occur giving rise to an Asp residue (Scheme 5), i. e. the major product of the deamidation reaction. However, since Intermediate I only has one carbonyl functionality (unlike the succinimide ring), its hydrolysis cannot lead to the other deamidation product, the iso-Asp. Direct hydrolysis of the Asn side chain will also result in Asp formation, exclusively. Indeed, there are many instances where only Asp is formed as a result of deamidation [52]. Energy values in gas phase (normal) and solution (italics) are given in kcal/mol. B. PEPTIDE FRAGMENTATION AT ASP RESIDUES Asp can also undergo peptide chain cleavage through formation of an anhydride intermediate and by means of a mechanism similar to peptide bond cleavage at Asn. The Asp side chain carboxylic acid group is a better candidate for nucleophilic attack than the Asn side chain NH 2 , since the Asp side chain is expected to be present as carboxylate at physiological pH. However, the effective pK a of the acid may depend substantially on the molecular environment. In the present work, the reaction has been modeled with the carboxylic acid form of Asp rather than the carboxylate anion. The calculated barrier may be considered as an upper limit for the process since the carboxylate anion would be a stronger nucleophile. The Asp side chain attacks the backbone carbonyl and a cyclic tetrahedral intermediate (Asp-Intermediate I) forms in the initial step (Scheme 8), much like the case in Asn (Scheme 3). The ring closure step is followed by the actual cleavage step (Scheme 8). Cleavage products are similar to those for Asn, except an anhydride ring forms at one of the peptide tail ends, instead of a succinimide ring (Scheme 4). The overall energetics of peptide bond cleavage at Asp is depicted in Figure 12. Although the activation barrier for the initial cyclization step is higher (30.5 kcal/mol) than the following cleavage step (16 kcal/mol), the latter is rate-determining in contrast with the Asn case. Energy values in gas phase (normal) and solution (italics) are given in kcal/mol. Analysis of the energetics of backbone fragmentation at Asp residues might shed light on whether fragmentation is more likely to occur subsequent to deamidation of Asn into Asp, rather than cleavage at Asn itself. It is important to emphasize the difference in activation barriers (taken roughly as the relative free energy of the highest TS) of Asn (47.6 kcal/mol, Figure 9) and Asp (38.0 kcal/mol, Figure 12) cleavage. Peptide bond cleavage is energetically much more favorable for Asp. From the energetic data acquired in this study and previous computational studies done on deamidation of Asn [21,22], it is therefore reasonable to suggest that in relatively flexible peptides, backbone cleavage at Asn residues may be preceded by deamidation into an Asp. CONCLUSION In this study, mechanisms leading to non-enzymatic peptide bond cleavage at Asn and Asp have been investigated using computational methods. Mechanism and energetics of peptide fragmentation at Asn has been comparatively analyzed with previous calculations on deamidation of Asn [21]. The cyclization step was shown to be the rate determining step for backbone cleavage at Asn (Figure 9). Although concerted cyclization for deamidation and cleavage have comparable activation barriers, previous computational studies have shown that deamidation does not necessarily involve such a reaction step. Direct hydrolysis of the Asn side chain [22] and cyclic imide formation via a tautomerization route [21] have lower activation barriers. These processes are water assisted and therefore require the presence of water molecules. Backbone cleavage is unlikely to be competitive with deamidation in peptides with access to solvent molecules. An important conclusion of this study is the energetics of backbone cleavage at Asp residues. The peptide fragmentation barrier is indeed much lower (approximately 10 kcal/mol) for Asp than that for Asn. We therefore, suggest that cleavage at Asn residues takes place after an Asn residue has deamidated into an Asp. Since the cleavage products differ for Asn (Scheme 4) and Asp (Scheme 8), experimental verification of this proposal appears to be quite feasible. VII. PRIMARY SEQUENCE DEPENDANCE OF ASPARAGINE DEAMIDATION RATES Previous chapters mainly focused on the mechanistic details, energetics and feasibility of the formerly suggested succinimide-mediated deamidation route, which involves the Asn n +1 residue. As a result the most plausible pathways for deamidation have been outlined. A recent experimental study on pentapeptides has shown that Asn deamidation rates are directly related to primary structure (peptide sequence) near Asn, with a more prominent effect from the carboxyl-side (n +1) residue. This is in accord with the succinimide mechanism; however, the effect of the identity of the n +1 residue is unclear. The objective of the following study is to explore the correlation between experimental deamidation rates and primary structure of peptides and proteins, in order to identify the factor(s) causing this dependence. Quantummechanical calculations were performed in order to compare activation barriers for systems with different n +1 residues. On the other hand, molecular dynamic simulations were used to better understand the differences in the dynamic structure. In addition to the previously suggested succinimide-mediated deamidation pathway that was used as a benchmark during this study, an alternative route for deamidation -involving the Asn amino-side-has been proposed here for the first time. This paper is still a work in progress and the final form of the paper is intended for publication in Biochemistry. INTRODUCTION Non-enzymatic asparagine (Asn) deamidation occurs spontaneously under physiological conditions [1][2][3][4][5] and is known to limit the lifetime of peptides and proteins [6][7][8]. The conversion of Asn to aspartate (Asp) has been suggested to go through a stepwise pathway involving a cyclic imide [9,10] (Scheme 1). Detection and isolation of succinimide derivatives as well as isomerization of Asp to iso-aspartate (iso-Asp) during some peptide deamidation experiments has supported this idea [9][10][11][12][13][14][15][16]. The succinimide-mediated Asn deamidation mechanism has been subject to several computational studies [17][18][19][20][21][22][23][24]. A recent computational study has shown that the ratedetermining step of this mechanism is the hydrolysis of the cyclic intermediate [24]; this explains the relatively long lifetime of the succinimide intermediate and its detection/isolation during peptide deamidation experiments. However, there are many cases where an intermediate was not observed and deamidation did not lead to iso-Asp formation [5]; this has been explained through the presence of another mechanism, direct hydrolysis, which was shown to be operating at ease under physiological conditions [24]. Direct hydrolysis is the conversion of the Asn side chain amide into a carboxylate through a noncyclic concerted step. It is well-known that the hydrolysis of amides is catalyzed in the presence of acid or base [25]; however, it was shown that direct hydrolysis is at least as feasible as the succinimide-mediated route even under physiological conditions [24]. All Asn residues in peptides and proteins are inherently unstable and are prone to deamidation. The conversion of the neutral amide side chain (Asn) into a negatively charged carboxylate (Asp) causes substantial conformational changes in peptides and proteins. There are many reports that these changes markedly affect protein function or stability or both [26]. Robinson et al. have studied the primary sequence dependence of deamidation rates on pentapeptides [27]. They found that deamidation rates could be varied over a wide range by changing primary sequence. The timed processes of protein turnover, development, and aging have been suggested as possible roles for deamidation [28]. It has been hypothesized that deamidation serves as a molecular clock for the timing of biological processes [29]. Robinson et al. have measured the rates of deamidation of 64 peptides between 3 to 13 residues in length [27]. They showed that a pentapeptide model, where Asn is the central amino acid, is adequate to analyze the primary sequence dependence of Asn deamidation. The model Gly-Xxx-Asn-Yyy-Gly was designed so that the charges on the peptide ends are moved away from the amide by the Gly residues, without making the peptide so long that it might adopt special three-dimensional configurations. The complete 800-pentapeptide set of all possible combinations of the sequences Gly-Xxx-Asn-Yyy-Gly and Gly-Xxx-Gln-Yyy-Gly, where Xxx and Yyy are any of the 20 naturally occurring amino acid residues were synthesized [27]. Deamidation rate was shown to be controlled primarily by the carboxyl side residue (Yyy) with smaller effects from the amino side residue (Xxx) (Figure 1). This is consistent with the succinimide reaction mechanism that was originally proposed to explain the unusually rapid deamidation rates of Asn-Gly sequences and iso-Asp formation [10,11]. Individual deamidation rates of these peptides revealed some remarkable characteristics (Table 1). Glycine (Gly) -with no interfering side chain-exhibited an unusually short halftime of 1 day; this was associated with the ease of succinimide ring formation. The additional methyl group in alanine (Ala) increased the half-time to 25 days; larger aliphatic groups increased the half-time even more, with the most extreme example being isoleucine (Ile), which has a mean half-time of about 320 days. Increase in steric hindrance was suggested as the reason for the difference in half-times. However, in light of previous computational studies, which have proved the catalytic effect of water molecules [23,24], the size of the hydrophobic side chain could also effect deamidation rates by inhibiting solvent accessibility; this would impair deamidation whether it proceeds through the cyclic intermediate or through direct hydrolysis. Significant deamidation of Asn-Pro (proline) sequences were not observed even after 1000 days. Since the backbone nitrogen is covalently bound to the side chain, Asn-Pro is not expected to deamidate through the succinimide mechanism. On the other hand, polar and charged side chains were shown to cause an increase in the deamidation rate. Histidine (His), which is expected to have similar steric hindrance to that of phenylalanine (Phe), deamidates 7 times faster. Similarly threonine (Thr), which is similar to valine (Val) in terms of shape and size, has a deamidation rate that is 5 times faster, possibly due to the presence of the OH group, which facilitates H-bonding with nearby solvent molecules, rather than repel them. The fact that the identity of the n +1 residue has a major effect on deamidation half-times of pentapeptides -while the n -1 residue's effect is comparably negligible-is anticipated due to the succinimide-mediated mechanism, where the initial cyclization step involves the attack of the carboxyl-side NH. The size and charge of the n +1 side chain is expected to affect the mode of attack, hence enhancing or inhibiting deamidation. However, there is no direct correlation between size and charge of the carboxyl side residue and deamidation half-times listed for pentapeptides (Figure 2). Phenylalanine, tryptophan (Trp) and tyrosine (Tyr) -three amino acids with aromatic side chains-have much lower deamidation half-times than Val, which bears a small isopropyl group. It is important to understand the factors that cause the differences in deamidation rates, keeping in mind that there may be several parameters operating simultaneously. The objective of this study is to get a deeper insight on primary sequence dependence of pentapeptide deamidation rates, using computational methods. The succinimide-mediated mechanism will be modeled with different n +1 residues and energetically compared, in an effort to see whether activation barriers are similar for different carboxyl-side residues. This requires quantum-mechanical calculations on a model compound which encompasses the carboxyl side residue. For this purpose, three end-blocked dipeptides, ACE-Asn-Yyy-NME (Scheme 2) -where Yyy is Gly, Ala and Val-were chosen and modeled with regard to the succinimide-mediated mechanism, in light of the knowledge gained from previous work [23,24]. Pentapeptide deamidation half-times (Table 1) are quite different for Gly, Ala and Val (1,25 and 250 days, respectively); this is expected to be reflected in deamidation energetics, in case the difference is caused by variance in activation barriers. However, there may be an alternative explanation for the large difference in rates of these three pentapeptides. We have shown before [23,24] that water molecules play a crucial role in deamidation mechanisms by assisting proton transfer and hydrolysis steps. Accordingly, the increase in deamidation halftimes for the three pentapeptides might be correlated to the increase in hydrophobicity of the three side chains (Gly, Ala and Val, respectively), since water access to the backbone NH may be increasingly hindered as the side chain gets larger. To explore this hypothesis, molecular dynamic (MD) simulations were performed on these pentapeptides, Gly-Gly-Asn-Yyy-Gly -where Yyy is Gly, Ala and Val-in order to investigate the difference in solvent distribution. Molecular dynamic simulations will also enable a comparison with respect to the per cent occurrence of the reactive conformers observed in quantum-mechanical calculations, also known as near attack conformers (NACs). For this purpose, MD simulations will be studied in detail, with respect to accessibility of solvent molecules, backbone structure, intramolecular H-bonds and water bridges. Apart from the results of the pentapeptide experiment, there are several experimental studies where the identity of the n -1 residue is influential on deamidation rate [30,31]. This may be due to a different mechanism that is in effect, which involves the amino-side of the Asn residue. Exploring the presence and/or feasibility of such a mechanism through quantummechanical calculations is an additional aim in this study. COMPUTATIONAL METHODOLOGY There are two main theoretical approaches in this study; energetic comparison among three different dipeptides (Scheme 2) will be facilitated by quantum-mechanical (QM) calculations utilizing the Density Functional Theory (DFT); the alternative route involving the Asn amino-side will also be explored with this methodology. In addition, molecular dynamic (MD) simulations will be used in order to investigate the time-dependent behavior of three pentapeptides (Gly-Gly-Asn-Yyy-Gly, where Yyy is Gly, Ala and Val) and their dynamic differences. Methodological details and practical aspects of the two methods are described below in detail. QM Calculations The same level of theory and basis set are used with aforementioned computational studies on deamidation [23,24]. Full geometry optimizations were performed in gas-phase -without any constraints-using the density functional theory (DFT) [32][33][34][35] at the B3LYP/6-31+G** level [36][37][38]. The use of this basis set and method in similar peptide systems is well established [23,24]. Stationary points were characterized by a frequency analysis. Zero-point energy and thermal corrections were attained using the ideal gas approximation and standard procedures. Local minima and first-order saddle points were identified by the number of imaginary vibrational frequencies. The species reached by each transition structure was determined by intrinsic reaction coordinate (IRC) calculations [39,40]. Free energies of activation (∆G ‡ ) are calculated as the difference of free energies between transition states and reactants (reactantwater complex) of each step. Energy values for gas-phase optimizations listed throughout the discussion include thermal free energy corrections at 298 K and 1 atm. All calculations have been carried out using the Gaussian 03 program package [41]. All distances and free energies listed in the discussion are in angstroms (Å) and kcal/mol, respectively. MD Simulations Molecular dynamic simulations have been performed using the AMBER 9.0 program package [42]. All simulations were performed in water, using the TIP3P water model. Calculations were employed at 300 K in the isobaric-isothermal (NPT) ensemble, using a single pentapeptide molecule in a cubic box with periodic boundary conditions. Each system contained approximately 12 600 atoms and 4180 water molecules. The step size was chosen to be 0.001 ps. Nonbonded interactions were cut off at 9.0 Å for the direct sum and Particle Mesh Ewald was employed to account for long-range electrostatic interactions. For each pentapeptide, a 5 ns simulation was performed and coordinates were saved every 0.2 ps for analysis. RESULTS AND DISCUSSION The first part of this study is based on quantum-mechanical calculations on the succinimidemediated pathway (Scheme 1) using a new and larger model compound (Scheme 2) that will mimic the Asn residue together with its carboxyl-side residue. In addition, calculations on the deprotonation of the backbone NH for different n +1 residues will be discussed. The second part of the discussion consists of analysis of molecular dynamic simulations performed on three pentapeptides, with significantly different deamidation half-times, in order to get a dynamic -rather than a static-perspective. Finally, an alternative deamidation mechanism, involving the Asn amino-side, will be discussed. I. ENERGETICS FOR DIFFERENT n +1 RESIDUES Primary sequence dependence of Asn deamidation was explored through the succinimidemediated mechanism, which was modeled with different n +1 residues (Scheme 2) in order to see the differences in activation barriers. For this purpose, three end-blocked dipeptides, ACE-Asn-Yyy-NME -where Yyy is Gly, Ala and Val-were used and modeled in light of the knowledge gained from previous computational studies on deamidation [23,24]. Succinimide formation was previously shown to proceed through several mechanisms; the tautomerization route (Scheme 3) was suggested to be the most plausible pathway [23,24]; this route consists of three steps, the first being the tautomerization of the Asn side-chain amide. Asn side-chain tautomerization is not expected to be noticeably effected by the n +1 side chain, at least in the case of peptides with considerable flexibility and large solvent CH 3 R R R R However, if the hydrophobicity of the carboxyl-side R group defers water access to the Asn side chain, this may effect the ease of tautomerization, hence deamidation half-times. Therefore, inside the three-dimensional structure of a protein, a large aliphatic group nearby the Asn residue may significantly decrease deamidation rates. The cyclization step leading up to the tetrahedral intermediate was previously proposed to be the rate-determining step for the formation of the succinimide intermediate (Scheme 3). This step is followed by water-assisted deamination to reveal the succinimide. In order to see the effect of the n +1 side chain on reaction barriers, the concerted cyclization step was modeled with no water, one-water and two-water assistance (Table 2). Water-assistance has lowered the activation barriers for all Asn-Yyy sequences, the catalytic effect of solvent molecules on deamidation rates has already been previously suggested [23,24]. The effect of the n +1 side chain is apparent in the cyclization barriers listed (Table 2); as the size of the Yyy side chain increases cyclization becomes more difficult, possibly due to steric hindrance. The experimental deamidation half-times listed for pentapeptides with Yyy as Gly, Ala and Val correspond to barrier differences of 1.9 and 3.3 kcal/mol for Gly/Ala and Gly/Val, respectively (calculated using the relationship δ∆G = -RTln(t 2 /t 1 ) where t is halftime, T=310K, and we assume the same number of assisting water molecules for all pentapeptide reactions). Calculated barrier differences -although consistent in terms of trenddo not reproduce the same results as experiment, this suggests that the rate difference does not solely depend on the difference in activation barriers for the three Asn-Yyy sequences under study. Nonetheless, molecular dynamic simulations will provide further information on solvent accessibility and per cent occurrence of near attack conformations (NAC). A current computational study has suggested [24] that the rate-determining step for the overall deamidation process is in fact the succinimide hydrolysis step (Scheme 1) and not the cyclization step leading to the formation of this intermediate (Scheme 3). However, the aforementioned study was performed on a smaller model system and it has been suggested that the identity of the n +1 residue might reverse the order of the rate-determining step, considering that the differences were not very significant. On the other hand, the effect of the carboxyl-side R group is not expected to increase the barriers for hydrolysis of the ring carbonyl and we have not carried out computations for such a reaction step. Thus, if hydrolysis is the rate-determining step in deamidation of pentapeptides, the role of the n +1 residue should probably be related to the change in water access to the reaction site, i.e. to the hydrophobicity of the side chain near the Asn residue. Deprotonation of the n +1 residue for different Asn-Yyy sequences The ease of deprotonation of the n +1 backbone NH is essential for deamidation studies. It could be considered as an important measure of susceptibility to cyclization; once the deprotonated structure is formed the cyclization step will be much easier [17]. A previous computational study has shown that the barriers for cyclization are as low as 10 kcal/mol for the anionic nitrogen case [17]. Quantum-mechanical calculations discussed in this section give an idea of how the increase in size of the hydrophobic side chain on the carboxyl-side of Asn will change activation barriers and the difference in free energy of deprotonation for the n +1 backbone NH, which are closely related to the deamidation process. However, there are dynamic aspects in reactions that cannot be accounted for with this methodology. For this purpose, MD simulations have been performed in order to better understand the differences in timedependent behavior of these systems in solution. II. MOLECULAR DYNAMIC SIMULATIONS OF PENTAPEPTIDES In this section, results of molecular dynamic (MD) simulations performed on three pentapeptides (Gly-Gly-Asn-Yyy-Gly, where Yyy is Gly, Ala and Val) will be discussed. MD simulations will be analyzed in terms of solvent accessibility, intramolecular water bridges, backbone structure, and per cent occurrence of near attack conformations (NAC). Intramolecular Hydrogen-bonds Information on intramolecular hydrogen-bonds between proton-acceptor and proton-donor sites is of substantial value; these interactions will be an indicator for the occurrence of reactive conformers (near-attack-conformers) throughout the course of the simulation. Prominent H-bonding interactions have been listed in Table 3; occurrence rates below 1% were discarded. Distances for H-bonds between proton-acceptor/proton-donor were taken as < 2.3 Å. Atomic numbering (Scheme 5) is identical for all three pentapeptides regardless of the identity of the n +1 residue. Intermolecular H-bonds are diverse for the three pentapeptides studied (Table 3). These peptides are quite small and rather flexible, therefore binding secondary structure is not expected, however, there seems to be a reasonable amount of long-lasting prominent intermolecular interactions, especially for the Ala and Val cases, as seen from the interaction between H 39 -O 16 . Long-lasting intramolecular H-bonds between sites on the peptide backbone suggest that a level of secondary structure exists. High per cent occurrence of Hbonds for a particular atom suggests that it is not readily available to interact with other groups, as is the case for H 32 . Nonexistence of H-bonds also provides valuable information; the absence of a H-bond between O 25 -H 32 is noteworthy. This set of data does not provide a meaningful explanation for the deamidation rate difference observed in the pentapeptide experiment, however, it is important to emphasize that H-bonding pattern is rather different for the three pentapeptides that only differ in terms of an R group on the n +1 residue. Intramolecular Water-bridges Previous computational studies on deamidation have revealed the importance of intramolecular water-bridges, which enable transfer of protons and catalyze steps in the deamidation process [23,24]. In light of this knowledge, MD simulations have been analyzed and long-lasting, prominent water-bridges among potential proton-accepting and donating sites were listed for all three pentapeptides. Per cent occurrences of water-bridges were evaluated for the entire simulation (Table 4). Distances for water bridges between protonacceptor/proton-donor were taken as < 2.3 Å. The significant per cent occurrence of water-bridges between O 25 -H 27 indicates that favorable structures for water-assisted side-chain tautomerization are frequently encountered for all three pentapeptides. The nonexistence of certain water-bridges throughout the simulation is also very relevant. The absence of a water bridge between O 25 -H 32 indicates that the reactive conformer (near-attack-conformers) for one of the prominent water-assisted cyclization mechanisms is not readily encountered. Pentapeptide MD simulations were also analyzed in terms of critical distances and dihedrals. The aim was to pinpoint differences in the evolution of three-dimensional structure for the three pentapeptides under study. Atoms with potential reactivity were particularly important; in light of the deamidation mechanisms previously studied [23,24], distances between nucleophilic atoms and electrophilic centers were analyzed. In addition, backbone dihedrals were studied, in order to deduce the three-dimensional structure of the peptide backbones as well as the relationship between Asn side-chain amide and the peptide backbone. Graphs shown in Figures 3 and4 are for Gly, Ala and Val, respectively. Distances are in Å, dihedrals are in degrees. a. Critical Distances In terms of the cyclization step in the succinimide-mediated deamidation reaction the timedependent behavior of the nucleophilic and electrophilic sites are of great importance. The evolution of the distances between C 24 -N 31 an O 25 -H 32 during the course of the simulation is given in Figure 3. An average distance of 4 Å is observed for C 24 -N 31 ; in terms of the nucleophilic attack leading to ring closure, there is no apparent effect of the difference in n +1 residues. In addition the O 25 -H 32 distances are rather large for proton transfer to occur. b. Critical Dihedrals Peptide backbone dihedrals are important in terms of understanding the orientation of reactive groups with respect to one another. For this purpose, critical dihedrals have been plotted for the three pentatpeptides (Figure 4). The most important observation in terms of the peptide backbone is the orientation of N 31 that can be seen in the evolution of the dihedral ψ 2 as well as the orientation of the Asn side-chain which is indicated by dihedral χ 1 (Figure 4). A newman projection (Scheme 6) better demonstrates the orientation of N 31 with respect to the Asn side-chain. The progress of dihedrals ψ 2 and χ 1 -throughout the MD simulation-indicate that N 31 and C 24 are never on the same side of the peptide backbone, except for a short instance for Gly (see χ 1 in Figure 4). This is true for all three pentapeptides and it is important to emphasize that if cyclization is to occur a major rotation must take place either on the peptide backbone (ψ 2 ) or on the Asn side chain (χ 1 ). III. ALTERNATIVE DEAMIDATION PATHWAY As mentioned previously, contrary to the results of the pentapeptide experiment, there are several experimental studies where the identity of the n -1 residue is influential on deamidation rate [30. 31]; this may be due to a different mechanism that is in effect. This part of the discussion will be based on exploring an alternative deamidation pathway, which involves the amino-side of the Asn residue. The feasibility of this mechanism will be compared, in terms of energetics, with the succinimide-mediated mechanism previously modeled [23,24]. Previous computational studies on the succinimide mechanism were performed on a smaller model compound (ACE-Asn-NME) [23,24]. For comparative purposes, the alternative mechanism investigated herein was modeled using the same model. Deamidation via the amino-side As depicted in Scheme 7, an alternative pathway, which goes through a cyclic intermediate previously reported [23,24]. In the second step, Int I undergoes deamination -with water-assistance-to lose an ammonia molecule and form a ring (Int II) that is more stable due to conjugation. The water molecule helps proton transfer from the ring -OH to the -NH 2 group. The transition state shows that proton transfer is incomplete and ejection of the NH 3 is yet to occur, as seen in the C-N distance, which is still quite short (1.580 Å) The deamination step is straightforward and enables better conjugation throughout the ring atoms (Figure 6); hence, relative energies reveal that Int II is approximately 10 kcal/mol more stable than Int I. The activation barrier for the deamination step of Int I (17 kcal/mol) is basically identical to the corresponding step in the succinimide-mediated deamination [23,24]. The last step of this mechanism is the hydrolysis of Int II to give a peptide with an Asp residue (Figure 7). This particular hydrolysis step requires more than one water molecule. In fact, the proton relay mechanism takes place through three water molecules. A solvent molecule attacks the ring carbonyl carbon (O-C distance 2.575 Å); the O-(C=O) bond on the ring lengthens (1.545 Å); meanwhile, through a proton relay, solvent molecules donate a H + to the backbone N (N-H distance 1.032 Å) which had originally lost a proton in the first step (Figure 5). In this way, the ring opens and the peptide backbone is restored. The side chain now bears a carboxylic acid group instead of the amide group, deamidation is complete. The energetics of the overall process is comparable with the succinimide mechanism previously studied [23,24]. 147 intermediates during deamidation reactions. The bottleneck of this process is therefore, proposed to be the hydrolysis of the succinimide intermediate. Another rather important finding was the competitiveness between direct hydrolysis and the imide-mediated route at neutral pH. It was shown that direct hydrolysis is as plausible as imide-mediated deamidation of Asn even in the absence of acid or base catalysis. In the third study, mechanisms leading to non-enzymatic peptide bond cleavage at Asn and Asp have been investigated and it has been suggested that backbone cleavage is unlikely to be competitive with deamidation in peptides with access to solvent molecules. However, the fact that peptide cleavage products at Asn residues are quite often encountered in experimental studies was explained through the ease of backbone cleavage near Asp. Peptide fragmentation barriers are much lower (approximately 10 kcal/mol) for Asp than those for Asn, it is suggested that cleavage at Asn residues may take place after an Asn residue has deamidated into an Asp. In other words, Asn cleavage may be a result of a deamidation reaction followed by a consequent cleavage of the Asp. This proposal may be subject to verification by experiments, since the cleavage products differ for Asn and Asp. The last study aimed to better understand the reasons behind the significantly different Asn deamidation rates in pentapeptides with different n +1 residues were explored. Calculated activation barriers for the cyclization step leading to the succinimide formation, which was previously suggested to be the rate-determining step, was consistent with the trend dictated by experimental half-times. In addition, deprotonation of the n +1 NH reproduced similar results, demonstrating that different R groups on the side-chain of the n +1 residues influence the ease of deprotonation. On the other hand, MD simulations of pentapeptides with the sequence Gly-Gly-Asn-Yyy-Gly, where Yyy is Gly, Ala and Val, revealed that near-attack-conformations, which are predicted from QM calculations, were not encountered during the simulation. In addition, some of the critical water-bridges and intramolecular H-bonds were non-existent during the simulations due to the absence of the correct orientation of reactive groups. This suggests that either a different mechanism for deamidation may be in effect or major backbone rotation is necessary to achieve the reactive conformation. CHAPTER IX FUTURE WORK IX. FUTURE WORK As a result of the studies mentioned herein, detailed mechanistic insights of several deamidation routes for Asn have been acquired. In light of the work done on Asn, one of the next goals is to investigate the mechanistic aspects of Gln deamidation, in order to rationalize the slower deamidation rate of Gln. This will require a quantum mechanical approach analyzing differences in reaction barriers as well as a molecular dynamic simulation which will enable a comparison of the dynamic differences between Asn and Gln. In this way, pinpointing the factor(s) that cause Gln deamidation to be substantially slower than Asn deamidation may be possible. The next challenge in the pursuit of understanding deamidation will be studying a protein which is known to deamidate, preferentially at two or more sites with the same primary structure but different deamidation rates due to secondary structure. This will enable a better understanding of the effect of secondary structure on deamidation. A good candidate for this study is the protein Bcl-x L . Bcl-xL is an antiapoptotic member of the Bcl-2 family, which inhibits apoptosis initiated by various cellular stresses, and has a pivotal role in the survival of tumor cells [1]. This protein is currently a major topic of interest [2,3]. It is known to deamidate at two of its Asn residues exclusively. Modeling the deamidation reaction in this system will require the use of molecular dynamics simulations and combined QM/MM potentials. Both of the above mentioned studies are in progress. La déamidation des protéines est un thème de grand intérêt qui a été le sujet de nombreuses études théoriques et expérimentales. La déamidation est un processus non-enzymatique et spontané qui convertit les résidus asparagines dans les protéines en acides aspartiques. Le changement de charge aboutit à des changements temporels de conformation dans les protéines et a été associé à la dégradation des protéines et au phénomène de vieillissement. Dans ce manuscrit, certains aspects mécanistiques de ce processus ont été étudiés et de nombreuses mises à jour ont été obtenues sur les mécanismes potentiels amenant à la déamidation. Ces mécanismes et leurs énergies sont présentés en détail. Une autre destinée possible des résidus asparagines, la coupure de la chaîne principale, est introduite et comparée au mécanisme de déamidation. Enfin, des tentatives pour comprendre l'effet des résidues adjacents dans la déamidation des asparagines sont élaborées et plusieurs idées pour un futur travail sont soulignés. Résumé (anglais): Deamidation of proteins is a topic of wide interest that has been subject to experimental and theoretical studies. Deamidation is a nonenzymatic and spontaneous process that converts asparagine residues in proteins into aspartic acid. The change in charge leads to time-dependent conformational changes in proteins and has been associated with protein degradation and ageing. In this manuscript, certain mechanistic aspects of this process have been investigated and many insights have been attained on potential mechanisms leading to deamidation. These mechanisms and their energetics have been presented in detail. Another potential fate of asparagine residues, backbone cleavage, has been introduced and compared with the deamidation mechanism. Finally, attempts to understand the effect of neighboring residues on Asn deamidation have been elaborated and several ideas for future work have been outlined. Mots-clés: déamidation ; asparagine ; hydrolyse des amides ; théorie de la fonctionnelle de la densité 10) with J[ρ] being the coulomb energy, T s [ρ] being the kinetic energy of the non-interacting electrons and E xc [ρ] being the exchange-correlation energy functional. The exchangecorrelation functional is expressed as the sum of an exchange functional E x [ρ] and a correlation functional E c [ρ], although it contains also a kinetic energy term arising from the kinetic energy difference between the interacting and non-interacting electron systems. Kinetic Energy term, being the measure of the freedom, and exchange-correlation energy, describing the change of opposite spin electrons (defining extra freedom to an electron), are the favorable energy contributions. The Coulomb energy term describes the unfavorable electron-electron repulsion energy and therefore disfavors the total electronic energy. xcU is the potential energy of exchange-correlation at intermediate coupling strength. The adiabatic connection formula can be approximated by: exchange energy of the Slater determinant of theKohn- a=0.04918, b=0.132, c=0.2533 and d=0.349. by the standard electrostatics in terms of the dielectric constant, ε, and the electric field perpendicular to the surface, F ur , generated by the charge distribution within studies on Asn deamidation have focused on modeling the aforementioned route without solvent assistance. The following study shows the favorable contribution of explicit solvent molecules on the imide-mediated mechanism of deamidation. In addition, a novel route which involves tautomerization of the Asn side chain amide into an amidic acid has been suggested as an alternative for the formation of the cyclic imide. This article has been published in the Journal of Physical Chemistry -A 110 (27), 8354-8365, 2006. 3 . 3 Scheme -3. Rate Determining Step for Deamidation in Neutral Media Suggested byKonuklar et al[14]. Figure - 1 .Figure - 2 . 2 O 122 Figure -1. Potential Free Energy Profile for the Concerted Cyclization Reaction Scheme - 5 . 40 RelativeFigure - 3 .Figure- 3 . 54033 Scheme -5. Mechanism for Stepwise Water Assisted Cyclization. Figure - 5 .Figure - 6 .Figure - 7 . 567 Figure -5. Potential Free Energy Profile for the Asparagine Side Chain Tautomerization without explicit H 2 O molecule they lower the barrier by stabilizing the transition state through a hydrogen bond network. Ring closure of the amidic acid tautomer can also occur through a stepwise mechanism with active water assistance (Figure-10). In the first step of the stepwise ring closure, the backbone NH proton is transfered to the Asn side chain through a water molecule, forming a zwitterionic intermediate, which then undergoes ring closure to form the tetrahedral intermediate. The overall free energy of activation for the stepwise ring closure of the amidic acid tautomer is 4.4 kcal/mol lower than the concerted ring closure with no water assistance (Figure-8). The favorable interactions in Figure-9 and the stepwise mechanism in Figure-10, which prove to be helpful in reducing the activation barrier, have been combined to model the stepwise ring closure of the amidic acid tautomer with three peripheral water molecules (Figure-11), where only one of the water molecules is actively involved in the reaction mechanism. The three waterassisted ring closure mechanisms (Figures-9, 10 and 11) have comparable barriers. It is imperative to indicate that the activation barriers associated with the ring closure of the amidic acid tautomer (Figures-8, 9, 10 and 11) cannot be directly compared with barriers of concerted (Figures-1, 2 and 3 )Figure - 8 . 50 RelativeFigure - 9 . 5 22Figure - 10 . 8 B3LYP/ 6 - 5 24Figure - 11 . 3850951086511 Figure -8. Potential Free Energy Profile for the Concerted Ring Closure in Amidic Acid Tautomer Scheme - 7 . 7 Scheme -7. Deamination (loss of NH 3 ) in the Tetrahedral Intermediate to yield the Figure - 12 . 1 B3LYP/ 6 -Figure - 13 .Figure - 14 . 12161314 Figure -12. Potential Free Energy Profile for the Deamination without explicit H 2 O molecules 4 and 11 ) 11 were shown to have overall barriers in the range of 34 -37 kcal/mol approximately 15 kcal/mol lower in energy than the concerted waterless mechanism previously proposed. The most probable mechanism for the formation of the tetrahedral intermediate is proposed to be the tautomerization route, nevertheless since the barrier differences between these three mechanisms is small, the other two should also be considered as competitive reaction pathways. SCHEME 1 : 1 SCHEME 1: Succinimide-mediated Deamidation of Asn residues. 3 3 Capasso et al. have proposed that deamidation of relatively unrestrained Asn residues goes through a succinimide intermediate (Scheme 1) the succinimide-mediated mechanism, deamidation reaction rates of ordinary Asn residues in physiological solvent conditions are largely affected by several factors. Firstly by the intrinsic acidity of the n +1 backbone nitrogen, which depends on inductive and electrostatic effects, and, therefore, upon peptide sequence. Second, by the amount of available conformational space and steric hindrance in the vicinity of the Asn residue, which may enhance or inhibit the formation of the cyclic intermediate. This is especially important in proteins where rearrangements in the protein three-dimensional structure may be necessary to allow proper alignment of the side chain and/or the backbone for ring closure. Finally, by the availability of water molecules or a proton donor, which is crucial for the decomposition of the cyclic tetrahedral intermediate that may otherwise revert to the open form. While protein structure chain amide was modeled, to set a benchmark for comparison with mechanisms involving a Chapter V 69 succinimide intermediate, which are thoroughly discussed thereafter. Finally, the energetics and feasibilities of different mechanisms leading to deamidation are discussed. Different number of explicit water molecules was used (0, 1 or 2) to analyze the effect of solvent on reaction mechanism and energetics. The initial structure for each system is the model compound or model compound-water complex (Figure 1). The following abbreviations are used throughout the discussion: asparagine (asn); succinimide intermediate (suc); tetrahedral intermediate (tet); amidic acid tautomer (taut); gemdiol (gem); aspartic acid (asp). Figure 1 . 1 Figure 1. Optimized structures model peptides with Asn residue. SCHEME 2: Direct Hydrolysis of Asn to Asp. Figure 2 .Figure 3 . 23 Figure 2. Optimized geometries and free energies of activation for the transition state of direct hydrolysis of Asn to Asp (asn → asp) with one (TS1-1H 2 O) and two (TS1-2H 2 O) water molecules, respectively. 73 SCHEME 4 : 74 SCHEME 5 : 734745 SCHEME 4: Succinimide Formation -Two-Step Mechanism (asn → tet → suc). Chapter V 75 SCHEME 6 : 756 SCHEME 6: Concerted Hydrolysis of the Succinimide Intermediate (suc → asp). Chapter V 76 SCHEME 7 : 767 SCHEME 7: Stepwise Hydrolysis of the Succinimide Intermediate (suc → gem → asp). Figure 4 . 4 Figure 4. Given that this is a hydrolysis reaction, transition state TS7-1H 2 O corresponds to an unassisted process and TS7-2H 2 O corresponds to a one-water assisted concerted hydrolysis reaction. The two transition state structures differ in several aspects. The unassisted hydrolysis of the imide (TS7-1H 2 O) is a four-centered concerted -yet asynchronous-transition state, where the proton transfer from the water molecule to the ring nitrogen has already occurred (N-H distance 1.126 Å), the lengthening in the C-N distance is substantial (1.684 Å), however the attacking -OH is still rather far (C-O distance 1.954 Å). The one-water assisted mechanism (TS7-2H 2 O), however, shows different geometrical features; in the six-centered transition state, the proton transfer is still not complete, the nucleophilic -OH group is quite close (C-O distance 1.670 Å) to the carbonyl under attack, but the ring opening is still premature (C-N distance 1.693 Å). Figure 4 . 4 Figure 4. Optimized geometries for the transition state of concerted hydrolysis of the succinimide intermediate into Asp (suc → asp), "unassisted" (TS7-1H 2 O) and one-water assisted (TS7-2H 2 O) mechanisms, respectively. 2H 2 O), while the second solvent molecule assists this process. The following ring opening step (TS9-1H 2 O) is also assisted by a water molecule. The two water case (TS8-2H 2 O and TS9-1H 2 O) is undoubtedly a better model to study the gemdiol mechanism, due to the water-assistance, which is absent in the former case (TS8-1H 2 O and TS9). This will be more apparent in the next section, when energetics of these reactions are comparatively discussed. The main difference in the geometries of the gemdiol formation transition states (TS8-1H 2 O and TS8-2H 2 O), is the extent of proton transfer. In the water-assisted case (TS8-2H 2 O), proton transfer from the attacking water molecule to the carbonyl oxygen is almost complete, contrary to the unassisted case (TS8-1H 2 O). However, the carbonyl C-O distance is longer in TS8-1H 2 O (1.313 Å as opposed to 1.293 Å in TS8-2H 2 O). A strong H-bond network can be seen in the water-assisted case (TS8-2H 2 O). In the second step, water-assistance (TS9-1H 2 O) has enhanced the extent of proton transfer compared to the unassisted case (TS9). Ring opening is slightly more achieved in TS9-1H 2 O (2.238 Å as opposed to 2.220 Å in TS9). Figure 5 . 5 Figure 5. Optimized geometries for the transition state of gemdiol-mediated hydrolysis of the succinimide intermediate into Asp (suc → gem → asp), "unassisted" (TS8-1H 2 O and TS9) and one-water assisted (TS8-2H 2 O and TS9-1H 2 O) mechanisms, respectively. Figure 6, does not include the hydrolysis of the imide (suc) into an Asp, since this requires a Figure 6 .Figure- 7 . 67 Figure 6. Reaction Coordinate for Deamidation -no water. and 7). The concerted asn → suc mechanism for the formation of the succinimide intermediate was not further investigated with two explicit water molecules since the barrier height is considerably higher than the stepwise routes (asn → tet → suc and asn → taut → tet → suc) as shown in Figures6 and 7. The tautomerization route (asn → taut → tet → suc) is the most plausible pathway for succinimide formation with two water assistance. The cyclization step is rate-determining for succinimide formation in both stepwise pathways (asn → tet → suc and asn → taut → tet → suc). Figure- 8 . 8 Figure-8. Reaction Coordinate for Deamidation -two water. is not the only fate of asparaginyl residues; many deamidation experiments have yielded fragmentation products as well. This chapter deals with the cleavage of the peptide backbone near Asn and Asp. Peptide fragmentation has been experimentally observed near these residues and the aim of this study is to propose a plausible pathway for backbone cleavage. The details of the mechanism and the energetics will also provide insight on the feasibility of cleavage near Asn and Asp. It is in fact important to predict the likelihood of deamidation preceding backbone cleavage at Asn residues. The following article has recently been submitted for publication in the Journal of Organic Chemistry. SCHEME 1 : 1 SCHEME 1: Succinimide-Mediated Deamidation Pathway. SCHEME 3: Side-Chain NH 2 Attack on Asn Backbone Carbonyl to Form a Cyclic Tetrahedral Intermediate. 2 OFigure 1 . 21 Figure 1. Optimized geometries and free energies of activation for the transition state of side chain NH 2 attack on backbone Asn carbonyl, concerted non-assisted (TS1) and concerted water-assisted (TS1-H 2 O) mechanisms, respectively. Energy values in gas phase (normal) and solution (italics) are given in kcal/mol. Figure 2 .Figure 3 . 23 Figure 2. Relative free energies for the concerted non-assisted cyclization step in deamidation [21] and backbone cleavage of Asn.Energy values in gas phase (normal) and solution (italics) are given in kcal/mol. SCHEME 4 : 2 O 42 SCHEME 4: Peptide Backbone Cleavage at Intermediate I. Figure 4 . 4 Figure 4. Optimized geometry and free energy of activation for the transition state of cleavage at Intermediate I, water-assisted (TS2-H 2 O).Energy values in gas phase (normal) and solution (italics) are given in kcal/mol. SCHEME 5B: Stepwise Ring Amide Hydrolysis at Intermediate I via a GemdiolIntermediate. Figure 5 . 5 Figure 5. Optimized geometry for the transition state of concerted ring amide hydrolysis, one water-assisted. Figure 6 . 6 Figure 6. Optimized geometries for the transition states of gemdiol-mediated stepwise ring amide hydrolysis, one water-assisted. Figure 7 . 7 Figure 7. Relative free energies for concerted and gemdiol-mediated ring amide hydrolysis at Intermediate I, water-assisted. SCHEME 6 :SCHEME 7 : 67 SCHEME 6: Deamination (NH 3 expulsion) at Intermediate II. Figure 8 . 8 Figure 8, together with optimized structures of transition states for these two reactions. The deamination transition state (TS6-H 2 O) shows that the ammonia is almost expelled (C-N bond distance 1.695 Å). In the cleavage case (TS7-H 2 O), however, the backbone lengthening has just started (C-N bond distance 1.583 Å). The cleavage route is energetically favorable by almost 6 kcal/mol. Figure 9 . 9 Figure 9. Relative free energies for peptide bond cleavage at Asn, water-assisted mechanism. SCHEME 8 : 8 SCHEME 8: Peptide Bond Cleavage Mechanism for Asp Residues Figure 10 .Figure 11 . 1011 Figure 10. Optimized geometries and free energies of activation for the transition state of side chain -OH attack on backbone Asp carbonyl, concerted waterless (Asp-TS1) and concerted one water-assisted (Asp-TS1-H 2 O) mechanisms, respectively. Energy values in gas phase (normal) and solution (italics) are given in kcal/mol. Figure 12 . 12 Figure 12. Relative free energies for peptide bond cleavage at Asp, water-assisted mechanism. 123SCHEME 1 : 1 SCHEME 1: Deamidation via a Succinimide Intermediate. Figure 1 . 1 Figure 1. Sequence dependence of Asn pentapeptide half-times, from reference 27. SCHEME 2 : 2 SCHEME 2: Model Peptide with Asn-Yyy Residue. The deprotonation reaction (Scheme 4 )SCHEME 4 : 5 : 445 SCHEME 4: Deprotonation of the n +1 backbone NH. SCHEME 6 : 6 SCHEME 6: Newman Projection for C 29 -C19 instead of a succinimide, was modeled. This mechanism involves the attack of the n -1 residue's backbone carbonyl to the Asn side chain carbonyl forming an intermediate with a six-membered ring (Int I). Consequently, deamination and hydrolysis of this intermediate leads to the formation of the aspartyl residue (Scheme 7).In the first step of this mechanism, the backbone carbonyl of the n -1 residue acts as a nucleophile and attacks the carbonyl carbon of the amide on the Asn side chain. Meanwhile, a water molecule assists the transfer of a proton from the NH adjacent to the aforementioned backbone carbonyl to the oxygen of the Asn side chain carbonyl. Proton transfer takes place prior to ring closure, indicating an asynchronous transition state (Figure5); the Asn backbone NH is essentially deprotonated in the transition state, while the side chain carbonyl carbon bears a proton (0.994 Å). In fact, proton transfer enhances the nucleophilic attack by twofold. First by forming a resonance structure where the backbone carbonyl carries a more negative charge and therefore is a much better nucleophile and secondly, by forming a more positive center on the side chain carbonyl carbon, further enhancing the nucleophilic attack. Meanwhile, ring closure is shown to be underway, as indicated by the O-C distance of 1.951 Å. As a result of this step the cyclic intermediate (Int I) forms. The barrier for ring closure is rather high (41.3 kcal/mol), but is comparable to succinimide formation barriers SCHEME 7 :Figure 5 .Figure 6 .Figure 7 . 7567 SCHEME 7: Alternative Pathway for Deamidation of Asn. O O - O N H O N H 2 C O O N H O O - H 2 C O N H L-Aspartyl residue L-Iso-Aspartyl residue Succinimide Intermediate L-Asparaginyl residue NH 3 H 2 O H 2 O Capasso et al. have proposed that the deamidation of relatively unrestrained Asn residues proceeds through a succinimide intermediate [10-12] (Scheme-2). Acylation of the amino group on the neighboring (n+1) carboxyl side residue by the β-carbonyl (side chain carbonyl) group of the L-Asn residue produces a five membered cyclic imide, namely the succinimide intermediate. The first step in the mechanism (Scheme-2) includes the cyclization and consecutive loss of an NH 3 . The succinimide intermediate then hydrolyzes at either one of the two carbonyls. Experimental findings indicate that the hydrolysis reaction gives L-Asp and L-β-Asp (L-iso-Asp) in a 3:1 ratio [13]. O H 2 C H N N H O O NH 2 O H 2 C H N N H O TABLE 1 : 1 . Mean Deamidation Half-Times for Asn-Yyy Sequences in Pentapeptides, from reference 27. Carboxyl-Side Residue a Half-Time (Days) Gly 1.2 His 10 Ser 16 Ala 25 Asp 32 Thr 47 Cys 54 Lys 59 Met 61 Glu 64 Arg 65 Phe 69 Tyr 81 Trp 99 Leu 130 Val 250 Ile 320 Pro > 1000 a Deamidation half-times at T=310 K. TABLE 2 : 2 Activation barriers for cyclization in Asn-Yyy sequences. Yyy Deamidation b ∆G ‡ a Half-times no H 2 O 1 H 2 O 2 H 2 O Gly 1 44.1 40.0 37.1 Ala 25 45.6 42.4 38.0 Val 250 48.0 47.2 40.3 a Experimental values (T=310 K), are in days [x]. b Activation barriers are in kcal/mol. TABLE 3 : 3 Per cent occurrence of intramolecular H-bonds among proton-accepting and donating sites. H-bonds % occurrence Gly Ala Val H 11 -O 16 3.8 6.4 15.8 H 18 -O 9 5.0 5.8 3.6 H 28 -O 37 2.5 - - H 32 -0 16 2.3 1.8 3.0 H 32 -O 37 12.9 3.9 1.6 H 32 -O 9 16.5 11.9 14.6 H 39 -O 16 5.4 34.2 29.5 H 39 -O 9 2.5 7.7 15.5 TABLE 4 : 4 Per cent occurrence of water-bridges among proton-accepting and donating sites. Water-bridges % occurrence Gly Ala Val H 11 -O 16 3.9 4.3 8 H 11 -O 40 4.2 1.4 1.5 H 18 -O 37 5.0 1.1 - H 18 -O 9 3.7 1.9 2.2 H 27 -O 25 13.5 16.8 12.8 H 27 -O 30 4.8 5.0 3.4 H 28 -O 16 6.9 4.9 3.4 H 28 -O 30 3.5 2.3 1.6 H 28 -O 37 6.3 - - H 32 -O 37 6.7 2.0 2.0 H 32 -O 9 8.1 7.5 1.7 H 39 -O 16 1.5 3.3 1.5 H 39 -O 9 3.7 1.5 - Backbone Structure (Near-Attack-Conformation) ACKNOWLEDGEMENTS S. Catak thanks the French Embassy in Ankara, Turkey, for the co-tutelle grant in France and the TUBITAK National Ph.D. scholarship. The authors thank CINES (Centre Informatique National de l'Enseignement Supérieur) for computer facilities (Project No. lct2636) and acknowledge the TUBITAK High Performance Computing Center for computational resources. S. Catak would also like to express gratitude to Dr. B. Balta, for fruitful discussions during the course of this study. ACKNOWLEDGEMENTS This research was supported by the PIA-BOSPHORUS project (Project No. PIA-TBAG-U/147 (105T275)) and the BAP project (Project No. 05HB501). Computational resources were provided by CINES (Centre Informatique National de l'Enseignement Supérieur) and the TUBITAK ULAKBIM High Performance Computing Center. S.C. acknowledges a TUBITAK Ph.D. Scholarship and a co-tutelle grant from the French Embassy in Ankara, Turkey. S.C. would like to thank Prof. Dr. Xavier Assfeld (Nancy-Université, Nancy, France) for mechanistic suggestions on the concerted succinimide formation. ACKNOWLEDGEMENTS Computational resources provided by CINES (Centre Informatique National de l'Enseignement Supérieur) and TUBITAK ULAKBIM High Performance Computing Center. S.C. acknowledges a TUBITAK Ph.D. Scholarship and a "co-tutelle" grant from the French Embassy in Ankara, Turkey. This research was supported by PIA-BOSPHORUS (Project No. PIA-TBAG-U/147 (105T275)) and BAP (Project No. 05HB501). The authors also acknowledge the sixth framework project COSBIOM (FP6-2004-ACC-SSA-2, Project No 517991) for travel and accommodation support. This research was supported in part by TUBITAK through TR-Grid e-Infrastructure Project. TR-Grid systems are hosted by TÜBİTAK ULAKBİM, Middle East Technical University, Pamukkale University, Çukurova University, Erciyes University, Boğaziçi University and İstanbul Technical University of Turkey. Visit http://www.grid.org.tr for more information. ACKNOWLEDGEMENTS Computational resources provided by CINES (Centre Informatique National de l'Enseignement Supérieur) and TUBITAK ULAKBIM High Performance Computing Center. S.C. acknowledges a TUBITAK National Ph.D. Scholarship and a co-tutelle grant from the French Embassy in Ankara, Turkey. This research was supported by PIA-BOSPHORUS (Project No. PIA-TBAG-U/147 (105T275)) and BAP (Project No. 05HB501). The authors also acknowledge the sixth framework project COSBIOM (FP6-2004-ACC-SSA-2, Project No 517991) for travel and accommodation support. CHAPTER VI NON-ENZYMATIC PEPTIDE BOND CLEAVAGE AT ASPARAGINE AND ASPARTIC ACID Computational Study on Non-Enzymatic Peptide Bond Cleavage at Asparagine and Aspartic Acid Saron Catak 1, 2 , Gérald Monard 1 , Viktorya Aviyente 2 , Manuel F. Ruiz-López 1 1 Equipe de Chimie et Biochimie Théoriques, UMR CNRS-UHP No. 7565, Nancy-Université, BP 239, 54506 Vandoeuvre-lès-Nancy, France. 2 Department of Chemistry, Bogazici University, 34342 Bebek, Istanbul, Turkey. ABSTRACT Non-enzymatic peptide bond cleavage at asparagine (Asn) and glutamine (Gln) residues has been observed during peptide deamidation experiments; cleavage has also been reported at aspartic acid (Asp) and glutamic acid (Glu) residues. Although, peptide backbone cleavage at Asn is known to be slower than deamidation, fragmentation products are often observed during peptide deamidation experiments. In this study, mechanisms leading to the cleavage of the carboxyl-side peptide bond of Asn and Asp residues were investigated using computational methods (B3LYP/6-31+G**). Single point solvent calculations at the B3LYP/6-31++G** level, were carried out in water, utilizing the integral equation formalism-polarizable continuum (IEF-PCM) model. Mechanism and energetics of peptide fragmentation at Asn were comparatively analyzed with previous calculations on deamidation of Asn. When deamidation proceeds through direct hydrolysis of the Asn side chain or through cyclic imide formation -via a tautomerization route-it exhibits lower activation barriers than peptide bond cleavage at Asn. The fundamental distinction between the mechanisms leading to deamidation -via a succinimide-and backbone cleavage was found to be the difference in nucleophilic entities involved in the cyclization process (backbone versus side chain amide nitrogen). If deamidation is prevented by protein three-dimensional structure, cleavage may become a competing pathway. Fragmentation of the peptide backbone at Asp was also computationally studied, in order to understand the likelihood of Asn deamidation preceding backbone cleavage. The activation barrier for backbone cleavage at Asp residues is much lower (approximately 15 kcal/mol) than that for Asn. This suggests that peptide bond cleavage at Asn residues is more likely to take place after it has deamidated into Asp. Cases where deamidation is prevented by protein three-dimensional structure, due to hindrance of backbone rotation caused by sterics, secondary structure and/or spatial distribution of neighboring residues, backbone cleavage of the Asn residue may ensue as a competitive reaction. CHAPTER VII PRIMARY SEQUENCE DEPENDANCE OF ASPARAGINE DEAMIDATION RATES Primary Sequence Dependence of Asn Deamidation Rates Saron Catak ABSTRACT Deamidation rates in peptides and proteins are known to be influenced by primary structure, i.e. the peptide sequence near the asparagine (Asn) residue. In a current study, Robinson et al. showed that Asn deamidation half-times for pentapeptides are primarily affected by the carboxyl-side (n +1) residue, with less influence from the amino (n -1) side. This is consistent with the previously suggested succinimide-mediated mechanism, which involves the attack of the carboxyl-side backbone NH to the Asn side-chain carbonyl to form a cyclic intermediate. However, the effect of the identity of the n +1 residue is unclear. This study uses computational techniques in an attempt to identify the dominant factor(s) causing the variation in deamidation rates of pentapeptides with different primary sequence. The succinimide-mediated mechanism previously studied with a smaller model peptide (ACE-Asn-NME) was modeled (B3LYP/6-31+G**) using a larger system (ACE-Asn-Yyy-NME) and different n +1 (Gly, Ala and Val) residues. The succinimide-mediated reaction barriers for all three systems were energetically compared, in an effort to see whether activation barriers are similar for Asn with different carboxyl-side residues. In a separate attempt to understand the dynamic differences between these systems, molecular dynamic (MD) simulations were performed on three pentapeptides, Gly-Gly-Asn-Yyy-Gly, -where Yyy is Gly, Ala and Val-with significantly different experimental deamidation half-times. MD simulations were analyzed in terms of solvent accessibility, backbone structure, intramolecular hydrogen-bonds, water bridges and per cent occurrence of potentially reactive conformations. In addition to the previously suggested succinimide-mediated deamidation pathway that was used as a benchmark during this work, an alternative route for deamidation -involving the Asn amino-side-has been proposed here for the first time. CONCLUSION The aim of this study was to better understand the factor(s) causing primary sequence dependence in Asn deamidation, in particular, the reasons behind the significantly different Asn deamidation rates in pentapeptides with different n +1 residues. Calculated activation barriers for the cyclization step leading to the succinimide formation, which was previously suggested to be the rate-determining step, was consistent with the trend dictated by experimental half-times. In addition, deprotonation of the n +1 NH reproduced similar results, demonstrating that different R groups on the side-chain of the n +1 residues influence the ease of deprotonation, indicating that deamidation may be effected as well. On the other hand, MD simulations of pentapeptides with the sequence Gly-Gly-Asn-Yyy-Gly, where Yyy is Gly, Ala and Val, revealed that near-attack-conformations, which are predicted from QM calculations, were not encountered during the simulation. In addition, some of the critical water-bridges and intramolecular H-bonds were non-existent during the simulations due to the absence of the correct orientation of reactive groups. This suggests that either a different mechanism for deamidation may be in effect or major backbone rotation is necessary to achieve the reactive conformation. CHAPTER VIII GENERAL CONCLUSION VIII. GENERAL CONCLUSION The first part of this study has shown that deamidation is catalyzed by water molecules and that several different mechanisms exist for the formation of the succinimide intermediate. Three different mechanisms were suggested for the cyclization step of the deamidation reaction, which was previously proposed to be the rate determining step. All water assisted cyclization reactions investigated were shown to have overall barriers in the range of 34 -37 kcal/mol approximately 15 kcal/mol lower in energy than the concerted waterless mechanism previously proposed. The most probable mechanism for the formation of the tetrahedral intermediate is proposed to be the tautomerization route. The effect of water molecules on the deamination step of the deamidation reaction has also been established, verifying that the cyclization step, with a substantially higher barrier for activation, is the rate determining step for the succinimide formation. It is also noteworthy that the involvement of water molecules in the deamination step has lowered the barrier to half. It can be suggested that water molecules in the vicinity of asparaginyl residues serve as a catalyst in deamidation reactions. Therefore, one may conclude that deamidation in proteins or enzymes will be more probable for those potential deamidating sites that exhibit the largest accessibility by solvent molecules. Main conclusions drawn from the second study are: 1) water assistance increases the rate of deamidation, by catalyzing both the formation and hydrolysis of the succinimide intermediate; 2) the tautomerization route has the lowest barrier for the formation of the succinimide intermediate regardless of the number of water molecules that assist the reaction, including the waterless mechanism; 3) hydrolysis of the succinimide intermediate is much more likely to go through a stepwise mechanism, where a gemdiol intermediate is formed. More importantly, the hydrolysis barriers are higher than those for succinimide formation. Previous calculations had shown that the cyclization step is the rate determining step for the formation of the succinimide intermediate, however, when the entire deamidation process is considered, the hydrolysis step is the actual rate determining step. The stepwise hydrolysis barrier is the rate determining step for the overall deamidation process, which is likely to proceed through the tautomerization route. This also explains the isolation of succinimide

01713886

en

2024/03/05 22:32:07

2018

https://inria.hal.science/hal-01713886v2/file/audio-source-separation.pdf

Antoine Liutkus Christian Rohlfing Antoine Deleforge AUDIO SOURCE SEPARATION WITH MAGNITUDE PRIORS: THE BEADS MODEL Keywords: audio probabilistic model, magnitude, phase, source separation 1 Audio source separation comes with the need to devise multichannel filters that can exploit priors about the target signals. In that context, experience shows that modeling magnitude spectra is effective. However, devising a probabilistic model on complex spectral data with a prior on magnitudes is non trivial, because it should both reflect the prior but also be tractable for easy inference. In this paper, we approximate the ideal donut-shaped distribution of a complex variable with approximately known magnitude as a Gaussian mixture model called BEADS (Bayesian Expansion Approximating the Donut Shape) and show that it permits straightforward inference and filtering while effectively constraining the magnitudes of the signals to comply with the prior. As a result, we demonstrate large improvements over the Gaussian baseline for multichannel audio coding when exploiting the BEADS model. I. INTRODUCTION Audio source separation aims at processing an audio mixture x composed of I channels (e.g. stereo I = 2) so as to recover its J constitutive sources s j [START_REF] Vincent | From blind to guided audio source separation: How models and side information can improve the separation of sound[END_REF]. It has many applications, including karaoke [START_REF] Liutkus | Adaptive filtering for music/voice separation exploiting the repeating musical structure[END_REF], upmixing [START_REF] Avendano | Frequency domain techniques for stereo to multichannel upmix[END_REF], [START_REF] Liutkus | Separation of music+ effects sound track from several international versions of the same movie[END_REF] and speech enhancement [START_REF] Barker | The third chime speech separation and recognition challenge[END_REF]. Usually, the sources are recovered by some time-varying filtering of the mixture, which is amenable to applying a I × I complex matrix G (f, t) to each entry x (f, t) of its Short-Term Fourier Transform (STFT) [START_REF] Vincent | Probabilistic modeling paradigms for audio source separation[END_REF], [START_REF] Benaroya | Audio source separation with a single sensor[END_REF], [START_REF] Avendano | Frequency domain techniques for stereo to multichannel upmix[END_REF]. Devising good filters requires either dedicated models for sources power spectrograms [START_REF] Ozerov | Multichannel nonnegative matrix factorization in convolutive mixtures for audio source separation[END_REF], [START_REF] Ozerov | A general flexible framework for the handling of prior information in audio source separation[END_REF] or machine learning methods to directly predict effective filters [START_REF] Nugraha | Multichannel audio source separation with deep neural networks[END_REF]. The theoretical grounding for these linear filtering procedures boil down to considerations about second-order statistics for the sources STFT coefficients s j (f, t) ∈ C. When seen from a probabilistic perspective, this is often translated as picking a complex isotropic Gaussian distribution [START_REF] Gallager | Circularly Symmetric Complex Gaussian Random Vectors -A Tutorial[END_REF] This work was partly supported by the research programme KAMoulox (ANR-15-CE38-0003-01) funded by ANR, the French State agency for research. Fig. 1: The BEADS probabilistic model for a complex variable with a magnitude close to b and squared uncertainty σ. Re for each s j (f, t) ∈ C, which is called the Local Gaussian Model (LGM) [START_REF] Arberet | Underdetermined instantaneous audio source separation via local Gaussian modeling[END_REF], [START_REF] Duong | Under-determined convolutive blind source separation using spatial covariance models[END_REF] and is depicted on Figure 1a. Although it enjoys tractability and easy inference, the main shortcoming of the LGM is that it gives highest probability to 0, which may appear as counterintuitive. Still, this comes as a consequence of the fact that stationarity makes all phases to be equally probable a priori, so that E [s j (f, t)] = 0. Combining the two first moments only, maximum entropy principles naturally lead us to pick the LGM [START_REF] Jaynes | Probability theory: The logic of science[END_REF]. However, this badly reflects the additional prior knowledge one often has on s j (f, t). The vast majority of methods use priors on its magnitude: |s j (f, t)| should be close to some b j (f, t) > 0 with squared uncertainty σ j (f, t). In that setting, the donut-shaped distribution shown in Figure 1b is much better than LGM because it gives its highest probability mass on the circle of radius b j (f, t). If σ = 0, we end up with the phase unmixing problem [START_REF] Deleforge | Phase unmixing: Multichannel source separation with magnitude constraints[END_REF]. However, even with some uncertainty σ > 0, such a distribution suffers from non-tractability. In particular, it is not stable with respect to additivity, nor allows for simple posterior inference that would lead to straightforward filtering procedures. In this paper, we translate our prior as a mixture of C identical Gaussian components evenly located over the circle of radius b j (f, t), yielding our proposed BEADS model (Bayesian Expansion Approximating the Donut Shape), depicted on Figure 1c. The most remarkable feature of BEADS is to allow both for straightforward filtering procedures while complying with priors on magnitude and possibly priors on phase. As such, it extends recent research on anisotropic modeling of complex spectra [START_REF] Magron | Phase-dependent anisotropic gaussian model for audio source separation[END_REF] by translating phase information into the choice of one particular component from the model. It thus appears as yet another way to incorporate phase information in source separation [START_REF] Sturmel | Phase-based informed source separation for active listening of music[END_REF]. We illustrate the BEADS model in an informed source separation (ISS) setting, where the true sources are available at a first coding stage, that allows to compute good models to be used for separation at a decoding stage [START_REF] Nikunen | Object-based Modeling of Audio for Coding and Source Separation[END_REF], [START_REF] Liutkus | Informed source separation : a comparative study[END_REF], [START_REF] Rohlfing | NMF-based informed source separation[END_REF]. As demonstrated already before [START_REF] Ozerov | Codingbased informed source separation: Nonnegative tensor factorization approach[END_REF], this ISS setup is interesting because the separation parameters can be encoded very concisely, leading to effective instances of spatial audio object coding [START_REF] Breebaart | Spatial audio object coding (SAOC)-the upcoming MPEG standard on parametric object based audio coding[END_REF]. II. PROBABILISTIC MODEL II-A. BEADS source model The BEADS model is expressed as follows: P [s j (f, t) | b j (f, t) , σ j (f, t)] = C c=1 π j (c | f, t) N (b j (f, t) ω c , σ j (f, t)) , (1) where N denotes here the complex isotropic Gaussian distribution 1 , ω = exp (i2π/C) is the C th root of unity and π j (c | f, t) is the prior for the phase of source j at time-frequency (TF) bin (f, t): it indicates the probability that s j (f, t) is drawn from component c and hence that its phase is close to ω c . While some phase unwrapping approach [START_REF] Magron | Model-based STFT phase recovery for audio source separation[END_REF] may be used to set this prior, we take it as uniform here. The parameter σ j (f, t) stands for the variance of each component. It may be understood as the expected squared error of our estimate b j (f, t) for the magnitude. We can note that the LGM is equivalent to C = 1 1 π I |Σ| exp -(x -µ) Σ -1 (x -µ) , where |Σ| is the determinant of Σ [START_REF] Gallager | Circularly Symmetric Complex Gaussian Random Vectors -A Tutorial[END_REF]. and b j (f, t) = 0, and that many beads tend to the donut shape, as shown in Figure 1d. Now, we consider the joint prior distribution of the J independent sources. We need to consider all the C J possible combinations for the components. Let N C be the set of the first C natural numbers. We write z (f, t) ∈ N J C for the J × 1 vector whose j th entry z j (f, t) ∈ N C is the actual component drawn for source j. We define π (c | f, t) as the probability of each combination: ∀c ∈ N J C , π (c | f, t) def = P [z (f, t) = c] = J j=1 π j (c j | f, t) , (2) where c j is the j th entry of c ∈ N J C . We have c π (c | f, t) = 1. The joint prior distribution of the sources is given by: P [s (f, t) | Θ] = c∈N J C π (c | f, t) N (ω c • b (f, t) , [σ (f, t)]) , (3) where ω c for c ∈ N J C denotes the J × 1 vector with entries ω cj , a•b denotes element-wise multiplication of the vectors a and b and [v] denotes the diagonal matrix whose diagonal is the vector v. In words, the prior distribution for the sources under the BEADS model is a Gaussian Mixture Model (GMM), with weights π (c | f, t) , which is reminiscent but different from the pioneering work in [START_REF] Rennie | Variational speech separation of more sources than mixtures[END_REF] that was limited to one source signal. II-B. Mixture likelihood and separation We take the mix as a convolutive mixture with the narrowband assumption, so that x (f, t) ≈ A (f ) s (f, t), where A (f ) is the I × J mixing matrix at frequency bin f . Further exploiting the BEADS model (1), we get the marginal model of the mixture given all parameters Θ as: P [x (f, t) | Θ] = c∈N J C π (c | f, t) N (x c (f, t) , Σ x (f, t)) , (4) with x c (f, t) = A (f ) (ω c • b (f, t)) Σ x (f, t) = A (f ) [σ (f, t)] A (f ) . (5) Now, the real advantages of the BEADS model is that we can straightforwardly obtain the joint posterior distribution of the sources as: P [s (f, t) | x, Θ] = c∈N J C π (c | f, t, x) N (µ c , Σ c ) , (6) where the posterior statistics µ c and Σ c for each combination c of the phases is: µ c (f, t) = G (f, t) (x (f, t) -x c (f, t)) + ω c • b (f, t) Σ c (f, t) = Σ (f, t) = [σ (f, t)] -G (f, t) A (f ) [σ (f, t)] , (7) Algorithm 1 BEADS decoder to update the phase configuration probabilities and perform separation. Input: parameters Θ, mixture x (f, t), prior π j , C. Initialization: Σ π ← 0, ŝ ← 0 For all c n ∈ N J C : 1) π (c | f, t, x) ← π (c | f, t) N (x (f, t) | x c (f, t) , Σ x (f, t)) 2) ŝ (f, t) += π (c | f, t, x) (ω c • b (f, t) -x c (f, t)) 3) Σ π (f, t) += Σ π (f, t) + π (c | f, t, x) Finalization: 1) ŝ (f, t) ← ŝ(f,t) Σπ(f,t) 2) ŝ (f, t) += G (f, t) x (f, t) with the J × I Wiener gain G (f, t) defined as: G (f, t) = [σ (f, t)] A (f ) A (f ) [σ (f, t)] A (f ) -1 . The minimum mean squared error (MMSE) estimate for the sources thus becomes: E [s (f, t) | x, Θ] = G (f, t) x (f, t) + c∈N J C π (c | f, t, x) (ω c • b (f, t) -x c (f, t)) . (8) In the case we know the true phases configuration z (f, t) or have estimated one at the decoder, this estimate (8) simplifies to ŝ (f, t) = µ z(f,t) . III. PARAMETERS ESTIMATION In this section, we only consider the informed case, where the true source signals s j (f, t) and the mixing matrices A (f ) are known at the coder, while the mixture and the parameters Θ = {b, σ, A} are known at the decoder. III-A. Decoder: posterior π (c | f, t, x) and separation We show how the probabilities π (c | f, t) for the phase configurations may be updated to yield their posterior π (c | f, t, x) that fully exploit the BEADS model for constraining the phases. Dropping the (f, t) indices for readability, we have: ∀c ∈ N J C , π (c | x) = π (c) N (x | x c , Σ x ) P [x] . (9) This posterior probability may hence be expressed up to a normalizing constant independent of c as: π (c | x) ∝ π (c | x) = π (c) exp -(x -x c ) Σ -1 x (x -x c ) , (10) which can straightforwardly be computed at the decoder with known parameters Θ = {b, σ}. π (c | x) is obtained by normalization after computation for all c ∈ N C J . Algorithm 1 summarizes the computations done at the decoder to perform estimation of the posterior probabilities for the phase configurations and separation. III-B. Coder: amplitudes b and errors σ The parameters to be learned at the coder are the amplitude priors b j (f, t) and the error models σ j (f, t). First, for saving bitrate and computing time, we only use BEADS for the F 0 frequency bands that have the highest energy in the mix, let them be F 0 , and simply pick b j (f, t) = 0 for others. Then, for f ∈ F 0 we compress them by picking a Nonnegative Tensor Factorization model [START_REF] Cichocki | Nonnegative matrix and tensor factorizations: applications to exploratory multi-way data analysis and blind source separation[END_REF]: b j (f, t) = k W b (f, k) H (t, k) Q b (j, k) if f ∈ F 0 0 otherwise, ( 11 ) in which case Θ b = {W b , H, Q b } are small nonnegative F 0 × K, T × K and J × K, respectively. Same thing for σ, where we further reduce the number of parameters of the model by taking the same activations H in both cases, but this time, we model all frequencies. In this section and for convenience, we will assume that the component z j (f, t) drawn for each source at each TF bin is known and equal to the one closest to s j (f, t) . This simplification has the advantage of strongly reducing the computational cost of the estimation algorithm. Indeed, the BEADS model ( 1) then reduces to: s j (f, t) | z j ∼ N b j (f, t) ω zj (f,t) , σ j (f, t) . We can define the relative source j (f, t) : j (f, t) def = s j (f, t) ω zj (f,t) ∼ N (b j (f, t) , σ j (f, t)) . (12) Provided z j (f, t) is correctly chosen as the component whose argument 2πzj (f,t) C is the closest to that of s j (f, t) , we can furthermore safely assume that the real part R ( j (f, t)) of j (f, t) is nonnegative. Now, we detail the learning procedure we propose for b and σ. The strategy is to alternatively fix each of them and learn the other one. Learning b j (f, t): Assume σ j (f, t) is kept fixed. The distribution [START_REF] Arberet | Underdetermined instantaneous audio source separation via local Gaussian modeling[END_REF] for the relative sources mean that we may estimate Θ b using a weighted Euclidean method: Θ b ← argmin Θ f,t,j |R ( j (f, t)) -b j (f, t | Θ)| 2 σ j (f, t) , which can be done with a classical weighted NTF [START_REF] Virtanen | Combining pitch-based inference and non-negative spectrogram factorization in separating vocals from polyphonic music[END_REF] scheme with the Euclidean cost function. Learning σ j (f, t): If b j (f, t) is fixed we see from (12) that j (f, t) -b j (f, t) has an isotropic complex Gaussian distribution with variance σ j (f, t). This means Θ σ can be estimated through: Θ σ ← argmin Θ f,t,j d IS | j (f, t) -b j (f, t)| 2 σ j (f, t | Θ) , where d IS (a b) is the classical Itakura-Saito divergence for two nonnegative scalars a and b. This optimization is classical in the audio processing literature [START_REF] Fitzgerald | On the use of the beta divergence for musical source separation[END_REF], [START_REF] Févotte | Nonnegative matrix factorization with the Itakura-Saito divergence. With application to music analysis[END_REF], [START_REF] Févotte | Algorithms for nonnegative matrix factorization with the beta-divergence[END_REF]. Considering that the activations H we take for σ are those for b, we only learn W σ and Q σ with it. IV. EVALUATION We evaluate the BEADS model through its performance for ISS, i.e. by displaying its average quality as a function of the bitrate required to transmit its parameters. To assess quality, we use BSSeval metrics [START_REF] Vincent | Performance measurement in blind audio source separation[END_REF]: SDR (Source to Distortion Ratio) and SIR (Source to Interference Ratio), both expressed in dB and higher for better separation. For normalization purpose, we compute δ-metrics, defined as the difference between the score and performance of oracle Wiener filtering, i.e. using true sources spectrograms [START_REF] Vincent | Oracle estimators for the benchmarking of source separation algorithms[END_REF]. The data consists of 10 excerpts of 30 s, taken from DSD100 database 2 . Each consists of J = 4 sources (vocals, bass, drums and accompaniment), sampled at 44.1 kHz. We generated either mono (I = 1) or stereo (I = 2) mixtures from these sources, through simple summation or anechoic mixing (delays+gains), respectively. STFT was conducted with 50 % overlap and a window size of 93 ms. We evaluated the following methods: • BEADS oracle: ŝ (f, t) = µ z(f,t) . • BEADS point using only the phase configuration ẑ that is most likely a posteriori. • BEADS as given in Algorithm 1. • Itakura-Saito NTF [START_REF] Liutkus | Informed source separation through spectrogram coding and data embedding[END_REF], with K components. Given all these methods and data, our extensive evaluation consisted in trying the methods with F 0 = 150 frequency bands for BEADS magnitudes, C = {8, 16} beads, and all methods were tried for K ∈ [8, 128] NTF components. We picked 16 quantization levels for all parameters. Results were smoothed using LOESS [START_REF] Cleveland | Locally weighted regression: an approach to regression analysis by local fitting[END_REF] and are displayed on Figure 2. An interesting fact we see on Figure 2 is that the oracle BEADS model significantly outperforms standard oracle Wiener filtering, even for very crude magnitude models b j (f, t). This can be seen by the fact that its δmetrics get positive even at very small bitrates. Then, we may notice that the δ-metrics appear as higher for mono than for stereo mixtures. In this respect, we should highlight that the absolute performance of oracle Wiener is of course higher for stereo (not shown on Figure 2), due to the knowledge of the mixing filters A (f ) ∈ C that alone bring good separation already and actually some information about the phase of the sources. Adding additional spectral knowledge in that case is then less important than in the mono case, where it is crucial. Now, we see a very clear improvement of BEADS as described in Algorithm 1 over classical NTF-ISS, of approximately 2 dB SDR and 5 dB SIR, at most bitrates. This significant boost in performance shows that BEADS helps a 2 http://sisec.inria.fr. lot in predicting the source signals by adequately handling priors on magnitudes, which is the main result for this study. Finally, Figure 2 also shows that the procedure for computing the phase posterior probabilities is not sufficient for correctly identifying the true phase configuration. This can be seen by the strong discrepancies between the BEADS point estimate and its oracle performance. While marginalization over the configuration as described in Algorithm 1 helps a lot in this respect, there is much room for improvement for parameter estimation of this model. V. CONCLUSION In this paper, we introduced BEADS as a convenient probabilistic model for complex data whose magnitude is approximately known. BEADS is a Gaussian Mixture Model where all components share the same variance and are scattered along a circle. While simple conceptually, BEADS comes with several advantages. First, it translates the delicate problem of modeling the phase into setting probabilities over a discrete set of components. Second, it allows for easy inference and, finally, it straightforwardly leads to effective filtering procedures. Although we demonstrated its potential in an audio-coding application, we believe it may also be useful in the blind separation setting when embedded in an Expectation-Maximization estimation procedure. The BEADS model combines advantages of both (C = 8)... ...and approximates the donut shape well (C = 16). 16 Fig. 2 : 162 Fig.2: BEADS for ISS on mono (left) or stereo (right) mixes. Metrics are δSDR (top) and δSIR (bottom). Units are kilobits/second/source (x-axis) and dB (y-axis).

01748519

en

2024/03/05 22:32:07

2018

https://hal.science/hal-01748519/file/L1Stationary_EHL_M-Pierre.pdf

El Haj Laamri email: [email protected] Michel Pierre email: [email protected] Stationary reaction-diffusion systems in L 1 Keywords: reaction-diffusion systems, nonlinear diffusion, cross-diffusion, global existence, porous media equation, weak solutions. 2010 MSC: 35K10, 35K40, 35K57 come Introduction and main results Our goal is to analyze the existence of solutions to stationary reaction-diffusion systems in L 1 -spaces. We first give a general abstract result for nonlinearities satisfying as many structural inequalities as the number of equations. This is done in the framework of abstract m-accretive operators in L 1 spaces for the diffusion part and the full system is controlled by a somehow "good" associated cross-diffusion system. We give several examples where this abstract result applies. Then, we provide a general result for more specific systems associated with general chemical reactions for which less structure holds for the nonlinearities. We denote by (Ω, µ) a measured space where µ is a nonnegative measure on the set Ω with µ(Ω) < +∞. We consider systems of the type (S)    ∀i = 1, ..., m, u i ∈ D(A i ) ∩ L 1 (Ω, dµ) + , h i (•, u) ∈ L 1 (Ω, dµ), u i + A i u i = h i (•, u 1 , ..., u m ) + f i (•) ∈ L 1 (Ω, dµ), (1) where for all i = 1, ..., m, -A i is a (possibly) nonlinear operator in L 1 (Ω, dµ) (generally a diffusion operator in applications), defined on D(A i ) ⊂ L 1 (Ω, dµ), -h i : Ω × R m → R is a nonlinear "reactive" term, -f i ∈ L 1 (Ω, dµ) + [:= {g ∈ L 1 (Ω, dµ) ; g ≥ 0, µ -a.e.}]. We are interested in nonnegative solutions u = (u 1 , ..., u m ) ∈ L 1 (Ω, dµ) +m . These systems are naturally associated with the evolution reaction-diffusion systems ∂ t u i (t) + A i u i (t) = h i (•, u(t)), t ∈ [0, +∞), i = 1, ..., m. ( When approximating these evolution systems by an implicit time discretization scheme, we are led to solving the following set of equations, for all small time interval ∆t ∀ i = 1, ..., m, n = 0, 1, ..., u n+1 i ∈ D(A i ), u n+1 i -u n i ∆t + A i u n+1 i = h i (•, u n+1 ) or u n+1 i + (∆t)A i u n+1 i = (∆t)h i (•, u n+1 i ) + u n i . This is exactly the system (S) with unknown u = u n+1 , up to trivially changing (∆t)A i into A i and (x, u) → [(∆t)h i (x, u) + u n i (x)] 1≤i≤m into (x, u) → [h i (x, u) + f i (x)] 1≤i≤m . The asymptotic steady states of the evolution system (2) are also the solutions u = (u 1 , ..., u m ) to the system {A i u i = h i (•, u), i = 1, ..., m, } which is also essentially included in the system (S) up to a slight change of the operators A i . The kind of operators A i we have in mind are : -The Laplacian u → -∆u on a bounded open set Ω of R N with various boundary conditions on ∂Ω. -More general elliptic operators u → -N i=1 ∂ x i N i=1 a ij (x)∂ x j u + b i u on the same Ω, with various boundary conditions as well. -Nonlinear operators of porous media type like u → -∆ϕ(u) where ϕ : R → R is an increasing function, again with different boundary conditions on ∂Ω. -Nonlinear operators of p-Laplacian type like u → -∆ p u := -∇ • |∇u| p-2 ∇u where p ∈ (1, +∞) and | • | is the euclidian norm in R N . -Classical perturbations and various associations of all of those. All these operators will satisfy the assumption (A) below which means that each A i is an m-accretive operator in L 1 (Ω, dµ) whose resolvents are compact and preserve positivity, namely the following, where I denote the identity : 1 (Ω, dµ), and for all λ ∈ (0, +∞), (A1) maccretivity : I + λA i is onto and (I + λA i ) -1 is nonexpansive, (A2) positivity : Ω sign -(u i )Au i ≥ 0, ∀ u i ∈ D(A i ), (A3) compactness : (I + λA i ) -1 : L 1 (Ω, dµ) → L 1 (Ω, dµ) is compact. (A)            ∀ i = 1, ..., m, A i : D(A i ) ⊂ L 1 (Ω, dµ) → L (3) Here I denotes the identity on L 1 (Ω, dµ) and in (A 2 ), we define sign -(s) := -sign + (-s), ∀s ∈ R where sign + (s) := 1, ∀ s ∈ [0, +∞), sign + (s) = 0, ∀ s ∈ (-∞, 0). This implies immediately that (I + λA i ) -1 L 1 (Ω, dµ) + ⊂ L 1 (Ω, dµ) + since then, for g ∈ L 1 (Ω, dµ) + and u i := (I + λA i ) -1 g 0 ≥ Ω sign -(u i )g dµ = Ω sign -(u i )[u i + λA i u i ] dµ ≥ Ω u - i dµ ⇒ u - i = 0 µ -a.e. For simplicity, we consider only single-valued operators A i , but everything would also work for multivalued m-accretive operators A i . The nonlinear reactive terms h i will preserve positivity as well. We assume that, for all i = 1, ..., m : (H1)        (1) h i : Ω × R m → R measurable ; (2) h i (•, R) := sup{|h i (•, r)|; |r| ≤ R} ∈ L 1 (Ω, dµ) + , ∀ R ∈ [0, +∞) ; (3) r ∈ R m → h i (•, r) is continuous ; (4) quasipositivity : h i (•, r 1 , ..., r i-1 , 0, r i+1 , ..., r m ) ≥ 0, ∀ r ∈ [0, +∞) m . (4) Moreover, the nonlinear reactive terms h i we are considering are those for which mass conservation or, more generally, mass dissipation or at least mass control holds for the associated evolution system [START_REF] Bebernes | Finite time blowup for semilinear reactive-diffusive systems[END_REF]. It is the case when the h i 's satisfy a relation like m i=1 h i (•, r) ≤ 0 for all r ∈ [0, +∞) m or more generally (H2) m i=1 a i h i (•, r) ≤ m i=1 b i r i +ω(•), ω ∈ L 1 (Ω, dµ) + , for some 0 ≤ b i < a i , i = 1, ..., m, r ∈ [0, +∞) m . ( 5 ) As we will see, this structure naturally implies an a priori L 1 -estimate on the solutions u i of the system (S) (see Proposition 2.2), together with "standard operators" A i by which we mean (A inf ) a ∞ := inf Ω A i u i dµ, u i ∈ D(A i ) ∩ L 1 (Ω, dµ) + > -∞ for each i = 1, ..., m. (6) This assumption essentially holds when 0 ∈ D(A i ), i = 1, ..., m. It also holds with most diffusion operators with non homogeneous boundary conditions (see Section 3) except a few ones : this is discussed in Remark 3.4. Actually, the main point will be to get a priori L 1 -estimates even on the nonlinearities h i (•, u) where u is solution of System (S). This will be satisfied if more structure is required on the nonlinear functions h i , namely (using the natural order in R m ) : (M )    There exist two m × m matrices M 0 , M 1 where M 0 is invertible, with nonnegative entries and such that M 0 h(•, r) ≤ M 1 r + Θ(•), ∀ r ∈ [0, +∞) m , Θ ∈ L 1 (Ω, dµ) +m . (7) Applying the matrix M 0 to the system (S) leads to the following set of m inequalities for an associated cross-diffusion system : M 0 u + M 0 Au = M 0 h(•, u) + M 0 f ≤ M 1 u + Θ + M 0 f, where we denote Au := (A i u i ) 1≤i≤m , h(•, u) := (h i (•, u)) 1≤i≤m and f := (f i ) 1≤i≤m . As proved in Proposition 2.3, this will imply an a priori estimate of h i (•, u), i = 1, ..., m in L 1 (Ω, dµ). More precisely, we will consider an approximate problem where the nonlinearities h i are replaced by truncated versions h n i . Then (M ) will imply that h n i (•, u n ) is bounded in L 1 (Ω, dµ) for the approximate solutions u n . The compactness of u n in L 1 (Ω, dµ) will then follow. Thus up to a subsequence, u n will converge in L 1 (Ω, dµ) m and µ-a.e. to some u and h n i (•, u n ) will also converge µ-a.e. to h i (•, u) for all i. But this is not sufficient yet to pass to the limit in the system since one essentially needs the convergence of h n i (•, u n ) in L 1 (Ω, dµ). It is the case if h n i (•, u n ) is uniformly integrable, by Vitali's lemma. This will hold if we add the following technical assumption which is some kind of compatibility condition between the various operators A i . It will be satisfied in the examples of Section 3. (Φ) There exist ϕ : [0, +∞) m → [0, +∞) continuous with lim |r|→+∞ ϕ(r) = +∞ and b ∈ (0, +∞) m such that Ω sign + (ϕ(u) -k)b • M 0 Au dµ ≥ 0, ∀ u ∈ D(A) ∩ L 1 (Ω, dµ) +m , ∀ k ∈ [0, +∞), D(A) := Π m i=1 D(A i ). ( 8 ) We now state our abstract result. It will be proved in Section 2 and applied to several explicit examples in Section 3. We refer to [START_REF] Laamri | Existence globale pour des systèmes de réaction-diffusion dans L 1[END_REF] where this kind of results and examples were already widely discussed and analyzed. Theorem 1.1 Assume that (A), (H1), (H2), (A inf ), (M ), (Φ) hold. Then the system (S) has a solution for all f ∈ L 1 (Ω, dµ) +m . As already explained, the assumption (M ) means that m independent inequalities hold between the m nonlinear functions h i . But many systems come with less than m such relations. The following 2 × 2 system satisfies only one relation of type (H2) and not (M ):    u 1 -d 1 ∆u 1 = (β 1 -α 1 )[u α 1 1 u α 2 2 -u β 1 1 u β 2 2 ] + f 1 =: h 1 (u 1 , u 2 ) + f 1 , u 2 -d 2 ∆u 2 = (β 2 -α 2 )[u β 1 1 u β 2 2 -u α 1 1 u α 2 2 ] + f 2 =: h 2 (u 1 , u 2 ) + f 2 , ∂ ν u 1 = 0 = ∂ ν u 2 on ∂Ω, (9) where for i = 1, 2, d i ∈ (0, +∞), α i , β i ∈ {0} ∪ [1, +∞), f i ∈ L 1 (Ω) + , Ω ⊂ R N equipped with the Lebesgue measure. This kind of nonlinearity appears in reversible chemical reactions with two species, namely α 1 A 1 + α 2 A 2 β 1 A 1 + β 2 A 2 . ( 10 ) Here (β 1 -α 1 )(β 2 -α 2 ) < 0. The nonlinearity h 1 and h 2 are quasipositive but satisfy the only one relation γ 2 h 1 + γ 1 h 2 = 0, γ i := |β i -α i |. It turns out that existence indeed holds for system (9) when f i ∈ L 1 (Ω) + for all i, as we prove it here. Actually we prove existence for a general class of such "chemical systems" when f i ∈ L 1 (Ω) + and also f i log f i ∈ L 1 (Ω). This is the second main result of this paper. Let us consider the system (CHS)      For all i = 1, ..., m, u i -d i ∆u i = (β i -α i ) k 1 Π m k=1 u α k k -k 2 Π m k=1 u β k k + f i , ∂ ν u i = 0 on ∂Ω, (11) where k 1 , k 2 ∈ (0, +∞) and, for all i = 1, ..., m, d i ∈ (0, +∞), α i , β i ∈ {0} ∪ [1, +∞) and f i ∈ L 1 (Ω) + where Ω is bounded regular open subset of R N and where    I := i ∈ {1, ..., m} ; α i -β i > 0 , J := j ∈ {1, ..., m}; β j -α j > 0 satisfy : I = ∅, J = ∅, I ∪ J = {1, ..., m}. (12) We denote by |I| (resp. |J|) the number of elements of I (resp. J). Theorem 1.2 Assume that f i ∈ L 1 (Ω) + , f i log f i ∈ L 1 (Ω) for all i = 1, ..., m. Assume also that |I| ≤ 2 (or |J| ≤ 2). Then there exists a nonnegative solution u ∈ W 2,1 (Ω) +m of (CHS) with h i (u) ∈ L 1 (Ω) for all i = 1, ..., m. If moreover m = 2, then the same result holds if f i ∈ L 1 (Ω) + , i = 1, 2 only. Remark 1.3 System (11) arises when modeling a general reversible chemical reaction with m species, according to the mass action law and to Fick's linear diffusion. In fact, it also contains systems written more generally as v i -d i ∆v i = λ i K 1 Π m k=1 v α k k -K 2 Π m k=1 v β k k + f i , where λ i ∈ R, λ i (β i -α i ) > 0, i = 1, ..., m. Indeed, we may go back to the exact writing of (11) by setting u i := (β i -α i )v i /λ i , k 1 := K 1 Π k [λ k /(β k -α k )] α k , k 2 := K 2 Π k [λ k /(β k -α k )] β k , f i := (β i -α i ) f i /λ i . On the other hand, it does not include systems like u 1 -d 1 ∆u 1 = +[u 3 1 u 2 2 -u 2 1 u 3 2 ] + f 1 (= u 2 1 u 2 2 [u 1 -u 2 ] + f 1 ), u 2 -d 2 ∆u 2 = -[u 3 1 u 2 2 -u 2 1 u 3 2 ] + f 2 . Here the condition λ i (β i -α i ) > 0 is not satisfied. Remark 1.4 Note that besides the 2 × 2 system (9), Theorem 1.2 contains as particular cases some favorite systems of the literature like h i (u) = (-1) i [u α 1 1 u α 3 3 -u β 2 2 ], i = 1, 2, 3, α 1 , α 3 , β 2 ∈ [1, +∞), h i (u) = (-1) i [u 1 u 3 -u 2 u 4 ], i = 1, 2, 3, 4. We may use Remark 1.3 to write them as in [START_REF] Laamri | Existence globale pour des systèmes de réaction-diffusion dans L 1[END_REF]. Analysis of systems of this kind may be found for instance in [START_REF] Laamri | Global existence of classical solutions for a class of reaction-diffusion systems[END_REF], [START_REF] Desvillettes | Global existence for quadratic systems of reactiondiffusion[END_REF], [START_REF] Pierre | Global existence for a class of quadratic reaction-diffusion systems with nonlinear diffusions and L 1 initial data[END_REF], [START_REF] Laamri | Global existence for reaction-diffusion systems with nonlinear diffusion and control of mass[END_REF], [START_REF] Goudon | Regularity analysis for systems of reaction-diffusion equations[END_REF], [START_REF] Cañizo | Improved duality estimates and applications to reaction-diffusion equations[END_REF], [START_REF] Caputo | Solutions of the 4-species quadratic reaction-diffusion systems are bounded and C ∞ -smooth[END_REF], etc. In fact Theorem 1.2 applies to quite general systems like, for instance, those obtained by multiplying the above 3 × 3 (resp. 4 × 4) example by u σ 1 1 u σ 2 2 u σ 3 3 (resp. u σ 1 1 u σ 2 2 u σ 3 3 u σ 4 4 ) where σ i ∈ [0, +∞). They lead to the following nonlinearities h i (u) = λ i [Π m k=1 u α k k -Π m k=1 u β k k ] , where m = 3 (resp. m = 4) and λ i (β i -α i ) > 0. Actually, Theorem 1.2 applies also to these same nonlinearities when m = 5. Indeed, since I ∪ J = {1, 2, 3, 4, 5}, it follows that I or J contains at most two elements. We believe that the result of Theorem := |x|, x ∈ Ω := B(0, 1) ⊂ R N , b ∈ (N/2, N -2), c ∈ (0, +∞): σ(r) := r -b + br + c, ∀ 0 < r ≤ 1, ⇒ σ -∆σ = f, f (r) := c + r -b + br + b(N -2 -b)r -(b+2) -b(N -1)r -1 , σ (1) = 0, where c is chosen large enough so that f ≥ 0. Note that f ∈ L p (Ω), p ∈ [1, N/(b + 2)). Let us now consider the solution (which exists by Theorem 1.2) of the system        u 1 , u 2 ∈ W 2,1 (Ω), u 2 2 -u 2 1 ∈ L 1 (Ω), u 1 -∆u 1 = u 2 2 -u 2 1 + f /2, u 2 -∆u 2 = -[u 2 2 -u 2 1 ] + f /2, ∂ ν u 1 = ∂ ν u 2 = 0 on ∂Ω. Then (u 1 + u 2 ) -∆(u 1 + u 2 ) = f, ∂ ν (u 1 + u 2 ) = 0 on ∂Ω. Thus u 1 + u 2 = σ. But u 1 , u 2 cannot be in L 2 (Ω) since σ is not in L 2 (Ω) by the choice of b > N/2. Remark 1.6 It is classical that an entropy structure holds in System (11) since m i=1 [log u i + µ i ]h i (u) = -[log k 1 Π m k=1 u α k k -log k 2 Π m k=1 u β k k ][k 1 Π m k=1 u α k k -k 2 Π m k=1 u β k k ] ≤ 0, when choosing µ i := [log k 1 -log k 2 ]/[m(α i -β i )]. Under the assumptions of Theorem 1.2, and in particular the LLogL assumption on the f i , we have the estimate Ω m i=1 [log u i + µ i ]u i + d i |∇u i | 2 u i ≤ Ω m i=1 [log u i + µ i ]f i ≤ Ω m i=1 f i log f i + (µ i -1)f i + u i , where, for the last inequality, we use the Young's inequality (75) with r = f i , s = log u i . But we will not use here this stucture in the proof of Theorem 1.2. Our strategy will consist in proving that the nonlinearity h i (u) is a priori bounded in L 1 (Ω). Then adequate compactness arguments allow us to pass to the limit in the approximate system. Note that the entropy inequality provides the extra information that ∇ √ u i ∈ L 2 (Ω) for the solutions obtained in Theorem 1.2 when f i log f i ∈ L 1 (Ω), i = 1, ..., m. Theorem 1.1 will be proved in Section 2, examples of applications are given in Section 3 and the proof of Theorem 1.2 is given in Section 4. Proof of Theorem 1.1 Since µ is fixed, we will most of the time more simply write L 1 (Ω), L 1 (Ω) + instead of L 1 (Ω, dµ), L 1 (Ω, dµ) + . We will use the natural order in R m namely [r ≤ r ⇔ r i ≤ ri , i = 1, ..., m] and we also denote r + := (r + 1 , ..., r + m ). Lemma 2.1 Let f = (f 1 , ..., f m ) ∈ L 1 (Ω) +m . We set h n i (x, r) := h i (x, r) 1 + n -1 m j=1 |h j (x, r)| , ∀ i = 1, ..., m, r ∈ R m , x ∈ Ω. ( 13 ) Assume that (A) and (H1) hold. Then the following approximate system (S n ) ∀i = 1, ..., m, u n i ∈ D(A i ) ∩ L 1 (Ω) + , u n i + A i u n i = h n i (•, u n ) + f i (•) in L 1 (Ω, dµ), (14) has a nonnegative solution u n = (u n 1 , • • • , u n m ). Proof. We consider the mapping T : v ∈ L 1 (Ω) m → w ∈ L 1 (Ω) m where w = (w 1 , ..., w m ) is the solution of ∀i = 1, ..., m, w i ∈ D(A i ), w i + A i w i = h n i (•, v + ) + f i (•) in L 1 (Ω, dµ), (15) where v + := (v + 1 , ..., v + m ). The mapping T is well defined since |h n i (x, r)| ≤ n for all (x, r) ∈ Ω × R m and f ∈ L 1 (Ω) m so that the solution is given by w i = (I + A i ) -1 h n i (•, v + ) + f i , i = 1, ..., m. Moreover h n i (•, v) + f i L 1 (Ω) ≤ n µ(Ω) + f i L 1 (Ω) =: M i . Let K i := (I + A i ) -1 {g ∈ L 1 (Ω) ; g L 1 (Ω) ≤ M i } , K := K 1 × ... × K m ⊂ L 1 (Ω) m . By assumption (A) and in particular (A3), K is a compact set of L 1 (Ω) m and K ⊃ T L 1 (Ω) m . On the other hand, we easily check that T is continuous. Thus T is a compact operator from L 1 (Ω) m into the compact set K ⊂ L 1 (Ω) m . By Schauder's fixed point theorem (see e.g. [START_REF] Zeidler | Nonlinear Functional Analysis and its applications, 1. Fixed Point Theorems[END_REF]), T has a fixed point u n . This means that u n is solution of (S n ). To prove the nonnegativity property of u n , we multiply the i-th equation by sign -(u n i ) and integrate to obtain, using Ω sign -(u i )A i u i ≥ 0 (see (A2) ), Ω (u n i ) -dµ ≤ Ω sign -(u n i )h n i (•, (u n ) + ) + sign -(u n i )f i dµ. By the nonnegativity of f i and the quasipositivity of h i assumed in (H1), the integral on the right is nonpositive so that Ω (u n i ) -dµ ≤ 0 or u n i ≥ 0 µ -a.e.. We will now progressively obtain estimates on u n independently of n. We start with the control of the total mass of u n . Proposition 2.2 Assume that (A), (H1), (H2), (A inf ) hold. Then there exists C ∈ (0, +∞) independent of n such that the solution u n of the approximate system (S n ) satisfies max 1≤i≤m u n i L 1 (Ω) ≤ C. Proof. According to (H2), we multiply each equation by a i and we add them to obtain m i=1 a i (u n i + A i u n i ) = m i=1 a i [h n i (•, u n ) + f i ] ≤ m i=1 [b i u n i + a i f i ]. We use (A inf ) to obtain n i=1 (a i -b i ) Ω u n i dµ ≤ -a ∞ m i=1 a i + m i=1 a i Ω f i dµ. Using a i > b i for all i yields the estimate of Proposition 2.2. We now prove that the nonlinearities are bounded in L 1 independently of n. Proposition 2.3 Assume that (A), (H1), (H2), (A inf ), (M ) hold. Then there exists C ∈ (0, +∞) independent of n such that the solution u n of the approximate system (S n ) satisfies max 1≤i≤m h n i (•, u n ) L 1 (Ω) ≤ C. Proof. According to the assumption (M ), let us multiply the system (S n ) by the matrix M 0 . As before, we denote Au n := (A 1 u n , ..., A m u n ), h n (•, u n ) := (h n 1 (•, u n ), ..., h n m (•, u n )) , f := (f 1 , ..., f m ). Then M 0 u n + M 0 Au n = M 0 h n (•, u n ) + M 0 f ≤ M 1 u n + Θ + M 0 f. (16) Since the entries of M 0 are nonnegative, and thanks to (A inf ) and the nonnegativity of u n , there exists C ∞ ∈ [0, +∞) m such that Ω [M 0 u n + M 0 Au n ]dµ ≥ -C ∞ which implies Ω [M 0 h n (•, u n ) + M 0 f ]dµ ≥ -C ∞ . We deduce Ω [M 1 u n + Θ -M 0 h n (•, u n )]dµ ≤ C ∞ + Ω [M 1 u n + Θ + M 0 f ]dµ ≤ D ∞ ∈ (0, +∞) m , the last inequality using also Proposition 2.2. As the function M 1 u n + Θ -M 0 h n (•, u n ) is nonnegative, these inequalities provide a bound for its L 1 (Ω) m -norm. It follows that M 0 h n (•, u n ) is also bounded in L 1 (Ω) m independently of n. Since M 0 is invertible, we deduce that h n (•, u n ) is itself bounded in L 1 (Ω) m : indeed if | • | denotes the euclidian norm in R m and • the induced norm on the m × m matrices, we may write |h n (•, u n )| = |M -1 0 M 0 h n (•, u n )| ≤ M -1 0 |M 0 h n (•, u n )|. Whence Proposition 2.3. Proposition 2.4 Assume that (A), (H1), (H2), (A inf ), (M ), (Φ) hold. Then, if u n is the solution of the approximate system (S n ), {h n (., u n )} n is uniformly integrable in Ω which means that, for all ε > 0, there exists δ > 0 such that for all measurable set E ⊂ Ω µ(E) < δ ⇒ E n i=1 |h n i (•, u n )|dµ ≤ ε for all n. Proof. By Proposition 2.3, we already know that h n (•, u n ) is bounded in L 1 (Ω) m . Let us prove that the extra condition (Φ) implies that it is even uniformly integrable. First, since u n i = (I + A i ) -1 (h n (•, u n ) + f i ), the compactness condition (A3) in (3) implies that, for all i = 1, ..., m, u n i belongs to a compact set of L 1 (Ω). Let us show that M 0 h n (•, u n ) is uniformly integrable. Since M 0 is invertible, this will imply that h n (•, u n ) is itself uniformly integrable and end the proof of Proposition 2.4. We will successively prove that (M 0 h n (•, u n )) + and (M 0 h n (•, u n )) -are uniformly integrable. Note that by the definition (13), h n (•, u n ) ≤ h(•, u n ). We may write (recall that the entries of M 0 are nonnegative) M 0 h n (•, u n ) ≤ M 0 h(•, u n ) ≤ M 1 u n + Θ ⇒ (M 0 h n (•, u n )) + ≤ (M 1 u n ) + + Θ. ( 17 ) Since u n is in a compact set of L 1 (Ω) m , so is (M 1 u n ) + and it is in particular uniformly integrable. We then deduce from the last inequality that (M 0 h n (•, u n )) + is uniformly integrable. To control (M 0 h n (•, u n )) -, we go back to ( 16) and, using ( 17), we rewrite it as follows M 0 u n + M 0 Au n + (M 0 h n (•, u n )) -= (M 0 h n (•, u n )) + + M 0 f ≤ (M 1 u n )) + + Θ + M 0 f. ( 18 ) According to (Φ), we multiply this inequality by sign + (ϕ(u n ) -k)b. By (Φ) on one hand and by the nonnegativity of u n , b and of the entries of M 0 on the other hand, we have Ω sign + (ϕ(u n ) -k)b • M 0 Au n dµ ≥ 0, Ω sign + (ϕ(u n ) -k)b • M 0 u n dµ ≥ 0, ∀ k ∈ [0, +∞). Combining with [START_REF] Pierre | Global existence for a class of quadratic reaction-diffusion systems with nonlinear diffusions and L 1 initial data[END_REF], we deduce Ω sign + (ϕ(u n ) -k)b • (M 0 h n (•, u n )) -dµ ≤ Ω sign + (ϕ(u n ) -k)b • (M 1 u n )) + + Θ + M 0 f dµ. (19) Since lim |r|→+∞ ϕ(r) = +∞, for all k ∈ [0, +∞), there exists R(k) such that [|r| ≥ R(k)] ⊂ [ϕ(r) ≥ k] and therefore [|u n | ≥ R(k)] ⊂ [ϕ(u n ) ≥ k] , µ -a.e.. (20) Let E ⊂ Ω be a measurable set. Then E b • (M 0 h n (•, u n )) -dµ ≤ E∩[|u n |≤R(k)] b • (M 0 h n (•, u n )) -dµ + E∩[|u n |≥R(k)] b • (M 0 h n (•, u n )) -dµ =: I n -+ I n + . If we denote ϕ R := max{ϕ(r); |r| ≤ R}, then R ∈ [0, +∞) → ϕ R ∈ [0, +∞) is a nondecreasing function such that [ϕ(r) ≥ k] ⊂ [|r| ≥ ψ(k)] where ψ(k) := inf ϕ -1 R ([k, +∞)) . By [START_REF] Schmitt | Existence globale ou explosion pour les systèmes de réaction-diffusion avec contrôle de masse[END_REF] and [START_REF] Rothe | Global solutions of reaction-diffusion systems[END_REF] and [ϕ(u n ) ≥ k] ⊂ [|u n | ≥ ψ(k)], we have I n + ≤ [ϕ(u n )≥k] b • (M 1 u n ) + + Θ + M 0 f dµ ≤ [|u n |≥ψ(k)] b • (M 1 u n ) + + Θ + M 0 f dµ. ( 21 ) Since u n lies in a compact set of L 1 (Ω) m and lim k→+∞ ψ(k) = +∞, then lim k→+∞ µ([|u n | ≥ ψ(k)]) = 0 uniformly in n. Thus, given ε ∈ (0, 1), there exists k = k ε large enough so that I n + ≤ ε/2 for all n. (22) We can also control I n -as follows. We remark that (see assumption [START_REF] Bebernes | Finite time blowup for semilinear reactive-diffusive systems[END_REF] in (H1)) |u n | ≤ R(k ε ) ⇒ |h n i (•, u n )| ≤ |h i (•, u n )| ≤ h i (•, R(k ε )). This implies that for some B ∈ (0, +∞), I n -= E∩[|u n |≤R(k)] b • (M 0 h n (•, u n )) -dµ ≤ E B m i=1 h i (•, R(k ε )) (23) We may choose δ small enough (independent of n) such that µ(E) ≤ δ ⇒ E m i=1 h i (•, R(k ε )) < ε/2B. Combining with I n + ≤ ε/2 proved above (see (22)), we deduce that b • (M 0 h n (•, u n )) -is uniformly integrable. Since b ∈ (0, +∞) m , this implies that (M 0 h n (•, u n )) -is itself uniformly integrable. We already know that (M 0 h n (•, u n )) + is uniformly integrable. Thus so is M 0 h n (•, u n ) and this ends the proof of Proposition 2.4. Proof of Theorem 1.1. We consider the solution u n of the approximate system (S n ) built in Lemma 2.1. By Propositions 2.2, 2.3, 2.4, up to a subsequence as n → +∞, we may assume that u n converges in L 1 (Ω) m and µ-a.e. to some u ∈ L 1 (Ω) +m . Moreover, by definition of h n i (see [START_REF] Laamri | Global existence for reaction-diffusion systems with nonlinear diffusion and control of mass[END_REF] ) and the continuity property of h i assumed in (H1), h n i (•, u n ) converges µ-a.e. to h(•, u). Moreover h n (•, u n ) is uniformly integrable. By Vitali's Lemma (see e.g. [START_REF] Schmitt | Existence globale ou explosion pour les systèmes de réaction-diffusion avec contrôle de masse[END_REF] or [START_REF] Fonseca | Modern Methods in the Calculus of Variations : L p spaces[END_REF]), h n (•, u n ) converges also in L 1 (Ω) m to h(•, u). Since u n i = (I +A i ) -1 (h n (•, u n )+f i ) this implies that u i = (I + A -1 i (h(•, u) + f i ) which means that u is solution of the limit system (S) and this ends the proof of Theorem 1.1. Examples In all examples below, Ω is a bounded open subset of R N with regular boundary and equipped with the Lebesgue measure. Examples with linear diffusions and homogeneous boundary conditions We start with a simple example associated with the Laplacian operator with homogeneous boundary conditions. Corollary 3.1 Assume the nonlinearity h = (h 1 , ..., h m ) satisfies (H1), (H2), (M ). Let d i ∈ (0, +∞), f i ∈ L 1 (Ω) + , i = 1, ..., m. Then the following system has a solution    for all i = 1, ..., m; u i ∈ W 1,1 0 (Ω) + , h i (•, u) ∈ L 1 (Ω), u i -d i ∆u i = h i (•, u) + f i . (24) Proof. We consider the operators D(A i ) := {u ∈ W 1,1 0 (Ω) ; ∆u ∈ L 1 (Ω)}, A i u := d i ∆u. ( 25 ) It is classical that these operators A i satisfy the three conditions of (A) in (3) (see e.g. [START_REF] Brezis | Semilinear elliptic equations in L 1[END_REF]). Since 0 ∈ D(A i ), as already noticed just after ( 6), (A inf ) is also satisfied. And for (Φ), if we denote M 0 = [m ij ] 1≤i,j≤m , we choose ϕ(r) := m j=1 ( m i=1 m ij ) d j r j , b = (1, ..., 1) t ∈ (0, +∞) m . Note that m i=1 m ij > 0 for all j = 1, ..., m since m ij ≥ 0 and M 0 is invertible. In particular, lim |r|→+∞ ϕ(r) = +∞). Then, if u ∈ D(A), k ∈ (0, +∞), Ω sign + (ϕ(u) -k)b • M 0 Au = - Ω sign +   m i,j=1 m ij d j u j -k   ∆   m i,j=1 m ij d j u j   ≥ 0. ( 26 ) Then we may apply Theorem 1.1. Remark 3.2 As examples of functions h, we may for instance choose    m = 2, α i , β i ∈ {0} ∪ [0, +∞), i = 1, 2, λ ∈ (0, 1) h 1 (•, u 1 , u 2 ) = λu α 1 1 u α 2 2 -u β 1 1 u β 2 2 , h 2 (•, u 1 , u 2 ) = -u α 1 1 u α 2 2 + u β 1 1 u β 2 2 . We easily check that (H1) holds and that (H2) is satisfied with a i = 1, b i = 0 for all i and ω = 0 (in other words h 1 + h 2 ≤ 0). Moreover (M ) is satisfied with M 1 = 0, Θ = (0, 0) t and M 0 = 1 1 1 λ . Note that for λ = 1, M 0 is not invertible so that only one relation h 1 + h 2 ≤ 0 holds and Theorem 1.1 does not apply. This kind of systems is considered in Theorem 1.2. • Here is another example of a nonlinearity h = (h 1 , h 2 , h 3 ) which satisfies the corollary. m = 3, α i ∈ [1, +∞), i = 1, 2, h 1 = u 3 -u α 1 1 u α 2 2 = h 2 = -h 3 . The corresponding evolution problem is studied in [START_REF] Rothe | Global solutions of reaction-diffusion systems[END_REF] for N ≤ 5, [START_REF] Martin | Nonlinear reaction-diffusion systems[END_REF] for any dimension N with α 1 = α 2 = 1, and in [START_REF] Laamri | Global existence of classical solutions for a class of reaction-diffusion systems[END_REF] where (α 1 , α 2 ) ∈ [1, +∞) 2 . Here (H1) is obviously satisfied and so is (H2) with a i = 1, i = 1, 2, a 3 = 2, b i = 0, i = 1, 2, 3, ω = 0. Then (M ) is satisfied with Θ = (0, 0, 0) t and M 0 =   1 0 0 0 1 0 1 1 2   , M 1 =   0 0 1 0 0 1 0 0 0   . A main point here is that the dependence in u 3 is linear. When it is superlinear, the system does not fit any more into the scope of Theorem 1.1. It is however analyzed in Theorem 1.2. About more general linear diffusions Let us now make some comments on the following example where diffusions are more general than in Corollary 3.1.                  u i ∈ W 1,1 0 (Ω), h i (u 1 , u 2 ) ∈ L 1 (Ω), i = 1, 2, u 1 - N i,j=1 a ij ∂ x i x j u 1 = h 1 (u 1 , u 2 ) + f 1 , u 2 - N i,j=1 b ij ∂ x i x j u 2 = h 2 (u 1 , u 2 ) + f 2 , (27) where a ij , b ij ∈ R, N i,j=1 a ij ξ i ξ j , N i,j=1 b ij ξ i ξ j ≥ α|ξ| 2 , α ∈ (0, +∞), ∀ξ = (ξ 1 , ..., ξ N ) ∈ R N . ( 28 ) We consider the operators        D(A 1 )[resp. D(A 2 )] := v ∈ W 1,1 0 (Ω), N i,j=1 a ij ∂ x i x j v [resp. N i,j=1 b ij ∂ x i x j v] ∈ L 1 (Ω) , A 1 v := N i,j=1 a ij ∂ x i x j v, A 2 v := N i,j=1 b ij ∂ x i x j v. It is easy to see that the assumptions (A), (A inf ) are satisfied. Thus if (H1), (H2), (M ) are satisfied like in Corollary 3.1, then for the approximate solutions u n of this system, as defined in the proof of Theorem 1.1, it follows that u n , h n (•, u n ) are bounded in L 1 (Ω) 2 and u n lies in a compact set of L 1 (Ω) 2 . However, the extra condition (Φ) is not satisfied in general so that it is not clear whether h n (•, u n ) is uniformly integrable. Actually, we have to choose here a different strategy with does not seem to be generalized to the abstract setting of Section 2. It consists in looking at the equation satisfied by T k (u n 1 + ηu n 2 ) where T k is a cut-off function. It is then easy to pass to the limit in the nonlinear terms of the truncated approximate system since they are multiplied by T k (u n 1 + ηu n 2 ) which vanishes for u n i large for all i. Therefore a.e. convergence is sufficient to pass to the limit. The difficulty is then to prove precise estimates independent of n, in terms of η, in order to control the other terms as it done in the parabolic case (see [START_REF] Pierre | Weak solutions and supersolutions in L 1 for reaction-diffusion systems[END_REF], [START_REF] Pierre | Global Existence in Reaction-Diffusion Systems with Dissipation of Mass : a Survey[END_REF], [START_REF] Laamri | Global existence for reaction-diffusion systems with nonlinear diffusion and control of mass[END_REF]). This approach through cut-off functions T k is precisely developed in Section 4 to prove Theorem 1.2 and we refer the reader to this other approach without giving more details here. Examples with linear diffusions and Robin-type boundary conditions Now we analyze what happens for systems like (24) when the boundary conditions are different. If we replace the homogeneous boundary conditions of (24) by homogeneous Neumann boundary conditions, then the result is exactly the same. On the other hand, the situation is quite more complicated if the boundary conditions are of different type for each of the u i 's. This is actually connected with the content of the assumption (Φ). For simplicity, we do it only for a 2 × 2-system. Given a nonlinearity h satisfying (H1), (H2), (M ), we consider the following system with general Robin-type boundary conditions.            i = 1, 2, u i ∈ W 1,1 (Ω) + , h i (u 1 , u 2 ) ∈ L 1 (Ω), u i -d i ∆u i = h i (u 1 , u 2 ) + f i , λ i u i + (1 -λ i )∂ ν u i = ψ i on ∂Ω, λ i ∈ [0, 1], d i ∈ (0, +∞), ψ i ∈ [0, +∞), f i ∈ L 1 (Ω) + . (29) Corollary 3.3 Let (f 1 , f 2 ) ∈ L 1 (Ω) + × L 1 (Ω) + . Assume the nonlinearity h satisfies (H1), (H2), (M ). Assume moreover [0 ≤ λ 1 , λ 2 < 1] or [λ 1 = λ 2 = 1 and ψ 1 = ψ 2 = 0]. Then the system (29) has a solution. Remark 3.4 It is known that the case when one of the λ i is equal to 1 is different (see the analysis in [START_REF] Martin | Influence of mixed boundary conditions in some reaction-diffusion systems[END_REF]). For instance, finite time blow up may occur for the associated evolution problem when the boundary conditions are u 1 = 1, ∂ ν u 2 = 0 (see [START_REF] Bebernes | Finite time blowup for semilinear reactive-diffusive systems[END_REF], [START_REF] Bebernes | Finite-time blowup for a particular parabolic system[END_REF]). Then the operator A 1 does not satisfy (A inf ) as easily seen by considering (as in [START_REF] Martin | Influence of mixed boundary conditions in some reaction-diffusion systems[END_REF]) the following simple example: u σ (x) := cosh(σx)/ cosh(σ) on Ω := (-1, 1) with σ ∈ (0, +∞). Then u σ ≥ 0 and u σ = 1 on ∂Ω. But -Ω u σ = u σ (-1) -u σ (1) → -∞ as σ → +∞. Here we consider only the cases that directly fall into the scope of Theorem 1.1. A few other cases could be treated directly like λ 1 = λ 2 and positive data ψ 1 , ψ 2 or also λ 1 = 0 = λ 2 and ψ 1 = 0. Proof of Corollary 3.3. Here we define D(B i ) = {u ∈ W 2,1 (Ω); λ i u i + (1 -λ i )∂ ν u i = ψ i on ∂Ω}, B i u := -d i ∆u i . Then the closure A i of B i in L 1 (Ω) satisfies the assumption (A) (see [START_REF] Brezis | Semilinear elliptic equations in L 1[END_REF]). • Let us first assume 0 ≤ λ i < 1, i = 1, 2. Then, for u i ∈ D(B i ) Ω B i u i = - Ω d i ∆u i = - Ω d i ∂ ν u i = d i Ω (1 -λ i ) -1 (λ i u i -ψ i ) ≥ -d i (1 -λ i ) -1 Ω ψ i . This remains valid for A i by closure. Thus the assumption (A inf ) is satisfied. For the condition (Φ), we come back to (26) with the same ϕ, b. Let p q (r) be a standard approximation of sign + (r -k) like p q (r) = 0, ∀ r ∈ (-∞, k -1/q]; p q (r) = q (r -k) + 1, ∀ r ∈ [k -1/q, k]; p q (r) = 1, ∀ r ∈ [k, +∞). (30) Then, for u j ∈ D(B j ), j = 1, 2 and for V : = 2 i,j=1 m ij d j u j Ω p q (ϕ(u) -k)b • M 0 Bu = Ω p q (V )|∇V | 2 - ∂Ω p q (V )∂ ν V ≥ - ∂Ω p q (V )∂ ν V. - ∂Ω p q (V )∂ ν V = ∂Ω p q (V ) 2 i,j=1 m ij d j (1 -λ j ) -1 (λ j u j -ψ j ). If ψ j = 0 for all j = 1, ..., m, then -∂Ω p q (V )∂ ν V ≥ 0 and letting q → +∞, we obtain that B and therefore A satisfies the condition (Φ). When ψ j = 0 for some j, we only obtain Ω sign + (ϕ(u) -k)b • M 0 Bu ≥ - ∂Ω sign + (V -k) m i,j=1 m ij d j (1 -λ j ) -1 ψ j =: -η(V, k). The point is that we can slightly modify the proof of Theorem 1.1 to get the same conclusion with this weaker estimate from below. Indeed, in the present case, we have to add the term η(V n , k) in the inequality [START_REF] Rothe | Global solutions of reaction-diffusion systems[END_REF] where V n = 2 i,j=1 m ij d j u n j . Since η(V n , k) tends to 0 as k → +∞ uniformly in n, the rest of the proof remains unchanged if we choose k ε large enough so that η(V, k ε ) < ε also. • Assume now that λ 1 = λ 2 = 1. Then we make the same choice as in (26) and we see that - Ω sign + (ϕ(u) -k)b • M 0 Au ≥ - ∂Ω sign + ( 2 i,j=1 m ij d j ψ j -k)∂ ν ( 2 i,j=1 m ij d j u j ) ≥ 0, since u j ≥ 0, u j = 0 on ∂Ω imply that ∂ ν u j ≤ 0 on ∂Ω. For the same reason, (A inf ) holds since - Ω ∆u j = - ∂Ω ∂ ν u j ≥ 0. Examples with nonlinear diffusions Let us now consider nonlinear diffusions. • We start with porous media type equations.    For i = 1, ..., m, u i ∈ L 1 (Ω) + , ϕ i (u i ) ∈ W 1,1 0 (Ω), h i (u) ∈ L 1 (Ω), u i -∆ϕ i (u i ) = h i (u) + f i , (31) where ϕ i : [0, +∞) → [0, +∞) is continuous, increasing with ϕ i (0) = 0, lim s→+∞ s -(N -2) + /N ϕ i (s) = +∞. Corollary 3.5 Let (f 1 , • • • , f m ) ∈ (L 1 (Ω) + ) m . Assume the nonlinearity h satisfies (H1), (H2), (M ). Then the system (31) has a solution. Proof. We naturally define D(A i ) := {u i ∈ L 1 (Ω) ; ϕ i (u i ) ∈ W 1,1 0 (Ω), ∆ϕ i (u i ) ∈ L 1 (Ω) }, A i u i := -∆ϕ i (u i ). It is classical that the operators A i satisfy the assumptions (A), (A inf ) (see [START_REF] Brezis | Semilinear elliptic equations in L 1[END_REF], [START_REF] Ph | Équations d'évolution dans un espace de Banach et applications[END_REF]). To check that (Φ) is satisfied, we consider r ∈ [0, +∞) m → ϕ(r) := m i,j=1 m ij ϕ j (r j ) and b := (1, ..., 1) t . Then Ω sign + (ϕ(u) -k)b • M 0 Au = - Ω sign + ( m i,j=1 m ij ϕ j (u j ) -k)∆   m i,j=1 m ij ϕ j (u j )   ≥ 0. We then apply Theorem 1.1. • A second example with nonlinear diffusions is the following where p ∈ (1, +∞) :    For i = 1, 2, u i ∈ W 1,p-1 0 (Ω) + , f i ∈ L 1 (Ω), d i ∈ (0, +∞), u i -d i ∆ p u i = h i (u 1 , u 2 ) + f i , (32) where for v ∈ W 1,p-1 (Ω), ∆ p v := ∇ • (|∇v| p-2 ∇v). Corollary 3.6 Let (f 1 , f 2 ) ∈ L 1 (Ω) + ×L 1 (Ω) + . Assume the nonlinearity h satisfies (H1), (H2), (M ). Then the system (32) has a solution. Proof. Here we define for i = 1, 2 D(B i ) := {v ∈ W 1,p 0 (Ω) ∩ L 2 (Ω) ; ∆ p v ∈ L 2 (Ω)}, B i (v) := -d i ∆ p v. And the operators A i are defined as the closure of B i in L 1 (Ω). Then the assumptions (A), (A inf ) are satisfied (see e.g. [START_REF] Herrero | Asymptotic behaviour of the solutions of a strongly nonlinear parabolic problem[END_REF]). Let us prove that (Φ) holds. For p q defined as in (30) and ûi ∈ D(B i ), i = 1, 2, we have - Ω p q (û 1 + û2 )(∆ p û1 + ∆ p û2 ) = Ω (∇û 1 + ∇û 2 )(|∇û 1 | p-2 ∇û 1 + |∇û 2 | p-2 ∇û 2 )p q (û 1 + û2 ). The mapping r ∈ R N → |r| p is convex. Therefore its gradient r ∈ R N → p|r| p-2 r ∈ R N is monotone, which means (r 1 -r 2 ) • (|r 1 | p-2 r 1 -|r 2 | p-2 r 2 ) ≥ 0 , ∀ r 1 , r 2 ∈ R N . We apply this with r 1 := ∇û 1 (x), r 2 := -∇û 2 (x), x ∈ Ω to deduce after integration on Ω: - Ω p q (û 1 + û2 )(∆ p û1 + ∆ p û2 ) ≥ 0. And letting q → +∞ gives -Ω sign + (û 1 + û2 -k)(∆ p û1 + ∆ p û2 ) ≥ 0 and by closure Ω sign + (û 1 + û2 -k)(A 1 û1 + A 2 û2 ) ≥ 0 ∀ u 1 ∈ D(A 1 ), ∀ u 2 ∈ D(A 2 ). ( 33 ) For the condition (M ), we choose b := (1, 1) t , ϕ(r 1 , r 2 ) := c 1 r 1 + c 2 r 2 , c 1 := {(m 11 + m 21 )d 1 } 1/(p-1) , c 2 := {(m 21 + m 22 )d 2 } 1/(p-1) where M 0 := (m ij ) 1≤i,j≤2 . Then for u 1 ∈ D(A 1 ), u 2 ∈ D(A 2 ), u = (u 1 , u 2 ), Ω sign + (ϕ(u) -k)b • M 0 Au = Ω sign + (c 1 u 1 + c 2 u 2 -k) 2 i,j=1 m ij d j A j u j = Ω sign + (c 1 u 1 + c 2 u 2 -k)[A 1 (c 1 u 1 ) + A 2 (c 2 u 2 )] ≥ 0, the last inequality coming from (33) applied with û1 := c 1 u 1 , û2 := c 2 u 2 . This ends the proof of Corollary 3.6. • To end this section, let us comment on the following system which is a model of situations where the operators A i are very different from each other :            p = 2 u 1 ∈ W 1,p-1 0 (Ω) + , u 2 ∈ W 1,1 0 (Ω) + , h 1 , h 2 , f 1 , f 2 ∈ L 1 (Ω), u 1 -∆ p u 1 = h 1 (u 1 , u 2 ) + f 1 , u 2 -∆u 2 = h 2 (u 1 , u 2 ) + f 2 . (34) As a consequence, the compatibility condition (Φ) is not generally satisfied. However, if the nonlinearity h satisfies (H1), (H2), (M ), then for the approximate solution u n defined in Section 2, h n (•, u n ) is bounded in L 1 (Ω) × L 1 (Ω) and therefore u n lies in a compact set of L 1 (Ω) × L 1 (Ω). Thus we may assume that, up to a subsequence, u n converges to some u in L 1 (Ω) × L 1 (Ω) and a.e. so that h n (•, u n ) converges a.e. to h(•, u). Unfortunately we do not know whether we can pass to the limit in the approximate version of system (34). And it is not clear how the use of cut-off function T k as in the next section can help. We leave this as an open problem. Proof of Theorem 1.2 For each k = 1, ..., m, we define σ k := min{α k , β k }, γ k := |α k -β k | so that (see [START_REF] Laamri | Global existence of classical solutions for a class of reaction-diffusion systems[END_REF] for the definition of I, J ): h k (u) = (β k -α k ) Π m =1 u σ B(u), B(u) := k 1 Π i∈I u γ i i -k 2 Π j∈J u γ j j . (35) We first solve the approximate system with the bounded data f n i := inf{f i , n}, n ∈ N. Lemma 4.1 There exists a nonnegative solution u n ∈ ∩ p∈[1,∞) W 2,p (Ω) +m of    For i = 1, ..., m, u n i -d i ∆u n i = (β i -α i )Π m k=1 (u n k ) σ k B(u n ) + f n i in Ω, ∂ ν u i = 0 on ∂Ω. (36) Moreover, we have γ j u n i + γ i u n j -∆(γ j d i u n i + γ i d j u n j ) = γ j f n i + γ i f n j , ∀ i ∈ I, j ∈ J, (37) and u n is bounded in L 1+η (Ω) m for some η > 0. Proof. By the abstract Lemma 2.1, for ε ∈ (0, 1), there exists a regular nonnegative solution u ε of        For i = 1, ..., m, u ε i -d i ∆u ε i = (β i -α i ) Π m k=1 (u ε k ) σ k B(u ε ) 1 + ε Π k=1 m (u ε k ) α k + Π m k=1 (u ε k ) β k + f n i in Ω, ∂ ν u ε i = 0 on ∂Ω. (38) Indeed the nonlinearity is here quasi-positive and uniformly bounded by max i γ i /ε. Then by multiplying the equations in i ∈ I, j ∈ J respectively by γ j , γ i , we have (γ j u ε i + γ i u ε j ) -∆(γ j d i u ε i + γ i d j u ε j ) = γ j f n i + γ i f n j . (39) If d := max 1≤k≤m d k , this implies (since u ε i , u ε j ≥ 0) d -1 [γ j d i u ε i + γ i d j u ε j ] -∆(γ j d i u ε i + γ i d j u ε j ) ≤ γ j f n i + γ i f n j , ∂ ν (γ j d i u ε i + γ i d j u ε j ) = 0 on ∂Ω. We deduce that γ j d i u ε i + γ i d j u ε j ≤ d -1 I -∆ -1 (γ j f n i + γ i f n j ). (40) Since the f n i are bounded by n (fixed), using the nonnegativity of u ε , we obtain that the u ε i are bounded in L ∞ (Ω) independently of ε. By standard elliptic regularity results applied to the equation (38) in u i , they are also bounded in ∩ p∈[1,∞) W 2,p (Ω). They are therefore included in a compact set of L ∞ (Ω) as → 0. We can then easily pass to the limit as ε → 0 and obtain the convergence of a subsequence of u ε toward a solution u n of (36). Then the identities (37) follow by passing to the limit in (39). But (40) remains also valid at the limit and implies that u n i , i = 1, ..., m are bounded in L 1 (Ω) independently of n. Next we may rewrite (37), using d := min 1≤k≤m d k : d -1 [γ j d i u n i +γ i d j u n j ]-∆(γ j d i u n i +γ i d j u n j ) = γ j [(d -1 d i -1)u n i +f n i ]+γ i [(d -1 d j -1)u n j +f n j ], ∀ i ∈ I, j ∈ J, (41) together with ∂ ν (γ j d i u n i + γ i d j u n j ) = 0 on ∂Ω. Since the right-hand side is bounded in L 1 (Ω), this implies that γ j d i u n i + γ i d j u n j is bounded in L p (Ω) for all p ∈ [1, N/(N -2) + ) (see e.g. [START_REF] Brezis | Semilinear elliptic equations in L 1[END_REF]). This ends the proof of Lemma 4.1. In order to rewrite the relations (41), we denote g n k := (d -1 d k -1)u n k + f n k ≥ 0, ∀ k = 1, ..., m. (42) When f k log f k is assumed to be in L 1 (Ω) for all k = 1, ..., m, then sup n∈N max 1≤k≤m Ω g n k + | g n k log g n k | < +∞. (43) This is due to the "L LOG L" assumption on the f i 's and on the L 1+η -bound on the u n i stated in Lemma 4.1. Next we introduce the solutions G n k of d -1 G n k -∆G n k = g n k in Ω, ∂ ν G n k = 0 on ∂Ω, G n k ≥ 0, k = 1, ..., m, (44) so that the relations (37), (41) may be rewritten γ j d i u n i + γ i d j u n j = γ j G n i + γ i G n j , ∀ i ∈ I, j ∈ J. ( 45 ) Our goal is to prove that the nonlinearity of the system (36) is bounded in L 1 (Ω) m independently of n. It will imply enough compactness on u n to pass to the limit. The following lemma will provide the key estimate. Lemma 4.2 Under the assumptions of Theorem 1.2, there exists θ n ∈ W 2,1 (Ω) +m (unique) such that    γ j d i θ n i + γ i d j θ n j = γ j G n i + γ i G n j , ∀ i ∈ I, j ∈ J, k 1 Π i∈I (θ n i ) γ i = k 2 Π j∈J (θ n j ) γ j (or B(θ n ) = 0), ∂ ν θ n k = 0 on ∂Ω, ∀ k = 1, ..., m. (46) Moreover sup n max 1≤k≤m ∆θ n k L 1 (Ω) < +∞. ( 47 ) We postpone the proof of this lemma and we end the proof of Theorem 1.2. Without loss of generality, we assume that 1 ∈ I. Using the equation in u n 1 given in the approximate system (36), and the fact that B(θ n ) = 0, we have u n 1 -θ n 1 -d 1 ∆(u n 1 -θ n 1 ) + γ 1 Π m k=1 (u n k ) σ k [B(u n ) -B(θ n )] = ρ n , ρ n := f n 1 -θ n 1 + d 1 ∆θ n 1 . (48) Now the main point is that B(u n ), B(θ n ) have the property that B(u n ) = b n (•, u n 1 ), B(θ n ) = b n (•, θ n 1 ), (49) where r → b n (•, r) is a increasing function. More precisely, we deduce from the relations (45), (46) that γ i d j θ n j = γ j G n i + γ i G n j -γ j d 1 θ n 1 , ∀ j ∈ J, γ 1 d i θ n i = γ i d 1 θ n 1 + γ 1 G n i -γ i G n 1 , ∀i ∈ I. γ i d j u n j = γ j G n i + γ i G n j -γ j d 1 u n 1 , ∀ j ∈ J, γ 1 d i u n i = γ i d 1 u n 1 + γ 1 G n i -γ i G n 1 , ∀i ∈ I. Plugging this into B(u n ), B(θ n , recalling that B(v) = k 1 Π i∈I v γ i i -k 2 Π j∈J v γ j j , we indeed obtain (49) by setting b n (x, r) := k 1 Π i∈I (δ i d 1 r + F n i (x)) γ i -k 2 Π j∈J (F n j (x) -δ j d 1 r) γ j , ∀ r ∈ [r n -, r n + ], (50) where we define    δ k := γ k /(γ 1 d k ), ∀ k = 1, ..., m, F n i := d -1 i G n i -δ i G n 1 , ∀ i ∈ I, F n j := δ j G n 1 + d -1 j G n j , ∀ j ∈ J, r n -(x) := max i∈I [F n i (x)] -/(δ i d 1 ), r n + (x) := min j∈J F n j (x)/(δ j d 1 ). ( 51 ) We multiply the equation (48) by sign(u n 1 -θ n 1 ) and we integrate. Then we use the two following properties Ω sign(u n 1 -θ n 1 )[u n 1 -θ n 1 -d 1 ∆(u n 1 -θ n 1 )] ≥ 0, sign(u n 1 -θ n 1 )[B(u n ) -B(θ n )] = sign(u n 1 -θ n 1 )[b n (•, u n 1 ) -b n (•, θ n 1 )] = |b n (•, u n 1 ) -b n (•, θ n 1 )|, to obtain γ 1 Ω Π m k=1 (u n k ) σ k |b n (•, u n 1 ) -b n (•, θ n 1 )| ≤ Ω |ρ n |. Since ρ n is bounded in L 1 (Ω) by Lemma 4.2, and since b n (•, θ n 1 ) ≡ 0, this implies that the nonlinearity of the equation in u n 1 u n 1 -d 1 ∆u n 1 + γ 1 Π m k=1 (u n k ) σ k b n (•, u n 1 ) = f n i in Ω, is bounded in L 1 (Ω) independently of n. This implies that ∆u n 1 is bounded in L 1 (Ω). Going back to (45), it follows that ∆u n k is bounded in L 1 (Ω) m for all k = 1, ..., m as well. Up to a subsequence, we may deduce that u n k converges for all k to some u k in L 1 (Ω) and a.e. and that ∇u n k converges in L 1 (Ω) N to ∇u k . Note also that by Fatou's lemma, Π k (u k ) σ k B(u) ∈ L 1 (Ω). Now to end the passing to the limit, we look at the equation satisfied by T R (v n i ) where v n i := u n i + ε j =i u n j and ε ∈ (0, 1) and the T R are C 2 -cut-off functions satisfying T R (s) = s if s ∈ [0, k -1], T R (s) = 0 if s ≥ k, T R (s) ≤ k, ∀s ∈ [0, +∞), 0 ≤ T R (s) ≤ 1, T R (s) ≤ 0 for all s ≥ 0. ( 52 ) -∆T R (v n i ) = -T R (v n i )∆v n i -T R (v n i )|∇v n i | 2 ≥ -T R (v n i )∆(u n i + ε j =i u n j ). Coming back to the system (36), we have that for all ψ ∈ C ∞ Ω + , Ω ∇ψ∇T R (v n i ) ≥ Ω ψT R (v n i )    Π k (u n k ) σ k B(u n )[ β i -α i d i + ε j =i β j -α j d j ] + f n i -u n i d i + ε j =i f n j -u n j d j    . We know that, up to a subsequence, u n converges in L 1 (Ω) m and a.e. to some u ∈ W 1,1 (Ω) m . We may pass to the limit along this subsequence in the above inequality. Indeed the nonlinear terms on the right converge a.e. and T R ( v n i )Π k (u n k ) σ k B(u n ) is uniformly bounded, while the f n k , u n k , k = 1, ..., m converge in L 1 (Ω). Moreover, ∇T R (v n i ) = T R (v n i )∇v n i converges in L 1 (Ω) to ∇T R (v i ), v i := u i + ε j =i u j . At the limit, we obtain the same inequality without the superscript n . Then, we let ε go to 0 to obtain Ω ∇ψ∇T R (u i ) ≥ Ω ψT R (u i ) Π k (u k ) σ k B(u) β i -α i d i + f i -u i d i . And now we let R → +∞ to obtain Ω ∇ψ∇u i ≥ Ω ψ Π m k=1 (u k ) σ k B(u) β i -α i d i + f i -u i d i , or equivalently ∀i = 1, ..., m, Ω ψu i + d i ∇ψ∇u i ≥ Ω ψ {Π k (u k ) σ k B(u)(β i -α i ) + f i } . ( 53 ) Throughout the rest of the proof, we will assume that γ j G n i + γ i G n j ≡ 0. By maximum principle applied to the equation (44) defining the G n k , a n ij := inf(γ j G n i + γ i G n j ) > 0. Also for the rest of the proof, choosing n 0 large enough, we fix c ∈ (0, +∞) such that 0 < c < a n ij /[γ j d i + γ i d j ], ∀ (i, j) ∈ I × J, ∀n ≥ n 0 . (55) This definition of c will be used only in STEPS 6 and 7 of the proof. For simplicity, we now drop the superscript ' n ' in the rest of the proof. STEP 2: Existence of θ n satisfying (46). Again we assume (without loss of generality) that 1 ∈ I so that we may use (50), (51). Thus, for all x ∈ Ω, the function r ∈ [r -(x), r + (x)] → b(x, r) is increasing where r -, r + are defined in (51). Moreover b(x, r -(x)) = -k 2 Π j∈J (F j (x) -d 1 δ j r -(x)) γ j ≤ 0, b(x, r + (x)) = k 1 Π i∈I (d 1 δ i r + (x) + F i (x)) γ i ≥ 0. Thus there exists a unique θ 1 (x) ∈ [r -(x), r + (x)] such that b(x, θ 1 (x)) = 0. Since the function (x, r) → b(x, r) is regular, by the implicit function theorem, so is x → θ 1 (x). The function θ k are then uniquely determined from the first line of (46) which we rewrite as: θ i := d 1 δ i θ 1 + F i , ∀ i ∈ I; θ j := F j -d 1 δ j θ 1 , ∀j ∈ J. (56) It remains to prove that ∂ ν θ k = 0 on ∂Ω for all k = 1, ..., m. This will be a consequence of the following computation (see (60) ). k 1 Π i∈I θ γ i i = k 2 Π j∈J θ γ j j , which implies log k 1 + i∈I γ i log θ i = log k 2 + j∈J γ j log θ j . (57) Differentiating this leads to i∈I γ i ∇θ i θ i = j∈J γ j ∇θ j θ j . (58) Inserting (56) in this formula, we obtain ∇θ 1 in terms of the θ k , namely d 1 ∇θ 1 A = j∈J γ j ∇F j θ j - i∈I γ i ∇F i θ i , A := m k=1 γ k δ k θ k . ( 59 ) Since ∇F k • ν = 0 on ∂Ω for all k = 1, ..., m, it follows from this identity that ∇θ 1 • ν = 0 on ∂Ω as well. And by (56) it also follows that ∇θ k • ν = 0 on ∂Ω, ∀ k = 1, ..., m. (60) Differentiating once more (58) gives i∈I γ i ∆θ i θ i - γ i |∇θ i | 2 θ 2 i = j∈J γ j ∆θ j θ j - γ j |∇θ j | 2 θ 2 j , or also, using again (56) and the definition of A in (59), d 1 A∆θ 1 = i∈I - γ i ∆F i θ i + γ i |∇θ i | 2 θ 2 i + j∈J γ j ∆F j θ j - γ j |∇θ j | 2 θ 2 j . ( 61 ) Our goal is to estimate the L 1 -norm of ∆θ 1 . We remark that, if we denote α k := γ k δ k /Aθ k , k = 1, ..., m, then 0 ≤ α k ≤ 1, m k=1 α k = 1. But the relation (61) may be rewritten d 1 ∆θ 1 = i∈I α i [- ∆F i δ i + |∇θ i | 2 δ i θ i ] + j∈J α j [ ∆F j δ j - |∇θ j | 2 δ j θ j ]. (62) According to the definition of F k in (51) and to the definition of G k (or more precisely of G n k ) in ( 44) and (43), we know that ∆F k L 1 (Ω) , k = 1, ..., m is bounded in terms of the data (independently of n). Thus i∈I -α i ∆F i δ i + j∈J α j ∆F j δ j L 1 (Ω) ≤ C, (63) where C is independent of n. We also have that Ω ∆θ 1 = ∂Ω ∂ ν θ 1 = 0. Inserting this into (62) and (63) gives Ω j∈J α j |∇θ j | 2 δ j θ j ≤ Ω i∈I α i |∇θ i | 2 δ i θ i + C, (64) where again C does not depend on n. Therefore, it is sufficient to bound the right-hand side of (64) to obtain a bound on ∆θ 1 L 1 (Ω) and this will end the proof of Lemma 4.2 (since an L 1 -bound on ∆θ 1 implies an L 1 -bound on ∆θ k for all k = 1, ..., m). STEP 4 : A bound from below on the θ k . The previous step indicates that one has to bound |∇θ i | 2 /θ i in L 1 (Ω) for all i ∈ I. The identity (59) says that c 2 i 0 Ω |∇F i 0 | 2 a + |F i 0 | ≤ Ω | f i 0 | log | f i 0 | + | f i 0 |[a -1 d -1] + 1, where f i 0 = d -1 F i 0 -∆F i 0 = g i 0 /d 1 -δ i 0 g 1 as indicated in (70). We conclude from the estimate (43) that ∇F i 0 θ i 0 √ θ 1 A L 2 (Ω) ≤ C where C is independent of n. (73) STEP 6: More bounds from below on the θ k . We prove here the two following facts, where the real number c was defined in (55): Ω = Ω I ∪ Ω J , Ω I := {x ∈ Ω ; θ i (x) > c, ∀i ∈ I}, Ω J := {x ∈ Ω ; θ j (x) > c, ∀j ∈ J}, sup i∈I θ i ≥ c 1 > 0, for some c 1 ∈ (0, ∞) independent of n. (74) For the first part of (74), assume by contradiction that there exists x ∈ Ω \ (Ω I ∪ Ω J ). Then there exists (i, j) ∈ I × J such that, for this x: θ i (x) ≤ c and θ j (x) ≤ c, or equivalently, γ j d i θ i (x) ≤ γ j d i c, γ i d j θ j (x) ≤ γ i d j c. Let us add these last two inequalities. Using the first line of (46) and the definition of c in (55), we have γ j G i (x) + γ i G j (x) = γ j d i θ i (x) + γ i d j θ j (x) ≤ [γ j d i + γ i d j ]c < inf{γ j G i + γ i G j }. And this is a contradiction. Whence the first statement of (74). For the second one, let us first note that sup Now let x ∈ Ω J so that, by the previous statement, θ j (x) ≥ c for all j ∈ J. We then use the second line of (46) to obtain k 2 c j∈J γ j ≤ k 2 Π j∈J θ j (x) γ j = k 1 Π i∈I θ i (x) γ i ≤ k 1 [sup i∈I θ i ] i∈I γ i . This implies that sup i∈I θ i ≥ c 1 := min{c, k 2 c j∈J γ j /k 1 [ i∈I γ i ] -1 }. Whence the second statement of (74). STEP 7: End of the proof of Lemma 4.2. This is where we use that I (for instance) has at most two elements. Indeed, let us go back to the expression of ∇θ 1 / √ θ 1 in (65). We already know by STEP 5 that all terms indexed by j ∈ J are bounded in L 2 (Ω). Since F 1 ≡ 0, there is at most one term indexed by i ∈ I, namely none if I = {1}, and only ∇F i 0 /(θ i 0 √ θ 1 A) if I = {1, i 0 }. If I = {1}, it immediately follows that ∇θ 1 / √ θ 1 is bounded in L 2 (Ω) independently of n. If I = {1, i 0 }, then sup i∈I θ i = sup{θ 1 , θ i 0 }. It follows from the second line of (74) in STEP 6 that (72) holds. Consequently, as proved in Remark 4.4, ∇F i 0 /(θ i 0 √ θ 1 A) is bounded in L 2 (Ω) independently of n. Having controlled all terms in (65), we can conclude that ∇θ 1 / √ θ 1 is itself also bounded in L 2 (Ω) independently of n. By symmetry, this also holds for ∇θ i 0 / θ i 0 . This implies that the right-hand side of (64) in STEP 4 is bounded independently of n and ends the proof of Lemma 4.2. Let us now state the following technical lemma which was used in two places in the previous proof. If moreover f ≥ 0, then Ω |∇F | 2 F ≤ Ω f log + f + (d -1 -1)f + de -1 . Proof. For the first inequality of the lemma, let us first remark that, if we set F := a F , then by homogeneity, If f ≥ 0, then F ≥ 0 by maximum principle. We multiply the equation in F by log(F + ε) and we integrate by parts. Then, Ω dF log(F + ε) + |∇F | 2 F + = Ω f log(F + ε). (76) We apply the Young's inequality (75) with r := f (x), s := log(F (x) + ε). Then f log(F + ε) ≤ (f log f -f ) + F + ε, so that, using (76), we deduce [F ≥1] dF log(F + ε) + Ω |∇F | 2 F + ε ≤ Ω f log f -f + F + ε -d [F ≤1] F log(F + ε). STEP 3 : 3 Differentiating B(θ) = 0. The condition B(θ) = 0 means x∈Ω I θ i (x) ≥ inf x∈Ω I θ i (x) ≥ c, ∀i ∈ I. Lemma 4 . 5 45 Let F ∈ W 2,1 (Ω) such that, for some d ∈ (0, +∞)dF -∆F = f, ∂ ν F = 0 on ∂Ω with f, f log |f | ∈ L 1 (Ω).For a ∈ (0, +∞), we haveΩ |∇F | 2 a + |F | ≤ Ω |f | log |f | + |f |[(ad) -1 -1] + 1. Ω |∇F | 2 aΩ|∇ F | 2 1+ 22 + |F | = a | F | .Let us now multiply the equation d F -∆ F = f /a by sign( F ) log(| F | + 1). We obtainΩ | F | log(| F | + 1) + |∇ F | 2 1 + | F | = 1 a Ω f sign( F ) log(| F | + 1) ≤ 1 a Ω |f | log(| F | + 1).Now we use the Young's convexity inequality∀ r ∈ [0, +∞), ∀ s ∈ R, rs ≤ (r log r -r) + e s .(75)We apply it with r := |f (x)|, s := log(| F (x)| + 1) to deduce Ω |f | log(| F | + 1) ≤ Ω |f |[log |f | -1] + | F | + 1. From the equation in F , we also derive d Ω | F | ≤ Ω |f |/a. The first inequality of Lemma 4.5 follows. Now we let ε → 0 and we use[F ≥1] F log F ≥ 0, f log f ≤ f log + f, Ω dF = Ω f, x log x ≥ -e -1 , ∀ x ∈ (0, 1),to deduce the second estimate of Lemma 4.5. 1.2 holds even if |I| and |J| > 2. But it is not clear how to extend our main Lemma 4.2 to the general case. We leave this as an open problem. Theorem 1.2 provides a solution u such that h i (u) ∈ L 1 (Ω). It is easy to write down explicit examples where the nonlinear terms Π k u α k k , Π k u β k k are not separately in L 1 (Ω). For instance, let N > 4 and let us introduce the following function σ where r Remark 1.5 But, on the other hand, by passing directly to the limit as n → +∞ in (37), we have that for i ∈ I, j ∈ J Ω ψ(γ j u i + γ i u j ) + ∇ψ∇[γ j d i u i + γ i d j u j ] = Ω ψ(γ j f i + γ i f j ). This implies that all inequalities in (53) are actually equalities so that, for all i = 1, ..., m, This ends the proof of Theorem 1.2. Proof of Lemma 4.2. STEP 0 : The case m = 2. For simplicity, we drop the index 'n'. We assume I = {1}, J = {2}. Then the system (46) is equivalent to It is easily seen that the equation in θ 1 has a unique regular solution (see STEP 2 for more details) and differentiating this equation gives If γ = 1, we immediately have By (42) and (44), we know that ∆G is bounded in L 1 (Ω) independently of n as soon as the f k are (only) in L 1 (Ω) + for all k. Therefore so is ∆θ 1 . If γ = 1, then integrating (54) and using ∂ ν θ 1 = 0, we obtain Again, the last integral is bounded independently of n if the f k are (only) assumed to be in L 1 (Ω). Therefore so is the first integral. But by positivity, this implies that θ γ-2 (Ω) independently of n. So is ∆θ 1 by going back to (54). And finally the same holds for ∆θ 2 = ∆(G -δθ 1 ). We now come back to the general situation m ≥ 2 and with the LLogL assumptions on the f k . STEP 1 : Let us first treat the trivial case when there exists (i 0 , j 0 ) ∈ I × J such that γ j 0 G n i 0 + γ i 0 G n j 0 ≡ 0 (i.e. G n i 0 ≡ 0 ≡ G n j 0 ). Then, by the first line of (46), θ i0 ≡ 0 ≡ θ j 0 . Using again the first line of (46), we deduce that θ n k = G n k for all k = 1, ..., n. Thus, the conclusion of the lemma is obvious in this case.

00174870

en

2024/03/05 22:32:07

2007

https://hal.science/hal-00174870/file/2007-32.pdf

I Hlaváček email: [email protected] A A Novotny email: [email protected] J Soko Lowski email: [email protected] A Żochowski Energy change in elastic solids due to a spherical or circular cavity, considering uncertain input data In the paper we consider topological derivative of shape functionals for elasticity, which is used to derive the worst and also the maximum range scenarios for behavior of elastic body in case of uncertain material parameters and loading. It turns out that both problems are connected, because the criteria describing this behavior have form of functionals depending on topological derivative of elastic energy. Therefore in the first part we describe the methodology of computing the topological derivative with some new additional conditions for shape functionals depending on stress. For the sake of fulness of presentation the explicit formulas for stress distribution around cavities are provided. Introduction In the paper we consider topological derivative of shape functionals for elasticity, which is used to derive the worst and also the maximum range scenarios for behavior of elastic body in case of uncertain material parameters. It turns out that both problems are connected, because the criteria describing this behavior have form of functionals depending on topological derivative of elastic energy. Therefore in the first part we describe the methodology of computing the topological derivative with some new additional conditions for shape functionals depending on stress. For the sake of fulness of presentation the explicit formulas for stress distribution around cavities are provided. Topological Derivative The topological derivative T Ω of a shape functional J (Ω) is introduced in [START_REF] Soko Lowski | On topological derivative in shape optimization[END_REF] in order to characterize the infinitesimal variation of J (Ω) with respect to the infinitesimal variation of the topology of the domain Ω. The topological derivative allows us to derive the new optimality condition for the shape optimization problem: J (Ω * ) = inf Ω J (Ω) . The optimal domain Ω * is characterized by the first order condition [START_REF] Soko | Introduction to Shape Optimization[END_REF] defined on the boundary of the optimal domain Ω * , dJ(Ω * ; V ) ≥ 0 for all admissible vector fields V , and by the following optimality condition defined in the interior of the domain Ω * : T Ω * (x) ≥ 0 in Ω * . The other use of the topological derivative is connected with approximating the influence of the holes in the domain on the values of integral functionals of solutions, what allows us to solve a class of shape inverse problems. In general terms the notion of the topological derivative (TD) has the following meaning. Assume that Ω ⊂ IR N is an open set and that there is given a shape functional J : Ω \ K → IR for any compact subset K ⊂ Ω. We denote by B ρ (x), x ∈ Ω, the ball of radius ρ > 0, B ρ (x) = {y ∈ IR N | y -x < ρ}, B ρ (x) is the closure of B ρ (x), and assume that there exists the following limit T(x) = lim ρ↓0 J (Ω \ B ρ (x)) -J (Ω) |B ρ (x)| which can be defined in an equivalent way by T(x) = lim ρ↓0 J (Ω \ B ρ (x)) -J (Ω) ρ N The function T(x), x ∈ Ω, is called the topological derivative of J (Ω), and provides the information on the infinitesimal variation of the shape functional J if a small hole is created at x ∈ Ω. This definition is suitable for Neumann-type boundary conditions on ∂B ρ . In several cases this characterization is constructive, i.e. TD can be evaluated for shape functionals depending on solutions of partial differential equations defined in the domain Ω. For instance, TD may be computed for the 3D elliptic Laplace type equation, as well as for extremal values of cost functionals for a class of optimal control problems. All these examples have one common feature: the expression for TD may be calculated in the closed functional form. As we shall see below, the 3D elasticity case is more difficult, since it requires evaluation of integrals on the unit sphere with the integrands which can be computed at any point, but the resulting functions have no explicit functional form. In the particular case of energy functional we obtain the closed formula. In section 5 we compare the results of the present paper with the formulae for 2D elasticity. The main contribution of the present paper is the procedure for computations of the topological derivatives of shape functionals depending on the solutions of 3D elasticity systems. Therefore it constitutes an essential extension of the results given in [START_REF] Soko Lowski | On topological derivative in shape optimization[END_REF] for the 2D case. Problem setting for elasticity systems We introduce elasticity system in the form convenient for the evaluation of topological derivatives. Let us consider the elasticity equations in IR N , where N = 2 for 2D and N = 3 for 3D,    div σ(u) = 0 in Ω u = g on Γ D σ(u)n = T on Γ N (1) and the same system in the domain with the spherical cavity B ρ (x 0 ) ⊂ Ω centered at x 0 ∈ Ω, Ω ρ = Ω \ B ρ (x 0 ),        div σ ρ (u ρ ) = 0 in Ω ρ u ρ = g on Γ D σ ρ (u ρ )n = T on Γ N σ ρ (u ρ )n = 0 on ∂B ρ (x 0 ) (2) where n is the unit outward normal vector on ∂Ω ρ = ∂Ω ∪ ∂B ρ (x 0 ). Assuming that 0 ∈ Ω, we can consider the case x 0 = 0. Here u and u ρ denote the displacement vectors fields, g is a given displacement on the fixed part Γ D of the boundary, T is a traction prescribed on the loaded part Γ N of the boundary. In addition, σ is the Cauchy stress tensor given, for ξ = u (eq. 1) or ξ = u ρ (eq. 2), by σ(ξ) = D∇ s ξ , (3) where ∇ s (ξ) is the symmetric part of the gradient of vector field ξ, that is ∇ s (ξ) = 1 2 ∇ξ + ∇ξ T , (4) and D is the elasticity tensor, D = 2µII + λ (I ⊗ I) , (5) with µ = E 2(1 + ν) , λ = νE (1 + ν)(1 -2ν) and λ = λ * = νE 1 -ν 2 (6) being E the Young's modulus, ν the Poisson's ratio and λ * the particular case for plane stress. In addition, I and II respectively are the second and fourth order identity tensors. Thus, the inverse of D is D -1 = 1 2µ II - λ 2µ + N λ (I ⊗ I) , The first shape functional under consideration depends on the displacement field, J u (ρ) = Ωρ F (u ρ ) dΩ , F (u ρ ) = (Hu ρ • u ρ ) p , ( 7 ) where F is a C 2 function, p ≥ 2 is an integer. It is also useful for further applications in the framework of elasticity to introduce the yield functional of the form J σ (ρ) = Ωρ Sσ(u ρ ) • σ(u ρ ) dΩ , (8) where S is an isotropic fourth-order tensor. Isotropicity means here, that S may be expressed as follows S = 2mII + l (I ⊗ I) , where l, m are real constants. Their values may vary for particular yield criteria. The following assumption assures, that J u , J σ are well defined for solutions of the elasticity system. (A) The domain Ω has piecewise smooth boundary, which may have reentrant corners with α < 2π created by the intersection of two planes. In addition, g, T must be compatible with u ∈ H 1 (Ω; IR N ). The interior regularity of u in Ω is determined by the regularity of the right hand side of the elasticity system. For simplicity the following notation is used for functional spaces, H 1 g (Ω ρ ) = {ψ ∈ [H 1 (Ω ρ )] N | ψ = g on Γ D }, H 1 Γ D (Ω ρ ) = {ψ ∈ [H 1 (Ω ρ )] N | ψ = 0 on Γ D }, H 1 Γ D (Ω) = {ψ ∈ [H 1 (Ω)] N | ψ = 0 on Γ D } , here we use the convention that eg., in our notation H 1 g (Ω ρ ) stands for the Sobolev space of vector functions [H 1 g (Ω ρ )] N . The weak solutions to the elasticity systems are defined in the standard way. Find u ρ ∈ H 1 g (Ω ρ ) such that, for every φ ∈ H 1 Γ D (Ω ρ ), Ωρ D∇ s u ρ • ∇ s φ dΩ = Γ N T • φ dS (9) We introduce the adjoint state equations in order to simplify the form of shape derivatives of functionals J u , J σ . For the functional J u they take on the form: Find w ρ ∈ H 1 Γ D (Ω ρ ) such that, for every φ ∈ H 1 Γ D (Ω ρ ), Ωρ D∇ s w ρ • ∇ s φ dΩ = - Ωρ F u (u ρ ) • φ dΩ, (10) whose Euler-Lagrange equation reads        div σ ρ (w ρ ) = F u (u ρ ) in Ω ρ w ρ = 0 on Γ D σ ρ (w ρ )n = 0 on Γ N σ ρ (w ρ )n = 0 on ∂B ρ (x 0 ) , ( 11 ) while v ρ ∈ H 1 Γ D (Ω ρ ) is the adjoint state for J σ and satisfies for all test functions φ ∈ H 1 Γ D (Ω) the following integral identity: Ωρ D∇ s v ρ • ∇ s φ dΩ = -2 Ωρ DSσ(u ρ ) • ∇ s φ dΩ. ( 12 ) which associated Euler-Lagrange equation becomes        div σ ρ (v ρ ) = -2div (DSσ ρ (u ρ )) in Ω ρ v ρ = 0 on Γ D σ ρ (v ρ )n = -2DSσ ρ (u ρ )n on Γ N σ ρ (v ρ )n = -2DSσ ρ (u ρ )n on S ρ (x 0 ) = ∂B ρ (x 0 ) . ( 13 ) Remark 1 We observe that DS can be written as DS = 4µmII + γ (I ⊗ I) (14) where γ = λlN + 2 (λm + µl) (15) Thus, when γ = 0, the boundary condition on ∂B ρ (x 0 ) in eq. ( 13) becomes homogeneous and the yield criteria must satisfy the constraint m l = - µ λ + N 2 , ( 16 ) which is naturally satisfied for the energy shape functional, for instance. In fact, in this particular case, tensor S is given by S = 1 2 D -1 ⇒ γ = 0 and 2m + l = 1 2E , ( 17 ) which implies that the adjoint solution associated to J σ can be explicitly obtained such that v ρ = -(u ρ -g). Main result We shall define the topological derivative of the functionals J u , J σ at the point x 0 as: T J u (x 0 ) = lim ρ↓0 dJ u (ρ) d(|B ρ (x 0 )|) , (18) T J σ (x 0 ) = lim ρ↓0 dJ σ (ρ) d(|B ρ (x 0 )|) . ( 19 ) Now we may formulate the following result, giving the constructive method for computing the topological derivatives: Theorem 1 Assume that (A) is satisfied, then T J u (x 0 ) = - 1 2(N -1)π [ 2(N -1)πF (u) + Ψ(D -1 ; σ(u), σ(w))] x=x 0 , (20) T J σ (x 0 ) = - 1 2(N -1)π [ Ψ(S; σ(u), σ(u)) + Ψ(D -1 ; σ(u), σ(v))] x=x 0 , (21) where w, v ∈ H 1 Γ D (Ω) are adjoint variables satisfying the integral identities ( 10) and ( 12) for ρ = 0, i.e. in the whole domain Ω instead of Ω ρ , that is Ω D∇ s w • ∇ s φ dΩ = - Ω F u (u) • φ dΩ. ( 22 ) Ω D∇ s v • ∇ s φ dΩ = -2 Ω DSσ(u) • ∇ s φ dΩ. ( 23 ) for all test functions φ ∈ H 1 Γ D (Ω). Some of the terms in (20), (21) require explanation. The function Ψ is defined as an integral over the unit sphere S 1 (0) = {x ∈ IR N | x = 1} of the following functions: Ψ(S; σ(u(x 0 )), σ(u(x 0 ))) = S 1 (0) Sσ ∞ (u(x 0 ); x) • σ ∞ (u(x 0 ); x) dS (24) Ψ(D -1 ; σ(u(x 0 )), σ(v(x 0 ))) = S 1 (0) σ ∞ (u(x 0 ); x) • D -1 σ ∞ (v(x 0 ); x) dS (25) Ψ(D -1 ; σ(u(x 0 )), σ(w(x 0 ))) = S 1 (0) σ ∞ (u(x 0 ); x) • D -1 σ ∞ (w(x 0 ); x) dS (26) The symbol σ ∞ (u(x 0 ); x) denotes the stresses for the solution of the elasticity system (2) in the infinite domain IR N \ B 1 (0) with the following boundary conditions: • no tractions are applied on the surface of the ball, S 1 (0) = ∂B 1 (0); • the stresses σ ∞ (u(x 0 ); x) tend to the constant value σ(u(x 0 )) as x → ∞. In this notation σ ∞ (u(x 0 ); x) is a function of space variables depending on the functional parameter u(x 0 ), while σ(u(x 0 )) is a value of the stress tensor computed in the point x 0 for the solution u. The dependence between them results from the boundary condition at infinity listed above. The method for obtaining such solutions (and u ∞ ), based on [START_REF] Kachanov | Handbook of Elasticity Solutions[END_REF], is discussed in the next section. In order to derive the above formulae (20), (21) we calculate the derivatives of the functional J u (ρ) with respect to the parameter ρ, which determines the size of the hole B ρ (x 0 ), by using the material derivative method [START_REF] Soko | Introduction to Shape Optimization[END_REF]. Then we pass to the limit ρ ↓ 0 using the asymptotic expansions of u ρ with respect to ρ. For the functional J u the shape derivative with respect to ρ is given by J u (ρ) = Ωρ F u (u ρ ) • u ρ dΩ - Sρ(x 0 ) F (u ρ ) dS, (27) and in the same way for the state equation: Ωρ D∇ s u ρ • ∇ s φ dΩ - Sρ(x 0 ) D∇ s u ρ • ∇ s φ dS = 0, ( 28 ) where u ρ is the shape derivative, i.e. the derivative of u ρ with respect to ρ, [START_REF] Soko | Introduction to Shape Optimization[END_REF]. After substitution of the test functions φ = w ρ in the derivative of the state equation, φ = u ρ in the adjoint equation, we get J u (ρ) = - Sρ(x 0 ) [F (u ρ ) + D∇ s u ρ • ∇ s w ρ ] dS = - Sρ(x 0 ) [F (u ρ ) + σ(u ρ ) • D -1 σ(w ρ )] dS, (29) and similarly for J σ J σ (ρ) = - Sρ(x 0 ) [Sσ(u ρ ) • σ(u ρ ) + D∇ s u ρ • ∇ s v ρ ] dS = - Sρ(x 0 ) [Sσ(u ρ ) • σ(u ρ ) + σ(u ρ ) • D -1 σ(v ρ )] dS. (30) Observe, that both matrices D -1 and S are isotropic, and therefore the corresponding bilinear forms in terms of stresses are invariant with respect to the rotations of the coordinate system. Now we exploit the fact, that dJ u (ρ) d(|B ρ (x 0 )|) = 1 2(N -1)πρ N -1 dJ u dρ , and use the existence of the asymptotic expansions for u ρ in the neighborhood of B ρ (x 0 ), namely u ρ = u(x 0 ) + u ∞ + O(ρ 2 ). (31) In addition, u ∞ is proportional to ρ, u ∞ IR N = O(ρ), on the surface S ρ (x 0 ) of the ball. The expansion of σ(u ρ ) corresponding to (31) has the form σ(u ρ ) = σ ∞ (u(x 0 ); x) + O(ρ). (32) It may be proved, that w ρ and v ρ have similar expansions. Using the formulae (31),(32) we may justify the following passages to the limit: lim ρ↓0 1 ρ N -1 Sρ(x 0 ) σ(u ρ ) • D -1 σ(v ρ ) dS = Ψ(D -1 ; σ(u(x 0 )), σ(v(x 0 ))), lim ρ↓0 1 ρ N -1 Sρ(x 0 ) σ(u ρ ) • D -1 σ(w ρ ) dS = Ψ(D -1 ; σ(u(x 0 )), σ(w(x 0 ))), lim ρ↓0 1 ρ N -1 Sρ(x 0 ) Sσ(u ρ ) • σ(u ρ ) dS = Ψ(S; σ(u(x 0 )), σ(u(x 0 ))), lim ρ↓0 1 ρ N -1 Sρ(x 0 ) F (u ρ ) dS = 2(N -1)πF (u(x 0 )). This completes the proof of the theorem. The main difficulty lies in the computation of the values of the functions denoted above as Ψ(S; σ(u(x 0 )), σ(u(x 0 ))), Ψ(D -1 ; σ(u(x 0 )), σ(w(x 0 ))) and Ψ(D -1 ; σ(u(x 0 )), σ(v(x 0 ))), which, in general, is difficult to obtain in the closed form, in contrast with the two dimensional case. Therefore we can approximate them using numerical quadrature. It is possible, because we may calculate the values of integrands at any point on the sphere. This makes the computations more involved, but does not increase the numerical complexity in comparison to evaluating single closed form expression. Remark 2 The tensor S in the definition of J σ may, in fact, be arbitrary, not only isotropic. However, it is difficult to imagine such a need for the isotropic material. Anyway, in the general case, we would have to transform S according to the known rules for the fourth order tensor, connected with the rotation of the reference frame. Topological derivatives in 3D elasticity The shape functionals J u , J σ are defined in the same way as presented in section 2.2 with the exception, that J σ is now the energy stored in a 3D elastic body (see remark 1). The weak solutions to the elasticity system as well as adjoint equations are defined also analogously to the section 2.2. Then, considering the expansions presented in Appendix A.2, we may state the following result [START_REF] Novotny | Topological Sensitivity Analysis for three-dimensional linear elasticity problems[END_REF] (see also [START_REF] Garreau | The Topological Asymptotic for PDE Systems: The Elasticity Case[END_REF]): Theorem 2 The expressions for the topological derivatives of the functionals J u , J σ have the form T J u (x 0 ) = -F (u) + 3 2E 1 -ν 7 -5ν (10(1 + ν)σ(u) • σ(w) -(1 + 5ν)trσ(u)trσ(w)) x=x 0 , (33) T J σ (x 0 ) = 3 4E 1 -ν 7 -5ν 10(1 + ν)σ(u) • σ(u) -(1 + 5ν)(trσ(u)) 2 x=x 0 . ( 34 ) Topological derivatives in 2D elasticity For the convenience of the reader we recall here the results derived in [START_REF] Soko Lowski | On topological derivative in shape optimization[END_REF] for the 2D case. The principal stresses associated with the displacement field u are denoted by σ I (u), σ II (u), the trace of the stress tensor σ(u) is denoted by trσ(u) = σ I (u) + σ II (u). The shape functionals J u , J σ are defined in the same way as presented in section 2.2, with the tensor S isotropic (that is similar to D). The weak solutions to the elasticity system as well as adjoint equations are defined also analogously to the section 2.2. Then, from the expansions presented in Appendix A.1, we may formulate the following result [START_REF] Soko Lowski | On topological derivative in shape optimization[END_REF]: Theorem 3 The expressions for the topological derivatives of the functionals J u ,J σ have the form T J u (x 0 ) = -F (u) + 1 E (a u a w + 2b u b w cos 2δ) x=x 0 = -F (u) + 1 E (4σ(u) • σ(w) -trσ(u)trσ(w)) x=x 0 (35) T J σ (x 0 ) = -η(a 2 u + 2b 2 u ) + 1 E (a u a v + 2b u b v cos 2δ) x=x 0 = -η(4σ(u) • σ(u) -(trσ(u)) 2 ) + 1 E (4σ(u) • σ(v) -trσ(u)trσ(v)) x=x 0 (36) Some of the terms in ( 35), (36) require explanation. According to eq. ( 15) for N = 2, constant η is given by η = l + 2 m + γ ν E . (37) Furthermore, we denote a u = σ I (u) + σ II (u), b u = σ I (u) -σ II (u), a w = σ I (w) + σ II (w), b w = σ I (w) -σ II (w), a v = σ I (v) + σ II (v), b v = σ I (v) -σ II (v). (38) Finally, the angle δ denotes the angle between principal stress directions for displacement fields u and w in (35), and for displacement fields u and v in (36). Remark 3 For the energy stored in a 2D elastic body, tensor S is given by eq. (17), γ = 0 and η = 1/(2E). Thus, since v = -(u -g), we obtain the following well-known result T J σ (x 0 ) = 1 2E 4σ(u) • σ(u) -(trσ(u)) 2 x=x 0 (39) Compare these expressions to the 3D case. Their simplicity comes from the fact, that on the plane the rotation of one coordinate system with respect to the other is defined by the single value of the angle (here δ). This is a purely 2D phenomenon and it makes the explicit computations possible. Uncertain input data In reality, the values of input data(loading, material parameters) are guaranteed only in some given intervals. One of the simplest remedy is to apply the worst scenario or maximum range scenario method [START_REF] Hlaváček | Uncertain Input Data Problems and the Worst Scenario Method[END_REF]. In what follows, we present the methods for the traction problem (1) with ∂Ω = Γ N and the criterion corresponding to the topological derivatives (34) or (39), respectively. Traction problem in 3D-elasticity Let us consider a bounded domain Ω ⊂ R 3 with Lipschitz boundary ∂Ω ≡ Γ, occupied by a homogeneous and isotropic elastic body. Let the body be loaded by surface forces T ∈ [L ∞ (Γ)] 3 and the body forces be zero. We introduce sets of admissible uncertain input data as follows : (i) Lamé coefficients λ ∈ U λ ad = λ, λ , 0 ≤ λ < λ < ∞, µ ∈ U µ ad = µ, µ , 0 < µ < µ < ∞; (ii) surface loading forces T i ∈ U T i ad = τ ∈ L ∞ (Γ) : τ Γp ∈ C (0),1 Γ p , |τ | ≤ C 1 , |∂τ /∂s j | ≤ C 2 a.e. on Γ, j = 1, 2 , where Γ = P p=1 Γ p , Γ k ∩ Γ m = ∅ for k = m, i = 1, 2, 3, s j are local coordinates of the surface Γ p and C 1 , C 2 are given constants, T ≡ (T 1 , T 2 , T 3 ) ∈ U T ad = {T i ∈ U T i ad , i = 1, 2, 3 and Γ T dS = 0, Γ x × T dS = 0}. Finally, we define U ad = U λ ad × U µ ad × U T ad and A ≡ {A, T }, A = {λ, µ}. We will consider the following criterion-functional based on the topological derivative associated to the energy shape functional (34) Φ(A, σ) = σ T H(A)σ where σ ≡ σ(y) is the stress tensor of a full body at the center y ∈ Ω of a spherical cavity, H(A) = 3(1 -ν) 4E (Λ 1 + 10(1 + ν) 7 -5ν Λ 2 ), (40) Λ 1 = 1 3 I ⊗I, Λ 2 = II -Λ 1 , ν is the corresponding Poisson's constant and E the Young's modulus. Note that ν = λ 2(λ+µ) , E = µ(3λ+2µ) λ+µ . Continuous dependence of the criterion on the input data Our main result of the present section is given by the following theorem Theorem 4 Let A n ∈ U ad , A n → A in R 2 × [L ∞ (Γ)] 3 as n → ∞. Then Φ(A n , σ(A n )) → Φ(A, σ(A)). Proof. The estimate (47) follows from the Cauchy-Schwartz inequality and the boundedness of sets U λ ad , U µ ad . To justify (48), we write a(A; u, u) ≥ 2µ Ω ε ij (u)ε ij (u)dx and use the Korn's inequality Ω ε ij (u)ε ij (u)dx ≥ c u 2 1,Ω ∀u ∈ V 0 (see e.g. [7]-Lemma 7.3.3). Lemma 2 Let λ n ∈ U λ ad , µ n ∈ U µ ad , λ n → λ and µ n → µ as n → ∞. Then ν n → ν and T * ij (ν n ) → T * ij (ν) in [L 2 (Γ)] 3 . ( 49 ) Proof. Since λ n + µ n ≥ λ + µ > 0, ν n = λ n 2(λ n + µ n ) → λ 2(λ + µ) = ν. We infer that s ij k (ν n ) → s ij k (ν) in L 2 (Γ), k = 1, 2, 3. (50) Indeed, we have λ n /κ n → λ/κ, µ n /κ n → µ/κ and u ij0 (ν n ) -u ij0 (ν) H 1 (Γ) ≤ C|ν n -ν| → 0, so that (50) holds. Since the field w ij is independent of A, we arrive at (49). Lemma 3 Let λ n ∈ U λ ad , µ n ∈ U µ ad , λ n → λ and µ n → µ as n → ∞ and u * ij (A n ) ∈ V 0 . Then u * ij (A n ) → u * ij (A) in [H 1 (Ω)] 3 . Proof. For brevity, let us denote T * n = T * ij (ν n ), T * = T * ij (ν), u * n = u * ij (A n ), u * = u * ij (A). By definition, we have a(A n ; u * n , v) = Γ T * n v dS (51) a(A; u * , v) = Γ T * v dS (52) for all v ∈ [H 1 (Ω)] 3 . Let us consider also solutions ûn ∈ V 0 of the following problem a(A; ûn , v) = Γ T * n v dS ∀v ∈ [H 1 (Ω)] 3 . (53) From ( 53) and (52) we obtain a(A; ûn -u * , v) = Γ (T * n -T * )v dS. Proof. We may write |σ kl (A n ) -σ kl (A)| = | Γ T n • (c klij (A n )G ij y (A n ))dS -Γ T • (c klij (A)G ij y (A))dS| ≤ | Γ T n • ((c klij (A n ) -c klij (A))G ij y (A n ))dS| +| Γ T n • (c klij (A)(G ij y (A n ) -G ij y (A)))dS| +| Γ (T n -T ) • (c klij (A)G ij y (A))dS| ≡ I 1 + I 2 + I 3 , where I 1 ≤ Γ C A n -A 0,∞ |G ij y (A n )|dS → 0 and I 2 ≤ Γ C|G ij y (A n ) -G ij y (A)|dS → 0 due to Proposition 1 and the boundedness of T n in [L ∞ (Γ)] 3 . I 3 tend to zero by assumption. Proof of Theorem 1. We have |Φ(A n , σ(A n )) -Φ(A, σ(A))| ≤ |σ(A n ) T H(A n )(σ(A n ) -σ(A))| +|σ(A n ) T (H(A n ) -H(A))σ(A)| +|(σ(A n ) T -σ(A) T )H(A)σ(A)| = J 1 + J 2 + J 3 . By Proposition 2 we infer that J 1 and J 3 tend to zero. We also use the continuity of the function A → H(A), which follows from Lemma 2 and the convergence E n = 2µ n (1 + ν n ) → 2µ(1 + ν) = E ≥ 2µ > 0. As a consequence, J 2 tends to zero, as well. The worst scenario and the maximum range scenario Suppose that we wish to be "on the safe side", taking uncertain input data A and T in consideration. Then we solve either the worst scenario problem A 0 = arg max A∈U ad Φ(A, σ(A)) (59) or the maximum range scenario problem: find (i) A 0 according to (59) and (ii) A 0 = arg min A∈U ad Φ(A, σ(A)). (60) In other words, we seek exact upper and lower bounds of the criterion functional (see the monograph [START_REF] Hlaváček | Uncertain Input Data Problems and the Worst Scenario Method[END_REF] for applications of problem (60) within the frame of the fuzzy set theory). Theorem 5 Problems (59) and ( 60) have at least one solution. Proof. The set U ad is compact in R 2 × ( Traction problem in 2D-elasticity Let us consider a plane elasticity, i.e., either the case of plane strain or that of plane stress. It is well-known, that both cases have the same stress-strain relations, where only the coefficient λ varies It is either λ or λ , see [START_REF] Novotny | Topological Sensitivity Analysis for three-dimensional linear elasticity problems[END_REF]. λ = Eν (1 + ν)(1 -2ν) for plane strain, whereas λ = λ * = Eν 1 -ν 2 for plane stress. Let us consider a bounded domain Ω ⊂ R 2 with a Lipschitz boundary ∂Ω ≡ Γ, occupied by a homogeneous and isotropic elastic body, loaded only by surface loads T ∈ [L ∞ (Γ)] 2 . Assume that λ ∈ U λ ad , µ ∈ U µ ad and T i ∈ U T i ad , i = 1, 2, with U λ ad , U µ ad and U T i ad defined in section 1. Moreover, assume that the forces T are in equilibrium, i.e. Γ T dS = 0, Γ (x 1 T 2 -x 2 T 1 )dS = 0. ( 61 ) We define U T ad = {T ≡ (T 1 , T 2 ) : T i ∈ U T i ad , i = 1, 2, T satisfy (61)}, U ad = U λ ad × U µ ad × U T ad , A = {λ, µ}, A = {A, T } and introduce the criterion-functional based on the topological derivative associated to the energy shape functional (39) Φ(A, σ) = σ T H(A)σ, (62) where σ ≡ σ(y) is the stress tensor of a full body at the center y ∈ Ω of a circular cavity, and H(A) = (K + µ) 2Kµ (Λ 1 + 2Λ 2 ), (63) where K = λ + µ is the bulk modulus. Continuous dependence of the criterion on the input data The main result of the present section will be represented by an analogue of Theorem 1 as follows. Theorem 6 Let A n ∈ U ad , A n → A in R 2 × [L ∞ (Γ)] 2 as n → ∞. Then Φ(A n , σ(A n )) → Φ(A, σ(A)). For the proof we shall employ the following integral representation formula, analogous to (41), namely ∂u i ∂y j (y) = Γ T • G ij y dS, i, j ∈ {1, 2}. (64) By Reciprocity theorem, we obtain Γ (T • u * ij -T * ij • u)dS = 0. (71) Then ( 69) and (71) yield ∂u i ∂y j (y) = Γ T • (u * ij -u ij )dS + Γ u • (s ij -T * ij )dS. The last integral vanishes by virtue of normalization conditions, since s ij -T * ij = -w ij . As a consequence, we arrive at the formula (64), where G ij y = u * ij -u ij . (72) Now we may go on in proving Theorem 3 as in the prooof of Theorem 1. We establish an analogue of Lemma 1, where the subspace V 0 is defined by V 0 = v ∈ [H 1 (Ω)] 2 : Γ v dS = 0, Γ (x 1 v 2 -x 2 v 1 )dS = 0 . For the Korn's inequality in V 0 , see e.g. Section 10.2.2 in [START_REF] Nečas | Mathematical Theory of Elastic and Elasto-plastic Bodies : An Introduction[END_REF]. As far as an analogue of Lemma 2 is concerned, we use the formula ν = λ 2(λ + µ) for plane strain and ν = λ * λ * + 2µ for plane stress. It is readily seen that s ij ≡ s ij (ν), i.e., it does not depend on the modulus E. Then we can prove that λ * n → λ * , ν n → ν and s ij (ν n ) → s ij (ν) in L 2 (Γ)] 2 as ν n → ν , since λ n (K n + 2µ n )/κ 0n → λ(K + 2µ)/κ 0 (73) and λ n K n /κ 0n → λK/κ 0 for K n = λ n + µ n , λ n ∈ U λ ad , µ n ∈ U µ ad , A n → A. The field w ij is independent of A, so that we arrive at T * ij (ν n ) → T * ij (ν) in L 2 (Γ)] 2 . An analogue of Lemma 3 can be proved in the same way as Lemma 3. We infer that u * ij (A n ) → u * ij (A) in [H 1 (Ω)] 2 . ( 74 ) Using again (73), we observe that u ij (A n ) → u ij (A) in L 2 (Γ)] 2 . (75) Combining (72) with (74), the Trace theorem and (75), we obtain G ij y (A n ) → G ij y (A) in L 2 (Γ)] 2 . ( 76 ) Theorem 3 follows in a way parallel to the proof of Theorem 1, from (76), the uniform convergence of surface loads on Γ and the continuity of the function A → H(A). The worst scenario and the maximum range scenario Both the worst scenario problem (59) and the maximum range scenario problem (60) have at least one solution. This assertion is a consequence of Theorem 3 and the compactness of the set U ad in R 2 × 2 i=1 P p=1 C(Γ p ). • for i = 1 σ rr 1 (ξ ρ ) = σ I (ξ) 14 -10ν ρ 3 r 3 - ρ 5 r 5 + 14 -10ν -10(5 -ν) ρ 3 r 3 + 36 ρ 5 r 5 sin 2 θ sin 2 ϕ , (84) σ rθ 1 (ξ ρ ) = σ I (ξ) 14 -10ν -5ν + 5(1 + ν) ρ 3 r 3 -12 ρ 5 r 5 sin 2θ sin 2 ϕ , (85) σ rϕ 1 (ξ ρ ) = σ I (ξ) 14 -10ν -5ν + 5(1 + ν) ρ 3 r 3 -12 ρ 5 r 5 sin θ sin 2ϕ , (86) • for i = 2 Remark 4 It is important to mention that the stress distribution for i = 1, 2 was obtained from a rotation of the stress distribution for i = 3. In addition, the derivation of this last result (for i = 3) can be found in [START_REF] Kachanov | Handbook of Elasticity Solutions[END_REF], for instance. p )), so that the assertion follows from Theorem 1. r 5 -r 5 r 5 555 14 -10ν + 25(1 -2ν) cos 2θ sin 2 ϕ , (87)σ θϕ 1 (ξ ρ ) = σ I (ξ) 14 -10ν -5ν + 5(1 -2ν) ρ 3 r 3 + 3 ρ 5 r 5 cos θ sin 2ϕ ,(88)σ ϕϕ 1 (ξ ρ ) = σ 1 (ξ) 56 -40ν -20ν + (11 -10ν) ρ 3 r 3 + 9 ρ 5 r 5 + 28 -20ν + 5(1 -2ν) cos 2θ sin 2 ϕ , r 5 5 sin 2 θ , (101) where σ I (ξ), σ III (ξ) and σ III (ξ) are the principal stress values of tensor σ(ξ), for ξ = u, ξ = w or ξ = v associated to the original domain without hole Ω. 1,Ω ∀u ∈ V 0 ,(48)whereV 0 = {v ∈ [H 1 (Ω)] 3 : Γ v dS = 0, Γ v × x dS = 0}. Γ (s ij • u -T • u ij )dS, (69)which follows by differentiating the so-colled Somigliana's identityu i (y) = Γ (T • u i y -u • s i y )dS,(70)where ∂s i y /∂y j = -s ij . Acknowledgments 4 Conclusions We have seen that the worst case and maximal range scenario problems are solvable with criterions of energy-based topological derivative. The same methodology, considering topological derivatives of different shape functional may be applied to derive similar analysis for criteria dependent for example on displacement (kinematic constraints) and yield constraints. Proof is based on the formulas ([7]-Theorem 10.1.1) and where and r = x -y. Since (κ(A n )) -1 -→ (κ(A)) -1 and the components u ij0 k are bounded on Γ, The vector field u * ij (A) is the displacement solving the first boundary value problem with zero body forces and the equilibriated surface loading where and represents a rigid body displacement such that (e i denote unit vectors in the directions of Cartesian coordinates). The field w ij is uniquely determined by the conditions shown. Inserting (43) in (45), we observe that since u ij0 , λ κ and µ κ are independent of the modulus E. Lemma 1 Let us define Inserting v := ûn -u * and using Lemma 1, we infer that so that ûn -u * 1,Ω → 0 follows from Lemma 2. We can show that Indeed, (51) and Lemma 1 yield that so that (55) follows from Lemma 2. We can use ( 51), ( 53) and Lemma 1 to obtain Then ( 55) and ( 56) yield The convergence u * n → u * in [H 1 (Ω)] 3 follows from the triangle inequality, (54) and (57). as n → ∞. Proof. Since by (42) we have the assertion follows from Lemma 3, the Trace theorem and (44). Proposition 2 Let the stress components at the point y be We can construct the vector function G ij y in a way parallel to that of the proof of Theorem 10.1.1 in [START_REF] Nečas | Mathematical Theory of Elastic and Elasto-plastic Bodies : An Introduction[END_REF]. First, we consider the well-known Kelvin's solution where κ 0 = 4πµ(K + µ), r = x -y and define u ij = -∂u i y /∂y j . The corresponding surface forces on Γ are then We can find that Let us construct the rigid body translation Note that the field w ij is uniquely determined by conditions (67). If we define the forces T * ij are in equilibrium, i.e., they satisfy conditions (61). There exists a unique displacement field u * ij , which solves the first boundary value problem of elasticity with zero body forces and surface loads T * ij and satisfies the normalization conditions Next, we assume that the field u fullfils conditions (68) as well and consider the so-called Love's formula ∂u i ∂y j (y) = A Stress distribution around cavities We present in this appendix the analytical solution for the stress distribution around a circular (N = 2) and spherical (N = 3) cavities respectively for two and three-dimensional linear elastic bodies. A.1 Circular cavity Considering a polar coordinate system (r, θ), we have the following expansion for the stress distribution σ(ξ ρ ) around a free boundary circular cavity (σ rr (ξ ρ ) = σ rθ (ξ ρ ) = 0 on ∂B ρ (x 0 )), with where the angle θ u = θ and θ w = θ + δ, with δ denoting the angle between principal stress directions for displacement fields u and w in (35). In addition, the following expansion for σ(v ρ ) satisfying the boundary condition on ∂B ρ (x 0 ) given by σ rθ (v ρ ) = 0 and σ rr (v ρ ) = -2γσ θθ (u ρ ), holds where the angle θ v = θ + δ, with δ denoting the angle between principal stress directions for displacement fields u and v in (36). Finally, where σ I (ξ) and σ II (ξ) are the principal stress values of tensor σ(ξ), for ξ = u, ξ = w or ξ = v associated to the original domain without hole Ω. A.2 Spherical cavity Let us introduce a spherical coordinate system (r, θ, ϕ). Then, the stress distribution around the spherical cavity B ρ (x 0 ) is given by σ rr (ξ ρ ) = σ rr 1 (ξ ρ ) + σ rr 2 (ξ ρ ) + σ rr 3 (ξ ρ ) + O(ρ) , σ rθ (ξ ρ ) = σ rθ 1 (ξ ρ ) + σ rθ 2 (ξ ρ ) + σ rθ 3 (ξ ρ ) + O(ρ) , σ rϕ (ξ ρ ) = σ rϕ 1 (ξ ρ ) + σ rϕ 2 (ξ ρ ) + σ rϕ 3 (ξ ρ ) + O(ρ) , σ θθ (ξ ρ ) = σ θθ 1 (ξ ρ ) + σ θθ 2 (ξ ρ ) + σ θθ 3 (ξ ρ ) + O(ρ) , σ θϕ (ξ ρ ) = σ θϕ 1 (ξ ρ ) + σ θϕ 2 (ξ ρ ) + σ θϕ 3 (ξ ρ ) + O(ρ) , σ ϕϕ (ξ ρ ) = σ ϕϕ 1 (ξ ρ ) + σ ϕϕ 2 (ξ ρ ) + σ ϕϕ 3 (ξ ρ ) + O(ρ) , where ξ ρ = u ρ , ξ ρ = w ρ or ξ ρ = v ρ ; σ rr i (ξ ρ ), σ rθ i (ξ ρ ), σ rϕ i (ξ ρ ), σ θθ i (ξ ρ ), σ θϕ i (ξ ρ ) and σ ϕϕ i (ξ ρ ), for i = 1, 2, 3, are written, as:

01749050

en

2024/03/05 22:32:07

2011

https://hal.univ-lorraine.fr/tel-01749050/file/BAO.Lei.SMZ1114.pdf

Lei Bao Christophe Schuman Jean-Sébastien Lecomte Marie-Jeanne Philippe Xiang Zhao Liang Zuo Claude Esling Study of deformation mechanisms in titanium by interrupted rolling and channel die compression tests Keywords: titanium, deformation mechanism, twinning, gliding, orientation, EBSD Titanium, Rolling, Rotation flow field, Gliding, Twinning, Texture Twin, Twin variants, Twin growth, Schmid factor, Titanium Titanium, Twinning, Double twinning, Variant selection, interrupted in situ SEM/EBSD orientation determination Titanium and its alloys are widely used in aviation, space, military, construction and biomedical industry because of the high fracture strength, high ductility and good biocompatibility. The mechanisms of plastic deformation in titanium have been studied in detail, especially deformation twinning since it has a great influence on the ductility and fracture strength. In this study, an interrupted "in situ" SEM/EBSD investigation based on a split sample of commercial titanium T40 was proposed and performed in rolling and channel die compression. This approach allows to obtain the time resolved information of the appearance of the twin variants, their growth, the interaction between them and the interaction with the grain boundaries or twin boundaries. With the orientation data acquired by the EBSD technique, we calculated the Schmid factor, crystallographic geometry, and plastic energy associated with each variant of primary twins, secondary twins and double twins to investigate the lattice rotation, the activation of twins, the growth of twins, and the variant selection criterion. In this observation, two types of twin systems were activated: {10-12} tension and {11-22} compression twins. Secondary twins were also activated, especially the twin variants with the highest Schmid factors (e.g. higher than 0.4). The growth of the two types of twin is quite different. The {11-22} twin shows Multiple Variants System (MVS) whereas the {10-12} twin shows Predominant Variant System (PVS). The twinning occurs in grains that have particular orientations. Generally, the reorientation induced by the twinning aligns the c-axis of the twinned part to the stable rolling texture orientations, so that no further secondary twinning can be induced. The secondary twinning occurs only when the primary twinning orientates the c-axis of the primary twins far away from the stable orientations. For twinned ~ II ~ grains, the lattice rotation of the matrix is similar to that of the grains having a similar crystallographic orientation but without any twin. Two sets of double twins were observed in this study, classified as C-T1 and T1-C double twins respectively. All the variants of C-T1 and T1-C double twins were classified into three groups: A, B and C according to the crystallographic symmetry. The misorientations of theses three groups with respect to the matrix are 41.34°, 48.44° and 87.85°. Strong variant selection took place in double twinning. In C-T1 double twins, 78.9% variants belong to group B whereas in T1-C double twins, 66.7% variants belong to group C. The plastic energy and Schmid factor both play important roles in the variant selection of double twinning. Geometrical characteristics, like the common volume or strain accommodation do not contribute significantly to the variant selection. ~ III ~ Résumé Le titane et ses alliages sont largement utilisés dans les domaines aéronautique, spatial, de l'armement, du génie civil, dans des applications commerciales et biomédicales en raison de sa résistance à la rupture élevée, d'une bonne ductilité et d'une grande biocompatibilité. Les mécanismes de la déformation plastique du titane ont été étudiés en détail par le passé, particulièrement sur l'étude de la déformation par maclage car il a une grande influence sur les propriétés mécaniques. Une méthode d'essais "in situ" en EBSD basée sur des tôles polies et colées ensemble a été développée dans cette étude et utilisée en laminage et en compression plane. Avec cette méthode, des mesures EBSD sont effectuées à chaque étape de la déformation dans la même zone comprenant un grand nombre de grains. Par conséquent, l'information sur l'orientation de ces grains à chaque l'étape de la déformation est mesurées. Le maclage apparait dans les grains qui ont des orientations particulières. En règle générale, la réorientation induite par le maclage aligne l'axe c de la partie maclées vers les orientations stables de la texture de laminage, de sorte qu'aucun autre maclage secondaire peut être induit. Le maclage secondaire se produit uniquement lorsque le macle primaire envoie l'axe c loin des orientations stables. Pour les grains maclés, la rotation du réseau de la matrice est semblable à celle des grains ayant une orientation cristallographique identique mais sans macles. Deux types de systèmes de macles ont été activés au cours de la déformation à la température ambiante: des macles de tension (10)(11)(12) et des macles de compression (11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22). Dan le maclage primaire, les résultats montrent que les variantes de macle ayant des facteurs Schmid supérieurs à 0.4 ont une bonne chance d'être actifs. Les comportements des deux types de maclage sont complètement différents. Dans la déformation en compression, les macles (11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22) montrent le comportement de type ~ IV ~ multiplication des variants (Multiply Variants System: MVS) alors que les macles (10)(11)(12) montrent le type de maclage prédominant (Predominant Twin System: PTS). Cette étude présente deux types de macles doubles dénommées C-T1 (= macle primaire de Compression et macle secondaire de Tension) et T1-C (= macle primaire de Tension et macle secondaire de Compression). Tous les variants sont classés seulement en trois groupes: A, B et C par symétrie cristallographique. Les désorientations de ces 3 groupes par rapport à l'orientation de la matrice sont respectivement de 41.34°, 48.44° et 87.85°.Une forte de sélection de variant se déroule dans le maclage double. Pour les macles doubles CT, 78.9% des variantes appartiennent à la B et pour T1-C, 66.7% des variantes appartiennent à C. Le facteur de Schmid joue un rôle prépondérant dans la sélection des variants des macles doubles. Les caractéristiques géométriques, associant " volumes communs " et l'accommodation de la déformation ne contribuent pas de manière significative à la sélection des variants. ~ V ~ would like to give my hearted thanks to all the members at this laboratory for their kind help. I would like to sincerely thank my supervisors, Prof. Claude Esling, Dr. Christophe Shuman at University of Metz, and Prof. Xiang Zhao at Northeastern University for guiding me into this research field and for their constant help and support on my research work. Every progress in this work coagulated their care and enlightenment. Special thanks should be given to Prof. Marie-Jeanne Philippe for her enriching ideas and the fruitful discussions. I am also deeply indebted to Dr. Jean-Sébastien Lecomte, who is a friendly co-worker of outstanding practical ability and active thinking. At last, I would like to give my hearted thanks to Dr. Yudong Zhang for her constant help not only on my research work but also on my daily life in France. As an illuminating guider, she exhibits the characters of being reliable, modest, serious and sympathetic. I did extremely enjoy working with them all. During my study, I always received direct help from the French Ph. D students at LETAM, especially Pierre Blaineau and Jean-Christophe Hell. Their kind care, selfless help and the deep friendship between us have made up of large part of the support and enjoyment of my study. I will cherish our friendship deeply in my mind. Last but not least, I would like to give my hearted thanks to my mother and my wife. Their deep love, understanding, constant support and encouragement over the years are the great impetus to my study. Metals with hexagonal close-packed structure The hexagonal structure materials such as titanium and magnesium are especially interesting because of their properties. The properties of titanium are particularly appreciated by the aerospace and biomedical industry. Magnesium is applied in automotive, computers or sports equipment. Zirconium is studied for its use in nuclear reactors. However, since their slip systems are not as sufficient as cubic's, so the twinning becomes more common in these materials. It becomes important to know the characteristics of deformation, including the activity of twinning, shear critical resolved on different slip systems... These relevant issues on hexagonal structure materials are the subject of many research works from 1950s until recent years [START_REF] Schmid | Plasticity of Crystals, FA Hughes and Co[END_REF]. Hexagonal close-packed crystal structure The atom positions in the hexagonal close-packed structure are shown in Figure 1-1. If atoms are assumed to be hard spheres, the closest arrangement in an atom plane produces a series of hexagonal placed closely. The stacking sequence of close-packed atom planes one upon another produces is ABAB (Figure1-1). In an ideal closed-packed structure, the axial ratio = ⁄ = 8 3 ⁄ ≈ 1.633. The coordination number of an ideal hexagonal close-packed structure is 12, as same as the FCC structure. However, no pure metal has the ideal 1.633 axial ratio. The pure metals with axial ratio higher than 1.633, have 6 nearest atoms in the basal plane; in the other case, the metals with axial ratio lower ~ 3 ~ than 1.633, have 6 nearest atoms, three above the basal plane and three under basal plane [START_REF] Hume-Rothery | The structure of metals and alloys[END_REF][START_REF] Christian | The theory of transformation in metals and alloys[END_REF]. Because the Miller indices of the crystallographic planes and directions from a same family appear quite dissimilar, and in order to avoid the possibility of confusion and the inconvenience, A Miller-Bravais indices were developed to describe the crystallographic planes and directions in hexagonal system [START_REF] Taylor | X-ray Metallography[END_REF][START_REF] Reed-Hill | Physical metallurgy principles[END_REF][START_REF] Barrett | Structure of metals[END_REF]. This Miller-Bravais indices base on a 4-axis system, where the coplanar vectors a 1 , a 2 and a 3 are at 120 to each other and vector c is perpendicular to the plane consists of vectors a 1 , a 2 and a 3 . (Figur 1-2) The vector a 3 is redundant since a 3 =-(a 1 + a 2 ). In the Miller-Bravais indices, a crystallographic direction d will have 4 indices [uvtw], such as d=ua 1 +va 2 +ta 3 +wc, and then the crystallographically equivalent directions have similar indices. Twinning modes Twinning in hexagonal metals is of great important because the limited slip modes in these metals make twinning a necessary way to accommodate the deformation. Unlike the slip mode, twinning products a specific new orientation in crystal by shear. The geometry description of twinning shear is illustrated in Figure1-3. All points of the lattice on the upper side of plane K 1 are displaced in the direction  1 by an amount of u 1 proportional to their distance above K 1 . The plane containing  1 and the normal to K 1 is called the plane of shear S. And it is evidently that all vectors in that plane through which is normal to plane S are unchanged in length, although rotation. The plane K2 in Figure 123conventionally called the second undistorted plane. K1 is neither rotated nor When a crystal is completely converted to a twin, all directions lying in the initially acute sector between K 1 and K 2 are shortened, while all directions lying in the obtuse sector are lengthened. Since that twin can been seen as a specific type of crystalline structure relationship between the twin part and the matrix, we can describe twin in the same way as describe two crystalline structure relationship, with a rotation angle and a rotation axis. {10-12}, {11-21} and {11-22} twins in titanium are illustrated in Figure1-4. In hexagonal metals, many types of twinning are exhibited, and the type can be related to the c/a ratio of the metal. Generally speaking, the lower c/a ratio, the greater the variety of twinning exhibited. Figure1-4: The {10-12}, {11-21} and {11-22} twins in titanium. ~ 7 ~ The observed type of twinning in hexagonal metals is showed in Table 1-1. Twinning in {10-12} planes occurs in all hexagonal metals, because this twinning has the lowest shear. Titanium, zirconium and hafnium (c/a<1.633) exhibit {10-12} twinning in tension of c axis, {11-22} twinning in compression and in some case, {11-21} twinning in tension [START_REF] Rosi | Mechanism of plastic flow in titanium at low and high temperatures[END_REF][START_REF] Chin | On the microstructure of extruded rods and drawn wires of beryllium[END_REF][START_REF] Conrad | Mechanisms of fatigue in metals under coal liquefaction conditions[END_REF]. Magnesium and beryllium (c/a<1.633) shows only {10-12} twinning in tension [START_REF] Kelley | The deformation characteristics of textured magnesium[END_REF][START_REF] Wonsiewicz | Analysis of local necking in a biaxially stretched sheet[END_REF][START_REF] Mahajan | Deformation twinning in metals and alloys[END_REF][START_REF] Chin | On the microstructure of extruded rods and drawn wires of beryllium[END_REF]. Zinc and Cadmium (c/a>1.633) exhibit {10-12} twinning in compression [START_REF] Price | On dislocation loops formed in zinc crystals during low temperature pyramidal glide[END_REF]. The calculation of shear {10-12} of twinning is illustrated in Figure 1-5, in the schematic Where =c/a. The formulas to calculate shear of observed twin mode from [START_REF] Christian | Deformation twinning[END_REF] are given in Table 1-2. The relationship between the shear and the c/a ratio of various types of twinning is illustrated in Figure 1-6. Because the twinning only can provide shear in single direction, the load direction and crystal orientation influence the activation of twinning greatly. The twins with positive slope in Figure 1-6 ({11-22} and {10-11}) only activate under the compression force in the direction of c axis. On the other hand, the twins with negative slope ({10-12} and {11-21}) only activate under the tension force in the direction of c axis. Table 1-2: Shear of various twin modes [START_REF] Christian | Deformation twinning[END_REF]] Twin modes Shear {10-12}<10-1-1> (3 -)/√3 {10-11}<10-1-2> (4 -9)/4√3 {11-21}<11-2-6> 1 ⁄ {11-22}<11-2-3> ( -)/ 1-3 1-1 1-2 ~ 9 ~ Slip modes The crystallographic slip is a mechanism that operates in all crystalline materials, metals and metal alloys, calcite or crystalline polymers. This mechanism has already been observed before the twentieth century. Metalworkers observed deformed lines or streaks regular on polycrystals under an optical microscope, and they called them "slip lines". In fact, later observed in the Scanning Electron Microscope, these slip lines were actually steps. The formation of these steps is a direct result of the mechanism of deformation of parts of the crystal (or polycrystal) slide over each other on well defined crystallographic planes (slip systems). This mechanism is due to the movement of dislocations in these slip planes. A dislocation can be 2) [11.3] (11.1) [11.6] Range of hexagonal metals The relationship between the shear and the c/a ratio of various types of twinning. ~ 10 ~ activated when a stress applied to the crystal. Under sufficient stress, the dislocations glide through the crystal. It produces a small displacement (Burgers vector b ⃑ ) on the surface. When a dislocation slips, the volume of metals remains unchanged, because the shift occurs by shear between parallel planes of the crystal. In the face-centered cubic lattice, the Burgers vectors are in the direction of <110>, and <111> in the bcc. In materials with hexagonal structures, there are several families of slip systems with Burgers vectors of type <a> and <c+a>. A slip system is defined by a glide plane and a direction slip contained in this plan. Table 1-3 and Figure 1-7 below show the different families of slip system operating in hexagonal structures. ~ 11 ~ Hexagonal metals In this section, we provide a presentation of the main works related to crystal orientation, evolution of texture and different deformation mechanisms in titanium, zirconium, magnesium and zinc. Titanium The plastic deformation in titanium have been studied all the time and especially during rolling (hot and cold). First, it should be mentioned the for the rapid development of texture and the transitions in texture. The formation of the stable end texture is thought to be due to slip. In 1994, Kailas et al. (Prasad, Biswas et al. 1994) studied the influence of initial texture on the instability of the microstructure of titanium during compression at temperatures between 25 and 400C. They found that at strain rates  1 s -1 , both sets of specimens, in the rolling direction specimens and in the long transverse direction specimens, exhibited adiabatic shear bands, but the intensity of shear bands was found to be higher in the rolling direction specimens than in the long transverse direction specimens. At strain rates  ~ 13 ~ 0.1s -1 the material deformed in a micro structurally inhomogeneous fashion. For the rolling direction specimens, cracking was observed at 100 °C and at strain rates  0.1 s -1 . This is attributed to dynamic strain aging. Such cracking was not observed in the long transverse specimens. The differences in the intensity of adiabatic shear bands and that of dynamic strain aging between the two sets of test specimens are attributed to the strong crystallographic texture present in these plates. Subsequently, in 1997, Lebensohn and Canova (Lebensohn and Canova 1997) proposed a self-consistent model to simulate the evolution of texture in titanium and apply to the rolling. This model accounts for crystallographic textures and grain morphologies, as well as for the phase correlation, both in space and orientation. In their experiment, the two phases, (α + β) Ti alloys, exhibit specific morphologic and crystallographic correlations. Their study showed that the model leads to better texture predictions when all these correlations are accounted for. In 1999, Singh et al. (Singh, Bhattacharjee et al. 1999) ~ 18 ~ Zirconium In 1991, Tomé et al. [START_REF] Tome | A model for texture development dominated by deformation twinning: application to zirconium alloys[END_REF] propose a new method for modeling grain reorientation due to twinning, they deal with tension and compression in zirconium alloys. This new model, called "Volume Transfer Scheme", a part of the volume of the crystal transferred to the twinned position directly in the Euler space. Their model predicts the evolution of texture more accurately than traditional models when the twinning is the predominant mechanism of deformation. However, the disadvantage of this model is that you cannot take into account the work hardening. In 1993, Lebensohn et al. (Lebensohn and Canova 1997) present an approach anisotropic viscoplastic self-consistent model for the plastic deformation of polycrystals. This approach is different from that presented earlier by Molinari et al. in his formulation 'stiffness'. The self-consistent model is particularly suitable for highly anisotropic materials such as hexagonal. The authors applied their model to predict the evolution of texture in rolled zirconium and get better results after compared their prediction with experiments. In 1994 and 1995, Philippe et al. [START_REF] Philippe | Modelling of texture evolution for materials of hexagonal symmetry--I. Application to zinc alloys[END_REF] propose a model of the evolution of texture in hexagonal materials. The authors describe a first step in the evolution of texture and microstructure during cold rolling in zinc, subsequently, Philippe et al. [START_REF] Philippe | Modelling of texture evolution for materials of hexagonal symmetry--II. application to zirconium and titanium [alpha] or near [alpha] alloys[END_REF] extended their studies to titanium and zirconium, taking both the sliding plastic modeling and also twinning into account. The authors performed simulations using models of Sachs and Taylor and compared their results to the available literature. Fortunately, the simulation results were in good agreement with the experiments for reductions between 0 and 80% ~ 19 ~ reduction. [START_REF] Francillette | Experimental and predicted texture evolutions in zirconium alloys deformed in channel die compression[END_REF][START_REF] Francillette | Experimental and predicted texture evolutions in zirconium alloys deformed in channel die compression[END_REF] performed "channel die" compression tests on polycrystalline zirconium at room temperature, at the plastic deformation of 40%. They tested 5 samples with different initial texture. They simulated the evolution of texture using a crystal plasticity model of self-consistent, the simulation agreed with experiments very well. In 2000, Kaschner and Gray [START_REF] Kaschner | The influence of crystallographic texture and interstitial impurities on the mechanical behavior of zirconium[END_REF] studied the influence of texture on the mechanical behavior of zirconium. They evaluate the response of the material in compression according to the predominant orientation of the c-axis of the hexagonal mesh for several test temperatures and strain rate. They found that the compressive-yield responses of both high-purity (HP) crystal-bar and lower-purity (LP) zirconium depend on the loading orientation relative to the c-axis of the hcp cell, the applied strain rate, which varied between 0.001 and 3500 s -1 , and the test temperature, which varied between 77 and 298 K. The rate of strain hardening in zirconium was seen to depend on the controlling defect-storage mechanism as a function of texture, strain rate, and temperature. The substructure evolution of HP zirconium was also observed to be a function of the applied strain rate and test temperature. The substructure of HP zirconium was seen to display a greater incidence of deformation twinning when deformed at a high strain rate at 298 K or at 77 K. In 2001, Sanchez et al. [START_REF] Sanchez | Torsion texture development of zirconium alloys[END_REF] Orientation imaging microscopy in a scanning electron microscope and defect analysis via transmission electron microscopy are used to characterize the defect microstructures as a function of initial texture, deformation temperature and plastic strain. Finally, they found that the observed deformation mechanisms are correlated with measured mechanical response of the material. Magnesium In this section, we propose a review of literature on textures and their evolution in magnesium. In 2000, Kaneko et al. [START_REF] Kaneko | Effect of texture on the mechanical properties and formability of magnesium wrought materials[END_REF] ~ 22 ~ examined the effect of texture on the mechanical properties of AZ31 magnesium. They found that strong texture is observed in magnesium wrought materials in which the basal plane is oriented parallel to the extrusion or rolling direction. As a result, tensile strength of magnesium wrought materials increased by 15 to 20% due to texture hardening at room temperature. The AZ31 alloy sheet is highly anisotropic at room temperature with high r-value above 4, resulting that forming limits in biaxial tension are much lower than those in uniaxial tension. However, this anisotropy decreases with increasing forming temperature and no texture hardening is found at 473 K. In this respect, formability of magnesium alloys sheets in terms of Erichsen and conical cup values is remarkably poor at room temperature but appreciably improves with increasing temperature. Sheet forming of magnesium alloys is practically possible only at the high temperature range where plastic anisotropy disappears. In 2006, Yi et al. [START_REF] Yi | Deformation and texture evolution in AZ31 magnesium alloy during uniaxial loading[END_REF][START_REF] Yi | Deformation and texture evolution in AZ31 magnesium alloy during uniaxial loading[END_REF]) studied the behavior of magnesium alloy AZ31 under a uniaxial loading (tension and compression). The specimens were cut at 0°, 45° and 90° to the direction of extrusion. They highlight the various modes of deformation acting under the direction of the initial texture. They found that the activity of the basal <a> slip and the tensile twinning exert a significant effect on the mechanical anisotropy during tension, while the importance of the <c + a> slip increases during compression. Helis et al. [START_REF] Helis | Microstructure evolution and texture development during high-temperature uniaxial compression of magnesium alloy AZ31[END_REF] [START_REF] Watanabe | Effect of temperature of differential speed rolling on room temperature mechanical properties and texture in an AZ31 magnesium alloy[END_REF] studied the texture evolution in AZ31 magnesium and ductility in different conditions of rolling symmetrical. They found that the strain rate was inversely proportional to the square of the grain size and to the second power of stress. The activation energy was close to that for grain boundary diffusion at 523-573 K, and was close to that for lattice diffusion at 598-673 K. From the analysis of the stress exponent, the grain size exponent and activation energy, it was suggested that the dominant diffusion process was The as-received material was hot-rolled and then annealed commercial pure titanium sheet (mean grain size is 10 m) of 1.5 mm thickness with the composition given in table 2-1. Heat treatment In order to obtain a coarse grain microstructure with equiaxed grains, a grain growth annealing was performed on so some samples at 800°C for 2 hours, and then the grain had grown to 200 m after the annealing treatment. Samples preparation The samples were firstly mechanically polished with silicon carbide sandpaper (600 # , 1200 # , 2400 # until 4000 # sandpaper (Struers standard)) and then electrolytically-polished in a solution of 200 ml perchloric acid in 800 ml methanol at 17V (30 seconds) at a temperature of 20°C . Electron Back Scattered Diffraction (EBSD) This part introduces our most important experimental methods and equipment -Electron Back Scattered Diffraction (EBSD) in detail. We will ~ 34 ~ explain how an EBSD system works, describes the experiments that can be performed and how to undertake them, and finally outlines the basic crystallography needed for EBSD. In this work, A Field emission JEOL-6500F FEG-SEM with EBSD camera and HKL CHANNEL5 software is used to perform EBSD measurement and analysis. HKL CHANNEL5 uses a modular approach to its various. All software modules interact seamlessly with one another and form a powerful and expressive suite with which to perform microstructural characterization. Therefore the HKL CHANNEL5 Flamenco allows image collection, versatile EBSD analysis and phase identification all within a single program. Introduction EBSD is a technique which allows crystallographic information to be obtained from samples in the scanning electron microscope (SEM). In EBSD a stationary electron beam strikes a tilted crystalline sample and the diffracted electrons form a pattern on a fluorescent screen. This pattern is characteristic of the crystal structure and orientation of the sample region from which it was generated. The diffraction pattern can be used to measure the crystal orientation, measure grain boundary misorientations, discriminate between different materials, and provide information about local crystalline perfection. When the beam is scanned in a grid across a polycrystalline sample and the crystal orientation measured at each point, the resulting map will reveal the constituent grain morphology, orientations, and boundaries. This data can also be used to show the preferred crystal ~ 35 ~ orientations (texture) present in the material. A complete and quantitative representation of the sample microstructure can be established with EBSD. Basics of EBSD The principal components of an EBSD system are shown in ~ 38 ~ The crystal orientation is calculated from the Kikuchi band positions by the computer processing the digitized diffraction pattern collected by the CCD camera. The Kikuchi band positions are found using the Hough transform. The transform between the coordinates (x, y) of the diffraction pattern and the coordinates (,) of Hough space is given by: = cos + sin A straight line is characterized by , the perpendicular distance from the origin and θ the angle made with the x-axis and so is represented by a single point (,) in Hough space. Kikuchi bands transform to bright regions in Hough space which can be detected and used to calculate the original positions of the bands. Using the system calibration, the angles between the planes producing the detected Kikuchi bands can be calculated. These are compared with a list of inter-planar angles for the analyzed crystal structure to allocate Miller indices to each plane. The final step is to calculate the orientation of the crystal lattice with respect to coordinates fixed in the sample. His whole process takes less than a few milliseconds with modern computers. Basic crystallography for EBSD Crystal orientation A crystal orientation is measured with respect to an orthogonal coordinate system fixed in the sample. The sample system is normally aligned with The relationship between a crystal coordinate system and the sample system is described by an orientation matrix G. A direction measured in the crystal system r c is related to the same direction measured in the sample system r s by: = The rows of the matrix G are the direction cosines of the crystal system axes in the coordinates of the sample system. Misorientation The orientation between two crystal coordinate systems can also be defined by the angle-axis pair [uvw]. One coordinate system can be superimposed onto the other by rotating by an angle  around the common axis [uvw] (Figure 234). Because it is an axis of rotation, the direction [uvw] is the same in both coordinate systems. The angle-axis pair notation is normally used to describe grain boundary misorientations. The orientation between two coordinate systems can also be defined by a set of three successive rotations about specified axes. These rotations are called the Euler angles  1 , ,  2 and are shown in Figure 2345, which shows the rotations necessary to superimpose the crystal coordinate system (red) onto the sample system (blue). The first rotation  1 is about the z axis of the crystal coordinate system. The second rotation is  about the new x-axis. The third rotation is  2 about the new z-axis. The dotted lines show the positions of the axis before the last rotation. Note that the orientation can also be defined by an equivalent set of Euler angles which superimpose the sample coordinate system onto the crystal coordinate system. 2-5 ~ 41 ~ ~ 42 ~ In-situ EBSD test and interrupted "in-situ" EBSD test After automated EBSD systems are used in combination with other equipment within the scanning electron microscope (SEM), it is possible to perform in-situ tension test in the sample chamber of SEM and apply EBSD measurements simultaneously. While a changes in a single EBSD pattern could be observed during in-situ deforming of a sample, so much greater insight can be gained by EBSD scan data during an experiment. This chapter briefly introduces in-situ EBSD tension test and interrupted "in-situ" EBSD test used in this study. In-situ EBSD test The arrangement of an in-situ EBSD test is illustrated in The EBSD measurement are taking during the in-situ experiment.In order to get high quality Kikuchi patterns, the sample must be maintained at the standard 70 tilt during the experiment. This requires good compatibility between the SEM,EBSD camera and deformation stage. ~ 43 ~ Interrupted "in-situ" EBSD test Although in-situ EBSD test is an excellent method in collecting orientation data of grains during deformation, it requests some strict conditions such as low deformation speed, small measure area and so on; most important is that tension test is the only deformation mode it can provide. Therefore, an interrupted "in situ" EBSD investigation method was proposed and applied. In this method, we concentrate on sufficient amount of grains and perform EBSD measurement prior and after the deformation in the same zone, therefore, we can acquire the detailed orientation information of these grains in the interrupted step of deformation and identify the deformation modes during the deformation process. This method can be applied in rolling, tension, compression and some other deformation modes. Experiment arrangement The samples were cold rolled or compressed in channel die in several passes respectively, first to a certain amount of deformation and then to a further amount of deformation, and continue to more and more deformation. To perform interrupted "in situ" measurement, a 500×300 mm In this chapter, we continued this study on the deformation modes in rolling and channel die compression at room temperature, to provide information on the lattice rotation in the course of the deformation. The role of twinning, the formation and the evolution of mechanical twins is studied with the method of interrupted "in situ" SEM/EBSD measurements. A Introduction The mechanism of plastic deformation has been studied in some detail in the past [Yoo (1981) } compression twins are activated during plastic deformation at room temperature. Due to the compacity ratio c/a <1.633 in titanium, prismatic glide is the easiest one at room temperature but basal and pyramidal glide were also observed [Pochettino, Gannio, Edwards and Penelle (1992)]. However the previous studies concerning the texture and deformation modes were mostly performed after a certain amount of deformation either by of X-ray diffraction XRD or by transmission electron microscopy (TEM). Therefore the initial orientation of the individual grains and the evolution of the orientation flow during deformation were not documented. Moreover, after a certain amount of deformation, twinning and gliding were both active and interacted with each other. Thus it was difficult to resolve the specific orientation condition to activate each deformation mode (either twinning or gliding). Therefore, the present work is devoted to these aspects, providing the lack of information in the literature. In order to follow the evolution of individual orientations during the deformation and to determine the effect of initial orientation on the deformation modes, an interrupted "in situ" EBSD investigation method was proposed. In this ~ 48 ~ method, we follow a sufficient number of grains and perform EBSD measurement in the same area, prior to and after the deformation. Thus it is possible to acquire detailed orientation information of these grains in the interrupted step of deformation and identify the active deformation modes. Experimental The as-received material was hot-rolled and then annealed commercial purity titanium sheet of 1.5 mm thickness with the composition given in table 1. In order to obtain a twin-free microstructure with equiaxed grains, a grain growth annealing was performed at 750°C for 2 hours. After the annealing, the samples were mechanically and then electrolytically-polished in a solution of 200 ml perchloric acid in 800 ml methanol at 17V (30 seconds) and at a temperature of 5°C before deformation. Then, the samples were cold rolled or compressed in channel die in two passes respectively, first to 10 % and then to 20% reduction. To perform interrupted "in situ" measurement, a 500×300 mm 2 area was carefully polished and marked out with four micro-indentations. The orientation of all the grains in this Deformation in rolling The orientation map of the grain growth annealed sample shown in Fig. 2 ~ 50 ~ The lattice rotation was studied after each rolling pass. The orientation of each individual grain was carefully determined, so that the lattice rotation of each grain could be brought in relation to its own orientation as well as to that of its neighbouring grains. It was found that { 2 > axis [Philippe, Esling and Hocheid(1988) } twinning was predominant at this stage of deformation. This result is reasonable considering that the initial orientation favors the activation of this compression twin. Whereas when rolling continues to 20%, { 2 1 10 } tension twinning was remarkably increased (Fig. 4 (c)). The orientation analysis could be studied from the microscopic point of view of the crystal reorientation step by step, in terms of the rotation flow field. A small arrow is plotted in the Euler space between the initial grain orientation and the final grain orientation. This field of small arrows offers a graphical representation of the rotational flow field. The flow field can be defined and plotted in the Euler space, and represents an efficient tool to describe the texture evolution through modeling, e.g. [START_REF] Clement | Eulerian simulation of deformation textures[END_REF], Bunge andEsling (1984), Knezevic, Kalidindi andFullwood(2008). In order to identify the activated glide systems corresponding to the traces observed, the possible traces of all possible glide planes [Partridge (1967)] are calculated in the crystal coordinate system, using the orientation data of the related grains and comparing with the observed traces. Consequently, basal <a>, prismatic <a> and pyramidal <a> or <c+a> glide systems are identified in this work. Deformation in channel die compression In channel die compression, only one type of twin -{ 2 1 10 } tension twin is observed and the amount of twinned grains is very low, only 1.07% of the observed grains, which cannot be clearly resolved from the misorientationangle distribution diagram in Fig. 7. Compared with rolling, numerous slip traces are observed in a great number of grains in EBSD maps after channel die compression. Using the above trace comparison method, basal <a>, prismatic <a> and pyramidal <a> or <c+a> glide are identified. A statistical set of 100 randomly selected grains with slip traces is studied. The occurrence of various glide systems in the studied grains is listed in Table 2. It is seen that among the activated glide systems, 11% are basal <a>, 51% are prismatic <a> and 38% are pyramidal <c+a> or <a>. The Schmid factors for the three glide systems were calculated, and the calculation indicated that most grains have high Schmid factors for pyramidal <c+a> glide system but low Schmid factors for ~ 55 ~ basal <a> and prismatic <a>. This is due to the strong texture which means a majority of grains belong to one main orientation mode. Table .2 Activated glide systems in 100 randomly selected grains. Activated glide system frequency Basal <a> 11% Prismatic <a> 51% Pyramidal <a> or <c+a> 38% ~ 59 ~ pole. The pole can be seen as a dislocation source and the partial dislocations are produced continuously in the planes parallel to the twin plane during deformation. If the path in which the partial dislocations move is short, the dislocations will easily be blocked, pile up and react to the source disabling it. Deformation twins always have a lenticular shape, since the interface can deviate from the twinning plane without greatly increasing the twin-interface energy [Partridge (1967)]. In addition, the strain energy increase should also be taken into account, approximately equal to (c/r)µS 2 , where c and r represent the thickness and length of the twin, µ the shear modulus and S the twinning shear. Hence, secondary twins are already confined in a small volume due to the lenticular shape and cannot provide enough free path to develop a higher order twinning. In fact, in titanium alloys it is difficult to induce twinning at room temperature once the size of the matrix drops below the range of about 10 µm. A major benefit of the interrupted "in situ" method is that we can follow the deformation process step by step. For example, in the grains having their initial c-axis close to ND for which secondary twinning occurs inside the primary twins, we can clearly discriminate the initial matrix from the primary twins thanks to the in-situ orientation information. In this case the primary twinned area is much larger than the remaining matrix and thus represents the "new matrix" for possible subsequent secondary twinning. The effect of the neighboring grains slightly modifies the orientation in the vicinity of the grain boundary. This leads to a larger spread in the orientation measured in the vicinity of the grain boundary when the ~ 60 ~ neighboring grain is strongly misoriented with respect to the considered grain. Compared with rolling, channel die compression showed a simple deformation mode including only { 2 1 10 } tension twinning and various gliding. The primary reason is the simple stress condition applied in channel die compression. Therefore, we used the channel die compression results to clarify the effect of grain size on twinning. 357 grains with orientation favorable to the activation of { 2 1 10 } twinning were selected and divided into three groups according to their diameter. Group 1: 0 to 10 µm (221 grains); Group 2: 10 to 20 µm (129 grains); Group 3: 20 to 30 µm (7 grains). The calculated percentage of grains with twins for the three groups is shown in Fig. 11. It is clear that with the similar orientation (c axis tilted 70º~90º from normal direction ND), no twin occurs in the grains smaller than 10m. With the increase of the grain size, the occurrence of the twin increases. Hence, grain size is an important factor affecting twin activation. The reason can be understood from geometrical and energetical considerations we introduced above. ~ 61 ~ Conclusion Twinning occurs in grains having specific orientations. Generally, the reorientation induced by twinning aligns the c-axis of the twinned part to the stable rolling texture orientation, so that no further secondary twinning can be induced. Secondary twinning occurs only when the primary twinning orientates the c-axis of the primary twins far away from the stable orientations (this is generally the case for the { 2 } primary twin results in a reorientation of the c-axis of the secondary twin to a stable orientation. Only a little amount of second order twin could be observed and twinning of higher than second order was not found. The rotation of the matrix-part of the grains having twins is similar to that of the non-twinned grains with similar orientation. The twinned part of a grain can be considered as a new grain. When twins grow within the grain, Abstract: The present work was undertaken to provide information, lacking in the literature, on the lattice rotation and the role of twinning during cold rolling of commercial purity titanium (T40). The proposed method consists of determining the individual rotation of the grains induced by low to intermediate deformation (up to 30% in thickness reduction) and following the rotation field using electron backscattered diffraction (EBSD) measurements in a high resolution FEG SEM at different steps of deformation (10 and 20 %). We have especially studied the formation, the evolution and the role of mechanical twins. According to the former research, during the deformation at room temperature, three different types of twin systems were activated: { 2 1 10 } tensile twinning, { 2 2 11 } compression twinning and -to a small extent -{ 1 2 11 } tensile twinning, depending on the grain orientation. Secondary twins (often { 2 1 10 } within { 2 2 11 } twins) were activated in the grains oriented favourably for this secondary twinning. This resulted in a heterogeneous microstructure in which grains were refined in some areas. It also induced re-orientation of the c-axes to stable orientations. No twins of higher order than the second order twins could be found. Introduction Introduction The titanium textures observed at room temperature in the hexagonal close-packed (HCP) structure are inherited to some extent from their prior texture in the body centred cubic (BCC) structure [1] . However most of the ~ 66 ~ research effort concentrated on the strong deformation textures that Titanium [2-4] , like other HCP metals, develops during the rolling at room temperature, that lead to a pronounced plastic anisotropy of the polycrystalline materials [5] . The mechanical response of HCP metals is strongly dependent on the combination of active deformation modes: slip and twinning. The specific deformation mechanisms depend on the c/a ratio, the available deformation modes, the critical resolved shear stress (CRSS) for slip and the twin activation stress, as well as the imposed deformation relative to the crystallographic texture. For pure titanium, { 0 1 10 } < 0 2 11 > slip is the primary deformation mode. This slip mode alone, however, cannot accommodate the imposed strain in the grains of a polycrystalline aggregate, because it cannot provide 5 independent slip systems [4, 6-9] . Additional deformation mechanisms such as pyramidal planes with <c+a> Burger's vector or twinning usually have to be activated. Chun et al. [10] studied the effect of deformation twinning on microstructures during cold rolling of commercially pure (CP) titanium. The primary twinning systems activated were { 2 In order to describe the texture evolution, different models of polycrystalline plasticity are used. Polycrystal plasticity models are routinely employed to predict deformation textures. Wu et al. [11] employed a new ~ 67 ~ Taylor type of crystal plasticity model to predict the texture evolution and anisotropic stress-strain curves in -titanium. The main features of this model include: (i) incorporation of slip inside twins as a significant contributor to accommodating the overall imposed plastic deformation; and (ii) extension of slip and twin hardening laws to treat separately the hardening behaviour of the different slip families (prismatic <a>, basal <a> and pyramidal <c+a>) using hardening parameters that are all coupled to the extent of deformation twinning in the sample. Proust et al. [12] developed a model which takes into account the texture evolution associated with twin reorientation and the effect of the twin barriers on dislocation propagation. The role of the twins as barriers to dislocations was explicitly incorporated into the hardening description via geometrically necessary dislocations (GNDs) and a directional Hall-Petch mechanism. However, with these complex models, the lack of direct experimental information on slip and twinning systems imposes difficulties for the related modelling practices. Experimental The as-received material was hot-rolled and then annealed commercial pure titanium sheet of 1.5 mm thickness with the composition is given in table 1. In order to obtain a microstructure with a mean grain size of 30 µm, a grain growth anneal was performed at 750°C for 2 hours. After annealing, the samples were mechanically ground and then electrolytically-polished in a solution of 20 ml perchloric acid in 80 ml methanol at 17V (30 seconds) and a temperature of 5°C before the cold rolling. Then, the samples were cold rolled in two passes, first to 10 % and then to 20% reduction. The cold rolling was performed in the transverse direction of the former hot rolling in order to induce significant reorientations of the C-axes of the grains. To Results Initial microstructure and texture The orientation map of the grain growth annealed sample shown in Fig. 2 reveals a completely recrystallized microstructure. No twins were observed as metals with HCP structure do not undergo recrystallization twinning. The{0002}-pole figure (PF) in Fig. 3 shows two strong maxima at ±35° tilted from ND towards TD, the setting of the coordinate systems, and the definition of the Euler angles being in accordance to Bunge's convention (see e.g., Ref. [13]). The { 0 1 10 } PF displays the maximum pole densities parallel to RD. Lattice rotation fields and texture development The orientation analysis could be studied from the microscopic viewpoint of the crystal reorientation step by step, in terms of the lattice rotation flow field. A small arrow is plotted between the initial grain orientation and the final grain orientation. This field of small arrows offers a graphical representation of the flow field. The orientation flow field can be defined and plotted in the Euler space, and represents an efficient tool to describe the texture evolution through modeling, e.g. [START_REF] Clement | Eulerian simulation of deformation textures[END_REF] [13] , [START_REF] Bunge | Texture development by plastic deformation[END_REF] [14] . In the present case of hexagonal material, due to the particular importance of the c-axes, we choose to plot the small arrows linking the initial orientation and the final orientation in the two dimensional pole figures of the c-axes. For the further discussion, it was of interest to plot separately the rotation flow field of the grains having no twinned part on the one hand (Fig. 5 (a)) and the rotation flow field of the matrix part of the grains presenting twinned parts inside the grains (Fig. 5 (b)). Both orientation flow fields are similar, but for a smaller amplitude of the rotation of the matrix of twinned grains, as compared to the grains without twins. The orientation analysis could be also studied from the macroscopic Interestingly it is found that tension twins and compression twins can coexist in one and the same grain (Fig. 9). From studies of the literature, we could not find an explanation for this coexistence which was initially unexpected. Effect of the orientation of the neighbouring grains: heterogeneous deformation In large grains, different domains in the one and the same grain may undergo different lattice rotations. Fig. 12 presents a large grain (green color) in which several parts have experienced different lattice rotations. ~ 78 ~ The numbers inside the small neighboring grains give the misorientations of the c-axes with respect to the c-axis corresponding to the mean orientation of the large green grain. We can study here the respective rotations of the domains inside the large green grain, notably in the neighbourhood of the grain boundary, and thus estimate the influence of the neighbouring grains. In any case the rotation of the grain interior is smaller than that of the part close to the grain boundary, especially when the neighbouring grains are highly misoriented with respect to the investigated domain. Discussion Comparison of the rotation of the matrix part of the grains containing twins with the rotation of the untwinned grains indicates that the lattice rotation in the two cases are similar, even if the amplitude of the rotation in the non-twinned part of the twinned grains is relatively smaller. The c-axis of this secondary twin orientates towards to the stable orientation. We could hardly observe the presence of any third-order twin, even after much higher deformation. This can be easily understood by the relation between the geometrical and energetical characteristics of twinning (and by the mean free path necessary for the formation of a twin). In fact, in titanium alloys it is difficult to induce twinning once the grain size in the matrix drops below the range of about 10 µm. The great benefit of the present method is that we can follow the deformation process step by step. For example, in the grains having their initial c-axis close to ND for which secondary twinning occurs inside the primary twins, we can clearly discriminate the initial matrix from the ~ 80 ~ primary twins thanks to the in-situ orientation information. In this case the primarily twinned area is much larger than the remaining matrix and thus represents the "new matrix" for the subsequent secondary twinning. The effect of the neighboring grains slightly modifies the orientation in the vicinity of the grain boundary, and leads to a larger spread in orientation when the neighboring grain is strongly misoriented with respect to the considered grain. The detailed experimental study of the complex twinning conditions in hexagonal materials may be helpful to the implementation of mechanical twinning in models and codes of polycrystalline plasticity. A thorough study of the implementation of mechanical twinning in a Grain Interaction Model and the application to magnesium alloys [16] can be read with interest in this special volume of AEM. Conclusion Twinning occurs in grains that have particular orientations. Generally, the reorientation induced by twinning aligns the c-axis of the twinned part to the stable rolling texture orientations, so that no further secondary twinning can be induced. Secondary twinning occurs only when the primary twinning orientates the c-axis of the primary twins far away from the stable orientations (this is generally the case for the { 2 of the deformation energy of each variant should be adopted as the main criterion in predicting the variant selection, and the Schmid factor used as an additional second criterion. We also extended our discussion to the various twinning behaviors of {11-22} and {10-12} primary twins. The {11-22} twin shows a behaviour of multiply twin variants system (MVS) and the {10-12} twin shows predominant variant system (PVS) and these mainly result from the orientation relationship between stress components and the parent grain. I Introduction Twinning is a particularly important deformation mode in  titanium [1][2][3][4][5][6]. Twinning process strongly depends on the crystallographic orientation of the matrix. The simple Schmid factor allows to calculate the resolved shear stress on the twin system from the applied macroscopic stress. Titanium exhibits two major twinning systems at room temperature, {10-12} twin and {11-22} twin. The {10-12} system is referred to as tension twin ({11-22} as a compression twin) because it only activates under tension (compression) load along the c axis of the matrix [7]. Hence, Schmid's law loses effectiveness of selecting twin type because of the directionality of twinning, and it is not clear whether Schmid's law is an applicable criterion This work is concerned with experimental verification of Schmid' law as a criterion for selecting twin variants, and extend to a study of the twin growth as well. In order to trace the evolution of individual orientations during the deformation and the growth of twin, an interrupted "in situ" EBSD investigation method [8] was proposed and applied. In this method, we concentrate on a sufficient amount of grains and perform EBSD measurement on these grains in each step of the deformation. Therefore, we can acquire the detailed orientation information of these grains in the interrupted deformation step. Experimental The material used in this investigation is a cold-rolled commercially pure titanium T40 sheet with 1062 ppm (wt.) oxygen. The sheet was annealed at 800°C for 2 hours to allow the deformed microstructure to be fully recrystallized. The final average grain size is 200µm. A four-stepped compression test was performed at room temperature at the rate of 0.5mm/min. The total thickness reduction after each step is 8%, 16%, 24% and 35%, respectively. The EBSD orientation measurement was performed on the same sample area after each deformation step with a JEOL 6500F field-emission-gun SEM equipped with EBSD acquisition camera and HKL channel 5 software. In order to obtain statistically representative results, 100 twinned grains are randomly selected to analyze the twin variant selection role and twin ~ 86 ~ growth behaviors. To eliminate the effect of grain size, the selected grains have a similar size. Results Initial texture The selection of twin variants In the present study, after 8% deformation, both {10-12} tension twin and {11-22} compression twin are spotted in some grains but {11-22} ~ 87 ~ compression twin is more frequent. This result is coherent with the initial texture. Theoretically, there are six twin variants in each twin family [1], but in one initially un-deformed grain, only a limited number of variants are active in each twinning mode. Due to the crystallographic symmetry, the maximum number of variants for the {10-12} tension twin is two, whereas that for the {11-22} compression twin can reach four, depending on the proximity of the c-axis to the applied force. With the increase of the amount of deformation, twinning occurs in more grains. The growth of twin Figure 2 shows the EBSD maps of two typical grains at the initial state, 8% and 16% deformation, one containing {11-22} compression twin (Figure 2 (a)) and the other {10-12} tension twin (Figure 2 (c)). ~ 90 ~ In order to study the mechanisms that lead to the selection of specific twin variants, the misorientation between the variants of the respective {11-22} and {10-12} twins is calculated and listed in Table 1. in the studied grains is 0.43 and the minimum is 0.32. From the results above, it can be deduced that Schmid law is determinant for the activation of the twin variants. However, still in some cases, variants with non-highest SF (NSF<1) are activated. Their activation may be attributed to the microscopic local stress that is deviated from the macroscopic load. In these cases, the Schmid factor of the active twin system with respect to the local stress may be the highest. ~ 92 ~ Table 1 The misorientation angle and axis between each pair of variants of respectively {11-22} twin and {10-12} twin. {11-22} twin variant (11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22) variant (-12-12) variant (-2112) variant (-1-122) variant variant (2-1-12) variant (11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)) 0° 60.00° [11][12][13][14][15][16][17][18][19][20][21] 77.29° 51.20° 77.29° [8-1-70] 60.00° [-1-12-1] variant (-12-12) 0° 60.00° [-12-1-1] 77.29° [8-7-10] 51.20° 77.29° ~ 94 ~ Conclusions In this work, we performed an analysis on twin variant selection and twin growth in polycrystalline titanium during small reduction compression by "interrupted in situ" EBSD measurement. The following results can be concluded: 1. Schmid' law is an appropriate criterion for twin variants selection. Twin variants with Schmid factors higher than 0.4 have a good chance to be activated. In compression Introduction Mechanical twinning is a particularly important deformation mode in  titanium [START_REF] Vedoya | Plastic Anisotropy of Titanium, Zirconium and Zircaloy 4 Thin Sheets[END_REF]; Philippe et al. (1995); [START_REF] Fundenberger | Modelling and prediction of mechanical properties for materials with hexagonal symmetry (zinc, titanium and zirconium alloys)[END_REF]Zaefferer, (2003)]. It also offers a means to control the properties of titanium such as ductility and fracture strength [START_REF] Kocks | The importance of twinning for the ductility of HCP polycrystals[END_REF]; Partridge (1967); [START_REF] Mahajan | Deformation twinning in metals and alloys[END_REF]]. At room temperature, three major twinning systems are commonly observed in titanium: the {10-12} twin, the {11-21} twin and the {11-22} twin [Philippe et al. (1988)]. The which indicated that the variants with higher resolved shear stress are more likely to be selected. However in the case of secondary twinning, Barnet [Barnet et al. (2008)] have shown that the variant selection does not follow Schmid's law. [START_REF] Capolungo | Variant selection during secondary twinning in Mg-3%Al[END_REF]] have shown that in magnesium, Schmid's law is not the only criterion controlling the variant selection, shared volumes and primary to secondary accommodation shears are prevalent. In this paper, some calculations of Schmid factors, crystallographic geometry, and plastic energy associated with double twinning were performed to investigate the variant selection in double twinning and ~ 98 ~ reveal the relevant microscale features responsible for the formation of secondary twins. A study on the influences of the twinning on the texture evolution was also carried out. Moreover, an "interrupted in situ SEM/EBSD orientation determination" [Bao et al (2010b)] was adopted. This approach allows to obtain the time resolved information on the appearance of the twin variants, their growth, the interaction between them and the interaction with the grain boundaries or twin boundaries. Experimental The material used was a commercial pure titanium T40 sheet of 1.5 mm thickness with the composition given in table 1. First, the sheet was annealed at 800°C for 2 hours to allow the deformed microstructure to be fully recrystallized with a final average grain size of about 200µm. The samples with the dimension of 15mm×10mm were prepared, mechanically polished and further electrolytically polished at a temperature of 5°C in a solution of 200 ml perchloric acid in 800 ml methanol at 17V ( 30seconds) for subsequent SEM/EBSD measurements. On the polished surface, a 1250×950 m 2 area was selected and delimited by four microindentations. The microstructure and OIM of this selected area was measured by SEM/EBSD before and after each channel die compression step (0, 8, 16, 24 and 35% reduction). The surface with the selected area ~ 99 ~ was firmly stuck with another polished sample (sandwich like) to avoid any surface sliding during the compression, in order to maintain a good surface quality. The channel die compression layout is illustrated in Fig. 1. SEM/EBSD measurements were performed with a JEOL 6500F fieldemission-gun SEM equipped with EBSD acquisition camera with a step size of 0.6 µm at 15KV. The data were acquired and processed with the HKL Channel 5 software from HKL technology. In order to obtain statistical representation of the results, 100 twinned grains were randomly selected to analyze the twin variant selection. To eliminate the effect of the grain size, only the grains having similar sizes ranging from 180m to 220m were selected. Results Microstructure and texture The OIM of the initial sample shown in Fig. 2 (a) reveals a completely recrystallized microstructure. From the pole figures (PF) in Fig. 2 (c), it can be found that the initial texture is characterized by two strong maxima at In Table 2, ~ 105 ~ Variant selection of the double twinning Since the T2 twin (the {11-21} tension twin) was not found in this study, only the combination of T1 and C was investigated in the experimental study on variant selection for double twinning. To perform a statistical analysis, 100 grains with double twins were systematically studied. C-T1 and T1-C double twins were analyzed separately despite they induce the same misorientation by symmetrically equivalent rotations, namely Group A (41.3° about <1-543>), Group B (48.4° about <5-503>) and Group C (87.9° about <4-730>). The frequency of the occurrence of each variant group in both C-T1 and T1-C double twins was presented in Fig. 4. It could be seen that the Group B (78.9% in the frequency) predominated over the other two in C-T1 double twin, the Group A was 20% and the Group C was nearly inactivated (1.8%). In the case of the T1-C double twin, the dominant variant group was the Group C (66.7% in the frequency), whereas Group A was 33.3%. ~ 106 ~ Schmid factor analysis Since the Schmid factor (SF) plays an important role in twin variant selection [START_REF] Lebensohn | A study of the stress state associated with twin nucleation and propagation in anisotropic materials[END_REF]], this study first examined the effect of the Schmid factor on the variant selection in the primary and secondary twinning, respectively. For each selected grain, the geometric SF was calculated to examine the resolved stress applied on the twin plane and along the twin shear direction, given the applied macroscopic compressive stress. Primary twinning The ranking of the SF corresponding to the active variants of primary twins was shown in Fig. 5 (a) in the form of a histogram. Normalized Schmid factor (NSF) [START_REF] Capolungo | Nucleation and ~ 126 ~ growth of twins in Zr: A statistical study[END_REF]] was adopted in the present work to investigate whether the Schmid's law is conclusive for the variant selection. In a given grain, the NSF of this grain equals to the SF of the active variants ~ 108 ~ Plastic energy analysis A twin variant will be activated if the energy of deformation which is used to create the twin was sufficient and the internal energy of the material would decrease with this operation. We have considered here that the material is an ideal (i.e., no strain hardening) rigid-plastic body to calculate the energy of deformation. Because the elastic energy is restored when the twin is created, we restrict to the plastic energy of deformation which is calculated by the equation: ( Where ' ij is the critical resolved shear stress required (=shear stress  expressed in the sample frame) to activate the twinning system and  ij is the corresponding twinning deformation. In the case of channel die compression, the deformation is equivalent to in-plane compression and the compressive force is applied in the sample normal direction (the third axis 33). In a grain, the stress applied to a twinning system is composed of the macroscopic applied stress and an additional local stress resulting from the interaction of the considered grain with the neighboring grains. Since we restrict to relatively small deformation degrees, we neglect the local stress resulting from the interaction with the neighboring grains. The stress applied to a twinning system is thus restricted to the macroscopic compressive stress  33, which corresponds to the Sachs (or static model) hypothesis. The twinning system will be active when the resolved shear stress reaches the corresponding critical value' 33 . When the twinning system is active, the corresponding deformation energy expressed in the macroscopic coordinate system is given by the above Eq. ( 1). We introduce the grain size effect by expressing the critical resolved shear stress according to a Hall Petch (HP) type equation: L k   0   (2) with 0  and k constants, 0  representing the stress when the length of the grain is infinite. L is the free path of the twin before encountering an obstacle (grain boundary, precipitate or other twins). Then the deformation energy can be expressed as: 33 33 0 33 0 ) (      L k L k W     (3) In Eq. ( 3), 0  and k are unknowns. Taking into account that twinning is activated when the size of a grain exceeds a certain value below which only ~ 110 ~ crystal glide can be activated, we can deduce that the second term of the equation is dominant. Rearranging Eq. ( 3) we obtain: L k W 33 33 0      (4) In the right hand term of Eq. ( 4), 33  and L , are accessible to the experiment. In the following we will mainly focus on this term, where u, v and w are the displacement components and x, y, and z are the coordinates in the sample system, was first expressed in an orthonormal reference frame defined by the related twinning elements. The unit vector normal to the twinning plane, the unit vector normal to the shear plane and the unit vector in the twinning direction define this reference frame. In this frame the displacement gradient tensor has a particularly simple form: ~ 111 ~ = 0 0 0 0 0 0 0 0 (6) With = √ for the (10)(11)(12) twin and = ( ) for the (11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22) twin where = ⁄ ratio of titanium, the displacement gradient tensor for the two types of twins can be obtained as: = ⎝ ⎜ ⎛ ⎠ ⎟ ⎞ = 0 0 0 0 0 0 0 0 = ⎩ ⎪ ⎨ ⎪ ⎧ 0 0 0.218 0 0 0 0 0 0 {112 2} 0 0 0.175 0 0 0 0 0 0 {101 2} (7) Through coordinate transformation, this displacement gradient tensor can be expressed in the crystal coordinate system (here we choose the orthonormal reference system set to the hexagonal crystal basis and the setting follows the Channel 5 convention, i.e. e 2 //a 2 and e 3 //c). With the Euler angles measured by SEM/EBSD that represent a set of rotations from the sample coordinate system to the orthonormal crystal basis, this tensor can be further transformed into the macroscopic sample coordinate system. If G is the coordinate transformation matrix from the macroscopic sample coordinate system to the orthonormal twin reference system, the displacement gradient tensor with respect to the sample coordinate system can be expressed as: ( ) = ( ) (8) Thus the deformation tensor in the macroscopic sample coordinate system can be obtained as the symmetrized displacement gradient: = + (9) ~ 112 ~ With Eq. ( 9), the energy term in Eq. ( 4) can thus be calculated. The energy term in Eq. ( 4) has been calculated for all the examined grains. Here, the ranks referring to the decreasing order of energy term corresponding to each twin variant of primary and secondary twins were calculated separately. Further, for each energy term rank, the frequency of being selected is calculated and plotted in Fig. 7. The results indicate that in the case of primary twin, the prediction is correct in 85% of the cases using the energy term as a variant selection criterion, and in 95% of the cases of secondary twin. The variant selection strongly depends on the energy because the free path length for twin variant is included in this calculation. According to the previous study, normally the selection of twin variant is dependent on the grain shape. In the equiaxed grains, the free path for each variant is almost the same, so several variants can form together in one grain. However, in most instances of elongated grains, only one variant can be activated because of the longer free path. Also for the case of elongated grains, although the activation of the twin variant changes the dimensions of the grain, it does not change the free path length of this variant. Thus this variant can form repeatedly as long as it does not create conditions more favorable for another variant. Under such conditions, the appearing twins can continue to grow until all the parent grain is completely twinned. In most cases, primary twins that accommodate the secondary twins usually show the appearance of lamellae, i.e. secondary twins always form in parent grains of uneven shape (primary twin). The conditions of being active of secondary twin variants are thus largely dependent on the free path. This is also the reason why the energy term is L 33  L 33  ~ 113 ~ highly accurate in view of predicting the variant selection of secondary twinning. Geometric features of double twins analysis The geometric features of the combination of C and T1 twins were illustrated in Fig. 8, where the twin elements of primary and secondary twin, twin plane (TP), shear direction (SD) and shear plane (SP), were plotted as poles in a pole figure, in a reference frame bound to the primary twinning system, as seen in Fig. that in the magnesium, the growth potential is strongly related to the angles of shear plane normal (SPN) and SD between primary and secondary twins because primary twins very efficiently grow along the SPN and SD and the growth of the secondary twins is primarily limited by the lengths of the Fig. 7. Frequency of energy term ranks ("1" is the highest and "6" is the lowest) corresponding to the active variants of secondary twin. ~ 114 ~ primary twin along these directions. Therefore, the variants of double twins with small angles between SPN and SD of primary and secondary twins could easily be activated from the point of view of the growth potential. In the case of titanium, the corresponding angles were summarized in table 3. The C-T1 double twin agrees well with this theory (Fig. 8 (a)). Group B has the lowest angles of TP and SD between primary and secondary twin (see Table 3), i.e. the growth of secondary twin variants belonging to group B suffer the least limitation from their primary twins. In Fig. 4, it is seen that group B was dominant and took the proportion up to 78.9%. However, this theory seems less convincing in the case of the T1-C double twin. In Table 3 it can be deduced that group B should be predominant according to the theory. Our experiment, on the contrary, exhibited a reverse result. Also in Fig. 4, the Group C took a high proportion of 66.7% and group B was not spotted. shear in the adjacent grain [START_REF] Yoo | Interaction of slip dislocations with twins in HCP metals[END_REF]]. On OIM micrographs, it appears that this twin shear crosses the grain boundaries and continues in the adjacent grain. In our observation, however, the most common case is that twins do not cross the boundaries. Therefore, aiming at these twins straddling boundaries, a statistical analysis of the angle between the twin planes on each side of the boundary was carried out. From the results displayed in Fig. 9, it can be deduced that for those twins whose twin planes do not deviate beyond 20°, there is a high tendency to cross the boundary. On the contrary, those twins with larger deviation, generally terminate at the boundary. Discussion The occurrence of twinning is governed by various factors, such as the orientation, the size, the shape of parent grain, the boundaries, and even the slip activity [Yoo (1981)]. Due to symmetries of the crystal lattice, several potential twins are in competition, so that it is necessary to clarify the mechanisms of the variant selection process. In this study, as verified by the experimental results, the deformation energy gave an excellent accuracy up to 85%, the Schmid Factor (SF) a still acceptable accuracy of 50%, as the variant selection criterion. The reason of high accuracy of the deformation energy criterion, as compared to the SF criterion, can be explained by the fact that the calculation not only involves the deformation energy associated with the creation of a twin variant, but also the effect of the size and shape of the parent grain. This is quite effective in the case of a non-homogeneous microstructure or non- ~ 122 ~ equiaxed parent grain. When a twin forms in a grain, it cuts the parent grain and thus changes the shape and dimensions of this grain, generally subdividing the initial parent grain into three domains. The twin can be considered as a new grain with its own dimensions and crystallographic orientation. When a twin forms, the size of the parent grain is modified and thus the critical stress on each possible twin variant will change according to the HP law. As a result, variants which did not have a sufficient level of stress through the SF can nevertheless be activated in a newly created domain, despite the orientation of the initial parent grain not having changed. This explains why in equiaxed grains, several variants can appear. However, in the elongated grains, although the activation of the twin variant changes the dimension of the grain, it does not change the length of the free path of this variant. Thus this variant can form repeatedly, at least as long as it does not create conditions more favorable for another variant. In such conditions, the twins activated can continue to grow until the whole grain is twinned. A twin can thus be regarded as a new grain, which is a slightly different point of view when considering secondary twinning. Generally, this new grain (i.e. the primary twin) presents an elongated shape (at least at the early stage of its formation). Thus there will generally be only one activated variant in this existing primary twin. In fact, this concept strongly depends on the size of the twin, since, as previously discussed, when a grain is fully twinned, there can be several secondary twin variants thereafter. With respect to a homogeneous microstructure, the geometric SF that is the factor transforming the applied stress into the resolved shear stress on ~ 123 ~ the twin plane and along the twin direction, should in principle be conclusive in the variant selection. But the local stress tensor effectively applied to the grain does not coincide with that derived from the macroscopic applied stress tensor. Thus, the geometric SF thus may be not pertinent in some cases, notably when in one parent grain, the first and second highest SFs of variants are very close (NSF very close to 1). As seen in Fig. 5 and Fig. 6, the high frequency of the rank 2 SF probably results from this deviation between the local stress tensor and macroscopic applied stress tensor. Moreover, the high proportion of NSF range from 0.8 to 1.0 (not including 1.0) in Fig. 5 (b) and Fig. 6 (b) also supports this interpretation. It could be inferred that the true frequency of rank 2 SF should be about 15 to 20% lower than the calculated value in Fig. 5 and Fig. 6. If it was possible to calculate the true local stress state, the accuracy using SF in variant selection could increase dramatically. It can therefore be concluded that the deformation energy can be adopted as an effective criterion in variant selection, and SF should not be completely abandoned but used as a complementary criterion in the case of homogeneous microstructure. Conclusion In this study, an investigation of the effect of various factors on twin variant selection in double twins was performed. Twinning affects the texture evolution by the reorientation of the crystallographic lattice from the former stable orientation of the parent grain into the new stable orientation of the twin. The stable orientations depend on the deformation mode, like rolling or channel die compression for example. Prospects The following are some suggestions for future work based on the findings, conclusions and problems identified in the course of the present work. In order to study the deformation mechanisms in titanium, many efforts are required on twinning, gliding and the interaction between them. In the present study, we focused on the twinning and extended the discussion a little to the gliding. Therefore, some prospects could be suggested as follows: 1. The interrupted "in situ" EBSD method is an effective way to study the deformation twinning in detail. With this approach, various deformation modes could be studied, such as shear, tensile test and ECAP So far, the deformation in this approach is limited within 35% Le maclage apparait dans les grains qui ont des orientations particulières. En règle générale, la réorientation induite par le maclage aligne l'axe c de la partie maclées vers les orientations stables de la texture de laminage, de sorte qu'aucun autre maclage secondaire peut être induit. Le maclage secondaire se produit uniquement lorsque le macle primaire envoie l'axe c loin des orientations stables. Pour les grains maclés, la rotation du réseau de la matrice est semblable à celle des grains ayant une orientation cristallographique identique mais sans macles. Deux types de systèmes de macles ont été activés au cours de la déformation à la température ambiante: des macles de tension (10)(11)(12) et des macles de compression (11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22). Dan le maclage primaire, les résultats montrent que les variantes de macle In this study, an interrupted "in situ" SEM/EBSD investigation based on a split sample of commercial titanium T40 was proposed and performed in rolling and channel die compression. This approach allows to obtain the time resolved information of the appearance of the twin variants, their growth, the interaction between them and the interaction with the grain boundaries or twin boundaries. With the orientation data acquired by the EBSD technique, we calculated the Schmid factor, crystallographic geometry, and plastic energy associated with each variant of primary twins, secondary twins and double twins to investigate the lattice rotation, the activation of twins, the growth of twins, and the variant selection criterion. In this observation, two types of twin systems were activated: {10-12} tension and {11-22} compression twins. Secondary twins were also activated, especially the twin variants with the highest Schmid factors (e.g. higher than 0.4). The growth of the two types of twin is quite different. The {11-22} twin shows Multiple Variants System (MVS) whereas the {10-12} twin shows Predominant Variant System (PVS). The twinning occurs in grains that have particular orientations. Generally, the reorientation induced by the twinning aligns the c-axis of the twinned part to the stable rolling texture orientations, so that no further secondary twinning can be induced. The secondary twinning occurs only when the primary twinning orientates the c-axis of the primary twins far away from the stable orientations. For twinned grains, the lattice rotation of the matrix is similar to that of the grains having a similar crystallographic orientation but without any twin. Two sets of double twins were observed in this study, classified as C-T1 and T1-C double twins respectively. All the variants of C-T1 and T1-C double twins were classified into three groups: A, B and C according to the crystallographic symmetry. The misorientations of theses three groups with respect to the matrix are 41. 1). Le plan de symétrie sera le plan de maclage. Le taux de cisaillement induit est donné par la géométrie du système de maclage (plan et direction cristallographique de maclage) dans la maille cristalline. Étape 4 : on place l'assemblage dans de l'acétone pour dissoudre la colle. On revient à l'étape 0 où l'on fait un EBSD. Et le cycle peut recommencer. Conclusion Générale Le présent travail vise à améliorer la compréhension de la contribution des macles de déformation lors de la déformation plastique d'un alliage de titane T40, ainsi que de l'établissement d'un critère de sélection de variant de macles. À partir des données expérimentales et des études théoriques, les conclusions importantes suivantes peuvent être tirées: Le maclage se produit dans les grains ayant des orientations spécifiques. Généralement, lors du laminage et de la compression plane, le maclage de compression se produit dans les grains qui ont leur axe c parallèle à la force de compression (ou légèrement incliné); le maclage de tension se produit dans les grains avec leur axe c perpendiculaire la force de compression. La partie maclée d'un grain peut être considérée comme un nouveau grain. Lorsque les macles grandissent dans le grain, elles peuvent consommer pratiquement toute la matrice. Dans ce cas, la zone principale maclées est beaucoup plus grande que la matrice restante et représente donc Le maclage affecte l'évolution de la texture par la réorientation de la maille cristallographique de l'ancienne orientation stable du parent vers la nouvelle orientation stable de la macle. Les orientations stables dépendent du mode de déformation, comme le laminage ou la compression plane par exemple. ~ 1 ~ 1 : 11 Chapter Basic understanding and review of the literature on the plastic deformation of hexagonal materials his chapter is devoted to introducing the basic concepts and definitions essential to the understand of the present work topic. It also proposed a review of the literature on plastic deformation mechanism in materials with hexagonal crystal structure, especially in titanium. At first, an introduction of the concepts about the hexagonal close-packed structure and a detailed description about deformation modes, twinning and slipping are proposed. Then we provide a review on some hexagonal metals often studied in the field of research in materials science.T ~ 2 ~ Figure 1 - 1 : 11 Figure 1-1: Atoms in the hexagonal close-packed structure. Figure 1 - 2 : 12 Figure 1-2: Four-axis system in Miller-Bravais indices. Figure 1 - 3 : 13 Figure 1-3: The geometry description of twinning shear. Figure 1 - 5 : 15 Figure 1-5: Schematic description of twinning in Titanium. Figure 1 - 7 : 17 Figure 1-7: The families of slip system operating in hexagonal structures. describe the evolution of texture in titanium alloy Ti-10V-4.5Fe-1.5Al during rolling and annealing. The rolling and re-crystallisation textures obtained in the study are compared with those of other β titanium alloys and bcc metals and alloys such as tantalum and low carbon steel. More recently,Chun et al. in 2005 (Chun, Yu et al. 2005), studied the effect of deformation twinning on microstructure and the evolution of texture during cold rolling. They found that for low to intermediate deformation up to 40% in thickness reduction, the external strain was accommodated by slip and deformation twinning. In this stage, both compressive {11-22} and tensile {10-12} twins, as well as, secondary twins and tertiary twins were activated in the grains of favorable orientation, ~ 14 ~ and this resulted in a heterogeneous microstructure in which grains were refined in local areas. For heavy deformation, between 60 and 90%, slip overrode twinning and shear bands developed. The crystal texture of deformed specimens was weakened by twinning but was strengthened by slip, resulting in a split-basal texture in heavily deformed specimens. Also in 2005, Bozzolo et al.(Bozzolo, Dewobroto et al. 2005) examined the microstructure and texture in titanium alloy rolled to 80% in order to study the grain enlargement and the effects of dynamic recrystallization. They found that Recrystallization of 80% cold-rolled sheets and subsequent grain growth lead to equiaxed microstructures. The texture obtained at the end of primary recrystallization is very close to that of the cold-rolled state, with the maximum value of the orientation distribution function at {0°, 35°, 0°}. The orientations developing during grain growth correspond to a broad peak centered around {0°, 35°, 30°} which is a minor component in the initial texture. The disappearing orientations are widely scattered throughout orientation space and present two major disadvantages in the growth competition. The grain boundaries remaining after extended grain growth are characterized by an increasing proportion of misorientations below 30° and random rotation axes. In 2001, the work on the behavior of titanium before and after Equal channel angular pressing (ECAP) was made. Stolyarov et al.[START_REF] Stolyarov | Microstructures and properties of ultrafine-grained pure titanium processed by equal-channel angular pressing and cold deformation[END_REF]) interested in the microstructure and properties of ultrafine grain structure in pure titanium past ECAP, which improved mechanical properties: grain size obtained by ECAP alone is about 260 nm.The strength of pure Ti was improved from 380 to around 1000 MPa by the two-step process. It is also reported the microstructures, microhardness, ~ 15 ~ tensile properties, and thermal stability of these Ti billets processed by a combination of ECAP and cold deformation. In 2002, Shin et al.[START_REF] Shin | Shear strain accommodation during severe plastic deformation of titanium using equal channel angular pressing[END_REF] studied the mechanisms to accommodate the shear induced by titanium in the ECAE. They observed the activity of twinning system and identify them. TEM analysis of the twins revealed that their twin plane is {10-11} and the twins are accompanied with dislocations on non-basal planes.These results suggest that the severe plastic deformation imposed on titanium via ECA pressing is accommodated mainly by the {10-11} twinning, rather than dislocation slips commonly observed in pressing of other metals such as aluminum and steel. The twinning modes that might accommodate the severe strain were proposed based on the dislocation slip systems observed during the pressing. In 2003, Kim et al. also observed deformation twinning in pure titanium during ECAE by means of EBSD measurements. In 2003, Shin et al. (Shin, Kim et al. 2003) analyzed the microstructure of titanium samples after ECAE. They highlight the twinning systems and ensure slip deformation. Transmission electron microscopy (TEM) revealed that the strain imposed by pressing was accommodated mainly by {10-11} deformation twinning. During the second pass, the deformation mechanism changed to dislocation slip on a system which depended on the specific route. For route C (in which the shear is fully reversed during successive passes by 180° rotation of the sample between passes), prism (a) and pyramidal (c+a) slip occurred within alternating twin bands. For route B (in which the sample is rotated 90° after the first pass), prism a slip was the main deformation mechanism. For route A (in which the billet is not rotated between passes), deformation was controlled by basal a slip and micro-twinning in alternating ~ 16 ~ twin bands. They indicated that the variation in deformation behavior was interpreted in terms of the texture formed during the first pass and the Schmid factors for slip during subsequent deformation. More recently, in 2006, Perlovich et al. (Perlovich, Isaenkova et al. 2006) studied the titanium rods after ECAE at 200°C and 400°C, and they found heterogeneity of deformation, microstructures and textures through the thickness of stem.More generally in 2001, Bache and Evans[START_REF] Bache | Impact of texture on mechanical properties in an advanced titanium alloy[END_REF], proposed a study on the influence of texture on the mechanical properties of a titanium alloy. They had the conclusions that highly textured, uni-directionally rolled Ti 6/4 plate demonstrates significantly different in monotonic strength characteristics according to the direction of the principal stress relative to the predominant basal plane texture. Loading in the transverse orientation, perpendicular to basal planes preferentially lying co-incident with the longitudinal-short transverse plane, promotes a relatively high yield stress and ultimate tensile strength (UTS). Under strain controlled fatigue loading, the longitudinal orientation was found to offer the optimum cyclic response.The differences in mechanical behavior have been related to the ability to induce slip in the various plate orientations. In addition, stress relaxation is encouraged under cyclic loading parallel to the longitudinal direction due to the preferential arrangement of {10-10} prismatic planes. In 2003, Zaefferer(Zaefferer 2003) studied the activity of deformation mechanisms in different titanium alloys and its dependence with regard to their compositions. The main results are: in TiAl6V4, <a> basal slip has a lower critical resolved shear stress,  c , than prismatic slip. <c+a> pyramidal glide shows a very high  c , which is up to two times larger than that for prismatic slip. Nevertheless, ~ 17 ~ <c+a> glide systems were only rarely activated and twinning systems were never activated. Therefore, deformation with c-components may be accommodated by β-phase deformation or grain boundary sliding. The observed c-type texture is due to the strong basal glide. In T40,  c for <c+a> glide is up to 13 times higher than that for prismatic glide. However, <c+a> glide and twinning were strongly activated, leading to the observed t-type texture. In T60, the high oxygen content completely suppressed twinning and strongly reduced <c+a> glide. The less developed t-type texture is due to the combination of <c+a> and basal glide. In 2007, Wu et al.(Wu, Kalidindi et al. 2007) simulate the evolution of texture as well as the behavior of titanium during large plastic deformations. A new crystal plasticity model has been formulated to simulate the anisotropic stress-strain response and texture evolution for α-titanium during large plastic strains at room temperature. The major new features of the model include: (i) incorporation of slip inside twins as a significant contributor to accommodating the overall imposed plastic deformation; and (ii) extension of slip and twin hardening laws to treat separately the hardening behavior of the different slip families (prismatic <a>, basal <a> and pyramidal <c + a>) using hardening parameters that are all coupled to the extent of deformation twinning in the sample. Reasonable agreement between model predictions and experimental measurements has been observed for both the anisotropic stress-strain responses and the evolved deformation textures in three different monotonic deformation paths: (1) simple compression along ND; (2) simple compression along TD; and (3) simple shear in the RD-TD plane. influenced by temperature and grain size. It was demonstrated that the notion of effective diffusivity explained the experimental results. In 2005, Hartig et al.[START_REF] Hartig | Plastic anisotropy and texture evolution of rolled AZ31 magnesium alloys[END_REF] measured the texture of the rolled AZ31 and perform simulations by the crystal plasticity model self-consistent viscoplastic. They considered that anisotropy of plastic flow stresses can be explained by the off-basal character of the texture and the activation of the ~ 25 ~ prismatic slip in addition to the basal, pyramidal slip and the (01-1-2) <011-1> twinning system, discussed the results obtained by comparing the simulation with the experience. Gehrmann et al. (Gehrmann, Frommert et al. 2005) studied the texture on the plastic deformation of magnesium AZ31 in compression flat at 100 and 200 ° C. The measured flow curves and the microstructure investigation reveal that plastic deformation of magnesium at these temperatures is generally inhomogeneous and dominated by the appearance of shear bands. However, if the initial texture is chosen such that the formation of a basal texture is slowed down or even suppressed, substantial ductility can be achieved at temperatures as low as 100 °C. The texture development due to crystallographic slip can be reasonably modelled by a relaxed constraints Taylor simulation and yields information on the activated slip systems. Walde et al. (Walde and Riedel 2005; Walde and Riedel 2007) simulated the texture evolution during rolling of AZ31 with finite element code ABAQUS / Explicit, in which they have implemented a model self-consistent viscoplastic. This model is able to describe the softening behavior of the magnesium alloy AZ31 during hot compression and allows us to simulate the development of the typical basal texture during hot rolling of this alloy. Their simulations are quite consistent with experimental measurements of texture. devoted to introducing the materials, samples preparation, equipment and experimental techniques used in the present work, especially the SEM/EBSD technique and analysis. Then, we provide a detailed description of the interrupted "in situ" experiment arrangement used in rolling and channel die compression tests. Figure2- 1 Figure 2 - 1 : 121 Figure 2-1: Components of an EBSD system. Figure 2 - 2 : 22 Figure 2-2: EBSD geometry. texture measurements on rolled sheet materials. The x axis is parallel to the rolling direction of the sample (RD), the y axis parallel to the transverse direction (TD) and the z axis parallel to the normal direction (ND) (Figure2-3). Figure 2 - 3 :Figure 2 - 232 Figure 2-3: Relationship between crystal and sample coordinate systems. α 1 ,  1 and  1 are the angles between the crystal direction [100] and RD, TD and ND respectively. Figure 2 - 4 : 24 Figure 2-4: Two interpenetrating lattices can be realigned by a single rotation about a common axis [uvw] by an angle . In the figure the axis is the common [111] direction and the rotation angle 60°. Figure 2 - 5 : 25 Figure 2-5: Definition of the Euler angles. Figure 2 - 9 , 29 including mini tensile testing machine installed in the chamber of SEM, a load and displacement recording for controlling the tensile speed, loading, unloading and recording the force and displacement data, an automated EBSD systems to collect Kikuchi patterns, a computer to save Kikuchi patterns collected and index them with special software. Figure 2 - 6 : 26 Figure 2-6: An in-situ EBSD tension test figures and orientation flow fields. Figure 2 - 7 : 27 Figure 2-7: The preparation of samples and arrangement of the rolling and channel die compression tests. ; McDarmaid, Bowen and Partridge (1984);[START_REF] Vedoya | Plastic Anisotropy of Titanium, Zirconium and Zircaloy 4 Thin Sheets[END_REF];Philippe, Serghat, Houtte, and Esling (1995); Panchanadeeswaran, Doherty and Becker (1996); Kalidindi, Bhattacharyya and Doherty (2004); Prasannavenkatesan, Li, Field and Weiland (2005); Merriman, Field and Trivedi (2008); Skrotzki, Toth, Kloden, Brokmeier and Arruffat-Massion (2008); Quey, Piot and Driver (2010)]. Many efforts have been made in studying deformation twinning and gliding in single and polycrystalline metals [Akhtar (1975); Akhtar, Teghtsoonian and Cryst (1975); Kalidindi (1998); Field, True, Lillo and Flinn (2004); Jiang, Jonas, ~ 47 ~ Mishra, Luo, Sachdev and Godet (2007); Jiang and Jonas (2008)]. The mechanical response of titanium, like other HCP metals, is strongly dependent on the combination of active deformation modes: gliding and twinning. The specific deformation mechanisms depend on the c/a ratio, the available deformation modes, the critical resolved shear stress (CRSS) for gliding, the twin activation stress as well as the imposed deformation relative to the crystallographic texture. According to previous work, polished area was measured by SEM/EBSD before and after each deformation step. The rolling and channel die compression layout is illustrated in Fig. 1. Both sheets of the sandwich were firmly stuck together ~ 49 ~ to avoid any surface sliding during the rolling in order to maintain a good surface quality. EBSD measurements were performed with a JEOL-6500F SEM with a step size of 0.4 µm. The evolution of grain orientation during deformation will be presented later both by pole figures and orientation flow fields. (a) reveals a completely recrystallized microstructure with an average grain size of 10 µm. The {0002}-pole figure (PF) in Fig. 2 (b) shows two strong maxima at ±30° tilted from ND towards TD, whereas the { 0 1 10 } PF displays the maximum pole densities parallel to RD. In this work, the setting of the coordinate systems and the Euler angles {1,, 2} are defined according to Bunge's convention, seeBunge, Esling and Muller (1980). Fig. 1 : 1 Fig. 1: The preparation of samples and arrangement of the rolling and channel die compression tests. misorientation between the twin and its matrix corresponds to a 65° rotation about their common < ]) were most frequently observed (Fig 3). The respective amount of these two types of twinning was further presented by means of the misorientation-angle distributions in Fig. 4. It is seen that after 10% rolling (Fig. 4 (b)), 65° misorientation occurs most frequently, suggesting that { 2 2 11 Fig. 2 : 2 Fig. 2: EBSD orientation mapping (a) and {0002}, {10-10} pole figures (b). Fig. 5 : 11 } 11 } 51111 Fig. 5: Rotational flow field in grains showing no twins (a). Lattice rotation field in untwinned part (or matrix) of grains showing twins (b). Fig. 6 : 6 Fig. 6: Initial grain with c axis close to the ND (dark red) (a); 10% deformation: { 2 2 11 } compression twins outlined by red lines (b); 20% deformation: a part of { 2 2 11 } twins undergoes secondary { 2 1 10 } tension twinning, (c). outlined by blue twinning boundaries delimiting the tension twin (in yellow colour). Fig. 7 : 11 } 711 Fig. 7: Misorientation-angle distribution, before compression (a) and after 20 % compression (b). Fig. 9 : 11 } 911 Fig. 9: (0002) Pole Figure delimitating schematically the orientation domains of the c-axes of the grains in which the indicated twinning is expected to be activated (theoretical expectations) Fig. 10 10 Fig. 10 Schematical reorientation of c axes by { 2 1 10 } twinning (blue) and { 2 2 11 } twinning (red arrow). Fig. 11 : 11 Fig.11: Percentage of twinned grains in the three different grain-size groups. twins. Secondary twins, mainly of the tensile type, were also activated. The present work was undertaken to provide information, lacking in the literature, on the lattice rotation and the role of twinning during cold rolling of commercial purity titanium (T40). The method proposed consists of determining the individual rotation of the grains induced by low to intermediate deformation (up to 30% in thickness reduction) and to following the rotation field using electron backscattered diffraction (EBSD) measurements in a high resolution FEG SEM at different steps of deformation(10 and 20 %). We have especially studied the formation, the evolution and the role of mechanical twins. Additional work on ~ 68 ~ identification of slip systems by two different approaches (deformation experiments on single crystals as well as numerical ab initio and molecular dynamics calculations) are being developed in parallel. follow the rotation of the individual grains during the deformation, a 500×300 m 2 area was carefully polished and marked out with four microindentations. The orientation of all the grains in this polished area (about 800 grains) was measured by SEM/EBSD before and after each deformation step. The rolling layout is illustrated in Fig.1. Both sheets of the sandwich were firmly stuck together to avoid any surface sliding during the rolling in ~ 69 ~ order to maintain a good surface quality. EBSD measurements were performed with a JEOL-6500F SEM with a step size of about 0.77 µm. The evolution of grain orientation during deformation will be presented later both by pole figures and lattice rotation fields. Fig. 1 Fig. 2 ( 2 2 11 } 12211 Fig. 1 Successive cold-rolling was performed on sandwich samples. The inner surface was initially polished and the orientation of the grains was measured by EBSD before and after each rolling step. Fig. 4 4 Fig. 4 Misorientation-angle distributions of samples deformed to (a) 0%, (b) 10% and (c) 20% reduction. 2 1 10 } 2 1 10 } 2 2 11 } 210210211 viewpoint of development of crystallographic texture, and presented in the classical representation of pole figures. ~ 73 ~ Orientation analysis indicates that at 10% reduction, the c-axis of the grains with { twins (tension along c-axis) is oriented close to the rolling direction, as shown in Fig. 6 (a); whereas that of the grains with { 2 2 11 } twins (compression along c-axis) is oriented close to the sample normal direction, as shown in Fig. 6 (b). These results are coherent with the theory (Fig. 7part in each grain is systematically oriented close to a stable orientation belonging to the rolling texture component, however, the { 2 2 11 } twinning leads the c-axis of the twinned part oriented close to the rolling direction i.e. to an unstable orientation. As shown in Fig. 8, the { twinning gives an 84.78°misorientation of c-axis and illustrated with blue arrow in this figure. Likewise, the { twinning gives a 64.62°misorientation of c-axis and shown with red arrow. Green area delimits the stable orientation belongs to the rolling texture (characterized by c axes tilted 30 from ND to TD). Form the figure, we can see clearly that the { 2 1 10 } twinning transfer the Fig. 5 (Fig. 7 ( 57 Fig. 5 (a) Lattice rotation field in grains showing no twins. (b) Lattice rotation field in untwinned part or matrix of grains showing twins. Fig. 6 ( 2 1 10 } 2 2 11 } 6210211 Fig. 6 (a) {0002} -Pole figure of grains having { 2 1 10 } twin and (b) {0002} Pole Figure of grains having { 2 2 11 } -Twin (10% deformation). Fig. 8 2 2 11 } 2 2 11 } 2 1 10 } 2 2 11 } 8211211210211 Fig. 8 Misorientation of c axes caused by { 2 1 10 } twinning (blue arrow) and { 2 2 11 } twinning (red arrow). Fig. 12 12 Fig. 12 Effect of the neighboring grains on the lattice rotation. (a) Misorientation of the c-axes of neighboring grains (numbers inside the neighboring grains) with respect to the large grain they surround and reorientation of the c-axis in the regions A, B…in the neighborhood of the grain boundary (color code in the inset). (b) Illustration of the reorientation of the c axes in the various boundary regions A, B…of the large green grain. 2 1 10 } 210 This tends to show that the main effect of twinning is to create a newly oriented zone (a new grain) and does not induce additional deformation mechanisms in the remaining matrix part of the grain. In other words, the deformation mechanisms in the matrix part of the twinned grains remain the same as those in the untwinned grains. The reorientation induced tends to orientate the c-axes close to stable orientations. Thus, there is no tendency for secondary twinning to occur within such twins.The secondary twinning only takes place in the { 2 2 11 } compression twins whose c-axes are orientated far away from the stable texture orientation.In such a case, the new and major twins appearing inside the { twins. This explains why when deformation is increased from 10% to 20% the amount of { tension twin drastically increases. Chapter 4 : 4 reorientation of the c-axis of the secondary twin to a stable orientation. Only a little amount of second order twinning could be observed and twinning of higher than second order was never found.~ 81 ~The rotation of the matrix-part of the grains having twins is similar to that of the non-twinned grains with similar orientation. The twinned part of a grain can be considered as a new grain. When twins grow within the grain, they can consume almost the whole matrix. Special attentions should be paid when determining the twinned volume fraction. Thanks to the EBSD measurement, a strong increase of the twinned volume could be demonstrated. This contradicts the conventional judgement that the twinned part is always the smaller part in a twinned grain, as concluded by optical microscopy. Only step by step EBSD orientation mapping allows an unambiguous determination of the twinned volume fraction. The confirmation that the order of twinning never exceeds the second order is very useful for the modelling of plasticity in polycrystalline metals, such that the order allowed for twins should be restricted to only the first and second order (also called double twins).~ 83 ~ Variant selection in primary twins, secondary twins and double twins n this chapter, with the purpose of studying the mechanisms governing the selection of specific twin variants, we performed some calculations of Schmid factor SF, deformation energy factor and some geometrical factors in primary and secondary twins. We came to the conclusion that the Schmid and energy factor plays an important role in variant selection during primary and secondary twinning. It is suggested that rank (or level) in the selection of twin variants. The development of an efficient activation criterion is essential when employed to model the deformation behavior of ~ 85 ~ materials with twinning as their important deformation mechanism, such as HCP metals. Figure 1 1 Figure1shows the {0002} and {10-10} pole figures (PFs) of the as-annealed Ti40 sheet. It is seen that the initial texture is characterized by two strong maxima at ±30° tilted from ND towards TD in the {0002} PF and the Figure. 1 : 1 Figure. 1: The initial texture presented in the pole figures of {0002} and {10-10}. Figure 2 ( 2 b) and (d) indicate the SF for each twin variant (active variants are highlighted in red). It appears that the growth of {11-22} compression twin variants and {10-12} tension twin variants exhibit different features. After {11-22} compression twin lamellae form, they grow and expand rapidly along the shear direction. Subsequently, the rapid expansion causes the growing variants to intersect with each other. The collision between lamellae of different variants blocks the respective growth of one another. The subsequent deformation progressed by the formation of new twin lamellas of the same variants (or repeated twin nucleation) in the un-deformed matrix contoured by the already formed twin lamellae with the progress of the deformation (Figure 2(a)). ~ 88 ~ However, for the {10-12} tension twin, repeated twin nucleation occurs at the very beginning of the deformation. The twin lamellae also rapidly grow in the shear direction until they intersect the grains boundaries. Different from the compression twinning, it accommodates the subsequent deformation mainly by thickening the already formed twin lamellae, which transforming the whole initial grain into twin, as shown in Figure 2 (c). Discussion In this study, {11-22} compression twinning and {10-12} tension twinning show quite different twin variant selection and growth characters. {11-22} twin always exhibits three or four variants in one grain and these variants possess large misorientation with respect to each other. We call this multiply variants system (MVS), as shown in Figure2(a). However, {10-12} twin exhibit only one or two variants. We call it predominant variant system (PVS), as shown in Figure2 (c). After growth, they consume almost the whole volume of the initial grain. Figure 2 2 Figure 2 (a) and (c) show EBSD maps of two grains at the deformation steps of initial, 8%, 16% , (a) with {11-22} compression twin variants and (c) with {10-12} tension twin variants. (b) and (d) indicate the Schmid factors for each twin variant (active variants are highlighted in red). Figure 3 Figure 4 , 34 Figure4, the Schmid factors of all the active variants SF a lie in the range from 0.32 to 0.5 (X axis in Figure4(a)) and the NSF that is equal to one has the highest occurrence. The average Schmid Factor of all active twins (SF a ) Figure 3 3 Figure 3 Schematic presentation of twin variants of {11-22} twin (a) and {10-12} twin (b). Figure 4 4 Figure 4 Distribution of NSFs (a) and their frequency (b). , when the c-axis of the initial grain is close to the compression load, the SFs of the six variants are close and high, showing MVS. 3. In contrast, {11-22} twin always exhibit only one variant per grain. This predominant twin variant grows fast until almost no matrix is left, showing PVS. and {11-21} systems are referred to as tension twin ({11-22} as a compression twin) because it is only activated under tension (compression) load along the c axis of the parent crystal[Rosi et al. (1956)]. Once a twin forms, the crystal lattice of the twin has a misorientation with respect to the former parent crystal lattice, misorientation which is represented by a rotation depending on the twin systemThis orientation change accelerates the occurrence of the secondary twinning inside the primary twin due to a more favorable orientation. The combination of primary and secondary twins is called double twin. In other words, a double twin can be regarded as the structure of a secondary twin inside a pre-existing primary twin[Barnet et al. (2008);[START_REF] Capolungo | Variant selection during secondary twinning in Mg-3%Al[END_REF]]. In titanium, the double twins combined of the {10-12} tension twin and the {11-22} compression twin are commonly observed in rolling and compression deformation[Bao et al (2010a)]. At present, it is well established that the variant selection in the primary twin generally follow Schmid law[START_REF] Lebensohn | A study of the stress state associated with twin nucleation and propagation in anisotropic materials[END_REF]], Fig 1 : 1 Fig 1: Schematic description of the channel-die set-up used. Fig. 2 (b) exhibits the same area selected as shown in Fig. 2 (a), but after 8% reduction. The boundaries of {10-12} tension twins were marked in blue and the boundaries of {11-22} compression twins in red. Two types of primary twins and two types of double twins were spotted: the {10-12} tension twin (T1 twin) and the {11-22} compression primary twin (C twin), Fig. 2 . 2 Fig. 2. (a). OIM of the initial microstructure of the T40 titanium. (b). OIM of the microstructure after 8% deformation. Red line represents {11-22} twin boundaries and blue line represents {10-12} twin boundaries. (c). {0002} and {10-10} pole figures of the initial texture. figure with a symbol and a color code corresponding to every set of double twin (Fig.3). Note that this is a pole figure referenced in a XYZ coordinate frame, i.e. it is not used to describe an orientation with respect to the sample frame, but to represent the misorientation with respect to the parent grain. This is a direct and effective way to study the texture Fig. 3 . 3 Fig. 3. The {0002} pole figure of all possible double twin variants. The color code relates the secondary twin variants (six symbols) to their respective primary twin variants (filled circle). Each primary twin variant and the six possible secondary twin variants inside are presented with a same color. Fig. 4 . 4 Fig. 4. Frequency of occurrence of double twin variant group A, B and C. ( SF a ) divided by the highest SF (SF h ) of the six possible variants. If NSF=1, it means that the active twin variant is the one with the highest SF; if NSF<1, another twin variant is activated instead of the one with the highest SF. The frequency of the NSF was displayed in Fig.5 (b). The interest of NSF is that it provides the information to what extent the SF of the active variants deviate from the highest SF in the case of an active variant with non-highest SF. As shown in Fig.5, it can be seen that Schmid's law gives a 50% accuracy in predicting variants selection of the primary twinning. The majority of the primary twins form on the variants with the first or second rank of SF.~ 107 ~Secondary twinningThe SFs of the secondary twins in the C-T1 and T1-C double twin were examined separately and plotted with different color in Fig.6. Fig.6(a) exhibited the frequency of each SF rank in the form of histograms and Fig.6(b) showed the frequency of NSF associated with variants of secondary twins. It is seen that the accuracy of SF in predicting variants selection of the secondary twinning declines to 40%. Note that the variants having the second rank of SF in the C-T1 secondary twins still have a very high proportion (see red bar in Fig.6 (a)). Fig. 5 . 5 Fig. 5. (a) Frequency of SF ranks (here "1" is the highest and "6" the lowest) corresponding to the active variants of primary twin and (b Frequency of NSF associated with variants of primary twins. Fig. 6 . 6 Fig. 6. (a) Frequency of SF ranks (here "1" is the highest and "6" the lowest) corresponding to the active variants of secondary twin in C-T1 (red) and T1-C (blue) double twin and (b) is the frequency of NSF associated with variants of secondary twins L 33  33 . Clearly, the length (L) of the free path of a twin lamella in a grain can be visualized with its boundary traces on the sample observation plane. The maximum longitudinal length of the twin lamella appearing on the sample observation plane is determined as L for each twin variant. In the present work 8 . 8 The poles corresponding to the secondary twin elements were represented in stereographic projection by symbols defined in the figure caption of Fig.8 . In each variant group there are two geometrically equivalent variants, one was represented by a filled symbol and the other an open symbol. Martin et al. [Martin et al. (2010)] suggested Fig. 8 : 8 Fig. 8: Stereographic plots of the twin plane normal, shear direction and shear plane normal for group A, B and C in the C-T1 and T1-C double twins are represented with different symbols. Fig. 9 . 4 . 94 Fig. 9. Frequency of the angles between the twin planes on each side of the boundary. Fig. 11 . 11 Fig. 11. The {0002} and {10-10} pole figures measured at (a) initial, (b) 8%, (c) 16%, (d) 24% and (e) 35% deformation degree. 5 .Chapter 5 : 55 All possible misorientations corresponding to double twin combinations of {10-12}, {11-21} and {11-22} were calculated with respect to the sample coordinate frame. This leads to ~ 124 ~ the crystallographic characterization of each double twin variant. The main conclusion can be summarized as follows:1. The twinning order does affect the resulting misorientation induced by double twinning, even though they are identical in misorientation angle and have symmetrically equivalent axes. All the double twin variants are classified into 15 orientation variant groups rather than 10 geometric variants groups.2. Strong variant selection takes place in the C-T1 and the T1-C double twinning. The Group B is predominant with respect to the two others in the C-T1 double twin (78.9%). In the case of the T1-C double twin, the predominant variant group is the Group C (66.7%).3. Schmid factor analysis was performed on the primary twin and secondary twin respectively. SF gave an accuracy of about 50% for predicting the primary twin variant selection and about 40% for the secondary twin variant selection. The relative inaccuracy is probably due to the deviation between the local stress tensor and macroscopic applied stress tensor. 4. A new calculation associated with deformation energy was described to assess the influence of deformation energy on the variant selection. It gave an excellent accuracy up to 85% for predicting the primary twin variant selection and about 95% for the secondary twin variant selection. The twins formed will induce in the adjacent grains variants to be activated across the boundary as long as the angles between the twin ~ 125 ~ planes are not beyond 20°. No influence of geometric features was found on the variant selection in the double twinning. 6. It is suggested that deformation energy rank of each variant should be adopted as a main criterion in predicting variant selection, and the Schmid factor used as an additional second criterion. 7. Twinning affects the texture evolution by the reorientation of the crystallographic lattice from the former stable orientation of the parent grain into the new stable orientation of the twin. The stable orientations depend on the deformation mode, like rolling or channel die compression for example. ~ 127 ~ Conclusions and prospects his chapter lists the main conclusions obtained in the present work and some suggestions for future work based on the findings, conclusions and problems identified in the course of the present work. The present work attempts to improve the understanding of the contribution of deformation twinning to the plastic deformation of hexagonal T40 titanium alloy, as well as establishing a criterion for the twin variants selection. From the experimental data and theoretical investigations, the following important conclusions can be drawn: Twinning occurs in grains having specific orientations. Generally, in rolling and compression, compression twinning occurs in the grains with their c axis close to the compressive force; tension twinning occurs in the grains with their c axis perpendicular to the compressive force. The twinned part of a grain can be considered as a new grain. When twins grow within the grain, they can consume almost the whole matrix. In this case the primary twinned area is much larger than the remaining matrix and thus represents the "new grain" for possible subsequent secondary twinning. Special attention should be paid when determining the twinned volume fraction. With the EBSD technique, a large twinned volume fraction could be demonstrated. This contradicts the conventional judgement that the twinned part is always smaller in a twinned grain than the remaining part of the parent grain, as was generally concluded from optical microscopy. Only step by step EBSD orientation mapping allows an unambiguous determination of the twinned volume fraction. In rolling and compression, it appears that the growth of {10-12} tension twin variants and {11-22} compression twin variants exhibit different characteristics. {10-12} tension twins usually exhibit only one variant per ~ 129 ~ grain. Even if another variant was activated, it would be rapidly absorbed by the first variant. This predominant twin variant grows very fast until almost no original parent is left, showing predominant variant system (PVS). This is because the {10-12} tension twins form under the resolved compressive force that is almost perpendicular to the c axis of the parent grain. In this situation, only two variants have a high Schmid factor and the misorientation between the two variants is only 9.98°. Hence, in a given grain, {10-12} twin lamellae of these two variants can easily merge into one another. In contrast, it is easy for the {11-22} compression twin to activate more than one variant, and they collide with each other and block growing, showing multiply variants system (MVS). This is because that as a compression twin, {11-22} twin forms under the resolved compressive force that is parallel to the c axis of the matrix. Since all the six {11-22} twin variants have symmetrical orientation relations by a 60° rotation around the c axis, the six variants should have the same Schmid factor under a compressive force parallel to the c axis. In our experimental observations, for the grains with {11-22} compression twins, the applied compressive force always deviates to some extent from the c axes, therefore usually three or four variants have a relatively high Schmid factors and these variants are activated simultaneously. PVS is inclined to occur in elongated grains and MVS is inclined to occur in equiaxed grains. The order of the twinning does affect the resulting misorientation induced by double twinning, even though they are identical in misorientation angle and have symmetrically equivalent axes. The set of all the double twin ~ 130 ~ variants are classified into 15 orientation variant groups rather than 10 geometric variants groups.In this study, two sets of double twins were observed, C-T1 double twins and T1-C double twins respectively. All the variants of these two sets of double twins are classified into 3 groups by symmetrically equivalent rotation with respect to the parent crystal, namely Group A (41.3° about <1-543>), Group B (48.4° about <5-503>) and Group C (87.9° about <4-730>). Strong variant selection takes place in these two double twinning systems. The Group B is predominant with respect to the two others in the C-T1 double twin (78.9%). In the case of the T1-C double twin, the predominant variant group is the Group C (66.7%).Schmid Factor (SF) analysis was performed on the primary twin and secondary twin respectively. SF gave an accuracy of about 50% for predicting the primary twin variant selection and about 40% for the secondary twin variant selection. The relative inaccuracy is probably due to the deviation between the local stress tensor and the macroscopic applied stress tensor. A new calculation associated with deformation energy was described to assess the influence of the deformation energy on the variant selection. It gave an improved accuracy up to 85% for predicting the primary twin variant selection and about 95% for the secondary twin variant selection.The twins formed will induce in the adjacent grains variants to be activated across the boundary as long as the angles between the twin planes are not beyond 20°. No influence of geometric features was found on the variant ~ 131 ~ selection in the double twinning. It is suggested that deformation energy rank of each variant should be adopted as a main criterion in predicting variant selection, and the Schmid factor used as an additional second criterion. 2 . 3 . 23 due to the poor EBSD index ratio after larger plastic deformation. A ~ 132 ~ solution for this problem would be quite useful in the view of future work. The in situ deformation (tensile and shear tests) experiments in the chamber of SEM/EBSD with microgrids deposited on the sample surface would be another interesting approach to study the local strain (thus stress) distribution in polycrystalline titanium during deformation. Further studies could be concentrated on dislocations and slips, with TEM technique. Additional detailed investigation on deformation mechanisms could be performed using Burger vector identification methods in the TEM. Résumé Le titane et ses alliages sont largement utilisés dans les domaines aéronautique, spatial, de l'armement, du génie civil, dans des applications commerciales et biomédicales en raison de sa résistance à la rupture élevée, d'une bonne ductilité et d'une grande biocompatibilité. Les mécanismes de la déformation plastique du titane ont été étudiés en détail par le passé, particulièrement sur l'étude de la déformation par maclage car il a une grande influence sur les propriétés mécaniques. Une méthode d'essais "in situ" en EBSD basée sur des tôles polies et colées ensemble a été développée dans cette étude et utilisée en laminage et en compression plane. Avec cette méthode, des mesures EBSD sont effectuées à chaque étape de la déformation dans la même zone comprenant un grand nombre de grains. Par conséquent, l'information sur l'orientation de ces grains à chaque l'étape de la déformation est mesurées. ayant des facteurs Schmid supérieurs à 0.4 ont une bonne chance d'être actifs. Les comportements des deux types de maclage sont complètement différents. Dans la déformation en compression, les macles (11-22) montrent le comportement de type multiplication des variants (Multiply Variants System: MVS) alors que les macles (10-12) montrent le type de maclage prédominant (Predominant Twin System: PTS). Cette étude présente deux types de macles doubles dénommées C-T1 (= macle primaire de Compression et macle secondaire de Tension) et T1-C (= macle primaire de Tension et macle secondaire de Compression). Tous les variants sont classés seulement en trois groupes: A, B et C par symétrie cristallographique. Les désorientations de ces 3 groupes par rapport à l'orientation de la matrice sont respectivement de 41.34°, 48.44° et 87.85°.Une forte de sélection de variant se déroule dans le maclage double. Pour les macles doubles CT, 78.9% des variantes appartiennent à la B et pour T1-C, 66.7% des variantes appartiennent à C. Le facteur de Schmid joue un rôle prépondérant dans la sélection des variants des macles doubles. Les caractéristiques géométriques, associant " volumes communs " et l'accommodation de la déformation ne contribuent pas de manière significative à la sélection des variants. Summary Titanium and its alloys are widely used in aviation, space, military, construction and biomedical industry because of the high fracture strength, high ductility and good biocompatibility. The mechanisms of plastic deformation in titanium have been studied in detail, especially deformation twinning since it has a great influence on the ductility and fracture strength. Figure 1 Figure 2 2 . 1 . 3 . 2 . 122132 Figure 1 Illustration du maclage Figure 3 : 3 Figure 3: Synoptique d'un essai interrompu : Étape 0 après polissage, on le ''nouveau grain'' pour d'éventuels maclages secondaires ultérieurs. Une attention particulière doit être accordée lors de la détermination de la fraction volumique maclée. Avec la technique EBSD, une fraction volumique maclée plus grande peut être trouvée. Ceci contredit l'opinion selon 20 laquelle la partie maclée est toujours plus petite dans un grain maclé que dans la partie restante du grain parent, comme cela a été généralement conclu par microscopie optique. Seule la technique EBSD en suivant étape par étape l'évolution de la cartographie d'orientation permet une détermination sans équivoque de la fraction volumique maclée. Lors du laminage et de la compression plane, il apparaît que la croissance des variants de macles de tension {10-12} (notées T1) et des variants de macles de compression {11-22} (notées C) présentent des caractéristiques différentes. Les macles de tension {10-12} présentent généralement un seul variant par grain. Même si un autre variant a été activé, il sera rapidement absorbé par le premier variant. Ce variant de macle prédominant croit très vite jusqu'à ce qu'il ne reste plus de matrice ; ceci montre le système de variant prédominant (PVS). C'est parce que les macles de tensions {10-12} se sont formées avec une force de compression parallèle à DN alors que les axes c du grain parent sont dans DT (perpendiculaire à DN). Dans cette situation, seuls deux variants ont un haut facteur de Schmid et la désorientation entre les deux variants n'est que de 9,98 °. Ainsi, dans un grain donné, ces deux variants de macles {10-12} peuvent facilement se fondre l'un avec l'autre. En revanche, il est facile pour les macles de compression {11-22} d'activer plus d'un variant. Ceux-ci entrent en contact les uns avec les autres et se bloquent. C'est le système de variants multiples (MVS). C'est parce que les macles de compression {11-22} sont formées avec une force de compression parallèle à DN qui est parallèle à l'axe c de la matrice. Comme tous les six variants de macle {11-22} ont des relations d'orientation symétrique par une rotation de 60 ° autour de l'axe c. Les six variants devraient avoir le même facteur Schmid sous une force de compression parallèle à l'axe c. Dans nos observations expérimentales, pour les grains ayant des macles de compression {11-22}, la force de compression appliquée dévie toujours dans une certaine mesure des axes c des grains, donc seulement trois ou quatre variants auront des facteurs Schmid relativement élevés. Ces variants seront activés simultanément. PVS est enclin à se produire dans les grains allongés et MVS est enclin à se produire dans les grains équiaxes. L'ordre d'apparition du maclage affecte la désorientation résultante induite par le maclage double (maclage secondaire), même si elles sont identiques en angle de désorientation et ont des axes symétriquement équivalents.L'ensemble de tous les variants de macles secondaires sont classés en 15 groupes de variant d'orientation plutôt qu'en 10 groupes de variants de 85% pour prédire la sélection de variant de macle primaire et 95% environ pour la sélection du variant de macle secondaire.Les variants de macle formées dans un grain pourront induire dans les grains adjacents des variants qui vont être activés à travers le joint de grain tant que les angles entre les plans de macle ne sont pas au-delà de 20 °. Il n'a pas été trouvé d'influence des caractéristiques géométriques sur la sélection de variant dans le maclage double. Il est suggéré de prévoir que le calcul de l'énergie de déformation de chaque variant doit être adopté comme principal critère de sélection de variant, et que le facteur de Schmid soit utilisé comme un critère supplémentaire secondaire. Table 1 - 1 : 11 Types of twinning in hexagonal metals Metals c/a Types of twinning Cd 1.886 {10-12} Zn 1.586 {10-12} Mg 1.624 {10-12}, {11-21} Zr 1.593 {10-12}, {11-21}, {11-22} Ti 1.588 {10-12}, {11-21}, {11-22} Table 1 - 1 Slip Systems Slip plane and direction Basal <a> {0001}<11-20> Prismatic <a> {10-10}<1-210> Pyramidal <a> {10-11}<1-210> Pyramidal <c+a> {10-11}<11-2-3> Pyramidal <c+a> {2-1-12}<-2113> 3: Slip systems in h.c.p. metals Table Ⅱ Ⅱ -1: Chemical composition of commercially pure titanium T40 Elements H C N O Fe Ti Composition ppm (wt.) 3 52 41 1062 237 Balance Table 1 1 Chemical composition of commercially pure titanium T40 Element H C N O Fe Ti Composition (ppm (wt.)) 3 52 41 1062 237 Balance Table 1 1 Chemical composition of commercially pure titanium T40 Element H C N O Fe Ti Composition (ppm (wt.)) 3 52 41 1062 237 Balance Table 1 1 Chemical composition of commercially pure titanium T40 Element H C N O Fe Ti Composition (ppm (wt.)) 3 52 41 1062 237 Balance Table 3 3 : Angles between twinning elements associated with variants groups of double twins C-T1 double twins Angle between Angle between Angle between Variant groups primary and primary and primary and secondary twin's TP secondary twin's SD secondary SPN A 84.1° 103.5° 30° B 27.4° 23.7° 30° C 66.9° 123.6 90° T1-C double twins Angle between Angle between Angle between Variant groups primary and primary and primary and secondary twin's TP secondary twin's SD secondary SPN A 84.1° 76.5° 30° B 27.4° 23.6° 30° C 66.9° 55.3° 90° Acknowledgements This work was carried on at Laboratoire d'Etude des Textures et Application aux Materiaux (LETAM CNRS FRE 3143), University of Metz in France. First of all, I Acknowledgement This work was supported by the Federation of Research for Aeronautic and Space (Fédération de Recherche pour l'Aéronautique et l'Espace Thème Matériaux pour l'Aéronautique et l'Espace : project OPTIMIST (optimisation de la mise en forme d'alliage de titane). Acknowledgement This study was supported by the Federation of Research for Aeronautic and Space (Fédération de Recherche pour l'Aéronautique et l'Espace Thème Matériaux pour l'Aéronautique et l'Espace : project OPTIMIST (optimisation de la mise en forme d'alliage de titane). THESE

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2011

https://hal.univ-lorraine.fr/tel-01749173/file/Li.Zongbin.SMZ1146.pdf

Keywords: Ferromagnetic shape memory alloys, Ni-Mn-Ga, Microsructure, Twinning, Martensitic transformation, EBSD Since the large magnetic-field induced strain in Ni-Mn-Ga ferromagnetic shape memory alloys was reached through the reorientation of martensite variants, the microstructural configuration and crystallographic correlation of constituent variants have strong influence on the activation of the magnetic shape memory effect. In this work, the microstructural and crystallographic features of Ni-Mn-Ga alloys were thoroughly studied. For the 5M martensite in a Ni 50 Mn 28 Ga 22 alloy, the modulated superstructure information was applied for the first time to perform EBSD measurements, which enables a deeper insight into the microstructural characters. Consequently, four types of twin-related variants (A, B, C and D) were unambiguously revealed, while only two variants can be identified if using the simplified tetragonal non-modulated crystal structure for the Kikuchi pattern indexation. Based on the exact local orientations of martensite variants and crystallographic calculations, the twinning elements and the twin interface planes were fully determined. Furthermore, with no residual austenite, the most favorable orientation relationship governing martensitic transformation was revealed to be the Pitsch relation, i.e. (101) A //(1 2 5 ) 5M and [10 1 ] A //[ 5 5 1] 5M . The auto orientation mappings on 7M martensite in a Ni 50 Mn 30 Ga 20 alloy were realized by using the incommensurate superstructure information, which provided an alternative means for verifying the crystal structure information. Four types of alternately distributed martensite variants (A, B, C, and D) in one martensite colony were determined to be twin-related: A and C (or B and D) possess type I twin relation, A and B (or C and D) type II twin, and A and D (or B and C) compound twin. All the twin interfaces are in coincidence with the respective twinning plane (K 1 ). The energetically favorable orientation relationship between the austenite and the 7M martensite was revealed to be the Pitsch relation with (101) A //(1 2 10 ) 7M and [10 1 ] A //[10 10 1] 7M . Notably, the ambiguity of the geometrically favorable orientation relationships was resolved by examining the lattice discontinuity caused by the phase transformation and the structural modulation. In a Ni 54 Mn 24 Ga 22 alloy with NM martensite, four types of plates are identified locally and each plate consists of nano-scaled paired fine variants. Totally, eight orientation variants are found in one martensite colony. The paired fine variants in each plate were found to be compound twin related with the {112} Tet as the twinning plane and the <11 1 > Tet the twinning direction. The inter-plate interfaces are close to {1 1 2} Tet plane but with ~3° deviation, while the interfaces of two paired fine variants are in good agreement with {112} Tet twinning plane. In a Ni 53 Mn 22 Ga 25 alloy with transformation temperatures around room temperature, the microstructure evolution in continuous austenite-7M-NM transformation was observed for the first time. The initially formed self-accommodated 7M martensite was characterized by a diamond-shaped morphology composed of four variants. 7M martensite was revealed to be an intermediate phase and thermodynamically metastable and it possesses an independent crystal structure rather than nanotwinned combination of NM martensite. The transformation from the 7M to the NM martensite is realized by lattice distortion following the (001) 7M //(112) Tet and [100] 7M //[11 1 ] Tet relation, which is accompanied by the thickening of the 7M plates. The role of 7M martensite in bridging the austenite to NM martensite transformation is to relieve the large lattice mismatch between austenite and NM martensite and to avoid the formation of the incoherent NM plate interfaces that represent insurmountable energy barrier. Résumé Depuis qu'une grande déformation induite par le champ magnétique dans les alliages à mémoire de forme ferromagnétiques Ni-Mn-Ga a été atteinte grâce à la réorientation des variantes de martensite, la configuration de la microstructure et la corrélation cristallographique des variantes constitutives ont une forte influence sur l'activation de l'effet de mémoire de forme magnétique. Dans ce travail, les caractéristiques microstructurales et cristallographiques des alliages Ni-Mn-Ga ont été soigneusement étudiées. Pour la martensite 5M dans un alliage de Ni 50 Mn 28 Ga 22 , l'information de la superstructure modulée a été appliquée pour la première fois à la réalisation de mesures EBSD, ce qui permet une compréhension plus profonde des caractères microstructuraux. En conséquence, quatre types de variantes (A, B, C et D) en relation de macle ont été clairement révélés ; en revanche, seulement deux variantes peuvent être identifiées en cas d'utilisation de la structure cristalline tétragonale simplifiée non modulée pour l'indexation des clichés de Kikuchi. Basé sur les orientations locales exactes des variantes de martensite et sur des calculs cristallographiques, les éléments de maclage et les interfaces des macles ont été entièrement déterminés. Par ailleurs, sans austénite résiduelle, la relation d'orientation la plus favorable de transformation martensitique a été révélée être la relation Pitsch, c'est-à-dire (101) A //(1 2 5 ) 5M et [10 1 ] A //[ 5 5 1] 5M . Des cartes d'orientation sur la martensite 7M dans un alliage Ni 50 Mn 30 Ga 20 ont été réalisées automatiquement en utilisant l'information de superstructure cristalline incommensurable, qui ont fourni un moyen alternatif pour la vérification de l'information sur la structure cristalline. Quatre types de variantes de martensite (A, B, C et D) distribués alternativement dans une colonie de martensite ont été déterminés être en relation de maclage. A et C (ou B et D) est une macle de type I, A et B (ou C et D) une macle de type II, et A et D (ou B et C) une macle de type composé. Toutes les interfaces entre les variantes sont respectivement en coïncidence avec le plan de maclage (K 1 ). La relation d'orientation énergétiquement favorable entre l'austénite et la martensite 7M s'est révélée être la relation de Pitsch (101) A //(1 2 10 ) 7M and [10 1 ] A //[10 10 1] 7M . Notamment, l'ambiguïté de la relation d'orientation géométriquement favorable a été résolue par l'examen de la discontinuité de la maille causée par la transformation de phase et la modulation de la structure de martensite. Dans un alliage Ni 54 Mn 24 Ga 22 avec la martensite non modulée (NM), quatre types de platelets sont identifiés localement et chaque platelet est constitué de variantes fines nanométriques jumelées. En tout, huit variantes d'orientation sont trouvées dans une colonie de martensite. Les variantes fines jumelées dans chaque platelet ont été trouvées être en relation de maclage composé avec {112} Tet comme plan de maclage et <11 1 > Tet comme direction de maclage. Les interfaces inter-platelets sont proches de {1 1 2} Tet mais avec ~ 3° de l'écart, tandis que les interfaces des deux variantes fines jumelées sont en bon accord avec le plan {112} Tet , i.e. le plan de maclage. Dans un alliage Ni 53 Mn 22 Ga 25 avec les températures de transformation autour de la température ambiante, l'évolution de la microstructure dans la transformation continue austénite-7M-NM a été observée pour la première fois. La martensite 7M auto-accommodée qui se forme initialement a été caractérisée par une morphologie en forme de losange composée de quatre variantes. La martensite 7M a été révélée comme étant une phase intermédiaire et thermodynamiquement métastable. La martensite 7M possède une structure cristalline indépendante plutôt que d'être la combinaison de martensite NM nanométrique en relation de macle. La transformation de la martensite 7M en celle NM est réalisée par la distorsion du réseau suite à la relation de (001) 7M // (112) Tet et [100] 7M //[11 1 ] Tet , qui est accompagnée par l'épaississement des platelets de 7M. Le rôle de la martensite 7M en établissant un pont lors de la transformation martensitique de l'austénite à la martensite NM est de relaxer le grand désaccord des mailles entre l'austénite et la martensite NM et d'éviter la formation d'interfaces incohérentes de platelets de martensite NM qui représentent une barrière énergétique infranchissable. Mots-clés: alliages à mémoire de forme ferromagnétiques; Ni-Mn-Ga; microstructure; maclage; transformation martensitique; EBSD. Ni-Mn-Ga Ni-Mn-Ga Ni-Mn-Ga EBSD Ni 50 Mn 28 Ga 22 A, B, C, D EBSD Pitsch (101) A //(1 2 5 ) 5M [10 1 ] A //[ 5 5 (001) 7M // (112) Tet [100] 7M //[ 11 1 ] Tet Ni-Mn-Ga Chapter 1 Literature Review General introduction In human history, the development of science and technology has always been marked by the innovations and the evolutions in the use of materials. New advanced materials are always the stepping stones to human progress and have brought great convenience to the daily life of people. Among the numerous advanced materials, the so-called smart materials have the unique attractiveness due to their special properties to sense and respond to the environment around them in a predictable manner. They can allow their shape to be changed in different external field, like electric, magnetic or temperature field, transforming one form of energy into another. Hence, they have wide applications as actuators or sensors in various domains [1][2][3][4], such as medical, civil engineering, aerospace and marine industries. Piezoelectric ceramics, magnetostrictive materials and shape memory alloys are the main smart materials groups that have been utilized for practical applications. Piezoelectric ceramics are the most widely used smart materials. They present two distinct abilities due to the electro-mechanical coupling [5], i.e. becoming electrically charged when subjected to a mechanical stress and making an expansion or contraction when voltage is applied. Piezoelectric materials have high response frequency, on the order of tens of 100 kHz, making them ideal for precise and high speed actuation. The most common piezoelectric material used in these days is lead-zirconate-titanate (PZT) [6]. Nevertheless, piezoelectric ceramics have some limitations. They tend to be brittle and the induced strain in these materials is relatively small. The best piezoelectric ceramics only exhibit a strain of about 0.19% [5]. Magnetostrictive materials such as Terfenol-D are similar to piezoelectric ceramics, but these materials respond to magnetic field rather than electric field. When a magnetic field is applied, the electron spin tries to align in the field direction and the orbit of that electron also tends to be reoriented, which results in considerable lattice distortion, hence magnetostriction. Magnetostrictive materials can operate at high frequencies up to 10 kHz, but they also have the same drawbacks of small output strain. The leading magnetostrictive material (Tb 0.3 Dy 0.7 )Fe 2 , shows a field-induced strain of about 0.24%. Moreover, they are also expensive to produce and highly brittle. Conventional shape memory alloys (SMAs) are another class of smart materials that can produce very high recoverable strains as a result of reversible martensitic transformation. The most common commercially available shape memory alloy is Ni-Ti. Ni-Ti alloys can produce strains up to 8% [2] and also can be deformed easily and biocompatible. When these shape memory alloys are cooled to martensite, the martensite phase has a multi-variant structure and, in general, almost no noticeable shape change is observed between the high-temperature and low-temperature phases owing to the small difference between the volumes of the unit cells and to the self-accommodation of martensite variants. Application of a uniaxial stress breaks this arrangement and the twin boundaries between martensite variants are activated to move. Those variants with favorable orientation with respect to the stress grow at the expense of other variants. This results in a macroscopic shape change. On heating the alloy to parent phase, the crystallographically reversible transformation causes the recovery of the original macroscopic shape. However, a major inconvenience for the practical application of the thermal-induced shape memory effect was the low working frequency (less than 1Hz) inherent to the thermal control of the effect. Recently, a significant breakthrough in the research of shape memory alloys came about with the discovery of ferromagnetic shape memory alloys (FSMAs). The realization of large field-induced strains by the external magnetic field at high working frequencies has aroused intensive research attention to these materials [7]. So far, the reported magnetic-field-induced strain (MFIS) reached up to ~10% [8], which is an order of magnitude comparative to the conventional shape memory alloys and much higher than the strains generated in piezoelectric and magnetostrictive materials. Meanwhile, the possibility of controlling the shape change by the application of magnetic field enables the relatively higher working frequency (KHz) than that of conventional shape memory alloys [9]. Due to these features, the ferromagnetic shape memory alloys are taken as the potential candidates for a new class of magnetic actuator materials. The magnetic-field-induced strain effect, also termed as magnetic shape memory effect, in ferromagnetic shape memory alloys can be achieved through different mechanisms driven by external magnetic field, which includes (1) the rearrangement of ferromagnetic martensite variants by twinning and detwinning, such as Ni-Mn-Ga [10] and Fe-Pd alloys [11]; (2) the phase transition from the paramagnetic parent phase to ferromagnetic martensite, such as Fe-Mn-Ga alloys [12]; and (3) the reverse phase transition from the antiferromagnetic martensite to ferromagnetic parent phase, such as Ni(Co)-Mn-In [13] and Ni(Co)-Mn-Sn alloys [14]. Among the ferromagnetic shape memory alloys, the most significant magnetic-field-induced strain was found in Ni-Mn-Ga alloys and thus these materials have become the most attractive ones in the family of the ferromagnetic shape memory alloys. The magnetic shape memory phenomenon in Ni-Mn-Ga alloys has been demonstrated in two of the martensitic structures with the phase transformation close to the ambient temperature [8,15]. Martensitic transformation and twin relationship between variants Since the fundamental condition to the shape memory phenomena is the occurrence of a thermoelastic martensitic transformation, it would be useful to have a brief overview on this transformation. Many solids undergo a solid-to-solid phase transformation. Solid state transformations are usually of two types: diffusional and displacive. Diffusional transformations take place by long range diffusion resulting from thermally activated atomic movements. The new phase is of a different chemical composition compared with that of the parent. Displacive (diffusionless) transformations, on the other hand, do not require long-range diffusion during the phase transformation. Only small atomic movements usually less than the inter-atomic distances are needed. The atoms are rearranged into a new structure in a cooperative manner, with no change of the chemical composition and their atomic arrangements. Martensitic transformation is a kind of shear-dominant diffusionless solid-state phase transformation and occurs by nucleation and growth of the new phase from the parent phase [16]. Typically, upon cooling, the high temperature phase (austenite) with higher symmetry transforms to a low temperature phase (martensite) with lower symmetry through a first-order phase transition. There is a rigorous crystallographic connection between the lattices of the initial and final phases. The martensitic transformation is called thermoelastic, when it is thermally reversible and associated with mobile interfaces between the parent and martensitic phases with small transformation temperature hysteresis [17,[START_REF] Otsuka | Shape memory materials[END_REF]. In general, there are four characteristic temperatures defining the martensitic transformation process. The forward transformation start and finish temperatures from austenite to martensite are called M s and M f , respectively; while the inverse transformation start and finish temperatures from martensite to austenite are called A s and A f , respectively. Usually, the transformation temperatures differ on heating and cooling during the transformation. There is a hysteresis associated with phase transformation. The transformation temperatures depend mainly on the alloy composition and processing history. Microstructural defects, degree of order and grain size of the parent phase can also alter the transformation temperatures by several degrees [START_REF] Macqueron | [END_REF]. Martensitic transformation is usually accompanied by a change in the form of the transformed region which manifests itself in a characteristic relief on the surface where the martensite plate appears. Moreover, numerous physical properties show different changes with the occurrence of martensitic transformation [17]. During the transformation, a latent heat associated with the transformation is absorbed or released depending on the transformation direction. The two phases also have different resistance due to their different crystallographic structures, thus the phase transformation is associated with a change in the electrical resistivity. These changes allow for the measurement of the transformation temperatures by differential calorimetry and electrical resistivity; respectively. In addition, changes in specific volume, mechanical properties and magnetization et al. also allow the transformation temperatures to be determined [START_REF] Delaey | Diffusionless Transformations[END_REF][START_REF] Webster | [END_REF]. The martensite phase usually takes the form of plates and the interface separating the martensite from the parent phase is called the habit plane. A careful analysis of the surface relief reveals that any vector in the habit plane is left unrotated and undistorted during the transformation [START_REF] Delaey | Diffusionless Transformations[END_REF]. The habit plane is thus essentially "undistorted" and the shape change resulted from the martensitic transformation is an "invariant plane strain". To illustrate the transformation mechanism, in 1924, Bain proposed that the change in the structure from FCC lattice to BCC lattice could be achieved by a simple homogeneous deformation, as illustrated in Fig. 1.1 [22]. A unit cell of the BCT structure is drawn within two FCC cells. Then, the transformation to a BCC unit cell can be achieved by the compression along the z axis and the expansion along the x and y axes. This homogeneous deformation is often called the Bain distortion or Bain strain. Although Bain distortion simply explains how the BCC unit cell can be transformed from FCC unit cells with minimum atomic movement, in fact, the Bain deformation alone would cause enormous transformation strains depending on the volume transformed and a non-invariant plane strain can be achieved. To fulfill the requirements of an invariant plane, Wechsler, Lieberman and Read [23], and Bowles and Mackenzie [24] independently developed the so called "phenomenological crystallographic theory of martensitic transformation" to explain the shape change of martensitic transformation. Although the mathematical treatment is slightly different in the two theories, they are essentially equivalent [25]. In the phenomenological theories, the shape deformation is artificially decomposed into following three components: a lattice deformation, i.e. Bain strain, a lattice invariant shear (slip, twinning or stacking faults), and a rigid body rotation. The pure lattice strain transforms the parent lattice into the product lattice and the combination of lattice invariant shear will, in general, leave one plane undistorted but rotated to a new position; a rigid body rotation is applied to bring back the undistorted plane into the original position. The phenomenological theories have successfully predicted crystallographic data associated with the martensitic transformation, such as the habit plane, orientation relationship, and magnitude of shape deformation et al., in various alloy system, which were confirmed by experimental measurements. Since martensitic transformation is diffusionless completed by atomic movement in a coordinated manner, it follows that the austenite and the martensite lattices should be intimately related and lead to a reproducible orientation relationship between them. The orientation relationship can be described by specifying the parallelism between certain crystallographic planes and directions. Representative orientation relationships of martenstic transformation are given below (the subscript A represents austenite and M martensite): Bain relation [22]: (001) A // (001) M and [100] A // [1 1 0] M Kurdjumov-Sachs (K-S) relation [26]: (111) A // (011) M and [1 1 0] A // [1 1 1] M Nishiyama-Wassermann (N-W) relation [27,28]: (111) A // (011) M and [ 2 11] A // [0 1 1] M Greninger-Troiano (G-T) relation [29]: (111) A about 1° from (011) M and [1 1 0] A about 2.5° from [1 1 1] M Pitsch relation [30]: (110) A // (1 1 2 ) M and [1 1 0] A // [ 1 1 1 ] M Burgers relation (BCC to HCP) [START_REF] Delaey | Diffusionless Transformations[END_REF]: (111) A // (0001) M and [1 1 0] A // [1 2 10] M Orientation relationships of the two phases change from one alloy system to another, and within a given alloy system from one composition to another. Under different orientation relationships, the number of induced martensite variants is different. As the crystal lattice of the martensite phase usually has lower symmetry than that of the parent austenite phase, elastic strains associated with the transformation accompany the nucleation and growth of the martensite. The elastic strains increase with the increasing martensite fraction. To compensate the transformation strains, different oriented variants are formed from the same parent phase. For keeping the lattice continuity, the neighboring martensite variants usually develop a twin relationship to each other. Such twins make a substantial contribution to the lattice-invariant deformation [START_REF] Christian | The theory of transformations in metals and alloys PERGAMON[END_REF]. The classical definition of twinning describes that the twin and the matrix lattices are related by reflection with respect to a plane or by a rotation of 180° about an axis [START_REF] Christian | [END_REF]. These two types of twin often designated as "reflection" twins and "rotation" twins. By convention [33], a twinning mode is defined by six elements: (1) K 1 -the twinning or composition plane that is the invariant (unrotated and undistorted) plane of the simple shear; (2) 1 -the twinning direction or the direction of shear lying in K 1 ; (3) K 2 -the reciprocal or conjugate twinning plane, the second undistorted but rotated plane of the simple shear; (4) 2 -the reciprocal or conjugate twinning direction lying in K 2 ; (5) P -the plane of shear that is perpendicular to K 1 and K 2 and intersects K 1 and K 2 in the direction 1 and 2 , respectively; (6) s -the magnitude of shear. The geometrical configuration of K 1 , K 2 , 1 , 2 and P is shown in Fig. 1.2. According to the rationality of the Miller indices for K 1 , K 2 , 1 and 2 , crystal twins are usually classified into three categories: type I twin (K 1 and 2 are rational), type II twin (K 2 and 1 are rational), and compound twin (K 1 , K 2 , 1 and 2 are all rational). Twins are usually formed by a homogeneous simple shear of the matrix lattice when the material is strained, in which the resulting structure is identical to that of the matrix, but differently oriented. This type of twinning is termed as deformation twinning which is an important plastic deformation mechanism in many materials. On the other hand, twin structures are often found in the products of many martensitic transformations, i.e. transformation twin. Transformation twin produces highly organized structures with alternative twin lamellae of fixed thickness ratios. In many martensites of shape memory alloys, the twin boundaries are highly glissile. Upon the application of a load, the twin boundaries between martensite variants are easily activated to move, thus to accommodate the induced strain, resulting in detwinning of martensite variants with the growing of some energetically favored variants at the expense of others. Ni-Mn-Ga Ferromagnetic Shape Memory Alloys Introduction Ni-Mn-Ga ferromagnetic shape memory alloys have emerged as a promising new class of smart materials due to the large magnetic-field-induced strains. They can be driven at higher actuation frequency than conventional shape memory alloys. The Ni-Mn-Ga alloys have been already studied for more than 50 years when the initial work exhibited Ni 2 MnGa alloy as one of the Heusler alloys having the formula X 2 YZ [34][35][36][37]. Soltys was the first to concentrate only on the Ni-Mn-Ga alloy system [38]. In 1984, Webster et al. firstly investigated the martensitic transformation and magnetic order of Ni 2 MnGa [START_REF] Webster | [END_REF]. More systematic studies in the 1990's were carried out by Kokorin et al. [39,40], and by Chernenko et al. [41], when Ni-Mn-Ga alloys were investigated as potential shape memory alloys. In 1996, Ulakko et al. firstly demonstrated a 0.19% field-induced strain in a Ni 2 MnGa single crystal sample at 265 K under a magnetic field of 8kOe [7]. Since then, the research interest on magnetic shape memory alloys grew rapidly. In 2000, Murray et al. achieved a 6% large magnetic-field-induced strain in a non-stoichiometric Ni-Mn-Ga single crystal with a five-layered martensitie [15]. In 2002, a much larger strain close to 10% was reported by Sozinov et al. [8] in the single crystal of a seven-layered martensite, which is the largest magnetic field induced strain obtained in ferromagnetic shape memory alloys. Due to the high fabrication cost of single crystals, the research was also directed towards polycrystalline materials. However, the magnetic field induced strain is almost zero in fine grained alloys. Innovative routes of fabrication, such as films, textured polycrystals and foams, were designed to improve the magnetic field induced strain in polycrystalline alloys. An strain of 0.065% was reported for 0.1-1 m thick films on a 10 m thick Mo substrate [42]. By preparation of highly textured bulk alloy, Gaitzsch et al. [43] observed 1% field-induced strain in a polycrystalline alloy. By producing a porous material, the magnetic field induced strain of 0.12% was reported [44]. After a certain training, the magnetic field induced strain in the magnetic shape-memory alloy foams reached 8.9% [45]. So far, numerous experimental and theoretical studies have been performed on Ni-Mn-Ga alloys, such as crystal structure, phase transformation, magnetic properties, magnetic shape memory effect, mechanical behavior, martensite stability and alloying et al., and interesting phenomena have been revealed. Magneto-structural transformation in Ni-Mn-Ga alloys Ni 2 MnGa alloy is one of the Heusler alloys with the occurrence of both the paramagentic-ferromagnetic and the thermoelastic martensitic transformation on cooling. The so-called Heusler alloys are highly ordered intermetallics having the common formula X 2 YZ with a L2 1 order. On cooling from the melt state, Ni-Mn-Ga alloys firstly transform into the B2' phase at about 1100 . Below the ordering temperature of about 750-800 , the Heusler phase (austenite) with L2 1 long range order is formed. The austenite of Ni-Mn-Ga alloys has a cubic structure with space group Fm 3 m (No. 225) [46], where Ni atoms occupy 8c (0.25, 0.25, 0.25) Wyckoff sites, Mn and Ga atoms 4a (0, 0, 0) and 4b (0.5, 0.5, 0.5) sites, respectively [46], as shown in Fig. 1.3. ). This magnetic transition is one of the main limits when considering the service temperatures of the magnetic shape memory effect. The T C of the stoichiometric Ni 2 MnGa was reported to be 376K [START_REF] Webster | [END_REF]. The T C of off-stoichiometric Ni-Mn-Ga alloys is less sensitive to the chemical composition [41], having a value of around 370K for a wide range of compositions. Ni-Mn-Ga alloys are a special case of shape memory alloy and thus they exhibit thermoelastic martensitic transformation on cooling from the parent phase. In some near-stoichiometric alloys, prior to the martensitic transformation at low temperatures, premartensitic transformation occurs and the austenite transforms to micromodulated three-layered (3M) martensite [46]. The premartensitic transformation is characterized by anomalies in the elastic [47][48][49], thermal [47,48], resistivity [50,51] and magnetic properties [48,[50][51][52]. Inelastic neutron scattering experiments performed on stoichiometric Ni 2 MnGa showed the existence of a soft [ 0] TA 2 phonon mode over a wide temperature interval. An important observation in these measurements was that the TA 2 phonon branch at a wave vector of 1/3 incompletely condenses at the premartensitic transition temperature, which is well above the martensitic transition temperature [53,54]. This sudden soft mode phonon freezing is the typical character of the premartensite phase and can only be observed in the premartensite phase. Weak spots (or peaks) were found using diffraction techniques by electron [47,49], neutron [46], and x-ray [55], which can be indexed using a propagation vector of (1/3, 1/3, 0) with cubic symmetry. Based on Landau theory, first principle simulation and phenomenological model have revealed that premartensitic transformation was a kind of weak first order transformation, which is driven by magnetoelastic interaction [56,57]. The martensitic transformation in Ni-Mn-Ga alloys may result in several martensites. The most frequently observed are five-layered modulated (5M) martensite, seven-layered modulated martensite (7M) and non-modulated martensite (NM). The kind of martensite which appears on cooling depends on the composition, but the stability of the structures seems to be always the same. 5M martensite is the most unstable one; 7M martensite is in the middle and the NM martensite is the most stable one [58]. The alloys displaying a direct transformation to NM martensite typically have martensitic transformation temperatures close or above their Curie temperatures [59,60]. The martensitic transformation temperatures (T M ) of Ni-Mn-Ga alloys are highly dependent on the chemical composition [41,61,62]. Generally, the increase of the martensitic transformation temperature can be described as a linear function with the average number of valence electrons per atom (electron concentration) e/a. Suppose that the number of valence electrons for Ni (3d 8 4s 2 ), Mn (3d 5 4s 2 ) and Ga (4s 2 4p 1 ) atoms are 10, 7 and 3, respectively. The e/a can be calculated as follows: In addition to the occurrence of martensitic transformation, there also exists a first order inter-martensitic transformation (IMT) in some alloys that transforms one type of martensite to another [63][64][65][66][67]. Depending on the composition and the thermal history of the alloys [63,66], typical transformation path of inter-martensitic transformation on cooling was reported as from modulated martensite to final NM martensite, such as 5M-7M-NM or 7M-NM. Thus, NM martensite is the ground state of the alloys. The inter-martensitic transformations are usually accompanied by anomalies in the calorimetric, mechanical, magnetic and electrical resistivity both on cooling and heating process [63]. Moreover, the occurrence of inter-martensitic transformation has a certain influence on their practical applicability. In this aspect, it means that the working range of ferromagnetic shape memory alloys could be restricted to a temperature range where inter-martensitic transformations are not observed, i.e. the inter-martensitic transformation may set the lower limit for the service temperature of the magnetic shape memory effect [68]. Crystal structure of martensite Among the three types of martensite formed after martensitic transformation, the NM martensite has a tetragonal crystal structure with space group I4/mmm (No. 139) [69], while 5M and 7M martensites possess monoclinic superstructure subjected to periodic shuffling along the (110) A [1 1 0] A system. Shuffling means that transformation shear occurs periodically on the (110 ) A plane in [1 1 0] A or [ 1 10] A direction. In electron diffraction patterns, the lattice modulation is reflected by the satellite spots between two main spots. The distance between the two main reflections is divided into five parts by four satellite spots or seven parts by six satellite spots for 5M and 7M martensite; respectively [70]. Conventionally, the diffraction data is interpreted as that the superstructure is composed of five consecutive subcells for 5M martensite and seven consecutive subcells for 7M martensite. Generally, two models were commonly used to represent layered martensitic structure. The first one is the long-period stacking order approach [71], which is a well-known method to describe the close-packed layered martensite structure and widely accepted in the studies on Ni-Al alloys and Cu-based shape memory alloys. This method is based on the uniform shear occurring on each basal plane, which is originated from the {110} A . It is necessary to use Zhdanov notation to explicitly indicate the stacking sequence, such as (5 2 ) 2 for sever-layered martensite. The second approach is to consider the layered structure as a modulation function superimposed onto the basic structure [72]. The deviations of atoms from their ideal positions are defined by the modulation function. The distinction between these two methods is that the long-period stacking order model assumes uniform shear between two neighboring atomic planes, which is the case only when considering the modulation function with the zero-order harmonic coefficients [73]. In such a context, the stacking order model is the simple case of the modulation function model and it is valid only for a commensurate modulated structure [73]. In Ni-Mn-Ga alloys, recent investigations confirmed that the superstructure of modulated martensite can be commensurate or incommensurate [74,75], thus, the lattice modulation model is more appropriate. In spite of the numerous structural characterization works, the crystal structures of modulated martensites are still in dispute. Martynov et al. [72,76] proposed that the structural modulation could be interpreted with a sinusoidal function applied to the (001) planes of the distorted martensite lattice. The displacement of the jth plane from the regular position in the shuffling direction can be expressed as: sin(2 /L) + sin(4 /L) + sin(6 /L) j A j B j C j (1.2) where L is the modulation period; A, B and C are constants to achieve the best fit with the observed intensities of the main and satellite spots. For 5M martensite (L=5), A = -0.06, B = 0.002, C = -0.007 [72]. For 7M martensite (L=7), A = 0.083, B=-0.027, C=0 [76]. Ignoring the lattice modulation, Wedel et al. used simplified tetragonal crystal structure (I4/mmm) and orthorhombic crystal structure (Fmmm) to describe the 5M and 7M martensites [77]. By elastic neutron scattering measurement, Zheludev et al. [78] found that the 5M modulation of a near-stoichiometric Ni 2 MnGa alloy was an incommensurate one, in which the modulation was along wave vectors ( M , M , 0), M = 0.43. Brown et al. [46] performed structural refinement for the thermally induced Ni 2 MnGa martensite based on powder neutron diffraction and suggested a commensurate seven-layered modulation with orthorhombic crystal structure (Pnnm). However, this structure was more likely to be an incommensurate 5M modulation. Righi et al. applied the superspace theory to fit powder X-ray diffraction patterns and demonstrated that the 5M modulation in Ni-Mn-Ga alloys can be commensurate or incommensurate [74,79]. It was revealed that the incommensurate 5M modulated structure (Pnmn) composed of seven consecutive subcells related to the orthorhombic structure for the stoichiometric Ni 2 MnGa alloy [79]; while the Mn-rich alloy has commensurate structure with five consecutive subcells related to the monoclinic lattice basis (I2/m) [74]. Glavatskyy et al. reported the refinement results of powder neutron diffraction pattern for the 5M martensite in a non-stoichiometric alloy and also suggested the commensurate monoclinic superstructure but with the different space group (P2/m) [80]. For 7M martensite, it was reported by Righi et al. that the crystal structure was monoclinic (P2/m) and the superlattice was incommensurate one composed of ten subcells [75]. Recently, Kaufmann et al. proposed that the so-called 7M long-period structure is simply composed of nanotwinned tetragonal NM martensite lamellae with (5 2 ) 2 stacking sequence, ruling out the existence of the independent modulated structure [81]. Further confirmation and clarification of the most appropriate crystal structure for the modulated martensites are still needed. Ferromagnetism of Ni-Mn-Ga alloys The ferromagnetism of Ni-Mn-Ga alloys is mainly from the contribution of the magnetic moment of Mn atoms. Ni atoms only carry a small magnetic moment and the magnetic moment of Ga is negligible [START_REF] Webster | [END_REF]. From the magnetization measurements, Webster et al. [START_REF] Webster | [END_REF] reported that the total magnetic moment of the cubic Heusler phase of Ni The ferromagnetism of Ni-Mn-Ga alloys is interesting because Mn is antiferromagnetic in its pure element state. The change from the antiferromagnetic behavior of pure Mn to the ferromagnetic behavior of Ni-Mn-Ga alloys is due to the increased distance between the Mn sites in the L2 1 structure compared with the distance between Mn atoms in pure Mn, which changes the Mn-Mn exchange interaction from antiferromagnetic to ferromagnetic [83]. Magnetocrystalline anisotropy is an intrinsic property of a material in which the magnetization favors preferred directions (easy directions). In Ni-Mn-Ga alloys, the easy axis of magnetization of the parent phase was reported to be <100> A [84]. The easy axis of magnetization of modulated martensites, i.e. 5M and 7M, correspond to the shortest axis of the distorted austenite, i.e. b-axis of the superlattice. NM martensite has the easy magnetization plane that perpendicular to the c-axis. Magnetocrystalline anisotropy energy, defined as the work necessary to rotate the magnetization from the easy axis to the hard axis with an applied magnetic field, is an important parameter for the achievement of giant magnetic-field-induced strain effects in Ni-Mn-Ga alloys. Magnetocrystalline anisotropy energy is usually expressed by the magnetocrystalline anisotropy constants, which can be acquired by measuring magnetization curves along different crystalline directions. It is reported that the magnetocrystalline anisotropy constant of the austenite for Ni-Mn-Ga alloys is relatively low, of the order of 10 3 J/m 3 , whereas the anisotropy constant of the martensite is increased by two orders of magnitude [85]. J/m 3 and K 2 =0.9×10 5 J/m 3 referring to the hard and mid-hard axes; and for Ni 50.5 Mn 30.4 Ga 19.1 NM martensite K 1 =-2.3×10 5 J/m 3 and K 2 =0.55×105 J/m 3 . It should be noted that the magnetic anisotropy constants are temperature and composition dependent [86,87]. Mechanism of magnetic field-induced strain The macroscopic shape memory effects induced by magnetic field in Ni-Mn-Ga alloys are the results of field-induced twin variants reorientation [10]. The twin boundary motion can occur when the difference in magnetization energy ( E mag ) between different martensite variants exceeds the elastic energy needed for twin boundary motion [88]: E mag 0 tw (1.3) where 0 is the reorientation strain and tw is the twinning stress. If the material is magnetized to saturation perpendicularly to the easy axis of one variant and along the axis of the adjacent variant, the difference of the magnetic energy is equal to the difference of the magnetocrystalline anisotropy energy (K u ). The condition for twin boundary motion under magnetic field can be rewritten as [88]: K u / 0 > tw + ( ext ). (1.4) The term in bracket, ext , is the external applied stress applied in the direction perpendicular to the magnetic field direction. This equation describes the usual setup of an actuator. When an external magnetic field is applied, the variants with their magnetization vectors aligning along the applied field are more energetically favorable and they will increase the volume fraction at the expense of the other variants through the twin boundary motion, leading to the macroscopic shape change of the sample, namely magnetic field induced strain [58]. The schematic illustration of the rearrangement of the martensite variants under a magnetic field is shown in Fig. 1.4. So far, large strains up to ~6% and ~10% have been reported in 5M [15] and 7M martensite [8], while the strain induced by magnetic field in NM martensite is negligible [89]. In general, the shape change achieved by the rearrangement of martensite variants under an applied field is not recovered when the magnetic field is switched off. To recover the strain induced by magnetic field, one possibility is to rotate the magnetic field [90]. Another possibility is to apply a external stress perpendicular to the direction of the field [91]. Assume that in the absence of the magnetic field, the martensite is at single-variant stage and it is favored by the stress. On applying a magnetic field, if the maximum magnetic stress (K u / 0 ) overcomes the sum of the twinning stress and the external stress, the stress-favored variant transforms to a different variant which is favored by the magnetic field, resulting in a large shape change. When the magnetic field is switched off, if the twinning stress is very low and the external stress exceeds the twinning stress, fully reversible behavior can be obtained. The condition for full reversibility of the magnetic shape memory effect can be written as [88] K u / 0tw > ext > tw (1.5) This equation reflects the fact that high magnetic anisotropy (K u ) and low twinning stress ( tw ) are requisites for the occurrence of magnetic shape memory. It also shows that too large external stresses will inhibit twin boundary motion. In principal, the magentocrystalline anisotropy energy can be increased by increasing the saturation magnetization and Curie temperature (T C ). It is reported that the Curie temperature and the saturation magnetization of Ni-Mn-Ga alloys are less sensitive to the composition [41,92]. The easiest way to increase the magnetocrystalline anisotropy energy is to choose an operating temperature (T O ) significantly lower than T C since the magnetocrystalline anisotropy energy increases with decreasing temperature below T C [91]. Similarly, twinning stress also increases with decreasing temperature below transformation temperatures [93]. Therefore, it is critical to understand and quantify the relationship between magentocrystalline anisotropy energy, twinning stress, operating temperature, T C and transformation temperatures to select materials and operating temperatures for obtaining high blocking stress (the external stress level above which magnetic field induced reorientation is not possible) and magentocrystalline anisotropy energy. In many cases, the observed the twinning stress exceeds the stress induced by magnetic field due to the interlocked variants with various orientation. [90]. The magnetic field induced strain increased from the initial 6% to 9.7%. Crystallographic orientation relationship of Ni-Mn-Ga alloys As the field-induced effect originates from the reorientation of martensite variants through a detwinning and twinning process, the microstructural configuration and crystallographic correlation of constituent martensite variants have strong influence on the activation of the magnetic shape memory effect (output strain and dynamic response). Thus, comprehensive crystallographic knowledge on the microstructural features of martensite variants and variant boundaries is desired, from which a clear image of structure-property relation is built up and also necessary indications for further precise control and improvement of properties are presented. So far, constant efforts have been devoted to the study of orientation relationships between martensite variants by means of X-ray diffraction (XRD) [97] and transmission electron microscopy (TEM) [98][99][100]. Mogylnyy et al. successfully determined the twinning mode in a Ni-Mn-Ga single crystal with 5M martensite by X-ray diffraction methods [97]. Han et al. [98,99] investigated the orientation relationship between the martensite variants in 5M and 7M martensite by TEM. It was found that there exists a twin relationship between the variants in both 5M and 7M martensite. The twinning plane in 5M martensite is {1 2 5 } 5M , while that in 7M martensite is {1 2 7 } 7M . More systematic TEM investigation on twin types and twinning elements in 5M and 7M martensite were presented by Nishida et al. [100]. However, the XRD analysis suffers a limit on acquiring simultaneously the spatial microstructural information of the measured orientations, which prevents the identification of orientation-microstructure correlation. In contrast, the TEM analysis enables one to reveal the variant morphology and the inter-variant orientation relationships, but the exploitable area is too local. Hence, it is difficult to obtain a global orientation image of multi-variants. It has also been noted that for the incommensurate modulation, the number of the observed satellites plus the main reflection does not conform to the number of the subcells in the superstructure [75]. Apparently, the habitual interpretation of the structural modulation according to TEM observation is not suited for the determination of the lattice constants and hence the orientations of the martensite variants [100], and needs to be reconsidered. Moreover, due to the limitation of TEM observation, the crystallographic nature of inter-variant interfaces and their statistical distributions has also not been well addressed. were fully determined and the inter-plate interfaces were found to be close to {112} Tet twinning plane with 2.6° deviation. In modulated martensite, some pioneer work has been done to characterize the global microstructures and variant orientations using approximate structure information [103][104][105][106][107]. For instance, the simplified tetragonal crystal structure was used to approximate the 5M modulated monoclinc superstructure, considering that the monoclinic angle of the modulated structure is very close to 90° and the difference between the basis vector a and b is very small. As a consequence of that, crystallographic nature was not fully disclosed. Especially, the variant number was not detected accurately. Further efforts should be made for the full disclose of the microstructural features and the crystallographic characteristics concerning the number and the shape of the martensite variants, the inter-variant orientation relationships and the twin interfaces. One the other hand, the key factor behind the magnetic shape memory effect in Ni-Mn-Ga alloys is the martensitic transformation below a certain temperature. The microstructural configuration of martensite variants and their crystallographic correlation results directly from the martensitic transformation that follows certain specific orientation relationships (ORs) between the parent phase and the product phase. Thus, the determination of possible phase transformation orientation relationships in Ni-Mn-Ga alloys is of practical importance for their microstructure control and theoretical interest for insight into their martensitic transformation process. However, in most of the alloys, the transformation is complete. It is difficult to obtain a mixed microstructure consisting of austenite and martensite, thus to determine the orientation relationship between the two phases. An alternative option for the determination of the transformation OR is to calculate the orientations of parent austenite from the orientations of the transformed martensite variants under an assumed OR [108], and then to identify the most favorable OR. Recently, a specific OR between austenite and NM martensite, namely the K-S relation with substantial advance in the study of martensitic transformation in Ni-Mn-Ga alloys [102]. Further efforts should be made on complete description of the phase transformation OR between austenite and modulated martensie. Content of the present work During the last twenty years, extensive investigations have been devoted to understanding the shape memory behaviors of Ni-Mn-Ga alloys. However, as Ni-Mn-Ga ferromagnetic shape memory alloys are newly emerging materials, there are still many fundamental issues that need to be further explored. Some obtained results are still in controversial due to the limitations of the characterization techniques. Electron backscatter diffraction (EBSD) measurement, as a promising tool for crystallography investigation, directly correlates the spatial crystallographic orientations with the morphological characters and allows a larger scale of measurement on the bulk sample. However, due to the complexity of structural modulation of the modulated martensite phase, there are seldom EBSD characterizations by application of superstructure information. Consequently, many crystallographic natures have not been correctly revealed. Based on such a background, the present work was designed to perform a thorough crystallographic investigation on different kinds of martensite in Ni-Mn-Ga alloys, i.e. 7M, 5M and NM martensite, based on EBSD orientation and microstructural examination techniques. The superstructure information of modulated martensites was applied for the orientation determination. The main content of the present work will focus on: (1) Precisely determine variant number, the orientation relationships of adjacent variants and the variant interface planes of 5M and 7M martensite by using the precise superstructure information for the EBSD auto-orientation mapping. (2) Determine the most favorable orientation relationship between austenite and modulated martensites based on the experimentally acquired orientation data and crystallographic calculation. (3) Examine the morphology and crystallographic features of NM martensite by EBSD measurements and crystallographic calculations. (4) Study the self-accommodation mechanism of modulated martensite. Reveal the thermodynamic stability and crystal structure nature of modulated martensite. Clarify the role of the modulated martensite in the transformation from the austenite to NM martensite. Chapter 2 Experimental and calculation methods Alloy preparation Ni-Mn-Ga polycrystalline alloys with different nominal composition were dedicatedly prepared, aiming to obtain different kinds of martensite at room temperature. High purity elements Ni (99.97 wt.%), Mn (99.9 wt.%) and Ga (99.99 wt.%) were used as raw materials to prepare as-cast ingot. The alloys were melted in arc-melting furnace with water-cooled copper crucible under argon atmosphere. The weight of each target ingot was about 70-80g. Prior to be melted, the pure raw materials were weighed by an electronic balance according to the designed nominal composition with the precision of 0.01g. To reduce the loss of Mn caused by strong evaporation during melting, the pure Mn was placed at the bottom of the crucible before melting. Each alloy was melted for four times and the electromagnetic stirring was applied during melting process for the composition homogenization. The final as-cast ingot had a button-shaped appearance, as shown in To further reduce the composition inhomogeneity in the prepared alloys, the alloys were sealed in a vacuum quartz tube and then homogenized at 900°C for 24h. Finally, they were quenched in the water by breaking the quartz tube. Sample preparation The samples for compressive testing with dimension of 6mm×5mm were cut by electrical discharge wire-cutting. The small cylindrical samples were mechanically polished with SiC grinding paper. To prepare the samples for the powder X-ray diffraction (XRD) measurement, parts of the homogenized alloys were crushed and ground into powder. To release the stress introduced by grinding, the powder was sealed in vacuum quartz tubes and annealed at 600 for 5 hours. Rectangular parallelepiped bulk samples for microstructure observation and electron backscatter diffraction (EBSD) measurements were cut out of the as-cast button or suction cast rod by electrical discharge wire-cutting. For the preparation of suitable observation plane, all the samples were first mechanically polished with SiC grinding paper. Then they were electrolytically polished in a solution of 20% nitric acid in methanol at room temperature with the voltage of 12V for 30 seconds. The samples for transition electron microscope (TEM) observation were mechanically thinned to ~100 m. Then the thin foils were electrolytically thinned in a twin-jet device at room temperature with the same solution as mentioned above. Characterization methods Optical microscopy The microstructure observations after polishing were firstly performed with OLYMPUS BX61 optical microscope equipped with polarized light. The microstructure evolution during inverse martensitic transformation for the alloys with the martensitic transformation temperatures around room temperature was also observed with the optical microscope. The sample was firstly pre-cooled into the full martensite state and then subjected to the optical microscope observation at room temperature. Mechanical property testing The mechanical properties of the as-cast alloys and the suction-cast alloys were measured by compressive testing using a CMT5305 Electronic Universal Testing Machine. The load was applied by controlling the displacement with a compressing rate of 0.2mm/min. The sample was loaded till fractured. Differential scanning calorimetry The forward and backward martensitic phase transformation temperatures for the prepared alloys were measured by differential scanning calorimetry (DSC). A Q100 DSC device, TA Instruments, was used in the present work. During DSC experiments, two crucibles were placed next to each other. One crucible was empty functioning as a reference and the other was filled with the sample to be measured. The two crucibles were heated and cooled under the same experimental setting. The heat flux in and out of a specimen at a specific temperature was determined and recorded by comparing the sample temperature with that of the reference. The heating and cooling rates for the measurement were set to be 10 /min under a constant flow of Argon. The martensitic transformation temperatures were determined by tangent method using the heat flux peaks in the DSC curve during heating and cooling. X-ray diffraction The crystal structures of the alloys were determined by means of X-ray diffraction (XRD). The XRD measurements were preformed in a PANalytical X'Pert Pro MPD diffractometer with a heating and cooling stage. The stress-free powder samples were used for the measurements. The XRD patterns were measured in the 2 range of 20-120° with a step size of 0.0334° using Cu K radiation. The lattice parameters of the alloys were determined from the experimental XRD patterns by fitting the diffraction profiles using Powder Cell software [109]. Scanning electron microscopy For detailed microstructural analyses, the field emission gun scanning electron microscope (SEM) Jeol JSM 6500 F was used. Additionally to the ability of detecting secondary electrons, the SEM is also equipped with a backscattered electron detector, an energy dispersive X-ray spectrometry (EDS, BRUKER, Germany) and an electron backscatter diffraction camera. The morphological features were characterized by secondary electron (SEI) imaging and backscattered electron (BSE) imaging. The composition of the alloy was verified by EDS. Electron backscatter diffraction (EBSD) was applied for the orientation acquisition. In the present work, the orientation measurements were performed in a field emission gun scanning electron microscope (Jeol JSM 6500 F) with EBSD acquisition camera and Channel 5 software. The working voltage was set at 15KV. The detailed crystal structure information was input into the Channel 5 software to construct the corresponding database for Kikuchi indexing. The orientation data were manually and automatically acquired. The beam control mode was applied for automatic orientation mapping. To ensure the indexation accuracy using the monoclinic superlattice for 7M and 5M martensite, the minimum number of detected bands is set to be 10 and 15, respectively. The maximum allowed MAD that represents the mean angular deviation between the calculated and the experimental EBSD pattern was set to be 1. For the case of 5M martensite, as the monoclinic angle is very close to 90°, it is unavoidable that there exist misindexations in the orientation map. For the determination of correct crystallographic orientation and variant number, we firstly judged and acquired the orientation data manually, then replaced the misindexations in the map by the post treatment. transforming the macroscopic sample coordinate system to the orthonormal crystal coordinate system. The transformation between two coordinate systems can also be represented by the following rotation matrix G and its inverse matrix: where the {i, j, k} and {a M , b M , c M } are respectively the basis vectors of the orthonormal crystal system and the monoclinic crystal system, as shown in Fig. 2.2. 1 2 1 2 1 2 1 2 1 1 2 1 2 1 2 Let (h M , k M , l M ) and [u M , v M , w M ] be the respective Miller indices of a given lattice plane and direction in the monoclinic crystal system. Their corresponding coordinates (n 1 , n 2 , n 3 ) and [u, v, w] in the orthonormal crystal system are then given by: 1 2 3 cos 1 0 sin sin 1 0 0 1 0 0 M M M M M M a c n h n k b n l c (2.2) 1 1 1 sin 0 0 0 0 cos 0 M M M M M M M u u a v b v a c w w (2.3) Fig. 2.2 Schematic representation of the monoclinic crystal system and the orthonormal crystal system. Misorientation calculation The misorientation between two crystals (described by the orthonormal reference systems) is defined by sets of rotations from one of the symmetrically equivalent coordinate systems of one crystal to another equivalent coordinate system of the other crystal. Let us consider two adjacent variants 1 and 2. The misorientation between them can be expressed in matrix notation [101]: 1 1 1 2 i j g S G G S (2.4) where g is the misorienation matrix; G 1 and G 2 are the rotation matrices transforming the orthonormal sample coordinate system to the orthonormal coordinate system set to the lattice basis of the respective variants; S i and S j are the symmetry elements; the superscript "-1" denotes the inverse of a matrix. If we denote = (d , d , d ) = ( , , ) 2sin 2sin 2sin d g g g g g g . (2.7) (2) = 180° 33 11 22 1 2 3 1 1 1 (d , d , d ) ( , , ) 2 2 2 d g g g with d = max( d ,i = 1, 2,3) d > 0, by convention i m, sgn(d ) = sgn(g ) m i m i i m (2.8) (3) =0° 1 2 3 (d , d , d ) (1, 0, 0) d (by convention) (2.9) Chapter 3 Characterization of Ni-Mn-Ga martensites 5M martensite For Ni-Mn-Ga ferromagnetic shape memory alloys, the crystallographic features of modulated martensite (including the number of constituent variants, the inter-variant orientation relationship and the geometrical distribution of variant interfaces) determine the attainability of the shape memory effect. In this section, a comprehensive microstructural and crystallographic investigation has been conducted on a bulk polycrystalline Ni 50 Mn 28 Ga 22 alloy. As a first attempt, the orientation measurements by EBSD using the precise information on the commensurate 5M modulated monoclinic superstructure (instead of the simplified non-modulated tetragonal structure) were successfully performed to identify the crystallographic orientations of martensite variants. Consequently, the morphology of the modulated martensite, the orientation relationships between adjacent variants and the twin interface planes were unambiguously determined. Based on the correct orientation data of the martensite variants acquired from EBSD measurement, the favorable orientation relationship (OR) between austenite and 5M martensite was revealed by detailed crystallographic calculation with no residual austenite. Crystal structure of 5M martensite The off-stoichiometric Mn-rich Ni-Mn-Ga polycrystalline alloy, with nominal composition of Ni 50 Mn 28 Ga 22 (at. %), was prepared by arc-melting. By means of energy-dispersive X-ray analysis (SEM/EDX), the composition of the alloy was verified to be Ni It is noted that, in the pioneering EBSD orientation measurements on the same kind of alloys [103][104][105][106][107], the 5M modulated martensite was usually approximated by a non-modulated tetragonal structure (I4/mmm, No. 139) [77] as the monoclinic angle is very close to 90° and the difference between a 5M and c 5M /5 is very small. Using this simplified structure (a=b=4.226Å, c=5.581Å), the XRD pattern was also recalculated in a similar way, as shown in Fig. 3.1c. It is evident that the diffraction peaks related to the monoclinic distortion, the orthorhombic distortion and the lattice modulation (arrowed positions) do disappear in the recalculated pattern. This indicates that the approximation of the 5M modulated superstructure with a non-modulated tetragonal structure may lead to an incorrect EBSD orientation identification, due to the loss of the important structure modulation information. Microstructural features of 5M martensite Orientation identification of 5M martensite Orientation relationships between martensite variants With the correct orientation data of the individual martensite variants determined by EBSD measurements, the inter-variant orientation relationships can be further calculated in terms of misorientation angle/axis, i.e. a set of misorientation angles around the corresponding rotation axes. By taking into account possible combinations of the four types of variants A, B, C and D in Fig. 3.3a, the complete sets of misorientation angles ( ) and the corresponding rotation axes (d) were calculated according to Eq. (2.4) and are shown in Table 3.1. Owing to the monoclinic symmetry, there are two sets of distinct misorientations Based upon the minimum shear criterion, the complete twinning elements (K 1 -the twinning plane; 1 -the twinning direction; K 2 -the reciprocal or conjugate twinning plane; 2 -the reciprocal or conjugate twinning direction; P-the plane of shear; s-the amount of shear) of the above three types of twins were unambiguously determined using a general method recently developed [33] and are displayed in Table 3.2. It is seen that type I twin (rational K 1 and 2 ) and type II twin (rational K 2 and 1 ) have the same magnitude of shear s, but K 1 and K 2 , and 1 and 2 are interchanged. Therefore, type I twin and type II twin can be regarded as conjugate or reciprocal to each other [START_REF] Christian | [END_REF]. Among the three types of twins, compound twin (rational K 1 , K 2 , 1 and 2 ) possesses the smallest twinning shear. Geometrically, all these twin relationships can be equivalently expressed by a minimum misorientation between two twinned variants. This minimum misorientation may be used as a simple criterion to judge possible twinning relationships between two variants for post EBSD orientation analysis, i.e. ~86° around the [501] 5M direction for type I twin, ~94° around the normal of the (105) 5M plane for type II twin, and ~180° around the [ 5 01] 5M direction or ~180° around the normal of the (105) 5M plane for the compound twin. Characters of twin interface planes Because magnetic-field induced strains in Ni-Mn-Ga alloys are realized by the reorientation of martensite variants through interface motion, the crystallographic nature of the interface planes is of major significance for the efficiency of the magnetic shape memory effect. Here, the twin interfaces between neighbouring variants were further analyzed using the indirect two-trace method [113]. The Miller indices of individual twin interface planes were unambiguously determined from the orientation data of adjacent variants and the trace vectors of twin interfaces in the sample coordinate system. To achieve statistical reliability, five groups of twin interface normals were calculated and the mean values are shown in Table 3.3. For above three types of twins, the calculated interface planes are in coincidence with their respective twinning planes (K 1 plane) with a slight deviation, being close to (1 2 5 ) 5M for type I twin, (1.0569 2 4.7155 ) 5M for type II twin and (105) 5M for compound twin. Therefore, all these three types of interfaces can be considered as coherent interfaces; hence they should have the highest mobility and reversibility as compared with the general incoherent high angle boundaries in polycrystalline materials. Table 3.3 Mean values of twin interface normals calculated by indirect two-trace method [113] and expressed in the orthonormal crystal coordinate system and their deviations from the ideal interface plane normals. According to the accurate orientation identification of the martensite microstructure (Fig. Clearly, the geometrical combination of the four differently oriented variants in the plate microstructure is not an optimal solution to the maximum shape memory effect. The existence of a large amount of compound twin interfaces could be considered as one of the major reasons for a high actuation stress required for these alloys [43,96,106]. In this connection, the compound twin interfaces should be eliminated to achieve a large strain output. Owing to the very small twinning shear of compound twins, the compound twin interfaces might be removed readily by detwinning via mechanical training, with the training force applied in the inverse twinning direction, i.e. [5 01] 5M direction. One can further conceive that, for the most ideal case, only two variants, with either type twin or type twin relationship, are left in the plate microstructure after the complete detwinning of the compound twins. In such a circumstance, the shape memory process would be achieved by the detwinning or twinning of one variant with respect to the other variant. As the two types of twins have the same twinning shear, the intensity of the external actuating field would be the same for the two cases. The angle between the easy magnetization directions of the two variants is 86.04° in type I twin relation and 93.96° in type II relation. To obtain the maximum strain output under a minimum actuating field, the field should be applied in the easy magnetization direction of either of the two variants. Hereafter, we will refer to this average subcell of the 5M modulated martensite as the unit cell of a so-called "1M martensite", and denote the austenite, 1M martensite and 5M martensite by putting the symbols "A", "1M" and "5M" as the subscript in this section, respectively. Orientation relationship between austenite and 5M martensite Since the martensitic transformation is diffusionless and realized by coordinate displacement of atoms, some specific ORs between the parent and product phases are required to minimize the lattice discontinuity across the phase boundary. In most cases, the determination of these ORs are rendered to find a plane and in-plane direction parallelism by making use of the coexistence of the retained parent austenite and the product martensite. However, for the present Ni 50 Mn 28 Ga 22 alloy, the martensitic transformation is complete at room temperature, i.e. no residual austenite. It is not possible to make a direct determination of the OR between the austenite and the martensite. Therefore, verifying the austenite orientations that are calculated from the orientations of the martensite variants induced from the same initial austenite under an assumed OR could be an alternative solution to deduce the transformation OR [108]. If the austenite orientations calculated from all individual martensite variants inherited from the same parent grain share a common orientation, the assumed OR could be the one that governs the martensitic transformation, and the resultant common orientation is the orientation of the initial austenite grain. The austenite orientations ( l A G ) with respect to the sample coordinate system can be recalculated by the following equation expressed in matrix notation [108]: 1 ( ) l k i j A M M A G G S T S (3.1) where k M G represents the measured orientation of the kth martensite variant with respect to the sample coordinate system; T is the rotation matrix transforming the orthonormal crystal coordinate system fixed to the monoclinic martensite lattice to the austenite lattice basis under the given OR; j A S (j = 1, 2, ..., 24) and i M S (i = 1, 2) are the respective cubic (austenite) and monoclinic (martensite) symmetry elements. When T is fixed, a total of 48 different austenite orientations may be generated from one martensite variant with Eq. (3.1) on account of the symmetry elements of cubic and monoclinic basis, but they are not all symmetrically independent and can be further incorporated into the physically distinct orientations. By a survey of the literature, those widely addressed Bain [22], K-S [26], N-W [27,28] and Pitsch [30] ORs are presumed as possible ORs between the parent austenite and the product martensite in the present work, as listed in Table 3.4. The assumed plane and in-plane direction parallelisms are firstly used to specify the ORs between austenite and 1M martensite. (111) A //(011 ) 1M & [10 1 ] A //[ 1 1 1] 1M N-W relation (111) A //(011) 1M & [11 2 ] A //[0 1 1] 1M Pitsch relation (101) A //(1 2 1 ) 1M & [10 1 ] A //[ 1 1 1] 1M Based on the above considerations, we calculated the orientations of parent austenite from the measured orientations of locally adjacent four twin-related martensite variants, using the possible ORs listed in Table 3.4. Here, four twin-related variants in one broad plate were treated as a variant group. To achieve statistical significance, the variant orientation data measured from six different groups (denoted as g1, g2…… g6) were served as initial input data. For easy visualization, the calculated austenite orientations under a given OR were plotted in the {001} standard stereographic projection in the macroscopic sample coordinate frame. As the resultant martensite of the present material possesses a modulated crystal structure, the information on structural modulation should be taken into account to evaluate the deviation from the ideal OR between two phases. The structural modulation can be considered as an atomic reshuffling in each subcell of the supercell with respect to the "averaged unit cell". Clearly, this modulation results in a certain angular deviations of the same indexed planes and directions in each subcell of 5M martensite with respect to those in the averaged unit cell. Based on the determined Pitsch OR between the two phases, the theoretical number of martensite variants induced from the same austenite grain can be predicted. The possible orientations of martensite variants with respect to the sample coordinate system are given by the following equation: 1 ( ) k l j i M A A M G G S T S (3.2) Due to the cubic symmetry of the austenite and the monoclinic symmetry of the martensite, there are at most 24 martensite variants inherited from the same austenite grain under the Pitsch OR. As former EBSD analysis has revealed that only 4 variants appear in one variant group and they are twin-related one another, these 24 variants can be divided into 6 groups. Thus, the formation of self-accommodated martensite in one initial austenite grain is realized firstly by combination of 4 twin-related variants in an individual group and then by combination of different groups over the entire grain. Such a configuration of the product microstructure should ensure a minimum lattice discontinuity between the parent austenite and the martensite. Summary The crystal structure, microstructural features, twin relationships of 5M Considering the minimum misorientation angles between the calculated austenite orientations, the most favorable OR governing transformation from the austenite to 5M martensite was revealed to be the Pitsch relation with (101) A //(1 2 5 ) 5M and [10 1 ] A //[ 5 5 1] 5M with no residual austenite. Under such an OR, at most 24 variants could be induced from the same austenite grain after the martensitic transformation. 7M Martensite 7M martensite, as the other kind of modulated martensite in Ni-Mn-Ga alloys, can generate much larger magnetic field induced strain (i.e. ~10%) than that of 5M martensite. So far, the crystal structure studies of these materials conducted by TEM have suffered from uncertainties in determining the number of subcells of modulated superstructure, i.e. commensurate and incommensurate, and consequently improper interpretations of orientation correlations of martensite variants. In this section, the microstructural and crystallographic characteristics of 7M martensite in a polycrystalline Ni 50 Mn 30 Ga 20 alloy were investigated by EBSD with the application of the correct incommensurate superstructure information of 7M martensite (7M(IC)). The orientation relationships of adjacent martensite variants and their twin interface characters in the incommensurate 7M martensite were unambiguously determined. With the accurate orientation measurement on inherited martensitic variants, the local orientations of parent austenite grains were predicted using four classical ORs for the martensitic transformation. Furthermore, a specific OR between austenite and 7M martensite was unambiguously determined by considering the magnitude of discontinuity between the lattices of the product and parent phases and the structural modulation of the incommensurate 7M modulated martensite. Phase transformation temperatures and mechanical property Polycrystalline Ni-Mn-Ga alloy with nominal composition of Ni 50 Mn 30 Ga 20 (at.%) was prepared. For the reduction of microcracks and porosity, the ingots were remelted and suction cast into a chilled copper. The actual composition was determined to be Ni 49.6 Mn 30.4 Ga 19.9 by energy dispersive X-ray spectrometry (EDS) measurements. The phase transformation temperatures were measured by differential scanning calorimetry (DSC). The martensitic transformation start temperature (M s ) and finish temperature (M f) were determined to be 88.7 and 78.3 , and the reverse transformation start temperature (A s ) and finish temperature (A f ) were 87.7°C and 97.2°C, respectively. The compressive stress-strain curves of the as-cast alloy and the suction-cast alloy are presented in Fig. 3.8. For the button without suction cast, the maximum compressive strength and compressive strain are 385MPa and 8.6%, respectively; while for the rod after suction casting, the maximum strength and strain reach the corresponding values of 705MPa and 10.1%, respectively. The compression strength and the deformation capacity of the suction cast alloy are increased by 83.1% and 17.4%; respectively, compared with those obtained by common casting. This could be attributed to the smaller grain size and the less microcracks and porosity in the as-suctioned rod than that in the as-cast button due to the faster cooling rate of suction-cast. its indexation with the incommensurate 7M modulation (7M(IC)) structure model. Crystal structure of 7M martensite Determination of twin relationships and twin interfaces of 7M martensite To determine the orientation relationships between two adjacent variants, the misorientations were calculated according to Eq.(2.4) using the orientation data of the corresponding variants in Fig. 3.11a and are shown in Table 3.6. For each variant pair, there exist two sets of misorientations ( /d) owing to the monoclinic point symmetry. Among the two sets of misorientations, if taking possible experimental errors into account, there are only one 180° rotation between A and C (or B and D) and between A and B (or C and D), but two 180° rotations between A and D (or B and C) with their rotation axes perpendicular to each other. This suggests that all the variant pairs are twin related according to the classical definition of twins [START_REF] Christian | [END_REF]112]. For variant pair A:C, the 180° rotation axis is close to the normal of the rational plane (1 2 10 ) 7M (with 0.41° deviation) of the monoclinic lattice, indicating that the two variants are reflection of each other with respect to this rational plane; while for the variant pair A:B, the 180° rotation axis is close to the rational direction [10 According to the minimum shear criterion [33], the complete twinning elements of the above three types of twins were unambiguously determined, as displayed in are interchanged [START_REF] Christian | [END_REF], as the case of the 5M martensite. It is noted that the local variant number and the twin types of the 7M martensite are in consistent with those of 5M martensite, which should be due to the fact that both 7M and 5M martensite possess the monoclinic crystal structure. However, the twinning shear of the 7M martensite for each kind of twin is much larger than the corresponding value of the 5M martensite. This means that for the 7M martensite, larger force is needed for variant reorientation. Since the magnetic-induced strains of Ni-Mn-Ga alloys are achieved by the reorientation of martensite variants through the motion of variant interfaces, insight into the crystallographic nature of these interface planes is surely of theoretical interest and practical significance. Here, by using the indirect two-trace method [113], the Miller indices of individual twin interface planes were unambiguously determined. To achieve statistical reliability, the results were collected from five groups of variants with different orientations, and the mean values are showed in Since the martensitic transformation is complete at room temperature for Ni 50 Mn 30 Ga 20 alloy, the orientation relationship (OR) between austenite and 7M martensite was determined by the indirect method, as the case of the 5M martensite. Orientation relationship between austenite and 7M martensite The classical Bain [22], K-S [26], N-W [27,28] and Pitsch [30] ORs listed in Table 3.4 are presumed as possible ORs between the parent austenite and the product martensite. The assumed plane and in-plane direction parallelisms are firstly used to specify the ORs between the austenite and the 1M martensite. Based on the above considerations, the orientations of the austenite were calculated according to Eq. orientations calculated from each martensite pair were estimated, as shown in Table 3.9. Indeed, among all the selected variant groups, both K-S and Pitsch ORs deliver the smallest deviation angle and there is almost no difference in the resultant austenite orientations, suggesting that they could be considered as two possible transformation ORs. This result is different from that of the 5M martensite, where the Pitch OR is favorable without any ambiguity. It is seen from Table 3.10 that the deformation components that represent the discontinuity between the lattices of austenite and 1M martensite under the K-S and Pitsch ORs are not the same. As compared to the K-S OR, the Pitsch OR involves a relatively small lattice discontinuity (only the last 2 shear components in Table 3.10 are slightly elevated) for the austenite to 1M martensite transformation. Considering that an elongation or contraction requires volume change but a simple shear does not, the deformation by a simple shear may occur more easily. In this sense, the Pistch OR has somewhat energetically advantageous over that of the K-S OR for the austenite to 1M martensite transformation. Furthermore, the information on structural modulation should be exploited to discriminate the energetically more favorable OR. For the incommensurate 7M modulation, the tenfold superstructure can be produced in such a way that a set of ten consecutive average unit cells of the 1M martensite are sheared into the corresponding waved subcells of the 7M martensite by a monoclinic angle. Certainly, the same indexed planes and directions -expressed respectively in the 5 distinct subcells of the 7M martensite and in the average unit cell of the 1M martensite -exist somewhat angular deviations between them, as shown in Fig. 3.17. It is seen that the angular deviations both for plane (Fig. 3.17a) and in-plane direction (Fig. Once an assumed OR is proved to be valid for the martensitic transformation, we can predict the possible number of martensite variants within one original austenite grain according to Eq. (3.3). Because of the cubic symmetry of austenite and the monoclinic symmetry of the 7M martensite, there are at most 24 physically distinct martensite variants inherited from the same austenite grain under the Pitsch OR. As the EBSD analysis has revealed that only 4 variants appear in each variant colony (Fig. 3.11a) and they are twin-related one another, these 24 variants can be divided into 6 groups. Here, we may envisage that the formation of self-accommodated martensite variants in one initial austenite grain is realized firstly by combination of 4 twin-related variants in an individual group and then by combination of different groups over the entire grain, ensuring the minimum transformation strain and hence the lowest transformation energy consumed. Summary It has been demonstrated that the EBSD technique can be used as an advanced tool for unambiguously determining the orientation relationships of martensite Such an approach to determine the OR from measured orientations of martensite variants -irrespective of the presence of retained austenite -can be easily adapted to various martensitic transformations that produce martensite with modulated superstructure. Non-modulated martensite NM martensite only exhibits a negligible magnetic shape memory strain, however, it is reported that NM martensitic alloys possess better mechanical properties than modulated martensite and some of them have good magnetocaloric effect due to the co-occurrence of the magneto-structural transition. By proper composition modification to obtain the higher martensitic transformation temperatures, NM martensitic alloys also display the potential as high-temperature shape memory alloys. In this section, the non-modulated Ni 54 Mn 24 Ga 22 alloys were prepared by arc-melting and suction-cast. The grain sizes were refined through suction-cast, making improvements on the mechanical properties. The morphology and the crystallographic features were further analyzed by electron backscatter diffraction (EBSD) and crystallographic calculation. Phase transformation temperatures and crystal structure Compressive properties The compressive stress-strain curves of the as-cast alloy and the as-suctioned alloy are shown in Fig. 3.20. For the as-cast button, the maximum compressive strength and compressive strain are 670MPa and 8.5%, respectively. However, for the as-suctioned rod, the corresponding values are 990MPa and 18%, respectively. Obviously, the better mechanical properties can be achieved through suction-cast. This could be attributed to the relatively small grain size, less microcracks and porosity in the as-suctioned rod than those in the as-cast button. of the neighboring grain. It is also found that some cracks appear among the martensite plates, as shown in Fig. 3.21b, suggesting that the cracks can also nucleate and propagate inside the martensite plates, leading to intragranular fracture. The coordinate frame (X 0 -Y 0 -Z 0 ) refers to the macroscopic sample coordinate frame. For the paired fine lamellae in each plate, they were found to be compound twin related according to the misorientation calculations. An example of the misorientation calculation results corresponding to two fine lamellae in plate P1 was given in Table 3.11. It is seen that there are two sets of ~180° rotation and the corresponding rotation axes are perpendicular to each other, indicating that the two variants are twin-related. Microstructure One ~180° rotation axis is near the normal of {112} Tet with 0.40° deviation and the other is near <11 1 > Tet with 0.32° deviation. The twinning elements are unambiguously determined according to the minimum shear criterion and displayed as follows: K 1 = {112} Tet ; K 2 = {11 2 } Tet ; 1 = <11 1 > Tet ; 2 = <111> Tet ; P = {110} Tet ; s = 0.393. Determination of inter-plate and inter-lamellar interface The interfaces between the neighboring plates (marked as red line in Fig. 3.23b) were determined using the indirect two-trace method [113]. Table 3.12 shows the calculated indices of 14 inter-plate interfaces expressed in the orthonormal basis. The mean interface indices were determined to be {0.565129, -0.499292, 0.656763}. In the tetragonal basis, the mean value of the determined inter-plate interface plane is {1, -0.883501, 1.998241} Tet , which is 3.15° from the {1 1 2} Tet plane, indicating that the inter-plate interface is not fully coherent. The interfaces between the neighboring lamellae (marked as green line in Fig. 3.23b) were also calculated. The calculated results show that the twin interface planes are in good agreement with the {112} Tet twinning plane thus coherent. Summary Ni 54 Mn 24 Ga 22 alloys were prepared by arc-melting and suction-cast. The two processed alloys have the identical tetragonal crystal structure, but the grain sizes were refined through suction-cast, resulting in an increase of transformation hysteresis and an improvement of mechanic property. The fractures occur mainly along the austenite grain boundaries. The adjacent plates have the misorientation of 80°~85° around <110> Tet axes. Locally, there are four types of martensite plates and each plate consists of paired fine variants, thus totally eight variants can be found in a martensite colony. The paired fine variants in each plate were found to be compound twin related with the {112} Tet as the twinning plane and the <11 1 > Tet the twinning direction. The inter-plate interfaces are close to {1 1 2} Tet plane but with ~3° deviation, while the interfaces of two paired fine variants are in good agreement with {112} Tet twinning plane. Chapter 4 Austenite-7M-NM transformation 4.1 Formation of self-accommodated 7M martensite Generally, in a transformation from high-symmetry austenite to low-symmetry martensite, more than one martensite variants can be induced in the same austenite grain to minimize the macroscopic transformation strain. Since the crystal structures of two phases are different, the elastic strain is generated by crystal lattice mismatch between martensite and its parent phase, as well as between differently oriented variants of martensite. Reducing the strain energy is an essential factor in the nucleation and growth processes of martensitic transformation, which can be achieved through the formation of unroated and undistorted phase boundaries and the self-accommodation of martensite variants. In this section, we selectively prepared the polycrystalline Ni-Mn-Ga alloys with the nominal composition of Ni 53 Mn 22 Ga 25 , for the observation of the morphologic characters associated with martensite transformation at room temperature. Detailed microstructural investigation on the co-existed austenite and incommensurate 7M modulated martensite was performed by EBSD measurements. The formation of diamond-shaped martensite was evidenced at the beginning of the transformation and the mechanism of self-accommodation during nucleation and growth was further discussed. Martensitic transformation temperatures and crystal structure The Then the "diamond" gradually grows into the paired plates, as shown in Fig. 4.5b. Occasionally, it was also found that the growth of the martensite variants can be realized by the formation of the fork configuration (A:D or B:C pair), as shown in Fig. 4.5c, where one thin plate bifurcates away from the other plate. The adjacent two bifurcated thin plates also tend to form into a spear. With the abovementioned two types of growth manner for the "diamond", a large number of martensite plates form and they always keep spear configuration adjacent to the austenite matrix, as shown in stacking faults on the basal plane can be recognized in the martensite plates. The occurrence of the stacking faults should be as a kind of complementary shear [117] to accommodate the transformation strain and to achieve the formation of an invariant plane between the austenite and the martensite. Confirmation of the orientation relationship between austenite and 7M martensite By crystallographic calculation using the orientation data of the two phases, we confirmed the energetically favorable OR between the austenite and 7M martensite to Experimental determination of habit plane The habit plane is believed to be an invariant plane unrotated and undistorted on the macroscopic scale. In the present study, the habit plane between the austenite and the 7M martensite was determined by the indirect two-trace method [113]. To achieve statistical reliability, sixteen habit plane normals were calculated. The calculation results, expressed in the coordinate frame of austenite and listed in Calculation based on the crystallographic phenomenological theory In general, the martensitic transformation tends to proceed in a self-accommodated manner to minimize the total transformation strain when no external stress is applied. For further analyzing the martensitic transformation crystallography and the self-accommodation mechanism of 7M martensite, the crystallographic phenomenological theory of martensitic transformation [23] was used to perform the theoretical calculation, which has been well applied to the prediction of crystallographic parameters in many shape memory alloys [118][119][120][121][122]. This prediction is based on the assumption that the interface between austenite and its product phase is invariant on a macroscopic scale. In the present austenite (cubic)-7M martensite (monoclinic) system, the lattice invariant shear for the calculation was supposed to occur on the basal plane in either of the two opposite crystallographic directions, which correspond to {101} A <10 1 > A system in the parent phase. As the 7M plate possesses the modulated structure, the structure itself (lattice modulation) provides the main contribution of invariant shear and the remaining is balanced by the stacking faults. The faulted nature of martensite plate has been revealed by TEM observation, as illustrated in Fig. 4.7. By inputting the lattice parameters of austenite and martensite for the phenomenological calculation, the results show that there are 24 pairs of habit plane normal and shape deformation direction due to the symmetry of the cubic system, corresponding to 24 variants. The 24 variants are divided into 6 groups and each group is composed of four twin-related variants distributed around one of the {101} A pole. As an example, Table 4.2 shows the habit plane normals, shape deformation directions and shape deformation matrices of variant A, B, C and D in the group around (101) A pole. The magnitude of the shape deformation for each variant is 0.09596, but the shape deformation direction is different. The theoretical habit plane normal was predicted to be {0.720332, 0.692379, 0.041627} A , which is consistent with experimental results with 2.11° deviation. Apparently, the shape deformation matrix of one single variant is far from the identity matrix, and the magnitude of deformation, i.e. 0.09596, is quite large. Supposing that four variants constitute a diamond group with equal volume fraction (1/4), the total shape deformation matrix can be approximated by the method of summation. Taking (101) A group as an example, the total shape deformation matrix is The total shape deformation matrix is quite close to the identity matrix, thus the transform strain are effectively cancelled out by the combination of the four variants. Hence, the "diamond" as a whole is an energetically feasible, self-accommodated combination when formed from the parent phase. Further compensation of transformation strain can be achieved by the combination of different variant groups. The total shape deformation matrix of all the 6 variant groups is The total shape deformation matrices of type and type twin pair are quite close to the identity matrix compared with the deformation matrix of a single variant. Thus, they are self-accommodated. However, the total shape deformation matrix of the compound twin is not close to the identity matrix. Thus, the compound twin can not effectively accommodate the transformation strain. The self-accommodation is a process that minimize the transformation strain hence the strain energy. This process should guarantee an invariant interface between austenite and martensite. Considering the diamond model and without losing coherence at the phase interfaces, the initial growth is through the coordinated mutual movement of the four habit planes into the austenite matrix. This expansion will be blocked due to the accumulated elastic strain caused by the increased volume fraction of the transformed martensite, as elastic strain energy is proportional to the volume of the martensite [123]. Then the possible route of further growth for the diamond can be through the extension of either type twin (A:C or B:D) or compound twin (A:D or B:C), if the chemical driving force is sufficient. As compound twin is not self-accommodated, it is not favorable that the growth of a variant group is through the extension of compound twin. Then, the spears (type twin) should be responsible for the variant growth. Comparing the total shape deformation matrix over variants A and C (or B and D) with that over variants A, B, C and D, there still exist some residual unaccommodated elastic strain [124]. This residual strain can be further accommodated by introducing the type twin pair, as the situation in Fig. 4.5a and b, which renders the growth of martensite through the forward progression of the spear (type twin). As the martensite "diamond" only consists of type and compound twin systems, the type twin should be a secondary twin after further shear of variant pair A and D or B and C during the growth process. Locally, type twin and type twin are bridged by compound twin and the compound twin interface has always appeared with its neighboring Type I and Type II twin interfaces as a whole in the final martensite microstructure. If the chemical driving force cannot overcome the elastic barrier to induce the forward movement of spears, the martensite tends to form much thinner plates for the reduction of the elastic strain energy, as the thin plate bifurcates from the other plate in Fig. 5c. Summary The structural and microstructural characters from austenite to 7M martensite in a analysis. Results showed that the modulated martensite has its own crystal structure with an incommensurate 7M modulation, other than the nanotwin combined structure. Based upon the XRD measurement and the concept of adaptive phase, Kaufmann et al. [81] examined the co-existing austenite, 7M and NM martensite in an epitaxial Ni-Mn-Ga film. They concluded that the 7M modulated martensite can be simply constructed from nanotwinned lamellae of a tetragonal martensite phase with (5 2 ) 2 stacking sequence, ruling out the existence of the independent modulated structure. The 7M modulated martensite phase built up from microscopic twins may evolve into the final NM martensite phase by thickening the nanotwin width through branching [81]. Notably, the controversy on long-period modulated structure remains unclarified for many alloy systems. Indeed, it is very difficult to make a direct validity discrimination of the above two structure models by either diffraction or thermodynamic measurements. On one hand, the formation of the modulated Lattice modulation and nanotwin combination The The two structure models of the modulated phase are further tested with the adaptive phase criteria [130]. The lattice constants of the 7M (IC) structure and the nanotwin combination are expressed in the cubic austenite coordinate system and compared with those calculated using the equation (a ad = c Tet + a Teta A ; b ad = a A ; c ad = a Tet ) proposed by the nanotwin combination theory [130]. The results are given in Table 4.3. Clearly, the two sets of lattice constants also possess excellent agreement with the calculated ones. This indicates that the verification criterion of the nanotwin combination theory is not sufficient to discriminate the two structures for the modulated phase. give the same results concerning the types of twins, but some differences appear in the twinning elements, as detailed in Table 4.4. EBSD measurements on the coexistence of three phases Inter-plate interface To visualize the atomic match on the plate interface, the atomic correspondences of the type I twin interface is constructed under the two structure models and displayed in Fig. 4.12. For the 7M(IC) structure (Fig. 4.12a), some of the atoms on the interface are slightly deviated from their exact equilibrium position due to the structure modulation. The plate interface is basically coherent and thus should have low interfacial energy. This coherence surely renders a good mobility in the twinning and detwinning process. The twinning configuration (the twinning shear plane and direction parallel to the plate interface) ensures that if only two variants (plates) exist in the material, one variant can be easily reoriented to the other by twinning or detwinning through the plate interface movement. This feature has been well observed in many magnetic field driven shape memory experiments in Ni-Mn-Ga alloys [8,89,90]. In contrast, if the modulated martensite has the (5 2 ) 2 nanotwin combination structure, although the crystal structure differs slightly from the 7M(IC) structure, the plate interfacial feature and the in-plate structure (atomic shuffling in 7M(IC) plates and nanotwins in nanotwin combined plates) will be very different. As shown in Fig. To further quantify the atomic misfits at the plate interface under the two structure models, the average atomic displacement ( r ) of the atoms on the plate interface from their equilibrium positions was calculated using the L2 norm: % 100 ) r ( N 1 r N 1 i 2 i . (4.7) Here, N represents the number of the atomic layers in one modulation period (N = 20 for the 7M(IC) structure and N = 14 for nanotwin combination structure); r i is the amount of atomic displacement of the i th atom on the plate interface from its equilibrium position. The results show that the average displacement is 1.52% for 7M(IC) structure and 3.81% for the nanotwin combination structure. It is clear that the deviation of the nanotwin combination at the plate interface is much larger than that of the 7M(IC) structure. In consequence, a high plate interfacial energy resulting from the atomic displacement for the nanotwin combination model could be expected. The thickening of the NM plate to reduce the specific interface displayed in Fig. 4.10a indirectly proves that the plate interface of the nanotwin combined martensite should possess high interfacial energy. Furthermore, the large mismatch on the plate interfaces under the nanotwin combination model surely imposes significant constraints on the variant reorientation. For plate reorientation to achieve shape memory effect, it has to overcome two barriers. Firstly, no uniform magnitude of shear and shear system available for an easy coordinated atom displacement in the plate reorientation that is essential to a reversible external field induced shape change. Secondly, the coherent nanotwin interfaces that are densely distributed inside the plate act as pins to the movement of the plate interface. Under such circumstances, the plate reorientation resistance is inevitably enhanced. The loss of magnetic field induced shape change in tetragonal NM martensite is also well recognized in many Ni-Mn-Ga alloys [89]. Such experimental evidences indirectly provide support to the above analysis. The main argument for the nanotwin combination model is that the nanotwin combined structure could offer reduced lattice mismatch with respect to the lattice of the parent austenite [130]. Actually, we will see that the 7M(IC) model offers even smaller lattice mismatch. The lattice misfits of the two structure models with respect to the cubic austenite are expressed in stress-free strain tensors [130]. The non-zero terms in these tensors ( 11 , 22 , 33 , ) are shown in Table 4.5. It is evident that the nanotwin combination has a higher lattice mismatch than the 7M(IC) modulated structure. All the above results have demonstrated that the modulated martensite in the Ni-Mn-Ga alloy has its own crystal structure, other than the nanotwin combination of the tetragonal NM martensite. The transformation from the modulated martensite to the final NM martensite is realized by further lattice distortion, and this might significantly degrade the magnetic field-induced shape memory performance. Summary In summary, we present experimental evidence for the thermodynamic stability and the crystal structure of the long-period modulated martensite in Ni-Mn-Ga alloys. It is proved that the modulated martensite is an intermediate state between the austenite to the NM martensite. It possesses its own crystal structure, instead of the nanotwins of the NM martensite (tetragonal simple structure) proposed by the nanotwin combination theory. The 7M(IC) structure can generate a reduced number of local variants and more favorable configuration for twinning and detwinning. This is essential to the attainability of the magnetic field induced shape change. The present results provide unambiguous evidence to clarify the long debated issue concerning the nature of the long-period modulated phases. plates stretch in roughly the same direction. In general, the plates of the 7M martensite are thinner than those of the NM martensite. When approaching the NM martensite, some 7M plates tend to thicken and the width of 7M plates is close to that of NM plates, indicating that the transformation from 7M to NM martensite is accompanied by a reduction of the specific area of the NM plate interface. This tendency suggests that the plate interfacial energy of the NM martensite is higher than that of the 7M martensite. By thickening the plates, the total interfacial area is reduced and thus the interfacial energy is lowered. The coordinate frame (X 0 -Y 0 -Z 0 ) refers to the macroscopic sample coordinate system. For the tetragonal NM martensite as analyzed above, the neighboring fine lamellae in each plate are also found to be compound twin related with the (112) Tet as the twinning plane and the [11 1 ] Tet the twinning direction. The full twinning elements are given in Table 4.6. The inter-lamellar interfaces (as marked with green line in Fig. 4.13a) were determined to be the twinning planes ((112) Tet ) and thus coherent. At the inter-plate interfaces of NM martensite (as marked with red line in Fig. 4.14a), two thick lamellae and two thin lamellae from neighboring plates intersect. When the thick lamellae meet, i.e. L1 and L3, as illustrated at the right upper corner of Fig. 4.14a, they appear to have (1 1 2) Tet [ 1 11] Tet twin relationship but with a certain degrees of deviation in the twinning plane and in the twinning direction. The angular deviations calculated using the experimental orientation data are 5.91° between the twinning planes and 3.89° between the twinning directions, respectively. Further calculation manifests that the plate interface of the NM martensite is oriented close to the (112) Tet planes of the thick lamellae with ~3° deviation. Using (112) Tet Orientation relationship between 7M martensite and NM martensite To further precisely reveal the transformation process and the atomic arrangements of the 7M and the NM plate interfaces, the OR between the 7M and the NM martensites were determined. Calculations show that the 7M martensite and the NM martensite possess a specific OR with (001) 7M //(112) Tet and [100] 7M //[11 1 ] Tet , i.e. the shuffling plane in one 7M variant is parallel to the twinning plane of its corresponding NM lamellae and the shuffling direction is parallel to the twinning direction. The OR secures a one-to-one correspondence between the 7M plate and the NM plate, as displayed in Transformation mechanism from 7M martensite to NM martensite The present experimental results allow evidencing the transformation sequence from austenite to NM martensite bridged by the 7M martensite and the microstructure evolution during this phase transition. The intrinsic OR between the 7M and NM martensites with (001) 7M //(112) Tet and [100] 7M //[11 1 ] Tet indicates that the structure change from the 7M (monoclinic) to the NM (tetragonal) can be viewed as further reshuffling of the atoms on the (001) 7M along the [100] 7M or the [ 1 00] 7M direction. As the crystal structure of the 7M is different from that of the NM, the reshuffling should also be accompanied with certain distortion of these planes and their interplanar distance. Consequently, one 7M variant finally transforms to one NM plate composed of paired twin-related fine variants. Four 7M variants correspond to 8 NM lamellae. Based on such an ideal OR between the two phases, the plate interfaces of the 7M and the NM martensite was reconstructed to further reveal the structure change at the plate interface during this phase transition. As an example, Fig. 4.16 displays the atomic correspondence of the plate interface between variant A and C (type I twin) of 7M martensite and the plate interface between plate P1 and P3 of NM martensite following the transformation OR. It is seen from Fig. 4.16a that although some of the atoms on the interface are slightly deviated from their exact equilibrium position due to the structure modulation, the 7M plate interface is basically coherent. Therefore, the interfacial energy should be relatively low. However, when the 7M plates transforms to the NM, the misfit at the interface becomes non neglectable, as shown in Fig. 4.16b. The plate interface thus becomes non-coherent. This atomic misfit will surely result in high interfacial energy and thus contributes as energy barrier to the transformation. It is due to this high atomic misfit that the 7M to NM transformation is accompanied by the thickening of the NM martensite plates, which can reduce the specific interfacial area of the NM martensite and thus lower the total interfacial energy of NM plates, as evidenced in Fig. 4.13. Therefore, the 7M modulated structure is advantageous in the plate interfacial energy. Another energetically advantage of the 7M modulated structure is that it offers smaller lattice distortion with respect to the parent lattice. This lattice distortion can be quantified with the corresponding stress-free strain tensors [130] for the transformation from the austenite to the 7M martensite and from the austenite directly to the NM martensite. The non-zero terms in these tensors ( 11 , 22 , 33 , ) are shown in Table 4.7. It is clear that the lattice distortion accompanying the transformation from the cubic austenite to the tetragonal martensite is much larger than that occurring in the transformation from the austenite to the 7M martensite. This lattice distortion gives rise to volume dependent elastic energy that contributes as energy barrier to the transformation. Together with the large plate interfacial energy, this volume dependent elastic energy imposes insurmountable barriers to the transformation from the austenite to the normal tetragonal martensite (NM martensite). Therefore, the formation of the intermediate 7M modulated phase is unavoidable in bridging the parent austenite to the final NM martensite and thus a two-step lattice distortion is experienced in the transformation from the austenite to the NM martensite. From the microstructural configurations of the two kinds of martensite, it is clear that the reorientation of the 7M plates (variants) that gives rise to the shape memory effect is much easier than those of the NM. For 7M, there are 2 advantages. First, 7M plate corresponds to one orientation variant and the plate interface is coherent. Second, the shear system possessing a uniform and much smaller twinning shear is parallel to the plate interface. If we can obtain only two alternately distributed variants, the magnetic field can easily reorient one variant to the other by twinning or detwinning through plate interface movement. In contrast, the situation in the NM configuration is very different. Two NM plates correspond to four orientation variants (lamellae). The inter-plate interfaces and the lamellar interfaces are in an inter-pinned position. Moreover, the twinning system is not consistent with the plate interface and the twinning shear of the lamellar variants is much higher. Clearly, the pinning of the interfaces, the large atomic mismatch at the plate interface and the much larger twinning shear all act as plate or lamellae reorientation resistance. In such a context, it is not difficult to understand why modulated martensite of Ni-Mn-Ga alloys offers superior magnetic field induced reversible shape change; whereas the tetragonal martensite does not as have been demonstrated by many experiments [89]. As the 7M modulated martensite of Ni-Mn-Ga alloys is essentially important for the field induced shape memory applications, its thermodynamic metastablility has become a critical issue for the development of the Ni-Mn-Ga alloys. Its existing temperature window is of particular importance. Measures should be taken to enlarge this temperature window. From the examinations of the transformation path and the possible energy barriers in this study, it is seen that the lattice misfit between the 7M and NM (Table 4.7) and the atomic mismatch on the NM plate interfaces can be the controlling factors to postpone the 7M to NM transformation, hence enlarge the existing temperature range of 7M. This could be achieved by adjusting the composition or by alloying new elements. Summary In summary, we clarified the roles of the Ni-Mn-Ga 7M modulated martensite in The available temperature window for the stable existence of the 7M martensite depends on the energy barriers related to the lattice mismatch between the 7M and the NM martensite and the atomic misfit on the plate interfaces of the NM martensite. Perspective Since Ni-Mn-Ga ferromagnetic shape memory alloys are newly developed system, there are still some research margins. In these alloys, giant magnetic field induced strains were only obtained in the bulk single crystals with modulated structure. Due to the complexity of producing single crystal, the high fabricating cost and the severe composition segregation hinder their practical application. In contrast, the manufacture of polycrystalline alloys is much simpler and easier to be implemented in practical production, but the randomness of the crystallographic orientation of the polycrystalline alloys leads to the loss of the magnetic shape memory performance. Texturation of polycrystalline alloys has become a promising way to improve the shape memory performance comparable to that of the single crystal. To achieve this goal, it is necessary to introduce innovative fabricating routes or external field training processes. Moreover, the low blocking stress and the intrinsic brittleness of Ni-Mn-Ga alloys act as a great hindrance to the practical applications. To increase the blocking stress and improve the ductility, appropriate alloying is quite necessary. Since the inter-variant interfaces are crucial for the magnetic shape memory effect, further work should pay more attention on revealing the details of inter-variant interface by considering the lattice modulation. Therefore, detailed TEM or HRTEM characterizations of the interface structure are needed. Résumé Etendu Les alliages à mémoire de forme ferromagnétiques sont de nouveaux matériaux L'écart de la maille des deux modèles de structure par rapport à celle de l'austénite cubique ont été calculées et exprimées en tenseurs de déformations sans contraintes ( 11 , 22 , 33 , ) [9]. Les termes non nuls de ces tenseurs sont indiqués dans le tableau 5. Il est évident que la combinaison de nano-macles a un décalage de maille supérieur à celui de la structure modulée 7M(IC). Tous les résultats ci-dessus ont montré que la martensite modulée dans l'alliage Ni-Mn-Ga a son propre structure cristalline, autre que la combinaison de nano-macles de la martensite NM tétragonale. Tableau 5 Tenseurs de déformations entre la matrice et les phases produites pour la transformation de l'austénite en 7M et de l'austénite en phase adaptive (structure de combinaison de nano-macles). Austenite -7M modulation Austenite -nanotwin combination Fig. 1 . 1 . 11 Fig. 1.1. Bain distortion (FCC to BCC) of martensite. The transformation involves the contraction of the parent phase along the z direction, and the expansion in x and y directions, respectively. Fig. 1 . 2 12 Fig.1.2 The geometrical configuration of K 1 , K 2 , 1 , 2 and P. Fig. 1 . 3 13 Fig.1.3 Crystal structure of Ni 2 MnGa L2 1 cubic austenite 1 ) 1 Chernenko et al.[41] carefully investigated the composition dependence of martensitic transformation temperatures of Ni-Mn-Ga alloys and concluded that (i) at a constant Mn content, Ga addition lowers T M ; (ii) Mn addition at a constant Ni content increases T M and (iii) substitution of Ni by Mn at a constant Ga content lowers T M . Depending on the martensitic transformation temperatures and transformation latent heat, these alloys have been conventionally classified into three groups[61].Group I (e/a<7.55) is composed of alloys with transformation temperatures well below the room temperature and the Curie point. Alloys in Group II (7.55 e/a 7.7) have the transformation temperatures around room temperature and below the Curie point, while the alloys in Group III (e/a>7) have the transformation temperature above the Curie point. Fig 1 . 4 14 Fig 1.4 Illustration of the rearrangement of martensite variants under a magnetic field in ferromagnetic shape memory alloys (FSMAs) [58]. With the mature of SEM/EBSD technique, large-scale spatially resolved orientation examination has made the correlation between microstructure and crystallographic orientation possible, which overcomes the above mentioned limitations. The merits of EBSD measurements lie on that, firstly, it provides an alternative means for verifying the crystal structure information, as the EBSD-based orientation determination requires the complete crystal structure information including the lattice constants and the atomic position of each atom in the unit cell; secondly, it enables the automatic orientation mapping of individual martensite variants, which correlates the crystal structure and orientation information with the morphologic features on an individual variant basis; thirdly, it allows an unambiguous determination of the orientation relationships of adjacent variants and the twin interface planes, and thus a full crystallographic analysis on a bulk sample with statistical reliability. Cong et al. systematically investigated the twinning relationship of NM martensite in a Ni 53 Mn 25 Ga 22 alloy by means of SEM/EBSD [101, 102]. The twinning elements ( 111 ) 111 A //(101) Tet and [1 1 0] A //[11 1 ] Tet , was predicted by Cong et al. based on experimental measurements and crystallographic calculations, representing a Fig 2 . 2 1a. To reduce microcracks and porosity, some ingots were remelted and then fast solidified by suction casting into a chilled copper mold with 6mm in diameter, through which a rod was obtained, as shown in Fig 2.1b. Fig 2 . 1 21 Fig 2.1 Macroscopic appearance of prepared alloys. (a) as-cast; (b) suction-cast. 50 . 1 501 Mn 28.3 Ga 21.6 (at. %). According to the DSC measurements, the austenite to martensite transformation started at 43.8 °C (M s ) and finished at 30.0 °C (M f ) upon cooling. The reverse transformation started at 41.1 °C (A s ) and finished at 51.3 °C (A f ) upon heating. Fig. 3 . 3 Fig. 3.1a shows the powder X-ray diffraction (XRD) pattern of the Ni 50 Mn 28 Ga 22 Fig. 3 . 3 Fig.3.1 XRD patterns of Ni 50 Mn 28 Ga 22 alloy at room temperature: (a) measured; (b) recalculated according to the commensurate 5M superstructure; (c) recalculated according to the non-modulated tetragonal crystal structure. The insets show the unit Fig. 3 . 3 Fig. 3.2a shows a typical backscattered electron (BSE) image of the 5M martensite taken at room temperature. According to the gray-level contrasts in the image, the microstructural features can be characterized as alternatively distributed broad plate with 10-20 micrometers in width and separated by inter-plate boundaries.Moreover, the plates with light mean contrast consist of sub-plates and those with dark mean contrast consist of blocks. It is seen from Fig.3.2b -the zoomed image of the marked frame in Fig.3.2a -that the broad plates are composed of pairs of thin lamellae with the thickness in the nanometer range. In each pair, one plate is thicker than the other. The thin lamellae are bound by long and short inter-lamellar interfaces.The bending at the short inter-lamellar interfaces is accompanied with contrast change in the BSE image. The short interfaces mark the sub-plates interfaces and the block interfaces. According to the BSE contrast of nanoplates, there are always four types of martensite variants that are locally interconnected. As the martensite is a mono-phase and has a homogeneous chemical composition, the contrast changes in the BSE image should be attributed to the orientation variations. This will be clarified by the Fig. 3 . 3 Fig.3.3 EBSD maps of fine lamellae reconstructed according to orientation identification with (a) the commensurate 5M superstructure and (b) the non-modulated tetragonal crystal structure. There are four types of variants (designated as A, B, C and D) in (a), but only two types of variants in (b). The coordinate frame (X 0 Y 0 Z 0 ) refers to the macroscopic sample coordinate system. Fig. 3 . 4 34 Fig.3.4 Kikuchi patterns showing the orientation difference across bended fine lamellae: (a) and (b) acquired from two measured positions marked in the inset of BSE image in Fig. 3.4 (a); (c) and (d) recalculated patterns using the commensurate 5M superstructure; (e) and (f) recalculated patterns using the non-modulated tetragonal crystal structure. A set of three Euler angles with respect to the sample coordinate system are listed in each recalculated pattern. ( angle-axis) for each variant pair. Among them, variant pairs A:C and B:D, A:B and C:D, A:D and B:C have almost identical sets of rotations. It is seen that each pair of variants possesses at least one 180° rotation, suggesting that all the variant pairs are twin related according to the classical definition of twinning[START_REF] Christian | [END_REF]112]. By further transforming the rotation axes into the monoclinic crystal basis, it demonstrates that for variant pair A:C (or B:D) the 180° rotation axis is close to the normal of the rational plane (1 2 5 ) 5M (with 0.39° deviation), whereas for the variant pair A:B (or C:D) the 180° rotation axis is close to the rational direction [ 55 1] 5M (with 0.33° deviation). As for the variant pair A:D (or B:C), two 180° rotation axes exist that are perpendicular to each other, being close to the normal of the rational plane (105) 5M (with 0.23° deviation) and the rational direction [ 5 01] 5M (with 0.15° deviation), respectively. The small deviations from the rational planes/direction with low indices may be attributed to the experimental inaccuracy caused by the EBSD measurements.Following the rationality criterion of the Miller indices of the twinning elements for different types of twin[START_REF] Christian | [END_REF]112], one could infer that variant pair A:C (or B:D) has twin relationship of type I, A:B (or C:D) of type II twin, and A:D (or B:C) of compound twin. 3.3a), the bended thin lamellae are revealed as martensite variants connected by compound twin interfaces. Locally, a compound twin interface intersects its neighboring type and type twin interfaces. This morphological feature of bent lamellae with changed orientation could result from the self-accommodation of thin lamellae during martensitic transformation to reduce the transformation strain and minimize the resistance. However, such a configuration does not favor the global reorientation of variants through twin boundary movement under an actuating magnetic field, since compound twin interface represents a kind of interface pin. Moreover, the (105) 5M plane that contains the [010] 5M direction (i.e. the b axis -the easy magnetization direction) is the invariant twinning plane of compound twins. As two twin-related crystals are in mirror symmetry with respect to the twinning plane (105) 5M , their b axes are parallel to each other. Thus, it becomes almost impossible to trigger the reorientation of one crystal with respect to the other through the motion of the compound twin interface under a unidirectional actuating magnetic field, i.e. the compound twin interface is inert. Fig. 3 . 5 . 35 Fig. 3.5. Illustrations of (a) the unit cell of cubic austenite, (b) the supercell of monoclinic 5M modulated martensite consisting of five subcells (outlined by dash lines), and (c) the reduced average unit cell (ignoring the lattice modulation). Fig. 3 . 3 Fig. 3.5a and b illustrate the lattice correspondence between austenite and 5M martensite. The high-temperature austenite possesses a cubic L2 1 Heusler structure (Fm-3m, No. 225) with the lattice parameter a A =5.84Å [21, 114]. The 5M martensite (a 5M =4.226Å, b 5M =5.581Å, c 5M =21.052Å and =90.3°) has a monoclinic superstructure (P2/m, No. 10) and the supercell of 5M martensite can be decomposed into five consecutive subcells along the c-axis (denoted as C 1 , C 2 ……C 5 in Fig.3.5b). The monoclinic crystallographic axes of 5M martensite align along the Fig. 3 . 3 Fig. 3.6. {001} standard stereographic projections of austenite orientations in the macroscopic sample coordinate frame calculated from the four martensite variants in variant group g1 under Bain, K-S, N-W and Pitsch relations, respectively. The common poles are enclosed in squares. The angular deviations correspond to Pitsch OR due to structural modulation are shown in Fig. 3.7. It is seen that the angular deviation for both (1 2 1 ) 1M plane and [ 1 1 1] 1M direction increases with the difference in the monoclinic angle between the "averaged unit cell" and each subcell. The structural modulation generates ~1-2.6° angular deviations for the corresponding plane and ~1-2.5° deviations for the in-plane direction in each subcell. Fig. 3 . 7 37 Fig. 3.7 Angular deviations of the same indexed (1 2 1 ) 1M and [ 1 1 1] 1M in each subcell from those of the average unit cell. martensite and phase transformation OR in a bulk polycrystalline Ni 50 Mn 28 Ga 22 alloy were investigated. The crystal structure analysis has shown that the martensite possesses a commensurate 5M modulated structure, the superlattice of which is composed of 5 subcells. The microstructure of the 5M martensite can be characterized by broad plates with alternatively distributed fine lamellar-shaped variants. The correct and accurate EBSD orientation indexing was made using the precise crystal structure information. From the microstructure reconstructed with the individually measured orientations, the four types of martensite variants A, B, C and D with distinct orientations were revealed. All the variants are twin related, i.e. the variant pair A and C (or B and D) are in type I twin relation, A and B (or C and D) in type II twin relation, and A and D (or B and C) in compound twin relation. Moreover, the twin interface planes were determined to be in coincidence with their respective twinning planes (K 1 ). Based on the local orientations of the individual martensite variants measured by EBSD, the orientations of the parent austenite were calculated using the assumed ORs between the cubic austenite and the monoclinic martensite. Fig 3 . 8 38 Fig 3.8 Compressive stress-strain curves of the as-cast and the as-suction alloys. Fig. 3 . 3 Fig.3.9 displays the measured and recalculated powder X-ray diffraction (XRD) patterns of the Ni 50 Mn 30 Ga 20 suction-cast alloy at room temperature. The profile of the measured XRD pattern (Fig.3.9a) is consistent with that reported by L. Righi et al.[75], suggesting that the alloy may have an incommensurate 7M modulated (7M(IC)) structure. The superlattice consists of 10 unit cells along the c-axis, belonging to the monoclinic space group P2/m (No.10). With the information of the atomic coordinates in the superlattice (as shown in Appendix II)[75], the measured XRD pattern was Fig. 3 . 3 Fig. 3.9 XRD patterns of Ni 50 Mn 30 Ga 20 suction-cast alloy at room temperature: (a) measured; (b) recalculated using the incommensurate 7M modulation (7M(IC)) structure model. The inset shows the schematic illustration of superstructure. 3. 2 . 3 MicrostructureFig. 3 . 233 Fig. 3.10a shows the typical backscattered electron (BSE) image of the Ni 50 Mn 30 Ga 20 suction-cast alloy taken at room temperature. It is clearly seen that the alloy presents the plate-like morphological features. The martensite plates are clustered in colonies in equiaxed shape and one colony or several colonies are located within an original austenite grain, which is different from the microstructure of 5M martensite. For the 5M martensite, the martensite colonies are in lamellar shape. Further TEM observations manifest that the 7M martensite plates possess stacking faults as internal sub-structures, as shown in Fig. 3.10b. Fig. 3 . 3 Fig. 3.11 (a) Orientation map of Ni 50 Mn 30 Ga 20 alloy taken at room temperature. Four types of martensite variants with different colours are designated as variant A, B, C, and D, respectively. The coordinate frame X 0 -Y 0 -Z 0 refers to the microscopic sample coordinate frame. (b) Kikuchi line pattern acquired from one of the variants; and (c) 10 1] 7M (with 0.62° deviation), suggesting that the two variants have a rotation of 180° with each other about this rational direction. Following the definition of twinning[START_REF] Christian | [END_REF]112], one can deduce that variant pair A:C (or B:D) are type I twin; while A:B (or C:D) type II twin.As for the variant pair A:D (or B:C), one of the 180° rotation axes is close to the normal of the rational plane (1 0 10) 7M (with 0.25° deviation) and the other is close to the rational direction [10 0 1] 7M (with 0.39° deviation), indicating that the variants A:D (or B:C) are compound twins to each other. Fig. 3 .Fig. 3 . 33 Fig. 3.12a and b illustrate the lattice correspondences between the cubic L2 1 structure of the austenite and the monoclinic structure of the incommensurate 7M modulated martensite. The high temperature austenite possesses the cubic (Heusler) (3. 1 ) 1 by using the measured orientations of inherited martensite variants. To achieve statistical significance, the EBSD orientation data of seven groups of martensite variants (numbered as g1, g2, …, g7), were measured from different colonies and served as initial input data and the results were presented in the {001} standard stereographic projection of the cubic austenite in the macroscopic sample coordinate frame. As an example, Fig.3.13 displays the three {001} austenite poles calculated from the respective martensite variants A, B, C and D from the same variant group g1. Fig. 3 . 3 Fig. 3.13 {001} standard stereographic projections of austenite calculated from the martensite variants A ( ), B ( ), C ( ) and D ( ) in variant group g1 under (a) Bain, (b) K-S, (c) N-W and (d) Pitsch relations, respectively, in the macroscopic sample coordinate frame The common austenite orientations are enclosed in the open squares. Fig 3 . 14 Fig. 3 . 3143 Fig 3.14 Schematic stacking sequences of the respective planes in austenite and 1M martensite viewed along [1 0 1 ] A and [ 1 1 1] 1M . Fig. 3 . 3 Fig.3.16 Illustration of lattice deformation to achieve the transformation from austenite to 1M martensite under (a) K-S OR and (b) Pitsch OR. The solid line represents the unit cell of austenite and the dashed line represents the average unit cell of 1M martensite. 3.17b) increase with the increased difference in monoclinic angle between the average unit cell of the 1M martensite and each individual subcell of the 7M martensite. Obviously, the K-S and Pitsch ORs differ from each other in the plane deviation as the [10 1 ] A //[ 1 1 1] 1M direction parallelism holds for the two ORs. As shown in Fig. 3.17a, the K-S OR ((011) 1M ) possesses larger deviation than that of the Pitsch OR ((1 2 1 ) 1M ), which means that the K-S OR requires larger atomic reshuffling to achieve the structural modulation. This result may further confirm that the Pitsch OR, i.e. (101) A //[1 2 1 ] 1M and [10 1 ] A //[ 1 1 1] 1M between the austenite and the 1M martensite, or (101) A //[1 2 10 ] 7M and [10 1 ] A //[10 10 1] 7M if referred to the 7M martensite, is more energetically favorable than the K-S OR for the martensitic transformation. Fig 3 . 17 317 Fig 3.17 Angular deviations of the same indexed (a) (1 2 1 ) 1M (Pitsch OR) and (011) 1M (K-S OR) and (b) [ 1 1 1] 1M in 5 subcells (C 1 , C 2 , C 3 , C 4 and C 5 ) with distinct monoclinic angle from those of the average unit cell. variants and the crystallographic characteristics of twin interfaces for materials having modulated superstructure. Detailed analyses on the incommensurate 7M martensite of Ni 50 Mn 30 Ga 20 alloy showed that there exist four types of twin-related martensite variants (A, B, C, and D) that are alternately distributed. All the pairs of variants can be categorized into three twinning modes: variants A and C (or B and D) are in type I twin relation, variants A and B (or C and D) in type II twin relation and variants A and D (or B and C) in compound twin relation. All the twin interfaces are in coincidence with the respective twinning plane (K 1 ). Based on the local orientations of individual martensite variants measured by EBSD system, the orientations of the parent austenite were evaluated using the assumed ORs. By a detailed crystallographic analysis, the energetically favorable OR governing the austenite to incommensurate 7M martensite transformation was revealed to be the Pitsch relation with (101) A //(1 2 10 ) 7M and [10 1 ] A //[ 10 10 1] 7M . Under this determined OR, at most 24 physically distinct martensite variants may be resulted from an initial austenite grain during the martensitic transformation. Notably, in the present work, the first attempt has been made to resolve the ambiguity of the geometrically favorable ORs by examining the lattice discontinuity caused by the phase transformation and the structural modulation. Fig. 3 .Fig. 3 . 33 Fig. 3.18 DSC curve of the as-cast alloy and the as-suctioned Ni 54 Mn 24 Ga 22 alloy Fig. 3 . 3 Fig. 3.19 Powder X-ray diffraction patterns measured at room temperature. (a) as-cast alloy; (b) suction-cast alloy. The inset shows the illustrations of the crystal structure. Fig. 3 .Fig. 3 . 33 Fig. 3.20 Compressive stress-strain curves of as-cast alloy and as-suctioned alloy Fig. 3 . 3 Fig. 3.21 shows the secondary electron images of the fractured surface after uniaxial compressive loading of the suction-cast rod. It can be seen in Fig.3.21a that Fig. 3 . 3 Fig.3.22 (a) Orientation micrograph of Ni 54 Mn 24 Ga 22 suction-cast alloy; (b) band contrast image; (c) misorientation angle distribution covering the measured region; (d) distribution of the corresponding rotation axes with misorientation angle of 80°-85°. Fig. 3 . 3 Fig. 3.22 displays the EBSD measurement results of the suction-cast alloy taken at the room temperature. In the orientation map in Fig. 3.22a, the plates are colored according to their orientation. It is seen that some plates are straight, while others are bent with slight orientation variation. Fig. 3.22b displays the band contrast image of Fig. 3 . 3 Fig. 3.23 (a) BSE image obtained in a martensite colony; four kinds of plates are numbered as P1, P2, P3, P4; (b) Zoomed image of the squared region showing the paired fine lamellae in the martensite plates. The inter-plate and inter-lamellar interfaces are marked with red and green lines, respectively. off-stoichiometric Ni-Mn-Ga polycrystalline alloy with nominal composition of Ni 53 Mn 22 Ga 25 (at.%) was prepared. The composition of the alloy was verified by energy dispersive X-ray spectrometry (EDS) to be Ni 53.4 Mn 21.8 Ga 24.8 . Fig.4.1 shows the measured DSC curves for the Ni 53 Mn 22 Ga 25 alloy on heating and cooling the sample. The forward martensitic transformation start temperature (M s ) and finish temperature (M f ) were determined to be 13.5 and 4.9 , respectively; the inverse martensitic transformation start temperature (A s ) and finish temperature (A f ) were 16.3 and 30.1 , respectively. In addition, much smaller peaks appear in the DSC curve on the cooling and the heating at the temperatures below -10°C (as arrowed in the figure), indicating the possibility of further transformation of the already formed martensite. It is noted that the martensitic transformation temperatures are around room temperature, suggesting that the co-existence of austenite and martensite at room temperature is possible. DSC curves also indicate that the Curie temperature (T c ) of the alloy is ~68.8 on cooling and ~74.6 on heating, respectively. The decrease of the magnetic transition temperatures with respect to that of the stoichiometric Ni 2 MnGa alloy should be attributed to the excess of Ni [116]. Fig. 4 . 4 Fig.4.1 DSC curves of Ni 53 Mn 22 Ga 25 alloy on heating and cooling. Fig. 4 . 2 4 . 1 . 2 42412 Fig.4.2 Powder XRD patterns of Ni 53 Mn 22 Ga 25 alloy measured at (a) 25°C; (b) -30°C and (c) -120°C. Fig. 4 . 4 Fig.4.3 (a) BSE image of co-existing austenite and 7M martensite. (b) EBSD micrograph of martensite "diamond" composed of four variants (denoted as A, B, C and D) surrounded by austenite. The coordinate frame (X 0 -Y 0 -Z 0 ) referring to the sample coordinate frame. Fig. 4 .Fig. 4 . 4 444 Fig. 4.4 shows the microstructure evolution of the martensite "diamond" during Fig. 4 . 4 Fig.4.5 shows the orientation micrographs reflecting the further growth of the martensite variants from the "diamond". It is seen in Fig.4.5a that the growth of the "diamond" proceeds by the elongation through the forward motion of spears, i.e. A:C or B:D pair. This elongation results in the formation of type II twin behind the spears. Fig. 4 . 4 Fig.4.5d. Locally, a cluster of four types of martensite variants, i.e. A, B, C and D, constitutes one variant group. It is usually observed that there are several variant groups formed in the same austenite grain. An example is shown in Fig. 4.6. Fig. 4 . 4 Fig.4.5 EBSD orientation map with co-existing two phases: (a) elongation of martensite "diamond"; (b) formation of paired plates from the "diamond"; (c) formation of the fork configuration by bifurcating; (d) formation of a large number of martensite plates keeping spear configuration adjacent to the austenite matrix. Fig. 4 . 6 46 Fig.4.6 Formation of various variant groups (G1, G2, G3 in the figure) in an austenite grain. Fig. 4 . 4 Fig.4.7 TEM bright field image showing the internal sub-structure of the 7M martensite plate. The corresponding SAED patterns are shown in the lower right corner. be the Pitsch relations, i.e. (101) A //(1 2 10 ) 7M and [10 1 ] A //[10 10 1] 7M , as we have found in Ni 50 Mn 30 Ga 20 alloy. The corresponding pole figures are shown in Fig. 4.8, which is plotted using the orientation data of the two phases in Fig.4.5b. Fig. 4.8a displays the {101} A pole figure of austenite and the {1 2 10 } 7M pole figures of the four martensite variants in the macroscopic sample coordinate system. It is noted that from the four martensite variants, one {1 2 10 } 7M pole of each variant is overlappedand they are all common to one {101} A pole of the austenite, as marked in Fig.4.8a, suggesting {101} A //{1 2 10 } 7M . Accordingly, the four variants share a common <10 10 1> 7M pole that is in common with one <10 1 > A pole of the austenite, as shown in Fig.4.8b, indicating <10 1 > A //<10 10 1> 7M . Therefore, the orientation relationships between the austenite and the 7M martensite are confirmed to be Pitsch OR with {101} A //{1 2 10 } 7M and <10 1 > A //<10 10 1> 7M , which is well consistent with our previous result. Fig 4 . 8 48 Fig 4.8 Identification the Pitsch OR from the corresponding pole figures of the two 3 )( 3 ) 33 This combination of all six self-accommodation groups gives the minimum transformation strain energy. Now, we pay our attention to the more local area in a self-accommodated variant group and the total deformation matrices for the three types of twin pairs, i.e. A:C (or B:D) pair, A:B (or C:D) pair and A:D (or B:C) pair, are calculated.(1) Type-twin pair (A:C) : Compound twin pair (A:D): Ni 53 53 Mn 22 Ga 25 alloy with the martensitic transformation temperatures around room temperature were investigated. The formation of characteristic diamond-like martensite microstructure with four variants (A, B, C and D) during the martensitic transformation was evidenced. As revealed by EBSD measurements, the martensite "diamond" consists of type I twin (A:C and B:D pair) and compound twin (A:D and B:C pair); the long ridge of martensite "diamond" corresponds to type I twin interface and the short ridge to compound twin interface. The "diamond" thus can be regarded as being built up of two spears (A:C and B:D pair) placed back to back. The favourable way for the "diamond" growth is through the forward progression of spears. The OR of the martensitic transformation was further confirmed to be Pitsch relation according to the crystallographic calculation. The habit plane normals were determined experimentally to be {0.736130, 0.673329, 0.068855} A by means of indirect double-trace method, which is consistent with results of phenomenological theory with 2.11° deviation. Further calculation indicates that the characteristic four variants in a "diamond" group clustered around one {101} A pole and the elastic strains around martensite were effectively cancelled out by making such a group. Both A:C (or B:D) and A:B (or C:D) variant pairs are self-accommodated, whereas A:D (or B:D) variant pair is not. martensite variants is very sensitive to local constraints. On the other hand, different characterization techniques require different sample geometries and sizes that represent different internal constraints. Clearly, significant inconsistency may result from different experimental techniques and the characterization information (global and local) suffers from a lack of generality. So far, numerous experimental studies of modulated structures have been performed almost exclusively by diffraction techniques. The role of the micrstructural correlations between martensite plates has seldom been taken into account by either experimental examination or the nanotwin combination theory. These microstructural features surely have a strong influence on the stability of the modulated martensites and their functionality as shape-memory materials. In this context, insight into the microstructure-property correlation may bring convincing clues to clarify the structure of long-period modulated martensite. Under such a background, the polycrystalline bulk Ni 53 Mn 22 Ga 25 alloy having martensitic transformation near room temperature is selected as an ideal testing material for the following reasons. The alloy displays a transformation sequence from the austenite to the modulated martensite and then the NM martensite during continuous cooling detected by X-ray diffraction (XRD) measurements, and islands of modulated martensite and NM martensite coexist in some initial austenite grains when kept at room temperature. Using the local electron backscatter diffraction (EBSD) orientation determination technique, the crystal structure nature of the long-period modulated martensite was unambiguously clarified through microstructural and crystallographic examination. XRD pattern of the modulated phase in Ni 53 Mn 22 Ga 25 alloy measured at -30 , as shown in Fig. 4.9a, is first solved and further refined with the incommensurate 7M modulation structure model (hereafter denoted 7M(IC) in this section) [75]. It showed that the modulated phase has a monoclinic long-period superstructure (P2/m, No. 10) with the lattice constants a 7M = 4.222 Å; b 7M = 5.537 Å; c 7M = 41.982 Å, and = 92.5°. The unit cell of the combined nanotwins was artificially constructed based on the lattice constants of the tetragonal NM unit cell (a Tet = b Tet = 3.879Å, c Tet = 6.511Å) and the (5 2 ) 2 stacking sequence of (112) Tet twins [81]. The resolved lattice constants of the adaptive phase (with a structure of combined nanotwins) [81, 130] are a ad = 4.257 Å, b ad = 5.486 Å, c ad = 29.446 Å, and ad = 94.2°. The calculated atomic coordinates of the nanotwined superstructure is shown in Appendix III. The XRD patterns are recalculated using the two structure models and are displayed in Fig. 4.9b and c. It is seen that the two structures possess very close pattern to the measured one. Close examination reveals that the 7M(IC) model delivers a slightly better fit to the measured profile. The distinguishable difference between the two structures appears in the secondary minor peaks (as arrowed in the figure) around the three main diffraction peaks in the 2 range from 40 to 50°. Fig. 4 . 4 Fig.4.9 XRD patterns for the modulated martensite: (a) measured at -30°C; (b) recalculated pattern with the 7M(IC) superstructure; (c) recalculated with the tetragonal nanotwin combined unit cell. The insets show the unit cells of the two structures. Fig. 4 .Fig. 4 .Fig. 4 . 444 Fig.4.10 EBSD map showing (a) the coexistence of austenite, 7M(IC) and NM martensite within an initial austenite grain, and (b) four 7M variants designated as A, B, C and D where the coordinate frame (X 0 -Y 0 -Z 0 ) refers to the macroscopic sample coordinate system. Fig. 4 . 4 Fig.4.11 (a) Measured EBSD Kikuchi pattern of the modulated martensite; (b) recalculated with the 7M (IC) structure; (c) recalculated with the nanotwin combination structure. Fig. 4 . 4 Fig. 4.10b presents the EBSD orientation micrograph corresponding to Fig.4.10a, 4.12b, the two orientation plates correspond to four tetragonal NM orientation variants that are in lamellar shape. The in-plate variants are twin related. The twinning elements of the(112) Tet nanotwin in the plate are determined to be K 1 =(112) Tet ; K 2 =(11 2 ) Tet ; 1 =[11 1 ] Tet ; 2 =[111] Tet and the twinning shear s=0.344. The twinning shear is almost twice as that of the 7M(IC) twins (type I and type II). The inter-lamellar interfaces are the(112) Tet twinning planes and coherent. Due to this in-plate twin configuration, the atomic coherency at the plate interface is greatly deteriorated. At the plate interface, two thick lamellae and two thin lamellae from neighboring plates intersect, as illustrated in Fig.4.12b. When the thick lamellae meet, they appear to have a (1 1 2) Tet[ 1 11] Tet twin relationship but with some degrees of deviation in the twinning plane and twinning direction, i.e. 4.97° between (112) Tet planes and 2.62° between the[111] Tet directions. Using the two(112) Tet planes as reference, the closest planes of the thin lamellae from the neighboring plates are (010) Tet . The angular deviation between the two planes is 11.44°. Thus, the parallelism of the planes is far from perfect. Fig. 4 . 4 Fig.4.12 Atomic correspondences of type twin interface of the modulated martensite viewed along the [210] 7M direction constructed under (a) the 7M (IC) structure model; (b) the nanotwin combination structure model. For a clear representation, only Mn atoms are displayed. 4. 3 Fig. 4 . 34 Fig.4.13a shows the EBSD phase micrograph covering the co-existing austenite, incommensurate 7M martensite and NM martensite in the same initial austenite. The 7M and the NM martensite are in plate-shape and the neighboring 7M and NM Fig. 4 . 4 Fig. 4.13b shows the EBSD orientation micrograph corresponding to Fig.4.13a.The three phases are colored according to their orientations. It is seen that for the 7M martensite there are 4 orientation variants, whereas for the NM martensite, it seems that one plate corresponds to one orientation variant but in reality it is composed of alternatively distributed two fine lamellae, as displayed in Fig.4.14a. Of the two lamellae, one is thinner than the other so that the thickness of the thinner one is beyond the resolution of the present automated EBSD orientation mapping.Representing the orientation of NM plate with the orientation of the thicker lamellae, one can find 4 plate variants, as shown in Fig.4.13b. Considering that one NM plate is composed of two orientation variants, we can obtain, in total, 8 orientation variants for NM martensite. Fig 4 . 4 Fig 4.13 (a) EBSD phase index map of the coexisted austenite, 7M(IC) and NM martensite in the same austenite grain; (b) EBSD orientation map, four 7M variants are designated as A, B, C and D; four NM plates are nominated as P1, P2, P3 and P4. Fig. 4 . 4 Fig.4.14 (a) Zoomed BSE image of the squared region in Fig.4.13b showing the fine lamellae in plate P1 and P3 of NM martensite. The paired fine lamellae in two plates are denoted as L1, L1 , L3 and L3 , respectively, as shown in the inset schema. L1 and L3 represent the thicker lamellae, and L1 and L3 the thinner lamellae. The red and green lines represent the traces of inter-plate interface and (112) Tet twinning interface. (b) {112} Tet pole figure of the two thick lamellae (L1 and L3) and {010} Tet pole figure of the two thin lamellae (L1 and L3 ). The red line represents the trace of the plate interface normal. (c) <20 1 > Tet pole figure of the four fine lamellae in plate P1 and P3. figure) and Fig. 4.14c (direction pole figure). The projection of the plate interface normal (the red straight line) is displayed in the plane pole figure (Fig. 4.14b). The corresponding plane and direction poles are enclosed in the open squares in the pole figures and their deviations are given in Table 4.6. It is seen that the parallelisms are far from perfect, especially those of the planes. With such large degrees of deviations, elastic displacement of atoms from their equilibrium positions in vicinity of the interface area could be expected. This large mismatch on the plate interfaces surely imposes significant constraint on the transformation direct from austenite to NM martensite and this might be the reason for the formation of the 7M martensite at the intermediate state of the transformation to mitigate the transformation constraint. Fig. 4 . 4 Fig.4.15 BSE image showing the transition from 7M variant A to NM plate P1. Fig. 4 . 4 Fig.4.16 Atomic projection of (a) the type twin interface in [210] 7M and their cross section atomic correspondences for 7M martensite; (b) the corresponding inter-plate interface of NM martensite and their cross section atomic correspondences. Chapter 5 ( 2 )( 3 )( 4 ) 5234 the transformation from the cubic austenite to the tetragonal NM martensite through crystal structure, crystallographic orientation and microstructure examinations. In this transformation, the lattice mismatch between the cubic austenite and the tetragonal NM martensite and the formation of incoherent NM plate interfaces represent as an insurmountable energy barrier for a direct transformation from the austenite to the NM martensite. The formation of 7M modulated martensite is evidenced as an intermediate step in this transformation by introducing an intermediate crystal structure that greatly mitigates the large lattice mismatch and by forming coherent plate interfaces. During this two step transformation, one austenite variant will gives rise to 4 twin related 7M orientation variants and one 7M variant will result in 2 twin related NM variants, correspondingly 8 NM orientation variants in total. The transformation from the 7M to the NM martensite is realized by lattice distortion following the (001) 7M //(112) Tet and [100] 7M // [11 1 ] Tet OR, which is accompanied by the degradation of the atom coherency in the vicinity of the NM plate interface and the complete change of twin configuration. This microstructure change is well correlated with the experimentally observed field induced shape memory performance degradation from 7M to NM. The available temperature window for the stable existence of the 7M martensite depends on the energy barriers related to the lattice mismatch between the 7M and the NM martensite and the atomic misfit on the plate interfaces of the NM martensite. Conclusion and perspective Conclusion In this dissertation, four off-stoichiometric Ni-Mn-Ga polycrystalline alloys were deliberately prepared. The microstructure and martensitic transformation features were thoroughly investigated. Through this dissertation, it has been demonstrated that the EBSD technique can be used as an advanced characterization tool for accurate crystallographic analyses for materials having modulated superstructure. The conclusions of the present study are summarized as follows: (1) Ni 50 Mn 28 Ga 22 alloy: The martensite possesses a monoclinic 5M modulated superstructure, the superlattice of which is composed of 5 subcells. The microstructure of the 5M martensite can be characterized by broad plates with alternatively distributed fine variants. EBSD measurements using the monoclinic superstructure information revealed four twin-related variants A, B, C and D with distinct orientations in one broad plate, i.e. the variant pair A:C (or B:D) has the relation of type I twin, A:B (or C:D) type II twin, and A:D (or B and C) compound twin. The variant interfaces were revealed to be the corresponding twinning planes (K 1 ). Based on the local orientations of the individual martensite variants measured by EBSD system and crystallographic calculation, the more favorable transformation OR from austenite to 5M martensite was revealed to be the Pitsch relation with (101) A //(1 2 5 ) 5M and [10 1 ] A //[ 5 5 1] 5M with no residual austenite. Ni 50 Mn 30 Ga 20 alloy: It is confirmed by Kikuchi pattern indexation that 7M martensite possesses an incommensurate monoclinic superstructure and the superlattice is composed of ten subcells. From a parent austenite grain, one or several martensite colonies are inherited, each consisting of 4 types of twin-related variants (A, B, C, and D) according to EBSD measurements. All the pairs of variants can be categorized into three twinning modes: variants A and C (or B and D) are in type I twin relation, variants A and B (or C and D) type II twin, and variants A and D (or B and C) compound twin. All the twin interfaces are in coincidence with the respective twinning plane (K 1 ). Furthermore, based on the local orientations of individual martensite variants measured by EBSD system and detailed crystallographic analysis, the energetically favorable OR governing the austenite to incommensurate 7M martensite transformation was revealed to be the Pitsch relation with (1 0 1) A //(1 2 10 ) 7M and [1 0 1 ] A //[10 10 1] 7M . Under this determined OR, at most 24 physically distinct martensite variants may be resulted from an initial austenite grain during the martensitic transformation. Notably, in the present work, the first attempt has been made to resolve the ambiguity of the geometrically favorable ORs by examining the lattice discontinuity caused by the phase transformation and the structural modulation. Ni 54 Mn 24 Ga 22 alloy: The alloys are composed of self-accommodated NM martensitic plates with tetragonal crystal structure. The adjacent plates have the relationship of 80°~85° rotation around the <110> Tet axes. Locally, four types of plates are identified and each plate consists of paired fine variants. Totally, eight orientation variants are found in one martensite colony. The paired fine variants in each plate were found to be compound twin related with the {112} Tet as the twinning plane and the <11 1 > T the twinning direction. The inter-plate interfaces are close to {1 1 2} Tet plane but with ~3° deviation, while the interfaces of two paired fine variants are in good agreement with {112} Tet twinning plane. Ni 53 Mn 22 Ga 25 alloy: At room temperature, austenite and martensite co-exist in Ni 53 Mn 22 Ga 25 alloy. The formation of the characteristic diamond-like martensite microstructure with four variants (A, B, C and D) during the austenite-7M martensite transformation was evidenced. As revealed by EBSD measurements, the martensite "diamond" consists of type I twin (A:C and B:D pair) and compound twin (A:D and B:C pair); the long ridge of martensite "diamond" corresponds to type I twin interface and the short ridge to compound twin interface. The "diamonds" finally transforms into martensite plates. The favorable way for the "diamond" growth is through the forward of spears. Crystallographic calculation manifests that the characteristic four variants in a "diamond" group clustered around one {101} A pole and the elastic strains around martensite were effectively cancelled out by making such a group. Both A:C (or B:D) and A:B (or C:D) variant pairs are self-accommodated; while A:D (or B:D) variant pair is not.The microstructure with co-existing three phases (austenite, modulated martensite and final NM martensite) was observed in some austenite grains. Three phases are located with a fixed adjacency, i.e. austenite -modulated martensite -NM martensite.The long-period 7M martensite occurs on cooling as a thermodynamically metastable phase that is intermediate between the parent austenite and the final stable NM martensite. The modulated martensite phase possesses an independent crystal structure, rather than the nanotwin combination of the normal non-modulated martensite proposed by the nanotwin combination theory. The modulated structure of 7M martensite provides reduced local number of variants and favors twinning and detwinning configuration. This is essential to the attainability of the magnetic field induced shape change.The role of 7M martensite in the transformation from the cubic austenite to the tetragonal NM martensite has been clarified. In this transformation, the lattice mismatch between the cubic austenite and the tetragonal NM martensite and the formation of the incoherent NM plate interfaces represent as an insurmountable energy barrier for a direct transformation from the austenite to the NM martensite. The formation of 7M modulated martensite is evidenced as an intermediate step in this transformation by introducing an intermediate crystal structure that greatly mitigates the large lattice mismatch and by forming coherent plate interfaces. During this two step transformation, one austenite variant will gives rise to 4 twin related 7M orientation variants and one 7M variant will result in 2 twin related NM variants, correspondingly 8 NM orientation variants in total. The transformation from the 7M to the NM martensite is realized by lattice distortion following the (001) 7M // (112) Tet and [100] 7M // [11 1 ] Tet OR, which is accompanied by the degradation of the atom coherency in the vicinity of the NM plate interface and the complete change of twin configuration. This microstructure change is well correlated with the experimentally observed field induced shape memory performance degradation from in NM state. 1 .Fig. 1 11 Fig.1Représentations schématiques de (a) de la cellule élémentaire de l'austénite cubique, (b) de la supercellule monoclinique de la martensite modulée 5M constituée de cinq sous-cellules (décrites par les traits pointillés), et (c) de la cellule réduite unitaire moyenne (en ignorant les modulations du réseau). Fig. 2 2 Fig.2 (a) L'image ESB des plaques de martensite larges dans un grain initial d'austénite; (b) micrographie d'orientation des fines lamelles. L'orientation l AGG l d'un grain d'austénite par rapport au repère orthonormé de l'échantillon peut être exprimée par la relation suivante en matrices: représente l'orientation mesurée du k ème variant de la martensite par rapport au repère orthonormé de l'échantillon ; T est la matrice de rotation qui transforme le repère orthonormé lié à la base du réseau monoclinique de la martensite à celui cubique de l'austénite en respectant la OR donnée ; j A S (j = 1, 2, ..., 24) et 7 sont les éléments de symétrie de rotation du système cristallin cubique et du système cristallin monoclinique, respectivement. En se basant sur une étude bibliographique, les ORs de Bain, de KS, de NW et de Pitsch sont présumées comme les ORs possibles de la transformation dans le présent travail. Pour des raisons de simplicité, les ORs exprimées en parallélisme des plans et des directions dans les plans sont d'abord vérifiées entre l'austénite et la martensite 1M moyenne (Fig.1c). Fig. 3 {001} 3 Fig.3 {001} Projections stéréographiques standard des orientations de l'austénite calculées à partir des variantes de martensite A ( ), B ( ), C ( ) et D ( ) dans la variante du groupe g1 sous les hypothèses d'OR de Bain, KS, N-W et Pitsch, respectivement. Les orientations communes de l'austénite sont encadrées. 2 .Fig. 4 24 Fig.4 Représentation schématique de (a) maille de l'austénite; (b) supercellule de martensite 7M, où chaque cellule élémentaire (martensite 1M) est indiquée en pointillés et les cinq cellules élémentaires avec des constantes distinctes sont désignées comme C 1 , C 2 ... C 5 ; (c) cellule moyenne de la martensite modulée. Fig. 5 5 Fig.5 (a) Une image typique en contraste d'ESB de l'alliage polycristallin Ni 50 Mn 30 Ga 20 . (b) Cartographie d'orientation EBSD déterminée localement sur une colonie. Le système de coordonnées (X 0 -Y 0 -Z 0 ) se réfère au système de coordonnées macroscopiques de l'échantillon. OR de Bain classique, celles de KS, de NW et de Pitsch sont présumés comme OR possibles entre l'austénite parent et la martensite 7M. Les parallélismes des plan présumés et de directions dans le plan sont d'abord utilisés pour spécifier les ORs entre l'austénite et la martensite 1M (Fig. 4c). Fig. 6 6 Fig. 6 Figures de pôles {0 0 1} de l'austénite calculées pour les variantes A ( ), B ( ), C ( ) and D ( ) de la martensite sous l'hypothèse des ORs de (a) Bain, (b) K-S, (c) N-W et (d) Pitsch, respectivement. Les pôles correspondant à l'orientation commune de l'austénite sont encadrées. Fig. 7 Tableau 4 . 74 Fig.7 Correspondances atomiques des plans respectifs dans l'austénite et la martensite 1M sous l'hypothèse (a) relation de K-S avec (111) A //(011) 1M et [10 1 ] A //[ 1 1 1] 1M ; (b) relation de Pitsch avec (101) A //(1 2 1 ) 1M et [10 1 ] A //[ 1 1 1] 1M . Fig. 8 8 Fig. 8 Écarts angulaires de (a) plans (1 2 1 ) 1M (relation de Pitsch) et (011) 1M (relation de K-S) et (b) direction [ 1 1 1] 1M dans les 5 sous cellules (C1, C2, …, C5) par rapport à ceux de la cellule moyenne 1M. 3 . 3 Caractéristiques cristallographiques de la martensite NM. L'alliage Ni 54 Mn 24 Ga 22 avec la martensite non-modulée à la température ambiante a été préparé. Les diagrammes de diffraction de rayons X de poudre mesurés à la température ambiante démontrent que l'alliages possède seulement la martensite non-modulée avec une structure cristalline tétragonale. Les spectres peuvent être indexés par le groupe d'espace I4/mmm (n ° 139) [10]. Les constantes de maille de l'alliages ont été déterminés comme a T =b T = 3.853 Å et c T =6.625 Å, respectivement. Des mesures EBSD ont révélé que les plaques de martensite adjacentes ont la relation d'orientation de 80° ~ 85° de rotation autour d'axes <110> T . Des observations additionnelles en MEB à fort grossissement ont montré que les plaques NM sont composées de lamelles mince et il y a quatre types de plaques de martensite (notées P1, P2, P3 et P4) dans une colonie de martensite selon l'orientation des minces interfaces lamellaires, comme représenté sur la fig. 9. Les lamelles fine apparaissent en paires, dans lesquelles l'une est plus épaisse que l'autre. Ainsi, dans une colonie de martensite, on peut toujours trouver huit variantes d'orientation. Les variantes fines appariées dans chaque plaque ont été trouvées en relation de macle avec le plan {112} T comme plan de macle et <11 1 > T comme direction de maclage. Les interfaces inter-plaques sont proches du plan {1 1 2} T , mais avec un écart de ~ 3 °, tandis que les interfaces de deux paires de variantes fines appariées sont en bon accord avec le plan de macle {112}. Fig. 9 9 Fig.9 (a) Image en électrons rétrodiffusés obtenue dans une colonie de martensite; quatre sortes de plaques sont numérotés P1, P2, P3, P4; (b) Image agrandie de la région délimitée par le rectangle qui montre les lamelles fines appariées dans les plaques de martensite. Les interfaces inter-plaques et inter-lamellaires sont marquées avec des lignes rouge et verte, respectivement. Fig. 10 ( 10 Fig.10 (a) Micrographie EBSD de martensite en forme de « diamant » composée de quatre variantes notées A, B, C et D. (b) Formation des platelets appariés à partir du « diamant ». 5 . 5 Clarification de la structure et la stabilité de longue période martensite moduléeUne apparente contradiction dans l'interprétation de la structure martensitique de longue période a été soulevée récemment dans les alliages à mémoire de forme ferromagnétiques Ni-Mn-Ga. Righi et al.[4] ont fait une étude détaillée sur la martensite modulée 7M par application de l'approche du superespace à l'analyse de diffraction de rayons X (XRD) de poudres. Les résultats ont montré que la martensite modulée possède sa propre structure cristalline avec une modulation incommensurable 7M (désignée ci-après 7M (IC)). En se basant sur des mesures DRX et le concept de la phase d'adaptation[12], Kaufmann et al.[13] ont examiné la co-existence de l'austénite, de la martensite 7M et NM dans des films épitaxiaux de Ni-Mn-Ga. Ils ont conclu que la martensite 7M modulée peut être tout simplement construite à partir de variantes nanomaclées de la martensite tétragonale MN avec la séquence d'empilement (5 2 ) 2 , ce qui exclut l'existence d'un structure indépendante modulée.En effet, il est très difficile de faire une discrimination directe de la validité de ces deux modèles de structure. Jusqu'ici, de nombreuses études expérimentales de structures modulées ont été réalisées presque exclusivement par des techniques de diffraction. Le rôle des corrélations microstructurales entre les platelets de martensite a rarement été pris en compte soit par l'examen expérimental soit par le modèle de la combinaison de nanomacles. Ces caractéristiques microstructurales existant dans la nature ont sans doute une forte influence sur la stabilité et les fonctionnalités de la martensite modulée. Dans un tel contexte, l'alliage polycristallin massif Ni 53 Mn 22 Ga 25 ayant une température de transformation martensitique voisine de la température ambiante est sélectionné en tant que matériau d'essai idéal. L'alliage affiche une séquence de transformation de l'austénite à la martensite modulée puis la martensite NM au cours du refroidissement continu, détectée par des mesures de diffraction de rayons X (XRD) et des îlots de martensite modulée et martensite NM coexistent dans certains grains initiaux d'austénite, lorsqu'ils sont conservés à la température ambiante. Le diagramme XRD de la phase modulée dans l'alliage Ni 53 Mn 22 Ga 25 mesuré à -30°C, comme représenté sur la fig.11a, est tout d'abord résolu puis affiné ensuite avec le modèle de structure 7M(IC) [4] en utilisant le logiciel PowderCell [14]. Il est montré que la phase modulée possède une superstructure monoclinique à longue période (P2/m, No. 10) avec les constantes de réseau a 7M = 4.222 Å; b 7M = 5.537 Å; c 7M = 41.982 Å, and = 92.5°. La cellule élémentaire de la combinaison des nanomacles a également été construite artificiellement sur la base des constantes de réseau de la cellule élémentaire tétragonale NM (a T = b T = 3.879Å, c T = 6.511Å) et la séquence d'empilement (5 2 ) 2 des macles (112) T [13]. Les constantes de réseau résolues pour la phase d'adaptation sont a ad = 4.257 Å, b ad = 5.486 Å, c ad = 29.446 Å, et ad = 94.2°. Les diagrammes DRX sont recalculés en utilisant les deux modèles de structure et sont présentés dans les Fig.11b et c. On voit que les deux structures possèdent un diagramme de diffraction très proche de celui mesuré. Un examen attentif révèle que le modèle 7M(IC) fournit un ajustement au profil mesuré légèrement meilleur. La différence distinguable entre les deux structures apparaît dans les pics secondaires mineurs correspondants (marqués par une flèche dans la figure) localisés près des trois pics de diffraction principaux dans le domaine 2 entre 40 et 50°. Fig. 11 11 Fig.11 Diagrammes XRD pour la martensite modulée: (a) mesuré à -30 ° C; (b) recalculé avec la superstructure 7M(IC); (c) recalculé avec la cellule élémentaire tétragonale de nanomacles combinées. Les encadrés dans les figures montrent les cellules élémentaires des deux structures. Fig. 12 ( 12 Fig.12 (a) Micrographie EBSD des phases qui montre l'austénite, la marteniste 7M (IC) et la martensite NM coexistant dans un même grain d'austénite; (b) micrographie d'orientation EBSD, où les quatre variantes de 7M sont désignées A, B, C et D et les quatre platelets de NM sont nommés P1, P2, P3 et P4. Le système de coordonnées (X0 Y0-Z0-) se réfère au système de coordonnées de l'échantillon macroscopique. Fig. 13 13 Fig.13 Correspondance atomique à l'interface de macle de type de la martensite modulée vue le long de la direction [210] 7M construite sous l'hypothèse (a) du modèle de la structure 7M(IC) et (b) du modèle de la structure de combinaison de nano-macles. Pour une représentation claire, uniquement les atomes de Mn sont dessinés. 6 .Fig. 14 614 Fig. 14 Image en électrons rétrodiffusés (BSE) montrant la transition de la variante A de 7M au Fig. 15 (Tableau 6 22 15622 Fig. 15 (a) La projection de l'interface atomique de macle de type I dans la direction [210] 7M et les correspondances atomiques dans la section transversale pour la martensite 7M ; (b) l'interface correspondante entre les platelets de martensite NM et les correspondances atomiques dans la section transversale. ( 2 ) 2 L'alligae Ni 50 Mn 30 Ga 20 : Il est confirmé par l'indexation de clichés de Kikuchi que la martensite 7M possède une superstructure incommensurable monoclinique et le super-réseau est composé de dix sous-cellules. Dans un grain d'austénite parente, une ou plusieurs colonies de martensite se forment, chacune constituée de 4 variantes (A, B, C et D) qui sont en relation de macle l'une par rapport à l'autre. Toutes les paires de variantes peuvent être classées en trois modes de macle: les variantes A et C (ou B et D) forment des macles de type I, les variantes A et B (ou C et D) des macles de type II, et les variantes A et D (ou B et C) des macles composées. Toutes les interfaces sont en coïncidence avec le plan respectif de maclage (K1). En outre, en se basant sur les orientations locales des différents variantes martensitiques mesurées par EBSD et sur les analyses cristallographiques détaillées, la relation d'orientation la plus favorable énergétiquement régissant la transformation de l'austénite en martensite 7M incommensurable a été identifiée être la relation de Pitsch avec (1 0 1) A //(1 2 10 ) 7M et [1 0 1 ] A //[ 10 10 1] 7M . Notamment, l'ambiguïté de la relation d'orientation géométriquement la plus favorable a été résolue par l'examen de la discontinuité du réseau causée par la transformation de phase et par la modulation structurale. (3) L'alliage Ni 54 Mn 24 Ga 22 : L'alliage se compose de platelets de martensite NM auto-accomodés ayant une structure cristalline tétragonale. Les platelets adjacents ont la relation de désorientation 80° ~ 85° autour des axes <110> T . Localement, quatre types de platelets sont identifiés et chaque platelet se compose de paires de variantes fines. Totalement, huit variantes d'orientation se trouvent dans une colonie de martensite. Les variantes fines en paires dans chaque platelet ont été trouvées en relation de macle composée avec le plan {112} T comme plan de maclage et la direction <11 1 > T comme direction de maclage. Les interfaces inter-platelets sont proches du plan {112} T , mais avec ~ 3 ° d'écart, tandis que les interfaces de deux paires de variantes fines sont en bon accord avec le plan de maclage {112} T . (4) L'alliage Ni 53 Mn 22 Ga 25 : A température ambiante, l'austénite et la martensite co-existent dans l'alliage Ni 53 Mn 22 Ga 25 . La formation de microstructures caractéristiques de martensite en forme de « diamant » avec quatre variantes (A, B, C et D) au cours de la transformation de l'austénite en martensite modulée 7M a été mise en évidence. Comme révélé par des mesures EBSD, le « diamant » est constitué des variantes de martensite en relation de macle de type I (A: C et B: D) et de macles composées (A: D et B: C); la longue crête du « diamant » correspond à l'interface de macle de type I et la courte crête à l'interface de macle composée. Les diamants se transforment finalement en platelets de martensite. Des calculs cristallographiques montrent que les quatre variantes caractéristiques dans un groupe de diamant sont autour d'un plan {101} A et les déformations élastiques issues de la transformation ont été effectivement annulées par la formation du groupement. Les paires de variantes A: C (ou B: D) et A: B (ou C: D) sont toutes les deux auto-accommodées, tandis que les paires A: D (ou B: D) ne le sont pas. La microstructure avec les trois phases (austénite, martensite modulé et martensite NM final) co-existantes a été observée dans certains grains d'austénite. Les trois phases sont situées avec une contiguïté fixe, c.-à-d. austénite -martensite modulé -martensite NM. La martensite 7M de période longue se produit lors du refroidissement en tant qu'une phase thermodynamiquement métastable qui est intermédiaire entre l'austénite parente et la martensite NM finale stable. La phase martensite modulée possède une structure cristalline indépendante, plutôt que la combinaison de nano-macles de martensite normale non-modulée comme proposée par la théorie de combinaison de nano-macles. La structure modulée de la martensite 7M offre un nombre de variantes locales réduit et la configuration favorable pour maclage et démaclage. Cela est essentiel pour réaliser le changement de forme induit par un champ magnétique. Le rôle de la martensite 7M dans la transformation de l'austénite cubique en martensite tétragonale NM a été clarifié. Dans cette transformation, le désaccord de maille entre l'austénite cubique et la martensite NM tétragonale et la formation des interfaces inter-platelet NM incohérentes représentent une barrière d'énergie insurmontable pour une transformation directe de l'austénite en martensite NM. La 2.3.6 Transmission electron microscopy TEM is capable of imaging at a significantly higher resolution than optical microscopy and SEM, which enables the research to examine fine detail of the microstructure. Here, a Philips CM 200 LaB 6 cathode TEM was used to observe the stacking faults inside the martensite plates. A Gatan MSC 792 CCD camera was used to acquire the images. The working voltage was set at 200KV. 2. 4 Crystallographic calculation method 2.4.1 Coordinate transformation between orthonormal reference system and monoclinic system By convention, individual orientations acquired by EBSD measurement are represented by a set of rotations expressed in Euler angles ( 1 , , 2 ) [110] a M in i-O-k plane, cos cos sin sin cos cos sin sin cos cos sin sin G sin cos cos sin cos sin sin 1 cos cos cos 2 1 cos sin 2 sin sin 2 cos sin cos (2.1) In this study, the orthonormal crystal coordinate system is linked to the monoclinic crystal coordinate system by setting c M //k, b M //j and Table 3 . 3 1 Misorientation angles ( ) and rotation axes (d) between variants A, B, C, and D in Fig. 3.3a. The rotation axes refer to the orthonormal crystal coordinate frame set to the monoclinic martensite lattice basis. Variant pair Misorientation angle [°] d 1 Rotation axis, d d 2 d 3 A:C 86.80 -0.70846 0.00241 -0.70575 179.81 0.48490 -0.72659 -0.48676 B:D 86.79 179.83 -0.71242 -0.48214 -0.00221 0.72661 -0.70175 0.48947 93.32 0.70870 0.00098 0.70551 A:B 179.92 -0.51312 -0.68633 0.51543 C:D 93.09 179.64 0.70882 -0.51205 0.00435 -0.68778 0.70538 0.51455 A:D 179.64 179.89 0.70487 -0.70933 -0.00098 0.00317 -0.70933 -0.70487 179.88 0.70978 0.00079 0.70443 B:C 179.91 -0.70443 -0.00102 0.70978 Table 3 . 3 2 Twinning elements of commensurate 5M modulated martensite in Ni 50 Mn 28 Ga 22 . K 1 (1 2 5 ) (1.0569 2 4.7155 ) (1 0 5) K 2 (1.0569 2 4.7155) (1 2 5) (1 0 5) 1 [ 5.2504 5 0.9499] [ 5 5 1] [ 5 0 1] 2 [5 5 1 ] [5.2504 5 0.9499 ] [5 0 1] P (1 0.0514 5.2568) (1 0.0514 5.2568) (0 1 0) s 0.1387 0.1387 0.0074 Type (A:C and B:D) Type (A:B and C:D) Compound (A:D and B:C) Table 3 . 3 Transformation OR Plane and in-plane direction parallelism Bain relation (001) A //(010) 1M & [010] A //[101] 1M K-S relation 4 A selection of possible ORs between austenite and martensite. Note that the Miller indices of planes and in-plane directions for product martensite with monoclinic structure are referred to the average unit cell illustrated in Fig. 3 .5c. Table 3 . 3 5 Minimum misorientation angles between the austenite orientations calculated from variant A and from other three variants in six variant groups under Bain, K-S, N-W and Pitsch OR. Group No. Variant pair Bain (°) K-S (°) N-W (°) Pitsch (°) A:B 3.45 0.75 2.09 0.75 g1 A:C 3.52 1.61 1.74 0.50 A:D 0.44 1.58 2.44 0.25 A:B 3.81 0.34 2.34 0.33 g2 A:C 3.30 1.89 1.86 0.66 A:D 1.07 1.81 2.86 0.81 A:B 3.33 0.66 1.80 0.66 g3 A:C 3.22 1.72 1.62 0.81 A:D 0.41 1.90 2.66 0.20 A:B 3.53 0.48 1.83 0.48 g4 A:C 3.42 1.44 1.50 0.70 A:D 0.49 1.52 2.46 0.26 A:B 3.20 0.79 1.81 0.79 g5 A:C 3.22 1.56 1.44 0.86 A:D 0.84 1.38 2.38 0.40 A:B 3.70 0.43 4.17 0.43 g6 A:C 3.66 1.90 3.94 0.36 A:D 0.40 1.98 2.86 0.28 Table 3 . 3 Variant pair Misorientation angle [°] d 1 Rotation axis, d d 2 d 3 A:C 82.6319 -0.728809 -0.00337 -0.684708 179.745 -0.452053 0.751082 0.481169 B:D 82.9978 179.8 -0.725021 -0.456351 -0.002634 0.74897 -0.688722 0.480404 97.783 0.723592 0.003766 0.690218 A:B 179.675 -0.520057 -0.65749 0.545204 C:D 96.5924 179.909 0.719889 0.518203 -0.001061 0.66528 0.694089 -0.537465 A:D 179.219 179.507 0.724602 -0.689145 0.004305 -0.006812 0.689154 0.724592 179.588 0.723417 0.003106 0.690404 B:C 179.644 -0.690403 -0.003594 0.723416 6 Misorientation angles ( ) and rotation axes (d) among four types of variants (A, B, C, and D) in Fig. 3.11a, where the coordinates of rotation axes refer to an orthonormal crystal coordinate frame fixed to the monoclinic martensite lattice basis. Table 3 . 3 7. It is seen that the variant pair A:C (or B:D) does possess the rational K 1 and 2 (type I twin), A:B (or C:D) the rational 1 and K 2 (type II twin), and A:D (or B:C) the rational K 1 , 1 , K 2 and 2 (compound twin). Among the three twin types, the compound twin type has the smallest twinning shear, while the other two types possess the twinning shear of the same order. Type I twin and type II twin are conjugate or reciprocal to each other with a common s, but K 1 and K 2 , and 1 and 2 Table 3 . 3 7 Full twinning elements of Ni 50 Mn 30 Ga 20 7M martensite twin variants. All the indices are expressed in the incommensurate 7M superlattice basis. Elements Type (A:C / B:D) Type (A:B / C:D) Compound (A:D / B:C) K 1 (1 2 10 ) (1.0621 2 9.3785 ) (1 0 10) K 2 (1.0621 2 9.3785) (1 2 10) (1 0 10) 1 [10.5541 10 0.9446] [10 10 1] [10 0 1] 2 [10 10 1 ] [10.5541 10 0.9446 ] [10 0 1] P (1 0.057 10.5699) (1 0.057 10.5699) (0 1 0) s 0.2299 0.2299 0.0135 Table 3 . 3 8. It can be seen that for the three types of twin interfaces, the calculated interface planes are in coincidence with their respective twinning planes (K 1 plane) within reasonable deviation, being close to {1 2 10 } 7M for type I twin, {1.0621 2 9.3785 } 7M for type II twin and {1 0 10} 7M for compound twin. Therefore, all these three types of twin interfaces can be considered as coherent interfaces. Obviously, individual movement of such interfaces would generate the smallest atomic mismatch, hence the highest mobility and reversibility as compared with other types of boundaries. This constitutes the positive necessities for the fast dynamic response to the actuating field in these materials. Table 3 . 3 8 Mean values of twin interface normals in the orthonormal crystal coordinate frame calculated by indirect two-trace method[113] and their deviations from the related K 1 planes. Twin type Variant pair Twin interface normal Deviation from related K 1 plane Type A:C B:D (0.46507, -0.75038, -0.46972) (0.46842, -0.74778, -0.47054) 1.09° deviation from {1 2 10 } 1.14° deviation from {1 2 10 } Type A:B C:D (0.48386, -0.74886, -0.45287) 0.27° deviation from {1.0621 2 9.3785 } (0.49928, -0.74497, -0.44242) 1.02° deviation from {1.0621 2 9.3785 } Compound A:D B:C (0.72807, 0.01186, 0.68540) (0.72073, 0.02296, 0.69284) 0.73° deviation from {1 0 10} 1.35° deviation from {1 0 10} Table 3 . 3 Group No. Variant pair Bain (°) K-S (°) N-W (°) Pitsch (°) A:B 6.68 0.38 2.05 0.39 g1 A:C 6.78 0.33 1.38 0.30 A:D 0.72 0.30 1.74 0.21 A:B 7.47 1.14 2.31 1.14 g2 A:C 7.39 1.85 2.51 1.52 A:D 0.35 0.60 1.43 0.32 A:B 6.46 0.27 2.00 0.27 g3 A:C 6.71 0.89 1.54 0.57 A:D 0.47 0.81 1.50 0.49 A:B 7.38 1.30 3.18 1.27 g4 A:C 7.81 1.41 2.49 1.31 A:D 1.13 1.12 1.98 1.03 A:B 6.43 0.27 1.80 0.27 g5 A:C 6.43 0.72 1.22 0.42 A:D 0.83 0.89 1.89 0.53 A:B 6.13 0.89 1.09 0.89 g6 A:C 5.52 1.50 1.02 1.36 A:D 0.58 1.09 1.15 0.93 g7 A:B 6.69 0.24 1.89 0.25 9 Minimum misorientation angles between austenite orientations calculated from martensitic variant pairs (A:B, A:C and A:D) in seven variant groups under Bain, K-S, N-W and Pitsch relations. Table 3 . 3 10 Components of lattice deformation for austenite to 1M martensite transformation under K-S OR and Pitsch OR. K-S Pitsch Dilation in [111] A -0.4834% Dilation in [101] A -0.3923% Dilation in [1 2 1] A 0.0531% Dilation in [0 1 0] A -0.0428% Dilation in [10 1 ] A 0.2664% Dilation in [10 1 ] A 0.2664% Shear in (111) A [ 1 01] A 0.0701 Shear in (101) A [ 1 01] A 0.0051 Shear in (1 2 1) A [10 1 ] A 0.0910 Shear in (0 1 0) A [10 1 ] A 0.1149 Shear in (10 1 ) A [1 2 1] A 0.0018 Shear in (10 1 ) A [0 1 0] A 0.0045 Table 3 . 3 11 Misorientation calculation results of the two fine lamellae in plate P1 expressed in the orthonormal basis. Misorientation Rotation axes angle (°) d1 d2 d3 79.48 -0.705085 0.709088 -0.007046 100.53 0.711234 -0.702951 0.002353 113.69 -0.763601 0.002162 0.645685 114.56 0.002151 0.759855 -0.650089 126.09 0.005053 -0.862696 -0.505697 Table 3 . 3 No. n 1 n 2 n 3 1 0.564232 -0.503713 0.654153 2 0.577535 -0.474879 0.664036 3 0.582612 -0.484456 0.652583 4 0.531444 -0.52204 0.667114 5 0.577133 -0.491117 0.652473 6 0.579255 -0.506573 0.63863 7 0.576666 -0.487242 0.655783 8 0.553357 -0.498681 0.667168 9 0.580171 -0.499824 0.6431 10 0.557753 -0.508412 0.65607 11 0.549901 -0.503883 0.666116 12 0.56102 -0.516914 0.646573 13 0.563209 -0.507294 0.652264 14 0.555645 -0.483403 0.676447 Mean value 0.565129 -0.499292 0.656763 12 Coordinates of the inter-plate interface normal n expressed in the orthonormal crystal basis. Table 4 4 n = i i n n is {0.736130, 0.673329, 0.068855} A , which is near to {110} A with 4.7° deviation. .1, manifest that the habit planes between the austenite and the 7M martensite are irrational. The mean values of the calculated habit plane normals determined using the formula Table 4 . 4 1 Coordinates (n 1 , n 2 , n 3 ) of the calculated habit plane normals and the mean habit plane. All the indices are expressed in the coordinate frame of austenite. No. n 1 n 2 n 3 1 0.738894 0.665652 0.104611 2 0.738351 0.669307 0.082866 3 0.756754 0.650963 0.059750 4 0.723577 0.687154 0.065253 5 0.742925 0.662117 0.098304 Table 4 . 4 2 Habit plane normals, shape deformation directions and shape deformation matrices of variant A, B, C and D in the group around (101) A pole calculated according to the phenomenological theory. The results are expressed in the austenite basis. Variant Habit plane normal Shape deformation direction Shape deformation matrix 0.951753 -0.002901 -0.050195 A [0.692379, 0.041627, 0.720332] [-0.726173, 0.039664, 0.686367] 0.002635 1.000158 0.002742 0.045603 0.002742 1.047444 1.047444 0.002742 0.045603 B [0.720332, 0.041627, 0.692379] [0.686367, 0.039664, -0.726173] 0.002742 1.000158 0.002635 -0.050195 -0.002901 0.951753 1.047444 -0.002742 0.045603 C [0.720332, -0.041627, 0.692379] [0.686367, -0.039664, -0.726173] -0.002742 1.000158 -0.002635 -0.050195 0.002901 0.951753 0.951753 0.002901 -0.050195 D [0.692379, -0.041627, 0.720332] [-0.726173, -0.039664, 0.686367] -0.002635 1.000158 -0.002742 0.045603 -0.002742 1.047444 Table 4 . 4 3 Lattice constants of the modulated martensite in the frame of 7M(IC) and nanotwin combination, in comparison with those calculated according to the adaptive phase criteria (a ad = c Tet + a Teta A ; b ad = a A ; c ad = a Tet ). Noted that all the lattice parameters are expressed in the cubic parent phase coordinate system. Lattice constants 7M(IC) (Å) Nanotwin combination (Å) Theoretical (Å) a 6.082 6.200 6.186 b 5.823 5.761 5.811 c 5.537 5.486 5.486 Table 4 . 4 4 Twinning elements of three types of twins in Ni 53 Mn 22 Ga 25 7M martensite under the 7M(IC) and nanotwin combination models. Model Type (A:C / B:D) Type (A:B / C:D) Compound (A:D / B:C) K 1 (1 2 10 ) (1.0632 2 9.3676 ) (1 0 10) K 2 (1.0632 2 9.3676) (1 2 10) (1 0 10) 7M(IC) 1 2 [10.5719 10 0.9428] [10 10 1 ] [10 10 1] [10.5719 10 0.9428 ] [10 0 1] [10 0 1] P (1 0.0589 10.5888) (1 0.0589 10.5888) (0 1 0) s 0.1883 0.1883 0.0113 K 1 (1 2 7 ) (1.1023 2 6.2839 ) (1 0 7) K 2 (1.1023 2 6.2839) (1 2 7) (1 0 7) Nanotwin 1 [ 7.6497 7 0.9072] [ 7 7 1] [ 7 0 1] combination 2 (7 7 1 ) [7.6497 7 0.9072 ] [7 0 1] P (1 0.0973 7.6813) (1 0.0973 7.6813) (0 1 0) s 0.2458 0.2458 0.0239 Table 4 . 4 5 Strain tensors between matrix and product phases for austenite to 7M and austenite to adaptive phase (nanotwin combination structure) transformation. Austenite -7M modulation Austenite -nanotwin combination 11 0.0275 0.0360 22 -0.0472 -0.0559 33 0.0207 0.0210 -0.0446 -0.0750 Table 4 . 4 6 Orientation relationships between the NM fine lamellae connected by inter-plate interface between plate P1 and P3. The orientation data of the thin lamellae were acquired manually by EBSD measurement. Orientation relationship L1/L3 twin-related K 1 =(112) Tet ;K 2 =(11 2 ) Tet ; L3/L3 twin-related 1 =[11 1 ] Tet ; 2 =[111] Tet ; s=0.344 L1/L3 (1 1 2) L1 5.91° from (1 1 2) L3 [20 1 ] L1 1.62° from [20 1 ] L3 L1 /L3 (010) L1 11.05° from (010) L3 [20 1 ] L1 1.07° from [20 1 ] L3 L1 /L3 (010) L1 2.70° from (1 1 2) L3 [20 1 ] L1 0.97° from [20 1 ] L3 Table 4 . 4 7 Strain tensors between matrix and product phases for austenite to 7M, austenite to NM and 7M to NM transformation. Austenite -7M Austenite -NM 7M-NM 11 0.0275 -0.0560 0.0083 22 -0.0472 0.1204 -0.0092 33 0.0207 -0.0560 0.0002 -0.0446 0 0.0252 ad = 4.257 Å; b ad = 5.486 Å; c ad = 29.446 Å; ad = 94.2°; space group: PM (No. 6) Ni 2c -0.0965 0.25 0.6429 Appendix III: atomic coordinates of nanotwinned 7M superstructure Ni 2c 0.3553 0.25 0.7143 Appendix II: atomic coordinates of incommensurate 7M Ni 2c -0.1930 0.25 0.7857 Ni 2c 0.2588 0.25 0.8571 superstructure Ni 2c -0.1206 0.25 0.9286 Atom type Wyck. position x y z Mn a 7M = 4.2651 Å; b 7M = 5.5114 Å; c 7M = 42.365 Å; = 93.27°; space group: P2/m (No. 10) 1a 0 0 0 Mn 1b 0.4518 0.5 0.0714 Atom type Wyck. position Mn 1a x -0.0965 y 0 z 0.1429 Mn Mn 1a 1b 0 0.3553 0 0.5 0 0.2143 Mn Mn 1c 1a 0 -0.1930 0 0 0.5 0.2857 Mn Mn 2m 1b 0.066 0.2588 0 0.5 0.1004 0.3571 Mn Mn 2m 1a -0.02139 -0.1206 0 0 0.1996 0.4286 Mn Mn 2m 1b -0.0848 0.5 0 0.5 0.3000 0.5 Mn Mn 2m 1a 0.1055 -0.0482 0 0 0.4002 0.5714 Mn Mn 2n 1b 0.605 0.4035 0.5 0.5 0.0501 0.6429 Mn Mn 2n 1a 0.5083 -0.1447 0.5 0 0.1502 0.7143 Mn Mn 2n 1b 0.4096 0.3070 0.5 0.5 0.2496 0.7857 Mn Mn 2n 1a 0.5475 -0.2412 0.5 0 0.3499 0.8571 Mn Mn 2n 1b 0.5414 0.3794 0.5 0.5 0.4505 0.9286 Ga Ga 1b 1b 0 0 0.5 0.5 0 0 Ga Ga 1f 1a 0 0.4518 0.5 0 0.5 0.0714 Ga Ga 2n 1b 0.066 -0.0965 0.5 0.5 0.1004 0.1429 Ga Ga 2n 1a -0.02139 0.3553 0.5 0 0.1996 0.2143 Ga Ga 2n 1b -0.0848 -0.1930 0.5 0.5 0.3000 0.2857 Ga Ga 2n 1a 0.1055 0.2588 0.5 0 0.4002 0.3571 Ga Ga 2m 1b 0.605 -0.1206 0 0.5 0.0501 0.4286 Ga Ga 2m 1a 0.5083 0.5 0 0 0.1502 0.5 Ga Ga 2m 1b 0.4096 -0.0482 0 0.5 0.2496 0.5714 Ga Ga 2m 1a 0.5475 0.4035 0 0 0.3499 0.6429 Ga Ga 2m 1b 0.5414 -0.1447 0 0.5 0.4505 0.7143 Ni Ga 2l 1a 0.5 0.3070 0.75 0 0.5 0.7857 Ni Ga 2j 1b 0.5 -0.2412 0.75 0.5 0 0.8571 Ni Ga 4o 1a 0.1056 0.3794 0.25 0 0.0501 0.9286 Ni Ni 4o 2c 0.0075 0.5 0.25 0.25 0.1502 0 Ni Ni 4o 2c -0.0920 -0.0482 0.25 0.25 0.2496 0.0714 Ni Ni 4o 2c 0.0511 0.4035 0.25 0.25 0.3499 0.1429 Ni Ni 4o 2c 0.0395 -0.1447 0.25 0.25 0.4505 0.2143 Ni Ni 4o 2c 0.5663 0.3070 0.75 0.25 0.1004 0.2857 Ni Ni 4o 2c 0.4786 -0.2412 0.75 0.25 0.1996 0.3571 Ni Ni 4o 2c 0.4151 0.3794 0.75 0.25 0.3000 0.4286 Ni Ni 4o 2c 0.6055 0 0.75 0.25 0.4002 0.5 Ni 2c 0.4517 0.25 0.5714 a Tableau 1 1 Angles de désorientation ( ) et les axes de rotation (d) entre les variantes A, B, C et D dans la figure3(a). Les axes de rotation sont répérés au repère orthonormé de cristal placé sur le réseau monoclinique de la martensite. Eléments de maclage de la martensite modulée 5M dans Ni 50 Mn 28 Ga 22 . sont nécessaires pour éliminer la discontinuité de réseau. Ces ORs ont été généralement déterminées en tirant parti de la coexistence de l'austénite résiduelle et la martensite produite. Pourtant, pour le présent alliage étudié, comme la Variant pair Misorientation angle [°] d 1 Rotation axis, d d 2 d 3 A:C 86.80 -0.70846 0.00241 -0.70575 179.81 0.48490 -0.72659 -0.48676 B:D 86.79 179.83 -0.71242 -0.48214 -0.00221 0.72661 -0.70175 0.48947 93.32 0.70870 0.00098 0.70551 A:B 179.92 -0.51312 -0.68633 0.51543 C:D 93.09 179.64 0.70882 -0.51205 0.00435 -0.68778 0.70538 0.51455 A:D 179.64 179.89 0.70487 -0.70933 -0.00098 0.00317 -0.70933 -0.70487 179.88 0.70978 0.00079 0.70443 B:C 179.91 -0.70443 -0.00102 0.70978 Tableau 2 Type (A:C and B:D) Type (A:B and C:D) Compound (A:D and B:C) K 1 (1 2 5 ) (1.0569 2 4.7155 ) (1 0 5) K 2 (1.0569 2 4.7155) (1 2 5) (1 0 5) 1 [ 5.2504 5 0.9499] [ 5 5 1] [ 5 0 1] 2 [5 5 1 ] [5.2504 5 0.9499 ] [5 0 1] P (1 0.0514 5.2568) (1 0.0514 5.2568) (0 1 0) s 0.1387 0.1387 0.0074 Comme la transformation martensitique est réalisée par le déplacement coordonné d'atomes, certaines relations d'orientation (ORs) spécifiques entre la phase parent et la phase produit Afin de quantifier les écarts pour une OR présumée, les angles désorientation minimum entre deux orientations calculées de l'austénite ont été estimés. En effet, parmi tous les groupes de variantes choisies, Pitsch présente le plus petit angle de déviation. Par conséquent, la OR de Pitsch, c'est à dire (101) A //(1 2 1 ) 1M et [10 1 ] A //[ 1 1 1] 1M entre l'austénite et la martensite 1M, ou (101) A //(1 2 5 ) 5M et [10 1 ] A //[ 5 5 1] 5M si raaportée à la martensite 5M, devrait être considérée comme la OR la plus favorable pour la transformation. donnée ont été tracées dans la projection stéréographique standard {001} de l'austénite dans le repère macroscopique de l'échantillon. A titre d'exemple, la figure 3 affiche les résultats du calcul d'un groupe de variantes, où une orientation de l'austénite est représentée par trois pôles {001}. On voit qu'il y a une orientation distincte calculée pour l'austénite à partir de la martensite sous l'hypothèse de la relation de Bain, et deux sous KS, NW et Pitsch, respectivement. Si la OR présumée est vraie pour la transformation, tous les trois pôles {001} austénite d'une variante de martensite devraient correspondre à celles des trois autres variantes dans le même En se basant sur les considérations qui précèdent, nous avons calculé les orientations de l'austénite parente à partir des orientations mesurées des quatre variantes de martensite qui sont localement liées en relation de macle. Ici, un ensemble de quatre variantes en relation de macle dans un platelet large est traité comme un groupe de variantes. Les orientations calculées de l'austénite sous une OR groupe de variantes. Il apparaît que les relations de KS et Pitsch donnent le moins de dispersion parmi les pôles {001} respectifs pour une orientation commune de l'austénite. Tableau 3 3 Éléments de maclage de la martensite 7M modulée de l'alliage Ni 50 Mn 30 Ga 20 . Tous les indices sont exprimés en utilisant le modèle de supermaille composée de 10 sous cellules. Elements Type-(A:C or B:D) Type-(A:B or C:D) Compound (A:D or B:C) K 1 (1 2 10 ) (1.0621 2 9.3785 ) (1 0 10) K 2 (1.0621 2 9.3785) (1 2 10) (1 0 10) 1 [10.5541 10 0.9446] [10 10 1] [10 0 1] 2 [10 10 1 ] [10.5541 10 0.9446 ] [10 0 1] P (1 0.057 10.5699) (1 0.057 10.5699) (0 1 0) comme cela apparaît dans les agrandissements des carrés sur les figures 6 (b) et (d). Pour quantifier la qualité de l'accord des relations d'orientation, les angles de désorientation minimum entre les orientations des diverses solutions calculées pour l'austénite ont été estimés. Les résultats ont montré que l'OR de KS et celle de Pitsch donnent le plus petit angle de déviation et qu'il n'y a presque pas de différence entre les deux ORs, ce qui suggère que ces deux ORs pourraient être considérées comme possibles pour la transformation. Pour continuer à discriminer les deux relations et déterminer celle qui serait la plus favorable pour la transformation, les déformations de réseau pendant la transformation sous l'hypothèse des deux relations (cf. figure 7) ont été examinées. La déformation du réseau est décomposée en dilatation ou contraction et en cisaillement, comme indiqué dans la figure 7. Les quantités de déformation sous l'hypothèse des deux relations ont été calculées et reportées dans le tableau 4. Dans le cas de KS, l'allongement ou la contraction est assigné dans [111] A , [1 2 1] A et [10 1 ] A . Le cisaillement simple dans (111) A le long de [ 1 01] A , dans (1 2 1) A le long de [10 1 ] A et 'écart de plan, dans la mesure où les deux relations d'orientation possèdent le même parallélisme entre les directions. La OR de KS provoque un écart plus grand que celle de Pitsch, ce qui signifie que la OR de KS exige un plus grand réarrangement atomique pour réaliser la modulation de la structure. Ce résultat confirme également que la relation d'orientation de Pitsch, c'est à dire (101) A //(1 2 10 ) 7M et [10 1 ] A //[10 10 1] 7M entre l'austénite et la martensite, est plus favorable que celle de KS pour la transformation martensitique. dans (10 1 ) A le long de [1 2 1] A . Pour celle de Pitsch, l'allongement ou la contraction est assigné dans [101] A , [0 1 0] A and [10 1 ] A et le cisaillement simple dans (101) A le long de [ 1 01] A , dans (0 1 0) A le long de [10 1 ] A et dans (10 1 ) A le long de [0 1 0] A , respectivement. Par rapport à la relation de KS, celle de Pitsch implique une distorsion de réseau relativement petite (seulement les 2 derniers éléments de cisaillement sont légèrement élevés) pour la transformation de l'austénite à la martensite 1M moyenne. Considérant que l'allongement ou la contraction exige un changement de volume, et que le cisaillement simple n'exige pas, la déformation par un cisaillement simple peut se produire plus facilement. De ce point de vue, la relation de Pistch est énergiquement avantageuse par rapport à celle de KS pour la transformation de l'austénite en martensite 1M. Afin de mieux distinguer les quantités de déformation sous l'hypothèses des deux relations, les écarts entre les plans (1 2 1 ) (Pitch)/(011) (KS) et les directions [ 1 1 1] (Pitch et KS) de chaque souscellule 7M de la supermaille par rapport au ceux de la cellule moyenne 1M sont évalués. Comme montré dans la figure 8, la relation de KS et celle de Pitsch diffèrent l'une de l'autre seulement par l Acknowledgements This work is financially supported by the National Natural Science Foundation of China (Grant No. 50820135101), the Ministry of Education of China (Grant Nos. 2007B35, 707017 and IRT0713), the Fundamental Research Funds for the Central Universities of China (Grant No. N090602002), the CNRS of France (PICS No. 4164) and the joint Chinese-French project OPTIMAG (N°ANR-09-BLAN-0382). I would like to give my sincere thanks to these institutions. I also gratefully acknowledge the French Embassy in Beijing for providing a grant for my study in France. This work is completed at LEM3 (former LETAM, University of Metz, France) and the Key Laboratory for Anisotropy and Texture of Materials (Northeastern University, China). I had the honor to work with numerous colleagues in two labs and I would like to give my hearted thanks for their kind help. I would like to sincerely thank to the reviewers, Prof. Y. Fautrelle and Prof. Z. Q. Hu, for taking time out of their busy schedules to review my dissertation and provide constructive suggestions and comments. I would like to give my special thanks to my supervisors, Prof. Claude Esling, Dr. Yudong Zhang at University of Metz, and Prof. Liang Zuo at Northeastern University, not only for their support, ideas, guidance, and organizational help during the last three years, but also making me a better person and scientist by setting high standards and good examples. I would also like to thank Prof. Xiang Zhao and Dr. Changshu He, for encouraging, supporting, and mentoring me during my PhD study. I would like to thank all my group members who treated me with dignity and respect. Finally, I want to thank my parents, friends and family, especially my wife Ms. Xiaoliang Li, for their continuous support and understanding Avec la méthode ci-dessus, nous avons calculé les orientations de l'austénite à partir des orientations mesurées de variantes martensitiques, en utilisant les quatre ORs classiques. Pour faciliter la visualisation, les orientations calculées de l'austénite sont montrées dans la figures de pôles {001} dans le repère macroscopique de l'échantillon (figure 6). Si la OR assumée est effectivement celle de la transformation, les pôles {001} de l'austénite calculés à partir de différentes variantes martensitiques devraient tomber en coïncidence dans la figure de pôle. Apparemment, la relation d'orientation de KS et celle de Pitsch peuvent donner lieu au meilleur accord parmi les figures de pôles {001} correspondantes pour une orientation d'austénite commune, Clarification of structure and stability of long-period modulated martensite As one of the common characteristics of the -phase alloys [71], Heusler alloys [70,125] and piezoelectric materials [126], the transformation process from the parent austenite to the product martensite produces long-period lattice-modulated structures. The thermodynamic stability and the crystallographic nature of this modulated structures have been questioned [81], because the product martensite was found to have a non-modulated (NM) structure in several alloy systems [65,[127][128][129][130]. By analyzing the crystal lattice mismatch between the austenite and the NM martensite, Khachaturyan et al. [130,131] interpreted that the modulated structure with a plate-like macroscopic shape is actually the combination of twin-related variants of the normal NM martensite phase, wherein the twin domain has a lamellar shape in the order of a few atomic layers. The main argument for this structure model is that the geometrical requirement for an invariant habit plane between the parent and the product lattices can be fulfilled, which lowers the volume-dependent elastic energy. If the long-period modulated martensite is composed of nanotwin-related lamellae with a simple NM structure, the lattice constants of the modulated martensite phase should satisfy some specific relations with those of the parent austenite and the NM martensite [130,131]. Such relations have been taken as the criteria for the verification of the nanotwin combined structure (i.e. adaptive phase). For instance, excellent fits between the measured and calculated lattice constants were found in Ni-Al, Fe-Pd and lead-based ferroelectric perovskites [130,131]. However, the validity of this concept is strongly questioned in lead titanate [126], because the Raman spectroscopy measurement results demonstrated that the modulated phase possesses an independent crystal structure. Recently, this contradiction arose in the newly developed Ni-Mn-Ga ferromagnetic shape memory alloys. Righi et al. [75] made a detailed study on the crystal structure of the so-called seven-layered (7M) modulated martensite by application of the superspace approach to the powder X-ray diffraction (XRD)

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2012

https://hal.univ-lorraine.fr/tel-01749215/file/DDOC_T_2012_0049_SADIQ.pdf

ACKNOWLEDGEMENTS First of all, I would like to thank Dr. Mohammed Cherkaoui for providing me an opportunity to work under his supervision. His support, guidance and motivation have always been a great inspiration during the entire work. I would like to thank Dr. Karim Inal, Dr. Olaf Van der Sluis, Dr. El Mostafa Daya, Dr. Raphael Pesci, Dr. Suriyakan Kleitz and Mme. Stephanie Blanc who accepted my request to be the part of my PhD committee. I strongly appreciate their suggestions and remarks. I would like to thank the whole techno-group of Schlumberger, Paris, for providing me an opportunity to work on such a challenging project. I would like to thank Jean-Sebastien Lecomte for his help in Nanoindentation, Claude Guyomard for the die design, Olivier Naegelen for the sample casting and Marc Wary for the polishing and etching processes. I am also thankful to Abderrahim Nachit and Ammar for their help in the electrical characterization of the solder alloys. I am thankful to Dr. Min Pei for his help in the coarsening models and David Macel for his support in wettability testing. My experience at University of Lorraine was made more pleasant by my colleagues Wei, Mathieu, Bhasker, Malek and Liaqat, Sajid, Rafiq, Armaghan, Aamer, Jawad, Khuda Bux, Mohsin and Fahd. I would like to thank all of them. I would like to offer special thanks to my brother Muhammad Arif for all his help and support in editing and organizing my thesis. I would like to thank the whole team of University of Lorraine who helped me in everything to make this period more interesting and enjoyable. Ag 3 Sn RE RE RE K Q d d t T R T RE d d RE RE K K R E RE Q Q RE PILE-UP AREA CALCULATIONS c c A b h pu i A a i c ic i i b A h Ch a r op pu S E A A op pu F H A A 8.7 CREEP CHARACTERIZATIONS BY NANOINDENTATION b t CHAPTER 9 WETTABILITY TESTING Wettability of Solders Wetting is the behavior of a liquid to wet a solid surface. This behavior is very important to the interconnection of electronic packages, especially in soldering, for making highly strong bonding between different solid components. In order to attain successful soldering, a certain degree of wetting of the molten solder on the solid substrate surface is required. So generally, a wetting or solderability test is used to measure, The initial wettability of the component termination materials The wetting properties of a newly developed solder A proper metallurgical bond is always necessary to form before analyzing the wetting performance of any solder alloy. This bond, of course, varies for different substrates. An interfacial reaction takes place which forms certain amount of IMCs or in some cases an Inter Metallic Layer (IML) at the two interfaces. These IMCs or IML works as an adhesion layer between the substrate and the solder and hence keep them firmly together. The wettability of any lead-free solder alloy varies with changing its chemical composition and the substrate. For SnPb, the wettability is good over many metallic substrates including copper. Thus, changing from SnPb to lead-free, wettability is an important concern. The substitute must have sufficient wettability under severe service conditions and for different soldering processes such as reflow and wave soldering. Wettability Measurement Methods There are mainly 2 methods for the wettability measurements; Spread-area test (good for reflow soldering) and the wetting balance test (good for the wave soldering). They are described below. Spread-Area Test A solder disc (e.g. 6 mm in diameter and 1 mm in thickness) is coated with flux and melted, and then allowed to solidify on a substrate (e.g. Cu). When a bond is formed, the free energy is reduced and hence the solder changes its shape [START_REF] Wu | Properties of lead-free solder alloys with rare earth element additions[END_REF]]. This change in shape causes an increase in the contact area which indicates the wetting behavior of solder. In some cases, the ratio of the as-bonded area to this new area (after soldering) is taken as the wettability of solder. This wetting is given by wetting (or contact angle), c , and is calculated using the Young Dupre's equations as 9.1 and 9.2 [START_REF] Wu | Properties of lead-free solder alloys with rare earth element additions[END_REF] Cos where SF , LF , and SL refer to the surface tensions of the substrate/flux, liquid (solder)/flux, and substrate/liquid (solder) interfaces, respectively, as shown in Figure 9.1 [START_REF] Wu | Properties of lead-free solder alloys with rare earth element additions[END_REF]]. For a good wetting, the contact or wetting angle should be small (i.e. LF , and SL smaller and SF larger). The spread area test is a good approximation for the reflow soldering process. The solder disc, in the spread area test, has a similar shape to the layer of solder paste by screen-printing or in similar process. Besides, the solder disc in the spread area test and the solder paste in reflow soldering undergo the same process of heating. They are heated above the melting point and are then allowed to solidify. Wetting Balance Test Wetting balance test is the second important technique to evaluate the solder wettability. In this method, the coupon (for example Cu) is dipped into the molten solder, present inside the crucible at a temperature more than its melting point. The molten solder climbs up the Cu coupon due to wetting force exerted on it as shown in Figure 9.2 [START_REF] Wu | Properties of lead-free solder alloys with rare earth element additions[END_REF]]. As spread area test was a good approximation of reflow soldering, this wetting balance test is a good approximation of the wave soldering in which the substrate is brought into contact with the molten solder. The height of the solder onto the Cu coupon depends on many parameters including the soldering temperature, alloy composition and wetting time. Generally, the wetting F Vg p Where p is the perimeter of the coupon, LF the surface tension of the solder in contact with the flux, c the contact angle, the density of the solder, g the acceleration due to gravity, and V is the immersed volume. A typical wetting curve is shown in Figure 9.3 which represents different types of wettings for solders, in which the wetting time, t w , is the time at which the solder contact angle to the coupon is 90°, as shown at point B. Wetting occurring in a short wetting time is considered to be good. Hence, the wetting force and time of each solder system are revealed by the wetting balance test. The wettability of lead-free solders becomes crucial when high solder joint reliability is needed. As presented in equation 9.4, the contact angle is one of the most important parameter in characterizing the solderability (wettability) of any lead-free solder. A smaller contact angle, c , and thus higher cos c is representing good wettability. Ideally, cos c can reach to 1 which makes c to be 0° which is the best possible wettability. But, in practice, it is somewhere in the range of 40 to 50°, as shown in Figure 9.4, depending on the temperature applied, flux used and many other experimental parameters. On the other hand, a contact angle of more than 90° is not recommended for good joining. A schematic of contact angle more than 90° is provided in In [START_REF] Wu | Properties of lead-free solder alloys with rare earth element additions[END_REF]], the impact of RE elements, mainly Ce and La, has been studied for the wettability performance on SnPb, SnAg, SnCu and SAC. It has been reported that they can all be soldered successfully using the rosin mildly activated (RMA) flux. The wettability of SnPb solder is quite good in comparison to SnAg, SnCu and SAC alloys. In any case, over the copper substrates, SAC alloys perform well whereas the RE dopings have shown significant improvements in the wetting performance of all SnAg, SnCu and SAC alloys [START_REF] Wu | Properties of lead-free solder alloys with rare earth element additions[END_REF]]. In most of the cases, the addition of RE elements help to reduce the wetting angle. It can be seen that the amount of RE elements needs to be optimized for the best wetting performance, Figure 9.6. An extensive experimental study on wettability study on Sn3.5Ag eutectic and Sn-3.5Ag-RE doped alloys was made by [START_REF] Wu | Properties of lead-free solder alloys with rare earth element additions[END_REF]. The wetting curves of SnAg, SnAg-0.25RE and SnAg-0.5RE are compared with that of SnPb in Figure 9.6. It can be seen that all four solders have roughly the same wetting time. However, the wetting force of SnPb is the highest among the four alloys. The effect of adding RE elements (mainly Ce and La) is clearly demonstrated by a significant increase in the wetting force of Sn3.5Ag-0.25RE alloy as getting closer to the SnPb solder. However, an excessive amount of RE addition can lower the wetting force. In the same study, this effect was also applicable to the wetting angle, such that SnAg-0.25RE has the lowest wetting angle among SnAg, SnAg-0.25RE and SnAg-0.5RE. Sample Preparation And Experimental Conditions The equipment used for the wetting balance test was METRONELEC as shown in Figure 9.7, a French manufacturer of soldering equipments. The copper foils used as substrates were 0.025mm thick and 12mm wide. They were purchased from the "Good Fellows", France, with a purity of 99.9%. The samples were cleaned with acetone and dried. Later on, they were dipped into 20 % HNO 3 solution for 20 seconds and then placed in the flux for about 1 second. The flux used was resin mildly activated (RMA) and was manufactured by "Metaux Blancs Ouvres" with fluxing code BC 310.15. The immersion depth into the crucibles, for the copper substrate, was 3mm as shown in Figure 9.8. The wetting time used was 10s and the wetting temperatures applied were 250°C and 260°C. Cos The angle as directly linked to surface tensions is thus representative of the wetting quality. For a smaller contact angle, and thus better wettability, LV should be as small as possible. This is the main function performed by adding the RE elements which decrease the surface tension and this reduces the contact angle as performed during the experiments. This is important to mention that the contact angle is only conceivable for the alloys in the molten state. Meniscography Principle As described earlier, after immersing the Cu coupon into the molten solder bath, the surface tensions are highest at the solder/flux interface. The measurement of the resultant force is representative of the meniscus and, consequently, of the wetting angle and of the solderability. The principle set forth above is described in equation 9. Where is the mass density of the solder alloys, which is provided in Table 9.1, P is the perimeter of the coupon immersed into the solder crucible and g is the acceleration due to gravity (9.81 m/s 2 ). Thus F is the measured force from which the contact angle is deduced. Wettability Measuring Parameters Generally, there are 3 important parameters for the wettability measurements. These are the surface tension, wetting force and the contact (or wetting) angle. These parameters with required description and evaluation are provided below. Flux-Solder Surface Tension In order to determine the wettability in terms of wetting force and contact angles, it is necessary to calculate the surface tension, LV . The results for surface tension are given in RE is in excess (>0.25%), is mainly because of the oxidation resistance during wetting due to the strong affinity of RE to oxidize. Oxidation at the solder surface inhibits the molten solder from contacting with the solid substrate, which usually causes non-wetting behavior. But when the oxide is massive, or the oxide quickly generates, the flux cannot perfectly remove the oxides, and the wettability will degrade. This has also been confirmed by the work performed by [START_REF] Zhou | Investigation on properties of Sn 8Zn 3Bi lead-free solder by Nd addition[END_REF]]. They showed (after TGA analysis) that 0.1% of Nd is the optimum amount as the surface tension increases if Nd is more than 0.1% which finally increases the contact angle. Similarly [START_REF] Shi | Effects of small amount addition of rare earth Er on microstructure and property of SnAgCu solder[END_REF]] have suggested the maximum RE (Er) doping as 0.1-0.4% and [Wang el al. 2002] have demonstrated RE (Ce) as 0.25-0.5% in SnAg alloys. In the work performed by [START_REF] Noh | Effects of cerium content on wettability, microstructure and mechanical properties of Sn Ag Ce solder alloys[END_REF], they stated that RE is liable to oxidation [START_REF] Huang | Creep behavior of eutectic Sn-Ag lead-free solder alloy[END_REF] due to its strong affinity towards oxygen and have suggested 0.3 % of Ce as the optimum doping in SAC alloys. The Summary Extensive work has been performed on the wettability testing of SAC and SAC-La doped alloys at 2 different temperatures of 250°C and 260°C. It was noticed that increasing temperatures make better wettability for the same composition. This is dedicated to the decreasing effects of surface tension at elevated temperatures. For the SAC-La doped alloys, the surface tension decreased due to RE dopings at both 250°C and 260°C as compare to the SAC alloy. RE doping further increases the wetting forces from 5.7 mN (SAC) up to 6.7 mN (SAC-0.25La). SAC-0.5La has smaller wetting force than SAC-0.25La which means an optimization of RE elements is to be specified for better alloy compositions. Wetting or contact angles measurements were made on the basis of the surface tensions already evaluated. An appreciable decrease in the contact angles was investigated and once again SAC-0.25La has a better (smaller) contact angle than SAC and SAC-0.5La alloys. I am extremely thankful to Dr. Raphael Pesci for all his contributions, support and motivation during the entire experimental work. fr attitude gave me the real motivation to work in a friendly environment. I am thankful to Dr. El-Mostafa Daya and Dr. Jean-Marc Raulot from LEM3, University of Lorraine, France for their enriching discussions and valuable remarks. Figure 9 . 1 : 91 Figure 9.1: Schematic diagram of the spread area test Figure 9.2: Coupon geometry for wetting balance test on molten solder alloy[START_REF] Wu | Properties of lead-free solder alloys with rare earth element additions[END_REF] Figure Figure 9.5. Figure 9 9 Figure 9.3: Variation of wetting force with time [METRONELEC] Figure 9 9 Figure 9.6: Wetting curves of Sn-3.5Ag-RE solders compared with Sn-Pb solder[START_REF] Wu | Properties of lead-free solder alloys with rare earth element additions[END_REF] Figure Figure 9.7: METRONELEC wettability setup Figures 9 . 9 Figures 9.10-9.16 for different compositions and at different temperatures. Good reproducibility Figure 9 .Figure 9 Figure 9.10: SAC surface tension vs. wetting time at 250°C -9.20. The average values with 5% error bars are given in Figure 9.21. Figure 9 . 9 Figure9.17: SAC wetting force vs. wetting time at 250°C oxides is also confirmed during our EDS mapping on the free surface of unpolished specimens. Figure 9 . 9 Figure 9.22: SAC contact angle vs. wetting time at 250°C ], ..........................................................................(9.1) SF SL LF Cos c c SF SL .............................................................................(9.2) LF Table 9 . 9 1: Densities for SAC and SAC-La alloys Alloy Density (mg/mm 3 ) SAC 7.45 SAC-0.05 7.453 SAC0.25 7.46 SAC-0.5 7.48 CHAPTER 6 MECHANICAL PROPERTIES

01749219

en

2024/03/05 22:32:07

2012

https://hal.univ-lorraine.fr/tel-01749219/file/DDOC_T_2012_0052_ZHANG.pdf

Keywords: Magnetic Field, Phase Transformation, Magnetic Dipolar Interaction, Texture,Orientation Relationship In this work, the influence of the magnetic field on diffusional phase transformation in high purity Fe-C alloys has been investigated theoretically and experimentally. The magnetic field induced microstructural features and crystallographic orientation characteristics have been thoroughly studied in three different carbon content alloys: Fe-0.12C, Fe-0.36C and Fe-1.1C alloys. Magnetic field induces different aligned and elongated microstructures along the field direction, namely aligned and elongated pearlite colonies in Fe-0.12C alloy and elongated proeutectoid ferrite grains in Fe-0.36C alloy, due to the two scaled magnetic dipolar interaction. Magnetic field increases the amount of ferrite in hypoeutectoid alloys and this field effect becomes more pronounced with the increase of the carbon composition. Magnetic field inhibits the formation of Widmanstätten ferrite by introducing additional driving force to ferritic transformation and thus reducing the need for low energy interface which is required to overcome the transformation barriers during slow cooling process. Magnetic field promotes the formation of abnormal structure by increasing the driving force of transformation from carbon-depleted austenite to ferrite and it enhances the spheroidization of pearlite due to its influence on accelerating carbon diffusion resulting from increased transformation temperature, together with its effect on increasing the relative ferrite/cementite interface energy. The field induced enhancement of carbon solution in ferrite is evidenced through the WDS-EPMA measurements for the first time. Ab-initio calculations reveal that the presence of an interstitial carbon atom in bcc Fe modifies the magnetic moments of its neighboring Fe atoms. This leads to the decrease of the demagnetization energy of the system and makes the system energetically more stable under the magnetic field. Due to the atomic-scaled magnetic dipolar interaction, magnetic field favors the nucleation and growth of the ferrite grains with their distorted <001> direction parallel to the transverse field direction, and thus induces the enhancement of the <001> fiber component in the transverse field direction. This field effect is related to the crystal lattice distortion induced by carbon solution and its impact becomes stronger with the increase of the carbon content and the field intensity. Three ORs between pearlitic ferrite and cementite have been found in present work, namely Isaichev (IS) OR and two close Pitsch-Petch (P-P) ORs. Magnetic field hardly changes the types of the appearing ORs, but it considerably increases the occurrence frequency of the P-P2 OR, especially in Fe-1.1C alloy, by favoring the nucleation of ferrite. II Résumé Dans ce travail, l'influence du champ magnétique sur la transformation de phase diffusionnelle dans des alliages Fe-C de haute pureté a été étudiée théoriquement et expérimentalement. Les caractéristiques microstructurales et celles d'orientations cristallographiques induites par le champ magnétique ont été soigneusement étudiées dans trois alliages Fe-C à différents taux de carbone, à savoir Fe-0.12C, Fe-0.36C, Fe-1.1C. Le champ magnétique induit différentes microstructures alignées et allongées le long de la direction du champ, à savoir des colonies de perlite alignées et allongées dans l'alliage Fe-0.12C et des grains allongés de ferrite proeutectoïde dans l'alliage Fe-0.36C, en raison de l'interaction magnétique dipolaire à deux différentes échelles. Le champ magnétique augmente la quantité de ferrite dans les alliages hypoeutectoïdes et cet effet du champ est plus prononcé avec l'augmentation du taux de carbone. Le champ magnétique inhibe la formation de ferrite de Widmanstätten en introduisant une force motrice supplémentaire pour la transformation ferritique et réduisant ainsi la nécessité de l'interface de faible énergie qui est requise pour surmonter les barrières de transformation durant le processus de refroidissement lent. Le champ magnétique favorise la formation de la structure anormale en augmentant la force d'entraînement de la transformation de l' austénite appauvrie en carbone en ferrite et il améliore la sphereoidization de perlite en raison de son influence sur l'accélération de la diffusion de carbone entraînée par l'augmentation de la température de transformation, ainsi que son effet sur l'augmentation de l'énergie relative de l'interface ferrite /cémentite. L'augmentation de la solubilité du carbone dans la ferrite induite par le champ est mise en évidence à travers les mesures WDS-EPMA pour la première fois. Des calculas ab-initio montrent que la présence d'un atome de carbone interstitiel dans Fe C modifie les moments magnétiques des atomes de Fe voisins. Ceci conduit à la diminution de l'énergie de démagnétisation du système et rend le système énergétiquement plus stable dans le champ magnétique. En raison de l'interaction dipolaire magnétique à l'échelle atomique, le champ magnétique favorise la nucléation et la croissance des grains de ferrite ayant leur direction <001> distordue parallèle à la direction du champ transversal, et induit donc l'augmentation de la composante de fibre <001> dans le sens transversal par rapport à la direction du champ. Cet effet du champ est relié à la distorsion du réseau cristallin induite par une solution de carbone et son impact devient plus fort avec l'augmentation de la teneur en carbone et l'intensité du champ. Trois relations d'orientations (OR) entre la ferrite perlitique et la cémentite ont été trouvées dans ce travail, à savoir l'OR Isaichev (IS) et deux OR proches des OR Pitsch-Petch (PP). Le champ magnétique ne modifie guère les types d'OR qui apparaissent, mais il augmente considérablement la fréquence d'occurrence des OR P-P2, en particulier dans l'alliage Fe-1.1C, en favorisant la nucléation de la ferrite. Mots General introduction The earliest known descriptions of magnets and their properties were around 2500 years ago, from Greece, India, and China, when man happened to notice the lodestones and their affinity to iron. Since then, people were undergoing a long period from being curious about its magnetic power, and attempting to uncover the physical essence, to fully understand it and even to establish corresponding theory to make use of it for every aspect of daily life and scientific researches. Magnetic field is considered as a powerful tool for studying the properties of matter, because they couple directly to the electronic charge and magnetic moments of the protons, neutrons, and electrons of which matter is made up. In fact, all the matters are magnetic, but some materials are much more magnetic than the others. The origin of magnetism lies in the orbital and spin motions of electrons and the way by which the electrons interact with one another. The magnetic behavior of materials can be classified into five major groups: diamagnetism, paramagnetism, ferromagnetism, ferrimagnetism and antiferromagnetism (The ordering of magnetic moments in diamagnetism, paramagnetism, ferromagnetism, ferrimagnetism and antiferromagnetism materials have been shown in Figure 1.1). Diamagnetic substances are composed of atoms which have no net magnetic moments-all the orbital shells are filled and there are no unpaired electrons. Many everyday materials, e.g., water, wood, glass and many elements, e.g., H 2 , Ar, Cu, Pb, are diamagnetic. When a diamagnetic material is exposed to an external field, it magnetized: a negative magnetization is produced. If the field is removed, the magnetization of diamagnetic material vanishes. For paramagnetic materials, some of the atoms or ions have a net magnetic moment due to unpaired electrons in partially filled orbitals. The individual magnetic moments do not interact magnetically, so the total magnetization is zero without a field. In the presence of a field, there is a partial alignment of the atomic magnetic moments in the direction of the field, resulting in a net positive magnetization and positive susceptibility. In addition, the efficiency of the field in aligning the moments is opposed by the randomizing effects of thermal agitation. This results in a temperature dependent susceptibility, known as the Curie Law. Unlike paramagnetic materials, the atomic moments in ferromagnetic materials exhibit very strong interactions. These interactions are produced by electronic exchange forces and result in a parallel or anti-parallel alignment of atomic moments. Ferromagnetic materials exhibit parallel alignment of moments resulting in large net magnetization even in the absence of a magnetic field. Even though electronic exchange forces in ferromagnets are very large, thermal energy eventually overcomes the exchange and produces a randomizing effect. This occurs at a particular temperature called the Curie temperature (Tc). Below Tc, the ferromagnet is ordered and above it, disordered. The saturation magnetization goes to zero at the Curie temperature. For ferrimagnetic materials, the magnetic moments of the atoms on different sublattices are antiparallel. However, the opposing moments are unequal and a resolved spontaneous magnetization remains. If the opposing moments are exactly equal, the net moment is zero. This type of magnetic ordering is called antiferromagnetism. Diamagnetic Paramagnetic Ferromagnetic Ferrimagnetic Antiferromagnetic Figure 1.1 Magnetic ordering in magnetic materials. H Under the magnetic field, the field effect on different materials will be verified according to their magnetic behaviors, this underlie the essence of the magnetic field effect in material science. As the development of the magnetic field theories and technique the application of the high magnetic field is greatly encouraged. High magnetic field as a clean, non-contacting and powerful source of energy can affect atomic behaviors, such as atom arrangement, matching and migration and thus exert powerful influence on microstructures and properties of materials. Consequently, more and more meaningful results and new phenomena together with economic profits are brought, which in turn inspires stronger desire to conduct more widening and deepening research. Recently, magnetic field has been introduced to solid phase transformations of metallic materials; especially Fe-C based alloys, for the purpose of microstructure control. As is known, steels are the most widely used materials related to every industry in daily live. Any small improvement in properties of Fe-C alloys means a big progress in human being and a huge amount of interests. Together with a bright prospective based on mature knowledge, the study of phase transformation of Fe-C based alloys under high magnetic field has attracted special attention from all over the world, and will become a popular topic in the following years. Basic mechanisms of the magnetic field influence on solid phase transformations in steels Solid phase transformations in steels are often referred to the decomposition of austenite, which is usually classified into three groups: diffusionless, semidiffusional and diffusional transformations, from a viewpoint of atom moment. The diffusionless phase transformation is known as martensitic transformation which occurs without long-range diffusion of neither Fe nor C atoms; semi-diffusional transformation refers to the bainitic transformation, when only C atoms diffuses; as for diffusional transformation, it usually corresponds to the pearlitic and ferritic transformations, in which cases, both Fe and C atoms can fulfill a long-range diffusion. In terms of these solid phase transformations in steels, the phases involved are with different magnetisms. There are three equilibrium phases with great engineering importance located within different carbon composition and temperature range. One is high temperature austenite; the others are ferrite and cementite, which are considered as basic components to the product of the austenite decomposition. High temperature phase austenite is paramagnetic; ferrite is ferromagnetic below ~770°C and cementite is paramagnetic at the formation temperature but becomes ferromagnetic below ~210°C. Due to the natural magnetic difference of parent and product phases, the transformation process and the resultant microstructure could be modified by an external magnetic field. This is because that magnetic field could modify the energy terms due to its magnetization effect during the phase transformation and thus affect the phase transformation both thermodynamically and kinetically. Take the phase transformation from austenite to ferrite for an example, when an external magnetic field was applied, the phases under the magnetic field will be magnetized and their Gibbs free energy will be changed by introducing so-called intrinsic magnetization energy. This magnetization energy is also denoted as the "magnetic Gibbs free energy", it can be written as dM H µ V M ∫ 0 0 0 , where V is the volume of the phase, 0 µ the permeability of free space (vacuum), 0 H the magnetic field strength in free space and M the induced magnetization per unit volume. Therefore, for a transformation from austenite to ferrite in the presence of a magnetic field, the total Gibbs free energy change M G γ α γ → + ∆ will contain two terms: the "chemical Gibbs free energy difference" C G ∆ and the "magnetic Gibbs free energy difference" M G ∆ as follows: M C M G G G γ α γ → + ∆ = ∆ + ∆ (1.1) 0 0 0 0 0 0 0 ( ( ) ( ) ( ) M M M M M G Hd M Hd M H dM dM α γ α γ α γ α γ µ µ µ ∆ = - - = - - ∫ ∫ ∫ ∫ (1.2) where superscript and subscript α and γ denote ferrite and austenite and M the magnetization. As the magnetization of ferrite is higher than that of austenite at all the transformation temperature concerned, Temperature T 0 T 0 M Gibbs free energy G γ γ γ γ G α α α α G γ γ γ γ Μ Μ Μ Μ G α α α α Μ Μ Μ Μ Figure 1. 2 Schematics of the Gibbs free energy of γ and α without and with a magnetic field, γ-austenite, α-ferrite, M-magnetic field. The Gibbs free energy difference between the parent and the product phases is also denoted as the phase transformation driving force which deal with the nucleation and growth process of the product phases, thus magnetic field affects not only the phase transformation thermodynamics but also the transformation kinetics such as modifying the transformation rate. In addition to the thermodynamic and kinetic effects, there also exists magnetization and demagnetization effect of the magnetic field resulting from the strong magnetic interaction among the magnetic moments of Fe atoms when placed in a magnetic field. It is called magnetic dipolar interaction. Each Fe atom can be treated as a magnetic dipole. When an external magnetic field is applied, the atomic moments tend to align along the field direction, as schematized in Figure 1.3. Then, there exists the dipolar interaction between neighboring dipoles. They attract each other along the field direction (magnetization), but repel each other in the transverse field direction (demagnetization). This dipolar interaction has mainly microstructural effect as it works on the nucleation and the growth process of ferrite during the transformation from austenite to ferrite, especially when ferrite is ferromagnetic with high magnetization. By minimizing the demagnetization energy (repelling among magnetic moments), it θ FD r may bring about grain shape anisotropy by preferential grain growth in the field direction or crystallographic texture by selective grain nucleation or growth. Solid phase transformations in steels under magnetic field As mentioned above, since the magnetic natures of the parent and product phases involved in solid phase transformation in steels are usually different, the phase transformation process is expected to be modified by the application of magnetic field and the field influence is of great importance to be investigated. In 1960's, studies on the effects of the magnetic field on austenitic decomposition began with the diffusionless martensitic transformation. As for martensitic transformation, the transformation temperature is relatively low and transformation can only be studied under low magnetic fields. As soon as the maturity of superconducting technique, the generation of direct current high magnetic field was available. From then on, a number of high temperature phase transformations under magnetic field are studied including semi-diffusional transformations and diffusional transformations. Although the studies on diffusional phase transformation under the magnetic field had a relatively late start, the results on this issue are far more fruitful than the other two. Diffusionless transformation-Martensitic transformation Martensitic transformation is the representative diffusionless transformation in ferrous alloys which involves austenite with face-centered-cubic crystal structure transforms to carbon oversaturated martensite with basically body-centered-cubic or body-centered-tetragonal crystal structure depending on carbon oversaturation degree. In martensitic transformation, the parent austenite is paramagnetic; whereas the martensite is ferromagnetic. The induced magnetization of martensite is much higher than that of austenite. When the magnetic field is applied, the Gibbs free energy of austenite does not change much but that of martensite is greatly lowered. Thus, the phase equilibrium between parent and product phases is changed by magnetic field. For many years, the investigations on martensitic transformation under the magnetic field were focused on the effect of the magnetic field on increasing the Ms temperature and enhancing the amount of transformed martensite. In early 1960's, studies on athermal martensitic transformation reported that magnetic field promoted martensitic transformation and increased the Ms temperature in ferrous alloys. Krivoglaz et al. [1] based on their thermodynamic analysis suggested that the effect of magnetic field on martensitic transformations be due only to the magnetostatic (Zeeman) energy. They proposed following formula to estimate the shift of Ms: q T V H M T / 0 ⋅ ⋅ ⋅ = δ δ δ (1.3) where M δ is the magnetization difference between the product and the parent phases, H the strength of applied magnetic field, V δ volume change between the two phases and q the latent heat of transformation. The calculated shift of Ms for Fe-Ni alloys fitted well the experimental results [2,3] . Based on this, Kakeshita et al. [4][5][6][START_REF] Kakeshita | Magnetoelastic martensitic transformation in an ausaged Fe-Ni-Co-Ti alloy[END_REF][START_REF] Kakeshita | Magnetic field-induced martensitic transformations in Fe-Ni-C invar and non-invar alloys[END_REF][START_REF] Kakeshita | Magnetic Field-Induced Martensitic Transformation in Single Crystals of Fe-31.6 at%Ni Alloy[END_REF][START_REF] Kakeshita | Magnetic field-induced transformation from paramagnetic austenite to ferromagnetic martensite in an Fe-3.9Mn-5.0C (at%) alloy[END_REF][START_REF] Kakeshita | Effect of magnetic fields on martensitic transformations in alloys with a paramagnetic to antiferromagnetic transition in the austenitic state[END_REF][START_REF] Kakeshita | Magnetic field-induced martensitic transformations in a few ferrous alloys[END_REF][START_REF] Kakeshita | A new model explainable for both the athermal and isothermal natures of martensitic transformations in Fe-Ni-Mn alloys[END_REF][START_REF] Kakeshita | Effect of magnetic fields on athermal and isothermal martensitic transformations in Fe-Ni-Mn alloys[END_REF][START_REF] Kakeshita | Effects of static magnetic field and hydrostatic pressure on the isothermal martensitic transformation in a Fe-Ni-Mn alloy[END_REF][START_REF] Kakeshita | Effects of magnetic field and hydrostatic pressure on martensitic transformation process in some ferrous alloys[END_REF][START_REF] Kakeshita | Effect of magnetic field and hydrostatic pressure on martensitic transformation and its kinetics[END_REF][START_REF] Kakeshita | Martensitic transformations in some ferrous and non-ferrous alloys under magnetic field and hydrostatic pressure[END_REF][START_REF] Kakeshita | Effects of hydrostatic pressure and magnetic field on martensitic transformations[END_REF][START_REF] Kakeshita | Magnetic field-induced martensitic transformation and giant Magnetostriction in Fe-Ni-Co-Ti and ordered Fe 3 Pt shape memory alloys[END_REF][START_REF] Kakeshita | Time-dependent nature of displacive transformations in Fe-Ni and Fe-Ni-Mn alloys under magnetic field and hydrostatic pressure[END_REF] carried out more systematical and deeper studies on this issue. They used pulsed magnetic fields to induce larger increases in Ms and noticed that there existed a critical strength of magnetic field to effectively induce martensitic transformation at the temperatures above Ms, and the higher the temperature was, the stronger the critical strength became. They also found that the experimentally measured increases in Ms temperature of the Fe-Pt ordered and disordered alloys [5,[START_REF] Kakeshita | Magnetic field-induced martensitic transformation and giant Magnetostriction in Fe-Ni-Co-Ti and ordered Fe 3 Pt shape memory alloys[END_REF], Fe-Ni-Co-Ti thermo-elastic alloys [START_REF] Kakeshita | Magnetoelastic martensitic transformation in an ausaged Fe-Ni-Co-Ti alloy[END_REF] and Fe-Ni alloys [6,[START_REF] Kakeshita | Magnetic Field-Induced Martensitic Transformation in Single Crystals of Fe-31.6 at%Ni Alloy[END_REF] were not in agreement with the calculated results using Krivoglaz's formula (Eq. 1. 3). They proposed a more accurate expression for the magnetic Gibbs free energy change by clarifying the unknown effects which Krivoglaz [1] did not find under low field as the high field susceptibility and forced volume magnetostriction effects and gave the new estimation formula of the increase of Ms as follows [START_REF] Kakeshita | Effect of magnetic field and hydrostatic pressure on martensitic transformation and its kinetics[END_REF]: H have been calculated for Fe-Pt alloys [5], Invar Fe-Ni [6], non-Invar Fe-Ni-C [START_REF] Kakeshita | Magnetic field-induced martensitic transformations in Fe-Ni-C invar and non-invar alloys[END_REF] and Invar Fe-Mn-C alloys [START_REF] Kakeshita | Magnetic field-induced transformation from paramagnetic austenite to ferromagnetic martensite in an Fe-3.9Mn-5.0C (at%) alloy[END_REF], which are in good agreement with the experimental ones. The authors also clarified the influence of composition [6], grain boundaries [START_REF] Kakeshita | Magnetic Field-Induced Martensitic Transformation in Single Crystals of Fe-31.6 at%Ni Alloy[END_REF], crystal orientations [START_REF] Kakeshita | Magnetic field-induced martensitic transformations in Fe-Ni-C invar and non-invar alloys[END_REF][START_REF] Kakeshita | Magnetic Field-Induced Martensitic Transformation in Single Crystals of Fe-31.6 at%Ni Alloy[END_REF], Invar characteristics [START_REF] Kakeshita | Magnetic field-induced martensitic transformations in Fe-Ni-C invar and non-invar alloys[END_REF], thermo-elastic nature [START_REF] Kakeshita | Magnetoelastic martensitic transformation in an ausaged Fe-Ni-Co-Ti alloy[END_REF][START_REF] Kakeshita | Magnetic field-induced martensitic transformation and giant Magnetostriction in Fe-Ni-Co-Ti and ordered Fe 3 Pt shape memory alloys[END_REF] and austenitic magnetism [START_REF] Kakeshita | Effect of magnetic fields on athermal and isothermal martensitic transformations in Fe-Ni-Mn alloys[END_REF] on the magnetic field-induced martensitic transformations. Results showed that Eq. (1.4) suited well under the above conditions. Thus, the propriety of the newly derived equation is quantitatively verified. B H H w H H s M M Ms G Ms G c c p h c ⋅ ⋅ ∂ ∂ ⋅ + ⋅ ⋅ - ⋅ ′ ∆ - = ∆ - ∆ ) / ( ) 2 / 1 ( ) ( ) ' ( ) ( 0 2 ε χ (1. As for the influence of the magnetic field on the kinetics of martensitic transformation, Kakeshita et al. [START_REF] Kakeshita | A new model explainable for both the athermal and isothermal natures of martensitic transformations in Fe-Ni-Mn alloys[END_REF][START_REF] Kakeshita | Effect of magnetic fields on athermal and isothermal martensitic transformations in Fe-Ni-Mn alloys[END_REF] conducted their investigation in Fe-Ni-Mn alloys. Their results suggest that, under high magnetic fields, the originally isothermal kinetics of martensitic transformation can be changed to the athermal one and the kinetics in both case can be evaluated from the same equation. They also suggested [START_REF] Kakeshita | Effects of static magnetic field and hydrostatic pressure on the isothermal martensitic transformation in a Fe-Ni-Mn alloy[END_REF] that the shift of the nose temperature and incubation time increase with increasing magnetic field. The martensitic morphologies under the magnetic field have also been studied in a variety of Fe-based alloys. Many studies showed that the structure of the magnetic field-induced martensite did not show much difference [4][5][6][START_REF] Kakeshita | Magnetoelastic martensitic transformation in an ausaged Fe-Ni-Co-Ti alloy[END_REF][START_REF] Kakeshita | Magnetic field-induced martensitic transformations in Fe-Ni-C invar and non-invar alloys[END_REF][START_REF] Kakeshita | Magnetic Field-Induced Martensitic Transformation in Single Crystals of Fe-31.6 at%Ni Alloy[END_REF][START_REF] Kakeshita | Magnetic field-induced transformation from paramagnetic austenite to ferromagnetic martensite in an Fe-3.9Mn-5.0C (at%) alloy[END_REF][START_REF] Kakeshita | Effect of magnetic fields on martensitic transformations in alloys with a paramagnetic to antiferromagnetic transition in the austenitic state[END_REF][START_REF] Kakeshita | Magnetic field-induced martensitic transformations in a few ferrous alloys[END_REF][START_REF] Kakeshita | A new model explainable for both the athermal and isothermal natures of martensitic transformations in Fe-Ni-Mn alloys[END_REF][START_REF] Kakeshita | Effect of magnetic fields on athermal and isothermal martensitic transformations in Fe-Ni-Mn alloys[END_REF][START_REF] Kakeshita | Effects of static magnetic field and hydrostatic pressure on the isothermal martensitic transformation in a Fe-Ni-Mn alloy[END_REF][START_REF] Kakeshita | Effects of magnetic field and hydrostatic pressure on martensitic transformation process in some ferrous alloys[END_REF][START_REF] Kakeshita | Effect of magnetic field and hydrostatic pressure on martensitic transformation and its kinetics[END_REF][START_REF] Kakeshita | Martensitic transformations in some ferrous and non-ferrous alloys under magnetic field and hydrostatic pressure[END_REF][START_REF] Kakeshita | Effects of hydrostatic pressure and magnetic field on martensitic transformations[END_REF][START_REF] Kakeshita | Magnetic field-induced martensitic transformation and giant Magnetostriction in Fe-Ni-Co-Ti and ordered Fe 3 Pt shape memory alloys[END_REF][START_REF] Kakeshita | Time-dependent nature of displacive transformations in Fe-Ni and Fe-Ni-Mn alloys under magnetic field and hydrostatic pressure[END_REF]; whereas the amount of martensite was increased by the magnetic field [6,[START_REF] Kakeshita | Magnetic field-induced martensitic transformations in a few ferrous alloys[END_REF]. The increase in the amount of martensite is very meaningful and important, as it will result in further strengthening, considering that martensite is a hard phase, whereas austenite is a soft phase. Semi-diffusional transformation-Bainitic transformation The study on the effects of magnetic field on bainitic transformation has a relatively late start, and only a few researches have been done. In this case, the parent austenite is paramagnetic and the product bainite is ferromagnetic like martensite. In this content, the effect of magnetic field on bainitic transformation is similar to the case of martensitic transformation, on both thermodynamic and kinetic aspects. Grishin [START_REF] Grishin | Structure and properties of structural steels after isothermal treatment in a magnetic field[END_REF] made an investigation on the transformation from austenite to bainite under a magnetic field based on three different structural steels. It revealed that the bainitic transformation was accelerated by magnetic field. The acceleration effect by field was also observed in Fe-0.52C-0.24Si-0.84Mn-1.76Ni-1.27Cr-0.35Mo-0.13V alloy by microstructural observation. As shown in Figure 1.4 [START_REF] Bhadeshia | Bainite in Steels[END_REF], the amount of the transformed bainite (black) was greatly increased under the 10T magnetic field. Later, effects of magnetic fields on transformation temperature, transformation behavior and transformed microstructure have been investigated for bainitic transformation in a Fe-3.6Ni-1.45Cr-0.5C steel [START_REF] Ohtsuka | Effects of a high magnetic field on bainitic transformation in Febased alloys[END_REF][START_REF] Ohtsuka | Effects of strong magnetic fields on bainitic transformation[END_REF]. It was found that the Bs temperatures increase with the increase of the applied magnetic field. Bainitic transformation is accelerated by the applied magnetic field. Although elongated and aligned microstructures were observed for austenite to ferrite transformation in a Fe-0.4C alloy, but no elongation or alignment of transformed structure has been observed for transformations to bainite and lath martensite. Diffusion-controlled phase transformations under magnetic field Compared with diffusionless and semi-diffusional transformation, the diffusion-controlled phase transformations in steels are more complicated and intricate, due to the variety of the proeutectoid phases diversified by carbon composition. As the diffusional phase transformations usually happen in relatively high temperature range, experimental studies are difficult to conduct, especially when high magnetic field is difficult to be obtained. Studies on this subject began from the theoretical simulation calculations instead. Only until 1980's, magnetic field was tried to be applied to the diffusion-controlled transformation. Later, with the development of magnetic field technology, more experiments on diffusioncontrolled transformations under magnetic field have been carried out in various Fe based alloys under the instruction of the calculated results. Since then, more and more new phenomena have been uncovered and verified. In the meantime, corresponding theories have been explored for better understanding of magnetic field influential mechanisms. Unlike the martensitic transformation and bainitic transformation, in which the magnetic field effects were largely from energy aspects and work on the transformation thermodynamic and kinetic, but less on morphology, the influence of magnetic field on diffusional phase transformation were more widely and complicated, which also involved with the magnetic dipolar interaction and thus leads to the effects from both crystallographic and morphologic point of view. New Fe-C phase equilibrium under the magnetic field When people tried to understand the Fe-C solid phase transformation under magnetic field, they started from understanding the field effects on phase equilibriums. Even in earlier days, when experimental studies on the effects of magnetic field on diffusion-controlled solid phase transformation in steel were not available, thermodynamic calculations were carried out to explore field effects, and later the calculation results were applied to guide the experimental studies. The influence of magnetic field on phase equilibrium first expressed as the shift of phase transformation temperature. In 1965, Sadovskii et al. [START_REF] Sadovskii | Magnetic field and phase transformations in steel[END_REF] mentioned that with the increasing intensity of the magnetic field, the amount of the magnetic phase formed increases and the range of the existence of the phase increases with the degree of magnetization in a way similar to the effect of pressure. It was pointed out that the change in the temperature of the phase transformation under the influence of the magnetic field is expressed by the equation: q JT dH dT 0 ∆ = (1.5) where H is the intensity of the field; ∆J is the difference in the magnetization of the phase intervening in the transformation; T 0 is the temperature of phase equilibrium; q is the heat of transformation. Later, Ghosh et al. [START_REF] Ghosh | Phase tranformation of steel in magnetic field[END_REF] investigated the change of TTT diagram using a similar equation during austenite to ferrite transformation. This analysis did not take the magnetization of austenite into account, thus, it is only suitable for low magnetic field. When magnetic field is strong, the ignorance of the magnetization of austenite was quite a deviation from real situation, let alone the influence brought by composition changes. Although the equation is useful in a limited range, it is a very instructive calculation of the effects of magnetic field on solid phase equilibrium of iron based alloys. The studies of phase diagram are meaningful and necessary due to its guiding role in designing heat treatment plan and developing new materials. The basic idea of building up a new Fe-C phase diagram is to determine the new phase equilibrium temperature and equilibrium composition under the external magnetic field. In terms of this, the change of the Gibbs free energy induced by the magnetic field of related phases has to be evaluated by using appropriate mode and magnetic parameters. From last 90's, studies on the Fe-C phase diagram under the field have been launched by researchers from Korea and Japan. Considering that the absence of high-field magnetic susceptibility data around Tc makes it impossible to construct the Fe-C phase diagram under the high magnetic field based on the experimental data, they used the molecular field theory, which is capable to provide a theoretical method to estimate the high-field magnetization in this temperature region. As a result, former studies were based on the Weiss molecular field theory together with the Curie-Weiss law to evaluate the change in Gibbs free energy of individual phases involved with the applied magnetic field. Meanwhile, experimental measurements are important and needed to verify the validation of the calculated results. In term of this, experimental measurements of phase transformation temperature and examination on transformed microstructure had been made as powerful and direct methods to offer comparative study. The first comprehensive study on the prediction of the new Fe-C phase diagram under the magnetic field was carried out by Joo et al [START_REF] Joo | An effect of high magnetic field on phase transformation in Fe-C system[END_REF]. Considering the disagreament of magnetic susceptibility of ferrite from Curie-Weiss law and experimental data, the magnetization calculated by Weiss theory is used instead of susceptibility to calculate the magnetic Gibbs free energy of ferrite. In this work, the influence of the magnetic field on all phases was concerned, the Fe-C phase diagram were calculated under various applied magnetic fields and the eutectoid temperature, eutectoid composition as well as the γ/α transformation temperature were determined. Results showed that both Ac 1 and Ac 3 temperatures increase as the magnetic field is applied, while the Ac m temperature change is almost independent of the field. As a result, eutectoid point was shifted to high carbon and high temperature side, as seen in Figure 1.5. equilibrium transformation for various applied magnetic fields [START_REF] Joo | An effect of high magnetic field on phase transformation in Fe-C system[END_REF]. It is regarded for the first time the simulation of the Fe-C phase diagram under the field. This study is quite limited, because the magnetic Gibbs free energy of ferrite is obtained by mathematic method without interpreting the inner physic relation between M, H and T. Moreover, the prediction of the magnetic Gibbs free energy at any field strength is not obtainable. This limits the application of this study to some extent. Despite of this, the results obtained from this work are of great importance and meaning. It showed that the applied magnetic field could affect the phase equilibrium temperature as well as the eutectoid composition, and thus change the microstructure at room temperature. This enables the application of magnetic field as a method of microstructure control and materials properties optimization. In addition, the shift of eutectoid point by magnetic field enables the possibility of improving mechanical properties by increasing carbon content without hypereutectoid transformation. Joo et al. [START_REF] Joo | An effect of a strong magnetic field on the phase transformation in plain carbon steels[END_REF] measured the γ-α transformation temperature under 10T magnetic field by thermo-dilatometer. It turned out that the increase of the γ-α transformation temperature in pure iron and the eutectoid temperature calculated on the basis of molecular field theory are in good agreement with measured results [START_REF] Joo | An effect of high magnetic field on phase transformation in Fe-C system[END_REF]. Similar calculations using Weiss molecular field theory to calculate Gibbs free energy change of the phases were also carried out by Choi et al. [START_REF] Choi | Effects of a strong magnetic field on the phase stability of plain carbon steels[END_REF], Guo and Enomoto [START_REF] Enomoto | Influence of magnetic field on the kinetics of proeutectoid ferrite transformation in iron alloys[END_REF][START_REF] Guo | Influence of magnetic fields on γ-α equilibrium in Fe-C(-X) alloys[END_REF] as well as Hao and Ohtsuka [START_REF] Ohtsuka | Effects of a high magnetic field on bainitic transformation in Febased alloys[END_REF][START_REF] Ohtsuka | Effects of strong magnetic fields on bainitic transformation[END_REF][START_REF] Hao | Effect of High Magnetic Field on Phase Transformation Temperature in Fe-C Alloys[END_REF][START_REF] Hao | Quantitative Characterization of the Structural Alignment in Fe-0.4C Alloy Transformed in High Magnetic Field[END_REF][START_REF] Hao | Structural elongation and alignment in a Fe-0.4C alloy by isothermal ferrite transformation in high magnetic fields[END_REF][START_REF] Ohtsuka | Alignment of ferrite grains during austenite to ferrite transformation in a high magnetic field[END_REF][START_REF] Ohtsuka | Effects of magnetic field and prior austenite grain size on the structure formed by reverse transformation from lath martensite to austenite in an Fe-0.4C alloy[END_REF][START_REF] Ohtsuka | Structural control of Fe-based alloys through diffusional solid/solid phase transformations in a high magnetic field[END_REF] in Fe-C system. Choi et al. [START_REF] Choi | Effects of a strong magnetic field on the phase stability of plain carbon steels[END_REF] combined the calculation studies with experimental examination and proved that magnetic field can also increase the carbon content in eutectoid composition and solubility in ferrite. Guo and Enomoto even further enlarged their studies to Fe-C-Mn and Fe-C-Si alloys [START_REF] Enomoto | Influence of magnetic field on the kinetics of proeutectoid ferrite transformation in iron alloys[END_REF][START_REF] Guo | Influence of magnetic fields on γ-α equilibrium in Fe-C(-X) alloys[END_REF] by taking into account the interaction free energy of the alloying elements as well as the influence of the alloying elements on Curie temperature and the magnetic moments of the iron atoms. Considering the Curie-Weiss law can hardly offer reliable susceptibility data of ferrite around Curie temperature, these calculations were very approximate around and above the Curie temperature. However, valuable results were obtained that the α-γ transformation temperature is raised 1-3°C per Tesla depending on the alloy composition and the intensity of the applied field, whereas the γ-δ transformation temperature is decreased about 0.4°C per Tesla. They also predicted that, at about 100T pure bcc iron may be more stable than the fcc iron at all temperature range. Hao and Ohtsuka [START_REF] Hao | Effect of High Magnetic Field on Phase Transformation Temperature in Fe-C Alloys[END_REF] revealed that: It was found that when magnetic field is lower than 10T, the transformation temperature for pure Fe from austenite to ferrite has a linear relationship with magnetic field strength, increasing about 0.8°C per Tesla. For eutectoid transformation in Fe-0.8C alloy, similar relationship exists, the transformation temperature increases about 1.5°C per Tesla. Hao and Ohtsuka [START_REF] Hao | Effect of High Magnetic Field on Phase Transformation Temperature in Fe-C Alloys[END_REF] used the thermocouple and a digital recorder to measure the transformation temperature. It was indicated that the measured transformation temperature data are not consistent with calculation results using Weiss molecular field theory and moreover, using the experimental measured susceptibility under a low magnetic field to conduct the simulation calculations for high magnetic field is not appropriate. It was noted that Weiss model allows basic calculations for ferromagnetism and offers relatively accurate magnetization data of bcc Fe below Tc, however, it still had several shortcomings [START_REF] Guo | Influence of magnetic fields on γ-α equilibrium in Fe-C(-X) alloys[END_REF], which limited the application of Weiss model around and above Tc, especially when the applied magnetic field is high. Thus, new models were needed for simulation calculation of Fe-C equilibrium under the magnetic field. To solve this problem, Zhang [START_REF] Zhang | Calculation of magnetization and phase equilibrium in Fe-C binary system under a magnetic field[END_REF] revised the Weiss model by substituting the molecular field coefficient λ with a short-range-ordering coefficient γ around and above Tc and applied this revised mode to the susceptibility of bcc Fe above Tc, which turned out to be in good agreement with the ones measured experimentally. Moreover, the electronic band model is used to calculate the temperature variation of susceptibility of fcc Fe. Based on these, the magnetization and further the magnetic Gibbs free energy of ferrite and austenite are calculated, thus the Fe-C phase diagram under the magnetic field was calculated, the shift of Ae 3 temperature and the eutectoid temperature and composition are predicted. It is found that in the phase diagram the magnetic field enlarges the ferritic phase area and shrinks the austenitic one. With pure iron, the Ae 3 is raised by about 7°C under a magnetic field of 10T. This result is in good agreement with the measured values by Ohtsuka et al. [START_REF] Hao | Effect of High Magnetic Field on Phase Transformation Temperature in Fe-C Alloys[END_REF]. Generally, for Fe and Fe-0.8C the average γ-α transformation temperature measured experimentally was easily compared with the calculated results based on Weiss molecular field theory. The most accepted result was that the change of γ-α transformation temperature is almost proportional to the magnetic field when the ferrite phase is ferromagnetic at the transformation temperature (as is the case in Fe-0.8C), whereas it was proportional to the square of the magnetic field when the ferrite phase is paramagnetic (as is the case in pure iron). However, for intermediate pro-eutectoid steel, of which the (γ+α) region is large, this method is no longer valid [START_REF] Fukuda | Magnetic field dependence of γ-α equilibrium temperature in Fe-Co alloys[END_REF] and it would have been meaningless to define an average γ/α transformation temperature in order to compare it with a calculated γ/α equilibrium temperature due to the large hysteresis of the transformations in these alloys. Considering this, another approach was proposed to discuss the magnetic field dependence of diffusional phase transformation temperatures, taking into account the existence of temperature hysteresis due to the large driving force required for transformation [START_REF] Farjami | Effect of Magnetic Field on γ-α Transformation in Fe-Rh Alloys[END_REF]. More recently, this new approach was applied to study the nonequilibrium α-γ and γ-α transformation temperature separately in Fe-Rh alloys [START_REF] Farjami | Effect of Magnetic Field on γ-α Transformation in Fe-Rh Alloys[END_REF], Fe-C-Mn alloys [START_REF] Garcin | Experimental evidence and thermodynamics analysis of high magnetic field effects on the austenite to ferrite transformation temperature in Fe-C-Mn alloys[END_REF] and Fe-Ni alloys [START_REF] Garcin | Thermodynamic analysis using experimental magnetization data of the austenite/ferrite phase transformation in Fe-xNi alloys (x= 0, 2, 4 wt%) in a strong magnetic field[END_REF][START_REF] Garcin | In situ characterization of phase transformations in a magnetic field in Fe-Ni alloys[END_REF]. Together with the development of new analysis approach, new devices to monitor high temperature phase transformations under magnetic field had emerged. Rivoirard and Garcin [START_REF] Garcin | Experimental evidence and thermodynamics analysis of high magnetic field effects on the austenite to ferrite transformation temperature in Fe-C-Mn alloys[END_REF][START_REF] Garcin | Kinetic effects of magnetic field on the γ/α interface controlled reaction in iron[END_REF][START_REF] Rivoirard | High temperature dilatation measurements by in situ laser interferometry under high magnetic field[END_REF] developed a high magnetic field in situ dilatometer, ranging from room temperature up to 1500K, using a high resolution Michelson laser interferometer. This new devices were applied to monitor the austenite to ferrite transformation under a magnetic field of 16T in pure iron. Then, measured Ar 3 temperatures were compared with the calculated results. In their studies, experimental magnetization data, measured up to 3.5 T and 1100 K by high-sensitivity magnetometer [START_REF] Gaucherand | Magnetic susceptibility of high-Curie-temperature alloys near their melting point[END_REF], were used to calculate the magnetic contribution to the Gibbs free energy instead of the calculated one from Weiss molecular field theory. In this approach, calculated Ar 3 temperatures were found to be in good agreement with the experimental ones measured by dilatometer. Besides the modification of γ-α transformation temperature, the eutectoid point shift has also been calculated and experimentally evidenced in hypereutectoid carbon steel [START_REF] Zhang | Effect of a high magnetic field on eutectoid point shift and texture evolution in 0.81C-Fe steel[END_REF][START_REF] Zhang | Shift of the eutectoid point in the Fe-C binary system by a high magnetic field[END_REF]. Microstructural observation, which indicates formation of ferrite in the hypereutectoid carbon steel under field, evidences the shift of the eutectoid point in the Fe-C system. It shifts under a 12 T magnetic field from 0.77 wt. %C to 0.8287 wt. %C. Furthermore, Zhang et al. [START_REF] Zhang | Effect of a high magnetic field on eutectoid point shift and texture evolution in 0.81C-Fe steel[END_REF][START_REF] Zhang | Shift of the eutectoid point in the Fe-C binary system by a high magnetic field[END_REF] propose a general and comprehensive calculation method by combining the well-established statistical thermodynamic models with the magnetism theory to predict this eutectoid point shift (both in carbon composition and temperature scales) as shown in Figure 1.6. The eutectoid carbon content and temperature calculated without and with a 12 T magnetic field is 0.779 wt. %C; 725.71 °C and 0.847 wt. %C; 754.68 °C. The eutectoid carbon contents without and with the magnetic field-as calculated-appears to be very close to the values determined experimentally. It shows that the equations proposed offer a relatively accurate prediction of the eutectoid point shift under a high magnetic field in both carbon content and temperature scales. New microstructural features under the magnetic field The most remarkable microstructural feature induced by magnetic field is the formation of the aligned and elongated microstructures during transformation between austenite and ferrite. Studies on this subject have been drawing increasing attention as it may lead to the control of the microstructure and thus the texture and the mechanical properties. The first observation of the aligned microstructure was reported by Shimotomai [START_REF] Shimotomai | Aligned of two-phase structures in Fe-C alloys[END_REF] in Fe-0.1C alloy and Fe-0.6C alloy during the α to γ inverse transformation. The chains or columns of the paramagnetic austenite were found along the field direction in the ferromagnetic ferrite phase during the ferrite to austenite inverse transformation in 8T magnetic field, as shown in Figure 1.7. They contributed the formation of the alignment to the dipolar interactions of the magnetic moments between the pairs of paramagnetic austenite nuclei which are regarded as magnetic hole in ferrite matrix. Later, the alignment of the γ nuclei under the field effect was even noticed above the Curie temperature during γ-α inverse transformation [START_REF] Maruta | Magnetic field-induced alignment of steel microstructures[END_REF]. Later, Shimotomai and Maruta studied the magnetic field-induced alignment structures in Fe-0.6C alloy, Fe-0.2C-0.2Si-1.3Mn-0.1Ti alloy and Fe-0.1C-2.0Si-2.0Mn [START_REF] Maruta | Magnetic field-induced alignment of steel microstructures[END_REF][START_REF] Maruta | Alignment of two-phase structures in Fe-C alloys by application of magnetic field[END_REF][START_REF] Shimotomai | Formation of aligned two-phase microstructures by applying a magnetic field during the austenite to ferrite transformation in steels[END_REF] during the γ-α transformation. A special experimental setup was designed, which features a combination of roller dice for hot deformation and a superconducting magnet for applying a magnetic field during the phase transformation. The concept is characterized by deforming steels prior to the transformation under the field in order to introduce more nucleation sites for transformation. In their studies, the formation mechanism of the aligned structures has been discussed from the point of view of nucleation and growth of the ferrite grains. They observed the ferrite nucleation sites under the field using SEM and drew a conclusion that no matter the nucleation site is at the grain boundary or in the austenite grain interiors, the long axis of the ferrite grain is always parallel to the field direction, as shown in Figure 1.8. Further analysis on ferrite morphology showed that the ferrite particles nucleated and grew along the magnetic field direction are mostly ellipsoidal in shape and this is determined by a competition between the magnetic field energy that favors an elongated shape and the interfacial energy that favors a spherical shape. Moreover, it was reported that, though ferrite grains were elongated and aligned by applied field, no preferred crystallographic orientation of ferrite is developed. Shimotomai and Maruta considered a combination of prior rolling and transformation in an external field is essential for yielding an aligned structure during γ-α transformation. However, a similar aligned structures were reported by Ohtsuka et al in a Fe-0.4C alloy without prior deformation during continuous slow cooling under the 10T magnetic field [START_REF] Ohtsuka | Alignment of ferrite grains during austenite to ferrite transformation in a high magnetic field[END_REF]. As seen from Figure 1.9, each ferrite grain is elongated and these grains are distributed head to tail along the field direction. The effects of magnetic field strength, cooling rate and austenite grain size on the transformed structure in a magnetic field were observed for Fe-Mn-C-Nb alloy, but no elongation or alignment of ferrite grains has been observed [START_REF] Ohtsuka | Alignment of ferrite grains during austenite to ferrite transformation in a high magnetic field[END_REF]. On this basis, the existence of alloying elements may totally eliminate this elongation and alignment morphology. Later, Zhang et.al [START_REF] Zhang | Thermodynamic and kinetic characteristics of the austenite-to-ferrite transformation under high magnetic field in medium carbon steel[END_REF][START_REF] Zhang | New microstructural features occurring during transformation from austenite to ferrite under the kinetic influence of magnetic field in a medium carbon steel[END_REF][START_REF] Zhang | A new approach for rapid annealing of medium carbon steels[END_REF] found that the aligned structure formed during slow cooling process rather than fast cooling process, when they studied the austenite decomposition in a hot-rolled 42CrMo steel under the magnetic field. They suggested that when cooling rate is 10°C/min, due to the inhomogeneous deformation during rolling and the dipolar attraction between ferrite nuclei, the microstructure of alternately distributed ferrite grains and pearlite colonies along the field direction is obtained. In the case of cooling at 46°C/min, the nucleation at high temperature is greatly inhibited; as a result, the nucleation is postponed at lower temperature, when more sites inside the austenite grains besides grain boundaries are available. Consequently, the microstructure is characteristic of randomly distributed ferrite grains and pearlite colonies with smaller sizes, seen in Figure 1.11. The magnetic field-induced grain elongation mechanism was analyzed from physical point of view [START_REF] Zhang | Magnetic-fieldinduced grain elongation in a medium carbon steel during its austenitic decomposition[END_REF]. It revealed that the grain elongation is the results of the opposing contributions from the atomic dipolar interaction energy of Fe atoms and the interfacial energy. The effect of magnetic field on the formation of elongated and aligned pearlite was investigated by Song et al [START_REF] Song | Effects of high magnetic field strength and direction on pearlite formation in Fe-0.12%C steel[END_REF]. It was reported that in the Fe-0.12C alloy, the pearlite was elongated and aligned along the field direction during its diffusional decomposition under the 12T magnetic field. Moreover, this tendency increased with increasing magnetic field strength and this field effect is dependent on the specimen orientation with respect to the field direction. It should be noticed that, several works indicates that the nucleation and growth of ferrite grains happen preferentially along the field direction, however, no Recently, magnetic field-induced microstructures features were investigated from crystallographic point of view. Zhang et al. applied a 12T magnetic field to a medium plain carbon steel during the diffusional decomposition of austenite and investigated the effect of a high magnetic field on the distribution of misorientation angles, grain boundary characteristics and texture formation in the ferrite produced [START_REF] Zhang | Grain boundary characteristics and texture formation in a medium carbon steel during its austenitic decomposition in a high magnetic field[END_REF]. It was reported that magnetic field can cause a considerable decrease in the frequency of low-angle misorientation and an increase in the occurrence of low Σ coincidence boundaries, especially the Σ3 of ferrite [START_REF] Zhang | Grain boundary characteristics and texture formation in a medium carbon steel during its austenitic decomposition in a high magnetic field[END_REF]. They attributed this to the elevation in the transformation temperature caused by the magnetic field and, therefore, the reduction of the transformation stress. It is also found that magnetic field enhances the <001> texture component along the transverse field direction due to the dipolar interaction between the magnetic moments of Fe atoms. Significance and content of this work The investigations on phase transformation under the magnetic field have been carried out for more than fifty years. Many meaningful experimental phenomena have been discovered and some related theories have been developed. Nowadays, as a new promising technique, magnetic field has been widely applied in materials processing. However, most of the former studies on the effect of the magnetic field on diffusional phase transformation are lack of regularity, and are not systematic either. The fundamental theories of magnetic field influence on phase transformation are still in need to be addressed. As alloying elements affect the phase transformation in steels to a large extent, the fundamental theories on field influential mechanism are hard to establish based on the existing studies, most of which were conducted in Fe-based alloys with considerable alloying elements. With the development of the magnetic field generator technique, the application of magnetic field becomes more and more extensive. Deeper understanding of field influential mechanism is necessary and imperative. Based on such background, the present work has been carried out as a fundamental research to explore the magnetic field influential mechanism on diffusional phase transformation and also enrich the existing phase transformation theory. To examine the field effect without the involvement of impurities and alloying elements and to obtain a systematic result, three high purity Fe-C alloys in both hypo and hyper eutectoid composition range were prepared as experimental materials. The field induced microstructural features and the crystallographic characteristics have been thoroughly investigated. The main content of present work can be summed up as follows: (1) Experimentally examine the microstructural features induced by magnetic field in both hypo and hyper eutectoid Fe-C alloys: Magnetic field induced aligned and elongated microstructures are analyzed through a comparative study in Fe-0.12C alloy and Fe-0.36C alloy. Microstructural modifications by magnetic field due to its thermodynamic influence on phase equilibrium in hypo and hyper eutectoid Fe-C alloys, respectively. (2) Experimentally examine the modification of carbon solubility in ferrite under the magnetic field, conduct calculations on magnetic moments of the related atoms and underline the physical essence of the field influential mechanism using magnetic field dipolar interaction. (3) By means of SEM/EBSD to examine the effect of the magnetic field on crystallographic orientation characteristics. The field induced preferred orientation distribution of ferrite and then the texture formation mechanism is investigated. The types of the orientation relationships in pearlite and the corresponding occurrence frequency have been examined. Chapter 2 Experimentals and Calculations Materials preparation The materials used in this work are three high purity Fe-C alloys with different carbon content, namely, Fe-0.12C alloy, Fe-0.36C alloy and Fe-1.1C alloy. To minimize the involvement of impurities during the preparation, the alloys were prepared by vacuum induction melting using high purity constituent elements in EPM laboratory of Northeastern University, China. The preparation of the high purity Fe-C alloys involves following two steps. (1) Preparation of high purity cast iron as high carbon-content constituent for carbon content tuning of the alloys. The high purity (99.99%) electrolytic iron was melted in a high purity graphite crucible (carbon 99.99%) by repeated vacuum induction melting, allowing carbon to diffusion into the iron melt from the graphite to produce cast iron. The chemical composition of the obtained cast iron is shown in Table 2.1. (2) Preparation of the final alloys with assigned carbon content. The cast iron obtained in step (1) as high carbon constituent was melted with high purity (99.99%) electrolytic iron in water cooled copper crucible by vacuum induction levitation melting. By varying the quantity of the cast iron, the final Fe-C alloys with assigned carbon contents (0.12C, 0.36C and 1.1C in wt.%) were obtained. Figure 2.1 shows the top view of one high purity Fe-C alloy ingot. Pre-treatment of high purity Fe-C alloys In order to study the influence of the magnetic field on phase transformation without the interference from the inhomogeneity of the initial microstructure, the three high purity alloys were pre-treated to obtain the homogeneous and equilibrium microstructures for the subsequent magnetic field heat treatments. First, the ingots were multidirectionally forged to homogenize the microstructure. The multidirectional forging was performed by first forging the ingots into cubes and then further forging the cubes along their three perpendicular edge directions to resume the initial cube form of the work pieces. This process was repeated 4 times. Finally the cube was forged along one of its diagonal direction into a pancake shape. The multidirectional forging was conducted within the temperature range from 1273K to 1073K without annealing between the forging steps. After forging, the work pieces were air cooled to the room temperature. To avoid the involvement of the outside oxide coating and decarburized layer in the specimens, a plate of 70mm×70mm×35mm in size was cut out from the centre of each forged pancake by electrical spark cutting. Then, the plates were homogeneously annealed in vacuum at set temperature for 10 hours to homogenize the composition. (The annealing temperature is 1373K for Fe-0.12C alloy and Fe-0.36C alloy; 1323K for Fe-1.1C alloy). After that, full annealing was performed to obtain equilibrium microstructures. The alloys were fully austenitized for 45min, and cooled to 973K at 0.3K/min for proeutectoid and eutectoid transformation, and then cooled to room temperature at The corresponding phase equilibrium temperatures of the three alloys were calculated with Thermo-Calc Software. For the Fe-0.12C alloy and the Fe-0.36C alloy, their Ae 3 temperatures are 1135K and 1069K, respectively; for the Fe-1.1C alloy, Ae cm temperature is 1128K. Magnetic field heat treatment equipment and experiments In this work, all the field heat treatments were conducted at EPM laboratory of Northeastern University, China. The magnetic field heat treatment furnace is set in a 12T cryo-cooled superconducting magnets with a bore sized 100mm in diameter (Figure 2.2 shows the photo (a) and the schema (b) of the magnetic field heat treatment furnace). The magnetic field ramping time from 0T to 12T is 39 minutes. For this magnetic field heat treatment furnace, the highest heating temperature is 1473K, the maximum heating rate is 5K/min and the maximum cooling rate is For field heat treatments, the magnetic field was applied during the whole heating, isothermal holding and cooling process. For Fe-0.12C and Fe-1.1C alloys, the 12T magnetic field was applied, while for Fe-0.36C alloy, both 8T and 12T magnetic fields were applied. The non-field heat treatments for comparison were carried out in the same furnace using the same heat treatment parameters only without switching on the magnetic field. During the field heat treatment, the specimens were kept in the centre (zero magnetic force) area with one of their length directions parallel to the field direction, as shown in To obtain an appropriate observation surface for microstructural examinations, the treated specimens were first mechanically polished using SiC grinding papers from 800 # to 2000 # and then polished using diamond liquid with the size of the diamond particle from 3µm to 1 µm. For optical microstrutural examinations and carbon content measurements, the specimens were further etched with 4% Nital at room temperature for several seconds. Specimen cutting and specimen geometry For SEM microstructural and EBSD crystallographic orientation examinations, the specimens were electronic polished in 8% perchloric acid ethanol solution at room temperature at the voltage of 30V for 10~15 seconds after mechanical polishing (paper polishing up to 2000 # ). It should be emphasized that the specimen preparation procedures of the non-field and field treated specimens (electrolytic polishing or etching, and rinsing) were strictly controlled to ensure identical preparation conditions, so that the output results from the non-field and the field treated specimens are comparable. Optical microstructural examinations The optical microstructure observations were performed with an Lamellar spacings of pearlite 20 areas Wavelength-dispersive spectroscopic (EPMA) analysis The carbon concentration of the proeutectoid ferrite in Fe-0.36C alloy was measured by means of wavelength-dispersive spectroscopy using a Shimadzu 1610 electron probe microanalyzer (WDS-EPMA). Six standard samples with carbon content ranging from 0.0075% to 0.978% were used to set the calibration curve for the WDS measurements. Both specimens treated without and with the magnetic field were examined during the same specimen loading to secure the identical measurement conditions for the non-field and field treated specimens. 20 proeutectoid ferrite areas in each specimen were randomly selected and measured to reach a global representation of the results. Hardness tests The hardness of proeutectoid ferrite in Fe-0.36C alloy was tested using a micro-indenter (diamond Vickers indenter Leitz, Wetzlar, Germany) with an applied load of 25g for 20s. The mean hardness was averaged over 8 measurements for each specimen. Scanning electron microscopic and crystallographic orientation analysis In this work, a field emission gun scanning electron microscope-Jeol JSM 6500F (the photo and the schematic illustration are displayed in Figure 6 In this work, two different beam controlled modes were applied for different orientation analysis purposes. (1) Automated orientation mapping for massive orientation acquisition and for correlated microstructure and texture analyses. The automated orientation mapping was performed to examine the orientation distribution of ferrite. As is known, ferrite is a simple-crystal-structured phase. The (b) (a) orientation determination of ferrite by EBSD is usually efficient and accurate guaranteed by the attainable high-quality Kikuchi patterns form bulk ferrite grains and even from lamellar pearlitic ferrite. This ensures the orientation determination of ferrite from a large measurement area can be performed in the automatic mode within a short time. In this work, the automated mappings were performed on the whole cross section of the specimens at different step sizes, namely 0.5µm, 1µm, 2µm and 3µm. As the crystal structure of cementite is relatively complicated and it is subject to a high level of internal constraint, within the same acquisition time for one Kikuchi pattern from ferrite, the acquired Kikuchi patterns of cementite is of very poor quality. Therefore, the measurement points on cementite always remain as non-indexed zero solution. The orientation information of pearlite colonies was only from their pearlitic ferrite. In this way, each pearlite colony is simply indentified as a ferrite grain. Finally, only the texture of ferrite (pearlitic and proeutectoid) was analyzed. (2) Interactive mode for individual orientation measurements. To investigate the orientation relationships (ORs) between ferrite and cementite in pearlite and in abnormal structure, individual orientations of ferrite and its neighboring cementite were measured. Due to the above-mentioned reasons for the nature of cementite together with the geometrical constrains imposed by the anisotropic shape of pearlitic cementite (thin lamellar: ~20nm in width) and the anisotropic shape of the electron beam when it touches the specimen surface under EBSD measurement chamber geometry (as shown in To obtain a representative result of occurrence frequency of the appearing ORs, Determination of orientation relationships between ferrite and cementite In this work, the basic principle of the OR determination is to verify the direction-direction and plane-plane parallelisms between ferrite and cementite using the ORs between ferrite and cementite reported in literature. The utilized ORs are summarized in Table 2.5. Table 2.5 Summary of the well-known and new ORs. Well-known ORs Expressed in Conventional way New ORs [START_REF] Zhang | New insights into crystallographic correlations between ferrite and cementite in lamellar eutectoid structures obtained by SEM-FEG/EBSD and an indirect two-trace method[END_REF] Expressed in close-packed plane and in-plane direction BAG F C F C ] 111 //[ ] 010 [ ] 0 1 1 //[ ] Unknown The direction-direction and plane-plane parallelism verifications were performed by first transforming the corresponding direction vectors or plane normal vectors in the lattice basis of cementite (orthorhombic) to that of ferrite (cubic) by coordinate transformation using the determined orientation of ferrite and cementite by EBSD and then comparing them with the corresponding ferrite direction vectors or plane normal vectors to verify the parallelism between them to conclude the possible OR. The full OR determination process is detailed as follows. (1) Setting of coordinate systems. Three orthonormal coordinate systems have been chosen for convenience in addition to the crystal basis for ferrite and cementite. One is referred to the sample coordinate system. The second is to the crystal of ferrite that should be the same as the lattice basis of ferrite and the third one is to the crystal of cementite in a way that the corresponding basis vectors are parallel to lattice basis vectors (i. ). The difference between the two bases is that the crystal coordinate system of cementite is orthonormal, whereas the lattice basis is not. (2) Construction of coordinate transformation matrices. The coordinate transformation matrix j G (j=F or C, where F denotes ferrite and C cementite) from the sample coordinate system to the ferrite/cementite coordinate system can be constructed using the Euler angles φ 1 , Φ, φ 2 of ferrite/cementite with respect to the sample coordinate system determined by EBSD measurements.           Φ Φ Φ Φ - Φ + - Φ + Φ Φ - - Φ - = cos 2 1 2 1 2 1 2 1 2 1 1 2 1 2 1 2 1 2 1 ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ j G (2.1) Then, the coordinate transformation matrix from ferrite crystal coordinate system to cementite crystal coordinates system ∆G F→C (as illustrated in Figure 2.9) can be calculated in matrix notation: [ ] [ ] [ ] [ ] [ ] C C C F F F S G G S G ⋅ = ∆ ⋅ ⋅ → (2.2) [ ] [ ] [ ] [ ] [ ] C C F F C F S G G S G ⋅ ⋅ ⋅ = ∆ - - → 1 1 (2.3) where F S or C S is the rotational symmetry matrix of crystal system of ferrite or cementite. Figure 2.9 Coordinate system transformations from ferrite to cementite. ( 3) Coordinate transformation of vectors If one intends to transform the direction vectors or plane normal vectors in the cementite lattice basis to that of ferrite, the coordinate transformations start from cementite lattice basis between cementite crystal coordinate system, and followed by cementite crystal coordinate system between ferrite coordinates system (lattice basis). For a direction vector [u v w] in cementite lattice basis, its corresponding direction vector in cementite coordinate system v C is where i e uv (i=1,2,3) is the unit vector of cementite crystal coordinate system and a, b, c are the crystal lattice parameters of cementite. Then, the same vector in ferrite coordinate system F v , can be calculated with the transformation matrices F C F C v G v → = ∆ (2.5) For a normal vector of a plan (h k l) in cementite lattice basis, the coordinate transformation is more complicated. First, its normal vector in cementite lattice c V V V Λ Λ Λ = = = v v v v v v uuv uuv uuv (2.7) where V is the volume of the unit cell built on the three cementite crystal lattice vectors , , a b c uv uv uv . ( ) ( ) ( ) V a b c b c a c a b Λ Λ Λ = ⋅ = ⋅ = ⋅ v v v v v v v v v (2.8) Thus, (2.9) With equation (2.6) and (2.9), hkl g uuuv can be obtained: And so the same vector in ferrite coordinate system can be calculated by equation (2.5). (4) Angle between two vectors As soon as the direction vectors or plane normal vectors of cementite are transformed into the same coordinates system with the corresponding vectors of ferrite, they can be compared. We suppose 1 v uv and 2 v uu v are two vectors in ferrite coordinates system. As it is known, ( ) 1 2 1 2 1 2 cos v v v v v v ⋅ = ⋅ ⋅ uuv uv uu v uu v uv uu v $ (2.12) Then, ( ) 1 2 1 2 1 2 cos v v v v v v ⋅ = ⋅ uv uu v uv uu v $ uuv uu v (2.13) The angle ∆δ between two vectors can be calculated as following equation: ( ) 1 2 1 2 1 2 arc cos arc cos v v v v v v δ   ⋅   = =   ⋅   uv uu v uv uu v $ uuv uu v ∆ ∆ ∆ ∆ (2.14) In the present work, if ∆δ ≤ 3° for the parallel directions and plane normals from ferrite and cementite, we consider that direction and plane parallelisms are fulfilled and the OR between ferrite and cementite is confirmed. For the present study, the habit plane between ferrite and cementite in each OR was determined by the "indirect two-trace method" [START_REF] Zhang | Indirect two-trace method to determine a faceted low-energy interface between two crystallographically correlated crystals[END_REF]. Ab-initial calculations In order to interpret the working mechanism of the magnetic field on carbon solution in ferrite, the magnetic moments of bcc Fe and carbon atoms and the atomic magnetic dipolar interaction energy of Fe clusters without and with an interstitial carbon atom were evaluated. The calculations were carried out within the framework of density functional theory (DFT) using the Vienna ab-initio simulation package (VASP) [START_REF] Kresse | Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set[END_REF][START_REF] Kresse | From ultrasoft pseudopotentials to the projector augmented-wave method[END_REF]. The interaction between ions and electrons was described by the projector augmented wave method (PAW) [START_REF] Blöchl | Projector augmented-wave method[END_REF], and the exchange correlation potential was treated by the generalized gradient approximation (GGA) [START_REF] Perdew | Generalized gradient approximation made simple[END_REF]. The pseudopotentials with 3d 7 4s 1 and 2s 2 2p 2 as respective valence states for Fe and carbon and a spin polarized representation of the electronic charge densities that allows for collinear description of magnetic moments were used. The kinetic energy cutoff was set to be 400eV and a Monkhorst-Pack [START_REF] Monkhorst | Special points for Brillouin-zone integrations[END_REF] grid was employed to In section 3.1, the magnetic-field-induced aligned and elongated microstructures were studied through a comparative examination of Fe-0.12C alloy and Fe-0.36C alloy. The magnetic field influential mechanism on the aligned and elongated microstructures was theoretically analyzed using the magnetic dipolar mode and discussed as a function of carbon content. In section 3.2, the area fraction of the transformed ferrite was measured in both Fe-0.12C alloy and Fe-0.36C alloy. The effect of the magnetic field on modifying the amount of ferrite has been studied and discussed as a function of carbon content. Furthermore, the effect of the magnetic field on Widmänstatten ferrite has been studied in Fe-0.36C alloy. In section 3.3, the effects of the magnetic field on the abnormal structure and the spheroidization of pearlite have been investigated in Fe-1.1C alloy. The morphology features as well as the crystallographic characteristics of the abnormal structure and the pearlite were studied. For Fe-0.36C alloy, the transformed microstructures without and with the magnetic field are also composed of proeutectoid ferrite and pearlite, as shown in Magnetic-field-induced aligned and elongated microstructures Discussion The field induced elongated and aligned microstructure during the proeutectoid ferritic transformation has been focused and studied in some iron-based alloys [34, 36, 50-53, 56, 58, 59]. It can be qualitatively explained by the magnetic dipole model [START_REF] Shimotomai | Formation of aligned two-phase microstructures by applying a magnetic field during the austenite to ferrite transformation in steels[END_REF][START_REF] Zhang | Grain boundary characteristics and texture formation in a medium carbon steel during its austenitic decomposition in a high magnetic field[END_REF]. It is known that, under the magnetic field, the magnetic moments tend to align along the field direction, as schematically illustrated in Figure 3.3. Thus, there exists a dipolar interaction between the neighboring moments. The dipolar interaction energy E D can be expressed as follows: 0 1 12 2 12 1 2 3 12 [3( )( ) ] 4 D E m e m e m m r µ π = - ⋅ ⋅ -⋅ v v v v v v (3.1) where 0 µ is the vacuum magnetic permeability; 12 e v is a unit vector parallel to the line joining the centers of the two dipoles; 12 r is the distance between two dipoles 1 m v and 2 m v . If the magnetic moments align along the field direction and m 1 = m 2 =m, 12 r =r, the dipolar interaction energy E D can be considered in the following two situations: H Magnetic moment (1) The two magnetic dipoles are aligned in the magnetic field direction, as shown in Figure 3.4(a). Then the dipolar interaction energy between these two magnetic dipoles can be calculated as: 2 0 3 2 D m E r µ π = - (3.2) since E D is negative (E D <0), the magnetic dipolar interaction between them is attractive, as a results, the magnetic dipoles tend to attract each other along the field direction. (a) The two magnetic dipoles are aligned along the magnetic field direction. (b) The two magnetic dipoles are aligned in the transverse magnetic field direction. (2) The two magnetic dipoles are aligned in the transverse magnetic field direction, as shown in Figure 3.4(b). Then the dipolar interaction energy between these two magnetic dipoles is resulted in: 2 0 3 4 D m E r µ π = (3.3) It is seen that E D is positive (E D >0), which implies the magnetic dipolar interaction between them would make them repel each other in this direction. (b) 2 m v 1 m v r H 1 m v 2 m v r H (a) As a result, under the magnetic field, magnetic dipoles attract each other along the field direction but repel each other along the transverse field direction due to the dipolar interaction. During the austenite to ferrite and pearlite transformation, the magnetic dipolar interaction works in two scales: atomic-scale and micro-scale. In the atomic-scale, each Fe atom in ferrite grains carrying a magnetic moment can be regarded as a magnetic dipole. The magnetic dipolar interaction between them makes them attract each other along the field direction and repel each other in the transverse field direction. To minimize the demagnetization energy (caused by the repulsion) Fe atoms tend to align along the field direction. In this case, the effect of the magnetic field is mainly on grain growth process, which favors the elongation of the ferrite grains along the field direction. However, the elongation leads to an increase of the interfacial area and hence an increase of the interfacial energy which opposes the elongation and favors a spherical shape. The counterbalance between these two effects determines the final elongation degree of the grains [START_REF] Shimotomai | Formation of aligned two-phase microstructures by applying a magnetic field during the austenite to ferrite transformation in steels[END_REF][START_REF] Zhang | Magnetic-fieldinduced grain elongation in a medium carbon steel during its austenitic decomposition[END_REF]. In the microscale, each magnetized ferrite grain can be regarded as a magnetic dipole. Due to the same magnetic dipolar interaction, new ferrite nuclei tend to nucleate next to the existing ones along the field direction and form ferrite chains to reduce the demagnetization energy. In this way, the nucleation of the ferrite is affected and the alignment is induced. As the alignment of both Fe atoms and the ferrite nuclei can reduce the demagnetization energy, they are energetically favored by the magnetic field. This is the reason for the formation of the elongated and aligned microstructure under the magnetic field. In the present work, the carbon content of the Fe-0.12C alloy is relatively low. In this case, proeutectoid ferrite forms at high temperature, which is far above the Curie temperature, T C . One could imagine that at the early stage of proeutectoid ferritic transformation which is above the T C , the magnetic dipolar interaction in both atomic-scale and micro-scale is weak, as the induced magnetization of ferrite and austenite are very close. Hence, ferrite nuclei form randomly at the prior austenite boundaries and grow uniformly. When the temperature drops down to the T C , ferrite becomes ferromagnetic. As a consequence, the magnetic dipolar interaction in the two scales becomes stronger. Thus, both nucleation and grain growth in the subsequent cooling process is greatly influenced by the magnetic field. The new ferrite nuclei tend to form preferentially next to the existing ferrite grains along the field direction to form ferrite chains. It should be mentioned that as the carbon content of the alloy is low, the relative amount of proeutectoid ferrite is high and ferrite should be distributed densely in the specimen. This ensures small spacing between the existing ferrite grains and the new ferrite nuclei and hence strong micro-scale magnetic dipolar interaction between ferrite grains. Meanwhile, the existing ferrite grains preferentially grow along the field direction under the atomic dipolar interaction to form elongated grains. Since ferrite is in large quantity, its growth along the field direction may hinder the growth of the newly formed ferrite along that direction. Therefore, the growth of the new ferrite nuclei is restricted and occurs along the transverse field direction. As a result, the ferrite grains in two different elongation orientations are formed: the ferrite grains transformed at the early stage are elongated along the field direction, whereas those transformed at the late stage are elongated in transverse field direction. As ferrite is carbon depleted, the preferential nucleation and growth of proeutectoid ferrite along the field direction can cause the diffusion of carbon atoms in the field transverse direction. Hence, the remaining austenite between ferrite chains, especially next to the field-direction-elongated ferrite grains, is rich in carbon and provides perfect sites for the formation of pearlite. When temperature drops below Ar 1 , this carbonrich austenite located between ferrite chains decomposes into pearlite. As the For the Fe-0.36C alloy, magnetic field elongates the proeutectoid ferrite grains in the field direction through atomic-scaled dipolar interaction during grain growth process. This demonstrates magnetic field effect on inducing the elongated and aligned microstructures is carbon-content dependent. Magnetic-field-induced phase fraction modification of ferrite Results The measurement results of ferrite area fraction of Fe-0.12C alloy and Fe-0.36C alloy treated without and with the magnetic field are shown in Figure 3.5. It shows that the magnetic field increases the amount of ferrite in both alloys. The field-induced increment in area fraction of ferrite is 6.0% in Fe-0.12C alloy and 14.0% in Fe-0.36C alloy. In this content, magnetic field enhances the formation of ferrite by increasing its area fraction and this field effect becomes more pronounced with the increase of the carbon content. The area fraction of Widmänstatten ferrite in Fe-0.36C alloy is measured and the result is given in Table 3.1. It is found that the amount of the Widmänstatten ferrite is greatly decreased with the application of the magnetic field. Discussion When magnetic field is applied to proeutectoid ferritic phase transformation, both parent austenite and product ferrite can be magnetized and thus their Gibbs free energy is lowered according to their magnetization. The decrease in Gibbs free energy is expressed as M d B G M v v ⋅ - = ∆ ∫ 0 0 (3.4) where M is the magnetization of a certain phase and 0 B is the induction of the applied magnetic field. Due to the fact that the magnetization of the product ferrite is higher than that of the parent austenite within the whole transformation temperature range, decrease in the Gibbs free energy of ferrite is higher. Hence, the energy difference between the parent austenite and the product ferrite is increased by the magnetic field. As a consequence, an extra energy term is introduced by magnetic field as a driving force to the austenite to ferrite phase transformation. Therefore, the phase transformation is accelerated by the magnetic field. Previous simulation studies [28-33, 43, 68] have shown that magnetic field modifies the Fe-C phase equilibrium by shifting the α⁄α+γ and the γ⁄α+γ boundaries in the Fe-C phase diagram towards the high carbon content and high temperature side, as shown in Figure 3.6. Moreover, it is proved that the field-influence on the γ⁄α+γ boundaries is more pronounced. According to the 'Metallurgical Level Law', the amount of ferrite at eutectoid transformation temperature is proportional to the difference between the carbon solubility in austenite and the carbon content of the steel and inversely proportional to the carbon solubility difference between austenite and ferrite at the same temperature, i.e. ( ) ( ) ' ' ' F P F P C P P ---, as schematically illustrated in Figure 3.6. Therefore, the amount of ferrite was increased by the magnetic field due to its influence on phase equilibrium. The effect of the magnetic field on modifying the phase transformation temperature has been also discussed as the function of carbon content in Fe-based alloys. It is has been revealed by Garcin et al. [START_REF] Garcin | Experimental evidence and thermodynamics analysis of high magnetic field effects on the austenite to ferrite transformation temperature in Fe-C-Mn alloys[END_REF] that, with the increase of the carbon content of the alloys, the increment of the Ae 3 temperature rises. However, the increment of the Ae 1 temperature does not change with the carbon content. As a result, the field effect on enhancing the area fraction of ferrite becomes more pronounced when the carbon content increases, as proved by this work. F' F P P' C In addition, during the transformation from the parent austenite to ferrite, sometimes when the carbon content is suitable and the cooling rate is low, a kind of ferrite in acicular shape, known as Widmanstätten ferrite, of which the formation follows a K-S orientation relationship with the parent austenite, forms instead of the normal equiaxed ferrite. The formation of the Widmanstätten ferrite is considered to be resulted from the need to reduce the formation energy barrier when the dirving force is low [START_REF] Wang | Effect of a high magnetic field on the formation of Widmänstatten ferrite in Fe-0.52C[END_REF]. According to the solid-state phase transformation theory, during the austenite to ferrite transformation, the Gibbs free energy changes related to uniform nucleation of a new phase can be express as ε σ V S G V G V + + ∆ - = ∆ (3.5) where V is the volume of the product phase, V G ∆ is the volume Gibbs free energy difference between the parent phase and the product phase. S is the surface area of the nuclei, σ is the interfacial energy between the parent phase and the product phase and ε is the elastic strain energy of the new phase. V G V∆ is the driving force of the transformation, whereas, both σ S -the total interfacial energy-and ε V -the total elastic strain energy-are considered as the transformation barrier. When austenite transforms into Widmanstätten ferrite, the K-S OR guarantees a low energy semi-coherent interface between the parent austenite and the product ferrite that greatly reduces the transformation barrier related to the interfacial energy. Considering of that, it is clear that the formation of Widmanstätten ferrite is due to the need of the reduction of the interfacial energy by forming coherent or semicoherent interface, when the driving force is insuffient in the case of slow cooling. As we mentioned before, when an external magnetic field is applied, an additional transformation driving force item is introduced by magnetic field. ε σ V S G G V G M V + + ∆ + ∆ - = ∆ ) ( (3.6) Consequently, the need of forming low energy interface to compensate the insuffient driving force is reduced. Thus, the chance to form Widmanstätten ferrite is considerably lowered resulting in the reduced amount in the transformed microstructure of Fe-0.36C alloy under the magnetic field. As Widmanstätten ferrite is brittle and it is harmful to the ductility of steels, the field effect in reducing the amount of this ferrite is positive for its practical application. Summary Magnetic field increases the phase fraction of ferrite due to its thermodynamic influence on phase equilibrium. This field effect becomes more pronounced with the decreased austenite to ferrite transformation temperature that corresponds to the increased carbon content of steels. Magnetic field reduces the formation of Widmanstätten ferrite by introducing an additional transformation driving force. Magnetic field-induced microstructure features in Fe-1.1C alloy Results The microstructures of the Fe-1.1C alloy treated without and with the magnetic field and the corresponding pole figures of ferrite are displayed in Figure 3.7. As seen from the micrograph in Figure 3.7(a), without the magnetic field, the microstructure is composed of pearlite and a small amount of abnormal structure [START_REF] Mcquaid | Effect of quality of steel on case-carburizing results[END_REF] (indicated by the arrow) that consists of coarse proeutectoid cementite distributed along the initial austenite grain boundaries and a border of ferrite surrounding it. From the pole figures in Figure 3.7 (a), it is seen that ferrite (ferrite in abnormal structure and pearlitic ferrite) does not display any preferred crystallographic orientation. When the 12T magnetic field is applied, as shown in Figure 3.7 (b), the microstructural constituents of the alloy remain the same (pearlite plus abnormal structure) and the crystallographic orientations of ferrite stays random, indicating that the magnetic field has no special effect on modifying the components of the microstructure and the crystallographic texture of ferrite. However, the magnetic field exhibits clear influence on the amount of the microstructural constituents, the morphology of the pearlite and the occurrence of the orientation relationships between ferrite and cementite in pearlite. Table 3.2 displays the total area percentage of the abnormal structure obtained without and with the 12T magnetic field. It is seen that the total area percentage of the abnormal structure is increased under the magnetic field, indicating that the magnetic field promotes the formation of the abnormal structure. Nevertheless, no specific OR between the cementite and the ferrite in the abnormal structure has been found either in the non-field-or in the field-treated specimen. This is different from what Chairuangsri et al. [START_REF] Chairuangsri | Abnormal ferrite in hyper-eutectoid steels[END_REF] reported. In their work, they found specific orientation relationships (ORs) between the cementite and ferrite in the abnormal structure that are close to the Pitsch-Petch OR and the Bagaryatsky OR found in pearlite. 3.3. It is seen that the amount of spheroidal pearlite is increased under the magnetic field. This indicates that the magnetic field promotes the spheroidization of cementite in pearlite. In addition, the average pearlite lamellar spacing is found enlarged from 0.47µm in the non-fieldtreated specimen to 0.9 µm in the field-treated specimen. This phenomenon has been observed in other steels and has been thoroughly analyzed [START_REF] Zhang | Microstructural features induced by a high magnetic field in a hypereutectoid steel during austenitic decomposition[END_REF][START_REF] Zhang | The effects of thermal processing in a magnetic field on grain boundary characters of ferrite in a medium carbon steel[END_REF]. It is clear (c) Z X Y//FD that the area percentage of the abnormal structure and the morphology of the pearlitic cementite strongly depend on the orientation of the abnormal structures and the lamellar pearlite with respect to the observation plane (i.e. at which angle they intersect the observation plane). As there are no preferential crystallographic orientations in either the non-field-or the field-treated specimen, and the abnormal structures and the pearlite colonies are randomly selected with a substantial amount, the corresponding results on the abnormal structure and the pearlite could be considered only from the magnetic field. Discussion For a fully austenitized hypereutectoid steel, proeutectoid cementite first precipitates along the original austenite grain boundaries when the temperature drops to the range between Ar cm and Ar 1 and then pearlite composed of lamellar cementite and ferrite forms from the remaining austenite when the temperature drops below Ar 1 , during a slow cooling. The formation of the ferrite border around the proeutectoid cementite that constitutes the so-called abnormal structure is due to the fact that the formation of the proeutectoid cementite along the initial austenite grain boundaries that is carbon enriched generates a carbon depleted zone in the vicinity of the proeutectoid cementite. If the depleted carbon cannot be resupplied from the interior of the austenite grain through carbon diffusion, the local composition will become hypoeutectoid, hence this part of austenite will transform into ferrite when the temperature is below Ar 3 , forming the abnormal structure. The increase of the amount of abnormal structure by the magnetic field may be related to the influence of the magnetic field on phase equilibrium between austenite and ferrite. It is known that magnetic field promotes the formation of phases with higher induced magnetization during a transformation. In the case of the austenite to ferrite transformation, ferrite has higher magnetization than austenite; hence its formation is promoted by the magnetic field. In the present work, once the proeutectoid cementite forms along the prior austenite boundaries, the carbon depleted austenite in its vicinity obtain a higher driving force under a magnetic field to transform to ferrite [START_REF] Choi | Effects of a strong magnetic field on the phase stability of plain carbon steels[END_REF][START_REF] Garcin | In situ characterization of phase transformations in a magnetic field in Fe-Ni alloys[END_REF][START_REF] Garcin | Kinetic effects of magnetic field on the γ/α interface controlled reaction in iron[END_REF][START_REF] Zhang | Thermodynamic and kinetic characteristics of the austenite-to-ferrite transformation under high magnetic field in medium carbon steel[END_REF]. As the magnetic field also shifts the eutectoid point to high carbon content side [START_REF] Joo | An effect of a strong magnetic field on the phase transformation in plain carbon steels[END_REF][START_REF] Zhang | Shift of the eutectoid point in the Fe-C binary system by a high magnetic field[END_REF] the amount of ferrite obtained is also increased, as widely observed in hypoeutectoid steels after its transformation from austenite to ferrite [START_REF] Choi | Effects of a strong magnetic field on the phase stability of plain carbon steels[END_REF][START_REF] Enomoto | Influence of magnetic field on the kinetics of proeutectoid ferrite transformation in iron alloys[END_REF][START_REF] Garcin | Experimental evidence and thermodynamics analysis of high magnetic field effects on the austenite to ferrite transformation temperature in Fe-C-Mn alloys[END_REF][START_REF] Zhang | Thermodynamic and kinetic characteristics of the austenite-to-ferrite transformation under high magnetic field in medium carbon steel[END_REF]. Consequently, the amount of the abnormal ferrite surrounding the proeutectoid cementite is increased under the magnetic field. For lamellar pearlite, spheroidization is a natural tendency, as in this process, the total interfacial area between ferrite and lamellar cementite can be reduced and then the total system becomes thermodynamically stable [START_REF] Tian | Kinetics of pearlite spheroidizations[END_REF]. The influence of the magnetic field on spheroidization of pearlite may be related to two factors. First, spheroidization of pearlite is a carbon diffusion process [START_REF] Tian | Kinetics of pearlite spheroidizations[END_REF][START_REF] Nam | Accelerated spheroidization of cementite in high-carbon steel wires by drawing at elevated temperatures[END_REF][START_REF] Zhang | Effect of deformation on the evolution of spheroidization for the ultra high carbon steel[END_REF][START_REF] Kamyabi-Gol | Spheroidizing Kinetics and Optimization of Heat Treatment Parameters in CK60 Steel Using Taguchi Robust Design[END_REF]. As magnetic field elevates the proeutectoid transformation temperature [START_REF] Joo | An effect of high magnetic field on phase transformation in Fe-C system[END_REF][START_REF] Garcin | Experimental evidence and thermodynamics analysis of high magnetic field effects on the austenite to ferrite transformation temperature in Fe-C-Mn alloys[END_REF][START_REF] Zhang | Shift of the eutectoid point in the Fe-C binary system by a high magnetic field[END_REF][START_REF] Zhang | Microstructural features induced by a high magnetic field in a hypereutectoid steel during austenitic decomposition[END_REF], pearlite forms at higher temperature under a magnetic field. High temperature favors carbon diffusion. Therefore, the spheroidization of pearlite that is realized by cementite fragmentation and granulation through carbon diffusion is enhanced by the magnetic field. Second, it has been revealed that magnetic field increases the relative interfacial energy between ferrite and cementite [START_REF] Garcin | Kinetic effects of magnetic field on the γ/α interface controlled reaction in iron[END_REF][START_REF] Zhang | High temperature tempering behaviors in a structural steel under high magnetic field[END_REF], as the field has different impact on the magnetization of the boundary area and the grain interior. It is known that the Gibbs free energy of a crystal is lowered when it is magnetized. The degree of magnetization depends on the perfectness of the crystal in crystalline materials. As the boundary area possesses a high density of defects, the magnetization induced by a magnetic field is limited with respect to that of grain interior. The Gibbs free energy drop in grain interior is much higher than that in boundary area. As a result, the relative boundary energy is raised. Consequently, reducing the total boundary area thus the total boundary interfacial energy by spheroidization is energetically favored by the magnetic field. Chapter 4 Magnetic Field-Enhanced Carbon Solution in Ferrite Introduction Strengthening has been widely required as one of the necessary properties for the deveploment of high-performance structural materials. The most common strengthening mode refers to solid solution strengthening by adding various alloying elements, such as carbon in iron. As a promising technique for microstructure modification and texture control [START_REF] Ohtsuka | Alignment of ferrite grains during austenite to ferrite transformation in a high magnetic field[END_REF][START_REF] Shimotomai | Formation of aligned two-phase microstructures by applying a magnetic field during the austenite to ferrite transformation in steels[END_REF][START_REF] Zhang | Magnetic-fieldinduced grain elongation in a medium carbon steel during its austenitic decomposition[END_REF][START_REF] Zhang | Grain boundary characteristics and texture formation in a medium carbon steel during its austenitic decomposition in a high magnetic field[END_REF][START_REF] Bacaltchuk | Effect of magnetic field applied during secondary annealing on texture and grain size of silicon steel[END_REF][START_REF] Li | Effects of high magnetic field annealing on texture and magnetic properties of FePd[END_REF], high magnetic field has shown its capacity to modify the solubility of phases in Fe-C alloy system, hence demonstrating great potential to enhance the effect of solid solution strenthening of steels. According to thermodynamic calculations on phase equilibrium [28-32, 39, 68], the solubility of carbon in ferrite is increased under an applied magnetic field. Moreover, imposition of magnetic field increases the proeutectoid ferrite transformation temperature [START_REF] Joo | An effect of a strong magnetic field on the phase transformation in plain carbon steels[END_REF][START_REF] Garcin | Experimental evidence and thermodynamics analysis of high magnetic field effects on the austenite to ferrite transformation temperature in Fe-C-Mn alloys[END_REF][START_REF] Garcin | Kinetic effects of magnetic field on the γ/α interface controlled reaction in iron[END_REF][START_REF] Zhang | Grain boundary characteristics and texture formation in a medium carbon steel during its austenitic decomposition in a high magnetic field[END_REF] and enhances the phase fraction of the transformed ferrite [START_REF] Choi | Effects of a strong magnetic field on the phase stability of plain carbon steels[END_REF][START_REF] Enomoto | Influence of magnetic field on the kinetics of proeutectoid ferrite transformation in iron alloys[END_REF][START_REF] Garcin | Experimental evidence and thermodynamics analysis of high magnetic field effects on the austenite to ferrite transformation temperature in Fe-C-Mn alloys[END_REF][START_REF] Zhang | Thermodynamic and kinetic characteristics of the austenite-to-ferrite transformation under high magnetic field in medium carbon steel[END_REF]. As ferrite is carbon-depleted and cementite is carbon-enriched in Fe-C system, a net increase in the amount of ferrite under a magnetic field requires a balanced carbon repartition bewteen the ferrite and cementite phases to maintain the fixed carbon content of the material. Hardness test of ferrite 7 has demonstrated that under a magnetic field ferrite becomes harder, indicating that the hardness increase is due to the increased carbon solution. Until now, however, there has been no direct experimental evidence for this assertion and the underlying physical mechanism of the field-induced carbon solubility remains to be addressed. In the this chapter, the carbon concentrations of ferrite in the Fe-0.36C alloy treated without and with a 12T magnetic field were measured by means of wavelength-dispersive spectroscopy using a Shimadzu 1610 electron probe microanalyzer (WDS-EPMA). The hardness of proeutectoid ferrite was tested using a micro-indenter with a load of 25g. The mean hardness was calculated by averaging over 8 measurements for each specimen. In order to interpret the physical mechanism of the magnetic field on carbon solution in ferrite, the magnetic moments of bcc Fe and carbon atoms and the atomic magnetic dipolar interaction energy of Fe clusters without and with an interstitial carbon atom were evaluated. The calculations were carried out within the framework of density functional theory (DFT) using the Vienna ab-initio simulation package (VASP) [START_REF] Kresse | Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set[END_REF][START_REF] Kresse | From ultrasoft pseudopotentials to the projector augmented-wave method[END_REF]. The interaction between ions and electrons was described by the projector augmented wave method (PAW) [START_REF] Blöchl | Projector augmented-wave method[END_REF], and the exchange correlation potential was treated by the generalized gradient approximation (GGA) [START_REF] Perdew | Generalized gradient approximation made simple[END_REF]. The pseudopotentials with 3d 7 4s 1 and 2s 2 2p 2 as respective valence states for Fe and carbon and a spin polarized representation of the electronic charge densities that allows for collinear description of magnetic moments were used. The kinetic energy cutoff was set to be 400eV and a Monkhorst-Pack [START_REF] Monkhorst | Special points for Brillouin-zone integrations[END_REF] grid was employed to sample the Brillouin zone: 4×4×4 kpoints sampling within a supercell containing 54 Fe atoms (27 bcc Bravais cells) and 1 carbon atom located in the octahedral interstice of the bcc Bravais cell at the center of the supercell. For comparison, the same calculations were performed on the same supercell but without the carbon atom. For both specimens, the measurement carbon contents are higher than the theoretical values. This may come from two factors: the residual ethanol left from etching and rinsing during specimen preparation and the carbon contamination inside the EPMA column during the WDS measurements. Since the experimental conditions for the non-field and field treated specimens were strictly controlled to be identical, the carbon increments resulting from these two factors could be regarded as the same for the two specimens. Thus, the carbon content difference between the two specimens should be attributed to the magnetic field, as indirectly proved by the hardness tests (Table 4.3). Under the 12T magnetic field, the hardness of ferrite increases from HV 0.025 88.1 to 102.1 by 15.9%, suggesting the solid solution strengthening of excessive carbon atoms in ferrite. In this regard, the fieldinduced increase in the amount of carbon-depleted ferrite is actually balanced by the elevation of their carbon content to maintain the fixed carbon content of the alloy. µ Β /atom). Starting from the third neighbors, the magnetic moments of Fe atoms resume to that in the carbon free cell. As a consequence of the modified magnetic moments, the magnetic interaction between these atoms is changed. Results Discussion It is known that when a magnetic field is applied, the moments of Fe atoms tend to align along the field direction. Each Fe or carbon atom can be regarded as a magnetic dipole and the interaction energy E D between two magnetic dipoles can be calculated by Eq.(3.1) as discussed in Chapter 3. Then, the interaction energy E D can be decomposed into two opposite contributions: the one giving rise to magnetization is negative, whereas the other resulting in demagnetization is positive. Clearly, the atomic structural configuration that enhances the magnetization contribution is preferred by the system. given by integrating the dipolar interaction of all atom pairs in the cluster, under the consideration that starting from the third neighbors, the Fe moments in the carbon containing cell are the same as those of their counterparts in the carbon free cell. Accordingly, the magnetic interaction energies over the atom cluster without and with an interstitial carbon atom were calculated to be , respectively. The magnetic interactions in the two cases are both positive, indicating that the demagnetization contribution is more important than that of magnetization. However, when a carbon atom occupies the octahedral interstice, the interaction energy is decreased, i.e. the system becomes energetically more stable. This means that the magnetic field favors the solution of carbon atoms in the bcc Fe through reducing the demagnetization energy. In this way, the carbon content in ferrite is increased when a magnetic field is applied during the austenite to ferrite transformation. Summary The carbon content increase in ferrite induced by magnetic field has been experimentally demonstrated through the WDS-EPMA for the first time. According to the Ab-initio calculations, the modified Fe magnetic moments originate from the carbon solution. When the magnetic moments of Fe atoms are aligned under an external magnetic field, the demagnetization energy due to atomic dipolar magnetic interaction is reduced with the carbon solution. This underlies the physical mechanism of field-enhanced carbon solution in ferrite. mechanism was analyzed as the function of magnetic field intensity and carbon content. Results The inverse pole figures of the proeutectoid ferrite in Fe-0.12C alloy treated without and with the 12T magnetic field are shown in Figure 5.1. The sample coordinate system is shown in Figure 5.2. It can be seen, there is no obvious preferential orientation of ferrite in the Fe-0.12C alloy. The inverse pole figures of ferrite in Fe-0.36C alloy treated without and with the application of the magnetic field are shown in Figure 5.3 (under the same sample coordinate system in Figure 5.2). In the Fe-0.36C alloy, an enhancement of <001> fiber component along the ND (transverse field direction) appears in the field treated specimens. Though this enhancement in 8T specimen is not obvious, it becomes much stronger when field is increased to 12 T. Based on the above results, it is seen that the field induced ferrite texture is related to the field intensity and the carbon content of the alloys. Discussion It is known that, when carbon atoms dissolve into ferrite, they occupy preferentially the octahedral interstices in the ferrite which are flattened in the <001> direction. As the atomic spacing between the two summate Fe atoms of the octahedral interstice in the <001> direction is smaller than the diameter of the This will generate the lattice distortion and thus distortion energy [START_REF] Zhang | Grain boundary characteristics and texture formation in a medium carbon steel during its austenitic decomposition in a high magnetic field[END_REF]. Since under the magnetic field, Fe atoms attract each other along the field direction and repels each other along the transverse field direction (the atomic dipolar interaction), the lattice distortion can be reduced by the increased atom spacing in the transverse field direction. If the distorted <001> direction is along the transverse field direction, the nucleation and growth of such ferrite grains will be energetically favored by the magnetic field. In this way, the <001> fiber component along the transverse field direction is enhanced. Obviously, this field effect is strongly related to the degree of lattice distortion of ferrite. Generally, the more the crystal lattice is distorted, the larger the distortion energy is. In this case, the need to reduce the distortion energy by favoring the growth of grain with their distorted <001> direction parallel to the transverse field direction would be increased. Consequently, the field effect appears stronger and the intensity of this <001> fiber component along the transverse field direction becomes enhanced. The degree of the lattice distortion is determined by two factors. One is the thermal expansion of the lattice which is temperature dependent and the other is the amount of oversaturated carbon atoms in ferrite when the transformation is non-equilibrium. For the Fe-0.36C alloy, the austenite to ferrite transformation happens at relatively lower temperature compared with that of the Fe-0.12C alloy. Carbon diffusion is restrained, and then more oversaturated carbon atoms will be left in the formed ferrite. Moreover, thermal expansion that the ferrite lattice can reach is also reduced. Thus, higher degree of lattice distortion could be expected in the Fe-0.36C alloy than in the Fe-0.12C alloy. As a result, the effect of the magnetic field on the texture of the proeutectoid ferrite is enhanced for the Fe-0.36C alloy, resulting in a visible enhancement of <001> fiber texture along the transverse field direction. Orientation relationships of pearlite under the magnetic field Introduction Pearlitic transformation, regarded as the most classic solid-solid state transformation, has been widely studied under the effect of magnetic field. The magnetic field influences on pearlitic transformation kinetics, mechanism of the pearlite formation and structure evolutions have been fruitful. It has been proved that magnetic field shows considerable effects on pearlitic transformation. It elevates the transformation temperature [START_REF] Zhang | Calculation of magnetization and phase equilibrium in Fe-C binary system under a magnetic field[END_REF][START_REF] Garcin | Experimental evidence and thermodynamics analysis of high magnetic field effects on the austenite to ferrite transformation temperature in Fe-C-Mn alloys[END_REF][START_REF] Zhang | Shift of the eutectoid point in the Fe-C binary system by a high magnetic field[END_REF][START_REF] Zhang | Thermodynamic and kinetic characteristics of the austenite-to-ferrite transformation under high magnetic field in medium carbon steel[END_REF], increases eutectoid carbon composition [START_REF] Zhang | Shift of the eutectoid point in the Fe-C binary system by a high magnetic field[END_REF] and modifies the morphology of pearlite [START_REF] Zhang | Microstructural features induced by a high magnetic field in a hypereutectoid steel during austenitic decomposition[END_REF]. Nowadays, the investigations on the mechanism of the pearlitic transformation have been widely conducted from crystallographic point of view [START_REF] Zhang | New insights into crystallographic correlations between ferrite and cementite in lamellar eutectoid structures obtained by SEM-FEG/EBSD and an indirect two-trace method[END_REF][START_REF] Zhang | Indirect two-trace method to determine a faceted low-energy interface between two crystallographically correlated crystals[END_REF]. Several orientation relationships (ORs) between pearlitic ferrite and cementite have been consistently reported such as Isaichev OR [START_REF] Isaichev | Orientation between cementite and ferrite[END_REF], Bagaryatsky OR [START_REF] Bagaryatsky | Veroyatnue mechanezm raspada mar-tenseeta[END_REF] and Pitsch-Petch OR [START_REF] Petch | The orientation relationships between cementite and α-iron[END_REF][START_REF] Pitsch | Der Orientierungszusammenhang zwischen Zementit und Ferrit im Perlit[END_REF]. Recently, new ORs have been confirmed by Zhang et.al [START_REF] Zhang | New insights into crystallographic correlations between ferrite and cementite in lamellar eutectoid structures obtained by SEM-FEG/EBSD and an indirect two-trace method[END_REF]. However, the effect of the magnetic field on the ORs of pearlite has been less studied yet. Based on this, in this section, the ORs of pearlite in three high purity Fe-C alloys treated without and with the magnetic field have been examined by means of SEM/EBSD. The effect of the magnetic field on pearlitic ORs and their corresponding occurrence frequency are analyzed. Results The type of the ORs of pearlite found in this work has been summarized in Table 5.1. Three ferrite/cementite ORs were detected, namely, Isaichev OR denoted IS OR, and two near Pitsch-Petch ORs denoted P-P1 OR and P-P2 OR respectively [START_REF] Zhang | New insights into crystallographic correlations between ferrite and cementite in lamellar eutectoid structures obtained by SEM-FEG/EBSD and an indirect two-trace method[END_REF]. The types of the appearing ORs are the same in all the three alloys without and with the application of the magnetic field. It is noticed that, without the presence of the magnetic field, the IS OR is the most favorable OR in low carbon content steel (Fe-0.12C alloy), whereas, with the increase of the carbon content of the alloy, P-P1 OR tends to increase in the appearing number (Fe-0.36C alloy: 46.7% and Fe-1.1C alloy: 60.0%) at the expense of mainly P-P2 OR. For all the alloys, the occurrence of P-P2 OR is much lower than the other two ORs. When the magnetic field is applied, the occurrence frequency of the appearing ORs has been modified and it is found that the effect of the magnetic field is varied according to the carbon content of the alloys. For alloy with very low carbon (Fe-0.12C alloy), the effect of the magnetic field is rather limited indicating by a slight decrease in IS OR and small increase in both P-P1 and P-P2. When the carbon content increases, the effect of the magnetic field becomes pronounced. Magnetic field shows a consistent effect on increasing the number of P-P2 OR. This field effect is especially noticeable in Fe-1.1C alloy. Discussion It is known that when fcc austenite decomposes into dual-phase pearlite which consists of bcc ferrite and orthorhombic cementite, there exists two transformation barriers: one is the transformation strain energy caused by the lattice misfit at the austenite/ferrite [START_REF] Garcin | Kinetic effects of magnetic field on the γ/α interface controlled reaction in iron[END_REF] and austenite/cementite interfaces [START_REF] Zhou | Ferrite: Cementite crystallography in pearlite[END_REF]; the other is the interfacial energy of the ferrite/cementite interface which depends on the atom misfit at the ferrite/cementite connecting plane [START_REF] Zhang | Crystallography and morphology of Widmanstätten cementite in austenite[END_REF]. To minimize the total transformation energy barrier, special ORs between the ferrite and the cementite in pearlite that ensure a low misfit interface between the parent and the product phase and a low misfit habit plane connecting the product phases are required. In terms of the three ORs obtained in the present work (IS, P-P1 and P-P2 OR), they all possess a common feature of closed-packed plane parallelism between the ferrite and the cementite, namely: {103} C //{101} F . Since the interplanar spacing of the {103} C and of the {101} F are both close to that of the closed-packed plane {111} A of austenite [START_REF] Zhang | New insights into crystallographic correlations between ferrite and cementite in lamellar eutectoid structures obtained by SEM-FEG/EBSD and an indirect two-trace method[END_REF], the transformation strain at the austenite/ferrite and the austenite/cementite interface are minimized and hence leads to the minimum transformation strain energy. In addition, it has been illustrated [START_REF] Zhang | New insights into crystallographic correlations between ferrite and cementite in lamellar eutectoid structures obtained by SEM-FEG/EBSD and an indirect two-trace method[END_REF] that atoms from both the pearlitic ferrite and the pearlitic cementite are well matched at the connecting interface, which guarantees small interfacial energy under the three ORs in general. In view of the strain and the interfacial atomic mismatch, the IS and the P-P1 OR have the lowest formation energy barrier, so they are the more energetically favored and appear in large numbers. However, for the P-P2 OR, there is a 3.5° deviation in plane parallelism between the pearlitic ferrite and the pearlitic cementite at the connecting interface, therefore its occurrence is reduced, much less than the other two. It has also been found that different ORs correspond to different nucleation situations. Previous work [START_REF] Zhang | New insights into crystallographic correlations between ferrite and cementite in lamellar eutectoid structures obtained by SEM-FEG/EBSD and an indirect two-trace method[END_REF] has suggested that the IS OR could occur with either pearlitic ferrite or cementite nucleating first; the P-P1 OR happens when pearlitic ferrite and cementite nucleate simultaneously; while the P-P2 OR appears when pearlitic ferrite nucleates prior to pearlitic cementite. In this point of view, the frequency of P-P2 OR is expected to be low in high carbon alloys, as it is difficult to offer large low carbon content area in high carbon alloys for ferrite nucleation first. This is consistent with the results in this study. In a hypereutectoid steel, with relatively high carbon content, austenite should possess high carbon content when austenite to pearlite transformation takes place. This composition characteristic is in favor of the formation of cementite or the simultaneous formation of cementite and ferrite. This is in good accordance with the present observation. As displayed in Table 5.2, the P-P1 and the IS OR account for the majority of the occurrence in the three ORs when the magnetic field is not applied. However, when the magnetic field is applied, the occurrence of the P-P2 OR is increased. As mentioned above, magnetic field promotes the formation of high magnetization phases [START_REF] Zhang | Microstructural features induced by a high magnetic field in a hypereutectoid steel during austenitic decomposition[END_REF]. In the pearlitic transformation temperature range, ferrite is ferromagnetic with higher magnetization and cementite is paramagnetic Chapter 6 Conclusion and Perspectives Conclusion In this dissertation, the effect of the magnetic field on diffusional phase transformation has been thoroughly investigated in high purity Fe-C alloys theoretically and experimentally. Three high purity Fe-C alloys with different carbon content from both hypo-and hyper-eutectoid range were deliberately prepared. The effect of the magnetic field on microstructures features and crystallographic orientation characteristics of the transformed microstructures have been analyzed. The main achievements and conclusions have been drawn as follows. Magnetic field induces new morphology features of microstructures. (1) Due to the magnetic dipolar interaction, magnetic field induces the elongated and aligned microstructures in hypo-eutectoid alloys. This field effect is carbon-content dependent. The magnetic dipolar interaction works in two scales: atomic-and micro-scale. In the atomic scale, the magnetic dipolar interaction affects the grain growth process and results in the elongation; in the micro-scale, the magnetic dipolar interaction influences the nucleation process and introduces the alignment. (2) Magnetic field promotes the formation of ferrite due to its thermodynamic influence on increasing eutectoid carbon composition. This field effect becomes more pronounced with the increase of the carbon content. Magnetic field inhibits the formation of Widmanstätten ferrite by introducing the additional driving force. (3) Magnetic field promotes the formation of the abnormal structure by increasing the driving force of the transformation from carbon-depleted austenite to ferrite. There is no specific OR between ferrite and cementite in abnormal structure. Magnetic field enhances the spheroidization of pearlite through combination effect of enhanced carbon diffusion resulting from the elevation of the transformation temperature and the increased relative ferrite/cementite interface energy from the magnetization difference between boundary areas and grain interiors. Magnetic field enhances carbon solution in ferrite. Carbon solution in ferrite affects the neighboring Fe magnetic moments, this leads to a decrease in the demagnetization energy which caused by the atomic dipolar magnetic interaction and makes the system more stable under the magnetic field. The field-induced carbon content enhancement offers a new possibility of material strengthening. Magnetic field modifies the crystal orientation distribution of ferrite and affects the orientation relationship of pearlite. (1) Magnetic field favors the nucleation and the growth of the ferrite grains with their distorted <001> direction parallel to the transverse field direction due to the atomic-scaled magnetic dipolar interaction, this leads to the enhancement of the <001> fiber component in the transverse field direction. This field effect is carbon content dependent. For low carbon content alloy (Fe-0.12C alloy), it is greatly reduced due to the reduced carbon oversaturation in ferrite and elevated formation temperature. For Fe-0.36C alloy, a noticeable enhancement of <001> fiber component along the transverse field direction is detected under the 12T magnetic field. At the meantime, this field effect is also strongly related to the field intensity, as the enhancement of <001> fiber component along the transverse field direction becomes more pronounced with the increase of the magnetic field intensity. (2) Magnetic field can hardly change the type of the appearing ORs in pearlite. However, magnetic field promotes the nucleation of high magnetization phasepearlitic ferrite and thus increases the occurrence of the P-P2 OR that corresponds to the situation that pearlitic ferrite nucleates first. This enhancement of the magnetic field on the occurrence of the P-P2 OR is more pronounced in high carbon content alloys. Perspectives Up to date, the application of magnetic field has been popular in many areas of materials science. As the maturity of the magnetic field theory, plenty of magnetic field-induced phenomena have been discovered and well explained. However, better understanding of materials behavior under high magnetic field is still in need. Meanwhile, more physical phenomena are waiting to be revealed. Moreover, magnetic data, such as magnetic moment, Tc temperature, magnetic anisotropy, magnetostriction and so on, are waiting to be completed. New devices thus need to be developed for experimental measurements. 本研究从利用 -clés: Champ magnétique, Transformation de phase, Interaction Dipolaire Magnétique, Texture, Relation d'Orientation -0.12C 合金, Fe-0.36C 合金 和 Fe-1.1C 合金)扩 散型固态相变过程中显微组织形貌以及晶体学特征进行了系统的实验研究和理论解析。 研究发现,由于微观尺度和原子尺度上的磁偶极子相互作用, 强磁场能诱发亚共析钢中沿 磁场方向排列和伸长的显微组织形貌的形成, 并且根据合金含碳量的不同, 显微组织形貌呈现不 同特点。含碳量很低的 Fe-0.12C 合金中, 珠光体团呈沿磁场方向排列和伸长的趋势, 而 Fe-0.36C 合金中, 先共析铁素体呈沿磁场方向排列和伸长的形貌。由于强磁场对相平衡的影响, 使共析点 含碳量升高, 导致强磁场下相变产物中铁素体含量明显增加,并且强磁场对铁素体转变的促进 作用随合金含碳量的增加而增强。此外还发现,强磁场能明显抑制先共析魏氏体的形成。这是 由于强磁场能提高铁素体相变驱动力,从而降低了依靠形成先共析魏氏体借助低能界面来克服 相变阻力的需要。另外,强磁场通过提高铁素体相变驱动力,促进晶界渗碳体周围贫碳奥氏体 的分解, 促进 Fe-1.1C 合金中反常组织的形成。同时,由于强磁场能提高珠光体相变温度, 促进 碳原子扩散明显加快球化过程, 并通过提高铁素体/渗碳体界面能, 增加球化驱动力, 导致强磁场 下 Fe-1.1C 合金中珠光体的球化趋势显著增加。 IS, P-P1 和 P-P2 取向关系。强磁场能促进高磁化率相-铁素体相的形核, 从而增加铁素体相优先形核的几率, 提高与铁素体相优先形核相对应的 P-P2 取向关系的出现几 率。磁场的这一作用在 Fe-1.1C 合金中最为明显。但是, 强磁场对合金中出现的珠光体晶体学 取向关系的类型并没有明显影响。 关键词 关键词 关键词 关键词:强磁场,相变,磁偶极子作用,织构,取向关系 Chapter 1 Literature Review 1. absolute value of the Gibbs free energy difference between the two phases is increased. This leads to the new phase equilibrium and phase stability under the magnetic field by modifying the phase equilibrium temperature and equilibrium composition, as shown in Figure1.2. Figure 1 . 3 A 13 Figure 1.3 A pair of magnetic dipoles in a magnetic field. FD: magnetic field direction. Figure 1 . 4 A 14 Figure 1.4 A Fe-0.52C-0.24Si-0.84Mn-1.76Ni-1.27Cr-0.35Mo-0.13V (mass) alloy austenitized at 1000°C and transformed isothermally to bainite, followed by helium quenching to ambient temperature. (a) 0T and (b) 10T [23]. Figure 1 . 5 15 Figure 1.5 Fe-C phase diagram associated with the α/γ+α, γ/α and γ/Fe 3 C Figure 1 . 6 16 Figure 1.6 Eutectoid carbon content and temperature as a function of the magnetic field induction [49]. Figure 1 . 7 17 Figure 1.7 Microstructures of Fe-0.1C alloy (a) and Fe-0.6C alloy (b) subjected to the α/γ inverse transformation under 8T magnetic field [50]. (The dark spots are γ phase followed by quenching to a new martensite, while the light regions represent the annealed α phaseinitial martensitethe. The arrow indicates the direction of the magnetic field). Figure 1 . 8 18 Figure 1.8 SEM micrographs of Fe-0.1C alloy under 12T magnetic field (a) a ferrite particle nucleated inside an austenite grain, (b) ferrite particles nucleated at the grain boundary of austenite. Figure 1 . 9 19 Figure 1.9 Microstructures of Fe-0.4C alloy during continuous slow cooling from γ to α transformation under 10T magnetic field (The field direction is vertical) [36]. Figure 1 . 10 110 Figure1.[START_REF] Kakeshita | Magnetic field-induced transformation from paramagnetic austenite to ferromagnetic martensite in an Fe-3.9Mn-5.0C (at%) alloy[END_REF] The degree of elongation of ferrite grains as a function of isothermal holding time and temperature in a magnetic field of 10T in Fe-0.4C alloy[START_REF] Hao | Structural elongation and alignment in a Fe-0.4C alloy by isothermal ferrite transformation in high magnetic fields[END_REF]. Figure 1 . 11 111 Figure 1.11 Microstructures after heating at 880°C for 33 min and cooling at 10°C /min with magnetic field of (a) 6T, (b) 10T, (c) 14T and (d) cooling at 46°C /min with 14T magnetic field.(the magnetic field direction and the rolling direction are vertical in the pictures) [56]. (d) preferred crystallographic orientations in the microstructure were formed[START_REF] Ohtsuka | Alignment of ferrite grains during austenite to ferrite transformation in a high magnetic field[END_REF][START_REF] Shimotomai | Aligned of two-phase structures in Fe-C alloys[END_REF][START_REF] Maruta | Magnetic field-induced alignment of steel microstructures[END_REF][START_REF] Maruta | Alignment of two-phase structures in Fe-C alloys by application of magnetic field[END_REF][START_REF] Shimotomai | Formation of aligned two-phase microstructures by applying a magnetic field during the austenite to ferrite transformation in steels[END_REF][START_REF] Zhang | Thermodynamic and kinetic characteristics of the austenite-to-ferrite transformation under high magnetic field in medium carbon steel[END_REF][START_REF] Zhang | New microstructural features occurring during transformation from austenite to ferrite under the kinetic influence of magnetic field in a medium carbon steel[END_REF][START_REF] Zhang | A new approach for rapid annealing of medium carbon steels[END_REF]. Figure 2 . 1 21 Figure 2.1 Photo of the ingot of the Fe-C alloy. 23 .Figure 2 . 2 2322 Figure 2.2 Photo (a) and schema (b) of magnetic field heat treatment furnace. Figure 2 .Figure 2 . 3 223 Figure 2.3 Heat treatment parameters for Fe-C alloys (a) Fe-0.12C (b) Fe-0.36C and (c) Fe-1.1 C. The respective Ae 3 and Ae cm temperatures are calculated by Thermo-Calc, but the Ae 1 temperatures are from the common accepted theoretical value (1000K). For the final heat treatments, small sized specimens were cut out from the center of the plates after the pre-heat treatment. The dimensions of the specimens are 7mm×7mm×1mm for the Fe-0.12C alloy and the Fe-0.36C alloy; and 10mm×10mm×3mm for the Fe-1.1C alloy, as illustrated in Figure2.4. Figure 2 . 4 24 Figure 2.4 Illustration of specimens for heat treatments (a) and the sizes of the specimens (b) for Fe-0.12C alloy and Fe-0.36C alloy; (c) for Fe-1.1C alloy. OLYMPUS/BX61 microscope (The observation surface and the sample reference frame are shown in Figure 2.5). Phase fraction of the transformed microstructures and lamellar spacing of pearlite were analyzed with analySIS and averaged over certain number of analyzed areas. The details of the measurement information are displayed in Table 2.3. Figure 2 . 5 25 Figure 2.5 Schema of the observation area of the specimen. )-with EBSD acquisition camera and Oxford-HKL Channel 5 software was used for microscopic and crystallographic orientation analyses. The working voltage was set at 15KV and the working distance was 15 mm. For microscopic observations, the morphology of pearlitic cementite was examined and the area of spherical pearlite was measured in 20 randomly selected areas. The crystallographic orientation analyses were performed for texture analyses and individual orientation measurements. The orientation data were obtained by acquiring and indexing the electron back-scatter diffraction Kikuchi patterns. The orientations are represented in the form of three Euler angles (φ 1 , Φ, φ 2 ) in Bunge notation. The crystal structure data of ferrite and cementite for EBSD orientation measurements, are given in Table 2.4. Figure 2 . 6 26 Figure 2.6 Photo (a) and schema (b) of the field emission gun SEM-Jeol JSM 6500 F. Figure 2 . 7 )Figure 2 . 8 . 2728 Figure 2.8. It is seen the raw pattern in Figure 2.8 (a) clearly displays the line details and recalculated pattern in (b) matches with the raw pattern, demonstrating that our strategies work well in ensuring the acquisition quality and the orientation determination quality. Figure 2 . 7 27 Figure 2.7 Geometry relation between the electron beam and the specimen surface under EBSD measurement condition. (The electron beam is elongated along the Y direction in the specimen coordinate system). Figure 2 . 8 28 Figure 2.8 Kikuchi pattern of cementite (a) and the one superposed with the recalculated pattern (b). vectors of the cementite reciprocal lattice basis which can be calculated by the following equations: Figure 2.10). For comparison, the same calculations were performed on the same supercell but without the carbon atom. Figure 2 . 10 210 Figure 2.10 Supercell containing 54 Fe atoms and 1carbon atom. Figure 3 . 3 Figure 3.1 shows the microstructures of the Fe-0.12C alloy treated without and with the 12T magnetic field. It is seen that the transformed microstructures are both composed of proeutectoid ferrite (white) and pearlite (dark). Without the field, most ferrite grains and pearlite colonies are equiaxed. Though some others have elongated shapes, their major axes are randomly oriented. With the application of the 12T magnetic field, the pearlite colonies are obviously elongated along the field direction and moreover, many proeutectoid ferrite grains are aligned in chains along the field direction. Some of the ferrite grains are also elongated but with two elongation orientations: one is along the field direction and the other is perpendicular to the field direction. It can be seen that most of the elongated pearlite colonies are located between the ferrite chains. Figure 3 . 1 31 Figure 3.1 Microstructures of Fe-0.12C alloy without (a) and with (b) a 12T magnetic field. The arrow indicates the magnetic field direction. Figure 3 . 2 . 32 Figure 3.2. There are two kinds of proeutectoid ferrite with different morphologies: one is equiaxed, the other is acicular, known as Widmänstatten ferrite. Without the field, most of the proeutectoid ferrite is of Widmänstatten type, stretching through the thickness of the specimen. The proeutectoid ferrite and the pearlite colonies are basically randomly distributed, whereas with the field, the proeutectoid ferrite grains are obviously elongated along the field direction. Figure 3 . 2 32 Figure 3.2 Microstructures of Fe-0.36C alloy without (a) and with (b) a 12T magnetic field. The arrow indicates the magnetic field direction. Figure 3 . 3 33 Figure 3.3 Schematic of the magnetic moments under the magnetic field. Figure 3 . 4 34 Figure 3.4 Illustration of the dipolar interaction between magnetic dipoles. Figure 3 . 5 35 Figure 3.5 Ferrite area fraction of Fe-0.12C alloy and Fe-0.36C alloy treated without and with the 12T magnetic field. Figure 3 . 6 36 Figure 3.6 Effects of magnetic field on Fe-C phase equilibrium [49]. Figure 3 . 7 Table 3 . 2 Figure 3 . 8 373238 Figure 3.7 Optical micrographs and the corresponding pole figures of ferrite of Fe-1.1C alloy treated (a) without, (b) with a 12T magnetic field (the abnormal structure is indicated by the arrow; the field direction is horizontal) and (c) the corresponding sample coordinate system. Figure 3 . 8 38 Figure 3.8 SEM secondary electron micrographs of transformed microstructure in Fe-1.1C alloys treated (a) without and (b) with a 12T magnetic field, the field direction is horizontal. Figure 4 . 4 Figure 4.1 shows the microstructures of the specimens treated without and with the 12T magnetic field. The microstructures of both specimens are composed of proeutectoid ferrite (white) and pearlite (dark). Figure 4 . 1 41 Figure 4.1 Microstructures of Fe-0.36C alloy treated without (a) and with (b) a 12T magnetic field. The arrow indicates the magnetic field direction. Figure 4 . 2 42 Figure 4.2 Octahedral interstice and its first and second neighboring Fe atoms in bcc structure. Figure 4 . 4 Figure 4.2 illustrates an atom cluster that contains one octahedral interstice and its first and second neighboring Fe atoms in the bcc structure. When a carbon atom is present, it occupies the octahedral interstice. Suppose that the magnetic moments of the Fe atoms and the carbon atom are parallel to the field direction under the Figure 5 . 1 Figure 5 . 2 5152 Figure 5.1 Inverse pole figures of ferrite in Fe-0.12C alloy treated without (a) and with (b) a 12T magnetic field. Figure 5 . 3 53 Figure 5.3 Inverse pole figures of the ferrite in Fe-0.36C alloy (a) 0T, (b) 8T and (C) 12T the carbon atom displaces the two Fe atoms in the opposite direction. Table 2 . 1 21 Chemical composition of cast iron, %. C S O N Fe 4.39 0.001 0.0025 0.0003 Bal. Table 2 . 3 23 Information of microstructural measurements. Fe-C alloys Measurements No. or area of measurements Fe-0.12C Area precentage of proeutectoid ferrite 15 areas Fe-0.36C Area precentage of proeutectoid ferrite 15 areas Area precentage of Widmanstätten ferrite whole cross section(7mm×1mm) Fe-1.1C Area percentage of abnormal structure whole cross section(10mm×3mm) Table 2 . 4 24 Crystal structure data of ferrite and cementite. Phase Crystal structure/ Lattice parameters Space group (No.) No. Atom Wyckoff Atomic position X Y Z Occupancy Ferrite bcc a=0.28665 nm. Im 3 m (229) 1 Fe 2a 0 0 0 1 Cementite orthorhombic 1 C 4c 0.881 0.25 0.431 1 a=0.5090 nm b=0.6748 nm Panm (62) 2 Fe 4c 0.044 0.25 0.837 1 c=0.4523 nm 3 Fe 8d 0.181 0.063 0.837 1 Table 3 . 1 31 Area fraction of Widmänstatten ferrite in Fe-0.36C alloy treated without and with magnetic field. Magnetic field Area fraction of Widmänstatten ferrite, % 0T 48.44 12T 24.36 Table 3 . 3 33 Area percentage of the spherical pearlite in Fe-1.1C alloy treated without and with a 12T magnetic field. Magnetic field Area percentage of the spherical pearlite, % 0T 12.4 12T 27.1 Table 4 . 4 1 and Table 4.2, respectively. It is seen that the magnetic field increases the amount of proeutectoid ferrite by 14% and the carbon content in proeutectoid ferrite by 35.7%. Table 4 . 1 41 Area percentages of proeutectoid ferrite in Fe-0.36C alloy treated without and with the 12T magnetic field (Number of measurements: 10). Magnetic field Area percentage of proeutectoid ferrite, (Standard deviation) Increment 0T 57 (±1.1)% -- 12T 65 (±1.2)% 14% Table 4.2 Carbon contents of proeutectoid ferrite in Fe-0.36C alloy treated without and with the 12T magnetic field (Number of measurements: 20). Magnetic field Carbon content of proeutectoid ferrite, (Standard deviation) Increment 0T 0.14(±0.017)% -- 12T 0.19(±0.024) % 35.7% Table 4 . 3 43 Mean hardness of proeutectoid ferrite in Fe-0.36C alloys treated without and with the 12T magnetic field (Number of measurements: 8). Magnetic field Mean hardness of proeutectoid ferrite, HV 0.025 (Standard deviation) Increment 0T 88.1 (±2.71) -- 12T 102.1 (±6.86) 15.9% Considering that the magnetic natures of Fe and carbon atoms are not the same, introducing a carbon atom into bcc Fe structure may modify the magnetic interaction between local atoms. The magnetic moments of Fe and carbon atoms calculated from ab-initio simulations are summarized in Table 4.4 and 4.5. Table 4 . 4 44 Calculated magnetic moments carried by Fe atoms.The ab-initio calculations were carried out using a supercell consisting of 54 Fe atoms. Magnetic moments, (µ Β /atom) Lattice constant, (Å) 2.151 2.81199 Table 4 . 5 45 Calculated magnetic moments carried by Fe and carbon atoms.The ab-initio calculations were carried out using a supercell consisting of 54 Fe atoms with one carbon atom in the octahedral interstice of the centre bcc Bravais cell.It is seen that without the carbon atom in the bcc Fe structure, the magnetic moments of Fe atoms are identical. When a carbon atom is introduced, the magnetic moments of Fe atoms vary with the neighborhood order to the carbon atom. The first (closest) neighbors of the Fe atoms to the carbon atom possess a smaller moment (1.615µ Β /atom) with respect to that (2.151 µ Β /atom) of the Fe atoms in the carbon free cell, whereas the second neighbors carry slightly larger moment(2.178 Magnetic moments, (µ Β /atom) Lattice constant, (Å) First neighbor 1.615 Fe Second neighbor 2.178 Others 2.151 2.84018 Carbon -0.119 Table 5 . 1 51 Orientation relationships in pearlite.Although no change in the type of the appearing ORs is obtained, the corresponding occurrence frequency of each OR is evidently varied by the carbon content and the application of magnetic field, as illustrated in Table5.2. IS OR P-P1 OR P-P2 OR (103) / /(101) C F (103) / /(110) C F (103) / /(011) C F [010] / /[111] C F [010] / /[1 13] C F [311] / /[1 11] C F Habit plane Habit plane Habit plane (101) / /(1 12) C F (001) / /(125) C F ( ) 001 C . 3 5 ° from ( 1 ) 25 F Table 5 . 2 52 Occurrence frequency of the appearing ORs in different Fe-C alloys. Carbon content Magnetic field IS OR P-P1 OR P-P2 OR Fe-0.12C 0T 43.8% 37.5% 18.8% 12T 40.0% 40.0% 20.0% 0T 46.7% 46.7% 6.7% Fe-0.36C 12T 46.7% 40.0% 13.3% 0T 33.3% 60.0% 6.7% Fe-1.1C 12T 26.7% 53.3% 20.0% Acknowledgements This work is financially supported by the National Natural Science Foundation of China (Grant No. 50971034, 50901015 and 50911130365), the Program for Changjiang Scholars and Innovative Research Team in University (Grant No. IRT0713), the "111" Project (Grant No. B07015), the CNRS of France (PICS No. 4164) and the joint Chinese-French project OPTIMAG (N°ANR-09-BLAN-0382). I would like to give my sincere thanks to these institutions. I also gratefully acknowledge the CHINA SCHOLARSHIP COUNCIL for providing the scholarship to support my PhD study in France. This work is completed at LEM3 (former LETAM, University of Lorraine, France) and the Key Laboratory for Anisotropy and Texture of Materials (Northeastern University, China). I had the honor to work with numerous colleagues in two labs and I would like to give my hearted thanks for their kind help. I would like to thank the reviewers, Prof. Y. Fautrelle and Prof. Z. M. Ren for taking time out of their busy schedules to review my dissertation and provide constructive suggestions and comments. I would like to give my special thanks to my supervisors, Prof. Claude Esling, Prof. Yudong Zhang at University of Lorraine, Prof. Liang Zuo and Prof. Xiang Zhao at Northeastern University, not only for their support, ideas, guidance, and organizational help during the last three years, but also making me a better person and scientist by setting high standards and good examples. I would like to thank all my group members who treated me with dignity and respect. Last but not least, I want to thank my parents, friends and family, especially my husband Mr. Lijun Zhang for his constant support, understanding and encouragement. growth of the pearlite is restricted between the ferrite chains, the elongated pearlite colonies along the field direction are finally obtained. As for the Fe-0.36C alloy, the proeutectoid ferrite transformation temperature is much lower than that of the Fe-0.12C alloy due to the increased carbon content. Though the ferrite transformation temperature is still above the T C , the difference between them is reduced. Thus, the two-scaled magnetic dipolar interaction functions and affects the nucleation and growth of the proeutectoid ferrite, as most proeutectoid ferrite is transformed around and below T C . However, as the carbon content is high, the relative amount of proeutectoid ferrite is low. The interspacing between ferrite grains is large. As the magnetic dipolar interaction energy E D is proportional to r -3 (seen from Eq. 3.1), thus the micro-scaled dipolar interaction is reduced. As a result, transformed ferrite is mainly elongated in the field direction. No obvious field direction alignment is obtained compared with the case of Fe-0.12C alloy. Summary Magnetic field induces the elongated and aligned microstructures due to the effect of the magnetic dipolar interaction. During the austenitic decomposition process, the magnetic dipolar interaction works in two scales: atomic-scale and micro-scale. In atomic-scale, the grain growth process is influenced and the elongation of the nucleated grains in the field direction is resulted. In the microscale, the nucleation of the ferrite along the field direction is favored and the alignment is induced. For the low carbon content Fe-0.12C alloy, the elongated pearlite colonies along the field direction are induced by the chained elongated ferrite grains in the field direction due to the effects of the atomic-and micro-scaled magnetic dipolar interaction on the nucleation and the growth of the ferrite around and below the T C . Summary Magnetic field promotes the formation of the abnormal structure through its influence on increasing the driving force of the carbon-depleted austenite to ferrite transformation. No specific OR between the cementite and the ferrite in the abnormal structure is found in the present study. Magnetic field enhances the spheroidization of pearlite through enhanced carbon diffusion resulting from the elevation of the transformation temperature and the increased relative ferrite/cementite interface energy from the magnetization difference between boundary areas and grain interiors. The field induced texture has been noticed and studied during magnetic annealing process in Fe based alloys. Martikainen et al. [START_REF] Martikainen | Observations on the effect of magnetic field on the recrystallization in ferrite[END_REF] reported the increase of the <001> texture component along the field direction and attributed this texture formation to the anisotropic magnetization in different crystallographic directions, as <100> direction is the easiest magnetization direction. Later, Zhang et al. [START_REF] Zhang | Grain boundary characteristics and texture formation in a medium carbon steel during its austenitic decomposition in a high magnetic field[END_REF] studied the effect of the magnetic field on texture of the ferrite during austenitic decomposition process and found the field-induced enhancement of <001> component along the transverse field direction in a medium carbon steel (0.49C). However, the similar texture component was not obtained in a 42CrMo steel [START_REF] Zhang | New microstructural features occurring during transformation from austenite to ferrite under the kinetic influence of magnetic field in a medium carbon steel[END_REF]. This reveals the magnetic field influential mechanism on texture formation works differently and this field effect remains to be addressed. In this chapter, the field-induced texture of ferrite was studied in two hypoeutectoid alloys (Fe-0.12C alloy and Fe-0.36C alloy), the magnetic field influential In addition, it is clear that the intensity of the <001> fiber component along the traverse field direction is directly related to the field intensity. For the present Fe-0.36C alloy, the visible enhancement of the <001> fiber texture requires high field intensity, i.e., 12T. Summary Due to the atomic-scaled magnetic dipolar interaction, magnetic field favors the nucleation and the growth of the ferrite grains with their distorted <001> direction parallel to the transverse field direction, and thus induces the enhancement of the <001> fiber component in the transverse field direction. This field effect is carbon content dependent. For low carbon content alloy (Fe-0.12C alloy), it is greatly reduced due to the reduced carbon oversaturation in ferrite and elevated formation temperature. At the meantime, this field effect is also strongly related to the field intensity, as the enhancement of <001> fiber component along the transverse field direction becomes more pronounced with the increase of the magnetic field intensity. For Fe-0.36C alloy, a noticeable enhancement of <001> fiber component along the transverse field direction is detected under the 12T magnetic field. with lower magnetization. The formation of ferrite is thus promoted by the magnetic field. Therefore, some low carbon concentration areas in austenite would transform to ferrite with the help of magnetic field before the formation of cementite. Consequently, the occurrence of the P-P2 OR, which corresponds to pearlitic ferrite nucleates first is increased by the magnetic field. Summary To minimum the transformation barriers, pearlitic ferrite and cementite follows specific ORs which guarantees the small transformation stains and the low atomic misfits, during the pearlite formation and growth process. Though magnetic field is able to offer additional driving force for this transformation and considerably raise the transformation temperature, it hardly overcomes the transformation strain energy barrier and the interfacial energy barrier to offer new ORs between the ferrite and the cementite in pearlite. As a result, the magnetic field has little influence on the type of the ORs appearing. As magnetic field favors the nucleation of the high magnetization phasepearlitic ferrite, the occurrence of the P-P2 OR that corresponds to the situation that pearlitic ferrite nucleates first, is increased by the magnetic field. This enhancement of the magnetic field on the occurrence of the P-P2 OR is more pronounced in high carbon content alloys, i.e. Fe-1.1C alloy.

01749243

en

2024/03/05 22:32:07

2012

https://hal.univ-lorraine.fr/tel-01749243/file/DDOC_T_2012_0173_CHENTOUF.pdf

Examinateur Rapporteur Rapporteur Les alliages intermétalliques riches en fer du système fer-aluminium, Fe 3 Al, ont des caractéristiques très intéressantes pour des applications mécaniques à haute température. Ils possèdent, comme la plupart des composés intermétalliques, une résistance mécanique élevée, une bonne résistance à l'oxydation ainsi qu'une faible densité. Cependant, les principales raisons qui limitent leurs applications sont leur fragilité à température ambiante et une forte diminution de leur résistance pour des températures supérieures à 550°C. Un aspect intéressant de ces alliages est leur comportement envers les métaux de transition. Certains éléments, comme Ti, peuvent augmenter la stabilité de la phase D0 3 , en augmentant la transition D0 3 /B2 vers des températures plus élevées. La situation est moins claire dans le cas du Zr. En effet, malgré l'effet bénéfique du dopage en Zr sur la cohésion des joints de grains et la ductilité, il n'existe pas de données expérimentales concernant son effet sur la stabilité de la structure D0 3 du composé Fe 3 Al. Ce travail de thèse vise à étudier l'effet de ces deux métaux de transitions Ti et Zr sur les propriétés du composé intermétallique D0 3 -Fe 3 Al en utilisant des calculs pseudopotentiels ab initio basées sur la théorie de la fonctionnelle de la densité (DFT). Deux principaux thèmes ont été abordées: (i) la compréhension du rôle de ces deux métaux de transition en termes de stabilité de la phase D0 3 à la lumière de leur site préférentiel dans la structure D0 3 -Fe 3 Al (ii) le comportement du Ti et Zr dans le joint de grains Σ5 (310) [001] ainsi que leur effet sur la stabilité structurale de cette interface. Un élément important pour étudier ces aspects est de prendre en compte l'effet de la température. Cela nécessite un traitement de type dynamique moléculaire des atomes dans la supercellule. La technique dynamique moléculaire ab initio (AIMD) résout ces problèmes en combinant des calculs de structure électronique avec la dynamique à une température finie. Ainsi, notre étude a été menée à la fois en utilisant des calculs ab initio statiques à 0K ainsi que par la prise en compte de l'effet de la température jusqu'à 1100K (Dynamique Moléculaire Ab Initio). the light of their site preference in the D0 3 -Fe 3 Al structure (ii) the behaviour of Ti and Zr transition metals in the ∑5 (310) [001] grain boundary and their effect on the structural stability of this interface. An important issue when studying these aspects is to take into accounts the effect of temperature. This requires a molecular dynamics treatment of the atoms in the supercell. The technique known as ab initio molecular dynamics (AIMD) solves these problems by combining 'on the fly' electronic structure calculations with finite temperature dynamics. Thus, our study was conducted both using the conventional static ab initio calculations (0K) as well as by taking into account the effect of temperature (Ab Initio Molecular Dynamics). To my parents Acknowledgment I would like to express my gratitude to the many people who have supported me as I completed my graduate studies and dissertation. First, I would like to sincerely thank my advisor, Pr. Thierry GROSDIDIER, for providing continuous support and instructive guidance throughout my doctoral studies. I feel lucky to have an adviser like him with great patience and understanding to the difficult conditions that I had during this thesis. His effort, encouragement and advices are greatly appreciated. I gratefully acknowledge Pr. Hafid AOURAG who introduced me to research and I will always be highly grateful to him for his commitment to my academic success. I also would like to thank my co-director Dr. Jean-Marc RAULOT for the many interesting discussions we had, and for his ability to inspire thoughts and suggestions on a large variety of topics, ranging from fundamentals of theoretical condensed matter physics to applications and computational issues. Introduction Fe 3 Al-based intermetallic compounds are promising materials for structural applications at high temperature. Their advantageous properties originate from their low density and their high corrosion resistance in oxidizing and sulfidizing environments. At ambient and intermediate temperatures, Fe 3 Al shows higher strength than other single-phase iron alloys due to its ordered D0 3 superlattice structure. However, at about 550 °C, disordering of the D0 3 structure as well as a sharp drop in the flow stresses occur in the binary stoichiometric Fe 3 Al compound, causing detrimental effects to this material with regard to structural applications. An interesting aspect of these alloys is their behavior towards transition metal impurities. Some elements like Ti increase the stability of the D0 3 by increasing the D0 3 /B2 transition towards higher temperature. The situation is less clear for Zr addition, indeed, despite the beneficial effect of small Zr addition on the grain boundary cohesion and ductility; there is no experimental data available concerning its effects on the stability of the D0 3 -Fe 3 Al compounds. In this thesis the effect of the Ti and Zr transition metals on the D0 3 -Fe 3 Al intermetallic compounds has been investigated by means of ab initio PseudoPotentials numerical simulations based on Density Functional Theory. Two main issues will be addressed (i) the understanding of the role of the these two transition metals in terms of stability of the bulk at the light of their site preference in the D0 3 -Fe 3 Al structure (ii) knowing that the ductility of iron aluminides is affected by grain boundary brittleness and that experimental information on segregation at grain boundary (G.B.) is hardly available because of the resolution of the measuring tools, the behaviour of Ti and Zr transition metals in the ∑5 (310) [001] grain boundary will then be studied to point out their effect on the structural stability of this interface. The particular ∑5 (310) [001] grain boundary has been selected because of its short period and the presence of a single type of (310) planes in the D0 3 structures. These two factors make it much easier a numerical simulation by keeping the calculation time within reasonable limit. It is also reasonable to consider however that, due to its high degree of coincidence, this specific grain boundary can be representative of a wide range of boundaries in Fe 3 Al alloys. Thus, the behavior depicted here for Ti and Zr atoms with respect to their neighbor atoms should not change drastically with the nature of the grain boundary. In principle, and this is the approach emphasized in this thesis, a theory attempting to realistically describe of the structural properties (stability of the D0 3 structure and grain boundary relaxation) needs to take into account the effect of temperature. This requires a molecular dynamics treatment of the atoms in the supercell. The technique known as ab initio molecular dynamics (AIMD) solves these problems by combining 'on the fly' electronic structure calculations with finite temperature dynamics. Thus, our study was conducted both using the conventional static ab initio calculations (0K) as well as by taken into account the effect of temperature (Ab Initio Molecular Dynamics). The most important element in an AIMD calculation is the representation of the electronic structure. The calculation of the exact ground-state electronic wave-function is intractable, and approximations must be used. In the present study, we choose to use pseudopotentials methods implemented in the Vienna Ab Initio Simulation Package (VASP). It is one of the most powerful ab initio DFT pseudopotential-based packages available at present. It has been already applied to a wide range of problems and materials, including bulk systems, surfaces and interfaces. This thesis is organized as follows. In Chapter I we provide a literature review concerning the effect of alloying elements on the properties of FeAl based intermetallics compounds. In Chapter II an outline of the theoretical background that serves as a foundation of our calculations is given. The implementation of Ab Initio Molecular Dynamics within the framework of plane wave pseudopotential density functional theory is given in detail. A short introduction to density functional and pseudopotential theory will be given in the second part of this chapter. Chapter III and IV are devoted to the application of these methods and the use of tools described in the previous section. In Chapter III, the results of the static ab initio calculations of the substitutions of the Ti and Zr transition metals in the bulk as well as the ∑5 grain boundary are presented. Chapter IV gives the results of Ab Initio Molecular Dynamic calculations that was set up to investigate the effect of temperature on the structural stabilities of the two transition metals impurities in the bulk and ∑5 (310) [001] grain boundary of the D0 3 -Fe 3 Al compounds. Finally, we conclude the document and summarize the basic insights Chapter I Background for the iron aluminides based intermetallics analysis I. Intermetallic compounds Intermetallic compounds can be simply defined as ordered alloy phases formed between two or more metallic elements. These materials have different crystal structures from those of their based metallic constituents metallic components and exhibits long-range ordered superlattices. In comparison with conventional metallic materials, intermetallic compounds have the advantages of high melting point and high specific strength, which make them promising high temperature structural materials for automotive, aircraft, and aerospace applications. The ordered nature of intermetallic compounds generates high temperature properties due to the presence of long-range-ordered superlattice, which reduce dislocation mobility and diffusion processes at elevated temperatures [1,2]. Aluminides based intermetallics are low density materials which are distinctly different from conventional solid-solution alloys. For example, In this chapter, we give survey of the literature concerning the effect of the alloying elements on the properties of FeAl based intermetallics compounds. After a brief introduction to the properties of the intermetallics and the more especially the intermetallic compounds based on the FeAl system, in Section I and II, the different strengthening mechanisms of the alloying elements in the bulk are presented in Section III. In Section IV the effect of additions at the grain boundaries are also discussed with a description of the different tools for the characterization of the grain boundaries. Ni 3 Al exhibits an increase in yield strength with increasing temperature, whereas conventional alloys exhibit a general decrease in strength with temperature [3,4]. Nickel and iron aluminides also possess sufficiently high concentration of aluminium, thus formation of a continuous and adherent alumina scale on the external surface of the material could always be achieved. In contrast, most of the alloys and capable of operating above 700 °C in oxygen-containing environments contain less than 2 wt. % aluminium, and invariably contain high concentration of chromium for oxidation protection with chromia. Nickel and iron aluminides therefore could provide excellent oxidation resistance at temperatures ranging from 1100 to 1400 °C owing to their high aluminium contents and high melting points [4]. II. Iron aluminides Among the big family of intermetallic compounds, the Fe-Al, Ni-Al and Ti-Al systems are attracting most of the attention. The FeAl system is attractive because of specific features. Due to their excellent oxidation resistance _first noted in the 1930s_ iron aluminides have been the subjected to extensive studies with respect to structural and functional applications [6]. In addition to their superior oxidation and sulfidation resistance, iron aluminides also offer the advantages of low material cost, Consisting of non-strategic elements, their density is also lower in comparison with stainless steels. Therefore they have long been considered for applications in the automotive and petrochemical industries as well as conventional power plants and coal conversion plants, fot components such as, shofts, pipes as well as coatings for heat exchangers and molten soft applications [7]. However, their poor ductility at room temperature and significant drop in strength above 600 °C together with inadequate high temperature creep resistance has limited their potential for structural applications. The phase diagram of the binary Fe-Al system, according to Kubaschewski [8], is shown in Fig. I-1. The solid solubility of Al in f.c.c. γ-Fe is limited to 1.3 at.% at 1180 °C. In contrast, in the disordered b.c.c. α-Fe (A2) up to 45 at.% Al can be dissolved at high temperature (1310 °C). Between 0 and 54 at.% Al two ordered compounds exist. The D0 3 -ordered Fe 3 Al is stable at compositions around 27 at.% Al and from room temperature to 550°C (830K). Above 550°C the ordered Fe 3 Al with D0 3 structure transforms to an imperfectly ordered B2 (α 2 ') structure, which ultimately changes to a disordered solid solution, A2 (α). On the other hand, FeAl exists with B2 structure and is stable from about 36-48 at% Al, and the transition from B2 (α 2 (I)) to A2 occurs well above 1100°C. In contrast to the newer diagrams by Massalski [9] the ordered α 2 phase field has been subdivided into three separate modifications, α 2 ' and α 2 (I) region at lower and α 2 (h) one at higher temperatures. The subdivision is a result of measurements by Köster and Gödecke [10] who recorded energy evolution as well as expansion coefficients and elastic moduli as function of composition and temperature. In the present work, we are studying the Fe3Al stochiometric alloy with the D03 structure. According to the phase diagram in Fig. I-1, it is expected that, at this composition, the structure encounters the change at about 550°C (850K). Following by the change at 790°C (1063K) Figure.I-1 Fe-rich part of the Fe-Al system according to Kubaschewski [8]. In addition to the phase boundaries for γ (disordered A1), α (disordered A2), Fe 3 Al (ordered D0 3 ) and FeAl (α 2 ; ordered B2) additional lines are shown for the Curie temperature (T c ), for different variants of α 2 and the area in which the so-called 'k-state' is observed. During the last decade, efforts have been made to enhance room-temperature ductility, hightemperature strength, and high-temperature creep resistance by alloying of iron aluminides. Two approaches, namely, solid-solution strengthening and precipitation strengthening, were considered for strengthening of iron aluminides. Elements such as Nb, Cu, Ta, Zr, B and C were considered for precipitation strengthening; while Cr, Ti, Mn, Si, Mo, V and Ni were added into iron aluminides for solid solution strengthening. In general, the addition of elements either for precipitation strengthening or solid solution strengthening to improve high temperature tensile strength and creep resistance resulted in low room temperature tensile elongations [11]. It is well established that the fracture of iron-based intermetallics is often brittle in nature occurring by cleavage and/or intergranular (i.e occurring along grain boundaries) fracture. In this context, it is important to determine the effect of ternary element additions both on (i) bulk strengthening as well as (ii) grain boundary "softening". III. Bulk strengthening III.1 Strengthening by solid-solution hardening The Fe-Al-Cr system is an example where, within a large area of compositions only, solidsolution hardening is possible. In Fig. I-2 the isothermal section at 1000 °C is shown [12]. At this temperature the phase boundary between α-(Fe,Al) and FeAl has not been determined and the term α-(Fe,Al)/FeAl is used here and also otherwise in this chapter when no distinction between the two phases is made. Complete solid solubility between α-(Fe,Al)/FeAl and α-Cr exists and according to the 1000 °C isotherm solid-solution hardening is the only strengthening mechanism available from the phase diagram for compositions up to 50 at.% Al. Only at higher Al contents the possibility of precipitating a second phase, e.g. hexagonal Al 8 Cr 5 , exists. There are a number of Fe-Al-X systems, e.g. with X=Si, V, Mn, Co, Ni, Cu, Zn, where an extended solid solubility for X in α-(Fe,Al)/FeAl exists. Figure.I-2 Isothermal section of the Fe-Al-Cr system at 1000 °C [12]. The phase boundary between α-(Fe,Al) and FeAl has not been determined and is therefore given by a broken line. It has been reported that the addition of Cr, which is beneficial for increasing the room temperature ductility [13,14], does not show any effect on yield stress at 600 °C for alloys with 25 at.% Al [15]. This is not corroborated by the results of Stein et al. [16] who investigated the effect of solid-solution hardening for various alloying additions for alloys containing 26 at.% Al (Fig. I-3). At 600 °C an increase of the yield stress is observed by adding 2 at.% of Ti, V, Cr, or Mo. If higher amounts are added a further increase of the yield stress is only observed for Mo while for V and Cr even a decrease of the yield stress is reported [16]. At 700 °C the yield stress increases continuously with the amount of alloying addition for Ti, V and Mo while for Cr only an addition of 2 at.% increases the yield stress. At 800 °C again a continuous increase of the yield stress with the amount of alloying addition is observed for Ti, V and Mo within the investigated composition range while the addition of Cr has no marked effect on the yield stress at this temperature (Fig. I-3). Figure.I-3 0.2%-yield stress (in compression; 10 -4 s -1 deformation rate) at 600, 700 and 800 °C for Fe-26Al with additions of 2 and 4 at.% X (X = Cr, V, Mo, Ti) [16] and Fe-28Al-5Mo [17]. Black symbols denote B2-type ordering while grey symbols denote D0 3 /L2 1 -type ordering at the respective temperature. Several experimental and theoretical studies have focused on the structural and magnetic properties of Fe-Cr-Al with the D0 3 -type structure. X-ray, neutron, magnetization and Mossbauer effect [18] studies which have been carried out on Fe 3-x Cr x Al alloys with x < 0.6 showed that chromium atoms occupy preferentially FeI-sites and enter also Al-positions. Their magnetic moments are small, if any, and they diminish the value of the neighbouring iron atoms by roughly 0.1 u B per chromium atom. Ready et al. [20] have found that Cr couple antiferromagneticaly with the Fe atoms and occupy the FeI site. More recently the self-consistent TB-LMTO calculations [19] confirm the result of Ready and al. indicating that a strong exists preference of the FeI-site occupation by chromium in Fe 3 Al. The TB-LMTO calculations [19] show also the effect of the surroundings on the magnetic moment. They confirm negative magnetic moment of chromium found experimentally in [18]. III.2. Strengthening by incoherent precipitates III.2.1. Precipitation of intermetallic phases In many Fe-Al-X systems the solid solubility for the third element within the Fe-Al phases is limited and the possibility of strengthening Fe-Al-based alloys by precipitation of another intermetallic compound exists. In several systems this intermetallic phase is a Laves phase, e.g. in the Fe-Al-X systems with X=Ti, Zr, Nb and Ta. In order to study the effect of precipitates on strengthening, the Fe-Al-Zr system may be considered as a prototype system as only limited solid solubility for Zr in the Fe-Al phases has been found, which is independent from temperature, at least between 800 and 1150 °C [21]. Fig. I-4 presents the partial isothermal section of the Fe-corner at 1000 °C, which is shown by means of example, for the phase equilibria in the temperature range 800-1150 °C. The isothermal section reveals that for α-(Fe,Al)-based alloys strengthening by a Laves phase is possible in this temperature range while FeAl-based alloys may be strengthened by precipitates of the tetragonal phase (Fe,Al) 12 Zr (τ 1 ). As the solubility for Zr in α-(Fe,Al)/FeAl does not increase with temperature, no possibility exists for generating fine and evenly distributed precipitates from a solid solution. Figure.I-4 Partial isothermal section of the Fe-Al-Zr system at 1000 °C [21]. The two ternary compounds, which are in equilibrium with the Fe-Al phases, are either a Laves phase (λ, with λ 0 denoting the cubic C15 structure and λ 1 denoting the hexagonal C14 structure) or the tetragonal ThMn 12 -type phase (Fe,Al) 12 Zr (τ 1 ). A set of experiments with the laves phases in the Fe-Al system have been done. However, the dependence of their mechanical properties on the chemical composition is not yet understood. Thus a deeper understanding of the structure stability and mechanical properties of the laves phases is essential to control the material properties of the iron aluminides. Recently, experimental investigations of the pure lave phases Fe 2 Nb [22] and (FeAl) 2 Nb [23] have been performed and the phase equilibria in the respective ternary systems has been studied. In addition to experiments, the structural properties of the Fe 2 Nb laves phase (C14 Hexagonal structure) has been investigated by quantum-mechanical ab initio calculations [24]. Fig. I-5 shows the results of the calculated single crystalline elastic constants tensor C ij , which gives direct insight into the directional dependence of the Young modulus. It indicats a rather strong elastic anisotropy, i.e. deviations from an ideal sphere. The derived Young's modulus at T=0K by using selfconsistent crystal homogenization method is about 250 ± 22 GPa. The authors have determined also the site preference of Al between the Fe sublattices of the C14 structure using a combined experimental and theoretical approach. Figure.I-5 Quantum-mechanically calculated directional dependence of single-crystalline Young's modulus of Fe 2 Nb with the hexagonal C14 structure. Shape deviations from an ideal sphere identify elastic anisotropy of the studied Laves phase compound. In the ab initio investigation, the site preference has been investigated at both low and elevated temperatures, making use of CALPHAD-like statistical sublattice models to determine the configurational entropy. The resulting free energies are shown for (Fe 0.75 Al 0.25 ) 2 Nb in Fig. I-6 (considering a double-layer antiferromagnetic structure). The x axis at the bottom/top indicates the fraction of 2a/6h sites occupied by Al. The authors show that the 2a site has the lowest solution free enthalpy already at T = 0 K (see Fig. I-6). With increasing temperature, the larger number of sites and thus configurations in the 6h sublattice and the corresponding gain in configurational entropy make the occupation of this sublattice more and more attractive. However, even for temperatures up to 1500 K the minimum of the free enthalpy curve is above x = 0.25 (the value corresponding to the statistical distribution), yielding a net-preference to the 2a sites. Figure.I-6 The ab initio calculated solution free enthalpies for (Fe 0.75 Al 0.25 ) 2 Nb at different temperatures. The black dots indicate results of ab intitio calculations, solid lines combine these results with a sublattice regular solution model [24]. III.2.2. Precipitation of carbides Besides hardening by precipitation of intermetallic phases, carbides could also act as strengthening phases. Fig. I-7 shows two partial isothermal sections at 800 and 1000 °C of the Fe corner of the Fe-Al-C system [25]. At both temperatures α-(Fe,Al) and FeAl are in equilibrium with the cubic K-phase Fe 3 AlC. The solid solubility for carbon in α-(Fe,Al)/FeAl changes only slightly between 800 and 1200 °C but is considerably lower at lower temperatures, e.g. drops from about 1 at.% C at 1000 °C to about 50 ppm at 320 °C [26]. This leads to the precipitation of fine needle shaped precipitates of the k phase at the grain boundaries during cooling. As the carbon diffusivity is even high at ambient temperatures, these precipitates at the grain boundaries are found at room temperature in all alloys of appropriate compositions even after quenching and they do strongly affect mechanical properties at low temperatures [26]. The effect of k phase precipitates on the mechanical behaviour of Fe-Al-based alloys with Al contents between 25 and 30 at.% has been studied in detail by Schneider et al. [27]. Figure.I-7 Partial isothermal sections of the Fe-Al-C system at (a) 800 (b) and 1000 °C [15]. The exact course of the α-(Fe,Al)/FeAl phase boundary has not been determined within the ternary system and therefore only its position in the binary Fe-Al system is indicated by a bar on the Fe-Al axis. To control the precipitation end microstructures for the carburizing process of the Fe-Al alloys, it is necessary to rely on the thermodynamical properties of the iron riche phases Fe-Al-C system, and to know the fundamental properties of these phases. Several experimental and theoretical informations are present in the literature about the k carbide. The k phase is associated to the The stoichiometric Fe 3 AlC has, in fact, never been observed. Experimentally, the stoichiometry proposed for k is Fe 4-y Al y C x where 0.8<y<1.2 and 0<x<1 [28]. Other results indicate that the composition of the different synthesized compounds is probably close to Fe 3 AlC x=1/2 [29,30]. In addition, the experimental magnetic nature of the compound (ferro-or nonmagnetic) is not yet well established. Since the investigations of Morral (1934) [31], it has been stated several times that the Kappa phase is ferromagnetic. The given Curie temperature values would lie between 125 [32] and 290 °C [33]. However, the investigations of Parker et al. [34] indicate that the k phase might not be magnetic. Later, the investigations of Andryushchenko et al. [28] seem to have confirmed these observations. These authors have observed that the distribution of aluminium on the corners of the cube and of iron on the faces of the cube is apparently not perfect. Antisites' defects (aluminium atoms on iron sites and reciprocally iron on aluminium sites) seem to be at the origin of the reduced magnetic moment. Ohtani et al. [35] have published a Fe-Al-C phase diagram based on ab initio calculations within an all electron approach, and Maugis et al. [36] have discussed the relative stability of various phases in aluminium-containing steels, through ab initio calculations using the VASP package. More recently, Connétable et al. [37] have investigated the influence of the carbon on different properties of the Fe 3 Al system using ab intio calculations. The authors have found that the insertion of the carbon atom decreases the magnetism of the iron atoms and modifies strongly the heat capacity and the elastic constant in k-phase compared to the Fe 3 Al-L1 2 structure. The interactions between the Fe and the C are is the main origin of these modifications. Kellou et al. have also investigated the structural and thermal properties of Fe 3 AlC k-carbide [78]. The authors show that The C addition has the highest effect in strengthening the cohesion of the Fe 3 Al base between several additions. These authors have found that the bulk modulus (166GPa) and cohesive energy (5.7eV/ atom) of the Fe 3 AlC (k-carbide) phase has been found to the highest from all the investigated Fe 3 AlX compounds (X=. H, B, C, N, O) [78]. III.3. Strengthening by coherent precipitates In the Fe-Al-Ni system a miscibility gap between disordered α-(Fe,Al) (A2) and ordered NiAl (B2) exists at temperatures below about 1200 °C [38]. The lattice mismatch of both phases is sufficiently small so that it is possible to produce very fine-scale coherent two-phase microstructures of disordered α-(Fe,Al) (A2) + ordered (Ni,Fe)Al (B2). Except for the Fe-Al-Ni system, the mechanical properties of the coherent two-phase microstructures have not been studied in detail. The coherent precipitates have a strong strengthening effect and microstructures can be varied such that the hard (Ni,Fe)Al phase is either the matrix or the precipitate and in both cases a strengthening effect has been achieved. The deformation behaviour of ternary Fe-Al-Ni alloys at high temperatures has been studied [39]. These studies have been extended to quaternary Fe-Al-Ni-Cr alloys and first results, especially on the creep behaviour of these alloys, are reported by Stallybrass et al. [40]. III.4. Strengthening by order An additional possibility for strengthening of Fe-Al based alloys is to stabilise the D0 3 structure with respect to the B2 structure to higher temperatures. Nishino et al. [41,42] have determined the D0 3 -B 2 transformation temperatures in (Fe 1-x M x ) 3 Al with M= Ti, V, Cr, Mn and Mo. In particular, the transformation temperatures T 0 for M= Ti and V increase rapidly with increasing x, reaching T 0 values as high as 1300 K for x=0.15 (approx. 11 at.% Ti) and x=0. 25 (approx. 19 at.% V). Anthony and Fultz [43] have reviewed the solute effects on T 0 in Fe 3 Al and also measured the changes in T 0 for a large number of solutes only in the dilute limit (see Fig. I-9). Figure.I-9 Effect of ternary concentrations on ∆ ି [43]. Among the transition elements, the addition of Ti gives rise to the sharpest increase in T 0 at the rate of 55 K/at.% Ti [43,41]. Likewise, the additions of V and Mo increase T 0 but only at the rates of 35 K/at.% V [43,41], and 25 [44,42] or 30 [43] K/at.% Mo. An initial rise in T 0 for M=Ti, V and Mo tends to moderate at a higher composition. An approximately linear dependence of T D03-B2 on ternary concentration is observed for all except for two elements of the transition metals. The two exceptions, Nb and Ta, have a limited solubility in Fe 3 Al, and the increase in T D03-B2 saturates at a 1% concentration [43] (as seen from Fig. I-9). In contrast, the Cr, Hf and the Zr additions have been reported to have no significant effect on T 0 [43], except for a slight increase in T 0 reported by Mendiratta and Lipsitt [45]. It is worthwhile mentioning here that an increase in the D0 3 -B2 transformation temperature T 0 can lead to an improvement in the high-temperature strength of Fe 3 Al-based alloys [2,46]. Nishino et al. [42] have indeed demonstrated that a peak in hardness extends to higher temperatures in parallel with the increase in T 0 for M= Ti, V and Mo. At present, there is no clear understanding of why these solutes have their characteristic effects on the transformation temperature. The effect of solute atom on T 0 D0 3 -B 2 is expected to be related to its c istallographic site preference in the D0 3 structure. selectively [47]. The elements to the left of Fe in the periodic table, i.e. Ti, V, Cr and Mn, substitute for the FeI site, while those to the right, i.e. Ni and Co, substitute for the FeII site. III.4.1. Site preference Such a selective site substitution has also been found for Fe 3 Ga based alloys [48,49], where the D0 3 phase is always stabilized although Fe 3 Ga forms an L1 2 phase, unlike Fe 3 Si and Fe 3 Al, in the low-temperature equilibrium state. Figure.I-10 The unit cell of D0 3 ordered Fe 3 Al. The site preference of transition elements in Fe 3 Al may to follow that of Fe 3 Si and Fe 3 Ga, as supported by the band calculations [50]. Nevertheless, the site preference data on Fe 3 Al are still incomplete. While Ti, Mo [44,43,51] and probably Mn [51] occupy the FeI site, Mossbauer experiments for the substitution of V [52] and Cr [53] provided tentative exceptions to the above trend. Other results of Mossbauer experiments have reported that Cr occupies preferentially the FeI site and also enters the Al site [54]. More recently, Reddy et al. [20], by using the ab initio calculations found, that the Cr can occupy the FeI and FeII sites with nearly equal energies. However, the authors [20] confirmed that the Cr couple anti-ferromagnticaly with Fe atoms and prefer to occupy the FeI site. The results of Reedy et al. [20] also show that the V occupies the FeI site (see Fig. I-11). Comparatively, the Co and Ni, though the difference is small between the FeI and FeII occupation, it has been found that these two impurities prefer the FeII site [20]. Furthermore, X-ray analysis [41] and ab initio calculations [55] have indeed demonstrated the FeI site selection of V atoms. The situation is less clear concerning an element such as Zr for which there is no data on site preference in D0 3 -Fe 3 Al. Figure.I-11 The energy gain/loss when the FeI /FeII sites are replaced by various 3d transitionmetal atoms. The negative energies correspond to the gain in energy while the positive energy corresponds to a lowering of the overall binding energy with respect to pure clusters [20]. After all, we believe that the site preference plays an important role in stabilizing the D0 3 phase of Fe 3 Al-based alloys. Therefore, it is of the utmost interest to determine the site preference and ab-initio calculations coupled with temperature effect analysis are a good tool for this. III.4.2. Solute effects on D0 3 ordering Although pseudopotential calculations [56] predicted some of the solute effects on the transformation temperature T 0 , further questions should be addressed as to why certain transition elements cause their own characteristic increases in T 0 . Fortnum and Mikkola [44] suggested that the difference between solutes in raising T 0 is most probably caused by the difference between their electronic structures and/or atomic sizes. Anthony and Fultz [43] have shown that the solute effect on T 0 is related to the difference between the metallic radii of a solute atom and an Al atom: the closer the metallic radius of a ternary solute to that of Al, the greater its effectiveness in raising T 0 , as shown in Fig. I-12. A qualitative support for this atomic size argument is provided by their lattice parameter results: additions of Mo, W and Ta are effective both in raising T 0 as well as in increasing the lattice parameter. Figure.I-12 The showing the relationship between the efficiency of a ternary additive in raising ∆ ି , and the absolute value of the difference in metallic radii between that ternary element and Al [43]. However, the atomic size effect inevitably assumes the FeI site occupation of any solute atom, regardless of the site preference rule, and is also in conflict with the lattice contraction for M=V, Taking the site preference into consideration, an attention is directed to the variation of electron concentration, as proposed by Nishino et al. [41,49]. The results of Reddy et al. [20] suggest that it is the sign of magnetic coupling that determines the preferential location of the impurity atoms. Impurities to the left of Fe couple antiferromagnetically to Fe and prefer FeI sites while the impurities to the right (Co and Ni) couple ferromagnetically and prefer FeII sites. For the case of Cr, whereas the small difference between energies when substituted in FeI/FeII sites, it is found to couple antiferromagnetically to Fe and ocuppy the FeI site (Table . I-1). However, these calculations have been carried out at 0 Kelvin, and the nature of coupling changes with increasing the temperature has not been investigated. Table.I-1 The bond length (BL), binding energy (BE), and the nature of coupling in various 21immers. The corresponding bond lengths in the relaxed 35-atom clusters are also given [20]. Dimer III. 5 Objective of our work for bulk analysis The above analysis of the effect of the transition metals in raising the temperature of transition In this framework, the purpose of the present work is to compare the behviours of Ti and Zr in the bulk of Fe 3 Al. In particular we aim at determining: The temperature dependence of the site preference of Ti and Zr. The effect of temperature on the stability of the D0 3 phase of the pure as well as Ti and Zr doped Fe 3 Al. The effect of temperature on the structural properties of the pure Fe 3 Al as well as the Ti and Zr doped compounds. IV. Effect of alloying elements on ductility IV. 1 Boron addition and grain boundary strength In the case of many intermetallic alloys, small boron additions modify their ambient temperature properties. In fact, these alloys, which present an intrinsic intergranular brittleness in their 'pure' state, change their fracture mode, when boron-doped. In some cases -like in the B-doped Ni 3 Al alloys -the fracture becomes ductile. In other cases, like in FeAl-B2 alloys, even in the B-doped alloys a brittle fracture is observed, it takes place cleavage in a transgranular manner. If the first (intergranular) type of room temperature brittleness of intermetallic alloys is commonly considered as an intrinsic one, the second one (transgranular) seems in fact to be due to an extrinsic embrittling action of atomic hydrogen, created during the oxidation reaction on the sample surface. 2Al+3H 2 O→Al 2 O 3 +6H (Eq. I-1) This phenomenon, first identified by Liu et al. [60], is known as the 'environmental effect'. The boron effect in intermetallic alloys is typically attributed to its intergranular segregation. This hypothesis is a simple conclusion of experimental measurements of some intergranular boron enrichment, mainly in Ni 3 Al alloys, by the Auger Electrons Spectrometry (AES) method [START_REF] Fraczkiewicz | A[END_REF][START_REF] Hondros | Physical metallurgy[END_REF]. This hypothesis was confirmed by the experimental results of Fraczkiewicz et al. [START_REF] Gay | [END_REF]. Figure.I-15 Effect of temperature on the intergranular concentration of boron. Fe-45Al+400 appm B alloy; annealing during 24 h [START_REF] Gay | [END_REF]. The authors [START_REF] Gay | [END_REF] have used AES to measure the concentration dependence of boron segregation at grain boundaries of polycrystalline Fe-40 at.% Al base intermetallics. In a series of alloys containing different (80-2000 at. ppm) contents of boron, a maximum grain boundary concentration of about 13 at.% boron was measured for the bulk concentrations above 800 at. ppm boron, i.e. close to its solid solubility limit under these conditions. The Fowler approach [64] was used to fit the experimental results providing them with the values of the Gibbs free energy of segregation, ∆G B 0 =-41 kJ/mol, and the Fowler interaction parameter zw=+96 kJ/mol [START_REF] Gay | [END_REF]. The numerical values of thermodynamic parameters describing this system were corrected later by the same authors [START_REF] Mckamey | Physical metallurgy and processing of intermetallic compounds[END_REF]. In fact, a strong non-equilibrium segregation was still present in the studied materials under the applied conditions of the heat treatment. After a prolonged annealing, however, all boron remaining at the grain boundaries can be considered to be in segregation equilibrium. The true equilibrium grain boundary fractions of boron, E X B GB , are shown in Table . I-2. Based on this correction, the values of ∆G B 0 ranging from -30 to -34 kJ/mol, and zw ranging from +220 to 320 kJ/mol were obtained. More recently, by means of ab initio pseudopotential calculations [START_REF] Mckamey | [END_REF], the comparison between the formation energies of boron insertions in the bulk and at a ∑5 grain boundary shows that the boron atoms prefer to segregate at the grain boundary of the FeAl intemetallic compound. Table.I-2 Measured dependence of grain boundary atomic fraction of boron, M X B GB , in polycrystalline Fe-40 at.% Al alloy at 673 K on bulk boron concentration, c B [START_REF] Gay | [END_REF] and corresponding equilibrium values of grain boundary atomic fraction, E X B GB [START_REF] Mckamey | Physical metallurgy and processing of intermetallic compounds[END_REF]. GB (3) X B GB (2) X B GB (1) X B GB (1) IV. 2. Transition metal additions The room temperature ductility of Fe 3 Al can also be significantly improved with Cr alloying [13,14] together with increasing the Al-content from the stochiometric 25 towards 28 at.% [START_REF] Mckamey | Physical metallurgy and processing of intermetallic compounds[END_REF][START_REF] Mckamey | [END_REF]. A small addition of Zr (up to about 1 at.%) are also beneficial for the ductility by increasing the grain boundary strength and by trapping of hydrogen by zirconium rich precipitates [67]. Furthermore, the results of a combined experimental and finite element modeling simulations of intergranular fracture indicates that the 0.5% of Zr to the ternary Fe-28%Al-5%Cr alloy increases the intrinsic fracture resistance at room temperature [68]. IV. 3 Modelling approach and objectives of our G.B. simulations After all, we believe that the grain boundaries are the key parameters determining the macroscopic mechanical properties of iron aluminides, and must therefore be characterized as regards of their intrinsic structural properties or the influence of the alloying elements. To this purpose, experimental techniques _High-Resolution Transmission Electron Microscopy (HRTEM) and Auger Electron Spectroscopy, among others_ are particularly well adapted to yield structural information. However, Fe-Al samples (high-purity bicrystals prepared in wellcontrolled conditions, for instance), because of their extreme sensitivity to impurities, are very difficult to obtain and manipulate, which certainly contributes to limit experiments. In complementary manner, atomic-scale simulations can offer a valuable way of investigating both the structure and thermodynamics of model interfaces. This must yield matter for comparison with available or future experimental results and/or provide with new insights for understanding of the related mechanisms. In performing atomic-scale simulations, special care has to be taken of the choice of the potential-energy model. Ab initio methods provide the reputedly most accurate state-of-the-art potentials but require a high computational power compared to other semi-emperical (e.g. tightbinding) or empirical (e.g., embedded atom method, EAM) models. In spite of the continuous enhancement of the available computer power, the high computational cost of the ab initio calculations limits the size of tractable systems to about 50 transition-metal atoms, hindering comprehensive grain boundary studies that have to include point defect thermodynamics, chemical and segregation effects, in addition to more common studies generally limited to a few specific grain boundary variants. In particular, in order to determine the ground state properties of a given grain boundary, the configurational atomic phase space that has to be investigated must include in-plane rigid-body translations (RBT's) and local composition that can be different from the bulk one (if segregation occurs). Owing to the difficulty of this task, there is no example of ab initio study embracing the full problem, including both the chemical and translational degrees of freedom, a deficiency that still makes relevant the use of the empirical potentials. Although alloy interfacial segregation is a well-known phenomenon, deeper insight into it was recently gained from the ab initio, explaining in particular the effect the grain boundary segregation of boron and sulphur on grain boundary cohesion [69] and co-segregation of boron, titanium and oxygen at the grain boundaries of α-iron [70]. Segregated gallium was found to draw charge from the surrounding aluminium atoms, thus, to reduce the cohesion of aluminium [71]. On the basis of the local density functional equations, several phenomena were also determined: the structure and electronic properties of boron and sulphur at the coherent twin boundary in ferritic iron [72], the embrittlement of the same boundary induced by phosphorus segregation [73], hydrogen segregation [74] or the effect of boron [75] on the cohesion of iron. The first-principles quantum mechanical calculations showed that large bismuth atoms weaken the interatomic bonding by pushing apart the copper atoms at the interface [76] The density-functional theory was further applied to study the geometric and magnetic structures of fully relaxed symmetrical tilt {013} grain boundary in iron and {012} grain boundary in nickel. In both cases, enhancements of the local magnetic moments of the atoms in the grain boundary plane were found. Calculated values of the segregation enthalpy of silicon and tin at these grain boundaries were in good agreement with experiment [77]. Concerning more specifically the atomic simulation of the grain boundary segregation in intermetallics, except for some studies by Besson et al. [79] and Raulot et al. [64], they were done Ni-Al. Concerning the modeling works carried out on Fe-Al based alloys, the majority of these calculations were always carried out without taking into account the vibrational effect related to the effect of temperature. In this context the goals of the present modeling work on the ∑5 (310)[001] in D0 3 -Fe 3 Al are done to determine: The site preference of the two transition metals (Ti and Zr) between different configurations on the grain boundary interface. The effect of the transition metals on the stability of the grain boundary. The effect of the relaxation on the structural deformations of the grain boundary interface. The effect of temperature on structural relaxation of the grain boundary. The temperature dependence of the site preference of the two transition metals Ti and Zr. Chapter II Theoretical tools In this Chapter we review the theoretical models that were used in our simulations. Two major theories will be described, the Ab Initio Molecular Dynamics and the Density Functional Theory. In the first part, the focus is on the temporal evolution of a molecular system. In particular, the methods considering the nuclei as classical particles are described. However, the goal is not to list in an exhaustive manner the different methods of molecular dynamics but to clarify the context within which the ab initio molecular dynamics takes place. In the second part, the Density Functional Theory 'DFT' used in the context of our simulations is described. Finally we detail the computational techniques that were used to carry out our pseudopotential calculations through the VASP (Vienna Ab initio Simulation Package) code. Part A: Ab Initio Molecular Dynamics I.1. Introduction Modern theoretical methodology, aided by the advent of high speed and massively parallel computing, has advanced to a level that the microscopic details of chemical processes in condensed phases can now be treated on a relatively routine basis. One of the most commonly used theoretical approaches for such studies is the Molecular Dynamics (MD) method, in which the classical Newtonian equations of motion for a system are solved numerically starting from a specified initial state and subject to a set of boundary conditions appropriate to the problem. MD methodology allows both equilibrium thermodynamic and dynamical properties of a system at finite temperature to be computed. The quality of a MD calculation rests largely on the method by which the forces are specified. In many applications, these forces are computed from an empirical model or "force field", an approach that has enjoyed tremendous success in the treatment of systems ranging from simple liquids and solids to polymers and biological systems including proteins, membranes, and nucleic acids. Since most force fields do not include electronic polarization effects [1] and can treat chemical reactivity only through specialized techniques [2], it is often necessary to turn to the methodology of ab initio MD (AIMD). AIMD is a rapidly evolving and growing technique that constitutes one of the most important theoretical tools developed in the last decades. In an AIMD calculation, finite-temperature dynamical trajectories are generated by using forces obtained directly from electronic structure calculations performed "on the fly" as the simulation proceeds. Thus, AIMD permits chemical bond breaking and forming events to occur and accounts for electronic polarization effects [3,4]. AIMD has been successfully applied to a wide variety of important problems in physics and chemistry and is now beginning to influence biology as well. In numerous studies, new physical phenomena have been revealed and microscopic mechanisms elucidated that could not have been uncovered by using empirical methods, often leading to new interpretations of experimental data and even suggesting new experiments to be perform. Figure. II-1 Ab initio molecular dynamics unifies approximate ab initio electronic structure theory (i.e solving Schrodinger's wave equation numerically using, for instance, Hartree-Fock theory or the Local Density Approximation (LDA) within Kohn-Sham theory) and classical molecular dynamics (i.e solving Newton's equation of motion numerically for given interaction potential as reported by Fermi, Pasta, Ulam, and Tsingou for one-dimensional anharmonic chain model of solids and published by Alder and Wainwright for the three-dimensional hard-sphere model of fluids [5]). I.2. Quantum Molecular Dynamic I.2.1. Deriving Classical Molecular Dynamics The starting point of the following discussion is non-relativistic quantum mechanics as formalized via the time-dependent Schrödinger equation ݅ℏ డ డ௧ φሺሼ ሽ, ሼ ூ ሽ; ݐሻ = ܪφሺሼ ሽ, ሼ ூ ሽ; ݐሻ (Eq. A-1) in its position representation in conjunction with the standard Hamiltonian for the electronic ሼݎ ሽ and nuclear ሼܴ ூ ሽ degrees of freedom. ܪ = - ℏ ଶ ܯ2 ூ ∇ ூ ଶ ூ - ℏ ଶ 2݉ ∇ ଶ + ݁ ଶ ห -ห - ழ ݁ ଶ | ூ -| + ூ, ݁ ଶ ܼ ூ ܼ ห ூ -ห ூழ = - ℏ ଶ ܯ2 ூ ∇ ூ ଶ ூ - ℏ ଶ 2݉ ∇ ଶ + ܸ ି ሺሼ ሽ, ሼ ூ ሽሻ = -∑ ℏ మ ଶெ ∇ ூ ଶ ூ + ܪ ሺሼ ሽ, ሼ ூ ሽሻ (Eq. A-2) The goal of this section is to derive classical molecular dynamics [7][8]9] starting from Schrodinger's wave equation and following the elegant route of Tully [10,11]. To this end, the nuclear and electronic contributions to the total wave function φሺሼݎ ሽ, ሼܴ ூ ሽ; ݐሻ, which depends on both the nuclear and electronic coordinates, have to be separated. The simplest possible form is a product ansatz φሺሼ ሽ, ሼ ூ ሽ; ݐሻ ≈ ψሺሼ ሽ; ݐሻχሺሼ ூ ሽ; ݐሻ ݔ݁ ቂ ℏ ݐ݀ ′ ܧ ሺݐ ′ ሻ ௧ ௧ బ ቃ (Eq. A-3) Where the nuclear and electronic wave functions are separately normalized to unity at every instant of time, i.e ۦχ; |ݐχ; ۧݐ = 1 and ۦψ; |ݐψ; ۧݐ = 1, respectively. In addition, a convenient phase factor ܧ = ݀݀ ψ * ሺሼ ሽ; ݐሻ χ * ሺሼ ூ ሽ; ܪ‪ሻݐ ψሺሼ ሽ; ݐሻχሺሼ ூ ሽ; ݐሻ (Eq. A-4) was introduced at this stage such that the final equations will look nice; ݀݀ refers to the integration over all i=1,… and I=1,… variables ሼ ሽ and ሼ ூ ሽ, respectively. It is mentioned in passing that this approximation is called a one-determinant or single-configuration ansatz for the total wavefunction, which at the end must lead to a mean-field description of the coupled dynamics. Note also that this product ansatz (excluding the phase factor) differs from the Born-Oppenheimer ansatz [12,13] for separating the fast and slow variables φ ை ሺሼ ሽ, ሼ ூ ሽ; ݐሻ = ∑ ψ ሺሼ ሽ, ሼ ூ ሽሻχ ሺሼ ூ ሽ; ݐሻ ∞ ୀ (Eq. A-5) even in its one-determinant limit, where only a single electronic state k (evaluated for the nuclear configurationሼܴ ூ ሽ) in included in the expansion. Inserting the separation ansatz Eq. (A-3) into Eqs. (A-1)-(A-2) yields (after multiplying from the left by 〈 ψ| and 〈 χ| and imposing energy conservation 〉ܪ〈݀ ݐ݀ ≡ 0 ⁄ ) the following relations ݅ℏ డψ డ௧ = ∑ ℏ మ ଶ ∇ ଶ + ሼ ݀χ * ሺሼ ூ ሽ; ݐሻܸ ି ሺሼ ሽ, ሼ ூ ሽሻχሺሼ ூ ሽ; ݐሻሽψ (Eq. A-6) ݅ℏ డχ డ௧ = -∑ ℏ మ ଶெ ∇ ூ ଶ ூ χ + ሼ ݀ψ * ሺሼ ሽ; ܪ‪ሻݐ ሺሼ ሽ, ሼ ூ ሽሻ ψሺሼ ሽ; ݐሻሽχ (Eq. A-7) This set of coupled equations defines the basis of the Time-Dependent Self-Consistent Field (TDSCF) method introduced as early as 1930 by Dirac [14]. Both electrons and nuclei move quantum-mechanically in a time-dependent effective potential (or self-consistently obtained average fields) obtained from appropriate averages (quantum mechanical equation values〈… 〉) over other class of degree of freedom (by using the nuclear and electronic wavefunctions, respectively). Thus, the single-determinant ansatz Eq. (A-3) produces, as already anticipated, a mean-field description of the coupled nuclear-electronic quantum dynamics. This is the price to pay for the simplest possible separation of electronic and nuclear variables. The next step in the derivation of classical molecular dynamics is the task to approximate the nuclei as classical point particles. How can this be achieved in the framework of the TDSCF approach, given one quantum-mechanical wave equation describing all nuclei. A well-known route to extract classical mechanics from quantum mechanics in general starts with rewriting the corresponding wavefunction χሺሼ ூ ሽ; ݐሻ = ܣሺሼ ூ ሽ; ݐሻ expሾ݅ܵሺሼ ூ ሽ; ݐሻ/ℏሿ (Eq. A-8) In terms of an amplitude factor A and a phase S which are both considered to be real and A>0 in this polar representation [15]. After transforming the nuclear wave function in Eq. (A-7) accordingly and after separating the real and imaginary parts, the TDSCF equation for the nuclei డௌ డ௧ + ∑ ଵ ଶெ ሺ∇ ூ ܵሻ ଶ ூ + ݀ ψ * ܪ ψ = ℏ ଶ ∑ ଵ ଶெ ∇ మ ூ (Eq. A-9) డ డ௧ + ∑ ଵ ெ ሺ∇ ூ ܣሻ ூ ሺ∇ ூ ܵሻ + ∑ ଵ ଶெ ܣሺ∇ ூ ܵሻ ଶ = 0 ூ (Eq. A-10) is (exactly) re-expressed in terms of the new variables A and S. This so-called "quantum fluid dynamical representation" Eqs. (A-9),(A-10) can actually be used to solve the time-dependent Schrodinger equation [16]. The relation for A, Eq. (A-10), can be rewritten as continuity equation [15] with the help of identification of the nuclear density |χ| ଶ ≡ ܣ ଶ as directly obtained from the definition Eq. (A-8). This continuity equation is independent of ℏ and ensures locally the conservation of the particle probability |χ| ଶ associated to the nuclei in the presence of a flux. More important for the present purpose is a more detailed discussion of the relation for S, Eq. (A-9). This equation contains one term that depends on ℏ, a contribution that vanishes if the classical limit డௌ డ௧ + ∑ ଵ ଶெ ሺ∇ ூ ܵሻ ଶ ூ + ݀ ψ * ܪ ψ = 0 (Eq. A-11) Is taken as ℏ →0; an expansion in terms of ℏ would lead to a hierarchy of semi-classical methods. The resulting equation is now isomorphic to equations of motion in the Hamilton-Jacobi formulation [17] డௌ డ௧ + ܪሺሼ ூ ሽ, ሼ∇ ூ ܵሽሻ = 0 (Eq. A-12) of classical mechanics with the classical Hamilton function ܪሺሼ ூ ሽ, ሼ ூ ሽሻ = ܶሺሼ ூ ሽሻ + ܸሺሼ ூ ሽሻ (Eq. A-13) Defined in terms of (generalized) coordinates ሼ ூ ሽ and their conjugate momenta ሼ ூ ሽ. With the help of the connecting transformation ூ ≡ ∇ ூ ܵ (Eq. A-14) the Newtonian equation of motion ሶ ூ = -∇ ூ ܸሺሼ ூ ሽሻ corresponding to Eq. (A-11) ௗ ௗ௧ = -∇ ூ ݀ ψ * ܪ or ܯ ூ ሷ ூ ሺݐሻ = -∇ ூ ݀ ψ * ܪ ψ (Eq. A-15) = -∇ ூ ܸ ா ሺሼ ூ ሺݐሻሽሻ (Eq. A-16) can be read off. Thus, the nuclei move according to classical mechanics in an effective potential ܸ ா due to the electrons. This potential is a function of only the nuclear positions at time t as a result of averaging ܪ over the electronic degrees of freedom, i.e. computing its quantum expectation value ܪ|‪ψۦ |ψۧ, while keeping the nuclear positions fixed at their instantaneous values ሼ ூ ሺݐሻሽ. However, the nuclear wave function still occurs in the TDSCF equation for the electronic degrees of freedom and has to be replaced by the positions of the nuclei for consistency. In this case the classical reduction can be achieved simply by replacing the nuclear density |χሺሼ ூ ሽ; ݐሻ| ଶ in equation Eq. (A-6) in limit ℏ→0 by a product of delta functions Π ூ ߜ൫ ூ -ݐܫ centered at the instantaneous positions ݐܫ of the classical nuclei as given by Eq. (A-15). This yields e.g. for the position operator ݀ χ * ሺሼ ூ ሽ; ݐሻ ூ χሺሼ ூ ሽ; ݐሻ ℏ→ ሱۛሮ ூ ሺݐሻ (Eq. A-17) the required expectation value. This classical limit leads to a time-dependent wave equation for the electrons ݅ℏ ߲ψ ݐ߲ = - ℎ ଶ 2݉ ∇ ଶ ψ + ܸ ି ሺሼ ሽ, ሼ ூ ሺݐሻሽሻψ = ܪ ሺሼ ሽ, ሼ ሺݐሻሽሻψሺሼ ሽ, ሼ ூ ሽ; ݐሻ (Eq. A-18) Which evolve self-consistently as the classical nuclei are propagated via Eq. (A-15). Note that now ܪ and thus ψ depend parametrically on the classical nuclear position ሼ ூ ሺݐሻሽ at the time t through ܸ ି ሺሼ ሽ, ሼ ூ ሺݐሻሽሻ. The approach relying on solving Eq. (A-15) together with Eq. (A-18) is sometimes called "Ehrenfest molecular dynamics" in honor of Ehrenfest who was the first to address the question of how Newtonian classical dynamics can be derived from Schrodinger's wave equation [18]. In the present case this leads to a hybrid or mixed approach because only the nuclei are forced to behave like classical particles, whereas the electrons are still treated as quantum objects. I.2.2. « Ehrenfest » Molecular Dynamics Although the TDSCF approach underlying Ehrenfest molecular dynamics clearly is a mean-field theory, transitions between electronic states are included in this scheme. This can be made evident by expanding the electronic wavefunction Ψ (as opposed to the total wave function φ according to Eq. (A-5)) in terms of many electronic states or determinants Ψ k Ψሺሼ ሽ, ሼ ூ ሽ; ݐሻ = ∑ ܿ ሺݐሻΨ ሺሼ ሽ, ሼ ூ ሽሻ ஶ ୀ (Eq. A-19) with complex coefficients ሼܿ ሺݐሻሽ. In this case, the coefficients ሼ|ܿ |)ݐ( ଶ ሽ) (with ∑ |ܿ |)ݐ( ଶ ≡ 1) describe explicitly the time evolution of the populations (occupations) of the different states ሼ݇ሽ whereas interferences are included via the ሼܿ * ܿ ≠ ݇ሽ contributions. One possible choice for the basis functions ሼΨ ሽ is the adiabatic basis obtained from solving the time-independent electronic Schrodinger equation ܪ (ሼ ሽ; ሼ ூ ሽΨ = ܧ (ሼ ூ ሽ)Ψ (ሼ ሽ; ሼ ூ ሽ) (Eq. A-20) where ሼ ୍ ሽ are the instantaneous nuclear positions at time t according to Eq. (A-15). Thereby, the a priori construction of any type of potential energy surface is avoided from the outset by solving the time-dependent electronic Schrodinger equation "on the fly". This allows one to compute the force from ∇ ூ ܪ〈 〉 for each configuration ሼ ூ ()ݐሽ generated by molecular dynamics. The corresponding equations of motion in terms of the adiabatic basis Eq. (A-20) and the time-dependent expansion coefficients Eq. (A-19) ܯ ூ ሷ ூ )ݐ( = -∑ |ܿ |)ݐ( ଶ ∇ ூ ܧ -∑ ܿ * ܿ ܧ( -ܧ )݀ ூ , (Eq. A-21) ݅ℏܿሶ )ݐ( = ܿ ܧ)ݐ( -݅ℏ ∑ ܿ )ݐ( ሶ ூ ݀ ூ ூ, (Eq. A-22) Where the coupling terms are given by ݀ (ሼ ூ ()ݐሽ) = ݀ Ψ * ∇ ூ Ψ ூ (Eq. A-23) with the property d ୍ ୩୩ ≡ 0. The Ehrenfest approach is thus seen to include rigorously nonadiabatic transitions between different electronic states Ψ and Ψ within the framework of classical nuclear motion and the mean-field (TDSCF) approximation to the electronic structure. The restriction to one electronic state in the expansion Eq. (A-19), which is in most cases the ground state Ψ , leads to ܯ ூ ሷ ூ )ݐ( = -∇ ூ ۦΨ ܪ| |Ψ ۧ (Eq. A-24) ݅ℏ డΨ బ డ௧ = ܪ Ψ (Eq. A-25) note that ܪ is time-dependent via the nuclear coordinates ሼ ூ ()ݐሽ. A point worth mentioning here is that the propagation of the wavefunction is unitary, i.e. the wavefunction preserves its norm and the set of orbitals used to build up the wavefunction will stay orthonormal. Ehrenfest molecular dynamics is certainly the oldest approach to "on the fly" molecular dynamics and is typically used for collision-and scattering-type problems. However, it was never in widespread use for systems with many active degrees of freedom typical for condensed matter problems (although a few exceptions exist [19,20] but here the number of explicitly treated electrons is fairly limited with the exception of [21]). I.2.3. « Born-Oppenheimer » Molecular Dynamics An alternative approach to include the electronic structure in molecular dynamics simulations consists in straightforwardly solving the static electronic structure problem in each molecular dynamics step given the set of fixed nuclear positions at that instance of time. Thus, the electronic structure part is reduced to solving a time-independent quantum problem, e.g. by solving the time-independent Schrodinger equation, concurrently to propagating the nuclei via classical molecular dynamics. Thus, the time-dependence of the electronic structure is a consequence of nuclear motion, and not intrinsic as in Ehrenfest molecular dynamics. The resulting Born-Oppenheimer molecular dynamics method is defined by ܯ ூ ሷ ூ )ݐ( = -∇ ூ ݉݅݊൛ൻψ หܪ หψ ൿൟ (Eq. A-26) ܧ ψ = ܪ Ψ (Eq. A-27) for the electronic ground state. A deep difference with respect to Ehrenfest dynamics concerning the nuclear equation of motion is that the minimum of ܪ〈 〉 has to be reached in each Born-Oppenheimer molecular dynamics step according to Eq. (A-26). In Ehrenfest dynamics, on the other hand, a wavefunction that minimized ܪ〈 〉 initially will also stay in its respective minimum as the nuclei move according to Eq. (A-24). A natural and straightforward extension [22] of ground-state Born-Oppenheimer dynamics is to apply the same scheme to any excited electronic state ψ k without considering any interferences. In particular, this means that also the "diagonal correction terms" [23] ܦ (ሼ ூ ()ݐሽ) = - ݀ ψ * ∇ ூ ଶ ψ (Eq. A-28) are always neglected; the inclusion of such terms is discussed for instance in ( Refs. [10,11]). These terms renormalize the Born-Oppenheimer or "clamped nuclei" potential energy surface E k of a given state Ψ (which might also be the ground state Ψ ) and lead to the so-called "adiabatic potential energy surface" of that state [23]. Whence, Born-Oppenheimer molecular dynamics should not be called "adiabatic molecular dynamics", as is sometime done. It is useful for the sake of later reference to formulate the Born-Oppenheimer equations of motion for the special case of effective one-particle Hamiltonians. This might be the Hartree-Fock approximation defined to be the variational minimum of the energy expectation value ൻψ หܪ หψ ൿ given a single Slater determinant ψ = det൛ψ ൟ subject to the constraint that the ܯ ூ ሷ ூ )ݐ( = -∇ min൛ൻψ หܪ ுி หψ ൿൟ (Eq. A-33) 0 = ܪ- ுி ψ + ∑ ߉ ψ (Eq. A-34) for the Hartree-Fock case. A similar set of equations is obtained if Hohenberg-Kohn-Sham density functional theory [24,25] is used, where ܪ ுி has to be replaced by the Kohn-Sham effective one-particle Hamiltonian ܪ ுி . Instead of diagonalizing the one-particle Hamiltonian an alternative but equivalent approach consists in directly performing the constraint minimization according to Eq. (A-29) via nonlinear optimization techniques. Early applications of Born-Oppenheimer molecular dynamics were performed in the framework of a semi empirical approximation to the electronic structure problem [26,27]. But only a few years later an ab initio approach was implemented within the Hartree-Fock approximation [28]. Born-Oppenheimer dynamics started to become popular in the early nineties with the availability of more efficient electronic structure codes in conjunction with sufficient computer power to solve "interesting problems". Undoubtedly, the breakthrough of Hohenberg-Kohn-Sham density functional theory in the realm of chemistry -which took place around the same time -also helped a lot by greatly improving the "price/performance ratio" of the electronic structure part [29,30]. A third and possibly the crucial reason that boosted the field of ab initio molecular dynamics was the pioneering introduction of the Car-Parrinello approach [31]. This technique opened novel avenues to treat large-scale problems via ab initio molecular dynamics and catalyzed the entire field by making "interesting calculations" possible, see also the closing section on applications. I.2.4. « Car-Parrinello » Molecular Dynamics A non-obvious approach to cut down the computational expenses of molecular dynamics which includes the electrons in a single state was proposed by Car and Parrinello in 1985 [31]. In retrospect it can be considered to combine the advantages of both i.e. they can be integrated on the time scale given by nuclear motion. However, this means that the electronic structure problem has to be solved self-consistently at each molecular dynamics step, whereas this is avoided in Ehrenfest dynamics due to the possibility to propagate the wavefunction by applying the Hamiltonian to an initial wavefunction (obtained e.g. by one selfconsistent diagonalization). The basic idea of the Car-Parrinello approach can be viewed to exploit the quantum-mechanical adiabatic time-scale separation of fast electronic and slow nuclear motion by transforming that into classical-mechanical adiabatic energy-scale separation in the framework of dynamical systems theory. In order to achieve this goal the two-component quantum/classical problem is mapped onto a two-component purely classical problem with two separate energy scales at the expense of losing the explicit time-dependence of the quantum subsystem dynamics. Furthermore, the central quantity, the energy of the electronic subsystemൻψ หܪ หψ ൿ evaluated with some wavefunction ψ , is certainly a function of the nuclear positionsሼ ூ ሽ. But at the same time it can be considered to be a functional of the wavefunction ψ and thus of a set of oneparticle orbitals ൛ψ ൟ. Now, in classical mechanics the force on the nuclei is obtained from the derivative of a Lagrangian with respect to the nuclear positions. This suggests that a functional derivative with respect to the orbitals, which are interpreted as classical fields, might yield the force on the orbitals, given a suitable Lagrangian. In addition, possible constraints within the set of orbitals have to be imposed, such as e.g. orthonormality. and that the constraints are holonomic [32]. Following this route of ideas, generic Car-Parrinello equations of motion are found to be of the form ܯ ூ ሷ ூ )ݐ( = - డ డ ൻψ หܪ หψ ൿ + డ డ ‪ሽݏݐ݊݅ܽݎݐݏ݊ܿ‪ሼ (Eq. A-38) ߤ ψሷ )ݐ( = - ఋ ఋψ * ൻψ หܪ หψ ൿ + ఋ ఋψ * ‪ሽݏݐ݊݅ܽݎݐݏ݊ܿ‪ሼ (Eq. A-39) where ߤ (= ߤ) are the "fictitious masses" or inertia parameters assigned to the orbital degrees of freedom; the units of the mass parameter ߤ are energy times a squared time for reasons of dimensionality. ݏݐ݊݅ܽݎݐݏ݊ܿ = ݏݐ݊݅ܽݎݐݏ݊ܿ (൛ψ ൟ, ሼ ூ ሽ) (Eq. A-40) might be a function of both the set of orbitals ൛ψ ൟ and the nuclear positions ሼ ூ ሽ .These dependencies have to be taken into account properly in deriving the Car-Parrinello equations following from Eq. (A-35) using Eqs.( A-36)-(A-37). According to the Car-Parrinello equations of motion, the nuclei evolve in time at a certain (instantaneous) physical temperature ∝ ∑ ܯ ூ ሶ ூ ଶ ூ , whereas a "fictitious temperature" ∝ ∑ ߤ ൻψሶ หψሶ ൿ is associated to the electronic degrees of freedom. In this terminology, "low electronic temperature" or "cold electrons" means that the electronic subsystem is close to its instantaneous minimum energy ݉݅݊ ൛ψ ൟ ൻψ หܪ หψ ൿ, i.e. close to the exact Born-Oppenheimer surface. Thus, a ground-state wave function optimized for the initial configuration of the nuclei will stay close to its ground state also during time evolution if it is kept at a sufficiently low temperature. I.3. Integration of the equations of motion In the classical MD [9,7,33], the trajectory for the various components of the system is generated by integrating the Newton equations of motion, which, for each particle i, write: ቐ ܯ ூ ௗ మ (௧) ௗ௧ మ = ݂ ூ )ݐ( ݂ ூ )ݐ( = - డ(ோ ) డ (௧) (Eq. A-41) V(x) is the potential energy function of the N-particle system, which only depends upon the Cartesian coordinates { ሬሬԦ ூ }. Eqs.(A-41) are integrated numerically using an infinitesimal timestep, δt, to guarantee the conservation of the total energy of the system. Hoping to generate exact trajectories over long times is, however, illusory, considering that the Newton equations of motion are solved numerically, with a finite time-step. The exactness of the solution of Eqs.(A-41) is, nevertheless, not as crucial as it would seem. What really matters in reality is the correct statistical behavior of the trajectory to ensure that the thermodynamic and dynamic properties of the system be reproduced with a sufficient accuracy. This pivotal condition is fulfilled only if the integrator employed to propagate the motion possesses the property of symplecticity [34][35]. A so-called symplectic propagator conserves the invariant metric of the phase space, Γ. As a result, the error associated with this propagator is bound: lim ೞ →∞ ቀ ଵ ቁ ∑ ቚ ఢ(ఋ௧)ିఢ() ఢ() ቚ ୀଵ ≤ ߝ ெ (Eq. A-42) Here, n step denotes the number of steps of the simulation, ߳(0) ≡ (ܪ ூ , ௫ ; 0), the initial total energy of the equilibrated system, and ߝ ெ , the upper bound for energy conservation-viz. 10 -4 constitutes an acceptable value. Assuming that the time-step is limited, integration of the equations of motion does not lead to an erratic growth of the error associated with the conservation of the total energy, which may affect significantly the statistical behavior of MD over long times. In order to clarify the statistical equivalence between the numerical solution and the exact solution of the equations of motion, it is useful to recall some points of hamilton's approach and relationship with statistical mechanics. I.3.1. Hamiltion's point of view and statistical mechanics The Hamiltonian of a system with N particles moving under the influence of a potential function ܷ is defined as (ܪ ே , ே ) = ∑ మ ଶெ + ே ூୀଵ ܷ( ே ) (Eq. A-43) Were ሼ ூ ሽ are the momenta of particles defined as ܲ ሬԦ ூ = ܸܯ ሬԦ ூ . ே ( ே ) is the union of all positions (or momenta) ሼ ଵ , ଶ , … , ே ሽ. The forces on the particle are derived from the potential ூ ( ே ) = - డ( ಿ ) డ (Eq. A-44) The equations of motion are according to Hamilton's equation ሶ ூ = డு డ = ெ (Eq. A-45) ሶ ூ = - డு డ = - డ డ = ூ ( ே ) (Eq. A-46) From which we get Newton's second law using ூ = ܯ ூ ሶ ூ ܯ ூ ሷ ூ = ூ ( ே ) (Eq. A-47) The equations of motion can also be derived using the Lagrange formalism. The Lagrange function is ܮ൫ ே , ሶ ே ൯ = ∑ ଵ ଶ ܯ ூ ሶ ூ ଶ -ܷ( ே ) ே ூୀଵ (Eq. A-48) And the associated Euler-Lagrange equation ௗ ௗ௧ డ డ ሶ = డ డ (Eq. A-49) leads to the same final results. The two formulations are equivalent, but the ab initio molecular dynamics literature almost exclusively uses the Lagrangian techniques. I.3.2. Microcanonical Ensemble The equations of motion are time reversible (invariant to the transformation t → -t) and the total energy is a constant of motion డா డ௧ = డு൫ ಿ , ሶ ಿ ൯ డ௧ = 0 (Eq. A-50) These properties are important to establish a link between molecular dynamics and statistical mechanics. The latter connects the microscopic details of a system the physical observables such as equilibrium thermodynamic properties, transport coefficients, and spectra. Statistical mechanics is based on the Gibbs' ensemble concept. That is, many individual microscopic configurations of a very large system lead to the same macroscopic properties, implying that it is not necessary to know the precise detailed motion of every particle in a system in order to predict its properties. It is sufficient to simply average over a large number of identical systems, each in a different configuration; i.e. the macroscopic observables of a system are formulated in term of The thermodynamic variables that characterize an ensemble can be regarded as experimental control parameters that specify the conditions under which an experiment is performed. Now consider a system of N particles occupying a container of volume V and evolving under Hamilton's equation of motion. The Hamiltonian will be constant and equal to the total energy E of the system. In addition, the number of particles and the volume are assumed to be fixed. Therefore, a dynamical trajectory (i.e. the positions and momenta of all particles over time) will generate a series of classical states having constant N, V, and E, corresponding to a microcanonical ensemble. If the dynamics generates all possible states then an average over this trajectory will yield the same result as an average in a microcanonical ensemble. The energy conservation condition, ܪ൫ ே , ሶ ே ൯ which imposes a restriction on the classical microscopic states accessible to the system, defines a hypersurface in the phase space called a constant energy surface. A system evolving according to Hamilton's equation of motion will remain on this surface. The assumption that a system, given an infinite amount of time, will cover the entire constant energy hypersurface is known as the ergodic hypothesis. Thus, under the ergodic hypothesis, averages over a trajectory of a system obeying Hamilton's equation are equivalent to averages over the microcanonical ensemble. In addition to equilibrium quantities also dynamical properties are defined through ensemble averages. Time correlation functions are important because of their relation to transport coefficients and spectra via linear response theory [36,37]. The important points are: by integration Hamilton's equation of motion for a number of particles in a fixed volume, we can create a trajectory; time averages and time correlation functions of the trajectory are directly related to ensemble averages of the microcanonical ensemble. I.3.3. The molecular dynamics propagators Several approaches for integrating numerically the Newton equations of motion (Eq. A-41) are currently available. Among them, three will be detailed here. Unquestionably the simplest, the Verlet algorithm relies upon the knowledge of the triplet ሼ ூ ,)ݐ( ூ ݐ( -,)ݐߜ ூ ()ݐሽ , where ூ )ݐ( = ሷ ூ )ݐ( denotes the acceleration of particles I [38]. Modifying the positions of the particles is achieved through a Taylor expansion of the position ݐ -ݐߜ and at ݐ + ,ݐߜ leading to: ூ ݐ( + )ݐߜ = 2 ூ )ݐ( -ூ ݐ( -)ݐߜ + ூ )ݐ( ݐߜ ଶ (Eq. A-51) Which implies possible errors in ݐߜ( ସ ). It is worth noting that the velocities, ூ )ݐ( = ሶ ூ )ݐ( = ݀ ூ )ݐ( ݐ݀ ⁄ , do not appear explicitly in this scheme. They cancel out in the Taylor expansion of ݔ ூ ݐ( + )ݐߜ and ݔ ூ ݐ( -. )ݐߜ Though unnecessary for describing the trajectory, their evaluation is an obligatory step for computing the kinetic energy, () , which depends upon the sole momentum variables, p, and, consequently, the total energy of the system, according to: ூ )ݐ( = (௧ାఋ௧)ି (௧ିఋ௧) ଶ ఋ௧ (Eq. A-52) At each time-step, the associated error is in ݐߜ( ଶ ). The so-called leap-frog algorithm, derived from the preceding one, makes use the ሼ ூ ,)ݐ( ூ ݐ( -,)2ݐߜ )ݐ(ܫ triplet. The origin of its name appears clearly in the writing of the algorithm: ቐ ூ ݐ( + )ݐߜ = ூ )ݐ( + ூ ቀݐ + ఋ௧ ଶ ቁ ݐߜ ூ ቀݐ + ఋ௧ ଶ ቁ = ூ ቀݐ - ఋ௧ ଶ ቁ + ூ ݐߜ)ݐ( (Eq. A-53) In practice, the first step is the computation of ூ ݐ( + ݐߜ 2) ⁄ , from which ܸ ூ )ݐ( is deduced, which is a requisite for evaluating the term, , following: ቐ ூ ݐ( + )ݐߜ = ூ )ݐ( + ூ ݐߜ)ݐ( + ଵ ଶ ூ ݐߜ)ݐ( ଶ ூ ݐ( + )ݐߜ = ூ )ݐ( + (௧)ା (ାఋ௧) ଶ ݐߜ (Eq. A-54) This scheme involves the two following steps: ூ ቀݐ + ఋ௧ ଶ ቁ = ூ )ݐ( + ଵ ଶ ூ ݐߜ)ݐ( (Eq. A-55) From which the thermodynamic forces, ݂ ூ , and accelerations, ܽ ூ , at time ݐ + ݐߜ can be evaluated. It ensures that: ூ ݐ( + )ݐߜ = ூ ቀݐ + ఋ௧ ଶ ቁ + ଵ ଶ ூ ݐ( + ݐߜ)ݐߜ (Eq. A-56) The kinetic energy may then be deduced at time ݐ + ,ݐߜ while the potential energy, , is computed in the force loops. I.3.4. Extended System Approach In the framework of statistical mechanics all ensembles can be formally obtained from the microcanonical NVE ensemble -where particle number, volume and energy are the external thermodynamic control variables -by suitable Laplace transforms of its partition function. Thermodynamically this corresponds to Legendre transforms of the associated thermodynamic potentials where intensive and extensive conjugate variables are interchanged. In thermodynamics, this task is achieved by a "sufficiently weak" coupling of the original system to an appropriate infinitely large bath or reservoir via a link that establishes thermodynamic equilibrium. The same basic idea is instrumental in generating distribution functions of such ensembles by computer simulation. Additional degrees of freedom that control the quantity under consideration are added to the system. The simulation is then performed in the extended microcanonical ensemble with a modified total energy as a constant of motion. This system has the property that after the correct integration over the additional degrees of freedom has been performed the distribution function of the targeted ensemble is recovered. Two important special cases are: thermostats and barostats, which are used to impose temperature instead of energy and / or pressure instead of volume as external control parameters [7,9,39,40,41,42,43]. I.3.4.1. Barostats Keeping the pressure constant is a desirable feature for many applications of molecular dynamics. The concept of barostats and thus constant-pressure molecular dynamics was introduced in the framework of extended system dynamics by Andersen [42]. His method was devised to allow for isotropic fluctuations in the volume of the supercell. An extension of Andersen's method consists in allowing for changes of the shape of the computational cell to occur as a result of applying external pressure [41], including the possibility of non-isotropic external stress; the additional fictitious degrees of freedom in the Parrinello-Rahman approach [41] are the lattice vectors of the supercell. These variable-cell approaches make it possible to study dynamically structural phase transitions in solids at finite temperatures. The basic idea to allow for changes in the cell shape consists in constructing an extended Lagrangian where the lattice vectors a 1 , a 2 and a 3 of the simulation cell are additional dynamical variables. Using the 3 × 3 matrix h = [a 1 , a 2 , a 3 ] (which fully defines the cell with volume det h) the real-space position R I of a particle in this original cell can be expressed as ூ = ℎ ூ (Eq. A-57) where ூ is a scaled coordinate with components ܵ ,௨ ∈ [0,1] that defines the position of the ith particle in a unit cube (i.e. Ω ௨௧ = 1) which is the scaled cell [41]. The resulting metric tensor ܩ = ℎ ௧ ℎ converts distances measured in scaled coordinates to distances as given by the original coordinates. The variable-cell extended Lagrangian can be postulated ܮ = ∑ ଵ ଶ ܯ ூ ே ூ ൫ ሶ ூ ௧ ܩ ሶ ூ ൯ -ܷ(ℎ, ே ) + ଵ ଶ ܹ ݎܶ ℎ ሶ ௧ ℎ ሶ -Ω (Eq. A-58) with additional nine dynamical degrees of freedom that are associated to the lattice vectors of the supercell h. Here, p defines the externally applied hydrostatic pressure, W defines the fictitious mass or inertia parameter that controls the time-scale of the motion of the cell h. In particular, this Lagrangian allows for symmetry-breaking fluctuations -which might be necessary to drive a solid-state phase transformation -to take place spontaneously. The resulting equations of motion read ܯ ூ ሷ ூ,௨ = -∑ డ൫,ௌ ಿ ൯ డ ,ೡ (ℎ ௧ ) ௩௨ ିଵ ଷ ௩ୀଵ -ܯ ூ ∑ ∑ ܩ ௨௩ ିଵ ܩ ሶ ௩௦ ܵ ሶ ூ,௦ ଷ ௦ୀଵ ଷ ௩ୀଵ (Eq. A-59) ܹ ሷ ௨௩ = Ω ∑ (Π ௨௦ ௧௧ -ߜ ௨௦ )(ℎ ௧ ) ௦௩ ିଵ ଷ ௦ୀଵ (Eq. A-60) Where the total internal stress tensor Π ௨௦ ௧௧ = ଵ Ω ∑ ܯ ூ ൫ܵ ሶ ூ ௧ ܵܩ ሶ ூ ൯ ௨௦ ூ + Π ௨௦ (Eq. A-61) Is the sum of the thermal contribution due to the nuclear motion at finite temperature and the internal stress tensor Π derived from the interaction potential. A modern formulation of barostats that combines the equation of motion also with thermostats (see next section) was given by Martyna et al. [43]. I.3.4.2. Thermostats Standard molecular dynamics generates the microcanonical or NVE ensemble where in addition the total momentum is conserved [9]. The temperature is not a control variable and cannot be preselected and fixed. But it is evident that also within molecular dynamics the possibility to control the average temperature (as obtained from the average kinetic energy of the nuclei and the energy equipartition theorem) is welcome for physical reasons. A deterministic algorithm of achieving temperature control in the spirit of extended system dynamics [42] by a sort of dynamical friction mechanism was devised by Nosé and Hoover [40,44]. Thereby, the canonical or NVT ensemble is generated in the case of ergodic dynamics. It is well-known that the standard Nosé-Hoover thermostat method suffers from non-ergodicity problems for certain classes of Hamiltonians, such as the harmonic oscillator [44]. A closely related technique, the so-called Nosé-Hoover-chain thermostat [45], cures that problem and assures ergodic sampling of phase space even for the pathological harmonic oscillator. This is achieved by thermostatting the original thermostat by another thermostat, which in turn is thermostatted and so on. In addition to restoring ergodicity even with only a few thermostats in the chain, this technique is found to be much more efficient in imposing the desired temperature. The underlying equations of motion read ܯ ூ ሷ ூ = -∇ ூ ܧ ௌ -ܯ ூ ξ ሶ ଵ ሶ ூ (Eq. A-62) ܳ ଵ ξ ሷ ଵ = ൣ∑ ܯ ூ ሶ ூ ଶ -݃݇ ܶ ூ ൧ -ܳ ଵ ξ ሶ ଵ ξ ሶ ଶ (Eq. A-63) ܳ ξ ሷ = ቂܳ ିଵ ξ ሶ ିଵ ଶ -݇ ܶቃ -ܳ ξ ሶ ξ ሶ ାଵ (1 -ߜ ) ݁ݎ݁‪ℎݓ ݇ = 2, … , ܭ (Eq. A-64) By inspection of (Eq. A-22) it becomes intuitively clear how the thermostat works: ξ ሶ ଵ can be considered as a dynamical friction coefficient. The resulting "dissipative dynamics" leads to non-Hamiltonian flow, but the friction term can acquire positive or negative sign according to its equation of motion. This leads to damping or acceleration of the nuclei and thus to cooling or heating if the instantaneous kinetic energy of the nuclei is higher or lower than k B T which is preset. As a result, this extended system dynamics can be shown to produce a canonical ensemble in the subspace of the nuclear coordinates and momenta. In spite of being non-Hamiltonian, Nosé-Hoover (-chain) dynamics is also distinguished by conserving an energy quantity of the extended system; see (Eq. A-67). The desired average physical temperature is given by T and g denotes the number of dynamical degrees of freedom to which the nuclear thermostat chain is coupled (i.e. constraints imposed on the nuclei have to be subtracted). It is found that this choice requires a very accurate integration of the resulting equations of motion (for instance by using a high-order Suzuki-Yoshida integrator [46]). The integration of these equations of motion is discussed in detail in Ref. [46] using the velocity Verlet algorithm. One of the advantages of the velocity Verlet integrator is that it can be easily used together with higher order schemes for the thermostats. The choice of the "mass parameters" assigned to the thermostat degrees of freedom should be made such that the overlap of their power spectra and the ones of the thermostatted subsystems is maximal [46]. The relations ܳ ଵ = ಳ ் ఠ మ (Eq. A-65) ܳ = ಳ ் ఠ మ (Eq. A-66) Assures this if ߱ is a typical phonon or vibrational frequency of the nuclear subsystem (say of the order of 2000 to 4000 cm -1 ). There is a conserved energy quantity in the case of thermostatted molecular dynamics. This constant of motion reads ܧ ௦ ே் = ∑ ଵ ଶ ܯ ሶ ଶ + ܷ( ே ) + ∑ ଵ ଶ ܳ ξ ሶ ଶ + ∑ ݇ ܶξ + ݃݇ ܶξ ଵ ୀଶ ୀଵ ூ (Eq. A-67) For Nosé-Hoover-chain thermostatted molecular dynamics. Part B: The Electronic Structure Methods II. 1. Introduction Up to this point, the electronic structure method to calculate the ab initio forces ∇ ூ ۦΨ ܪ| |Ψ ۧ was not specified in detail. It is immediately clear that ab initio molecular dynamics is not tied to any particular approach, although very accurate techniques are of course prohibitively expensive. It is also evident that the strength or weakness of a particular ab initio molecular dynamics scheme is intimately connected to the strength or weakness of the chosen electronic structure method. Over the years a variety of different approaches were combined with molecular dynamics. The focus of the present review is classical molecular dynamics in conjunction with Hohenberg-Kohn-Sham density functional theory [1,2]. In the following, only those parts of density functional theory are presented that impact directly on our static ab initio and molecular dynamics calculations. II. 2. Density Functional Theory The total ground-state energy of the interacting system of electrons with classical nuclei fixed at positions ሼܴ ூ ሽcan be obtained min Ψ బ ሼۦΨ ܪ| |Ψ ۧሽ = min ൛φ ൟ ܧ ௌ ൣ൛φ ൟ൧ (Eq. B-1) as the minimum of the Kohn-Sham energy [1, 2] ܧ ௌ ൣ൛φ ൟ൧ = ܶ ௦ ൣ൛φ ൟ൧ + ݀ ܸ ௫௧ ሺሻ ݊ሺሻ + ଵ ଶ ܸ݀ ு ሺሻ݊ሺሻ + ܧ ௫ ሾ݊ሿ (Eq. B-2) which is an explicit functional of the set of auxiliary functions ൛φ ሺሻൟ that satisfy the orthonormality relation ൻφ หφ ൿ = ߜ . This is a dramatic simplification since the minimization with respect to all possible many-body wavefunctions ሼΨሽ is replaced by a minimization with respect to a set of orthonormal one-particle functions, the Kohn-Sham orbitals ൛φ ൟ . The associated electronic one body density or charge density ݊ሺሻ = ∑ ݂ |φ ሺሻ| ଶ (Eq. B-3) is obtained from a single Slater determinant built from the occupied orbitals, where ሼ݂ ሽ are integer occupation numbers. The first term in the Kohn-Sham functional (Eq. B-1) is the kinetic energy of a non-interacting reference system ܶ ௦ ൣ൛φ ൟ൧ = ∑ ݂ ർφ ቚ- ଵ ଶ ∇ ଶ ቚφ (Eq. B-4) consisting of the same number of electrons exposed to the same external potential as in the fully interacting system. The second term comes from the fixed external potential ܸ ௫௧ () = -∑ | ି| + ∑ ห ି ห ூழ ூ (Eq. B-5) in which the electrons move, which comprises the Coulomb interactions between electrons and nuclei and in the definition used here also the internuclear Coulomb interactions; this term changes in the first place if core electrons are replaced by pseudopotentials. The third term is the Hartree energy, i.e. the classical electrostatic energy of two charge clouds which stem from the electronic density and is obtained from the Hartree potential ܸ ு ሺሻ = ݀′ ሺ ′ ሻ |ି′| (Eq. B-6) which in turn is related to the density via ∇ ଶ ܸ ு ሺሻ = -4ߨ݊ሺሻ (Eq. B-7) Poisson's equation. The last contribution in the Kohn-Sham functional, the exchange-correlation functional ܧ ௫ ሾ݊ሿ, is the most intricate contribution to the total electronic energy. The electronic exchange and correlation effects are lumped together and basically define this functional as the remainder between the exact energy and its Kohn-Sham decomposition in terms of the three previous contributions. The minimum of the Kohn-Sham functional is obtained by varying the energy functional Eq. B-1) for a fixed number of electrons with respect to the density Eq. B-2) or with respect to the orbitals subject to the orthonormality constraint. This leads to the Kohn-Sham equations ቄ- ଵ ଶ ∇ ଶ + ܸ ௫௧ ሺሻ + ܸ ு ሺሻ + ఋா ೣ ሾሿ ఋሺሻ ቅ φ ሺሻ = ∑ ߉ φ ሺሻ (Eq. B-8) ቄ- ଵ ଶ ∇ ଶ + ܸ ()ቅ φ () = ∑ ߉ φ () (Eq. B-9) ܪ ௌ φ () = ∑ ߉ φ () (Eq. B-10) which are one-electron equations involving an effective one-particle Hamiltonian ܪ ௌ with the effectif potential ܸ . Note that ܪ ௌ nevertheless embodies the electronic many-body effects by virtue of the exchange-correlation potential ఋா ೣ [] ఋ() = ܸ ௫ () (Eq. B-11) A unitary transformation within the space of the occupied orbitals leads to the canonical form ܪ ௌ φ = ߳ φ (Eq. B-12) of the Kohn-Sham equations, where ሼ߳ ሽ are the eigenvalues. In conventional static density functional or "band structure" calculations this set of equations has to be solved self-consistently in order to yield the density, the orbitals and the Kohn-Sham potential for the electronic ground state [3]. The corresponding total energy (Eq. B-2) can be written as ܧ ௌ = ∑ ߳ - ଵ ଶ ݀ ܸ ு ()݊() + ܧ ௫ [݊] - ݀ ఋா ೣ [] ఋ() ݊() (Eq. B-13) where the sum over Kohn-Sham eigenvalues is the so-called "band-structure energy". II. 3. Energy functionals Crucial to any application of density functional theory is the approximation of the unknown exchange and correlation functional. Those exchange-correlation functionals that will be considered in the implementation part, belong to the class of the "Generalized Gradient Approximation" [4] ܧ ௫ ீீ [݊] = ݀ ݊()ߝ ௫ ீீ ൫݊(); ∇݊()൯ (Eq. B-15) where the unknown functional is approximated by an integral over a function that depends only on the density and its gradient at a given point in space The combined exchange-correlation function is typically split up into two additive terms ߝ ௫ and ߝ for exchange and correlation, respectively. In the simplest case it is the exchange and correlation energy density ߝ ௫ (݊) of an interacting but homogeneous electron gas at the density given by the "local" density n(r) at space-point r in the inhomogeneous system. This simple but astonishingly powerful approximation [5] is the famous local density approximation LDA [6] (or local spin density LSD in the spin-polarized case [7]). The self-interaction correction [8] SIC as applied to LDA was critically assessed for molecules in Ref. [9] with a disappointing outcome. A significant improvement of the accuracy was achieved by introducing the gradient of the density as indicated in Eq. B-14) beyond the well-known straightforward gradient expansions. These so-called GGAs (also denoted as "gradient corrected" or "semilocal" functionals) extended the applicability of density functional calculation to the realm of chemistry. Another considerable advance was the successful introduction of "hybrid functional" [10,11] that includes to some extent "exact exchange" [12] in addition to a standard GGA. Although such functional can certainly be implemented within a plane wave approach [13], they are prohibitively time-consuming. A more promising route in this respect are those functional that include higher-order powers of the gradient (or the local kinetic energy density) in the sense of a generalized gradient expansion beyond the first term. Promising results could be achieved by including Laplacian or local kinetic energy terms [14], but at this stage a sound judgment concerning their "prize/performance ratio" has to await further scrutinizing tests. The "optimized potential method" (OPM) or "optimized effective potentials" (OEP) are another route to include "exact exchange" within density functional theory. Here, the exchange-correlation functional ܧ ௫ ைெ = ܧ ௫ [൛φ ൟ] depends on the individual orbitals instead of only on the density or its derivatives. II. 4. The plane wave pseudopotential method The Kohn Sham equation, (Eq. B-12), can be solved in practice by expanding the Kohn Sham orbitals in a finite set of basis functions. The Schrodinger equation then transforms into an algebraic equation for the expansion coefficient which may be solved by various wellestablished numerical methods. Among these methods, we limit our discussion to the Plane Wave (PW) basis set. Plane waves are the exact eigenfunctions of the homogeneous electron gas. Therefore, plane waves are the natural choice for a basis expansion of the electron wave functions for simple metals where the ionic cores can be viewed as rather small perturbations to the homogeneous electron gas ("nearly free electron" metals). Plane waves are orthonormal and energy-independent. Hence, upon a basis set expansion the Schrodinger equation transforms into a simple matrix eigenvalue problem for the expansion coefficients. A further advantage of plane waves is that they are not biased to any particular atom. Any region in space is treated on an equal footing so that calculations do not have to be corrected for a basis set superposition error. Since plane waves do not depend on the positions of the atoms, the Hellmann-Feynman theorem can be applied directly to calculate atomic forces. Even for a non-complete basis set the Pulay terms are identical zero. In practical calculations only plane waves up to a certain cutoff wave vector are included in the basis set. The convergence of the calculations with respect to the basis set size is therefore controlled by a single parameter and can be checked simply by increasing the length of the cutoff wave vector. However, due to the nodal structure of the valence wave functions in the core region of the atoms a prohibitively large number of plane waves would be needed for a good representation of these fast oscillations. For plane wave approaches to be of practical use we have to replace the Coulomb potential of the electron-nucleus interaction by pseudopotentials. By introducing pseudopotentials we are able to achieve two goals: First, we can remove the core electrons from our calculations. The contribution of the core electrons to the chemical bonding is negligible but they contribute most to the total energy of the system (typically a thousand times more than the valence electrons). Hence, the removal of the core electrons from the calculation means that total energy differences between ionic configurations can be taken between much smaller numbers so that the required accuracy for the total energy calculations will be much less demanding than in an all-electron calculation. Second, by introducing pseudopotentials we replace the true valence wave functions by so-called pseudo wave functions which match exactly the true valence wave functions outside the ionic core region but are nodeless inside. These pseudo wave functions can be expanded using a much smaller number of plane wave basis states. A further advantage of pseudopotentials is that relativistic effects can be incorporated easily into the potential while further treating the valence electrons non-relativistically. In spite of introducing pseudopotentials, the number of basis functions N pw needed for an accurate calculation is still an order of magnitude larger than for approaches using localized orbitals. This disadvantage, however, is more than compensated by the possibility to evaluate many expressions with the help of the Fast Fourier Transform (FFT) algorithm. The most time consuming step in solving the single-particle Schrodinger equations is to apply the Hamilton operator to the valence wave functions. In a traditional matrix representation of the Hamilton operator this step scales quadratically with the number of basis functions. With plane waves and the FFT algorithm this operation reduces to a N pw ln(N pw ) scaling. Hence, for large systems the use of plane wave basis functions will become much more efficient than localized basis sets. Furthermore, the total charge density and the Hartree potential are easily calculated in a plane wave representation. II.4.1. Plane waves II.4.1.1. Supercell Although we have simplified the complicated many-body problem of interacting electrons in the Coulomb potentials of fixed nuclei to a set of single-particle equations, the calculation of the one-electron wave functions for an extended (or even infinite) system is still a formidable task. To make the problem tractable we assume that our system of interest can be represented by a box of atoms which is repeated periodically in all three special directions. The box shall be described by three vectors a 1 , a 2 , and a 3 . The volume of the box is given by Ω = ଵ . ( ଶ ݔ ଷ ) (Eq. B-16) The three vectors define a lattice in real space. General lattice vectors T are multiples of the primitive lattice vectors: = ܰ ଵ ଵ + ܰ ଶ ଶ + ܰ ଷ ଷ (Eq. B-17) Where N 1 , N 2 , N 3 can be any integer number. The box can be, for example, either the primitive unit cell of a crystal or a large supercell containing a sufficient number of independent atoms to mimic locally an amorphous solid or a liquid phase. By using supercells also atomic point defects, surfaces or isolated molecules can be modeled as illustrated schematically in II.4.1.2. Fourier representations The translational symmetry of the atomic arrangements can now be exploited to reduce the computational cost for solving the Kohn-Sham equations. The effective potential (as well as the electron density) is a periodic function with the periodicity of the lattice, i.e. ܸ ( + ) = ܸ () (Eq. B-18) for any lattice vector T of the form of (Eq. B-17). Therefore V eff can be expanded into a Fourier series ܸ () = ∑ ܸ ()݁ , ܸ () = ଵ Ω ܸ ()݁ ି ݀ ଷ Ω (Eq. B-19) The sum runs over all vectors G which fulfill the condition G.T= 2π M for all lattice vectors T with M being an integer number. The vectors G form a lattice, the so-called reciprocal lattice, which is generated by the three primitive vectors b 1 , b 2 , b 3 defined by [15] . = 2ߨߜ , ݅, ݆ = 1,2,3 (Eq. B-20) The volume of the unit cell of the reciprocal lattice is given by ଵ . ( ଶ ݔ ଷ ) = (ଶగ) య Ω (Eq. B-21) II.4.1.3. Bloch's Theorem The solutions of a single-particle Schrödinger equation with a periodic potential are by no means themselves necessarily periodic. However, the eigenstates can be chosen in such a way that associated with each wave function φ is a wave vector k to hold φ( + ) = ݁ φ() (Eq. B-22) for every lattice vectors T (Bloch's theorem). From now on all eigenstates of the single-particle Schrödinger equation will be labeled with its corresponding vector k. From the form of the exponential factor in (Eq. B-22) it is obvious that the values of k can be restricted to within one unit cell of the reciprocal lattice. By convention this unit cell is usually taken to be the first Brillouin Zone (BZ) [15]. Different solutions to the same vector k will be labeled with the band index j. Bloch's theorem is often stated in an alternative form. The property in (Eq. B-22) is equivalent to the statement that all eigenfunctions φ kj of a single-particle Schrödinger equation with a periodic potential can be written as a periodic function u kj modulated by a plane wave with wave vector k [15]: φ () = ݁ ݑ () (Eq. B-23) This allows us to restrict the calculation of the eigenfunctions to within one unit cell. The form of the eigenfunctions in all other unit cells is determined by applying (Eq. B-22). From now on we will assume that the eigenfunctions are normalized with respect to a single unit cell: |φሺሻ| ଶ ݀ ଷ = 1 Ω (Eq. B-24) Since the functions u kj are periodic they can be expanded in a set of plane waves. Together with the exponential prefactor we get φ ሺሻ = ∑ ܿ ݁ ሺାሻ (Eq. B-25) Before we make use of the plane wave expansion of the wave functions we write the Kohn-Sham equations of density functional theory in the notation of Bloch-states: ቀ- మ ଶ ∆ + ܸ ሺሻቁ φ ሺሻ = ߳ φ ሺሻ (Eq. B-26) with ܸ () = ܸ ௫௧ () + ܸ ு [݊()] + ܸ ௫ [݊()] (Eq. B-27) and ݊() = 2 Ω (ଶగ) య ∑ ቚφ ()ቚ ଶ Θ(ܧ ி -߳ )݀ ଷ (Eq. B-28) V ext , V H and V xc are the external potential of the nuclei, the Hartree and the exchange-correlation potential, respectively. By the factor of 2 in (Eq. B-28) we take the electron spin into account. Θ is a step function which is one for positive and zero for negative arguments. E F is the Fermi energy up to which single-particle states have to be occupied. The Fermi energy is defined by the number of electrons N e in the unit cell: ݊()݀ ଷ = ܰ Ω (Eq. B-29) For an insulator the Fermi energy lies in a band gap. Hence, at each k-point exactly N e /2 bands will be occupied. For metals one or more bands cross the Fermi energy so that the number of occupied states will change between k-points. II.4.1.4. k-Point Sampling By making use of Bloch's theorem we have transformed the problem of calculating an infinite number of electronic states extended infinitely in space to one of calculating a finite number of eigenstates at an infinite number of k-points which are extended over a single unit cell. At first glance this seems to be only a minor improvement since still an infinite number of calculations are needed for the different k-points. However, the electronic wave functions at k-points which are close together will be very similar. Hence it is possible to represent the wave functions of a region of k-space by the wave function at a single k-point. We thus define a regular mesh of N kpt k-points and replace the integral over the Brillouin zone by a discrete sum over the chosen kpoint mesh: Ω (ଶగ) య … Θ(ܧ ி -߳ )݀ ଷ → ଵ ே ೖ ∑ ݂ (Eq. B-30) The f kj are occupation numbers which are either one or zero. Several schemes to construct such k-point meshes have been proposed in the literature [16][17] The convergence of all calculations with respect to the basis set size can be tested simply increasing step by step the plane wave cutoff energy. The electron density in Fourier representation is given by Since we have truncated the wave functions at a maximum wave vector it is obvious from Eq. B 34 that the electron density has only non this cutoff wave vector. In Fourier space the calculation of the H simple. It is given by sually only a small number of k-points is required to get good converged results. For increasing size of the supercell the volume of the Brillouin zone becomes smaller and smaller (see Eq. B 22). Therefore, with increasing supercell size less and less k-points are needed. From a certain supercell size on it is often justified to use just a single k-point, which is usually taken to be k=0 point approximation). For metallic systems, on the other hand, much denser k are required in order to get a precise sampling of the Fermi surface. In these cases the convergence with respect to the k-point density can often be accelerated by introducing fractional occupation numbers [18][19]. II.4.1.5. Fourier representation of the Kohn-Sham equations In a plane wave representation of the wave functions the Kohn-Sham equations assume a particular simple form. If we insert (Eq. B-25) into (Eq. B-26), multiply from left with and integrate over r we get the matrix eigenvalue equation ቀ మ ቁ ‖ ‖ ଶ ߜ ᇱ ܸ ሺ ᇱ െ ሻܿ ൌ ߳ ܿ In practical calculations the Fourier expansion (Eq. B-25) of the wave functions is truncated by keeping only those plane wave vectors (k+G) with a kinetic energy lower than a g ቀ మ ଶ ቁ ‖ ‖ ଶ ܧ ௪ The convergence of all calculations with respect to the basis set size can be tested simply increasing step by step the plane wave cutoff energy. electron density in Fourier representation is given by ݊ሺሻ ൌ ଶ ே ೖ ∑ ݂ ∑ ൫ܿ ᇲ ି ൯ * ᇱ ܿ ᇲ Since we have truncated the wave functions at a maximum wave vector it is obvious from Eq. B 34 that the electron density has only non-vanishing Fourier components up to twice the length of this cutoff wave vector. In Fourier space the calculation of the Hartree potential is particularly The convergence of all calculations with respect to the basis set size can be tested simply by (Eq. B-33) Since we have truncated the wave functions at a maximum wave vector it is obvious from Eq. Bvanishing Fourier components up to twice the length of artree potential is particularly (Eq. B-34) ܸ ு ሺሻ ൌ 4ߨ݁ ଶ ሺሻ ‖‖ మ As the electron density, the Hartree potential has a finite Fourier expansion. To calculate the exchange-correlation potential we have to Fourier transform the electron density to real-space, evaluate the given functional and Fourier transform back the result. II.4.1.6. Fast Fourier Transformation (FFT) The main advantage of working with plane waves is that the evaluation of various expres-sions can be speeded up significantly by using FFTs. In particular, since the wave functions and the electron density have a finite Fourier representation this can be done without any loss in accuracy, as long as we use in our real-space Fourier grid twice as many grid points in each spacial direction than the number of points in the Fourier space grid [20]. For example, the calculation of the electron density according to Eq. B-34 scales quadratically with the number N pw of plane waves. However, if we Fourier transform the wave functions to real-space (which scales with N pw ln(N pw )), calculate ቚφ ()ቚ ଶ on the real-space Fourier grid (N pw scaling) and then Fourier transform back the result we significantly reduce the computational cost. Along the same arguments we can also reduce the number of calculations for the evaluation of the term ∑ ܸ ( ᇱ -)ܿ in Eq. B-32 from a ܰ ௪ ଶ to a N pw ln(N pw ) scaling. II.4.2. Pseudopotentials II.4.2.1. Norm conserving Pseudopotentials The norm-conserving pseudopotential approach provides an effective and reliable means for performing calculations on complex molecular, liquid and solid state systems using plane wave basis sets. In this approach only the chemically active valence electrons are dealt with explicitely. The inert core electrons are eliminated within the frozen-core approximation, being considered together with the nuclei as rigid non-polarizable ion cores. In turn, all electrostatic and quantum-mechanical interactions of the valence electrons with the cores, as the nuclear Coulomb attraction screened by the core electrons, Pauli repulsion and exchange and correlation between core and valence electrons, are accounted for by angular momentum dependent pseudopotentials. These reproduce the true potential and valence orbitals outside a chosen core region but remain much weaker and smoother inside. The valence electrons are described by smooth pseudo orbitals which play the same role as the true orbitals, but avoid the nodal structure near the nuclei that keeps the core and valence states orthogonal in an all-electron framework. The respective Pauli repulsion largely cancels the attractive parts of the true potential in the core region, and is built into the therefore rather weak pseudopotential. This pseudoization of the valence wavefunctions along with the removal of the core states eminently facilitates a numerically accurate solution of the Kohn-Sham equations and the Poisson equation, and enables the use of plane waves as an expedient basis set in electronic structure calculations. By virtue of the norm-conservation property and when constructed carefully pseudopotentials present a rather marginal approximation, and indeed allow for an adequate description of the valence electrons over the entire chemically relevant range of systems. • Pseudopotentials should be additive and transferable. Additivity can most easily be achieved by building pseudopotentials for atoms in reference states. Transferability means that one and the same pseudopotential should be adequate for an atom in all possible chemical environments. This is especially important when a change of the environment is expected during a simulation, like in chemical reactions or for phase transitions. • Pseudopotentials replace electronic degrees of freedom in the Hamiltonian by an effective potential. They lead to a reduction of the number of electrons in the system and thereby allow for faster calculation or the treatment of bigger systems. • Pseudopotentials allow for a considerable reduction of the basis set size. Valence states are smoother than core states and need therefore less basis functions for an accurate description. The pseudized valence wavefunctions are nodeless (in the here considered type of pseudopotentials) functions and allow for an additional reduction of the basis. This is especially important for plane waves. Consider the 1s function of an atom φ ଵୗ )ܚ( ~ ݁^(-ܼ * )ݎ (Eq. B-35) with Z* ≈ Z , the nuclear charge. The Fourier transform of the orbital is φ ଵୱ (۵) ~ 16π ఱ మ ⁄ ۵ మ ା మ (Eq. B-36) From this formula we can estimate the relative cutoffs needed for different elements in the periodic table. • Most relativistic effects are connected to core electrons. These effects can be incorporated in the pseudopotentials without complicating the calculations of the final system. II.4.2.1.1. Hamann-Schluter-Chiang conditions Norm-conserving pseudopotentials are derived from atomic reference states, calculated from the atomic Kohn-Sham equation (Eq. B-4). This equation is replaced by a valence electron only equation of the same form (-∇ ଶ + ܸ ௩ ) หφ ௦ 〉 = ߳̂ หφ ௦ 〉 (Eq. B-37) Hamann, Schluter, and Chiang [21] proposed a set of requirements for the pseudo wavefunction and pseudopotential. The pseudopotential should have the following properties 1. Real and pseudo valence eigenvalues agree for a chosen prototype atomic configuration. ϵ = ϵ̂ 2. Real and pseudo atomic wave functions agree beyond a chosen core radius r c . φ )ݎ( = φ ௦ )ݎ( for ݎ ≥ ݎ 3. The integrals from 0 to R of the real and pseudo charge densities agree for R ≥ r c for each valence state (norm conservation). ൻφ ௦ หφ ௦ ൿ ோ = ൻφ หφ ൿ ோ for ܴ ≥ ݎ where ൻφ ௦ หφ ௦ ൿ ୖ = r ଶ | ୖ φ ௦ ሺrሻห dr ଶ (Eq. B-38) 4. The logarithmic derivatives of the real and pseudo wave function and their first energy derivatives agree for r ≥ r c . Property 3) and 4) are related through the identity െ ଵ ଶ ቂሺrφ ௦ ሻ ଶ ୢ ୢ ୢ ୢ୰ lnφ ௦ ቃ ୖ ൌ r ଶ | ୖ φ ௦ ห dr ଶ (Eq. B-39) They also gave a recipe that allows to generate pseudopotentials with the above properties: 1. V ሺଵሻ ሺrሻ ൌ V ሺrሻ ቂ1 െ f ଵ ቀ ୰ ୰ ౙౢ ቁ ቃ (Eq. B-40) r cl : core radius ≈ 0.4 -0.6 R max , where R max is the outermost maximum of the real wave function. 2. V ሺଶሻ ሺrሻ ൌ V ሺଵሻ ሺrሻ c ୪ f ଶ ቀ ୰ ୰ ౙౢ ቁ (Eq. B-41) determine c l so that ϵ ୪ ൌ ϵ ො ୪ in ቀെ∇ ଶ V ሺଶሻ ሺrሻቁ w ሺଶሻ ሺrሻ ൌ ϵ ො w ሺଶሻ ሺrሻ (Eq. B-42) 3. φ ௦ (r) = γ ቂ w (ଶ) (r) + δ r ାଵ f ଷ ቀ ୰ ୰ ౙౢ ቁቃ (Eq. B-43) where γ l and δ l are chosen such that φ ௦ (r) → φ (r) for r ≥ r ୡ୪ and γ ଶ ቚw (ଶ) (r) + δ r ାଵ f ଷ ( ୰ ୰ ౙౢ )ቚ ଶ ݎ݀ = 1 (Eq. B-44) 4. Invert the Schrodinger equation for ϵ ො φ ௦ (r) to get V ௩ (r) 5. Unscreen V ௩ (r) to get V ௦ (r) V ௦ (r) = V ௩ (r) -V ு (݊ ௩ ) -V ௫ (݊ ௩ ) (Eq. B-45) where V H (ρ v ) and V xc (ρ v ) are the Hartree and exchange and correlation potentials of the pseudo valence density. Hamann, Schluter and Chiang chose the following cutoff functions f ଵ (x) = f ଶ (x) = f ଷ (x) = exp (-x ସ ). These pseudopotentials are angular momentum dependent. Each angular momentum state has its own potential that can be determined independently from the other potentials. It is therefore possible to have a different reference configuration for each angular momentum. This allows it for example to use excited or ionic states to construct the pseudopotential for l states that are not occupied in the atomic ground state. The total pseudopotential in a solid state calculation then takes the form ܸ ௦ )ݎ( = ∑ ܸ ௦ ܲ)ݎ( (Eq. B-46) where L is a combined index {l,m} and P L is the projector on the angular momentum state {l,m}. II.4.2.1.2 Bachelet-Hamann-Schluter (BHS) form Bachelet et al. [22] proposed an analytic fit to the pseudopotentials generated by the HSC recipe of the following form V ୮ୱ (r) = V ୡ୭୰ୣ (r) + ∑ ∆V ୧୭୬ (r) (Eq. B-47) V ୡ୭୰ୣ (r) = - ೡ ୰ ൣ ∑ c ୧ ୡ୭୰ୣ erf (ඥα ୧ ୡ୭୰ୣ r) ଶ ୧ୀଵ ൧ (Eq. B-48) ∆V ୧୭୬ (r) = ∑ (A ୧ + ଷ ୧ୀଵ r ଶ A ୧ାଷ )exp (-α ୧ r ଶ ) (Eq. B-49) The cutoff functions were slightly modified to be f 1 (x) = f 2 (x) = f 3 (x) = exp(-x 3.5 ). They generated pseudopotentials for almost the entire periodic table (for the local density approximation), where generalizations of the original scheme to include spin-orbit effects for heavy atoms were made. Useful is also their list of atomic reference states. BHS did not tabulate the A i coefficients as they are often very big numbers but another set of numbers C i , where C ୧ = -∑ A Q ୧ ୀଵ (Eq. B-50) and A ୧ = -∑ C Q ୧ ିଵ ୀଵ (Eq. B-51) with ܳ = ൞ 0 ݅ > ݈ ൣ ܵ -∑ ܳ ଶ ିଵ ୀଵ ൧ ଵ ଶ ⁄ ݅ = ݈ ଵ ொ ൣ ܵ -∑ ܳ ܳ ିଵ ୀଵ ൧ ଵ ଶ ൗ ݅ < ݈ (Eq. B-52) Where ܵ = ݎ ଶ ߮ ߮)ݎ( )ݎ( ஶ ݎ݀ and ߮ )ݎ( = ቊ ݁ ିఈ మ , ݅ = 1,2,3 ݎ ଶ ݁ ିఈ మ , ݅ = 4,5,6 (Eq. B-53) II.4.2.1.3. Kerker Pseudopotetials Also in this approach [23] pseudopotentials with the HSC properties are constructed. But instead of using cutoff functions (f 1 , f 2 , f 3 ) the pseudo wavefunctions are directly constructed from the all-electron wavefunctions by replacing the all-electron wavefunction inside some cutoff radius by a smooth analytic function that is matched to the all-electron wavefunction at the cutoff radius. The HSC properties then translate into a set of equations for the parameters of the analytic form. After having determined the pseudo wavefunction the Schrdinger equation is inverted and the resulting potential unscreened. Note that the cutoff radius of this type of pseudopotential construction scheme is considerably larger than the one used in the HSC scheme. Typically the cutoff radius is chosen slightly smaller than R max , the outermost maximum of the all-electron wavefunction. The analytic form proposed by Kerker is φ ௦ )ݎ( = ݎ ାଵ ݁ () (Eq. B-54) with )ݎ( = ݎߙ ସ + ݎߚ ଷ + ݎߛ ଶ + ߜ (Eq. B-55) The term linear in r is missing to avoid a singularity of the potential at r = 0. The HSC conditions can be translated into a set of equations for the parameters α, β, γ, δ. II.4.2.1.4. Trouiller-Martins Pseudopotentials The Kerker method was generalized by Trouiller and Martins [24] to polynomials of higher order. The rational behind this was to use the additional parameters (the coefficients of the higher terms in the polynomial) to construct smoother pseudopotentials. The Trouiller-Martins wavefunctions has the following form φ ௦ )ݎ( = ݎ ାଵ ݁ () (Eq. B-56) with )ݎ( = ܿ + ܿ ଶ ݎ ଶ + ܿ ସ ݎ ସ + ܿ ݎ + ܿ ଼ ݎ ଼ + ܿ ଵ ݎ ଵ + ܿ ଵଶ ݎ ଵଶ (Eq. B-57) and the coefficients c n are determined from • norm-conservation • For n=0…4 ௗ φ ೞ ௗ ฬ ୀ = ௗ φ ௗ ቚ ୀ (Eq. B-58) • ௗφ ೞ ௗ ฬ ୀ = 0 (Eq. B-59) II.4.2.1.5. Kinetic Energy Optimized Pseudopotentials This scheme is based on the observation that the total energy and the kinetic energy have similar convergence properties when expanded in plane waves. Therefore, the kinetic energy expansion is used as an optimization criteria in the construction of the pseudopotentials. Also this type [25] uses an analytic representation of the pseudo wavefunction within r c φ ௦ )ݎ( = ∑ ܽ ݆ ݍ( )ݎ ୀଵ (Eq. B-60) where j l (qr) are spherical Bessel functions with i-1 zeros at positions smaller than r c . The values of q i are fixed such that ′ ( ) ( ) = φ ′ ( ) φ ( ) (Eq. B-61) The conditions that are used to determine the values of a i are: • φ ௦ is normalized • First and second derivatives of φ are continuous at r c • ∆E K ({a i }, q c ) is minimal ܧ∆ = - ݀ ଷ ݎφ ௦ * ∇ ଶ ߖ ௦ - ݀ ݍݍ ଶ หφ ௦ )ݍ( ห ଶ (Eq. B-62) ∆E K is the kinetic energy contribution above a target cutoff value q c . The value of q c is an additional parameter (as for example r c ) that has to be chosen at a reasonable value. In practice q c is changed until it is possible to minimize ∆E K to a small enough value. II.4.2.2. Pseudopotentials in the Plane Wave Basis With the methods described in the last section we are able to construct pseudopotentials for states l = s, p, d, f by using reference configurations that are either the ground state of the atom or of an ion, or excited states. In principle higher angular momentum states could also be generated but there physical significance is questionable. In a solid or molecular environment there will be wavefunction components of all angular momentum character at each atom. The general form of a pseudopotential is ܸ (, ′ ) = ∑ ∑ ܸ ܲ)ݎ( (߱) ୀିଵ ∞ ୀ (Eq. B-63) where P lm (ω) is a projector on angular momentum functions. A good approximation is to use ܸ () = ܸ () for ݈ > ݈ ௫ (Eq. B-64) With this approximation one can rewrite ܸ (, ′ ) = ܸ ()ܲ (߱) ∞ + [ܸ () -ܸ ()]ܲ (߱) ∞ = ܸ () ܲ (߱) + ߜܸ () ∞ ܲ (߱) ∞ = ܸ () + ∑ ߜܸ ()ܲ (߱) ∞ (Eq. B-65) where the combined index L = {l, m} has been used. The pseudopotential is now separated into two parts; the local or core pseudopotential V c (r) and the non-local pseudopotentials δV l (r)P lm (ω). The pseudopotentials of this type are also called semilocal, as they are local in the radial coordinate and the nonlocality is restricted to the angular part. The contribution of the local pseudopotential to the total energy in a Kohn-Sham calculation is of the form ܧ ୀ ܸ ()݊()݀ (Eq. B-66) It can easily be calculated together with the other local potentials. The non-local part needs special consideration as the operator in the plane wave basis has no simple structure in real or reciprocal space. There are two approximations that can be used to calculate this contribution to the energy. One is based on numerical integration and the other on a projection on a local basis set. II.4.2.2.1. Gauss-Hermit Integration The matrix element of the non-local pseudopotential ܸ (, ′ ) = 1 Ω න ݀݁ ି ∆ܸ ()݁ ′ = ∑ ݀ ∞ 〈|ܻ 〉 ఠ ݎ ଶ ∆ܸ ሺሻ〈ܻ |′ 〉 ఠ (Eq. B-67) where 〈. |. 〉 ఠ stands for an integration over the unit sphere. These integrals still depend on r. The integration over the radial coordinate is replaced by a numerical approximation ଶ ݂ሺሻ݀ ≈ ∑ ݓ ݂ሺ ሻ ∞ (Eq. B-68) The integration weights w i and integration points r i are calculated using the Gauss-Hermit scheme. The non-local pseudopotential is in this approximation ܸ ሺ, ′ ሻ ൌ 1 Ω ݓ ∆ܸ ሺ ሻ〈|ܻ 〉 ఠ 〈ܻ |′ 〉 ఠ ൌ ∑ ଵ Ω ∑ ݓ ∆ܸ ሺ ሻܲ * ()ܲ ( ′ ) (Eq. B-69) Where the definition for the projectors P ܲ () = 〈ܻ | 〉 ఠ (Eq. B-70) has been introduced. The number of projectors per atom is the number of integration points (5 -20 for low to high accuracy) multiplied by the number of angular momenta. For the case of s and p non-local components and 15 integration points this accounts to 60 projectors per atom. The integration of the projectors can be done analytically ܲ ሺሻ ൌ න ܻ * ݁)ݓ( ݀߱ ఠ = න ܻ * ߨ4)ݓ( ݅ ݆ ( ) ܻ ′ * ܻ)ݓ( ′ ()݀߱ ′ ୀିଵ ∞ ୀ ఠ = 4ߨ݅ ݆ ( )ܻ ൫ܩ ൯ (Eq. B-71) where the expansion of a plane wave in spherical harmonics has been used. j l are the spherical Bessel functions and ܩ the angular components of the Fourier vector G. II.4.2.2.2. Kleinman-Bylander Scheme The other method is based on the resolution of the identity in a local basis set ∑ | ߯ ఈ 〉〈߯ ఈ | ఈ ൌ 1, (Eq. B-72) where {χ α } are orthonormal functions. This identity can now be introduced in the integrals for the non-local part ܸ (, ′ ) = න ܻ|ۦ݀ ۧ ఠ ݎ ଶ △ ܸ ሺሻܻۦ |′ۧ ఠ ∞ ൌ ∑ ∑ ݀ ∞ ߯|ۦ ఈ ߯ۦۧ ఈ |ܻ ۧ ఠ ݎ ଶ ∆ܸ ሺݎሻൻܻ ห߯ ఉ ൿ ఠ ൻ߯ ఉ ห′ൿ ఈ,ఉ (Eq. B-73) and the angular integrations are easily performed using the decomposition of the basis in spherical harmonics ߯ ఈ ሺݎሻ ൌ ߯ ఈ ሺሻܻ ሺ߱ሻ (Eq. B-74) This leads to ܸ ሺ, ′ ሻ ൌ ܺ|ۦ ఈ ۧ න ݀߯ ఈ ሺሻݎ ଶ ∆ ∞ ఈ,ఉ ܸ ሺሻܺ ఉ ሺሻൻ߯ ఉ ห′ൿ ൌ ∑ ∑ ߯|ۦ α ۧ ఈ,ఉ ∆ܸ ఈఉ ൻ߯ ఉ ห′ൿ (Eq. B-75) which is the non-local pseudopotential in fully separable form. The coupling elements of the pseudopotential ∆ܸ ఈఉ ൌ ݀߯ ఈ ሺሻ ∞ ݎ ଶ ∆ܸ ሺሻ߯ ఉ ሺሻ (Eq. B-76) are independent of the plane wave basis and can be calculated for each type of pseudopotential once the expansion functions χ are known. The final question is now what is an optimal set of basis function χ. Kleinman and Bylander [26] proposed to use the eigenfunctions of the pseudo atom, i.e. the solutions to the calculations of the atomic reference state using the pseudopotential Hamiltonian. This choice of a single reference function per angular momenta guarantees nevertheless the correct result for the reference state. Now assuming that in the molecular environment only small perturbations of the wavefunctions close to the atoms occur, this minimal basis should still be adequate. The Kleinman-Bylander form of the projectors is ∑ ห ఞ ಽ 〉〈∆ ಽ ఞ ಽ | 〈ఞ ಽ ∆ ಽ ఞ ಽ 〉 ൌ 1, (Eq. B-77) where χ L are the atomic pseudo wavefunctions. The plane wave matrix elements of the non- local pseudopotential in Kleinman-Bylander form is ܸ ሺ, ′ ሻ ൌ 〈 |∆ ಽ ఞ ಽ 〉〈∆ ಽ ఞ ಽ | ′〉 〈ఞ ಽ ∆ ಽ ఞ ಽ 〉 (Eq. B-79) Generalizations of the Kleinman-Bylander scheme to more than one reference function were introduced by Blöchl [27] and Vanderbilt [28]. They make use of several reference functions, calculated at a set of reference energies. In transforming a semilocal to the corresponding Kleinman-Bylander (KB) pseudopotential one needs to make sure that the KB-form does not lead to unphysical "ghost" states at energies below or near those of the physical valence states as these would undermine its transferability. Such spurious states can occur for specific (unfavorable) choices of the underlying semilocal and local pseudopotentials. They are an artefact of the KB-form nonlocality by which the nodeless reference pseudo wavefunctions need to be the lowest eigenstate, unlike for the semilocal form [29]. Ghost states can be avoided by using more than one reference state or by a proper choice of the local component and the cutoff radii in the basic semilocal pseudopotentials. The appearance of ghost states can be analyzed by investigating the following properties: • Deviations of the logarithmic derivatives of the energy of the KB-pseudopotential from those of the respective semilocal pseudopotential or all-electron potential. • Comparison of the atomic bound state spectra for the semilocal and KBpseudopotentials. • Ghost states below the valence states are identified by a rigorous criteria by Gonze et al. [29]. II.4.2.3. Non-linear Core Correction The success of pseudopotentials in density functional calculations relies on two assumptions: the transferability of the core electrons to different environments and the linearization of the exchange and correlation energy. The second assumption is only valid if the frozen core electrons and the valence state do not overlap. However, if there is significant overlap between core and valence densities, the linearization will lead to reduced transferability and systematic errors. The most straightforward remedy is to include "semi-core states" in addition to the valence shell, i.e. one more inner shell (which is from a chemical viewpoint an inert "core level") is treated explicitely. This approach, however, leads to quite hard pseudopotentials which call for high plane wave cutoffs. Alternatively, it was proposed to treat the non-linear parts of the exchange and correlation energy E xc explicitely [30]. This idea does not lead to an increase of the cutoff but ameliorates the above-mentioned problems quite a bit. The method of the non-linear core correction dramatically improves results on systems with alkali and transition metal atoms. For practical applications, one should keep in mind that the non-linear core correction should only be applied together with pseudopotentials that were generated using the same energy expression. II.4.2.4. Ultrasoft Pseudopotentials Method For norm-conserving pseudopotentials the all-electron wavefunction is inside some core radius replaced by a soft nodeless pseudo wavefunction, with the crucial restriction that the PS wavefunction must have the same norm as the all-electron wavefunction within the chosen core radius; outside the core radius the pseudo and all-electron wavefunction are identical. It is well established that good transferability requires a core radius around the outermost maximum of the all-electron wavefunction, because only then the charge distribution and moments of the allelectron wavefunction are well reproduced by the pseudo wavefunctions. Therefore, for elements with strongly localized orbitals (like first-row, 3d, and rare-earth elements) the resulting pseudopotentials require large plane wave basis sets. To work around this problem, compromises are often made by increasing the core radius significantly beyond the outermost maximum of the all-electron wavefunction. But this is usually not a satisfactory solution because the transferability is always adversely affected when the core radius is increased, and for any new chemical environment, additional tests are required to establish the reliability of such soft pseudopotentials. An elegant solution to this problem was proposed by Vanderbilt [31]. In his method, the normconservation constraint is relaxed and to make up for the resulting charge deficit, lo-calized atom-centered augmentation charges are introduced. These augmentation charges are defined as the charge difference between the all-electron and pseudo wavefunctions, but for convenience they are pseudized to allow an efficient treatment of the augmentation charges on a regular grid. The core radius of the pseudopotential can now be chosen around the nearest neighbor distance; independent of the position of the maximum of the all-electron wavefunction. Only for the augmentation charges a small cutoff radius must be used to restore the moments and the charge distribution of the all-electron wavefunction accurately. The pseudized augmentation charges are usually treated on a regular grid in real space, which is not necessarily the same as the one used for the representation of the wavefunctions. The relation between the ultrasoft pseudopotential method and other plane wave based methods was discussed by Singh [32]. Tools The most important factors determining the level of theory of a quantum-mechanical computer experiment are the choice of an exchange-correlation functional, the choice of a basis-set for the expansion of the Kohn-Sham orbitals, charge-and spin densities and potentials, and the algorithm adopted for solving the Kohn-Sham equations and for calculating energies, forces and stresses. The degree to which the chosen functional accounts for many-electron correlations and the completeness of the basis-set determine the accuracy of the calculation, the numerical algorithms are decisive for its efficiency. Our calculations, both static and ab initio molecular dynamics, were performed using the Chapter III Static ab initio calculations (0K) In this chapter, the results of the static calculations of the substitutions of Ti and Zr transition metals in the bulk as well as at the ∑5 (310)[001] grain boundary are presented. To this end, the chapter is divided into three sections. In section I, the computational details of our static calculations are given. Section II gives the results of the energetic of the point defects in the bulk of the D0 3 -Fe 3 Al structure. Emphasis is also given on the importance of using relaxation when determining formation energy of calculation for site preference configurations. In the last section, the behaviour of the two transition metals Ti and Zr in the ∑5 (310) grain boundary are discussed. The formation energies, the interface energies and the electronic charge density transfer associated with the presence of the impurity has been investigated. I. Computational details I.1. Computational method Our calculations were performed within the Vanderbilt-type UltraSoft PseudoPotential (USPP) [1] and the formalism of Density Functional Theory as implemented in the Vienna Ab initio Simulation Package (VASP) [2,3]. The electronic wave functions were expanded in plane waves with a kinetic energy cutoff of 240 eV. The USPPs employed in this work explicitly treated eight valence electrons for Fe (4s 2 3d 6 ), three valence electrons for Al (3s 2 3p 1 ) and four valence electrons for both Ti (4s 2 3d 2 ) and Zr (5s 2 4d 2 ). The spin polarization was taken into account for all calculations. The Generalized Gradient Approximation (GGA-PW91) was employed for evaluation of the exchange -correlation energy with the Perdew and Wang version [4,5]. The Brillouin zone integrations were performed using Γ centered Monkhorst-Pack [6] k-point meshes, and the Methfessel-Paxton [7] technique with a 0.3 smearing of the electrons levels. Tests were carried out for Fe 3 Al unit cell (four atoms by cell) using different k-point meshes to ensure the absolute convergence of the total energy to a precision of 10 -3 eV/atom. As a result, the k-mesh for the Fe 3 Al unit cell was adapted using (16x16x16) k-points. Depending on the structure and the size of the cell, the number of k-points changes as a consequence of the resultant modification of the Brillouin zone size. For total energy calculations of the Fe 3 Al supercells with 32 atoms (2x2x2 unit cell) and 108 atoms (3x3x3 the unit cell), the (8x8x8) and (4x4x4) k-points mesh were chosen, respectively. In the case of grain boundary configurations, the Monkhorst-Pack grid was adapted to the supercell parameters using (4x2x5) k-points. The ground state atomic geometries were obtained by the minimizing the Hellman-Feyman forces using a conjugate -gradient algorithm until force on each atom reaches a convergence level of 0.1 Å/atom and for an external pressure lower than 0.3 GPa. I.2. Structural properties All the results presented below were obtained employing the computational settings described in the previous paragraph. However, for the Fe 3 Al-D0 3 unit cell additional calculations were also conducted with alternative settings to gauge the overall accuracy of the reported results. Specifically, test calculations were performed employing the Local-Density Approximation (LDA) as well as the GGA, in order to compare the two approximations and determine the influence of the analytical representation of the exchange-correlation functional. The values are given in Table . III-1 along with the available theoretical and experimental data for D0 3 -Fe 3 Al. Figure. III-1 The total energy within LDA and GGA according to the pseudopotential calculations. Not surprisingly, the GGA functional gives a larger value for the equilibrium lattice constant compared to the LDA functional, as it is well known that in most cases the frequent over binding for transition metals and their compounds in LDA is corrected with GGA. Furthermore, a fairly good agreement is found for the lattice parameter and the bulk modulus, which are calculated with the USPP-GGA, when compared with the values from experiment [13]. The present value of lattice parameter 5.76Å is also comparable to the other theoretical values obtained with GGA exchange-correlation functional [13][14]15]. In summary, the results show that the used ultarsoft pseudopotential and the GGA exchange-correlation approximation reproduce successfully the structural properties of the D0 3 -Fe 3 Al structure. I.3. Energetics In this section, we first define the energies used in our calculations. To examine the preferential site occupations of the transition metal atoms both in the bulk and at the grain boundary, their formation energies E f were evaluated in different sites by: E f = E solid (Fe 3 AlX) + E Fe or Al -E solid (Fe 3 Al) -E X (Eq.III-1) for substitutional impurities and by E f = E solid (Fe 3 AlX) -E solid (Fe 3 Al) -E X (Eq.III-2) for interstitial impurities. The formation energy, for the case of vacancies, was evaluated using the following equation [16] E f = E solid (Fe 3 Alv) + E Fe or Al -E solid (Fe 3 Al) (Eq.III-3) In these equations, X and v represent the T.M. atoms and vacancies, respectively. E solid (Fe 3 AlX) (solid referring to bulk as well as G.B.) is the energy of the supercell containing one substitutional or interstitial impurity and E solid (Fe 3 Al) is the energy of the pure supercell without defect. E Fe or Al (Fe or Al being the atom which is substituted) and E X are the calculated total energies for pure metals in their equilibrium lattices -bcc Fe, fcc Al, hcp Ti and hcp Zr. Note 1 Ref. [10] 2 Ref. [11] 3 Ref. [12] 4 Ref. [13] 5 Ref. [14] 6 Ref. [15] that, as it is very unlikely that Ti and Zr reside on interstitial sites in the bulk, Eq. III-2 will only be used for the analysis of Ti and Zr interstitials within the G.B. The grain boundary energy γ GB for the undoped system is defined as: γ GB = (E GB -E bulk ) / 2A (Eq.III-5) E GB and E bulk are the total energies of the grain boundary and bulk supercells, respectively, and A is the area of the interface (the factor of ½ is needed to account for the presence of two symmetrically equivalent grain boundaries per simulation cell). In our computer simulations, the energies E GB and E bulk are calculated for simulations blocks consisting of an equal number of atoms of each species. For the impurity doped system, the G.B. energy is evaluated from: γ GB = (E GBX -E Fe3AlX ) / 2A (Eq.III-6) In this formulation, as presented in [17,18], E GBX and E Fe3AlX are the total energies of the supercells with impurity-doped G.B. and bulk Fe 3 Al, respectively. II. Point defects in bulk Fe 3 Al We first investigate the properties of point defects in the bulk of the D0 3 Fe 3 Al structure. Because of their atomic size, Ti and Zr must occupy substitutional sites rather than interstitial ones within the bulk Fe 3 Al. The three types of possible point defects-i.e substitution impurity or vacancy formation on Al, FeI, FeII sites (see Fig. III-2)-were modeled within a supercells containing 32 and 108 atoms. It must be noted that, when substituting an impurity atom (Ti or Zr) on a site of the 32 atoms supercell, the corresponding impurity concentration is about 3 at.%. For the case of 108 atoms supercell, the impurity concentration is about 1 at. %. II.1. Importance of relaxation In order to have some insight in the effect of relaxation and underline the importance of this relaxation for the calculation of intermetallics, Fig. III-3 compares the formation energy of the substitution defects calculated by using unrelaxed and relaxed supercells (with 32 atoms). It is clear that the relaxation leads to an overall reduction in the formation energies. Additionally, though the curves have the same profile, it appears clearly that the differences in formation energies determined using the unrelaxed and relaxed supercells are lower for substitutions of Ti than for Zr. On a first approach, this difference may be attributed to the differences in size between the Ti and Zr atoms (the Ti atom is smaller than the Zr one). What is important to stress here is essentially the magnitude of the differences. While the overall differences under the relaxed and unrelaxed modes are in the range 15 to 50 % for the various Ti substitutions, the relaxation decreases the formation energy by more than 140 % in the case of the substitution of a Zr atom on a Fe II site. For the case of vacancies, the relaxation is more pronounced when the vacancy is created in the Al site. This can also be related to the difference in size between the Fe and Al atoms. Knowing that the Al atom is greater than the Fe one, the void created when the vacancy is produced in the Al site is larger and, consequently, the relaxation is also more important. Thus, in the following, only data obtained from relaxed configurations are treated. Figure. III-3 Energy profile of point defects formation energies for unrelaxed and relaxed D0 3 -Fe 3 Al supercells. II.2. The site preference of point defects in the bulk D0 3 -Fe 3 Al The values of formation energies for substitutions and vacancies in relaxed supercells are given in 1.97 2.44 1.71 From Table III-2, it can be seen that, though the vacancies occur with positive formation energies in the three different configurations, the lower formation energy correspond to the substitution on the FeII site. This tend agrees well with the conclusion of Mayer et al. [19] obtained by ab-initio pseudopotential method. Our value of the formation energy of a vacancy in FeII sublattice (1.09 eV) is also comparable to (1.18±0.04 eV) obtained by Schaefer et al. [20] from positron annihilation experiments. Comparatively, Jiraskova et al. [21], who investigated the Mössbauer spectra of a Fe 72 Al 28 compound, found that the Fe vacancies appear on the FeI sublattice which is in conflict with our result and that obtained by Mayer et al [19]. Since the results of Jiraskova were obtained at ambient temperature, unlike our calculations and those of Mayer et al. [19] which were carried out at 0K, the difference in site occupation of vacancies can be related to the effect of temperature. To check this possibility, we have calculated the defect energies of the vacancies on the FeI and FeII sites at 300K, using the Ab Initio Molecular Dynamics. The defect energies are defined as a change in energy of the pure cluster when an impurity replaces the FeI or the FeII sites, namely, ܧ ௗ = ܧሺ݀ሻ -ܧ (Eq. III-7) ܧሺ݀ሻ and ܧ are the energies of the supercell with and without transition metal impurities, respectively. The preferential site then corresponds to the case where the energy is gained by replacing the FeI/FeII sites. More details about the calculations of the temperature dependence of the defect energies will be presented in Chapter IV. However, some results are presented here in Table . III-3 together with the results of the defects energies calculated at 0K. The values of the defect energies are more important than that of the formation energies. This is because the value of the total energy of pure iron in its equilibrium lattices (bcc Fe) has not been subtracted as defined in Eq. III-2. As seen from Table . III-3, the FeII site remains the favoured site occupation of vacancies even at 300K. This means that the temperature changes in the range 0→300K does not modify the stability of vacancies in the bulk D0 3 -Fe 3 Al intermetallic compound. Here, both the static ab initio and molecular dynamics at 300K results are in conflict with the Mossbauer conclusions. This indicates that the disagreement with the experimental results does not be related to the effect of temperature but is certainly related to the high sensitivity of these alloys to the vacancies. This is because the difference between the formation energies of 3% and 1% of vacancies calculated in supercells with 32 and 108 atoms is important. From Table . III-2, one can see that the formation energies at different sites are reduced by ~60% when the concentration of vacancies is reduced. From 1% (at 108 atoms) to 3% (at 32 atoms), additionally, the difference between the formation energies of vacancies on the FeI and FeII sites is also reduced with the concentration of the vacancies. The difference between the formation energies is about 0.6 eV for the 3% of vacancies while for the case of the concentration of 1%, the difference is only about 0.03 eV. The calculated formation energies for the substitutions in the relaxed scheme are also shown in Table III-2. For the three configurations of Ti substitution, negative formation energies are obtained. This means that these defects are stable and, in other words, that 1 at% as well as 3 at% of Ti impurities are miscible in Fe 3 Al. This is consistent with the solubility data obtained experimentally [22] and from phase diagram calculations available in the literature [23]. The calculation also shows that the most stable configuration for Ti is to reside on the FeI site, with the lower effective formation energies for both types of calculation (i.e. concentration). This is the correct prediction of the experimentally observed behavior [24]. For the case of Zr, the most favoured configuration is also the substitution on a FeI site. However, contrary to Ti, at 3% concentration all the formation energies are positive, indicating a cost in energy to introduce Zr on a substitution site in the bulk. It is interesting to notice that these values tend to diminish when the calculation is carried out with 108 atoms and that the concentration of defects decreases. This is consistent with experimental observation indicating that the solubility of Zr is below 0.1% and that precipitation of a second phases such as Laves phase Zr(Fe,Al) 2 and the ߬ ଵ phase Zr(Fe,Al) 12 by adding small amounts of Zr [25]. III. Impurity segregation at grain boundaries III.1. Crystal structures and location of structural defects The atomic structure of the ∑5 (310)[001] symmetric tilt grain boundary in III.2. Site preference and effect of Ti and Zr on the grain boundary cohesion The calculated impurity formation energies of the two transition metals in the grain boundary are given in Table III-4 for each type of defect. atoms which preferred to be inserted along a ∑5 (310) grain boundary at locations where they could be surrounded by Fe near neighbors [26]. There is however here a clear difference in the behaviors of the Ti and Zr impurity atoms. The formation energy is positive for both configurations if we consider the presence of Ti interstitial, indicating thereby that it costs more energy to insert Ti at the grain boundary interface. Comparatively, the atomic configuration becomes more stable (-0.18 eV) when Zr is inserted at a (1) site (i.e. iron rich configuration). As for B in FeAl [26], the negative values of formation energy (-0.18 eV) obtained here for Zr indicates that this atom is more stable when inserted within Fe neighbors at the grain boundary than in the bulk of the material. It is important to recall that small additions of these two atoms tend to bring some ductility in iron aluminides [27][28]29]. In the case of substitutions, the results given in Table III- The comparison of the formation energies within the bulk material and within the grain boundary shows that Ti is generally stable with the same order both in the bulk and within grain boundary interface. However, for case of the Zr impurity which is clearly not stable in the bulk, prefers to segregate at the grain boundary with lower formation energies Thus, the effect of Zr at grain boundary has really to be taken into account to understand the overall properties of these ternary iron aluminides. Comparing Al and Fe sites, it is clear that substitution of the transition metals on Al sites are never the favoured configurations. For the Fe substitutions, as for the bulk results, the Ti impurity always prefers to reside on the FeI sites rather than the FeII ones. The situation is less pronounced for Zr. Indeed while Fe sites are always preferred to Al ones, the FeII are favoured when the substitution of Zr atom is lo the second plane (-0.78 eV). Finally, it is also interesting to note that, for both transition metal substitution, the most stable is obtained for substitution on a FeI site located in the first plane away from the interface (-1.32 eV for Ti/ the Ti impurity always prefers to reside on the FeI sites rather than the FeII ones. The situation is less pronounced for Zr. Indeed while Fe sites are always preferred to Al ones, the FeII are favoured when the substitution of Zr atom is located at the G.B. interface ( 0.78 eV). Finally, it is also interesting to note that, for both transition metal substitution, the most stable is obtained for substitution on a FeI site located in the first plane 1.32 eV for Ti/-0.97 for Zr). Impurity formation energies (in eV) for different substitution sites in (a) Titanium doped systems and (b) Zirconium-doped systems. 87 the Ti impurity always prefers to reside on the FeI sites rather than the FeII ones. The situation is less pronounced for Zr. Indeed while Fe sites are always preferred to Al ones, the FeII are cated at the G.B. interface (-0.51 eV) and within 0.78 eV). Finally, it is also interesting to note that, for both transition metal substitution, the most stable is obtained for substitution on a FeI site located in the first plane Impurity formation energies (in eV) for different substitution sites in (a) Titanium- The influence of the segregated impurities on the interfacial energetic has been calculated for different substitutional configurations using Eq. III-6. Our calculated results are listed in Table III for Zr (0.29 J/m 2 ), respectively. It can be noticed also that the interface energies for the Zr-doped grain boundaries are systematically lower than for the Ti-doped ones [Fig. III-7]. Thus the principal trend that can be drawn here is that zirconium is more cohesion enhancer than titanium for ∑5 (310) Fe 3 Al-grain boundary. Indeed, the presence of Zr instead of Ti on the various sites systematically decreases further the interface energy by 0.02 / 0.03 J/m 2 ; this is to say by 5% to 8%. Table III-5 The calculated interface energies (in J/m 2 ) for impurities in substitutional sites. In order to investigate the interaction between the impurities and their adjacent atoms, the formation energies are calculated for the first nearest neighbor vacancies to 1FeI substituted by transition metal impurities (i.e. the most stable configuration for the doped grain boundary). For comparison, we have calculated the formation energies for these vacancies in clean-grain boundary. The results of the formation energies are listed in Tables III-6 and III-7 for Ti and Zr impurities. From Table III-6, it can be seen that, for the Ti impurity, the formation energy increased only when the FeII is replaced by a vacancy (1.15 eV) compared to that calculated in the clean grain boundary (1.02 eV). This means that it is more expensive to remove a FeII atom in the presence of Ti impurity which indicates that Ti strengthens the interactions with their FeII first neighbor. This may originate from the antiferromagnetic coupling that forms between Ti and its FeII atoms [30]. However, for the case of Zr impurity (see III.3. Charge density distribution In order to gain new insight at the microscopic level of the bonding charge density at the grain boundary from that in the bulk, we have calculated the charge density difference both for the bulk system and the grain boundary. The charge density difference is defined as the difference between the total charge density in the solid and the superposition of each charge density placed at lattice sites, namely, III.4. Impurities induced bonding charge density To understand the effect of impurities on the bonding charge properties of the grain boundary we consider the redistribution of bonding charge induced by the impurity atom when placed at the substitutional site (1FeI). This configuration is considered because it is the most stable one between the different tested configurations. The bonding charge properties can be best described by the difference in charge density between the pure and impurity-doped grain boundaries, namely, ∆ρ(r) = ρ(Fe 3 AlX) -∑ρ(Fe) -∑ρ(Al) -ρ(X) (Eq.III-8) The impurity-induced bonding charge density in ( 004 As in the clean grain boundary, the closest atoms to the interface have different charge density distributions because they moved from their initial positions by the effect of relaxation. This will be discussed in the following sections. III. 5. The relaxation of the clean grain boundary In this section, the effect of relaxation on the structural deformation of the ∑5(310) grain boundary will be discussed. After having examined all the configurations of substitutions, it has been found that the behaviour of impurities can be classified in three different categories according to their distances from the G.B. interface. In this section, three representative configurations will be presented, namely, the impurities substitutions within the G.B. interface, substitutions in the first plane from the interface and substitutions on the second plane from the interface. All the configurations are presented in Appendix B. IV. Summary and Conclusion In this chapter, the effects of Ti and Zr transition metal impurities located in the bulk as well as at the ∑5 (310)[001] tilt grain boundary in the D0 3 -Fe 3 Al intermetallic compound were studied by means of static ab-initio calculations. The main conclusions are as follows: -Relaxation is extremely important to determine accurate formation energies and determine the stability of the point defects. For example, 140% error in formation energy is obtained when considering a Zr substitution on a FeII site using a cell that is not relaxed. -In the bulk, the calculated formation energies reveal that the FeII sites are the preferential sites for vacancies while both Ti and Zr prefer to reside on FeI sites. -The interface energy of a clean ∑5 (310) interface has been found to be (0.36 J/m 2 ). The lowest formation energies for the T.M. substitutions have always been obtained on FeI sites on the first plane away from the exact grain boundary interface. Ti and Zr impurities are found to reduce the interface energies on various sites of this ∑5 (310) grain boundary by an average of 14% and 22%, respectively. However, significant differences between the behavior of Zr and Ti atoms were revealed. In particular, Ti can reside in bulk and grain boundary configurations with the same order of stability. Comparatively, Zr is stable within the grain boundary both as an insertion and as a substituting element (on FeI and FeII sites). Also, the creation of FeII vacancies in a Ti doped boundary is energetically costly while it favored in a Zr-doped one. -The expansion of the G.B region occurs as an effect of the relaxation in the pure grain boundary. The presence of the impurities in the G.B interface does not affect the relaxation of the grain boundary. For the case of substitutions in the first and second plane, the important displacements correspond to the first nearest neighbours atoms of the impurities while the G.B. region was not affected. The Ti impurity prefers to reside in iron rich environment while the Zr impurity tends to relax to the grain boundary interface. Chapter IV Effect of temperature on the structural stabilities of Ti and Zr in the bulk and the ∑5 grain boundary: Ab Initio Molecular Dynamics study. In this chapter the results of the AIMD calculations of Ti and Zr substitutions in the bulk and at the ∑5 grain boundary are given. This chapter is divided in three major sections. In the first section (section I), the calculation details together with the preliminary calculations are presented. Section II gives the results of the temperature dependence of the site preference of the two transition metals and their effect on the structural properties of the bulk D0 3 -Fe 3 Al. The pair distribution functions were also calculated in different temperatures to get insight on the phase stability of the D0 3 structure with the transition metals additions at higher temperatures. In Section III, the defect energies of Ti and Zr transition metals are calculated in different configurations within the ∑5 grain boundary to compare their stability with the bulk. In a second part of section III, the effects of temperature on the structural relaxation of the grain boundary is given. Finally, the limitations of the CSL model at higher temperature are also discussed. I. Calculation details I. 1. Computational methods In our molecular dynamic simulations, the atomic forces were calculated from Density Functional Theory (DFT) as implemented in the VASP (Vienna Ab initio Simulation Package) code [1,2]. The calculations are based on the Generalized Gradient Approximation (GGA) [3,4] and made use of the Ultra Soft Pseudo-Potential approach USPP [5]. The Verlet algorithm was used to integrate the Newton's equations of motion with a time step of ∆t=5 fs and each simulation were allowed to turn for a total of 200 steps. To ensure that any memory of the initial configuration is completely erased, the first 100 time steps were reserved for equilibrating the system and were discarded from the subsequent analysis. We have used a cutoff energy of 250 eV which was found to be required in order to reach convergence in total energy. The valence states used are 4s 2 3d 6 for Fe, 3s 2 3p 1 for Al, 4s 2 3d 2 for Ti and 5s 2 4d 2 for Zr. The large dimension of the supercells used in our calculation (108 atoms in bulk supercell and 80 atoms in the ∑5 (310) G.B. supercell) allowed us to limit the sampling of the Brillion zone to the Γ point. The simulation were performed in the canonical ensemble NVT (particle Number, Volume and Temperature are fixed). I. 2. Preliminary calculations Preliminary test calculations were performed at each temperature in order to obtain the lattice parameter of the D0 3 structure at zero pressure. In a first step, the lattice parameter of the Fe 3 Al-D0 3 supercell was dilated/compressed to the equilibrium lattice constant at the zero pressure. From this value of lattice parameter, a new calculation was carried out to obtain the new pressure. In the second step, the resulting value of pressure -which is close to zero-was reintroduced in the graph of the evolution of the lattice parameter with the pressure to obtain a more accurate lattice parameter as shown in the The pressure difference was found smaller than 1.5 kBar in the simulated temperature range from 100 to 1100K. Such a small pressure difference in our simulation should not bring significant errors for the comparison of structural properties at different temperatures. Figure. IV-1. The evolution of pressure as function of expansion/compression of the lattice parameter (at temperature T=100K). The diamond and squares symbols represent the calculated pressure and the statistical uncertainty, respectively. I.3. Energetic In order to examine the temperature dependence of the site preference of Ti and Zr transitions metals, their defect energies were calculated both in the bulk and the ∑5 (310)[001] over a wide range of temperatures (100-1100 K). The defect energies are defined as change in energy of the pure supercell when an impurity replaces an atom in the supercell, namely, ܧ ௗ = ܧ ௦ௗ ݁ܨ( ଷ )݈ܺܣ -ܧ ௦ௗ ݁ܨ( ଷ )݈ܣ (Eq.IV-1) Where X represent the T.M. impurities. ܧ ௦ௗ ݁ܨ( ଷ )݈ܺܣ ܽ݊݀ ܧ ௦ௗ ݁ܨ( ଷ )݈ܣ (solid referring to bulk as well as G.B.) are the energies of the supercell with and without transition metal impurities, respectively. The preferential site then corresponds to the case where the energy is gained by replacing an atom. Unlike the formula of the formation energy used in the previous chapter (Chapter III, section I-2) the energies of the pure elements (Fe, Al, Ti and Zr) are not introduced in the formula of the defect energy. Thus, the comparisons (i) between the defect energies of the two transition metals and (ii) between substitutions on the Fe and Al sites are not possible. 5.767Å It is important to recall that the formation energy is defined as ܧ = ܧ ௦ௗ ݁ܨ( ଷ )݈ܺܣ + ܧ ி -ܧ ௦ௗ ݁ܨ( ଷ )݈ܣ -ܧ (Eq. IV-2) In this equation, E Fe or Al (Fe or Al being the atom which is substituted) and E X are the calculated total energies for pure metals in their equilibrium lattices -bcc Fe, fcc Al, hcp Ti and hcp Zr. In the following, we will only examine the replacement of FeI or FeII sites by transition metal impurities. Indeed, the case of substitutions on the Al site was not treated since the comparison is not possible but also as we have found, from the static calculations (at 0K), that the substitutions on the Al sites are not favourable both in the bulk and at the grain boundary ∑5 (310)[001]. II. Transition metal impurities in the bulk D0 II.1. Site preference of the Ti and Zr substitutions The defect energies of the two transition metals were calculated using a atoms of Fe and 27 atoms of Al), i.e. 3x3x conditions. By substituting an impurity in this supercell, the impurity concentration is about 1% (exactly 0.92%). The results of the defect energies are summarized in Table substitutions. To help and indicate better the trends, t substitution in FeI and FeII sites are also listed and presented in Fig. The defect energies of the two transition metals were calculated using a system of 108 d 27 atoms of Al), i.e. 3x3x3 the D0 3 -Fe 3 Al unit cell, under periodic boundary conditions. By substituting an impurity in this supercell, the impurity concentration is about 1% The results of the defect energies are summarized in Ti Zr The replacement of Ti on the FeI site leads to a significant gain in energy (-1.62~-2.55 eV) on the overall temperature range. Comparatively, the replacement on the FeII site is energetically more expensive (+0.05~-0.7 eV). This also indicates that, over the entire tested temperature range, the Ti impurity is always more stable when seated on the FeI site. For the case of the Zr impurity, the gains in energy are even more substantial: (-2.04~-2. Bond lengths (Å) Temperature (K) Ti-Al Ti-FeI FeII-Al FeII-FeI Pure-Fe 3 Al Ti-doped Fe 3 Al 2,4 2,45 2,5 2,55 2,6 2,65 2,7 2,75 2,8 Bond lengths (Å) Temperature (K) Zr-Al Zr-FeI FeII-Al FeII-FeI Pure-Fe3Al Zr-doped Fe3Al To quantify the interactions between the Zr impurity and their adjacent atoms at higher temperatures the defect energies were calculated for the first neighbour vacancies to FeII site substituted by Zr impurity. We have chosen to perform these calculations in the most stable configurations; i.e. substitutions on the FeII site at 900K and 1000K. For comparison, the defect energies for these vacancies were calculated in the pure supercell. This method is more detailed in the previous chapter (Chapter III, Section III-1). The results of defect energies are presented in Table IV-2 together with the relative difference between the defect energies of vacancies in doped and pure Fe 3 Al. The defect energies of vacancies in Ti-doped Fe 3 Al are also listed. The results of the defect energies shows that the vacancies are more favoured in the Zr doped Fe 3 Al than in the pure supercell. Their defect energies of vacancies decrease by (-39%) and (-66%) when the vacancies are created in the Al and the FeI sites, respectively, at 900K. This suggests that the presence of Zr on a FeII site weakens both the FeAl and the FeFe bonds. Despite the small increase in the defect energy of vacancy at 1000K, it remains low compared to that in the case of Ti additions. One can see from Table IV-2 that it is energetically more expensive (about +60%) to create a vacancy on the FeI site when the FeII is replaced by the Ti impurity. Comparatively, the defect energy of vacancy created on the Al site is (-65%) smaller than that created in the pure-Fe 3 Al, at 900K. Further the Ti-Al average bond length is about 2.6 Å compared to Ti-Fe average bond length of 2.45 Å. The occupancy of a FeII site by a Ti, therefore, not only leads to a reduced interactions but also induces strain on the neighbouring Al atoms. Ti atoms, therefore, occupy FeI site even at higher temperatures. The above results reveal that the behaviour of the Zr impurity differs from that of the Ti one. At higher temperature (about ~900K), while Ti impurity remains stable on a FeI site, the Zr impurity can reside on both the FeI and/or the FeII sites with the same order of stability. From Fig. IV-5 it can be seen that a small decrease in the lattice parameter takes place between 100 and 300K before the curves level off at 400K. This small decrease may be due to the changes of magnetic moment at these temperatures. According to a theoretical calculation [7], II.2. Structural and stability results II.2.1. Equilibrium lattice parameters performed by TB-LMTO code, it was found that there is a close relationship between Fe magnetic moment and lattice parameter in the FeAl intermetallic compounds. On other hand, magnetization measurements [8] and neutron diffraction analysis [9] have reported magnetization decrease with temperature below 170 K in D0 3 -Fe 3 Al. Fig. IV-6 shows our calculated magnetic moment of Fe atoms as function of the temperature. In agreement with the experimental results [8,9] a small decrease in the magnetic moment is observed below 200K. Another interesting point is the increase in the magnetic moment that occurs smoothly from about 600K and increasing sharply at 800K. This is related to the structural disorder that occurs in the D0 3 after 830K. This is reflected also on the lattice parameter. From Fig. IV-5, it can be seen that, from about 300K the lattice parameter increases gradually with temperature up to 5, 75 5,76 5,77 5,78 5,79 5,8 5,81 5,82 5,83 5,84 5,85 Lattice parameters (Å) Temperature (K) 700K. Between 700 and 900K the rate of increase of the lattice parameter is slightly larger. Thus, in agreement with the experimental result, there is a direct relationship between the magnetic and structural properties in the D0 3 -Fe 3 Al intermatallic compounds. Figure. IV-6. the temperature dependence of the total magnetic moment in the D0 3 -Fe 3 Al. Magnetic moment (µ µ µ µ B ) Temperature (K) -0,2 0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 10 2 x∆ ∆ ∆ ∆L/L Temperature (K) Present work Experiment [10] Debye model [11] From Fig. IV-7 it can be seen that our results are very close to the experimental data which indicates the validity of the AIMD used method. Our results are also comparable with the pioneer results obtained by Seletskaia et al. [11] using the Debye model to take into account the effect of temperature. At low temperatures, the calculated fractional lengths obtained using the Debye model are consistent with our results as well as the experimental data. However, an increasing divergence is observed when the temperature increases. This is due to the growing contribution of the optical phonons to the thermal expansion, since, the anharmonic effects (the vibrational contributions) are neglected in the Debye model. Considerable effort has lately gone into understanding the role of lattice vibrations in thermodynamics of chemical ordering in binary substitunational alloys [12,13]. It has been suggested [14,15] in this connection that the contribution of vibrational entropy to the phase stability in alloys can be quite significant. Nix and Shockley [16] first suggested that the state of order could affect the lattice vibrations through a change in the Debye temperature. Suprisingly enough, the vibrational entropy contribution has remained largely neglected as compared to the configurational entropy for quite some time. Several theoretical calculations [17,18] followed, emphasizing its significance. This indicates that the approach used in the present work is rather powerful to take into account the effect of the temperature in this kind of material. Lattice parameters (Å) Temperature (K) Fe3Al_pure Fe3AlTi Fe3AlZr It is clear that the addition of Ti and Zr leads to an increase on the lattice parameter. It appears also that the effect of Zr addition on the calculated lattice parameter is more important than that of Ti one. This difference is related to the difference in size between the Zr and Ti atoms (the Ti atom is smaller than the Zr one). What is important to stress here is the profile of the curves above 800K. For the case of Ti substitution, the lattice parameter increases linearly up to higher temperature. Comparatively, the rate of increase becomes lower of the pure Fe 3 Al compound and Zr doped compound. This suggests that the Zr impurity does not affect the structural stability of pure Fe 3 Al, from about 800K, despite the large increase that it brings to the lattice parameter. for the pure Fe 3 Al between 900 and 1000K. There is a cross over between the curves corresponding to the Ti and Zr doped compounds between 1000 and 1100K. This means that the In order to explore, in more detail, the effect of these transition metals on the stability of the D0 3 structure we will examine, in the following section, the pair distribution functions for the pure Fe 3 Al structure and those for Ti and Zr-doped Fe 3 Al, repectively. -0,2 0 0,2 0,4 0,6 0,8 1 1,2 1,4 10 2 x II.2.2. Pair distribution function The Pair Distribution Function PDF or pair correlation function, g(r), is a very suitable measure to analyze the structure of material. Its variation with temperature and pressure gives information about the structural phase transition. Considering an homogeneous repartition of the particles in space, the g(r) represents the probability to find a particle in the shell dr at the distance r of another particle (Fig. . By discretizing the space in intervals dr (Fig. , it is possible to compute for a given atom the number of atoms dn(r) at a distance between r and r + dr of this atom: dn(r) = ே g(r) 4π r 2 dr (Eq.IV-4) where N represents the total number of particles, V the volume and where g(r) is the radial distribution function. In this notation the volume of the shell of thickness dr is approximated ( V shell = ସ ଷ π(r + dr) 3 - ସ ଷ πr 3 ≈4π r 2 dr). By distinguishing the chemical species it is possible to compute the partial radial distribution functions g αβ (r): g αβ (r)= ௗ ഀഁ () ସగ మ ௗఘ ഀ with ρ α = ே ഀ = ே୶ ഀ (Eq.IV-5) where c α represents the concentration of species α. These functions give the density probability for an atom of the α species to have a neighbour of the β species at a given distance r. In the following, we have computed the g αβ (r) evolution with temperature (T=100-1000K) in order to investigate the effect of temperature on the structural phase stability of the D0 From the comparison of each profile with the corresponding g(r) at 100K, it can be seen that all the peaks of the g(r) become broader when the temperature increases. This is due to the larger thermal motion of atoms in the supercell. For the case of the g Fe-Fe (r), the noticeable feature is that the characteristic double-peak structure in the range 2.5 Å-3 Å disappears as the temperature reaches 800K. The first peak is located at around 2.5 Å ( √ଷ ସ ܽ, the distance between the FeI and its eight first near negibours FeII atoms). This peak decreases and widens gradually with increasing temperature. However, the peak position is almost unchanged meaning that the distance between the first near neighbours Fe atoms remains constant. The second peak is located at around 2.87 Å ( ଶ , the distance between the FeII and its second near neighbours FeII atoms). With the increase of temperature, its position shows a slight shift towards 3 Å and its height decreases obviously. In the Al-Al PDF curve [Fig. ], the first peak position is around 4.1 Å ( √ଶ ଶ ܽ, the distance between the first near neighbours Al atoms in the D0 3 -Fe 3 Al structure). Its height decreases with the increase in temperature. Furthermore, it can be noticed that a new peak appears around 3.9 Å in the profile of g AlAl (r) from about 800 K. The appearance of the new peak in the g Al-Al (r) indicates that the distances between the first near neighbour atoms decreases as a consequence of the large displacement of the Al atoms in the supercell. This means also that the order of the D0 3 superlattice is affected from this temperature. This result is remarkable as it II.2.2.2. Pair distribution functions for doped Ti and Zr-Fe 3 Al Based on the results obtained from the defect energies calculations, the radial distribution functions were calculated at first time for the Ti and Zr substitutions on the FeI site in the D0 3 structure. For the case of Zr, the pair distribution functions were also calculated for the FeII substitutions at higher temperatures for comparison knowing that the stability of Zr on the FeI and FeII becomes equivalent at higher temperatures. This indicates that the stability of the D03 structure is not affected by the presence of Zr on the FeI site. Knowing that the stability of Zr on the FeII increases site for higher temperatures and tends to be equivalent to that of FeI substitutions, the pair distribution functions were also calculated for the case when the Zr is placed on the FeII site. The the calculated g AlAl (r) for Zr substitutions on the III. Transition metals segregation in ∑5 (310) [001] grain boundary After having determined the differences in the behaviour between of the two transition metals (Ti and Zr) in the bulk, in the following, their stabilities in the ∑5 (310) [001] will be treated. The calculations were performed using a supercell with 80 atoms. More details about the method used for the construction of the grain boundary and the choice of the size of the supercell are Unlike the bulk, each calculation has been turned with 400 dynamic molecular steps in order to gain a compromise between the size of the supercell and the relative running time. The 300 first steps were reserved for equilibrating the system and were discarded from the subsequent analysis. III.1. Site preference of Ti and Zr in the ∑5(310)[001] Because the large number of configurations to be taken into account in the ∑5 (310) [001] grain boundary and the AIMD consuming calculation time, the defect energies of the two transition metal impurities were determined only for three temperatures (300 K, 600 K and 900 K). The defect energies of the two transition metals in the grain boundary were calculated using the Eq. IV-1. However, unlike the bulk, the defect energies were also calculated for the transition metals substitutions on the Al site in addition to the substitutions on the FeI and FeII site. -3,99 - These calculations were carried out to determine the site preference of the transition metals between the substitutions on the Al configurations and then to examine the relaxation of the grain boundary. It is important to recall that the comparisons (i) between the defect energies of the two transition metals and (ii) between substitutions on the Fe and Defect energies (eV) - -0,05 -0,54 -0 Defect energies (eV) 0FeII 1FeII The calculated defect energies of Ti impurity in the ∑5 (310) [001] grain boundary at 300 K, 600 K and 900 K temperatures, for substitutions on (a) sites, by taking into account the distance from the G.B. - ) 600 (K) 900 (K) -4,31 -4,85 - Temperature (K) (b) Ti on the FeI site 1FeI 2FeI -0,61 -1,05 -6,44 -0,94 -3,77 0,43 -3,33 -1,18 -1,38 300 (K (K) Defect energies (eV) 4,76 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 0 Defect energies (eV) 0FeII 1FeII Ab Initio Molecular Dynamics calculations The calculated defect energies of Zr impurity in the ∑5 (310) [001] grain boundary at 300 K, 600 K and 900 K temperatures, for substitutions on (a) Al sites (b) FeI sites and (c) FeII sites, by taking into account the distance from the G.B. - shows that for the three temperatures, the defect energies are generally smaller for the configurations of the FeI substitutions, except for the three configurations 2FeII at 300K (-3.33 eV), 1FeII at 600K (-3.77 eV) and 0FeII at 900K (-6.44 eV) when the Ti impurity occupy the FeII sites with lower defects energies. The fact that the Ti impurity occupy the FeII site near the interface may be related to the effect of the environment knowing that the Ti impurity prefers to occupy the site where is surrounded only by the Fe atoms as first neighbours (i.e. iron rich environment). Note also that, contrary to the bulk, the geometry of the grain boundary has greatly changed as an effect of the temperature which led to changes in the first nearest neighbours of the Ti impurity. The structural environment of the impurities will be discussed below (Section III-2). On the other hand, The classification of the most favourable configurations for the three temperatures, by taking into account the distance from the interface, is Fe substitution in: the second plane at 300K (2FeII)→ the first plane from the interface at 600K (1FeII)→ the G.B interface at 900K (0FeII). This indicates that the stability of the Ti impurity changes with increasing the temperature. While for intermediate temperatures (300K) is stable within configuration close to the bulk, by increasing temperature Ti impurity tends to occupy configuration in the grain boundary interface. In another words, this means that, it takes more temperature for the Ti impurity to be stable at the G.B. interface. For the case of Zr substitutions, it can be seen from interface. Whereas for the case of Ti substitutions, a higher temperature (~900K) was needed to cause its relaxation to the grain boundary. Now, the comparison between the defect energies for substitutions in the FeI and the FeII sites [Figs. and (b)] shows that for intermediate temperature (300K) the Zr impurity prefer to occupy the FeI sites. However, for higher temperatures, the Zr impurity tends to occupy the FeI and FeII sites with nearly equal energies. This indicates that, like in the bulk, the stability of Zr on the FeII site increases with increasing temperature. From the calculated defect energies, it is clear that the two transition metals have different behaviours. While Ti is more stable at the bulk and it takes a higher temperature to relax to the grain boundary, the Zr impurity prefers to segregate at the grain boundary with lower defect energies even at intermediate temperature. III.2. The effect of temperature on the structural relaxation of ∑5 grain boundary In this section, the influence of temperature on the structural deformations of the Ti and Zrdoped grain boundaries will be treated. Before attempting to model the geometrical relaxation of the grain boundary induced by the impurity segregation, it is important to examine the relaxation of the pure grain boundary structure. III.2.1. Relaxation of the clean grain boundary To quantify the relaxation of the atoms we have calculated their displacements in the supercell. The displacements were calculated as difference between the average positions of the last 100 MD steps. We have considered only the last 100 steps after having tested that the movements of the atoms become almost to a stable configuration after relaxation towards the final positions. Clearly one can see that the relaxations with temperature of the grain boundary structure are important when compared to that at 0K. For the temperatures 300 and 600K, the displacements of the atoms are in range 0.2-1.6 Å. Comparatively, for the case of relaxation at 0K, the displacements of the atoms are in between 0 and 0.35Å, i.e. on average 70% smaller than that at 300K. This is due to the large vibrations of atoms as an effect of temperature. Increasing the temperature increases the internal energy and this is reflected in an increase in the average motion of the atoms in the system. For the case of relaxations at 900K, it can be seen that the displacements are more important, and can reach up to 2.5 Å (≈ to the distance between the Fe nearest neighbours atoms in Fe 3 Al) in the interface region. results presented in the literature. Some authors consider indeed, from experimental results of grain boundary relaxation, that pronounced relaxation takes place when the temperature is higher than T 0 ≅ 0.4 T m or so (T m is the melting temperature) [19]. Note, that for the case the Fe 3 Al intemetallic compounds T 0 ≅ 614K ≅ 0.4x1536K. Thus is consistent with our estimation showing a shop change between 600 and 900K. This also indicates that significant local disordering only occurs above this temperature. Also, according to results of dynamic molecular simulation for bicrystals with CSL structure becomes disordered above this temperature [20,21]. This signifies that T 0 is the transformation temperature at which the structure of the grain boundary transforms from CSL structure to disordered structure. Thus in the following, the study will be limited to the structural relaxation at the intermediate temperature of 300K. Figure .IV-21 Average displacements of the atoms in the relaxed supercells at 0K, 300K, 600K and 900K. The relative differences are marked in the graph. As reported by T.S. Ke et al. [22], the coincidence lattice grain boundary constructed on the basis of geometrical considerations can not be stable when the temperature increases, because atomic overlaps or crowding of atoms will be produced in the boundary plane so that atomic readjustment or rigid translations will take place to reduce the energy. Then, although the periodicity of the boundary can be maintained, it is expected that the coincidence sites will no longer coincide with atoms. Especially when the misorientation is large or when there are deviations from coincidence site lattice misorientations, the disordered grain boundary region formed should be complicated. III.2.2. Relaxation of the doped grain boundary at 300K Now we turn to the structures of the doped grain boundaries with the substitutional Ti and Zr transition metal impurities. First of all, we examine the displacement of the impurities in different substitutional configurations. The first point to be made is that the displacements of the impurities at 300K are more important than that at 0K. From Fig.V-23 it can be seen that the displacements of the impurities at 0K are in between 0.1 and 0.6 Å. Comparatively, while for the case of 300K, the amplitudes of displacements is between 0.35 and 1.1 Å. On the other hand, it can be seen that the most significant displacements take place when impurities are incorporated in the first and second planes from the grain boundary interface. These displacements are in between 0.8 and 1.1 Å, which are closer to the distance between two parallel planes (0.92 Å) in the grain boundary supercell. This suggests that the impurities tend to relax to the grain boundary interface. It important to recall that, at 0K, it has been found that the largest displacements of the impurities correspond to the configurations of the substitutions on the FeII sites (Figs. IV-23 (c) and (d)). The reason for this is that the impurities, particularly the Ti impurity, in these configurations (poor iron environment, 4FeI+4Al) tend to relax to an iron rich environment. However, at 300K, it can be seen from Figs. IV-23 (a) and (b), that the displacements of the impurities are important for substitutions on the FeII sites as well as for Al and FeI sites. Additionally, in the first and second planes from the interface, the displacements of the impurities on the Al and FeI sites are larger than that on the FeII ones, despite the fact that these configurations are initially richer in iron. Knowing that the atoms in the grain boundary supercell have moved from their initial positions, it very much possible that the environment around the impurities has changed due to the large displacement of the first nearest neighbours. In this case, it is likely that the impurities also tried to move towards a new iron rich environment. To check this possibility we have analyzed the environment (the first nearest neighbours) corresponding to each configuration after relaxation. and 2FeII), the structural environments becomes similar to that of the bulk. For the case of Ti substitutions, the comparison between the defect energies of the three configurations (i.e. G.B interface, 1 st plane and 2 nd plane) for each type of substitutions (i.e. Al, FeI and FeII) shows that the most stable configurations correspond to the substitutions with an iron rich environments. The three most stable configurations are, depending on the nature of substitutions, substitution on the 0Al site within the G.B. interface with (5FeII+3FeI) as first neighbours, substitution on the 1FeI site in the first plane from the interface with (3FeII-2FeI-2Al) as first neighbours and substitution on the 2FeII site in the second plane from the interface with (3FeII-3FeI-2Al) as first neighbours. On another hand, comparison between the substitutions on the FeI and FeII sites reveals that the most stable configuration corresponds to substitutions on the FeII site in the second plane from the interface (2FeII) with the lower defect energies. This indicates that the Ti, in addition to the nature of the environment, prefers to occupy a configuration close to the bulk. Comparatively, for the case of Zr substitutions, the most stable configurations correspond to the substitutions on the G.B. interface (0Al, 0FeI and 0FII). Except for the case of substitution on the 0FeI site with (5FeII-3Al), the two other configurations correspond to the substitutions with iron rich environments. This indicates that, contrary to the Ti impurity, the Zr atom prefers to reside in the configurations within the G.B. interface whatever the nature of its environment. Configurations IV. Summary and conclusion In this chapter Ab Initio Molecular Dynamic study was set to investigate the effect of temperature on the structural stabilities of Ti and Zr impurities in the bulk and ∑5 (310)[001] of the D0 3 -Fe 3 Al intermetallic compound. The results of defect energies indicate that the stability of Ti on the FeI site increase with temperature. Comparatively, Zr impurities have the tendency to occupy nearly equally the FeI and FeII sites at higher temperatures. The calculated defect energies of vacancies created in the first nearest neighbours to the impurities allow to draw the conclusion that the Ti impurity strengthens the interaction with their Fe first neighbour atoms. However, the Zr additions reduce the interactions with their first nearest neighbours when compared to Fe-Al and Fe-Fe bonds in the pure Fe 3 Al. The calculated pair distribution functions in the pure Fe 3 Al reveal that the structural stability of the D0 3 is affected from about 800K. We show also that, in agreement with the experimental results, there is a direct relationship between the magnetic moment and the lattice parameter of the D0 3 Fe 3 Al intermetallic compounds. The structural disorder of the D0 3 starting from about 800K alters the values of magnetic moment of the Fe atoms. In agreement with the the trends observed experimentally, it is found here that the 1% of Ti increases the stability of the D0 3 structure up to 1000K. Comparatively, the Zr addition does not affect the stability of the D0 3 structure. The calculated defect energies of the impurities in the ∑5(310) grain boundary show that the Ti impurity prefers to reside in an iron rich environment further away the interface at intermediate temperature of 300K. It becomes stable at the G.B. interface only at high temperature (900K). The results show also that, contrary to the Ti impurity, the Zr segregate at the G.B. interface with the lower defect energies even at intermediate temperatures. The relaxation of the ∑5 (310) grain boundary at the temperatures of 300K and 600K is 70% more important than that occurred at 0K. However, the structural geometries of the interface remain unchanged. In agreement with the theoretical assumptions and experimental data for metals, the relaxation of the grain boundary becomes pronounced only above 0.4xT m (T m melting temperature). At the intermediate temperature of 300K, the segregation of impurities in the ∑5 (310) grain boundary affects its relaxation in an irregular manner and a local disorder occurs in the grain boundary interface. Conclusion Combining low density with creep and corrosion resistance, the D0 3 ordered iron aluminides are promising materials for a wide variety of medium to high-temperature structural applications. However, their application is still limited because of their room-temperature intergranular brittleness and their low strength and creep resistance at higher temperatures. Efforts to improve the high-temperature mechanical properties have often included ternary additions to Fe 3 Al in order to extend the temperature range over which the D0 3 phase is stable up to higher temperatures. In this work, the effects of Ti and Zr transition metals on the stability of the D0 3 structure were studied by means of ab-initio calculations. Knowing that, in these compounds, small additions of transition elements, such as Zr, can strengthen the grain boundary cohesion, comparison of the behaviour of these two transition metals when are placed at the ∑5 (310) [001] has been then investigated. Our study was performed both by using the static and Ab Initio Molecular Dynamics to take into account the effect of temperature. The results obtained based on the static ab initio calculations can be summarized as follows : The analysis of site occupancy in the bulk confirms, consistently with the previous literature results, that FeII sites are the preferential sites for vacancies. Comparatively, both Ti and Zr prefer to reside on FeI sites. The positive formation energies calculated for all Zr substitutions suggest however that Zr has very little miscibility in the bulk D0 3 -Fe 3 Al while the 3% of Ti impurities are miscible in Fe 3 Al. The interface energy of a clean ∑5 (310) interface has been found to be (0.36 J/m 2 ). The presence of transition metal impurities on various sites is found to reduce the interface energies by about 14% and 22% for Ti and Zr, respectively. The maximum expected reductions are obtained when the transition metals are located on a FeII site in the first plane away from the exact interface. This suggests that both Ti and Zr doped grains boundaries are more stable than the parent 'clean' grain boundary. The interface energies for the Zr-doped grain boundaries are systematically lower than for the Ti-doped boundary. So that the stability induced with Zr is more important than that of Ti. The interface energy depends also on the relaxation of the multilayers of the G.B interface: the larger the interface expansion, the higher the interface energy. At the grain boundary, the analysis of interstitial configurations indicates that the most favorable sites for Zr and Ti is the one for which this transition metal impurity interacts with only Fe atoms as first neighbors. Interestingly, the negative values of formation energy (-0.18 eV) obtained for Zr indicates that this atom is more stable when inserted on such a site at a ∑5 (310) grain boundary than in the bulk of the material. Comparatively, Ti is clearly more stable within the bulk than inserted at the grain boundary. Contrary to Ti which is not stable as an insertion, the results indicates that Zr is stable within the grain boundary both as an insertion and as a substituting element (on FeI and FeII sites). This brings a high potential for introducing Zr at the grain boundary with a large domain of stability in terms of the exact location within a relaxed grain boundary. The bonding charge distribution in the boundary region is different from that in the bulk due to the different atomic rearrangement. For the clean grain boundary, the bonding normal to the grain boundary develops between the FeI atoms in the (001) planes. Whereas, in the (002) plane, the accumulation of interstitial bonding charge across the neighbours FeII pair is parallel to the grain boundary. The T.M. impurities (when substituted) in the most stable configuration are found to reduce the general bonding with their first neighbor FeI atoms in the (004) plane and enhance the bonding charge normal to the interface between the FeII atoms in the pure FeII plane. Therefore the beneficial effect of the impurities on the cohesion of the ∑5 (310) [001] grain boundary may originate from the FeII-FeII covalent bonding normal to the interface in the pure FeII planes which holds the two grains together. Selected results from the ab initio molecular dynamics are summarized below: The analysis with temperatures (100-1100 K) of the site preference for substitutions of Ti and Zr impurities on FeI and FeII sites of the Fe 3 Al has revealed interesting differences between the two transition metal elements. Ti is more stable on a FeI site and the stability of this site increases with temperature. Comparatively, Zr impurities have the tendency to occupy nearly equally the FeI and FeII sites at high temperature. The calculated bond lengths between the impurities and their first nearest neighbours show that the Ti impurity induces a strain on their neighbouring Al atoms when it is placed on the FeII site. Complementary, it was found that it is rather more expensive to create a vacancy on the Fe site first nearest neighbour to the Ti impurity than on the Al one. This indicates that the Ti impurities strengthen the interactions with their first Fe nearest neighbours atoms. However, for the case of Zr additions, the vacancies are favoured both on the Fe and Al sites. Therefore, the Zr impurities reduce the interactions with their first nearest neighbours in the Fe 3 Al structure. The appearance of new peaks in the Al-Al pair distributions function, calculated for the pure The relaxation of the grain boundary becomes significant as an effect of temperature. It was found that the relaxation at intermediate temperatures is 70% more important than that occurred at 0K. However, although the large displacement of the atoms, the structural stability of the ∑5 (310)[001] grain boundary maintains unchanged. From about 600K the relaxation becomes pronounced, 90% more important than that at 0K, and the disorder occurs in the grain boundary supercell. Based on the theoretical and experimental assumptions this temperature corresponds to the temperature transformation T 0 ≅0.4xT m (T m , melting temperature) of the CSL grain boundary structure to a disordered structure. Thus, in our study, the analysis of the structural geometries induced by the segregated impurities has been limited to the intermediate temperature of 300K. It has been found that the segregated impurities (Ti and Zr) affect the relaxation process of the grain boundary. The relaxation of both Ti and Zr doped-grain boundaries occurs in irregular manners. However, despite the bigger size of the Zr impurity, it has been found that, the distortions created by Ti addition are more important than that produced in the case of Zr segregation. The reason is because the Ti impurity tends to strengthen the interactions with their first neighbours Fe atoms. On a more general basis, the work carried out in this PhD brings additional information on the complex field dealing with the understanding of the effect of transition metal additions in intermetallic coumpounds. Indeed, it is worth remembering that the iron aluminide based intermetallic compounds still presents several open questions, despite the numerous experimental and theoretical studies in the literature. For example, it is well known that their ductility and fracture toughness can be modified by addition of ternary elements and that the associated grain boundary segregation can change the fracture mode in several compounds of the FeAl systems. However, little is known from the microscopical point of view on dislocation nucleation, mobility or pile-up at grain boundaries and the influence of interstitial and substitution solutes. A lot of work is still required before mastering the modeling of crack tip plasticity or the atomistics of brittle fracture in the presence of ductilizing additives. The type of work carried out in this thesis is just one of the numerous stages to be done to understand, in the future with the combination of the experimental investigation, the effect of segregated elements on the ductility/hardening of the FeAl intermetallic compounds. Our work, using first principal density functional theory calculation, has shown that Ti and Zr are two transition metals having significantly different effects in Fe 3 Al. Their site preferences with temperature have been determined in the bulk and for a specific ∑5 grain boundary. Such information on the location of site defects and the way different transition metals relax a grain boundary is the type of important information that will be required by mechanical engineers and metallurgists dealing with dislocation interactions to understand plasticity (or lack of plasticity) in these alloys. Fig. 1 The description of tilt GBs in the framework of edge dislocations. A model which has received much success in the interpretation of experimental data is the coincidence site lattice (CSL) model [4,5]. This is the model upon which the present work is based. Thus it seems necessary to explain in more detail the basic idea of the CSL. The coincidence site lattice In general the CSL represents lattice points of two discrete lattices that coincide if one imagines that the two lattices interpenetrate each other. If both crystals have the same crystal structure and are aligned without any misorientation or translation then the CSL would be the underlying lattice of the material. Introducing a misorientation between the two lattices described by a rotation matrix R will only give CSLs for certain misorientation angles and axes. For a better understanding of the underlying mathematics let us recall here that the position of an atom i can only be found in real space if its coordinates (x i , y i , z i ) as well as the unit vectors, e x , e y and e z , spanning out the actual coordinate system are known. Here r i = (x i ,y i ,z i ) only gives the coordinates of atom i with respect to a certain coordinate system and its real space vector would be given by r i = x i e x + y i e y + z i e z . The same is true for two misoriented crystals that do interpenetrate each other. Here this means that for either of the two crystals the internal coordinates (x i , y i , z i ) of any atom i are exactly the same when bearing in mind that they are defined with respect to each coordinate system. This now offers a simple way of mathematically expressing the basic equation for coincidence, namely writing ݎ ଶ ൌ ܴ. ݎ ଵ ൌ ݎ ଵ ݐ ଵ (Eq. 1) where r L1 represents the internal coordinates of a certain lattice point with respect to coordinate system No. Here one should note that the CSL is confined to descrite lattice points. In order to find the vectors that span out the CSL, Eq. 1 needs to be rewritten and thus one obtains Eq. 2. Inserting the translation vectors of lattice No.1 in Eq. 1 would then give the CSL vectors. ܴ . ݎ ଵ െ ݎ ଵ ൌ ሺܴ െ 1ሻ ݎ ଵ ൌ ݐ ଵ ⇔ ݎ ଵ ൌ ሺܴ ିଵ െ 1ሻ ݐ ଵ (Eq. 2) Extending the discussion to any point r L1 and r L2 within the lattices leads to the concept of the O-lattice [6]. It is a more general concept than the CSL and the CSL is a sub-lattice of the Olattice. There are several different approaches to describe a misorientation relationship between two coordinate system. In this work the misorientation relationship is expressed by a rotation axis ሾ,ܪ ,ܭ ܮሿand a rotation angle θ. For certain misorientation relationships between two crystals depending on the rotation axis and angle well-defined CSL lattices exist that are characterized by a single parameter namely their ∑ values. Since this work only deals with cubic materials, the further discussion is restricted to cubic materials. Well-defined misorientations between both crystals can for instance be expressed by a rational Rodrigues vector ρ ൌ ሾ,ܪ ,ܭ ܮሿ , where m, n, H, K and L are integers [6]. Here the misorientation angle is not explicitly set but rather defined through the distinct set of m, n, H, K and L. In terms of rotation axis and angle the Rodrigues vector represents a rotation about an axis ሾ,ܪ ,ܭ ܮሿ by θ where θ is given by ݊ܽݐ ቀ θ ଶ ቁ ൌ ܪ√ ଶ ܭ ଶ ܮ ଶ (Eq. 3) Furthermore the ratio of the volume of the primitive CSL with respect to the atomic volume of the material is characterized by ∑ and formally given by ∑ ൌ n ଶ m ଶ . ሺH ଶ K ଶ L ଶ ሻ (Eq. 4) Eq. 3 and 4 can be used as master equations to calculate the misorientation angle θ and ∑ for a chosen rotation axis ሾH, K, Lሿ and m and n. If ∑ as given by Eq. 4 is an even number, ∑ is then to be divided by 2β with β being the smallest integer to find the largest odd ∑ number. The CSL scheme itself is not unique concerning the misorientation path. Theoretically a given CSL can be generated by 24 different misorientation paths for cubic materials due to the symmetry operations possible for cubic materials [7] Concerning the nature of GBs some general statements can be made with respect to the CSL procedure if one restricts the discussion to one distinct CSL misorientation representation out of the 24 possible. Generally it can then be stated that a pure twist GB follows the rule that its GB normal and the misorientation axis are parallel. For pure tilt GBs a similar rule exists, namely that the GB normal and the tilt axis are perpendicular to each other. These rules are rather simple and once again illustrated by Fig. 3 (a) and (b) and for any pure twist or tilt GB the rule will apply for at least one of the 24 misorientation representations. Résumé en français Les alliages intermétalliques riches en fer du système fer-aluminium, Fe I. Détails de calculs I.1. Méthodes de calculs Nos calculs ont été effectués en utilisant les pseudopotentiels de Vanderbilt (UltraSoft USPP) [1] II. Résultats des calculs statiques à 0K II. 1. Défauts ponctuels dans le bulk-Fe 3 Al Dans cette section les résultats des défauts ponctuels dans le bulk D0 3 -Fe 3 Al seront présentés. Les trois types de configurations de substitution des impuretés ou formation de lacunes II.1.1. Importance de la relaxation Afin d'avoir un aperçu sur l'effet de relaxation et de souligner son importance sur les calculs des intermétalliques, Fig. [13] et à partir de calculs de diagramme de phase disponibles dans la littérature [14]. Le calcul montre également que la configuration la plus stable pour les Ti correspond à la substitution dans le site FeI, avec les plus basses énergies de formation efficace les deux types de calcul (de concentration par exemple). Il s'agit de la prévision correcte du comportement observé expérimentalement [15]. Pour le cas de Zr, la configuration la plus favorisée est également la substitution sur un site II.2.2. Sites préférentiels et l'effet de Ti et Zr sur la cohésion des joints de grains Les énergies de formations des deux métaux de transition ( III.2. Site préférentiel de Ti et Zr dans le joint de grains Σ5 (310) [001] A cause du grand nombre de configurations à prendre en compte dans le joint de grains Σ5 (310) [001] le temps prohibitif de calcul AIMD, les énergies de défauts des deux métaux de transition ont été déterminés que pour trois températures (300 K, 600 K et 900 K). Les énergies de défauts des deux métaux de transition dans les joints de grains ont été calculées en utilisant l'Eq. 6. Cependant, contrairement au bulk, les énergies de défauts ont été également Fe 3 3 AlC structure, with the Strukturbericht Designation E2 1 (a perovskite-type structure). This carbide is based on the fcc ordered structure Fe 3 Al-L1 2 where the iron atoms are located in the center of each face, and the aluminium atoms sit on the corners of the cube (see Fig. I-8). The carbon atom occupies the central octahedral interstitial position formed by the six iron atoms as first nearest neighbours. Figure.I- 8 8 Figure.I-8 Conventional cell of the k-Fe 3 AlC carbide. The carbon atom is represented in white in octahedral position, the aluminium atoms in grey and the iron atoms in dark-grey. Fe 3 3 Al crystallizes in a D0 3 -type structure shown in Fig. I-10. In this structure, there are two inequivalent Fe sites with specific neighbor configurations, which are named FeI and FeII sites. The former has eight Fe nearest neighbors in an octahedral configuration, and the latter has four Fe and four Al nearest neighbors in a tetrahedral configuration. It is known in Fe 3 Si, which is isomorphous with Fe 3 Al, that transition-metal impurities occupy the FeI or the FeII site Figure.I-13Lattice parameters of the D0 3 phase in (Fe 1-x M x ) 3 Al as a function of composition x for M= Ti, V, Cr, Mn and Mo. The D0 3 single-phase state is obtained in the range shown by the solid lines[41]. Fig. I-14 shows the D0 3 -B 2 transformation temperatures T 0 in (Fe 1-x M x ) 3 Al with M=Ti, V, Cr, Mn and Mo as a function of the average electron concentration e/a. The D0 3 -B 2 transformation temperature always increases at the values of e/a lower than 6.75 for Fe 3 Al since, the Fe atoms are substituted by the transition elements with less than half-filled d states. As mentioned in Section III-4.2, the increasing rate of T 0 for the addition of Mo is lower than that of V. In contrast, as shown in Fig. I-14, the curve of the Mo addition almost coincides with that of the V addition in the range of the D0 3 single-phase state when plotted as a function of e/a. The electron concentration effect further demonstrates that the curves for M= V and Mo are rather close to that for M= Ti, despite the sharpest increase in T 0for the Ti addition. Consequently, the variation of the electron concentration plays a dominant role in determining the values of T 0 . The situation is less clear for M= Cr, since the site preference of Cr is not as clear as in M= Ti, V and Mo. Finally, it was also suggested that the electron concentration effect could predict the increase in T 0 even for the additions of two or more kinds of transition elements, if these complex alloys have a D0 3 single phase structure. Figure.I- 14 D0 3 - 143 Figure.I-14 D0 3 -B2 transformation temperatures in (Fe 1-x M x ) 3 Al as a function of average electron concentration (e/a) for M=Ti, V, Cr, Mn and Mo [49]. D0 3 3 →B 2 (T c ) and consequently the strength of the Fe 3 Al intermetallic shows that the increases in T c is related to the increases in the ordering of the D0 3 superlattice caused by specific site substitutions by the solutes. The most effective solutes in raising T c , (Ti , V and Mo) have been found to occupy the FeI site in the D0 3 structure. The situation is different and in contradictions for Cr additions. Whereas for the case of Zr, there is no experimental data about it site preference in the D0 3 Fe 3 Al. Additionally, the majority of the calculations for the intermetallic compounds are performed at zero tempreture. The effect of temperature rarely used because of the time consuming calculations. Stein et al. have measured the flexural fracture strains of as-cast samples as a function of temperature in four-point bending tests, in order to test their Britle-Ductile Temperature Transition BDTT. The results show that the BDT temperatures of the ternary alloys strongly increase with increasing volume fraction of second phase τ1. The effect of the Al content on the BDT of the investigated ternary Fe-Al-Zr alloys is qualitatively very similar to that of binary Fe-Al alloys even in the presence of 50 vol% of second phase τ1. At Al contents below 40at.% there is only a very weak dependence of the temperature range of the BDT on Al content, whereas the alloy series with 40at.% Al shows a strong increase of the BDT temperatures. Figure.I- 16 16 Figure.I-16 Flexural fracture strains as a function of temperature for two series of Fe-Al-Zr alloys (a) with varying Al content in Fe-Al+50 vol% Laves phase alloys and (b) with varying volume fraction of the second phase in Fe-40AlCx vol% t1 phase (the arrows indicate fracture strains above 3%). one-particle orbitals ψ are orthonormal ർψ ቚψ = ߜ . The corresponding constraint minimization of the total energy with respect to the orbitals min ൛ൻψ หܪ หψ ൿൟห ቄർψ ቚψ ೕ ୀఋ ೕ ቅ (Eq. A-29) can be cast into Lagrange's formalism ܮ = -ൻψ หܪ หψ ൿ+∑ ߉ (ർψ ቚψ , δ ) (Eq. A-30) where ߉ are the associated Lagrangian multipliers. Unconstrained variation of this Lagrangian with respect to the orbitals ఋ ఋψ * = 0 (Eq. A-31) leads to the well-known Hartree-Fock equations ܪ ுி ψ = ∑ ߉ ψ (Eq. A-32) The diagonal canonical form ܪ ுி ψ = ߳ ψ is obtained after a unitary transformation and ܪ ுி denotes the effective one-particle Hamiltonian. The equations of motion corresponding to Eqs.(A-26)-(A-27) read Car and Parrinello postulated the following class of Lagrangians [ 31 ] 31 ൿ -ൻψ หܪ หψ ൿ + ݏݐ݊݅ܽݎݐݏ݊ܿ (Eq. A-35) to serve this purpose. The corresponding Newtonian equations of motion are obtained from the associated Euler-Lagrange equations mechanics, but here for both the nuclear positions and orbitals; note ψ * = 〈ψ | ensemble averages. Statistical ensembles are usually characterized by fixed values of thermodynamic variables such as energy, E; temperature, T; pressure, P; volume, V; particle number, N; or chemical potential µ. One fundamental ensemble is called the microcanonical ensemble and is characterized by constant particle number, N; constant volume, V ; and constant total energy, E, and is denoted the NVE ensemble. Other examples include the canonical or NVT ensemble, the isothermal-isobaric or NPT ensemble, and the grand canonical or µV T ensemble. Thus, Eqs. (B-8)-(B-10) together with Eqs. (A-33)-(A-34) define Born-Oppenheimer molecular dynamics within Kohn-Sham density functional theory. The functional derivative of the Kohn-Sham functional with respect to the orbitals, the Kohn-Sham force acting on the orbitals, the connection to Car-Parrinello molecular dynamics, see Eq. A-39. Fig. II-1. It is essential to make the supercells large enough to prevent the defects, surfaces or molecules in neighboring cells from interacting appreciably with each other. The independence of the configurations can be checked systematically by increasing the volume of the supercell until the computed quantity of interest has converged. Figure. II- 2 . 2 Figure. II-2. Schematic illustration of a supercell geometry (a) for a vacancy in a bulk crystalline solid, (b) surface, and (c) for a isolated molecule. The boundaries of the supercells are shown by dashed lines. 61 points 61 is required to get good converged results. For increasing size of the supercell the volume of the Brillouin zone becomes smaller and smaller (see Eq. Bints are needed. From a certain point, which is usually taken to be k=0 point approximation). For metallic systems, on the other hand, much denser k-point meshes a precise sampling of the Fermi surface. In these cases the point density can often be accelerated by introducing Sham equations assume a 26), multiply from left with we get the matrix eigenvalue equation ᇱ (Eq. B-31)25) of the wave functions is truncated by) with a kinetic energy lower than a given cutoff (Eq. B-32) Generalized Gradient Approximation formulated by Perdew-Wang functional (GGA-PW91) and the UltraSoft PseudoPotential (USPP) method and plane wave basis set. These calculations were carried out using the Vienna Ab initio Simulation Package (VASP). The VASP code is developed at the Institute fur Materialphysik at the University of Wien by Kresse, Furthmüller and Hafner. More details about the calculation methods will be given in the beginning of the two following chapters. Fig. III- 1 1 Fig. III-1 shows the total energy plotted, as function of the volume of the unit cell, for the GGA and LDA functionals. The solid lines are the result of fit to Birch-Murnaghan equation of state[8]. The equilibrium lattice constants and bulk modulli were determined from these fitted curves. Figure. III- 2 2 Figure. III-2 (a) The un-doped structure of the bulk Fe 3 Al and the three point defects substitutions as well as vacancies on (b) the Al site (c) the FeI site and (d) the FeII site. Fe 3 3 Al was obtained using geometrical rules of the Coincidence Site Lattice model (CSL). An overview about the CSL theory is given in Appendix A. Fig. III-4 gives a view of the resultant cell showing the symmetry of the ∑5 (310) [001] grain boundary. Along the [001] direction, the supercell contains in fact four alternating (001) layers separated by the fourth lattice constant, a 0 . The four layers consist of two pure FeII layers and two mixed FeI/Al layers as shown in Fig. III-4 (b). The cell size was chosen in order to preserve a large amount of bulk crystal between the two interfaces visible in Fig. III-4 (a), and, thereby, reasonable energy convergence. Several calculations were made preliminarily in order to estimate the sufficient number of planes.Following these energy convergence calculations, it was estimated that 20 planes parallel to the grain boundary plane were required. This configuration leads to a total of 80 atoms per calculation cell. It must also be noted that, considering a grain boundary region having a thickness of five atomic planes, the local concentration of impurities at the grain boundary when substituting one atom is about 1.25 at.%. Figure. III- 4 4 Figure. III-4 Atomic structure of the Fe 3 Al ∑5 (310) [001] grain boundary (a) viewed along the [001] direction (b) viewed along the (130) direction. The D0 3 structure being more complex than the B2 one [26], the number of G.B. defects to be taken into account is significantly larger. The location of the G.B. defects is given in Fig.III-5. Figure. III- 5 5 Figure. III-5 Typical (a) substitutional sites and (b) the interstitial sites within the G.B.interface. 4 are also plotted in Figs. III-6 (a) and III-6 (b) by taking into account the distance from the G.B interface (Fig. III-5). For comparison the formation energies of the transition metals substitutions in the bulk (supercell with 108 atoms) are represented in Figs. III-6 (a) and III-6 (b). Figure. III- 6 6 Figure. III-6 Impurity formation energies (in eV) for different substitution sites in (a) Titanium doped systems and (b) Zirconium : Static ab initio calculations(0K) Figure. III- 8 Figure. III- 9 : 89 Figure. III-8 Difference charge density of bulk Fe (b) the (001) pure FeII plane. Positive (negative) contours represent contours of increased (decreased) charge density. Contours start from ±0.25 )] planes the charge density is also 9 (a) and (b) shows the charge density difference for the grain boundary ∑5(310)[001] 5 (310) clean grain boundary, on (a) the (004) FeI/Al mixed plane and (b) the (001) pure FeII plane. Positive (negative) contours represent contours of increased (decreased) charge density. Contours start from ±0.25 e/(Å) 3 and 9 (a) that in the grain boundary a depletion of up of charge density at the ionality in FeI atoms further away from the interface has changed as results of misorientation of the crystals by an angle of 36.8° (the angle of the ∑5 (310) grain boundary), except for one atom in the third plane from the interface who try to create bonding nearest FeI atom in the first plane from the grain boundary interface. The bonding charge distribution in the boundary region is different from that in the bulk due to interstitial bonding charge between FeI atoms across the boundary plane [Fig. III-9 (a)], increases the covalent bonding normal to the grain boundary. This indicates that the Fe atoms have an The charge density difference on the (001) plane in Fig. III-9 (b) shows a different charge distribution between the nearest neighbour FeII-FeII atoms across the grain boundary plane. The bonding parallel to the interface develops between the FeII (refer to rectangle) atoms, which contributes very little, if any, to the grain boundary cohesion. This accumulation is within very thin range and extends only about 0.2 Å away from the grain boundary plane. In the first plane from the interface, it can be seen that the FeII atoms [Fig. III-9 (b)] have different charge density distributions. This is related (due) to the effect of relaxation as will be demonstrated in the Section III-5. During the relaxation the atoms moves from their initial positions, thus their charge density distribution in the vertical sections to the interface will be different. This is because in the vertical section [Fig. III-9 (b)] only a portion of total charge density are represented. Figure. III- 10 10 Figs. III-10(a) and III-10(c) represent the redistribution of bonding charge density for doped Tigrain boundary and the Figs III-10(b) and III-10(d) are for doped Zr-grain boundary. Fig. III-11 (a) represents the calculated magnitude of displacements of the atoms. The magnitudes of the displacements of the atoms are obtained for each configuration by subtracting the un-relaxed atomic positions from the relaxed ones. The positions of the various atoms (Al, FeI and FeII) are given on the x axes depending on their location (n) away from the exact interface at n=0. Due to the cell symmetry 0 and -/+10 label the two interfaces in the supercell. From Fig. III-11, it can be seen that the larger values of displacements correspond to atoms located in the vicinity of the interface. This is particularly true for the first and second planes labeled -/+1, -/+2 as well as -/+8 and -/+9. Note also that, the rate of the displacement decay away from the grain boundary interfaces (or towards the bulk). The displacement of each atom from its initial position is also represented by solid arrow in Fig. III-11 (b). The large displacement of the atoms in the first and second planes from the grain boundary interface is clearly visible with the largest arrows. As pointed out in previous study by Wolf et al.[33], the large atomic displacement seen in the grain boundary interface is mainly due to the strong repulsive forces between the counter atoms in both sides of the mirror plane. This gives rice to a shift of the first and second plane parallel to the interface causing an expansion of the grain boundary. It was found that the grain boundary expansion affects significantly the kinetics of formation and migration of point defects as well as the interaction between lattice dislocations and grain boundaries[34, 35]. Figure. III- 11 III. 5 . 115 Figure. III-11 (a) The calculated displacements of the atoms in the relaxed pure-grain boundary as function of distance from (0 and -10 label two interfaces in the supercell). These displacements are represented with solid arrows in (b) III.5. The relaxation of the doped grain boundary Figure. III- 12 12 Figure. III-12 The displacement of the impurities (a) Ti and (b) Zr in different configurations of substitutions. As seen from Figs. III-12 (a) and (b), for both Ti and Zr impurities the largest displacements correspond to the substitutions on the FeII sites. This may be related to the environment effect.In the FeII substitutions, the impurities are surrounded by four FeI and four Al atoms. Based on the assumption that both Ti and Zr impurities prefer to reside on the FeI site where are surrounded by eight FeII atoms, their large displacements reveals that the impurities tend to relax to an iron rich configuration. Figure. III- 13 13 Figure. III-13 (a) The displacements of the atoms in the relaxed supercell with solid arrows for the case of Zr substitution on the 2Al site. These displacements are represented with solid arrows in (b). Figure. III- 14 14 Figure. III-14 (a) The displacements of the atoms in the relaxed supercell with solid arrows for the case of Ti substitution on the 2Al site. These displacements are represented with solid arrows in (b) For the case of Figure. III-15 (a) The displacement of the atoms in the relaxed supercell with Ti substitutions on the 0Al site as function of the positions of the planes parallel to the grain boundary interface. These displacements are represented with solid arrows in (b). Figure. III- 16 16 Figure. III-16 (a) The displacement of the atoms in the relaxed supercell with Zr substitutions on the 0Al site as function of the positions of the planes parallel to the grain boundary interface. These displacements are represented with solid arrows in (b). Figure . . Figure. III-17 (a) The displacement of the atoms in the relaxed supercell with Ti substitutions on the 1FeI site as function of the positions of the planes parallel to the grain boundary interface. These displacements are represented with solid arrows in (b). Figure. III- 18 18 Figure. III-18 (a) The displacement of the atoms in the relaxed supercell with Zr substitutions on the 1FeI site as function of the positions of the planes parallel to the grain boundary interface. These displacements are represented with solid arrows in (b). Figure . . Figure. III-19 (a) The displacement of the atoms in the relaxed supercell with Ti substitutions on the 2FeII site as function of the positions of the planes parallel to the grain boundary interface. These displacements are represented with solid arrows in (b). Figure. III- 20 20 Figure. III-20 (a) The displacement of the atoms in the relaxed supercell with Zr substitutions on the 2FeII site as function of the positions of the planes parallel to the grain boundary interface. These displacements are represented with solid arrows in (b). Fig. IV-1 from the calculations at 100 K. The deduced lattice parameter at 100K is indicated by red arrow in Fig. IV-1. Figure. IV- 2 . 2 Figure. IV-2. The energies differences (in eV) between the substitutions Fig. shows the calculated distances between the Zr atom, when placed in the FeII site, and its first neighbour FeI and Al. As seen in Fig.IV-4, both the Zr-FeI and Zr-Al bond lengths are larger when compared to the FeII-FeI and FeII-Al ones in the pure supercell over the entire temperature range. This may be related to the size effect, knowing that the atomic radius of Figure. IV- 3 . 3 Figure. IV-3. The bond lengths in the relaxed pure and Ti-doped Fe 3 Al supercells. Figure. IV- 4 . 4 Figure. IV-4. The bond lengths in the relaxed pure and Zr-doped Fe 3 Al supercells. Fig. IV- 5 5 Fig. IV-5 represent the calculated lattice parameters of D0 3 -Fe 3 Al in the temperature interval 100 -1100 K with the increment ∆T=100 K. The value obtained at 0K from the static ab initio calculations are also ploted (5.76Å). Figure. IV- 5 . 5 Figure. IV-5. The calculated lattice parameters at different temperatures for pure D0 3 -Fe 3 Al. The error bars represent the statistical uncertainty. Fig. IV- 7 7 Fig. IV-7 shows the temperature dependence of the fractional length change ܶ(ܮ/ܮ∆ ) = )ܶ(ܮ[ -ܶ(ܮ ܶ(ܮ/]) ) where the reference of temperature T=300K, together with the previously experimental[10] and theoretical[11] data. Figure. IV- 7 . 7 Figure. IV-7. Fractional length change data of D0 3 -Fe 3 Al. Fig. IV- 8 Figure. IV- 8 . 88 Fig. IV-8 shows the temperature dependence of the lattice parameter with Ti and Zr substitutions in FeI sites, respectively. The lattice parameters of the pure Fe 3 Al are also represented for comparison. Fig. IV- 9 9 Fig. IV-9 shows the variation of the fractional length ܶ(ܮ/ܮ∆ ) as a function of temperature for the pure Fe 3 Al, Ti-doped Fe 3 Al and Zr-doped Fe 3 Al when the transition metals are placed in on a FeI site. Figure. IV- 9 . 9 Figure. IV-9. Fractional length change data of D0 3 -Fe 3 Al, Ti-doped Fe 3 Al and Zr-doped Fe 3 Al (Ti and Zr are placed on the FeI site). As seen from Fig.IV-9, the values of the fractional length for Ti-doped Fe 3 Al are smaller than that of the pure and Zr-doped Fe 3 Al. This means that the thermal expansion of Ti-Fe 3 Al is lower than that of the pure and Zr-doped Fe 3 Al. Interestingly, the variation of the Ti doped compound linear up to the highest temperature of 1100K while a change in slope is visible for the case of pure and Zr-doped-Fe 3 Al compounds. It is clearly visible, from Fig. IV-9, that the evolution of the three fractional lengths remain rather close up to 800K. The discontinuity starts to take place and Zr on thermal expansion of the Fe 3 Al are different reflecting their different effects on the stability of the D0 3 -Fe 3 Al above 800K. These trends are consistent with the results depicted in Fig. IV-8 for the lattice parameters. Figure. IV- 10 . 10 Figure. IV-10. Space discretization for the evaluation of the radial distribution function. 3 - 3 Fig. IV-11 shows the PDFs g(r) in D0 3 -Fe 3 Al for the Fe-Fe [Fig. IV-11(a)], Al-Al [Fig. IV-11(b)] and Fe-Al [Fig. IV-11(c)] pairs. points to a direct correlation with the experimentally observed transition phase (D0 3 -B 2 ) that occurs around 820K in the Fe 3 Al compound. As shown in Fig. IV-11 (c), the disappearance of the second peak from 800K is also noticed in the g FeAl (r) pair distribution function. Figure. IV- 11 . 11 Figure. IV-11. Pair distribution functions (a) Fe-Fe (b) Al-Al and (c) Fe-Al pairs for the pure D0 3 -Fe 3 Al, in the temperature range of 100-1000K. Figs Figs. IV-12 and IV-14 represent the PDFs g(r) in D0 3 -Fe 3 Al with Ti and Zr substitutions in the FeI site, respectively, for (a) Al-Al, (b) Fe-Fe and (c) Fe-Al pairs. Figure. IV- 12 . 12 Figure. IV-12. Pair distribution functions (a) Fe-Fe (b) Al-Al and (c) Fe-Al pairs for the Tidoped D0 3 -Fe 3 Al (Ti on FeI site), in the temperature range of 100-1000 K. Figure. IV- 13 . 13 Figure. IV-13. The g AlAl (r) for pure and Ti-doped D0 3 Fe 3 Al (on the FeI site) at 600K, 800K and 1000K. Figure. IV- 14 .Fe 3 143 Figure. IV-14. Pair distribution function (a) Fe-Fe (b) Al-Al and (c) Fe-Al pairs for the Zrdoped D0 3 -Fe 3 Al (Zr on the FeI site), in the temperature range of 100-1000K. FeI figure, the profile curves of the two PDFS shows that there is no significant difreneces between the substitutions on the FeI and FeII sites. This indicates that the presence of Zr whether on a FeI site or a FeII site has no effect on the stability of the D0 3 -structure. Figure IV- 15 . 15 Figure IV-15. the gAlAl(r) for pure and Zr-doped D0 3 Fe 3 Al on the FeI site and FeII sites at 600K, 800K and 1000K. described in the previous chapter (Chapter III, Section III-1). The configurations of the transition metals substitutions considered in our calculations are given in the Fig. IV-16. These configurations are grouped in three categories (i) three substitutions in the grain boundary interface (ii) three substitutions in the first plane from the interface and (iii) three substitutions in the second plane from the interface. Figure. IV- 16 . 16 Figure. IV-16. Sketch showing the nine substitutional sites located within three different planes the within ∑5 (310)[001] grain boundary. Al sites are not possible. The results are represented in Figs. IV-17 and IV-18 for substitutions on the (a) Al sites (b) FeI sites and (c) FeII sites, by taking into account the distance from the G.B. interface. For comparison, the values obtained from the static calculations at 0K (Chapter III) are also recalled. Figure. IV- 17 . 17 Figure. IV-17. The calculated defec boundary at 300 K, 600 K and 900 K t and (c) FeII sites, by taking into account the distance from the G.B. ∑5 (310) [001] grain n (a) Al sites (b) FeI sites sites, by taking into account the distance from the G.B. interface. Figure. IV- 18 . 18 Figure. IV-18. The calculated defect energies of Zr boundary at 300 K, 600 K and 900 K t and (c) FeII sites, by taking into account the distance from the G.B. ∑5 (310) [001] grain n (a) Al sites (b) FeI sites and (c) FeII sites, by taking into account the distance from the G.B. interface. As shown in Fig. IV-17 (a) for Ti on Al configuration, for the three temperatures, the most stable configuration corresponds to substitutions in the G.B. interface. For the case of Fe configurations, a comparison between the substitutions of Ti on the FeI and FeII sites [Figs. IV-17 (b) and (c)] Fig.IV-18 that the most favorable configurations correspond to the substitutions in the G.B. interface (0Al, 0FeI and 0FeII) for the three temperatures. Contrary to the case of Ti configurations at the grain boundary, the Zr impurity prefers to reside in the G.B. interface even at intermediate temperature. It is important to recall that, from the static calculations at zero temperature [Fig. IV-18], the most favorable configurations were found to be the substitutions in the first plane from the grain boundary interface (1Al, 1FeI and 1FeII) for the two transition metals. Therefore, the small increase in temperature (to the intermediate temperature) leads to the migration of Zr impurity to the G.B. For example, Fig. IV-19 represents the positions of atoms during the relaxation, at 300K, for (a) 400 molecular dynamic (MD) steps (b) the last 200 MD steps and (c) the last 100 MD steps. It is clear from Fig.IV-19 (b) that the relaxation of atoms is less important when the 200 first steps were removed. The effect of relaxation is even less important when only the last 100 steps are considered [Fig. IV-19 (c)]. This suggests that the atoms have relaxed to their final positions after the 300 first steps of thermalization of the system. Fig.IV- 20 20 Fig.IV-20 represents the displacements of the atoms calculated from the static calculations at 0Kand that from AIMD calculations for the three different temperatures 300K, 600K and 900K. It is important to note before further analysis that the scale indicating the magnitude of the displacement vectors are rather different between 0K and 300K-600K and 900K. Figure.IV- 19 . 19 Figure.IV-19. The atomic positions in pure grain boundary ∑5 (310) for (a) 400 MD steps (b) the last 200 MD steps and (c) the last 100 MD steps. Figure.IV- 20 . 20 Figure.IV-20. The displacement of the atoms from in the relaxed grain boundary supercell from their initial positions (a) at 0K (b) at 300K (c) at 600K and (d) at 900K. Fig. IV-21 gives the relative difference between the average displacements at each temperature and the average positions in the relaxed grain boundary at 0K. On can see that the displacement of atoms at 900K are very important, about ~90%. This indicates that a local disorder occurs in the grain boundary suprecell at this temperature. This trends is consistent with the experimental and theoretical Fig. IV-22 gives the relaxed pure grain boundary (a) at 0K and (c) at 300K. The displacements of the atoms from their initial positions are represented in Fig. IV-22 (b). At first sights, the magnitude of the displacement seems such that the center of the simulation box (away from the G.B) has a different aspect. This different aspect is in fact partially related the fact that the image of the grain boundary in Fig. IV-22 (c) correspond to the superposition of four different planes contained in the depth of the simulation box and that even the small displacement of the atom position leads to a higher number of atoms visible in Fig. IV-22 (c). However, comparison between Fig.IV-(a) and (c), shows that despite the apparent important relaxation of the atoms from their initial positions, the global structure of the ∑5 (310) is preserved. In particular, the misorientation angle of the grain boundary is not affected. Figure. IV- 22 . 22 Figure. IV-22. (a) The initial positions of the relaxed grain boundary at 0K (b) the displacement of the atoms from the initial positions in the relaxed grain boundary at 300K and (c) the final positions of the atoms in the relaxed grain boundary at 300K. Fig. V-23 (a) and (b) shows the calculated displacement of the impurities in different substitutional configurations. In Fig. V-23 the configurations are classified based on their positions from the grain boundary interface. For comparison, the calculated displacements of Ti and Zr impurities at 0K are also represented in Fig V-23 (c) and (d) respectively. Figure. IV- 23 . 23 Figure. IV-23. The displacements of the impurities (a) Ti (b) Zr at 300K, (c) Ti and (d) Zr at 0K on different substitutional configurations. Fig. IV- 24 24 Fig. IV-24 represents the structural environment distributions for the different configurations of the Ti and Zr substitutions. For comparison, the structural environments of the impurities in the un-relaxed grain boundary are also represented. In this figure, the calculated defect energies that were obtained in Section III.1 (Chapter IV) are also listed to get more insights about the most stable configurations. Figure.IV- 23 . 23 Figure.IV-23. Snapshots of the structural environment of the impurities before and after relaxation in the ∑5 (310)[001] grain boundary. Figure. V- 24 . 24 Figure. V-24. The left plots (a), (b) and (c) are the relaxed grain boundary geometries with Ti substitution on the most stable configurations 0Al, 1FeI and 2FeII, respectively. The right plots (d), (e) and (f) the displacement of each atom in the grain boundaries represented in (a), (b) and (c) from its initial Ab Initio Molecular Dynamics calculationsThe left plots (a), (b) and (c) are the relaxed grain boundary geometries with Ti on the most stable configurations 0Al, 1FeI and 2FeII, respectively. The right plots (d), (e) and (f) the displacement of each atom in the grain boundaries represented in (a), (b) and (c) from its initial positions. 143 Figure. V-25. The left plots (a), (b) and (c) are the relaxed grain boundary geometries with Zr substitution on the most stable configurations 0Al, 1FeI and 2FeII, respectively. The right plots (d), (e) and (f) the displacement of each atom in the grain boundaries represented in (a), (b) and (c) from its initial Ab Initio Molecular Dynamics calculationsThe left plots (a), (b) and (c) are the relaxed grain boundary geometries with Zr substitution on the most stable configurations 0Al, 1FeI and 2FeII, respectively. The right plots (d), (e) and (f) the displacement of each atom in the grain boundaries represented in (a), (b) and (c) from its initial positions. 144 The left plots (a), (b) and (c) are the relaxed grain boundary geometries with Zr substitution on the most stable configurations 0Al, 1FeI and 2FeII, respectively. The right plots (d), (e) and (f) the displacement of each atom in the grain boundaries represented in (a), (b) and (c) from its initial Fe 3 3 Al, from about 800K indicates that the stability of the D0 3 -Fe 3 Al structure is affected from this temperature and local disorder occurs. For the case of Ti substitutions, the pair distributions function g Al-Al (r) maintains unchanged up to 1000K. This indicates that, in agreement with the experimental observations, the 1% of Ti increases the stability of the D0 3 structure up 1000K.However, for the case of Zr substitutions, the analysis of the pair distribution function shows that the Zr additions have no effect on the stability of the D0 3 both when are placed on the FeI and FeII sites.The calculated defect energies of the impurities in the ∑5(310)[001] grain boundary shows that the Ti impurity prefers to occupy the iron rich configurations further away the interface at intermediate temperature of 300K. The relaxation of Ti impurity to the G.B. interface occurs only at high temperature (at 900K). Comparatively, the Zr impurities occupy the configurations at the G.B interface with the lower defect energies even at intermediate temperatures. Fig. 2 2 Fig. 2 (a) Two interpenetrating lattices, misoriented by 36.87° /001 forming a dichromatic pattern, viewed down 001. (b) GB generated from the dichromatic pattern in (a). (c) coincidence site lattice, ∑=5, generated by the misorientation shown in (a) and (b). (d) GB formed by a misoriented of 22.62°/001, which gives a CSL of ∑=13. The coincidence sites are denoted by solid symbols throughout. [8] Fig. 3 3 Fig. 3 Schematic of twist and tilt GB geometries. (a) illustrates the generation scheme of symmetrical twist GBs and (b) illustrates the generation scheme of symmetrical tilt GBs. possibles dans les sites Al, FeI et FeII (voir Fig. 1.) ont été modélisés dans des supercellules contenant 32 et 108 atomes. Il faut noter que, en cas de substitution d'une d'impureté (Ti ou Zr) sur un site de la supercellule à 32 atomes, la concentration de défaut correspondante est d'environ 3% at. Pour le cas de la supercellule à 108 atomes, la concentration de défaut est d'environ 1 % at. Fig. 1 . 1 Fig. 1. (a) La structure D0 3 -Fe 3 Al non-dopée et les trois défauts de substitutions ainsi que les lacune sur les sites (b) Al (c) FeI et (d) FeII. FeI. Cependant, contrairement au Ti, à une concentration de 3% de toutes les énergies de formation sont positives, indiquant un coût en énergie pour introduire Zr sur un site de substitution dans le bulk. Il est intéressant de noter que ces valeurs ont tendance à diminuer lorsque le calcul est effectué avec 108 atomes c-à-d lorsque la concentration de défauts diminue. Ceci est cohérent avec l'observation expérimentale indiquant que la solubilité de Zr Fig. 3 3 Fig. 3 La structure atomique du joint de grains Σ5 (310) [001] dans l'intermétallique D0 3 -Fe 3 Al (a) vue le long de la direction [001] (b) vue le long de la direction (130). La structure D0 3 étant plus complexe que celle du B2 [17], le nombre de configuration de défauts à prendre en compte dans le joint de grains est beaucoup plus important. Les configurations des substitutions des défauts dans le joint de grains est donnée dans la Fig. 4. A partir la Figure. 4, est clairement visible que neufs configurations de substitution sont présentes au sein du joint de grains Σ5 (310). Elles sont regroupées en trois catégories en fonction de leur distance de l'interface: (i) trois substitutions au niveau de l'interface 0Al, 0FeI et 0FeII (ii) trois substitutions dans le premier plan suivant l'interface 1Al, 1FeI et 1FeII, et (iii) trois substitutions suivant le deuxième plan de l'interface, 2Al 2FeI et 2FeII. Dans la Fig. 4 (b) les deux sites correspondant à l'insertion sont présentés. Ils correspondent à l'emplacement (1) avec seulement des atomes FeII comme premiers voisins et sur le site (2) avec à la fois les atomes Al et FeI comme premiers proches voisins. Fig. 4 . 4 Fig. 4. Les différentes configurations (a) de substitution et (b) sites interstitiels dans l'interface du joint de grains. a) et 5 A 4 A 5 - 545 titre de comparaison les énergies de formation de (supercellule avec 108 atomes) sont repr que, entre les deux types d'interstitiels (1 et 2), la configuration la plus favorable est le site interstitiel (1) pour les deux métaux de transitions Ti . Ceci peut être expliqué simplement comme une conséquence des effets de onfiguration les impuretés sont entourés que par des atomes FeII voisins. Le même comportement a été obtenu dans FeAl L'atome du Bohr préfère être inséré au joint de grains Σ5 (310) dans a cependant ici une nette différence dans les comp Ti et Zr. L'énergie de formation est positive pour les deux configurations, si l'on dans des sites interstitiels, indiquant qu'il en coûte plus pour insérer Ti à l'interface des joints de grains. Comparativement, la configuration atomique 0,18 eV) où Zr est inséré dans un site (1) (c-à-d une configuration riche le cas des additions du B dans FeAl [26], les valeurs négatives de 0,18 eV) obtenues ici pour Zr indique que cet atome est plus stable rsqu'il est inséré dans une configuration riche au niveau du joint de grains Il est important de rappeler que de petites additions de ces deux atomes tendent à aluminiures de fer [18-19, 20]. substitutions, les résultats présentés dans le Tableau III reportés sur les Figures. 5 (a) et 5 (b) en prenant en compte la distance de l'interface (Fig. titre de comparaison les énergies de formation des métaux de transition dans l (supercellule avec 108 atomes) sont représentées dans les Figures. 5 (a) et types d'interstitiels (1 et 2), la métaux de transitions Ti e conséquence des effets de onfiguration les impuretés sont entourés que par des atomes FeII voisins. Le même comportement a été obtenu dans FeAl dopé avec dans des configurations a cependant ici une nette différence dans les comportements des Ti et Zr. L'énergie de formation est positive pour les deux configurations, si l'on qu'il en coûte plus d'énergie pour insérer Ti à l'interface des joints de grains. Comparativement, la configuration atomique d une configuration riche l [26], les valeurs négatives de 0,18 eV) obtenues ici pour Zr indique que cet atome est plus stable rsqu'il est inséré dans une configuration riche au niveau du joint de grains que dans le bulk. de petites additions de ces deux atomes tendent à améliorer la présentés dans le Tableau III sont également tance de l'interface (Fig. 4). métaux de transition dans le bulk ésentées dans les Figures. 5 (a) et 5 (b). FeIIFig. 5 . 5 Fig. 5. Énergies de formation (a) le système dopé avec Fig. 6 .III. 1 . 61 Fig. 6. Les énergies d'interface (en J/m2) calculées pour les joint de grains dopé et non-dopé. Fig. 7 7 Fig.7 Les différences des énergies (en eV) entre les substitutions dans des sites FeI et FeII à un gain significatif de l'énergie (-1,62 ~ -2,55 ment sur le site FeII 0,7 eV). Cela indique également que, pour toutes les températures testées, l'impureté Ti est toujours plus stable dans un site FeI. Pour le cas 2,65 eV) et (-1,29 ∼ eV) sur les sites FeI et FeII, respectivement. Le point le plus intéressant révélé par ces données est l'évolution en fonction de la température de la stabilité relative entre les différents Ti Zr sites pour les deux métaux de transitions. La différence de comportement entre les deux éléments est plus illustrée dans la Fig. 7. Pour Ti, la préférence du site FeI que FeII est confirmée avec une différence d'énergie dans la gamme de -1,16 à -2,27 eV. En outre, la pente négative représenté par les points de données dans la Fig. 7 indique une tendance pour augmenter la stabilité du site FeI que le site FeII lorsque la température augmente. Les tendances observées pour Zr sont assez différentes. Premièrement, l'ampleur en termes de différence d'énergie entre les deux sites ne dépasse pas 0,75 eV (-0,75 max à 100K). Deuxièmement, alors que FeI est favorisée à basse température, l'augmentation de la température a tendance de stabiliser de plus en plus le site FeII. Finalement, avec la configuration énergétique très stable pour le site FeII à 900 K (-2,55 eV) et 1000 K (-2,28 eV), les préférences pour les sites FeI et FeII sont très proches. L'analyse en températures des sites préférentiels (100-1100 K) pour les substitutions des impuretés Ti et Zr sur les sites de la FeI et FeII de la supercellule Fe 3 Al a révélé des différences de comportements intéressantes entre les deux éléments de métaux de transition. Le Ti est plus stable sur un site FeI et la stabilité sur ce site augmente avec la température. Comparativement, le Zr a tendance à occuper avec le meme ordre de stabilité les sites FeI et FeII à haute température. calculées pour les substitutions des deux métaux de transitions dans le site d'Al en plus de la substitution sur le site FeI et FeII. Ces calculs ont été effectués pour déterminer les sites préférentiels des métaux de transitions entre les substitutions dans les configurations d'Al et ensuite examiner la relaxation des joints de grains. Il est important de rappeler que les comparaisons (i) entre les énergies de défauts des deux métaux de transition et (ii) entre des substitutions sur les sites FeI et Al ne sont pas possibles. Les résultats sont représentés dans les figures. 8 et 9 pour les substitutions sur (a) les sites Al (b) les sites FeI et (c) les sites FeII, en tenant compte de la distance de l'interface du joint de grains. A titre de comparaison, les valeurs obtenues à partir des calculs statiques à 0K (chapitre III) sont également rappelées. Fig. 8 8 Fig.8 Energies de défauts pour le Ti pour différentes température 300, 600, 900K sur différents sites (a) Al (b) FeI et (c) FeII en tenant compte des distances suivant l'interface. Energies de défauts pour le Ti pour différentes température 300, 600, 900K sur différents sites (a) Al (b) FeI et (c) FeII en tenant compte des distances suivant l'interface. 179 Energies de défauts pour le Ti pour différentes température 300, 600, 900K sur différents sites (a) Al (b) FeI et (c) FeII en tenant compte des distances suivant l'interface. Fig. 9 9 Fig. 9 Energies de défauts pour le différents sites (a) Al (b) FeI et (c) FeII en tenant compte des distances suivant l'interface. Energies de défauts pour le Zr pour différentes température 300, 600, différents sites (a) Al (b) FeI et (c) FeII en tenant compte des distances suivant l'interface. température 300, 600, 900K sur différents sites (a) Al (b) FeI et (c) FeII en tenant compte des distances suivant l'interface. size of the supercell the volume of the Brillouin zone becomes smaller and smaller (see Eq. B22). Therefore, with increasing supercell size less and less k supercell size on it is often justified to use just a single k (k-point approximation). For metallic systems, on the other hand, much denser k are required in order to get convergence with respect to the k fractional occupation numbers [18 II.4.1.5. Fourier representation of the Kohn In a plane wave representation of the wave functions the Kohn particular simple form. If we insert (Eq. B expሺെ݅ሺ ᇱ ሻݎሻ and integrate over ∑ ቀ . Within this approximation the electronic states at only a finite number of k-points are needed to calculate the charge density and hence the total energy of the solid. The error induced by this approximation can be reduced systematically by increasing the density of the k-point mesh. For insulators it turns out that usually only a small number of k మ ଶ In practical calculations the Fourier expansion (Eq. B keeping only those plane wave vectors ( value E pw : Table . . III-1. Lattice parameters and the bulk modulus for Fe 3 Al-D0 3 . a(Å) B(Gpa) Present Work USPP-GGA 5.76 143 USPP-LDA 5.59 157 Theory FPLAPW-GGA-sp [Gonzales et al. 2002] 1 5.77 - USPP-PBE [Connétable et al. 2008] 2 5.76 159 MBPP-PBE-sp [Lechermann et al. 2002] 3 5.78 151 MBPP-CAPZ-sp [Lechermann et al. 2002] 5.60 192 PWSCF-PBE [Kellou et al. 2010] 4 5.78 139 Experiment 5.79 [Nishino et al. 1997] 5 144 [Leamy et al. 1967] 6 Table . III-2 . The calculated effective formation energies (in eV) for point defects in D0 3 -ordered bulk Fe 3 Al. ܧ ் ܧ ܧ ௩ Al FeI FeII Al FeI FeII Al FeI FeII Present work Relaxed (108 atoms) -0.46 -1.01 -0.39 0.45 -0.14 0.39 0.94 0.34 0.31 Relaxed (32 atoms) -0.20 -0.72 -0.06 0.70 0.17 0.76 2.26 1.72 1.09 Theory [19] Unrelaxed (32 atoms) Table . . III-3. The defect energies of vacancies on the FeI and FeII sites. ܧሺ݀ሻ and ܧ are the energies of the supercell with and without transition metal impurities, respectively. ܧ ௗ = ܧሺ݀ሻ -ܧ (eV) FeI FeII 0K Supercell with 32 atoms (3 at.%) 10.59 9.78 Supercell with 108 atoms (1 at.%) 7.85 7.55 300K Supercell with 108 atoms (1 at.%) 9.55 8.89 Table . III-4 . The calculated effective formation energies (in eV) for relaxed grain boundary supercells. E f (in eV) Insertions Substitutions G.B. Interface First plane from G.B. Second plane form G.B. (1) (2) 0Al 0FeI 0FeII 1Al 1FeI 1FeII 2Al 2FeI 2FeII Ti-doped GB 0.31 0.90 0.05 -0.86 -0.63 -0.44 -1.32 -1.21 0.31 -0.98 -0.93 Zr-doped GB -0.18 2.09 0.49 -0.47 -0.51 -0.04 -0.97 -0.90 0.05 -0.66 -0.78 As seen in Table III-4, between the two types of interstitials (1 and 2), the most favorable configuration is the interstitial site (1) for both Ti and Zr doped grain boundaries. This may be explained simply as a consequence of surroundings effects: in this configuration the impurities are surrounded only by FeII neighboring atoms. The same behavior was obtained in FeAl for B -5 and plotted in Fig. III-7. Our calculation indicates that the values of interface energy for a clean ∑5 (310) [001] D0 3 -Fe 3 Al grain boundary is 0,37 J/m 2 . Comparatively, this is about three This suggests that the alloying elements Ti and Zr can stabilize the grain boundary in D0 3 -Fe 3 Al. The maximum expected reductions are obtained when the transition metals are on a FeII site locate in the first plane away from the exact interface: 14% for Ti (0.32 J/m 2 ) and 22% times lower than the values obtained on a ∑5 (310) [001] iron aluminide grain boundary for the FeAl B2 state (1,12 J/m 2 ) [26]. Except for one case (0Al for Ti/ γ GB =0.38 J/m 2 ), it can be seen that Ti and Zr impurities lower the interface energy when compared for a clean-grain boundary (0.37 J/m 2 ). Table. III-7), the formation energy increases slightly (~+2%) when a vacancy is produced on an Al site. 0,45 0,4 γ γ ) γ γ (J/m 2 0,3 0,35 Ti Zr Clean-GB 0,25 0,2 0Al 1Al 2Al 0FeI 1FeI 2FeI 0FeII 1FeII 2FeII Table III-6. The calculated formation energies (in eV) for vacancies in clean-grain boundary and Ti doped grain boundary. E f E f v (Clean GB) v (Ti doped GB) ϑ Al 3.40 3.26 ϑ FeI 1.44 1.08 ϑ FeII 1.02 1.15 Relative difference -3% -25% +13% Table III - 7 . III7 The calculated formation energies (in eV) for vacancies in clean-grain boundary and Zr doped grain boundary. E f E f v (Clean GB) v (Zr doped GB) ϑ Al 2.35 2.37 ϑ FeI 1.43 0.75 ϑ FeII 1.11 1.01 Relative difference +2% -47% -9% Table . . IV-1. The defect energies (in eV) when the FeI/FeII sites are replaced by the Ti and Zr transition metals. T (K) FeI FeII 100 -1.79 0.05 200 -1.62 -0.46 300 -2.54 -0.56 400 -1.90 -0.65 500 -1.77 -0.34 600 -2.16 -0.48 700 -1.98 -0.55 800 -1.99 0.16 900 -2.35 -0.08 1000 -2.55 -0.70 1100 -1.71 -0.31 Table d d indicate better the trends, the energies differences between the substitution in FeI and FeII sites are also listed and presented in Fig.IV-2.The defect energies (in eV) when the FeI/FeII sites are replaced by the Ti and Zr 111 system of 108 atoms (81 Al unit cell, under periodic boundary conditions. By substituting an impurity in this supercell, the impurity concentration is about 1% The results of the defect energies are summarized in Table IV-1 for Ti and Zr he energies differences between the The defect energies (in eV) when the FeI/FeII sites are replaced by the Ti and Zr Ti Zr FeII E d (FeI)-E d (FII) FeI FeII E d (FeI)-E E d (FeII) 0.05 -1.84 -2.04 -1.29 -0.75 0.75 0.46 -1.16 -2.34 -1.62 -0.72 0.72 0.56 -1.98 -2.35 -1.76 -0.59 0.59 0.65 -1.25 -2.62 -1.93 -0.69 0.69 0.34 -1.43 -2.30 -1.60 -0.7 0.7 0.48 -1.68 -2.36 -1.69 -0.67 0.67 0.55 -1.43 -2.23 -1.71 -0.52 0.52 0.16 -2.15 -2.29 -1.79 -0.5 0.5 0.08 -2.27 -2.08 -2.55 +0.47 +0.47 0.70 -1.85 -2.65 -2.28 -0.37 0.37 0.31 -1.40 -2.58 -2.14 -0.31 0.31 Temperature (K) The energies differences (in eV) between the substitutions on on FeI and FeII sites. Table . IV-2. . The calculated defect energies (in eV) for vacancies in pure Fe 3 Al as well as Ti and Zr-doped Fe 3 Al. The relative difference between the defect energies of vacancies in the pure and doped Fe 3 Al are also presented. Pure-Fe 3 Al Ti-doped Fe 3 Al Zr-doped Fe 3 Al 900K Vacancies on Al sites (ߴ ) -2.64 -7.61 (-65%) -4.18 (-39%) Vacancies on FeI sites (ߴ ிூ ) 1.14 2.79 (+59%) 0.38 (-66%) 1000K Vacancies on Al sites (ߴ ) -8.16 -7.12 (+13%) -8.08 (+1.2%) Vacancies on FeI sites (ߴ ிூ ) -4.61 -1.78 (+61%) -4.24 (+8%) 1, r L2 the internal coordinates of that lattice point with respect to coordinate system No.2 and finally t L1 the coordinates of a translation vector of lattice No.1 with respect to coordinate system No.1. Thus any lattice point r L1 of lattice No.1 that satisfies Eq. III-4 is a lattice point of lattice No.1 and No.2 and therefore represents a coincident point. . For instance one obtains a ∑5 CSL by a 36.87° [001] rotation (see Fig. 2) as well as by a 53.13° [001] rotation. These two representations are linked by the 270° [001] symmetry operation of the cubic lattice. So far aGB has yet not appeared in the discusssion of the CSL scheme and thus the plain CSL scheme is therefore only associated with misorienting two crystals. Utilizing the CSL concept to misorient two crystals, the final step to geometrically generate symmetrical CSL twist GBs, symmetrical CSL tilt GBs as well as asymmetrical CSL tilt GBs is to define the GB plane that will separate the two crystals from each other. This means that either one of the crystals only exists on one side of the GB, thus here the discussion of interpenetrating crystals comes to an end. Obviously the scheme itself is rather theoretical since for instance in sample preparation of experiments such a working sequence of misorienting interpenetrating crystals can only be regarded as imaginary. 3 Al, ont des caractéristiques très intéressantes pour des applications mécaniques à haute température. Ils possèdent, comme la plupart des composés intermétalliques, une résistance mécanique élevée, une bonne résistance à l'oxydation ainsi qu'une faible densité. Cependant, les principales raisons qui limitent leurs applications sont leur fragilité à température ambiante et une forte diminution de leur résistance pour des températures supérieures à 550°C. Un aspect intéressant de ces alliages est leur comportement envers les métaux de transition. Certains éléments, comme Ti, peuvent augmenter la stabilité de la phase D0 3 , en augmentant la transition D0 3 /B2 vers des températures plus élevées. La situation est moins claire dans le cas du Zr. En effet, malgré l'effet bénéfique du dopage en Zr sur la cohésion des joints de grains et la ductilité, il n'existe pas de données expérimentales concernant son effet sur la stabilité de la structure D0 3 du composé Fe 3 Al. Ce travail de thèse vise à étudier l'effet de ces deux métaux de transitions Ti et Zr sur les propriétés du composé intermétallique D0 3 -Fe 3 Al en utilisant des calculs pseudopotentiels ab initio basées sur la théorie de la fonctionnelle de la densité (DFT). Deux principaux thèmes ont été abordées: (i) la compréhension du rôle de ces deux métaux de transition en termes de stabilité de la phase D0 3 à la lumière de leur site préférentiel dans la structure D0 3 -Fe 3 Al (ii) le comportement du Ti et Zr dans le joint de grains Σ5 (310) [001] ainsi que leur effet sur la stabilité structurale de cette interface. Un élément important pour étudier ces aspects est de prendre en compte l'effet de la température. Cela nécessite un traitement de type dynamique moléculaire des atomes dans la supercellule. La technique dynamique moléculaire ab initio (AIMD) résout ces problèmes en combinant des calculs de structure électronique avec la dynamique à une température finie. Ainsi, notre étude a été menée à la fois en utilisant des calculs ab initio statiques à 0K ainsi que par la prise en compte de l'effet de la température jusqu'à 1100K (Dynamique Moléculaire Ab Initio). et basé sur formalisme de la théorie fonctionnelle de la densité telle que implémentée dans le code VASP (Vienna Ab initio Simulation Package)[2, 3]. Les fonctions d'onde électroniques ont été étendues sur une base d'ondes planes avec une énergie de coupure de 240 eV. Les USPPs employés dans ce travail explicitement traitent huit électrons de valence pour le cas du Fe (4s 2 3d 6 ), trois électrons de valence pour Al (3s 2 3p 1 ) et quatre électrons de valence à la fois pour Ti (4s 2 3d 2 ) et Zr (5s 2 4d 2 ). Tous les calculs ont été spin polarisés. Où X et ν représentent les métaux de transitions et lacunes, respectivement. E solid (Fe 3 AlX) (solide, indique bulk ainsi que joint de grains) est l'énergie de la supercellule contenant une impureté et E solid (Fe 3 Al) est l'énergie de la supercellule sans défaut. E Fe ou Al (Fe ou Al étant l'atome qui est substitué) et E X sont les énergies totales calculées pour les métaux purs dans leurs réseaux d'équilibre Fe-bcc, Al-fcc, Ti-hcp et Zr-hcp. Notant que, comme il est très improbable que Ti et Zr occupent des sites interstitiels dans le bulk, l'Eq. 2 ne sera utilisée que pour l'analyse des interstitiels dans le joint de grains. Dans cette équation, telle que présentée dans [8, 9], E GBX et E Fe3AlX sont les énergies totales des supercellules joint de grains et bulk-Fe 3 Al dopés, respectivement. E f = E solid (Fe 3 AlX) -E solid (Fe 3 Al) -E X (Eq.2) pour les configurations d'insertion. L'énergie de formation, pour le cas des lacunes, a été évaluée en utilisant l'équation suivante [7] : E f = E solid (Fe 3 Alv) + E Fe ou Al -E solid (Fe 3 Al) (Eq.3) γ GB = (E GBX -E Fe3AlX ) / 2A (Eq.5) I.2. Energies Dans cette section, nous définissons d'abord les énergies utilisées dans nos calculs. Pour examiner les site préférentiels des métaux de transition à la fois dans le bulk et au joint de grains, leur énergies de formations E f ont été calculées dans différents configurations en utilisant l'équation suivante: E f = E solid (Fe 3 AlX) + E Fe ou Al -E solid (Fe 3 Al) -E X (Eq.1) pour les configurations de substitution et par L'approximation du gradient généralisé (GGA-PW91) a été utilisée pour décrire l'énergie d'échange et de corrélation avec la version de Perdew et Wang [4, 5] . Les intégrations de la zone de Brillouin ont été effectuées en utilisant une grille d'échantillonnage de Monkhorst-Pack [6] . Les tests ont été effectués pour la cellule unité Fe 3 Al (quatre atomes par cellule) en utilisant différents nombre de points k dans la grille k pour assurer la convergence de l'énergie totale avec une précision de 10 -3 eV/atome. Par conséquence, la grille pour la maille unité a été adaptée en utilisant Fe 3 Al (16x16x16) points-k. Selon la structure et la taille de la cellule, le nombre de points-k change comme conséquence de la modification de la taille de zone de Brillouin. Pour les calculs d'énergie totale de la supercellules Fe 3 Al avec 32 atomes (2x2x2 cellule unité) et 108 atomes (3x3x3 de la cellule unitaire), les grilles ont été générées avec (8x8x8) et (4x4x4) points-k, respectivement. Dans le cas du joint de grains, la grille Monkhorst-Pack a été adaptée pour les paramètres en utilisant (4x2x5) k-points. Les géométries atomiques de l'état fondamental ont été obtenues par la minimisation des forces Hellman-Feyman utilisant un algorithme de gradient conjugué. L'énergie d'interface γ GB pour le système non dopé est définie comme: γ GB = (E GB -E bulk ) / 2A (Eq.4) E GB et E bulk sont les énergies totales des supercellules du joint de grains et bulk, respectivement. A est la surface de l'interface (le facteur de ½ est nécessaire pour rendre compte de la présence de deux joints de grains symétriquement équivalents dans la suprcellule). Dans nos simulations, les énergies E GB et E bulk sont calculées pour les blocs de simulations, comprenant un nombre égal d'atomes de chaque espèce. Pour le système dopé, l'énergie d'interface est calculée en utilisant: II.1.2. Les sites préférentiels des défauts ponctuels dans le bulk D0 3 -Fe 3 Al : 2. compare les énergies de formations des impuretés calculées en utilisant des supercellules relaxées et non relaxées (avec 32 atomes). Il est bien clair que la relaxation conduit à une réduction globale des énergies de formation. En outre, bien que les courbes aient le même profil, il apparaît clairement que les différences dans les énergies de formation calculées en utilisant les supercellules relaxée et non relaxée sont plus basses pour entre les énergies de formations calculés dans des supercellules relaxées et non relaxées sont dans la fourchette de 15 à 50% pour les différentes substitutions du Ti diverses, la relaxation diminue l'énergie de formation de plus de 140% dans le cas de la substitution du Zr sur un site FeII. Pour le cas des lacunes, la relaxation est plus prononcée lorsque la lacune est créée dans le site Al. Cela peut aussi être lié à la différence de taille entre les atomes de Fe et Al. Sachant que l'Al est plus grand que le Fe, le vide créé lorsque la lacune est produite dans le site Al est plus grand et, par conséquent, la relaxation est également plus importante. Ainsi, dans la suite, seuls les résultats obtenus à partir des configurations détendues seront traitées.Les valeurs des énergies de formation pour les substitutions et les lacunes dans les supercellules relaxées sont données dans le Tableau. I Les résultats obtenus par Mayer et al. Les énergies de formation (en eV) pour des défauts ponctuels dans le bulk D0 3le Tableau. I, bien que la lacune se produise avec des énergies de formations positives dans les trois différentes configurations, la plus basse énergie de formation correspond à la substitution sur le site FeII. Cette tendance est en accord avec la conclusion de Mayer et al.Fe 72 Al 28 , ont trouvé que les lacunes Fe apparaissent sur le sous-réseau FeI, ce qui est en conflit avec notre résultat et celui obtenu par Mayer et al[10]. Puisque les résultats de Jiraskova ont été obtenus à température ambiante, contrairement à nos calculs et ceux de Mayer et al.[10] qui ont été réalisées à 0K, la différence de l'occupation des lacunes peuvent être liés à l'effet de la température. Pour vérifier cette possibilité, nous avons calculé les énergies de défaut des lacunes dans les sites FeI et FeII à 300K, en utilisant la dynamique moléculaire ab initio. Les énergies de défauts sont définis comme les différences dans l'énergie du système pur quand une impureté remplace le FeI ou FeII, à savoir, et E 0 sont les énergies de la supercellule avec et sans impuretés, respectivement. Le site préférentiel correspond alors au cas où l'énergie est acquise en remplaçant les sites FeI / FeII. Plus de détails sur les calculs sur la dépendance en température des énergies défaut seront présentés dans la section III. Cependant, certains résultats sont présentés ici dans le Tableau. II ainsi que les résultats des énergies de défauts calculées à 0K. Les valeurs des énergies de défauts sont plus importantes que celle des énergies de formation. C'est parce que la valeur de l'énergie totale du fer pur dans son réseau d'équilibre (bcc Fe) n'a pas été soustrait tel que défini dans l'Eq. 2.Comme le montre le Tableau. II, le site FeII reste le site privilégié des lacunes, même à 300K.Cela signifie que les changements de température dans la gamme 0 → 300K ne modifie pas la stabilité des lacunes dans le bulk du composé intermétallique D0 3 -Fe 3 Al. Ici, les deux résultats obtenus par les deux méthodes ab initio statique et ab initio dynamique moléculaire à 300K sont en conflit avec les conclusions Mossbauer. Cela indique que le désaccord avec les résultats expérimentaux ne doit pas être lié à l'effet de la température, mais est certainement liée à la haute sensibilité de ces alliages aux lacunes. C'est parce que la différence entre les énergies de formation de 3% et 1% des lacunes calculées dans des supercellules à 32 et 108 atomes est importante. D'après le Tableau. I, on peut voir que les énergies de formation sur différents sites sont réduites par environ 60% lorsque la concentration des lacunes diminue. En outre, de 1% (à 108 atomes) à 3% (à 32 atomes), la différence entre les énergies de formation des lacunes dans les sites FeI et FeII est également réduite avec la concentration des lacunes. La différence entre les énergies de formation est d'environ 0,6 eV pour les 3% de lacunes alors que pour le cas de la concentration de 1%, la différence n'est que d'environ 0,03 eV.Les énergies de formation calculées pour les substitutions dans la supercellule relaxée sont également indiquées dans le Tableau. I. Pour le cas du Ti, les trois configurations de substitution donnent des énergies de formation négatives. Cela signifie que le Ti est un défaut stable et/ou en autres termes, les 1% ainsi que 3% de Ti sont miscibles dans Fe 3 Al. Ceci est cohérent avec les données de solubilité obtenues expérimentalement Tableau. I Fe 3 Al. ܧ ் ܧ ܧ ௩ Al FeI FeII Al FeI FeII Al FeI FeII Present work Relaxed (108 atoms) -0.46 -1.01 -0.39 0.45 -0.14 0.39 0.94 0.34 0.31 Relaxed (32 atoms) -0.20 -0.72 -0.06 0.70 0.17 0.76 2.26 1.72 1.09 Theory [19] Unrelaxed (32 atoms) 1.97 2.44 1.71 Figure. 2. Profil des énergies de formation des défauts ponctuels calculées dans des supercellules relaxées et non relaxées -1 -0,5 0 0,5 1 1,5 2 2,5 3 3,5 E f (eV) X_Al X_FeI X_FeII Ti-unrelaxed Ti-relaxed Zr-unrelaxed Zr-relaxed v-unrelaxed v-relaxed D'après E d = E (d)-E 0 (Eq. 6) En utilisant une supercellule à 32 atomes [10] sont également représentés dans le Tableau. I E(d) pour comparaison. les substitutions de Ti que celles du Zr. Dans une première approche, cette différence peut être reliée à des différences de taille entre le Ti et Zr (le Ti est plus petit que le Zr). Ce qui est important à souligner ici est essentiellement l'ampleur des différences. Alors que l'ensemble des différences [10] obtenu par la méthode ab-initio pseudopotentiel. La valeur de l'énergie de formation d'une lacune dans le sous-réseau FeII (1,09 eV) est également comparable à (1,18 ± 0,04 eV) obtenue par Schaefer et al. [11] en utilisant la méthode d'annihilation de positons. Comparativement, Jiraskova et al. [12] , qui ont utilisé les mesures Mössbauer d'un composé Ti et Zr) calculée dans différentes configurations du joint de grains ∑5 (310) [001] sont données dans le Tableau II. Tableau. III Les énergies de formations (en eV) calculées pour différentes configurations dans la supercellule relaxée du joint de grains. Tableau III, on peut voir que, configuration la plus favorable est le site interstitiel (1) pour les deux et Zr. Ceci peut être expliqué simplement comme un l'environement: dans cette configuration les impuretés sont entourés que par des atomes FeII comme premier proches voisins. Le même comportement a été obtenu dans FeAl du B, L'atome du Bohr préfère être inséré au joint de grains riches en Fer [26]. Il y a cependant ici une nette différence dans les comp atomes Ti et Zr. L'énergie de formation est positive pour les deux configurations, si l'on considère la présence de Ti dans des sites pour insérer Ti à l'interface des joints de grains. Comparativement, la configuration atomique devient plus stable (-0,18 eV) où Zr est inséré dans un site (1) ( en Fer). Comme pour le cas des additions du l'énergie de formation (-0,18 eV) obtenues ici pour Zr indique que cet atome est plus stable lorsqu'il est inséré dans une configuration riche au niveau du joint de grains Il est important de rappeler que ductilité des aluminiures de fer [ Dans le cas des substitutions, les résultats reportés sur les Figures. 5 ( A partir du E f (in eV) Insertions Substitutions G.B. Interface First plane from G.B. Second plane form G.B. (1) (2) 0Al 0FeI 0FeII 1Al 1FeI 1FeII 2Al 2FeI 2FeII Ti-doped GB 0.31 0.90 0.05 -0.86 -0.63 -0.44 -1.32 -1.21 0.31 -0.98 -0.93 Zr-doped GB -0.18 2.09 0.49 -0.47 -0.51 -0.04 -0.97 -0.90 0.05 -0.66 -0.78 La comparaison des énergies de formation des deux métaux de transitions lorsqu'ils sont s le joint de grains montrent que le Ti est généralement stable avec le bulk et au niveau du joint de grains. Toutefois, pour le cas de l'impureté Zr qui n'est certainement pas stable dans le bulk, préfère ségréger le énergies de formation négatives. Ainsi, l'effet de Zr au joint de grain doit pris en compte pour comprendre les propriétés globales du ces aluminiures de fer.Fe, il est clair que la substitution des métaux de tran sites d'Al n'est jamais la configuration favorable. Pour les substitutions comme pour le bulk, le Ti préfère toujours résider dans des sites FeI plutôt que des sites La situation est moins prononcée pour Zr. En effet alors que les sites Fe sont toujours favorables aux Al, le FeII sont plus favorable lorsque le Zr est situé à l'interface 0,51 eV) et dans le deuxième plan (-0,78 eV). Enfin, il est également intéressan substitutions des deux métaux de transition, la configuration la la substitution sur un site FeI dans le premier plan suivant l'interface ( des impuretés sur les énergies interfaciales a été calculé pour différentes isant l'Eq. 6. Nos résultats sont groupés dans le Tableau IV et . Nos calculs indiquent que les valeurs de l'énerg Ti est généralement stable avec le joint de grains. Toutefois, pour le cas de as stable dans le bulk, préfère ségréger le joint de grains Ainsi, l'effet de Zr au joint de grain doit être pris en compte pour comprendre les propriétés globales du ces aluminiures de fer. métaux de transition sur les . Pour les substitutions dans des sites Fe, comme pour le bulk, le Ti préfère toujours résider dans des sites FeI plutôt que des sites FeII.La situation est moins prononcée pour Zr. En effet alors que les sites Fe sont toujours plusFeII le joint de grains-pur Σ5 (310) [001] est 0,37 J/m 2 . Comparativement, ce sont environ trois fois inférieur aux valeurs obtenues pour le joint de grains Σ5 (310) [001] des aluminure de fer avec la structure B2-FeAl (1,12 J/m2)[26]. A part pour le cas (0Al pour Ti / γGB = 0,38 J/m 2 ), il est bien clair que la présence des impuretés Ti et Zr diminue l'énergie d'interface par rapport celle du joint de grains pur (0,37 J/m2). Ceci suggère que le Ti et Zr peuvent stabiliser le joint de grains du composé intermétallique D0 3 -Fe 3 Al. Les réductions maximales attendues sont obtenus lorsque les métaux de transition sont sur un site FeII situer dans le premier plan suivant l'interface exacte: 14% pour Ti (0,32 J/m2) et 22% pour Zr (0,29 J/m2), respectivement. Il peut être remarqué aussi que les énergies d'interface pour le joint de grains dopé avec Zr sont systématiquement plus faibles que pour celui dopé avec du Ti [Fig.6].Ainsi, la tendance principale qui peut être tirée ici est que la stabilité apportée par la présence du Zr est plus importante que celle du titane. 174 (b) Zr G.B. doped 0.45 0..39 0..39 0.05 -0.04 -0.14 0.51 -0.66 -0.78 -0.9 -0.97 Al FeI Interface Interface plane_1 plane_1 plane_2 plane_2 Bulk substitution Bulk substitution Tableau IV Les énergies d'interface calculé (en J/m2) pour différentes configuration de (en eV) pour différents sites de substitution dans zirconium. substitution. Clean-GB γ GB des deux métaux de transitions lorsqu'ils sont 0.37 Ti doped GB Zr doped GB G.B. Interface 0Al 0.38 0.36 0FeI 0.36 0.34 0FeII 0.34 0.31 First plane from G.B. 1Al 0.36 0.33 1FeI 0.34 0.31 1FeII 0.32 0.29 Second plane from G.B. 2Al 0.36 0.34 2FeI 0.36 0.33 2FeII 0.33 0.30 interface du joint de , il est également intéressant de deux métaux de transition, la configuration la plus stable l'interface (-1,32 eV été calculé pour différentes ultats sont groupés dans le Tableau IV et . Nos calculs indiquent que les valeurs de l'énergie d'interface pour Energies de défauts (en eV) pour les substitutions du Ti et Zr dans des sites Les différences des énergies (en eV) entre les substitutions dans des sites FeI et FeII nt de Ti sur le site FeI mène à un gain significatif de l'énergie ( eV) sur la tout l'intervalle de température. Comparativement, le remplacement sur le site FeII est énergétiquement plus coûteux (0,05 ~ -0,7 eV). Cela indique également que, pour toutes les températures testées, l'impureté Ti est toujours plus stable dans un site FeI. Pour le cas d'impureté Zr, les gains en énergie sont encore plus importantes: (-2,04 ∼ -2,65 eV) sur les sites FeI et FeII, respectivement. Le point le plus intéressant révélé par ces données est l'évolution en fonction de la température de la stabilité relative entre les différents Energies de défauts (en eV) pour les substitutions du Ti et Zr dans des sites Les différences des énergies (en eV) entre les substitutions dans des sites FeI et FeII 177 Ti Zr FeII E d (FeI)-E d (FII) FeI FeII E d (FeI)-E d d (FeII) 0.05 -1.84 -2.04 -1.29 -0.75 0.75 0.46 -1.16 -2.34 -1.62 -0.72 0.72 0.56 -1.98 -2.35 -1.76 -0.59 0.59 0.65 -1.25 -2.62 -1.93 -0.69 0.69 0.34 -1.43 -2.30 -1.60 -0.7 0.7 0.48 -1.68 -2.36 -1.69 -0.67 0.67 0.55 -1.43 -2.23 -1.71 -0.52 0.52 0.16 -2.15 -2.29 -1.79 -0.5 0.5 0.08 -2.27 -2.08 -2.55 +0.47 +0.47 0.70 -1.85 -2.65 -2.28 -0.37 0.37 0.31 -1.40 -2.58 -2.14 -0.31 0.31 Temperature (K) Ti_1FeI Ti_0FeII Zr_1FeI jury, Pr. Ghouti MERAD and Dr. Jacques LACAZE, as it is a great honor for me to have them to evaluate my work. This thesis has been carried out in the Framework of the French/Algerien CMEP PHC Tassili project N° 12053TL and the Eiffel Excellence Scholarship N° 690532C. I would like to acknowledge EGIDE (Centre Français pour l'Accueil et les Échanges Internationaux) for the Financial support. I present also my sincere thanks to Pr. (1 st and 2 nd planes) are more important in the bulk region than that for the configurations when the impurities are placed in the G.B. interface. However the expansion of the interface for these configurations (substitutions in the 1 st and 2 nd plane) is lower when compared to that produced both in the clean interface and the doped G.B. with impurities substitutions in the G.B. interface. In terms of energy, these configurations correspond to the lower interface energies (listed in Table III-7 and presented in Fig. III-5). This is in agreement with the results of Shiga et al. [36]. These authors show that, the G.B. interface varies depending on the multilayer relaxation of the G.B. interface: the larger the interface expansion, the higher the interface energy. Appendix A In this Appendix, the calculated displacements of the atoms in the doeped-grain boundary supercell will be presented. It is important to recall that the displacement of the atoms is calculated as difference between the final positions of the atoms in the relaxed supercell and the initial positions in the un-relxed supercell. The following left plots represent the calculated displacements of the atoms. The positions of the various atoms (Al, FeI and FeII) are given on the x axes depending on their location (n) away from the exact interface at n=0. Due to the cell symmetry 0 and -/+10 label the two interfaces in the supercell. In the right plots, the displacements are represented by solid arrows. Appendix B This appendix will give a brief general introduction into the field of two-dimensional defects in materials and the Coincidence Site Lattice theory. Generally one speaks of an interface being present in a system if the physical properties discontinuously change across that interface. Phase boundaries i.e. exist between phases of different state of order. For example this is true for liquid-solid or liquid-gaseous interfaces. Limiting the discussion to internal interfaces in solid state materials one differentiates between either phase or grain boundaries. Phase boundaries separate grains of different phases. This i.e. could be grains having different lattice structure. Grain boundaries (GBs) separate almost stress free grains having the same lattice structure but different orientation. Throughout history the structure of GBs was seen quite differently. First models by Rosenhain [1] understood a GB as being an amorphous disordered region separating grains. A succeeding model by Mott [2] assumed that GBs are two-dimensional defects where zones of good lattice match and zones of bad lattice match exist. A different approach is to discuss GBs in the framework of dislocations. The Read-Shockley model [3] tried to explain GBs on the basis of dislocations. Such an approach is well suited for low-angle GBs. Here it is well known that low-angle GBs can form by arrays of lattice dislocations. Although discrete lattice dislocations are having long-range stress fields, dislocation theory is able to show that by linear superposition of GB dislocation stress fields, GBs can be formed that only exhibit a short range stress field. For instance Fig. 1 demonstrates how stabile almost stress free GBs can be formed. One of the major drawbacks of this model is the non-inclusion of any rigid body translations between crystals [9].

01749405

en

2024/03/05 22:32:07

2012

https://hal.univ-lorraine.fr/tel-01749405/file/DDOC_T_2012_0326_SHASHKOV.pdf

Keywords: déformation plastique, dynamique des dislocations, effet Portevin -Le Chatelier, maclage, émission acoustique, systèmes dynamiques, auto-organisation, analyse statistique, analyse multifractale Recent studies of plastic deformation using high-resolution experimental techniques testify that deformation processes are often characterized by collective effects that emerge on a mesoscopic scale, intermediate between the scale of individual crystal defects and that of the macroscopic sample. In particular, the acoustic emission (AE) method reveals intermittency of plastic deformation in various experimental conditions, which is manifested by the property of scale invariance, a characteristic feature of self-organized phenomena. The objective of the dissertation was to study the inherent structure of AE for different mechanisms of plastic deformation, to examine its dependence on the strain 2 rate and strain hardening of the material, and to understand the relationships between short time scales related to organization of defects and those relevant to the continuous approach of plasticity. The study was performed on AlMg and Mg-based alloys, the plastic deformation of which is accompanied by a strong acoustic activity and controlled by different physical mechanisms: the Portevin-Le Chatelier (PLC) effect in the first case and a combination of twinning and dislocation glide in the second case. Application of a technique of continuous AE recording ("data streaming") allowed proving that the apparent behavior, discrete or continuous, of AE accompanying the PLC effect depends on the time scale of observation and the physical parameters surveyed. However, unlike the traditional view, it appears that AE has an intermittent character during both stress serrations and macroscopically smooth flow. Using methods of the theory of nonlinear dynamical systems, such as the multifractal analysis, a tendency to a transition between the scale-invariant dynamics and the behaviors characterized by intrinsic scales was detected during work hardening. Finally, we proved that the power-law statistical distributions persist in wide ranges of variation of parameters conventionally used to individualize acoustic events. This result is of general importance because it applies to all avalanche-like processes emerging in dynamical systems. Résumé Les études récentes de la déformation plastique à l'aide de techniques expérimentales à haute résolution témoignent que les processus de déformation sont souvent caractérisés par des effets collectifs qui émergent à une échelle mésoscopique, intermédiaire entre celle de défauts cristallins et celle d'une éprouvette macroscopique. Notamment, la méthode de l'émission acoustique (EA) révèle, dans divers conditions expérimentales, l'intermittence de la déformation plastique, qui se manifeste par une propriété de l'invariance d'échelle, caractéristique de phénomènes d'auto-organisation. L'objectif de la thèse a été d'étudier la structure inhérente de l'EA pour différents mécanismes de déformation plastique, d'examiner sa dépendance à la vitesse de déformation et à l'écrouissage du matériau, et d'appréhender les liens entre les petites échelles de temps, liées à l'organisation des défauts, et celles qui relèvent de l'approche continue de la plasticité. L'étude a été réalisée sur des alliages AlMg et des alliages base Mg, dont la déformation plastique est accompagnée d'une forte activité acoustique et contrôlée par differénts mécanismes physiaues : l'effet Portevin-Le Chatelier (PLC) dans les premiers et une combinaison du maclage et du glissement des dislocations dans les deuxièmes. L'utilisation de la technique d'enregistrement continue de l'EA ("data streaming") a permis de montrer que le comportement apparent -discrète ou continue -de l'EA accompagnant l'effet PLC dépend de l'échelle de temps d'observation et du paramètre physique étudié. Cependant, contrairement à une vision traditionnelle, il se trouve que l'EA a un caractère intermittent pendant l'écoulement macroscopiquement lisse tant que pendant l'instabilité macroscopique de la déformation plastique. Grace aux méthodes d'analyse issues de la théorie des systèmes dynamiques non linéaires, telles que l'analyse multifractale, une tendance à la transition entre la dynamique invariante d'échelle et les comportements caractérisés par des échelles intrinsèques a été trouvée lors de l'écrouissage des matériaux. Enfin, nous avons prouvé que les distributions statistiques en loi puissance persistent dans des larges intervalles de variation des paramètres, conventionnellement utilisés pour individualiser les événements acoustiques. Ce résultat est d'une importance générale car il s'applique à tous les processus avalancheux émergeant dans différents systèmes dynamiques. Introduction The plasticity of crystalline materials results from the motion of defects of crystal structure -dislocations, twins, point defects, and so on. Until recently, the research in plasticity was divided into two major parts, that of the microscopic motions of defects and that of the macroscopic behavior of the material. The latter was considered as resulting from averaging over local random fluctuations of the distribution and mobility of defects, dislocations par excellence, which statistically compensate each other and give rise to spatially uniform and continuous plastic flow. Since the 1980th, however, many studies showed that the ensemble of crystal defects represents an example of nonlinear dissipative dynamical systems in which the interaction between various constituents may lead to self-organization phenomena. The properties of the collective dynamics appear to be common for dynamical systems of different origin, coming from various fields, such as physics, mechanics, chemistry, biology,.... [START_REF] Nicolis | Self-organization in nonequilibrium systems : from dissipative structures to order through fluctuations[END_REF][START_REF] Haken | Synergetik[END_REF]. Each example of collective effects, interesting by itself, is also interesting as a representative of a class of phenomena characterized by universal behavior. The complex dynamics of such systems is often associated with the property of scale invariance, or self-similarity, which manifests itself through power-law relationships. It was found that self-organization of dislocations concerns both spatial and temporal behavior and results in formation of dislocation structures and/or intermittency of plastic deformation, characterized by non Gaussian statistics and invalidating the averaging operations. These phenomena give rise to a very complex problem because depending on the material and deformation conditions, the collective dynamics of defects may show up at different scales and lead to various collective effects, displaying both universal and unique properties. The main challenges for investigations of this problem are to determine the limits of the continuous approach to the description of plasticity and to find a link between the elementary mechanisms of plastic deformation and the macroscopic behavior of deforming materials. Understanding such multiscale behavior is especially important nowadays because the technological developments are turning towards micro-and nanosystems with dimensions comparable to the scales imposed by collective processes in the dislocation system. In this framework, the present doctoral research will mostly concern the problem of the intermittency of plastic deformation of crystals. The intermittent collective motion of defects generates jumps in the plastic strain rate, which are characterized by nonrandom statistics and in particular, by power-law statistical distributions. Such properties were first identified for different mechanisms giving rise to macroscopic plastic instabilities, mostly the Portevin-Le Chatelier effect (PLC) -jerky flow in dilute alloys caused by interaction between dislocations and impurities [START_REF] Portevin | [END_REF][START_REF] Kubin | Dislocations in Solids[END_REF]. In tensile tests with constant strain rate, this effect displays a complex spatiotemporal behavior, associated with repetitive strain localization in deformation bands and concomitant abrupt variations of the deforming stress. Various approaches to the analysis of serrated deformation curves were proposed [START_REF] Lebyodkin | [END_REF]6,7,8,9,10,11]. They all showed that spatiotemporal patterning corresponds to nontrivial dynamical regimes. In particular, the dynamical [12] and statistical [START_REF] Lebyodkin | [END_REF] analyses testified to the existence of a deterministic chaos [13] in some range of strain rates, and a transition, at higher strain rates, to self-organized criticality (SOC) [14] which is generally considered as a paradigm of avalanche-like processes. These two modes demonstrate different statistics of amplitudes and durations of serrations: distributions with characteristic peaks in the case of chaos and power-law distributions in the case of SOC. It should be noted, however, that chaos is also associated with scale invariance reflected in the geometry of the phase trajectory of the corresponding dynamical system. Moreover, application of the multifractal analysis [15] revealed scale-invariant behaviors in the entire strain-rate range in which the PLC effect is observed. Recently, acoustic emission (AE) technique was applied to study the PLC effect in an AlMg alloy [START_REF] Bougherira | Etude des phénomènes d'auto-organisation des ensembles de dislocations dans un alliage au vieillissement dynamique[END_REF][START_REF] Lebyodkin | [END_REF]18]. AE stems from transient elastic waves generated within a material due to localized changes of microstructure, therefore, it reflects the motion of group of defects. Surprisingly, power-law distributions were found for amplitudes of AE events in all experimental conditions. This result indicates, on the one hand, that plastic deformation is an inherently intermittent, avalanche-like process at mesocopic scales relevant to AE, and, on the other hand, that scale invariance may not spread out to the macroscopic scale. Furthermore, another group of investigations showed that "regular" plastic flow is also intermittent although the jumps in the strain rate are smaller than those leading to macroscopic stress serrations. These jumps can however be evaluated using highresolution extensometry or the AE technique. The AE statistics was studied rather in detail during macroscopically homogeneous deformation of some pure crystalline solids, such as ice single and polycrystals and copper single crystals, and displayed a persistent power-law character [19,20]. This conclusion was later confirmed in experiments on local extensometry during plastic flow of Cu single crystals [21,22]. Power-law statistics were also found for the AE accompanying twinning of single crystals of Zn and Cd [23], as well as for stress serrations observed during compression of microscopic pillars of pure metals [24]. All the above-described results led to the growing recognition of a ubiquitous character of self-organization phenomena in dislocation ensembles. Moreover, various data on small-scale intermittency in pure single crystals provided approximately the same value for the power-law exponent, thus giving rise to a "universality" conjecture [25]. However, different exponent values were reported for AE in ice polycrystals [26]. Besides, the exponents obtained for AE accompanying the PLC effect were found to depend on strain rate and microstructure [27,[START_REF] Lebyodkin | [END_REF]18]. These differences raise a question on the relationship between the general laws governing the collective dislocation dynamics at relevant mesoscopic scales and the role of specific mechanisms of plasticity. Furthermore, as can be seen from the above-said, the range of scale invariance is limited from above. However, the transition to the macroscopic scale is not well understood because the macroscopic behaviors include not only smooth deformation curves but also jerky flow, with statistics depending on the experimental conditions. On the other hand, the scale invariance must also break at small scales because of the limited experimental resolution. So, the classical analysis of EA considers as elementary the acoustic events at the time scale inferior to 1 ms. However, a recent study under conditions of the PLC effect showed that these "elementary" events may possess a fine structure detected by the multifractal analysis [28]. It can be supposed that such temporal structures characterize short-time correlations between the motions of defects, which have not been studied up to date. With these questions in view, the objective of the present doctoral research was to study the intrinsic structure of AE at different time scales and for different mechanisms of plasticity; to characterize the relationships between the correlations of deformation processes at very short time scales, corresponding to "elementary" acoustic events, and the long-time correlations, up to the macroscopic scale of the deformation curve; and to examine the influence of strain and/or strain rate on the observed statistical behavior. Mg and Al based alloys were chosen as the main objects of this study. Both these alloys exhibit a highly cooperative character of plastic deformation, leading to strong acoustic activity which is governed by distinct microscopic mechanisms, -respectively, mechanical twinning and the PLC effect. In order to get a comprehensive description of complex behavior the data treatment combined various methods, including the statistical, spectral, and multifractal analyses. The thesis contains six chapters. The first chapter reviews literature data concerning the problem of intermittency of plasticity, gives examples of similar behaviors in solid state physics, introduces notions of nonlinear dynamical systems, and describes the mechanisms of the PLC effect and twinning. Experimental details on the materials, testing, and data recording are given in Chapter 2. Chapter 3 introduces the relevant types of data analyses. The results of investigations are presented in Chapters 4 to 6. Chapter 4 dwells on the effect of the parameters of individualization of acoustic events on their apparent statistics. This study treats a general question arising in experimen-tal investigations of avalanche processes of different nature: are the results of statistical analysis not affected by the superposition of the analyzed events? Indeed, the superposition might occur for various reasons, for example, because of an almost instantaneous emergence of avalanches or because of insufficient resolution of individual events. Understanding this influence is an indispensable basis for the quantitative evaluation of the critical exponents characterizing scale invariance. However, there are virtually no pertinent studies in the literature, or at least this question is only considered theoretically. Chapter 5 utilizes a technique of continuous AE recording to compare the nature of AE during jerky and smooth flow in an AlMg alloy. As a matter of fact, the AE accompanying unstable plastic deformation is usually considered to be composed of discrete bursts associated with the motion of large dislocation ensembles, giving rise to stress serrations, and a continuous emission generated during macroscopically smooth plastic flow. This traditional viewpoint however contradicts the observation of intermittency of plastic flow in smoothly deforming pure materials. A minute examination of AE records reported in Chapter 5 allows overcoming this contradiction and provides a basis for the subsequent analysis of AE statistics. The chapter is completed with first results of similar investigations on Mg alloys. The last chapter describes the results of the statistical and multifractal analysis of AE in Al and Mg alloys. The general conclusions and discussion of the perspectives for future investigations complete the dissertation. Chapter 1 BACKGROUND The current research into self-organization of crystal defects and collective deformation processes explores various directions and is developing into a large field of investigation. Without trying to give an exhaustive overview of related problems, this chapter will treat some aspects which concern the above-formulated objectives of the doctoral study and are necessary for understanding its results. We first outline observations of scale-invariant behavior, associated with power laws, during plastic flow of crystalline solids. In order to position the plasticity phenomena within a more general problem of avalanche-like processes in physics and mechanics, we further present examples of powerlaw statistical behavior in solid state physics and introduce some general concepts of nonlinear dynamical systems and theoretical frameworks for explanation of power-law statistics. This general representation is followed by a more detailed description of the Portevin-Le Chatelier effect, which will be one of the main focuses of investigation. After briefly introducing the macroscopic manifestations and the microscopic mechanism of the effect, we present experimental observations and computer modeling of the complexity of its spatiotemporal behavior, with an accent put on statistical properties and application of AE method to experimental investigation of jerky flow. The chapter ends with the description of the phenomenon of twinning -another mechanism of plastic deformation studied in this dissertation and characterized by strong AE displaying features qualitatively different from those observed for the PLC effect. Intermittency and power-law scaling in plastic flow Plastic deformation of crystalline solids is governed by the multiplication and motion of crystal defects -dislocations, twins, point defects, and so on. Consequently, it is intrinsically heterogeneous and discontinuous at the microscopic scale. By contrast, the macroscopically smooth plastic deformation of crystals is conventionally viewed as homogeneous and continuous plastic "flow." This is understood as a result of averaging over independent motions of a very large number of defects contained in the material. Indeed, the typical densities of dislocations in deformed samples are of the order of magnitude of 10 10 per cm 2 . The approximate sense of this viewpoint which supposes that the interaction between defects can be neglected has been recognized for a long time. On the one hand, application of the electron microscopy to investigation of dislocations revealed formation of complex spatial structures (e.g., [29,30]). However, the development of this spatial heterogeneity was observed during macroscopically smooth deformation and investigated regardless of the problem of discontinuity of plastic flow, although the numerical models proposed for explanation of the spatial aspect of selforganization of dislocations are also promising as to understanding its temporal aspect [31,32,33]. On the other hand, due to utilization of elaborated experimental techniques, several observations of sporadic bursts in plastic strain rate were reported very early, even before the occurrence of the concepts of plastic deformation through motion of defects (e.g., [34,35]). These bursts were however considered as resulting from large stochastic fluctuations in the system of defects. Finally, in some cases the continuous plastic flow becomes unstable at macroscopic scale and serrated deformation curves are observed. The jerky flow may be caused by various microscopic mechanisms which have been extensively studied, e.g., the Portevin-Le Chatelier (PLC) effect controlled by interaction of dislocations with solutes [START_REF] Portevin | [END_REF][START_REF] Kubin | Dislocations in Solids[END_REF][START_REF] Cottrell | Dislocations and plastic flow in crystals[END_REF][START_REF] Estrin | [END_REF], thermomechanical instability caused by insufficient heat evacuation from samples deformed at very low temperatures [38,39], twinning [40,41], martensitic transformations [START_REF] Ogata | AIP Conference Proceedings[END_REF][START_REF] Zhang | [END_REF], and so on. The observation of jerky flow testifies that at least in certain conditions, interaction between defects may lead to intermittent plastic flow caused by short-time cooperative motion of large groups of defects. The creation of the theory of nonlinear dissipative systems which are characterized by self-organization [START_REF] Haken | Synergetik[END_REF][START_REF] Nicolis | Self-organization in nonequilibrium systems : from dissipative structures to order through fluctuations[END_REF] made it possible to analyze discontinuous plastic deformation from the viewpoint of collective processes. Such investigations started in 1980th, mostly using the example of the PLC effect, which displays complex spatiotemporal behavior associated with strain localization within deformation bands and stress serrations of various kinds. A detailed description of this phenomenon will be given in § 1.5. Essentially, various approaches to the analysis of stress drops showed that spatiotemporal patterning corresponds to nontrivial dynamical regimes [START_REF] Lebyodkin | [END_REF]6,7,8,9,10,11]. In particular, power-law Fourier spectra of series of stress serrations and power-law statistical distributions of their amplitudes and durations were found in a range of high strain rates [START_REF] Lebyodkin | [END_REF]6,7,8]. The power-law dependences are equivalent to scale-free behavior, as can be easily seen from the relationship p(kx) ∝ (kx) β ∝ k β x β ∝ k β p(x) which displays self-similarity upon scaling. This scalefree behavior bears witness to possible manifestation of self-organized criticality. The concept of SOC was introduced by P. Bak et al. [44] as a general framework for explaining the avalanche-like phenomena in spatially extended dynamical systems and, more specifically, the flicker noise characterized by 1/f power density spectrum. Although some authors contest the application of SOC to explain the 1/f -noise [45], it is widely used to model earthquakes and dry friction -the phenomena presenting certain similarities with jerky flow. It should be noted that such a hypothesis is consistent with the infinite number of degrees of freedom of the dislocation ensembles. At lower strain rates histograms with characteristic peaks were observed. They were shown (e.g., [START_REF] Kubin | Dislocations in Solids[END_REF]7,8]) to neither be associated with stochastic behavior, but with the so-called deterministic chaos [START_REF] Bergé | Order within chaos, towards a deterministic approach to turbulence[END_REF]. Interestingly, this observation means a drastic reduction in the number of degrees of freedom because chaos occurs in low-dimensional systems. It is also characterized by scale invariance which is reflected in the so-called fractal geometry (see § 3.3 and Appendix) of the attractor of phase trajectories of the dynamical system. Observing scale invariance in the distributions of stress serrations at the macroscopic level, suggests extending the analysis to finer event scales by using more sensitive techniques. Recently, acoustic emission (AE) technique was used to investigate statistics of the PLC effect [START_REF] Bougherira | Etude des phénomènes d'auto-organisation des ensembles de dislocations dans un alliage au vieillissement dynamique[END_REF][START_REF] Lebyodkin | [END_REF]18]. Surprisingly, it was found that acoustic emission is characterized by power-law statistics of event size for all strain rates. Moreover, a similar statistical behavior of AE was found for both jerky flow and macroscopically smooth plastic flow in the same tests. Such persistence of statistical behavior testifies to the invariant nature of deformation processes during both stable and unstable deformation, and implies an inherently intermittent scale-invariant character of plastic activity at a mesoscopic scale relevant to AE. It is noteworthy that this conjecture is conform with the earlier observations of power-law statistics of series of electric signals accompanying stress serrations during low-temperature catastrophic dislocation glide or twinning in pure metals [START_REF] Bobrov | [END_REF]48]. Although the above research testifies to an important role of self-organization of dislocations, the PLC effect is often considered as a particular, exotic case. Nevertheless, application of AE technique to pure materials, either displaying serrations caused by twinning (Cd, Zn) or deforming by smooth dislocation glide (ice, Cu), showed that the intermittency of plastic deformation is rather the rule than the exception and results from an avalanche-like collective dislocation motion [19,49,21,25,23]. An example of power-law statistical distributions of acoustic energy bursts in ice single crystals deformed by creep is illustrated in Fig. 1.1. The results of AE studies were recently confirmed using another sensitive method based on high-resolution extensometry [21,22]. In this case, power-law distributions were found for local strain-rate bursts detected during plastic flow of Cu single crystals. All these results bear evidence to the intermittent scale-invariant character of macroscopically uniform plastic activity, albeit at sizes of the local strain-rate bursts much smaller than in jerky flow. Finally, serrated deformation curves and power-law distributions of serration amplitudes were recently observed in the compression of microscopic pillars of various pure Figure 1.1: Statistical distributions of acoustic energy bursts recorded in ice single crystals under constant stress. [49] metals (Fig. 1.2) [24,50,51]. These works provide a direct proof of collective dislocation dynamics, which shows up on the deformation curves when the size of the deformed specimens is small enough, so that their plastic deformation cannot be considered anymore as a result of averaging over many independent plasticity events. The collection of all the above-described results led to the growing recognition of a ubiquitous character of self-organization phenomena in dislocation ensembles. 1 The various data on small-scale intermittency in pure single crystals provided approximately the same value for the power-law exponent. Namely, distributions p(E) ∝ E β with β ≈ -(1.5 ÷ 1.8) were reported for the energy E of AE events recorded during deformation of single crystals of various materials with hexagonal and cubic crystal structure [19,20,23]. A similar β-range was found for jerky displacements in the tests on micropillars. As the abrupt displacement takes place at approximately the same force level, it determines the produced mechanical work and, therefore, also characterizes the energy dissipated in the process of plastic deformation. The entirety of these data gave Figure 1.2: Left: intermittency of the shear-stress vs. shear-strain curve for Ni sample with [ 269] orientation; the numbers designate the pillar diameter. Right: distribution of slip events plotted in logarithmic coordinates. Open circles -data for a single sample with diameter ∼ 20 µm, solid circles -aggregate data from several samples. [24] rise to a "universality" conjecture [25]. Concurrently, the value β * ≈ -1.35 was found for the amplitude distribution of AE in ice polycrystals [26]. This value corresponds to an even higher estimate of the exponent β for the energy distribution, as can be illustrated with the aid of the approximation suggested in [20], that the energy dissipated during an acoustic event is proportional to the square of its peak amplitude. Using this simplification, the relationship β = (β * -1)/2 can be easily deduced (see, e.g., [22]) and gives an estimate β ≈ -1.2. Much lower values, β ≈ -(2 ÷ 3), which in addition depend on strain rate and evolve with work hardening, were reported for AE accompanying the PLC effect [START_REF] Bougherira | Etude des phénomènes d'auto-organisation des ensembles de dislocations dans un alliage au vieillissement dynamique[END_REF][START_REF] Lebyodkin | [END_REF]18]. Taking into account that at the macroscopic scale of stress serrations the power-law distributions were only found at high strain rates, with exponents varying in rather large intervals for various samples {β ≈ -(1 ÷ 1.7) was typically reported for amplitude distributions of stress serrations}, the questions on the relationship between the general laws governing the collective dislocation dynamics and on the role of specific mechanisms of plasticity remain open. Analogues in physics The observations of power-law statistics in plasticity are reminiscent of similar features in nonlinear phenomena of various natures, often referred to as "crackling noise". Numerous investigations following the development of the theory of dynamical dissipative systems proved that the emergence of scale invariant behavior is one of the fundamental properties of self-organization phenomena in systems consisting of a large number of interacting elements. Their behavior is expressed by a jerky response to smoothly changing external conditions. Well known examples in physics and material science include the Barkhausen noise in magnetic materials [54], vortex avalanches in type II superconductors [55], charge density waves [56], fracture [57,58], martensitic transformations [59], dry friction [60], earthquakes [61], and so on. All these phenomena are characterized by avalanche-like relaxation processes alternating with periods of slow loading, and are characterized by power-law distributions of sizes and durations of the avalanches. Several examples of such behavior are presented below in some detail. The Barkhausen effect is an instability responsible for the jerky character of magnetization of ferromagnets [62]. Discovered in 1919, it provided direct experimental evidence for the existence of magnetic domains. Let us consider a ferromagnet below the Curie temperature. In a zero magnetic field it is divided into domains whose magnetic moments tend to disorder, in order to compensate each other and to minimize the internal energy which is the lowest in the unmagnetized state. When an external magnetic field is applied to the material, its magnetization takes place through displacement of the domain boundary walls. If the crystal structure were free of defects, the domain walls would move in infinitesimal magnetic fields. In reality they interact with various pinning centers, such as dislocations, polycrystalline grains, stacking faults, surface roughness, etc. In particular, this explains the existence of permanent magnets, i.e., materials having a spontaneous magnetic moment in the absence of external magnetic field, because their full demagnetization is impeded by pinning of the domain walls. The other consequence of pinning is the Barkhausen effect which is caused by jump-like displacements of domain walls. The corresponding jumps in magnetization can be observed, e.g., with the help of inductive techniques. Recently, the development of magneto-optical methods allowed direct (in situ) observations of the intermittent motion of domains walls [63]. However, the most reliable experiments in terms of signal statistics are still based on inductive measurements. A typical example of such measurements is presented in Fig. 2. This analogy is rather profound as both phenomena are related to the problem of collective pinning. It is not surprising that the same authors who studied the phenomenon of plastic instability also suggested a model of unstable magnetization processes [START_REF] Mccormick | [END_REF]. However, most of works treating in some way the analogy between plasticity and magnetization problems mostly considered the motion of individual domain walls through pinning centers (e.g., [66]). An avalanche dynamics of magnetic flux, characterized by power-law statistics, was also observed in type II superconductors [55,67] and in discrete superconductors -Josephson junction arrays [68,[START_REF] Ishikaev | New Developments in Josephson Junctions Research[END_REF]. Magnetization of these materials is determined by the dynamics of a system of vortices (called, respectively, Abrikosov or Josephson vortices) carrying magnetic flux. There are evident qualitative similarities between the magnetic flux flow in such materials and the plastic flow. Indeed, both vortices and dislocations are linear "defects" which are subjected to external forces (Lorentz force [START_REF] Landau | The Classical Theory of Fields[END_REF] and Peach-Koehler force [START_REF] Friedel | Dislocations[END_REF], respectively), pinning forces, and mutual interaction forces. It should be noted, however, that the case of dislocations presents more complexity. In particular, the magnetic vortices are oriented in the direction of the applied magnetic field and their interactions are mostly reduced to repulsion, whereas the interactions between dislocations depend on their type, Burgers vector, mutual orientations, bounding to certain slip systems, and so on, and include annihilation and multiplication [START_REF] Friedel | Dislocations[END_REF]. Perhaps, the most famous example of avalanche-like behavior is rock fracture. Whereas the above examples are typically characterized by self-similar statistics over only 1-3 orders of magnitude of the measured variable, because of strong limitations of the sample size in the laboratory conditions, in geology the avalanches can reach the size of giant earthquakes. Consequently, the distribution of sizes of seismic events obeys a power law, known as the Gutenberg-Richter law, over more than eight orders of magnitude [61]. The dynamics of earthquakes is generally related to a stick-slip mechanism, which involves sliding of a plate of the earth crust along another plate [60]. Their key feature is that the friction force, which acts along the fault line between the plates, decreases with increasing the slip velocity. As argued in [START_REF] Lebyodkin | [END_REF], the mechanism responsible for such behavior is similar to that responsible for the property of negative strain-rate sensitivity of stress under conditions of the PLC effect. Application of AE technique to statistics of deformation processes The sudden changes in the internal structure of materials (cracking, motion of dislocation pile-ups, phase transitions, twinning, etc. ) under action of external forces lead to emission of sound waves. This phenomenon has been known since long ago because the sound is sometimes audible, as in the well-known case of mechanical twinning during bending of a tin bar ("tin cry"). The recording of the acoustic signal provides information about these processes, taking place on various spatial and temporal scales. The regular application of the AE technique as an in situ method of investigation has started since a comprehensive study by Kaiser [73,[START_REF] Kaiser | Untersuchung über das Auftreten von Geräuschen beim Zugversuch[END_REF]. It has proven to be a powerful tool to study the mechanisms of plastic deformation and failure in wide time and space ranges, from microscopic scales corresponding to the motion of groups of defects to the macroscopic scale of the deforming sample (see, e.g., reviews [START_REF] Mathis | Acoustic Emission, chapter Exploring Plastic Deformation of Metallic Materials by the Acoustic Emission Technique[END_REF][START_REF] Heiple | [END_REF] and references therein). In particular, simple estimations show that a measurable stress drop of a few hundredth of MPa requires cooperative motion of the order of 10 5 dislocations through the sample cross-section, whereas the motion of a few hundred dislocations through a polycrystalline grain can provoke a measurable acoustic event [77]. Indeed, significant AE accompanies even macroscopically smooth deformation curves of pure materials [21]. Consequently, this technique is particularly useful for the study of the collective motion of dislocations. In spite of a wide application of AE to monitoring microstructural changes in deforming materials, its interpretation is not direct because the signal is affected by the transfer function of the AE sensor, the sound reflections, the modulations by the propagating medium, etc... In some cases, comparison of AE data with microstructural studies may provide information on the possible sources of elastic waves. In other cases, the interpretation depends on the physical assumptions. Dislocations-based models of AE usually consider three mechanisms: (i) relaxation of stress fields caused by dislocation motion; (ii) annihilation of dislocations; (iii) "bremsstrahlung" radiation from accelerated dislocation [START_REF] Heiple | [END_REF]. The estimates of the energy released by these sources prove a preponderant role of the first type of AE sources (see, e.g., [START_REF] Mathis | Acoustic Emission, chapter Exploring Plastic Deformation of Metallic Materials by the Acoustic Emission Technique[END_REF]). A model of such source was proposed by Rouby et al. [78], who considered the motion of a straight dislocation line at a constant velocity. Pursuing this approach, Richeton et al. [79] deduced the following relationship for the amplitude of the acoustic wave generated by a dislocation avalanche with a total dislocation length L and the average velocity v: A(t) = 3C 2 T 4πC 2 L ρb D 2 Lv (1.1) where C T and C L are, respectively, the transverse and longitudinal wave velocities, ρ is the density of the material, b is the Burgers vector, and D is the distance between the AE source and the sensor. The term Lv corresponds to the rate of area sweeping by dislocations: Lv = dS/dt. Normalized by volume and combined with the Orowan equation [80] for plastic strain rate, dε dt = kρ m bv, where ρ m is the density of mobile dislocations and k is a geometrical factor, this relationship shows that A(t) ∼ dε/dt. In statistical investigations the maximum amplitudes A of acoustic events are usually examined. Integration of Eq. 1.1 under assumption that the avalanche velocity v exponentially decays with time leads to a proportionality relationship between A and the strain increment caused by the dislocation avalanche: A ∼ ε. Its feasibility is confirmed, e.g., by experiments on ice single crystals, which showed proportionality between the global AE activity and plastic deformation [19]. Weiss et al. [21] provide arguments that this relationship should be valid in a more general case, e.g., for a decay following a power-law, as could be expected for avalanche-like behavior. It is often supposed that the radiated acoustic energy is a more relevant characteristic of deformation processes, also applicable to non dislocation sources, e.g., twinning or cracking [81,20,82]. This characteristic is considered to scale with A 2 , as follows from the estimate of the energy dissipated at the source by a single screw dislocation [83]: E = KL 2 b 2 v 2 (1.2) where K is given by a combination of material constants, and its comparison with Eq 1.1. As specified in § 3.1, this approach is used for the statistical analysis in the present study. Theoretical approaches to power-law statistics The theory of nonlinear dynamical systems can be found in many books and journal reviews, starting from the classical ones [START_REF] Nicolis | Self-organization in nonequilibrium systems : from dissipative structures to order through fluctuations[END_REF][START_REF] Haken | Synergetik[END_REF]. The most striking feature of such systems is that they self-organize. The result of self-organization may be a very complex behavior which cannot be understood in terms of summation of random or periodic motions. Three aspects of complexity, i.e., fractals, deterministic chaos, and self-organized criticality, have already been mentioned above. Since SOC is characteristic of infinite dimensional systems and leads to power-law statistics, it seems to be relevant to the studied problem and will be described below in some detail. Alternative interpretations of power laws will be also presented. At the same time, the correlation between the system elements may drastically reduce the effective number of degrees of freedom controlling its dynamics. It was shown in the case of the PLC effect that this reduction may lead to a transition from SOC to low-dimensional chaos [7,84]. Moreover, it is known that a system composed of many oscillators may reach perfect synchronization leading to simple periodic behavior, when all oscillators move in phase [85]. Another realization of synchronization can occur through propagation of a kind of switching waves [86], giving rise to the so-called "relaxation oscillations" which are characterized by alternation of fast and slow kinetics [START_REF] Andronov | Theory of Oscillators[END_REF]. Relationships between SOC and these dynamical modes will be shortly discussed. Finally, another remarkable feature of dynamical systems is universality, i.e., similar behavior of systems governed by distinct mechanisms and even coming from different fields of science, which allows classifying various systems into classes of universality. Thanks to this feature, simple models, like the ones presented below, often prove usefulness for understanding and modeling real behaviors. Self-organized criticality Despite a vast literature on self-organized criticality there is no clear definition of this concept, which is usually explained using examples of specific dynamical systems. The SOC concept was proposed by Bak et al. to explain the behavior of a simple cellular automaton model of a "sandpile" [44]. It was shown that the sandpile reaches a kind of critical state, characterized by power-law correlations, similar to second-order phase transitions. The salient feature of the behavior is that in contrast to phase transitions, the sandpile reaches the critical state spontaneously, without fine tuning of an order parameter. A simple sandpile model can be demonstrated using a square grid, each element of which is assigned an integer variable z(x, y) representing the number of sand grains accumulated on this site. At each time step a grain is placed on a randomly selected site. If z(x, y) reaches a critical value (equal to 4 in 2D case) on a given site, the grains are redistributed among its nearest neighbors or, eventually, leave the system through the grid boundaries. This redistribution may trigger chain processes leading to formation of an avalanche, with size s defined as the total number of "toppling" sites and duration T given by the number of time steps during which the chain process is developed. It can be said that the system performs avalanche-like transitions between different metastable states. The avalanche behavior is characterized by several power laws which describe the fractal structure of avalanche patterns, the probability densities P (s) and P (T ), the relationship s(T ) between the variables, and the Fourier spectra of time series representing the time evolution of system variables. The last feature explains why SOC is often considered as a possible mechanism of 1/f -noise. More specifically, the dependencies demonstrate a cut-off at large scale because of the finite dimensions of the system. For example, P (s) is generally described by a relationship: P (s) = s βs f c (s/s 0 ), (1.3) where β s usually varies between -1 and -2, and the cut-off scale s 0 is associated with the linear system size. The cellular automata present stochastic models of SOC. Another kind of models is based on a deterministic approach. The so-called spring-block models were proposed to explain the Gutenberg-Richter statistical law for earthquakes [START_REF] Chen | [END_REF]89,60]. Such models usually consider an array of blocks connected to their neighbors by springs. The blocks are pulled across an immobile plate by a driving plate to which they are also connected with the aid of springs. In this case the role of the parameter z is played by the local force acting on a given block. Moving the driving plate leads to adding a small force to each block. As mentioned in §1.2, the nonlinear friction law between the blocks and the immobile plate (see Fig. 1.4) results in a stick-slip behavior. If the local force exceeds the friction threshold, the corresponding block slips and the resulting rearrangement of local forces may trigger a chain process. Again, it was found that the dynamics of the system is described by power laws, providing that the driving rate is low. This simple scheme demonstrates three basic ingredients controlling the dynamics of such models: the threshold friction, separation of a slow time scale (loading) and a fast time scale (avalanche-like relaxation), and spatial coupling between blocks. Using 2D and 3D block-spring models, a good agreement was found between the simulated power-law exponents and the values following from earthquakes catalogues. As the earthquakes are related to accumulation of stresses in the earth crust, there is a direct analogy between this natural phenomenon and the plastic deformation of solids [START_REF] Lebyodkin | [END_REF][START_REF] Lebyodkin | [END_REF]. Consequently, the block-spring schemes are of special interest for modeling the intermittent plastic flow. In particular, they were successfully used to simulate the PLC behavior, as will be discussed in § 1.5.4. Alternative explanations of power-law behavior Although SOC was proposed as a general framework for explanation of numerous observations of power-law statistics, the systems for which it has been clearly established are rare. Various alternative models were proposed in the literature to explain such critical-like behavior within the concept of "plain old criticality", by adding some uncertainty. For example, Sethna et al. [90] studied a random field Ising model to explain the Barkhausen effect [90]. The Ising model considers a lattice of magnetic spins and represents a classical model for a first-order transition in a ferromagnet below Curie temperature: when the magnetic field passes through zero, the equilibrium magnetization reverses abruptly. By adding disorder in the form of a random magnetic field the authors did not only reproduce magnetic hysteresis, instead of a sharp transition, but also found scale-free fluctuations of magnetization for certain values of disorder. Another model giving power laws without parameter tuning was proposed by Sornette [91]. It is based on the slow sweeping of a control parameter [91]. The idea of this approach can be illustrated using a fiber bundle made of N independent parallel fibers with identically distributed independent random failure thresholds. The bundle is subjected to a force F . The ratio F/N plays the role of the control parameter whose value changes because when the applied force F is increased, more and more fibers break down. Again, the numerical simulation reveals power-law distributions of rupture sizes for a low loading rate. These models are conform with a general consideration by Weissman [45], who examined various mechanisms of 1/f -noise and concluded that it can occur because of smeared kinetics features. In particular, a system described by characteristic rates which are determined by a large number of independent factors should have a log-normal distribution. If this distribution is wide enough, its flat top might provide many octaves of power-law dependence. The author suggests that the 1/f -noise may be viewed as an extreme limiting case of many types of broadened kinetics. Role of finite loading rate and overlapping The models presented above bear evidence that the scale-invariant statistics may be due to various mechanisms. Therefore, determination of the underlying mechanism necessitates a detailed statistical analysis and a careful measurement of various critical exponents and relationships between them. This requirement raises above all the question of reliability of experimental determination of power-law exponents. As a matter of fact, the above models suggest either a very slow driving rate or, in the ideal case, halting the loading during the propagation of an avalanche. Consequently, no two avalanches will propagate simultaneously. The experiment cannot always approach this ideal situation, so that avalanches can overlap in time and in space and be recorded as a single event. Few works concerned a possible effect of the avalanche superposition on the critical exponents. Durin and Zapperi [92] experimentally measured the Barkhausen noise in soft ferromagnetic materials and distinguished two cases: when the exponent β T describing the avalanche duration probability tends to -2 for zero rate of the external magnetic field, such material shows a linear increase in β s and β T with an increase in the field rate; in the case when β T (0) > -2, the slopes are independent of the field rate. White and Dahmen [93] proposed a theoretical explanation of this behavior by considering a linear superposition of avalanches. According to their model, the decrease in the slopes of the power-law dependencies in the former case occurs because small events are absorbed into larger ones. In the latter case the superposition is strong starting from very low field rates, so that no changes are observed when the rate is varied. The authors also analyzed theoretically the case β T < -2 and found that the exponents must be rate independent for small enough rates, but a crossover to one-dimensional percolation on the time axis should occur when the rate is increased. To our knowledge, the effect of overlapping on the statistics has not been explored in the case of AE investigations of plasticity. Recent data on the distributions of AE amplitudes during the PLC effect [18] showed similar β s values for low and intermediate strain rates and somewhat flatter dependences for higher strain rates, in qualitative agreement with the above predictions. However, even at small strain rates the PLC instability results in merging of AE events during stress serrations, the latter being short on the macroscopic time scale but quite long in comparison with the individual AE events. This merging causes distinct bursts in the duration of AE events, so that a clear power-law behavior is actually observed only for AE amplitudes. Moreover, any investigation of plasticity represents a fundamental problem: the behavior is never statistically stationary because of the microstructural changes which are at the origin of the work hardening of the material. As a result, the values of the critical exponents evolve during deformation. Since there exist no two samples with identical microstructure, and besides, the microstructure evolution depends on various factors including the strain rate, a rigourous comparison of the results obtained at different strain rates is hardly possible. To study the effect of overlapping, another strategy was adopted in the present study (see Chapter 4). Namely, the criteria used to extract events from a signal were varied, so that various sets of events were extracted from each of the recorded signals and analyzed. Relation to other dynamical regimes As mentioned above, the analysis of stress serrations has led to a conjecture that the PLC effect presents a rare example of a transition from an infinite (SOC) to low (chaos) dimensional behavior when the driving rate is decreased. Another known example of a transition from a scale-free to a chaotic state concerns the hydrodynamic turbulence [94]. It can be conjectured that spatially extended dynamical systems may find themselves in various regimes which present different manifestations of complexity. The complexity associated with the deterministic chaos is related to the sensitivity to initial conditions, which can be quantified using the so called Lyapunov exponents. To illustrate it, one can take an infinitesimal initial distance between two phase trajectories and consider the evolution of its projections on the principal directions. In the limit of small time, it is expressed in terms of e λ i t , where λ i is the Lyapunov exponent in the ith direction. Chaos occurs when one λ i becomes positive, which reflects instability in the sense of the exponential divergence of trajectories, while the dynamics remains stable in other directions. Although the language of phase trajectories is impractical for the description of infinite dimensional systems, SOC is often associated with almost zero Lyapunov exponents which reflect a slower, power-law divergence. Another nontrivial dynamical regime is associated with the phenomenon of collective synchronization in a system of coupled oscillators, which spontaneously lock to a common phase, despite initially different phases of individual oscillators [85]. In fact, this phenomenon is modeled using the same lattice models as those proposed for SOC, i.e., characterized by a threshold dynamics, the separation of slow and fast time scales, and spatial coupling. Perez et al. [86] showed that such models allow for a transition between the scale-free and synchronized behaviors. Their dynamics is governed by the strength of the spatial coupling and the degree of nonlinearity of the driving force. More specifically, when the nonlinearity of the driving force is strong and the coupling strength is weak, large avalanches repetitively sweep the whole system, giving rise to relaxation oscillations. Moving away from these conditions leads to a gradual transition to a discrete distribution of a few avalanche sizes, coexistence of a discrete and a continuous distribution, and finally, power-law behavior. It seems natural to suggest from these two examples that the dynamical chaos is related to synchronization of various elements of the dynamical system. However, since chaos is commonly studied in low dimensional systems, such a relationship has not been examined so far. 1.5 Portevin-Le Chatelier effect General behavior The PLC effect is a plastic instability observed in dilute alloys and caused by interaction between dislocations and solute atoms. It was discovered in the early part of the 20th century [95,96] but continues attracting great attention of researchers up to now. One of the fundamental reasons for this interest is that the PLC effect offers a striking example of complex spatiotemporal dynamics of dislocations [START_REF] Kubin | Dislocations in Solids[END_REF]. Indeed, as will be seen below, it is one of the cases when the heterogeneity of plastic flow shows up at the macroscopic scale, which requires collective behavior of large numbers of dislocations. Besides, understanding of this behavior presents high practical interest because of undesirable detrimental effects on the formability of materials widely used in industry, e.g., steels and aluminum alloys [START_REF] Estrin | Continuum Models for Materials with Microstructure[END_REF]. Most of research into the PLC effect has been realized in the geometry of uniaxial tensile tests. In this case, the effect consists in repetitive localization of plastic strain rate within transversal deformation bands, which may either propagate along the tensile axis or be immobile. The static bands have a life time about milliseconds [START_REF] Schwarz | [END_REF]). The traces of the deformation bands which usually have width from fractions to several millimeters can be observed on the side surface of specimens using an optical microscope or even with the naked eye [99]. In the tests with a constant velocity of the mobile grip, i.e., constant imposed overall strain rate εa , the elastic reaction of the "deformation machinespecimen" system on the unstable plastic flow of the specimen gives rise to abrupt stress variations on stress-time or stress-strain curves (see Fig. 1.5). 2As can be recognized in Fig. 1.5, the serrations usually onset after macroscopically uniform deformation to a certain critical strain ε cr [102,103]. Figure 1.6 represents a typical dependence of the critical strain on the imposed strain rate. It can be seen that when ε is increased ε cr increases in the region of high strain rates (the so-called "normal behavior") but decreases at low strain rates ("inverse behavior"). Figure 1.6 also indicates a strain-rate range of observation of instability. As a matter of fact, the PLC effect occurs in limited domains of temperature and strain rate. The morphology of stress variations strongly depends on the experimental conditions, as illustrated in Fig. 1.5 for variation of εa at room temperature. In general, the shape of the deformation curves also depends on the material composition, microstructure, and sample dimensions. Nevertheless, several generic types of behavior have been found (e.g., [105]). Usually three persistent types called A, B, and C are distinguished, depending on the shape of the deformation curve and the spatial pattern of deformation bands [START_REF] Kubin | Dislocations in Solids[END_REF]. A salient feature is the gradual transition between different behaviors, which takes place when either T or εa is varied from one to another boundary of the respective range of instability. At a given temperature, type A behavior occurs close to the upper bound of the strain-rate interval of existence of the PLC effect, typically above 10 -3 s -1 for AlMg alloys. It is characterized by irregular stress fluctuations without visible characteristic amplitude, which are associated with deformation bands usually nucleating near one specimen end and propagating quasi-continuously along the tensile axis. Some stress fluctuations are likely to result from fluctuations of the width and velocity of a propagating band. Besides, the band propagation through the sample is followed by a phase when the stress increases until a new band is nucleated, upon which the stress falls towards the envelope deformation curve -the imaginary prolongation of the smooth curve prior to ε cr . Such a pattern of stress drops preceded by stress rises gave rise to a name of "locking serrations" for type A behavior. Type B instability is observed around εa = 10 -4 s -1 and is characterized by rather regular stress drops, such that the stress oscillates around the envelope curve. Each stress drop results from a static deformation band nucleated ahead of the previous band, for which reason this regime is often referred to as "hopping propagation" of a band. As the localized plastic deformation results in the material work hardening within the band, the serrations often form groups corresponding to the hopping band propagation along the entire specimen and separated by periods of smoother deformation, during which the stress is increased to finally trigger a new band. When the strain rate is further decreased, the transition to type C instability occurs upon approaching εa = 10 -5 s -1 . This behavior is associated with "unlocking serrations" -sharp stress drops below the envelope curve. Like type B serrations such stress drops display a characteristic size. They are usually attributed to randomly nucleated bands, although the analysis of serrations bears evidence to existence of some correlation [106]. Finally, mixed behaviors are observed at intermediate strain rates. Microscopic mechanism Since the dislocation mechanism of plasticity is governed by thermally activated motion of dislocations through obstacles [START_REF] Friedel | Dislocations[END_REF], it is characterized, like any other activated process, by a positive dependence of the driving force on the rate of the process: a higher force is necessary to sustain a higher rate. The occurrence of the PLC instability is generally associated with a negative value of the strain-rate sensitivity (SRS) of the applied stress σ: S = ∂σ ∂ln ε < 0. The inversion of the sign of S is attributed to dynamic strain aging (DSA) of dislocations [107, 108], engendering a recurrent process of pinningunpinning of dislocations from solute atoms. The DSA mechanism can be schematically described as follows. On microscopic scale the motion of dislocations is discontinuous: their free motion lasts very short duration and alternates with long arrest on local obstacles, such as forest dislocations, during the waiting time for thermal activation. The impurity atoms diffuse to dislocations during t w and additionally anchor them. This additionnal pinning stress is thus determined by the competition between two time scales, the waiting time t w and the solute diffusion time t a , which depend on the strain rate and temperature. The PLC effect occurs when the two time scales are comparable. This is illustrated in Fig. 1.7(a) which schematically represents the σ( ε)-dependence for a given temperature. If the strain rate is very high, t w is much smaller than t a and the dislocations move as if there were no impurity atoms. In the opposite limit (t w ≫ t a ) the dislocations are constantly saturated with solute atoms and move together with solute clouds. In both cases a normal positive-slope σ( ε)-dependence is observed for thermally activated process, as shown in the figure for the intervals ε < ε1 and ε > ε2 . However, the left-hand side of the curve corresponds to a higher stress level. Consequently, a negative slope occurs in the interval ε1 and ε2 , where the concentration C of solutes on dislocations diminishes with an increase in ε. The overall behavior is thus represented by a N-shaped σ( ε)-dependence. The interval of instability covers several orders of magnitude of ε (the scheme in Fig. 1.7(a) would look realistic if the ε-axis was supposed to be logarithmic). Such a stress-rate dependence leads to an instability in the form of relaxation oscillations [START_REF] Andronov | Theory of Oscillators[END_REF], providing that the imposed strain rate value finds itself in the range between ε1 and ε2 . Indeed, as demonstrated by Penning [109], when the loading brings the system to the threshold stress, ε undergoes a jump from the "slow" to the "fast" positive-slope branch of the characteristic N-curve. In velocity-driven experiments (constant εa ) the elastic reaction of the deformation machine converts the strain rate jump into drastic unloading, which terminates with a backward ε jump to the slow branch. The alternation of the slow loading and abrupt stress drops gives rise to a saw-toothed deformation curve, as illustrated in Fig. 1.7(b). Such a cyclic behavior is similar, for example, to the Gann effect in a medium with a negative differential resistance [110]. Different mathematical descriptions of this behavior use variants of the following constitutive equation composed of three additive terms: σ = σ h (ε) + S 0 ln( ε ε0 ) + σ DSA {1 -exp(- Ω(ε) ετ p )}, (1.4) where the first term reflects the strain hardening, S 0 is the positive SRS in the absence of DSA, and the third term is due to DSA [111,112]. In the last term, σ DSA is the maximum pinning stress corresponding to saturation of dislocations with solutes, τ is the characteristic diffusion time of solutes, p is equal to 2/3 for the bulk diffusion [113] or 1/3 for the diffusion in dilocation cores [114], Ω(ε) = bρ m ρ -1/2 f is the quantity introduced in [112] to describe the "elementary plastic strain", i.e., the strain produced when all mobile dislocations are activated and move to the next pinned configuration (b and ρ m are the Burgers vector and density of mobile dislocations, ρ f is the density of forest dislocations). This equation allows explaining the existence of a critical strain for the onset of the PLC effect. Kubin and Estrin [112] considered the evolution of Ω with deformation to find the conditions when the SRS becomes negative. Alternatively, the evolution of τ due to generation of vacancies during deformation was examined [115]. These analyses provided qualitative explanations of the normal and inverse behaviors of ε cr but the quantitative predictions were quite loose, especially for the inverse behavior. Recently, several attempts were made to improve the quantitative predictions. Some of these approaches are based on taking into account the strain dependencies of either σ DSA or the solute concentration on dislocations, the latter leading to a modification of the argument of the exponential function in Eq. 1.4 [116,117]. Mazière and Dierke [118] showed that the agreement with experimental results can be improved by replacing the condition S < 0 by a stronger condition leading to an exponential growth of instability. This question remains a matter of debate. Observations of complex behavior Complexity on the macroscopic scale of stress serrations The qualitative prediction of plastic instability associated with propagating PLC bands and saw-toothed deformation curves was one of the important successes of the microscopic models. However, the experimental observations display more diverse and complex patterns. Some aspects of this complexity were mentioned above (see also Introduction). Various observations are shortly summarized below. During the last few decades quite a number of different approaches were proposed to quantify the complexity of serrated deformation curves, e.g., dynamical analysis [12], statistical analysis [START_REF] Lebyodkin | [END_REF], wavelet analysis [119], multifractal analysis [START_REF] Lebyodkin | Multifractal analysis of unstable plastic flow[END_REF], and so on, each method highlighting one or another aspect of behavior. The entirety of results proved a correlated nature of stress serrations, with correlations strongly depending on the material and experimental conditions (temperature, strain rate, microstructure). In particular, power-law statistical distributions of the stress drop size and duration, as well as power-law Fourier spectra of deformation curves were found for type A instability [START_REF] Lebyodkin | [END_REF]7,84]. The analysis of critical exponents allowed for a conjecture of SOC-type behavior. However, this hypothesis remains a matter of debate. For example, Annathakrishna and Bharathi [START_REF] Ananthakrishna | [END_REF] pointed out that SOC models require a slow driving rate, while the behavior of the PLC effect is peculiar in this sense: whereas power-law statistics are observed at high εa -values, the decrease in εa leads to peaked histograms. These authors suggested that the critical behavior at high εa is similar to that in the hydrodynamic turbulence [94]. For type B serrations the emergence of deterministic chaos was proved [12,7,84]. In this case the statistical distributions of stress drops acquire an asymmetrical peaked shape. Finally, type C serrations are characterized by near Gaussian distributions. However, even these serrations are not completely uncorrelated, as revealed by the multifractal analysis [106]. As a matter of fact, multifractal scaling was found for stress-time curves in the entire strain-rate range of instability [84,122,[START_REF] Lebyodkin | Multifractal analysis of unstable plastic flow[END_REF]. Recent studies revealed that power-law distributions may appear even under conditions of type C behavior [18]. Unexpected in such conditions, this conclusion concerns stress fluctuations that occur on a smaller scale than the deep type C serrations. Indeed, amplifying the deformation curves, one can distinguish two distinct scales of stress drops at low strain rates. The mechanism responsible for the occurrence of small drops attracted little attention so far. Usually, they are considered as "noise" caused by the material heterogeneity. This point of view is confirmed, e.g., by their dependence on the surface treatment [123]. Nevertheless, such fluctuations show a nontrivial statistical behavior. When they are not disregarded, the statistical analysis results in bimodal histograms with two distinct peaks [11,[START_REF] Bougherira | Etude des phénomènes d'auto-organisation des ensembles de dislocations dans un alliage au vieillissement dynamique[END_REF]. Application of the analysis separately to the two groups of events shows that whereas the large stress drops follow rather symmetrical peaked distributions, power-law behavior with an exponent between -1 and -1.5 is found for small serrations (Fig. 1.8) [START_REF] Bougherira | Etude des phénomènes d'auto-organisation des ensembles de dislocations dans un alliage au vieillissement dynamique[END_REF]18]. This observation proves their non-random nature. Therefore, it reveals the multiscale character of deformation processes and adds interest to investigations with a higher resolution, e.g., with the aid of the AE technique. Complexity on a mesoscopic scale of AE Until recently, the AE study of the PLC effect was mostly concentrated on the analysis of average characteristics, such as the AE count rate [124,125,126]. Here, a count is recorded each time when the amplitude of an acoustic oscillation exceeds a given threshold value, usually set at the level of the background noise, as described in § 2.4. This confrontation of the discrete and continuous AE under conditions of the PLC effect has been questioned recently. First, the amplitudes of acoustic events were found to vary in the same range during smooth and jerky flow in an AlMg alloy, whereas bursts were observed for their durations and related characteristics such as the count rate or energy [START_REF] Bougherira | Etude des phénomènes d'auto-organisation des ensembles de dislocations dans un alliage au vieillissement dynamique[END_REF]. It was supposed that the PLC band is formed through a process of synchronization of glide events in a similar amplitude range, due to propagation of elastic waves, which leads to merging of the corresponding acoustic events. This conjecture is consistent with the optical observations of the complex evolution of the forming deformation bands [127,128], although it should be remarked that the optical methods applied were limited by millisecond time scale. It is also conform to the results of the multifractal analysis of individual waveforms recorded during jerky flow in an AlCu alloy [28], which proved that the single AE events may not be elementary but possess a fine structure. On the other hand, in Vinogradov and Lazarev [129] application of the data streaming technique, which allows recording the entire AE signal (see § 2.3), proved a continuous character of AE accompanying the PLC effect in α-brass. However, only type A behavior has been examined in this work. The statistical analysis of AE during the PLC effect started several years ago [27,[START_REF] Bougherira | Etude des phénomènes d'auto-organisation des ensembles de dislocations dans un alliage au vieillissement dynamique[END_REF][START_REF] Lebyodkin | [END_REF]18]. As already mentioned above in this Chapter, one of the major results of these works is that AE is characterized by power-law statistics of event size in all experimental conditions. Moreover, the AE events extracted separately during stress serrations and during smooth intervals revealed very close distributions. These results led to a conjecture that the deformation processes relevant to the mesoscopic scale (uncovered through AE measurements) have a similar nature during both stress serrations and smooth plastic flow. This unexpected result was also confirmed due to application of the multifractal analysis to the series of AE events, which testified that the temporal correlations characterizing the AE show no peculiarities associated with the PLC effect [START_REF] Lebyodkin | [END_REF]. At the same time, the exponents of the power-law distributions of AE amplitudes are clearly different from those observed for pure materials and are not unique. In particular, the values of the exponents depend on εa and evolve in the course of deformation. Numerical modeling The numerical models of the PLC effect aimed at reproducing the observed complex behavior of serrated deformation curves by allowing for strain heterogeneity and spatial coupling between differently strained regions in the deforming specimen. The general framework for phenomenological models was proposed by Zbib and Aifantis who suggested a spatial coupling term in the form of the second gradient of strain [130]. It appears as either an additional internal stress, σ = σ h (ε) + F ( ε) + c ∂ 2 ε ∂ 2 x (1.5) or in the form proposed by Jeanclaude and Fressengeas [131] and describing the dislocation transport due to double cross-slip of dislocations [START_REF] Friedel | Dislocations[END_REF], σ = σ h (ε) + F ( ε -D ∂ 2 ε ∂ 2 x ) (1.6) where F ( ε) is the above described N-shaped characteristic given by the sum of the 2nd and the 3rd terms in Eq. 1.4, c is a constant having an elastic nature, and D is a diffusion-like coefficient. Various mechanisms governing the coupling via internal stresses were examined in the literature: plastic strain incompatibility, which must by compensated by an elastic strain and thus gives rise to internal stresses [132], elastic fields of dislocations [START_REF] Canova | Large Plastic Deformations[END_REF], nonlocal strain hardening [130], rotation of the specimen axis in the case of single crystals [START_REF] Hähner | [END_REF], variation of the specimen cross-section, leading to a triaxial character of stress [START_REF] Bridgman | Studies in large plastic flow and fracture[END_REF]. The comparison of various mechanisms showed that long-range incompatibility stresses play a preponderant role in the most frequent case of polycrystals [START_REF] Hähner | [END_REF][START_REF] Lebyodkin | [END_REF], although the transport of dislocations may manifest itself on Similar ideas, based on the combination of the nonlocal approach with the abovedescribed microscopic model of DSA, were used to implement 3D finite-element models (e.g. [140,141,142,137]). Whereas 3D models are very expensive with regard to com-putation time, they provide an opportunity to avoid explicit conjectures on the nature of spatial coupling. For example, Kok et al. suggested a model of the PLC effect in polycrystals, in which spatial coupling appears due to incompatibility of plastic strains between differently oriented adjacent grains, provided that dislocation glide is considered in various slip systems characteristic of fcc metals [140]. This model described salient spatiotemporal features of the PLC effect, as well as the power-law and chaotic dynamical regimes. The entirety of these results proves that the dynamics of the PLC effect is essentially determined by two factors: the negative strain rate sensitivity, stemming from DSA and leading to instability of the homogeneous plastic flow, and strain heterogeneity, which tends to disappear due to spatial coupling but is recurrently resumed because of the instability. Twinning Mechanisms and dynamics In some cases plastic deformation may proceed by twinning. The term "twinning" describes a situation when two crystals with the same crystal lattice but different orientation adjoin each other and are separated by an interface. The crystallography of twins is described in various manuals [START_REF] Hirth | Theory of dislocations[END_REF][START_REF] Friedel | Dislocations[END_REF]. A simple scheme is presented in Fig. 1.11, where the open circles denote the atomic positions in a perfect crystal lattice and the black circles show atoms in twinned positions, so that the upper half of the crystal is a mirror reflection of the original lattice. It is noteworthy that although the shear of the two parts of the crystal is macroscopic, this scheme shows that it can be realized due to shifting each atom over only a part of the interatomic distance. Such shift in one atomic plane may be produced by a displacement of a partial disclocation with a Burgers vector smaller than a translation vector of the lattice. In contrast, dislocation glide involves motion of perfect dislocations which shift the lattice by a complete translation vector and do not create an interface. As the energy of a dislocation is proportional to b 2 , the partial dislocation carries less energy and seems to provide a favorable mechanism of plasticity. However, the interfaces occurring during deformation of real crystals are not perfect and their creation needs much energy. Consequently, twinning requires high local overstresses and occurs when dislocation glide is impeded for some reason. For example, it is often observed at low temperatures, even in materials with high symmetry, e.g., face-centered cubic and body-centered cubic crystals which possess many slip systems [41]. The low-temperature twinning is usually explained by a faster increase of the flow stress for perfect dislocations than for partial dislocations. Furthermore, twinning is important in materials with a limited number of slip systems, such as hexagonal closepacked crystals [START_REF] Mathis | Acoustic Emission, chapter Exploring Plastic Deformation of Metallic Materials by the Acoustic Emission Technique[END_REF]. It should be underlined that twinning and dislocation glide are interdependent processes. On the one hand, piling-up of dislocations provides internal stress concentration necessary for twin nucleation. On the other hand, twinning locally changes orientation of the crystal lattice and makes it more favorable for slip[144]. The deformation (or mechanical) twins occur in the form of thin layers confined by two boundaries. In this sense, there is a similarity between twinning and unstable plastic flow, because both are associated with a localized shear. Various twinning mechanisms were suggested in the literature. Perhaps, the most known is the Cottrell-Bilby pole mechanism which considers formation of a twin layer due to rotation of a twinning dislocation around a fixed point, so that the dislocation shifts with each turn to an adjacent crystallographic plane [START_REF] Hirth | Theory of dislocations[END_REF]. Among other, such mechanisms were considered as the breakaway of partial dislocations from sessile dislocation configurations, nucleation of partial dislocations due to splitting and double cross-slip of dislocations near dislocation pile-ups [START_REF] Hirth | Theory of dislocations[END_REF], autocatalytic mechanism of twin nucleation similar to martensite transformation [145], and so on. Even this incomplete list shows that different models often aim at describing different aspects of the process, which is likely to include several mechanisms. Indeed, numerous experimental observations proved that nucleation of twinning dislocations and their motion leading to formation of interfaces is a very fast process (the twin tip velocity can reach hundreds of meters per second) [146,147,41], in agreement with the need of high local overstresses for its triggering. This can hardly be explained by the pole mechanism, which can however allow for the subsequent twin widening with velocities about 10 -3 cm/s [41]. Twinning is known to be accompanied with high-amplitude discrete AE and, therefore, it offers another model system for the present study. Moreover, as it involves fast nucleation, multiplication, and motion of dislocations, twinning possesses features of avalanche processes. In particular, it can give rise to serrations on deformation curves. The jerky flow occurs as a rule at low temperatures [40] but is also observed at room temperatures [82]. Finally, it is generally suggested that the fast twin nucleation process is responsible for the occurrence of AE during twinning, whereas the slow twin growth is virtually silent [START_REF] Mathis | Acoustic Emission, chapter Exploring Plastic Deformation of Metallic Materials by the Acoustic Emission Technique[END_REF]. In the present thesis twinning was studied using Mg polycrystalline alloys with hcp crystal structure. There are three main slip planes in Mg (Fig. 1.12): basal (0 0 0 1), prismatic {1 1 0 0}, and first-order pyramidal plane {1 0 1 1}, which provide a total of four independent slip systems. According to the von Mises criterion [148], plastic deformation of polycrystals necessitates operation of five independent slip systems. The dislocations in the second-order pyramidal plane have a very large Burgers vector and their activation was only observed above 200°C [149]. Consequently, twinning is found to be an important deformation mode of Mg alloys at room temperature. The main twinning system in a Mg crystal deformed by tension is {1 0 1 2} 1 0 1 1 , which provides extension along the c-axis of the hexagonal lattice. However, as the polycrystalline grains are differently constrained, twins were also observed in other planes, mainly in {1 0 1 1} and {1 0 1 3} [150]. Figure 1.12: Slip systems in the hexagonal close-packed lattice. The prismatic and firstorder pyramidal planes are hatched. The second-order pyramidal plane is outlined by a dash-and-dot line. AE studies Acoustic emission in single and polycrystals of magnesium and its alloys was studied in a number of works (e.g., [151,[START_REF] Heiple | [END_REF]152,153,154]). However, all these papers were devoted to investigation of the overall AE behavior as a function of material and experimental conditions: material composition, orientation of single crystals, strain rate, and temperature. It was proven that both deformation twinning and dislocation glide contribute to the AE. To our knowledge, no statistical investigation was carried out on this material, except for a paper [155] included in this thesis. Several works on the statistics of AE in other hexagonal metals (cadmium and zinc) appeared recently [82,23]. Figure 1.13 displays examples of stress-strain curves for Zn-0.08%Al single crystals with different orientations with regard to the tension direction. The deep serrations observed on the deformation curves testify to intense twinning in these crystals. Usually, there are too few stress drops before fracture, so that the statistical analysis of their amplitudes is hardly possible. In contrast, such materials manifest a strong AE, the statistics of which was studied in the cited papers. exponent ranging from -1.4 to -1.6. This β-value is similar to the data for ice single crystals deformed by basal glide [20]. Moreover, no difference was found in [82] between the distributions obtained for the stage of easy basal glide and for the subsequent stage characterized by intense twinning, as illustrated in Fig. 1.14 for a Cd single crystal. This result testifies that from the viewpoint of complex dynamical systems, twinning and dislocation glide may belong to the same universality class. Another interesting result consists in the observation of two distinct AE waveforms depicted in Fig. 1.15. This result will be discussed in § 5.2.3 in relation to the present study. However, some aspects are noteworthy here. Based on a similarity with the data obtained on sapphire single crystals [156], the authors supposed that the abrupt events (upper figure) are generated by dislocation avalanches and the complex patterns (bottom figure) are indicative of twinning. In the latter case, the initial high-frequency burst is supposed to correspond to the abrupt twin nucleation and the next low-frequency behavior is associated with the twin growth. The last statement contradicts the abovementioned opinion that the twin growth generates virtually no AE [START_REF] Mathis | Acoustic Emission, chapter Exploring Plastic Deformation of Metallic Materials by the Acoustic Emission Technique[END_REF]. On the other hand, by analyzing the evolution of the ratio between AE peak amplitude and AE root mean square amplitude (Fig. 1.16), which takes on higher values for short events, the authors found that such events indeed dominate during the basal slip, whereas the second type of waveforms becomes preponderant when twinning occurs (see Fig. 1.16). Whatever the exact nature of each of these types of events, their separation allowed for an important observation that they mutually trigger each other. It can be concluded that twinning and dislocation glide are interdependent processes not only on the macroscopic time scale, associated with the work hardening, but also on a mesoscopic scale related to avalanche deformation processes. AlMg alloys. Tensile tests were carried out using aluminium alloys of 5000 series with 5wt.% and 3wt.% of Mg. The average grain size in these two kinds of alloys was about 4-6 µm and 30 µm, respectively (Figs. 2.1). Specimens with a dog-bone shape and gauge parts 30 × 7 × 1mm 3 (Al5Mg) or 25 × 6.8 × 2.5mm 3 (Al3Mg) were cut from cold-rolled sheets in two orientations, along and across the rolling direction, and tested either in as-rolled conditions or after annealing for 2h at 400 • C, followed by water quenching. Such heat treatment is well-known in literature. It is aimed at dissolution of secondphase inclusions and obtaining a virtually homogeneous solid solution with a uniform distribution of magnesium atoms [157]. For both alloys, it led to partial recrystallization and resulted in grain size about twice as large as it was initially (Figs. 2.2). Mg alloys. A thorough study was realized using MgZr alloys which are known to be prone to twinning. Samples with different Zr content were taken as it strongly influences on the grain size. Specimens with a rectangular cross-section of 5 × 5mm 2 and gauge length of 25 mm were prepared from material with 0.04wt%, 0.15wt%, and 0. Mechanical testing The present work presents data obtained in tensile tests. The AlMg samples were deformed in a highly sensitive screw-driving Zwick/Roell 1476 machine controlled by the software package testExper. Either 10kN or 100kN load cell was used depending on the specimens cross-section, in order to warrant the maximum resolution. The tests were carried out at a constant crosshead speed V , i.e. in the hard-machine configuration (the machine stiffness was approximately 10 7 N/m). This loading mode is known to be characterized by complex deformation curves and acoustic emission, for which various nontrivial dynamical regimes have been found (see Chapter 1) [12,[START_REF] Lebyodkin | [END_REF]82]. It is also worth reminding the analogy between the PLC effect in the hard machine and the dynamical "stick-slip" models of earthquakes, which was suggested in several works [START_REF] Lebyodkin | [END_REF]6] and adds interest to the analysis of AE from the general viewpoint of investigation of avalanchelike processes. Since the PLC effect displays various types of behavior depending on the imposed strain rate and/or temperature, V was varied in a wide range corresponding to the nominal applied strain rate (referred to the initial specimen length) in the range εa = 2 × 10 -5 s -1 ÷ 2 × 10 -2 s -1 . All tests were performed at room temperature. The choice of the acquisition time for recording the stress-time curves was a compromise between the Acoustic emission measurements Since the present work was aimed at investigating the intrinsic structure of the global AE signals generated in the deformed sample, this allowed using not very long specimens and a single acoustic sensor in each experiment. Spatial localization of the sources of AE events, i.e., mapping the sites where local plastic processes take place, will be a task for future work. The AE recording equipment utilized in the present study was based on the systems which allow for continuous sampling of the AE signal arriving from the piezoelectric transducer. This made possible a comprehensive post-processing of both the complete stored signal and the waveforms of individual acoustic events. Since such continuous recording results in huge data files, the entire data stream was only gathered for high enough strain rates, εa > 2 × 10 -4 s -1 . The signal was recorded piece-wise in the slower tests. In all cases, the equipment additionally used a standard procedure of picking out acoustic events ("hits") during the entire test, applying some preset criteria (without data streaming), as described below (see §2.4). In the experiments on AlMg and MgZr alloys, the signal from the transducer was pre-amplified by 40 Individualization of acoustic events As follows from the discussion of the possible overlapping of avalanche processes in § 1.4.3, the problem of extraction of individual events from a continuous signal is very important for application of quantitative analyses to experimental data. For example, acoustic events may either closely follow each other or be generated almost simultaneously in different locations in the sample, and be recorded as a single event. It is not clear a priori how such overlap of individual events would affect the results of the statistical analysis. On the other hand, each AE event might give rise to several echoes due to sound reflections from interfaces and consequently, be recorded as a few separate events. Another source of errors stems from insufficient resolution of the individual events against noise. All these factors are affected by the criteria utilized to identify the events within the acoustic signal. Up to now, the sensitivity of the apparent statistics to these criteria has not been verified experimentally, although the problem is general and concerns a vast range of dynamical systems of different nature which are characterized by depinning transitions and avalanche-like behavior. Since the data streaming leads to huge data files (many hundreds of gigabytes), some standard procedures are used in most of AE applications to extract significant acoustic events in real time, without recording the continuous signal itself. In order to ensure the continuity of the present results with literature data, as well as the conformity of the results obtained by different methods and for different datasets, we applied the same approach to the events identification in a continuously stored AE signal. Namely, the record is looked through and the procedure makes use of four preset parameters (Fig. 2.6): • The threshold voltage U 0 . This parameter is aimed at cutting off the part of the acoustic signal below the noise level. An event is considered to start when the signal surpasses U 0 . • The hit definition time HDT. The event is considered to have come to an end if the signal remains below U 0 longer than during HDT. • The peak definition time PDT which determines the event peak amplitude A. Namely, the software detects the local maxima of the signal and compares them to the current value of its absolute maximum. The current global maximum is recorded as the event peak amplitude A if it has not been exceeded during a period equal to the PDT. Otherwise, it is assigned the new value and the time counting is restarted. In what follows, this parameter is supposed to be equal to half HDT, if not stated explicitly otherwise. • The hit lockout time (HLT), or dead time. After detecting the end of an event, no measurements is performed during HLT in order to filter out sound reflections. The HLT is triggered at the end of HDT. Consequently, the sum of HDT and HLT represents the minimum time between the end of one event and the start of the next one. • The duration δ; • The dissipated energy E computed as the integral of the squared signal amplitude over the event duration: E = ´tb +δ t b U 2 (t)dt; • The count-rate defined as the number of crossings of the acoustic signal over the noise threshold. It is obvious that the choice of U 0 , HDT, and HLT may influence on the identification of the AE events and, therefore, on the apparent statistical distributions of their characteristics. The conventional AE studies apply the following rules of thumb to set the time parameters. In one approach, a high value is chosen for HDT in order to include all sound reflections into the event. HLT can then be taken small. The disadvantage of this approach is that the event duration and the related parameters, such as the AE energy, are loosely determined. In the opposite case, a small HDT is set in order to separate the hit from the sound reflections which are then cut off by choosing a large HLT. This method is not free from drawbacks either, because it results in a lost of a part of the useful signal, particularly, the "aftershocks" that may follow the initial plastic event [82]. In any case, the criteria of "smallness" or "largeness" are approximate. These difficulties justify the need for investigation of the influence of the parameters of the events individualization on their statistics, which was undertaken in the present thesis. Chapter 3 Data analysis Statistical analysis The statistical analysis proved to be rather useful for studying plastic deformation because of numerous examples of intermittent deformation behavior which is associated with non-gaussien, power-law statistical distributions of plasticity processes (see Chapter 1). The statistical analysis of AE accompanying various physical processes is usually applied to the events amplitudes (e.g., [57,82,[START_REF] Bougherira | Etude des phénomènes d'auto-organisation des ensembles de dislocations dans un alliage au vieillissement dynamique[END_REF]) because the duration and other allied characteristics, in particular, the dissipated energy obtained by integration over the AE event duration, do not properly reflect the properties of the source signal. Indeed, the time characteristics may be affected by the properties of the transfer function of the piezoelectric transducer, reflections and interference of sound waves, and so on. An additionnal reason for such a restriction concerning the PLC effect is the merging of acoustic events during stress serrations, which was conjectured in [START_REF] Bougherira | Etude des phénomènes d'auto-organisation des ensembles de dislocations dans un alliage au vieillissement dynamique[END_REF] and will be justified by explicit proofs in the present thesis. It is found that the events merging results in bursts in their apparent durations, whereas the amplitude range remains essentially unchanged. For these reasons, we will pay the most attention to the analysis of the amplitude distributions. Two approaches to the statistical analysis of AE amplitudes are reported in the literature. The first approach is based on the qualitative analogy between plastic deformation of solids and seismic processes. Analysis of a large number of seismic records proved that the earthquakes statistics obeys the Gutenberg-Richter relationship between their magnitude M, a characteristic roughly corresponding to the logarithmic peak amplitude of AE events, and the number N of earthquakes greater than or equal to magnitude M : log 10 N = a -(b × M) [61]. Here, the seismic b-value characterizes the power-law scaling. As is readily seen, a disadvantage of such a direct projection of the methods of seismology on the study of AE during plastic deformation is that power-law behavior is expected a priori. Another approach to the analysis of AE statistics was proposed by Weiss et al. who studied plastic deformation in pure materials (e.g., [20]). The authors argued that the squared value of the peak amplitude A of an acoustic wave provides a physically based measure of plastic activity, reflecting the energy E dissipated by the plastic processes in the deforming sample: E ∼ A 2 . Both approaches were used in the present thesis, and led to consistent results. In what follows, the results obtained by the latter method are presented.The following procedure was used to compute distributions of (squared) amplitudes and durations of acoustic events. For a given quantity x, its probability density function P (x) is expressed as P (x) = 1 δx δN (x) N , where N is the total number of data in the statistical sample and δN(x) the number of events corresponding to x-value in the interval [x -δx/2, x + δx/2]. To handle the statistics of rare events, e.g., events with high amplitude, the method of variable bin sizes was used. As a rule of thumb, when an initial-size bin contains less than a preset minimum number of events, it is merged with the next bins until this minimum number is reached (cf. [START_REF] Lebyodkin | [END_REF]6]). In the present work, it was chosen to be equal to 5. In the case when the data were distributed according to a power-law in a certain interval of value, the power-law exponent was determined as the slope of the corresponding linear dependence in double logarithmic coordinates using the least-squares method. Recently, a new generalized method, based on the Monte-Carlo technique, was proposed to evaluate the closeness of experimental statistics to a power law [158]. The test calculations showed that the evaluation of the exponent and its uncertainty by these two methods provide similar results. In order to examine the strain dependence of the power-law dependences, the time intervals where both the AE and the deformation curves look visually steady were first searched for. This partition was then refined by varying the width of the intervals and repeating the calculation of the distribution functions until finding intervals where the magnitudes of the exponents remained constant within the error determined by the least-squares method. Fourier analysis Another well-known fundamental tool for processing time series is Fourier transformation which allows determining the frequency spectrum by decomposing the analyzed signal f (t) into the sum of harmonic functions {e ikt }: f (t) = +∞ k=-∞ f k e ikt , (3.1) where f k is the contribution of the frequency k, given by the following relationship: f k = 1 2π ˆπ -π f (t)e -ikt dt (3.2) For numerical analysis of discrete time series, the respective discrete form of Fourier transformation (DFT) reads: F m = N -1 n=0 f n e -2πi N mn (3.3) where m = 0, 1, ..., N -1. Equations (3.3) translate the mathematical meaning of the transform but the direct computation using this definition is often too slow. Fast Fourier transformation (FFT) is usually applied in practice [159]. In the thesis, the MATLAB realisation of this method was utilized to determine the spectra of the typical AE waveforms observed in the experiment. This analysis is particularly efficient when the analyzed signal is composed of several harmonics. Then, the output of the computation provides their frequencies and intensi-ties. However, the FFT applications are not limited to these cases. In fact, the Fourier transformation often provides important pieces of information on the nature of complex signals. For example, white noise possesses a continuous spectrum, where all harmonics have the same amplitude. The spectra of deterministic signals describing non-periodic motion are also continuous, however, various harmonics give different contributions. For example, the spectrum of a chaotic signal typically decreases with increasing frequency, with broad peaks or narrow lines often superposed on this background [START_REF] Kubin | Dislocations in Solids[END_REF]. In the case of SOC, the continuous spectrum is described by a power-law S(f ) ∼ 1/f α , with the exponent α in the range between 1 and 2 [44]. It is worth noting that in practice, these idealized patterns are perturbed by the contribution from noise, which reduction is often a difficult task. In addition to FFT analysis of the AE waveforms, the Fourier spectral analysis was also applied to the continuously recorded signal in order to examine the evolution of the emitted energy and the characteristic frequency of the signal. The details of the procedure can be found in the original paper by Vinogradov [160]. The data set was divided into half-overlapping windows of 4096 data points. After subtraction of the laboratory noise which was recorded before each test, the power spectral density function P (f ) of each subset was computed using FFT, and two characteristics were obtained from the P (f ): the sound "energy" E E = ˆfmax f min P (f )df (3.4) and the median frequency f med : ˆfmed 0 P (f )df = ˆ∞ f med P (f )df (3.5) The so-obtained time dependences of energy and median frequency were smoothed over ten to thousand points, depending on the applied strain rate. As argued in [160], the variations in the average energy and median frequency reflect the variations in the degree of correlation in the motion of dislocations generating the AE. Multifractal analysis The multifractal (MF) analysis [15] has been widely used to detect self-similarity in natural systems with complex dynamics. The applications concern the treatment of problems or, somewhat more often, in the field of materials science could be cited. In particular, it was successfully used to characterize the serrated deformation curves under conditions of the PLC effect [84,[START_REF] Lebyodkin | Multifractal analysis of unstable plastic flow[END_REF]. The approach described in [START_REF] Lebyodkin | Multifractal analysis of unstable plastic flow[END_REF] is accepted in this thesis. A short outline of the method, necessary for understanding the data analysis, is given below. A more detailed description including the basic concepts of fractal dimensions is presented in Appendix 1. The application of the MF analysis requires defining the so-called local probability measure. This quantity is constructed to characterize the intensity of the signal locally, in a time window, and allows detecting the scaling law when the window width is varied. As the papers [84,106,[START_REF] Lebyodkin | [END_REF] dealt with other kinds of time series, in particular, with stress fluctuations, one issue to be discussed here concerns the choice of the measure for the analyzed signals. In the present case, two kinds of time series have been dealt with: the raw AE signal U(t) measured "continuously" with a given sampling time dt, i.e., at t = jdt (j is the serial number) and the series of amplitudes of AE events, A j . Note that in the latter case, the index j does not order the recorded data points but the extracted AE events characterized by the amplitude and time of occurrence. The same calculation procedure was utilized in both cases. In what follows, both time series will be designated as ψ j (j = 1...N, where N is the total number of data points in the respective series). The calculations were performed using the fixed-size box-counting algorithm [START_REF] Falconer | Fractal Geometry, Mathematical Foundations and Applications[END_REF]. For this purpose, the analyzed time interval T is covered by a grid with division δt (see µ i (δt) = n k=1 ψ k N j=1 ψ j , (3.6) where n is the number of data points in the i th interval. Using this definition for the measure, the value of δt is varied and the scaling of the partition functions      Z q (δt) = i µ q i , q = 1 Z 1 (δt) = i µ i lnµ i , q = 1 (3.7) is studied for several real q-values. It should be noted for clarity that the series A j do not fill the time interval continuously. Obviously, only nonempty intervals corresponding to nonzero measure are present in these sums. As can be easily shown (e.g., [START_REF] Lebedkina | [END_REF]), in the trivial case of self-similarity of a constant signal, Z q is proportional to δt q-1 and the dependences log(Z q )/(q -1) vs log(δt) are represented by straight lines with the same slope equal to 1 for all q = 1 (Z q ∝ log(δt) for q = 1). A pure stochastic or a periodic signal on average tend to this case due to homogeneously filling the time axis above a certain characteristic scale (the magnitude of the period or the average spacing). A fractal signal would also be characterized by a unique slope. However, its value, which is called fractal dimension, differs from unity. As insignificant as it could seem, this quantitative difference in the scaling law reflects very complex behavior. For multifractal objects, most often met in real complex systems, the slopes of the straight lines depend on q. In this case, the following relationships are fulfilled:      Z q (δt) = δt (q-1)D(q) Z 1 (δt) = D(1)lnδt (3.8) where D(q) are the generalized fractal dimensions (designation D q will be also used when needed to facilitate reading) . Besides the spectra of generalized dimensions, an equivalent description in terms of singularity spectra f (α) [169] was used as well. Here, the singularity strength α (Lipchits-Hölder index) describes one more feature of self-similar structures, namely, singular behavior of the local measure, expressed as a scaling law: µ i ∼ δt α . (3.9) For multifractal objects, the exponent α can take on a range of values corresponding to different regions of the heterogeneous object analyzed. The f -value gives the fractal dimension of the close-to-uniform subset corresponding to close singularity strength between α and α + dα. By representing a heterogeneous object as consisting of interpenetrating fractal subsets, this description clarifies the physical meaning of the MF analysis. At the same time, the singularity spectra are more difficult to calculate. For the sake of clariness, more details are taken to Appendix 1. Essentially, the MF analysis allows uncovering the presence of correlations between both amplitudes and times of occurrence of events in the signal through their scaling behavior, and characterizing the heterogeneity of the scaling properties over a range of subsets of events. Focusing on a given subset is obtained through the choice of q: for example, large positive q-values tend to select large measures in the partition function, while large negative q-values allow highlighting small events. Variation of q over a set of real numbers thus provides continuous characterization of heterogeneity, the property known as a "mathematical microscope". A wide spectrum of the values of D, f , or α indicate a substantial shift in the correlation characteristics between large events on the one hand, and small events on the other hand. Note that most natural fractals are multifractals, since (homogeneous) fractality is a more demanding property than (heterogeneous) multifractality. In the case of real signals, the above scaling laws are satisfied in bounded ranges of δt. Indeed, experimental data always possess characteristic scales implied by the size of the system, the dimensions of its structural elements, the length of the time interval analyzed, the resolution of the equipment, the experimental noise, and so on. In order to reduce noise, two approaches were applied. First, the test calculations showed that the MF analysis of an entire AE signal leads to stochastic-type behavior. By applying a threshold U 0 and gradually increasing its height, it was found that multifractal behavior can already be detected for the data exceeding 24 dB, in spite of the incomplete suppression of the noise. This threshold level was usually applied. A complementary method of noise reduction made use of wavelet analysis described in the next section. The MF spectra were plotted by varying q in a large interval from - Wavelet analysis Wavelet transformation is a powerful tool of signal processing, whose principles and numerical implementation have been described in detail in many books and original papers (e.g., Refs. [START_REF] Daubechies | Ten Lectures on Wavelets[END_REF][START_REF] Percival | Wavelet Methods for Time Series Analysis[END_REF]). Only basic concepts, which are of relevance to the subsequent analysis, will be recapitulated in this section. Wavelet transformation decomposes a signal into constituent parts with different frequencies and evaluates the contribution of each part on different time scales [START_REF] Daubechies | Ten Lectures on Wavelets[END_REF]. It is defined through relationship: W (a, b) = 1 √ 2 ˆ∞ -∞ f (t)ψ( t -b a )dt, (3.10) where a is a scale factor, b is a time shift, and ψ is a soliton-like function, known as "mother" wavelet, which satisfies a following condition: 1. Take a wavelet and compute the coefficient from relationship 3.10. ˆ∞ -∞ ψ(t)dt = 0. ( 3 2. Shift the wavelet to the right and repeat step 1 until the whole signal is covered. 3. Scale (stretch) the wavelet and repeat steps 1 through 2. 4. Repeat steps 1 through 3 for all scales. In the present work, we used Matlab discrete wavelet transformation (DWT) based on dyadic scales. It is applied to series with length equal to a power of 2, and the wavelets are stretched, or dilated, by a factor of 2. In each level of analysis, the analyzed signal is decomposed into two parts: the approximation (low-frequency components) and the details (high-frequency components), as shown schematically in Fig. 3.2. The 66 decomposition can be iterated, so that the signal becomes represented as a sum of many components with specified sets of frequencies. It is worth noting that for many real signals, the low-frequency content is the most essential. This remark concerns, for example, the human voice. Indeed, although the voice cleared of the high-frequency components sounds differently, the content of the speech remains recognizable. On the contrary, the removal of the low-frequency components results in a voice sounding like noise. Thus, the wavelet transformation filtering the high-frequency components may uncover the inherent elements of the signal. Chapter 4 Role of superposition on the statistics of AE events In this Chapter, the effect of the parameters of individualization of acoustic events on their statistics is examined using various materials the plastic deformation of which is accompanied by strong acoustic activity and controlled by different microscopic mechanisms: a combination of twinning and dislocation glide in hexagonal MgZr alloys and the PLC effect in face-centered cubic AlMg alloys. These materials present a particular interest for the analysis because on the one hand, as pointed out in Chapter 1, both the waveforms of the individual acoustic events and the overall AE behavior have different signatures in the case of twinning or dislocation glide [82,[START_REF] Vinogradov | [END_REF]. On the other hand, recent investigations reveal a persistent power-law character of the amplitude statistics of AE in different experimental conditions, although the corresponding exponent may vary with conditions. Such robustness allows analyzing the sensitivity of the experimental estimates to the event identification criteria. A detailed analysis of the AE statistics in relation with the relevant deformation processes will be the scope of the next chapters. To study the effect of the parameters applied to individualize acoustic events on their apparent statistics, the continuous acoustic signal was recorded using "data streaming" technique. As described in Chapter 2, in order to ensure conformity of the results obtained by different methods and for different datasets, the events identification in a continuously stored AE signal was designed to mimic the standard procedure used by the acoustic system to extract AE hits in the course of measurement. Each of the three parameters, U 0 , HDT, and HLT, was varied in a wide range of magnitude while keeping the other parameters constant. The entirety of experiments showed that for all samples, and similar to literature on various materials (e.g., [152,174,124]), nonstationary AE activity is observed at small strains, in the region of the elastoplastic transition. It further decreases and displays roughly stationary behavior, which allows performing the statistical analysis in a steady-state range. The conformity of the statistics for the events which are either extracted from the continuously recorded signal or detected by the equipment using preset parameters was first verified as illustrated in Fig. 4.1. where the statistical analysis was performed for this specimen were selected for t >120 s. An example of comparison of the results of calculation of the statistical distributions is illustrated in Fig. 4.1(c). In this test, the threshold U 0 was chosen to be equal to 16.48 mV for the continuously measured signal, which corresponds to the logarithmic threshold of 45 dB in the case of real-time measurement. The time parameters were set at HDT = 800 µs and HLT = 100 µs. Importantly, despite the discreteness of the logarithmic measure allowed by the standard procedure, the series of events selected with the aid of two different methods coincide with high accuracy. Furthermore, it can be seen in Fig. 4.1(c) that both methods result in power-law dependences over more than three orders of magnitude of A 2 and give close values for the exponent β determined as a least-square estimate of the slope of the dependences in double-logarithmic coordinates: β = -1.80 ± 0.03 and β = -1.78 ± 0.03 for the "continuous" and "discrete" methods, respectively. This verification justifies the application of the continuously stored signals to the investigation of the effect of the conventional parameters of identification of AE events on their statistics. We further proceed to a detailed examination of their role for two kinds of alloys. Using families of such curves, the dependences of β on U 0 , HDT, and HLT were traced in large ranges of variation of each parameter. recognized that β relatively strongly depends on U 0 only in a narrow range U 0 < 10 mV (even here, all changes in β do not exceed 0.2). The rate of the dependence falls with increasing U 0 . A weak or no dependence (within the error bars) is observed for U 0 in the range from 10 mV to the maximum value of 90 mV, above which the amount of data becomes too small for the statistical analysis. Most probably, the initially higher β-value is explained by the merging of successive AE events when U 0 is low, so that the fraction of smaller events is decreased and the apparent power law is flatter than the true one. Indeed, such merging can take place for low U 0 values if the individual AE hits are linked to each other due to the presence of a (quasi)continuous background, e.g., noise, as illustrated in Fig. 4.4. By admitting this conjecture, the end of the fast changes in β can be explained by U 0 exceeding the continuous background level, so that the AE events become essentially isolated. The merge of such isolated hits will be weaker and will depend on the relationships between all identification parameters and the temporal arrangement (clustering) of the hits, so that some further slight (if any) evolution of β with increasing U 0 may be attributed to the diminishing merge of successive events. MgZr alloys This suggestion is consistent with the changes observed when HDT is reduced from 800 µs to 50 µs: since less effective overlapping of successive hits is expected for the lower HDT, the initial fall in the dependence becomes sharper and β quickly saturates at an approximately constant value. The weak dependences which can be detected in the almost saturated region may also be due to the influence of the statistics depletion. In any case, an important conclusion following from this figure is that the effect observed is weak in a wide U 0 -range. Furthermore, for each HDT, a threshold value can be found above which its influence is insignificant. Another interesting observation is that whereas two of the three kinds of samples display close β-values, the curve obtained for the material with the largest grain size goes separately. In spite of this quantitative difference, all the curves have the same shape described above. Thus, although the different power law indicates a different specific structure of the AE signal in Mg0.04%Zr, a similar effect of U 0 is found. Such robustness testifies that the quantitative difference between the materials with different microstructure is not due to artefacts of the AE method but reflects physically sound changes in the correlation of the deformation processes (see, e.g., [154]). The physical consequences of this and similar observations will be discussed in the next chapters. The companion Fig. 4.3(b) represents the effect of HDT for the same tests and for two U 0 -levels: one selected just above the range of the sharp β(U 0 )-dependence (17 mV) and another taken in the far saturation region (67 mV). It can be recognized that the effect of HDT is weak in both cases and almost negligible for U 0 = 17 mV. Some variation of β in a narrow range of small HDT values can be detected for U 0 = 67 mV and is most likely due to the statistics depletion. As a whole, the analysis is quite robust against HDT variation. As far as the HLT is concerned (see Fig. 4.3(c)), no significant dependence was found over the whole range studied, from 0 to 1 ms. The entirety of these results testifies that the power laws observed for MgZr alloys are robust with regard to the criteria used to select the AE events. AlMg alloys The case of AlMg alloys is particularly delicate for performing such analysis because the corresponding AE commonly has a low intensity and is known to be characterized The colours show the events detected for two choices of noise thresholds, U 0 , and the same choice of HDT which is deliberately taken very large. Application of a large threshold U 01 gives four separate events (magenta colour) with relatively short duration. Decreasing U 0 leads to merging of consecutive events (blue colour). The number of detected events and the resulting stack of amplitude values do not change significantly but the apparent durations increase drastically. by strong overlapping and merging of neighboring hits [START_REF] Bougherira | Etude des phénomènes d'auto-organisation des ensembles de dislocations dans un alliage au vieillissement dynamique[END_REF]. On the other hand, these features allow for attacking the investigated problem in the conditions of very noisy signals. Additional interest to this material is caused by the presence of transitions between different types of the PLC effect, which lead to different behavior of the instability at the macroscopic scale. First of all, it is worth noting that the β-range found for the Al5Mg alloy (typically from -2 to -3) lies remarkably below the similar range for MgZr alloys (-1.5 to -2). This difference is much more significant than any variation in β observed upon modification of the events identification criteria. Furthermore, β-values are almost always higher for the as-delivered specimens (circles) than for the annealed ones (squares). Taking into account that annealing leads to an increase in the average grain size, this observation is consistent with the above-mentioned grain-size effect observed for MgZr alloys. Importantly, the change in β upon annealing is usually more significant than the uncertainty of β determination caused by the influence of the identification criteria. More specifically, the first row of Fig. 4.5 represents β(U 0 )-dependences. For the high strain rate (Fig. 4.5(a)) and large HDT and HLT values (blue color), β displays behavior similar to the case of Mg alloys, namely a fast initial drop followed by almost no dependence. The fast initial change, which was attributed above to linking of successive AE events by the agency of a continuous background, disappears when HDT is reduced, so that β becomes nearly constant in the entire U 0 -range. It is natural to suppose that when the imposed strain rate is decreased, the measurable AE events which require motion of powerful enough dislocation ensembles become rarer and better separated (the on-off time ratio becomes larger). 2 As a result, the fast initial change does not occur for lower εa (Fig. 4.5(b) and (c)). The β(U 0 )-dependences obtained for the lowest εa (Fig. 4.5(c)) using a large HDT show a somewhat opposite tendency: a slow increase with U 0 . This trend is also consistent with the discussed framework. Indeed, the increase in U 0 leads to discarding small events and decaying big compound events into two or more (big) components, thus resulting in a larger fraction of big events and an apparently higher β-value. This effect is illustrated in Fig. 4.6 which demonstrates the change in the slope of the power-law for the truncated distribution. It should be noted that alongside with the left truncation, obviously stemming from scrapping small hits, some truncation from the right is also seen. It is a common effect for a limited observation time, as it is caused by the casual lost of the rare large events when they find themselves within a HLT interval. The companion curves in Fig. 4.5(c) display the behavior for a reduced value of HDT and demonstrate that the increasing trend in β(U 0 )-dependences is suppressed when the AE events are better separated due to a more appropriate choice of HDT. In this case, the U 0 increase only leads to (left) illustrated by the two upper curves in Fig. 4.5(e), which correspond to an as-delivered sample deformed at εa = 2 × 10 -4 s -1 . This fall can be explained if one recalls that the small HDT may lead to erroneously taking one of the first local maxima of an event as its peak amplitude. This error would enhance the fraction of small events and reduce β correspondingly. Note that the strongest variations on the β(HDT )-curves obtained for MgZr alloys were observed in the same HDT-range (see Fig. 4.3(b)). Thus, the value of 40 µs determines the lower limit for selecting HDT. The following growth of the exponent, more significant for the higher strain rates, is obviously due to the abovediscussed effect of AE events merging. The optimum HDT value does not seem to be the same for each strain rate. However, as the β(HDT )-dependences are rather weak above HDT=40 µs, this influence can be neglected in most cases. The third row of Fig. 4.5 illustrates the effect of HLT for two choices of HDT. As could be expected, the power-law exponent strongly depends on HLT for small HDT values, thus confirming the above discussion of the effect of HDT. However, for HDT >= 50 µs, the influence of HLT is insignificant, similar to the data for MgZr case. Its effect mostly consists in the statistics impoverishment which should not influence on the scale-invariant distributions, provided that enough data remain in the dataset. The effect of HDT and HLT on AE amplitude statistics was also verified in the usual (without data stream) tests on Al3%Mg samples, in which series of acoustic events were recorded using preset parameters. In these experiments, U 0 was set at 27 dB -the value corresponding to the noise level for the free-running deforming machine -and the effect of changing the HDT and HLT by a factor of ten was checked at different strain rates. Such tests are illustrated in Fig. 4.7 which presents examples of AE amplitude statistics for three AlMg samples. All samples were deformed in the same experimental conditions but the time settings used to detect the AE events were different. The analyzed data are normalized with regard to the average over the respective dataset and all three dependences fall onto one master curve, probably except for some deviations from the power law for the largest events. This and similar results obtained for different εa testify that the power law observed for Al3Mg alloys is quite robust against the variation of Conclusion In summary of this chapter, power-law statistics are found for the amplitudes of AE events accompanying plastic flow of various materials in different experimental conditions. The data obtained confirm the hypothesis that the plastic deformation is inherently intermittent, critical-type process at the scale relevant to AE [82,21,27,26]. The major result of the above investigation is that the criteria used to identify the individual AE events weakly influence on the apparent AE statistics. This conclusion has been verified using Mg and Al alloys which are characterized by distinct deformation mechanisms and display different AE behaviors. A very week influence is detected for Mg alloys, in consistence with the literature data reporting well separated abrupt AE hits accompanying mechanical twinning. More specifically, almost no effect was discerned when HDT and HLT were varied. Somewhat stronger, albeit a weak effect is observed in a wide range of the voltage threshold U 0 . The only case when a relatively strong influence on the power-law is found corresponds to a narrow range of the lowest U 0 values and is most likely due to aggregation of successive AE events. A less favorable situation for the application of the AE method is found under con-ditions of the PLC effect which is known to generate lasting AE events, most likely due to merge of many hits because of successive triggering of many dislocation ensembles during either repetitive formation of deformation bands at low strain rates or propagation of deformation bands at high strain rates. Nevertheless, the influence of the event identification parameters is rather weak in wide ranges of parameters even in this case. In particular, the variations in the β-value caused by the variation of the identification parameters are typically bounded in a range of several tenths, which is much narrower than the difference (about 1 to 1.5) between β-values observed for AlMg and MgZr alloys. This robustness not only justifies the quantitative estimates of the critical indices but also provides an additional proof of the scale-invariant statistics of AE during plastic deformation. Indeed, one of the consequences of the variation of the event identification parameters is the cutoff of a part of data from the entire statistical sample. The robustness of the statistics against such cutoff is consistent with the scale invariance reflected in the power-law dependences. Chapter 5 Multiscale study of AE during smooth and jerky flow Since the pioneer work by Kaiser [73] in 1953, the AE technique is used as a powerful tool to study plastic deformation in various materials. However, before the occurrence of the data streaming technique these studies were based on the extraction of individual acoustic events, as illustrated in the previous chapter, followed by either an analysis of average characteristics over a series of events or, only recently, analysis of their statistics. The data streaming method has opened possibilities for an investigation of AE from Plastic instability in an Al5Mg alloy As described in Chapter 1, very recent investigations [START_REF] Bougherira | Etude des phénomènes d'auto-organisation des ensembles de dislocations dans un alliage au vieillissement dynamique[END_REF][START_REF] Lebyodkin | [END_REF]18,129] questioned the conventional point of view that in the case of the PLC effect, the burst-like AE events are caused by the motion of large dislocation ensembles giving rise to stress serrations, whereas smaller-size dislocation avalanches occur randomly during the macroscopically smooth plastic flow and generate virtually continuous AE. In the present research the data streaming technique is applied to study the nature of AE accompanying the jerky flow. On the one hand, it is aimed at providing insight into the kinematics of formation of deformation bands in a wide strain-rate range corresponding to various types of behavior of the PLC effect. On the other hand, the nature of AE is compared during jerky and smooth plastic flow. AE patterns The material used for this study was the Al5Mg alloy (see Chapter 2). As shown in Fig. 5.1, it presents mechanical behavior typical of aging alloys. The plastic deformation starts with a Lüders plateau, which is generally interpreted as due to propagation of a deformation band through the gauge length of the specimen, a process associated with unpinning of dislocations from their solute atmosphere in a statically aged material. The Lüders plateau is followed by the PLC effect, governed by dynamical strain aging of dislocations. All three conventional types of deformation curves, C, B, and A, were found as εa was varied. The solution treatment led to some softening of the material, resulting in a lower yield point and enhanced ductility, but did not visibly affect the plastic instability. In the case of high and intermediate strain rate values, the plastic instability started practically immediately after the end of the Lüders plateau. At the lowest strain rate the onset of type C instability was observed after a significant critical strain ε cr around 15%. Besides the deep type C serrations, the initial portions (below ε cr ) of deformation curves displayed stress drops described in § 1.5.1 which had much lower amplitude and frequency of occurrence (cf. [27]). The analysis of AE for the low strain rate will be presented with most details because it responds in the best way to the purpose of comparison between AE generated before and during jerky flow. rates. The high resolution of the elongation and force makes it possible to observe the first deviation from linear elastic behavior, which begins very early with regard to the conventional yield strength. It can be recognized that AE is almost absent before this "true elastic limit". Some sporadic AE events recorded in this region may be considered as due to occasional breakthrough of dislocation pile-ups. To be precise, such events may also be due to noise pickup. However, the verification records during idling showed that noise events with such amplitude arrive much rarer. The AE activity manifestly starts increasing after this initial period. Both the AE activity and intensity grow abruptly in the region corresponding to a slight inflection on the deformation curve, which is showed with higher resolution in the figure inset. Such inflection was characteristic of the studied Al5Mg alloy and was not observed for samples with exactly the same size and geometry made from other materials. Thus, it is likely to correspond to enhancing microplastic deformation in this material. This observation confirms that the recorded AE is generated by the (correlated) motion of dislocations. It also attests the AE as a sensitive indicator of the onset of dislocations movements. The data streaming reveals that already at this stage the AE appears at all strain rates in the form of discrete bursts superimposed on a continuous signal, as illustrated in Fig. 5.3. At the lowest strain rate (Fig. 5.3(c)) short isolated bursts with a rise time of several microseconds and duration of several tens of microseconds appear above the continuous noise level. Such bursts are also detected at higher εa . At the same time, a tendency to increasing AE is observed, as could be expected from the intensification of deformation processes required to sustain the higher imposed strain rate. Therewith, the increase is not uniform. In particular, long-duration complex events are formed due to clustering and merging of individual bursts. The merge is relatively weak at εa = 2 × 10 -4 s -1 (Fig. 5.3(b)), so that many of these events have a short rise time, thus allowing to distinguish the main initial shock. At the highest strain rates, the large bursts have much longer wave fronts, durations approaching a millisecond, and often a complex structure, all this indicating that such a burst reflects a developing process of deformation (Fig. 5.3(a)). Besides, the overall increase in the frequency of small pulses adds to the continuous-type signal exceeding the noise level. These patterns already contain many aspects of the signals observed at the later deformation stages. More examples presenting details of the typical waveforms will be given below. is weak and similar to that observed before elasto-plastic transition, whereas very strong acoustic events with durations reaching fractions of a second accompany the serrations. Behavior during the Lüders plateau Consecutive zooming of such events show that they appear as almost stationary at a millisecond scale (Fig. 5. 5(b)). Behavior in the PLC conditions Figure 5.6 illustrates the overall evolution of the AE signal recorded during deformation of an annealed specimen at εa = 2 × 10 -5 s -1 . As can be seen in Fig. 5.6(a), both the activity and intensity of AE diminish after the end of the Lüders phenomenon. However, the qualitative features of the signal do not change: it is composed of a quasi-continuous background with amplitude varying above the noise level, superimposed with larger-amplitude bursts. What attracts attention is that these bursts are not correlated with the deep stress serrations caused by the PLC effect. Indeed, most of them occur during the initial plastic deformation characterized by small stress serrations. Moreover, even the continuous background level decreases during deformation, in spite of the onset of the PLC instability [124]. Important information on this counterintuitive behavior is obtained from consecutive zoom in steps uncovering AE at various scales. burst-like character. The burst-like events occur across the deformation curve, i.e., not only at the instants of stress drops but also during smooth plastic flow between them. As can be seen in Fig. 5.6(b), the small stress drops observed at the initial deformation stage are characterized by acoustic events with relatively high amplitude. However, the instants of stress drops are not exceptional: AE events with similar amplitudes also occur during smooth plastic flow. Moreover, the correlation between AE and stress drops becomes weaker in the course of deformation. Namely, the difference between the amplitude of the acoustic events accompanying type C serrations and that of neighboring events either becomes less significant, as illustrated by the example of Fig. 5.6(c), or even completely disappears. The latter behavior is similar to the data on Al3Mg alloy studied in [START_REF] Bougherira | Etude des phénomènes d'auto-organisation des ensembles de dislocations dans un alliage au vieillissement dynamique[END_REF], where no difference was found between AE events extracted with the help of a standard acoustic equipement during stress drops and during intervals between them. Further increasing the time resolution allows two principal waveforms of the observed AE events to be distinguished. ration varying from hundreds of microseconds to tens or even hundreds of milliseconds, on a background from which some short bursts can be separated, as illustrated in Figs. 5.8(a) and 5.8(b). Like in the case of the pulses accompanying the Lüders plateau (Fig. following each other. In particular, small stress drops are often accompanied by dense sequences of short bursts which can still be isolated from each other. Similarly, their more frequent occurrence often precedes deep stress bursts. The beginning of a very long AE event usually just precedes an abrupt stress drop, thus testifying to a developing deformation process. Another example is given in Fig. 5.8(b), which reveals a short higher-amplitude burst superimposed on a long event. Furthermore, this suggestion is fully consistent with results [28] in which individual long-duration AE events observed during the PLC effect in a AlCu alloy were shown to possess a complex correlated structure revealed by multifractal analysis. A quantitative analysis of such short-time correlations which confirms the complex nature of these signals will be presented in Chapter 6. The samples not subjected to solution treatment demonstrated the same AE patterns for each scale of observation as those illustrated in Figs. (5.6-5.8). Visual inspection of the overall AE intensity level also did not reveal noticeable difference. However, the maximum amplitude of AE bursts appears to reach values up to twice as high as in the case of annealed specimens. For this reason statistical analyses were performed to reveal possible quantitative changes caused by the thermal treatment of the material. 2 The statistical analysis was performed for one parameter set optimized using tests with the so-called Hsu-Nielsen source (pencil lead break): the amplitude threshold was chosen equal to 27 dB, HDT = 300 µs, HLT = 40µs, and PDT = 40 µs. Power-law probability functions were found for both kinds of samples almost over almost the entire range of AE amplitudes, except for the largest events which showed tendency for an increased probability. An example of a comparison of such dependences is given in Fig. 5.9 for the time interval from 2000 s to 4000 s, which corresponds to the region before onset of the PLC effect. It can be seen that the slope of the power-law function is steeper in the case of the annealed sample. In other words, annealing leads to an increasing probability of smaller events. Consecutive analysis for similar time intervals along the entire deformation curve showed that this tendency does not persist during the test: the exponent β for the annealed specimens approaches the value for the as-delivered specimens, which changes little with strain hardening (check in Chapter 6), so that the difference between the dependences for annealed and as-delivered samples gradually decreases and finally vanishes beyond ε cr . The overall behavior of AE on the time scale of the test duration (see Fig. Therewith, although the stress drops are often associated with rather intense AE events, events with similar amplitude are also observed in the intervals between stress drops. Finally, at the highest strain rate bursts with not very high amplitude are more difficult to isolate, so that a virtually continuous AE signal on which rare strong bursts are superimposed is observed at a similar time scale (see Fig. 5.11(b)). Spectral analysis As described in the previous section, visual examination of AE signals allows distinguishing the main waveforms of acoustic signals accompanying plastic deformation in AlMg. Additional quantitative information can be derived from spectral analysis of both individual waveforms and coarse signals. Figure 5.13 illustrates typical waveforms with the respective power spectra. Herewith, no significant evolution in spectra of individual events was revealed during deformation. More delicate analysis based on the multifractal formalism and aimed at revealing possible changes will be presented in the next chapter. The portions of the close-to-noise signals selected at different strains give spectral shapes close to the Fourier spectrum of the noise recorded before the loading start, represented in Fig. 5.13(a). The comparison of the power spectra obtained for various types of events (Fig. 5.13(a)-(d)) reveals a persistent peak around 320 kHz, which can therefore be supposed to reflect the properties of the sound propagation in the investigated media for the given specimen geometry. The width and height of this peak, however, depend on the event type. For example, the spectrum of a burst-like event followed by a dozen of regular oscillations with decaying amplitude is mostly determined by this peak, as shown in Fig. 5.13(b). Merging and overlapping of such "elementary" bursts gives rise to complex waveforms and leads to occurrence of new peaks and resultant broadening of spectra (see Fig. 5.13(c),(d)). illustrate that besides strong discontinuities caused by large AE bursts, practically all stress drops give rise to distinct rises in acoustic energy and simultaneous falls in median frequency. The reduction in f med is usually attributed to an enhanced correlation in dislocation processes and strain localization and therefore, reflects highly cooperative processes (see, e.g., [129,178]). It should be noted that the changes in E and f med begin well before the stress drop. The closer analysis allows relating this effect to the AE increase prior to the drop, as was specified above. The existence of such predecessors of "catastrophes" also agrees with the data of correlation analysis of series of AE events in [18]. Finally, at high strain rates the evolution of E and f med also displays fluctuations on the scale of test duration but, as the almost continuous character of AE complicates distinguishing the individual events, accurate analysis on the scale of one band propagation is difficult (see Fig. Discussion The overall behavior of AE observed in the present work is consistent with the results of earlier studies on the PLC effect interpreted from the viewpoint of generation of AE by multiplication and motion of dislocations (e.g., [124,125,18]). Further discussion will be developed within the same framework of the dislocation mechanism of AE. However, the possible role of cracking of second-phase particles in the overall AE signal, which is usually disregarded in the literature on AE accompanying the PLC effect, should be mentioned. Indeed, post-mortem electron microscopy of fracture surfaces, although displaying ductile dimpled fracture, revealed some sparse broken inclusions, mainly in the non-annealed specimens (Fig. 5.17). Besides the scarcity of such sites found in the microscopy investigation, some other experimental observations suggest a minor role of cracks. In particular, the exponents of the power-law distributions determined in the present work are close to the data for an AlMg alloy in which no cracks were observed [18], and much higher than the typical values reported for AE caused by cracking (e.g., [179]), which are similar to those found for dislocation glide or twinning in pure materials (e.g., [21]). However, the role of inclusions as AE sources during plastic deformation of aging alloys remains an open question and would require a special study. As was underlined in Chapter 1, the conventional vision of AE accompanying the 3), whereas the increasing density of obstacles in the work hardened material would reduce the free path of mobile dislocations. As argued in [124], the waveform of such bursts is mainly determined by the properties of sound propagation in the material. More precisely, their leading front time is supposed to arise from different propagation speeds of different (bulk and surface) modes of stress waves, whereas the rear front of these bursts may reflect the developing deformation process and last for a long duration. On the other hand, the merging of many burst-like events during the development of stress drops leads to a nearly continuous appearance of such composed events when the time scale is further refined (Fig. 5.8). The occurrence of long composed events explains the usually reported observation of drastic count-rate bursts accompanying stress serrations [124,[START_REF] Bougherira | Etude des phénomènes d'auto-organisation des ensembles de dislocations dans un alliage au vieillissement dynamique[END_REF], whereas the AE amplitude shows lesser or no bursts. As suggested in [START_REF] Bougherira | Etude des phénomènes d'auto-organisation des ensembles de dislocations dans un alliage au vieillissement dynamique[END_REF], the merging of AE events reflects synchronization of the deformation processes which are responsible for large stress serrations. Indeed, at low enough strain rate the internal stress caused by strain incompatibility produced by a deformation band efficiently relaxes during slow reloading following the corresponding stress drop. Therefore, when the threshold of instability is reached at some site in the crystal the neighboring sites are also close to the threshold. In particular, this suggestion is confirmed by the observation of an increasing number of AE bursts preceding the large event (cf. [124,18]), which is quantitatively confirmed by the results of spectral analysis. Eventually the local dislocation glide may trigger propagation of plastic activity and the formation of a deformation band. The band development is stopped later on by the fall in stress, which moves the system state far from conditions of instability. When εa is increased short AE bursts are still distinguishable on the relevant time scale both during relatively silent intervals corresponding to deformation band propagation and at the background of more intense AE during band nucleation. It can thus be suggested that the "elementary" plasticity events in the investigated material are essentially collective avalanche-like processes, similar at all strain rates, in agreement with the earlier observations of the same AE waveforms for different types of serrations [124]. At the same time growth of the overall plastic activity required to sustain the faster loading results in enhanced merging and superposition of AE events and gives rise to virtually continuous AE. It can also be suggested that these changes are in fact more profound and reflect changes in the correlations between the elementary processes, which lead to a transition to a different dynamical pattern on the macroscopic scale. Indeed, at high strain rate not only AE but also the (type A) stress serrations are characterized by power-law statistics, whereas peaked distributions are observed for type B and type C serrations [START_REF] Lebyodkin | [END_REF]84,18]. This transition fits in the above-described framework. Namely, when the imposed strain is increased the internal stresses caused by strain heterogeneity do not have enough time to relax so that most dislocation ensembles are constantly close to the threshold of instability. Consequently, an increase in the strain rate leads to a transition from distinct serrations caused by repetitive synchronization of dislocations to a critical-type behavior characterized by stress fluctuations of all sizes. The other important observation follows from a comparison of AE during smooth and jerky flow for a given strain rate, including the regions before the onset of the PLC effect. It suggests that AE has an essentially intermittent nature over the whole deformation curve, although the AE events show different synergy effects displaying series of individual bursts during smooth plastic flow and a high degree of correlation for jerky flow. This conclusion is consistent with the results of a concurrent multifractal analysis of series of stress serrations and series of AE events in [START_REF] Lebyodkin | [END_REF], as well as with the observation [27] of the closeness of AE statistics calculated separately for AE events gathered during stress serrations and between them. It can thus be conjectured that the elementary processes of plastic deformation are the same not only for different types of serrations but also for a macroscopically uniform flow. However, it may not be universal for all materials. Specifically, essentially continuous AE at all scales was observed in [129] for the PLC effect in α-brass under type A conditions. The persistent nature of power-law statistical distributions of AE amplitudes supports the entirety of the above results, leading to the conjecture of an intrinsically intermittent character of the dislocation processes in the investigated material. One feature of the observed distributions is, however, unusual and deserves a special discussion. Namely, the power-law statistics characterizing the dynamics of various real systems usually manifest a cut-off at the large scale of the analyzed variable. Alongside with various specific mechanisms of cut-off, it is caused by general reasons, such as the limitations imposed on the size of avalanches by the system dimensions, the impossibility of waiting long enough to accumulate sufficient statistics for rare large events, and so on. In contrast, the present data often manifest an enhanced probability of large AE events (see Fig. 5.9). This behavior sheds light on the effect of grain boundaries on the collective dislocation dynamics. Indeed, being effective obstacles to dislocation motion, the grain boundaries may play a dual role. On the one hand, the stress concentration caused by dislocation pile-ups may trigger dislocation sources in the neighboring grains and, therefore, promote large dislocation avalanches. Such a situation seems to be at the origin of the observation in [26] of an increase in the power-law exponent in ice polycrystals, i.e., an increase in the probability of larger AE events, in comparison with single crystals of the same material. On the other hand, the β-values reported in [26] and similar works {β ≈ -(1.3 ÷ 1.7)} are much higher than those found in the case of the PLC effect {β ≈ -(2 ÷ 3)}. In [START_REF] Bougherira | Etude des phénomènes d'auto-organisation des ensembles de dislocations dans un alliage au vieillissement dynamique[END_REF], this difference was ascribed to a more important role of other hardening mechanisms caused by forest dislocations and solutes. The present data provide more arguments confirming this hypothesis. First, although a detailed comparison with the data of [START_REF] Bougherira | Etude des phénomènes d'auto-organisation des ensembles de dislocations dans un alliage au vieillissement dynamique[END_REF] is not possible because the Al5Mg alloy investigated in the present chapter has a different Mg content, initial dislocation density, and grain structure, the observation of a similar β-range for two alloys confirms the limitation of avalanche size in such materials. Furthermore, a tendency for an increasing probability of large dislocation avalanches was observed in [START_REF] Bougherira | Etude des phénomènes d'auto-organisation des ensembles de dislocations dans un alliage au vieillissement dynamique[END_REF], but this effect is considerably stronger in the present study. Both these observations may be explained within the framework of the discussed hypothesis, taking into account the small grain size in the investigated material, compared with the typical grain size about 30÷70 µm in the alloy studied in [START_REF] Bougherira | Etude des phénomènes d'auto-organisation des ensembles de dislocations dans un alliage au vieillissement dynamique[END_REF]. Indeed, the decrease in grain size may reinforce the role of grain boundaries in promoting the transfer of plastic activity to neighboring grains, at least for strong dislocation avalanches. The discussed hypothesis also agrees with both the above-described effect of β decreasing on doubling the grain size by thermal treatment and its disappearance after work hardening of the material. Finally, it can be suggested that this effect is not specific of AlMg alloys: a similar effect of grain size was reported in Chapter 4 for MgZr alloys. In summary, among the literature on the intermittency of plastic flow, the phenomenon of jerky flow is usually considered as an exotic case, because of the huge instabilities resulting in macroscopic stress fluctuations. Moreover, these fluctuations show scale-free power-law distributions only at high strain rates, whereas characteristic scales appear for slow deformation. The results of the AE study presented above show that the large stress drops are accompanied with bursts in the duration of acoustic events but, rather counter intuitively, the amplitudes of these events are confined to the same amplitude range as in the absence of macroscopic instabilities. The latter observation indicates some general limitations of the collective dislocation dynamics. In [21], it was argued that the dislocation avalanche size is mainly limited by the sample dimensions in the case of single crystals of ice and pure metals. The present data testify that the grain size and the dislocation microstructure (e.g., forest dislocations) may cause important limiting effects. Within this limited range, the amplitudes of AE events obey power laws, thus confirming a ubiquitous nature of intermittency and unifying the cases of smooth and jerky flow. At the same time, the power-law exponents are larger than the values found in the case of pure materials, and depend on the material microstructure. As far as the bursts in duration are concerned they may be caused by the synchronization of dislocation avalanches, which is realized as a propagation process, similar to relaxation oscillations [86]. Consequently, although the traditional representation of AE using duration-dependent characteristics displays bursts during stress drops, dense successions of acoustic events may lead to virtually continuous appearance of the AE signal itself, provided that the proper time and voltage resolutions are chosen. Twinning and dislocation glide in Mg alloys As described in § 1.6.1 plastic deformation of hcp metals is essentially governed by twinning and dislocation glide. It is known that in some cases twinning manifests itself through macroscopic stress serrations, e.g., at low temperatures or in single crystals. However, deformation curves obtained for polycrystals tested at room temperature are often macroscopically smooth. Up to date, the question on the relative role of twins and dislocations in the total plastic deformation and in the concomitant AE remains a controversy. In several works a high potential of the analysis of the event waveform for separation of AE events was demonstrated [START_REF] Vinogradov | [END_REF]82]. For this reason, application of the data streaming technique seems promising. This paragraph presents the first results of investigation on the AE accompanying deformation of Mg alloys, using the approach described above for the case of the PLC effect. MgZr alloys 5.18(a) ). However, the microstructural analysis of deformed samples reveals a large number of differently oriented twins (Fig. 5.19). That twinning plays a significant role in the plastic flow of this alloys is also confirmed by the character of deformation curves. Indeed, the tensile curve presented in Fig. 5.18(a) demonstrates a substantial strain hardening, which can be attributed to mechanisms related to twins, e.g., by taking into account that twin boundaries are strong obstacles to the motion of dislocations. Further, magnification shown in Fig. 5.18(b) allows discerning irregular stress fluctuations which may be produced by twins. The data streaming records show that the AE is essentially burst-like in this material. Indeed, magnification of a signal portion appearing as continuous on a global scale of Fig. 5.18(a) displays a large number of isolated bursts with strongly varying amplitudes, as illustrated in Fig. 5.18(b). The first acoustic events occur during the elasto-plastic transition. The AE sharply increases, reaches a maximum, and gradually decreases to a constant average level which persists up to the specimen failure. However, strong AE bursts are observed during the whole test. is not the same. The comparison of two figures shows that the amplitudes of the bursts described by the similar waveform vary in a remarkably wide range. The shape of the rear front of these events does not show a smooth transient but is rather complex. Consequently, the corresponding spectra are large and strongly vary. Nevertheless, they possess a generic feature, namely an intense peak in the interval 200-300 kHz, which dominates the spectrum. Another waveform, presented in Fig. 5.20(d) and, occurs much rarer. It possesses a relatively small amplitude and often a large duration, up to a millisecond. However, in this case too, the signal shows abrupt fluctuations giving rise to a wide spectrum, which often consist of several peaks with similar intensity. Moreover, some of these events follow the short bursts. Taking into account the presence of abrupt fluctuations in the long event, it can be suggested that the deformation processes with different kinetics, giving rise to two different kinds of AE responses, mutually trigger each other. In spite of the fact that the waveforms and their spectra remain qualitatively similar in the course of deformation, inspection of the time evolution of the average energy and median frequency reveals clear quantitative trends, in addition to fluctuation caused by strong AE events. An example of such dependencies is presented in Fig. 5.21. It can be seen that the AE energy rises rapidly from the onset of plastic deformation, passes a maximum, and decreases to an approximately constant level, while the median frequency shifts gradually to higher frequencies. Similar trends were also observed in the case of the AlMg alloy (see § 5.1.2), although in a less pronounced manner. AZ31 alloy As described in Chapter 2, these samples were prepared using different procedures resulting in two different microstructures. Figure 5.22 presents examples of deformation curves and the accompanying AE signals for both kinds of samples deformed at εa = 1 × 10 -3 s -1 . It can be seen that the specimen with a heterogeneous grain structure (sample s1) is harder than that with a uniform grain structure (sample s2). In both cases the acoustic response shows similar patterns in the elastic and elastoplastic regions, characterized by very strong bursts. However, the behaviors becomes completely different after the yield. For sample s1 the AE almost vanishes and represents rare highamplitude bursts on a continuous background (Fig. 5.22(a)). In contrast, the response of the sample s2 gradually decreases with strain and keeps a high burst-like activity up to fracture (Fig. 5.22(b)). The postmortem microstructure analysis revealed substantial differences in the nature of plastic deformation of the two kinds of alloys, as shown in Fig. 5.23. No twins and no significant changes in the grain size have been found in sample s1 (compare with the initial grain structure in Fig. 2.4). In the second case, many large twins are observed, so that their presence makes impossible the analysis of the grain structure in sample s2. almost reducing it to one narrow peak. The spectra of long events are more complex. Besides peaks located in approximately the same frequency range (350-400 kHz), they also present rather high peaks at different frequencies. However, in comparison with the spectra obtained for long events in MgZr samples (Fig. 5.18(d)), these components are shifted to the low-frequency part of the spectrum ( Fig. 5.24(f)), thus indicating the absence of abrupt fluctuations in the corresponding waveforms. The PSD analysis of the entire AE signals was complicated in these tests because of the low resolution of the equipment. Nevertheless, its results confirm the same qualitative trends as discussed above. Discussion The observed overall evolution of AE is similar to the literature data reported for magnesium alloys of different chemical composition (e.g., [152,174]). Mathis et al. [152] distinguished three stages of AE evolution corresponding to distinct mechanisms of deformation: • At the beginning of deformation, basal slip (easy glide system in Mg) and primary {10 12} < 10 11 >twins in the grains unfavorably oriented for slip are found. Both this processes are considered to be effective sources of sound waves, contributing to the initial strong AE. • The maximum of AE corresponds to twining activation in other slip systems. µs for long events) were found in Tymiak et al. [156] for sapphire single crystals. In this work the long events were only observed in crystals displaying twinning. Therefore, these data provided an explicit proof that such events are generated by twins. The authors suggested that the sharp transient is due to twin nucleation and the slowly decaying part is controlled by its growth, whereas the events displaying only sharp transients may be due to dislocation avalanches. Based on the similarity of the observed waveforms, a similar hypothesis was adopted in [82] to interpret the results obtained for single crystals of hexagonal metals. An opposite conclusion was drawn in Vinogradov et al. [START_REF] Vinogradov | [END_REF] for Cu alloys containing various percentage of Ge. It was shown that twinning occurs only in the material with a high Ge content. These aloys present sharp AE bursts with duration about 100 µs, while the samples with a low Ge content deform by dislocation glide and are characterized by essentially continuous emission, in which individual bursts can hardly be isolated. However, the authors admit that some of the short bursts can be due to dislocation avalanches. The existence of diverse hypotheses on the waveforms generated by dislocation avalanches is not surprising, in view of the above results of investigation on Al5Mg alloy, which prove that both sharp transients and almost continuous signals can occur during dislocation glide. It can be suggested that the kinematics of deformation processes may considerably vary even in the same test and under conditions of one mechanism of plasticity, because of the variation of local conditions controlling spatial correlations. Obviously, the characteristic waveforms may also strongly depend on the studied material. It is also noteworthy that analysis of the correlation between different kinds of AE events, reported by Richeton et al. [82], indicated that twinning and dislocation glide may mutually trigger each other. The results obtained in the present study are consistent with the hypothesis that twins are responsible for essentially burst-like behavior displaying short transients and amplitudes reaching very high values. This conclusion directly follows from the comparison of two types of AZ31 samples, one deformed by dislocation glide (Fig. 5.22(a)) and the other showing intense twinning (Fig. 5.22(b)). It is also confirmed by the observation of burst-like behavior in MgZr alloys characterized by twinning. The occurrence of strong bursts at the beginning of deformation in Fig. 5.22(a) proves that dislocation avalanches can also produce very intense events, at least at small strains, when the dislocation can move over long distances. However, these events usually last much longer than those associated with twinning. The short duration and the distinct isolation of AE events from each other in the case of twinning bears witness that the deformation processes involving twinning provide more efficient relaxation of the constrained microstructure, and lead to an abrupt interruption of the localized fast deformation. Another feature that could be marked out is that Fourier spectra of the waveforms observed for Mg alloys are usually remarkably wider than those found for AlMg (cf. Figs. 5.13, 5.20, and 5.24). This difference reflects a more complex shape of the waveforms for Mg alloys and may be due to the conditions for plastic flow being more constraining in hcp than in fcc metals. This view is consistent with the conjecture of a complex nature of the processes determining the shape of an individual burst in a Mg alloy, which includes mutually triggering twinning and dislocation glide. Such coupling may be due to various mechanisms. For example, the elastic wave generated because of formation of a twin may trigger other twins, even in remote regions, whereas the concomitant reorientation of the crystal lattice facilitates the motion of dislocations within the twin region. Finally, it is noteworthy that the PSD analysis revealed a persistent feature of plastic deformation consisting in gradual increasing of the median frequency f med during deformation and observed both for Mg and Al alloys. Such changes are usually attributed to a decrease in the dislocation mean free path because of the increasing density of obstacles to their motion. Indeed, the respective reduction of the mean free flight time of dislocations would result in an increase in f med . The same reasoning can also be applied to twinning. Another aspect of this behavior which was discussed in § 5.1.2 should also be noted. Namely, the higher f med means a weaker correlation of the processes giving rise to AE. Therefore, the observed increase in the average median frequency suggests progressive stochastization of deformation processes on the global time scale. This question, concerning the changes in the correlations between deformation processes in the course of work hardening, will be addressed in the next chapter using statistical and multifractal analyses of AE on different time scales. Chapter 6 Statistical and multifractal analysis This chapter presents the results of the statistical and multifractal analysis of AE accompanying plastic deformation of the materials used in the study. The investigation is performed at various strain rates, various stages of deformation, and on various time scales. Until recently, only statistical analysis of AE was used to reveal information about the collective dislocation dynamics [20,82,18,[START_REF] Bougherira | Etude des phénomènes d'auto-organisation des ensembles de dislocations dans un alliage au vieillissement dynamique[END_REF]. In the case of the PLC effect [START_REF] Bougherira | Etude des phénomènes d'auto-organisation des ensembles de dislocations dans un alliage au vieillissement dynamique[END_REF]18] our attention was attracted by the observation of considerable changes in the indices of power-law statistical distributions during deformation of an Al3%Mg alloy. The increase in the slope of the dependencies indicates a tendency to a transition from scale invariance to a behavior characterized by an intrinsic scale. To verify this hypothesis we performed a similar analysis for a Al5%Mg alloy as well as for MgZr alloys. The investigation is corroborated by the multifractal analysis. It should be noted in this relation that the statistical distributions only characterize the probability of plastic activity with a given intensity during the test duration, but they do not provide information on the relative arrangement of plastic events. The multifractal analysis has the advantage of uncovering the presence of correlations and characterizing their scaling properties. Statistical analysis AlMg alloy The analysis was performed for Al5%Mg samples in both as-delivered and annealed strain, for four strain-rate values. As said above, at the highest strain rate the AE initially displays virtually continuous signals with superimposed large discrete bursts, so that only a few events are extracted by the applied numerical procedure. However, the AE signal acquires a more discrete character after some work hardening, providing enough data for statistical analysis. The results shown in Fig. 6.2(a) were obtained on statistical samples containing about 1000 events, which allowed for reliable detection of the power law. At least ten times larger statistical samples were typically obtained for lower strain rates, except for the quasi-elastic region before the Lüders plateau and the late intervals preceding the specimen rupture. The data for lower strain rates, represented in Fig. 6.2(b-d), testify that for a given εa the exponent β evolves during deformation. Both the overall behavior and the β-range are similar to the data for the Al3%Mg alloy [START_REF] Bougherira | Etude des phénomènes d'auto-organisation des ensembles de dislocations dans un alliage au vieillissement dynamique[END_REF]. Depending on εa the value of β varies between -1.8 and -2.4 at the onset of plastic deformation. Then it decreases and varies from -2 to less than -3. Finally, an inverse trend (an increase in β) occurs after some deformation at εa = 2 × 10 -5 s -1 . The comparison of the data obtained in similar strain intervals for different strain rates reveals a tendency to a decrease in β with decreasing εa , in the strain-rate range corresponding to type A and type B behavior (Fig. 6.2(a-c)). At this stage of investigation it is difficult to say whether such a trend is meaningful. It qualitatively agrees with the influence of the rate of change of the magnetizing field on the power-law statistics of the Barkhausen effect, as well as with theoretical predictions of the role of overlapping [93] (see § 1.4.3). Importantly, this trend does not show up when εa is further reduced (cf. Figs. 6.2(c) and (d)). It can thus be conjectured that correct values of β, unaffected by the overlapping, are obtained for εa ≤ 2 × 10 -4 s -1 . This observation explains the remarkable robustness of β with regard to the events individualization parameters, which was reported in [START_REF] Bougherira | Etude des phénomènes d'auto-organisation des ensembles de dislocations dans un alliage au vieillissement dynamique[END_REF] for a similar εa -range. The data of Fig. 6.2 make more exact the observation noticed in Chap. 5 in regard to the difference between β-values for as-delivered and solution treated samples. Namely, the slope of the power-law dependence is generally steeper for annealed samples, i.e., annealing leads to increasing the probability of smaller events, although in the case of type C instability this difference decreases during the test or even vanishes. The difference between β-values for as-delivered and annealed samples is conform with the conjecture that the grain boundaries may promote powerful avalanches due to triggering new dislocation sources in the neighboring grains. Indeed, the thermal treatment leads to growth of grain size and reduction of stress concentration on grain boundaries, which would reduce the triggering effect. The observation of a decrease in this difference after some deformation in the tests at εa = 2 × 10 -5 s -1 does not seem to contradict this conjecture, because the low strain rate provides more time for relaxation of local overstresses on the grain boundaries. Consequently, the accumulation of the dislocation density would put in the forefront the forest dislocations as obstacles to slip, and reduce the role of grain boundaries. MgZr alloys As reported in Chapter 4, power-law statistical distributions of AE were observed for all Mg alloys studied in the dissertation. Multifractal analysis General approach Two kinds of time series were used to reveal the multifractal properties of AE signals. One approach was based on the analysis of series of peak amplitudes of the events extracted from a signal (cf. [START_REF] Lebyodkin | [END_REF][START_REF] Bougherira | Etude des phénomènes d'auto-organisation des ensembles de dislocations dans un alliage au vieillissement dynamique[END_REF]). This method allows analyzing large time intervals. It also provides a way to filter the events that do not belong to the detected multifractal set. For this purpose, the calculation of the partition functions Z q (δt) (see § 3.3) is repeated several times, for different choices of the threshold which serves for cutting off events with either the lowest or the highest amplitudes. In the second approach, the time series is represented by the AE signal itself. Such time series are usually limited by short time intervals (< 5 s) to provide a reasonable computation time, but this limit can be considerably widened through removal of a noise component below a threshold. This approach is particularly useful for the analysis of individual waveforms, and also in the case of high strain-rate data for which the extraction of individual events is impeded because of their merging. examples may be found in [START_REF] Lebyodkin | Multifractal analysis of unstable plastic flow[END_REF]. The removal of noise allows detecting the multifractality in an interval of large enough δt, from 30 ms to about 1 s (Fig. 6.5(b)). As this linear behavior on log(Z q )/(q -1) vs. log δt dependencies. Consequently, truncation was mostly used in the present dissertation, due to its simplicity. The research with the aid of wavelet analysis will be continued in the future. AlMg alloy Analysis in large time intervals As reported in Chapter 1, application of the MF analysis to series of stress serrations revealed multifractality of deformation curves at all strain rates [84,106,103,[START_REF] Lebyodkin | [END_REF]. In [START_REF] Lebyodkin | [END_REF], multifractal behavior was also found for the corresponding series of amplitudes of AE events, except for the highest strain rate, εa = 6 × 10 -3 s -1 . In this last case the failure of the MF analysis was explained by a strong overlapping of AE events, which made it difficult to resolve individual bursts. For this reason we mostly used direct processing of AE signals when it was possible, in order to avoid errors associated with events individualization. linear log(Z q )/(q-1) vs. log δt dependencies are found over intervals covering noticeably more than an order of magnitude of δt. The dependencies deviate from straight lines when δt is decreased to a scale corresponding to separate events. The upper scaling limit is related to the finite length of the analyzed time series. Using families of such dependencies, spectra of generalized dimensions, D(q), and singularity spectra, f (α), were calculated for different εa . Smooth MF spectra were found for all applied strain rates. It should be noted, however, that at low and intermediate strain rates, there exist time windows during which multifractality was not detected. of which, selected beyond ε cr , is shown in Fig. 6.8(b)). It can be seen that the spectra gradually expand and deteriorate in the course of deformation. Furthermore, the multifractality is usually not detected for the latest portions of deformation curves. That the branches corresponding to negative q-values (right-hand parts of singularity spectra) are particularly sensitive to any deviation from fractal behavior and are difficult to obtain in a reliable way, is a general problem for the treatment of real signals because the negative qs correspond to the subsets with the poorest statistics (see, e.g., [122,106]). Nevertheless, the increase in the spectra width in the range of q > 0 indicates an increasing heterogeneity of the signal. In order to verify the conclusions on the evolution of MF spectra, the calculations were also made for series of amplitudes of AE events. Figure 6.9 presents the resulting singularity spectra for time intervals including those processed in Fig. 6.7 (larger intervals are taken to provide statistically significant numbers of extracted AE events). As could be expected, the spectra do not coincide with those in Fig. 6.7. However, they fall into a similar range of singularity strength α. Importantly, the qualitative effect of the work hardening is the same for the two kinds of time series. Despite the increase in AE activity with increasing εa , the overlapping of AE events at intermediate strain rates seems insignificant during the macroscopically smooth parts of the deformation curves. Taking into account that the test duration diminishes accord- ingly to εa , both approaches to MF analysis of AE can be applied comfortably. Moreover, in these conditions the numerical procedure allows for processing time intervals covering several sequences of stress serrations, associated with the relay-race propagation of type B deformation bands. Examples of MF spectra for εa = 2 × 10 -4 s -1 are given in Fig. 6.10. For one segment, T=[520s; 700s], results obtained using both kinds of time series are presented. It can be seen that the entire AE signal yields in this case an apparently distorted spectrum (open circles), thus questioning the suggestion, based on the visual examination of the signal, that the events are weakly overlapped. In contrast, the series of amplitudes in the same time interval produces a smooth spectrum (solid circles). In any case, the comparison of MF spectra obtained for different time segments using the same approach proves that their evolution with deformation is similar to that found for type C behavior. In particular, the AE is depressed on the latest stages of deformation, so that many stress serrations do not show an acoustic response, and the multifractality completely disappears. For illustration purposes, the dependencies marked by open squares show the results of formal calculation of MF spectra in this last case. It can be recognized that no smooth curve is found. Besides the treatment of large time intervals, the relatively high AE activity observed at intermediate strain rates allows for scaling examination on a scale of one "period" of that the spectrum has a similar shape and width as its counterpart in Fig. 6.10, determined for δt ≈[10 s; 100 s]. Although there is a gap between these two δt-ranges, it is likely to be artifact of the truncation procedure, which imposes the small-range limit of scaling. Indeed, it was verified by repeating the analysis using intervals with intermediate lengths. Thus, the observed similarity testifies to the same mechanism of correlations operating in a rather wide time range, from milliseconds to tens of seconds, although further verification of this conjecture is needed. Unfortunately, the number of AE events occurring during reloadings decreases with deformation, which makes difficult a systematic analysis. The further increase in εa leads to strong overlapping of acoustic events. Nevertheless, multifractal scaling is found for type A instability, too. Although the evolution of spectra with strain makes difficult evaluation of the effect of εa , the comparison of MF spectra obtained in similar strain intervals for different εa shows a trend to more heterogeneous behavior (wider spectra) at higher strain rates. This observation agrees with the results of analysis of stress serrations in the literature [84,8,122]. It can be illustrated using the example of a test at 6 × 10 -3 s -1 , which displays type A serrations at the beginning of the test and a progressive transition to type B serrations. Figure 6.12 shows the shapes of the analyzed AE signals and the results of the MF analysis. The AE initially appears to be essentially continuous on the scale of the figure (signal 1). The transition to type B instability is also reflected in AE, as a transition to more discrete behavior (signals 2 and 3). The comparison of MF spectra for the signals 1 and 2 shows a higher spectrum width for type A behavior, although it corresponds to an earlier deformation stage. Finally, the signal 3 selected on a late deformation stage does not possess a MF spectrum, in consistence with the above results for type B instability. Interestingly, the deterioration of correlations illustrated by the case of signal 3 concerns the coarse time scale (T = 3 s) of Fig. 6.12, but the correlations persist on a finer time scale. Indeed, figure 6.13 presents results of analysis in a much shorter interval (T = 0.1 s) corresponding to reloading between two stress serrations. The approach is similar to that used in Fig. 6.11, with the only difference that the latter corresponds to a much lower driving velocity, for which the reloading time is about 2 s. The characteristic waveforms observed at the lowest strain rate are presented in Fig. 6.14 (see also Chap. 5). The small stress fluctuations below ε cr are accompanied by merging sequences of closely following events (Fig. 6.14(a)). The deep PLC serrations generate long waveforms with a millisecond duration (Fig. 6.14(b)), which apparently present a finer structure but are usually extracted as single events by the standard AE methods. The macroscopically smooth regions of the deformation curve between two successive stress drops usually display separate short events. However, sequences of events are also observed and present interest for the analysis. The corresponding pattern is shown in Fig. 6.14(c). Figure 6.15 shows the MF spectra which testify to the presence of multifractal scaling in all signals presented in Fig. 6.14. Scaling behavior was found over δt intervals between a few microseconds and 0.1 ms for the first two waveforms and between 60µs and 1 ms for the last signal. It seems important that the AE activity is weak at the low strain rate and the above-described events are followed by periods without activity, when only noise is present. Consequently, increasing the analyzed time interval leads to disappearance of scaling. Scaling occurs again for long enough time intervals containing several stress serrations, as described in the previous paragraph. Thus, at this strain rate the AE is not globally multifractal: the detected correlations correspond to either a separate event (or cluster of events) or to rather long series of events. . Individual bursts with a short rising time, like the one displayed in Fig. 6.16(a), are usually observed during reloading parts of deformation curves. Such type of signals was studied in detail in [28]. Its structure is obviously not multifractal because it involves two distinct scales corresponding to the burst itself and the background signal. It was shown that some deviations from the trivial scaling occur, seemingly because of the presence of some fine structure during the burst decay. However, no smooth MF spectra were found. 6.17(a)). They exhibit approximately straight segments at small time scales and gradually converge to unity slope for δt above several hundreds of microseconds. The corresponding singularity spectrum presented in Fig. 6.18 (open circles) testifies to multifractality of the considered signal, but reveals strong imperfections even for q > 0. By selecting several values of threshold U tr and truncating the signal below the threshold, it was possible to uncover approximate scaling at larger scales, as illustrated in Fig. 6.17(b), at the same time causing degradation of the small-scale scaling. Linear segments were found in a similar δt-range for all trial values U tr =0.5 mV, 0.75 mV, and 1 mv, but the corresponding spectra vary considerably, depending on U tr (Fig. 6.18). Thus, a good approximation of the true MF spectrum was not found. Nevertheless, the obtained results testify with certainty to the presence of correlations in the treated signal. The calculation separately for two time domains yields good spectra for the small scale range (Fig. 6.19, open circles), proving the existence of short-range correlations in the signal structure. Multifractal behavior for positive q-values is found for the next time interval as well (solid circles), but the negative q branches of the spectra are strongly distorted. The latter behavior suggests two hypotheses: (1) taking into account that the slopes only slightly differ for q > 0, the second linear segment may be due to a slow deviation from the scaling law established for the first segment, thus reflecting a decay of correlations on larger time scales; [START_REF] Haken | Synergetik[END_REF] there is a crossover between two scaling domains. The latter situation might reflect a change in the physical mechanism of correlation. It is also noteworthy that similar to the low strain-rate case and in spite of the higher overall AE activity, the analysis in time intervals intermediate between the scale of the individual waveforms and that of the series of events often did not reveal scaling behavior. Finally, under conditions of type A behavior at εa 6 × 10 -3 s -1 , the signals are essentially continuous, similar to that presented in Fig. 6.16(c), and yield similar MF spectra. Some of the treated signals also showed a crossover between two scaling laws but this was rather exception than a rule. In contrast to the cases of lower strain rates, multifractality was found for all time scales. This universality may indicate formation of a globally correlated behavior, in consistence with the conjecture of self-organization to a critical state. However, this hypothesis needs a very accurate verification because, as follows from above, whereas the detection of multifractality is relatively simple, reliable quantitative determination of spectra and their comparison could only be made in rare cases. Further investigation, perhaps, using different methods of analysis is needed. Mg alloys The behavior of the AE accompanying deformation of MgZr alloys differs from that for AlMg but it is also characterized by multifractal correlations. The salient features observed are similar for all samples. After truncation of the latter background (U tr equal to 3 mV was used), calculations yield smooth spectra for the entire q-range (Fig. 6.20, open symbols), indicating that the main signal is characterized by a unique physical mechanism of correlation. At the same time, the relatively large span of α-values qualifies the signal as a complex heterogeneous structure. This initial AE activity is followed by a period of very intense non-stationary emission ascribed to the occurrence of primary twins. In this case the analysis was only performed during short time intervals, as described below, because of the non-stationary character of the AE signal. This stage is succeeded by a stage of dominant secondary twinning, giving rise to a stationary pattern of incessant burst-like AE, as shown on bottom of Fig. 6.20. This dense pattern gives a narrower MF spectrum in the range of positive qs, which testifies to a more uniform AE activity (Fig. 6.20, solid symbols). However, the spectrum is obtained after truncation of a significant lowamplitude part of the signal, using a threshold of 50 mV, which leads to deterioration of the negative q-branch. Finally, similar to experiments on AlMg, scaling fails at late stages of deformation. As discussed in concern with Fig. 6.16(a), the MF method cannot be applied to into account that the spectrum in Fig. 6.20 was obtained after truncation of a large background, it is clear that the low-amplitude and high-amplitude parts of this signal do not belong to the same set of events. However, it does not mean that the lowamplitude component is uncorrelated. Indeed, the treatment of short segments (about 1 ms) between intense bursts proved that the background signal also gives multifractal spectra with relatively small width (α min ≈ 0.8). Moreover, a similar check in the range of basal glide showed that the background, having in this case a much lower level, cannot be attributed to random noise either. It reveals scaling of partition functions, resulting in a narrow MF spectrum (α min ≈ 0.9). It can be concluded on the whole that the deformation processes are essentially correlated, but these correlations cannot be described by unique scaling dependencies and may be due to various mechanisms. governed by two competing processes: that of the generation of microstructural heterogeneities because of the intermittent localized deformation and that of plastic relaxation of the resulting incompatibility stresses (which is also realized through motion of dislocations). It is likely that this picture should also apply to mesoscopic-scale processes during macroscopically smooth plastic flow of pure materials. It would be of interest to realize a similar investigation in this case, too, whereas only the statistical method was applied to such data so far. The case studied in the dissertation is peculiar in the sense that the deformation of DSA alloys is unstable. One consequence of the macroscopic instability is that it may give rise to characteristic scales, associated with the ideal case of cyclic relaxation oscillations, according to the N-shaped SRS function (see § 1.5.2), or, in other words, with the tendency to synchronization of dislocations. This is a possible reason why scaling is not found all over the test duration for type C and type B behaviors but there exist time windows of non-fractal behavior. Second, the plastic instability leads to emergence of correlations in a range of short time scales (from microseconds to milliseconds), associated with the development of catastrophic processes of plastic instability. In this case it is natural to suggest that the correlations may be governed not only by the (fast) changes in the internal stress field but also by a direct impact of elastic waves, involving dislocations to a chain process. Actually, such a pattern corresponds well to the general statement that multifractal time sequences may be generated by cascade processes [162]. It is more surprising that the language of cascade processes also applies to long-range time scales. It can also be conjectured that there is no principal difference between these two cases, although at low and intermediate strain rates there is a "gap" in the observation of scaling between the short and long ranges. Indeed, this gap disappears when the strain rate is increased. Another interesting observation for small scale behavior concerns the crossover between two slopes illustrated in Fig. 6.19. It implies that an additional mechanism of correlations may act on small scales, for example, the mechanism of double cross-slip of dislocations, as was recently justified theoretically [137]. It is noteworthy that investigation of heterogeneous distributions of dislocation densities in ice single crystals [180] led to a similar conjecture on a possible effect of this mechanism on short-range spatial correlations in dislocation arrangements. This analogy raises a question of a relationship between self-organized dynamics dislocations and the resulting dislocation microstructure. Effect of work hardening Two competing factors seem to determine the evolution of the statistics and multifractal scaling of AE during deformation. On the one hand, work hardening creates obstacles to the motion of mobile dislocations and must cause deterioration of correlations between deformation processes. On the other hand, it leads to homogenization of the internal stress field and, therefore, will promote synchronization of dislocation avalanches. The increase in the probability of low-amplitude AE events which is reflected in the decrease in the exponent β of power-law statistical distributions (Fig. 6.2), suggests stochastization of the dislocation dynamics. The stochastization might also be responsible for the failure of multifractal scaling at late stages of deformation. However, the conjecture of stochastization cannot allow for the totality of observations. First, it would lead to collapse of multifractal behavior into the trivial scaling with unity fractal dimension, whereas experiment shows an increase in the width of MF spectra with deformation. A possible explanation of this observation evokes the tendency to synchronization of dislocations under conditions of the PLC effect. Indeed, it may lead to emergence of distinct scales which would enhance the heterogeneity. Moreover, it may eventually cause the failure of scaling and, therefore, provides an alternative interpretation of the final non-fractal behavior. Second, attention should be drawn to the inverse trend observed for β at large strains in the tests at εa = 2 × 10 -5 s -1 . A similar change was reported in [START_REF] Bougherira | Etude des phénomènes d'auto-organisation des ensembles de dislocations dans un alliage au vieillissement dynamique[END_REF] for a region close to fracture and attributed to localization of the PLC bands because of the developing necking. Indeed, the localization enhances simultaneity of slip, which results in the superposition of AE events and a higher probability of large amplitude bursts. In the present case this change was observed to take place before the beginning of necking. To explain this effect, it should be taken into account that as far as the slow loading provides conditions for efficient plastic relaxation of internal stresses, the resulting homogenization would promote the effect of synchronization. Finally, it is noteworthy that synchronization may also contribute to the tendency to an increased probability of the largest events, observed for type B and type C behavior (Fig. 6.1). Magnesium based alloys The results obtained for MgZr alloys are qualitatively similar to those for AlMg. First, the slope of the power-law statistical distributions of AE amplitudes increases (β decreases) with deformation. Second, the multifractal scaling fails at large strains. At the same time, the width of MF spectra does not increase during deformation, in contrast to the above-described behavior of the PLC effect. Actually, the data obtained for MgZr may be interpreted using the conjecture of stochastization of deformation processes. This also concerns the behavior on short time scales, namely, the observation of a gradually increasing contribution of low-amplitude events which are characterized by narrow MF spectra. In the case of AlMg the stochastization was explained by a decaying correlation between dislocations, because of the increasing density of obstacles to slip. It is not obvious whether the same logic can be applied to twinning. However, as far as the twinning may be considered from the viewpoint of the motion of twinning dislocations, it can be suggested that work hardening results in reduction of the width of the twin nuclei, which are supposed to be the main AE sources during twin formation, while the further twin growth may not generate acoustic waves. This conjecture is indirectly confirmed by the observation of thinner twins in secondary twinning systems which start operating after some deformation [154]. However, the post-mortem microscopy can only provide indirect proofs because it reveals the ultimate size of twins. In situ investigations will perhaps be able to verify the discussed hypothesis. Besides, the growing density of twins may cause an increase in the contribution of the dislocation glide which would lead to efficient relaxation of internal stresses and to a decrease in the correlation of the deformation processes. General conclusions and perspectives for future research Since the 1980th the application of the concepts of nonlinear dynamical systems to plasticity problems revealed a variety of complex behaviors which cannot be predicted by the traditional theory of dislocations. In particular, using the AE technique evidence was found for an intermittent collective motion of dislocations on a mesoscopic scale, which obeys power-law statistics, characteristic of avalanche-like dynamics. Furthermore, the values of the power-law exponents were observed not to vary considerably for single crystals of various pure materials and to be similar to critical exponents describing avalanche processes in other fields of physics, e.g., the motion of domain walls in magnetic materials or the martensitic transformations. The observed universality of dynamics allows generalizing some of the findings obtained in the study of the collective motion of dislocations to other dynamical systems. At the same time, recent investigations on polycrystals and alloyed materials showed limitations of the concept of universality. In the present dissertation, the effect of the experimental conditions and microstructure on the statistical and fractal properties of AE was studied using alloys characterized by different mechanisms of deformation. Alongside with the statistical investigation, the evolution of the acoustic signal during deformation was examined and the possible mechanisms of the observed changes were discussed. The main results of this research are shortly summarized below. Power-law amplitude distributions of AE accompanying plastic flow in AlMg and MgZr alloys were obtained in different experimental conditions. An important result of statistical analysis is that the criteria used to extract individual AE events have little effect on the apparent statistics. A very weak effect was found for MgZr, in consistence with the literature data showing that the materials deforming by twinning generate well separated abrupt AE hits. A less favorable situation for the statistical analysis of AE appears under conditions of the PLC effect. The entirety of the results obtained in the dissertation show that this difficulty is caused by the plastic instability being related to the tendency to synchronization of the dynamics of dislocations, which leads to localization of deformation on the macroscopic scale and merging of acoustic events on a mesoscopic scale. Nevertheless, even in this case the effect of the criteria used to identify AE events is not crucial and does not prevent from detecting the changes in the power-law exponents when the experimental conditions are modified. It should be noted that this result is of general importance for the study of avalanche processes in various fields of science. Statistical analysis of the amplitudes of AE events showed that the slope of the power-law distributions depends on the microstructure, particularly, the grain size and the microstructural changes induced by deformation and related to accumulation of dislocations and twins. More specifically, these results provided a direct proof of an important role of local stress concentrations on grain boundaries. Indeed, reduction of the grain size in both kinds of materials led to an increase in the probability of higheramplitude AE events. This effect can be explained by an efficient transfer of plastic activity to neighboring grains, promoting formation of powerful avalanches. On the contrary, the forest hardening results in an increase in the probability of lower-amplitude AE events, which indicates gradual weakening of correlations between dislocations, i.e., stochastization of plastic deformation processes. The conjecture of stochastization is also confirmed by the observation of a progressive increase in the median frequency of the acoustic signal during deformation, usually associated with a decrease in the mean free path of the defects generating the AE, which was also observed for both kinds of When viewed on a traditionally used time scale, which does not resolve the structure of individual AE bursts, their amplitudes are indistinguishable during the periods of smooth plastic flow and at the instants of stress drops. However, the synchronization of the dynamics of dislocations at the moments of stress drops leads to generation of AE events with millisecond durations, which exceed by more than an order of value the durations of the hits observed during smooth flow. The application of the multifractal analysis to continuous AE records permitted us to quantitatively characterize the correlations between deformation processes in various time scale ranges. Under conditions of the PLC effect, the time correlations observed in a very wide range, from about a hundred of milliseconds to hundreds of seconds, are most probably governed by internal stresses. Importantly, evidence was found that synchronization of dislocations leads to emergence of a distinct time scale associated with abrupt stress drops (types B and C of behavior) and corresponding to a microsecond range. Observation of a crossover in the multifractal scaling suggests that in this case, besides the changes in the internal stress field (and perhaps a direct impact of elastic waves), another mechanism of correlations may be present, e.g., the transfer of plastic activity due to double cross-slip of dislocations. Finally, in the case of MgZr alloys, the MF analysis revealed the existence of two scale ranges of amplitudes of AE events, which correspond to different correlations as revealed by MF spectra. High-amplitude bursts are generally attributed to mechanical twinning and are usually considered in AE studies of plastic deformation of such materials. However, we found that the low-amplitude events are by no means random noise, because they also show correlations leading to emergence of MF spectra. The nature of the two different sets of AE events is not clear yet. However, as the dislocation pile-ups are believed to yield lower-amplitude AE bursts than the twins, we hope that such a quantitative analysis would help distinguishing the AE related to twins and dislocations. Perspectives for further research The present dissertation is one of the first works aiming at multiscale quantitative The way to do it can be illustrated by adding complexity to the recurrent procedure of generation of the Cantor set. Let us begin with assigning a uniform weight distribution to the segments of the Cantor set. The modified procedure starts from a unit segment with a weight µ = 1 and includes an additional rule: the weight of each segment obtained at the current generation step is shared equally between the two segments created from it at the next step. After m iterations, the set consists of 2 m segments with the same length l i = (1/3) m and weight µ i = (1/2) m . Thus, the following power-law relationship holds: µ i ∼ l α i . (A.2) The Lipschitz-Hölder index α is often called singularity strength because when α < 1, the local density diverges in the limit l → 0: µ i /l i ∼ l α-1 i . The weight µ is an example of a probability measure which makes it possible to describe the distribution of a physical quantity on a fractal support. In the considered case, two indices, D f and α, provide such a description (for Cantor set, α = ln2/ln3 = D f ). This simple case can be further generalized to allow for description of real heterogeneous self-similar objects. As demonstrated, e.g., in Ref. [162], the modification of the recurrent rules so that the segments are divided into unequal parts and the weights are assigned with unequal probabilities results in heterogeneous fractal sets, for which α takes on a continuous range of (non-negative) values corresponding to different regions of the set. The heterogeneous set may then be described by calculating fractal dimensions f (α) of the subsets corresponding to close values of the singularity exponent between α and α + dα: N(α) ∼ l -f (α) , (A.3) where N(α) is the number of segments in the given subset. In the general case, the 1. 3 . 3 The left figure displays a burst-like voltage signal induced in a coil of wire wound on a ferromagnet and the magnetization curve obtained by integration of this signal over time. The almost horizontal portions of the magnetization curve correspond to smooth motion of domain walls and their pinning on obstacles. The upward jumps reflect the moments when the domains configuration becomes unstable and suddenly change to a new state. The statistical analysis of the voltage bursts reveals power-law distributions of event sizes and durations (right figure). Figure 1 . 3 : 13 Figure 1.3: Barkhausen effect. Left: Voltage signal measured in annealed F e 73 Co 12 B 15 amorphous ribbon and its time integral showing a staircase magnetization curve. Right: Distribution of Barkhausen jump duration and amplitude for Si-Fe[64]. Figure 1 . 1 Figure 1.3 shows a striking similarity with jerky deformation curves and acoustic signals in Figs.1.1 and 1.2. This analogy is rather profound as both phenomena are Figure 1 . 4 : 14 Figure 1.4: The velocity-weakening slip-stick friction law[60]. Figure 1 . 5 : 15 Figure 1.5: Examples of deformation curves for Al -3at.%Mg samples, recorded at room temperature for three different εa -values and corresponding to three commonly distinguished types of the PLC effect [18]: type C ( εa = 2 × 10 -5 s -1 ), type B ( εa = 2 × 10 -4 s -1 ), and type A ( εa = 6 × 10 -3 s -1 ). The respective arrows indicate the critical strain ε cr for the onset of plastic instability. Figure 1 . 6 : 16 Figure 1.6: Variation of critical strain as a function of strain rate for Al -4.8%Mg alloy [104]. Figure 1 . 7 : 17 Figure 1.7: (a) A scheme explaining the occurrence of an N-shaped SRS as a result of competition between two microscopic mechanisms; (b) The resulting instability in the form of a saw-toothed deformation curve. Figure 1 . 8 : 18 Figure 1.8: Distribution of the amplitudes of (a) large stress drops and (b) low-amplitude serrations observed in an AlMg alloy at εa = 2 × 10 -5 s -1 [18]. Figure 1 . 1 9 illustrates typical results for an AlMg alloy. Similar to other materials, including those deforming smoothly, the AE occurs very early during the nearly elastic deformation. The count rate quickly increases and passes a maximum in the region of the elastoplastic transition which sometimes displays a Lüders plateau, as shown in Fig.1.9(a). The latter phenomenon is often observed in the same alloys which show the PLC effect. It is generally interpreted as due to propagation of a deformation band through the gauge length of the specimen, a process associated with unpinning of dislocations from their solute atmosphere in a statically aged material, whereas the PLC effect is governed by dynamical strain aging of dislocations[START_REF] Friedel | Dislocations[END_REF]. Figure 1 . 9 : 19 Figure 1.9: (a) Stress-strain curves and (b) the corresponding time dependencies of AE count rate for Al-1.5%Mg alloys deformed at room temperature and various strain rates: (1) εa = 2.67 × 10 -6 s -1 ; (2) εa = 1.33 × 10 -5 s -1 ; (3) εa = 5.33 × 10 -5 s -1 ; (4) εa = 1.33 × 10 -4 s -1 [124]. Figure 1 . 1 Figure 1.9(b) illustrates that the abrupt stress drops of type C are accompanied with bursts of AE count rate. When εa is increased, such a correlation disappears progressively. So, only a part of type B serrations is accompanied with such bursts.Moreover, the strongest AE is observed during the phase of nucleation of a new relay race of deformation bands, although this region is characterized by less abrupt stress fluctuations. In the case of type A behavior the AE bursts are rare and they seemingly correlate with stress humps associated with nucleation of new deformation bands. The observation of the correlation between the AE count rate and the distinct stress drops, particularly for type C serrations, led to a suggestion that the PLC instability gives rise to discrete burst-like AE events produced by large-size dislocation ensembles, in consistence with the macroscopic size of stress serrations, whereas smaller-size dislocation avalanches occur randomly during the macroscopically smooth plastic flow, and generate virtually continuous AE. This hypothesis was also corroborated by the observation of discrete acoustic events during serrated deformation, whereas continuous AE signals superimposed with discrete events were found to accompany the propagation of Lüders bands[124] (see Fig.1.10). It should be noted, however, that the acoustic equipment Figure 1 . 10 : 110 Figure 1.10: AE signal waveforms observed during (a) Lüders phenomenon and (b) PLC effect[124]. The total time interval length is 2.5 ms. a short-range scale [137]. Several 1D models using these ideas and representing the deforming sample as a chain of coupled blocks were proposed. McCormick and Ling considered solid blocks, all obeying the same constitutive equation and coupled via triaxial stresses [138]. The model successfully reproduced certain aspects of deformation curves of type A and type B, as well as deformation band propagation. Lebyodkin et al. used a similar model in which solid blocks were coupled by elastic springs [5, 6]. It reproduced not only the characteristic types of behavior but also the transition from a power-law to a bellshaped statistics. Ananthakrishna et al. developed a model which does not apply a phenomenological nonlinear constitutive law to each block but considers diffusion and mutual reactions of several dislocation densities, one of which corresponds to dislocations carrying solutes (see review [139] and references therein). The coupling between blocks was supposed to be due to the double cross-slip of dislocations. Although this model does not apply the same formulation of the DSA mechanism as that used in other models, it also reproduces various aspects of the PLC effect, including the transition from scale-free to chaotic behavior. Figure 1 . 11 : 111 Figure 1.11: Rearrangement of the crystal lattice due to twinning. The twinning plane and the corresponding shear direction are denoted by K 1 and η 1 . Figure 1 .Figure 1 . 13 : 1113 Figure 1.14 presents one of the main results of this investigation. It was shown that the statistical distributions of the energy of AE events obey a power law with an Figure 1 . 14 : 114 Figure 1.14: Probability density functions of AE energies for the stage of easy basal glide (stars) and the stage manifesting twinning (circles). Cd single crystal. Figure 1 . 15 : 115 Figure 1.15: The main types of AE waveforms observed in [82]. Figure 1 . 16 : 1 1161 Figure 1.16: Time evolution of the averaged ratio between the peak amplitude and square root of energy of AE events recorded for Cd single crystal [82]. Figure 2 . 1 : 21 Figure 2.1: Microstructure of Al-5wt.%Mg(left) and Al-3wt.%Mg(right) before solution treatment. Figure 2 . 2 : 22 Figure 2.2: Microstructure of Al-5wt.%Mg(left) and Al-3wt.%Mg(right) after solution treatment. Figure 2 . 3 : 23 Figure 2.3: Microstructure of Mg-0.35wt.%Zr (left) Mg-0.04wt.%Zr (right) alloys. Figure 2 . 4 : 24 Figure 2.4: Microstructure of AZ31 alloy: structure with a fraction of finer grains (left), structure with coarse grains (right). Figure 2 . 5 : 25 Figure 2.5: Scheme of acoustic sensor disposition for an AlMg sample (left) and an AZ31 sample (right). The AE was captured by piezoelectric transducers clamped to the specimen surface using silicon grease and a spring, to warrant a good acoustic contact. Most of experi-ments on AlMg alloys were done using a Micro-80 sensor with the operating frequency band 200-900 kHz and sensitivity of 57 V/(m/s) (dB), fabricated by Physical Acoustic Corporation. During tension tests, it was usually clamped to the wide specimen head above its gauge length, in order to avoid direct shocks when the deformation bands emerge on the surface (Fig.2.5). The control tests with the transducer fixed in the middle of the gauge length of the specimens did not show influence of the sensor location on the AE statistics. The same sensor location was used in all tensile tests on Mg alloys (Fig. 2.5). In these experiments, a miniaturized MST8S piezoelectric transducer (3mm diameter, frequency band from 50 to 600 kHz, sensitivity 55 dB (ref. 1V ef f )) was used, which helped keeping a good acoustic contact in spite of the specimen surface distortion during deformation. Figure 2 . 6 : 26 Figure 2.6: Scheme of event selection both the structures emerging in such systems and the temporal signals reflecting their evolution. The well-known examples are the thin-film morphology [161], dendritic solidification [162], dielectric breakdown [163], volcano activity [164], rainfalls [165], street traffic [166], and so on. In contrast, only sparse examples of its application to plasticity Figure 3 . 1 : 31 Figure 3.1: Example of AE time series. In this case, the measure is distributed on the time axis, the red line shows division into sections δt. Fig. 3 . 3 Fig.3.1) varied as a power of 2. The local measure µ i (δt) is defined as follows: 20 to 40. Recently, it was shown that the estimates of the scaling exponents systematically depart from the correct values for |q| > 10 [170]. Still, the corresponding curves are useful as sensitive indicators of any imperfectness of the linear trend. For this reason and because the present analysis is based on the relative changes occurring when either the experimental conditions or the scale of observation are varied, such data will also be used in further illustrations. . 11 ) 11 Note that the wavelet transformation uses a time-frequency description similar to Fourier transformation in that both are performed by taking the integral of the inner product between the signal and the analyzing function. However, in contrast to Fourier analysis, which is based on infinite trigonometric series, the wavelet transform uses a finite wavelet function. It is this property that allows revealing information on both time and frequency. The right choice of wavelet ψ makes possible examination of the local features of the signal. Computation of wavelet coefficients consists of four steps: Figure 3 . 2 : 32 Figure 3.2: Signal decomposition scheme. Figures 4.1 (a) and (b) compare the series of acoustic events detected by these two methods in tensile test performed on a specimen of the Mg0.35%Zr alloy. Intervals of approximately stationary behavior Figure 4 . 1 : 41 Figure 4.1: Examples of series of AE events which are either (a; red colour) extracted from the continuously recorded signal using the software developed in the present doctoral research or (b; blue colour) detected by the acoustic equipment during the test, using parameters preset before the measurement. In the latter case, the logarithmic values recorded in dB are converted to the linear scale in order to facilitate the comparison. (c) The corresponding statistical distributions. The red line is arbitrarily shifted to the left to avoid superposition of the curves. εa = 3.5 × 10 -4 s -1 . Figure 4 . 4 Figure 4.2 shows examples of the probability functions illustrating the effect of U 0 for one particular choice of time parameters, HDT = 50 µs and HLT = 100 µs, for the same Mg0.35%Zr alloy. It can be recognized that the statistics obeys power laws in a wide U 0range and the corresponding slopes are fairly robust. The main effect of the increase in U 0 consists in the reduction of the number of events and the corresponding limitation of the interval of A 2 because of the cutoff of the low-amplitude events. Nevertheless, some decrease in β can be also detected: β = -1.80±0.05 for U 0 = 9 mV and β = -1.87±0.02 for U 0 = 60.1 mV. Figure 4 . 4 3(a) represents β(U 0 )curves for three MgZr alloys and for two choices of the time parameters. It can be Figure 4 . 2 : 42 Figure 4.2: Effect of the voltage threshold on the statistics of the amplitudes of AE events for the same specimen. 1 -U 0 = 9 mV, 2 -U 0 = 15.3 mV, 3 -U 0 = 30.5 mV, 4 -U 0 = 60.1 mV. Figure 4 . 3 : 43 Figure 4.3: Effect of (a) U 0 , (b) HDT and (c) HLT settings on the power-law index β for MgZr specimens deformed at εa = 3.5 × 10 -4 s -1 . 1 and 2 -Mg0.35%Zr, 3 -Mg0.15%Zr, 4 -Mg0.04%Zr. HLT = 100 µs; HDT = 800 µs except for the case (2) where HDT=50 µs. (b) HLT = 0 µs; U 0 = 17 mV except for the case (2) where U 0 = 67 mV. (c) U 0 = 17 mV; HDT = 100 µs except for the case (2) where HDT = 20 µs. 73 Figure 4 . 4 : 44 Figure 4.4: Illustration of merging of AE events.The colours show the events detected for two choices of noise thresholds, U 0 , and the same choice of HDT which is deliberately taken very large. Application of a large threshold U 01 gives four separate events (magenta colour) with relatively short duration. Decreasing U 0 leads to merging of consecutive events (blue colour). The number of detected events and the resulting stack of amplitude values do not change significantly but the apparent durations increase drastically. Figure 4 . 4 Figure 4.5 summarizes the results of the analysis for as-delivered and annealed Al5Mg specimens deformed at different strain rates corresponding to the three distinct types of behavior of the PLC effect 1 . It can be seen that the influence of the identification parameters is stronger than in the case of MgZr alloys and the dependences are less monotonous. It is obvious that because of the effects of merging of AE events, the results might depend on such casual factors as the specific relationships between the selected time and voltage parameters, the level of the measurement noise, and the distribution of the on-off time periods in the AE signal, which reflect the distribution of the hits durations and their occurrence times. Therefore, we will only stop on some characteristic trends. Figure 4 . 5 : 45 Figure 4.5: Effects of U 0 , HDT and HLT parameters on the power-law index β for Al5Mg specimens deformed at (a), (d), (g): εa = 6 × 10 -3 s -1 ; (b), (d), (h): εa = 2 × 10 -4 s -1 ; (c), (e), (i): εa = 2 × 10 -5 s -1 . Circles and squares designate results for as delivered and annealed specimens, colours designate the different choices of the parameters which are kept constant, for (a), (b) and (c): blue -HDT = 300 µs, HLT = 300 µs, red -HDT = 30 µs, HLT = 100 µs; for (d), (e) and (f): blue -U 0 = 1.85 mV, red -U 0 = 3.4 mV (HLT = 300 µs); for (g), (h) and (i): blue -HDT = 10 µs, red -HDT = 50 µs (U 0 = 1.85 mV). Figure 4 . 7 : 47 Figure 4.7: Probability density function for squared amplitude of AE events collected in the same strain range for three different samples deformed at driving strain rate εa = 2 × 10 -5 s -1 . a microsecond scale corresponding to individual oscillations within an acoustic event to the scale of the mechanical test. In this chapter we present first results of such investigations[175], which combine an inspection of AE patterns on different time scales with Fourier analysis of both individual waveforms and entire AE signals, as well as statistical analysis of AE events. The most detailed study was realized under conditions of the PLC effect in an Al5Mg alloy. First results are also presented for Mg based alloys. The comparison of behaviors for different materials allows formulation of further directions of these investigations. Figure 5 . 1 : 51 Figure 5.1: Examples of deformation curves for three values of imposed strain rate corresponding to different types of the PLC instability: 2×10 -5 s -1 (type C); 2×10 -4 s -1 (type B); 2×10 -2 s -1 (type A). The two upper curves are deliberately shifted along the ordinate axis to better discern the shape of serrations. Figure 5 . 2 : 52 Figure 5.2: Illustration of the onset of AE in Al5Mg specimens: portions of deformation curves (top) and the accompanying acoustic events (bottom). Colors: blue -εa = 6 × 10 -3 s -1 ; red -εa = 2 × 10 -4 s -1 ; black -εa = 2 × 10 -5 s -1 . Figure 5 . 1 (Figure 5 . 3 : 5153 Figure 5.3: Examples of AE data streaming during macroscopically elastic parts of deformation curves. (a) εa = 6 × 10 -3 s -1 ; (b) εa = 2 × 10 -4 s -1 ; (c) εa = 2 × 10 -5 s -1 . Figure 5 . 4 : 54 Figure 5.4: Examples of Lüders plateau and the accompanying AE: (a) εa = 2×10 -2 s -1 ; (b) εa = 2 × 10 -4 s -1 ; (c) εa = 2 × 10 -5 s -1 . Figure 5 . 5 : 55 Figure 5.5: Example of an acoustic event accompanying a stress drop during the Lüders plateau. εa = 2 × 10 -4 s -1 . Figure 5 .Figure 5 . 6 : 556 Figure 5.6: Superposition of an entire load-time curve and the accompanying AE signal for an annealed AlMg specimen. The arrow indicates the critical strain ε cr for the onset of the PLC effect; (b) zoom in a time interval corresponding to the deformation stage below ε cr ; (c) zoom in a time interval beyond ε cr . εa = 2 × 10 -5 s -1 . Figure 5 . 7 : 57 Figure 5.7: Example of a short isolated AE burst at two magnifications. The front of the signal is as short as approximately 2 µs, the main shock lasting about 30 µs. Some aftershocks can also be seen after the main shock. Figure 5 . 8 : 58 Figure 5.8: (a), (b) Examples of two composed AE events with large duration; (c), (d) respective magnification of these events showing a virtually continuous character of AE at a sufficiently fine time scale. Figure 5 . 9 : 59 Figure 5.9: Probability density function for the squared amplitude of AE events for an annealed specimen (solid symbols, the slope β ≈ -2.9) and an as-delivered specimen (open symbols, β ≈ -2.5 ). The events were collected in the time interval [2000 s; 4000 s], corresponding to the strain range ε < ε cr . εa = 2 × 10 -5 s -1 . 5.6(a)) is similar in the entire εa -range. This is illustrated by Figs. 5.10(a) and 5.11(a), which represent AE for annealed specimens deformed at a εa = 2 × 10 -4 s -1 and εa = 2 × 10 -2 s -1 , corresponding to type B and type A behavior, respectively. The finer details shown in Figs. 5.10 and 5.11 demonstrate a nonuniform growth of the AE activity with increasing imposed strain rate, similar to the effect discussed above in relation to the AE accompanying microplastic deformation (Fig. 5.3). Nevertheless, all the above conclusions made for type C behavior remain in the case of type B serrations. For example, Fig. 5.10(c) shows that the increased AE activity does not prevent burstlike patterns on a time scale corresponding to several stress drops (cf. Fig. 5.6(c)). Figure 5 . 10 :Figure 5 . 11 : 510511 Figure 5.10: Superposition of an entire force-time curve of type B and the accompanying AE signal; (b), (c) consequtive zoom in steps presenting a sequence of serrations and individual stress drops. The arrows (1) and (2) correspond to Figs. 5.12(a) and (b), respectively. εa = 2 × 10 -4 s -1 . Figure 5 . 12 : 512 Figure 5.12: Examples of AE waveforms during nucleation (a) and propagation (b) of type B deformation bands; (c), (d) magnification of figures (a) and (b), respectively. Arrows indicate the sites magnified at the bottom figures. εa = 2 × 10 -4 s -1 . Figure 5 . 5 Figure 5.13 also indicates that in the case of events with large duration, recorded during deep stress drops, calculation of average characteristics, such as energy E and median frequency f med (see Chapter 3), usually gives higher E value and lower f med value than in the neighboring regions. The correlation between the increasing AE activity and stress drops is confirmed by the graphs of evolution of E and f med , which are presented in Figs. 5.14 and 5.15 for type C and type B behavior, respectively. Indeed, this representation allow detecting discontinuities in AE signal on a coarse scale corresponding to rather continuous appearance of the signal itself. Figures 5.14 and 5.15 Figure 5 . 13 : 513 Figure 5.13: Typical patterns of AE and their power spectral density (PSD) function: (a) noise signal during idling of the deformation machine, (b) burst during the period separating two regular series of stress drops, (c) event during reloading between two successive stress drops , (d) portion of a large event recorded during a stress drop. εa = 2 × 10 -4 s -1 . 5.16). In this case, local maxima of energy and local minima of the median frequency are found during the stress humps corresponding to nucleation of a new deformation band, but the stress fluctuations accompanying the deformation band propagation do not produce noticeable effects. Figure 5 . 14 : 514 Figure 5.14: Superposition of a portion of the deformation curve with the time evolution of the average AE energy E and the median frequency f med . Annealed specimen; εa = 2 × 10 -5 s -1 . Figure 5 . 5 Figure 5.15: (a) Evolution of the PSD function in terms of energy and median frequency; (b) Close up of a portion of the upper figure. Annealed specimen; εa = 2 × 10 -4 s -1 . Figure 5 . 16 : 516 Figure 5.16: Same as Fig. 5.15 for an as-delivered specimen deformed at εa = 2×10 -2 s -1 . Figure 5 . 5 Figure 5.17: SEM image of the fractured surface of a nonannealed AlMg specimen. Arrow shows a broken particle appearing in white color. Figure 5 . 5 Figure 5.18 presents results of simultaneous recording of a tensile curve and AE signal for a Mg0.04%Zr sample deformed at εa = 3.5 × 10 -4 s -1 . Qualitatively similar patterns were observed for alloys with other chemical compositions, although the corresponding difference in the grain size was reflected in their different strength and ductility. The deformation curves of the investigated materials are smooth on the global scale (Fig. Figure 5 . 18 : 518 Figure 5.18: Example of load vs. time curve and simultaneously recorded acoustic response for Mg0.04%Zr deformed at εa = 3.5 × 10 -4 s -1 . Figure 5 . 5 Figure 5.20 represents typical waveforms observed for the studied materials, and the respective power spectral density functions. The upper pattern presents a noise signal recorded before the test start. The respective Fourier spectrum displays a narrow peak around 120 kHz and another peak at an approximately double frequency. Such a shape of Fourier spectra was found for all noise-like signals extracted during the test in the Figure 5 . 19 : 519 Figure 5.19: Microstructure of polycrystalline Mg samples after deformation. Left: Mg0.35%Zr sample with an average grain size <d>=170µm; right: Mg0.04%Zr sample with an average grain size <d>=550µm. Figure 5 . 20 : 520 Figure 5.20: Examples of typical waveforms and their power spectral density functions of AE signals in MgZr alloys. (a)-noise-like pattern; (b) and (c) type 1 pattern; (d)type 2 pattern. Figure 5 . 21 : 521 Figure 5.21: Time evolution of AE energy E (top) and median frequency f med (bottom) for Mg0.35%Zr deformed at εa = 3.5 × 10 -4 s -1 . The deep drop in both E and f med around 240 s is due to accidental noise pick-up. Figure 5 . 22 : 522 Figure 5.22: Example of deformation curves and AE signals for AZ31 alloy. (a) sample s1; (b) sample s2. Figure 5 . 23 : 523 Figure 5.23: Microstructure of AZ31 specimens after tension test. Left: sample s1; right: sample s2. Figure 5 . 5 Figure 5.24: (a), (b) Examples of two AE events in AZ31 alloy; (c), (d) magnification of the same events; (e), (f) the corresponding Fourier spectra. state. As shown in Chapter 4, the choice of the event individualization parameters only weakly affects the apparent AE statistics. Thanks to this observation the statistical analysis in the present work was performed for one parameter set for both kinds of material: U 0 =2.2 mV (27 dB), HDT=HLT=300 µs. Except for the initial stage of deformation with the highest strain rate of 2 × 10 -2 s -1 , where the analysis was impeded by a very strong events overlapping, power-law probability functions were found for all deformation conditions and almost over the entire range of AE amplitudes, as illustrated in Fig.6.1. It can also be seen that the largest events show a tendency to an increased probability. The possible nature of this unusual deviation was discussed in § 5.1.3 and attributed to overstresses generated by dislocation pile-ups stopped at grain boundaries, which can trigger dislocation sources in neighboring grains. Figure 6 . 1 : 61 Figure 6.1: Probability density functions for squared amplitude of AE events, for an annealed Al5%Mg specimen deformed at εa = 2 × 10 -4 s -1 . Open circles: events extracted in the time interval T =[100s; 400s]; solid circles: T =[1000s; 1500s]. Figure 6 . 6 Figure 6.2 demonstrates the evolution of the slope of power-law distributions with Figure 6 . 2 : 62 Figure 6.2: Example of collation of deformation curves with the evolution of power-law indexes β for the AE energy distribution. The rectangles designate the time intervals corresponding to statistically stationary series of AE events; their heights give the error of β determination, as defined by the least square method. Data for two samples are displayed for each strain rate: blue -annealed specimens, red -as delivered specimens. (a) εa = 2 × 10 -2 s -1 ; (b) εa = 6 × 10 -3 s -1 ; (c) εa = 2 × 10 -4 s -1 ; (d) εa = 2 × 10 -5 s -1 . Figure 6 . 6 3 summarizes the data of statistical analysis for MgZr with different Zr content, the latter being the principal factor determining the grain size. The following event individualization parameters were used to extract AE events: HDT=50 µs, HLT=100 µs and U 0 =9.1 mV (≈39 dB). It can be recognized that in spite of the different mechanisms of deformation and different character of AE in Al and Mg alloys (cf. Chap. 5), the principal trends found in the previous paragraph for AlMg are also valid for MgZr. The main conclusions which can be drawn from this figure are as follows: (1) the power-law index β is higher than in the case of AlMg and varies in a range close to the values reported previously for single crystals of hexagonal metals[82,23]; (2) despite this difference with the DSA materials, the effect of the grain size and strain is the same, i.e., β decreases in the course of deformation, and the material with a smaller grain size shows a trend to an enhanced probability of large avalanches (flatter slope of the power-law function). This observation presents an essential difference with the case of single crystals, for which no strain dependence of β was reported. Figure 6 . 3 : 63 Figure 6.3: Load vs. time curves and variation of the power-law index β for the AE energy distribution. Blue -Mg0.04%Zr (grain size about 550 µm); red -Mg0.15%Zr (360 µm); magenta -Mg0.35%Zr (170 µm). Figure 6 . 6 Figure 6.4(а) presents examples of partition functions for two q-values for a noise signal recorded during idling of the deformation machine. All dependencies follow very closely the trivial unity slope, which qualifies the numerical procedure for the MF analysis. This figure also shows the effect of the finite size of the analyzed set, which manifests itself in the form of steps occurring when δt approaches the length of the total time interval. Figure6.4(b) displays the result of truncation of a part of the same signal below a threshold. It leads to gradual deviation of the dependencies in the limit of small δt, because of addition of voids to the initially continuous signal. Figures 6. 5 5 Figures 6.5(a) and (b) illustrate the masking effect of noise on a multifractal set and Figures 6.5(a) and (b) illustrate the masking effect of noise on a multifractal set and the result of truncation of the noise component. It is seen that the presence of noise may completely mask the multifractality (Fig. 6.5(a)). It is not a rule, though. Opposite Figure 6 . 4 : 64 Figure 6.4: Examples of partition functions for experimental noise: (а) total signal; (б) after removal of a part of the signal below 1 mV. The red, blue, and black color correspond to q =10, 5, and 0. Figure 6 . 5 : 65 Figure 6.5: Example of the effect of noise on the partition functions for a signal recorded during deformation of an AlMg sample and displaying multifractal features: (а) total signal; (b) after truncation of a part of the signal below 1 mV (the noise level in the tests on AlMg samples was about 1.5 mV). Figure 6 . 123 Figure 6 . 6 : 612366 Figure 6.7 shows MF spectra for AE signals recorded at different stages of deformation in a test conducted at εa = 2 × 10 -5 s -1 . The first interval is selected before ε cr in a range of macroscopically uniform plastic flow between two low-amplitude stress drops (see Fig.6.8(a)). It yields smooth MF spectra, thus testifying that the correlations of the dynamics of dislocations, which lead to the emergence of multifractral patterns, exist before the occurrence of strong self-organization effects associated with the macroscopic plastic instability. Figure 6.7 also presents MF spectra for two subsequent intervals, one Figure 6 . 7 : 67 Figure 6.7: Examples of MF spectra for three portions of an AE signal recorded at εa = 2 × 10 -5 s -1 : open circles -T=[2700s; 2900s], before ε cr , U>1.7 mV; solid circles -T=[5300; 5600], before ε cr , U>1.7 mV; open squares -T=[9650s; 10050s], beyond ε cr , U>1.5 mV. The deformation curve and the AE signal are illustrated in Fig. 6.8 for the first and the third intervals. Figure 6 . 8 : 68 Figure 6.8: Examples of portions of the deformation curve and the accompanying acoustic signal (only the positive half-waves of oscillations are shown) from an annealed Al5Mg specimen deformed at εa = 2 × 10 -5 s -1 : (a) and (b) correspond to the regions before and beyond ε cr , respectively. Red lines mark the analyzed time intervals. U tr =1.52 mV. Figure 6 . 9 : 69 Figure 6.9: Examples of singularity spectra for series of events amplitudes: open circles -T=[2100s, 3100s]; solid circles -T=[5000s, 6000s]; open squares -T=[9200s;10200s]. The event selection settings were as follows: U 0 =6 mV; HDT=HLT=300 µs. Figure 6 . 10 : 610 Figure 6.10: Examples spectra of (a) generalized dimension D(q) and (b) singularity spectra f (α) for portions of AE signal from an annealed Al5%Mg specimen deformed under type B conditions; εa = 2 × 10 -4 s -1 . Solid squares -T=[100s; 260s], before ε cr ; open circles -T=[520s; 700s], beyond ε cr ; open square: T=[1300s; 1600s]. The solid circles illustrate MF spectra for series of amplitudes in the interval T=[520s; 700s]. U tr =1.8 mV. Figure 6 . 6 11(b) displays the corresponding singularity spectrum for this short segment, found for a δt-range from about 40 ms to 0.6 s. It is noteworthy Figure 6 . 6 Figure 6.11: (a) Example of AE burst-like signal accompanying reloading between two stress serrations (the same test as in Fig. 6.10); (b) -the corresponding singularity spectrum obtained after cutting off the background noise below U tr = 1.5 mV. δt=[40ms; 0.6s]. Figure 6 .Figure 6 . 12 : 6612 Figure 6.12: Examples of AE signals and their singularity spectra for a specimen deformed at εa = 6 × 10 -3 s -1 . displays a smooth MF spectrum found for δt=[1 ms; 20 ms]. This result indicates that although work hardening leads to a progressive lost of correlations between deformation processes on a long-range time scale, shorter-time memory can still be present. Figure 6 . 13 : 613 Figure 6.13: Example of (a) AE signal accompanying a sequence of reloading followed by a type B serration at εa = 6 × 10 -3 s -1 ; (b) the corresponding singularity spectrum, revealed after truncation of the background below U tr = 0.8 mV. Figure 6 . 14 : 614 Figure 6.14: Examples of long AE events observed during deformation with εa = 2 × 10 -5 s -1 . Figure 6 . 15 : 615 Figure 6.15: Singularity spectra of acoustic events exhibiting a complex temporal structure at εa = 2 × 10 -5 s -1 . The notations (a), (b) and (c) correspond to Fig. 6.14. Figure 6 . 6 Figure 6.16 shows some typical examples of acoustic signals observed in an AlMg sample deformed at the εa = 2 × 10 -4 s -1 . Individual bursts with a short rising time, Figure 6 . 16 : 616 Figure 6.16: Examples of acoustic emission events observed during jerky flow of an annealed specimen deformed at εa = 2 × 10 -4 s -1 . The signal in Fig. 6.16(b), displaying a sequence of short consecutive bursts, often accompanies the phase of nucleation of a new relay race of type B deformation bands (see § 1.5.1, 1.5.3). The partition functions computed for the entire signal (without truncation) are shown in Fig. 6.17(a)). They exhibit approximately straight segments at Figure 6 . 6 Figure 6.16(c) shows a part of an AE event with large duration. Such events occur during stress drops and seemingly consist of overlapping consecutive bursts. In these Figure 6 . 18 : 618 Figure 6.18: Singularity spectra f (α) for the AE signal in Fig.6.16(b): open circlesentire signal; open squares -after truncation of the signal below U tr = 0.5 mV; solid circles -U tr = 0.75 mV; solid squares -U tr = 1 mV. Figure 6 . 6 Figure 6.19: (a) Partition functions and (b) singularity spectra for the AE signal from Fig.6.16(c). Open circles represent the spectrum for small time scales (3µs ≤ δt < 30µs); solid circles denote that for larger time scales (30µs ≤ δt < 1ms). Figure 6 . 6 20 presents examples of AE signals and MF spectra for different stages of deformation of a Mg35%Zr specimen. The upper example is taken from the initial part of deformation, which is usually interpreted as due to basal slip of dislocations, perhaps, accompanied by some twinning[152]. It displays strong discrete AE bursts separated by long periods with close to noise emission. Figure 6 . 20 : 620 Figure 6.20: Examples of AE signals (left panel) and corresponding singularity spectra f (α)(right panel) for Mg35%Zr specimen. Open symbols: T=[40s; 50s], U tr =3 mV; solid symbols: T=[120s; 160s], U tr =50 mV. The scaling dependences were found for δt ∈[0.3s; 4s] and δt ∈[2s; 16s], respectively.study a separate short burst because its fine structure is masked by a rapid decay function. To uncover correlations on short time scales the analysis was performed over segments of 10-20 ms. Such analysis was not possible on the stage of basal glide because of large distances between events. The results for the next two stages of deformation are illustrated in Fig.6.21. Since the short intervals contain only several high-amplitude bursts (which largely determine the spectra of Fig.6.20) the truncation of the lower part would leave insufficient amount of data for the analysis. Therefore, no thresholding was applied. The upper signal corresponding to primary twinning yields a smooth singularity spectrum (solid symbols) for a δt-range extending beyond the average distance (∼ 1 ms) between separate bursts, δt =[0.1 ms, 2 ms]. It is characterized by a rather big width (α min < 0.5), in consistence with a strongly clustered AE pattern. The bottom signal, representing a magnification of its counterpart in Fig.6.20, also shows nontrivial partition functions but does not provide a smooth spectrum (open symbols). Taking Figure 6 . 21 : 621 Figure 6.21: AE signals (left) and the corresponding MF spectra (right) in shorter time intervals for the same Mg35%Zr specimen. Solid symbols -T=[70.3s; 70.31s], δt ∈[0.1ms, 2ms]; open symbols -T=[140.54s; 140.56s], δt ∈[0.24ms, 4ms]. U tr = 0 mV. alloys.The data streaming technique allowed investigating the jerky flow of an AlMg al-loy on different timescales and provided various evidences to support the hypothesis of a relation between type C and type B PLC instability and the phenomenon of synchronization in dynamical systems. In particular, it allowed clarifying a contradiction existing in the literature on the PLC effect, with regard to discrete and continuous attributes of the accompanying AE. Namely, the observation of huge AE count rate bursts at the instants of type B or type C stress serrations has led to contraposition between discrete AE associated with the PLC instability and continuous AE during stable flow.Among other, this contraposition contradicts the experimental fact that the amplitudes of AE events do not show peculiarities during stress serrations. The analysis of continuously recorded AE signals in the present work proved that AE has a burst-like character during both stress serrations and smooth flow, with amplitudes of bursts varying in the same range. At the same time, the apparent behavior, discrete or continuous, of the AE accompanying stress serrations is found to depend on the scale of observation. analysis of plastic deformation with the help of continuously recorded AE signals. The results obtained testify to the occurrence of complex temporal patterns during plastic deformation of fcc and hcp metals and raise various questions related to the description of the collective dynamics of dislocations. Some of the possible directions of future research are listed below:-A detailed investigation of the role of cracking of second phase particles, using both the analysis of microstructure at different stages of deformation and the comparison of statistical properties of AE with the results obtained for cracking in brittle materials; -Investigation of the spatial aspect of collective processes with the aid of localization of AE sources not only in time but also in space, using two acoustic sensors. Reconstruction of spatio-temporal patterns of deformation processes in various materials. For this purpose, it would be of interest to combine the AE technique with high-resolution local extensometry methods; -Investigation of the initial stage of plastic deformation during elastoplastic transition. This transition is almost unstudied experimentally from the viewpoint of collective phenomena. It can be expected that the AE technique would be able to judge not only about the collective motion of dislocations but also about the processes of their multi-plication; -Statistical analysis of durations of AE events, which remained beyond of this dissertation; -Multifractal analysis of AE signal in pure single crystals, for which only data on the amplitude distributions are available in the literature. It should be underlined that the observation of power-law amplitude distributions does not provide sufficient criteria to verify or falsify the hypothesis of SOC, which is most often applied to interpret these data; -Investigation of AE accompanying deformation by compression. It might be especially interesting in the case of hcp materials which present a strong asymmetry of plastic flow, e.g., different twinning systems operate in tension and compression. Figure A. 1 : 1 Figure A.1: Example of a self-similar object: snowflake under microscope. 35 wt% of Zr, which had the average grain size of 550 µm, 360 µm, and 170 µm, respectively (cf. Fig.2.3). All samples were obtained as cast and annealed for 1h at 250 • C before deformation, in order to reduce the density of dislocations and twins formed during casting. Some tests were done on an industrial alloy AZ31 (2.9wt.% Al, 0.98 wt.% Zn, 0.29 wt.% Mn). Cylindrical samples 35 mm long and 6 mm in diameter were obtained by extrusion from melt using different regimes (extrusion speed of 8 m/min and 45 m/min, dB and recorded with the aid of the Euro Physical Acoustics sys- tem (PCI-2 18-bit A/D device fabricated by Physical Acoustic Corporation), with the sampling rate of 2 MHz or 1 MHz, respectively. 1 The AE measurements during defor- mation of AZ31 samples were carried out with the aid of DAKEL-CONTI-4 AE system developed by ZD RPETY -DAKEL Rpety (sampling frequency of 2 MHz; 4-channel data acquisition with 12-bit A/D converter for each channel), which allows for recording data in four channels with different amplification, in order to avoid overamplification and saturation of the measured voltage. The signals were pre-amplified by 26 dB; the total gain was varied between 26 and 80 dB. It is noteworthy that recent investigations using optical methods bear evidence to another generic behavior observed in various materials[22,52,53]. Namely, it is shown that the intermittent strain localization may self-organize in space so that to give rise to a kind of excitation waves. However, this aspect goes beyond the scope of the present thesis. It is noteworthy that the unstable plastic flow under conditions of a constant loading rate has been discovered by F. Savart et A. Masson as early as in 1830th[100,101]. However, this experimental scheme is out of the scope of the present work because it leads to specimen fracture after several strain jumps only. In the first tests on AlMg alloys, an older Physical Acoustics LOCAN 320 system was utilized, which did not provide data stream but recorded series of acoustic events, using preset parameters. It should be reminded that the set-ups used in the tests on different materials did not have the same total gain, so that the absolute U -ranges may differ in figures for different materials. The experimental observations explicitly confirming this expectation will be described in detail in Chap.[START_REF] Lebyodkin | [END_REF] This is a rather unusual mode of propagation of the Lüders band. However, its analysis goes beyond the scope of the doctoral research. A detailed statistical analysis of AE for various strain rates will be presented in the next chapter. Acknowledgments The work reported in present dissertation has been carried out in the Laboratoire d 'Etude des Microstructures et de Mécanique des Matériaux at the Université de Lorraine and in the Institute of Solid State Physics Russian Academy of Sciences during the years 2009-2012. First of all I wish to thank my advisors Mikhail Lebedkin and Vladimir Gornakov for their inspiring and active guidance during these years, without which completing this thesis would not have been possible. I would also like to thank Tatiana Lebedkina, she has repeatedly expressed her encouragement and moral support. The work presented in this thesis has been done in fruitful collaboration with Prof. František Chmelík and his collegues who I wish to thank: for kind hospitality and interesting discussions during my two visits to Charles University of Prague. My sincere thanks also goes to Prof. Joel Courbon and Dr. Benoit Devincre, who has accepted the heavy task of review of my work. I would like to thank the rest of my doctoral committee: Prof. Claude Fressengeas and Dr. Nikolay Kobelev for their encouragement, insightful comments, and hard questions. I also want to thank all the members of the LEM3 for interesting conversations and good atmospher. Finaly, I am grateful to my family and especially to my wife Alena for encouraging and supporting me during this project. truncation without renormalizing the slope of the power-law dependence. It should also be noted from this point of view that in all three figures, (a)-(c), the β(U 0 )-dependences corresponding to higher HDT (blue symbols) generally lie above their homologs for lower HDT (red symbols), in conformity with the discussed influence of the AE events merging on the seeming β-value. The above data prove that the choice of the time parameters may be quite important as it can entrain considerable changes in the power-low exponents. The β(HDT )dependences are displayed in the second row of Fig. 4.5. It is noteworthy that, as illustrated in Figs. [START_REF] Kubin | Dislocations in Solids[END_REF].5(a) and (d) for the high strain rate, choosing a small U 0 -value corresponding to the range of fast changes on the β(U 0 )-dependence may lead to a considerable shift of the β(HDT )-curve with regard to its counterpart for a higher U 0 -value. Such a shift is observed for the annealed sample (Figs. 4.5(d), squares). The difference is inessential, though, for the as-delivered sample. It is also weak for the low strain rate and practically negligible for the intermediate strain rate which is characterized by weak β(U 0 )-dependences. In spite of these quantitative changes, similar shapes of the curves are obtained for all three strain rates: first, β rapidly decreases approximately by 0.2 with increasing HDT from 10 µs to 40 µs, then it slowly grows. The initial fast fall may be absent, as Appendix A Multifractal analysis A.1 Fractals, fractal dimension An accurate mathematical description of the (multi)fractal analysis can be found in a number of books and reviews, e.g. [162]. The aim of this section is to provide a brief qualitative consideration highlighting the physical meaning of the concept of fractals and its usefulness for characterization of complex structures and signals. The name "fractals" was proposed by Benoît Manelbrot [START_REF] Mandelbrot | The Fractal Geometry of Nature[END_REF] to describe specific non-Euclidean geometrical constructions which present self-similar, or scale-invariant, patterns. Soon afterwards, it has been understood that various natural objects manifesting complex spatial structures or evolution patterns possess the property of self-similarity in some range of scales and may be described using concepts of fractal geometry. Selfsimilar objects are abundant in nature and are found in everyday life as well as in various fields of science, starting from the obvious self-similarity of the hierarchic structure of snowflakes (see Fig. However, the application of the equation (A.1) results in a fractional positive D 0 value, which is called fractal dimension and is usually denoted D f . Indeed, choosing segments of length l = (1/3) n to cover the Cantor set readily gives N(l) = 2 n and therefore, D f = -lnN(l)/lnl = ln2/ln3. This value is greater than the (zero) topological dimension of the Cantor set and smaller than that of the embedding space -the initial 1D segment. This property leads to various peculiar features of fractals. In particular, it follows that L ∼ l 1-D f , i.e., measuring the length of the Cantor set gives results depending on the scale of observation. The modification of the recursion rule will lead to generation of a set with a different value of D f . Thus, the fractal dimension allows not only checking whether the set is self-similar, but also characterizing it quantitatively. At the same time, the fractal dimension is a global characteristic of the occupancy of the space (a hypercube is either occupied or not), which disregards local properties, such as the events amplitudes or clustering of the events, the latter leading to inhomogeneous filling of the space and various occupancy of the hypercubes at a given scale of observation. This problem is attacked with the aid of the multifractal formalism described in the following section. A.2 Multifractals The description of natural objects usually requires more than one scaling index. The reason for this is that besides the underlying fractal geometry, characterized by the fractal dimension, they may carry a locally fluctuating physical property. In addition to dependence f (α), often called singularity spectrum, is a continuous function. Obviously, f varies in a range between 0 and 1 for a 1D signal. The spectrum f (α) degrades to a single point in the case of a uniform fractal. The singularity spectrum makes clear the physical meaning of the multifractal formalism but the above definitions do not provide a method to calculate it. A convenient numerical procedure was proposed in [183]. Using a normalized measure μi (l, q) = µ q i j µ q j , where q ∈ Z, the values of f (α) can be found from the following scaling relationships As presented in Section 1.3, there also exists an alternative description in terms of generalized dimensions D(q) which are found from the scaling laws Z q (l) = l (q-1)D(q) Z 1 (l) = D(1)lnl where the partition functions Z q (l) are defined by the relationships Z q (l) = i µ q i , q = 1 Z 1 (l) = i µ i lnµ i , q = 1 (A.6) D(q) is constant for simple fractals, while the decreasing D(q) is a signature of a multifractal object. The two kinds of multifractal spectra, D(q) and f (α), are related with each other by the Legendre transform: f (α) = qα -τ (q) and α = dτ (q)/dq, where τ (q) = (q -1)D(q).

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Keywords: quantication oue dans une requête est très limité par rapport au coût de l'évaluation globale. Conclusion & Perspectives Cette thèse est la première proposant une extension oue du langage SPARQL visant à améliorer son expressivité et à permettre i) d'interroger des bases de données RDF oues et ii) d'exprimer des préférences complexes sur la valeur des données et sur la structure du graphe. Les résultats présentés dans ce manuscrit sont prometteurs et montrent que le coût supplémentaire dû à l'introduction de conditions de recherche oues reste limité/acceptable. De nombreuses perspectives peuvent être envisagées. Une première perspective concerne l'extension des langages FURQL et FUDGE avec des préférences plus sophistiquées dont certaines font appel à des notions provenant du domaine de l'analyse des réseaux sociaux (centralité ou prestige d'un noeud) ou de la théorie des graphes (par exemple, clique, etc). Nous envisageons ensuite d'étudier d'autres types de requêtes quantiées plus complexes, par exemple trouver les auteurs ayant un article publié dans la plupart des revues de base de données renommées (ou plus généralement, trouver les x tels que x est relié (par un chemin) à Q n÷uds d'un type donné T satisfaisant la condition C). Les logiciels SURF et SUGAR peuvent également être améliorés an de les rendre plus conviviaux, ce qui pose la question de l'élicitation de requêtes oues complexes. Il vaut également la peine d'étudier la manière dont notre cadre pourrait être appliqué à la gestion de dimensions de qualité des données (par exemple, précision, cohérence, etc.) qui sont en général d'une nature graduelle. List of Tables Appendix A Sample of Queries Bibliography Résumé en français La publication de données ouvertes (éventuellement liées) sur le web est un phénomène en pleine expansion. L'étude des modèles et langages permettant l'exploitation de ces données s'est donc grandement intensiée ces dernières années. Récemment, le modèle RDF (Resource Description Framework) s'est imposé comme le modèle de données standard, proposé par le W3C, pour représenter des données du web sémantique [W3C, 2014]. RDF est un cas particulier de graphe étiqueté orienté, dans lequel chaque arc étiqueté (représentant un prédicat) relie un sujet à un objet. SPARQL [START_REF] Prud | SPARQL query language for RDF[END_REF] est le langage de requête standard recommandé par le W3C pour l'interrogation de données RDF. Il s'agit d'un langage fondé sur la mise en correspondance de patrons de graphe. Les travaux que nous présentons visent à introduire plus de exibilité dans le langage (SPARQL ici) en orant la possibilité d'intégrer des préférences utilisateur aux requêtes. Les motivations pour intégrer les préférences des utilisateurs dans les requêtes de base de données sont multiples. Tout d'abord, il semble souhaitable d'orir à l'utilisateur la possibilité d'exprimer des requêtes dont la forme se rapproche, autant que possible, de la formulation de la requête en langage naturel. Ensuite, l'introduction de préférences utilisateur dans une requête permet d'obtenir un classement des réponses, par niveau décroissant de satisfaction, ce qui est trés utile en cas d'obtention d'un grand nombre de réponses. Et enn, là où une requête booléenne classique peut ne retourner aucune réponse, une version à préférence (qui peut être vue comme une version relaxée et donc moins restrictive), peut permettre de produire des réponses proches des objets idéals visés. [START_REF] Bruno | Top-k selection queries over relational databases: Mapping strategies and performance evaluation[END_REF], Chomicki, 2002, Torlone and Ciaccia, 2002, Borzsony et al., 2001, Kieÿling, 2002, Tahani, 1977, Bosc and Pivert, 1995, Pivert and Bosc, 2012]. La littérature sur les requêtes à preférences dans le contexte de bases de données RDF n'est pas aussi abondante puisque cette question n'a commencé à attirer l'attention que récemment. La plupart des approches existantes sont des adaptations directes des propositions faites dans le contexte des bases de données relationnelles. En particulier, elles se limitent à l'expression de préférences sur les valeurs présentes dans les n÷uds. Dans un contexte de graphe RDF, la nécessité d'exprimer des conditions sur la structure des données, puis d'extraire les relations entre les ressources dans le graphe RDF, a motivé des travaux visant à étendre SPARQL et à le rendre plus expressif. Dans [START_REF] Kochut | SPARQLer: Extended SPARQL for semantic association discovery[END_REF], Anyanwu et al., 2007, Alkhateeb et al., 2009] et [START_REF] Pérez | nSPARQL: A navigational language for RDF[END_REF], les auteurs étendent principalement SPARQL en permettant d'interroger RDF à l'aide de patrons de graphe en utilisant des expressions régulières. Mais dans ces approches, le graphe RDF et les conditions de recherche restent non-ous (booléens). Le modèle RDF de base ne permet en eet de représenter nativement que des données de nature booléenne. Les concepts du monde réel à manipuler sont cependant souvent de nature graduelle. Il est donc nécessaire de disposer d'un langage plus exible qui prenne en compte des graphes RDF dans lesquels les données sont intrinsèquement décrites de façon pondérée. Les poids peuvent représenter des notions graduelles telles qu'une intensité ou un coût. Par exemple, une personne peut être l'amie d'une autre avec un degré fonction de l'intensité de la relation d'amitié. An de représenter ces informations, plusieurs auteurs ont proposé des extensions oues du modèle de données RDF. Cependant, les extensions oues de SPARQL qui peuvent être trouvées dans la littérature restent très limitées en termes d'expression de préférences. Notre objectif dans cette thèse est de dénir un langage de requête beaucoup plus expressif pour i) traiter des bases de données RDF oues et non oues et ii) exprimer des préférences complexes sur les valeurs des noeuds et sur la structure du graphe. Un exemple d'une telle requête est: trouver les acteurs a tels que la plupart des lms récents où a joué l'acteur a, sont bien notés et ont été recommandés par un ami proche de a. Nos contributions principales sont décrites dans la suite. Une extension oue de SPARQL avec des capacités de navigation oue Notre objectif dans la première contribution est d'étendre le langage SPARQL de façon à lui permettre d'exprimer des préférences utilisateur pour exprimer des requêtes exibles, portant sur des données RDF véhiculant ou non des notions graduelles. Tout d'abord, nous proposons une extension de la notion de patron de graphe, fondée sur la théorie des ensembles ous, que l'on nomme patron ou de graphe. Cette extension repose sur celle de patron de graphe SPARQL introduite dans [START_REF] Pérez | Semantics and complexity of SPARQL[END_REF] et [START_REF] Arenas | Querying semantic web data with SPARQL[END_REF]. Dans ces travaux, les auteurs dénissent un patron de graphe SPARQL dans un formalisme algébrique plus traditionnel que le formalisme introduit dans la norme ocielle. Un patron de graphe est récursivement déni comme étant soit un graphe contenant des variables, soit un graphe complexe obtenu par l'application d'opérations sur des patrons de graphe. Ensuite, on nous fondant sur cette notion de patron ou de graphe, nous proposons le langage FURQL qui est plus expressif que toutes les propositions existantes de la littérature, et qui permet: 1. d ( [START_REF] Mazzieri | A fuzzy semantics for semantic web languages[END_REF], Udrea et al., 2006, Mazzieri and Dragoni, 2008, Lv et al., 2008, Straccia, 2009, Udrea et al., 2010, Zimmermann et al., 2012]), dont le principe commun consiste à ajouter un degré dans [0, 1] à chaque triplet RDF, formalisé ou bien par l'encapsulation d'un degré ou dans chaque triplet ou bien par l'ajout au modèle d'une fonction associant un degré de satisfaction à chaque triplet (ces deux représentations sont sémantiquement équivalentes et présentent la même expressivité). Un degré attaché à un triplet s, p, o exprime à quel point l'objet o satisfait la propriété p sur le sujet s. Préférences oues Le langage FURQL est basé sur des patrons ous de graphe qui permettent d'exprimer des préférences oues sur les données d'un graphe ou F-RDF via des conditions oues (par exemple, l'année de publication d'un lm est récente ) et sur sa structure via des expressions régulières oues (par exemple, le chemin entre deux amis doit être court ). Syntaxiquement, le langage FURQL permet d'utiliser des patrons ous de graphe dans la clause where et des conditions oues dans la clause filter. La syntaxe d'une ex-pression oue de graphe est proche de celle de chemin, comme déni dans SPARQL 1.1 [Harris and Seaborne, 2013], permettant d'exhiber des n÷uds reliés par des chemins exprimés sous forme d'une expression régulière. On permet ici l'expression d'une propriété oue portant sur les n÷uds reliés. Une propriété d'un chemin concerne des notions classiques de la théorie des graphes ous [Rosenfeid, 2014] : la distance et la force de la connexion entre deux n÷uds, où la distance entre deux noeuds est la longueur du plus court chemin entre ces deux noeuds et la distance d'un chemin est dénie comme étant le poids de l'arc le plus faible du chemin. Ce travail a été publié dans les actes de la 25ème Conférence internationale IEEE sur les systèmes ous (Fuzz-IEEE 16), Vancouver, Canada, 2016. Requêtes quantiées structurelles oues dans FURQL La deuxième contribution traite de requêtes quantiées oues adressées à une base de données RDF oue. Les requêtes quantiées oues ont été étudiées de façon approfondie dans un contexte de bases de données relationnelles pour leur capacité à exprimer diérents types de besoins d'information imprécis, voir notamment [START_REF] Kacprzyk | FQUERY III +:a "human-consistent" database querying system based on fuzzy logic with linguistic quantiers[END_REF], Bosc et al., 1995], où elles servent à exprimer des conditions sur les valeurs des attributs des objets stockés. Cependant, dans le cadre spécique de RDF/SPARQL, les approches actuelles de la littérature traitant des requêtes quantiées considèrent des quanticateurs non-ous uniquement [START_REF] Bry | SPARQLog: SPARQL with rules and quantication[END_REF], Fan et al., 2016] sur des données RDF non-oues. Nous étudions une forme particulière de requête quantiée oue structurelle et montrons comment elle peut être exprimée dans le langage FURQL déni précédemment. Plus précisement, nous considérons des propositions quantiées oues du type QB X are A sur des bases de données RDF oues, où Q est le quanticateur qui est représenté par un ensemble ou et est soit relatif (par exemple, la plupart) soit absolu (par exemple, au moins trois), B est une condition oue, X est l'ensemble de noeuds dans le graphe RDF, et A désigne une condition oue. Un exemple d'une telle proposition quantiée oue est : la plupart des albums récents sont très bien notés. Dans cet exemple, Q correspond au quanticateur ou relatif la plupart, B est la condition oue être récent, X correspond à l'ensemble des albums présents dans le graphe RDF et A correspond à la condition oue être très bien noté. Conceptuellement, l'interprétation d'une telle proposition quantiée oue dans une requête FURQL peut être basée sur l'une des approches de la littérature proposées dans [Zadeh, 1983, Yager, 1984, Yager, 1988]. Son évaluation comporte trois étapes: 1. la compilation de la requête quantiée oue R en une requête non-oue R , 2. l'interprétation de la requête SPARQL R , 3. le calcul du résultat de R (qui est un ensemble ou) basé sur le résultat de R . Ce travail a été publié dans les actes de la 26ème Conférence internationale IEEE sur les systèmes ous (Fuzz-IEEE'17), Naples, Italie, 2017. Mise en ÷uvre et expérimentation Dans cette thèse, nous abordons également l'implantation du langage FURQL. Nous avons à cet eet considéré deux aspects: 1. le stockage de graphes ous (modèle de données étendu que nous considérons) et 2. l'évaluation de requêtes FURQL. Le premier point peut être résolu par l'utilisation du mécanisme de réication qui permet d'attacher un degré ou à un triplet, solution proposée dans [Straccia, 2009]. Concernant l'évaluation de requêtes FURQL, nous avons développé une couche logicielle permettant la prise en compte de requêtes FURQL, que l'on associe à un moteur SPARQL standard. Cette couche logicielle, appelé SURF, est composée principalement des deux modules suivants: • Dans une étape de prétraitement, un module de compilateur de requête FURQL produit les fonctions dépendantes de la requête qui permettent de calculer les degrés de satisfaction pour chaque réponse retournée, une requête SPARQL classique qui est ensuite envoyée au moteur de requête SPARQL pour récupérer les informations nécessaires pour calculer les degrés de satisfaction. La compilation utilise le principe de dérivation introduit dans [START_REF] Pivert | Fuzzy Preference Queries to Relational Databases[END_REF] dans un contexte de bases de données relationnelles qui consiste à traduire une requête oue en une requête non oue. • Dans une étape de post-traitement, un module de traitement des données oues qui calcule le degré de satisfaction pour chaque réponse renvoyée, classe les réponses et les ltre qualitativement si une alpha-coupe a été spéciée dans la requête oue initiale. Une preuve de concept de l'approche proposée, le prototype SURF, est disponible et téléchargeable à l'adresse https://www-shaman.irisa.fr/furql/. Pour évaluer les performances du prototype SURF que nous avons développé, nous avons eectué deux séries d'expériences sur diérentes tailles de bases de données RDF oues. Les premières expériences visent à mesurer le coût supplémentaire induit par l'introduction du ou dans SPARQL, et les résultats obtenus montrent l'ecacité de notre proposition. Les deuxièmes expériences, qui concernent des requêtes quantiées oues, montrent que le coût supplémentaire induit par la présence d'un quanticateur ou dans les requêtes reste très limité, même dans le cas de requêtes complexes. Requêtes quantiées structurelles oues dans FUDGE A la n de cette thèse, nous nous situons dans un cadre plus général: celui de bases de données graphe [START_REF] Angles | Survey of graph database models[END_REF]. Jusqu'à présent, une seule approche de la littérature, décrite dans [START_REF] Castelltort | Fuzzy queries over NoSQL graph databases: Perspectives for extending the Cypher language[END_REF], considère des requêtes quantiées oues dans un tel environnement, et seulement d'une manière assez limitée. Une limitation de cette approche tient au fait que seul le quanticateur est ou (alors qu'en général, dans une proposition quantiée oue de la forme QB X are A, les prédicats A et B peuvent également l'être). Nous proposons quant à nous d'étudier des requêtes quantiées oues impliquant des prédicats ous (en plus du quanticateur) sur des bases de données graphe oues. Nous considerons le même type de requête quantiée oue structurelle que celui considéré dans FURQL mais dans un cadre plus général. Cette contribution est basée sur notre travail décrit dans [Pivert et al., 2016e], dans lequel nous avons montré comment il est possible d'intégrer ces requêtes quantiées oues dans un langage nommé FUDGE, précédemment déni dans [Pivert et al., 2014a]. FUDGE est une extension oue de Cypher [START_REF] Cypher | Cypher[END_REF] qui est un langage déclaratif pour l'interrogation des bases de données graphe classiques. Une stratégie d'évaluation fondée sur un mécanisme de compilation qui dérive des requêtes classiques pour accéder aux données est également décrite. Elle s'appuie sur une surcouche logicielle au système Neo4j, baptisée SUGAR, dont une première version, décrite dans [Pivert et al., 2015[START_REF] Pivert | SUGAR: A graph database fuzzy querying system[END_REF], permet d'évaluer ecacement les requêtes FUDGE ne comportant pas de propositions quantiées. A cet eet, nous avons mis à jour ce logiciel, qui est une couche logicielle qui implémente le langage FUDGE sur le SGBD Neo4j, pour lui permettre d'évaluer des requêtes FUDGE contenant des conditions quantiées oues. Comme preuve de concept de l'approche proposée, le prototype SUGAR est disponible et téléchargeable à l'adresse www-shaman.irisa.fr/fudge-prototype. An de conrmer l'ecacité de l'approche proposée, nous avons eectué quelques expérimentations avec le prototype SUGAR en utilisant diérentes tailles de bases de données graphe oues. Les résultats obtenus sont prometteurs et montrent que le coût du traitement de la Introduction The relational model, introduced in 1970 by Edgar F. Codd [Codd, 1970], has been the most popular model for database management for many decades in academic, nancial and commercial pursuits. In this framework, data can be stored and accessed thanks to a database management system like Oracle, Microsoft SQL Server, MySQL, etc. However, in the recent decades, the traditional relational model faced new challenges, mainly related to the development of Internet. Data to be searched are more and more accessible on the Web (i.e., open environment) and never stop to increase in volume and complexity. As a solution, an alternative model, called NoSQL (Not only Structured Query Language), came to existence and has attracted a lot of attention since 2007. It aims to process eciently and store huge, distributed, and unstructured data such as documents, e-mail, multimedia and social media [Leavitt, 2010, Robinson et al., 2015]. Among NoSQL database systems, we may nd the famous Google's BigTable [START_REF] Chang | Bigtable: A distributed storage system for structured data[END_REF]], Facebook's Cassandra [START_REF] Lakshman | Cassandra: structured storage system on a P2P network[END_REF]], Amazon's Dynamo [START_REF] Decandia | Dynamo: amazon's highly available key-value store[END_REF]], LinkedIn's Project Voldemort, Oracle's BerkeleyDB [Berkeley, 2010] and mostly Graph Databases Systems (e.g., Neo4j 1 , Allegrograph 2 ,etc.), which are designed to store data in the form of a graph. In the last decade, there has been increased attention in graphs to represent social networks, web site link structures, and others. Recently, database research has witnessed much interest in the W3C's Resource Description Framework (RDF) [W3C, 2014], which is a particular case of directed labeled graph, in which each labeled edge (called predicate) connects a subject to an object. It is considered to be the most appropriate knowledge representation language for representing, describing and storing information about resources available on the Web. This graph data model makes it possible to represent heterogenous Web resources in a common and unied way, taking into consideration the semantic side of 1 http://www.neo4j.org/ the information and the interconnectedness between entities. The SPARQL Protocol and RDF Query Language (SPARQL) [START_REF] Prud | SPARQL query language for RDF[END_REF] is the ocial W3C recommendation as an RDF query language. It plays the same role for the RDF data model as SQL does for the relational data model and provides basic functionalities (such as, union and optional queries, value ltering and ordering results, etc.) in order to query RDF data through graph patterns, i.e., RDF graphs containing variables data. RDF data are usually composed of large heterogeneous data including various levels of quality e.g., over relevancy, trustworthiness, preciseness or timeliness of data (see [START_REF] Zaveri | Quality assessment for linked data: A survey[END_REF]). It is then necessary to oer convenient query languages that improve the usability of such data. A solution is to integrate user preferences into queries, which allows users to use their own vocabulary in order to express their preferences and retrieve data in a more exible way. This idea may be illustrated by an example of a real life scenario of movie online booking stated as follows: I want to nd a recent movie with a high rating. In order to process such a query, fuzzy predicates, such as recent and high which model user preferences, have to be taken into account during database querying. These terms are vague and their satisfaction is a question of degree rather than an all or nothing notion. Motivations for integrating user preferences into database queries are manifold [START_REF] Hadjali | Database preference queriesa possibilistic logic approach with symbolic priorities[END_REF]. First, it appears to be desirable to oer more expressive query languages that can be more faithful to what a user intends to say. Second, the introduction of preferences in queries provides a basis for rank-ordering the retrieved items, which is especially valuable in case of large sets of items satisfying a query. Third, a classical query may also have an empty set of answers, while a relaxed (and thus less restrictive) version of the query might be matched by some items. Introducing user preferences in queries has been a research topic for already quite a long time in the context of the relational database model. In the literature, one may nd many exible approaches suited to the relational data model: top-k queries [START_REF] Bruno | Top-k selection queries over relational databases: Mapping strategies and performance evaluation[END_REF], the winnow [Chomicki, 2002] and Best [START_REF] Torlone | Finding the best when it's a matter of preference[END_REF] operators, skyline queries [START_REF] Borzsony | The skyline operator[END_REF], Preference SQL [Kieÿling, 2002], as well as approaches based on fuzzy set theory [Tahani, 1977, Bosc and Pivert, 1995, Pivert and Bosc, 2012]. The literature about preference SPARQL queries to RDF databases is not as abundant since this issue has started to attract attention only recently. Most of these approaches are straightforward adaptations of proposals made in the relational database context. In particular, they are limited to the expression of preferences over the values present in the nodes. In an RDF graph context the need to query about the structure of data and then extract relationships between resources in the RDF graph, has motivated research aimed to extend SPARQL and make it more expressive. In [START_REF] Kochut | SPARQLer: Extended SPARQL for semantic association discovery[END_REF], Anyanwu et al., 2007, Alkhateeb et al., 2009] and [START_REF] Pérez | nSPARQL: A navigational language for RDF[END_REF], the authors mainly extend SPARQL by allowing to query crisp RDF through graph patterns using regular expressions but in these approaches, both the graph and the search conditions remain crisp (Boolean). However, in the real world, many notions are not of a Boolean nature, but are rather gradual (as illustrated by the example above), so there is a need for a exible SPARQL that takes into account RDF graphs where data is described by intrinsic weighted values, attached to edges or nodes. This weight may denote any gradual notion like a cost, a truth value, an intensity or a membership degree. For instance, in the real world, relationship between entities may be gradual (e.g., close friend, highly recommends, etc.) and an associated degree may express its intensity. A statement involving a gradual relationship is for instance an artist recommends a movie with a degree 0.8 (roughly, this movie is highly recommended by this artist). In order to represent such information, several authors proposed fuzzy extensions of the RDF data model. However, the fuzzy extensions of SPARQL that can be found in the literature appear rather limited in terms of expressiveness of preferences. Our aim in this thesis is to dene a much more expressive query language that i) deals with both crisp and fuzzy RDF graph databases and ii) supports the expression of complex preferences on the values of the nodes and on the structure of the graph. An example of such a query is most of the recent movies that are recommended by an actor, are highly rated and have been featured by a close friend of this actor. Contributions In this thesis, our main contributions are as follows. 1. We rst propose a fuzzy extension of the SPARQL query language that improves its expressiveness and usability. This extension, called FURQL, allows (1) to query a fuzzy RDF data model involving fuzzy relationships between entities (e.g., close friends), and (2) to express fuzzy preferences on data (e.g., the release year of a movie is recent ) and on the structure of the data graph (e.g., the path between two friends is required to be short ). A prototype, called SURF, has been implemented and some experiments have been performed that show that introducing fuzziness in SPARQL does not come with a high price. 2. We then focus on the notion of fuzzy quantied statements for their ability to express dierent types of imprecise and exible information needs in a (fuzzy) RDF database context. We show how a particular type of fuzzy quantied structural query can be expressed in the FURQL language that we previously proposed and study its evaluation. SURF has been extended to eciently process fuzzy quantied queries. It has been shown through some experimental results that introducing fuzzy quantied statements into a SPARQL query entails a very small increase of the overall processing time. 3. In the same way as we did with FURQL, we deal with fuzzy quantied queries in a more general (fuzzy) graph database context (RDF being just a special case). We study the same type of fuzzy quantied structural query and show how it can be expressed in an extension of the Neo4j Cypher query language, namely FUDGE, previously proposed in [Pivert et al., 2014a]. A processing strategy based on a compilation mechanism that derives regular (nonfuzzy) queries for accessing the relevant data is also described. Then, some experimental results are reported that show that the extra cost induced by the fuzzy quantied nature of the queries remains very limited. Structure of the thesis The remainder of the thesis is organized as follows: • Chapter 1 introduces background concepts and notations that are necessary to understand the rest of this thesis. We start with the RDF data model and SPARQL, which is the standard query language for RDF data, and briey touch upon fuzzy set theory. Readers familiar with RDF, SPARQL and fuzzy set theory may want to skip this chapter. • Chapter 2 discusses the state-of-the-art research work related to this thesis. We give a classied overview of approaches from the literature that have been proposed to make SPARQL querying of RDF data more exible. Then, we summarize the main features of these approaches and point out their limits. • Chapter 3 is devoted to the presentation of our rst contribution which consists of a fuzzy extension of the SPARQL query language. First, we dene the notion of a fuzzy RDF database. Second, we provide a formal syntax and semantics of FURQL, an extension of the SPARQL query language. To do so, we extend the concept of a SPARQL graph pattern dened over a crisp RDF data model, into the concept of a fuzzy graph pattern that allows: (1) to query a fuzzy RDF data model, and (2) to express fuzzy preferences on data (through fuzzy conditions) and on the structure of the data graph (through fuzzy regular expressions). • Chapter 4 is directly related to our second contribution that addresses the issue of integrating the notion of fuzzy quantied statements in the FURQL language introduced in Chapter 3 for querying fuzzy RDF databases. We rst recall important notions about fuzzy quantiers, and present dierent approaches from the literature for interpreting fuzzy quantied statements. Then, we introduce the syntactic format for expressing a specic type of fuzzy quantied structural query in FURQL and we show how they can be evaluated in an ecient way. • Chapter 5 provides a detailed architectural implementation of the SURF prototype and reports experimental results related to approaches described in the previous chapters. These results are promising and show the feasibility of the presented approaches. • Chapter 6 concerns fuzzy quantied queries in a more general (fuzzy) graph database context. We start by recalling important notions about graph databases, fuzzy graph theory, fuzzy graph databases, and the FUDGE query language which is a fuzzy extension of the Neo4j Cypher query language. We then discuss related work about fuzzy quantied statements in a graph database context and point out their limits. In this chapter, we consider again a particular type of fuzzy quantied structural query addressed to a fuzzy graph database. We dene the syntax and semantics of an extension of the query language Cypher that makes it possible to express and interpret such queries in the FUDGE language. A query processing strategy based on the derivation of nonquantied fuzzy queries is also proposed and some experiments are performed in order to study its performances. • Finally, we conclude the thesis by summarizing our main contributions. Then, we discuss our upcoming perspectives for future research in order to improve and extend the proposed approach. Publications & Softwares Parts of this thesis have been published as i) regular papers at the IEEE International Conference on Fuzzy Systems [Pivert et al., 2016c] [Pivert et al., 2017], at the International Conference on Scalable Uncertainty Management [Pivert et al., 2016e], and the ACM Symposium on Applied Computing [Pivert et al., 2016g], ii) as posters and demos at the IEEE International Conference on Research Challenges in Information Science [Pivert et al., 2016a[START_REF] Pivert | SUGAR: A graph database fuzzy querying system[END_REF]. [Pivert et al., 2016a] [ Pivert et al., 2016c] [ [START_REF] Pivert | Fuzzy quantied queries to fuzzy RDF databases[END_REF] Moreover, some works were published in French conferences: [START_REF] Pivert | FURQL : une extension oue du langage SPARQL[END_REF] The SURF prototype and the SUGAR prototype are available and downloadable respectively on the following web sites: Introduction Int his chapter, we introduce some background notions that will be used throughout the thesis. Section 1.1 presents the RDF graph data model, section 1.2 presents the SPARQL language used for querying this model and section 1.3 presents fuzzy set theory. Let us consider an album as a resource of the Web. Characteristics may be attached to the album, like its title, its artist, its date or its tracks. In order to express such a characteristic, the RDF data model uses a statement of the form of an RDF triple. Denition 1 provides a more formal denition. Denition 2 (RDF graph). An RDF graph is a nite set of triples of (U ∪B)×U ×(U ∪L∪B). An RDF graph is said to be ground if it does not contain blank nodes. An RDF graph can be modeled by a directed labeled graph where for each triple s, p, o , the subject s and the object o are nodes, and the predicate p corresponds to an edge from the subject node to the object one. RDF is then a graph-structural data model that makes it possible to exploit the basic notions of graph theory (such as, node, edge, path, neighborhood, connectivity, distance, in-degree, out-degree, etc.). Moreover, RDF provides a schema denition language called RDF Schema (RDFS), which allows to specify semantic deductive constraints on the subjects, properties and objects of an RDF graph. It permits to declare objects and subjects as instances of given classes, and inclusion statements between classes and properties. It is also possible to relate the domain and range of a property to classes. RDFS denes a set of reserved words from URI with its own predened semantics/vocabularies (i.e., RDFS vocabulary). Among RDFS vocabularies, we can mention the following list: • (rdf:type): represents the membership to a class; • (rdfs:subClassOf ): represents the subclass relationship between classes; • (rdfs:subPropertyOf ): represents the subclass relationship between properties; • (rdfs:domain): represents the domain of properties; • (rdfs:range): represents the range of properties; • (rdfs:Class): represents the meta-classes of classes; • (rdf:Property): represents the meta-classes of properties; • etc. RDF also declares entailment rules that make it possible to derive new triples from the explicit triples appearing in an RDF graph. Such implicit triples are part of the RDF graph even if they do not explicitly appear in it. They can be explicitly added to the graph. When all implicit triples are made explicit in the graph, then, the graph is said to be saturated. In this thesis, we only consider saturated RDF graph. A database which stores RDF graphs, containing statements of the form (subject-predicateobject), is called a triple store (or simply an RDF database ). There have been a signicant number of RDF databases over the last years mainly divided into two categories [START_REF] Faye | A survey of RDF storage approaches[END_REF]: • Native RDF stores implement their own database engine without reusing the storage and retrieval functionalities of other database management systems. Some examples of native RDF stores are AllegroGraph (commercial) 6 , Apache Jena TDB (open-source) 7 , etc. • Non-native RDF Stores use the storage and retrieval functionalities provided by other database management systems. Among the non-native RDF stores, we nd the Apache Jena SDB (open-source) using conventional relational databases 8 , etc. 1.2 Example 4 [Basic Graph Pattern] The albums featuring the artist Beyonce, with their names are described by the following graph pattern. ?artist dc:creator ?album . ?artist dc:title "Beyonce" . ?album dc:title ?name . According to the graph of Figure 1.1, two subgraphs that are isomorphic to this graph pattern may be found and they are given in Figure 1.3. A classical SPARQL query has the general form given in Listing 1.4, where the clause prefix is for abbreviating URIs (which will be omitted in the following examples), the clause select is for specifying which variables should be returned, the clause from denes the datasets to be queried, and the clause where contains the triple of the researched pattern. ..,distinct ...,limit ...,offset ...,projection ... #Modifiers Listing 1.4: Skeleton of a sparql query SPARQL also provides solution modiers, which make it possible to modify the result set by applying classical operators like order by for ordering the result set in ascending (asc (.) default ordering) or descending (desc(.)) order, distinct for removing duplicate answers, limit to limit the number of answers to a xed number (chosen by a user), projection to choose certain variables and eliminate others from the solutions, or offset to dene the position of the rst returned answers. Finally, the output of a SELECT SPARQL query is a set of mappings of variables which match the patterns in the where clause. Example 5 • Optional graph pattern: uses the clause optional and allows for a partial matching of the query. The query tries to match a graph pattern and does not discard a candidate answer when some part of the optional patterns is not satised. • Filter graph pattern: using the clause filter followed by an expression to select answers according to some criteria. This expression may contain classical operators (e.g., =, + , * , -, / , < , > , ≥ , ≤) and functions (e.g., isU RI(?x), isLiteral(?x), isBlank(?x), regex(?x, "A. * )). Example 8 • SELECT query: is equivalent to an SQL SELECT, used to return a set of variables from the query pattern using the select clause. For instance, all the aforementioned examples of SPARQL queries are of the SELECT form; • CONSTRUCT query: returns a single RDF graph by creating new triples that satisfy a specic template from the query pattern. Example 9 [CONSTRUCT query] Let us assume that, if a person X knows a person Y and if this latter (X ) knows a person Z, so, we can say that the rst person X knows the person Z or any person known by Y . Thus, we can create this relationship thanks to the following CONSTRUCT query. construct { ?x foaf:knows ?z . } where { ?x foaf:knows ?y . ?y foaf:knows ?z . } Listing 1.10: An example of a CONSTRUCT query • ASK query: is used to return a Boolean result: true if there exists at least one result that matches the query pattern and false otherwise. Example 10 [ASK query] The following query illustrates the use of the ASK query: Is Beyonce the name of the resource uri:beyonce ? ask { uri:beyonce dc:title "Beyonce" . } Listing 1.11: An example of an ASK query This query returns true since the resource uri:beyonce is indeed the artist Beyonce. • DESCRIBE query: is used to return a single RDF graph with information about the selected resources. Example 11 [DESCRIBE query] An example of a DESCRIBE query is given in Lsiting 1.12. describe uri:beyonce Listing 1.12: An example of a DESCRIBE query This query returns information about the ressource <uri:beyonce>, such as, its name, its age, its rating, its type, etc. Recently, SPARQL 1.1 [Harris and Seaborne, 2013] is a new version of SPARQL supporting new features, such as, property paths, update functionalities, subqueries, negation, value assignments, aggregates functions, etc. • Property paths: they are known as regular expressions tackled in [START_REF] Kochut | SPARQLer: Extended SPARQL for semantic association discovery[END_REF], Anyanwu et al., 2007, Pérez et al., 2008, Alkhateeb et al., 2009, Pérez et al., 2010 • Assignments: The value of a complex expression can be added to a solution mapping by binding a new variable to the value of this expression. The variable can then be used in the query and also can be returned in the result. The assignment is of the form: (expression as ?var). Example 16 [Query with assignment] The following query aims to return the albums released less than 6 years before 2017. Fuzzy Set Theory In the classical set theory, there are two possible situations for an element: to belong or to not belong to a subset. In 1965, Lot Zadeh [Zadeh, 1965] proposed to extend classical set theory by introducing the concept of gradual membership in order to model classes whose borders are not clear-cut. A fuzzy set is associated with a membership function which takes its values in the range of real numbers [0,1], that is to say that graduations are allowed and an element may belong more or less to a fuzzy subset. The theory of fuzzy sets has advanced applications in articial intelligence, computer science, decision theory, expert systems, robotics, etc. They also play an important role in expressing fuzzy user preferences queries to relational databases [Dubois andPrade, 1997, Pivert andBosc, 2012]. In the following, we rst give a formal denition and some characteristics of the notion a fuzzy set, and then the main operations over fuzzy sets are detailed. 1.3.1 Denition Let X be a classical set of objects called the Universe and x be any element of X. If A is a classical subset of X, the membership degree of every element can take only extreme values 0 or 1. This corresponds to the classical denition of a characteristic function : µ A (x) = 1 i x ∈ A, 0 otherwise. When A is a fuzzy subset of X [Zadeh, 1965] it is denoted by: A = {(x, µ A (x)), x ∈ X} with µ A :X → [0,1], where µ A (x) is a degree of membership (simply denoted degree in the following) that quanties the membership grade of x in A. The closer the value of µ A (x) to 1, the more x belongs to A. Therefore, we can have the three situations: µ A (x)=0 , 0 < µ A (x) < 1 , µ A (x)=1. where µ A (x)=0 means that x does not belong to A at all, 0 < µ A (x) < 1 if x belongs partially to A and µ A (x)=1 means that x belongs entirely to A. In practice, the membership function of A is of a trapezoidal shape (see Figure 1.4) and is expressed by the quadruplet (A -a, A, B, B + b). A = {µ A (x 1 )/x 1 , ..., µ A (x n )/x n }, It is worth mentioning that in practice the elements for which the degree equals 0 are omitted. Remark 1. A fuzzy subset of X is called normal if there exists at least one element x ∈ X such as µ A (x) = 1. Otherwise it is called subnormal. 1.3.2 Characteristics of a Fuzzy Set Several notions can be used to describe a fuzzy set. Among them we can cite. 1.3.2.1 Support,height and core The support of a fuzzy subset A in the universal set X, denoted by supp(A), is a crisp set that contains all the elements of X that have a strictly positive degree in A (i.e., which belong somewhat to A). More formally: supp(A) = {x | x ∈ X, µ A (x) > 0}. The core of a fuzzy subset A, denoted by core (A), is the crisp subset of X containing all the elements with a degree equal to 1 (i.e. that completely belong to A with degree equal to 1). More formally: core(A) = {x | x ∈ X, µ A (x) = 1}. Remark 2. Note that in the case of a crisp set, the support and the height collapse, since if x is somewhat in A it belongs (totally) to A. Example 19 Let us consider two fuzzy subsets A and B of the set X, with X= {x 1 , x 2 , x 3 , x 4 , x 5 }, A= {1/x 1 , 0.3/x 2 , 0.2/x 3 , 0.8/x 4 , 0/x 5 } and B= {0.6/x 1 , 0.9/x 2 , 0.1/x 3 , 0.3/x 4 , 0.2/x 5 }. The supports of the two subsets A and B are: supp(A) = {x 1 , x 2 , x 3 , x 4 }, supp(B) = {x 1 , x 2 , x 3 , x 4 , x 5 }. The core of these two subsets is as follows: core(A)= {x 1 }, core(B)= ∅. The height of a fuzzy subset A of X denoted by hgt(A) is the largest degree attained by any element of X that belongs to A. More formally: hgt(A) = sup x∈X µ A (x). A is said to be normalized i ∃ x ∈ X, µ A (x) = 1 which means that hgt(A) = 1. 1.3.2.2 α-cut The ordinary set of such elements x ∈ X having a membership degree larger or equal to a threshold α ∈]0, 1] is the α-cut (A α ) of the fuzzy subset A dened as: A α = {x | x ∈ X, µ A (x) ≥ α}. Example 20 Let us consider X={x 1 , x 2 , x 3 } and a fuzzy subset A={0.3/x 1 + 0.5/x 2 + 1/x 3 }, the α-cuts of this subset are as follows : A 0.5 = {x 2 , x 3 }, A 0.1 = {x 1 , x 2 , x 3 }, A 1 = {x 3 } The membership function of a fuzzy subset A can be expressed in terms of characteristic functions of its α-cuts according to the following formula: µ A (x) = sup α∈]0,1] min(α, µ Aα (x)) , where µ Aα (x) = 1 i x ∈ A α , 0 otherwise. The strict (or strong) α-cut of A, denoted by A ᾱ, contains all the elements in X that have a membership value in A strictly greater than α: A ᾱ = {x|x ∈ X, µ A (x) > α}. The following properties hold: • A0 = supp(A), • A1 = core(A), • α 1 > α 2 ⇒ A α 1 ⊆ A α 2 . It can easily be checked that: (A ∪ B) α = A α ∪ B α and (A ∩ B) α = A α ∩ B α . Operations on Fuzzy Sets Classical operations on crisp sets have been extended to fuzzy sets. These extensions are equivalent to classical operations of set theory when dealing with membership functions belonging to values 0 or 1. The most commonly used operations are presented hereafter and the interesting reader may refer to [Dubois, 1980]. 1.3.3.1 Complementation The complement of a fuzzy set A, denoted by Ā, is dened as: ∀x ∈ X, µ Ā(x) = 1 -µ A (x). Example 21 Let us consider the fuzzy subset A = {1/ x 1 + 0.3/x 2 + 0.2/x 3 + 0 .8/x 4 + 0/x 5 }. Its complement is Ā = {0/x 1 + 0.7/x 2 + 0.8/x 3 + 0.2/x 4 + 1/x 5 }. This operation is involutive, i.e., Ā = A (µ Ā(x) = µ A (x)). 1.3.3.2 Inclusion Let us consider two fuzzy sets A and B dened on X. If for any element x of X, x belongs less to A than B or has the same membership, then A is said to be included in B (A ⊆ B). Formally A ⊆ B if and only if: ∀x ∈ X, µ A (x) ≤ µ B (x). When the inequality is strict, the inclusion is said to be strict and is denoted by A ⊂ B. Obviously, A=B i A ⊆ B and B ⊆ A. 1.3.3.3 Intersection and union of fuzzy sets The intersection of two fuzzy subsets A and B in the universe of discourse X, denoted by A ∩ B, is a fuzzy set given by: µ A∩B (x) = (µ A (x), µ B (x)), where is a triangular norm (abbreviated t-norm ) and usually we take the minimum. The union of two fuzzy subsets A and B in the universe X (denoted by A ∪ B) is a fuzzy subset given by: µ A∪B (x) = ⊥(µ A (x), µ B (x)), where ⊥ is a triangular co-norm (abbreviated t-conorm ) and usually we take the maximun. The t-norms and t-conorms operators follow the properties showed in Table 1 A ∩ B = {0.6/x 1 , 0.3/x 2 , 0.1/x 3 , 0.3/x 4 , 0/x 5 }. The union of the two fuzzy subsets, taking ⊥ = max, is as follows: x 1 ,0.9/x 2 ,0.2/x 3 ,0.8/x 4 ,0.2/x 5 } A ∪ B = {1/ 1 ∧ x = x 0 ∨ x = x Commutativity x ∧ y = y ∧ x x ∨ y = y ∨ x Associativity x ∧ (y ∧ z) = (x ∧ y) ∧ z x ∨ (y ∨ z) = (x ∨ y) ∨ z Monotonicity if v ≤ w and x ≤ y then v ∧ x ≤ w ∧ y v ∨ x ≤ w ∨ y Remark 3. A t-norm is associated with a t-conorm (e.g., min/max) and they satisfy De Morgan's Laws. Later, compensatory operators, such as the averaging operators, have appeared useful for aggregating fuzzy sets, especially in the context of decision making [Zimmermann, 2011]. Averaging operators for intersection (resp., union) are considered to be more optimistic (resp., pessimistic) than t-norms (resp., t-conorms ). Let us also mention many other operators that may be used for expressing dierent kinds of trade-os, such as the weighted conjunction and disjunction [START_REF] Dubois | Weighted minimum and maximum operations in fuzzy set theory[END_REF], fuzzy quantiers [START_REF] Fodor | Fuzzy-set theoretic operators and quantiers[END_REF] (that are going to be explained in Chapter 4), or the non-commutative connectives described in [START_REF] Bosc | On four noncommutative fuzzy connectives and their axiomatization[END_REF]. 1.3.3.4 Dierence between fuzzy sets The dierence between two fuzzy sets A and B is dened as: ∀x ∈ X, µ A-B (X) = (µ A (x), µ B (x)) = (µ A (x), 1 -µ B (x)), which leads to: • µ A-B (x) = min(µ A (x), 1 -µ B (x)) with (x, y) = min(x, y), • µ A-B (x) = max(µ A (x) -µ B (x), 0) if (x, y) = max(x + y -1, 0) is chosen. Example 23 Consider the following fuzzy sets A= {1/a, 0.3/b, 0.7/c, 0.2/e } and B= {0.3/a, 1/c, 1/d, 0.6/e}. Using the minimum for the conjunction, one obtains { 0.7/a, 0.3/b, 0.2/e} for the dierence A -B, while A -B= { 0.7/a, 0.3/b} with the other choice. 1.3.3.5 Cartesian product of fuzzy sets The Cartesian product of the two fuzzy sets A and B, dened as: µ A×B (xy) = (µ A (x), µ B (x)), where is a triangular norm. Introduction Int he last years, with the rapid growth in size and complexity of RDF graphs, querying RDF data in a exible, expressive and intelligent way has become a challenging problem. In the following, we present the contributions from the literature that make SPARQL querying of RDF data more exible. Three categories of approaches may be associated with the following objectives: i) introducing user preferences into queries (which is directly related to this thesis), ii) relaxing user queries and iii) computing an approximate matching of two RDF graphs. These approaches are discussed further in the following sections. A part of this chapter related to introducing user preferences inside SPARQL queries was published in the form of a survey in the proceedings of the 31st ACM Symposium on Applied Computing (SAC'16). Preference Queries on RDF Data Introducing user preferences into queries has been a research topic for already quite a long time in the context of the relational database model. Motivations for integrating preferences are manifold [START_REF] Hadjali | Database preference queriesa possibilistic logic approach with symbolic priorities[END_REF]. First, it has appeared to be desirable to oer more expressive query languages that can be more faithful to what a user intends to say. Second, the introduction of preferences in queries provides a basis for rank-ordering the retrieved items, which is especially valuable in case of large sets of items satisfying a query. Third, a classical query may also have an empty set of answers, while a relaxed (and thus less restrictive) version of the query might be matched by some items. The literature about preference queries to RDF databases is not as abundant as in the relational context since this issue has started to attract attention only recently. In this section, we present an overview of approaches that have been proposed to extend SPARQL by integrating user preferences in queries, followed by a classication of these approaches into two categories according to their qualitative or quantitative nature. We rst present quantitative approaches (Subsection 2. 1.1), then qualitative ones (Subsection 2.1.2). Quantitative Approaches The quantitative approaches share the following principle: each involved preference is dened via an atomic scoring function allowing a score (aka., satisfaction degree) to be associated with each answer, making it possible to get a total ordering of the answers (i.e., tuple t 1 is preferred to tuple t 2 if the score of t 1 is higher than the score of t 2 ). Among the works which belong to the quantitative approaches we may nd those that are based on fuzzy set theory [Zadeh, 1965] and aim to a exible extension of the query language (SPARQL) [START_REF] Cheng | f-SPARQL: a exible extension of SPARQL[END_REF], Wang et al., 2012, Ma et al., 2016]. We can nd, also, those based on top-k querying of RDF data that aim to extend the SPARQL language with top-k queries [START_REF] Bozzon | Towards and ecient SPARQL top-k query execution in virtual RDF stores[END_REF], Bozzon et al., 2012, Magliacane et al., 2012, Wang et al., 2015]. Fuzzy set-based approach The standard version of the SPARQL query language supports only a few classical ways of retrieval, all based on Boolean logic. In order to meet user needs more eectively, [START_REF] Cheng | f-SPARQL: a exible extension of SPARQL[END_REF] proposes a syntactical fuzzy extension of SPARQL, called f-SPARQL (fuzzy SPARQL), which supports the expression of fuzzy conditions including (possibly compound) fuzzy terms, e.g., recent or young, and fuzzy operators, e.g., close to or at least, interpreted in a gradual manner. µ atleastY (x) =      0, if u ≤ w; u-w Y -w , if w < u < Y ; 1, if u ≥ Y. (?X θ FT ) | (?X θ Y)] [with α], where FT denotes a fuzzy term, θ denotes a classical operator (e.g., >, <, =, ≥, ≤, ! =), θ denotes a fuzzy operator (such as close to (around), at least, and at most ), and Y is a string, an integer or an other types allowed in RDF. The optional parameter [with α] species the smallest acceptable membership degree in the interval [0, 1]. Each f-SPARQL query is prexed by #FQ#. Example 25 The fuzzy query retrieve the name of the recent albums with Beyonce is formulated by the f-SPARQL listing 2.1. #FQ# select ?name where { ?artist dc:title "Beyonce". ?artist dc:creator ?album . ?album dc:title ?name. ?album dc:date ?date. filter (?date = recent).} Let us now assume that the database of the running example embeds a rating value for each album, through a property named dc:rate connecting an album (URI resource) to a rating value (a label). When a user wants to express preferences on several attributes (e.g., date, rating, ...), he/she may assign an importance to every partial preference. If no importance is specied, it is implicitly assumed that the partial degrees are aggregated by means of the triangular norm minimum that is commonly used in fuzzy logic to interpret the conjunction. In [START_REF] Cheng | f-SPARQL: a exible extension of SPARQL[END_REF], the authors propose to use a weighted mean in order to combine the partial scores coming from dierent atomic preference criteria: score(A) = n i=1 µ(A i ) × w(F i ) (2.2) where F = (F 1 , ..., F n ) is the set of filter conditions, A i is the property concerned by F i in the candidate answer A, µ(A i ) denotes the membership degree of the answer for F i , and w(F i ) denotes the weight assigned to F i , assuming that n i=1 w(F i ) = 1. Example 26 Consider the query retrieve the name of the recent (importance 0. It is also possible to apply a threshold α i to an atomic fuzzy condition F i (this threshold is associated with the underlying attribute in the select clause). Then, an answer is qualied only if its membership degree relatively to F i is at least equal to α i . Surprisingly, it does not seem that f-SPARQL makes it possible to specify a threshold on the global satisfaction degree. As in SQLF introduced in [Bosc and Pivert, 1995], two types of queries exist in f-SPARQL depending on the type of calibration: • a qualitative calibration in the case of exible queries (#fq#) (see Listing 2.2); • a quantitative calibration in the case of top-k exible queries (#top-k fq# with k (see Listing 2.3), and then, only the top-k answers are returned. The query type has to be declared before the select clause: #fq# (exible query) in the rst case, and #top-k fq# with k (top-k exible query) when a quantitative threshold is used. Example 27 Let us consider again the query from Example 26 and assume that a user only wants to get the 10 best answers. The authors of [START_REF] Cheng | f-SPARQL: a exible extension of SPARQL[END_REF] exhibit a set of translation rules to convert f-SPARQL queries into Boolean ones so as to be able to benet from the existing implementations of standard SPARQL. The same principle was initially proposed in [START_REF] Bosc | SQLf query functionality on top of a regular relational database management system[END_REF] in the context of relational databases (under the name derivation principle ) to process SQLf (fuzzy) queries. It aims to derive a crisp (SQL) query from an SQLf query involving a (global) qualitative threshold α in order to return only answers with satisfaction degree greater or equal to the α-cut. Dierent types of translation rules were used in [START_REF] Cheng | f-SPARQL: a exible extension of SPARQL[END_REF] depending on the the types of fuzzy terms (including simple atomic terms, e.g., recent, modied fuzzy terms, e.g., very recent, and compound fuzzy terms, e.g., popular and very recent ) and fuzzy operators. Some of the authors of [START_REF] Cheng | f-SPARQL: a exible extension of SPARQL[END_REF] proposed two variants of f-SPARQL. The First one, called fp-SPARQL (fuzzy and preference SPARQL) [START_REF] Wang | fp-Sparql: an RDF fuzzy retrieval mechanism supporting user preference[END_REF], involves an alternative way of (i) interpreting modied fuzzy terms (i.e., an atomic fuzzy term modied by an adverb such as extremely, rather, etc), and (ii) interpreting compound fuzzy conditions where atomic predicates are assigned a priority. The second query language, called SPARQLf-p [START_REF] Ma | SPARQL queries on RDF with fuzzy constraints and preferences[END_REF], makes it possible to express i) more complex conditions including fuzzy relations (e.g., physical health is a fuzzy relation between height and weight) besides fuzzy terms and fuzzy operators and, ii) multidimensional user preferences. From another point of view, the authors of [START_REF] Buche | Flexible querying of fuzzy RDF annotations using fuzzy conceptual graphs[END_REF], Buche et al., 2009, Buche et al., 2013] dened a exible querying system using fuzzy RDF annotations based on the notion of similarity and imprecision. This approach is beyond the scope of our work since it does not explicitly propose an extension of the SPARQL query language. Top-k-based approach Top-k -query approaches have been proposed for already many years in a relational database context (cf., the survey of [START_REF] Ilyas | A survey of topk query processing techniques in relational database systems[END_REF]). They have been useful in several application areas such as system monitoring, information retrieval, multimedia databases, sensor networks, etc. Top-k queries [START_REF] Bruno | Top-k selection queries over relational databases: Mapping strategies and performance evaluation[END_REF] are a popular class of queries that return only the k most relevant (best) tuples according to user's preferences. The attribute values of each tuple are associated with a value or score using a simple linear function. Top-k -queries can be viewed as a special case of fuzzy queries limited to conditions of the form: attribute constant The distance between an attribute value and the ideal value is computed by mean of a dierence (absolute value), after a normalization step (which yields domain values between 0 and 1). The overall distance is calculated by aggregating the elementary distances using a function which can be the minimum, the sum, or the Euclidean distance. The steps in the computation are the following: 1. using k, and taking into account both the chosen aggregation function and statistics about the considered relation, a threshold α over the global distance is deduced, 2. a Boolean query computing the desired α-cut or a superset of this α-cut is determined, 3. this query is processed and the score attached to every element of the result is calculated, 4. if at least k tuples with a score greater than or equal to α have been obtained, the k best are returned to the user; otherwise the procedure is run again (from step 2) using a lower threshold α. For eciently processing Top-k -queries in the context of relational databases, several algorithms have been proposed (e.g., Threshold Algorithm (TA) and No Random Access Algorithm (NRA) [START_REF] Fagin | Optimal aggregation algorithms for middleware[END_REF], the Best Position Algorithm [START_REF] Akbarinia | Best position algorithms for top-k queries[END_REF], LPTA [START_REF] Das | Answering top-k queries using views[END_REF], LPTA+ [START_REF] Xie | Ecient top-k query answering using cached views[END_REF] and IV-Index [START_REF] Xie | Ecient top-k query answering using cached views[END_REF]). In the Semantic Web community, top-k -queries have raised a growing interest in the last few years [START_REF] Bozzon | Extending SPARQL algebra to support ecient evaluation of top-k SPARQL queries[END_REF], Magliacane et al., 2012, Dividino et al., 2012, Wang et al., 2015] for alleviating information overload problems. A major challenge is to make the processing of such queries ecient in a SPARQL-like setting. Classical top-k -SPARQL queries can be expressed in SPARQL 1.1 by solution modiers, such as, order by and limit clauses, that respectively order the result set, and limit the number of results. Example 28 The top-k -SPARQL query of Listing 2.4 aims to nd the best ve oers of albums ordered by a function of user ratings and oer date where g 1 and g 2 are scoring functions. select ?album ?offer (g 1 (?rating) + g 2 (?date) AS ?score) where { ?album rdf:type mo:Album. ?album dc:rating ?rating. ?album dc:date ?date. ?album dc:hasOffers ?offer. } order by desc(?score) limit 5 Listing 2.4: Standard top-k -SPARQL-query Naive query processing then relies on a materialize-then-sort procedure which entails an evaluation of all the candidate answers (i.e., those satisfying the condition in the where clause), followed by a computation of the ranking function for each of them, even if only a small number (typically, k = 5) of answers is requested. As a consequence, this processing strategy produces poor performances especially in the case of a large number of answers matching the selected query. A smart processing should stop as soon as the top-k results are returned. In this respect, recent works have proposed solutions to optimize the evaluation of these queries. For instance, the authors of [START_REF] Bozzon | Towards and ecient SPARQL top-k query execution in virtual RDF stores[END_REF], Bozzon et al., 2012, Magliacane et al., 2012] introduced a SPARQL-RANK algebra which is an extension of the SPARQL algebra [START_REF] Pérez | Semantics and complexity of SPARQL[END_REF]] and an incremental rank-aware execution model for top-k -SPARQL queries. This algebra enables splitting the scoring function that may be interleaved with other binary operators. The general objective is to derive an optimized query execution plan and reduce as much as possible the evaluation to a restricted number of answers. [ [START_REF] Bozzon | Towards and ecient SPARQL top-k query execution in virtual RDF stores[END_REF] rst applied this algebra to the processing of top-k SPARQL queries addressed to virtual RDF datasets through query rewriting using the rank-aware relational algebra presented in [START_REF] Li | RankSQL: query algebra and optimization for relational top-k queries[END_REF]. Then, [START_REF] Bozzon | Extending SPARQL algebra to support ecient evaluation of top-k SPARQL queries[END_REF] proposed a detailed version of the SPARQL-RANK algebra, which can be applied to both RDBMS and native RDF datasets. They introduced a rank-aware operator denoted by ρ for evaluating a ranking criterion and redened unary and binary operators (such as, selection (σ), join ( ), union (∪), dierence (\) and left joint ( )) for processing the ranked set of mappings in this context. New algebraic equivalence laws involving this operator have also been proposed. Among these equivalence laws we may nd, pushing ρ over binary operators, splitting the criteria of a scoring function into a set of rank operators and using commutativity of ρ with itself. In [START_REF] Magliacane | Ecient execution of top-k SPARQL queries[END_REF], an incremental execution model for the SPARQL-RANK algebra is proposed and a rank-aware SPARQL query engine denoted by ARQ-RANK based on this algebra is implemented. This engine eciently improves the performance of top-k queries. Later, in [START_REF] Zahmatkesh | Towards a top-k SPARQL query benchmark generator[END_REF], the authors presented top-k DBPSB, an extension of DBPSB (DBpedia SPARQL benchmark) that makes it possible to automatically generate top-k queries from the queries of DBPSB and its datasets. According to [START_REF] Wang | Top-k queries on RDF graphs[END_REF], the SPARQL-RANK algebra proposed by [START_REF] Bozzon | Towards and ecient SPARQL top-k query execution in virtual RDF stores[END_REF], Bozzon et al., 2012, Magliacane et al., 2012] suers from frequent unnecessary input and output in the rank-join operation and this is seen as a drawback in the case of a large dataset. To deal with this issue, they proposed in [START_REF] Wang | Top-k queries on RDF graphs[END_REF] a graph-exploration-based method for eciently processing top-k queries in crisp RDF graphs. They introduced a novel tree index called an MS-tree. Based on this MS-tree, candidate entities are constructed (ranked and ltered) in an appropriate way and the process immediately stops as soon as possible (i.e., as soon as the top-k answers are generated). In case of complex scoring functions, a cost-model-based optimization method is used in order to improve the query processing performance. An evaluation of the approach with both synthetic and real-world datasets using SPARQL-RANK as a competitor is presented in the paper. The experimental results conrm that the model proposed in [START_REF] Wang | Top-k queries on RDF graphs[END_REF] signicantly outperforms SPARQL-RANK approach in case of large datasets to be cached in memory. From an RDF data model view, in [START_REF] Dividino | Ranking RDF with provenance via preference aggregation[END_REF] the authors introduce an approach for top-k querying RDF data annotated with provenance information. In this context, annotations may concern the origin, history, truthfulness, or validity of an RDF statement. An annotated RDF statement is considered as a tuple S= α : θ 1 ,. The statement #1 says that the artist TAL plays in the Le Grand Rex and this information has been published on 03.02.17, has 0.9 as a certainty degree, and was picked up from the Web site www.legrandrex.com. The presence of multiple independent annotation dimensions in the query can induce dierent rankings of answers. In this regard, [START_REF] Dividino | Ranking RDF with provenance via preference aggregation[END_REF] discusses the problem of preference aggregation (or judgement aggregation) and proposes a framework to aggregate all the annotation dimensions into a single joint ranking ordering using dierent aggregation methods. Finally, the authors of [START_REF] Dividino | Ranking RDF with provenance via preference aggregation[END_REF] perform top-k querying using these ranking methods in oine (i.e., available results) and online (i.e., the aggregation of streaming data) settings. Qualitative Approaches: Skyline-based Approaches In the relational database domain, qualitative approaches to preference queries have attracted a large interest, in particular skyline queries [START_REF] Borzsony | The skyline operator[END_REF], which aim to lter an n-dimensional dataset S according to a set of user preference relations and return only the tuples of S that are not dominated in the sense of Pareto order. Note that these approaches only yield a partial order, contrary to the quantitative ones. Let us consider two tuples t = (u 1 , . . . , u n ) and t = (u 1 , . . . , u n ) from S (reduced to the attributes on which a preference is expressed). The tuple t dominates (in the sense of Pareto order) the tuple t , denoted by t t , i t is at least good as t in all dimensions and strictly better than t in at least one dimension. This may be represented by: t t ⇔ ∀i ∈ {1, . . . , n}, t.u i i t .u i and ∃j ∈ {1, . . . , n} such that t.u j j t .u j (2.3) Example 30 Let us assume that a user is looking for an album to listen to, and prefers an album which is recent and high rated. For every preference: recent (resp. high rated), the higher the date (resp. rating) is, the more preferred the tuple is. Consider three albums A 1 (date 2015, rating 5.8), A 2 (date 2013, rating 4) and A 3 (date 2014, rating 8). Album A 1 is more recent and has a higher rating than A 2 . So, A 1 dominates A 2 . Nevertheless, A 1 does not dominate A 3 since A 1 is more recent than A 3 but has a worse rating than A 3 . Hence, the skyline result is {A 1 , A 3 }. In the literature, few works [START_REF] Siberski | Querying the semantic web with preferences[END_REF], Gueroussova et al., 2013] have dealt with the expression and evaluation of skyline queries in a SPARQL-like language. In [START_REF] Siberski | Querying the semantic web with preferences[END_REF], Siberski et al. extend SPARQL with a preferring clause in order to support the expression of multidimensional user preferences. This extension is based on the principle underlying skyline queries, i.e., it aims to nd the nondominated objects. The main syntax of this extension is as follows: select ... where ... { filter (A or B) } preferring P and P' ... and P* Listing 2.5: Extension of SPARQL using Skyline Two types of preferences may be distinguished: Boolean preferences where the answers that meet the condition are favored over those which do not, and scoring preferences (introduced by the keywords highest or lowest, where the elements with a higher value are favored over those with a lower value and vice versa). Example 31 Let us consider that a user has the following preferences: (P 1 ) prefer the artists rated excellent over the very good ones (Boolean preference), (P 2 ) prefer the artist's concert taking place between 9pm and 1am (Boolean preference) and (P 3 ) prefer the artist's concert taking place the latest (scoring preference) pro- vided that they are taking place between 9pm and 1am. In the absence of a skyline functionality, one would use the classical SPARQL query of Listing 2.6 that returns those artists satisfying the Boolean conditions, ordered according to the starting time of their concert. As we can see, a classical skyline query can be expressed in SPARQL with the clauses filter, order by and desc. However, the classical skyline query of Listing 2.6 also returns dominated artists, but only at the bottom of the list of answers. In the extended SPARQL version of [START_REF] Siberski | Querying the semantic web with preferences[END_REF], lines 5 to 7 of Listing 2.6 are replaced by: 5 preferring 6 ?rating = ft:excellent 7 and 8 (?startingTime >= 9pm && ?endingTime <= 1am) 9 cascade highest(?startingTime) Listing 2.7: Skyline extension of SPARQL [START_REF] Siberski | Querying the semantic web with preferences[END_REF] Lines 1 to 4 represent the graph patterns and hard constraints. Line 6 corresponds to preference P 1 , line 8 corresponds to P 2 , and line 9 corresponds to P 3 . The cascade clause in line 9 species that P 3 is evaluated if and only if two answers are equivalent with respect to P 2 . The authors of [START_REF] Siberski | Querying the semantic web with preferences[END_REF] gave the semantics and the implementation of the new constructs aimed to compute a skyline query with SPARQL and extended the SPARQL implementation ARQ in order to process these types of queries. Nevertheless, no optimization aspects are discussed in the paper. The approach proposed in [START_REF] Gueroussova | SPARQL with qualitative and quantitative preferences[END_REF] is based on [START_REF] Siberski | Querying the semantic web with preferences[END_REF]] and i) introduces user preferences in the filter clause, ii) replaces the cascade clause by a prior to clause in the spirit of Preference SQL [START_REF] Kieÿling | The preference SQL system-an overview[END_REF], iii) introduces new comparators for specifying atomic preferences: between, around, more than, and less than. The authors of [START_REF] Gueroussova | SPARQL with qualitative and quantitative preferences[END_REF] show that PrefSPARQL preference queries can be expressed in SPARQL 1.0 and SPARQL 1.1 using an optional clause or features available in SPARQL 1.1 such as not exists. Nevertheless, they do not deal with implementation issues and query processing/optimization aspects. In [START_REF] Rosati | Preference queries with ceteris paribus semantics for linked data[END_REF], the authors are interested also in qualitative preferences but the preferences are represented by means of a CP-net. A CP-net (network of conditional preferences) has been earlier suggested by [START_REF] Boutilier | CP-nets: A tool for representing and reasoning with conditional ceteris paribus preference statements[END_REF] for modeling relational database preference queries. It is a powerful graphical representation of statements that express conditional ceteris paribus (everything else being equal) preferences. Example 34 Let us consider the following ceteris paribus preferences on clothes: i) P 1 : black (b) jackets are preferred to white (w) jackets, ii) P 2 : black (b) pants are preferred to white (w) pants, iii) P 3 : if the jackets and the pants are of the same color, red (r) shirts are preferred to white (w) ones; otherwise, white shirts are preferred. These preferences are modeled by means of the CP-net depicted in The authors of [START_REF] Rosati | Preference queries with ceteris paribus semantics for linked data[END_REF] propose an RDF vocabulary to represent qualitative preference triples formulated under the ceteris paribus semantics. Inspired by [START_REF] Gueroussova | SPARQL with qualitative and quantitative preferences[END_REF], the authors of [START_REF] Rosati | Preference queries with ceteris paribus semantics for linked data[END_REF] present an algorithm to encode a CP-net into a standard SPARQL 1.1 query able to retrieve a ranked set of answers satisfying the user preferences. To the best of our knowledge, this work is the rst attempt to translate the semantics of a CP-net into a SPARQL query. Let us also mention that there exist some works (cf., [START_REF] Chen | Eciently evaluating skyline queries on RDF databases[END_REF]) that propose methods for the optimization of skyline queries in an RDF data context. Query Relaxation Nowadays, the size and the complexity of databases (including relational, semantic, etc.) increase over time at a sustained pace. In such circumstances, users when querying these databases do not have enough knowledge about their content and structure. So, they fail sometimes to formulate meaningful queries to get the expected result or even to avoid empty responses. In order to cope with these issues, some of the semantic Web systems include a query relaxation process for triple-pattern queries (i.e., adressed to data represented in the RDF format) sharing the same principle as the cooperative querying systems [START_REF] Gaasterland | Relaxation as a platform for cooperative answering[END_REF] [ Godfrey, 1997] [Chu et al., 1996] [Kleinberg, 1999] that operate on relational databases. These systems aim to automate the relaxation process of user queries when the selection criteria in the query do not make it possible to obtain answers that meet the user's needs. In a SPARQL/RDF setting, several works have been carried out [START_REF] Hurtado | A relaxed approach to RDF querying[END_REF], Hurtado et al., 2008, Huang et al., 2008, Poulovassilis and Wood, 2010, Calì et al., 2014, Frosini et al., 2017] that propose a relaxation framework for RDF data through RDFS entailment using information provided by a given ontology (see Figure 2.4) and being characterized by RDFS inferences rules (see Table 2.2). These rules enable a generalization of the SPARQL query in order to release its conditions in case of an empty result. Group A (Subproperty) (a,sp,b)(b,sp,c) (a,sp,c) (1) (a,sp,b)(x,a,y) (x,b,y) Group B (Subclass) (3) (a,sc,b)(b,sc,c) (a,sc,c) (2) (a,sc,b)(x,type,a) (x,type,b) (4) Group C (T yping) (a,dom,c)(x,a,y) (x,type,c) (a,range,d)(x,a,y) (y,type,d) x is an instance of a, then, x is an instance of b. [START_REF] Hurtado | A relaxed approach to RDF querying[END_REF], Hurtado et al., 2008] is interested in the relaxation of a conjunctive fragment of queries over RDF data (e.g., See [START_REF] Gutierrez | Foundations of semantic web databases[END_REF], Haase et al., 2004]). This type of queries has the following expression H ← B, where B is a graph pattern (i.e., a set of triples including URIs, literals, blanks nodes, and variables) and H = H 1 , ..., H n is a list of variables. It rstly aims to nd matchings from the graph pattern (i.e., the body of the query B ) to the data and, secondly, applies these matchings to the head of the query (H ) in order to get the nal answers. The authors propose to extend these conjunctive queries by introducing (one or several) relax clauses in the place of the optional clauses. This extension is detailed in the following example. Example 37 In order to avoid empty answers for some cases, a relaxation of some conditions using a specic ontology (see Figure 2.4) is needed. This ontology is represented in the form of an RDF graph based on an RDFS vocabulary that models documents along with properties that model dierent ways people contribute to them (e.g., as authors, editors, etc.). Thanks to this ontology, the following query may be generalized and relaxed in the following way: ?Z, ?Y ← {(?X, name , ?Z), relax {(?X, proceedingsEditorOf , ?Y)}}. The relax clause aims to return rstly editors of conference proceedings. Then, one can automatically rewrite the triple (?X, proceedingsEditorOf , ?Y) into (?X, editorOf, ?Y) or (?X, contributorOf, ?Y) since proceedingsEditorOf is a subproperty of editorOf and editorOf is a subproperty of contributorOf according to the ontology and rules from Table 2.2. So, the relaxed query allows to obtain people who are editors of a publication or in a more general way contributors to a document. The query relaxation strategy involves two types of relaxations: • simple type without using an ontology, which includes dropping triple patterns using the optional clause, replacing constants with variables in a triple pattern, etc. • more complex type using an ontology and inference rules, which includes: Type relaxation for example, following rule (4) from Predicate to range relaxation for example, using rule ( 6) from Table 2.2, the triple pattern (JohnRobert, editorOf, ?Y ) can be relaxed into (?Y, type, Publication) since we have (editorOf, range, Publication) ∈ cl(O). For the purpose of incrementally computing the relaxed answers to the query, an algorithm is presented, which eciently orders the answers according to how closely they meet the query conditions. In, [START_REF] Huang | Computing relaxed answers on RDF databases[END_REF] the authors points out that the approaches proposed in [START_REF] Hurtado | A relaxed approach to RDF querying[END_REF], Hurtado et al., 2008] may still be insucient. They propose a new similarity measure that requires computing the semantic similarity between the relaxed query and the original one. This measure makes it possible to reduce the number of answers as much as possible (or to the desired cardinality) and, then, ensure the quality of answers during the relaxation process. More recently, [START_REF] Reddy | Ecient approximate SPARQL querying of web of linked data[END_REF] proposed an extension of the work [START_REF] Huang | Computing relaxed answers on RDF databases[END_REF] to the web of linked data, where they dene an optimized query processing algorithm in which the relaxed queries are generated and answered on-the-y during query execution (at run time). This work diers from the approach of [START_REF] Huang | Computing relaxed answers on RDF databases[END_REF], which is dedicated only to centralized RDF repositories and aims to generate multiple relaxed queries and execute them sequentially one by one. Another related work is that by [START_REF] Dolog | Robust query processing for personalized information access on the semantic web[END_REF], Dolog et al., 2009], where the authors present user centered process for automatically relaxing over-constrained RDF queries. This relaxation is carried out by rewriting rules for making patterns optional, replacing value, replacing patterns or predicate and deleting patterns or predicate. Background knowledge about the domain of interest and the preferences of the user are taken into account during the query relaxation to rene and guide this process. From a dierent perspective, [START_REF] Poulovassilis | Combining approximation and relaxation in semantic web path queries[END_REF], introduce a framework wherein relaxations and approximations of regular path queries are combined in order to get a more exible querying of RDF data when the user lacks knowledge of their structure. [START_REF] Frosini | Flexible query processing for SPARQL[END_REF], Calì et al., 2014], rely on the work of [START_REF] Poulovassilis | Combining approximation and relaxation in semantic web path queries[END_REF] and propose a formal syntax and semantics of SP ARQL AR which is an extension of the query language SPARQL 1.1 (i.e., SPARQL with property path queries) with query approximation and query relaxation operators. A relaxation operator relies on RDF inference rules and follows the principle presented in [START_REF] Hurtado | Query relaxation in RDF[END_REF] and the approximation operator aims to transform a regular expression pattern P into a new expression pattern P using a set of edit operations (e.g., deletion, insertion and substitution). In [START_REF] Hogan | Towards fuzzy query-relaxation for RDF[END_REF], the authors base their relaxation framework on an industrial use-case from the European Aeronautic Defence and Space Company (EADS) that involve human observations which are presented in the form of natural language and may be imprecise Some contributions also address the problem of providing a guide for the user to relax his/her query. [START_REF] Elbassuoni | Query relaxation for entity-relationship search[END_REF] propose a novel approach for query relaxation based on statistical language models (LMs) for structured RDF data in an automated way. This approach generates a set of relaxation candidates which can be derived from the RDF data and also from external sources like ontologies and textual documents. From another angle, [START_REF] Fokou | Cooperative techniques for SPARQL query relaxation in RDF databases[END_REF], Fokou et al., 2017, Fokou et al., 2017] are inspired by some prior works in relational databases [Godfrey, 1997, Pivert and[START_REF] Pivert | [END_REF] and recommendation systems [Jannach, 2009] and they deal with the problem of explaining the failure of RDF queries in order to help the user to relax his/her query. In [START_REF] Fokou | Endowing semantic query languages with advanced relaxation capabilities[END_REF], the authors initially proposed an extension of SWDB (Semantic Web Database) query languages with new operators that allow to control the relaxation process. These operators describe the relaxation by specifying the part of the query to relax and the technique of relaxation to be used. Then, in [START_REF] Fokou | Cooperative techniques for SPARQL query relaxation in RDF databases[END_REF]] [Fokou et al., 2017] the authors addressed the problem of computing the Minimal Failing Subqueries (MFS) and the Maximal Succeeding Subqueries (XSSs) (i.e., which return non-empty answers) that are used to nd the parts of an RDF query that are responsible of the failure on the one hand, and the relaxed queries that are guaranteed to return a nonempty result on the other hand. Approximate Matching In the literature, the concept of graph isomorphism has been studied for a long time, cf., [Read andCorneil, 1977] [Fortin, 1996] [START_REF] Zhu | An approach for semantic search by matching RDF graphs[END_REF], De Virgilio et al., 2013, De Virgilio et al., 2015, Zheng et al., 2016]. [ [START_REF] Zhu | An approach for semantic search by matching RDF graphs[END_REF] introduces an approach for semantic search. The idea is to match RDF graphs in order to verify whether each candidate resource RDF graph matches the query RDF graph. The resource RDF graph is built up from a specic domain Web information and the query RDF graph corresponds to a user query. To do this, a new semantic similarity measure between two RDF graphs, based on an ontology has been dened. This measure takes into account the similarities between edges and also nodes. The approach proposed in [START_REF] Zhu | An approach for semantic search by matching RDF graphs[END_REF] only takes the similarity of nodes and edges into account in an RDF graph but ignores the structure formed by the nodes and the edges. To deal with this issue, [De Virgilio et al., 2013, De Virgilio et al., 2015] propose an approach dealing with approximate query answering in the context of large RDF data sets. This approach aims to measure the similarity between a portion of a (large) graph representing an RDF dataset and a sub-graph representing a query by applying substitutions and transformations to the paths of the latter. This operation is based on a scoring function that simulates the relevance of answers by taking into account two aspects: i) quality that measures how much the paths retrieved align with the paths in the query and ii) conformity that measures how much the combination of paths retrieved is similar to the combination of the paths in the query. A more recent work is [START_REF] Zheng | Semantic SPARQL similarity search over RDF knowledge graphs[END_REF], where the authors focus on the problem of Semantic SPARQL Similarity Search over RDF knowledge graphs. They propose a metric, semantic graph edit distance in order to measure the similarity between RDF graphs. This metric consider the graph structural, concept-level and semantic similarities in a uniform manner. Conclusion In this chapter, we reviewed several approaches from the literature that aim to query RDF data in a more expressive and exible way, either by introducing fuzzy user preferences, relaxing some preferences or applying approximate matching. We present a summary of these approaches in Table 2. 3. .3: Main features of the preference query approaches A rst observation concerns the limited expressiveness of the approaches. Indeed, all of them are straightforward adaptations of proposals made in the relational database context: they make it possible to express preferences on the values of the nodes, but not on the structure of the RDF graph (structural preferences may concern the strength of a path, the centrality of nodes, etc). Some of the relaxation approaches (e.g., [START_REF] Poulovassilis | Combining approximation and relaxation in semantic web path queries[END_REF], [START_REF] Calì | Flexible querying for SPARQL[END_REF] and [START_REF] Frosini | Flexible query processing for SPARQL[END_REF]), and approximation approaches (e.g., [De Virgilio et al., 2013], [De Virgilio et al., 2015] and [START_REF] Zheng | Semantic SPARQL similarity search over RDF knowledge graphs[END_REF]) have considered this issue but only in a crisp way. A second important remark is that all of the approaches presented above only deal with crisp RDF data. However, we believe that there is a real need for a exible SPARQL that takes into account RDF graphs where data is described by intrinsic weighted values, attached to edges or nodes. This weight may denote any gradual notion like a cost, a truth value, an intensity or a membership degree. The RDF data model should thus be enriched in order to represent gradual information, and new query languages should be dened. A rst step in this direction is the approach proposed in [START_REF] Cedeño | R2DF framework for ranked path queries over weighted RDF graphs[END_REF] where the authors propose an extension of the RDF model embedding weighted edges and an extension of SPARQL to support this feature, allowing new path predicates to express nodes reachability and the ability to express ranked queries. This approach takes the weights into account in order to rank the answers, but does not propose any means to express preferences in user queries. To the best of our knowledge, none of the existing approaches aims to dene a general purpose exible version of SPARQL to weighted RDF databases, which is the rst contribution of this thesis. Introduction Aswes In the literature, several types of approaches have been devoted to extending the SPARQL language among which: i) those that extend the research patterns with paths involving regular expressions, ii) those that consider fuzzy conditions. However, to the best of our knowledge, no approach cover both aspects at the same time. In this chapter, we intend to tackle this issue and we propose the FURQL query language which is a fuzzy extension of SPARQL that improves its expressiveness and usability. This In the following, in Section 3.1, we rst present the notion of the fuzzy RDF data model and then, in Section 3.2, we provide the syntax and the semantics of the FURQL query language. Fuzzy RDF (F-RDF) Graph The classical crisp RDF model is only capable of representing Boolean notions whereas real-world concepts are often of a vague or gradual nature. This is why several authors have proposed fuzzy extensions of the RDF model. Throughout the thesis, we consider the data model based on Denition 4 which synthesizes the existing fuzzy RDF models of literature ( [START_REF] Mazzieri | A fuzzy semantics for semantic web languages[END_REF], [START_REF] Udrea | Annotated RDF[END_REF], [START_REF] Mazzieri | A fuzzy semantics for the resource description framework[END_REF], [START_REF] Lv | Fuzzy RDF: A data model to represent fuzzy metadata[END_REF], [Straccia, 2009], [START_REF] Udrea | Annotated RDF[END_REF], [START_REF] Zimmermann | A general framework for representing, reasoning and querying with annotated semantic web data[END_REF]), whose common principle consists in adding a fuzzy degree to edges, modeled either by a value embedded in each triple or by a function associating a satisfaction degree with each triple, expressing the extent to which the fuzzy concept attached to the edge is satised. Example 38 [Fuzzy RDF triple] The corresponding fuzzy RDF triple ( Beyonce, recommends, Euphoria , 0.8) states that Beyonce, recommends, Euphoria is satised to the degree 0.8, which could be interpreted as Beyonce strongly recommends Euphoria. Denition 4 (Fuzzy RDF (F-RDF) graph ). A F-RDF graph is a tuple (T , ζ) such that (i) T is a nite set of triples of (U ∪ B) × U × (U ∪ L ∪ B), (ii) ζ is a membership function on triples ζ : T → [0, 1]. According to the classical semantics associated with fuzzy graphs, ζ(t) qualies the intensity of the relationship involved in the statement t. Intuitively, ζ attaches fuzzy degrees to the edges of the graph. Having a value of 0 for ζ is equivalent to not belonging to the graph. Having a value of 1 for ζ is equivalent to fully satisfying the associated concept. In the graph G M B of Figure 3.1, such edges appear as classical ones, i.e., with no degree attached. The fuzzy degrees associated with edges are given or calculated. A simple case is when, each degree is based on a simple statistical notion, e.g., the intensity of friendship between two artists may be computed as the number of their common friends over the total number of friends with respect to each artist. Remark 5. In the same way as the RDF graph, an F-RDF graph is said to be ground if it contains no blank nodes. Such a graph may be ground at the beginning or made ground e.g. by a skolemization procedure. In the following, we only consider ground fuzzy RDF graphs. JulioI Example 39 [Fuzzy RDF graph] Figure 3.1 is an example of a fuzzy RDF graph inspired by MusicBrainz 1 . This graph, denoted by G M B in the following, mainly contains artists and albums as nodes. For readability reasons, each URI node contains the value of its name instead of the URI itself. Literal values may be attached to an URI, like the age of an artist, the release date or the global rating of an album. The graph contains fuzzy relationships (e.g., friend, likes, recommends, memberOf ) as well as crisp ones (e.g., creator, date, . . . ). We limit our example to some entities including artists and albums and omit URI prexes to avoid overcrowding the gure. In order to create this graph, we started from a MusicBrainz nonfuzzy subgraph for which every relationship between nodes was Boolean and, then, we made it fuzzy by adding satisfaction degrees denoting the intensity of some relationships. Here for instance, • the degree associated with an edge of the form Art1friend → Art2 is the proportion of common friends (i.e., Boolean relationship) between Art1 and Art2 over the total number of friends of Art1 ; • the degree associated with an edge of the form Art -memberOf → Group is the number of years the artist stayed in this group over the number of years this group has been existing; • the degree associated with an edge of the form Art1likes → Art2 is the number of albums by Art2 that Art1 has liked over the total number of albums by Art2; • the degree associated with an edge of the form Artrecommends → Alb is the number of stars given by Art to Alb over the maximum number of stars. In the following, we rely on classical notions from fuzzy graph theory [Rosenfeid, 2014], which are the path, the distance and the strength (ST) of the connection between two nodes respectively given in Denitions 5,6 and 7. Denition 5 (Path between two nodes). Let G be an F-RDF data graph. Classically, a path p in G corresponds to a possibly empty sequence of triples (t 1 , • • • , t k , • • • , t n ) such that {t i | 1 ≤ i ≤ n} ⊆ G and for all 1 ≤ k ≤ n -1, the object of t k is the subject of t k+1 . Given two nodes x and y, P aths(x, y) denotes the set of cycle-free paths 2 in G connecting x to y, i.e., the set of paths of the form (t length(p) 1 , • • • , t k , • • • , t n ) such that x is the subject of t (3.1) where length(p) is the length of a path p in a fuzzy graph [Rosenfeid, 2014], dened by .2) The distance between two nodes is the length of the shortest path between these two nodes. length(p) = t∈p 1 ζ(t) . ( 3 Remark 6. In a crisp RDF graph (when ζ(t) ∈ {0, 1}), which is a special case of a fuzzy RDF graph, the distance between two nodes x and y given in Denition 6 is still valid and it expresses the number of edges between these nodes (which corresponds to the classical denition). Denition 7 (Strength between two nodes). The strength between two nodes x and y is dened by ST (x, y) = max p∈P aths(x,y) ST _path(p) (3.3) where ST _path(p) is the strength of the path connecting x and y in a fuzzy graph [Rosenfeid, 2014], dened by ST _path(p) = min({ζ(t)|t ∈ p} (3.4) The strength of a path is dened to be the weight of the weakest edge of the path. Example = min(0.8, 0.3, 0.5, 1) = 0.3. Thus, the strength between the pair of nodes (Beyonce, Euphoria) is ST(Beyonce, Euphoria)= 0.8. Here, the distance and the strength correspond to the same path, but it is of course not necessarily the case in general. Let us also mention that except for introducing the degree of truth within an RDF triple in case of imprecise information, several other extensions of RDF were proposed in the literature in order to deal with: • time ( [START_REF] Gutierrez | Introducing time into RDF[END_REF], [START_REF] Pugliese | Scaling RDF with time[END_REF], [START_REF] Tappolet | Applied temporal RDF: Ecient temporal querying of RDF data with SPARQL[END_REF]) to represent the validity periods of time of the information brought by the triple dened by an interval (containing the start and the end point of validity of this information), • trust [Hartig, 2009], used in case of uncertainty about the trustworthiness of the RDF triples. It is represented by a trust value which is either unknown or a value in the interval [-1, 1], where -1 encodes a full disbelief in the triple, 1 a total belief in the triple and 0 signies the lack of belief as well as the lack of disbelief; and, • provenance [START_REF] Dividino | Querying for provenance, trust, uncertainty and other meta knowledge in RDF[END_REF]: may contain information attached to an RDF triple (such as, origins/source (Where is this information from?), authorship (Who provided the information?), time (When was this information provided?), and others). Moreover, [START_REF] Udrea | Annotated RDF[END_REF] and [START_REF] Zimmermann | A general framework for representing, reasoning and querying with annotated semantic web data[END_REF] provided a single theoretical framework to handle the aforementioned extensions along with an extension of the RDF query language to deal with such a framework. FUzzy RDF Query Language (FURQL) In this section, we introduce the FURQL query language, and we formally study its expressiveness. FURQL is based on the notion of fuzzy graph pattern, which is a fuzzy extension of the SPARQL graph pattern notion introduced in [START_REF] Pérez | Semantics and complexity of SPARQL[END_REF] and [START_REF] Arenas | Querying semantic web data with SPARQL[END_REF] which present it in a more traditional algebraic formalism than the ocial syntax does [W3C, 2014]. In the following, we redened the associated syntax and semantics in order to introduce fuzzy preferences expressed over the F-RDF data model of Denition 4. Syntax of FURQL FURQL (Fuzzy RDF Query Language) consists in extending SPARQL graph patterns into fuzzy graph patterns. Before formally introducing the syntax of FURQL, we rst need to dene the notion of a fuzzy graph pattern. A fuzzy graph pattern allows to express fuzzy preferences on the entities of an F-RDF graph (through fuzzy conditions) and on the structure of the graph (through fuzzy regular expressions). It considers the following binary operators: and (SPARQL concatenation), union (SPARQL union), opt (SPARQL optional) and filter (SPARQL filter). We fully parenthesize expressions making explicit the precedence and association of operators. In the following, we assume the existence of an innite set V of variables such that V ∩ (U ∪ L) = ∅. By convention, we prex the elements of V by a question mark symbol. Let us rst dene the notion of a fuzzy regular expression. Denition 8 (Fuzzy regular expression). The set F of fuzzy regular expression patterns, dened over the set U of URIs, is recursively dened by: • is a fuzzy regular expression of F; • u ∈ U and '_' are fuzzy regular expressions of F; • if A ∈ F and B ∈ F then A|B, A.B, A * , A cond are fuzzy regular expressions of F. Above, denotes the empty pattern, the character '_' denotes any element of U, A|B denotes alternative expressions, A.B denotes the concatenation of expressions, A * stands for the classical repetition of an expression (the Kleene closure), A cond denotes paths satisfying the pattern A with a condition cond where cond is a Boolean combination of atomic formulas of the form: sprop is F term where sprop is a structural property of the path dened by the expression and F term denotes a predened or user-dened fuzzy term like short (see Figure 3.3). In the following, we limit the path structural properties to ST (see Denition 7) and distance (see Denition 6). Examples of conditions of this form are distance IS short and ST IS strong. We denote by A + the classical shortcut for A. • A fuzzy triple from (U ∪ V) × (U ∪ F ∪ V) × (U ∪ L ∪ V ) is a fuzzy graph pattern. • If P 1 and P 2 are fuzzy graph patterns then (P 1 and P 2 ), (P 1 union P 2 ) and (P 1 opt P 2 ) are fuzzy graph patterns. Fuzzy connectives include of course fuzzy conjunction ∧ (resp. disjunction ∨), usually interpreted by the triangular norm minimum (resp. maximum), but also many other operators that may be used for expressing dierent kinds of trade-os, such as the weighted conjunction and disjunction [START_REF] Dubois | Weighted minimum and maximum operations in fuzzy set theory[END_REF], mean operators, fuzzy quantiers [START_REF] Fodor | Fuzzy-set theoretic operators and quantiers[END_REF], or the non-commutative connectives described in [START_REF] Bosc | On four noncommutative fuzzy connectives and their axiomatization[END_REF]. Given a pattern P (which can be a fuzzy triple pattern in particular), var(P ) denotes the set of variables occurring in P . Example 42 [Fuzzy graph pattern] Let us consider P rec_low the fuzzy graph pattern dened by (?Art1, (f riend + ) distance is short .creator, ?Alb) AND (?Art1, recommends, ?Alb) AND ((?Alb, rating, ?r) FILTER (?r IS low)), of which Syntactically, FURQL naturally extends SPARQL, by allowing the occurrence of fuzzy graph patterns (which may contain fuzzy regular expressions) in the where clause and the occurrence of fuzzy conditions in the filter clause. A fuzzy regular expression is close to a property path, as dened in SPARQL 1.1 [Harris and Seaborne, 2013], but involves a fuzzy structural property (e.g. distance and strength over fuzzy graphs). The general syntactic form of a FURQL query is given in Listing 3. 1. a list of define clauses that makes it possible to dene the fuzzy terms. If a fuzzy term fterm has a trapezoidal function dened by the quadruple (A-a, A, B and B+b) meaning that its support is [A-a, B+b] and its core [A, B] , then the clause has the form define fterm as (A-a,A,B,B+b). If fterm is a decreasing function, like the term low of Figure 3.5, then, the clause has the form definedesc fterm as (δ,γ) (there is the corresponding defineasc clause for increasing functions). 1 definedesc low as (2, 8) 2 defineasc short as (3, 5) 3 select ?art1 where { 4 { ?art1 (friend+ | distance is short) ?art2 . 5 ?art2 creator ?alb . 6 ?alb rating ?r . 7 ?art1 recommends ?alb . } 8 filter (?r is low) 9 } cut 0.4 Listing 3.2: A FURQL query containing P rec_low In this example, the definedesc clause of line 1 denes the fuzzy term low of The pattern from lines 3 to 8 is the fuzzy pattern of Example 42. Line 9 species an α-cut of the fuzzy pattern with a satisfaction degree greater or equal to 0.4. Semantics of FURQL To dene the semantics of FURQL, we need to dene the semantics of fuzzy graph patterns. Intuitively, given an F-RDF data graph G, the semantics of a fuzzy graph pattern P denes a set of mappings, where each mapping (from var(P ) to URIs and literals of G) maps the pattern to an isomorphic subgraph of G. For introducing such a concept, the notion of satisfaction of a fuzzy regular expression must rst be dened. Denition 10 (Fuzzy regular expression matching of a path). Let G = (T , ζ) be an F-RDF graph and exp be a fuzzy regular expression. Let p = ( s 1 , p • exp is of the form . If p is empty then sat exp (p) = 1 else sat exp (p) = 0. • exp is of the form u ∈ U (resp. _). If p 1 is u (resp. any u ∈ U) then sat exp (p) = ζ( s 1 , p 1 , o 1 ) else 0. • exp is of the form f 1 .f 2 . Let P be the set of all pairs of paths (p 1 , p 2 ) s.t. p is of the form p 1 p 2 . One has sat exp (p) = max P (min(sat f 1 (p 1 ), sat f 2 (p 2 ))). • exp is of the form f 1 ∪ f 2 . One has sat exp (p) = max(sat f 1 (p), sat f 2 (p)). • exp is of the form f * . If p is the empty path then µ exp (p) = 1. Otherwise, we denote by P the set of all tuples of paths (p 1 , • • • , p n ) (n > 0) s.t. p is of the form p 1 • • •p n . One has sat exp (p) = max P (min i∈[1..n] (sat exp (p i ))). • exp is of the form f Cond where Cond is a fuzzy condition. sat exp (p) = min(sat f (p), µ Cond (p)) where µ Cond (p) denotes the degree of satisfaction of cond by p. Again, not satisfying is equivalent to getting a degree of 0. Denition 11 (Satisfaction of a fuzzy regular expression by a pair of nodes). Let G = (T , ζ) be an F-RDF graph and exp be a fuzzy regular expression. Let (x, y) be a pair of nodes of G. The statement the pair (x, y) satises exp with a satisfaction degree of sat exp (x, y) is dened by sat exp (x, y) = max p∈P aths(x,y) sat exp (p). Note that only cycle-free paths need to be considered in order to compute the satisfaction degree. Example Expression f 1 = (friend + ).creator is a fuzzy regular expression. A pair of nodes (x, y) satises f 1 if x has a friend-linked artist (an artist connected to x with a path made of friend edges), that created the album y. All of the pairs of nodes (EnriqueI, Justied), (Shakira, Buttery), (Beyonce, Euphoria), (Rihanna, Euphoria), (MariahC, Euphoria) and (Shakira, Euphoria), illustrated in Figure 3 = min(0.5, 1) = 0.5. Expression f 2 = (friend + ) distance is short .creator is a fuzzy regular expression. A pair of nodes (x, z) satises f 2 if x has a close friend artist y that created an album z, close meaning that x is connected to y by a short path made of friend edges (the term short is dened in Figure 3.3 on page 68). It is worth noticing that expression f 1 is a sub-expression of expression f 2 , so we are going to make use of the satisfaction degree of f 1 , denoted by sat f 1 , in order to calculate the satisfaction degree of f 2 , denoted by sat f 2 . According to the paths depicted in Figure 3.6: • the length of pair (EnriqueI, Justied) = 1/0.4 + 1 = 3.5, µ short (3.5) = 0.75 and sat f 1 (EnriqueI, Justied) = 0.4, then, sat f 2 (EnriqueI, Justied) = min(0.75, 0.4) = 0.4, • the length of pair (Shakira, Buttery)= 1/0.7 + 1 = 2.4, µ short (2.4) = 1 and sat f 1 (Shakira, Buttery) = 0.7, then, sat f 2 (Shakira, Buttery) = min(1, 0.7) = 0.7, • the length of pair (Beyonce, Euphoria) = 1/0.6 + 1/0.2 + 1 = 7.7, µ short (7.7) = 0 and sat f 1 (Beyonce, Euphoria) = 0.3, then, sat f 2 (Beyonce, Euphoria) = min(0, 0.3) = 0, • the length of pair (Rihanna, Euphoria) = 1/0.2 + 1 = 6, µ short (6) = 0 and sat f 1 (Rihanna, Euphoria)=0.2, then, sat f 2 (Rihanna, Euphoria) = min(0, 0.2) = 0, • the length of pair (MariahC, Euphoria)= 1/0.3 + 1/0.5 + 1= 6.33, µ short (6.33) = 0 and sat f 1 (MariahC, Euphoria) = 0.3, then, sat f 2 (MariahC, Euphoria) = min(0, 0.3) = 0, and • the length of pair (Shakira, Euphoria) = 1/0.5 + 1 = 3, µ short (3) = 1 and sat f 1 (Shakira, Euphoria) = 0.5, then, sat f 2 (Shakira, Euphoria) = min(1, 0.5) = 0.5. Then, the pairs of nodes (EnriqueI, Justied), (Shakira, Buttery) and (Shakira, Euphoria) are the only ones that match the fuzzy regular expression f 2 and their satisfaction degrees are sat f 2 (EnriqueI, Justied) = 0.4, sat f 2 (Shakira, Buttery) = 0.7 and sat f 2 (Shakira, Euphoria) = 0.5 respectively. Expression f 3 = (f riend+) ST >0. 65 .creator is a fuzzy regular expression. A pair of nodes (x, y) satises f 3 if x has a friend artist (an artist connected to x with a path made of friend edges which has a strength higher than 0.65), who created the album y. It is worth noticing that expression f 1 is a sub-expression of expression f 3 , so we are going to make use of the satisfaction degree of f 1 (denoted by sat f 1 ) in order to calculate the satisfaction degree of f 3 (sat f 3 ). The pair of nodes (Shakira, Buttery), shown in Figure 3.6, is the only one that matches the fuzzy regular expression f 3 with a non zero satisfaction degree: sat f 3 (Shakira, Buttery) = 0.7, where the strength between the pair of nodes (Shakira, Buttery)= min (0.7,1)= 0.7 and sat f 1 (Shakira, Buttery) = 0.7, then, sat f 3 (Shakira, Buttery) = min (0.7, 0.7) =0.7. Let us now come to the denition of a mapping. A mapping is a pair (m, d) where m : V → (U × L) and d ∈ [0, 1]. Intuitively, m maps the variables of a fuzzy graph pattern into a subgraph (answer) of the F-RDF data graph and d denotes the satisfaction degree associated with the mapping (the more satisfactory the subgraph, the higher the satisfaction degree). The expression m(t), where t is a triple pattern, denotes the triple obtained by replacing each variable x of t by m(x). The domain of a mapping m denoted by dom(m) is the subset of V for which m is dened. Two mappings m 1 and m 2 are compatible i for all ?v ∈ dom(m 1 ) ∩ dom(m 2 ), one has m 1 (?v) = m 2 (?v). Intuitively, m 1 and m 2 are compatible if m 1 can be extended with m 2 to obtain a new mapping m 1 ⊕m 2 and vice versa. Let M 1 and M 2 be two fuzzy sets of mappings. We dene the join, union, dierence and left outer-join of M 1 with M 2 as: Join M 1 M 2 ={(m 1 ⊕ m 2 , min(d 1 , d 2 )) | (m 1 , d 1 ) ∈ M 1 and (m 2 , d 2 ) ∈ M 2 and m 1 , m 2 are compatible}. The operation M 1 M 2 denotes the set of new mappings that result from extending mappings in M 1 with their compatible mappings in M 2 . Union M 1 ∪ M 2 ={(m, d) | (m, d) ∈ M 1 and m ∈ support(M 2 )} ∪ {(m, d) | (m, d) ∈ M 2 and m ∈ support(M 1 )} ∪ {(m, max(d 1 , d 2 )) | (m, d 1 ) ∈ M 1 and (m, d 2 ) ∈ M 2 } Here, ∪ corresponds to the classical set-theoretic union and support denotes the support of a fuzzy set of mappings and corresponds to the set of all elements of the universe of discourse whose their grade of membership is greater than zero. Dierence M 1 \M 2 ={(m 1 , d 1 ) | (m 1 , d 1 ) ∈ M 1 and ∀(m 2 , d 2 ) ∈ M 2 , m 1 and m 2 are not compatible}. M 1 \M 2 returns the set of mappings in M 1 that cannot be extended with any mapping in M 2 . Leftouterjoin M 1 M 2 = (M 1 M 2 ) ∪ (M 1 \M 2 ). A mapping m is in M 1 M 2 if it is the extension of a mapping of M 1 with a compatible mapping of M 2 , or if it is in M 1 and cannot be extended with any mapping of M 2 . Denition 12 (Mapping satisfying a fuzzy condition). Let m be a mapping and C be a fuzzy condition. Then m satises the fuzzy condition C with a satisfaction degree dened as follows, according to the form of C: • C is of the form bound(?x): if ?x ∈ dom(m) then m satises the condition C with a degree of 1, else 0. • C is of the form ?x θ c (where θ is a (possibly fuzzy) comparator and c is a constant): if ?x ∈ dom(m) then m satises the condition C with a degree of µ θ (m(?x), c), else 0. • C is of the form ?x θ ?y: if ?x ∈ dom(m) and ?y ∈ dom(m), then m satises the condition C with a degree of µ θ (m(?x), m(?y)), else 0. • C is of the form ?x is F term: if ?x ∈ dom(m) then m satises the condition C to the degree µ F term (m(?x)) (which can be 0). • C is of the form ¬C 1 or C 1 C 2 where is a fuzzy connective: we use the usual interpretation of the fuzzy operator involved (complement to 1 for the negation, minimum for the conjunction, maximum for the disjunction, etc [START_REF] Fodor | Fuzzy-set theoretic operators and quantiers[END_REF]). Denition 13 (Evaluation (interpretation) of a fuzzy graph pattern). The evaluation of a fuzzy graph pattern P over an F-RDF graph, denoted by P G is recursively dened by: • if P is of the form of a (crisp) triple graph pattern t ∈ (U ∪ V) × (U ∪ V) × (U × L × V) then P G = {(m, 1) | dom(m) = var(t) and m(t) ∈ G}, • if P is of the form of a fuzzy triple graph pattern t ∈ (U ∪ V) × F × (U × L × V) denoted by ?x, exp, ?y (where variables occur as subject and object) then P G = {(m, d) | dom(m) = {?x, ?y} and (m(?x), m(?y)) satises exp with a satisfaction degree d = sat exp (x, y)}. The case where the subject (resp. the object) of t is a constant of U (resp. U ∪ L) is trivially induced from this denition. • if P is of the form (P 1 and P 2 ) then P G = P 1 G P 2 G , • if P is of the form (P 1 opt P 2 ) then P G = P 1 G P 2 G , • if P is of the form (P 1 union P 2 ) then P G = P 1 G ∪ P 2 G , • if P is of the form (P 1 filter C) then P G = {(m, d) | m ∈ P G and m satises C to the degree of d}. Intuitively, expressions (P 1 and P 2 ), (P 1 union P 2 ), (P 1 opt P 2 ), and (P 1 filter C) refer to conjunction graph patterns, union graph patterns, optional graph patterns, and lter graph patterns respectively. Optional graph patterns allow for a partial match of the query (i.e., the query tries to match a graph pattern and does not omit a solution when some part of the optional pattern is not satised). Remark 7. Note that a crisp graph pattern is a special case of a fuzzy graph pattern where no fuzzy term or condition occurs (and thus, according to the previous denition, an answer necessarily has a satisfaction degree of 1). Example 45 [Evaluation of a fuzzy graph pattern] Let us recall the fuzzy graph pattern P rec_low from Example 42 dened by (?Art1, (f riend + ) distance is short .creator, ?Alb) AND (?Art1, recommends, ?Alb) AND ((?Alb, rating, ?r) FILTER (?r is low)), for which Figure 6.3 is a graphical representation. It can be represented as follows: P rec_low G M B = { ({?Art1 → EnriqueI , ?Alb → Justied, ?r → 6}, 0.33), ({?Art1 → Shakira , ?Alb → Buttery, ?r → 4}, 0.66)}. Note that the mapping {?art1 → Shakira, ?alb → Euphoria, ?r → 9} is excluded from the result of the evaluation of the pattern P rec_low since µ low_rating (9) = 0. Conclusion In this chapter, we have introduced a new query language named FURQL which is a fuzzy extension of SPARQL that goes beyond the previous proposals in terms of expressiveness inasmuch as it makes it possible i) to deal with crisp and fuzzy RDF data, and ii) to express fuzzy structural conditions beside more classical fuzzy conditions on the values of the nodes present in the graph. We rst presented the notion of a fuzzy RDF graph that makes it possible to model relationships between entities and then, we formalized a formal syntax and semantics of FURQL based on the notion of fuzzy graph pattern, which extends Boolean graph patterns introduced by several authors in a crisp querying context. Associated implementation issues and experiments will be presented in Chapter 5. In the following chapter, we propose to extend the FURQL query language to be able to express more sophisticated fuzzy conditions, namely fuzzy quantied statements. Introduction Fuzzyquant ied queries have been long recognized for their ability to express dierent types of imprecise and exible information needs in a relational database context. However, in the specic RDF/SPARQL setting, the current approaches from the literature that deal with quantied queries consider crisp quantiers only [START_REF] Bry | SPARQLog: SPARQL with rules and quantication[END_REF], Fan et al., 2016] over crisp RDF data. In the present chapter, we integrate fuzzy quantied statements in FURQL queries addressed to a fuzzy RDF database. We show how these statements can be dened and implemented in FURQL, which is a fuzzy extension of the SPARQL query language that we previously presented in Chapter 3. This work has been published in the proceedings of the 26th IEEE International Conference on Fuzzy Systems (Fuzz-IEEE'17), Naples, Italy, 2017. In the following, in Section 4.1 we rst present a refresher on fuzzy quantied statements in a relational database context, then, in Section 4.2 we introduce the syntactic format for expressing fuzzy quantied statements in the FURQL language and we describe their interpretation using dierent approaches from the literature. 4.1 Refresher on Fuzzy Quantied Statements In this section, we recall important notions about fuzzy quantiers, then, we present three approaches that have been proposed in the literature for interpreting fuzzy quantied statements. 4.1.1 Fuzzy Quantiers Fuzzy logic extends the notion of quantier from Boolean logic (e.g., ∃ and ∀) and makes it possible to model quantiers from the natural language such as most of, at least half, few, around a dozen, etc. In [Zadeh, 1983], the author distinguishes between absolute and relative fuzzy quantiers. Absolute quantiers refer to a number while relative ones refer to a proportion. Quantiers may also be increasing, as at least half , or decreasing, as at most three. An absolute quantier Q is represented by a function µ Q from an integer range to [0, 1] whereas a relative quantier is a mapping µ Q from [0, 1] to [0, 1]. In both cases, the value µ Q (j) is dened as the truth value of the statement Q X are A when exactly j elements from X fully satisfy A (whereas it is assumed that A is fully unsatised for the other elements). According to [Yager, 1988], fuzzy quantiers can be increasing (proportional) which means that if the criteria are all entirely satised, then the statement Q X are A is entirely true, and if the criteria are all entirely unsatised, then the statement Q X are A is entirely false. Moreover, the transition between those two extremes is continuous and monotonous. Therefore, when Q is increasing (e.g., most, at least a half ), function µ Q is increasing. Similarly, decreasing quantiers (e.g., at most two, at most a half ) are dened by decreasing functions. The characteristics of monotonous fuzzy quantiers are given in Table 4.1. Increasing quantier Decreasing quantier Calculating the truth degree of the statement Q X are A raises the problem of determining the cardinality of the set of elements from X which satisfy A. If A is a Boolean predicate, this cardinality is a precise integer (k), and then, the truth value of Q X are A is µ Q (k). If A is a fuzzy predicate, this cardinality cannot be established precisely and then, computing the quantication corresponds to establishing the value of function µ Q for an imprecise argument. µ Q (0) = 0 µ Q (0) = 1 ∃k such that µ Q (k) = 1 ∃k such that µ Q (k) = 0 ∀a, b, if a > b then µ Q (a) ≥ µ Q (b) ∀a, b, if a > b then µ Q (a) ≤ µ Q (b) Fuzzy quantied queries have been thoroughly studied in a relational database context, see e.g. [START_REF] Kacprzyk | FQUERY III +:a "human-consistent" database querying system based on fuzzy logic with linguistic quantiers[END_REF], Bosc et al., 1995] where they serve to express conditions about data values. The authors distinguished between two types of uses of fuzzy quantiers: • horizontal quantication (the quantier is used as a connective for combining atomic conditions in a where clause; this use was originally suggested in [START_REF] Kacprzyk | FQUERY III +:a "human-consistent" database querying system based on fuzzy logic with linguistic quantiers[END_REF]); • vertical quantication (the quantier appears in a having clause in order to express a condition on the cardinality of a fuzzy subset of a group, as in nd the departments where most of the employees are well-paid ). This is the type of use we make in our approach. Interpretation of Fuzzy Quantied Statements We now present dierent proposals from the literature for interpreting quantied statements of the type Q B X are A (which generalizes the case Q X are A by considering that the set to which the quantier applies is itself fuzzy) where X is a (crisp) referential and A and B are fuzzy predicates. 4. 1.2.1 Zadeh's interpretation Let X be the usual (crisp) set {x 1 , x 2 , . . ., x n } and n the cardinality of X. Zadeh [Zadeh, 1983] denes the cardinality of the set of elements of X which satisfy A, denoted by Σcount(A), as: Σcount(A) = n i=1 µ A (x i ) (4.1) The truth degree of the statement Q X are A is then given by µ(Q X are A) =        µ Q (Σcount(A)) (absolute), µ Q Σcount(A) n (relative) (4.2) One may notice, however, that a large number of elements with a small degree µ A (x) has a same eect as a small number of elements with a high degree µ A (x), due to the denition of Σcount. Example 46 Let us consider the following sets: X 1 = {0.9/x 1 , 0.9/x 2 , 0.9/x 3 , 0.8/x 4 , 0.8/x 5 , 0.7/x 6 , 0.6/x 7 }, X 2 = {1/x 1 , 1/x 2 , 0.3/x 3 , 0.2/x 4 , 0.1/x 5 , 0/x 6 , 0/x 7 }, X 3 = {1/x 1 , 1/x 2 , 1/x 3 , 1/x 4 , 1/x 5 , 0.8/x 6 , 0.3/x 7 }. and the quantier at least ve represented in Figure 4. As for quantied statements of the form Q B X are A (with Q relative), their interpretation is as follows: µ(Q B X are A) = µ Q Σcount(A ∩ B) Σcount(B) = µ Q x∈X (µ A (x), µ B (x)) x∈X µ B (x) (4.3) where denotes a triangular norm (for instance the minimum). Example 47 Let us evaluate the quantied statement Q B X are A where B={0.6/x 1 , 0.3/x 2 , 1/x 3 , 0.1/x 5 }, A={0.8/x 1 , 0.4/x 2 , 0.9/x 3 , 1/x 4 , 1/x 5 } and Q(x) = x 2 . Then, µ(Q B X are A) = µ Q ( 0.6+0.3+0.9+0+0.1 0.6+0.3+1+0+0.1 ) = µ Q ( 1.9 2 ) = µ Q (0.95) = 0.90. Yager's Competitive Type Aggregation The interpretation by decomposition described in [Yager, 1984] was originally limited to increasing quantiers. It was later generalized to all kinds of fuzzy quantiers in [Bosc et al., 1995], but hereafter, we consider the basic case where Q is increasing. The proposition Q X are A is true if an ordinary subset C of X satises the conditions c 1 and c 2 given hereafter: c 1 : there are Q elements in C, c 2 : each element x of C satises A. The truth value of the proposition: Q X are A is then dened as: µ(Q X are A) = sup C ⊆ X min(µ c 1 (C), µ c 2 (C)) (4.4) with µ c 1 (C) =        µ Q (|C|) if Q is absolute, µ Q |C| n if Q is relative (4.5) and .6) It has been shown in [Yager, 1984] that: µ c 2 (C) = inf x ∈ C µ A (x). ( 4 µ(Q X are A) = sup 1 ≤ i ≤ n min(µ Q (i), µ A (x i )). (4.7) where the elements of X are ordered in such a way that µ A (x 1 ) ≥ . . . ≥ µ A (x n ). Formula (4.7) corresponds to a Sugeno integral [Sugeno, 1974]. For quantied statements of the form QBX are A, the principle is similar. The statement is true if there exists a crisp subset C of X that satises the conditions c 1 and c 2 hereafter: c 1 : Q B X are in C, c 2 : each element x of C satises the implication (x is B) ⇒ (x is A). The truth value of the proposition: Q B X are A is then dened as: µ(Q B X are A) = sup C ⊆ X min(µ c 1 (C), µ c 2 (C)) (4.8) with µ c 1 (C) =            µ Q x∈C µ B (x) if Q is absolute, µ Q   x∈C µ B (x) x∈X µ B (x)   if Q is relative (4.9) and µ c 2 (C) = inf x ∈ C µ B (x) → µ A (x) (4.10) where → is a fuzzy implication (see e.g. [START_REF] Fodor | Fuzzy-set theoretic operators and quantiers[END_REF]). Notice that µ(Q B X are A) is undened when ∀x ∈ X, µ B (x) = 0 since this would result in a division by zero in Formula 4.9. Interpretation based on the OWA operator In [Yager, 1988], Yager considers the case of an increasing monotonous quantier and proposes an ordered weighted averaging operator (OWA) to evaluate quantications of the type Q X are A. It is shown in [Bosc et al., 1995] i) how it can be extended in order to evaluate decreasing quantications and ii) that this interpretation boils down to using a Choquet fuzzy integral. The OWA operator is dened in [Yager, 1988] as: OWA(x 1 , . . . , x n ; w 1 , . . . , w n ) = n i=1 w i × x k i (4.11) where x k i is the i th largest value among the x k 's and n i=1 w i = 1. Let n be the crisp cardinality of X. The truth value of the statement Q X are A is computed by an OWA of the n values µ A (x i ). The weights w i involved in the calculation of the OWA are given by w i =        µ Q (i) -µ Q (i -1) if Q is absolute, µ Q i n -µ Q i -1 n if Q is relative. (4.12) The aggregated value which is calculated is: OWA(µ A (x 1 ), µ A (x 2 ), . . . , µ A (x n ); w 1 , . . . , w n ) = n i=1 w i × c i (4.13) where c i is the i th largest value among the µ A (x k )'s. Example 48 Let us consider the sets X 1 , X 2 , and X 3 , and the quantier at least ve from Example 46. We have: w 1 = 0, w 2 = 0, w 3 = 1/3, w 4 = 1/3, w 5 = 1/3, w 6 = 0, w 7 = 0. We evaluate the statement at least ve elements of X 1 are A and we get the degree 0.83 (= 0.9 × 1/3 + 0.8 × 1/3 + 0.8 × 1/3). The same way, we get the degrees 0.2 for X 2 and 1 for X 3 . This interpretation corresponds to using a Choquet integral [Choquet, 1954], see also [START_REF] Murofushi | [END_REF]Sugeno, 1989, Grabisch et al., 1992]. As for statements of the form Q B X are A, Yager suggests to compute the truth degree of statements of the form Q B X are A by an OWA aggregation of the implication values µ B (x) → KD µ A (x) where → KD denotes Kleene-Dienes implication (a → KD b = max(1 -a, b)). Let X = {x 1 , . . . , x n } be such that µ B (x 1 ) ≤ µ B (x 2 ) ≤ . . . ≤ µ B (x n ) and n i=1 µ B (x i ) = d. The weights of the OWA operator are dened by: The implication values are denoted by c i and ordered decreasingly: w i = µ Q (S i ) -µ Q (S i-1 ), c 1 ≥ c 2 ≥ . . . ≥ c n . Finally: We rst order the elements of X such that µ B (x k 1 ) ≤ ... ≤ µ B (x kn ), e 1 = 0, e 2 = 0.1, e 3 = 0.3, e 4 = 0.6, e 5 = 1 and d = 2. Thus, we get S 1 = 0, S 2 = 0.05, S 3 = 0.2, S 4 = 0.5, S 5 = 1. µ(Q B X are A) = n i=1 w i × c i . µ Q (S 1 ) = 0, µ Q (S 2 ) = 0.025, µ Q (S 3 ) = 0.04, µ Q (S 4 ) = 0.25, µ Q (S 5 ) = 1. Therefore, the weights of the OWA operator are: w 1 = µ Q (S 1 ) -µ Q (S 0 ) = 0, w 2 = µ Q (S 2 ) -µ Q (S 1 ) = 0.025, w 3 = µ Q (S 3 ) -µ Q (S 2 ) = 0.04 -0.0025 = 0.0375, w 4 = µ Q (S 4 ) -µ Q (S 3 ) = 0.25 -0.04 = 0.21, w 5 = µ Q (S 5 ) -µ Q (S 4 ) = 1 -0.25 = 0.75. For each x i we calculate the implication value c i = max((1 -µ B (x i )), µ A (x i )) and these values are ordered decreasingly such that c 1 ≥ . . . ≥ c n . c 1 = max(0.4, 0.8) = 0.8, c 2 = max(0.7, 0.4) = 0.7, c 3 = max(0, 0.9) = 0.9, c 4 = max(1, 1) = 1, c 5 = max(0.9, 1) = 1. We reorder the implication values and we get c 1 = 1(c 4 ), c 2 = 1(c 5 ), c 3 = 0.9(c 3 ), c 4 = 0.8(c 1 ), c 5 = 0.7(c 2 ). Finally, the satisfaction degree using the OWA aggregation is: µ = (1) * 0 + 0.0025 * (1) + (0.375) * 0.9 + 0.21 * (0.8) + 0.75 * (0.7) = 0.73. FURQL with Fuzzy Quantied Statements In this section, we rst present some recent proposals from the literature for incorporating quantied statements into SPARQL queries, and then, we propose to integrate fuzzy quantied statements in the FURQL language. Related Work: Quantied Statements in SPARQL In an RDF database context, quantied statements have only recently attracted the attention of database community. In [START_REF] Bry | SPARQLog: SPARQL with rules and quantication[END_REF], Bry et al. propose an extension of SPARQL (called SPARQLog) with rst-order logic (FO) rules and existential and universal quantication over node variables. This query language makes it possible to express statements such as: for each lecture there is a course that practices this lecture and is attended by all students attending the lecture . This statement can be expressed in SPARQLog as follows: all ?lec ex ?crs all ?stu construct { ?crs uni:practices ?lec . ?stu uni:attends ?crs . } where { ?lec rdf:type uni:lecture . ?stu uni:attends ?lec . } More recently, in [START_REF] Fan | Adding counting quantiers to graph patterns[END_REF], Fan et al. introduced quantied graph patterns, an extension of the classical SPARQL graph patterns using simple counting quantiers on edges. Quantied graph patterns make it possible to express numeric and ratio aggregates, and negation besides existential and universal quantication. The authors also showed that quantied matching in the absence of negation does not signicantly increase the cost of query processing. However, to the best of our knowledge, there does not exist any work in the literature that deals with fuzzy quantied statements in the SPARQL query language, which is the main goal of the present chapter. Fuzzy Quantied Statements in FURQL In this subsection, we show how fuzzy quantied statements may be expressed in FURQL queries. We rst propose a syntactic format for these queries, and then we show how they can be evaluated in an ecient way. Syntax of a Fuzzy Quantied Query in FURQL In the following, we consider fuzzy quantied statements of the type Q B X are A over fuzzy RDF graph databases, where the quantier Q is represented by a fuzzy set and denotes either a relative quantier (e.g., most ) or an absolute one (e.g., at least three ), B is the fuzzy condition to be connected to a node x, X is the set of nodes in the RDF graph, and A denotes a (possibly compound) fuzzy condition. Example 50 [Fuzzy quantied statement] An example of a fuzzy quantied statements of the type Q B X are A is: most of the recent albums are highly rated. In this example, Q corresponds to the relative quantier most, B is the fuzzy condition to be recent, X corresponds to the set of albums present in the RDF graph, and A corresponds to the fuzzy conditions to be highly rated. Since the FURQL query language supports the expression of fuzzy preferences involving fuzzy structural properties (like for example, the distance and strength between two nodes over fuzzy graphs), fuzzy quantied structural queries can be expressed in the FURQL language and an example of such query is given hereafter. Example 52 [Fuzzy Quantied Structural Query in FURQL] We now consider a slightly more complex version of the above example by adding a fuzzy structural condition on the strength of the authors' recommendation: retrieve every artist (?art1) such that most of the recent albums (?alb) that he/she strongly recommends are highly rated and have been created by a young friend (?art2) of his/hers. The syntactic form of this query, denoted by R mostAlbums_ST , is given in Listing 4.3. 1 defineqrelativeasc most as (0.3,0.8) defineasc recent as (2010,2015) 2 defineasc high as (2,5) The interpretation of a fuzzy quantied statement in a FURQL query can be based on one of the formulas (4.3), ( 4.8), or (4.17). Its evaluation involves three stages : 1. the compiling of the fuzzy quantied query R into a crisp query denoted by R atBoolean , 2. the interpretation of the crisp SPARQL query R atBoolean , 3. the calculation of the result of R (which is a fuzzy set) based on the result of R atBoolean . Compiling The compiling stage translates the fuzzy quantied query R into a crisp query denoted by R atBoolean . This compilation involves two translation steps. First, R is transcripted into an intermediate query R at that allows to interpret the fuzzy quantied statement embedded in R. The query R f lat , whose general form 1 is given in List- ing 4.4, is obtained by removing the group by and having clauses from the initial query and adding the optional clause for the A part. This query aims to retrieve the elements of the B part of the initial query, matching the variables ?res and ?x, and possibly the elements of the A part of the initial query, matching the variable ?x, for which we will then need to calculate the nal satisfaction degree. select ?res ?X IB IA where { B(?res,?X) optional { A(?X) } } Listing 4.4: Derived query R at of R mostAlbums For each pair (?res, ?x), we retrieve all the information needed for the calculation of µ B and µ A , i.e., the combination of fuzzy degrees associated with relationships and node attribute values involved in B(?res,?x) and in A(?X), respectively denoted by I B and I A . Listing 4.5 of Example 53 below presents the derived query associated with the query R mostAlbums . The evaluation of R at is based on the derivation principle introduced by [START_REF] Pivert | Fuzzy Preference Queries to Relational Databases[END_REF] in the context of relational databases: R f lat is in fact derived into another query denoted by R atBoolean . The derivation translates the fuzzy query into a crisp one by transforming its fuzzy conditions into Boolean ones that select the support of the fuzzy statements. For instance, following this principle, the fuzzy condition ?year IS recent dened as defineasc recent as (2013,2016) becomes the crisp condition ?year > 2013 in order to remove the answers that necessary do not belong to the support of the answer. In the general case of a membership function having a trapezoidal form dened by a quadruple (a, b, c, d), the derivation introduces two crisp conditions ( ?var > a and ?var < d). Listing 4.6 of Example 53 below is an illustration of the derivation of the query R f lat . Crisp interpretation The previous compiling stage translates the fuzzy quantied query R embedding A fuzzy quantied statement and fuzzy conditions into a crisp query R atBoolean , whose interpretation is the classical Boolean one. For the sake of simplicity, we consider in the following that the result of R at , denoted by r at , is made of the quadruples (?res i , ?x i , µ Bi , µ Ai ) matching the query. Final result calculation The last stage of the evaluation calculates the satisfaction degrees µ B and µ A according to I B and I A . If the optional part does not match a given answer, then µ A = 0. The answers of the initial fuzzy quantied query R (involving the fuzzy quantier Q) are answers of the query R at derived from R, and the nal satisfaction degree associated with each element e can be calculated according to the three dierent interpretations mentioned earlier in Subsection 4. 1.2. Hereafter, we illustrate this using [Zadeh, 1983] and [Yager, 1988]'s approaches (which are the most commonly used for interpreting fuzzy quantied statements ). • Following Zadeh's Sigma-count-based approach (cf. Subsection 4. 1.2.1) we have: µ(e) = µ Q {(?res i ,?x i ,µ Bi ,µ Ai )∈ R at |?res i =e} min(µ Ai , µ Bi ) {(?res i ,?x i ,µ Bi ,µ Ai )∈ R at |res i =e} µ Bi (4.18) In the case of a fuzzy absolute quantied query, the nal satisfaction degree associated with each element e is simply Example 53 [Evaluation of a Fuzzy Quantied Query] Let us consider the fuzzy quantied query R mostAlbums of Listing 4.2. We evaluate this query according to the fuzzy RDF data graph G MB of Figure 4. 5. In order to interpret R mostAlbums , we rst derive the following query R at from R mostAlbums , that retrieves the artists (?art1) who recommended at least one recent album (corresponds to B(?art1,?alb) in lines 2 and 3), possibly (optional) highly rated and created by a young friend (corresponds to A(?alb) in lines 5 to 7). where µ p denotes the membership degree of the predicate p and ζ(t) denotes the membership value associated with the triple t (cf., Denition 4 on page 62). µ(e) = µ Q   {(?res i ,?x i ,µ Bi ,µ Ai )∈ R at |?res i =e} µ Ai   . For the sake of readability, the query of Listing 4.6 is a simplied version of the real derived query (cf. Listing A.1 in Appendix A). According to the fuzzy RDF data graph G MB of Figure 4.5, R at concerns three artists {JustinT, Shakira, Beyonce}. EnriqueI, Drake, Mariah and Rihanna do not belong to the result set of R at because EnriqueI, Drake and Mariah have not recommended any album made by any of their friends and Rihanna did not recommend any somewhat recent album. Then, the set of answers of the query R f lat , denoted by R f lat , is as follows: R f lat = { (?art1→ JustinT, ?alb→ One dance, µ B → 0.4, µ A → 0.3), (?art1→ JustinT, ?alb → Home, µ B → 0.1, µ A → 0.6), (?art1→ Shakira, ?alb → Euphoria, µ B → 0.1 , µ A → 0.07), (?art1→ Shakira, ?alb → Butterfly, µ B → 0.2, µ A → 0), (?art1→ Shakira, ?alb → Justified, µ B → 0.3, µ A → 0.4), (?art1→ Beyonce, ?alb → Home, µ B → 0.4, µ A → 0.3)}. Finally, assuming for the sake of simplicity that µ most (x) = x, the nal result of the query R mostAlbums evaluated on G MB using Formula 4.18 is: R mostAlbums = { ({?art1 → JustinT }, 0.80), ({?art1 → Beyonce }, 0.75), ({?art1 → Shakira }, 0.62)}. • Using Yager's OWA-based approach, for each element e returned by R at we calculate µ(e) = {(?res i ,?x i ,µ Bi ,µ Ai )∈ R at |?res i =e} w i × c i . (4.19) Let us consider condition B = {µ B 1 /x 1 , ..., µ Bn /x n } such that µ B 1 ≤ ... ≤ µ Bn , condition A = {µ A 1 /x 1 , ..., µ An /x n } and d = n i=1 µ B i . The weights of the OWA operator are dened by w i = µ Q (S x i ) -µ Q (S x i-1 ) with S x i = i j=1 µ B j d The implication values are denoted by c x i = max(1 -µ B i , µ A i ) and ordered decreasingly such that c 1 ≥ . Then, with µ most (x) = x, we get µ Q (S Euphoria ) = 0.17, µ Q (S Buttery ) = 0.5 and µ Q (S Justied ) = 1. Therefore, the weights of the OWA operator are: W 1 = µ Q (S Euphoria ) -µ Q (S 0 ) = 0.17, W 2 = µ Q (S Buttery ) -µ Q (S Euphoria ) = 0. 33, and W 3 = µ Q (S Justied ) -µ Q (S Buttery ) = 0.5. The implication values are: c Euphoria = max(1 -0.1, 0.07) = 0.9, c Buttery = max(1 -0.2, 0) = 0.8, and c Justied = max(1 -0.3, 0.36) = 0.7. Thus, c 1 = 0.9, c 2 = 0.8 and c 3 = 0.7. Finally, we get: µ(Shakira) = 0.17 × 0.9 + 0.33 × 0.8 + 0.5 × 0.7 = 0.15 + 0.26 + 0.35 = 0.77. Finally, assuming for the sake of simplicity that µ most (x) = x, the nal result of the query R mostAlbums evaluated on G MB using Formula 4.19 is: R mostAlbums = { ({?art1 → Shakira }, 0.77), ({?art1 → JustinT }, 0.66), ({?art1 → Beyonce }, 0.6) }. Conclusion In this chapter, we have investigated the issue of integrating fuzzy quantied structural queries of the type Q B X are A into the FURQL query language (a fuzzy extension of the SPARQL that we proposed in Chapter 3) aimed to query fuzzy RDF databases. We have dened the syntax and semantics of an extension of FURQL, that makes it possible to deal with such queries. A query processing strategy based on the derivation of nonquantied fuzzy queries has also been proposed using dierent interpretations from the literature previously discussed in Section 4. 1. The following chapter discusses implementation issues and presents some experiments. Introduction Chapters3and4cont ain the main contributions of the thesis which consist of the definition of the FURQL query language, which is a fuzzy extension of SPARQL with fuzzy preferences (including fuzzy quantied statements ) addressed to fuzzy RDF databases as well as crisp ones. In the present chapter, in Section 5.1, we describe a prototype implementation of FURQL built on top of a classical SPARQL engine and, then in Section 5.2, we present a performance evaluation of the prototype system using dierent sizes of fuzzy RDF databases. The main objective behind these experiments is to show that the extra cost due to the introduction of fuzziness remains limited/acceptable. rdf:subject, rdf:predicate, rdf:object and uri:degree that model respectively the type, the subject, the predicate, the object and the degree of the new statement. In order to create a fuzzy RDF database, we start from a nonfuzzy RDF subgraph database for which every relationship between nodes is Boolean and then, we make it fuzzy by adding satisfaction degrees denoting the intensity of some relationships using the reication mechanism (as illustrated in Example 55). Shakira MariahC friend Blank node Evaluation of FURQL Queries Concerning the evaluation of FURQL queries, two architectures may be thought of: • A rst solution consists in implementing a specic query evaluation engine inside the data management system. Figure 5.2 is an illustration of this architecture. The advantage of this solution is that optimization techniques implemented directly in the query engine should make the system very ecient for query processing. An important downside is that the implementation eort is substantial, but the strongest objection for this solution is that the evaluation of a FURQL query in a distributed architecture would imply having available a FURQL query evaluator at each SPARQL endpoint, which is not realistic at the time being. • An alternative more realistic architecture consists in adding a software add-on layer over a standard and possibly distant classical SPARQL engine (endpoint) which is the evaluation strategy that we adopted for processing FURQL queries. This software, called SURF 1 (Sparql with fUzzy quantieRs for rdF data), is imple- mented within the Jena Semantic Web Java Framework 2 for creating and manipulating 1. In a pre-processing step, the Query compiler module, produces the query-dependent functions that allow to compute the satisfaction degrees for each returned answer, a (crisp) SPARQL query which is then sent to the SPARQL query engine for retrieving the information needed to calculate the satisfaction degrees. The compilation uses the derivation principle introduced in [START_REF] Bosc | SQLf query functionality on top of a regular relational database management system[END_REF] in a relational database context that consists in translating a fuzzy query into a Boolean one. 2. In a post-processing step, the Score calculator module calculates the satisfaction degree for each returned answer, ranks the answers, and qualitatively lters them if an α-cut has been specied in the initial fuzzy query. SURF makes it possible to query FURQL queries (including quantied ones) as well as regular SPARQL queries. The dierent evaluation scenarios are presented hereafter. 1. For a FURQL query (that does not involve any quantied statement), the principle is simple, we rst evaluate the corresponding (crisp) SPARQL query returned by the Query compiler module (obtained using the derivation rules). For each tuple x from the result of the crisp SPARQL query, we calculate its satisfaction degree using the Score calculator module. Finally, a set of answers ranked in decreasing order of their satisfaction degree is returned. At the current time, Zadeh's approach [Zadeh, 1983] and Yager's OWA-based approach [Yager, 1988] have been implemented, and the choice of the interpretation to be used is made through the conguration tool of the system. Finally, we get a set of answers ranked in decreasing order of their satisfaction degree. The SURF GUI was created using Vaadin 3 , a web framework for Java under NetBeans IDE 8.2. It is mainly composed of two frames: • an input text area for entering and running a FURQL query, and • a table for visualizing the results of a query. Example 56 Figure 5.4 presents a screenshot of the SURF GUI, which contains the nal result of the evaluation of a FURQL query. Experimentations In order to demonstrate the performances of our approach in the case of fuzzy graph pattern queries, we ran two experiments in order to calculate the execution time of each step of the evaluation for FURQL queries with and without quantied statements and then to assess the cost of adding fuzzy preferences for each type of queries. In the following, we rst present the setup we used for the evaluation and then, we describe in detail each experiment. Experimental Setup All of the experiments were carried out on a personal computer running Windows 7 (64 bits) with 8GB of RAM. For these experiments, we used four dierent sizes of fuzzy RDF datasets containing crisp and fuzzy triples, as described in Table 5. 1. Our In the following, we rst present experiments on nonquantied FURQL queries (Section 5.2.2) and then on quantied ones (Section 5.2.3). Experiments for nonquantied FURQL Queries For this experiment, we considered dierent kinds of nonquantied FURQL queries (summarized in Table 5.2), based on the typology presented in [START_REF] Umbrich | Link traversal querying for a diverse web of data[END_REF]. Three types have been used. For each kind of queries, we consider two fuzzy subtypes: 1) a subtype for which a condition concerns a value, and 2) a subtype for which a condition concerns the intensity of the relationships. Such subtype is called Structural in the following. • Edge queries: that consist in retrieving an entity e by means of a pattern where e may appear either i) in the subject (denoted by edge-s ), ii) in the object (denoted by edge-o), or iii) both (denoted by edge-so). We consider in the following four edge queries of the form edge-so given in Figure 5. 5. Query Q 1.2 is a fuzzy edge query containing a fuzzy condition that aims to nd the recent albums recommended by an artist; Its corresponding crisp query, denoted by Q 1.1 , Q 1.3 Q 1.2 Q 1.4 Star query Q 2.1 , Q 2.3 Q 2.2 Q 2.4 Simple path query Q 3.1 , Q 3.3 Q 3.2 Q 4.4 (resp., Figure 5.8.(b)) presents the execution time in milliseconds of the processing of the edge queries (resp., star queries) from Table 5.2. Figure 5.8.(c) presents the execution time in milliseconds of the processing of the path queries from Table 5.2. The execution time is the elapsed time between submitting the query to the system and obtaining the query answers, it is measured in milliseconds using the system command time. A rst (and predictable) observation is that, for each crisp and fuzzy query presented in Table 5.2, the processing time of the overall process is proportional to the size of the dataset, the number of the results and the complexity of the query. It is straightforward to see that for all the crisp queries the query compiler and the score calculator modules do not play any role in the processing of the queries. Thus, the corresponding execution times in Figure 5.8 are equal to 0. In the case of fuzzy queries, these modules, which are directly related to the introduction of exibility into the query language, are strongly dominated in time by the crisp SPARQL evaluator (which includes the time for executing the query and getting the result set). As we can see in Figure 5.8, the time of the evaluation of the initial query by the SPARQL evaluator engine represents at least 89% of the overall process. Moreover, the FURQL compiling module takes so little time compared to the other two steps that it cannot even be seen in Figure 5.8. This time remains almost constant, and is then independent on the size of the dataset. As to the score calculation module, the time used for calculating the nal satisfaction degrees is slightly higher than the last step and is dependent on the size of the result set and the nature of the query. Comparing the pairwise queries (Q i.1 with Q i.2 and Q i.3 with Q i.4 ), we see that the processing time of the fuzzy query is slightly higher than that of its crisp version. The increase is 10% on average. Finally, the results obtained tend to show that introducing fuzziness into a SPARQL query entails a rather small increase of the overall processing time. According to our experimentations, it represents around 11% of the overall time needed for evaluating a FURQL query in the worst case. Q4 crisp complex complex crisp between two or four triple patterns. Each Q crisp contains three crisp conditions. These queries are detailed in Appendix A. In order to evaluate these queries, we used Yager's OWA-based interpretation. The results, depicted in Figure 6.8, present the execution time in milliseconds of the processing of the fuzzy quantied queries involving crisp conditions from Table 5.3 over the RDF datasets from Table 5.1 on page 103. Fuzzy quantied query involving fuzzy conditions We processed again four fuzzy quantied queries with fuzzy conditions (of the type Q B X are A) by changing each time the nature of the patterns corresponding to conditions B and A from simple to complex ones. Table 5.4 presents these queries. A complex pattern diers from a simple one by the number and the nature (including structural properties) of its statements. During these experiments, a complex pattern is composed of nine triple patterns at most, while a simple pattern contains between two and four triple patterns. For each complex pattern a fuzzy structural property (e.g., involving the notions of strength or distance) is involved. Each Q fuzzy contains three fuzzy conditions. These queries are detailed in Appendix A. The results of these experiments, using Yager's OWA-based interpretation, are depicted in Figure 5.10 that presents the execution time in milliseconds of the processing of the fuzzy quantied queries from Table 5.4 over the RDF datasets from Table 5.1 on page 103. Results interpretation A rst and obvious observation from Figure 5.9 and Figure 5.10 is that, for all the fuzzy quantied queries, the processing time taken by the overall process is proportional to the size of the dataset and the complexity of the pattern in the query. One can see that, the processing time taken by the compiling and the score calculation module, which are directly related to the introduction of exibility into the query language, are very strongly dominated by the time taken by the SPARQL evaluator (which includes the time for executing the query and getting the result set). As it is shown in Figure 5.9 and Figure 5.10, the time of the evaluation of the initial query by the SPARQL evaluator engine represents 99% on average of the overall process. Indeed, the FURQL compiling step takes so little time compared to the score calculation and the SPARQL evaluator modules that it cannot even be seen in Figure 5.9 and Figure 5.10. This time remains almost constant, and is independent on the size of the dataset while slightly increasing in the presence of complex patterns or fuzzy conditions. Moreover, the time needed for calculating the nal satisfaction degree in the score calculator module is relatively dependent on the size of the result set and the nature of the patterns. Again, these experimental results, even though preliminary, appear promising. They tend to show that introducing fuzzy quantied statements into a SPARQL query does not come with a high price (i.e., entails a very small increase of the overall processing time). Finally, this conclusion can be extended to the case of Zadeh's interpretation [Zadeh, 1983], inasmuch as it is even more straightforward, in terms of computation, than Yager's OWAbased approach [Yager, 1988]. Thus, the processing time of the score calculating step can only be smaller than in the case of Yager's OWA-based interpretation. Conclusion In this chapter, in Section 5.1, we discussed implementation issues related to the FURQL language and we presented an architecture which consists of a software add-on layer (called SURF) over the classical SPARQL engine. Then, in Section 5.2, we performed two set of experiments over dierent sizes of datasets in order to study the performances of our proposed approach. The rst experiments aimed to measure the additional cost induced by the introduction of fuzziness into SPARQL, and the results obtained show the eciency of our proposal. The second experiments, which concerned fuzzy quantied queries, show that the extra cost induced by the fuzzy quantied nature of the queries remains very limited, even in the case of rather complex fuzzy quantied queries. The results of the experiments performed in this chapter are summarized in Table 5. 5. Each cell of the table contains three values corresponding to the percentage of time devoted to the compilation, the crisp evaluation and the score calculation stages respectively. They show that in both experiments the compilation and the score calculation stages are strongly dominated by the crisp SPARQL evaluation. The latter represents at least 95% of the overall process. Thus, these results conrm the hypothesis that the extra cost due to the introduction of fuzziness remains limited/acceptable. Finally, these experiments are preliminary and more work is required to further assess FURQL by using dierent variety of queries (e.g., complex path queries of undetermined length) and considering large databases. Introduction Int he previous chapters, we addressed mainly the issue of dening an ecient approach for exible querying in a particular type of graph databases, namely RDF databases. This approach makes it possible to express fuzzy nonquantied and quantied queries into an extension of the SPARQL language. In the present chapter, we place ourselves in a more general framework: graph database [START_REF] Angles | Survey of graph database models[END_REF]]. An ecient approach for exible querying of fuzzy 6.1. Background Notions graph databases has been proposed in [START_REF] Pivert | On a fuzzy algebra for querying graph databases[END_REF]. This approach makes it possible to express only fuzzy nonquantied conditions. However, fuzzy quantied queries have a high potential in this setting since they can exploit the structure of the graph, beside the attribute values attached to the nodes or edges. So far, only one approach from the literature, described in [START_REF] Castelltort | Fuzzy queries over NoSQL graph databases: Perspectives for extending the Cypher language[END_REF], considered fuzzy quantied queries to graph databases but only in a rather limited way. This chapter is based on our work reported in [Pivert et al., 2016e], in which we showed how it is possible to integrate fuzzy quantied queries in a framework named FUDGE that was previously dened in [Pivert et al., 2014a]. FUDGE is a fuzzy extension of Cypher [START_REF] Cypher | Cypher[END_REF] which is a declarative language for querying (crisp) graph databases. This work is mostly related to the work presented in Chapter 4 in which we deal with the same type of fuzzy quantied structural query but in a more specic type of graph databases, called RDF database. The remainder of this chapter is organized as follows. Section 6.1 presents the dierent elements that constitute the context of the work. Section 6.2 discusses related work. In Section 6.3, we propose a syntactic format for expressing fuzzy quantied queries in the FUDGE language, and we describe its interpretation. Section 6.4 deals with query processing and discusses implementation issues. In Section 6.5, some experimental results showing the feasibility of the approach are presented. Background Notions In this section, we recall important notions about graph databases, fuzzy graph theory, fuzzy graph databases, and the query language FUDGE. 6.1.1 Graph Databases In the last few years, graph databases has attracted a lot of attention for their ability to handle complex data in many application domains, e.g., social networks, cartographic databases, bibliographic databases, etc [Angles andGutierrez, 2008, Angles, 2012]. They aim to eciently manage networks of entities where each node is described by a set of characteristics (for instance a set of attributes), and each edge represents a link between entities. A graph database management system enables managing data for which the data structure of the schema is modeled as a graph and data is handled through graph-oriented operations and type constructors [START_REF] Angles | Survey of graph database models[END_REF] [START_REF] Angles | Survey of graph database models[END_REF] for an overview), including the attributed graph (aka., property graph) aimed to model a network of entities with embedded data. In this model, nodes and edges can be described by data in attributes (aka., properties). Example 57 Figure 6.1 is an example of an attributed graph, inspired from DBLP 1 with crisp edges. Nodes are assumed to be typed. If n is a node of V , then T ype(n) denotes its type. In Figure 6.1, the nodes IJIS16 and IJIS10 are of type journal, the nodes IJIS16-p, IJIS10-p and IJIS10-p1 are of type paper, and the nodes Maria, Claudio and Susan are of type author. For nodes of type journal, paper and author, a property, called name, contains the identier of the node. Information about the title and the pages may be attached to node of type paper and information about the volume and the date may be attached to node of type journal. In Figure 6.1, the value of the property name for a node appears inside the node. Such a model may be extended into the notion of a fuzzy graph database where a degree may be attached to edges in order to express the intensity of any kind of gradual relationship (e.g., likes, is friends with, is about). In the following section, we introduce the notion of fuzzy graphs. Fuzzy Graphs A graph G is a pair (V, R), where V is a set and R is a relation on V . The elements of V (resp. R) correspond to the vertices (resp. edges) of the graph. Similarly, any fuzzy relation ρ on 6.1. Background Notions a set V can be regarded as dening a weighted graph, or fuzzy graph, see [Rosenfeld, 1975], where the edge (x, y) ∈ V × V has weight or strength ρ(x, y) ∈ [0, 1]. Having no edge between x and y is equivalent to ρ(x, y) = 0. A fuzzy data graph may contain both fuzzy edges and crisp edges as a fuzzy edge with a degree of 0 or 1 can be considered as crisp. Along the same line, a crisp data graph is simply a special case of fuzzy data graph (where ρ : V × V → {0, 1} is Boolean). We then only deal with fuzzy edges and data graphs in the following. An important operation on fuzzy relations is composition. Assume ρ 1 and ρ 2 are two fuzzy relations on V . Thus, composition ρ = ρ 1 • ρ 2 is also a fuzzy relation on V s.t. ρ(x, z) = max y min(ρ 1 (x, y), ρ 2 (y, z)). The composition operation can be shown to be associative : (ρ 1 • ρ 2 ) • ρ 3 = ρ 1 • (ρ 2 • ρ 3 ). The associativity property allows us to use the notation ρ k = ρ • ρ • . . . • ρ for the composition of ρ with itself k -1 times. In addition, following [Yager, 2013], we dene ρ 0 to be s. t. ρ 0 (x, y) = 0, ∀(x, y). Useful notions related to fuzzy graphs are those of strength and length of a path. These notions were previously used in Chapter 3 in the RDF context, Their denition, drawn from [Rosenfeld, 1975], is recalled hereafter. In other words, the strength of a path is dened to be the weight of the weakest edge of the path. Two nodes for which there exists a path p with ST (p) > 0 between them are called connected. We call p a cycle if n ≥ 2 and x 0 = x n . It is possible to show that ρ k (x, y) is the strength of the strongest path from x to y containing at most k links. Thus, the strength of the strongest path joining any two vertices x and y (using any number of links) may be denoted by ρ ∞ (x, y). Strength of a path. A path p in G is a sequence x 0 → x 1 → . . . → x n (n ≥ 0) such that ρ(x i-1 , x i ) > 0, 1 ≤ i ≤ n Length and distance. The length of a path p = x 0 → x 1 → . . . → x n in the sense of ρ is dened as follows: Length(p). Length(p) = n i=1 1 ρ(x i-1 , x i ) . ( 6 ( .3) It is the length of the shortest path from x to y. DB subgraph where variables can occur. An answer maps the variables to elements of DB. A fuzzy graph pattern expressed à la Cypher consists of a set of expressions (n1:Type1)-[exp]->(n2:Type2) or (n1:Type1)-[e:label]->(n2:Type2) where n1 and n2 are node variables, e is an edge variable, label is a label of E, exp is a fuzzy regular expression, and Type1 and Type2 are node types. Such an expression denotes a path satisfying a fuzzy regular expression exp (that is simple in the second form e) going from a node of type Type1 to a node of type Type2. All its arguments are optional, so the simplest form of an expression is ()-[]->() denoting a path made of two nodes connected by any edge. Conditions on attributes are expressed on nodes and edges variables in a where clause. Example 59 [Graph pattern] We denote by P the graph pattern: ing that its support is [A-a, B+b] and its core [A, B] , then the clause has the form define fterm as (A-a,A,B,B+b). If fterm is a decreasing function, then the clause has the form definedesc fterm as (δ,γ) meaning that the support of the term is [0, γ] and its core [0, δ] (there is the corresponding defineasc clause for increasing functions). 2. A match clause, which has the form match pattern where conditions that expresses the fuzzy graph pattern. Example 60 [FUDGE query] Listing 6.2 is an example of a FUDGE query. This pattern aims to retrieve the authors (au2) who have, among their close contributors (connected by a short path Length is short made of contributor edges), an author (au1) who published a paper (ar1) in IJWS12 and also published a paper (ar2) in a journal (j2) which has a high impact factor (i.value is high). The fuzzy terms short and high are dened on line 1. Figure 6.4 is a graphical representation of this pattern where the dashed edge denotes a path and information in italics denotes a node type or an additional condition on node or edge attributes. Related Work In the last decades, fuzzy quantied queries have proved useful in a relational database context for expressing dierent types of imprecise information needs [Bosc et al., 1995]. Recently, in a graph database context, such statements started to attract increasing attention of many researchers [Yager, 2013, Castelltort and Laurent, 2014, Castelltort and Laurent, 2015] since they can exploit the structure of the graph, beside the attribute values attached to the nodes or edges. In [Yager, 2013], R.R. Yager briey mentions the possibility of using fuzzy quantied queries in a social network database context, such as the question of whether most of the people residing in western countries have strong connections with each other and suggests In [START_REF] Castelltort | Extracting fuzzy summaries from NoSQL graph databases[END_REF], the same authors propose an approach aimed to summarize a (crisp) graph database by means of fuzzy quantied statements of the form Q X are A, in the same spirit as what [START_REF] Rasmussen | Summary SQL -A fuzzy tool for data mining[END_REF]] did for relational databases. Again, they consider that the degree of truth of such a statement is obtained by a sigma-count (according to Zadeh's interpretation) and show how the corresponding queries can be expressed in Cypher. More precisely, given a graph database G and a summary S = a[r ]>b, Q, the authors consider two degrees of truth of S in G dened as follows: truth 1 (S) = µ Q ( count(distinct S) count(distinct a) ) (6.4) truth 2 (S) = µ Q ( count(distinct S) count(distinct a[r ]>(?)) ) (6.5) They illustrate these notions using a database representing students who rent or own a house or an apartment. The degree of truth (in the sense of the second formula above) of the summary S = student [rent ]>apartment, most meaning most of the students rent an apartment (as opposed to a house) is given by the membership degree to the fuzzy quantier most of the ratio: (number of times a relationship of type rents appears between a student and an apartment) over (number of relations of type rents starting from a student node). The corresponding Cypher query is: A limitation of this approach is that only the quantier is fuzzy (whereas in general, in a fuzzy quantied statement of the form Q B X are A, the predicates A and B may be fuzzy too). The work the most related to that presented here is [START_REF] Pivert | Fuzzy quantied queries to fuzzy RDF databases[END_REF] described in Chapter 4, where we introduced the notion of fuzzy quantied statements in a (fuzzy) RDF database context. We showed how this statement could be expressed in the FURQL language (which is a exible extension of the SPARQL query language) that we previously proposed in [Pivert et al., 2016c]. Fuzzy Quantied Statements in FUDGE In this section, we show how a specic type of fuzzy quantied statements may be expressed in the FUDGE query language. We rst propose a syntactic format for these queries, then we show how they can be eciently evaluated. 6.3.1 Syntax of a Fuzzy Quantied Query In the following, we consider fuzzy quantied queries involving fuzzy predicates (beside the quantier) over fuzzy graph databases. The fuzzy quantied statements considered are of the same type as those used in Chapter 4 in the context of RDF databases. They are of the form Q B X are A, where the quantier Q is represented by a fuzzy set and denotes either an increasing/decreasing relative quantier (e.g., most ) or an increasing/decreasing absolute one (e.g., at least three ), where B is the fuzzy condition to be connected (according to a given pattern) to a node x, X is the set of nodes in the graph, and A is the fuzzy (possibly compound) condition. An example of such a statement is: most of the recent papers of which x is a main author, have been published in a renowned database journal. The general syntactic form of a fuzzy quantied query of the form Q B X are A in the FUDGE language is given in Listing 6. Example 61 [Fuzzy Quantied Query] The query, denoted by Q mostAuthors , that consists in retrieving every author (a) such that most of the recent papers (p) of which he/she is a main author, have been published in a renowned database journal (j) may be expressed in FUDGE as follows: 2. the interpretation of the crisp query Q derivedBoolean , 3. the calculation of the answers to Q based on the answers to Q derivedBoolean . Compiling The compiling stage translates the fuzzy quantied query Q into a crisp query denoted by Q derivedBoolean . This compilation involves two translation steps. First, Q is transcripted into a derived query Q derived whose aim is to retrieve the elements necessary to the interpretation of the fuzzy quantied statement from Q. The query Q derived , whose general form 3 is given in Listing 6.5, makes it possible to get the elements of the B part of the initial query, matching the variables res and x, for which we will then need to calculate the nal satisfaction degree. It is obtained by removing the with and having clauses from the initial query, and adding the optional match clause before the fuzzy graph pattern in condition A. The processing of Q derived is based on the derivation principle introduced by [START_REF] Pivert | Fuzzy Preference Queries to Relational Databases[END_REF] in the context of relational databases: Q derived is in fact derived into another query denoted by Q derivedBoolean . The derivation step translates the fuzzy query into a crisp one by transforming its fuzzy conditions into Boolean ones that select the support of the fuzzy statements. For instance, following this principle, the fuzzy condition p.year IS recent (where recent is dened as defineasc recent as (2013,2016)) becomes the crisp condition p.year > 2013 in order to remove the answers that do not belong to the support of the answer. Listing 6.7 of Example 62 below is an illustration of the derivation of the query Q derived . The derivation principle applied to the FUDGE language is detailed in [Pivert et al., 2015]. Crisp interpretation The previous compiling stage translates the fuzzy quantied query Q embedding fuzzy quantiers and fuzzy conditions into a crisp query Q derivedBoolean , that can be processed by a classical graph DBMS (e.g., Neo4j). For the sake of simplicity, we consider in the following that the result of Q derived , denoted by Q derived , is made of the quadruples (res i , x i , µ Bi , µ Ai ) matching the query. Final result calculation The last stage of the evaluation calculates the satisfaction degrees µ B and µ A according to I B and I A . If the optional part does not match a given answer, then µ A = 0. The answers of the initial fuzzy quantied query Q (involving the fuzzy quantier Q) are answers of the query Q derived derived from Q, and the nal satisfaction degree associated with each element e can be calculated according to the three dierent interpretations mentioned earlier in Section 4.1. Hereafter, we illustrate this using [Zadeh, 1983] and [Yager, 1988]'s approaches (which are the most commonly used when it comes to interpreting fuzzy quantied statements ). Following Zadeh's Sigma-count-based approach (cf. Subsection 4. 1.2.1) we have: µ(e) = µ Q {(res i ,x i ,µ Bi ,µ Ai )∈ Q derived |res i =e} min(µ Ai , µ Bi ) {(res i ,x i ,µ Bi ,µ Ai )∈ Q derived |res i =e} µ Bi (6.6) In the case of a fuzzy absolute quantied query, the nal satisfaction degree associated with each element e is simply µ(e) = µ Q   {(res i ,x i ,µ Bi ,µ Ai )∈ Q derived |res i =e} µ Ai   . Example 62 [Evaluation of a Fuzzy Quantied Query] Let us consider the fuzzy quantied query Q mostAuthors of Listing 6.4. We evaluate this query according to the fuzzy data graph DB of Figure 6. Finally, assuming for the sake of simplicity that µ most (x) = x, the nal result of the query Q mostAuthors evaluated on DB using Formula 6.6 is Using Yager's OWA-based approach (cf. subsection 4. 1.2.2), for each element e returned by Q derived we calculate µ(e) = {(res i ,x i ,µ Bi ,µ Ai )∈ Q derived |res i =e} w i × c i . The weights of the OWA operator are dened by Q mostAuthors = { µ(Peter) = µ most ( 0. w i = µ Q (S x i ) -µ Q (S x i-1 ) with S x i = i j=1 µ B j d . The implication values are denoted by c x i = max(1 -µ B i , µ A i ) and ordered decreasingly such that c 1 ≥ . . . ≥ c n . Example 63 In order to calculate µ(Maria) from Q derived , let us consider B (resp. A) the set of satisfaction degrees corresponding to condition B (resp. A) of element Maria as follows B={0.33/IJAR14, 0.6/IJIS16} and A={1/IJAR14, 6.4. About Query Processing The compilation uses the derivation principle introduced in [START_REF] Bosc | SQLf query functionality on top of a regular relational database management system[END_REF] in the context of relational databases. 2. In a post-processing step, the Score calculator module performs a grouping (according to the with clause of the initial query) of the elements, then calculates µ B , µ A and µ for each returned answer, and nally ranks the answers. For quantied queries of the type introduced in the previous sections (i.e., using relative quantiers), the principle is to rst evaluate the fuzzy query Q derivedBoolean derived from the original query. For each element x ∈ Q derivedBoolean , we return the satisfaction degrees related to conditions A and B, denoted respectively by µ A and µ B . The nal satisfaction degree µ can be calculated according to Formulas (4.3), (4.8) or (4.17) (presented in Chapter 4 Subsection 4. 1.2) using the values of µ B and µ A . At the current time, [Zadeh, 1983]'s approach and [Yager, 1988]'s OWA-based approach have been implemented, and the choice of the interpretation to be used is made through the system conguration tool. Finally, a set of answers ranked in decreasing order of their satisfaction degree is returned. As a proof-of-concept of the proposed approach, the FUDGE prototype is available at www-shaman.irisa.fr/fudge-prototype. A screenshot of this prototype is shown in Figure 6.7 which contains the nal result of the evaluation of the query Q mostAuthors of Example 61. The GUI is composed of two frames: • a central frame for visualizing the graph and the results of a query, and • an input eld frame (placed under the central one), for entering and running a FUDGE query. Figure 6.7: Screenshot of the FUDGE prototype 6.5 Experimental Results In order to conrm the eectiveness and eciency of the approach, we carried out some experiments on a computer running on Windows 7 (64 bits) with 8 Gb of RAM. The queries used in these experiments are based on the typology of [Angles, 2012] that considers three categories of queries: • Adjacency query: tests whether two nodes are adjacent (or neighbors) when there exists an edge between them or whether two edges are adjacent when they have a common node. • Reachability query: tests whether two given nodes are connected by a path. Two types of paths are considered: xed length paths, which contain a xed number of nodes and • The fourth query Q 4 (Listing 6.11), where A is a pattern matching, aims to nd the authors (a) such that most of the recent papers (p) of which they are main authors, have been published in a renowned database journal (j). The results of the processing of these queries over the RDF datasets from Table 6.1 are depicted in Figure 6.8 where Figure 6. The main result is that the processing time taken by the compiling and the score calculation stages, which are related to the introduction of exibility into the query language, are very strongly dominated by the time taken by the crisp Cypher evaluator. Moreover, the FUDGE compiling stage takes so little time compared to the other two stages that it cannot even be seen in Figure 6. Conclusion In this chapter, we have dealt with fuzzy quantied structural queries, addressed to fuzzy graph databases. We have rst dened the syntax and semantics of a fuzzy extension of the query language Cypher. This extension makes it possible to express and interpret such queries with dierent approaches from the literature. A query processing strategy based on the derivation of nonquantied fuzzy queries has also been proposed. Then, we updated the software SURF described in [Pivert et al., 2015[START_REF] Pivert | SUGAR: A graph database fuzzy querying system[END_REF] to be able to express such queries and performed some experiments using dierent sizes of fuzzy graphs in order to study its performances. The results of these experiments show that the cost of dealing with fuzzy quantication in a query is reasonable w.r.t. the cost of the overall evaluation. Conclusion The last decade has witnessed an increasing interest in expressing preferences inside database queries for their ability to provide the user with the best answers, according to his/her information need. Even though most of the work in this area has been devoted to relational databases, several proposals have also been made in the Semantic Web area in order to query RDF databases in a exible way. However, it appears that these approaches are mainly straightforward adaptations of proposals made in the relational database context: they make it possible to express preferences on the values of the nodes, but not on the structure of the RDF graph. Structural preferences are quite important in a graph database context and may concern the strength of a path, the distance between two nodes, etc. Moreover, these approaches consist of exible extensions of the SPARQL query language that only deal with crisp RDF data. In the real world, though, semantic Web data often carry gradual notions such as friendship in social networks, aboutness in a bibliographic context, etc. Such notions can be modeled by fuzzy sets, which leads to attaching a degree in [0, 1] to the edges of the graph. Motivated by these concerns, we addressed in this thesis the issue of ecient querying of (fuzzy) RDF data with the aim of extending the SPARQL query language so as to be able to express i) fuzzy preferences on data (e.g., the release year of a movie is recent ) and on the structure of the data graph (e.g., the path between two friends is required to be short ). and ii) fuzzy quantied preferences (e.g., most of the albums that are recommended by an artist, are highly rated and have been created by a young friend of this artist). To the best of our knowledge, this thesis is the rst attempt in this direction in which we provide solutions for these dierent issues. After motivating our work, we presented in Chapter 1 basic notions related to our thesis, namely the RDF data model, the SPARQL query language and fuzzy set theory. In Chapter 2 we provided an overview of the main proposals made in the literature that propose a exible extension of SPARQL based on user preferences queries, relaxation techniques and approximate matching. We discussed these approaches, classied them and pointed out their limits. Chapter 3 was dedicated to the denition of a fuzzy extension of SPARQL that goes beyond the previous proposals in terms of expressiveness inasmuch as it makes it possible i) to deal with both crisp and fuzzy RDF databases, and ii) to express fuzzy structural conditions beside more classical fuzzy conditions on the values of the nodes present in the RDF graph. The language, called FURQL, is based on the notion of fuzzy graph pattern which extends Boolean graph patterns introduced by several authors in a crisp querying context. Then, in Chapter 4 we proposed to integrate more complex conditions, namely, fuzzy quantied statements of the type Q B X are A into the FURQL language (addressed to fuzzy RDF database) previously introduced in Chapter 3. We dened the syntax and semantics of an extension of the FURQL query language, that makes it possible to deal with such queries. A query processing strategy based on the derivation of nonquantied fuzzy queries has also been proposed. These functionalities were successfully implemented using a prototype called SURF. Experimental results, described in Chapter 5, show the validity of the approach. In the case of fuzzy nonquantied queries, the results obtained indicate that introducing fuzziness into a SPARQL query comes with a very limited cost. And in the case of fuzzy quantied queries, the results show that the extra cost induced by the fuzzy nature of the queries remains also very limited, even in the case of rather complex fuzzy quantied queries. Finally, the last chapter was devoted to integrating fuzzy quantied queries in an extension of the Neo4j Cypher language, called FUDGE, (described in [START_REF] Pivert | On a fuzzy algebra for querying graph databases[END_REF][START_REF] Pivert | SUGAR: A graph database fuzzy querying system[END_REF]) in a more general (fuzzy) graph database context (of which fuzzy RDF databases are a special case). We rst proposed a syntactic format for expressing these queries in the FUDGE language, and we described their interpretation using dierent approaches from the literature. Then, we carried out some experimentations in order to assess the performances of the evaluation method. The results of these experiments show that the cost of dealing with fuzzy quantication in a query is very reasonable w.r.t. the cost of the overall evaluation. Future Work In this thesis, we have proposed a fuzzy extension of the SPARQL query language that makes it possible to express fuzzy structural conditions and fuzzy quantied statements in an ecient way. This work serves as a baseline and leaves some open questions to solve and sets the basis for further extensions. Dierent perspectives on short-term and long-term work have been identied and are outlined hereafter. Extend the FURQL and FUDGE languages with more sophisticated preferences In this thesis, we limited fuzzy structural properties to the distance and the strength where the distance between two nodes is the length of the shortest path between these two nodes and the strength of a path is dened to be the weight of the weakest edge of the path. It is also worth to consider other structural properties, like: • the centrality, the prestige and the inuence used in social networks analysis [START_REF] Rusinowska | Social networks: prestige, centrality, and inuence[END_REF]. For instance, the degree of centrality of a node measures the extent to which this node is connected with other nodes in a given social network. The question to answer is how central this node is in this network. The degree of prestige measures the extent to which a social actor in a network receives or serves as the object of relations sent by others in the network. Persons, who are chosen as friends by many others have a special position (prestige) in the group. • the clique which is one of the basic concepts of classical graph theory. Ronald R. Yager in [Yager, 2014] redened this notion in the case of a fuzzy graph. Moreover, we introduced a specic type of fuzzy quantied queries of the form: Q nodes, among those that are connected to a node x according to a certain pattern, satisfy a fuzzy condition c. An example of such a statement is most of the papers whose x is a main author, have been published in a renowned database journal. It would be interesting to study other types of fuzzy quantied queries, in particular, those that aim to nd the nodes x such that x is connected (by a path) to Q nodes reachable by a given pattern and satisfying a given condition c. An example of such a query is nd the authors x that had a paper published in most of the renowned database journals. And also those that aim to nd if there exists a path from x to a node satisfying c such that this path contains Q nodes (where Q is an absolute quantier). Make SURF and SUGAR more user-friendly The softwares that we developed make it possible to express fuzzy user preferences where the query is explicitly written in the syntax of the formal query language (FURQL for RDF data and FUDGE for graph data) and the fuzzy terms are dened in the query by a predened clause define. These softwares may be improved further in order to make them more user-oriented. One can think rst about proposing a way to help non-expert users dene fuzzy terms easily. There is, therefore, a denite need about developing a user interface in order to help casual users dene their preferences and the underlying fuzzy membership functions in a more easy way, following the work of [START_REF] Smits | ReqFlex: Fuzzy Queries for Everyone[END_REF] in which the authors described ReqFlex, an intuitive user interface to the denition of preferences and the construction of fuzzy queries in a relational context. Moreover, we may think also about integrating and analysing user proles in order to focus more on the user's interest and preferences and take into account the user's context in order to personalise the retrieved information. Add Quality-Related Metadata Another important perspective concerns the management of quality-related metadata [START_REF] Fürber | Using semantic web resources for data quality management[END_REF]. Since the RDF model that we used in this thesis makes it possible to model fuzzy notions, we can extend this model to represent data quality dimensions (e.g., accuracy, completeness, timeliness, consistency and so on). Indeed, these dimensions are of fuzzy nature and the values returned by the associated metrics may be viewed as satisfaction degrees. Then, it would be also worth investigating the way our framework FURQL could be extended: 1. to express fuzzy preferences queries concerning some quality dimensions. 2. to associate quality information with the answers to a query. This would make it possible to rank-order the answers according to their quality level (on one or several dimensions) and to warn the user about the presence of suspect answers, for instance. [START_REF] Bizer | The Berlin SPARQL Benchmark[END_REF], DBpedia SPARQL Benchmark (DBPSB) [START_REF] Morsey | DBpedia SPARQL BenchmarkPerformance Assessment with Real Queries on Real Data[END_REF], etc.) for data generator and benchmark queries in order to evaluate the performance of RDF stores. However, none of the existing benchmarks provides fuzzy RDF data or explicitly deals with fuzzy user preferences. For that purpose, in this thesis, we initially performed the evaluation of our approaches using a fuzzy RDF database inspired by Musicbrainz 4 with synthetic data generated by a script that allowed us to create datasets of dierent sizes. A future work would be to consider further evaluation using some of the existing real-world data benchmarks or ideally create our own Fuzzy RDF Benchmark. Obviously, many research problems remain open and this thesis is only a rst step which will help, we hope, to convince the databases community of the interest of using fuzzy logic for the exible/intelligent management of data in information systems. Example 2 [ 2 RDF graph] Let us consider an example of an RDF subgraph extracted from the MusicBrainz database 1 which is an open music encyclopedia that collects music metadata. The resource uri:lemonade is an album, entitled Lemonade. It was released in 2016, with genre R&B and rating 8.7. It was created by the resource uri:beyonce, named Beyonce, being 38 years old and a rating of 7. The resource uri:sorry and the resource uri:holdup entitled hold up are tracks of the latter resource. The resource uri:beyonce also created the resource uri:B'Day, entitled B'Day, that was released in 2006. Figure 1 . 1 Figure 1 . 1 : 1111 Figure 1.1: Sample RDF graph extracted from MusicBrainz In fact, RDF data may be represented by dierent syntaxes such as, RDF/XML (eXtensible Markup Language) 2 , N-Triples 3 , Notation 3 or N3 4 and Turtle (Terse RDF Triple Language) 5 , etc. Example 3 [RDF representations] Listing 1.1 is the RDF/XML representation corresponding to the resource uri:Lemonade from the RDF graph of Figure 1.1. rdf:resource=mo:album </rdf:type> 11 <dc:track> rdf:resource="uri:sorry" </dc:track> 12 <dc:track> rdf:resource="uri:hold up" </dc:track> 13 </rdf:Description> 14 </rdf:RDF> Listing 1.1: RDF/XML document In this listing, line 1 indicates an XML declaration and line 2 says that the following XML document is about RDF. Lines 2-4 declare namespaces which indicate the URI that will be used later. Lines 5-14, as the tag is closed in line 14, present the description of a resource in which lines 6-12 describe characteristics of this resource. Its corresponding N-Triples representation is given in Listing 1.2. Listing 1 . 3 : 13 A SPARQL Basic Graph PatternA graphical representation of this graph pattern is depicted in Figure1.2. Figure 1 . 2 : 12 Figure 1.2: A graphical representation of the graph pattern from Listing 1.3 Figure 1 . 3 : 13 Figure 1.3: Possible subgraphs from Figure 1.1 . 4. The set of t-norms (resp. t-conorms ) has an upper (resp. lower) element which is the minimum (resp. maximum) operator. Example 22 Let us come back to Example 19. The intersection of the two fuzzy subsets, taking = min, is as follows: Most fuzzy terms are assumed to be represented by a trapezoidal membership function (see for instance a possible representation of recent in Figure2.1). Figure 2 . 1 : 21 Figure 2.1: Membership function of recent Figure 2 . 2 : 22 Figure 2.2: Membership function of the fuzzy number at least Y Listing 2 . 1 : 21 An f-SPARQL query This query aims to retrieve from a music database the albums by Beyonce that have been recently released. If the MusicBrainz RDF database of Figure 1.1 on page 23 is queried, then the album entitled Lemonade belongs to the answer, with a satisfaction degree of 0.66, which corresponds to the degree of membership of value 2016 to the fuzzy term recent (see Figure 2.1). The other album from Figure 1.1, released in 2006, does not belong to the answer as it is not at all recent according to Figure 1.1. Figure 2 . 2 Figure2.3, where J, P and S are binary variables corresponding to the colors of the jacket, the pants and the shirt respectively. b ∧ b : r w w ∧ b : w r b ∧ w : w r w ∧ w : r w Figure 2 . 3 : 23 Figure 2.3: CP-net of Example 34 Figure 2 . 4 : 24 Figure 2.4: An RDFS Ontology [ or vague. They propose a conceptual framework to relax RDF queries relying on a matcher function (i.e., distance function) that assigns a relaxation score in [0,1] to a pair of values. tated in the previous chapter, RDF is a graph-based standard data model for representing semantic web information, and SPARQL is a standard query language for querying RDF data. Because of the huge volume of linked open data published on the web, these standards have aroused a large interest in the last years. extension allows(1) to query both crisp and fuzzy RDF data model, and (2) to express fuzzy preferences on values present in the graph as well as on the structure of the data graph, which has not been proposed in any previous fuzzy extension ofSPARQL. 62 3 . 1 . 31 Fuzzy RDF (F-RDF) GraphThis work has been published in the proceedings of the 25th IEEE International Conference on Fuzzy Systems (Fuzz-IEEE'16),Vancouver, Canada, 2016. Figure 3 . 1 : 31 Figure 3.1: Fuzzy RDF graph G M B inspired by MusicBrainz 41 [Distance and strength between two nodes] Let us consider the cycle-free paths from G M B connecting Beyonce to Euphoria, depicted in Figure 3.2, and let us compute the distance and the strength between the pair of nodes (Beyonce, Euphoria). The distance between the pair of nodes (Beyonce, Euphoria) is calculated as follows distance(Beyonce, Euphoria) = min (length(p 1 ), length(p 2 ), length(p 3 )), with length (p 1 )= 1/ζ (Beyonce, recommends, Euphoria )= 1/0 Figure 3 . 3 : 33 Figure 3.3: A possible representation of the fuzzy term short Figure 6 Figure 3 . 4 : 634 Figure 6.3 is a graphical representation. 2 .Figure 3 . 5 : 235 Figure 3.5: Representation of the fuzzy term low applied to a rating value Figure 3 . 5 , 35 Figure 3.5, and the following clause denes the fuzzy term short of Figure 3.3. Figure 3 . 6 : 36 Figure 3.6: Some paths from G M B . .6 , satisfy f 1 with the following satisfaction degrees:sat f 1 (EnriqueI, Justied) = min(ζ(EnriqueI, friend, JustinT), ζ(JustinT, creator, Justied)) = min(0.4, 1) = 0.4, sat f 1 (Shakira, Buttery) = min(ζ(Shakira, friend, MariahC), ζ(MariahC, creator, Buttery)) = min(0.7, 1) = 0.7, sat f 1 (Beyonce, Euphoria) = max(min(ζ(Beyonce, friend, Rihanna), ζ(Rihanna, friend, EnriqueI),ζ(EnriqueI, creator, Euphoria)), min(ζ(Beyonce, friend, MariahC), ζ(MariahC, friend, Shakira), ζ(Shakira, friend, EnriqueI), ζ(EnriqueI, creator, Euphoria))) = max(min(0.6, 0.2, 1), min(0.8, 0.3, 0.5, 1)) = 0.3, sat f 1 (Rihanna, Euphoria) = min(ζ(Rihanna, friend, EnriqueI), ζ(EnriqueI, creator, Euphoria)) = min(0.2, 1) = 0.2, sat f 1 (MariahC, Euphoria) = min(ζ(MariahC, friend, Shakira), ζ(Shakira, friend, EnriqueI), ζ(EnriqueI, creator, Euphoria)) = min(0.3, 0.5, 1) = 0.3, and sat f 1 (Shakira, Euphoria) = min(ζ(Shakira, friend, EnriqueI), ζ(EnriqueI, creator, Euphoria)) Figure 3 . 3 Figure3.8 gives the set of subgraphs of G M B satisfying the pattern P rec_low .The matching value of Art1 is either Shakira or EnriqueI who match the pattern P rec_low (i.e they are the only artists that have liked a low rated album created by another artist among their close friends).Note that (f riend + ) distance is short .creator is the fuzzy regular expression f 2 of Example 44 with sat f 2 (EnriqueI, Justied) = 0.4, sat f 2 (Shakira, Buttery) = 0.7 and sat f 2 (Shakira, Euphoria) = 0.5 and we consider µ low_rating (4) = 0.66, µ low_rating (6) = 0.33 and µ low_rating (9) = 0 dened in Figure3.5 on page 70.Then, the evaluation of the pattern P rec_low over the RDF graph G M B includes two mappings with their respective satisfaction degrees: Figure 3 . 8 : 38 Figure 3.8: Subgraphs satisfying P rec_low Figure 4 . 1 Figure 4 . 1 : 4141 Figure 4.1 gives two examples of monotonous decreasing and increasing fuzzy quantiers respectively. Figure 4 . 2 : 42 Figure 4.2: The fuzzy quantier at least ve us consider a quantied statement of the form Q B X are A from Example 47 and Q(x) = x 2 . Figure 4 . 3 : 43 Figure 4.3: Membership functions of Example 51 1 Figure 4 . 4 : 144 Figure 4.4: Membership function of the fuzzy term strong Figure 4 . 5 : 45 Figure 4.5: Fuzzy RDF graph G M B inspired by MusicBrainz 6 : 6 Query R atBoolean derived from R at This query returns a list of artist (?art1) with their recommended albums (?alb), satisfying the conditions of query R at , along with their respective satisfaction degrees µ B = min(µ recent (?alb), ζ(?art1, recommends, ?alb)) and µ A = min(µ high (?rating), µ young (?age), ζ(?art1, f riend, ?art2)). Figure 5 . 1 : 51 Figure 5.1: Reication of fuzzy triple of Example 55 ClientFigure 5 . 2 : 52 Figure 5.2: Implementation of a specic FURQL query evaluation engine Figure 5 . 3 53 Figure 5.3 illustrates this architecture. Figure 5 . 3 : 53 Figure 5.3: SURF software architecture 3 . 3 For a classical SPARQL query, we skip the Query compiler and Score calculator modules and the original query is transferred directly to the classical SPARQL engine. All the answers returned by the SPARQL engine are kept in the nal resultset with a satisfaction degree equal to1. Figure 5 . 4 : 54 Figure 5.4: Screenshot of SURF Figure 5 . 5 : 55 Figure 5.5: Edge query of the form edge-so Figure 5.8: Experimental results about the evaluation of FURQL queries Figure 5 . 9 : 59 Figure 5.9: Experimental results of Fuzzy Quantied queries involving crisp conditions Figure 5 . 5 Figure 5.10: Experimental results of Fuzzy Quantied queries involving fuzzy conditions and where n is the number of links in the path. The strength of the path is dened asST (p) = min i=1..n ρ(x i-1 , x i ). 4 ( 4 -[:author_of]->(ar1:paper), (ar1)-[:published]->(j1), au1)-[:author_of]->(ar2:paper), (ar2)-[:published]->(j2) 5 where j1.name="IJWS12" and j1.name <> j2.name Listing 6.1: Pattern expressed à la Cypher This pattern models information concerning authors (au2) who have, among their contributors, an author (au1) who published a paper (ar1) in IJWS12 and also published a paper (ar2) in another journal (j2). Figure 6 . 3 63 is a graphical representation of P. Figure 6 . 3 : 63 Figure 6.3: Pattern P 4 ( 5 ( 45 -[(contributor+)|Length is short]->(au1:author), au1)-[:author_of]->(ar1:paper), (ar1)-[:published]->(j1), 5 : 5 Derived query Q derived Such a query allows to retrieve the pairs {res, x} that belong to the graph and all the information needed for the calculation of µ B and µ A , i.e., the combination of fuzzy degrees associated with relationships and node attribute values involved in B(res,x) and in A(x), respectively denoted by I B and I A . The Listing6.6 of Example 62 below presents the derived query associated with the query Q mostAuthors . 2 . 2 In order to interpret Q mostAuthors , we rst derive the following query Q derived from Q mostAuthors , that retrieves the authors (a) who highly contributed to at least one recent paper (p) (corresponds toB(a,p) in lines 1 and 2) possibly (optional) published in a renowned database journal (corresponds to A(p) in lines 3 to 5). 1 match (a:author)-[author_of|ST IS strong]->(p:paper) 2 where p.year is recent 3 optional match (p)-[:published]->(j:journal), 4 (j)-[:impact_factor]->(i:impact_factor), (j)-[:domain]->(d:dom) 5 where i.value is high and d.name="database" 6 return a p µ A µ B Listing 6.6: Query Q derived derived from Q mostAuthors For the running example, Q derived returns the four answers {Peter, Maria, Claudio, Michel}. The authors Andreas, Susan and Bazil do not belong to the result of Q mostAuthors because Susan has not written a journal paper yet and Andreas and Bazil do not have a recent paper.For the running example, we then have Q derived (P eter) = {((0.2, 1)/IJAR14_p)}, Q derived (M aria) = {((0.33, 1)/IJAR14_p), ((0.6, 0.33)/IJIS16_p)}, Q derived (Claudio) = {((0.33, 1)/IJAR14_p), ((0.3, 0.07)/IJUFK15_p)}, and Q derived (M ichel) = {((0.3, 0.07)/IJUFK15_p)}. 2 0.2 ) = 1, µ(Maria) = µ most ( 0.66 0.93 ) = 0.71, µ(Claudio) = µ most ( 0.4 0.63 ) = 0.63, µ(Michel) = µ most ( 0.07 0.3 ) = 0.23}. the fuzzy set B = {µ B 1 /x 1 , ..., µ Bn /x n } such that µ B 1 ≤ ... ≤ µ Bn , the fuzzy set A = {µ A 1 /x 1 , ..., µ An /x n } and d = n i=1 µ B i . Figure 6 Figure 6 . 6 : 666 Figure 6.6 illustrates this architecture. p) are ( (p)-[:published]->(j:journal), 7 (j)-[:impact_factor]->(i:impact_factor), (j)-[:domain]->(d:dom) 8 where i.value is high and d.name="database" ) 9 return a Listing 6.11: Fuzzy quantied query with pattern matching Our experiments have been performed on a database inspired from DBLP containing crisp (e.g., published ) and fuzzy edges (e.g., contributor ). A java script have been developed to create random graph data of dierent sizes. In these experiments, four database sizes have been considered, see Table 6 . 1. 8.(a) (resp., Figure 6.8.(b)) presents the execution time in milliseconds using Zadeh's interpretation (resp., Yager's OWA-based interpretation). 8. This time remains almost constant, and is independent on the size of the dataset while slightly increasing in the presence of complex patterns or fuzzy conditions. As to the score calculation stage, it represents around 9% of the time needed for evaluating a fuzzy quantied FUDGE query. The time used for calculating the nal satisfaction degree is of course dependent on the size of the result set and the nature of the patterns. Figure 6.8: Experimental results of fuzzy quantied queries in FUDGE 'interroger un modèle de données RDF ou dans lequel les triplets sont porteurs de notions graduelles (dont le modèle RDF non ou est un cas particulier), et 2. d'exprimer des préférences oues portant non seulement sur les données mais également sur la structure du graphe, que celui-ci soit ou ou non. Modèle RDF ou Dans cette thèse, nous considérons un modèle de données, appelée F-RDF, qui synthétise les modèles RDF ous de la littérature Par exemple, le triplet ou Beyonce, recommande, Euphoria auquel est attaché le degré 0.8 indique que Beyonce, recommande, Euphoria est satisfait au niveau 0.8, ce qui peut être interprété comme Beyonce recommande fortement Euphoria. Les degrés ous peuvent être donnés ou calculés, matérialisés ou non. Dans sa forme la plus simple, un degré peut correspondre au calcul d'une notion statistique reétant l'intensité de la relation à laquelle le degré est attaché. Par exemple, l'intensité d'une relation d'amitié d'une personne p 1 vers une autre personne p 2 peut être calculée par la proportion d'amis communs par rapport au nombre total d'amis de p 1 . [ Pivert et al., 2016f] Pivert, O., Slama, O., and Thion, V. (2016f ). Requêtes quantiées oues structurelles sur des bases de données graphe. In Actes des Rencontres Franco- phones sur la Logique Floue et ses Applications (LFA'16), La Rochelle, France, pages 9-16. Denition 1 (RDF triple). Let U be the set of URIs, B the set of blank nodes, and L the set of literals. An RDF triple t:= s, p, o ∈ (U ∪ B) × U × (U ∪ L ∪ B) where the subject s denotes the resource being described, the predicate p denotes the property of the resource, and the object o denotes the property value. A triple t states that the subject s has a property p with a value o.Example 1 [RDF triple] For instance, the triple Beyonce, creator, Lemonade states that Beyonce has Lemonade as a creator property, which can be interpreted as Beyonce is a creator ofLemonade. 1 <?xml version="1.0"?> 2 <rdf:RDF xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#" 3 xmlns:mo="http://purl.org/ontology/mo/" 4 xmlns:dc="http://purl.org/dc/elements/1.1/"> 5 <rdf:Description rdf:about="uri:Lemonade"> :Lemonade> <http://www.w3.org/1999/02/22-rdf-syntax-ns#/type> <mo:album> . Listing 1.2: N-Triples le <uri:Lemonade> <http://purl.org/dc/elements/1.1/date> "2016" . <uri:Lemonade> <http://purl.org/dc/elements/1.1/title> "Lemonade" . <uri:Lemonade> <http://purl.org/dc/elements/1.1/rating> "8.7" . <uri:Lemonade> <http://purl.org/dc/elements/1.1/genre> "R & B" . <uri:Lemonade> <http://purl.org/dc/elements/1.1/track> <uri:sorry> . <uri:Lemonade> <http://purl.org/dc/elements/1.1/track> <uri:hold up> . <uri SPARQL: Crisp Querying of RDF data In order to eciently query RDF data, the SQL-like language SPARQL [Prud'hommeaux and Seaborne, 2008] is promoted by the W3C as a standard query language. It is a declarative query language based on graph pattern matching, in the sense that the query processor searches for sets of triples in the data graph that satisfy a graph pattern expressed in the query. A Basic Graph Pattern (BGP) is a basic building block of SPARQL, containing a set of triple patterns. A triple pattern is an RDF triple where variables may occur in the subject, predicate, or object position. Each variable is prexed by the question mark symbol. the results that do not match a given graph pattern. The second one uses theminus clause and aims to remove answers related to another pattern. Example 15 [Negation queries] The following query aims to nd the artists exp 1 and exp 2 are property path expressions, then, exp 1 |exp 2 and exp 1 /exp 2 are property path expressions, if exp is a property path expression, then, exp * , exp + , exp ? and ˆexp are property path expressions.where exp 1 |exp 2 denotes alternative expressions, exp 1 /exp 2 denotes a concatenation of exp 1 and exp 2 , exp * denotes a path that connects the subject and object of the path by zero or more matches of exp, exp + is a shortcut for exp * /exp and denotes a path that connects the subject and object of the path by one or more matches of exp, exp ? denotes a path that connects the subject and object of the path by zero or one matches of exp, ˆexp is an inverse path (from an object to the subject). Example 13 [Update query] The following query aims to add some infor- mation about a new artist "Ed Sheeran" in the default graph. insert data { ]. These works proposed more expressive languages and extended SPARQL by allowing path extraction queries (generally of uri:EdSheeran dc:title "Ed Sheeran" . uri:EdSheeran dc:age "26" . unknown length) within RDF datasets. Property paths came with the same principle uri:EdSheeran dc:rating "8" . } which is to allow for navigational querying over RDF graphs and are ocially integrated Listing 1.14: An example of an Update query in SPARQL 1.1. A property path is a possible path through a graph between two nodes. It can be of a variable length. A property path of length exactly 1 is a triple • Subqueries: The principle is the same as subqueries in SQL: a query may use the pattern. output of other queries for achieving complex results. Denition 3 (SPARQL property path expressions). SPARQL property path expressions Example 14 [Subqueries in SPARQL] Return a name (the one with the are recursively dened by: lowest sort order) for all the artists who are friend with beyonce and have a name. an IRI 9 is a property path expression that denotes a path of length one, select ?art ?minName where { uri:beyonce uri:friend ?y . { select ?art (min(?name) as ?minName) where { ?art uri:title ?name . } group by ?art } } Listing 1.15: Query involving a subquery • Negation: can be expressed in two ways. The rst uses the not exists clause and aims Example 12 [SPARQL property path query] Find the names of the artists to lter out who have issued no albums in 2015. that recommend albums made by friends or related friends of friends. select ?name where { select ?name where { ?artist dc:creator ?album . ?art1 dc:title ?name . ?art1 dc:recommends ?alb . filter not exists { ?artist dc:date "2005" . } } ?art2 dc:creator ?alb . ?art1 dc:friend+ ?art2 . } Listing 1.16: SPARQL query with negation (not exists) Listing 1.13: SPARQL query with property path The query that aims to retrieve names of artists having no albums is depicted in Listing 1.17. • Update functionalities: In addition to querying and manipulating RDF data, select ?name where { SPARQL 1.1 Update [Gearon et al., 2012] oers the possibility to modify the graph by ?artist dc:title ?name . adding/deleting triples, loading/clearing/creating/dropping an RDF graph and many minus { ?artist dc:creator ?album . } other facilities. } Listing 1.17: SPARQL query with negation (minus) if , when X is a nite set {x 1 , ..., x n }, is: Example 18 Let us consider the example of the predicate tall described in Ta-A more convenient notation ble 1.3. Tall can be dened by a Boolean condition (height ≥ 180). It corresponds to the crisp set (non fuzzy set) of Figure 1.5 and the result is in the third column of Table 1.3. Name height(cm) Memberships Crisp Fuzzy Chris 210 1 1 Marc 200 1 1 John 190 1 1 Tom 180 1 0.66 David 170 0 0.33 Tom 160 0 0 David 150 0 0 Table 1.3: Tall men However, it seems more natural to dene the predicate tall as a fuzzy set (cf., Fig- ure 1.6). The membership degrees associated with some individuals are shown in the fourth column of Table 1.3. Degree of membership 1 0 160 170 180 190 200 210 Height, cm Figure 1.5: Graphical representation of the predicate tall (crisp set) Degree of membership Degree of 1 membership 0.66 0.33 1 µ A 0 0 A -a 160 A 170 180 B 190 200 B + b 210 X Height, cm Figure 1.4: Trapezoidal membership function Figure 1.6: Graphical representation of the predicate tall (fuzzy set) Table 1 . 1 . 4: Properties of t-norm and t-conorm operators Property T-norm T-conorm Identity Table 2 . 2 ..,θ y with α being an RDF statement and θ 1 ,...,θ y being its annotations over a xed set Γ = {p 1 , ..., p y } of independent annotation dimensions. Example 29 Let us consider the RDF statement about music concerts shown in Table 2.1. Each statement is annotated by a set of dimensions Γ = {Time, Source, Certainty}. Dimensions Id Statement Time Source Certainty #1 TAL playsIn Le Grand Rex 03.02.17 www.legrandrex.com 0.9 #2 KUNGS playsIn L'OLYMPIA 15.01.17 www.fnacspectacles.com 0.7 #3 TAL hasRating 7 10.01.17 www.itunes.apple.com 0.5 #4 KUNGS hasRating 8 08.02.17 www.itunes.apple.com 0.5 1: The set of annotated RDF statements 32 In order to illustrate the form taken by skyline queries in PrefS-PARQL, let us consider again the query from Example 2.7. Listing 2.8 expresses this in PrefSPARQL. select ?artist ?concert where { ?artist dc:concert ?concert. ?concert dc:starts ?startingTime. ?concert dc:ends ?endingTime. ?artist dc:rating ?rating . preferring ( ?rating = ft:excellent and (?startingTime between (9pm, 1am) and ?endingTime between (9pm, 1am) prior to highest (?endingTime)))} Listing 2.8: Skyline query in PrefSPARQL Example 33 So as to illustrate conditional preferences, let us now assume that a user prefers a concert which takes place after 7:30pm on the weekdays and before 7pm during the weekends, formulated in Listing 2.9. This extension of SPARQL called PrefSPARQL supports not only the expression of qualitative preferences (skyline) but also conditional ones (if-then-else). A PrefSPARQL query returns a set of partially ordered tuples according to the satisfaction of the preferences. Example select ?concert where { ?concert dc:day ?D. ?concert dc:starts ?startingTime. preferring (if (?D = ``Saturday'' || ?D = ``Sunday'') then ?startingTime < 7pm else ?startingTime >= 7:30pm)} Listing 2.9: Conditional preference in PrefSPARQL Table 2 . 2 22 : RDFS Inferences Rules Example 36 The rule (4) from Table 2.2 states that if a is a subclass of b and Table 2 2 Predicate relaxation for example, using rule (2) from Table 2.2, the triple pattern (?X, proceedingsEditorOf, ?Y) can be relaxed into (?X, editorOf, ?Y) and then into (?X, contributorOf, ?Y) since we have (proceedingsEditorOf, sp, editorOf ) ∈ cl(O) and then (editorOf, sp, contributorOf ) ∈ cl(O); Predicate to domain relaxation for example, using rule (5) from Table 2.2, the triple pattern (a, p, b) can be relaxed into the triple pattern (a, type, c), since we have the triple pattern (p, dom, c) ∈ cl(O). .2 , the triple pattern (?X, type, ConferenceArticle) can be relaxed into (?X, type, Article) and then into (?X, type, Publication) since we have (ConferenceArticle, sc, Article) ∈ cl(O) and then (Article, sc, Publication) ∈ cl(O); and has as a principle to determine if two given graphs are the same; if they are, nd a matching (mapping) between them (i.e., which nodes However, all of the existing classical graph isomorphism algorithms do not t the semantic characteristics of RDF graphs (i.e., directed graphs with la- beled edges and nodes) [Carroll, 2002]. Then, an ecient semantic similar- ity measure based on RDF graph is required. Therefore, few approaches have proposed new techniques dealing with approximate querying over RDF data from one graph correspond to which nodes in the other. Similarity measures based on graph matching are commonly used in this context. Essentially, queries are represented as a graph (called the query graph ) and the aim is to nd an appropriate matching between the query graph and the resource graph. 1 and y is the object of t n . Example 40 [Path between two nodes] The (cy- cle free) paths between the nodes Beyonce and Euphoria from the fuzzy RDF graph G M B of Figure 3.1 are shown in Fig- ure 3.2. Beyonce recommends(0.8) Euphoria (p 1 ) Beyonce friend(0.6) Rihanna friend(0.2) EnriqueI creator Euphoria (p 2 ) Beyonce friend (0.8) MariahC friend (0.3) Shakira friend (0.5) EnriqueI creator Euphoria (p 3 ) dened by distance(x, y) = min p∈P aths(x,y) Figure 3.2: Cycle-free paths from G M B connecting Beyonce to Euphoria Denition 6 (Distance between two nodes). The distance between two nodes x and y is • If P is a fuzzy graph pattern and C is a fuzzy condition then (P filter C) is a fuzzy graph pattern. A fuzzy condition is a logical combination of fuzzy terms dened by: if {?x, ?y} ⊆ V and c ∈ (U ∪ L), then bound(?x), ?x θ c and ?x θ ?y are fuzzy conditions, where θ is a fuzzy or crisp comparator, if ?x ∈ V and F term is a fuzzy term then, ?x is F term is a fuzzy condition, if C 1 and C 2 are fuzzy conditions then (¬C 1 ) and (C 1 C 2 ) (where is a fuzzy connective) are fuzzy conditions. 1 , o 1 , ..., s n , p n , o n ) ⊆ G be a path of G.The statement p satises exp with a satisfaction degree of sat exp (p) is dened as follows, according to the form of exp (in the following, f , f 1 and f 2 are fuzzy regular expressions): Finally, the result of the query of Example 43 (Listing 3.2 on page 71) over G M B is the singleton {Shakira} which is m(?art1) in the mapping {?art1 → Shakira, ?alb → Buttery, ?r → 4}, i.e., the only mapping of P rec_low G M B having a satisfaction degree greater or equal to 0.4. ?Art1 (f riend + ) distance is short .creator ?Alb rating ?r recommends low Figure 3.7: Graphical representation of pattern P rec_low g 1 : EnriqueI friend(0.4) JustinT creator Justied rating 6 recommends(0.6) 0.33 g 2 : Shakira friend(0.7) MariahC creator Buttery rating 4 recommends(0.8) 0.66 Table 4 . 4 1: Characteristics of monotonous fuzzy quantiers Listing 4.5: Query R at derived from R mostAlbums Then, we evaluate the SPARQL query R f latBoolean given in Listing4.6, derived from the FURQL nonquantied query R f lat of Listing4.5. 1 select ?art1 ?alb µB µA where { 2 ?art1 recommends ?alb . ?alb date ?date . 3 filter ( ?date > 2010.0 ) 4 optional { 5 ?art1 friend ?art2 . ?art2 creator ?alb . 6 ?alb rating ?rating . ?art2 age ?age . 1 select ?art1 ?alb µB µA where { 2 ?art1 recommends ?alb . ?alb date ?date . 3 filter (?date is recent) 4 optional { 5 ?art1 friend ?art2 . ?art2 creator ?alb . 6 ?alb rating ?rating . ?art2 age ?age . 7 filter (?rating is high && ?age is young) } } . . ≥ c n . Example 54 In order to calculate µ(Shakira) from R at , let us consider B (resp. A) the set of satisfaction degrees corresponding to condition B (resp. A ) of element Shakira as follows B ={ 0.1/Euphoria, 0.2/Butterfly, 0.3/Justified} and A= { 0.07/Euphoria, 0/Butterfly, 0.4/Justified}. We have d = 0.6 and: S Euphoria = 0.1 0.6 = 0.17, S Buttery = 0.1 + 0.2 0.6 = 0.5, and S Justied = 0.1 + 0.2 + 0.3 0.6 = 1. RDF data is inspired by Musicbrainz 4 linked data (which is originally crisp), and for representing fuzzy information, we used the reication mechanism that makes it possible to attach fuzzy degrees to triples, as discussed earlier in Subsection 5.1.1 Table 5.1: Fuzzy RDF datasets Dataset Size Reied Triples DB 1 11796 triples 47185 triples DB 2 65994 triples 263977 triples DB 3 112558 triples 450393 triples DB 4 175416 triples 701665 triples A java script have been developed to create random fuzzy RDF data of dierent sizes. Table 5 . 5 2: Dierent types of FURQL queries Type crisp query Fuzzy Condition Fuzzy Structural Edge query Table 5 . 5 3: Set of fuzzy quantied queries with crisp conditions Query P B P A Conditions Q1 crisp simple simple crisp Q2 crisp complex simple crisp Q3 crisp simple complex crisp Table 5 . 5 4: Set of fuzzy quantied queries with fuzzy conditions Query P B P A Conditions Q1 fuzzy simple simple fuzzy Q2 fuzzy complex simple fuzzy Q3 fuzzy simple complex fuzzy Q4 fuzzy complex complex fuzzy Table 5 . 5 Average (0.59, 94.53, 4.89) (0.50, 94.86, 4.63) (0.12, 94.78, 5.10) (0.14, 99.38, 0.47) 5: Experimental results summarization DB2 DB3 DB4 (0.52, 93.34, 6.15) (0.51, 93.70, 5.79) (0.2, 94.40, 5.33) (0.49, 93.76, 5.75) (0.32, 94.29, 5.40) (0.23, 94.50, 5.27) (0.42, 92.76, 6,82) (0.15, 95.42, 4.43) (0,12, 94.78, 5.10) (0.01, 99.78, 0.21) (0.00, 99.95, 0.04) (0.00, 99.96, 0.04) DB1 Nonquantied queries (1.04, 96.66, 2.30) Nonquantied edge queries (0.97, 96.92, 2.12) Nonquantied star queries (0.87, 95.40, 3.73) Nonquantied path queries Quantied queries (0.55, 97.85, 1.60) Quantied queries with crisp conditions Quantied queries with fuzzy conditions . Among the existing systems, let us mention IJIS16 dans IJIS16_p Susan author_of Maria c o n t r ib u t o r IJIS10 where: Mai 2016} {volume: 30, IJIS10_p author_of d a n s {volume: 25, where: Avril 2010} {titre: An ..., pages: 81-98} Basil dans IJIS10_p1 {title: About ..., pages: 365-385} a u t h o r _ o f a u t h o r _ o f Claudio contributor {titre: A ...,pages: 287-325} AllegroGraph [allegrograph, 2017], InniteGraph [innitegraph, 2017], Neo4j [Neo4j, 2017] and Sparksee [sparksee, 2017]. Dierent models of graph databases have been proposed in the Figure 6.1: An Attributed graph inspired from DBLP literature (see .2) Clearly Length(p) ≥ n (it is equal to n if ρ is Boolean, i.e., if G is a nonfuzzy graph). We can then dene the distance between two nodes x and y in G as Distance(x, y) = min all paths p f rom x to y This query contains a list of define clauses for the fuzzy quantiers and the fuzzy terms declarations, a match clause for fuzzy graph pattern selection, a having clause for the fuzzy quantied statement denition, and a return clause for specifying which elements should be returned in the resultset. B(res, x) denotes the fuzzy graph pattern involving the nodes res and x and expressing the (possibly fuzzy) conditions inB. B(res, x) takes the form of a fuzzy graph pattern expressed à la Cypher by P B where C B (see Section 6.1.4). A(x) denotes the fuzzy graph pattern involving the node x and expressing the (possibly fuzzy) conditions inA. A(x) takes the form of a fuzzy graph pattern expressed à la Cypher by P A where C A (see 3. 1 define... in 2 match B(res, x) 3 with res having Q(x) are A(x) 4 return res Listing 6.3: Syntax of a fuzzy quantied query Section 6.1.4). Table 6 . 1 61 : Fuzzy graph datasets Dataset Size DB 1 700 nodes & 1447 edges DB 2 2100 nodes & 4545 edges DB 3 3500 nodes & 7571 edges DB 4 4900 nodes & 10494 edges http://franz.com/agraph/allegrograph/ https://musicbrainz.org/ http://www.w3.org/TR/rdf-syntax-grammar/ http://www.w3.org/2001/sw/RDFCore/ntriples/ http://www.w3.org/DesignIssues/Notation3 http://www.w3.org/TR/turtle/ http://www.franz.com/agraph/allegrograph/ http://jena.apache.org/ http://jena.apache.org/ 1.2. SPARQL: Crisp Querying of RDF data 1.3. Fuzzy Set Theory 2.1. Preference Queries on RDF Data Considering paths containing a cycle would not change the result of the following expressions (3.1) and (3.3). Hereafter, the define clauses are omitted for the sake of simplicity. https://www-shaman.irisa.fr/surf/ https://jena.apache.org https://vaadin.com/home https://musicbrainz.org/ http://www.informatik.uni-trier.de/~ley/db/ http://www.informatik.uni-trier.de/~ley/db/ Hereafter, the define clauses are omitted for the sake of simplicity. The general syntactic form of a fuzzy quantied query of the type Q B X are A in the FURQL language is given in Listing 4. 1. define ... select ?res where { B(?res,?x) group by ?res having Q(?x) are ( A(?x) ) } Listing 4.1: Syntax of a FURQL quantied query R The define clause allows to dene the fuzzy terms and the fuzzy quantier (denoted here by Q). Fuzzy quantiers are declared in the same way as fuzzy terms (see Subsection 3.2.1 of Chapter 3). The select clause species which variables ?res should be returned in the result set. The group by clause contains the variables (here ?res) that should be partitioned. Expression B(?res,?x) (in the where clause) denotes the fuzzy graph pattern, dened in the FURQL language (see Denition 9 on page 69), involving the variables ?res and ?x and expressing the (possibly fuzzy) conditions in B and expression A(?x) (in the having clause) denotes the fuzzy graph pattern involving the variable ?x that appears in A. Example 51 [Fuzzy Quantied Query in FURQL] The query, denoted by R mostAlbums , that aims to retrieve every artist (?art1) such that most of the recent albums (?alb) that he/she recommends are highly rated and have been created by a young friend (?art2) of his/hers may be expressed in FURQL as follows: 1 defineqrelativeasc most as (0.3,0.8), defineasc high as (2,5) 2 definedesc young as (25,40), defineasc recent as (2010,2015) 3 select ?art1 where { 4 ?art1 recommends ?alb . ?alb date ?date . where the defineqrelativeasc clause denes the fuzzy relative increasing quantier most of Figure 4.3.(c), the defineasc clauses dene the (increasing) membership functions associated with the fuzzy terms high and recent of Figure 4.3.(a) and (b), and the definedesc clause denes the (decreasing) membership function associated with the fuzzy term young of Figure 4.3.(d). In this query, ?art1 corresponds to ?res of Listing 4.2, ?alb corresponds to ?x of Listing 4.2, lines 4 to 5 correspond to B(?res,?x) of Listing 4.2 and lines 8 to 10 correspond to A(?x) of Listing 4.2. Implementation of FURQL In this section, we discuss implementation issues related to the FURQL query language. Two aspects have to be considered: i) the storage of fuzzy RDF graphs (see Subsection 5.1.1), and ii) the evaluation of FURQL queries with and without fuzzy quantied statements (see Subsection 5.1.2). Storage of Fuzzy RDF Graphs In this thesis we deal with fuzzy RDF graph, for which we need to attach fuzzy degrees to some edges in the RDF graph. Example 55 The fuzzy RDF triple ( Shakira, friends, MariahC , 0.7) states that Shakira, friends, MariahC is satised to the degree 0.7, which could be interpreted as Shakira is a close friend of MariahC. The representation of this fuzzy RDF triple using reication is given in Listing 5.1. The satisfaction degree 0.7 is given by the statement in Line 5. A possible graphical representation of this reication is depicted in Figure 5.1. The nodes in dashed lines represent reied nodes with the properties rdf:type, • Star queries (star-shaped queries): consist of three acyclic triple patterns that share the same node (called central node). The central node may appear in dierent positions; i.e., it can be the subject of the three triples patterns (denoted by star-s3 ), the object of three triples patterns (denoted by star-o3 ), the subject of a triple patterns and the object of the two others (denoted by star-s1-o2 ), or the subject of two triples patterns and the object of the remaining triple pattern (denoted by star-s2-o1 ). Again we used four queries of the form star-s2-o1 shown in Figure 5.6. • Path queries: consist of two or three triple patterns that form a path such that two triples share a variable. We may nd path shaped queries of length two or three. We consider in the following an example of a path shaped query of length three of the form given in Figure 5.7. Query Q 3.4 is a fuzzy structural simple path query containing a fuzzy structural condition that aims to nd every artist who has among his close friends an artist who created an album (cf., Listing 5.13). Its crisp counterpart, denoted by Q 3.3 , aims to nd every artist who has among his friends (with a friendship degree greater than 0.8) an artist who created an album (cf., Listing 5.12). select ?art1 where { ?art2 creator ?alb. ?alb date ?d . /* reification */ ?X1 subject ?art1. ?X1 predicate friend. ?X1 object ?art2. ?X1 degree ?degree. filter ( ?degree > 0.8 ) } Listing 5.12: Crisp strutural path query Q 3.3 defineasc strong as (0.7, 0.9) select ?alb where { ?art2 creator ?alb. ?alb date ?d. /* structural condition */ ?art1 (friend | ST is strong) ?art2 .} Listing 5.13: Fuzzy structural path query Q 3.4 We evaluated separately each type of queries over the dierent sizes of database given in Table 5.1 on page 103. The results of these queries are depicted in Figure 5.8. Figure 5.8.(a) Although these experimental results are preliminary observations, they appear very encouraging since they show that our approach does not entail any important overhead cost. 5.2.3 The main objective of these experiments is to assess the cost of each stage involved in the evaluation of fuzzy quantied queries and to show that the extra cost due to the introduction of fuzzy quantied statements remains limited/acceptable. Fuzzy quantied query involving crisp conditions In the rst experiment, we processed four fuzzy quantied queries with crisp conditions (of the type Q B X are A) by changing each time the nature of the patterns corresponding to conditions B and A from simple to complex ones. These queries are summarized in Table 5. 3. A complex pattern diers from a simple one by the number of its statements. Here, a complex pattern is composed of nine triple patterns at most, while a simple pattern has 6.1.3 Fuzzy Graph Databases We are interested in fuzzy graph databases where nodes and edges can carry data (e.g., keyvalue pairs in attributed graphs). So, we consider an extension of the notion of a fuzzy graph : the fuzzy data graph as dened in [Pivert et al., 2014a]. Denition 14 (Fuzzy data graph). Let E be a set of labels. A fuzzy data graph G is a quadruple (V, R, κ, ζ), where V is a nite set of nodes (each node n is identied by n.id), R = e∈E {ρ e : V × V → [0, 1]} is a set of labeled fuzzy edges between nodes of V , and κ (resp. ζ) is a function assigning a (possibly structured) value to nodes (resp. edges) of G. In the following, a graph database is meant to be a fuzzy data graph. The following example illustrates this notion. Example 58 [Fuzzy data graph] Figure 6.2 is an example of a fuzzy data graph, inspired from DBLP 2 with some fuzzy edges (with a degree in brackets), and crisp ones (degree equal to 1). In this example, the degree associated with A -contributor-> B is the proportion of journal papers co-written by A and B, over the total number of journal papers written by B. The degree associated with J -domain -> D is the extent to which the journal J belongs to the research domain D. Nodes are assumed to be typed. If n is a node of V , then T ype(n) denotes its type. In Figure 6.2, the nodes IJWS12, IJAR14, IJIS16, IJIS10 and IJUFK15 are of type journal, the nodes IJWS12-p, IJAR14-p, IJIS16-p, IJIS10-p, IJIS10-p1 and IJUFK15-p of type paper, and the nodes Andreas, Peter, Maria, Claudio, Michel, Bazil and Susan are of type author, the nodes named Database are of type domain and the other nodes are of type impact_factor. For nodes of type journal, paper, author and domain, a property, called name, contains the identier of the node and for nodes of type impact_factor, a property, called value, contains the value of the node. In Figure 6.2, the value of the property name or value for a node appears inside the node. 6. 1.4 The FUDGE Query Language FUDGE, based on the algebra described in [Pivert et al., 2015], is an extension of the Cypher language [START_REF] Cypher | Cypher[END_REF] propose any formal language for expressing such queries. A rst attempt to extend Cypher with fuzzy quantied queries in the context of a regular (crisp) graph database is described in [Castelltort and[START_REF] Castelltort | [END_REF][START_REF] Castelltort | [END_REF]. In [START_REF] Castelltort | Fuzzy queries over NoSQL graph databases: Perspectives for extending the Cypher language[END_REF], the authors take as an example a graph database representing hotels and their customers and consider the following fuzzy quantied query: In this query, a corresponds to res, p corresponds to x, lines 3 and 4 correspond to B and lines 6 to 8 correspond to A. According to the general syntax introduced in Listing 6.3, the variable a instantiates res and the variable p instantiates x. 6.3.2 where µ q denotes the membership degree of the predicate q and ρ e (x, y) denotes the weight of the edge (x, y). Let us consider Q derived (a) the set of answers of the query Q derived for a given author a. The set Q derived (a) provides a list of papers with their respective satisfaction degrees. This result set is of the form Then, with µ most (x) = x, we get µ Q (S IJAR14 ) = 0.35 and µ Q (S IJIS16 ) = 1. Therefore, the weights of the OWA operator are: The implication values are: Lastly, the nal result of the query Q mostAuthors evaluated on DB, given by Formula 6.7, is: 84, µ(Michel) = 0.7, µ(Maria) = 0.61}. About Query Processing For the implementation of these quantied queries, we updated the SUGAR software described in [Pivert et al., 2014a[START_REF] Pivert | SUGAR: A graph database fuzzy querying system[END_REF], which is a software add-on layer that implements the FUDGE language over the Neo4j graph DBMS. This software eciently evaluates FUDGE queries that contain fuzzy preferences, but its initial version did not support fuzzy quantied statements. The SUGAR software basically consists of two modules, which implement the Compiling and Final result calculation stages dened in Section 6. 3.2. These modules interact with a Neo4j engine, which implements the Crisp implementation stage dened in Section 6.3.2. 1. In a pre-processing step, the Query compiler module produces • the query-dependent functions that allow us to compute µ B , µ A and µ, for each returned answer, according to the chosen interpretation, and, • the (crisp) Cypher query Q derivedBoolean , which is then sent to the Neo4j engine for retrieving the information needed to calculate µ B and µ A . edges; and regular simple paths, which allow some node and edge restrictions (e.g., regular expressions). • Pattern matching query: graph pattern matching consists in nding all subgraphs of a data graph that are isomorphic to a graph pattern. During our experiments, we considered four queries with various forms of condition A. • The rst query Q 1 (Listing 6.8), where A is an adjacency pattern, aims to nd the authors such that most of the recent papers of which they are main authors, have been published in a journal. Sample of Queries The following listing is an example of a derived nonfuzzy query. • Q 4crisp : A fuzzy quantied query with complex pattern in B and complex pattern in A, involving crisp conditions (see Listing A.5), defineqrasc most AS (0,1) defineasc strong AS (0. ?art2 <uri:creator> ?alb2 . ?alb2 <uri:rating> ?rating2 . filter ( ?rating2 is high && ?age2 is young ) ) Listing A.7: A fuzzy quantied query Q 2f uzzy • Q 3f uzzy : A fuzzy quantied query with simple pattern in B and complex pattern in A, involving fuzzy conditions (see Listing A.8), defineqrasc most AS (0,1) defineasc recent AS (2014AS ( ,2016) ) definedesc young AS (25,32) defineasc high AS (3,6) defineasc high AS (2,5) select ?art1 where { ?art1 <uri:recommends> ?alb2 . ?alb2 <uri:date> ?date2 . filter ( ?date2 is recent )} group by ?art1 having most(?alb2) are ( ?art1 ( <uri:friend> | ST IS strong ) ?art2 . ?art2 <uri:age> ?age2 . ?art2 <uri:creator> ?alb2 . ?alb2 <uri:rating> ?rating2 . ?art2 <uri:rating> ?r2 . ?art2 <uri:memberOf> ?m2 . ?art2 <uri:gender> ?g2 . ?art2 <uri:type> ?t21 . ?alb2 <uri:type> ?t22 . filter ( ?rating2 is high && ?age2 is young ) ) Listing A.8: A fuzzy quantied query Q 3f uzzy • Q 4f uzzy : A fuzzy quantied query with complex pattern in B and complex pattern in A, involving fuzzy conditions (see Listing A.9). defineqrasc most AS (0,1) defineasc recent AS (2014AS ( ,2016) ) definedesc young AS (25,32) defineasc high AS (3,6) defineasc high AS (2,5) select ?art1 where { ?art1 (recommends | ST is strong) ?alb2 . ?alb2 <uri:date> ?date2 . ?art1 <uri:rating> ?r1 . ?art1 <uri:memberOf> ?m1 . ?art1 <uri:gender> ?g1 . ?art1 <uri:age> ?age1. ?art1 <uri:type> ?t11. filter ( ?date2 is recent )} group by ?art1 having most(?alb2) are ( ?art1 ( <uri:friend> | ST IS strong ) ?art2 . ?art2 <uri:age> ?age2 . ?art2 <uri:creator> ?alb2 . ?alb2 <uri:rating> ?rating2 . ?art2 <uri:rating> ?r2 . ?art2 <uri:memberOf> ?m2 . ?art2 <uri:gender> ?g2 . ?art2 <uri:type> ?t21 . ?alb2 <uri:type> ?t22 . filter ( ?rating2 is high && ?age2 is young ) ) Listing A.9: A fuzzy quantied query Q 4f uzzy